Statistical Field Theory and Networks of Spiking Neurons
aa r X i v : . [ q - b i o . N C ] S e p Statistical Field Theory and Networks of Spiking Neurons
Pierre Gosselin ∗ A¨ıleen Lotz † Marc Wambst ‡ September 2020
Abstract
This paper models the dynamics of a large set of interacting neurons within the framework of statisticalfield theory. We use a method initially developed in the context of statistical field theory [44] and lateradapted to complex systems in interaction [45][46]. Our model keeps track of individual interactingneurons dynamics but also preserves some of the features and goals of neural field dynamics, such asindexing a large number of neurons by a space variable. Thus, this paper bridges the scale of individualinteracting neurons and the macro-scale modelling of neural field theory.
Neural fields describes numerous patterns of brain activity, such as cognitive or pathologic processes. Thisapproach models large populations of neurons as homogeneous structures in which individual neurons areindexed by some spatial coordinates. Neural fields dynamics is usually studied in the continuum limit follow-ing Wilson, Cowan and Amari ([1][2][3][4][5][6][7][8][9]). Neural activity is represented through a population-averaged firing rate, a macroscopic variable generally assumed to be deterministic and the degrees of freedomof some underlying processes are aggregated to generate an effective theory with simpler variables.Neural fields theory, because it is a mean field approach that focuses on large scale effects, has a largerange of applications, and has been extended along various lines. This approach allows for travelling wavesolutions, possibly periodic (see [20][21] and references therein). Stochastic effects in firing rates or otherrelevant variables have been introduced [10][11][12][13][14] to model perturbations and diffusion patterns inthe pulse waves dynamics. The stochastic approach has also been used to study different regimes of mean fieldtheory and noisy transition between these regime ( see [15] ) . Neural network topology has been studied, alongwith spatial configurations’ effects (see [17], developments in [18], and references therein). Last but not least,the tools of Field theory have also been used to extend the Mean field approach [19][22][23][24][25][26][27].Indeed, mean field appears as the steepest descent approximation of a Statistical Field Theory. The statisticalfields involved in this formalism are directly related to the activity, i.e. the spike counts, at each point ofthe network. The field’s action encompasses fluctuations around the mean field. In such a setting, theperturbation expansion of the effective action allows to go further than the mean field approximation, asit keeps track of covariances between neural activity at different points. However, as for mean field theoryit remains at the collective level: it works with densities of activity and the field theory is built to recoverthe average mean field master equation plus some covariances in activity, rather than being designed on thebasis of microscopic features of the network. An alternate approach, also based on field theory, computesa partition function for the whole system of neurons in which the field represents the neurons’ activity (see[28][29] and references therein). The computation of the correlation functions yields results that go beyondthe mean field approximation. Yet, these models use simplifying assumptions at the microscopic level, suchas neglecting spatial indices and delays in interactions.Another branch of the literature considers neural processes as an assembly of individual interactingneurons. This alternate approach allows for a more precise account of the interrelation between neurons’ ∗ Pierre Gosselin: Institut Fourier, UMR 5582 CNRS-UGA, Universit´e Grenoble Alpes, BP 74, 38402 Saint Martin d’H`eres,France E-Mail: [email protected] † A¨ıleen Lotz: Cerca Trova, BP 114, 38001 Grenoble Cedex 1, France. E-mail: [email protected] ‡ Marc Wambst : Marc Wambst: IRMA, UMR 7501, CNRS, Universit´e de Strasbourg, 7 rue Ren´e Descartes, 67084 StrasbourgCedex, France. E-Mail: [email protected]
Individual dynamics and probability density of the system
Following [45][46], we describe a system of a large number of neurons (
N >>
We follow the description of [43] for coupled quadratic integrate-and-fire (QIF) neurons, but use the additionalhypothesis that each neuron is characterized by its position in some spatial range.Each neuron’s potential X i ( t ) satisfies the differential equation:˙ X i ( t ) = γX i ( t ) + J i ( t ) (1)for X i ( t ) < X p , where X p denotes the potential level of a spike. When X = X p , the potential is reset to itsresting value X i ( t ) = X r < X p . For the sake of simplicity, following ([43]) we have chosen the squared form γX i ( t ) in (1). However any form f ( X i ( t )) could be used. The current of signals reaching cell i at time t iswritten J i ( t ).Our purpose is to find the system dynamics in terms of the spikes’ frequencies. First, we consider the timefor the n -th spike of cell i , θ ( i ) n . This is written as a function of n , θ ( i ) ( n ). Then, a continuous approximation n → t allows to write the spike time variable as θ ( i ) ( t ). We thus have replaced: θ ( i ) n → θ ( i ) ( n ) → θ ( i ) ( t )The continuous approximation could be removed, but is convenient and simplifies the notations and compu-tations. We assume now that the timespans between two spikes are relatively small. The time between twospikes for cell i is obtained by writing (1) as: dX i ( t ) dt = γX i ( t ) + J i ( t )and by inverting this relation to write: dt = dX i γX i + J ( i ) (cid:0) θ ( i ) ( n − (cid:1) Integrating the potential between two spikes thus yields: θ ( i ) ( n ) − θ ( i ) ( n − ≃ Z X p X r dXγX + J ( i ) (cid:0) θ ( i ) ( n − (cid:1) Replacing J ( i ) (cid:0) θ ( i ) ( n − (cid:1) by its average value during the small time period θ ( i ) ( n ) − θ ( i ) ( n − J ( i ) (cid:0) θ ( i ) ( n − (cid:1) as constant in first approximation, so that: θ ( i ) ( n ) − θ ( i ) ( n − ≃ (cid:20) arctan (cid:18)q γJ ( i ) ( θ ( i ) ( n − ) X (cid:19)(cid:21) X p X r q γJ ( i ) (cid:0) θ ( i ) ( n − (cid:1) = " arctan X r J ( i ) ( θ ( i ) ( n − ) γ ! X r X p q γJ ( i ) (cid:0) θ ( i ) ( n − (cid:1) = arctan (cid:16) Xr − Xp (cid:17)r J ( i ) ( θ ( i )( n − ) γ J ( n ) ( θ ( n − ) γXrXp q γJ ( i ) (cid:0) θ ( i ) ( n − (cid:1) For γ normalized to 1 along and J ( n ) ( θ ( n − ) X r X p <<
1, this is: θ ( i ) ( n ) − θ ( i ) ( n − ≡ G (cid:16) θ ( i ) ( n − (cid:17) = arctan (cid:16)(cid:16) X r − X p (cid:17) q J ( i ) (cid:0) θ ( i ) ( n − (cid:1)(cid:17)q J ( i ) (cid:0) θ ( i ) ( n − (cid:1) (2)3he frequency ω i ( t ), or firing rate at t , is defined by the inverse time span (2) between two spikes: ω i ( t ) = 1 G (cid:0) θ ( i ) ( n − (cid:1) ≡ F (cid:16) θ ( i ) ( n − (cid:17) = q J ( i ) (cid:0) θ ( i ) ( n − (cid:1) arctan (cid:16)(cid:16) X r − X p (cid:17) q J ( i ) (cid:0) θ ( i ) ( n − (cid:1)(cid:17) The time interval between two spikes being considered small, we can write: θ ( i ) ( n ) − θ ( i ) ( n − ≃ ddt θ ( i ) ( t ) − ω − i ( t ) = ε i ( t ) (3)We added a white noise perturbation ε i ( t ) for each period to account for any internal uncertainty in thetimespan θ ( i ) ( n ) − θ ( i ) ( n −
1) which is independent from the instantaneous inverse frequency ω − i ( t ). Weassume these ε i ( t ) to have variance σ , so that equation (3) writes: ddt θ ( i ) ( t ) − G (cid:16) θ ( i ) ( t ) , J ( i ) (cid:16) θ ( i ) ( t ) (cid:17)(cid:17) = ε i ( t ) (4)The ω i ( t ) are computed by considering the overall current. Using the discrete notation, it is given by:ˆ J ( i ) (( n − J ( i ) (( n − κN X j,m ω j ( m ) ω i ( n − δ (cid:18) θ ( i ) ( n − − θ ( j ) ( m ) − | Z i − Z j | c (cid:19) T ij (( n − , Z i ) , ( m, Z j ))(5)The quantity J ( i ) (( n − i by neuron j during the short time span θ ( i ) ( n ) − θ ( i ) ( n − T ij (( n − , Z i ) , ( m, Z j )) isthe transfer function between cells j and i . It measures the level of connectivity between i and j . We assumethat: T ij (( n − , Z i ) , ( m, Z j )) = T (( n − , Z i ) , ( m, Z j ))The transfer function of Z j on Z i only depends on positions and times. It models the transfer functionas an average transfer between local zones of the thread. This transfer function is typically considered asgaussian or decreasing exponentially with the distance between neurons, so that the closer the cells, the moreconnected they are.The other terms arising in (5) can be justified in the following way: given the distance | Z i − Z j | betweenthe two cells and the signals’ velocity c , the signals arrive with a delay | Z i − Z j | c . The spike emitted by cell j at time θ ( j ) ( m ) has thus to satisfy: θ ( i ) ( n − < θ ( j ) ( m ) + | Z i − Z j | c < θ ( i ) ( n )to reach cell i during the timespan (cid:2) θ ( i ) ( n − , θ ( i ) ( n ) (cid:3) . This relation must be represented by a step functionin the current formula. However given our approximations, this can be replaced by: δ (cid:18) θ ( i ) ( n − − θ ( j ) ( m ) − | Z i − Z j | c (cid:19) as was done in (5). However, this Dirac function has to be weighted by the number of spikes emitted duringthe rise of the potential. This number is the ratio ω j ( m ) ω i ( n − that counts the number of spikes emitted byneuron j toward neuron i between the spikes n − n of neuron i . Again, this is valid for relatively smalltimespans between two spikes. For larger timespans, the frequencies should be replaced by their averageover this period of time.The sum over m and i is the overall contribution from any possible spikes of the thread to the current,provided these spikes arrive at i during the considered interval θ ( i ) ( n ) − θ ( i ) ( n − i , but depends also on i through ω i ( n − ω i ( t ) = q ˆ J ( i ) (( n − (cid:18)(cid:16) X r − X p (cid:17) q ˆ J ( i ) (( n − (cid:19) + υ i ( t ) (6)and J ( i ) (( n − υ i ( t ) accounts for the possible deviations from this relation,due to some internal or external causes for each cell. We assume that the variances of υ i ( t ) are constant,and equal to η , such that η << σ . Due to the stochastic nature of equations (4) and (6), the dynamics of a single neuron can be described bythe probability density for a path (cid:0) θ ( i ) ( t ) , ω − i ( t ) (cid:1) (see [45] and [46]): P (cid:16) θ ( i ) ( t ) , ω − i ( t ) (cid:17) = exp ( A i ) (7)where: A i = 1 σ Z (cid:18) ddt θ ( i ) ( t ) − ω − i ( t ) (cid:19) dt + Z (cid:0) ω − i ( t ) − G (cid:0) θ ( i ) ( t ) , J (cid:0) θ ( i ) ( t ) (cid:1)(cid:1)(cid:1) η dt (8)The integral is taken between some initial and final times. This time period depends on the time scale ofthe interactions. Actually, the minimization of (8) imposes both (3) and (6), so that the probability densityis centered around these two conditions, as expected. In other words, (3) and (6) are satisfied in mean. Aprobability density for the whole system is obtained by summing S i over all agents. We thus define: P (cid:16)(cid:16) θ ( i ) ( t ) , ω − i ( t ) (cid:17) i =1 ...N (cid:17) = exp ( − A ) (9)with: A = X i A i = X i σ Z (cid:18) ddt θ ( i ) ( t ) − ω − i ( t ) (cid:19) dt + Z (cid:0) ω − i ( t ) − G (cid:0) θ ( i ) ( t ) , J (cid:0) θ ( i ) ( t ) (cid:1)(cid:1)(cid:1) η dt (10) In [45][46], we show that the probabilistic description of the system (9) is equivalent to a statistical fieldformalism. In such a formalism, the system is collectively described by a field belonging to the Hilbertspace of complex functions. The arguments of these functions are the same as those describing an individualneuron.In our context, the field depends on the three variables ( θ, Z, ω ) and writes Ψ ( θ, Z, ω ). The field dynamicsis described by an action functional for the field and its associated partition function. This partition function,that reflects both collective and individual aspects of the system, allows to recover correlation functions foran arbitrary number of neurons.The field theoretic version of (8) is obtained using (10): a correspondence detailed in [45][46]) yields anaction S (Ψ) for a field Ψ ( θ, Z, ω ) and a statistical weight exp ( − ( S (Ψ))) for each configuration Ψ ( θ, Z, ω )of this field. The functional S (Ψ) is decomposed in two parts corresponding to the two contributions in (10).The first term of (10) is replaced by the corresponding quadratic functional in field theoretic : −
12 Ψ † ( θ, Z, ω ) ∇ (cid:18) σ ∇ − ω − (cid:19) Ψ ( θ, Z, ω ) (11)where σ is the variance of the errors ε i . 5he field functional corresponding to the second term of (8) is obtained by considering the statisticalweight for each dynamical variable i and taking into account the current induced by all j : V = 12 η X n X i (cid:0) ω − i ( n −
1) (12) − G J (cid:16) θ ( i ) ( n − , Z i (cid:17) + κN X j,m ω j ( m ) T ij (( n − , Z i ) , m, Z j ) ω i ( n − δ (cid:18) θ ( i ) ( n − − θ ( j ) ( m ) − | Z i − Z j | c (cid:19) with η <<
1. It is the constraint (6) imposed stochastically. Its continuous time equivalent is: V = 12 η Z dt X i ω − i ( t ) − G J (cid:16) θ ( i ) ( t ) , Z i (cid:17) + κN Z ds X j ω j ( s ) T ij (( t, Z i ) , s, Z j ) ω i ( t ) δ (cid:18) θ ( i ) ( t ) − θ ( j ) ( s ) − | Z i − Z j | c (cid:19) (13)The corresponding potential in field theory is obtained straightforwardly:12 η Z | Ψ ( θ, Z, ω ) | ω − − G J ( θ, Z ) + Z κN ω T (cid:16) Z, θ, Z , θ − | Z − Z | c (cid:17) ω (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z , ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ dω (14)We will write: T (cid:18) Z, θ, Z , θ − | Z − Z | c (cid:19) ≡ T ( Z, θ, Z )The field action is then the sum of (11), (14): S = −
12 Ψ † ( θ, Z, ω ) ∇ (cid:18) σ θ ∇ − ω − (cid:19) Ψ ( θ, Z, ω ) (15)+ 12 η Z | Ψ ( θ, Z, ω ) | ω − − G J ( θ, Z ) + Z κN ω ω (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z , ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, θ, Z ) dZ dω !! We can simplify (15) using that η <<
1. Actually, in that case, most field configurations Ψ ( θ, Z, ω ) havenegligible statistical weight. We can restrict the fields to those of the form:Ψ ( θ, Z ) δ (cid:16) ω − − ω − (cid:16) J, θ, Z, | Ψ | (cid:17)(cid:17) (16)where ω − ( J, θ, Z,
Ψ) satisfies: ω − (cid:16) J, θ, Z, | Ψ | (cid:17) = G J ( θ, Z ) + Z κN ω T (cid:16) Z, θ, Z , θ − | Z − Z | c (cid:17) ω (cid:16) J, θ, Z, | Ψ | (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z , ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ dω = G J ( θ, Z ) + Z κN ω T (cid:16) Z, θ, Z , θ − | Z − Z | c (cid:17) ω (cid:16) J, θ, Z, | Ψ | (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) × δ (cid:18) ω − − ω − (cid:18) J, θ − | Z − Z | c , Z , | Ψ | (cid:19)(cid:19) dZ dω (cid:19) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) = G J ( θ, Z ) + Z κN ω (cid:16) J, θ − | Z − Z | c , Z , Ψ (cid:17) T (cid:16) Z, θ, Z , θ − | Z − Z | c (cid:17) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ (17)The configurations Ψ ( θ, Z, ω ) that minimize the potential (14) can thus be considered: the field Ψ ( θ, Z, ω )is projected on the subspace (16) of functions of two variables. Therefore, we replace: ω − → ω − (cid:16) J, θ, Z, | Ψ | (cid:17) in (14) and the ”classical” effective action becomes: −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ σ ∇ θ − ω − (cid:16) J, θ, Z, | Ψ | (cid:17)(cid:19) Ψ ( θ, Z ) (18)Eventually, a more precise form can be given to the transfer function T ( Z, θ, Z ). We use a simplified versionof [47]. Appendix 6 shows that at the individual level and in first approximation, the transfer functions aremodelled by a product of a spatial factor T ( Z, Z ) and a function W of the frequencies ω ≡ ω (cid:16) J, θ, Z, | Ψ | (cid:17) and ω ≡ ω (cid:16) J, θ − | Z − Z | c , Z , | Ψ | (cid:17) . The function W is increasing in ω and decreasing in ω . Without lossof generality, we will consider W as an increasing function of (cid:16) ωω (cid:17) , so that: T ( Z, θ, Z ) = T ( Z, Z ) W (cid:18) ωω (cid:19) (19) We ultimately modify (18) by including collective terms to stabilize the number of active connexions. Suchterms correspond to some overall regulatory processes and do not appear at the individual level. In absenceof ”competition” between inhibitory and excitatory mechanisms, such a potential models the possibility fora system to come back to some minimal equilibrium activity.To do so, we modify the effective action by including a potential for maintaining and activating newconnections. We add to (18) the contribution: Z | Ψ ( θ, Z ) | U Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! (20)where U is a U shaped potential with U (0) = 0, so that U has a minimum for some positive value of (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . At this minimum, the value of U is negative. We write the expansion of (20) as: − ζ Z | Ψ ( θ, Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! + ∞ X n =2 ζ n Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! (21)The second term in (21) represents a global limitation to increase the overall number of connections andcurrents, so that we assume that: n X k =2 ζ k * k − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + > n > . The bracket hi denotes the expectation value of the product of fields.7he coefficients ζ n can be set to 0 for n > N , where N is an arbitrary threshold. The factor − ζ accountsfor a minimal number of connections maintained. It depends on external activity J . The contribution for n = 2 and the one proportional to − ζ can be gathered to rewrite the collective potential: ∞ X n =2 ζ ( n ) Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! (22)where ζ ( n ) = ζ n for n > ζ (2) = ζ − ζ . We assume that ζ (2) <
0, so that a nontrivial minimal collectivestate exists. The ”classical” action is thus: −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ σ ∇ θ − ω − (cid:16) J, θ, Z, | Ψ | (cid:17)(cid:19) Ψ ( θ, Z )+ ∞ X n =2 ζ ( n ) Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! (23)We can impose a constraint on the coefficients ζ ( n ) since we are interested in the relative magnitudes of thecoefficients σ θ and quantities such as ω − . As a consequence, we can impose, as a relative benchmark, that: ∞ X n =2 ζ ( n ) Z | Ψ ( θ, Z ) | n − Y i =1 *(cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) +! = Z | Ψ ( θ, Z ) | U Z *(cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) +! ≃ Appendix 1.1.2 and 1.1.3 show that the graphs perturbative expansion associated to (23) can be computedusing the propagator associated to the ”free action”: −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ σ ∇ θ − ω − (cid:0) ¯ J, Z, G (cid:1)(cid:19) Ψ ( θ, Z ) (24)where ω − (cid:0) ¯ J, Z, G (cid:1) is the static inversed frequency defined as the solution of the equation: ω − (cid:0) ¯ J, Z, G (cid:1) = G ¯ J ( Z ) + Z κN ω (cid:0) ¯ J, Z , G (cid:1) ω (cid:0) ¯ J, Z, G (cid:1) G (0 , Z ) T ( Z, θ, Z ) dZ ! (25)where ¯ J ( Z ) is the average over time of J ( θ, Z ) and G (0 , Z ) is the evaluation for θ = θ ′ of the Greenfunction G ( θ, θ ′ , Z ) of the operator: − ∇ θ (cid:18) σ ∇ θ − ω − ( J, θ, Z, (cid:19) (26)where ω − ( J, θ, Z,
0) is the inverse frequency given by (17) for Ψ ≡
0, i.e. ω − ( J, θ, Z,
Ψ) = G ( J ( θ, Z )).The solution of (25) and (26) is computed in Appendix 1.1.2. We find that for an external current decom-posed in a static and dynamic part ¯ J + J ( θ ): ω − ( J, θ, Z,
0) = G (cid:0) ¯ J ( Z ) + J ( θ ) (cid:1) ≃ G (cid:0) ¯ J ( Z ) (cid:1) = arctan (cid:16)(cid:16) X r − X p (cid:17) p ¯ J ( Z ) (cid:17)p ¯ J ( Z ) = 1¯ X r ( Z )and: G ( θ, θ ′ , Z ) = δ ( Z − Z ′ ) exp ( − Λ ( Z ) ( θ − θ ′ ))Λ ( Z ) H ( θ − θ ′ ) (27)8here: Λ ( Z ) = r π s(cid:18) σ ¯ X r ( Z ) (cid:19) + 2 ασ Λ ( Z ) = s(cid:18) σ ¯ X r ( Z ) (cid:19) + 2 ασ − σ ¯ X r ( Z )and H is the heaviside function: H ( θ − θ ′ ) = 0 for θ − θ ′ <
0= 1 for θ − θ ′ > ω − (cid:0) ¯ J ( Z ) , Z, G (0 , Z ) (cid:1) solves: ω − ( Z, G ) = G ¯ J ( Z ) + Z κN ω ( Z , G ) p π r(cid:16) σ ¯ X r ( Z ) (cid:17) + ασ ω ( Z, G ) T ( Z, θ, Z ) dZ (28)Once ω − ( Z, G ) is known, (24) implies that the effective action can be computed by considering the followingaction at the tree-order:Γ (Ψ) = −
12 Ψ † ( θ, Z ) ∇ θ (cid:18) σ ∇ θ − ω − (cid:0) ¯ J ( Z ) , Z, G (cid:1) + α (cid:19) Ψ ( θ, Z ) (29) − ζ Z | Ψ ( θ, Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! + ∞ X n =1 ζ n Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! In the sequel, for the sake of simplicity, the dependency in Z of ¯ X r ( Z ), Λ ( Z ), Λ ( Z ) will be implicit, sothat we will write: ¯ X r ( Z ) ≡ ¯ X r , Λ ( Z ) ≡ Λ, Λ ( Z ) ≡ Λ Appendix 2 computes the partition function with a source field Ω (cid:0) θ ( j ) (cid:1) . This partition function is obtainedthrough the sum of graphs induced by (29). The computations are performed in Appendix 2.3 and yield: Z (Ω) = X n > n ! Z ∆Ω † (cid:16) θ ( i ) f (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) Λ × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Λ (cid:18) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:19) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) ∆Ω (cid:16) θ ( i ) i (cid:17) n Where ∆Ω (cid:0) θ ( i ) (cid:1) is defined as: ∆Ω (cid:0) θ ( i ) (cid:1) = (cid:0) Ω (cid:0) θ ( i ) (cid:1) − Ω (cid:0) θ ( i ) (cid:1)(cid:1) , where Ω (cid:0) θ ( i ) (cid:1) is the source term corre-sponding to a null expectation value of the field , i.e. (cid:10) Ψ (cid:0) θ ( i ) (cid:1)(cid:11) = 0, and where the interaction verticesare: ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) = ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) − ¯ ζ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) (30)= n X l =2 X { k ,...,k l }⊂{ ,...,n } ,k j = i (cid:18) ¯Ξ ( l )1 (cid:16) Z i , (cid:8) Z k j (cid:9) , θ ( i ) i , θ ( i ) f (cid:17) − ζ ( l ) Λ l (cid:19) ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) = n X l =2 X { k ,...,k l }⊂{ ,...,n − } ,k j = i ¯Ξ ( l )1 (cid:16) Z i , (cid:8) Z k j (cid:9) , θ ( i ) i , θ ( i ) f (cid:17) (31)¯ ζ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) = n X l =2 X { k ,...,k l }⊂{ ,...,n − } ,k j = i ζ ( l ) Λ l and: ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) = X { k ,...,k l }⊂{ ,...,n − } ,k j = i ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , (cid:8) Z k j (cid:9)(cid:17) = P { k ,...,k l − }⊂{ ,...,n − } Q k j R θ ( i ) − θ ( kj ) i (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c dl k j p π r(cid:16) σ ¯ X r (cid:17) + ασ ! l +1 δ l R Ψ † (cid:0) θ ( i ) , Z i (cid:1) ∇ θ ( i ) ω − (cid:16) J, θ ( i ) , Z i , | Ψ | (cid:17) Ψ (cid:0) θ ( i ) , Z i (cid:1) dZ il Q i =1 δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) − l k j , Z k j (cid:1)(cid:12)(cid:12) δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) , Z i (cid:1)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) We also define: ¯Ξ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) = lim n →∞ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) (32)ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) = lim n →∞ ¯Ξ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) Appendix 2.4 computes the generating functional for connected correlation functions: W (Ω) = ln Z (Ω) Z (0) . Thisfunctional is found by defining the following operators: O ,n (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19) = − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Λ (cid:18) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:19) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) O , ∞ (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19) = − ¯ ζ + ∇ outθ ( j ) i Λ ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Λ (cid:18) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i (cid:19) (cid:16) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) O , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) = 1 G (cid:16) θ ( i ) i , θ ( i ) f (cid:17) = exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) Λ H (cid:16) θ ( i ) f − θ ( i ) i (cid:17) (33)and the products: O ( n )1 ,n (cid:16)(cid:16) θ ( i ) i , θ ( i ) f (cid:17) i (cid:17) = n Y i =1 O ,n (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19) O ( n )1 , ∞ (cid:16)(cid:16) θ ( i ) i , θ ( i ) f (cid:17) i (cid:17) = n Y i =1 O , ∞ (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19) (1 + O ,n ) ( n ) (cid:16)(cid:16) θ ( i ) i , θ ( i ) f (cid:17) i (cid:17) = n Y i =1 (cid:18) O ,n (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19)(cid:19) (1 + O , ∞ ) ( n ) (cid:16)(cid:16) θ ( i ) i , θ ( i ) f (cid:17) i (cid:17) = n Y i =1 (cid:18) O , ∞ (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19)(cid:19) h O , ∞ i by:exp ( h O , ∞ i ) = X n > n ! D (1 + O , ∞ ) ( n ) E n (34)To compute W (Ω) we also need the expectations of an operator A acting on the tensor products of fields ina product of background sources ∆Ω (cid:16) θ ( i ) i (cid:17) : h A i n = Z " n Y i =1 ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) A " n Y i =1 ∆Ω (cid:16) θ ( i ) i (cid:17) , n > h A i = 1For A acting on (cid:16) ∆Ω (cid:16) θ ( i ) i (cid:17)(cid:17) ⊗ n , the expectation h A i k , k < n for A symmetric is evaluated on the k firstvariables and defines an operator acting on (cid:16) ∆Ω (cid:16) θ ( i ) i (cid:17)(cid:17) ⊗ n − k . Using these definitions, the partition functionwith source rewrites: Z (Ω) = 1 + Z ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) ∆Ω (cid:16) θ ( i ) i (cid:17) + Z Y i =1 ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) (1 + O , ) (2) ∆Ω (cid:16) θ ( i ) i (cid:17) + X n > n ! Z n Y i =1 ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) (1 + O ,n ) ( n ) ∆Ω (cid:16) θ ( i ) i (cid:17) = 1 + X n > n ! D (1 + O ,n ) ( n ) E n ≃ h i + D (1 + O , ) (2) E + X n > n ! D (1 + O , ∞ ) ( n ) E n and we obtain the expression for the generating functional W (Ω): W (Ω) = ln h i + 12 D (1 + O , ) (2) E + X n > n − D (1 + O , ∞ ) ( n ) E n Now that W (Ω) is known, we can derive the formal expression for the effective action. It is computed usingthe relation between the background field and the source field:Ψ (cid:16) θ ( i )1 (cid:17) = δW (Ω) δ Ω † (cid:16) θ ( i )1 (cid:17) = Z θ ( i )1 D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G (cid:16) θ ( i )1 , θ ( i ) (cid:17) ∆Ω (cid:16) θ ( i ) (cid:17) dθ ( i ) Ψ † (cid:16) θ ( i )1 (cid:17) = δW (Ω) δ Ω (cid:16) θ ( i )1 (cid:17) = Z θ ( i )1 ∆Ω † (cid:16) θ ( i ) (cid:17) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G (cid:16) θ ( i ) , θ ( i )1 (cid:17) dθ ( i ) where G (cid:16) θ ( i )1 , θ ( i ) (cid:17) is the free propagator defined above in (33).To compute the effective action, we consider (168) evaluated at Ψ (cid:0) θ ( i ) (cid:1) , so that Ω Ψ ( θ ( i ) ) (cid:0) θ ( i ) (cid:1) = 0:11 D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n Ψ ( θ ( i ) ) G (cid:16) Ω Ψ ( θ ( i ) ) =0 (cid:16) θ ( i ) (cid:17)(cid:17) = Ψ (cid:16) θ ( i ) (cid:17)(cid:18) Ω † Ψ ( θ ( i ) ) =0 (cid:16) θ ( i ) (cid:17)(cid:19) G D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n Ψ ( θ ( i ) ) = Ψ † (cid:16) θ ( i ) (cid:17) Appendix 2.5 shows that the effective action then writes:Γ (Ψ) = h i + D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n (35) −
12 ln h i + 12 D (1 + O , ) (2) E + X n > n ! D (1 + O , ∞ ) ( n ) E n −
12 Ψ † (cid:16) θ ( i ) (cid:17) G − D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n − ( θ ( i ) ) Ψ (cid:16) θ ( i ) (cid:17) + H.C.
Appendix 4 computes the corrections to the zeroth order effective action (29) by using graphs expansion(the final formula is given in Appendix 4.3). This yields an expanded form of (29). We show that for weakvalues of the interaction parameters, the effective action takes the form:Γ (Ψ) (36)= −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17)(cid:19) + α (cid:19) Ψ ( θ, Z )+ X ζ ( n ) n ! Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) G (0 , Z j ) + Z θ ( i ) − θ ( j ) i | Zi − Zj | c dl j Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n + X Z n − † (cid:16) θ ( i ) f , Z i (cid:17) ˆ V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × (cid:20)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) D (cid:16) θ ( j ) f , θ ( j ) i , Z j , (cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (cid:21) n − where ω − ( J ( θ ) , θ, Z, G ) is solution of: ω − ( θ, Z ) = G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) G (0 , Z ) dZ V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) = 1 + (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) − (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17)(cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) × − ζ ( n ) Λ n + ∇ outθ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } ,θ ( i ) i ,θ ( i ) f (cid:17)(cid:16) θ ( i ) f − θ ( i ) i (cid:17) ! ˆΞ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i + exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) (cid:18) − ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) + (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) (cid:19) − (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) D (cid:16) θ ( j ) f , θ ( j ) i , Z j , (cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j = Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) ∇ outθ ( j ) i Λ θ ( j ) f − θ ( j ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j In these expressions, the values of ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) and ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) defined in(30) and (31) are approximated by:¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) ≃ Z θ ( i ) f θ ( i ) i n − X l =1 C ln − δ l − ω − (cid:16) J, θ ( i ) , Z i , | Ψ | (cid:17) Λ l l − Q m =1 δ (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) − | Z i − ¯ Z | c , ¯ Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) dθ ( i ) (37)ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) ≃ Z θ ( i ) f θ ( i ) i n − X l =1 C ln − δ l − ω − (cid:16) J, θ ( i ) , Z i , | Ψ | (cid:17) Λ l l − Q m =1 δ (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) − | Z i − ¯ Z | c , ¯ Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − ζ ( l ) Λ l | Ψ( θ,Z ) | = G (0 ,Z ) dθ ( i ) (38)where ¯ Z is the centre of the thread and depends only on Z i , θ ( i ) i , θ ( i ) f and ¯ Z . Appendix 4.4 shows that, for a wide range of parameters, the effective action has a minimum. The corre-sponding background field decomposes into a constant part Ψ and a contribution depending on the external13urrent. We show that for slowly varying currents J ( θ, Z i ), and for (cid:12)(cid:12) ζ ( n ) (cid:12)(cid:12) > ω − ( J ( θ ) , θ, Z, G ) the minimumof Γ (Ψ) given in (36) is reached for Ψ ( θ, Z ) = Ψ ( θ, Z ) + δ Ψ ( θ, Z ) and Ψ † ( θ, Z ) = Ψ † ( θ, Z ) + δ Ψ † ( θ, Z )where | δ Ψ ( θ, Z ) | << | Ψ ( θ, Z ) | and (cid:12)(cid:12) δ Ψ † ( θ, Z ) (cid:12)(cid:12) << (cid:12)(cid:12)(cid:12) Ψ † ( θ, Z ) (cid:12)(cid:12)(cid:12) . The fields Ψ ( θ, Z ) and Ψ † ( θ, Z ) describethe minimum of the potential: α Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) dZ i + X ζ ( n ) n ! G (0 , Z j ) + Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n This minimum exists for α << (cid:12)(cid:12) ζ (2) (cid:12)(cid:12) large. It is reached for a value X of R (cid:12)(cid:12) Ψ (cid:0) θ ( i ) , Z i (cid:1)(cid:12)(cid:12) dZ i . Upto an irrelevant phase, Ψ (cid:0) θ ( i ) , Z i (cid:1) = Ψ † ( θ, Z ) = q X V , where V is the volume of the thread. Appendix4.4.2 finds the expression for δ Ψ ( θ, Z ) and δ Ψ † ( θ, Z ). They satisfy the following first order equations:0 ≃ (cid:18) − (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19)(cid:19) + U ′′ ( X ) (cid:19) δ Ψ ( θ, Z ) − Z δ Ψ † ( θ , Z ) p X ∇ θ δω − ( J ( θ ) , θ , Z , G + X ) δ | Ψ ( θ, Z ) | ! Ψ ( θ , Z ) dθ dZ − (cid:0) ∇ θ ω − ( J ( θ ) , θ, Z, G + X ) (cid:1) Ψ ( θ, Z ) − Z p X ∇ θ δω − ( J ( θ ) , θ, Z, G + X ) δ | Ψ ( θ , Z ) | ! Ψ ( θ, Z ) δ Ψ ( θ , Z ) dθ dZ for δ Ψ ( θ, Z ), and:0 = 12 δ Ψ † ( θ, Z ) (cid:18) −∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19) + U ′′ ( X ) (cid:19) − Z δ Ψ † ( θ , Z ) p X ∇ θ δω − ( J ( θ ) , θ , Z , G + X ) δ | Ψ ( θ, Z ) | ! Ψ ( θ , Z ) dθ dZ for δ Ψ † ( θ , Z ). Appendix 4.4.2 shows that solutions are: δ Ψ † ( θ, Z ) = 0and: δ Ψ ( θ, Z ) = X n Z (cid:0) − C ¯Λ (cid:1) n +1 n Y l =1 exp (cid:16) − ¯Λ (cid:16) θ l − θ l +1 − | Z l − Z l +1 | c (cid:17)(cid:17) K Γ q ¯Λ K erf r K s θ l − (cid:18) θ l +1 + | Z l − Z l +1 | c (cid:19)! × ω ( J ( θ l ) , θ l , Z l , G + X ) } ˜ G ( θ n , θ n +1 , Z l ) (cid:0) ∇ θ ω − ( J ( θ n +1 ) , θ n +1 , Z l , G + X ) (cid:1) √ X n +1 Y l =2 dθ l dZ l with the convention that θ = θ . The constants K , Γ and ¯Λ depend on the parameters of the system. Thekernel ˜ G ( θ n , θ n +1 ) is computed in Appendix 4.4.2. It is given by:˜ G ( θ, θ ′ ) = exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ − σ ¯ X e ! ( θ − θ ′ ) − h ω − ( J ( θ ) ,θ,Z, G + X )2 i θθ ′ !p π r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ H ( θ − θ ′ ) (39) ≃ exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ − σ ¯ X e ! + (cid:20) ∇ θ ω − ( ¯ J,θ,Z, G + X ) (cid:21)! ( θ − θ ′ ) !p π r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ H ( θ − θ ′ )14here the upper bar on a quantity stands for the average computed over the period θ − θ ′ .The inverse frequency ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) is solution of: ω − ( θ, Z ) (40)= G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ( G (0 , Z ) + X ) dZ The field Ψ (cid:0) θ ( j ) , Z j (cid:1) is the - phase dependent - background field. In the trivial phase, it is equal to 0, sothat the effective action matches with the ”classical” one. In a non-trivial phase, Ψ (cid:0) θ ( j ) , Z j (cid:1) is not equal tozero and may be time dependent. It describes the accumulation of current, or signals, that shapes the longterm dynamics of frequencies. This explains the contribution ¯Λ in the first term.To conclude this section by expanding (40) in terms of current at the second order of approximation: ω − ( J, θ, Z ) = G [ J ( θ, Z ) , G + X ]+ κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) − ω ( θ, Z ) ω ( θ, Z ) G ′ [ J ( θ, Z ) , G ] ¯ G (0 , Z ) dZ = G (cid:2) J ( θ, Z ) , ¯ G (cid:3) − κN R T ( Z, Z ) (cid:16) ω (cid:16) θ − | Z − Z | c , Z (cid:17) − ω ( θ, Z ) (cid:17) G ′ (cid:2) J ( θ, Z ) , ¯ G (cid:3) ¯ G (0 , Z ) dZ ≃ G (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3) − R κT ( Z,Z ) N (cid:16) G h J (cid:16) θ − | Z − Z | c , Z (cid:17) , ¯ G i − G (cid:2) J ( θ, Z ) , ¯ G (cid:3)(cid:17) G ′ (cid:2) J ( θ, Z ) , ¯ G (cid:3) ¯ G (0 , Z ) dZ and: ω ( J, θ, Z )= 12 (cid:16) F (cid:2) J ( θ, Z ) , ¯ G (cid:3) + (cid:16)(cid:0) F (cid:2) J ( θ, Z ) , ¯ G (cid:3)(cid:1) +4 (cid:18)Z κN T ( Z, Z ) (cid:18) F (cid:20) J (cid:18) θ − | Z − Z | c , Z (cid:19) , ¯ G (cid:21) − F (cid:2) J ( θ, Z ) , ¯ G (cid:3)(cid:19) ¯ G (0 , Z ) dZ (cid:19) × F ′ (cid:2) J ( θ, Z ) , ¯ G (cid:3)(cid:1) (cid:17) where: ¯ G (0 , Z ) ≃ G (0 , Z ) + X and the constants C , K and ¯Λ that depend on the system are defined in Appendix 1.3.2.3.Incidentally, we note that a non-trivial minimum depending on the system parameters should allow forphase transition in the system of frequencies. This question is left for further work. The correlation functions can be found by computing the derivatives of the effective action at the classicalbackground field. This is done by using a graph expansion of the effective action. The two-points correlationfunction is defined by δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) . Appendix 3.1 shows that it satisfies a coupled equation with δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) .15ctually, defining:[∆Ω] = (cid:18) ∆Ω∆Ω † (cid:19) (41) K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) = K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K † , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K † , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) h X (cid:16) θ ( i )1 , θ ( i ) f (cid:17)i = (cid:18) h (1+ O , ) (2) i + P n > n − h (1+ O , ∞ ) ( n ) i n − h i + h (1+ O , ) (2) i + P n > n ! h (1+ O , ∞ ) ( n ) i n G (cid:19) − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) = X (cid:16) θ ( i )1 , θ ( i ) f (cid:17) ! the vector [∆Ω] (cid:16) θ ( i )1 (cid:17) satisfies the dynamic equation: δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = Z K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [∆Ω] (cid:16) θ ( i )2 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) dθ ( i )2 + h X (cid:16) θ ( i )1 , θ ( i ) f (cid:17)i (42)The operator valued coefficients: K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) = K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) are defined by: K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) = − Z ∆Ω † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) A (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + B (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 ! G d (cid:16) θ ( i )2 (cid:17) ′ +∆Ω (cid:16) θ ( i )1 (cid:17) Ψ † (cid:16) θ ( i )2 (cid:17)(cid:17) K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) = − Z G A † (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ + B † (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ ! G ∆Ω (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ +∆Ω † (cid:16) θ ( i )1 (cid:17) Ψ (cid:16) θ ( i )2 (cid:17)(cid:17) K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) = − Z A (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + B (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 ! G ∆Ω (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ +∆Ω (cid:16) θ ( i )1 (cid:17) Ψ (cid:16) θ ( i )2 (cid:17)(cid:17) K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) = K † , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) G is the operator whose kernel is defined in (33), and: A (cid:16) θ ( i )1 (cid:17) = G − ∗ P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) = A (cid:16) θ ( i )1 , θ ( i ) f (cid:17) B (cid:16) θ ( i )1 (cid:17) = G − ∗ (cid:0) O , (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) = B (cid:16) θ ( i )1 , θ ( i ) f (cid:17) and where the operators ¯ O ,n are defined by:(1 + O ,n ) G = (cid:0) O ,n (cid:1) ∗ G The symbol ∗ denotes the convolution product, so that we have:¯ O ,n = − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) ΛThe solution of (42) is given by: δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i )1 , θ ( i ) f (cid:17) (43)+ exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! (cid:18)Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ (cid:0) G − (cid:1) ∗ θ ( i )1 − (1 − exp ( − x )) (cid:18)Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) O G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ (cid:19) ×G − (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , ∞ (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ∗ θ ( i )1 − exp ( − x ) (cid:18)Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) O G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ (cid:19) × G − (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! ∗ θ ( i )1 ! × (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) Ψ ! (cid:16) θ ( i )1 (cid:17) h O , ∞ i is given by (34)): O = 1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) G − (cid:0) O , ∞ (cid:1) O = 1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) G − (cid:0) O , (cid:1) x = h O , ∞ i y = D (1 + O , ) (2) E z = h O , ∞ i − h i = h O , ∞ i and with the condition: θ ( i ) f < (cid:16) θ ( i )2 (cid:17) ′ < θ ( i )1 that is implied by the Heaviside functions in the integrals defining the interaction terms. The factor ¯ N (( θ i ))is given by:¯ N (( θ i )) ≃ * − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ × − (cid:16) (1 − exp ( − x )) x (cid:10)(cid:0) O , ∞ (cid:1)(cid:11) Ψ θ ( i ) + y exp ( − x ) (cid:10)(cid:0) O , (cid:1)(cid:11) Ψ θ ( i ) (cid:17)(cid:16) h i Ψ (cid:17) (cid:0) − x ) (cid:0) − z + ( y − x ) (cid:1)(cid:1) which is also approximated locally: δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! × Ψ † (cid:16) θ ( i ) f (cid:17) G − (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:0) G − (cid:1) θ ( i )1 − Ψ † (cid:16) θ ( i ) f (cid:17) O (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! × G − (1 − exp ( − x )) (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , ∞ (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! θ ( i )1 − Ψ † (cid:16) θ ( i ) f (cid:17) O (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! × G − exp ( − x ) (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! θ ( i )1 × (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! Ψ (cid:16) θ ( i )1 (cid:17) δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + "Z θ ( i )1 G − F Θ Ψ exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! "Z θ ( i ) f Ψ † G − F Θ − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! × "Z θ ( i ) f Ψ † G − (1 − exp ( − x )) F Θ (cid:0) O , ∞ (cid:1) − "Z θ ( i )1 G − F Θ (1 + O , ) Ψ exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! "Z θ ( i ) f Ψ † G − exp ( − x ) F Θ (1 + O , ) where: F = 1 + exp ( − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19) (44)Θ = (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) In the local approximation for θ ( i )1 ≃ θ ( i ) f , we thus obtain: δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + "Z θ ( i )1 G − F Θ Ψ θ ( i )1 Ψ † G − F Θ − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − (1 − exp ( − x )) F Θ (cid:0) O , ∞ (cid:1) − "Z θ ( i )1 G − F Θ (cid:0) O , (cid:1) Ψ θ ( i )1 Ψ † G − exp ( − x ) F Θ (cid:0) O , (cid:1) and:¯ N (( θ i )) ≃ * − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1) ((1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + ( y (1 + O , ) − x (1 + O , ∞ )))) + Ψ × − (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:16) (1 − exp ( − x )) x h (1 + O , ∞ ) i Ψ θ ( i ) + y exp ( − x ) h (1 + O , ) i Ψ θ ( i ) (cid:17)D ((1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + ( y (1 + O , ) − x (1 + O , ∞ )))) E Ψ δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) − = G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + F (cid:0) Ψ † G − Ψ (cid:1) ( α ( x ) − F x exp ( − x )) − ¯ ζ + ∇ outθ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) − F (cid:0) Ψ † G − Ψ (cid:1) ( β ( x ) − F exp ( − x )) y − ¯ ζ + ∇ outθ ( i ) i ′ Λ ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) + β ( x )( x + exp ( − x ) ( h i − x + ( y − x ))) This expression will be used below to compute the effective frequencies.
The frequencies can be found using the two-points correlation functions. We have found the solution for thebackground field. This solution is written Ψ (cid:0) θ ( j ) , Z j (cid:1) + δ Ψ ( θ, Z ). The second derivative: δ Γ (Ψ) δ Ψ † (cid:16) θ ( i ) i , Z i (cid:17) δ Ψ (cid:16) θ ( i ) i , Z i (cid:17) (45)evaluated at Ψ (cid:0) θ ( j ) , Z j (cid:1) + δ Ψ ( θ, Z ) yields the inverse Green function. Appendix 5 computes the quadraticpart Ψ † ( θ, Z ) ∇ θ N [Ψ] Ψ ( θ, Z ) of (45) and identifies N [Ψ] with the inverse effective frequency at position( θ, Z ) in time and space. It yields: ω − e ( J ( θ ) , θ, Z ) ≃ ω − (cid:16) J ( θ ) , θ, Z, G + (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( j ) , Z j (cid:17)(cid:12)(cid:12)(cid:12)(cid:17) (46)+ ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) + ω − (cid:16) J ( θ ) , θ, Z, | Ψ | (cid:17) where ω (cid:16) J (cid:0) θ ( i ) (cid:1) , θ ( i ) , Z i , ¯ G + | Ψ | (cid:17) is the solution of: ω (cid:16) θ ( i ) , Z (cid:17) = F J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) (cid:18) ¯ G (0 , Z ) + (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( j ) , Z (cid:17)(cid:12)(cid:12)(cid:12) (cid:19) dZ (47)The two other terms, ω − and ω − , are corrections due to the interactions in the system.First, function ω − is defined by its derivatives at G (0 , Z ): δ n ω − ( J ( θ ) , θ, Z, G ) δ n G (0 , Z ) = δ n ω − ( J ( θ ) , θ, Z, G ) δ n G (0 , Z ) ˇΞ ,l (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f , | Ψ | (cid:17) (48)where:ˇΞ ,l (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f , | Ψ | (cid:17) = ∞ X p =0 p ! ˆΞ ,p + l (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (cid:19) p ω (cid:16) J ( θ ) , θ, Z, R G (0 , Z j ) + (cid:12)(cid:12) Ψ (cid:0) θ ( j ) , Z j (cid:1)(cid:12)(cid:12) (cid:17) satisfies an equationsimilar to (47) whose coefficients are modified by ˇΞ ,l (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f , | Ψ | (cid:17) : ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) (49)= ˆ G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) × G (0 , Z ) + (cid:12)(cid:12)(cid:12)(cid:12) Ψ + δ Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! dZ ! The derivatives of ˆ G estimated at G (0 , Z ) are computed in Appendix 5. The function ˆ G is of order ¯ ζ , theaverage magnitude of the coefficients ¯ ζ n . For weak interaction, we show that:ˆ G ( n ) (cid:0) ¯ J + ¯ G (0 , Z i ) (cid:1) (50) ≃ ω ( J, θ, Z ) ¯Ξ (cid:0) ¯ J, | Z i − Z | (cid:1) κN T ( Z, Z ) ! n × − Z κN ω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ ! F ′ [ J, ω, θ, Z, Ψ] ! × (cid:16) ˇΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f , | Ψ | (cid:17)(cid:17) Second, frequency ω − ( J ( θ ) , θ, Z, Ψ) is given by: ω − ( J ( θ ) , θ, Z, Ψ)= F (cid:0) Ψ † G − Ψ (cid:1) × ( α ( x ) − F x exp ( − x )) (cid:16) ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) − ( β ( x ) − F exp ( − x )) y (cid:16) ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) ≃ − F (cid:16)p X U ′′ ( X ) δ Ψ ( θ, Z ) − ∇ θ ω − ( J ( θ ) , θ, Z, G + X ) X (cid:17) × ( α ( x ) − F x exp ( − x )) (cid:16) ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) − ( β ( x ) − F exp ( − x )) y (cid:16) ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) where F is given by (44) We look for a static solution of (46) for a constant background Ψ (cid:0) θ (1) , Z (cid:1) ≃ Ψ ( Z ) and constant cur-rent, i.e. J = ¯ J , ω ( θ, Z ) = ω ( Z ). For a static solution, (cid:0) Ψ † G − Ψ (cid:1) = 0, or equivalently: δ Ψ ( θ, Z ) = ∇ θ ω − ( J ( θ ) , θ, Z, G + X ) = 0, and:¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ out θ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i = 0 ω − e ( J ( θ ) , θ, Z ) = ω − (cid:0) ¯ J, Z, G (0 , Z ) + X (cid:1) + ω − (cid:0) ¯ J , Z, G (0 , Z ) + X (cid:1) where ω (cid:0) ¯ J ( Z ) , G (0 , Z i ) + X (cid:1) is solution of: ω ( Z ) = F (cid:18) ¯ J + κN Z T ( Z, Z ) ω ( Z ) ω ( Z ) W (cid:18) ω ( Z ) ω ( Z ) (cid:19) ¯ G (0 , Z i ) dZ (cid:19) (51)21nd: ¯ G (0 , Z i ) ≃ G (0 , Z i ) + X (52)Moreover, in the absence of external source, i.e. for J ( θ, Z ) = 0, the solution of (51) can be written ω ( Z ),which satisfies: ω ( Z ) = F (cid:18) κN Z T ( Z, Z ) ω ( Z ) ω ( Z ) W (cid:18) ω ( Z ) ω ( Z ) (cid:19) ¯ G (0 , Z i ) dZ (cid:19) (53)where N R dZ is normalized to 1. We linearize the dynamic equation for frequencies around some constant equilibrium. We will generalize theresult to a position-dependent equilibrium in the next paragraph.A linearized equation around static equilibrium can be found by considering:Ψ (cid:16) θ ( j ) , Z j (cid:17) = Ψ ( Z j ) + δ Ψ (cid:16) θ ( j ) , Z j (cid:17) where: (cid:12)(cid:12)(cid:12) δ Ψ (cid:16) θ ( j ) , Z j (cid:17)(cid:12)(cid:12)(cid:12) << | Ψ ( Z j ) | For a translation independent transfer function, i.e. T ( Z, Z ) = T ( Z − Z ), and d >> ω = G (cid:18) T W (1)¯Λ (cid:19) (54)where: T = κN Z T ( Z, Z ) dZ We will also assume that the transfer functions are symmetric, that is: T ( Z, Z ) = T ( Z , Z ) (55)To find the linearized equation for frequencies around the constant background (54), we first note that, given(19) and (55), one has: ∂∂ω ( θ, Z ) (cid:18) κN Z W (cid:18) ωω (cid:19) dZ (cid:19) ω ( θ,Z )= ω + ∂∂ω ( θ, Z ) (cid:18) κN Z W (cid:18) ωω (cid:19) dZ (cid:19) ω ( θ,Z )= ω = κN Z ( W ′ (1) − W ′ (1)) dZ = 0To write the linearized equation (46) around ω , we first solve (47). We start by finding a linearized equationfor ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) , considering the other terms in the right hand side of (46) as corrections. Theequation for ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) is: ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) = F J ( θ ) + Z κT ( Z, Z ) N ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) × G (0 , Z ) + (cid:12)(cid:12)(cid:12)(cid:12) Ψ + δ Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! dZ ! ω ( J, θ, Z ) = ω + Ω ( θ, Z ). A first approximation for small variation Ω ( θ, Z ) around ω allows to rewrite (47) as a linearized expansion around the solution of (51). The function F in (47) can beexpressed as: F J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) (56) × G (0 , Z ) + (cid:12)(cid:12)(cid:12)(cid:12) Ψ + δ Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! dZ ! ≃ F J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω + Ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω + Ω (cid:16) θ − | Z − Z | c , Z (cid:17) × (cid:18) ¯ G (0 , Z ) + Ψ δ Ψ (cid:18) θ − | Z − Z | c , Z (cid:19) dZ (cid:19)(cid:19) Similarly, the transfer function can be written: T ( Z, Z , ω, ω ) ≡ T ( Z, Z ) W (cid:18) ωω (cid:19) ≃ T ( Z, Z ) W J, θ, Z ) − Ω (cid:16) J, θ − | Z − Z | c , Z (cid:17) ω We also use the local linear approximation for δ Ψ ( θ, Z ) derived in Appendix 4.4.2: δ Ψ ( θ, Z ) ≃ N ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X ) − N ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X ) (57)The expansion of (56) for a non-static current is:Ω ( θ, Z ) = (cid:16) ˆ f + N (cid:17) ∇ θ Ω ( θ, Z ) + ˆ f ∇ Z Ω ( θ, Z ) + (cid:16) ˆ f − N (cid:17) ∇ θ Ω ( θ, Z ) + J (cid:16) θ ( i ) , Z i (cid:17) Γ (58)where we defined: ˆ f = W ′ (1) − W (1) c Γ , ˆ f = ( W (1) − W ′ (1)) Γ c Γ = κN X r Z | Z − Z | T ( Z, Z ) dZ ω ¯ G (0 , Z i ) Γ Γ = κ N X r R ( Z − Z ) T ( Z, Z ) dZ ω ¯ G (0 , Z i ) Γ Γ = F ′ (cid:18) κN Z T ( Z, Z ) W (1) dZ ¯ G (0 , Z i ) (cid:19) To include the corrective terms in (46), we rewrite this equation as: ω e ( J ( θ ) , θ, Z ) (59) ≃ ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) − (cid:16) ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) + ω − ( J ( θ ) , θ, Z, Ψ) (cid:17) × (cid:16) ω (cid:16) ¯ J, Z, G + | Ψ | (cid:17)(cid:17) The corrections are proportional to two terms ω − and ω − : ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) (cid:0) ω (cid:0) ¯ J, Z, G + X (cid:1)(cid:1) ω − ( J ( θ ) , θ, Z, Ψ) (cid:0) ω (cid:0) ¯ J, Z, G + X (cid:1)(cid:1) In the linear approximation, using (47) and (49), we can combine the two first terms in (59): ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) − (cid:16) ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17)(cid:17) (cid:0) ω (cid:0) ¯ J, Z, G + X (cid:1)(cid:1) ≃ ˆ ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) where ˆ ω satisfies:ˆ ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) = ( F − δF ) J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω + Ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω + Ω (cid:16) θ − | Z − Z | c , Z (cid:17) × (cid:18) ¯ G (0 , Z i ) + Ψ δ Ψ (cid:18) θ − | Z − Z | c , Z (cid:19) dZ (cid:19)(cid:19) with F − δF = F − ˆ GF . The factor ˇΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f , | Ψ | (cid:17) arising in the definition of ω − (see48) shows that ω − is a deformation of the effective frequency due to the interaction of the oscillations withthe global potential. As expected, the potential stabilizes the oscillations and thus mitigates the amplitudeof the waves.The second correction to the frequency results from the evolution of the background. Given (57), ω − ( J ( θ ) , θ, Z, Ψ) can be written in terms of frequencies: ω − ( J ( θ ) , θ, Z, Ψ) ≃ − F (cid:16)(cid:16)p X U ′′ ( X ) N + ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) X (cid:17) ∇ θ ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) − N ∇ θ ω (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17)(cid:17) × ( α ( x ) − F x exp ( − x )) (cid:16) ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) − ( β ( x ) − F exp ( − x )) y (cid:16) ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) To sum up, in the local approximation, frequencies are described by a wave equation whose form dependson the background field. This wave is deformed by the stabilization potential and the evolution of thebackground itself.
In the previous paragraph, we considered translation-invariant transfer functions. This hypothesis, althoughcorrect in first approximation, does not hold in general. Finite volume of the system or border conditions,for instance, may invalidate this hypothesis. We will thus consider transfer functions of the form T ( Z, Z ).To make things simpler, we will dismiss the corrections to the frequencies due to the potential and thebackground field.The derivation of the linearized expansion of (46) around ω ( Z ) is similar to that of (58) but now yields: σ θ ∇ θ ˆΩ ( θ, Z ) = g ( Z ) ˆΩ ( θ, Z ) − g ( Z ) ∇ θ ˆΩ ( θ, Z ) + g ( Z ) ∇ Z ˆΩ ( θ, Z ) (60)24here we defined:ˆΩ ( θ, Z ) = Ω ( θ, Z ) ω ( Z ) g ( Z ) = Γ ′ ( Z ) − Γ ( Z ) cσ θ + Γ ′ ( Z ) − Γ ( Z ) c g ( Z ) = Γ ′ ( Z ) − Γ ( Z ) σ θ + Γ ′ ( Z ) − Γ ( Z ) c Γ ( Z ) = κN X r Z | Z − Z | T ( Z, Z ) ω ( Z ) ω ( Z ) W (cid:16) ω ( Z ) ω ( Z ) (cid:17) dZ ω ( Z ) r π (cid:16) X r (cid:17) + π α Γ ( Z )Γ ′ = κN X r Z | Z − Z | T ( Z, Z ) ω ( Z ) ω ( Z ) W ′ (cid:16) ω ( Z ) ω ( Z ) (cid:17) dZ ω ( Z ) r π (cid:16) X r (cid:17) + π α Γ Γ = κ N X r R ( Z − Z ) T ( Z, Z ) ω ( Z ) ω ( Z ) W (cid:16) ω ( Z ) ω ( Z ) (cid:17) dZ ω ( Z ) r π (cid:16) X r (cid:17) + π α Γ Γ ′ ( Z ) = κ N X r R ( Z − Z ) T ( Z, Z ) ω ( Z ) ω ( Z ) W ′ (cid:16) ω ( Z ) ω ( Z ) (cid:17) dZ ω ( Z ) r π (cid:16) X r (cid:17) + π α Γ ( Z )Γ ( Z ) = G ′ κN Z T ( Z, Z ) W (cid:16) ω ( Z ) ω ( Z ) (cid:17) dZ r π (cid:16) X r (cid:17) + π α Note that equation (60) is a wave equation in an inhomogeneous medium.
A straightforward generalization of (58) can be derived by considering anisotropic transfer functions. Untilnow we have assumed that: Z ( Z − Z ) i ( Z − Z ) j T ( Z, Z ) ω ( Z ) ω ( Z ) W (cid:18) ω ( Z ) ω ( Z ) (cid:19) dZ = δ i,j where δ i,j is the Kronecker symbol. Relaxing this condition, we can replace f ( Z ) → f ij ( Z ), g ( Z ) → g ij ( Z ) = f ij ( Z )1+ f ( Z ) . Equation (58) becomes: ∇ θ Ω ( θ, Z ) = g ( Z ) Ω ( θ, Z ) + g ( Z ) ∇ θ Ω ( θ, Z ) + g ij ( Z ) ∇ Z i ∇ Z j Ω ( θ, Z )for distributions: f ( Z ) = ( ω W ′ (1) − W (1)) Γ ij Γ ij = κ N X r R ( Z − Z ) i ( Z − Z ) j T ( Z, Z ) dZ ω r π (cid:16) X r (cid:17) + π α Γ Implications
We compute the Green functions associated to equations (58) and (60) to assess the implications of the waveequations. The Green functions allow to find the propagation of an external signal at some particular pointsto the all thread.
The Green function of (58) and (60) are found using the usual Fourier representation. We will focus on theretarded Green functions that model the wave propagation initiated by a source.
Let us first consider (58). For g <
0, oscillations dampen over time at a rate | g | , and are magnified in timefor g >
0. Since we assumed oscillations of small magnitude around the equilibrium, this implies that ourmodel breaks down above a certain range of amplitudes. A non-linear mechanism of regulation is probablyinvolved at some point to drive the system back to the equilibrium. However, we are mainly interested inoscillatory patterns, and will assume | g | <<
1. As a consequence, equation (58) reduces to: ∇ θ Ω ( θ, Z ) = g ∇ Z Ω ( θ, Z ) + g Ω ( θ, Z ) (61)which is a Klein Gordon equation. We normalize it by setting g = 1 and write g = m . Using the Fourierrepresentation of (61), the retarded Green function of (58) is given by: G ( Z, Z ′ , t, t ′ ) = Z dk exp ( ik. ( Z − Z ′ ) − iω k ( t − t ′ )) ω k H ( t − t ′ ) (62)with ω k = √ k + m . The integral can be computed and we have: G ( Z, Z ′ , t, t ′ ) = H ( t − t ′ ) π δ ( t − t ′ ) − mJ (cid:18) m q ( t − t ′ ) − ( Z − Z ′ ) (cid:19)q ( t − t ′ ) − ( Z − Z ′ ) (63)where J is the n = 1 Bessel function. To inspect the implications of (63), it is sufficient to approximate(63) for small oscillations. This corresponds to g >> g , i.e. m >
1. As a consequence, we can expand √ k + m at the lowest order in k m , and write (62) as: G ( Z, Z ′ , t, t ′ ) ≃ Z dk exp (cid:16) ik. ( Z − Z ′ ) − i (cid:16) m + k m (cid:17) ( t − t ′ ) (cid:17) m H ( t − t ′ ) (64)up to terms of order m . Computing the Fourier transform in (64), the function G ( Z, Z ′ , t, t ′ ) can beapproximated by: G ( Z, Z ′ , t, t ′ ) = exp i m Z − Z ′ ) ( t − t ′ ) − m ( t − t ′ ) !! H ( t − t ′ ) (65)Equation (65) shows that the Green function G ( Z, Z ′ , t, t ′ ) represents the path integral of a particle underthe constant potential m . The Green function of equation (60) is a generalization of (63). It has been studied in the context of covariantquantum field theory, but (65) shows that we can produce a path integral formulation for the Green function.If g ( Z ) varies slowly with Z , the analog of (65) with non-constant coefficients is:26 ( Z, Z ′ , t, t ′ ) = Z exp i Z z ( t )= Zz ( t ′ )= Z ′ q g ( z ( s )) g ( z ( s )) (cid:18) dz ( s ) ds (cid:19) − p g ( z ( s )) ds Dz ( s ) H ( t − t ′ ) (66)where the sum is over paths z ( s ) starting from Z ′ and ending at Z in a time span of t − t ′ . The derivationof (66) is straightforward. Neglecting g ( Z ) as in the derivation of (61), (60) writes: σ θ ∇ θ ˆΩ ( θ, Z ) = g ( Z ) ˆΩ ( θ, Z ) + g ( Z ) ∇ Z ˆΩ ( θ, Z )then, cutting the time span t − t ′ into slices ∆ t such that g ( Z ) and g ( Z ) can be considered as constant ina domain of radius c ∆ t , the Green function for a time span ∆ t is given by a formula similar to (65), exceptthat g ( Z ) = 1: G ( z ( s + ∆ t ) , z ( s ) , ∆ t ) = exp i q g ( z ( s )) g ( z ( s )) z ( s + ∆ t ) − z ( s )) ∆ t − g ( z ( s )) ∆ t (67)The convolution of (67) over the time slices then yields (65). The Green function (65) allows to compute the diffusion of an external source along the thread by convolution.Assume an external source: J ( t, Z ) = exp ( − iω t ) δ ( Z − Z ) (68)which describes a signal located in Z , with frequency ω . Using (65), the amplitude Ω ( t, Z ) is:Ω ( t, Z ) = Z exp i m Z − Z ) ( t − t ′ ) − ω t − ( m − ω ) ( t − t ′ ) !! H ( t − t ′ ) dt ′ = exp ( − iω t − i √ m | ( m − ω ) | | Z − Z | + iπ ) p | ( m − ω ) | and for a signal including a whole range of frequencies:ˆ f ( t, Z ) = Z f ( ω ) exp ( − iω t ) dω (69)the corresponding response of the thread is:Ω ( t, Z ) = Z exp ( − iω t − i √ m | ( m − ω ) | | Z − Z | + iπ ) p | ( m − ω ) | f ( ω ) dω Assume that the range of frequencies in (69) is such that m − ω >
0. Then:Ω ( t, Z ) = Z exp ( − iω t − i √ m ( m − ω ) | Z − Z | + iπ ) p | ( m − ω ) | f ( ω ) dω = Z exp ( − iω ( t − √ m | Z − Z | )) p | ( m − ω ) | f ( ω ) dω exp (cid:16) − i ( √ m ) | Z − Z | + iπ (cid:17)p | ( m − ω ) | To simplify, we also assume that the frequencies of the signal satisfy | ω | << m , so that:Ω ( t, Z ) ≃ ˆ f (cid:0) t − √ m | Z − Z | , Z (cid:1) exp (cid:16) − i ( √ m ) | Z − Z | + iπ (cid:17) m (70)27t time t , the frequencies present the whole past history of the signal, which is thus recorded in the systemof oscillations. The result (70) can be extended for several independent sources located in two points Z , Z emitting some signal ˆ f ( t ) and ˆ f ( t ) with frequencies below m . In that case, the response is:Ω ( t, Z ) ≃ ˆ f (cid:0) t − √ m | Z − Z | (cid:1) exp (cid:16) − i ( √ m ) | Z − Z | + iπ (cid:17)p | ( m − ω ) | (71)+ ˆ f (cid:0) t − √ m | Z − Z | (cid:1) exp (cid:16) − i ( √ m ) | Z − Z | + iπ (cid:17)p | ( m − ω ) | As usual in waves dynamics, the response defined by (71) may present some interference phenomena, de-pending on ˆ f and ˆ f . Formula (71) may be useful to understand the implications of position-dependent coefficients in (66). Assumea thread divided in two regions, each characterized by some constant coefficients g and g . We also assumethat these regions are only connected via two ”entry points”. This can be modelled by g = 0 on the borderbetween the two regions, and g >> Z or Z cancel, due to large oscillations in the vicinity of the border.As a consequence, the paths contributing to the Green function have to cross at Z or Z , inducing someinterference phenomenon (71) on the transmitted signal.More generally, this dependency in Z along the paths impacts the result even for a simple signal (68).Actually, the contribution to the Green function (71) of the various paths reaching a point Z of the threadacquire a phase depending on the path and on the characteristic of the medium encountered. This maycreate some interference between these paths. One may conjecture that trained networks will present someparticular learned features in their transfer functions, i.e. the coefficients g ( Z ) and g ( Z ), that wouldproduce some constructive interference for some signals, and destructive for some others. ( l, m ) points correlation functions and probabilistic interpreta-tion Appendices 3.2 and 3.3 show how to compute the ( l, m ) points correlation functions by successive derivativesof (43). Neglecting the interactions, the correlation function is given by tensor powers of (cid:18) δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † ( θ ( i ) ) (cid:19) − .We give here the expressions for strong and weak background fields. Appendix 3.3.1 shows that, for strongbackground fields, the m -th tensor power of (43) becomes: δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) − ⊗ m δ l,m (72)= (cid:0) O , ∞ (cid:1) G Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ (cid:16) θ ( i )1 (cid:17) Ψ † (cid:16) θ ( i )2 (cid:17) ∇ θ ω − (cid:16) J (cid:16) θ ( i )2 (cid:17) , θ ( i )2 , Z, G (cid:17)! − ⊗ m ≃ (cid:0) O , ∞ (cid:1) G Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ (cid:16) θ ( i )1 (cid:17) Ψ † (cid:16) θ ( i )2 (cid:17) ∇ θ G (cid:16) J (cid:16) θ ( i )2 (cid:17) , θ ( i )2 , Z, G (cid:17)! − ⊗ m l, m ) points correlation function is: G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1) (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ )) (cid:16) θ ( i )1 , θ ( i ) f (cid:17)! −⊗ m δ l,m (73) ≃ G ⊗ m δ l,m = δ ( Z − Z ′ ) exp − r(cid:16) σ ¯ X r (cid:17) + ασ − σ ¯ X r ! ( θ − θ ′ ) ! Λ H ( θ − θ ′ )These formulas show that for strong background the interacting dynamics of 2 l cells is mediated by thecollective background, represented here by the source term ∆Ω that depends itself on the background fieldΨ. Appendix 3.3.1 computes the corrections to (72) and (73) due to the interactions. The general formulafor the 1PI vertex δ l +1 ,m +1 [∆Ω] (cid:16) θ ( i )2 (cid:17) δ [ΨΨ † ] (( θ ( i ) ) , ( θ ( i ) † )) is recursive: δ l +1 ,m +1 [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) = M ∗ (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) (74)+ M ∗ X r l, r ′ m r + r ′ C rl C r ′ m Z δ r,r ′ K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ l − r,m − r ′ [∆Ω] (cid:16) θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )2 = M ∗ (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) + M ∗ X r + ... + r l = ls + ... + s m = m r i , s i , r i + s i l ! r ! ...r p ! m ! s ! ...s p ′ ! × (cid:20) δ r ,s Kδ [ΨΨ † ] (cid:21) ∗ ... ∗ M ∗ δ r p ,s p ′ K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] ∗ X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) where: X , (cid:16) θ ( i )1 , θ ( i ) f (cid:17) ) = G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i )1 , θ ( i ) f (cid:17) K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) has a matricial form. It is defined by: K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) (75)= − Z h A (cid:16) θ ( i )1 (cid:17)i [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + h B (cid:16) θ ( i )1 (cid:17)i [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G d (cid:16) θ ( i )2 (cid:17) ′ + [∆Ω] (cid:16) θ ( i )1 (cid:17) [Ψ] † (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )2 (cid:17) ′ − θ ( i )2 (cid:19) = − Z (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + (cid:20) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G d (cid:16) θ ( i )2 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ + [∆Ω] (cid:16) θ ( i )1 (cid:17) [Ψ] † (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )2 (cid:17) ′ − θ ( i )2 (cid:19) with A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) and B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) given by: A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ≃ G − (1 − exp ( − x )) (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ≃ G − (cid:0) O , (cid:1) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ exp ( − x ) (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) and the matrices involved in the definition of K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) :[∆Ω] † = (cid:0) ∆Ω † , ∆Ω (cid:1) , h A (cid:16) θ ( i )1 (cid:17)i = A (cid:16) θ ( i )1 (cid:17) A † (cid:16) θ ( i )1 (cid:17) , h B (cid:16) θ ( i )1 (cid:17)i = B (cid:16) θ ( i )1 (cid:17) B † (cid:16) θ ( i )1 (cid:17) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 = (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 (cid:0) O , ∞ (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 = (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 (cid:0) O , (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) = A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) A ′ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ , θ ( i )1 (cid:19) (cid:20) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) = B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) B ′ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ , θ ( i )1 (cid:19) K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) are given by: δ r,r ′ K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) (76)= [Ψ] † (cid:16) θ ( i )2 (cid:17) δ r,r ′ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) + δ (cid:16) θ ( i ) † − θ ( i )2 (cid:17) δ r,r ′ − [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) − X Z δ t,t ′ (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ r − s − t,r ′ − s ′ − t ′ [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ s,s ′ [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + δ t,t ′ (cid:20) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ r − s − t,r ′ − s ′ − t ′ [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ s,s ′ [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) × "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G d (cid:16) θ ( i )2 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ Formula (74), (75) and (76) allow to find the 1PI n -th vertex. Appendix 3.3.2 performs the computation inthe strong field and weak field approximation. Appendix 3.3.2 shows that in the strong field approximation, the 1PI n -th vertex are: Z ∆Ω † (1 + O , ∞ ) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (1 + O , ∞ ) ∆Ω m × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)! and that the connected Green function are given by: F l,m (Ψ) Z ∆Ω † (1 + O , ∞ ) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (1 + O , ∞ ) ∆Ω m × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)! where F l,m (Ψ) is an increasing factor in the norm of the background field Ψ: F l,m (Ψ) ≃ m − X k =0 (cid:18) exp ( − ( x − α )) Z Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:19) p ≃ − exp ( − ( x − α )) R Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17) l, m ) correlation functions: δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i )2 (cid:17) − l δ l,m + inf( l,m ) X s =0 X k > X s > ... > s k > ,t > ... > t k > s i + t i > , P s i = l − s, P t i = m − s k Y i =1 F s i ,t i (Ψ) ! δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − s (77) × (cid:16) Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17)(cid:17) l − s − (cid:16) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:17) m − s − × (cid:0) y − x (cid:1) exp ( − x )2 δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − + (cid:16) Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17)(cid:17) (cid:16) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:17) × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)!! k where permutations over the 2 l points are implicit. In the weak field approximation, the correlation functions for l = m are negligible. Appendix 3.3.3 showsthat the 1PI n -th vertex is: δ l,l − ∆Ω (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ M ∗ (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) l X , + (cid:20) M ∗ δ , Kδ [ΨΨ † ] (cid:21) ∗ ... ∗ (cid:20) M ∗ δ , Kδ [ΨΨ † ] (cid:21) ∗ X , which can be approximated by: δ l,l − ∆Ω (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ ( − l G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l − ∗ ¯ O , ∞ ∗ G From the vertices, the connected correlation functions can be retrieved: G ( l ) C ≡ (cid:16) ( − l G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l − ⊗ ¯ O , ∞ ∗ G (cid:17) + ( − l X k > X l + ... + l k = l k Y m =1 X ∗ { lm }G ∗ { l m }G (cid:16) G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l m − ⊗ ¯ O , ∞ ∗ G (cid:17) where the sum over ∗ { l m }G denotes the sum over all possible convolutions between the blocks: (cid:16) G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l m − ⊗ ¯ O , ∞ ∗ G (cid:17) Two blocks are convoluted on at most one variable. The convolution are performed by insertion of apropagator G between the blocks. The expression for the connected correlation functions induces the fullcorrelation functions: G ⊗ l + X p,k ( − l − p X l k , P l n = l − p Y k G ( l n ) C G ⊗ p (78)= G ⊗ l + X p,k ( − l − p X l n , P l n = l − p Y n (cid:16)(cid:16) ( − l n G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l n − ⊗ ¯ O , ∞ ∗ G (cid:17) + ( − l n X k > X l + ... + l k = l n k Y m =1 X ∗ { lm }G ∗ { l m }G (cid:16) G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l m − ⊗ ¯ O , ∞ ∗ G (cid:17) ⊗ G ⊗ p Equations (77) and (78) may be interpreted in terms of joined probabilities for frequencies at different pointsof the thread. To explain this point, start with the two points correlation functions. At the zeroth order inperturbation, the function (cid:18) δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) (cid:19) − is the Green function of the operator: −∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G ) (cid:19) + α which is (27): G ( θ, θ ′ , Z ) = δ ( Z − Z ′ ) exp − r(cid:16) σ ¯ X r (cid:17) + ασ − σ ¯ X r ! ( θ − θ ′ ) ! Λ H ( θ − θ ′ )This function is the Laplace transform of the function ˆ G Z ( θ, θ ′ , ∆ n ): G ( θ, θ ′ , Z ) = Z ˆ G Z ( θ, θ ′ , ∆ n ) exp ( − α ∆ n ) dα The form of ˆ G Z ( θ, θ ′ , ∆ n ) is not necessary here.The function ˆ G Z computes the probability of a time interval θ − θ ′ for ∆ n spikes of the potential atpoint Z . The Laplace transform G ( θ, θ ′ , Z ) computes the probability of a time interval θ − θ ′ for a randomnumber of spikes ∆ n with average α . Since the of spikes’ frequency is ∆ nθ − θ ′ , G ( θ, θ ′ , Z ) computes the averageprobability of a frequency α ( θ − θ ′ ) of spikes. Computing the average h ( θ − θ ′ ) i confirms this point: G ( θ, θ ′ , Z ) = δ ( Z − Z ′ ) exp − r(cid:16) σ ¯ X r (cid:17) + ασ − σ ¯ X r ! ( θ − θ ′ ) ! Λ H ( θ − θ ′ ) ≃ δ ( Z − Z ′ ) exp (cid:0) − α ¯ X r ( θ − θ ′ ) (cid:1) Λ H ( θ − θ ′ )so that h ( θ − θ ′ ) i = α ¯ X r . The average inverse frequency is then α h ( θ − θ ′ ) i = X r .As a consequence, the expression of G ( θ, θ ′ , Z ) computed at α = 1 can be interpreted as the probability,at time θ + θ ′ , of a spikes’ frequency equal to θ − θ ′ . The same applies for higher order correlation functions.We show in Appendix 3 that δ l,l ∆Ω (cid:16) θ ( i )1 ,θ ( i )2 (cid:17) δ [ΨΨ † ] (( θ ( i ) ) , ( θ ( i ) † )) computes the transition probability of θ ( i ) † to (cid:0) θ ( i ) (cid:1) for i = 1 ...l for an average number of spikes of α , so that: δ l,l ∆Ω (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) α =1 = P (cid:18) ω (cid:18) Z , θ (1) + θ (1) † (cid:19) = 1 θ (1) − θ (1) † , ..., ω (cid:18) Z l , θ ( l ) + θ ( l ) † (cid:19) = 1 θ ( l ) − θ ( l ) † (cid:19) (79)computes the joined probability for a set of l frequencies at points Z ,..., Z l and times θ (1) + θ (1) † ,..., θ ( l ) + θ ( l ) † .Equation (79) can be rewritten in terms of density for the set of variables θ ( i ) = θ ( i ) + θ ( i ) † and ω ( i ) = θ (1) − θ (1) † : P (cid:18)(cid:16) ω ( i ) , θ ( i ) (cid:17) i l (cid:19) = l Y i =1 (cid:0) ω ( i ) (cid:1) ! δ l,l ∆Ω (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) α =1 (cid:18) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ ( θ ( i ) ) (cid:19) − ! m and G ⊗ m represent anindependent distribution for the frequencies at different point, and the corrective terms measure the mutualdependencies due to the interactions in the background field. Moreover, for l = m = 1, the probabilisticinterpretation is an alternate description to the frequencies’ local differential equation. We will examine (77)and (78) independently. In the strong field approximation, the fields Ψ † (cid:16) θ ( i )1 (cid:17) and Ψ (cid:16) θ ( i )2 (cid:17) in (77) can be written as functions of theset of variables θ ( i ) and (cid:0) ω ( i ) (cid:1) − as Ψ † (cid:18) θ ( i ) + ( ω ( i ) ) − (cid:19) and Ψ (cid:18) θ ( i ) − ( ω ( i ) ) − (cid:19) . We first consider l = 1and write the individual probabilities P (cid:0) ω ( i ) , θ ( i ) (cid:1) , up to some normalization factor: (cid:16) ω ( i ) (cid:17) P (cid:16) ω ( i ) , θ ( i ) (cid:17) = δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − (80)= (cid:0) O , ∞ (cid:1) ∗ G × Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ (cid:16) θ ( i )1 (cid:17) Ψ † (cid:16) θ ( i )2 (cid:17) ∇ θ G (cid:16) J (cid:16) θ ( i )2 (cid:17) , θ ( i )2 , Z, G (cid:17)! − The individual probability (80) can be written explicitly. First, the propagator arising in (80) is defined by: G ( θ, θ ′ ) = δ ( Z − Z ′ ) 1 p π exp − r(cid:16) σ ¯ X r (cid:17) + ασ − σ ¯ X r ! ( θ − θ ′ ) !r(cid:16) σ ¯ X r (cid:17) + ασ H ( θ − θ ′ ) (81)= δ ( Z − Z ′ ) exp (cid:0) − Λ ω ( i ) (cid:1) Λ H ( θ − θ ′ )where: Λ = s(cid:18) σ ¯ X r (cid:19) + 2 ασ − σ ¯ X r X r ≡ arctan (cid:16)(cid:16) X r − X p (cid:17) p ¯ J ( Z ) (cid:17)p ¯ J ( Z ) (82)The factor δ ( Z − Z ′ ) H ( θ − θ ′ ) will be skipped and reintroduced ultimately. Formula (81) is computedfor an average current over some timespan. This approximation can be relaxed and we can compute thefrequency at the average point θ ( i ) . As a consequence, (81) can be computed by setting ¯ J → J (cid:0) θ ( i ) (cid:1) in (82).This implies that we can replace Λ in the formula (82) by:Λ (cid:16) θ ( i ) (cid:17) = vuut σ ¯ X r (cid:0) θ ( i ) (cid:1) ! + 2 ασ − σ ¯ X r (cid:0) θ ( i ) (cid:1) with: 1¯ X r (cid:0) θ ( i ) (cid:1) ≡ arctan (cid:16)(cid:16) X r − X p (cid:17) q J (cid:0) θ ( i ) , Z (cid:1)(cid:17)q J (cid:0) θ ( i ) , Z (cid:1) G ( θ, θ ′ ) = exp (cid:18) − Λ ( θ ( i ) ) ω ( i ) (cid:19) Λ (83)Second, the quantity (cid:0) O , ∞ (cid:1) ∗ G = (1 + O , ∞ ) G in (80) is given by:(1 + O , ∞ ) G = − ¯ ζ + ∇ outθ ( j ) i Λ ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Λ (cid:18) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i (cid:19) (cid:16) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) G (84)It can be expressed in the set of variables θ ( i ) and ω ( i ) :(1 + O , ∞ ) G ≃ − ¯ ζ ∞ + ∇ outθ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) Λ (cid:16) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17)(cid:17) exp ˆΞ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) ω ( i ) − G (85)where: ˆΞ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) and: ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) are the average of ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) and ¯Ξ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) on the time span θ ( j ) f − θ ( j ) i .In first approximation, it depends on the mid-point θ ( i ) . This dependency arises as a function of the externalcurrent J (cid:0) θ ( i ) (cid:1) .As a consequence of (85) and (83), we obtain: (cid:0) O , ∞ (cid:1) ∗ G = − ¯ ζ ∞ + ∇ outθ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) Λ (cid:16) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17)(cid:17) exp ˆΞ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) ω ( i ) − × exp (cid:18) − Λ ( θ ( i ) ) ω ( i ) (cid:19) Λ (86)Using (64), the probability P (cid:0) ω ( i ) , θ ( i ) (cid:1) is given by: (cid:16) ω ( i ) (cid:17) P (cid:16) ω ( i ) , θ ( i ) (cid:17) (87)= − ¯ ζ ∞ + ∇ outθ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) Λ (cid:16) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17)(cid:17) exp ˆΞ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) ω ( i ) − × exp − Λ (cid:0) θ ( i ) (cid:1) ω ( i ) ! × Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ θ ( i ) − (cid:0) ω ( l ) (cid:1) − ! Ψ † θ ( i ) + (cid:0) ω ( l ) (cid:1) − ! × ∇ θ G J θ ( i ) + (cid:0) ω ( l ) (cid:1) − ! , θ ( i ) + (cid:0) ω ( l ) (cid:1) − , Z, G !! − ∇ θ G (cid:18) J (cid:18) θ ( i ) + ( ω ( l ) ) − (cid:19) , θ ( i ) + ( ω ( l ) ) − , Z, G (cid:19) , i.e. the variations of J (cid:0) θ ( i ) (cid:1) ,and the magnitude of Ψ, the denominator:1 + exp Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ θ ( i ) − (cid:0) ω ( l ) (cid:1) − ! Ψ † θ ( i ) + (cid:0) ω ( l ) (cid:1) − ! ×∇ θ G J θ ( i ) + (cid:0) ω ( l ) (cid:1) − ! , θ ( i ) + (cid:0) ω ( l ) (cid:1) − , Z, G ! cancels and P (cid:0) ω ( i ) , θ ( i ) (cid:1) has a pole ω ( i )0 (cid:0) J (cid:0) θ ( i ) (cid:1)(cid:1) . The average frequency is switched to this pole and adiscontinuity arises. The frequencies concentrate around this pole.We now consider the general case for l correlated frequencies. Up to a normalization factor such that: R P (cid:16)(cid:0) ω ( i ) , θ ( i ) (cid:1) i l (cid:17) l Q i =1 dω ( i ) = 1, the joint probability defined by (77) becomes: l Y i =1 (cid:16) ω ( i ) (cid:17) ! P (cid:18)(cid:16) ω ( i ) , θ ( i ) (cid:17) i l (cid:19) = l Y i =1 P (cid:16) ω ( i ) , θ ( i ) (cid:17) + l X s =0 X k > X s > ... > s k > ,t > ... > t k > s i + t i > , P s i = l − s, P t i = l − s k Y i =1 F s i ,t i (Ψ) ! s Y i =1 P (cid:16) ω ( i ) , θ ( i ) (cid:17)! × l − Y i = s +1 Ψ † (cid:18) θ ( i ) + (cid:16) ω ( i ) (cid:17) − (cid:19) l − Y i = s +1 Ψ (cid:18) θ ( i ) − (cid:16) ω ( i ) (cid:17) − (cid:19) × (cid:0) y − x (cid:1) exp ( − x )2 P (cid:16) ω ( l ) , θ ( l ) (cid:17) + Ψ † θ ( l ) + (cid:0) ω ( l ) (cid:1) − ! Ψ θ ( l ) − (cid:0) ω ( l ) (cid:1) − !! × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)!! k where permutations over the variables are implicit.In first approximation, we can replace Ψ † (cid:18) θ ( l ) + ( ω ( l ) ) − (cid:19) Ψ (cid:18) θ ( l ) − ( ω ( l ) ) − (cid:19) by Ψ † Ψ and for large Ψ ,the corrective term can be approximated by a function F (cid:16) Ψ † Ψ (cid:17) , so that (cid:18) l Q i =1 (cid:0) ω ( i ) (cid:1) (cid:19) P (cid:16)(cid:0) ω ( i ) , θ ( i ) (cid:1) i l (cid:17) is driven towards a uniform distribution by the correlations between the points. The first order correctionsdue to the background field uniformize the pattern of frequencies. In the weak field approximation, expression (78) for l = 1 becomes: (cid:16) ω ( i ) (cid:17) P (cid:16) ω ( i ) , θ ( i ) (cid:17) = (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) G (cid:16) θ ( i )1 , θ ( i ) f (cid:17) ≃ G = exp (cid:18) − Λ ( θ ( i ) ) ω ( i ) (cid:19) Λwhere (83) is used. We recover the probability for the frequency in the current J (cid:0) θ ( i ) (cid:1) . In the correctionfactor: (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) − x ) ( y − x )1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) can be included in the normalization factor. At the first order of approximation, we are left with: (cid:16) ω ( i ) (cid:17) P (cid:16) ω ( i ) , θ ( i ) (cid:17) ≃ (cid:18) y y − x O , (cid:19) exp (cid:18) − Λ ( θ ( i ) ) ω ( i ) (cid:19) ΛWriting explicitly O , in terms of ω ( i ) , we find: (cid:16) ω ( i ) (cid:17) P (cid:16) ω ( i ) , θ ( i ) (cid:17) (88) ≃ y y − x − ¯ ζ + ∇ outθ ( i ) i ′ Λ ¯Ξ , (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) Λ (cid:16) − ¯ ζ + ¯Ξ , (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17)(cid:17) exp ˆΞ , (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) ω ( i ) − × exp (cid:18) − Λ ( θ ( i ) ) ω ( i ) (cid:19) ΛIn a background field of small magnitude, the probabilities’ formula do not include any pole, and thefrequencies are not shifted by an external current. They rather include the minimal interactions ¯ ζ thatcorrespond to the minimal level of operating frequencies.Using (78) and (88), we directly deduce the joined probability for l frequencies: P (cid:18)(cid:16) ω ( i ) , θ ( i ) (cid:17) i l (cid:19) = l Y i =1 P (cid:16) ω ( i ) , θ ( i ) (cid:17) + X p,k ( − l − p × X l n , P l n = l − p Y n (cid:16)(cid:16) ( − l n (cid:17) ˆ P (cid:16)(cid:16) ω ( i (1) n ) , θ ( i (1) n ) (cid:17) , ..., (cid:16) ω ( i ( ln ) n ) , θ ( i ( ln ) n ) (cid:17)(cid:17) + ( − l n X k > X l + ... + l k = l n k Y m =1 X ∗ { lm }G ∗ { l m }G ˆ P (cid:16)(cid:16) ω ( i (1) n ) , θ ( i (1) n ) (cid:17) , ..., (cid:16) ω ( i ( lm ) m ) , θ ( i ( lm ) m ) (cid:17)(cid:17) G ⊗ p where the permutation over the variables (cid:0) ω ( i ) , θ ( i ) (cid:1) are implicit, and ∪ m (cid:26)(cid:16) i ( k ) m (cid:17) k =1 ,...l m (cid:27) = { , ..., n } . Thefunction ˆ P is defined by:ˆ P (cid:16)(cid:16) ω ( i (1) n ) , θ ( i (1) n ) (cid:17) , ..., (cid:16) ω ( i ( ln ) n ) , θ ( i ( ln ) n ) (cid:17)(cid:17) = P (cid:16) ω ( i (1) n ) , θ ( i (1) n ) (cid:17) × l n − Y s =2 P (cid:16) ω ( i ( s ) n ) , θ ( i ( s ) n ) (cid:17)! P (cid:16) ω ( i ( ln ) n ) , θ ( i ( ln ) n ) (cid:17) with: P (cid:16) ω ( i ) , θ ( i ) (cid:17) = Z (cid:0) O , ∞ (cid:1) G = (1 + O , ∞ ) G = P (cid:16) ω ( i ) , θ ( i ) (cid:17) + P (cid:16) ω ( i ) , θ ( i ) (cid:17) (cid:16) ω ( i ) , θ ( i ) (cid:17) = O , ∞ G = − ¯ ζ + ∇ outθ ( j ) i Λ ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Λ (cid:18) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i (cid:19) (cid:16) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) exp (cid:18) − Λ ( θ ( i ) ) ω (cid:19) Λ ≃ − ¯ ζ + ∇ outθ ( j ) i Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) Λ (cid:16) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17)(cid:17) exp ˆΞ , ∞ (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) ω − exp (cid:18) − Λ ( θ ( i ) ) ω (cid:19) Λ
10 Extensions
Several extensions of the formalism may be considered, the details are left for further research.
A first possible extension is to include inhibitory currents. This is done by introducing two different types ofcells, each defined by a different field. We write Ψ ( θ, Z, ω ) and Ψ (cid:16) θ, ˜ Z, ˜ ω (cid:17) for excitatory and inhibitoryneurons respectively. The influence of each type of cell on the other one is obtained through the actionsof the induced currents. If we assume that the transfer functions are identical for both type of fields, thecorresponding action terms for the frequencies are:12 η Z | Ψ ( θ, Z, ω ) | ω − − G J ( θ, Z ) + Z κN ω ω (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z , ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, θ, Z ) dZ dω − Z κN ˜ ω ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − ˜ Z (cid:12)(cid:12)(cid:12) c , ˜ Z , ˜ ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T (cid:16) Z, θ, ˜ Z (cid:17) d ˜ Z d ˜ ω and:12 η Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ, ˜ Z, ˜ ω (cid:17)(cid:12)(cid:12)(cid:12) ˜ ω − − G J ( θ, Z ) + Z κN ω ˜ ω (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z , ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, θ, Z ) dZ dω !! so that the action for the system writes: S = −
12 Ψ † ( θ, Z, ω ) ∇ (cid:18) σ θ ∇ − ω − (cid:19) Ψ ( θ, Z, ω ) −
12 Ψ † (cid:16) θ, ˜ Z, ˜ ω (cid:17) ∇ (cid:18) σ θ ∇ − ˜ ω − (cid:19) Ψ (cid:16) θ, ˜ Z, ˜ ω (cid:17) (89)12 η Z | Ψ ( θ, Z, ω ) | ω − − G J ( θ, Z ) + Z κN ω ω (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z , ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, θ, Z ) dZ dω − Z κN ˜ ω ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − ˜ Z (cid:12)(cid:12)(cid:12) c , ˜ Z , ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T (cid:16) Z, θ, ˜ Z (cid:17) d ˜ Z d ˜ ω η Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ, ˜ Z, ˜ ω (cid:17)(cid:12)(cid:12)(cid:12) ˜ ω − − G J ( θ, Z ) + Z κN ω ˜ ω (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z , ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, θ, Z ) dZ dω !! As in the section 3.2, we can project the fields on the frequency-dependent states. This leads to the followingaction: S = −
12 Ψ † ( θ, Z ) ∇ (cid:18) σ θ ∇ − ω − ( J, θ, Z, Ψ , Ψ ) (cid:19) Ψ ( θ, Z ) −
12 Ψ † (cid:16) θ, ˜ Z (cid:17) ∇ (cid:18) σ θ ∇ − ˜ ω − ( J, θ, Z, Ψ , Ψ ) (cid:19) Ψ (cid:16) θ, ˜ Z (cid:17) ω − ( J, θ, Z, Ψ , Ψ ) = G J ( θ, Z ) + Z κN ω (cid:16) J, θ − | Z − Z | c , Z , Ψ (cid:17) ω (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, θ, Z ) dZ − Z κN ˜ ω (cid:16) J, θ − | Z − Z | c , ˜ Z , Ψ (cid:17) ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − ˜ Z (cid:12)(cid:12)(cid:12) c , ˜ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T (cid:16) Z, θ, ˜ Z (cid:17) d ˜ Z ˜ ω − ( J, θ, Z, Ψ , Ψ ) = G J ( θ, Z ) + Z κN ω (cid:16) J, θ − | Z − Z | c , Z , Ψ (cid:17) T (cid:16) Z, θ, Z , θ − | Z − Z | c (cid:17) ˜ ω × (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ ! As in section 3.3, a collective potential can be added: ∞ X n =2 ζ ( n ) Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! where we define: | Ψ ( θ, Z ) | = | Ψ ( θ, Z ) | + | Ψ ( θ, Z ) | to model the fact that the equilibrium activity includes both excitatory and inhibitory activities. Until now, we have considered the dynamics of frequencies alone. The transfer functions were consideredin first approximation as depending on the frequencies. We show briefly how to generalize the model byincluding dynamic oscillations for the transfer function. We will work in constant background frequency tosimplify the formula, but the computations can be straightforwardly generalized to a position dependentbackground. The computations are presented in Appendix 2.To account for the dynamic nature of the transfer functions T ( Z, Z , ω, ω ), equation (58) must bemodified and associated with an evolution equation for T ( Z, Z , ω, ω ). Using (56), we replace T ( Z, Z , ω, ω )by a general function T ( Z, Z , θ ) that is a priori independent from frequencies. Around the equilibriumdefined by the background frequency ω , the function T ( Z, Z , θ ) then writes: T ( Z, Z , θ ) = T ( Z, Z ) + h ( Z, Z ) ˆ T ( Z, θ, Z )where T ( Z, Z ) is the transfer function in this equilibrium. The function ˆ T ( Z, θ, Z ) represents the fluctu-ations around this equilibrium. The expansion of G around ω becomes: G κN Z ω + Ω (cid:16) θ − | Z − Z | c , Z (cid:17) − Ω ( θ, Z ) ω r π (cid:16) X r (cid:17) + π α T ( Z, Z , ω, ω ) dZ ≃ ω + Γ Z Ω (cid:16) θ − | Z − Z | c , Z (cid:17) − Ω ( θ, Z ) ω r π (cid:16) X r (cid:17) + π α T ( Z, Z ) + h ( Z, Z ) ˆ T ( Z, θ, Z ) r π (cid:16) X r (cid:17) + π α As a consequence, equation (58) is replaced by: 39 θ ∇ θ Ω ( θ, Z ) = Ω ( θ, Z ) + Γ c ∇ θ Ω ( θ, Z ) − Γ ∇ Z Ω ( θ, Z ) − Γ c ∇ θ Ω ( θ, Z ) − Γ ˆ T ( Z, θ )where we defined: ˆ T ( Z, θ ) = Z h ( Z, Z ) ˆ T ( Z, θ, Z ) r π (cid:16) X r (cid:17) + π α and: Γ = κN X r Z | Z − Z | T ( Z, Z ) dZ ω r π (cid:16) X r (cid:17) + π α Γ Γ = κ N X r R ( Z − Z ) T ( Z, Z ) dZ ω r π (cid:16) X r (cid:17) + π α Γ Γ = G ′ κN Z T ( Z, Z ) dZ r π (cid:16) X r (cid:17) + π α Appendix 2 derives the dynamics for ˆ T ( Z, θ ) and yields a system of dynamic equations for (cid:16)
Ω ( θ, Z ) , ˆ T ( Z, θ ) (cid:17) : σ θ ∇ θ Ω ( θ, Z ) = Ω ( θ, Z ) − f ( Z ) ∇ θ Ω ( θ, Z ) + f ( Z ) ∇ Z Ω ( θ, Z ) − f ( Z ) ∇ θ Ω ( θ, Z ) (90)+Γ ∇ Z ˆ T ( Z, θ ) ∇ θ ˆ T ( Z, θ ) λ + U ( ω ) ∇ θ ˆ T ( Z, θ ) + U ( ω ) ˆ T ( Z, θ ) (91)= ρ ¯ C ( Z ) h ′ C ( ω ) − ρ (cid:16) D ( Z ) ˆ T ( Z ) h ′ D ( ω ) + ¯ C ( Z ) h ′ C ( ω ) (cid:17) λτ Ω (
Z, θ )+ ρD ( Z ) h ′ D ( ω ) (cid:0) Γ ∇ θ Ω (
Z, θ ) − (cid:0) Γ ∇ θ Ω (
Z, θ ) + Γ ∇ Z Ω (
Z, θ ) (cid:1)(cid:1) λτ with: ¯ C ( Z ) = 1 r π (cid:16) X r (cid:17) + π α Z h ( Z, Z ) C ( Z )¯ C ( Z ) = 1 r π (cid:16) X r (cid:17) + π α Z h ( Z, Z ) C ( Z ) ˆ T ( Z, Z )ˆ T ( Z ) = 1 r π (cid:16) X r (cid:17) + π α Z h ( Z, Z ) ˆ T ( Z, Z )Note that for a slowly varying ˆ T ( Z, θ ), formula (91) is similar to (58).
Appendix 1. Vertices of (18) involved in the computation of the n Green functions
To find the effective action associated to (18) and the collective term (22), we proceed in several steps. Thefirst one is to find the vertices involved in the computation of the Green functions. To do so, we will expand40he action (18) in series of field. This produces a series of an infinite series of vertices. However, given thatthe two points Green function are not symmetric by time reversal, we will show that only the 2 n first termsare involved in the computation of the 2 n Green functions. We will then estimate these vertices using therecursive relation (16) between frequencies depending on field. These results will be used in the next sectionto find the graph expansion of the system’s partition function.
We start with the two points Green function and prove (29). To do so, we will expand the action functional inseries of the field Ψ. The two points Green function will be computed by using the ”free” action’s propagatordefined by (18) and obtained by replacing ω − ( J, θ, Z,
Ψ) by ω − ( J, θ, Z, S = −
12 Ψ † ( θ, Z ) ∇ θ (cid:18) σ ∇ θ − ω − ( J, θ, Z, (cid:19) Ψ ( θ, Z ) (92)and the series in field will be considered, as usual, as a perturbation expansion.
Now, we compute the propagator associated to (92). We decompose the external current into a static and atime dependent parts ¯ J + J ( θ ) where ¯ J can be thought as the time average of the current. We will considerthat (cid:12)(cid:12) ¯ J ( Z ) (cid:12)(cid:12) > | J ( θ, Z ) | . At zeroth order in current J ( θ ), the function ω − ( J, θ, Z,
0) satisfies: ω − ( J, θ, Z,
0) = G (cid:0) ¯ J + J ( θ ) (cid:1) (93) ≃ G (cid:0) ¯ J ( Z ) (cid:1) = arctan (cid:16)(cid:16) X r − X p (cid:17) p ¯ J ( Z ) (cid:17)p ¯ J ( Z ) = 1¯ X r ( Z ) ≡ X r where the dependence in Z of ¯ X r will be understood. As a consequence ω ( θ, Z ) is thus approximativelyequal to ¯ X r . Under this approximation: S = − Ψ † ( θ, Z ) ∇ θ (cid:18) σ ∇ θ − X r (cid:19) Ψ ( θ, Z )and the Green function of the operator ∇ θ (cid:16) σ ∇ θ − X r (cid:17) is computed as: (cid:10) Ψ † ( θ, Z ) Ψ ( θ ′ , Z ) (cid:11) ≡ G (( θ, Z ) , ( θ ′ , Z ′ )) ≡ G ( θ, θ ′ , Z ) = δ ( Z − Z ′ ) Z exp ( ik ( θ − θ ′ )) σ k + ik X r + α dk (94)The right hand side of (94) can be computed as: Z exp ( ik ( θ − θ ′ )) σ k + ik X r + α dk = exp (cid:18) θ − θ ′ σ ¯ X r (cid:19) Z exp ( ik ( θ − θ ′ )) σ k + (cid:16) σ ¯ X r (cid:17) + α dk = 1 p π exp − r(cid:16) σ ¯ X r (cid:17) + ασ | θ − θ ′ | !r(cid:16) σ ¯ X r (cid:17) + ασ exp (cid:18) θ − θ ′ σ ¯ X r (cid:19) (95)and this is quickly suppressed for θ − θ ′ <
0. This is the direct consequence of non-hermiticity of operator.In the sequel, for σ ¯ X r <<
1, we can thus consider that: G ( θ, θ ′ , Z ) = δ ( Z − Z ′ ) 1 p π exp − r(cid:16) σ ¯ X r (cid:17) + ασ − σ ¯ X r ! ( θ − θ ′ ) !r(cid:16) σ ¯ X r (cid:17) + ασ H ( θ − θ ′ ) (96)41here H is the Heaviside function: H ( θ − θ ′ ) = 0 for θ − θ ′ <
0= 1 for θ − θ ′ > G ( J ( θ, Z )) = arctan (cid:16)(cid:16) X r − X p (cid:17) p J ( θ, Z ) (cid:17)p J ( θ, Z )For relatively high frequency firing rates, i.e., small periods of time between two spikes, we replace (94) bythe Green function of: ∇ θ (cid:18) σ ∇ θ − G ( J ( θ, Z )) (cid:19) ≃ ∇ θ (cid:18) σ ∇ θ − X r − J ( θ, Z ) G ′ (cid:0) ¯ J (cid:1)(cid:19) and ω − ( G Z (0 , G (( θ, Z ) , ( θ ′ , Z ′ )) = δ ( Z − Z ′ ) 1 p π exp − r(cid:16) σ ¯ X r (cid:17) + ασ − σ ¯ X r ! ( θ − θ ′ ) !r(cid:16) σ ¯ X r (cid:17) + ασ H ( θ − θ ′ ) × − p π G ′ (cid:0) ¯ J (cid:1)r(cid:16) σ ¯ X r (cid:17) + ασ Z θ ′ θ J ( θ ′′ , Z ) dθ ′′ as a consequence since J ( θ, Z ) is the deviation around the static part ¯ J , the corrective term vanishes quicklyas θ − θ ′ increases. Formula (96) allows to compute higher order contributions to the Green function of the action (18) by usinga graph expansion. Actually, writing ω − ( θ, Z ) for ω − ( J, θ, Z,
Ψ) when no ambiguity is possible, the higherorder contribution for the series expansion of ω − ( θ, Z ) in fields are obtained by solving recursively: ω − ( J, θ, Z ) = G J ( θ, Z ) + Z κN ω (cid:16) J, θ − | Z − Z | c , Z (cid:17) ω ( J, θ, Z ) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, θ, Z ) dZ dω (97)This will be done precisely in the next paragraph. For now, it is enough to note that given (97), the recursiveexpansion in ω − ( J, θ, Z ) of the potential term in (18):12 Ψ † ( θ, Z ) ∇ G J ( θ, Z ) + Z κN ω (cid:16) J, θ − | Z − Z | c , Z (cid:17) ω ( J, θ, Z ) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ) dZ Ψ ( θ, Z )(98)induces the presence of products in the series expansion of the two points Green function: m Y i =1 Z Ψ † (cid:16) θ ( i ) , Z i (cid:17) ∇ θ ( i ) k i Y k =1 l k Y l =1 n ( α ( l )) Y α ( l )=1 Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ ( i ) − (cid:12)(cid:12)(cid:12) Z i − Z (1) α ( l ) (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12) Z ( l − α ( l ) − Z ( l ) α ( l ) (cid:12)(cid:12)(cid:12) c , Z ( l ) α ( l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × dZ (1) α ( l ) ...dZ ( l k ) α ( l ) Ψ (cid:16) θ ( i ) , Z i (cid:17) dθ ( i ) dZ i (99)42ith n ( α ( l )) > n ( α ( l ′ )) for l > l ′ and m ∈ N . The function δ ( Z − Z ′ ) in (94) and the use of Wick’stheorem imply that all closed loop subgraphs drawn from this product reduce to a product of free Greenfunctions (96) of the following form (the gradient terms and the indices α ( l ) are not included and do notimpact the reasoning): Z Y i,k G θ ( i ) − X l n (cid:12)(cid:12)(cid:12) Z i − Z ( l )1 (cid:12)(cid:12)(cid:12) c , θ ( i +1) − X k n (cid:12)(cid:12)(cid:12) Z i +1 − Z ( k )1 (cid:12)(cid:12)(cid:12) c , Z δ ( Z − Z i ) δ ( Z − Z k ) dZ i dZ k Y i dθ ( i ) = Y i G θ ( i ) − X l n (cid:12)(cid:12)(cid:12) Z i − Z ( l )1 (cid:12)(cid:12)(cid:12) c , θ ( i +1) − X k n (cid:12)(cid:12)(cid:12) Z i +1 − Z ( k )1 (cid:12)(cid:12)(cid:12) c , Z Y i dθ ( i ) = Y i G (cid:16) θ ( i ) , θ ( i +1) , Z (cid:17)! Y i dθ ( i ) (100)by change of variable in the successive integrations. Moreover, the cancelation of G ( θ, θ ′ , Z ) for θ < θ ′ implies that this product is different from zero only for θ ( i ) < θ ( i +1) . As a consequence, for all closed loops θ < ... < θ ( i ) < θ ( i +1) < ...θ n = θ , the contribution for loop graphs (100) reduces to: Y i G ( θ , θ , Z ) = Y i G (0 , Z )with (see (96)): G (0 , Z ) = 1 r π (cid:16) σ ¯ X r (cid:17) + πασ As a consequence, the contribution of (99) to the two points Green function between an initial and finalstate: * Ψ † ( θ in , Z in ) Z m Y i =1 Ψ † (cid:16) θ ( i ) , Z i (cid:17) ×∇ θ ( i ) k i Y k =1 l k Y l =1 Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ ( i ) − (cid:12)(cid:12) Z i − Z (1) (cid:12)(cid:12) + ... + (cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12) c , Z ( l ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dZ (1) ...dZ ( l k ) × Ψ (cid:16) θ ( i ) , Z i (cid:17) dθ ( i ) dZ i Ψ ( θ fn , Z fn ) E (101)reduces to sums of the type: δ ( Z in − Z fn ) X p G ( θ in , θ , Z in ) G ( θ , θ , Z in ) ... G ( θ p , θ fn , Z in ) X n X n L ( p )1 ,...,L ( p ) n o n Y m =1 ( G (0 , , Z m )) l ( L ( p ) m ) (102)where n L ( p )1 , ..., L ( p ) n o is the set of all n -uplet of possible closed loops that can be drawn from the remainingvariables in (101) once p variables have been chosen.The result (102) is the same as if in (98) the potential had been expanded to the second order in Ψ andin all terms of higher order, | Ψ ( θ, Z ) | had been replaced by G (0 , Z ).Now, writing ω (cid:16) J, θ, Z, | Ψ | (cid:17) for ω and ω (0) = ω ( J, θ, Z,
0) (i.e. when we set Ψ ≡ −
12 Ψ † ( θ, Z ) ∇ θ (cid:18) σ θ ∇ θ − ω − (0) (cid:19) Ψ ( θ, Z ) (103)+ 12 Ψ † ( θ, Z ) X n> ∇ θ (cid:0) ω − (cid:1) ([ n ]) (0)[ n ]! ( G (0 , Z )) n Ψ ( θ, Z )+ X n> ∇ θ (cid:0) ω − (cid:1) ([ n − (0) | Ψ | [ n − G (0 , Z )) n − G ( θ, θ ′ , Z ) ! θ ′ = θ = −
12 Ψ † ( θ, Z ) ∇ θ (cid:18) σ θ ∇ θ − ω − (0) (cid:19) Ψ ( θ, Z ) + 12 Ψ † ( θ, Z ) X n> ∇ θ (cid:0)(cid:0) ω − (cid:1) ( G (0 , Z )) − ω − (0) (cid:1) Ψ ( θ, Z )+ (cid:16) ∇ θ (cid:16)(cid:0) ω − (cid:1) ([1]) ( G (0 , Z )) | Ψ | G ( θ, θ ′ , Z ) (cid:17)(cid:17) θ ′ = θ = −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ σ θ ∇ θ (cid:19) Ψ ( θ, Z ) + 12 | Ψ | δ h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) Ψ ( θ, Z ) (cid:17)i δ | Ψ | | Ψ( θ,Z ) | = G (0 ,Z ) where ( ω − ) ([ n ]) (0)[ n ]! is a short notation for: X l i Z n Y i =1 dZ (1) l i ...dZ ( l i ) l i δ n h ω − (cid:16) J, θ, Z, | Ψ | (cid:17)i n Q i =1 δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − Z (1) li (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12)(cid:12) Z li ( l − − Z ( li ) li (cid:12)(cid:12)(cid:12)(cid:12) c , Z ( l i ) l i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Ψ | =0 and ( ω − ) ([ n − (0) | Ψ | [ n − stands for: X l i Z n − Y i =1 dZ (1) l i ...dZ ( l i ) l i δ n − h ω − (cid:16) J, θ, Z, | Ψ | (cid:17)iQ i δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − Z (1) li (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12)(cid:12) Z ( l − li − Z ( li ) li (cid:12)(cid:12)(cid:12)(cid:12) c , Z ( l i ) l i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k li | Ψ | =0 × n − X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − Z (1) l j (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12) Z ( l − l i − Z ( l j ) l j (cid:12)(cid:12)(cid:12) c , Z ( l j ) l j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Similar notation is valid for ( ω − ) ([ n ]) ( G (0 , ,Z )) | Ψ | [ n − , the derivatives are evaluated at | Ψ ( θ, Z ) | = G (0 , , Z ).We have also used | Ψ | h δδ | Ψ | i as a shorthand for: X l Z dZ (1) l ...dZ ( l ) l ( k l )! ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − Z (1) l (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12) Z ( l − l − Z ( l j ) l (cid:12)(cid:12)(cid:12) c , Z ( l ) l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (104) × δδ (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − (cid:12)(cid:12)(cid:12) Z − Z (1) l (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12) Z ( l − l − Z ( l ) l (cid:12)(cid:12)(cid:12) c , Z ( l ) l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! .2 Higher order vertices involved in the effective action To compute the 2 n points Green functions, we proceed as for the two points function and consider a seriesexpansion of the potential in powers of Ψ ( θ, Z ). In products n Q i =1 | Ψ ( θ i , Z i ) | , n − k factors | Ψ ( θ i , Z i ) | are replaced by G (0 , , Z i ) at the higher orders. A derivation similar to (103) then shows that 2 n Greenfunctions are computed by using the expansion of the action: −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ σ θ ∇ θ (cid:19) Ψ ( θ, Z ) (105)+ 12 X n > k > | Ψ | k δ k [ k ]! δ k | Ψ | h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) Ψ ( θ, Z ) (cid:17)i! | Ψ( θ,Z ) | = G (0 ,Z ) where | Ψ | k δ k [ n ]! δ k | Ψ | generalizes (104) and stands for: X l i Z k Y i =1 (cid:16) dZ (1) l i ...dZ ( l i ) l i (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − Z (1) l j (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12) Z ( l − l i − Z ( l j ) l j (cid:12)(cid:12)(cid:12) c , Z ( l i ) l i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × δ k Q i δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ θ − (cid:12)(cid:12)(cid:12) Z − Z (1) li (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12)(cid:12) Z ( l − li − Z ( li ) li (cid:12)(cid:12)(cid:12)(cid:12) c , Z ( l i ) l i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k li Equation (105) can be shown recursively. To compute the 2 n correlation functions, the subgraphs with 2 k legs, k < n , are given by (105) at order 2 k . For k = n , the classical action yields a vertex:12 δ n [ n ]! δ n | Ψ | h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) Ψ ( θ, Z ) (cid:17)i! | Ψ( θ,Z ) | =0 | Ψ | n For k > n , a similar argument as in paragraph 1.1 in the vertex:12 δ k [ k ]! δ k | Ψ | h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) Ψ ( θ, Z ) (cid:17)i! | Ψ( θ,Z ) | =0 | Ψ | k k − n factor | Ψ ( θ, Z ) | have to be replaced by G (0 , , Z ). Summing over k , it means that the 2 n vertex iscomputed with:12 ∞ X l =0 δ l + n [ l + n ]! δ l + n | Ψ | h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) Ψ ( θ, Z ) (cid:17)i! | Ψ( θ,Z ) | =0 (cid:2) C ll + n (cid:3) ( G (0 , , Z )) l | Ψ | n where the symbol (cid:2) C ll + n (cid:3) reminds that among the product | Ψ ( θ , Z ) | ... | Ψ ( θ l + n , Z l + n ) | we sum over allthe C ll + n possibilities to replace l factor | Ψ ( θ j , Z j ) | by G (0 , , Z j ). Summing the series, we find for the 2 n vertices:= 12 n ! δ l + n [ l + n ]! δ l + n | Ψ | h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) Ψ ( θ, Z ) (cid:17)i! | Ψ( θ,Z ) | = G (0 ,Z ) | Ψ | n as requested. 45o compute the higher order corrections to the effective potential, it will be useful to write (105) withan other set of variables. We replace:Ψ θ − (cid:12)(cid:12)(cid:12) Z − Z (1) l i (cid:12)(cid:12)(cid:12) + ... + (cid:12)(cid:12)(cid:12) Z ( l − l i − Z ( l i ) l i (cid:12)(cid:12)(cid:12) c , Z ( l i ) l i by Ψ ( θ − l i , Z i ) where l i represents an arbitrary delay time. As a consequence, the 2 n -th vertex: V n = | Ψ | n δ n [ k ]! δ n | Ψ | h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) J, θ, Z, | Ψ | (cid:17) Ψ ( θ, Z ) (cid:17)i! | Ψ( θ,Z ) | = G (0 ,Z ) becomes (where ω − ( J, θ, Z ) stands for ω − (cid:16) J, θ, Z, | Ψ | (cid:17) when no confusion is possible): V n = 12 Z Ψ † ( θ, Z ) ∇ θ δ n − ω − ( J, θ, Z ) n − Q i =1 δ | Ψ ( θ − l i , Z i ) | n − Y i =1 | Ψ ( θ − l i , Z i ) | n − Y i =1 dZ i Ψ ( θ, Z ) dZdl i (106)+ Z G ′ ( Z ) δ n ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | n Y i =1 | Ψ ( θ − l i , Z i ) | n Y i =1 dZ i dZdl i + Z G ( Z ) δ n ∇ θ ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | n Y i =1 | Ψ ( θ − l i , Z i ) | n Y i =1 dZ i dZdl i with: G ′ ( Z ) = (cid:18) ∇ θ G ( θ, θ ′ , Z )2 (cid:19) θ = θ ′ However, the two last terms in (106) come from the backreaction of the n vertices on the whole system, andcan be neglected in first approximation. Actually in a neighborhood of the permanent regime, we have: G ( Z ) δ n ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | << δ n − ω − ( J, θ, Z ) n − Q i =1 δ | Ψ ( θ − l i , Z i ) | The neglected terms will be reintroduced later.We can thus consider that: V n = 12 Z Ψ † ( θ, Z ) ∇ θ δ n − ω − ( J, θ, Z ) n − Q i =1 δ | Ψ ( θ − l i , Z i ) | n − Y i =1 | Ψ ( θ − l i , Z i ) | n − Y i =1 dZ i Ψ ( θ, Z ) dZdl i (107)These terms are the coefficients obtained by the expansion of ω − ( J, θ, Z ) in powers of Ψ † ( θ, Z ) Ψ ( θ, Z ). Itis valid for | Ψ ( θ, Z ) | <
1. For | Ψ ( θ, Z ) | >
1, we can expand ω − ( J, θ, Z ) in powers of | Ψ( θ,Z ) | . Given theform of F and since arctan ( x ) = π − arctan (cid:0) x (cid:1) , the expansion is obtained by replacing the derivatives of F by those of − x F and by replacing ω with ω − .Formula (107) yields the vertices V n , n N , intervening in the computation of the 2 N correlationfunctions. We have to estimate the derivatives arising in (107), before computing the effective action. To compute the 2 n Green functions with vertices (107) and the graph expansion of the effective action, wefirst need to estimate the derivatives δ n ω − ( J,θ,Z ) n Q i =1 δ | Ψ( θ − l i ,Z i ) | appearing in (107). These derivatives can be computed46ecursively. To do so, we will need to approximate the results around some static solution. We define ¯ ω assolution of: ¯ ω − ( J, Z ) = G ¯ J ( Z ) + Z κN ¯ ω (cid:0) ¯ J, Z (cid:1) ¯ ω (cid:0) ¯ J, Z (cid:1) G (0 , , Z ) T ( Z, Z ) dZ ! (108)= G ¯ J ( Z ) + Z κN ¯ ω (cid:0) ¯ J, Z (cid:1) ¯ ω (cid:0) ¯ J, Z (cid:1) r π (cid:16) σ ¯ X r (cid:17) + πασ T ( Z, Z ) dZ where ¯ J ( Z ) is the average of J ( θ, Z ) over the full timespan. We also define: G ′ ( J, Z ) ≡ G ′ ¯ J ( Z ) + Z κN ¯ ω (cid:0) ¯ J, Z (cid:1) ¯ ω ( J, θ, Z ) G Z (0 , T ( Z, Z ) dZ ! (109)These quantities will be useful below. Now, we will find a recursive expansion for δ n ω − ( J,θ,Z ) n Q i =1 δ | Ψ( θ − l i ,Z i ) | . Using the recursive definition of ω − ( J, θ, Z ): ω − ( J, θ, Z ) = G J ( θ, Z ) + Z κN ω (cid:16) J, θ − | Z − Z | c , Z (cid:17) ω ( J, θ, Z ) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z | c , Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ) dZ (110)we first compute δω − ( J,θ,Z ) δ | Ψ( θ − l ,Z ) | : δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | (111)= δG J ( θ, Z ) + R κN ω (cid:18) J,θ − | Z − Z ′ | c ,Z ′ (cid:19) ω ( J,θ,Z ) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ δ | Ψ ( θ − l , Z ) | = κN ω (cid:16) J,θ − | Z − Z | c ,Z (cid:17) ω ( J,θ,Z ) T ( Z, Z ) G ′ [ J, ω, θ, Z, Ψ] δ (cid:16) l − | Z − Z | c (cid:17) − (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) G ′ [ J, ω, θ, Z,
Ψ]+ ω ( J,θ,Z ) R κN δω (cid:18) J,θ − | Z − Z ′ | c ,Z ′ (cid:19) δ (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z | c ,Z (cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ G ′ [ J, ω, θ, Z,
Ψ]1 − (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) G ′ [ J, ω, θ, Z,
Ψ]= ω ( J, θ − l , Z ) ˆ T ( θ, Z, Z , ω, Ψ) δ (cid:18) l − | Z − Z | c (cid:19) + Z δω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) δ | Ψ ( θ − l , Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T ( θ, Z, Z ′ , ω, Ψ) dZ ′ where we defined:ˆ T ( θ, Z, Z , ω, Ψ) (112)= 1 ω ( J, θ, Z ) κN T ( Z, Z ) G ′ [ J, ω, θ, Z,
Ψ]1 − (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) G ′ [ J, ω, θ, Z,
Ψ]47quation (111) shows that we also need δω ( J,θ,Z ) δ | Ψ( θ − l ,Z ) | to compute δω − ( J,θ,Z ) δ | Ψ( θ − l ,Z ) | . This is obtained by: δω ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = δF J ( θ, Z ) + R κN ω (cid:18) J,θ − | Z − Z ′ | c ,Z ′ (cid:19) ω ( J,θ,Z ) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ δ | Ψ ( θ − l , Z ) | (113)= ω ( J, θ − l , Z ) ˆ T ( θ, Z, Z , ω, Ψ) δ (cid:18) l − | Z − Z | c (cid:19) + Z δω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) δ | Ψ ( θ − l , Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T ( θ, Z, Z ′ , ω, Ψ) dZ ′ with:ˆ T ( θ, Z, Z ω, Ψ) = κN ω ( J, θ, Z ) T ( Z, Z ) F ′ [ J, ω, θ, Z, Ψ] ω ( J, θ, Z ) + (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) F ′ [ J, ω, θ, Z,
Ψ](114)Equation (113) and (114) define δω ( J,θ,Z ) δ | Ψ( θ − l ,Z ) | recursively. Actually, writing: δω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) δ | Ψ ( θ − l , Z ) | = Z ω (cid:18) J, θ − | Z − Z ′ | c − | Z ′ − Z ′′ | c , Z ′′ (cid:19) ˆ T (cid:18) θ − | Z − Z ′ | c , Z ′ , Z ′′ , ω, Ψ (cid:19) δ (cid:18) | Z − Z ′ | c + | Z ′ − Z ′′ | c − l (cid:19) dZ ′′ + Z δω (cid:18) J,θ − | Z − Z ′ | c − | Z ′− Z ′′ | c ,Z ′′ (cid:19) δ | Ψ( θ − l ,Z ) | ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c − | Z ′ − Z ′′ | c , Z ′′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T (cid:18) θ − | Z − Z ′ | c , Z ′ , Z ′′ , ω, Ψ (cid:19) dZ ′′ we have: δω ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = Z ω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) ˆ T ( θ, Z, Z , ω, Ψ) δ (cid:18) | Z − Z ′ | c − l (cid:19) dZ ′ + Z ω (cid:18) J, θ − | Z − Z ′ | c − | Z ′ − Z ′′ | c , Z ′′ (cid:19) ˆ T (cid:18) θ − | Z − Z ′ | c , Z ′ , Z ′′ , ω, Ψ (cid:19) × ˆ T ( θ, Z, Z ′ , ω, Ψ) δ (cid:18) | Z − Z ′ | c + | Z ′ − Z ′′ | c − l (cid:19) dZ ′ dZ ′′ + Z δω (cid:18) J, θ − | Z − Z ′ | c − | Z ′ − Z ′′ | c , Z ′′ (cid:19) δ | Ψ ( θ − l , Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c − | Z ′ − Z ′′ | c , Z ′′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) × ˆ T (cid:18) θ − | Z − Z ′ | c , Z ′ , Z ′′ , ω, Ψ (cid:19) ˆ T ( θ, Z, Z ′ , ω, Ψ) dZ ′ dZ ′′ δω ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = ∞ X n =1 Z ω J, θ − n X l =1 (cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12) c , Z ! n Y l =1 ˆ T θ − l − X j =1 (cid:12)(cid:12) Z ( j − − Z ( j ) (cid:12)(cid:12) c , Z ( l − , Z ( l ) , ω, Ψ × δ l − n X l =1 (cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12) c ! n − Y l =1 dZ ( l ) (115)and: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = ∞ X n =1 Z ω J, θ − n X l =1 (cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12) c , Z ! ˆ T (cid:16) θ, Z, Z (1) , ω, Ψ (cid:17) (116) × n Y l =2 ˆ T θ − l − X j =1 (cid:12)(cid:12) Z ( j − − Z ( j ) (cid:12)(cid:12) c , Z ( l − , Z ( l ) , ω, Ψ δ l − n X l =1 (cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12) c ! n − Y l =1 dZ ( l ) with the convention that Z (0) = Z and Z ( n ) = Z . We now use the static approximations (108) and (109).Actually, the values of ˆ T ( θ, Z, Z ω, Ψ) and ˆ T ( θ, Z, Z ω, Ψ) can be estimated for ¯ ω − (cid:0) ¯ J , Z (cid:1) . Moreover, inthe limit of small fluctuations, ¯ ω − (cid:0) ¯ J, Z (cid:1) , F ′ [ J, ¯ ω, Z, Ψ] and G ′ [ J, ¯ ω, Z, Ψ] can be approximated by theiraverage over Z , denoted ¯ ω − , ¯ F ′ and ¯ G ′ . Moreover for ¯ ω , both ˆ T and ˆ T can be considered independent of θ :ˆ T ( θ, Z, Z ¯ ω, Ψ) ≃ ˆ T ( Z, Z , ¯ ω )= κN ¯ ω − T ( Z, Z ) ¯ G ′ − ¯ G ′ ¯ ω R κN T ( Z,Z ′ ) dZ ′ r π (cid:16) σ Xr (cid:17) + πασ ˆ T ( θ, Z, Z ω, Ψ) ≃ ˆ T ( Z, Z , ¯ ω )= κN T ( Z, Z ) ¯ F ′ ¯ ω + ¯ F ′ R κN T ( Z,Z ′ ) dZ ′ r π (cid:16) σ Xr (cid:17) + πασ as a consequence ˆ T ( Z, Z , ¯ ω ) and ˆ T ( Z, Z , ¯ ω ) are functions of | Z − Z | denoted ˆ T ( | Z − Z | ). As a conse-quence (116) can be estimated by: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = ∞ X n =1 Z ω ( J, θ − l , Z ) ˆ T (cid:16)(cid:12)(cid:12)(cid:12) Z − Z (1) (cid:12)(cid:12)(cid:12)(cid:17) (117) × n Y l =2 ˆ T (cid:16)(cid:12)(cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12)(cid:12)(cid:17) δ l − n X l =1 (cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12) c ! × δ Z − Z − n X l =1 (cid:16) Z ( l − − Z ( l ) (cid:17)! n − Y l =1 dZ ( l ) and (115) is: δω ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = ∞ X n =1 Z ω J, θ − n X l =1 (cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12) c , Z ! n Y l =1 ˆ T (cid:16)(cid:12)(cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12)(cid:12)(cid:17) (118) × δ l − n X l =1 (cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12) c ! × δ Z − Z − n X l =1 (cid:16) Z ( l − − Z ( l ) (cid:17)! n − Y l =1 dZ ( l ) .3.2 Estimation of (117) and (115) close to the permanent regime1.3.2.1 General formula The series (117) can be computed by using the Fourier transform of the Diracfunctions: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = ∞ X n =1 Z ω ( J, θ − l , Z ) × ˆ T (cid:16)(cid:12)(cid:12)(cid:12) Z − Z (1) (cid:12)(cid:12)(cid:12)(cid:17) n Y l =2 ˆ T (cid:16)(cid:12)(cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12)(cid:12)(cid:17) × exp iλ cl − n X l =1 (cid:12)(cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12)(cid:12)!! × exp iλ . Z − Z − n X l =1 (cid:16) Z ( l − − Z ( l ) (cid:17)!! dλdλ n Y l =1 (cid:12)(cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12)(cid:12) d (cid:12)(cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12)(cid:12) dv l where the unit vectors v l are defined such that: Z ( l − − Z ( l ) = v l (cid:12)(cid:12)(cid:12) Z ( l − − Z ( l ) (cid:12)(cid:12)(cid:12) We also define λ . ( Z − Z ) = | λ | | Z − Z | cos ( θ ) λ .v l = | λ | cos ( θ l )The angles θ l are computed in the plane ( λ , Z − Z ) between the projection of v l and Z − Z . The angles ϕ l are defined as the angle between v l and the plane λ , Z − Z . As a consequence: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | (119)= ∞ X n =1 Z ω ( J, θ − l , Z ) × T ′′ ( λ + λ .v ) dv n Y l =2 Z T ′′ ( λ + λ .v l ) dv l exp ( iλcl + iλ . ( Z − Z )) dλdλ = δ ( | Z − Z | − cl ) ˆ T (cid:16)(cid:12)(cid:12)(cid:12) Z − Z (1) (cid:12)(cid:12)(cid:12)(cid:17) ω ( J, θ − l , Z )+ ( − n Z ω ( J, θ − l , Z ) × T ′′ ( λ + λ .v )2 dv n Y l =2 Z T ′′ ( λ + λ .v l )2 dv l exp ( iλcl + iλ . ( Z − Z )) dλdλ With the convention that for n = 1, the product n Q l =2 is set to be equal to 1. The functions T and T are thefourier transform of ˆ T H and ˆ T H respectively, and H is the heaviside function. Remark that the first termof (119) expresses the Dirac function δ ( | Z − Z | − cl ) as a Fourier transform:exp iλ cl − n X l =1 (cid:12)(cid:12)(cid:12) Z (0) − Z (1) (cid:12)(cid:12)(cid:12)!! × exp iλ . Z − Z − n X l =1 (cid:16) Z (0) − Z (1) (cid:17)!! dλdλ (cid:12)(cid:12)(cid:12) Z (0) − Z (1) (cid:12)(cid:12)(cid:12) d (cid:12)(cid:12)(cid:12) Z (0) − Z (1) (cid:12)(cid:12)(cid:12) dv l Some terms of (119) can be written in a useful form for the sequel:12 Z T ′′ ( λ + λ .v l ) dv l = π Z π T ′′ ( λ + | λ | cos ( θ l )) sin ( θ l ) dθ l = π Z − T ′′ ( λ + | λ | u ) du = 2 π ( T ′ ( λ + | λ | ) − T ′ ( λ − | λ | ))2 | λ |≡ ¯T ( λ, λ ) (120)50 T ′′ ( λ + λ .v l ) dv l = 2 π ( T ′ ( λ + | λ | ) − T ′ ( λ − | λ | ))2 | λ |≡ ¯T ( λ, | λ | ) (121)exp ( iλ . ( Z − Z )) dλ = exp ( i cos ( θ ) | λ | | Z − Z | ) sin ( θ ) | λ | d | λ | dθ (122)= exp ( iu | λ | | Z − Z | ) | λ | d | λ | du Using (120), (121) and (122), equation (119) becomes: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = ∞ X n =1 ( − n Z ω ( J, θ − l , Z ) × T ( λ + λ .v ) dv n Y l =2 Z T ( λ + λ .v l ) dv l exp ( iλcl + iλ . ( Z − Z )) dλdλ = − Z ω ( J, θ − l , Z ) × ¯T ( λ, | λ | )1 + ¯T ( λ, | λ | ) exp ( iλcl ) Z − exp ( iu | λ | | Z − Z | ) | λ | d | λ | dudλ = − Z ω ( J, θ − l , Z ) × ¯T ( λ, | λ | )1 + ¯T ( λ, | λ | ) exp ( iλcl ) (cid:18) | λ | | Z − Z | ) | Z − Z | | λ | (cid:19) d | λ | dλ (123)We remark that for even functions f , the following identity holds: Z + ∞ f ( | λ | ) 2 sin ( | λ | | Z − Z | ) | Z − Z | | λ | d | λ | = Z + ∞ f ( x ) exp ( ix | Z − Z | ) − exp ( − ix | Z − Z | ) i | Z − Z | xdx = Z + ∞ f ( x ) exp ( ix | Z − Z | ) i | Z − Z | xdx + Z −∞ f ( − x ) exp ( ix | Z − Z | ) i | Z − Z | xdx = Z + ∞−∞ f ( x ) exp ( ix | Z − Z | ) i | Z − Z | xdx so that (123) becomes: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | (124)= − Z ω ( J, θ − l , Z ) × ¯T ( λ, λ )1 + ¯T ( λ, λ ) λ i | Z − Z | exp ( iλcl + iλ | Z − Z | ) dλ dλ = − Z ω ( J, θ − l , Z ) × π ( T ′ ( λ + λ ) − T ′ ( λ − λ )) λ + π ( T ′ ( λ + λ ) − T ′ ( λ − λ )) λ i | Z − Z |× exp ( iλcl + iλ | Z − Z | ) dλ dλ Another simplification follows if we write T as a function of T : T ′ ( λ + λ ) − T ′ ( λ − λ ) = A A ( T ′ ( λ + λ ) − T ′ ( λ − λ ))Thus, setting: u = λ + λ v = λ − λ equation (124) becomes: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = − A iA | Z − Z | Z ω ( J, θ − l , Z ) × π ( T ′ ( u ) − T ′ ( v ))1 + 2 π ( T ′ ( u ) − T ′ ( v )) u − v (125) × exp (cid:16) i u cl + | Z − Z | ) + i v cl − | Z − Z | ) (cid:17) dλ dλ
51o compute (125), we study its two components independently: − A iA | Z − Z | Z ω ( J, θ − l , Z ) × π T ′ ( u )1 + 2 π ( T ′ ( u ) − T ′ ( v )) u − v (126) × exp (cid:16) i u cl + | Z − Z | ) + i v cl − | Z − Z | ) (cid:17) dudv and: A iA | Z − Z | Z ω ( J, θ − l , Z ) × π T ′ ( v )1 + 2 π ( T ′ ( u ) − T ′ ( v )) u − v (127) × exp (cid:16) i u cl + | Z − Z | ) + i v cl − | Z − Z | ) (cid:17) dudv In the integral (126), we first estimate the v integral using the residues theorem. The poles are solutions of:1 + 2 π T ′ ( u ) − T ′ ( v ) u − v = 0That is: v + 2 π T ′ ( v ) = u + 2 π T ′ ( u ) (128)with v = u . In the gaussian approximation for the transfer functions, T ( λ ) has the form: T ( λ ) = A exp (cid:18) − ν λ (cid:19) (cid:0) − erf (cid:0) i √ νλ (cid:1)(cid:1) (129)and its derivative satisfies: T ′ ( λ ) = − ν λ T ( λ ) − i √ ν As a consequence of these two identities, the solutions of (128) are given by: v (1 − πη T ( v )) = z (130)with: z = u (1 − πη T ( u ))To solve (130) it will be useful to expand T ( λ ) as a series expansion. In first approximation, one has (seeAbramovitz stegun): Im erf (cid:0) i √ νλ (cid:1) ≃ π + ∞ X k =1 exp (cid:0) − k (cid:1) sinh k √ νλk ≃ r νπ λ and: Im T ( λ ) ≃ A √ ν (cid:18) √ π exp (cid:18) − ν λ (cid:19) νλ (cid:19) > z > v ( πη T ( v )) = − z that is: ( Aπηv ) exp (cid:18) − η v (cid:19) (1 − erf ( i √ ηv )) = z for η << (cid:18) Aπηv (cid:19) exp (cid:18) − η v (cid:19) = z W k : v = vuut − η W k − z ( Aπ ) η ! for k >
0. They are approximatively equal to: v ≃ ± i vuut η ln u ( Aπ ) η ! + i (2 k + 1) π ! The terms involved in (126) can thus be evaluated at the poles. First, for ( Aπ ) << πη | T ( v ) | ≃ Aπη (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − η v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≃ Aπη exp ln √ u ( Aπ ) √ η !! = p ηu Asymptotically, for √ ηu >>
1, this formula justifies our previous approximation v (1 − πη T ( v )) ≃ − vπη T ( v ).For √ ηu <<
1, the solution is v = u and there is no pole. Second, we have: (cid:18) π ( T ′ ( u ) − T ′ ( v )) u − v (cid:19) ′ = − π T ′′ ( v ) u − v + 2 π ( T ′ ( u ) − T ′ ( v ))( u − v ) = − π T ′′ ( v ) u − v and (126) becomes: X k =0 πA A | Z − Z | Z ω ( J, θ − l , Z ) × ( u − v ) T ′ ( u )2 (1 + T ′′ ( v )) × exp i u cl + | Z − Z | ) − vuut η ln u ( Aπ ) η ! + i (2 k + 1) π ! | cl − | Z − Z || du Note that for ( Aπ ) η <<
1, we recover δ ( cl − | Z − Z | ) as neede in the lowest order approximation.The second integral (127) is obtained by inverting the role of u and v . It yields: − X k =0 πA A | Z − Z | Z ω ( J, θ − l , Z ) × ( u − v ) T ′ ( v )2 (1 + T ′′ ( u )) × exp − vuut η ln u ( Aπ ) η ! + i (2 k + 1) π ! ( cl + | Z − Z | ) + i v cl − | Z − Z | ) du and this can be neglected, since cl + | Z − Z | > Aπ ) η << δ ( cl + | Z − Z | ) = 0.Gathering the results for (126) and (127), we are left with: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | (131)= X k =0 πA A | Z − Z | Z ω ( J, θ − l , Z ) × ( u − v ) T ′ ( u )2 (1 + T ′′ ( v )) × exp i u cl + | Z − Z | ) − vuut η ln u ( Aπ ) η ! + i (2 k + 1) π ! | cl − | Z − Z || du T : T ′′ ( λ ) = − ν T ( λ ) + (cid:18) ν λ (cid:19) T ( λ ) + Ai (cid:0) √ ν (cid:1) λ v (1 − πη T ( v )) = u (1 − πη T ( u )) ≃ uπη T ( v ) ≃ v − u and this two equations imply that, for A (cid:0) √ η (cid:1) << π T ′′ ( v ) (132)= 1 − πη T ( v ) + 2 π (cid:16) η v (cid:17) T ( v ) + 2 πiA ( √ η ) v ≃ ( v − u ) ( − πηv ) ≃ i ( v − u ) r η C where C = r(cid:16) ln (cid:16) u ( Aπ ) η (cid:17) + i (2 k + 1) π (cid:17) . A consequence of (132) is that: − π T ′′ ( v ) u − v ≃ iπ √ ηC Moreover, for ( Aπ ) η <<
1, the function T ′ ( u ) can be replaced by the multiplication by i cl + | Z − Z | . Weare thus led to rewrite (131): δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | (133)= A A X k =0 ( cl + | Z − Z | )2 | Z − Z | Z ω ( J, θ − l , Z ) × T ( u ) √ ηC × exp i u cl + | Z − Z | ) − vuut η ln u ( Aπ ) η ! + i (2 k + 1) π ! | cl − | Z − Z || du ≡ A A Ξ ( | Z − Z | , l , ¯ ω ) ω ( J, θ − l , Z )Remark that, for ( Aπ ) η << vuut η ln u ( Aπ ) η ! + i (2 k + 1) π ! × exp − vuut η ln u ( Aπ ) η ! + i (2 k + 1) π ! | cl − | Z − Z || ≃ δ ( cl − | Z − Z | )so that one recovers the first order term.For ( Aπ ) η <<
1, Ξ ( | Z − Z | , l , ¯ ω ) is a function of | Z − Z | written Ξ ( | Z − Z | , ¯ ω ).Finally, the sum in (133) can be estimated in the following way:54 k =0 exp − vuut η ln u ( Aπ ) η ! + i (2 k + 1) π ! | cl − | Z − Z || = X k =0 exp − vuut η ln u ( Aπ ) η !vuut i (2 k + 1)ln (cid:16) u ( Aπ ) η (cid:17) π | cl − | Z − Z || ≃ C π Re Z exp (cid:18) − C r η √ ix | cl − | Z − Z || (cid:19) dx = C π Re Z exp (cid:18) − C r η (cid:0) x (cid:1) exp (cid:18) i x ) (cid:19) | cl − | Z − Z || (cid:19) dx with: C = vuut ln u ( Aπ ) η ! The upper bound of the integral is set to 1, in agreement with our approximation ln (cid:16) u ( Aπ ) η (cid:17) >>
1. Itamounts to neglect the poles for k >>
1, whose contributions are decreasing quicly with k as given byoscillatory integrals of frequencies proportional to k .By a change of variable, the last integral is also given by:2 C π Re Z exp (cid:18) − C r η (cid:16)p v + iv (cid:17) | cl − | Z − Z || (cid:19) (cid:18)p v + v √ v (cid:19) dv and we are left with the estimation for the first vertex: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | (134)= A A X k =0 ( cl + | Z − Z | )2 | Z − Z | Z ω ( J, θ − l , Z ) × T ( u ) √ ηC du C π Re Z exp (cid:18) − C r η (cid:16)p v + iv (cid:17) | cl − | Z − Z || (cid:19) (cid:18)p v + v √ v (cid:19) dv We can estimate the integral R dv in (134) by integrating between 0and + ∞ . 2 C π Re Z + ∞ exp (cid:18) − C r η (cid:16)p v + iv (cid:17) | cl − | Z − Z || (cid:19) (cid:18) p v − √ v (cid:19) dv = 2 C π Re Z + ∞ exp (cid:18) − C r η (cid:16)p v a + ivb (cid:17)(cid:19) (cid:18) p v − √ v (cid:19) dv with a = b = | cl − | Z − Z || . The last integral can be rewritten: − ∂ a C q η + C r η Z da C π Re Z + ∞ exp (cid:18) − C r η (cid:16)p v a + ivb (cid:17)(cid:19) dv = − ∂ a C q η + C r η Z da C πb Re Z + ∞ exp (cid:18) − C r η (cid:16)p b + v ab + iv (cid:17)(cid:19) dv ≃ − ∂ a C q η + C r η Z da C πb Re Z + ∞ exp (cid:18) − C r η (cid:18) − (cid:18)(cid:16) ab + i (cid:17) v + ab v (cid:19)(cid:19)(cid:19) dv (135)55or a ≃ b <<
1. We use that: − ∂ a C q η + C r η Z da C π Re Z + ∞ exp (cid:18) − C r η (cid:16) − (cid:16) ( a + ib ) v + a v (cid:17)(cid:17)(cid:19) dv = − ∂ a C q η + C r η Z da C π Re r aa + ib K C s a ( a + ib ) η ! (136)where K is a modified Bessel function, and that the following identity homds for K : r aa + ib K C s a ( a + ib ) η ! ≃ r aa + ib vuut π C q a ( a + ib ) η exp − C s a ( a + ib ) η ! for C >>
1. Then computing the integral R da in (136) yields: C r η Z da q aa + ib r π C q a ( a + ib ) η − Cη a + ib √ aη ( a + ib ) − Cη a + ib q aη ( a + ib ) exp − C s a ( a + ib ) η ! ≃ C r η q aa + ib r π C q a ( a + ib ) η − Cη a + ib √ aη ( a + ib ) exp − C s a ( a + ib ) η ! = − √ πa q C √ aη ( a + ib ) a + ib exp − C s a ( a + ib ) η ! for C >>
1. For a = b , this identity reduces to: − √ π − i p (1 + i ) r √ ηCa exp (cid:18) − Ca √ η √ i (cid:19) (137)The derivative arising in (136) can be estimated by: − ∂ a C q η r aa + ib vuut π C q a ( a + ib ) η exp − C s a ( a + ib ) η ! = − s(cid:18) − i (cid:19) √ πC a η exp (cid:16) − p (1 + i ) C q a η (cid:17)(cid:16) η (cid:17) r C q a η (1 + 2 i ) ((1 + i )) η s a η − (12 − i ) p (1 + i ) Ca ! ≃ s(cid:18) − i (cid:19) √ πC a η exp (cid:16) − p (1 + i ) C q a η (cid:17)(cid:16) η (cid:17) r C q a η (cid:16) (12 − i ) p (1 + i ) Ca (cid:17) = √ π s(cid:18) − i (cid:19) exp (cid:18) − q (1+ i ) η Ca (cid:19)q Ca √ η (cid:16) (12 − i ) p (1 + i ) (cid:17) (138)Gathering (137) and (138), we find that for C >>
1, For a = b = | cl − | Z − Z || , we find for (136):56 √ √ π (cid:18) √ (cid:19) exp (cid:18) − cos ( π ) C √ η | cl − | Z − Z || (cid:19)q C | cl −| Z − Z ||√ η cos cos (cid:0) π (cid:1) C √ η | cl − | Z − Z || ! = C p √ √ π exp (cid:18) − √ √ C √ η | cl − | Z − Z || (cid:19)q C | cl −| Z − Z ||√ η cos p √ C √ η | cl − | Z − Z || ! In the sequel, for ( Aπ ) η <<
1, we approximate: C = vuut ln u ( Aπ ) η ! ≃ vuut ln Aπ ) η ! Finally, the integral over u in (134) is: Z T ( u ) √ ηC exp (cid:16) i u cl + | Z − Z | ) (cid:17) du = 1 √ ηC ˆ T (cid:18) cl + | Z − Z | (cid:19) du so that, using that ˆ T (cid:18) cl + | Z − Z | (cid:19) = A A ˆ T (cid:18) cl + | Z − Z | (cid:19) The result for (134) is: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | ≃ √ √ π (cid:0) √ (cid:1) D exp ( − D | cl − | Z − Z || ) q C | cl −| Z − Z ||√ η cos ( D | cl − | Z − Z || ) × ( cl + | Z − Z | )2 | Z − Z | ˆ T (cid:18) cl + | Z − Z | (cid:19) where: D = 2 p √ C √ η We also write this result in a more compact form: δω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = A A Ξ ( | Z − Z | , l , ¯ ω ) ω ( J, θ − l , Z ) (139)for | Z − Z | cl <
1, and 0 otherwise, with:Ξ ( | Z − Z | , l , ¯ ω ) = √ √ π (cid:0) √ (cid:1) D exp ( − D | cl − | Z − Z || ) p D | cl − | Z − Z || cos ( D | cl − | Z − Z || ) (140) × ( cl + | Z − Z | )2 | Z − Z | ˆ T (cid:18) cl + | Z − Z | (cid:19) Similarly: δω ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | = Ξ ( | Z − Z | , l , ¯ ω ) ω ( J, θ − l , Z ) (141)The appearance of the cos ( D | cl − | Z − Z || ) in (140) is a consequence of our approximation computing theintegral between 0 and + ∞ . This approximation breaks down when the cos function becomes negative. Asa consequence for D | cl − | Z − Z || > π , we can set Ξ ( | Z − Z | , l , ¯ ω ) ≃ .4 Computation of the n -th Vertices in (107) The two previous paragraphs computed the firt order derivatives δω − ( J,θ,Z ) δ | Ψ( θ − l ,Z ) | . This can be used to computehigher order terms involved in (107). We first compute the 2-th vertex is given by: − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 δ h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) J, θ, | Ψ | (cid:17) Ψ ( θ, Z ) (cid:17)i δ | Ψ | | Ψ( θ,Z ) | = G (0 ,Z ) = − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 (cid:2) ∇ θ (cid:0) ω − ( J, θ, G (0 , Z )) (cid:1) + ω − ( J, θ, G (0 , Z )) ∇ θ (cid:3) + (cid:18) ∇ θ G ( θ, θ ′ , Z )2 (cid:19) θ = θ ′ δ h ∇ θ (cid:16) ω − (cid:16) J, θ, | Ψ | (cid:17)(cid:17)i δ | Ψ | | Ψ( θ,Z ) | = G (0 ,Z ) and given the results of the previous section, it writes:= − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 (cid:2) ∇ θ ω − ( J, θ, Z, G ) (cid:3) + s(cid:18) σ ¯ X r (cid:19) + 2 ασ − σ ¯ X r G (0 , Z ) Z A A Ξ ( | Z − Z | , l , ¯ ω ) ω ( J, θ − l , Z ) dZ dl ≃ − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 (cid:2) ∇ θ ω − ( J, θ, Z, G ) ∇ θ (cid:3) at the lowest order in perturbation theory. The inverse frequency ω − ( J ( θ ) , θ, Z, G (0 , Z )) is solution of: ω − ( J, θ, Z ) = G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) J, θ − | Z − Z | c , Z (cid:17) ω ( J, θ, Z ) W ω ( J, θ, Z ) ω (cid:16) J, θ − | Z − Z | c , Z (cid:17) G (0 , Z ) dZ ≃ G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) J, θ − | Z − Z | c , Z (cid:17) W (cid:18) ω ( J,θ,Z ) ω (cid:16) J,θ − | Z − Z | c ,Z (cid:17) (cid:19) dZ r π (cid:16) X r (cid:17) + π αω ( J, θ, Z ) (142)where we used (96). Recall that in first approximation, that will be used to compute higher order vertices,the solution is the constant: ω − ( J ( θ ) , θ, Z, G (0 , Z )) ≃ ¯ X r = G ¯ J + κN Z T ( Z, Z ) W (1) dZ r π (cid:16) X r (cid:17) + π α where ¯ J is the average of the external current. In this approximation, the 2-th vertex writes: − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 (cid:2) ω − ( G Z (0 , ∇ θ (cid:3) = − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 ¯ X r ∇ θ and the associated Green function is (96): G ¯ X r = exp − r(cid:16) σ ¯ X r (cid:17) + ασ L !p π r(cid:16) σ ¯ X r (cid:17) + ασ exp (cid:18) θ f − θ i σ ¯ X r (cid:19)
58e conclude by computing the derivative of the functionals ˆ T ( θ, Z, Z ω, Ψ) and ˆ T ( θ, Z, Z ω, Ψ) (112) and(114) arising in (113) and (114). They will be useful to obtain the higher order vertices. We find: δ ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17) δ | Ψ ( θ − l , Z ) | = 1 ω ( J, θ, Z ) κN T ( Z, Z ) G ′ h J, ω, θ, Z, | Ψ | i − (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) G ′ h J, ω, θ, Z, | Ψ | i = δ ln ω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17)! + δ ln G ′ [ J,ω,θ,Z, Ψ] δ | Ψ( θ − l ,Z ) | ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17) − (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) G ′ h J, ω, θ, Z, | Ψ | i + ˆ T (cid:16) θ, Z, Z , ω, | Ψ | (cid:17) ω ( J, θ, Z ) ˆ T (cid:16) θ, Z, Z , ω, | Ψ | (cid:17) ω ( J, θ − l , Z ) δ (cid:18) l − | Z − Z | c (cid:19) + ˆ T (cid:16) θ, Z, Z , ω, | Ψ | (cid:17) ω ( J, θ, Z ) Z δω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) δ | Ψ ( θ − l , Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T (cid:16) θ, Z, Z ′ , ω, | Ψ | (cid:17) dZ ′ δ ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17) δ | Ψ ( θ − l , Z ) | = ˆ T ( θ, Z, Z , ω, Ψ) δ ln ω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | + δ ln G ′ [ J,ω,θ,Z, | Ψ | ] δ | Ψ( θ − l ,Z ) | − R κN ω (cid:18) J,θ − | Z − Z ′ | c ,Z ′ (cid:19) ω ( J,θ,Z ) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ G ′ h J, ω, θ, Z, | Ψ | i + ˆ T ( θ, Z, Z , ω, Ψ) δ ln ω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | In the limit of small fluctuations around an equilibrium frequency ω , δ ln G ′ [ J,ω,θ,Z, | Ψ | ] δ | Ψ( θ − l ,Z ) | << G ′ h J, ω, θ, Z, | Ψ | i δ ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17) δ | Ψ ( θ − l , Z ) | ≃ T (cid:16) θ, Z, Z , ω, | Ψ | (cid:17) δ ln ω − ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | (143)with: 59 ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17) δ | Ψ ( θ − l , Z ) | = κN ω ( J, θ, Z ) T ( Z, Z ) F ′ h J, ω, θ, Z, | Ψ | i ω ( J, θ, Z ) + (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) F ′ h J, ω, θ, Z, | Ψ | i = δ ln ω ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17)! × R κN ω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ ! F ′ h J, ω, θ, Z, | Ψ | i − ω ( J, θ, Z ) ω ( J, θ, Z ) + (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) F ′ h J, ω, θ, Z, | Ψ | i + ω ( J, θ, Z ) δ ln F ′ [ J,ω,θ,Z, Ψ] δ | Ψ( θ − l ,Z ) | ˆ T ( θ, Z, Z ω, Ψ) ω ( J, θ, Z ) + (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) F ′ [ J, ω, θ, Z,
Ψ]+ ˆ T (cid:16) θ, Z, Z , ω, | Ψ | (cid:17) ω − ( J, θ, Z ) ˆ T ( θ, Z, Z , ω, Ψ) ω ( J, θ − l , Z ) δ (cid:18) l − | Z − Z | c (cid:19) + ˆ T (cid:16) θ, Z, Z , ω, | Ψ | (cid:17) ω − ( J, θ, Z ) Z δω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) δ | Ψ ( θ − l , Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T (cid:16) θ, Z, Z ′ , ω, | Ψ | (cid:17) dZ ′ = 2 δ ln ω ( J, θ, Z ) δ | Ψ ( θ − l , Z ) | ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17)! × R κN ω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ ! F ′ h J, ω, θ, Z, | Ψ | i ω ( J, θ, Z ) + (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) F ′ h J, ω, θ, Z, | Ψ | i + ω ( J, θ, Z ) δ ln F ′ [ J,ω,θ,Z, | Ψ | ] δ | Ψ( θ − l ,Z ) | ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17) ω ( J, θ, Z ) + (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) F ′ h J, ω, θ, Z, | Ψ | i For relatively high frequency, ω ( J, θ, Z ) > F ′ [ J, ω, θ, Z, Ψ] << δ ln G ′ [ J,ω,θ,Z, Ψ] δ | Ψ( θ − l ,Z ) | << δ ˆ T ( θ,Z,Z ω, Ψ) δ | Ψ( θ − l ,Z ) | can be discarded in first approximation. n -th vertex Close to the permanent regime, the 2 n point vertex contribution δ n ω − ( J,θ,Z ) n Q i =1 δ | Ψ( θ − l i ,Z i ) | can be computed using(116) and neglecting the derivatives of ˆ T ( θ, Z, Z , ω, Ψ). We assume that we have ranked the derivativessuch that l < l < ... < l n . An other simplification arises. Considering the vertex:60 n = 12 ( n )! Z Ψ † ( θ, Z ) δ n (cid:2)R Ψ † ( θ, Z ) ∇ θ ω − ( J, θ, Z ) Ψ ( θ, Z ) dZdθ (cid:3) n Q i =1 δ | Ψ ( θ − l i , Z i ) | (144) × n − Y i =1 | Ψ ( θ − l i , Z i ) | dl i Ψ ( θ, Z ) dZdθ | Ψ( θ,Z ) | = G (0 ,Z ) = 12 ( n − Z Ψ † ( θ, Z ) ∇ θ δ n − ω − ( J, θ, Z ) n − Q i =1 δ | Ψ ( θ − l i , Z i ) | | Ψ( θ,Z ) | = G (0 ,Z ) n − Y i =1 | Ψ ( θ − l i , Z i ) | dZ i Ψ ( θ, Z ) dZdθdl i + 12 n ! Z ∇ θ G δ n ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | | Ψ( θ,Z ) | = G (0 ,Z ) n Y i =1 | Ψ ( θ − l i , Z i ) | n Y i =1 dZ i dZdθdl i As for (107), we can neglect the second term, so that V n reduces to: V n = 12 ( n − Z Ψ † ( θ, Z ) ∇ θ δ n − ω − ( J, θ, Z ) n − Q i =1 δ | Ψ ( θ − l i , Z i ) | n − Y i =1 | Ψ ( θ − l i , Z i ) | n − Y i =1 dZ i dl i Ψ ( θ, Z ) dθdZ (145)The neglected contributions will be reintroduced in Appendix 4. The 2 n vertex contribution δ n ω − ( J,θ,Z ) n Q i =1 δ | Ψ( θ − l i ,Z i ) | can then be computed recursively. Assuming l < l < ... < l n and using (139) and (141), one has: δ n ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | = n X k =1 (cid:18) π ¯ ωl (cid:19) k X i
12 Ψ † ( θ, Z ) ∇ θ Ψ ( θ, Z ) n − Y i =1 | Ψ ( θ − l i , Z i ) | Ξ ( J, θ, l i , | Z i − Z | ) ¯ ω ! ≡
12 Ψ † ( θ, Z ) ∇ θ Ψ ( θ, Z ) n − Y i =1 | Ψ ( θ − l i , Z i ) | ¯Ξ ( J, θ, l i , | Z i − Z | ) ! (147)where ¯ ω is the average of ω ( J, θ − l i , Z i )on variable Z i . This represents the vertex of valence n issued from Z . Appendix 2 Computation of the graphs expansion and generatingfunctional for correlation functions
We compute the sum of graphs involved in the partition function with source term, i.e. the graphs deducedfrom the.interaction terms (147). This is done in several step. We first give the general form of these graphs.Then, we compute the factors arising from the vertices (147). This allows to find the full sum of graphs,that is, the generating function for correlation functions, then the sum of connected graphs, and ultimatelythe one particle irreducible graphs, which yields the effective action. The computation of the graphs is firstperformed without inertia coefficients, i.e. without considering the potential and setting ζ ( k ) = 0. Thesecoefficients are included in the computation subsequently. ζ l The 2 n -th point vertex Γ n contribution to the effective action is obtained by considering any 1 P I graphmade of arbitrary number m i of vertices V k i ( l i , Z i ) defined in (146), where k i n . Those graphs have noloop drawn between two legs of any of the vertices (these contributions are already taken into account bythe expansion around G (0 , Z )). They have P i m i k i − n segments.The absence of internal loops implies that the graphs associated to Γ n are made of n paths P i with l P i segment and: P l P i = P i m i k i − n . The segments are connected by the vertices V k i ( l i , Z i ). Thecontributions associated to these paths are products of 2 points Green functions. In the approximation ofconstant G (0 , Z ) = ¯ G (0 , k segments and of total length L between twopoints θ i and θ f is: exp − r(cid:16) σ ¯ X r (cid:17) + ασ − σ ¯ X r ! L ! p π r(cid:16) σ ¯ X r (cid:17) + ασ ! k where: ¯ X r = arctan (cid:16)(cid:16) X r − X p (cid:17) p ¯ J + ¯ G (0 , (cid:17)p ¯ J + ¯ G (0 , J , ¯ X r , ¯ G (0 ,
0) are the averages along the paths, since we sum over θ along the paths. Thesepaths are connected through the vertices V k i ( l i , Z i ) of valence k i n . These vertices connect one path atsome time θ and k others at time θ − l i with i = 1 , ...k −
1. To compute the sum of connected graphs, we62ave to add the graphs associated to all possible repartitions of vertices V k i ( l i , Z i ), k i n between thepaths P i . Each of this graph has to be summed over the time of insertion (i.e. θ , θ − l i ) for each vertex.To perform this computation, we consider the graph made of n paths for the points Z i with k ( i ) l vertices ofvalence 2 l with l = 1 , ..., n issued from the i -th path.This means that the total number of segments of this graph is: n X l =1 l n X i =1 k ( i ) l − n Equation (147) shows that the vertices issued from Z i induce factors of the form ¯Ξ ( J, θ, l i , | Z i − Z k | ) + ∇ θ ¯Ξ ( J,θ,l i , | Z i − Z k | ) G ′ , and a factor: G ′ = (cid:18) ∇ θ G ( θ, θ ′ , Z )2 (cid:19) θ = θ ′ = − s(cid:18) σ ¯ X r (cid:19) + 2 ασ − σ ¯ X r G (0 , Z )due to the insertion of Ψ † ( θ, Z ) ∇ θ Ψ ( θ, Z ) except at the final points of the graphs. Indeed, in:12 Ψ † ( θ, Z ) ∇ θ Ψ ( θ, Z ) n − Y i =1 | Ψ ( θ − l i , Z i ) | ¯Ξ ( J, θ, l i , | Z i − Z | ) ! the gradient ∇ θ | Ψ ( θ − l i , Z i ) | induces a nul contribution in the Green functions since: ∇ θ ( G ( θ , θ − l i , Z ) G ( θ − l i , θ , Z )) = 0The insertion Ψ † ( θ, Z ) ∇ θ Ψ ( θ, Z ) inserted between two points θ and θ yields terms of the form:( ∇ θ G ( θ , θ, Z )) G ( θ, θ , Z )that results in the presence of the term −G ′ ( Z ) ¯Ξ ( J, θ, l i , | Z i − Z | ), where G ′ ( Z ) = ∇ θ ( G ( θ , θ, Z )) θ = θ .On the other hand, the term G ( Z ) ∇ θ ¯Ξ ( J, θ, l i , | Z i − Z | ) with G ( Z ) = G ( θ, θ, Z ), has to be added andthe overall vertice yields a factor − G ′ ( Z ) ¯Ξ ( J, θ, l i , | Z i − Z k | ) + G ( Z ) ∇ θ ¯Ξ ( J, θ, l i , | Z i − Z k | ) ≡ ∇ θ G ( Z ) ¯Ξ ( J, θ, l i , | Z i − Z k | ) (148)In the sequel it is approximated by its value in the permanent regime −G ′ ( Z ) ¯Ξ ( J, θ, l i , | Z i − Z k | ) for thesake of simplicity.We consider the contributions of graphs without external legs. These ones are reintroduced ultimately.At a final point of graph θ ( i ) f , the insertion of Ψ † ( θ, Z ) ∇ θ Ψ ( θ, Z ) induces a term ∇ θ ( i ) i . There is anoverall factor of ( −
1) for each vertex, that can be accounted for by inserting rather G ′ at each vertices,except the initial one that includes a factor −∇ θ ( i ) i .The P i k ( i ) l vertices of valence l are associated to a factor (cid:16)P i k ( i ) l (cid:17) ! due to the development of theexponential. There is (cid:16)P i k ( i ) l (cid:17) ! Q i k ( i ) l ! ways to distribute these vertices between the n points. Once k ( i ) l vertices, l = 1; ..., n are attributed to a point i , there are (cid:16)P nl =1 k ( i ) l (cid:17) ! ways to order in time these vertices.63he factor associated to a vertex of valence l issued from i at time θ ( i ) is:¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) (149)= P { k ,...,k l − }⊂{ ,...,n − } Q k j R θ ( i ) − θ ( kj ) i (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c dl k j p π r(cid:16) σ ¯ X r (cid:17) + ασ ! l − Z Ψ † (cid:16) θ ( i ) , Z i (cid:17) ∇ θ ( i ) δ l − ω − (cid:0) J, θ ( i ) , Z i (cid:1) l − Q i =1 δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) − l k j , Z k j (cid:1)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17) dZ i | Ψ( θ,Z ) | = G (0 ,Z ) ≃ P { k ,...,k l − }⊂{ ,...,n − } Q k j R θ ( i ) q − θ ( kj ) i (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c dl k j p π r(cid:16) σ ¯ X r (cid:17) + ασ ! l ∇ θ ( i ) G ( Z ) δ l − R ω − (cid:0) J, θ ( i ) , Z i (cid:1) dZ i G ( Z ) l − Q i =1 δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) − l k j , Z k j (cid:1)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) with the convention ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) q , { Z j } j = i (cid:17) = 0 for n = 1. We also use the same convention as for (148): forany quantity X , ∇ θ ( i ) G ( Z ) X = G ( Z ) ∇ θ ( i ) X + G ′ ( Z ) X .The propagators induced by the vertices has beeen included in the definition of ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) .The functions depend implicitely on the border of the timespans h θ ( j ) f , θ ( j ) i i . Actually, the integrations R θ ( i ) − θ ( kj ) i (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c dl k j induces the presence of products of Heaviside functions H (cid:18) θ ( i ) f − θ ( k j ) i − | Z i − Z kj | c (cid:19) . We first start by computing the full sum of graphs without external legs and arising from all combination ofvertices between n initial points and n final points. The factors associated to each vertices have been foundin the previous paragraph. The P nl =1 k ( i ) l vertices implies the integration over θ ( i ) i < θ ( i )1 < ...θ ( i ) P nl =1 k ( i ) l < θ ( i ) f of the product of terms ¯Ξ ( l q )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) for q = 1 to P nl =1 k ( i ) l . The number of l q equal to l is k ( i ) l .Once an order l , l , ... is chosen, there are Q k ( i ) l ! ways to order the vertices satisfying this order. Then,summing over the various orders l , l , ... and over the k ( i ) l such that P nl =1 k ( i ) l = m is fixed, the global factorassociated to the vertices is: Z θ ( i ) i <θ ( i )1 <...θ ( i ) m <θ ( i ) f m Y q =1 X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) q , { Z j } j = i (cid:17)! δ (cid:16) θ ( i )1 − θ ( i ) i (cid:17) δ (cid:16) θ ( i ) m − θ ( i ) f (cid:17) dθ ( i ) q (150)The delta functions accounts for the fact that without external legs, two vertices are set at the borders ofthe interval. If we approximate P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) q , { Z j } j = i (cid:17) by its average on the interval h θ ( i ) i , θ ( i ) f i , that is: R θ ( i ) f θ ( i ) i P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) q , { Z j } j = i (cid:17) dθ ( i ) q θ ( i ) f − θ ( i ) i the sum of vertices for a pair of external points, denoted i becomes: R θ ( i ) f θ ( i ) i P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) f − θ ( i ) i m Z θ ( i ) i <θ ( i )1 <...θ ( i ) m <θ ( i ) f δ (cid:16) θ ( i )1 − θ ( i ) i (cid:17) δ (cid:16) θ ( i ) m − θ ( i ) f (cid:17) m Y q =1 dθ ( i ) q The contribution of i to the graphs is obtained by convoluting this quantity with the free propagator on theleft and on the right and adding the free propagator. The convolution by the propagator on the right and64n the left is: Z G (cid:16) θ ( i ) f , θ ( i ) f ′ , Z i (cid:17) R θ ( i ) f ′ θ ( i ) i ′ P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) f ′ − θ ( i ) i ′ m × Z θ ( i ) i ′ <θ ( i )1 <...θ ( i ) m <θ ( i ) f ′ δ (cid:16) θ ( i )1 − θ ( i ) i ′ (cid:17) δ (cid:16) θ ( i ) m − θ ( i ) f ′ (cid:17) m Y q =1 dθ ( i ) q G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ dθ ( i ) f ′ = Z G (cid:16) θ ( i ) f , θ ( i ) m , Z i (cid:17) Z θ ( i ) i <θ ( i )1 <...θ ( i ) m <θ ( i ) f R θ ( i ) m θ ( i )1 P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) m − θ ( i )1 m m Y q =1 dθ ( i ) q G (cid:16) θ ( i )1 , θ ( i ) i , Z i (cid:17) Replacing R θ ( i ) mθ ( i )1 P l ¯Ξ ( l )1 ( Z i ,θ ( i ) , { Z j } j = i ) dθ ( i ) θ ( i ) m − θ ( i )1 by its average over h θ ( i ) i , θ ( i ) f i it becomes: Z R θ ( i ) f θ ( i ) i P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) f − θ ( i ) i m Z θ ( i )1 <θ ( i )1 <...θ ( i ) m <θ ( i ) m m Y q =1 dθ ( i ) q exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) Λ = 1 m ! Z θ ( i ) f θ ( i ) i X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) ! m exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) Λ where we write: Z θ ( i ) f θ ( i ) i n X l =2 ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) = ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) (151)= n X l =2 X { k ,...,k l }⊂{ ,...,n } ,k j = i ¯Ξ ( l )1 (cid:16) Z i , (cid:8) Z k j (cid:9) , θ ( i ) i , θ ( i ) f (cid:17) Then summing over m , adding the free propagator and performing the product over i = 1 , ..., n yields thesum of graphs for the n paths between n initial points θ ( j ) i , Z ( j ) and n final points θ ( j ) f , Z ( j ) : Y i G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) + ∇ out θ ( i ) i G ′ X m> m ! (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) m = Y i ∇ out θ ( i ) i Λ (cid:16) exp (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n where ∇ outθ ( i ) f are understood to act outside the terms in the expression and with:Λ = r π s(cid:18) σ ¯ X r (cid:19) + 2 ασ Λ = s(cid:18) σ ¯ X r (cid:19) + 2 ασ − σ ¯ X r X n Y i G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) + ∇ out θ ( i ) i G ′ X m> m ! (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) m (152)= Y i ∇ out θ ( i ) i Λ (cid:16) exp (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n For later purposes, we can rewrite (152) in two alternative manners. First, we can keep a propagator infactor. In that case, the sum of graphs becomes, for m > Z G (cid:16) θ ( i ) f , θ ( i ) f ′ , Z i (cid:17) R θ ( i ) f ′ θ ( i ) i ′ P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) f ′ − θ ( i ) i ′ m × Z θ ( i ) i ′ <θ ( i )1 <...θ ( i ) m <θ ( i ) f ′ δ (cid:16) θ ( i )1 − θ ( i ) i ′ (cid:17) δ (cid:16) θ ( i ) m − θ ( i ) f ′ (cid:17) m Y q =1 dθ ( i ) q dθ ( i ) f ′ G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ = Z G (cid:16) θ ( i ) f , θ ( i ) m , Z i (cid:17) Z θ ( i ) i ′ <θ ( i )2 <...θ ( i ) m <θ ( i ) f X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)! R θ ( i ) m θ ( i ) i ′ P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) m − θ ( i ) i ′ m − × m Y q =2 dθ ( i ) q G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ ≃ Z X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)! R θ ( i ) f θ ( i ) i ′ P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) f − θ ( i ) i ′ m − × Z θ ( i ) i ′ <θ ( i )2 <...θ ( i ) m <θ ( i ) f m Y q =2 dθ ( i ) q exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) Λ G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ and for m = 1: Z X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) Λ G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ Adding the free propagator yields for the sum of graphs: X n Y i Z δ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17) + ∇ out θ ( i ) i ′ G ′ X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)! exp R θ ( i ) f θ ( i ) i ′ P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) f − θ ( i ) i ′ (153) × exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) Λ G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ X n Y i Z G (cid:16) θ ( i ) f , θ ( i ) f ′ , Z i (cid:17) G − (cid:16) θ ( i ) f ′ , θ ( i ) i ′ (cid:17) + ∇ out θ ( i ) i ′ G ′ X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)! δ (cid:16) θ ( i ) f ′ − θ ( i ) i ′ (cid:17) (154)+ ∇ out θ ( i ) i ′ G ′ X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)! X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) f ′ , { Z j } j = i (cid:17)! exp R θ ( i ) f ′ θ ( i ) i ′ P l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) θ ( i ) f ′ − θ ( i ) i ′ G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ dθ ( i ) f ′ ζ l As presented in the text, the effective action may be modified by including both inertia in frequency change,through a potential for maintaining and activating of new connections: Z Ψ † ( θ, Z ) (cid:0) ∇ θ (cid:0) ω − ( J ( θ ) , θ, Z, G ) (cid:1)(cid:1) Ψ ( θ, Z ) − ζ Z | Ψ ( θ, Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! + ∞ X n =1 ζ n n ! Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! (155)= −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ σ ∇ θ − ω − (cid:16) J, θ, Z, | Ψ | (cid:17)(cid:19) Ψ ( θ, Z ) + ∞ X n =2 n ! ζ ( n ) Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! with: ζ ( l ) = ζ l , l > ζ (2) = ζ − ζ ζ (1) = 0The second term represents the limitation in increasing the number of connections. This amounts to shiftthe vertices by + ζ . The factor − ζ accounts for a minimal number of connections maintained. It dependson external activity J .The first terms modify the 4-th vertices by − ζ . We write ¯Ξ ( l )1 ( Z i ) for ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) q , { Z j } j = i (cid:17) . The 2points propagator is modified by replacing α with: α + X k > k ! C k ζ ( k ) Λ k − = X k > l k − ζ ( k ) Λ k − which modifies the values of Λ and Λ . To obtain the contribution of the potential to the the vertices, weproceed as for the frequency. The vertices involved in the 2 n points correlation function are given by anexpansion of: ∞ X k =2 k ! ζ ( k ) Z | Ψ ( θ, Z ) | k − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! of order 2 n . The 2 n vertex is then δ nn Q j =1 δ | Ψ ( θ j , Z j ) | ∞ X k = n k ! ζ ( k ) Z | Ψ ( θ, Z ) | k − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) !! | Ψ( θ j ,Z j ) | = G (0 ,Z j ) n points on the system. ∞ X k = n k k ! ζ ( k ) Z | Ψ ( θ, Z ) | δ n − n − Q j =1 δ | Ψ ( θ j , Z j ) | k − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! | Ψ( θ j ,Z j ) | = G (0 ,Z j ) n − Y j =1 | Ψ ( θ j , Z j ) | + ∞ X k = n k ! ζ ( k ) Z G ( θ, θ, Z ) δ nn Q j =1 δ | Ψ ( θ j , Z j ) | k − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! | Ψ( θ j ,Z j ) | = G (0 ,Z j ) n Y j =1 | Ψ ( θ j , Z j ) | = ∞ X k = n k k ! C n − k − ζ ( k ) Λ k − n Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ X k = n k ! ζ ( k ) Z G ( θ, θ, Z ) C nk − n Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 12 ( n − ∞ X k = n k − n )! ζ ( k ) Λ k − n Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + 12 n ! ∞ X k = n Z G ( θ, θ, Z ) k ( k − n − ζ ( k ) Λ k − n n Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (156)The inclusion of the first term of (156) in the graphs expansion is straightforward. In terms of vertices,and given the overall − ζ ( l ) amounts to replace:¯Ξ ( l )1 ( Z i ) → ¯Ξ ( l )1 ( Z i ) − ζ ( l ) Λ l − With respect to the computation of graphs with n vertices, the terms ζ ( l ) , l n have to be replaced with: ζ ( l ) e = X k > l l ! k ! C lk ζ ( k ) Λ k − l = X k > l k − l )! ζ ( k ) Λ k − l For ζ ( k ) slowly varying and Λ >>
1, this is approximatively equal to ζ ( l ) . We keep the notation ζ ( l ) e → ζ ( l ) .The second term of (156) corresponds to the backreaction term that can be neglected at first. It will bereintroduced later.We note that if we write the potential as: Z | Ψ ( θ, Z ) | V (cid:18)Z | Ψ | (cid:19) the second term of (156) writes: Z G ( θ, θ, Z ) δ nn Q j =1 (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z i | c , Z i (cid:17)(cid:12)(cid:12)(cid:12) V (cid:18)Z | Ψ | (cid:19) (157) The computation of the sum of graphs without external legs is the same as before, the vertices are − ζ ( l ) Λ l − +¯Ξ ( l )1 ( Z i ), except that the last vertex among the m = P nl =1 k ( i ) l is replaced by − ζ ( l ) Λ l − + ¯Ξ ( l )1 ( Z i ) ∇ outθ ( i ) f Λ . For a68articular l , this induces an additional factor k ( i ) l . The P nl =1 k ( i ) l − n external legs: Y i P m> m ! R θ ( i ) i <θ ( i )1 <...θ ( i ) m <θ ( i ) f m − Q q =1 (cid:16)P l (cid:16) − ζ ( l ) Λ l − + ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) q , { Z j } j = i (cid:17)(cid:17)(cid:17) dθ ( i ) q × P l − ζ ( l ) Λ l − + ∇ outθ ( i ) i Λ ¯Ξ ( l )1 (cid:16) Z i , θ m , { Z j } j = i (cid:17)!! dθ m (158) × Λ n exp − Λ n X j =1 θ ( j ) f − n X j =1 θ ( j ) i We define:ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) = ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) − ¯ ζ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) = n X l =2 X { k ,...,k l − }⊂{ ,...,n − } ,k j = i ¯Ξ ( l )1 (cid:16) Z i , (cid:8) Z k j (cid:9) , θ ( i ) i , θ ( i ) f (cid:17) − ζ ( l ) (cid:16) θ ( i ) f − θ ( i ) i (cid:17) Λ l and: ¯ ζ n = n X l =2 X { k ,...,k l − }⊂{ ,...,n − } ,k j = i ζ ( l ) Λ l = n X l =1 C l − n − ζ ( l ) Λ l For example: ¯ ζ = ζ (2) Λ , ¯ ζ = ζ (3) Λ + 3 ζ (2) ΛIf we express ¯ ζ n as a function of the initial set of variables ζ ( l ) , we have:¯ ζ n = n X l =1 C ln P k > l k − l )! ζ ( k ) Λ k − l Λ l − = n X l =1 C ln X k > l k − l )! ζ ( k ) Λ k − so that: ¯ ζ = X k > k − ζ ( k ) Λ k − = ζ (2) Λ + X k > k − ζ ( k ) Λ k − and: ¯ ζ = X k > k − ζ ( k ) Λ k − + X k > k − ζ ( k ) Λ k − = 3 ζ (2) Λ + 3 X k > k − ζ ( k ) Λ k − + X k > k − ζ ( k ) Λ k − = 3 ζ (2) Λ + 3 X k > k − ζ ( k ) Λ k − + X k > Sup (1 , ( k − ζ ( k ) ( k − k − We will assume that ¯ ζ < ζ n > n >
2. This is possible under the conditions: X k > k − ζ ( k ) Λ k − < (cid:12)(cid:12) ζ (2) (cid:12)(cid:12) Λ < X k > k − ζ ( k ) Λ k − + X k > Sup (1 , ( k − ζ ( k ) k − k − that are satisfied for a certain range of the parameters, since: X k > Sup (1 , ( k − ζ ( k ) k − k − − X k > k − ζ ( k ) Λ k − = X k > ( Sup (1 , ( k − − ζ ( k ) k − k −
69s positive for ζ ( k ) large enough for k >
5. Replacing the terms ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) by their averageover the timespan θ ( j ) f − θ ( j ) i :ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17) ≡ D ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)E θ ( i ) f − θ ( i ) i θ ( i ) f − θ ( i ) i = R θ ( i ) f θ ( i ) i (cid:16)P l − ζ ( l ) Λ l − + ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17)(cid:17) dθ ( i ) θ ( i ) f − θ ( i ) i expression (158) writes: Y i X m> m ! (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) m − − ¯ ζ n + ∇ out θ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n = Y i X m> m ! (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) m − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (159) × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n so that the sum of graphs for an arbitrary number of external points becomes: X n n Y i =1 − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) × exp (cid:16) − Λ (cid:16)P nj =1 θ ( i ) f − P nj =1 θ ( i ) i (cid:17)(cid:17) Λ n (160)Given that ≃ G (0 , Z ), the hypothesis of the text, we may assume that ¯ ζ n > X n n Y i =1 (cid:18)Z (cid:16) δ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17) (161)+ − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ , θ ( i ) f (cid:17)(cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) Λ G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ and: X n n Y i =1 Z G (cid:16) θ ( i ) f , θ ( i ) f ′ , Z i (cid:17) G − (cid:16) θ ( i ) f ′ , θ ( i ) i ′ , Z i (cid:17) + − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) δ (cid:16) θ ( i ) f ′ − θ ( i ) i ′ (cid:17) (162)+ − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) f ′ (cid:17)(cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ , θ ( i ) f ′ (cid:17)(cid:17) ×G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ dθ ( i ) f ′ o The generating functional is obtained by including a source term
R (cid:0) Ω † (cid:0) θ ( i ) (cid:1) Ψ (cid:0) θ ( i ) (cid:1) + Ψ † (cid:0) θ ( i ) (cid:1) Ω (cid:0) θ ( i ) (cid:1)(cid:1) inthe action (155) Z Ψ † ( θ, Z ) (cid:0) ∇ θ (cid:0) ω − ( J ( θ ) , θ, Z, G (0 , Z )) (cid:1)(cid:1) Ψ ( θ, Z ) (163) − ζ Z | Ψ ( θ, Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! + ∞ X n =1 ζ n (2 n )! Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! + Z (cid:16) Ω † (cid:16) θ ( i ) (cid:17) Ψ (cid:16) θ ( i ) (cid:17) + Ψ † (cid:16) θ ( i ) (cid:17) Ω (cid:16) θ ( i ) (cid:17)(cid:17) The partition function can thus be expanded as a series expansion: Z (Ω) = 1 + X n > n ! n Y i =1 Z Ω † (cid:16) θ ( i ) f (cid:17) G n (cid:16) θ (1) f , θ (1) i , ..., θ ( n ) f , θ ( n ) i (cid:17) Ω (cid:16) θ ( i ) i (cid:17) The graphs G n (cid:16) θ (1) f , θ (1) i , ..., θ ( n ) f , θ ( n ) i (cid:17) are not the graphs computed in the previous paragraph. They arerather computed around Ω (cid:16) θ ( i ) i (cid:17) = 0. This value of the source term corresponds, after legendre transform,to the minimum of the effective action, i.e. a possible non nul expectation for the field Ψ (cid:0) θ ( i ) (cid:1) = (cid:10) Ψ (cid:0) θ ( i ) (cid:1)(cid:11) .The graphs computed in the previous paragraph are on the contrary computed for a null expectation, i.e. (cid:10) Ψ (cid:0) θ ( i ) (cid:1)(cid:11) = 0. This value of the field corresponds to a source term Ω (cid:0) θ ( i ) (cid:1) . Decomposing (163) as: Z Ψ † ( θ, Z ) (cid:0) ∇ θ (cid:0) ω − ( J ( θ ) , θ, Z, G (0 , Z )) (cid:1)(cid:1) Ψ ( θ, Z ) − ζ Z | Ψ ( θ, Z ) | (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! (164)+ ∞ X n =1 ζ n (2 n )! Z | Ψ ( θ, Z ) | n − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z i | c , Z i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ! + Z (cid:16) Ω † (cid:16) θ ( i ) (cid:17) (cid:16) θ ( i ) (cid:17) Ψ (cid:16) θ ( i ) (cid:17) + Ψ † (cid:16) θ ( i ) (cid:17) Ω (cid:16) θ ( i ) (cid:17)(cid:17) + Z (cid:18)(cid:16) Ω (cid:16) θ ( i ) (cid:17) − Ω (cid:16) θ ( i ) (cid:17)(cid:17) † (cid:16) θ ( i ) (cid:17) Ψ (cid:16) θ ( i ) (cid:17) + Ψ † (cid:16) θ ( i ) (cid:17) (cid:16) Ω (cid:16) θ ( i ) (cid:17) − Ω (cid:16) θ ( i ) (cid:17)(cid:17)(cid:19) Expanding in series of (cid:0) Ω (cid:0) θ ( i ) (cid:1) − Ω (cid:0) θ ( i ) (cid:1)(cid:1) and (cid:0) Ω (cid:0) θ ( i ) (cid:1) − Ω (cid:0) θ ( i ) (cid:1)(cid:1) t yields a series of expectations (cid:10)Q Ψ (cid:0) θ ( i ) (cid:1) Ψ † (cid:0) θ ( i ) (cid:1)(cid:11) computed with (cid:10) Ψ (cid:0) θ ( i ) (cid:1)(cid:11) which are precisely the graphs computed previously: This series Z (Ω) is thus ob-tained from (160) by multiplication by source terms and summing over n . Actually, the expansion underthe condition that h Ψ ( θ, Z ) i = (cid:10) Ψ † ( θ, Z ) (cid:11) = 0, the sum of graphs including a subtadpole graph, i.e. thegraphs that can be factored by Ω (cid:0) θ ( i ) (cid:1) or Ω † (cid:0) θ ( i ) (cid:1) at one end, cancel. Moreover, the sim of non connectedgraphs including factors Ω † (cid:16)(cid:0) θ ( i ) (cid:1) ′ (cid:17) and Ω (cid:0) θ ( i ) (cid:1) at each end can be factored in the partition function, andcan thus be discarded.We define ∆Ω (cid:0) θ ( i ) (cid:1) = (cid:0) Ω (cid:0) θ ( i ) (cid:1) − Ω (cid:0) θ ( i ) (cid:1)(cid:1) , so that: Z (Ω) = 1 + Z ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) ∆Ω (cid:16) θ ( i ) i (cid:17) + X n > n ! n Y i =1 Z ∆Ω † (cid:16) θ ( i ) f (cid:17) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) ∆Ω (cid:16) θ ( i ) i (cid:17) × exp (cid:16) − Λ (cid:16)P nj =1 θ ( i ) f − P nj =1 θ ( i ) i (cid:17)(cid:17) Λ n
71e can define ˆΞ , (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) = 0 and thus: Z ∆Ω † (cid:16) θ ( i ) f (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) Λ ∆Ω (cid:16) θ ( i ) i (cid:17) = Z ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) ∆Ω (cid:16) θ ( i ) i (cid:17) so that Z (Ω) can be written: Z (Ω) = X n > n ! Z n Y i =1 ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) ∆Ω (cid:16) θ ( i ) i (cid:17) that is: Z (Ω) = X n > n ! Z ∆Ω † (cid:16) θ ( i ) f (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) Λ × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Λ (cid:18) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:19) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) ∆Ω (cid:16) θ ( i ) i (cid:17) n ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) ≃ G ′ p π r(cid:16) σ ¯ X r (cid:17) + ασ ! l X { k ,...,k l − }⊂{ ,...,n − } ,k j = i Y k j = i Z θ ( i ) − (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c θ ( kj ) i dl k j ¯Ξ (cid:16) J, θ ( i ) , l k j , (cid:12)(cid:12) Z i − Z k j (cid:12)(cid:12)(cid:17) = X { k ,...,k l − }⊂{ ,...,n − } ,k j = i ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , (cid:8) Z k j (cid:9)(cid:17) ≃ P { k ,...,k l − }⊂{ ,...,n − } Q k j R θ ( i ) − θ ( kj ) i (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c dl k j p π r(cid:16) σ ¯ X r (cid:17) + ασ ! l − ∇ θ ( i ) G ( Z ) δ l − R ω − (cid:0) J, θ ( i ) , Z i (cid:1) Ψ (cid:0) θ ( i ) , Z i (cid:1) dZ il − Q i =1 δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) − l k j , Z k j (cid:1)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) The generating functional for connected correlation functions is given by: W (Ω) = ln Z (Ω) Z (0)We assume that − ¯ ζ n and ¯Ξ ,n grow approximatively at the same rate, so that − ¯ ζ n + ¯Ξ1 ,n ( Zj, { Zm } m = j,θ ( j ) i ,θ ( j ) f ) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ1 − ¯ ζ n + ¯Ξ1 ,n ( Zj, { Zm } m = j,θ ( j ) i ,θ ( j ) f ) θ ( j ) f − θ ( j ) i depends weakly on n and can be replaced by its limit for n → ∞ for n >
3. We also replace ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) by their limit ˆΞ , ∞ (cid:16) Z i , θ ( j ) i , θ ( j ) f (cid:17) and ¯Ξ , ∞ (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) . Asa consequence, for n > − ¯ ζ n + ∇ outθ ( j ) i Λ ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ≃ − ¯ ζ + ∇ outθ ( j ) i Λ ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Define: O ,n (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19) = − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Λ (cid:18) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:19) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) (165) O , ∞ (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19) = − ¯ ζ + ∇ outθ ( j ) i Λ ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Λ (cid:18) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i (cid:19) (cid:16) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) O , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) = 1 G = exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) Λ H (cid:16) θ ( i ) f − θ ( i ) i (cid:17) Remark that these operators depend on all variables θ ( j ) i , θ ( j ) f for i = j , mainly through Heaviside functions.We thus define: O ( n )1 ,n (cid:16)(cid:16) θ ( i ) i , θ ( i ) f (cid:17) i (cid:17) = n Y i =1 O ,n (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19) O ( n )1 , ∞ (cid:16)(cid:16) θ ( i ) i , θ ( i ) f (cid:17) i (cid:17) = n Y i =1 O , ∞ (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19) (1 + O ,n ) ( n ) (cid:16)(cid:16) θ ( i ) i , θ ( i ) f (cid:17) i (cid:17) = n Y i =1 (cid:18) O ,n (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19)(cid:19) (1 + O , ∞ ) ( n ) (cid:16)(cid:16) θ ( i ) i , θ ( i ) f (cid:17) i (cid:17) = n Y i =1 (cid:18) O , ∞ (cid:18) θ ( i ) i , θ ( i ) f , (cid:16) θ ( i ) i , θ ( i ) f (cid:17) j = i (cid:19)(cid:19) and for any operator A : h A i n = Z " n Y i =1 ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) A " n Y i =1 ∆Ω (cid:16) θ ( i ) i (cid:17) , n > h A i = 1For A acting on (cid:16) ∆Ω (cid:16) θ ( i ) i (cid:17)(cid:17) ⊗ n , the expectation h A i k , k < n for A symmetric is evaluated on the k firstvariables and defines an operator acting on (cid:16) ∆Ω (cid:16) θ ( i ) i (cid:17)(cid:17) ⊗ n − k .73ith these assumptions: Z (Ω) = 1 + Z ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) ∆Ω (cid:16) θ ( i ) i (cid:17) + Z Y i =1 ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) (1 + O , ) (2) ∆Ω (cid:16) θ ( i ) i (cid:17) + X n > n ! Z n Y i =1 ∆Ω † (cid:16) θ ( i ) f (cid:17) G (cid:16) θ ( i ) f , θ ( i ) i , Z i (cid:17) (1 + O ,n ) ( n ) ∆Ω (cid:16) θ ( i ) i (cid:17) = 1 + X n > n ! D (1 + O ,n ) ( n ) E n ≃ h i + D (1 + O , ) (2) E + X n > n ! D (1 + O , ∞ ) ( n ) E n and we obtain the expression for the generating functional W (Ω): W (Ω) = ln h i + 12 D (1 + O , ) (2) E + X n > n − D (1 + O , ∞ ) ( n ) E n The derivative of W (Ω) with respect to Ω † (cid:0) θ ( i ) (cid:1) and Ω (cid:0) θ ( i ) (cid:1) defines the background field: δW (Ω) δ Ω † (cid:16) θ ( i )1 (cid:17) = Z θ ( i )1 D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G ∆Ω (cid:16) θ ( i ) (cid:17) dθ ( i ) (166)and: δW (Ω) δ Ω (cid:16) θ ( i )1 (cid:17) = Z θ ( i )1 ∆Ω † (cid:16) θ ( i ) (cid:17) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G dθ ( i ) (167)and this yields the following equation for the classical field Ψ (cid:0) θ ( i ) (cid:1) : Z θ ( i )1 D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G (cid:16) θ ( i )1 , θ ( i ) (cid:17) ∆Ω (cid:16) θ ( i ) (cid:17) dθ ( i ) = Ψ (cid:16) θ ( i )1 (cid:17) (168)along with: Z θ ( i )1 ∆Ω † (cid:16) θ ( i ) (cid:17) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G (cid:16) θ ( i ) , θ ( i )1 (cid:17) dθ ( i ) = Ψ † (cid:16) θ ( i )1 (cid:17) To compute the effective action, we consider (168) evaluated at Ψ (cid:0) θ ( i ) (cid:1) , so that Ω Ψ ( θ ( i ) ) (cid:0) θ ( i ) (cid:1) = 0: − D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n Ψ ( θ ( i ) ) G (cid:16) Ω Ψ ( θ ( i ) ) =0 (cid:16) θ ( i ) (cid:17)(cid:17) = Ψ (cid:16) θ ( i ) (cid:17) Z Ω † (cid:16) θ ( i ) (cid:17) Ψ (cid:16) θ ( i ) (cid:17) + Z Ω (cid:16) θ ( i ) (cid:17) Ψ † (cid:16) θ ( i ) (cid:17) = (cid:16) ∆Ω (cid:16) θ ( i ) (cid:17)(cid:17) † D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G (cid:16) ∆Ω (cid:16) θ ( i ) (cid:17)(cid:17) + H.C. +Ψ † (cid:16) θ ( i ) (cid:17) Ω Ψ ( θ ( i ) ) =0 (cid:16) θ ( i ) (cid:17) + H.C. = 2 h i + D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n − Ψ † (cid:16) θ ( i ) (cid:17) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G − ( θ ( i ) ) Ψ (cid:16) θ ( i ) (cid:17) + H.C. and the effective action writes:Γ (Ψ) = h i + D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n (169) −
12 ln h i + 12 D (1 + O , ) (2) E + X n > n ! D (1 + O , ∞ ) ( n ) E n −
12 Ψ † (cid:16) θ ( i ) (cid:17) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G − ( θ ( i ) ) Ψ (cid:16) θ ( i ) (cid:17) + H.C.
The expectations appearing in (169) are functions of Ψ through (168). They can be estimated by usingaverages quantities. To so we will rewrite (168) in a compact form. Given our definitions, we have:
Z (cid:16) ∆Ω (cid:16) θ ( i ) (cid:17)(cid:17) † D (1 + O , ) (2) E G (cid:16) ∆Ω θ ( i ) (cid:17) = D (1 + O , ) (2) E Z (cid:16) ∆Ω (cid:16) θ ( i ) (cid:17)(cid:17) † X n > n − D (1 + O , ∞ ) ( n ) E n − (cid:16) ∆Ω θ ( i ) (cid:17) = X n > n − D (1 + O , ∞ ) ( n ) E n We also define the average h O , ∞ i by:exp ( h O , ∞ i ) = X n > n ! D (1 + O , ∞ ) ( n ) E n so that, for a background field of relatively small amplitude: X n > n − D (1 + O , ∞ ) ( n ) E n ≃ h O , ∞ i X n > n − D (1 + O , ∞ ) ( n − E n − ≃ h O , ∞ i (exp ( h O , ∞ i ) − − h O , ∞ i )and: X n > n ! D (1 + O , ∞ ) ( n ) E n ≃ exp ( h O , ∞ i ) − − h O , ∞ i − h O , ∞ i
75n average, we also have D (1 + O , ) (2) E ≃ (1 + O , ) rD (1 + O , ) (2) E and: X n > n − D (1 + O , ∞ ) ( n ) E n − ≃ (1 + O , ∞ ) (exp ( h O , ∞ i ) − − h O , ∞ i )where the operators (1 + O , ) and (1 + O , ∞ ) are understood as operators acting only the i -th variable,obtained by averaging over Z j and θ ( j ) where j = i . We define: x = h O , ∞ i y = D (1 + O , ) (2) E z = h O , ∞ i − h i = h O , ∞ i and (168) becomes:(1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) G (cid:16) θ ( i )1 , θ ( i ) (cid:17) ∆Ω (cid:16) θ ( i ) (cid:17) = Ψ (cid:16) θ ( i )1 (cid:17) (170)Equation (170) can be used to find defining equations for x , y , z . Actually, inverting (170):∆Ω (cid:16) θ ( i ) (cid:17) = (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) ! G ! − Ψ (cid:16) θ ( i )1 (cid:17) (171)The inverse of the operator is found by using (161) to write:(1 + O ,n ) G = (cid:0) O ,n (cid:1) ∗ G where ∗ denotes the convolution product and:¯ O ,n = − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) ΛAs a consequence, we write: (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) ! G ! − = G − ∗ − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! Computing the expectations of (1 + O , ∞ ), (1 + O , ) and O , ∞ in state (171). yields: x = (cid:28) G − (cid:20) − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19)(cid:21) A − Ψ (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) O , ∞ (cid:1) G (cid:12)(cid:12)(cid:12)(cid:12) G − (cid:20) − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19)(cid:21) A − Ψ (cid:29) y = (cid:28) G − (cid:20) − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19)(cid:21) A − Ψ (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) O , (cid:1) G (cid:12)(cid:12)(cid:12)(cid:12) G − (cid:20) − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19)(cid:21) A − Ψ (cid:29) z = (cid:28) G − (cid:20) − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19)(cid:21) A − Ψ (cid:12)(cid:12)(cid:12)(cid:12) ¯ O , ∞ G (cid:12)(cid:12)(cid:12)(cid:12) G − (cid:20) − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19)(cid:21) A − Ψ (cid:29) (172)with: A − Ψ = (cid:2)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:3) − Ψ76xpression (172) can be rewritten by using that the operators ¯ O (standing for 1+ ¯ O , , 1+ ¯ O , ∞ ...) decomposeas ¯ O = ¯ O ( C ) + ¯ O ( D ) ∇ (cid:12)(cid:12) ¯ O D (cid:12)(cid:12) < ¯ O C and ¯ O − ≃ ( ¯ O ( C ) − ¯ O ( D ) ∇ )( ¯ O ( C ) ) ≡ ¯ O † ( ¯ O ( C ) ) and [ A ] − can be obtained by:[ A ] − = (cid:2)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:3) − = (cid:16) O † , ∞ (cid:17) + exp ( − x ) (cid:16) − ¯ O † , ∞ + y (cid:16) O † , (cid:17) − x (cid:16) O † , ∞ (cid:17)(cid:17)(cid:16) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17)(cid:17) and the expressions for the expectations are: x = − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:16) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17)(cid:17) h Ψ | G − B † (cid:0) O , ∞ (cid:1) B | Ψ i y = − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:16) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17)(cid:17) h Ψ | G − B † (cid:0) O , (cid:1) B | Ψ i z = − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:16) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17)(cid:17) h Ψ | G − B † ¯ O , ∞ B | Ψ i with: B = (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) = (cid:0) O , ∞ (cid:1) (1 − x exp ( − x )) − exp ( − x ) ¯ O , ∞ + y (cid:0) O , (cid:1) exp ( − x )Now, use that for slowly varying field, we can replace: (cid:16)(cid:16) O † , ∞ (cid:17) (1 − x exp ( − x )) + exp ( − x ) (cid:16) − ¯ O † , ∞ + y (cid:16) O † , (cid:17)(cid:17)(cid:17) (cid:0) O , ∞ (cid:1) × (cid:0) (1 − x exp ( − x )) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1)(cid:1)(cid:1) ≃ (cid:16) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17)(cid:17) since the omitted terms are of second order in derivatives:. As a consequence: x ≃ − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17) h Ψ | G − (cid:0) O , ∞ (cid:1) | Ψ i y ≃ − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17) h Ψ | G − (cid:0) O , (cid:1) | Ψ i z ≃ − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17) h Ψ | G − ¯ O , ∞ | Ψ i These expressions allows a further simplification. We define: X = − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1) O ( C )1 , ∞ + exp ( − x ) (cid:16) − ¯ O ( C )1 , ∞ + y (cid:16) O ( C )1 , (cid:17) − x (cid:16) O ( C )1 , ∞ (cid:17)(cid:17)
77o that: x = X h Ψ | G − (cid:0) O , ∞ (cid:1) | Ψ i ≡ rX (173) y = X h Ψ | G − (cid:0) O , (cid:1) | Ψ i ≡ sXz = X h Ψ | G − ¯ O , ∞ | Ψ i ≡ tX The variable X satifies the equation: X = − rX ) (cid:0) − tX + (cid:0) s − r (cid:1) X (cid:1) O ( C )1 , ∞ + exp ( − rX ) (cid:16) − ¯ O ( C )1 , ∞ + sX (cid:16) O ( C )1 , (cid:17) − rX (cid:16) O ( C )1 , ∞ (cid:17)(cid:17) whose solution is approximatively: X ≃ − r (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) ! − t (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) + ( s − r ) (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) ! O ( C )1 , ∞ + exp − r (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) ! − ¯ O ( C )1 , ∞ + s (cid:16)
1+ ¯ O ( C )1 , (cid:17)(cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) − r (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) ! (174)= (cid:16) O ( C )1 , ∞ (cid:17) + exp − h Ψ |G − (
1+ ¯ O , ∞ ) | Ψ i (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) ! N (cid:16) O ( C )1 , ∞ (cid:17) + exp − h Ψ |G − (
1+ ¯ O , ∞ ) | Ψ i (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) ! D where: N = − h Ψ | G − ¯ O , ∞ | Ψ i + 12 (cid:16) h Ψ | G − (cid:0) O , (cid:1) | Ψ i − h Ψ | G − (cid:0) O , ∞ (cid:1) | Ψ i (cid:17)(cid:16) O ( C )1 , ∞ (cid:17) D = (cid:18) − ¯ O ( C )1 , ∞ (cid:16) O ( C )1 , ∞ (cid:17) + (cid:16) O ( C )1 , (cid:17) h Ψ | G − (cid:0) O , (cid:1) | Ψ i − (cid:16) O ( C )1 , ∞ (cid:17) h Ψ | G − (cid:0) O , ∞ (cid:1) | Ψ i (cid:19) In first approximation, this reduces to: X ≃ (cid:16) O ( C )1 , ∞ (cid:17) + exp − h Ψ |G − (
1+ ¯ O , ∞ ) | Ψ i (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) ! − h Ψ | G − ¯ O , ∞ | Ψ i + ( h Ψ |G − (
1+ ¯ O , ) | Ψ i −h Ψ |G − (
1+ ¯ O , ∞ ) | Ψ i ) (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) !(cid:16) O ( C )1 , ∞ (cid:17) + exp − h Ψ |G − (
1+ ¯ O , ∞ ) | Ψ i (cid:16)
1+ ¯ O ( C )1 , ∞ (cid:17) ! ¯ O ( C )1 , ∞ (cid:16) O ( C )1 , ∞ (cid:17) Appendix 3. Correlation functions
The correlation functions can be computed by successive derivatives of the effective action (169) estimatedat the background field. The first derivative (169) yields ∆Ω (cid:0) θ ( i ) (cid:1) as usual, apart from a constant term:78 Γ (Ψ) ∂ Ψ † (cid:0) θ ( i ) , Z i (cid:1) = ∂ ∆Ω † (cid:0) θ ( i ) , Z i (cid:1) ∂ Ψ † (cid:0) θ ( i ) , Z i (cid:1) ∂ Γ (Ψ) ∂ Ω † (cid:0) θ ( i ) , Z i (cid:1) (175)= 12 ∆Ω (cid:16) θ ( i ) (cid:17) − D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n G − ( θ ( i ) ) Ψ (cid:16) θ ( i ) , Z i (cid:17) = 12 " (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) G − Ψ (cid:16) θ ( i ) , Z i (cid:17) − " (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) G − ( θ ( i ) ) Ψ (cid:16) θ ( i ) , Z i (cid:17) This formula does not allow to compute the vacuum of the system, since (175) is identically nul for thevacuum Ψ (cid:0) θ ( i ) , Z i (cid:1) . We will derive Ψ (cid:0) θ ( i ) , Z i (cid:1) below through a graph expansion, but differentiating (175)yields the 2 points effective vertex, and successive derivatives will compute the higher order correlationfunctions. We will need δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i )2 (cid:17) and δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i )2 (cid:17) , the derivatives δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:16) θ ( i )2 (cid:17) and δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:16) θ ( i )2 (cid:17) being obtained by hermitianconjugation. We have: δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) ! G ! − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) (176)+ δ ∆Ω (cid:0) θ ( i ) (cid:1) δ Ψ (cid:16) θ ( i ) f (cid:17) Ψ † (cid:16) θ ( i ) (cid:17) + δ ∆Ω † (cid:0) θ ( i ) (cid:1) δ Ψ (cid:16) θ ( i ) f (cid:17) Ψ (cid:16) θ ( i ) (cid:17) ∆Ω (cid:16) θ ( i )1 (cid:17) − δ ∆Ω (cid:0) θ ( i ) (cid:1) δ Ψ (cid:16) θ ( i ) f (cid:17) δδ Ω ( θ ( i ) ) (cid:18) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − (cid:19) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − ∆Ω (cid:16) θ ( i )1 (cid:17) − δ ∆Ω † (cid:0) θ ( i ) (cid:1) δ Ψ (cid:16) θ ( i ) f (cid:17) δδ Ω † ( θ ( i ) ) (cid:18) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − (cid:19) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − ∆Ω (cid:16) θ ( i )1 (cid:17) where: δδ Ω ( θ ( i ) ) (cid:18) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − (cid:19) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − ∆Ω (cid:16) θ ( i )1 (cid:17)
79s a notation for the convolution: D (1 + O , ) (2) E + X n > n − D (1 + O , ∞ ) ( n ) E n − ∗ δδ Ω (cid:0) θ ( i ) (cid:1) D (1 + O , ) (2) E + X n > n − D (1 + O , ∞ ) ( n ) E n − ∗ ∆Ω (cid:16) θ ( i )1 (cid:17) and: δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i )2 (cid:17) = ∆Ω † (cid:16) θ ( i )1 (cid:17) δ ∆Ω (cid:0) θ ( i ) (cid:1) δ Ψ (cid:16) θ ( i ) f (cid:17) Ψ † (cid:16) θ ( i ) (cid:17) + δ ∆Ω † (cid:0) θ ( i ) (cid:1) δ Ψ (cid:16) θ ( i ) f (cid:17) Ψ (cid:16) θ ( i ) (cid:17) (177) − δ ∆Ω (cid:0) θ ( i ) (cid:1) δ Ψ (cid:16) θ ( i ) f (cid:17) δδ Ω ( θ ( i ) ) (cid:18) D (1 + O , ) (2) E † + P n > n − D (1 + O , ∞ ) ( n ) E † n − (cid:19) D (1 + O , ) (2) E † + P n > n − D (1 + O , ∞ ) ( n ) E † n − ∆Ω † (cid:16) θ ( i )1 (cid:17) − δ ∆Ω † (cid:0) θ ( i ) (cid:1) δ Ψ (cid:16) θ ( i ) f (cid:17) δδ Ω † ( θ ( i ) ) (cid:18) D (1 + O , ) (2) E † + P n > n − D (1 + O , ∞ ) ( n ) E † n − (cid:19) D (1 + O , ) (2) E † + P n > n − D (1 + O , ∞ ) ( n ) E † n − ∆Ω † (cid:16) θ ( i )1 (cid:17) These expression can be rewritten by using that the operators O (standing for 1+ O , , 1+ O , ∞ ...) decomposeas O = O ( C ) + O ( D ) ∇ | O D | < O C and O − ≃ ( O ( C ) − O ( D ) ∇ )( O ( C ) ) ≡ O † ( O ( C ) ) . We have: δδ Ω ( θ ( i ) ) (cid:18) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − (cid:19) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − ∆Ω (cid:16) θ ( i )1 (cid:17) = − δδ Ω ( θ ( i ) ) (cid:18) D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − (cid:19)(cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) ≃ ∆Ω † (cid:16) θ ( i ) (cid:17) × (cid:0)(cid:0) O , (cid:1) ∗ G (cid:1) (cid:16)(cid:0) O , (cid:1) θ ( i )1 ∗ G (cid:17) + (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) P n > h (1+ O , ∞ ) ( n − i n − ( n − (cid:16)(cid:0) O , ∞ (cid:1) θ ( i )1 ∗ G (cid:17)(cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) = ∆Ω † (cid:16) θ ( i ) (cid:17) G − ∗ (cid:0)(cid:0) O , (cid:1) ∗ G (cid:1) (cid:0) O , (cid:1) θ ( i )1 + (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) P n > h (1+ O , ∞ ) ( n − i n − ( n − (cid:0) O , ∞ (cid:1) θ ( i )1 D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) where the convolutions are understood. Thus, (176) and (177) write:80 ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = Z − ∆Ω † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) A (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + B (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 ! G +∆Ω (cid:16) θ ( i )1 (cid:17) Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) δ (cid:18)(cid:16) θ ( i )2 (cid:17) ′ − θ ( i )2 (cid:19)(cid:19) × δ ∆Ω (cid:16) θ ( i )2 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) dθ ( i )2 d (cid:16) θ ( i )2 (cid:17) ′ = Z − Z ∆Ω † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) A (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + B (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 ! G d (cid:16) θ ( i )2 (cid:17) ′ +∆Ω (cid:16) θ ( i )1 (cid:17) Ψ † (cid:16) θ ( i )2 (cid:17) δ ∆Ω (cid:16) θ ( i )2 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) dθ ( i )2 = Z K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ ∆Ω (cid:16) θ ( i )2 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) dθ ( i )2 where: A (cid:16) θ ( i )1 (cid:17) = G − ∗ P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) = A (cid:16) θ ( i )1 , θ ( i ) f (cid:17) B (cid:16) θ ( i )1 (cid:17) = G − ∗ (cid:0) O , (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) = B (cid:16) θ ( i )1 , θ ( i ) f (cid:17) Similar formula applies for the derivatives of ∆Ω † (cid:16) θ ( i )2 (cid:17) : Z δ ∆Ω † (cid:16) θ ( i )2 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) − A (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + B (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 ! G ∆Ω (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) +∆Ω (cid:16) θ ( i )1 (cid:17) Ψ (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )2 (cid:17) ′ − θ ( i )2 (cid:19)(cid:19) dθ ( i )2 d (cid:16) θ ( i )2 (cid:17) ′ = Z δ ∆Ω † (cid:16) θ ( i )2 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) − Z A (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + B (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 ! G ∆Ω (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ +∆Ω (cid:16) θ ( i )1 (cid:17) Ψ (cid:16) θ ( i )2 (cid:17)(cid:17) dθ ( i )2 = Z K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ ∆Ω † (cid:16) θ ( i )2 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) dθ ( i )2 δ ∆Ω † (cid:16) θ ( i )2 (cid:17) δ Ψ † (cid:16) θ ( i ) f (cid:17) − A † (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ + B † (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ ! G ∆Ω (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) +∆Ω † (cid:16) θ ( i )1 (cid:17) Ψ (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )2 (cid:17) ′ − θ ( i )2 (cid:19)(cid:19) dθ ( i )2 d (cid:16) θ ( i )2 (cid:17) ′ = Z δ ∆Ω † (cid:16) θ ( i )2 (cid:17) δ Ψ † (cid:16) θ ( i ) f (cid:17) − Z G A † (cid:16) θ ( i )1 (cid:17) (cid:0) O , ∞ (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ + B † (cid:16) θ ( i )1 (cid:17) (cid:0) O , (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ ! G ∆Ω (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ +∆Ω † (cid:16) θ ( i )1 (cid:17) Ψ (cid:16) θ ( i )2 (cid:17)(cid:17) dθ ( i )2 = Z K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ ∆Ω † (cid:16) θ ( i )2 (cid:17) δ Ψ † (cid:16) θ ( i ) f (cid:17) dθ ( i )2 Where G Ψ (cid:16) θ ( i )2 (cid:17) denotes the result of the action of G on Ψ: G Ψ (cid:16) θ ( i )2 (cid:17) ≡ ( G Ψ) (cid:16) θ ( i )2 (cid:17) As a consequence, defining:[∆Ω] = (cid:18) ∆Ω∆Ω † (cid:19) (178) K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) = K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K † , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) K † , (cid:16) θ ( i )1 , θ ( i )2 (cid:17) h X (cid:16) θ ( i )1 , θ ( i ) f (cid:17)i = (cid:18) h (1+ O , ) (2) i + P n > n − h (1+ O , ∞ ) ( n ) i n − h i + h (1+ O , ) (2) i + P n > n ! h (1+ O , ∞ ) ( n ) i n G (cid:19) − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) = X (cid:16) θ ( i )1 , θ ( i ) f (cid:17) ! Equations (176) and (177) can be written in the compact form of a dynamic equation: δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = Z K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [∆Ω] (cid:16) θ ( i )2 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) dθ ( i )2 + h X (cid:16) θ ( i )1 , θ ( i ) f (cid:17)i (179) The solution of (179) is given by a series expansion: δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = X n > Z n − Y k =1 K (cid:16) θ ( i ) k , θ ( i ) k +1 (cid:17)! h X (cid:16) θ ( i ) n , θ ( i ) f (cid:17)i n Y k =2 dθ ( i ) k (180)We can check recursively that the coefficients: K ( n )1 , K ( n )1 , K ( n )2 , K ( n )2 , ! have the form: 82 ( n )1 , = Z ∆Ω † (cid:18)(cid:16) θ ( i ) n (cid:17) ′ (cid:19) − A n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i ) n (cid:17) ′ ,θ ( i ) n + B n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:0) O , (cid:1) (cid:16) θ ( i ) n (cid:17) ′ ,θ ( i ) n !! G + C n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) Ψ † (cid:16) θ ( i ) n (cid:17) K ( n )2 , = − Z A † n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:0) O , ∞ (cid:1) θ ( i ) n , (cid:16) θ ( i ) n (cid:17) ′ + B † n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:0) O , (cid:1) θ ( i ) n , (cid:16) θ ( i ) n (cid:17) ′ ! ×G ∆Ω (cid:18)(cid:16) θ ( i ) n (cid:17) ′ (cid:19) d (cid:16) θ ( i ) n (cid:17) ′ + C † n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) Ψ (cid:16) θ ( i ) n (cid:17) K ( n )1 , = − Z A n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:0) O , ∞ (cid:1) θ ( i ) n , (cid:16) θ ( i ) n (cid:17) ′ + B n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:0) O , (cid:1) θ ( i ) n , (cid:16) θ ( i ) n (cid:17) ′ ! ×G ∆Ω (cid:18)(cid:16) θ ( i ) n (cid:17) ′ (cid:19) d (cid:16) θ ( i ) n (cid:17) ′ + C n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) Ψ (cid:16) θ ( i ) n (cid:17) K ( n )2 , = Z ∆Ω † (cid:18)(cid:16) θ ( i ) n (cid:17) ′ (cid:19) − A † n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i ) n (cid:17) ′ ,θ ( i ) n + B † n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:0) O , (cid:1) (cid:16) θ ( i ) n (cid:17) ′ ,θ ( i ) n !! G + C † n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) Ψ † (cid:16) θ ( i ) n (cid:17) where the coefficients A n , B n and C n are found recursively. To find these coefficients, we use that: K ( n )1 , K ( n )1 , K ( n )2 , K ( n )2 , ! (cid:18) K , K , K , K , (cid:19) = K ( n )1 , K , + K ( n )1 , K , K ( n )1 , K , + K ( n )1 , K , K ( n )2 , K , + K ( n )2 , K , K ( n )2 , K , + K ( n )2 , K , ! and note that the first coefficients A and B are defined by: A (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ∆Ω (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) = A (cid:16) θ ( i )1 (cid:17) B (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ∆Ω (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) = B (cid:16) θ ( i )1 (cid:17) A k and B k are: A k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) (181)= − Z G ∆Ω † (cid:18)(cid:16) θ ( i ) k (cid:17) ′ (cid:19) A k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (1 + O , ∞ ) (cid:16) θ ( i ) k (cid:17) ′ ,θ ( i ) k A (cid:16) θ ( i ) k (cid:17) + B k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (1 + O , ) (cid:16) θ ( i ) k (cid:17) ′ ,θ ( i ) k A (cid:16) θ ( i ) k (cid:17)! + C k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) Ψ † (cid:16) θ ( i ) k (cid:17) A (cid:16) θ ( i ) k (cid:17) − Z G A k (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) A † (cid:16) θ ( i ) k (cid:17) (1 + O , ∞ ) θ ( i ) k , (cid:16) θ ( i ) k (cid:17) ′ + B k (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) A † (cid:16) θ ( i ) k (cid:17) (1 + O , ) θ ( i ) k , (cid:16) θ ( i ) k (cid:17) ′ ! ∆Ω (cid:18)(cid:16) θ ( i ) k (cid:17) ′ (cid:19) + C k (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) A † (cid:16) θ ( i ) k (cid:17) Ψ (cid:16) θ ( i ) k (cid:17) = − A k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16) h (1 + O , ∞ ) A i >θ ( i ) k + h A (1 + O , ∞ ) i <θ ( i ) k (cid:17) − B k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16) h (1 + O , ) A i >θ ( i ) k + h A (1 + O , ) i <θ ( i ) k (cid:17) + C k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16) h A i ΩΨ >θ ( i ) k + h A i ΩΨ <θ ( i ) k (cid:17) B k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) (182)= − A k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16) h (1 + O , ∞ ) B i >θ ( i ) k + h B (1 + O , ∞ ) i <θ ( i ) k (cid:17) − B k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16) h (1 + O , ) B i >θ ( i ) k + h B (1 + O , ) i <θ ( i ) k (cid:17) + C k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16) h B i ΩΨ >θ ( i ) k + h B i ΩΨ <θ ( i ) k (cid:17) C k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) = − A k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16) h (1 + O , ∞ ) i >θ ( i ) k + h (1 + O , ∞ ) i <θ ( i ) k (cid:17) (183) − B k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16) h (1 + O , ) i >θ ( i ) k + h (1 + O , ) i <θ ( i ) k (cid:17) + C k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) (cid:16)(cid:16) h i ΩΨ >θ ( i ) k + h i ΩΨ <θ ( i ) k (cid:17)(cid:17) with: h A i ΩΨ >θ ( i ) k = Z Ψ † (cid:16) θ ( i ) k (cid:17) A (cid:16) θ ( i ) k (cid:17) h A i ΩΨ <θ ( i ) k = Z A † (cid:16) θ ( i ) k (cid:17) Ψ (cid:16) θ ( i ) k (cid:17) h B i ΩΨ >θ ( i ) k = Z Ψ † (cid:16) θ ( i ) k (cid:17) B (cid:16) θ ( i ) k (cid:17) h B i ΩΨ <θ ( i ) k = Z B † (cid:16) θ ( i ) k (cid:17) Ψ (cid:16) θ ( i ) k (cid:17) To compute the various brackets involved in (181), (182) and (183), we define the following ∆Ω averagevalues for an arbitrary operator M : h M i >θ ( i ) k = Z ∆Ω † (cid:18)(cid:16) θ ( i ) k (cid:17) ′ (cid:19) M (cid:16) θ ( i ) k (cid:17) ′ ,θ ( i ) k G ∆Ω (cid:16) θ ( i ) k (cid:17) (184) h M i <θ ( i ) k = Z ∆Ω † (cid:18)(cid:16) θ ( i ) k (cid:17) ′ (cid:19) M † (cid:16) θ ( i ) k (cid:17) ′ ,θ ( i ) k G ∆Ω (cid:16) θ ( i ) k (cid:17) h M i Ψ >θ ( i ) k = Z Ψ † (cid:18)(cid:16) θ ( i ) k (cid:17) ′ (cid:19) G − M (cid:16) θ ( i ) k (cid:17) ′ ,θ ( i ) k Ψ (cid:16) θ ( i ) k (cid:17) (185) h M i Ψ <θ ( i ) k = Z Ψ † (cid:18)(cid:16) θ ( i ) k (cid:17) ′ (cid:19) G − M † (cid:16) θ ( i ) k (cid:17) ′ ,θ ( i ) k Ψ (cid:16) θ ( i ) k (cid:17) Using (171):∆Ω (cid:16) θ ( i ) (cid:17) = (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) ! G ! − Ψ (cid:16) θ ( i )1 (cid:17) = G − ∗ − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! Ψ (cid:16) θ ( i )1 (cid:17) we can compute the ∆Ω averages (184) of M in terms of its Ψ averages (185) up to fourth order in derivatives: h M i >θ ( i ) k ≃ − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O C , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O C , ∞ + (cid:0) y (cid:0) O C , (cid:1) − x (cid:0) O C , ∞ (cid:1)(cid:1)(cid:1) ! (cid:10) ¯ M (cid:11) Ψ >θ ( i ) k ≃ h i Ψ * (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ (cid:10) ¯ M (cid:11) Ψ >θ ( i ) k h M i <θ ( i ) k ≃ − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O C , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O C , ∞ + (cid:0) y (cid:0) O C , (cid:1) − x (cid:0) O C , ∞ (cid:1)(cid:1)(cid:1) ! (cid:10) ¯ M (cid:11) Ψ <θ ( i ) k ≃ h i Ψ * (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ (cid:10) ¯ M (cid:11) Ψ <θ ( i ) k with ¯ M given by: M G = ¯ M ∗ G The previous formula imply that several quantities arising in (181), (182) and (183) can be computed: h (1 + O , ∞ ) A i >θ ( i ) k <θ ( i ) k = 1 h i Ψ * (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ × (cid:10)(cid:0) O , ∞ (cid:1) ¯ A (cid:11) Ψ >θ ( i ) k <θ ( i ) k h (1 + O , ∞ ) B i >θ ( i ) k <θ ( i ) k = 1 h i Ψ * (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ × (cid:10)(cid:0) O , ∞ (cid:1) ¯ B (cid:11) Ψ >θ ( i ) k <θ ( i ) k A (cid:16) θ ( i )1 (cid:17) = P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ≃ (1 − exp ( − x )) (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ¯ B (cid:16) θ ( i )1 (cid:17) = (cid:0) O , (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ≃ (cid:0) O , (cid:1) θ ( i )1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) We also define h M i Ψ θ ( i ) k = h M i Ψ >θ ( i ) k + h M i Ψ <θ ( i ) k . We can replace in average h M i Ψ >θ ( i ) k and h M i Ψ <θ ( i ) k by h M i Ψ θ ( i ) k in (181), (182) and (183). Moreover:Ψ † (cid:16) θ ( i ) k (cid:17) ∆Ω (cid:16) θ ( i ) k (cid:17) ≃ (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) D(cid:16)(cid:16) O † , ∞ (cid:17) + exp ( − x ) (cid:16) − ¯ O † , ∞ + y (cid:16) O † , (cid:17) − x (cid:16) O † , ∞ (cid:17)(cid:17)(cid:17)E Ψ >θ ( i ) k (cid:0)(cid:0) O C , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O C , ∞ + (cid:0) y (cid:0) O C , (cid:1) − x (cid:0) O C , ∞ (cid:1)(cid:1)(cid:1)(cid:1) = (cid:18) − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19)(cid:19) * (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ >θ ( i ) k ∆Ω † (cid:16) θ ( i ) k (cid:17) Ψ (cid:16) θ ( i )2 (cid:17) = (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) D(cid:16)(cid:16) O † , ∞ (cid:17) + exp ( − x ) (cid:16) − ¯ O † , ∞ + y (cid:16) O † , (cid:17) − x (cid:16) O † , ∞ (cid:17)(cid:17)(cid:17)E Ψ <θ ( i ) k (cid:0)(cid:0) O C , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O C , ∞ + (cid:0) y (cid:0) O C , (cid:1) − x (cid:0) O C , ∞ (cid:1)(cid:1)(cid:1)(cid:1) = * − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ <θ ( i ) k and, for the coefficients A (cid:16) θ ( i )1 (cid:17) , B (cid:16) θ ( i )1 (cid:17) : A (cid:16) θ ( i )1 (cid:17) (186)= G − ∗ P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) ≃ G − ∗ (1 − exp ( − x )) (1 + O , ∞ ) θ ( i )1 (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + ( y (1 + O , ) − x (1 + O , ∞ ))) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) B (cid:16) θ ( i )1 (cid:17) = G − ∗ (cid:0) O , (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) ≃ G − ∗ exp ( − x ) (1 + O , ) θ ( i )1 (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + ( y (1 + O , ) − x (1 + O , ∞ ))) ∗ G ∆Ω (cid:16) θ ( i )1 (cid:17) P n > n − D (1 + O , ∞ ) ( n − E n − (1 + O , ∞ ) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O C , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O C , ∞ (cid:1)(cid:19) (187) ≃ exp ( − x ) P n > h (1+ O , ∞ ) ( n − i n − ( n − (1 + O , ∞ ) θ ( i )1 (cid:16)(cid:16) O † , ∞ (cid:17) + exp ( − x ) (cid:16) − O † , ∞ + y (cid:16) O † , (cid:17) − x (cid:16) O † , ∞ (cid:17)(cid:17)(cid:17)(cid:0)(cid:0) O C , ∞ (cid:1) + exp ( − x ) (cid:0) − O C , ∞ + (cid:0) y (cid:0) O C , (cid:1) − x (cid:0) O C , ∞ (cid:1)(cid:1)(cid:1)(cid:1) ≃ (1 − exp ( − x )) (1 + O , ∞ ) θ ( i )1 (cid:16)(cid:16) O † , ∞ (cid:17) + exp ( − x ) (cid:16) − O † , ∞ + y (cid:16) O † , (cid:17) − x (cid:16) O † , ∞ (cid:17)(cid:17)(cid:17)(cid:0)(cid:0) O C , ∞ (cid:1) + exp ( − x ) (cid:0) − O C , ∞ + (cid:0) y (cid:0) O C , (cid:1) − x (cid:0) O C , ∞ (cid:1)(cid:1)(cid:1)(cid:1) As a consequence, the coefficients A k +1 , B k +1 , C k +1 can be written as a function of a coefficient N k +1 : A k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) ≃ h i Ψ * P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) + Ψ N k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) ≃ h i Ψ * (1 − exp ( − x )) (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) + Ψ N k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) B k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) ≃ h i Ψ * exp ( − x ) (cid:0) O , (cid:1) θ ( i )1 (cid:18) D (1 + O , ) (1) E (cid:0) O , (cid:1) + P n > n − D (1 + O , ∞ ) ( n − E n − (cid:0) O , ∞ (cid:1)(cid:19) + Ψ N k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) ≃ h i Ψ * exp ( − x ) (cid:0) O , (cid:1) θ ( i )1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) + Ψ N k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) C k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) ≃ N k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) Inserting this relation in (183) yields the recursive relation for N k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) :87 k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) ≃ − h i Ψ * (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) + Ψ × h i Ψ * (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ × (1 − exp ( − x )) (cid:10)(cid:0) O , ∞ (cid:1)(cid:11) Ψ h i Ψ (cid:10)(cid:0) O , ∞ (cid:1)(cid:11) Ψ θ ( i ) k + (cid:10)(cid:0) O , (cid:1)(cid:11) Ψ h i Ψ exp ( − x ) (cid:10)(cid:0) O , (cid:1)(cid:11) Ψ θ ( i ) k !) N k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) + * − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) + Ψ N k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) = − h i Ψ * (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) + Ψ × (1 − exp ( − x )) x h i Ψ (cid:10)(cid:0) O , ∞ (cid:1)(cid:11) Ψ θ ( i ) k + y h i Ψ exp ( − x ) (cid:10)(cid:0) O , (cid:1)(cid:11) Ψ θ ( i ) k ! N k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) + * − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) + Ψ N k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) and this formula factors as: N k +1 (cid:18)(cid:16) θ ( i ) l (cid:17) l k (cid:19) ≃ * − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ × − (cid:16) (1 − exp ( − x )) x (cid:10)(cid:0) O , ∞ (cid:1)(cid:11) Ψ θ ( i ) k + y exp ( − x ) (cid:10)(cid:0) O , (cid:1)(cid:11) Ψ θ ( i ) k (cid:17)(cid:16) h i Ψ (cid:17) (cid:0) − x ) (cid:0) − z + ( y − x ) (cid:1)(cid:1) N k (cid:18)(cid:16) θ ( i ) l (cid:17) l k − (cid:19) whose solution is: N n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) ≃ N (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) n − Y k =2 * − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ × − (cid:16) (1 − exp ( − x )) x (cid:10)(cid:0) O , ∞ (cid:1)(cid:11) Ψ θ ( i ) k + y exp ( − x ) (cid:10)(cid:0) O , (cid:1)(cid:11) Ψ θ ( i ) k (cid:17)(cid:16) h i Ψ (cid:17) (cid:0) − x ) (cid:0) − z + ( y − x ) (cid:1)(cid:1) This yields ultimately the solution of (179): 88 [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = Z K ( n )1 , (cid:16) θ ( i )1 , θ ( i ) n (cid:17) X (cid:16) θ ( i ) n , θ ( i ) f (cid:17) = X n > Z n − Y k =1 K (cid:16) θ ( i ) k , θ ( i ) k +1 (cid:17)! h X (cid:16) θ ( i ) n , θ ( i ) f (cid:17)i n Y k =2 dθ ( i ) k = X (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + X n > Z θ ( i )1 >θ ( i ) k ...>θ ( i ) n ¯ N n (cid:18)(cid:16) θ ( i ) l (cid:17) l n − (cid:19) (cid:16) ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ † (cid:16) θ ( i ) n (cid:17) − A (cid:16) θ ( i )1 (cid:17) Z θ ( i )2 ∆Ω † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) G (1 + O , ∞ ) + B (cid:16) θ ( i )1 (cid:17) Z θ ( i )2 ∆Ω † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) G (1 + O , ) !! × X (cid:16) θ ( i ) n , θ ( i ) f (cid:17) n Y k =2 dθ ( i ) k that can be approximated by: δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) (188) ≃ X (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + Z dθ ( i )2 exp Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! (cid:16) ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ † (cid:16) θ ( i )2 (cid:17) X (cid:16) θ ( i )2 , θ ( i ) f (cid:17) − A (cid:16) θ ( i )1 (cid:17) Z θ ( i )2 ∆Ω † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ − B (cid:16) θ ( i )1 (cid:17) Z θ ( i )2 ∆Ω † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ ! where the factor ¯ N (( θ i )) is given by:¯ N (( θ i )) ≃ * − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) + Ψ × − (cid:16) (1 − exp ( − x )) x (cid:10)(cid:0) O , ∞ (cid:1)(cid:11) Ψ θ ( i ) + y exp ( − x ) (cid:10)(cid:0) O , (cid:1)(cid:11) Ψ θ ( i ) (cid:17) h i Ψ (cid:0) − x ) (cid:0) − z + ( y − x ) (cid:1)(cid:1) and with condition: θ ( i ) f < (cid:16) θ ( i )2 (cid:17) ′ < θ ( i )1 due to the Heaviside functions in the integrals defining the interaction terms.89sing (186) and (187), the coefficients involved in (188) write:∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) = Z θ ( i )1 (1 + O , ∞ ) + exp ( − x ) ( − O , ∞ + y (1 + O , ) − x (1 + O , ∞ ))1 + exp ( − x ) (cid:0) − z + ( y − x ) (cid:1) ! G ! − Ψ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) = Z θ ( i )1 G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) !! Ψ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) A (cid:16) θ ( i )1 (cid:17) = Z θ ( i )1 G − (1 − exp ( − x )) (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) ! Ψ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) B (cid:16) θ ( i )1 (cid:17) = Z θ ( i )1 G − exp ( − x ) (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , (cid:1) θ ( i )1 (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) ! Ψ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) and G X (cid:16) θ ( i ) n , θ ( i ) f (cid:17) is equal to: G X (cid:16) θ ( i ) n , θ ( i ) f (cid:17) = G D (1 + O , ) (2) E + P n > n − D (1 + O , ∞ ) ( n ) E n − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) ≃ − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i ) n , θ ( i ) f (cid:17) In the local approximation X (cid:16) θ ( i )2 , θ ( i ) f (cid:17) ∝ δ (cid:16) θ ( i )2 − θ ( i ) f (cid:17) . Defining: O = 1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) G − (cid:0) O , ∞ (cid:1) (189) O = 1 (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) G − (cid:0) O , (cid:1) this yields the expanded form for the two points correlation functions (188): δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! (cid:18)Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ (cid:0) G − (cid:1) ∗ θ ( i )1 − (1 − exp ( − x )) (cid:18)Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) O G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ (cid:19) × G − (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , ∞ (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! ∗ θ ( i )1 − exp ( − x ) (cid:18)Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) O G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19) d (cid:16) θ ( i )2 (cid:17) ′ (cid:19) G − (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! ∗ θ ( i )1 ! × (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) Ψ ! (cid:16) θ ( i )1 (cid:17) δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ † (cid:16) θ ( i ) f (cid:17) G − (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − Ψ † (cid:16) θ ( i ) f (cid:17) O (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! ×G − (1 − exp ( − x )) (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) − Ψ † (cid:16) θ ( i ) f (cid:17) O (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! ×G − exp ( − x ) (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , (cid:1) θ ( i )1 (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! × (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1) (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! Ψ (cid:16) θ ( i )1 (cid:17) Thie previous expression for the two points correlation function can be rewritten more compactly: δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + "Z θ ( i )1 G − F Θ Ψ exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! "Z θ ( i ) f Ψ † G − F Θ − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! × "Z θ ( i ) f Ψ † G − (1 − exp ( − x )) F Θ (cid:0) O , ∞ (cid:1) − "Z θ ( i )1 G − F Θ (1 + O , ) Ψ exp Z θ ( i )1 θ ( i ) f ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! "Z θ ( i ) f Ψ † G − exp ( − x ) F Θ (1 + O , ) where: F = 1 + exp ( − x ) (cid:18) − z + 12 (cid:0) y − x (cid:1)(cid:19) (190)Θ = (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)
91n the local approximation for θ ( i )1 ≃ θ ( i ) f : δ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) f (cid:17) = G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + "Z θ ( i )1 G − F Θ Ψ θ ( i )1 Ψ † G − F Θ (191) − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − (1 − exp ( − x )) F Θ (cid:0) O , ∞ (cid:1) − "Z θ ( i )1 G − F Θ (cid:0) O , (cid:1) Ψ θ ( i )1 Ψ † G − exp ( − x ) F Θ (cid:0) O , (cid:1) The contributions into brackets can be neglected in first approximation since they include a factor exp ( − x ) × polynomial( x, y ).Actually decomposing: "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − (1 − exp ( − x )) F Θ (cid:0) O , ∞ (cid:1) (192)= "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − F Θ (cid:0) O , ∞ (cid:1) + "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − ( − exp ( − x )) F Θ (cid:0) O , ∞ (cid:1) we can regroup the second term of (191) and the first term of the right hand side of (192): "Z θ ( i )1 G − F Θ Ψ θ ( i )1 Ψ † G − F Θ − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − F Θ (cid:0) O , ∞ (cid:1) In average: (cid:0) O , ∞ (cid:1) Θ = (cid:18) x ( x + exp ( − x ) ( h i − x + ( y − x ))) (cid:19) So that: "Z θ ( i )1 G − F Θ Ψ θ ( i )1 Ψ † G − F Θ − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − F Θ (cid:0) O , ∞ (cid:1) ≃ "Z θ ( i )1 G − F Θ Ψ θ ( i )1 Ψ † G − F Θ × − x (cid:0) O , ∞ (cid:1) (cid:0) − x ) (cid:0) − x + (cid:0) y − x (cid:1)(cid:1)(cid:1) ( x + exp ( − x ) ( h i − x + ( y − x ))) (cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) ! In average this is equal to: ≃ "Z θ ( i )1 G − F Θ Ψ θ ( i )1 Ψ † G − F Θ − (cid:18) x ( x + exp ( − x ) (1 − x + ( y − x ))) (cid:19) (cid:18) − x ) (cid:18) − x + 12 (cid:0) y − x (cid:1)(cid:19)(cid:19)! which has the stated form. This corrective term can be given a more precise form: "Z θ ( i )1 G − F Θ Ψ θ ( i )1 Ψ † G − F α ( x ) (cid:0) O , ∞ (cid:1) + β ( x ) (cid:0) y (cid:0) O , (cid:1)(cid:1) ( x + exp ( − x ) ( h i − x + ( y − x ))) ! ≃ "Z θ ( i )1 G − Ψ θ ( i )1 Ψ † G − F h i α ( x ) (cid:0) O , ∞ (cid:1) + β ( x ) (cid:0) y (cid:0) O , (cid:1)(cid:1) ( x + exp ( − x ) ( h i − x + ( y − x ))) ! ( x ) = (cid:0) x + exp ( − x ) (cid:0) h i − x + (cid:0) y − x (cid:1)(cid:1)(cid:1) (1 − ( x + 1) exp ( − x )) − x (cid:18) − x ) (cid:18) − x + 12 (cid:0) y − x (cid:1)(cid:19)(cid:19) β ( x ) = (cid:0) x + exp ( − x ) (cid:0) h i − x + (cid:0) y − x (cid:1)(cid:1)(cid:1) exp ( − x )Moreover, gathering the second term of (192) and the last term of (191) yields in average: − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − ( − exp ( − x )) F Θ (cid:0) O , ∞ (cid:1) − "Z θ ( i )1 G − F Θ (cid:0) O , (cid:1) Ψ θ ( i )1 Ψ † G − exp ( − x ) F Θ (cid:0) O , (cid:1) ≃ − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − ( − exp ( − x )) F Θ (cid:0) O , ∞ (cid:1) + yx Z θ ( i )1 Ψ † G − exp ( − x ) F Θ (cid:0) O , (cid:1)! = − "Z θ ( i )1 G − F Θ (cid:0) O , ∞ (cid:1) Ψ θ ( i )1 Ψ † G − exp ( − x ) (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) x Θ F ! ≃ − "Z θ ( i )1 G − Ψ θ ( i )1 Ψ † G − F h i exp ( − x ) (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ( x + exp ( − x ) ( h i − x + ( y − x ))) ! ≃ − "Z θ ( i )1 G − Ψ θ ( i )1 Ψ † G − F h i exp ( − x ) (cid:0) y − x (cid:1) ( x + exp ( − x ) ( h i − x + ( y − x ))) ! which has again the expected form. Using that Ψ † is constant, we can write the corrective terms as: F (cid:0) Ψ † G − Ψ (cid:1) G − h i ( α ( x ) + F x exp ( − x )) (cid:0) O , ∞ (cid:1) + ( β ( x ) − F exp ( − x )) y (cid:0) O , (cid:1) + β ( x )( x + exp ( − x ) ( h i − x + ( y − x ))) ! (193)The two points correlation function (191) is thus: G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + F (cid:0) Ψ † G − Ψ (cid:1) G − h i ( α ( x ) + F x exp ( − x )) (cid:0) O , ∞ (cid:1) + ( β ( x ) − F exp ( − x )) y (cid:0) O , (cid:1) + β ( x )( x + exp ( − x ) ( h i − x + ( y − x ))) ! (194)The various terms arising in (194) can be found by using (161) which shows that for θ ( i )1 ≃ θ ( i ) f : (cid:0) O ,n (cid:1) = δ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17) + − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) × exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) Λso that: G − (cid:0) O ,n (cid:1) = G − + − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) δ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17) (195)and: G − (cid:0) O , ∞ (cid:1) = G − + − ¯ ζ + ∇ out θ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) δ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17) (196)93s a consequence, (194) rewrites: G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + F (cid:0) Ψ † G − Ψ (cid:1) ( α ( x ) − F x exp ( − x )) − ¯ ζ + ∇ outθ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) − F (cid:0) Ψ † G − Ψ (cid:1) ( β ( x ) − F exp ( − x )) y − ¯ ζ + ∇ outθ ( i ) i ′ Λ ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) + β ( x )( x + exp ( − x ) ( h i − x + ( y − x ))) (197) ( l, m ) at different points without interaction The expressions for correlation functions at m different points are directly given by: " G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + Z dθ ( i )2 exp Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! (cid:16) Ψ † (cid:16) θ ( i )2 (cid:17) X (cid:16) θ ( i )2 , θ ( i ) f (cid:17) − (1 − exp ( − x )) Z Ψ † (cid:16) θ ( i )2 (cid:17) ( O ) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19)! G − (cid:0) O , ∞ (cid:1) θ ( i )1 − exp ( − x ) Z Ψ † (cid:16) θ ( i )2 (cid:17) ( O ) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19)! G − (cid:0) O , (cid:1) θ ( i )1 ! (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! Ψ (cid:16) θ ( i )1 (cid:17) −⊗ m Where the notation − ⊗ m of an expression: [ X i ] −⊗ m represents the m tensor products: m Y i =1 [ X i ] − For weak background fields this reduces to: " G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i )1 , θ ( i ) f (cid:17) −⊗ m δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:16) θ ( i ) f (cid:17) − (198)= "(cid:18) G −
11 + ¯ O , ∞ (cid:19) θ ( i )1 ,θ ( i )2 + exp Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! (cid:18)Z Ψ † (cid:16) θ ( i )2 (cid:17) X (cid:16) θ ( i )2 , θ ( i ) f (cid:17) dθ ( i )2 − Z Ψ † (cid:16) θ ( i )2 (cid:17) ( O ) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G X (cid:18)(cid:16) θ ( i )2 (cid:17) ′ , θ ( i ) f (cid:19)! G − (cid:0) O , ∞ (cid:1) θ ( i )1 G − (cid:0) O , ∞ (cid:1) θ ( i )1 × (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! Ψ (cid:16) θ ( i )1 (cid:17) −⊗ m In this expression, the indivual elements interact directly with field and indirectly with other elements.Expression (198) can be written more explicitely, using (189). Actually, in strong background fields approx-imation, we have: O ≃ (cid:0) O , ∞ (cid:1) G − (cid:0) O , ∞ (cid:1) O ≃ (cid:0) O , ∞ (cid:1) G − (cid:0) O , (cid:1) X ≃ G − (cid:18)
11 + ¯ O , ∞ (cid:19) and the m -th tensor power of (198) becomes: δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:16) θ ( i )2 (cid:17) − ⊗ m δ l,m = "(cid:18) G −
11 + ¯ O , ∞ (cid:19) θ ( i )1 ,θ ( i )2 + exp Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! (199) × Z θ ( i )2 Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) G − (cid:0) O , ∞ (cid:1) d (cid:16) θ ( i )2 (cid:17) ′ − Z θ ( i )2 Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) G − d (cid:16) θ ( i )2 (cid:17) ′ !(cid:18) G − (cid:18)
11 + ¯ O , ∞ (cid:19) Ψ (cid:19) (cid:16) θ ( i )1 (cid:17)(cid:21) −⊗ m δ l,m ≃ "(cid:18) G −
11 + ¯ O , ∞ (cid:19) θ ( i )1 ,θ ( i )2 − exp Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! × Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) G − d (cid:16) θ ( i )2 (cid:17) ′ ! (cid:18) G −
11 + ¯ O , ∞ Ψ (cid:19) (cid:16) θ ( i )1 (cid:17) −⊗ m δ l,m given that Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) G − ≃
0. Factoring by G − (cid:16)
11+ ¯ O , ∞ (cid:17) , we can rewrite: G − (cid:18)
11 + ¯ O , ∞ (cid:19) − exp Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! × Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) G − d (cid:16) θ ( i )2 (cid:17) ′ ! (cid:18) G − (cid:18)
11 + ¯ O , ∞ (cid:19) Ψ (cid:19) (cid:16) θ ( i )1 (cid:17)! − = (cid:0) O , ∞ (cid:1) G − exp Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ (cid:16) θ ( i )1 (cid:17) Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) G − d (cid:16) θ ( i )2 (cid:17) ′ !! − G − : G − = −∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G ) + X (cid:19) we have, for σ θ << Z Ψ † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) G − d (cid:16) θ ( i )2 (cid:17) ′ ≃ − Ψ † (cid:16) θ ( i )2 (cid:17) (cid:0) O , ∞ (cid:1) ∇ θ ω − (cid:16) J (cid:16) θ ( i )2 (cid:17) , θ ( i )2 , Z, G (cid:17) = − Ψ † (cid:16) θ ( i )2 (cid:17) ∇ θ ω − (cid:16) J (cid:16) θ ( i )2 (cid:17) , θ ( i )2 , Z, G (cid:17) and the n points vertices for strong background field become: δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) − ⊗ m δ l,m (200)= (cid:0) O , ∞ (cid:1) G Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ (cid:16) θ ( i )1 (cid:17) Ψ † (cid:16) θ ( i )2 (cid:17) ∇ θ ω − (cid:16) J (cid:16) θ ( i )2 (cid:17) , θ ( i )2 , Z, G (cid:17)! − ⊗ m ≃ (cid:0) O , ∞ (cid:1) G Z θ ( i )1 θ ( i )2 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! Ψ (cid:16) θ ( i )1 (cid:17) Ψ † (cid:16) θ ( i )2 (cid:17) ∇ θ G (cid:16) J (cid:16) θ ( i )2 (cid:17) , θ ( i )2 , Z, G (cid:17)! − ⊗ m ( l, m ) at differentpoints To find the correlation functions including the interaction corrections, we first define the compact notation: δ l,m − ∆Ω (cid:0) θ ( i ) (cid:1) δ Ψ † (cid:0) θ ( i ) (cid:1) ...δ Ψ † (cid:0) θ ( i m ) (cid:1) δ Ψ (cid:0) θ ( i m + k ) † (cid:1) ...δ Ψ (cid:0) θ ( i m + l ) † (cid:1) = δ l,m − ∆Ω (cid:0) θ ( i ) (cid:1) δ (cid:2) ΨΨ † (cid:0)(cid:0) θ ( i ) † (cid:1) , (cid:0) θ ( i ) (cid:1)(cid:1)(cid:3) ≡ δ l,m ∆Ω (cid:0) θ ( i ) (cid:1) δ [ΨΨ † ]the number of variables is implicitly ( l, m −
1) in (cid:0)(cid:0) θ ( i ) † (cid:1) , (cid:0) θ ( i ) (cid:1)(cid:1) . The notation Ψ (cid:0) θ ( i k ) † (cid:1) and Ψ † (cid:0) θ ( i k ) (cid:1) represent Ψ (cid:0) θ ( i k ) † , Z i k (cid:1) and Ψ † (cid:0) θ ( i k ) , Z i k (cid:1) .Similarly: δ l,m − ∆Ω † (cid:0) θ ( i ) (cid:1) δ Ψ † (cid:0) θ ( i ) (cid:1) ...δ Ψ † (cid:0) θ ( i m ) (cid:1) δ Ψ (cid:0) θ ( i m + k ) † (cid:1) ...δ Ψ (cid:0) θ ( i m + l ) † (cid:1) = δ l,m − ∆Ω † (cid:0) θ ( i ) (cid:1) δ (cid:2) ΨΨ † (cid:0)(cid:0) θ ( i ) † (cid:1) , (cid:0) θ ( i ) (cid:1)(cid:1)(cid:3) ≡ δ l,m ∆Ω † (cid:0) θ ( i ) (cid:1) δ [ΨΨ † ]We will show recursively that δ l,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ ΨΨ † (( θ ( i ) ) , ( θ ( i ) † ))] satisfies the relation (the index m − m forthe sake of simplicity): δ l,m [∆Ω] (cid:0) θ ( i ) (cid:1) δ (cid:2) ΨΨ † (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:3) = Z K l,m (cid:16)(cid:16) θ ( i ) (cid:17) , θ ( i )1 (cid:17) δ l,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + h X l,m (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , (cid:16) θ ( i ) † (cid:17)(cid:17)(cid:17)i (201)96here the matrix K l,m (cid:16) θ ( i ) , θ ( i )1 (cid:17) and the vector (cid:2) X l,m (cid:0) θ ( i ) , (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:1)(cid:3) have to be determined. Thederivative of this equation with respect to Ψ allows to find recursively the matrix K l,m (cid:16) θ ( i ) , θ ( i )1 (cid:17) and thevector (cid:2) X l,m (cid:0) θ ( i ) , (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:1)(cid:3) : δ l +1 ,m [∆Ω] (cid:0) θ ( i ) (cid:1) δ (cid:2) ΨΨ † (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:3) = Z K l,m (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ l +1 ,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 (202)+ Z δK l,m (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δ l,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + δX l,m (cid:0) θ ( i ) , (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:1) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) As a consequence, K l +1 ,m (cid:16) θ ( i ) , θ ( i )1 (cid:17) = K l,m (cid:16) θ ( i ) , θ ( i )1 (cid:17) . Using the results for the two points correlationfunction, we have: K l,m (cid:16) θ ( i ) , θ ( i )1 (cid:17) = K (cid:16) θ ( i ) , θ ( i )1 (cid:17) (203)The recursive relation for (cid:2) X l,m (cid:0) θ ( i ) , (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:1)(cid:3) is also obtained from (202): X l +1 ,m (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , (cid:16) θ ( i ) † (cid:17)(cid:17)(cid:17) = Z δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δ l,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + δX l,m (cid:0) θ ( i ) , (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:1) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) = Z δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δ l,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + δδ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δX l − ,m (cid:0) θ ( i ) , (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:1) δ Ψ (cid:16) θ ( i ) l (cid:17) + Z δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) l (cid:17) δ l − ,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 = 2 Z δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δ l,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )2 + Z δ K (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δ Ψ (cid:16) θ ( i ) l (cid:17) δ l − ,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + δδ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δδ Ψ (cid:16) θ ( i ) l (cid:17) X l − ,m (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , (cid:16) θ ( i ) † (cid:17)(cid:17)(cid:17) = X r l C r +1 l +1 Z (cid:18) δδ Ψ (cid:19) r δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δ l − r,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + (cid:18) δδ Ψ (cid:19) l X ,m (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , (cid:16) θ ( i ) † (cid:17)(cid:17)(cid:17) And we obtain recursively: X l +1 ,m (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , (cid:16) θ ( i ) † (cid:17)(cid:17)(cid:17) = X r l C r +1 l +1 Z (cid:18) δδ Ψ (cid:19) r δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i l +1 ) † (cid:1) δ l − r,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + (cid:18) δδ Ψ (cid:19) l X ,m (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , θ ( i m +1 ) † (cid:17)(cid:17) Similarly, differentiating with respect to Ψ † (cid:0) θ ( i ) † (cid:1) yields:97 l +1 ,m +1 (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , (cid:16) θ ( i ) † (cid:17)(cid:17)(cid:17) (204)= X r l C rl Z (cid:18) δδ Ψ (cid:19) r − δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) l (cid:17) δ l − r,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + (cid:18) δδ Ψ (cid:19) l X ,m (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , θ ( i m +1 ) † (cid:17)(cid:17) = X r l C rl Z (cid:18) δδ Ψ (cid:19) r − δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ (cid:16) θ ( i ) l (cid:17) δ l − r,m [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + (cid:18) δδ Ψ (cid:19) l X r ′ m C r ′ m Z (cid:18) δδ Ψ † (cid:19) r ′ − δK (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ Ψ † (cid:16) θ ( i ) m (cid:17) δ ,m − r ′ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) = X r l, r ′ m r + r ′ C rl C r ′ m Z δ r,r ′ K (cid:16) θ ( i ) , θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ l − r,m − r ′ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 + (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) Using (203) and (201), we can write that: δ l − r,m − r ′ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) = M ∗ X l − r,m − r ′ (cid:16) θ ( i )1 , (cid:16)(cid:16) θ ( i ) (cid:17) , (cid:16) θ ( i ) † (cid:17)(cid:17)(cid:17) (205)where M is the matrix defined in (188): M (cid:16) θ ( i ) , θ ( i )1 (cid:17) (206)= δ (cid:16) θ ( i ) − θ ( i )1 (cid:17) + exp Z θ ( i θ ( i )1 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! (cid:16) ∆Ω (cid:16)(cid:16) θ ( i ) (cid:17)(cid:17) Ψ † (cid:16) θ ( i )1 (cid:17) − A (cid:16) θ ( i ) (cid:17) Z θ ( i )1 ∆Ω † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G + B (cid:16) θ ( i ) (cid:17) Z θ ( i )1 ∆Ω † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) (cid:0) O , (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G !! Expression (204) writes: X l +1 ,m +1 (cid:16) θ ( i ) , (cid:16)(cid:16) θ ( i ) (cid:17) , (cid:16) θ ( i ) † (cid:17)(cid:17)(cid:17) = (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) + X r + ... + r l = ls + ... + s m = m r i , s i , r i + s i l ! r ! ...r p ! m ! s ! ...s p ′ ! × (cid:20) δ r ,s Kδ [ΨΨ † ] (cid:21) ∗ ... ∗ (cid:20) M ∗ δ r p ,s p ′ Kδ [ΨΨ † ] (cid:21) ∗ X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) l + 1 , m + 1) correlations: δ l +1 ,m +1 [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) (207)= M ∗ (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) + M ∗ X r l, r ′ m r + r ′ C rl C r ′ m Z δ r,r ′ K (cid:16) θ ( i )1 , θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ l − r,m − r ′ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) dθ ( i )1 = M ∗ (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) + M ∗ X r + ... + r l = ls + ... + s m = m r i , s i , r i + s i l ! r ! ...r p ! m ! s ! ...s p ′ ! × (cid:20) δ r ,s Kδ [ΨΨ † ] (cid:21) ∗ ... ∗ (cid:20) M ∗ δ r p ,s p ′ Kδ [ΨΨ † ] (cid:21) ∗ X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) For the sequel, we will rewrite the kernel K (cid:16) θ ( i )1 , θ ( i )1 (cid:17) in a matricial form. We use that: A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ≃ G − (1 − exp ( − x )) (cid:0) O , ∞ (cid:1) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ≃ G − (cid:0) O , (cid:1) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ exp ( − x ) (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) and define the matrices:[∆Ω] † = (cid:0) ∆Ω † , ∆Ω (cid:1) , h A (cid:16) θ ( i )1 (cid:17)i = A (cid:16) θ ( i )1 (cid:17) A † (cid:16) θ ( i )1 (cid:17) , h B (cid:16) θ ( i )1 (cid:17)i = B (cid:16) θ ( i )1 (cid:17) B † (cid:16) θ ( i )1 (cid:17) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 = (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 (cid:0) O , ∞ (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 = (cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 (cid:0) O , (cid:1) θ ( i )2 , (cid:16) θ ( i )2 (cid:17) ′ (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) = A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) A ′ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ , θ ( i )1 (cid:19) (cid:20) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) = B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) B ′ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ , θ ( i )1 (cid:19)
99o write the kernel K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) (for two arbitrary times (cid:16) θ ( i )1 , θ ( i )2 (cid:17) and two associated points Z i Z i ) as: K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) = Z − h A (cid:16) θ ( i )1 (cid:17)i [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 (208)+ h B (cid:16) θ ( i )1 (cid:17)i [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G + [∆Ω] (cid:16) θ ( i )1 (cid:17) [Ψ] † (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )2 (cid:17) ′ − θ ( i )2 (cid:19)(cid:19) d (cid:16) θ ( i )2 (cid:17) ′ = − Z (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + (cid:20) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G d (cid:16) θ ( i )2 (cid:17) ′ + [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [Ψ] † (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ − θ ( i )1 (cid:19)(cid:19) d (cid:16) θ ( i )1 (cid:17) ′ so that its successive derivatives become: δ r,r ′ K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) = [Ψ] † (cid:16) θ ( i )2 (cid:17) δ r,r ′ [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) + δ (cid:16) θ ( i ) † − θ ( i )2 (cid:17) δ r,r ′ − [∆Ω] (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) − X Z δ t,t ′ (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ r − s − t,r ′ − s ′ − t ′ [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) × δ s,s ′ [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + δ t,t ′ (cid:20) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ r − s − t,r ′ − s ′ − t ′ [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ s,s ′ [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G d (cid:16) θ ( i )2 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ All the derivatives δ r ,s K (cid:16) θ ( i )1 ,θ ( i )1 (cid:17) δ [ΨΨ † ] are proportional to exp ( − x ). For background fields of magnitude greaterthan 1, the dominant term is given by the term: M ∗ (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) + (cid:20) M ∗ δ l,m Kδ [ΨΨ † ] (cid:21) X , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) (209)with: 100 , (cid:16) θ ( i ) , (cid:16) θ ( i m +1 ) † (cid:17)(cid:17) = G − D (
1+ ¯ O , ) (2) E + P n > n − D (
1+ ¯ O , ∞ ) ( n ) E n − h i + D (
1+ ¯ O , ) (2) E + P n > n ! D (
1+ ¯ O , ∞ ) ( n ) E n ! − (cid:0) θ ( i ) , (cid:0) θ ( i m +1 ) † (cid:1)(cid:1) ≃ G − (cid:18) − x ) ( − z + ( y − x ))(
1+ ¯ O , ∞ ) +exp( − x ) ( − ¯ O , ∞ + y (
1+ ¯ O , ) − x (
1+ ¯ O , ∞ )) (cid:19) (cid:0) θ ( i ) , (cid:0) θ ( i m +1 ) † (cid:1)(cid:1) We can skip the variables (cid:0) θ ( i ) , (cid:0) θ ( i m +1 ) † (cid:1)(cid:1) since X , (cid:0) θ ( i ) , (cid:0) θ ( i m +1 ) † (cid:1)(cid:1) is the kernel of an operator X , .To compute (cid:0) δδ Ψ (cid:1) l (cid:0) δδ Ψ † (cid:1) m X , in (209), we write the first order expansion of X , in exp ( − x ) :1 + exp ( − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ≃ O , ∞ + 1 − exp ( − x ) O , ∞ + 1 (cid:18) x − y + z (cid:19) − (cid:0) ¯ O , ∞ + 1 (cid:1) (cid:0) x (cid:0) ¯ O , ∞ + 1 (cid:1) − y (cid:0) ¯ O , + 1 (cid:1) + ¯ O , ∞ (cid:1)! so that: (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , ≃ G − (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m − exp ( − x ) O , ∞ + 1 (cid:18) x − y + z (cid:19) − (cid:0) x (cid:0) ¯ O , ∞ + 1 (cid:1) − y (cid:0) ¯ O , + 1 (cid:1) + ¯ O , ∞ (cid:1)(cid:0) ¯ O , ∞ + 1 (cid:1) !! The dominant term in field is then: (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , ≃ − (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m exp ( − x ) ! G − × (cid:18)(cid:18) x − y + z (cid:19) (cid:0) ¯ O , ∞ + 1 (cid:1) − − (cid:0) x (cid:0) ¯ O , ∞ + 1 (cid:1) − y (cid:0) ¯ O , + 1 (cid:1) + ¯ O , ∞ (cid:1) (cid:0) ¯ O , ∞ + 1 (cid:1) − (cid:19) Using that: (cid:18) δδ Ψ (cid:19) r (cid:18) δδ Ψ † (cid:19) s exp ( − x ) ≃ ( − r (cid:18) δδ Ψ † (cid:19) s Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) r exp ( − x ) ≃ ( − r + s X σ Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) r Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω s exp ( − x )+terms with powers of Ψ lowered by 2 101nd we obtain that: (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) m X , (210) ≃ ( − l + m +1 X σ G − Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m exp ( − x ) × (cid:18)(cid:18) x − y + z (cid:19) (cid:0) O , ∞ (cid:1) − − (cid:0) x (cid:0) O , ∞ (cid:1) − y (cid:0) ¯ O , + 1 (cid:1) + ¯ O , ∞ (cid:1) (cid:0) O , ∞ (cid:1) − (cid:19) ≃ ( − l + m G − Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m exp ( − x ) × (cid:18) y − x − z (cid:19) (cid:0) O , ∞ (cid:1) − We also rewrite K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) as: Z [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [Ψ] † (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ − θ ( i )1 (cid:19) d (cid:16) θ ( i )1 (cid:17) ′ − (cid:18)Z (cid:18)(cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [Ψ] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) × (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) G − (cid:0) O , ∞ (cid:1)! (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + (cid:20) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G d (cid:16) θ ( i )2 (cid:17) ′ ! d (cid:16) θ ( i )1 (cid:17) ′ At the lowest order in exp ( − x ) and in perturbation:0 ≃ δ l,m (cid:18) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [Ψ] † (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )1 (cid:17) ′ − θ ( i )1 (cid:19)(cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) − X Z (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) δ l,m [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [Ψ] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) × (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) G − (cid:0) O , ∞ (cid:1)! (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G d (cid:16) θ ( i )2 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ − x ) of the kernel’s derivatives is: δ l,m K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) (211) ≃ − X Z δ l,m (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) " (1 + O , ∞ ) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 + δ l,m (cid:20) B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G d (cid:16) θ ( i )2 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ − X Z (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [Ψ] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) × δ l,m (cid:18) ( − x ) ( − z + ( y − x )))(
1+ ¯ O , ∞ ) +exp( − x ) ( − ¯ O , ∞ + y (
1+ ¯ O , ) − x (
1+ ¯ O , ∞ )) G − (cid:0) O , ∞ (cid:1)(cid:19) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) G d (cid:16) θ ( i )2 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ δ l,m A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ ( − l + m +1 Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m ×G − exp ( − x ) (cid:0) O , ∞ (cid:1) θ ( i )1 (cid:0) (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:1) G ≃ ( − l + m +1 Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m G − × exp ( − x ) (cid:0) (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1)(cid:0) O , ∞ (cid:1) G δ l,m B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ ( − l + m Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ≃ ( − l + m Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m B ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ≃ ( − l + m Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m ×G − (cid:0) O , (cid:1) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ exp ( − x ) (cid:0) O , ∞ (cid:1) G and: δ l,m δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) G − (cid:0) O , ∞ (cid:1)! (cid:16) θ ( i )1 (cid:17) ′ θ ( i )1 ≃ ( − l + m (cid:0) y − x (cid:1) Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m × exp ( − x ) (cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) G − (cid:0) O , ∞ (cid:1)! (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 ≃ ( − l + m (cid:0) y − x (cid:1) exp ( − x ) Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m × (cid:0) O , ∞ (cid:1) G − (cid:0) O , ∞ (cid:1)! (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) δ l,m (cid:18) ( − x ) ( − z + ( y − x )))(
1+ ¯ O , ∞ ) +exp( − x ) ( − ¯ O , ∞ + y (
1+ ¯ O , ) − x (
1+ ¯ O , ∞ )) G − (cid:0) O , ∞ (cid:1)(cid:19) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ ( − l + m (cid:0) y − x (cid:1) exp ( − x ) [∆Ω] † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 × Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m As a consequence, for background fields of large magnitude, the dominant term of (211) is: δ l,m K (cid:16) θ ( i )1 , θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ − X Z δ l,m (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G d (cid:16) θ ( i )1 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ − X Z (cid:20) A ′ (cid:18) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19)(cid:21) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [Ψ] † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) × δ l,m (cid:18) ( − x ) ( − z + ( y − x )))(
1+ ¯ O , ∞ ) +exp( − x ) ( − ¯ O , ∞ + y (
1+ ¯ O , ) − x (
1+ ¯ O , ∞ )) G − (cid:0) O , ∞ (cid:1)(cid:19) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) G d (cid:16) θ ( i )1 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ Since A ′ (cid:18) θ ( i )1 , θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ≃ − x ), this leads to the following expression: δ l,m K (cid:16) θ ( i )1 , θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) (212) ≃ − exp ( − x ) X Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m × " G − (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) G [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 + 12 (cid:0) y − x (cid:1) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G d (cid:16) θ ( i )1 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ To find M ∗ δ l,m K (cid:16) θ ( i )1 ,θ ( i )1 (cid:17) δ [ΨΨ † ] X , (cid:16) θ ( i )1 , (cid:0) θ ( i ) † (cid:1)(cid:17) arising in (209), we compute the convolution between M andthe last factor in (212). For fields of large magnitude: A ′ (cid:18) θ ( i )1 , θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) ≃ B ′ (cid:18) θ ( i )1 , θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:19) < < = δ (cid:16) θ ( i )1 − θ ( i )1 (cid:17) + exp Z θ ( i )1 θ ( i )1 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! (cid:16) ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ † (cid:16) θ ( i )1 (cid:17) − A (cid:16) θ ( i )1 (cid:17) Z θ ( i )1 ∆Ω † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G + B (cid:16) θ ( i )1 (cid:17) Z θ ( i )1 ∆Ω † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) (cid:0) O , (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G !! ≃ δ (cid:16) θ ( i )1 − θ ( i )1 (cid:17) + exp Z θ ( i )1 θ ( i )1 ¯ N (cid:16)(cid:16) θ ( i ) (cid:17) dθ ( i ) (cid:17)! × ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ † (cid:16) θ ( i )1 (cid:17) − ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Z θ ( i )1 ∆Ω † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G ! ≃ δ (cid:16) θ ( i )1 − θ ( i )1 (cid:17) since: ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Z θ ( i )1 ∆Ω † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G ≃ ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ † (cid:16) θ ( i )1 (cid:17) and: ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ † (cid:16) θ ( i )1 (cid:17) − ∆Ω (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Z θ ( i )1 ∆Ω † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) (cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G ! is of order exp ( − x ). As a consequence: M ∗ δ l,m K (cid:16) θ ( i )1 , θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) X , (cid:16) θ ( i )2 , (cid:16) θ ( i ) † (cid:17)(cid:17) ≃ δ l,m K (cid:16) θ ( i )1 , θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) ≃ ( − l + m exp ( − x ) X Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m × " G − (cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) G [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 + 12 (cid:0) y − x (cid:1) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) "(cid:0) O , ∞ (cid:1) (cid:16) θ ( i )1 (cid:17) ′ ,θ ( i )1 G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) d (cid:16) θ ( i )1 (cid:17) ′ d (cid:16) θ ( i )1 (cid:17) ′ ≃ (cid:0) y − x (cid:1) ( − l + m exp ( − x ) X Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m × Z (cid:18) x (cid:19) [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:16) θ ( i )1 (cid:17) ≃ (cid:0) y − x (cid:1) ( − l + m exp ( − x ) X Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m × Z [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:16) θ ( i )1 (cid:17) (213)106nd (212) can be written: M ∗ δ l,m K (cid:16) θ ( i )1 , θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) (214) ≃ exp ( − x ) Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m × (cid:0) y − x (cid:1) Z [∆Ω] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [∆Ω] † (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) Gathering (210) and (213) leads to the strong background field approximation of (209): δ l,m ∆Ω α (cid:16) θ ( i )1 (cid:17) δ (cid:2) ΨΨ † (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:3) (215)= M ∗ (cid:18) δδ Ψ (cid:19) l − (cid:18) δδ Ψ † (cid:19) m − X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) + M ∗ δ l − ,m − K (cid:16) θ ( i )1 , θ ( i )1 (cid:17) δ [ΨΨ † ] X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) ≃ (cid:0) y − x (cid:1) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l − Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m − exp ( − x )+ 12 (cid:0) y − x (cid:1) exp ( − x ) Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l − Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m − × Z ∆Ω † Z ∆ΩAdditional contributions to (207) can be considered. We consider r products of (214). Using (215), eachterm of this product is given by: M ∗ δ r,s K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ exp ( − x ) Z ∆Ω † (cid:0) O , ∞ (cid:1) G δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l i Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G ∆Ω m i × (cid:0) y − x (cid:1) Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G [∆Ω] (cid:16) θ ( i )1 (cid:17) Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) (1 + O , ∞ ) ≃ exp ( − x ) Z G Z ∆Ω † (1 + O , ∞ ) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) r Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (1 + O , ∞ ) ∆Ω s × (cid:0) y − x (cid:1) [∆Ω] (cid:16) θ ( i )1 (cid:17) [∆Ω] † (cid:16) θ ( i )2 (cid:17) (1 + O , ∞ )where P l i = l and P m i = m .12 (cid:0) y − x (cid:1) Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0)(cid:0) θ ( i ) † (cid:1)(cid:1) (1 + O , ∞ ) [∆Ω] (cid:16) θ ( i )1 (cid:17) Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) (1 + O , ∞ ) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) ≃ (cid:0) y − x (cid:1) [∆Ω] (cid:16) θ ( i )1 (cid:17) [∆Ω] † (cid:16) θ ( i )2 (cid:17) P l i = l and P m i = m . The successive convolutions multiplied by X , (cid:16) θ ( i )1 , (cid:0) θ ( i ) † (cid:1)(cid:17) yield: (cid:18) (cid:0) y − x (cid:1)(cid:19) r exp ( − rx ) Z ∆Ω † (1 + O , ∞ ) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (1 + O , ∞ ) ∆Ω m × r Y m =1 (cid:20)Z (cid:16) [∆Ω] † ( θ ′ m ) [(1 + O , ∞ )] [∆Ω ( θ m )] (cid:17)(cid:21) the integral of the product being on the 2 r dimensional simplex θ ′ m < θ m +1 . In first approximation the sumof these expression can be replaced by: Z ∆Ω † (1 + O , ∞ ) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (1 + O , ∞ ) ∆Ω m × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)! That is, the expression (214) for δ l,m ∆Ω α (cid:16) θ ( i )1 (cid:17) δ [ ΨΨ † (( θ ( i ) ) , ( θ ( i ) † ))] is multiplied by the exponential. At the lowest order inexp ( − x ), the ( l, m ) points connected correlation functions are obtained by the convolution of this expressionwith (1 ,
1) propagators (cid:18) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ ( θ ( i ) ) (cid:19) − . At the lowest order in exp ( − x ), this leads to:12 (cid:0) y − x (cid:1) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − (cid:18)Z ∆Ω † (cid:0) O , ∞ (cid:1) G (cid:19) l − (cid:18)Z (cid:0) O , ∞ (cid:1) G ∆Ω (cid:19) m − + (cid:18)Z ∆Ω † (cid:0) O , ∞ (cid:1) G (cid:19) l (cid:18)Z (cid:0) O , ∞ (cid:1) G ∆Ω (cid:19) m ! × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)! ≃ (cid:0) y − x (cid:1) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − (cid:16) Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17)(cid:17) l − (cid:16) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:17) m − + (cid:16) Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17)(cid:17) l (cid:16) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:17) m (cid:19) × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)! ≃ (cid:0) y − x (cid:1) (cid:18)(cid:16) Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17)(cid:17) l (cid:16) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:17) m (cid:19) × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)! Additional contributions are obtained by convoluting the connected vertices of lowest order. Each of thisconvolution adds an additional factor:exp ( − x ) Z Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17) m − (cid:18) exp ( − x ) Z Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:19) p are identical. The number of such contributions, written A l,m,p , satisfies: A l,m,p = p A l − ,m,p + (cid:18) l + p − (cid:19) A l − ,m,p − (216)Actually, differentiation with respect to l of a connected graphs with l − , m external lines and p − p factors of the type: X l i ,m i = (cid:18) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ ( θ ( i ) ) (cid:19) − ! l i δ l ,m ∆Ω α (cid:16) θ ( i )1 (cid:17) δ [ ΨΨ † (( θ ( i ) ) , ( θ ( i ) † ))] (cid:18) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ ( θ ( i ) ) (cid:19) − ! m yieldstwo types of contributions to the graphs with l, m external lines. First, due to the heaviside functions inthe propagators, it yields in average p graphs with p − X l i ,m i .The second type of contribution has l + m external lines and p internal lines, p factors of the type: X l i ,m i = δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − l i δ l ,m ∆Ω α (cid:16) θ ( i )1 (cid:17) δ (cid:2) ΨΨ † (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:3) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − m and.one three points vertex: δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − l i δ l ′ ,m ′ ∆Ω α (cid:16) θ ( i )1 (cid:17) δ (cid:2) ΨΨ † (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1)(cid:3) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − m with l ′ + m ′ = 3. The insertion of this three points vertex arises in average at one of the l + p − lines atthe left of the initial graphs.Similarly: A l,m,p = p A l,m − ,p + (cid:18) m + p − (cid:19) A l,m − ,p − (217)Asymptotically, (216) and (217) yield that A l + m,p ≃ (cid:18) m + p − (cid:19) (cid:18) l + p − (cid:19) A l − ,m − ,p − and recursively: A l + m,p ≃ (cid:0) m + p − (cid:1) ! (cid:0) l + p − (cid:1) ! m ! l !Given that the sum over the permutations of points involved in the ( l, m ) correlation functions are m ! l !, thisyields for the connected correlation functions: F l,m (Ψ) (cid:18) (cid:0) y − x (cid:1) (cid:18)(cid:16) Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17)(cid:17) l (cid:16) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:17) m (cid:19) exp ( − x ) (cid:19) (218)with: F l,m (Ψ) = m − X k =0 (cid:0) m + p − (cid:1) ! (cid:0) l + p − (cid:1) !( l ! m !) (cid:18) exp ( − x ) Z Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:19) p l variables and m variables respectively is understood.For p → m + l , ( m + p − ) ! ( l + p − ) !( l ! m !) behaves as exp ( αp ) and: F l,m (Ψ) ≃ m − X k =0 (cid:18) exp ( − ( x − α )) Z Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:19) p ≃ − exp ( − ( x − α )) R Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17) Including the corrective factor F l,m (Ψ) leads to the connected correlation functions: F l,m (Ψ) Z ∆Ω † (1 + O , ∞ ) δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) l Z δ ∆Ω † (cid:16) θ ( i )1 (cid:17) δ Ψ † (cid:0) θ ( i ) (cid:1) (1 + O , ∞ ) ∆Ω m × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)! Ultimately, we can derive from (218) the ( l, m ) points correlation functions: δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − m δ l,m + inf( l,m ) X s =0 X k > X s > ... > s k > ,t > ... > t k > s i + t i > , P s i = l − s, P t i = m − s k Y i =1 F s i ,t i (Ψ) ! δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − s × (cid:16) Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17)(cid:17) l − s − (cid:16) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:17) m − s − × (cid:0) y − x (cid:1) exp ( − x )2 δ ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) − + (cid:16) Ψ † (cid:16)(cid:16) θ ( i )1 (cid:17)(cid:17)(cid:17) (cid:16) Ψ (cid:16)(cid:16) θ ( i )2 (cid:17)(cid:17)(cid:17) × exp (cid:0) y − x (cid:1) exp ( − x )2 (cid:18)Z [∆Ω] † (cid:16) θ ( i )2 (cid:17) [(1 + O , ∞ )] [∆Ω] (cid:19)!! k (219) For background of small magnitude, the correlation functions for l = m are negligible and the dominantcontribution while computing: (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) l − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (220)is obtained by replacing each derivative δδ Ψ δδ Ψ † of exp ( − x ) by 1 + ¯ O , ∞ = ν . Thus writing ddx for δδ Ψ δδ Ψ † ,the operator (cid:0) δδ Ψ δδ Ψ † (cid:1) l can be replaced by (cid:0) exp ( − νx ) ∂∂U + ∂∂x (cid:1) l where U = exp ( − x ). All references to the2 l points arising in the functional derivatives can be skipped and reintroduced at the end of the calculus.We thus compute: (cid:18) exp ( − νx ) ∂∂U + ∂∂x (cid:19) l U (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + U (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! U =exp( − νx ) (cid:0) exp ( − νx ) ∂∂U + ∂∂x (cid:1) l can be found recursively: (cid:18) − ν exp ( − νx ) ∂∂U + ∂∂x (cid:19) n +1 = (cid:18) exp ( − x ) ∂∂U + ∂∂x (cid:19) X C kn ( − ν ) k exp ( − kνx ) ∂ k ∂U k ∂ n − k ∂x n − k + X p a pn,k ( − ν ) k exp ( − kνx ) ∂ k ∂U k ∂ n − k − p ∂x n − k − p ! = X C kn +1 ( − ν ) k exp ( − kνx ) ∂ k ∂U k ∂ n +1 − k ∂x n +1 − k + X p a pn,k ( − ν ) k +1 exp ( − ( k + 1) νx ) ∂ k +1 ∂U k +1 ∂ n − k − p ∂x n − k − p + a pn,k ( − ν ) k exp ( − kνx ) ∂ k ∂U k ∂ n +1 − k − p ∂x n +1 − k − p − ka pn,k ( − ν ) k exp ( − kνx ) ∂ k ∂U k ∂ n − k − p ∂x n − k − p − X C kn k ( − ν ) k +1 exp ( − kνx ) ∂ k ∂U k ∂ n − k ∂x n − k and the coefficients a pn,k satisfy: a pn +1 ,k +1 = a pn,k + a pn,k +1 − ( k + 1) a p − n,k +1 − ( k + 1) C k +1 n δ p, Remark that the expression to differentiate in (220) rewrites:1 + exp ( − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) = 1 + U (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + U (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) = (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) + 1 − (
1+ ¯ O , ∞ )( − z + ( y − x ))( − ¯ O , ∞ + y (
1+ ¯ O , ) − x (
1+ ¯ O , ∞ )) (cid:0) O , ∞ (cid:1) + U (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) and its k th derivative with respect to U becomes:1 − (
1+ ¯ O , ∞ )( − z + ( y − x ))( − ¯ O , ∞ + y (
1+ ¯ O , ) − x (
1+ ¯ O , ∞ )) (cid:0) − (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) k (cid:0) O , ∞ (cid:1) + U (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) k +1 This is of order k in perturbation: it involves terms of order k . The dominant term in perturbation is thusfor k = 0 ,
1. As a consequence the operator (cid:0) − ν exp ( − νx ) ∂∂U + ∂∂x (cid:1) l reduces to: X C kl ( − ν ) k exp ( − kνx ) ∂ k ∂U k ∂ l − k ∂x l − k + X p a pl,k ( − ν ) k + p exp ( − kνx ) ∂ k ∂U k ∂ l − k − p ∂x l − k − p ≃ ∂ n ∂x n − lν exp ( − νx ) ∂∂U ∂ l − ∂x l − − ν X p exp ( − νx ) ∂∂U ∂ l − − p ∂x l − − p X q > p C pq = ∂ n ∂x n − lν exp ( − νx ) ∂∂U ∂ l − ∂x l − − ν X p ( − ν ) p C p +1 l exp ( − νx ) ∂∂U ∂ l − − p ∂x l − − p ∂ n ∂x n (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! − lν exp ( − νx ) ∂ l − ∂x l − (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − (cid:0) O , ∞ (cid:1) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − ν l − X p =1 ( − ν ) p C p +1 l exp ( − νx ) ∂ l − − p ∂x l − − p (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − (cid:0) O , ∞ (cid:1) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ∂ n ∂x n (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ! − ν l − X p =0 ( − ν ) p C p +1 l exp ( − νx ) ∂ l − − p ∂x l − − p (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − (cid:0) O , ∞ (cid:1) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + U (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) and: − ∂ l − − p ∂x l − − p ¯ O , ∞ (cid:0)(cid:0) O , ∞ (cid:1) + − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) + ∂ l − − p ∂x l − − p y (cid:0) O , (cid:1) + ( z − x ) (cid:0) O , ∞ (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − ∂ l − − p ∂x l − − p (cid:0) O , ∞ (cid:1) (cid:0) y − x (cid:1)(cid:0) O , ∞ (cid:1) + (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ≃ ∂ l − − p ∂x l − − p y (cid:0) O , (cid:1) + ( z − x ) (cid:0) O , ∞ (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − ∂ l − − p ∂x l − − p (cid:0) O , ∞ (cid:1) (cid:0) y − x (cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ≃ (cid:16)(cid:0) O , (cid:1) − (cid:0) − O , ∞ (cid:1) (cid:0) O , ∞ (cid:1)(cid:17) (cid:16)(cid:0) O , (cid:1) − (cid:0) O , ∞ (cid:1) (cid:17) l − − p (cid:0)(cid:0) O , ∞ (cid:1) + − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) l +1 − p −
12 ( l − − p ) ( l − − p )2 (cid:0) O , ∞ (cid:1) (cid:16)(cid:0) O , (cid:1) − (cid:0) O , ∞ (cid:1) (cid:17) l − − p (cid:0)(cid:0) O , ∞ (cid:1) + − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) l − p As a consequence, the dominant contribution is for p = l − (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) l − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) ≃ ( − ν ) l exp ( − νx ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − (cid:0) O , ∞ (cid:1) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0)(cid:0) O , ∞ (cid:1) + (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1)(cid:1) A (cid:16) θ ( i )1 (cid:17) << K (cid:16) θ ( i )1 , θ ( i )2 (cid:17) ≃ − Z h B (cid:16) θ ( i )1 (cid:17)i [∆Ω] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G ! d (cid:16) θ ( i )1 (cid:17) ′ d (cid:16) θ ( i )2 (cid:17) ′ + [∆Ω] (cid:16) θ ( i )1 (cid:17) [Ψ] † (cid:16) θ ( i )2 (cid:17) δ (cid:18)(cid:16) θ ( i )2 (cid:17) ′ − θ ( i )2 (cid:19) ≃ Z (cid:16) G − [Ψ] (cid:16) θ ( i )1 (cid:17) [Ψ] † (cid:16) θ ( i )2 (cid:17) − Z G − (cid:20)(cid:0) O , (cid:1) θ ( i )1 , (cid:16) θ ( i )1 (cid:17) ′ (cid:21) [Ψ] (cid:18)(cid:16) θ ( i )1 (cid:17) ′ (cid:19) [Ψ] † (cid:18)(cid:16) θ ( i )2 (cid:17) ′ (cid:19) G − "(cid:0) O , (cid:1) (cid:16) θ ( i )2 (cid:17) ′ ,θ ( i )2 G ! d (cid:16) θ ( i )2 (cid:17) ′ d (cid:16) θ ( i )2 (cid:17) ′ and in first approximation (207) becomes: δ l,l ∆Ω (cid:16) θ ( i )1 , θ ( i )2 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ M ∗ (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) l X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) + (cid:20) M ∗ δ , Kδ [ΨΨ † ] (cid:21) ∗ ... ∗ (cid:20) M ∗ δ , Kδ [ΨΨ † ] (cid:21) ∗ X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) For weak fields M ≃ X , ≃ G − . As a consequence: δ l +1 ,l ∆Ω (cid:16) θ ( i )1 (cid:17) δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) ≃ (cid:18) δδ Ψ (cid:19) l (cid:18) δδ Ψ † (cid:19) l X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) + (cid:20) δ , Kδ [ΨΨ † ] (cid:21) ∗ ... ∗ (cid:20) δ , Kδ [ΨΨ † ] (cid:21) ∗ X , (cid:16) θ ( i )1 , (cid:16) θ ( i ) † (cid:17)(cid:17) ≃ ( − l X p X u p + v p = l Y p Z δ u p ,u p ∆Ω † (cid:16) θ ( i )2 (cid:17) ′ δ [ΨΨ † ] (cid:0)(cid:0) θ ( i ) (cid:1) , (cid:0) θ ( i ) † (cid:1)(cid:1) (cid:0) O , ∞ (cid:1) G δ v p ,v p ∆Ω (cid:16) θ ( i )1 (cid:17) δ Ψ (cid:0) θ ( i ) (cid:1) × exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) − (cid:0) O , ∞ (cid:1) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) + l Y i =1 (cid:18) G − (cid:16) θ ( i )1 (cid:17) , (cid:16) θ ( i )1 (cid:17) ⊗ θ ( i ) † ) , (cid:16) θ ( i )2 (cid:17) − G − (cid:0) O , (cid:1) (cid:16) θ ( i )1 (cid:17) , ( θ ( i ) ) ⊗ G − (cid:0) O , (cid:1) ( θ ( i ) † ) , (cid:16) θ ( i )2 (cid:17) G (cid:19) l d (cid:16) θ ( i )1 (cid:17) d (cid:16) θ ( i )2 (cid:17) ≃ ( − l +1 (cid:0) G − ∗ (cid:0) O , ∞ (cid:1)(cid:1) ⊗ l G − ¯ O , ∞ where the symmetrisation over the variables is understood. To find the connected correlation functions, weproceed as for the strong field case. The connected vertices are convoluted with the propagators G . Theterm of lowest order in perturbation for the ( l, l ) correlation functions is: (cid:16) ( − l G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l − ⊗ ¯ O , ∞ ∗ G (cid:17) (cid:16) ( − l G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l − ⊗ ¯ O , ∞ ∗ G (cid:17) + X l + l = l (cid:16) ( − l G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l − ⊗ ¯ O , ∞ ∗ G (cid:17) ∗G ∗ ( − l G (cid:18)Z (cid:0) O , ∞ (cid:1) G (cid:19) ⊗ l − ¯ O , ∞ G ! + .... The convolution of the two blocks with G being performed on any external point of these blocks. Theprevious expression can be rewritten: (cid:16) ( − l G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l − ⊗ ¯ O , ∞ ∗ G (cid:17) + ( − l X k > X l + ... + l k = l k Y m =1 X ∗ { lm }G ∗ { l m }G (cid:16) G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l m − ⊗ ¯ O , ∞ ∗ G (cid:17) ≡ G ( l ) C where the sum over ∗ { l m }G denotes the sum over all possible convolutions between the different blocks: (cid:16) G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l m − ⊗ ¯ O , ∞ ∗ G (cid:17) Two blocks are convoluted on at most one variable. The convolution are performed by insertion of apropagator G between the blocks. The expression for the connected correlation functions induces the fullcorrelation functions: G ⊗ n + X p,k ( − l − p X l k , P l n = l − p Y k G ( l n ) C G ⊗ p = G ⊗ n + X p,k ( − l − p X l n , P l n = l − p Y n (cid:16)(cid:16) ( − l n G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l n − ⊗ ¯ O , ∞ ∗ G (cid:17) + ( − l n X k > X l + ... + l k = l n k Y m =1 X ∗ { lm }G ∗ { l m }G (cid:16) G ⊗ (cid:0)(cid:0) O , ∞ (cid:1) ∗ G (cid:1) ⊗ l m − ⊗ ¯ O , ∞ ∗ G (cid:17) ⊗ G ⊗ p Appendix 4. Estimation of 1PI graphs contributions and minimumof the effective action
The previous computations gave the form of the correlation functions once the minimum of the effectiveaction is known. To find this minimum we have to write directly the sum of 1PI graphs, and to find itsminimum. As before we first compute the sum without inertia coefficients, and then with the inclusion ofthese coefficients.
We write the sum of 1PI graphs. These are the graphs that cannot be factored into product of subgraphsbetween 2 k and 2 n − k points respectively, with k > min (cid:18) Z i , l ( i )1 , n Z k (1) j,i o , ..., l ( i ) p i , (cid:26) Z k ( pi ) j,i (cid:27)(cid:19) i =1 ,...,n ! = Y i ¯Ξ ( l )1 (cid:16) Z i , n Z k (1) j,i o , θ ( i ) i , θ ( i ) f (cid:17) .. ¯Ξ( l pi ) (cid:18) Z i , (cid:26) Z k ( pi ) j,i (cid:27) , θ ( i ) i , θ ( i ) f (cid:19) that are connected, and such that removing one of the vertices turns the graph in a not 1PI graph. The 1PIgraphs are those that can be factored by a minimal graph. The factorization is defined here by a convolutionproduct in the 2 n time variables between graphs with 2 n external vertices. The precise form of the minimalgraphs will be given below, but it is enough here to state that for p i = 1, the valence l is equal to n − p i >
1, one has P l ( i )1 = n . The factorization by a minimal graph amounts, in the computation of:1 m ! (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) m = 1 m ! Z θ ( i ) f θ ( i ) i X l ¯Ξ ( l )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) dθ ( i ) ! m to replace one of the factor ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ (cid:17) by ¯Ξ ( l )1 (cid:16) Z i , n Z k (1) j,i o , θ ( i ) i , θ ( i ) f (cid:17) . There are m possibil-ities to do so. Thus in the computation of the connected graphs, the term:1 m ! (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) m is replaced by: 1( m − ( l )1 (cid:16) Z i , n Z k (1) j,i o , θ ( i ) i , θ ( i ) f (cid:17) (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ (cid:17)(cid:17) m − (221)and similarly, the introduction of:¯Ξ ( l )1 (cid:16) Z i , n Z k (1) j,i o , θ ( i ) i , θ ( i ) f (cid:17) ... ¯Ξ( l pi ) (cid:18) Z i , (cid:26) Z k ( pi ) j,i (cid:27) , θ ( i ) i , θ ( i ) f (cid:19) implies a contribution:1( m − p i )! ¯Ξ ( l )1 (cid:16) Z i , n Z k (1) j,i o , θ ( i ) i , θ ( i ) f (cid:17) .. ¯Ξ( l pi ) (cid:18) Z i , (cid:26) Z k ( pi ) j,i (cid:27) , θ ( i ) i , θ ( i ) f (cid:19) (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ (cid:17)(cid:17) m − p i Ultimately, we have to take into account n external propagators. The dominant contribution at the lowestorder in perturbation is obtained by factoring these graphs:Γ (cid:16)(cid:16) Z i , (cid:16) θ ( i ) i , θ ( i ) f (cid:17)(cid:17) n (cid:17) = X p ,...,p n X l ( i )1 , ( Z k (1) j,i ) ,...,l ( i ) pi , ( Z k ( pi ) j,i )P i (cid:16) l ( i )1 + ... + l ( i ) pi (cid:17) = n × Y i,p i =0 (cid:16) exp ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) p i Y k =1 (cid:16) ¯Ξ ( l i )1 (cid:16) Z i , n Z k (1) j,i o , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) ∇ out θ ( i ) i Λ × Y i,p i =0 (cid:16) exp (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) ∇ out θ ( i ) i Λ × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n (cid:16)(cid:16) Z i , (cid:16) θ ( i ) i , θ ( i ) f (cid:17)(cid:17) n (cid:17) (222)= X i ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) ∇ out θ ( i ) i Λ × Y j = i (cid:16) exp (cid:16) ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) ∇ out θ ( j ) i Λ × Λ n exp − Λ n X j =1 θ ( j ) f − n X j =1 θ ( j ) i + X p ,...,p n X l ( i )1 , ( Z k (1) j,i ) ,...,l ( i ) pi , ( Z k ( pi ) j,i )P i (cid:16) l ( i )1 + ... + l ( i ) pi (cid:17) = n Y i,p i =0 exp (cid:16)(cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ (cid:17)(cid:17)(cid:17) p i Y k =1 ¯Ξ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , n Z k ( k ) j,i o , θ ( i ) i , θ ( i ) f (cid:17) ∇ out θ ( i ) i Λ × Y i,p i =0 (cid:16) exp (cid:16) ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) ∇ out θ ( i ) i Λ × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n The last sum is implicitly constrained by imposing that every propagator for Z j between θ ( j ) i and θ ( j ) f shouldbe connected to at least one other points Z k . The connected graphs can be approximated as before apart from some modification in the sum of connectedgraphs . By a similar reasoning that led to (221), the sum (159) is replaced by: X m> ( m − m ! ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) m − − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i + X m> m )! ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) m − − ζ ( n ) + ∇ outθ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ζ ( n ) + ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i The ( m −
1) factor in the first term holds for dispatching the insertion ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) at the m − ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) (with derivative) in last position. This leads to the following expression for the116um of vertices: − ζ ( n ) Λ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) +ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i and this expression can be also written for later purposes: − ζ ( n ) Λ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) + − ζ ( n ) Λ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) × exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − − ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) +ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) × exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i For ¯ ζ n >> ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) , this expression reduces toˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i and we will use this expression to estimate the various contributions, coming back later to the full expressionfor the vertices in order to find a precise expression for the effective action. The sum of one particle irreduciblegraphs becomes thus: 117 P I (cid:16)(cid:16) Z i , (cid:16) θ ( i ) i , θ ( i ) f (cid:17)(cid:17) n (cid:17) (223)= X i ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) × Y j = i (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n + X p ,...,p n X l ( i )1 , ( Z k (1) j,i ) ,...,l ( i ) pi , ( Z k ( pi ) j,i )P i (cid:16) l ( i )1 + ... + l ( i ) pi (cid:17) = n Y i,p i =0 exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) × p i Y k =1 ˆΞ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , n Z k ( k ) j,i o , θ ( i ) i , θ ( i ) f (cid:17)! − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i × Y i,p i =0 (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n with: ˆΞ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , n Z k ( k ) j,i o , θ ( i ) i , θ ( i ) f (cid:17) = ¯Ξ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , n Z k ( k ) j,i o , θ ( i ) i , θ ( i ) f (cid:17) − ζ ( l ) Λ l − (cid:16) θ ( i ) f − θ ( i ) i (cid:17) ˆΞ min (cid:18) Z i , l ( i )1 , n Z k (1) j,i o , ..., l ( i ) p i , (cid:26) Z k ( pi ) j,i (cid:27)(cid:19) i =1 ,...,n ! = Y i ˆΞ ( l ) (cid:16) Z i , n Z k (1) j,i o , θ ( i ) i , θ ( i ) f (cid:17) .. ˆΞ( l pi ) (cid:18) Z i , (cid:26) Z k ( pi ) j,i (cid:27) , θ ( i ) i , θ ( i ) f (cid:19) The sum of graphs can thus be decomposed in a sum of two terms:Γ P I (cid:16)(cid:16) Z i , (cid:16) θ ( i ) i , θ ( i ) f (cid:17)(cid:17) n (cid:17) = T ( n )1 + T ( n )2 (224) Having determined the sum of 1PI graphs we can write a series expansion of the effective action for anarbitrary field Ψ. Actually, the 2 n -th order contribution to the generating functional is obtained directly118rom (222): V n (cid:16) Ψ † (cid:16) θ ( i ) f , Z i (cid:17) , Ψ (cid:16) θ ( i ) i , Z i (cid:17)(cid:17) (225)= − n ! n Y i =1 Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) G − Γ (cid:16)(cid:16) Z i , (cid:16) θ ( i ) i , θ ( i ) f (cid:17)(cid:17) n (cid:17) × exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n G − Ψ (cid:16) θ ( i ) i , Z i (cid:17) dθ ( i ) f dθ ( i ) i dZ i where G − is the inverse propagator whose role is to remove external legs of the graphs. In this paragraph,we change the variable G − Ψ (cid:16) θ ( i ) i , Z i (cid:17) → Ψ (cid:16) θ ( i ) i , Z i (cid:17) and Ψ † (cid:16) θ ( i ) f , Z i (cid:17) G − → Ψ † (cid:16) θ ( i ) f , Z i (cid:17) . The impact ofthe factors G − will be computed once an expression for the effective action will be found.Using (224), we have Γ (cid:16)(cid:16) Z i , (cid:16) θ ( i ) i , θ ( i ) f (cid:17)(cid:17) n (cid:17) = P n > (cid:16) T ( n )1 + T ( n )2 (cid:17) , so that: V n (cid:16) Ψ † (cid:16) θ ( i ) f , Z i (cid:17) , Ψ (cid:16) θ ( i ) i , Z i (cid:17)(cid:17) = S + S where: S i = X n > Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) T ( n ) i n ! exp (cid:16) − Λ (cid:16)P nj =1 θ ( i ) f − P nj =1 θ ( i ) i (cid:17)(cid:17) Λ n Ψ (cid:16) θ ( i ) i , Z i (cid:17) dθ ( i ) f dθ ( i ) i (226)The generating functional is obtained by summing the lowest order contribution (222) and the terms (225).It has the form: −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ σ θ ∇ θ (cid:19) Ψ ( θ, Z ) + 12 | Ψ | δ h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) | Ψ ( θ, Z ) | (cid:17) Ψ ( θ, Z ) (cid:17)i δ | Ψ | | Ψ( θ,Z ) | = G (0 ,Z ) (227)+ α Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) + X n > V n (cid:16) Ψ † (cid:16) θ ( i ) f , Z i (cid:17) , Ψ (cid:16) θ ( i ) i , Z i (cid:17)(cid:17) As computed before: − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 δ h Ψ † ( θ ′ , Z ) ∇ θ (cid:16) ω − (cid:16) | Ψ ( θ, Z ) | (cid:17) Ψ ( θ, Z ) (cid:17)i δ | Ψ | | Ψ( θ,Z ) | = G (0 ,Z ) = − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 (cid:2) ∇ θ ω − ( J ( θ ) , θ, Z, G (0 , Z )) (cid:3) + 12 Ψ † ( θ ′ , Z ) δ h ∇ θ (cid:16) ω − (cid:16) | Ψ ( θ, Z ) | (cid:17)(cid:17)i δ | Ψ | Ψ ( θ, Z ) | Ψ( θ,Z ) | = G (0 ,Z ) ≃ − (cid:18) ∇ θ σ θ ∇ θ (cid:19) + 12 (cid:2) ∇ θ ω − ( J ( θ ) , θ, Z, G (0 , Z )) (cid:3) where ω − ( J ( θ ) , θ, Z, G (0 , Z )) is solution of: ω − ( θ, Z ) = G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) W (cid:18) ω ( θ,Z ) ω (cid:16) θ − | Z − Z | c ,Z (cid:17) (cid:19) dZ r π (cid:16) X r (cid:17) + π αω ( Z ) P n > V n (cid:16) Ψ † (cid:16) θ ( i ) f , Z i (cid:17) , Ψ (cid:16) θ ( i ) i , Z i (cid:17)(cid:17) is the sum of 1PI graphs Γ (cid:16)(cid:16) Z i , (cid:16) θ ( i ) i , θ ( i ) f (cid:17)(cid:17) n (cid:17) = P n > (cid:16) T ( n )1 + T ( n )2 (cid:17) (see (224)) multiplied by: × Λ n n ! exp − Λ n X j =1 θ ( i ) f − n X j =1 θ ( i ) i | Ψ | n As a consequence, the effective action as a sum of the classical action and two higher order terms:Γ (Ψ) = −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ σ θ (cid:0) ∇ θ − ω − ( J ( θ ) , θ, Z, G (0 , Z )) (cid:1)(cid:19) Ψ ( θ, Z ) + α Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) + S + S (228)We will see that for ζ Λ of order 1, the term S dominates. For this term, only the contributions for small n are relevant, the others are dampened. For ζ Λ <
1, both terms S and S a priori contribute, but thecontributions involving the derivatives are dominant in S since they are of order ˆΞ ( n − , while those comingfrom S are of order ˆΞ ( n )1 . We show below that S can be neglected with respect to S to find the saddlepoint. S for an arbitrary field Using the previous form (228) of the effective action, the first term S writes : S = X n > n ! Y i Ψ † (cid:16) θ ( i ) f , Z i (cid:17)! exp (cid:16) − Λ (cid:16)P nj =1 θ ( j ) f − P nj =1 θ ( j ) i (cid:17)(cid:17) Λ n (229) × exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) × ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i × Y j = i (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Y i Ψ (cid:16) θ ( i ) i , Z i (cid:17) The coefficients ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) .include product of heaviside functions H (cid:16) θ ( i ) − θ ( j ) − | Z i − Z j | c (cid:17) .We want to approximate the products of terms for j = i . We assume that − ¯ ζ n and ¯Ξ ,n grow approxima-tively at the same rate, so that − ¯ ζ n + ¯Ξ1 ,n ( Zj, { Zm } m = j,θ ( j ) i ,θ ( j ) f ) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ1 − ¯ ζ n + ¯Ξ1 ,n ( Zj, { Zm } m = j,θ ( j ) i ,θ ( j ) f ) θ ( j ) f − θ ( j ) i depends weakly on n and can be replacedby its limit for n → ∞ . We also replace in the product the terms ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) and¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) by their limit ˆΞ , ∞ (cid:16) Z i , θ ( j ) i , θ ( j ) f (cid:17) and ¯Ξ , ∞ (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) . As aconsequence: − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ≃ − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i × exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j ≃ Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i × exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j ≡ Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (230)where: H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) = − ¯ ζ (cid:16) exp (cid:16) ˆΞ , ∞ (cid:16) Z j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) = (cid:18) exp (cid:18)(cid:18) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i (cid:19) (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:19) − (cid:19) ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) The function H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) represents the average dependence of Z j in the whole system. We define:ˆΞ , ∞ (cid:16) Z i , { Z j } j = i (cid:17) = ˆΞ , ∞ (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) θ ( j ) f − θ ( j ) i (231)¯Ξ , ∞ (cid:16) Z i , { Z j } j = i (cid:17) = ¯Ξ , ∞ (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) θ ( j ) f − θ ( j ) i In (231) the quantities without time are averaged over θ ( i ) f − θ ( i ) i . In addition to that, we can also considerthe averages over the { Z j } j = i and define:ˆΞ , ∞ ( Z i ) = * ˆΞ , ∞ (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) θ ( j ) f − θ ( j ) i + { Z j,j = i } (232)¯Ξ , ∞ ( Z i ) = * ¯Ξ , ∞ (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) θ ( j ) f − θ ( j ) i + { Z j,j = i } ( n )1 ( Z i , { Z j,j = i } ) includes product of heaviside functions. Quantities ˆΞ ( n )1 ( Z i , { Z j,j = i } ),ˆΞ ,n ( Z i , { Z j,j = i } ) depend implicitly on θ ( i ) and θ ( j ) . Then, (229) writes: S = X n > X i,j n ! Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) (233) × ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × Y j = i Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j it allows to rewrites (229) by taking into account that there are n possibilities to attribute ˆΞ ( n )1 to a point: S ≃ X n > n − Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) (234) × ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + ∇ out θ ( j ) i Λ H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j n − S for an arbitrary field The contribution of S can be computed similarly to (234) under the same assumptions and approximations.The difference comes from the insertion of vertices at different points in (222). To each factor: exp (cid:16)(cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ (cid:17)(cid:17)(cid:17) p i Y k =1 ˆΞ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , n Z k ( k ) j,i o , θ ( i ) i , θ ( i ) f (cid:17)!!! is associated a contribution: Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) p i Y k =1 ˆΞ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17)! × − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) We first compute the contribution for p i = 1. Assuming p vertices ¯Ξ( l ( i ) ) (cid:16) Z i , n Z k ( k ) j,i o , θ ( i ) i , θ ( i ) f (cid:17) , the lowestpart in perturbation theory is obtained for P l ( i ) = n . by sharing the n points among p vertices. Once the p vertices are given, the connected contributions are obtained first by distributing the remaining n − p points.We attach l (1) − l (1)(1) of these points to Z ,... l ( p ) − l ( p )(1) to Z p . We have P (cid:16) l ( i ) − l ( i )(1) (cid:17) = n − p . There is: C l (1) − l (1)(1) n − p C l (2) − l (2)(1) n − p − (cid:16) l (1) − l (1)(1) (cid:17) ...C l ( p ) − l ( p )(1) n − p − (cid:16)(cid:16) l (1) − l (1)(1) (cid:17) + ...l ( p − − l ( p − (cid:17) = ( n − p )! (cid:16) l (1) − l (1)(1) (cid:17) ! ... (cid:16) l ( p ) − l ( p )(1) (cid:17) !122ossibilities. Then, we have p blocks each having vertices with valence l ( i )(1) and P l ( i )(1) = p . Each blockcontains l ( i ) − l ( i )(1) + 1 lines. To compute 1PI graphs, one has to consider l ( i )(1)
2. The number of verticesis P pi =1 (cid:18) − δ ,l ( i )(1) (cid:19) = p . Among these vertices p − p have valence 2 and p − p have valence 0. The last2 p − p have valence 1. As a consequence, p > p . The blocks have to be linked to form a loop of length p including all blocks. The blocks of valence 1are grouped in k blocks of valence 1 for a factor X r i C r p − p ...C r k p − p − r − ... − r k − r ! ...r k ! = (2 p − p )! X r i
1= (2 p − p )! P (2 p − p, k ) ≃ (2 p − p )! (2 p − p − k ) k − ( k − k blocks of valence 0. This yields a factor C kp − p and a factor:(2 p − p )! X k C kp − p (2 p − p − k ) k − ( k − p − p blocks of valence 0 and p − p blocks of valence 2. There are ( p − p − p − p )! possibleloops. Each block of valence 0 can be attached in l ( i ) +1 ways, each block of valence 1 in l ( i ) ways. The blocksof valence 0 are attached twice. This yields a factor: p Q i =1 (cid:18) δ ,l ( i )(1) (cid:16) l ( i ) − l ( i )(1) − (cid:17) + δ ,l ( i )(1) (cid:16) l ( i ) − l ( i )(1) (cid:17)(cid:19) δ ,l ( i )(1) .The factor is:( p − p − p − p )! (2 p − p )! X k C kp − p (2 p − p − k ) k − ( k − p Y i =1 (cid:18) δ ,l ( i )(1) (cid:16) l ( i ) − l ( i )(1) − (cid:17) + δ ,l ( i )(1) (cid:16) l ( i ) − l ( i )(1) (cid:17)(cid:19) δ ,l ( i )(1) N (cid:18) n, p, (cid:16) l ( i ) (cid:17) i =1 ,...,p (cid:19) ≃ X l ( i )(1) ( ,l (1) ) ( n − p )! (cid:16) l (1) − l (1)(1) (cid:17) ! ... (cid:16) l ( p ) − l ( p )(1) (cid:17) ! X p δ p − p X i =1 (cid:18) − δ ,l ( i )(1) (cid:19)! (235) × ( p − p − p − p )! (2 p − p )! X k C kp − p (2 p − p − k ) k − ( k − × p Y i =1 (cid:18) δ ,l ( i )(1) (cid:16) l ( i ) − l ( i )(1) − (cid:17) + δ ,l ( i )(1) (cid:16) l ( i ) − l ( i )(1) (cid:17)(cid:19) δ ,l ( i )(1) it can be approximated by the average for l (1)(1) = 1 and a number of terms such that p − p vertices havevalence 2, p − p have valence 0 and the last 2 p − p have valence 1., that is p !( p − p )!( p − p )!(2 p − p )! terms. Wecan also approximate: p Q i =1 (cid:18) δ ,l ( i )(1) (cid:16) l ( i ) − l ( i )(1) − (cid:17) + δ ,l ( i )(1) (cid:16) l ( i ) − l ( i )(1) (cid:17)(cid:19)(cid:16) l (1) − l (1)(1) (cid:17) ! ... (cid:16) l ( p ) − l ( p )(1) (cid:17) ! ≃ Q l ( i )(1) =2 (cid:0) l ( i ) − (cid:1) Q l ( i )(1) =1 l ( i ) Q l ( i )(1) =0 ( l ( i ) +1 ) l ( i ) (cid:0) l (1) − (cid:1) ! ... (cid:0) l ( p ) − (cid:1) ! ≃ Q l ( i ) (cid:0) l (1) − (cid:1) ! ... (cid:0) l ( p ) − (cid:1) ! N (cid:18) n, p, (cid:16) l ( i ) (cid:17) i =1 ,...,p (cid:19) ≃ ( n − p )! Q l ( i ) (cid:0) l (1) − (cid:1) ! ... (cid:0) l ( p ) − (cid:1) ! X p p !( p − p − X k C kp − p (2 p − p − k ) k − ( k − k can be replaced by its maximal value. Numerically, this corresponds to k ≃ p − p +12 so that: X k C kp − p (2 p − p − k ) k − ( k − ≃ ( p − p )!( p − p − (cid:0) p − p − (cid:1) ! (cid:0) p − p +12 (cid:1) ! (cid:0) p − p − (cid:1) p − p − (cid:0) p − p − (cid:1) !Similarly, the sum over p > p can be replaced by its maximal value, for p − p ≃ p and: X p ( p − p )!( p − p − (cid:0) p − p − (cid:1) ! (cid:0) p − p +12 (cid:1) ! (cid:16) p − p − p ) − (cid:17) p − p − (cid:0) p − p − (cid:1) ! ≃ Γ (cid:0) p + 1 (cid:1)(cid:0) Γ (cid:0) p + 1 (cid:1)(cid:1) (cid:18) p (cid:19) p ≃ exp (cid:16) p (cid:17) ≃ exp (cid:16) p (cid:17) and N (cid:16) n, p, (cid:0) l ( i ) (cid:1) i =1 ,...,p (cid:17) can be written: N (cid:18) n, p, (cid:16) l ( i ) (cid:17) i =1 ,...,p (cid:19) ≃ ( n − p )! p ! exp (cid:0) p (cid:1)(cid:0) l (1) − (cid:1) ! ... (cid:0) l ( p ) − (cid:1) !The contribution for p i > P i,k i l ( i ) k i = n , is obtained by a similar computation. The only difference isthat the blocks l ( i ) k (1) are attached and to produce 1PI graphs, one has to assume that P k l ( i ) k (1) = 0 , , p − ( P ( p i − p − p − p − p )!2 p − p (2 p − p )! X k C kp − p (2 p − p − k ) k k !by the following expression:( p − p ′ − p − p ′ )!2 p − p ′ (2 p ′ − p )! X k C kp − p ′ (2 p ′ − p − k ) k k !where p ′ = p − ( P ( p i − P ( p i − >>
1. We can thus discardthe contributions with p i > S (0)2 ≃ X n X { Z ki } n ! Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) × ( n − p )! p ! exp (cid:16) p (cid:17) X (0) n ( { Z k i } ) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j n − p where: X (0) n ( { Z k i } ) = X l ( i ) P pi =1 l ( i ) = n Y l ( i ) (cid:28) ¯Ξ( l ( i ) ) (cid:0) Z i , θ ( i ) , (cid:8) Z k j (cid:9)(cid:1)(cid:29)(cid:0) l ( i ) − (cid:1) ! C l ( i ) n − ¯Ξ( l ( i ) ) (cid:0) Z i , θ ( i ) , (cid:8) Z k j (cid:9)(cid:1)(cid:29) C l ( i ) n − = ˆΞ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) The contributions for P i,k i l ( i ) k i = n + m add a factor Q ǫ ( i ) ki C l ( i ) ki n − where P ǫ ( i ) k i = m , and, as before, thecontributions for p i > l ( i ) k i , n are selected to produce the factor (235) Thecontributions becomes: X ǫ + ... + ǫ p = m C ǫ i n − ( n − p )! p ! exp (cid:0) p (cid:1)(cid:0) l (1) − ǫ − (cid:1) ! ... (cid:0) l ( p ) − ǫ p − (cid:1) !Replacing m = P i l ( i ) − n , and estimating the sum by its maximal value for the ǫ i all equal to P i l ( i ) − np , weget: C P i l ( i ) − np n − ( n − p )! p ! exp (cid:0) p (cid:1)(cid:16) l (1) − P i l ( i ) − np − (cid:17) ! ... (cid:16) l ( p ) − P i l ( i ) − np − (cid:17) !The contribution becomes: S ( m )2 ≃ X n X l ( i ) , { Z ki } P pi =1 l ( i ) = n n ! Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) × ( n − p )! p ! exp (cid:16) p (cid:17) X ( m ) n ( { Z k i } ) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j n − p with: X ( m ) n ( { Z k i } ) = X l ( i ) P pi =1 l ( i ) = n + m Y l ( i ) C P i l ( i ) − np n − (cid:28) ¯Ξ( l ( i ) ) (cid:0) Z i , θ ( i ) , (cid:8) Z k j (cid:9)(cid:1)(cid:29)(cid:16) l ( i ) − P i l ( i ) − np − (cid:17) ! C l ( i ) n − = X l ( i ) P pi =1 l ( i ) = n + m Y l ( i ) C mp n − (cid:28) ¯Ξ( l ( i ) ) (cid:0) Z i , θ ( i ) , (cid:8) Z k j (cid:9)(cid:1)(cid:29)(cid:16) l ( i ) − mp − (cid:17) ! C l ( i ) n − The contributions of S ( m )2 to the 1PI graphs are obtained by removing double counting. Defining: Y ( m ) n ( { Z k i } ) = X ( m ) n ( { Z k i } ) − m − X k =0 Y ( k ) n ( { Z k i } ) ! X ( m ) n ( { Z k i } ) by Y ( m ) n ( { Z k i } ) in S ( m )2 . Rewriting: Y ( m ) n ( { Z k i } ) = X ( m ) n ( { Z k i } ) − m − X k =0 Y ( k ) n ( { Z k i } ) ! = X ( m ) n ( { Z k i } ) − m − X k =0 X ( k ) n ( { Z k i } ) + m − X k =0 k =0 , X ( k ) n ( { Z k i } ) X ( k ) n ( { Z k i } ) − ... = X ( m ) n ( { Z k i } ) m − Y k =0 (cid:16) − X ( k ) n ( { Z k i } ) (cid:17) The quantity X ( m ) n ( { Z k i } ) can be evaluated at the extremum l ( i ) = n + mp this yields that X ( m ) n ( { Z k i } ) is oforder: (cid:16) n − − mp (cid:17) ! (cid:16) mp (cid:17) ! (cid:16) n − − n + mp (cid:17) ! (cid:16) n + mp (cid:17) ! p (cid:28) ¯Ξ( l ( i ) ) (cid:16) Z i , θ ( i ) , (cid:8) Z k j (cid:9)(cid:17)(cid:29) The factor (cid:18) ( n − − mp ) ! ( mp ) ! ( n − − n + mp ) ! ( n + mp ) ! (cid:19) is maximal for p → m = n − n − ) ! ) ( n − ) ! < n > (cid:12)(cid:12)(cid:12) X ( m ) n ( { Z k i } ) (cid:12)(cid:12)(cid:12) << X ( m ) n ( { Z k i } ) m − Y k =0 (cid:16) − X ( k ) n ( { Z k i } ) (cid:17) ≃ X ( m ) n ( { Z k i } ) exp − m − X k =1 X ( k ) n ( { Z k i } ) ! The factors dampen quickly for m increasing, so that in first approximation, we can keep the contributionfor m = 0 Since C pn ( n − p )! n ! = p ! , we have the contribution: S ≃ X n X l ( i ) , { Z ki } P pi =1 l ( i ) = n n ! Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) × Y l ( i ) ˆΞ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) × ( n − p )! p ! exp (cid:0) p (cid:1)Q l ( i ) (cid:0) l ( i ) − (cid:1) ! × − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j n − p and using that:ˆΞ (cid:16) l ( i ) k (cid:17) (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) = − ζ (cid:16) l ( i ) k (cid:17) +1 (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + Z θ ( i ) f θ ( i ) i δ l ( i ) k +1 Ψ † (cid:0) θ ( i ) , Z i (cid:1) ∇ θ ( i ) i ω − (cid:0) J, θ ( i ) , Z i , G ( Z i ) (cid:1) Ψ (cid:0) θ ( i ) , Z i (cid:1) l ( i ) k +1 Q i =1 δ (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) − | Z i − Z j | c , Z j (cid:17)(cid:12)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) dZ i dθ ( i ) S ≃ X p exp (cid:0) p (cid:1) p ! Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) X l ( i ) exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17)(cid:0) l ( i ) − (cid:1) ! × ζ ( l ( i ) ) +1 + ∇ θ ( i ) i Z θ ( i ) f θ ( i ) i δ l ( i ) +1 Ψ † (cid:0) θ ( i ) , Z i (cid:1) ω − (cid:0) J, θ ( i ) , Z i , G ( Z i ) (cid:1) Ψ (cid:0) θ ( i ) , Z i (cid:1) l ( i ) k +1 Q i =1 δ (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) − | Z i − Z j | c , Z j (cid:17)(cid:12)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) dZ i dθ ( i ) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j l ( i ) − p = exp (cid:18) √ e Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) × Z θ ( i ) f θ ( i ) i δ Λ δ (cid:12)(cid:12) ¯Ψ ( θ, Z ) (cid:12)(cid:12) h U (cid:16)(cid:12)(cid:12) ¯Ψ ( θ, Z ) (cid:12)(cid:12) (cid:17) + ∇ θ ( i ) i ω − (cid:16) J, θ ( i ) , Z i , (cid:12)(cid:12) ¯Ψ ( θ, Z ) (cid:12)(cid:12) (cid:17)i dZ i dθ ( i ) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) where: (cid:12)(cid:12) ¯Ψ ( θ, Z ) (cid:12)(cid:12) = G (0 , Z )+ Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j As a consequence, at the lowest order in perturbation theory, or Λ >>
1, the contribution of S can beneglected compared to S to compute the saddle point field. Having shown that the contribution S can be neglected with respect to S , we can sum the expressions(234) for n > G − which leads thus to the following expression for (229):Γ (Ψ) ≃ −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G (0 , Z )) (cid:19)(cid:19) Ψ ( θ, Z ) + α Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) (236)+ X n > n − n Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) G − (cid:16) ¯ V ( n )1 + ¯ V ,n (cid:17) G − Ψ (cid:16) θ ( i ) i , Z i (cid:17) × Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) G − H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + ∇ out θ ( j ) i Λ H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) G − Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j n − V ( n )1 = − ζ ( n ) Λ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) ¯ V ,n = − ζ ( n ) Λ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) × exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − − ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) +ˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) The effective action is the sum of two terms according to the decomposition of the vertex into ¯ V ( n )1 + ¯ V ,n .We estimate the two contributions independently. Estimation of the term proportional to ¯ V ( n )1 Recall from (150) and equations below that the expres-sion: ˆΞ ( n )1 (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) stands for: Z θ ( i ) i <θ ( i ) <θ ( i ) f ˆΞ ( n )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) = Λ Z θ ( i ) i <θ ( i ) <θ ( i ) f exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) (cid:17)(cid:17) Λ ˆΞ ( n )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) − θ ( i ) i (cid:17)(cid:17) Λ= Λ G ∗ ˆΞ ( n )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) ∗ G (237)where: ˆΞ ( n )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17) ≃ n − Q k j =1 R θ ( i ) − θ ( kj ) i (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c dl k j p π r(cid:16) σ ¯ X r (cid:17) + ασ ! n − ζ ( n ) + ∇ θ ( i ) G ( Z ) δ n − R ω − (cid:0) J, θ ( i ) , Z i (cid:1) dZ i G ( Z ) n − Q i =1 δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) − l k j , Z k j (cid:1)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) The convolution of ¯ V ( n )1 with G − on both side is obtained in (154) and (161) and (162) with inertiacoefficients included. The convolution cancels the propagator exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) Λ , removes the integral and128ntroduces a δ function for the contribution proportional to ¯ V ( n )1 : G − ¯ V ( n )1 G − Z θ ( i ) i <θ ( i ) <θ ( i ) f G − − ζ ( n ) Λ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) G − = Λ − ζ ( n ) Λ n + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) f , { Z j } j = i (cid:17) δ (cid:16) θ ( i ) f − θ ( i ) i (cid:17) (238)The presence of the delta function also localizes the terms in Ψ (cid:16) θ ( j ) i , Z j (cid:17) that interact with Ψ (cid:16) θ ( i ) i , Z i (cid:17) inthe factor: Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) G − H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + ∇ out θ ( j ) i Λ H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) G − Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j Actually, we have: H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + ∇ out θ ( j ) i Λ H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) = (cid:16) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ + ¯Ξ , ∞ (cid:16) Z j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) and an expansion similar to (150) yields: H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + ∇ out θ ( j ) i Λ H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) = X n n Y i =1 Z G (cid:16) θ ( i ) f , θ ( i ) f ′ , Z i (cid:17) Λ δ (cid:16) θ ( i ) f ′ − θ ( i ) i ′ (cid:17) + Λ − ¯ ζ + ∇ out θ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) δ (cid:16) θ ( i ) f ′ − θ ( i ) i ′ (cid:17) + − ¯ ζ + ∇ out θ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) × (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) f ′ (cid:17)(cid:17) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17)(cid:17) G (cid:16) θ ( i ) i ′ , θ ( i ) i , Z i (cid:17) dθ ( i ) i ′ dθ ( i ) f ′ o which implies that: G − H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + ∇ out θ ( j ) i Λ H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) G − (239)= Λ δ (cid:16) θ ( i ) f ′ − θ ( i ) i ′ (cid:17) + Λ − ¯ ζ + ∇ out θ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) δ (cid:16) θ ( i ) f ′ − θ ( i ) i ′ (cid:17) +Λ − ¯ ζ + ∇ out θ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) f ′ (cid:17)(cid:17) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) G − ¯ V ( n )1 G − in (238) induces in first approximationa smeared distribution of Dirac function R θ ( i ) − θ ( j ) i | Zi − Zj | c dl j δ (cid:16) θ ( i ) i − l j − | Z i − Z j | c , Z j (cid:17) in the two first terms of theexpansion of (239). It leads to replace (239) by the contribution:Λ n − Z θ ( i ) − θ ( j ) i | Zi − Zj | c dl j Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) − ¯ ζ + ∇ out θ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) (240) × δ (cid:18) θ ( i ) i − l j − | Z i − Z j | c , Z j (cid:19) δ (cid:16) θ ( j ) f − θ ( j ) i (cid:17) + − ¯ ζ + ∇ out θ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) × (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) f ′ (cid:17)(cid:17) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17)(cid:17) Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ n − j At the lowest order in perturbation, (240) is approximatively equal to:Λ n − Z θ ( i ) − θ ( j ) i | Zi − Zj | c dl j Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − l j − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n − and the product with (238) yields the part of the effective action involving ¯ V ( n )1 :Ψ † (cid:16) θ ( i ) i , Z i (cid:17) G − exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) − ζ ( n ) Λ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) ×G − Ψ (cid:16) θ ( i ) i , Z i (cid:17) × Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) G − H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + ∇ out θ ( j ) i Λ H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) G − Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j n − = n − Q k j =1 R θ ( i ) − θ ( kj ) i (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c dl k j Λ n − − ζ ( n ) + δ n − R Ψ † (cid:0) θ ( i ) , Z i (cid:1) ∇ θ ( i ) ω − (cid:0) J, θ ( i ) , Z i (cid:1) Ψ (cid:0) θ ( i ) , Z i (cid:1) dZ in − Q i =1 δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) − l k j , Z k j (cid:1)(cid:12)(cid:12) δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) , Z i (cid:1)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) × (cid:18)Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) (cid:18) G − δ (cid:18) θ ( i ) i − l j − | Z i − Z j | c , Z j (cid:19) δ (cid:16) θ ( j ) f − θ ( j ) i (cid:17) + G − (cid:16) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) − (cid:17) + ∇ out θ ( j ) i Λ H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) G − Ψ (cid:16) θ ( i ) i , Z i (cid:17) dZ j n − ≃ n − Y k j =1 Z θ ( i ) q − θ ( kj ) i (cid:12)(cid:12)(cid:12)(cid:12) Zi − Zkj (cid:12)(cid:12)(cid:12)(cid:12) c dl k j − ζ ( n ) + δ n R Ψ † (cid:0) θ ( i ) , Z i (cid:1) ∇ θ ( i ) ω − (cid:16) J, θ ( i ) q , Z i (cid:17) Ψ (cid:0) θ ( i ) , Z i (cid:1) dZ in Q i =1 δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) − l k j , Z k j (cid:1)(cid:12)(cid:12) δ (cid:12)(cid:12) Ψ (cid:0) θ ( i ) , Z i (cid:1)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) × Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − l j − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n − Estimation of the term proportional to ¯ V ,n For the second contribution proportional to ¯ V ,n , weuse the fact that the series expansion of this term is of second order in interaction. As for the contribution130roportional to ¯ V ( n )1 , the convolution with G − on the right replaces the terms: − ζ ( n ) Λ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) (241)and: − ¯ ζ n (cid:16) θ ( i ) f − θ ( i ) i (cid:17) + ∇ out θ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) (242)by their derivatives evaluated at θ ( i ) f . These derivative are equal to their average in first approximation, sothat (241) and (242) can be replaced by: − ζ ( n ) Λ n + ∇ out θ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17)(cid:16) θ ( i ) f − θ ( i ) i (cid:17) and: − ¯ ζ n + ∇ out θ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:16) θ ( i ) f − θ ( i ) i (cid:17) Moreover, the left convolution by G − replaces one term in the series expansion of:exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − − ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) and the series expansion of: exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) by its derivative at θ ( i ) i or approximatively by its average. This amounts to replace those terms by theirderivative multiplied by the averaged term, that is:ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) − (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17)(cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) and: ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i × exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) (cid:18) − ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) + (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) (cid:19) − (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) As consequence, gathering the different terms, ¯ V ,n is replaced by:ˆ V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) = 1 + (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) − (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17)(cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) − ζ ( n ) Λ n + ∇ outθ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } ,θ ( i ) i ,θ ( i ) f (cid:17)(cid:16) θ ( i ) f − θ ( i ) i (cid:17) ! ˆΞ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i + exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) (cid:18) − ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) + (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) (cid:19) − (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i The corresponding factors multiplying ¯ V ,n : Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) G − (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i × exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) G − Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (cid:17) n − are localized, depending on the number of graphs issued from i reach j . For 1 link only, issued from¯Ξ ( n )1 (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) , the factor 1 is replaced by δ (cid:16) θ ( j ) f − θ ( j ) i (cid:17) . For 2 links ore more it remainsunchanged. The factor: (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i is replaced, as in the previous paragraph, by: (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i × ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i = (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ out θ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) G − (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ outθ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j ,θ ( j ) i ,θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i × exp (cid:16) − Λ (cid:16) θ ( j ) f − θ ( j ) i (cid:17)(cid:17) G − Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (cid:17) n − ≃ (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ out θ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − Effective action
Gathering the several contributions in the two previous paragraphs yields Γ (Ψ):Γ (Ψ)= −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G (0 , Z )) (cid:19)(cid:19) Ψ ( θ, Z ) + α Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) − X n − Z Ψ † ( θ, Z ) (cid:16) − ζ ( n ) + ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) , { Z j } j = i (cid:17)(cid:17) × Z θ ( i ) q − θ ( j ) i | Zi − Zj | c dl j Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n − Ψ ( θ, Z )+ X n − Z Ψ † (cid:16) θ ( i ) f , Z i (cid:17) ˆ V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ out θ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − or resumming the series expansion:Γ (Ψ) (243)= −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − (cid:18) J ( θ ) , θ, Z, G (0 , Z ) + Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( j ) , Z j (cid:17)(cid:12)(cid:12)(cid:12) (cid:19)(cid:19)(cid:19) Ψ ( θ, Z )+ α Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) + X ζ ( n ) n ! Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) G (0 , Z ) + Z θ ( i ) − θ ( j ) i | Zi − Zj | c dl j Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n + X Z n − † (cid:16) θ ( i ) f , Z i (cid:17) ˆ V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ out θ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − ω − (cid:0) J, θ ( j ) , G ( Z ) (cid:1) is solution of: ω − (cid:16) θ ( i ) , Z (cid:17) = G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ¯ G (0 , Z i ) dZ and with ˆ V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) defined by:ˆ V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) = 1 + (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) − (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17)(cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) × − ζ ( n ) Λ n + ∇ outθ ( i ) i Λ ¯Ξ ( n )1 (cid:16) Z i , { Z j,j = i } ,θ ( i ) i ,θ ( i ) f (cid:17)(cid:16) θ ( i ) f − θ ( i ) i (cid:17) ! ˆΞ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i + exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) (cid:18) − ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) + (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) (cid:19) − (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) × − ¯ ζ n + ∇ outθ ( i ) i Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Inclusion of backreaction terms
Once the effective action has be found, we can include the neglectedpart of the vertex (145) :12 n ! Z ∇ θ G δ n ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | | Ψ( θ,Z ) | = G (0 ,Z ) n Y i =1 | Ψ ( θ − l i , Z i ) | n Y i =1 dZ i dZdθdl i There is also a contribution to the potential givenThe computation of the corresponding graphs is identical to that of the previous paragraphs. Thedominant contributions of these vertices to the effective action modify S in (233) and (234) by a term: δS = X n > n ! Z G ( Z ) dZ × Z n Y j =1 ∇ θ G δ n ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | | Ψ( θ,Z ) | = G (0 ,Z ) (244) × Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j Actually, compared to (233), the term Ψ † (cid:16) θ ( i ) f , Z i (cid:17) Ψ (cid:16) θ ( i ) i , Z i (cid:17) and the factor n corresponding to the n possibilities to choose the point i , are replaced by R G ( Z ) dZ . These terms correspond to the back reaction134f the n points, including i , on the whole system. The action of the n − j on i in (233), i.e. the termˆΞ ( n )1 (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i ∇ outθ ( i ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i × exp (cid:16)(cid:16) − Λ + ˆΞ ,n (cid:16) Z i , { Z j } j = i (cid:17)(cid:17) (cid:16) θ ( i ) f − θ ( i ) i (cid:17)(cid:17) is thus replaced by: ∇ θ G δ n ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | − Z G δ n V (cid:16)R | Ψ | (cid:17) n Q j =1 δ | Ψ ( θ − l i , Z i ) | | Ψ( θ,Z ) | = G (0 ,Z ) The second term being the contribution of the potential given in (157). The contribution to the effectiveaction is thus: δS = X n > n ! Z G ( Z ) dZ × Z ∇ θ G δ n ω − ( J, θ, Z ) n Q i =1 δ | Ψ ( θ − l i , Z i ) | − Z G δ n V (cid:16)R | Ψ | (cid:17) n Q j =1 δ | Ψ ( θ − l i , Z i ) | | Ψ( θ,Z ) | = G (0 ,Z ) (245) × Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) + H (cid:16) Z j , ¯ Z, θ ( j ) f , θ ( j ) i (cid:17) ∇ out θ ( j ) i Λ Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n The contribution of (245) to the effective action (243) is equivalent to shift | Ψ ( θ, Z ) | and (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) i , Z i (cid:17)(cid:12)(cid:12)(cid:12) by G (0 , Z ).We will see below that this shift does not modify fundamentally the form of the vacuum. As a consequenceit is convenient to work with (243) in the sequel. Γ (Ψ)= −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17)(cid:19)(cid:19) Ψ ( θ, Z )+ α Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) + X ζ ( n ) n ! G (0 , Z ) + Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n + X Z n − † (cid:16) θ ( i ) f , Z i (cid:17) ˆ V ,n (cid:16) Z i , { Z j,j = i } , θ ( i ) i , θ ( i ) f (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ out θ ( j ) i Λ − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − ζ n +1 > n > ζ ( n +1) > ζ (2) <
0, the potential: α Z (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ ( i ) , Z i (cid:17)(cid:12)(cid:12)(cid:12) dZ i + X ζ ( n ) n ! G (0 , Z ) + Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n has a minimum for α << (cid:12)(cid:12) ζ (2) (cid:12)(cid:12) large. This minimum is reached for a value X of R (cid:12)(cid:12) Ψ (cid:0) θ ( i ) , Z i (cid:1)(cid:12)(cid:12) dZ i .Up to an irrelevant phase, Ψ (cid:0) θ ( i ) , Z i (cid:1) = Ψ † ( θ, Z ) = q X V where V is the volume of the thread.Moreover the operator O = ∇ θ σ θ (cid:0) ∇ θ − ω − ( J ( θ ) , θ, Z, G ) (cid:1) has positive eigenvalues. DevelopingΨ ( θ, Z ) = P a n Ψ n ( θ, Z ) where Ψ n ( θ, Z ) are the eigenstates of O , the definition of Ψ † ( θ, Z ) (see [45]and [46]) is given by: X ¯ a n Ψ † n ( θ, Z )where Ψ † n ( θ, Z ) are the eigenstates of the adjoint operator of O . As a consequence Z −
12 Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G ) (cid:19)(cid:19) Ψ ( θ, Z )is positive, and null for constant Ψ † ( θ, Z ) and Ψ ( θ, Z ). As a consequence, for (cid:12)(cid:12) ζ ( n ) (cid:12)(cid:12) > ω − ( J ( θ ) , θ, Z, G )the minimum of Γ (Ψ) is reached for Ψ ( θ, Z ) = Ψ ( θ, Z ) + δ Ψ ( θ, Z ) and Ψ † ( θ, Z ) = Ψ † ( θ, Z ) + δ Ψ † ( θ, Z )where | δ Ψ ( θ, Z ) | << | Ψ ( θ, Z ) | and (cid:12)(cid:12) δ Ψ † ( θ, Z ) (cid:12)(cid:12) << (cid:12)(cid:12)(cid:12) Ψ † ( θ, Z ) (cid:12)(cid:12)(cid:12) .Expanding the potential around Ψ ( θ, Z ) yields at the first order:Γ (Ψ) = − δ Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − (cid:16) J ( θ ) , θ, Z, G + X + p X (cid:0) δ (cid:0) Ψ † + δ Ψ (cid:1)(cid:1)(cid:17)(cid:19)(cid:19) Ψ ( θ, Z ) − δ Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17)(cid:19)(cid:19) δ Ψ ( θ, Z )+ 12 δ Ψ † ( θ, Z ) U ′′ ( X ) δ Ψ ( θ, Z ) ≃ − δ Ψ † ( θ, Z ) (cid:16) ∇ θ ω − (cid:16) J ( θ ) , θ, Z, G + X + p X (cid:0) δ (cid:0) Ψ † + δ Ψ (cid:1)(cid:1)(cid:17)(cid:17) Ψ ( θ, Z ) − δ Ψ † ( θ, Z ) (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19)(cid:19) δ Ψ ( θ, Z )+ 12 δ Ψ † ( θ, Z ) U ′′ ( X ) δ Ψ ( θ, Z )and this leads to the first order condition for δ Ψ ( θ, Z ):0 = 12 (cid:18) − (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19)(cid:19) + U ′′ ( X ) (cid:19) δ Ψ ( θ, Z ) − (cid:16) ∇ θ ω − (cid:16) J ( θ ) , θ, Z, G + X + p X (cid:0) δ (cid:0) Ψ † + δ Ψ (cid:1)(cid:1)(cid:17)(cid:17) Ψ ( θ, Z ) − δ Ψ † ( θ , Z ) p X ∇ θ δω − ( J ( θ ) , θ , Z , G + X ) δ | Ψ ( θ, Z ) | ! Ψ ( θ , Z ) ≃ (cid:18) − (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19)(cid:19) + U ′′ ( X ) (cid:19) δ Ψ ( θ, Z ) − Z δ Ψ † ( θ , Z ) p X ∇ θ δω − ( J ( θ ) , θ , Z , G + X ) δ | Ψ ( θ, Z ) | ! Ψ ( θ , Z ) dθ dZ − (cid:0) ∇ θ ω − ( J ( θ ) , θ, Z, G + X ) (cid:1) Ψ ( θ, Z ) − Z p X ∇ θ δω − ( J ( θ ) , θ, Z, G + X ) δ | Ψ ( θ , Z ) | δ Ψ ( θ , Z ) ! Ψ ( θ, Z ) dθ dZ δ Ψ † ( θ , Z ):0 = 12 δ Ψ † ( θ, Z ) (cid:18) −∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19) + U ′′ ( X ) (cid:19) − Z δ Ψ † ( θ , Z ) p X ∇ θ δω − ( J ( θ ) , θ , Z , G + X ) δ | Ψ ( θ, Z ) | ! Ψ ( θ , Z ) dθ dZ The solution for δ Ψ † ( θ, Z ) is: δ Ψ † ( θ, Z ) = 0 (246)This translates that there is no backward propagation of the signals. As a consequence, the equation for δ Ψ ( θ, Z ) rewrites:0 ≃ (cid:18) − (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19)(cid:19) + U ′′ ( X ) (cid:19) δ Ψ ( θ, Z ) (247) − (cid:0) ∇ θ ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) (cid:1) Ψ ( θ, Z ) − Z p X ∇ θ δω − ( J ( θ ) , θ, Z, G + X ) δ | Ψ ( θ , Z ) | δ Ψ ( θ , Z ) ! Ψ ( θ, Z ) dθ dZ Solving (247) amounts to find the Green function of the operator:¯ G − ( Z, θ, Z , θ ) = 12 (cid:18) − (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19)(cid:19) + U ′′ ( X ) (cid:19) δ ( θ − θ ) (248) − Z p X ∇ θ δω − ( J ( θ ) , θ, Z, G + X ) δ | Ψ ( θ , Z ) | Ψ ( θ, Z ) ≡ ˜ G − Z ( θ, θ ) δ ( Z − Z ) − Z p X ∇ θ δω − ( J ( θ ) , θ, Z, G + X ) δ | Ψ ( θ , Z ) | Ψ ( θ, Z )and the vacuum is given by:12 Z ¯ G − ( Z, θ, Z , θ ) (cid:0) ∇ θ ω − ( J ( θ ) , θ , Z , G + X ) (cid:1) Ψ ( θ , Z )The computation of ¯ G − ( Z, θ, Z , θ ) is done in two steps: ˜ G − Z ( θ, θ ) We first find the Green function of the operator:˜ G − Z ( θ, θ ) = 12 (cid:18) − (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G + X ) (cid:19)(cid:19) + U ′′ ( X ) (cid:19) δ ( θ − θ ) (249)To do so, we note that ˜ G − Z ( θ, θ ) has the form − ddθ (cid:0) a ddθ − b ( θ ) (cid:1) where a = σ θ and b ( θ ) = ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ).For any function d ( θ ), the following change of basis holds: − exp ( − d ( θ )) ddθ (cid:18) a ddθ − b ( θ ) (cid:19) exp ( d ( θ ))= − exp ( − d ( θ )) ddθ (cid:18) exp ( d ( θ )) (cid:18) a ddθ + a d ′ ( θ ) − b ( θ ) (cid:19)(cid:19) = − (cid:18)(cid:18) ddθ + d ′ ( θ ) (cid:19) (cid:18)(cid:18) a ddθ + a d ′ ( θ ) − b ( θ ) (cid:19)(cid:19)(cid:19) = − (cid:18)(cid:18) ddθ (cid:18)(cid:18) a ddθ + ad ′ ( θ ) − b ( θ ) (cid:19)(cid:19) − a d ′′ ( θ ) + (cid:16) a d ′ ( θ ) − b ( θ ) (cid:17) d ′ ( θ ) (cid:19)(cid:19) b as the average of b ( θ ) over some time span, and setting: d ′ ( θ ) = b ( θ ) − ¯ ba we have: exp ( − d ( θ )) (cid:18) − ddθ (cid:18) a ddθ − b ( θ ) (cid:19)(cid:19) exp ( d ( θ ))= − ddθ (cid:18)(cid:18) a ddθ − ¯ b (cid:19)(cid:19) − b ′ ( θ )2 − (cid:0) b ( θ ) + ¯ b (cid:1) (cid:0) b ( θ ) − ¯ b (cid:1) a As a consequence, the change of basis allows to decompose the initial operator under the form: − ddθ (cid:18)(cid:18) a ddθ − ¯ b (cid:19)(cid:19) for some constant ¯ b plus an additional term. This allows to find the inverse of the initial operator by a seriesexpansion: (cid:18) − ddθ (cid:18) a ddθ − b ( θ ) (cid:19)(cid:19) − ( θ ′ , θ )= exp ( − d ( θ ′ )) − ddθ (cid:18)(cid:18) a ddθ − ¯ b (cid:19)(cid:19) − b ′ ( θ )2 − (cid:0) b ( θ ) + ¯ b (cid:1) (cid:0) b ( θ ) − ¯ b (cid:1) a ! − ( θ ′ , θ ) exp ( d ( θ ))= exp ( − d ( θ ′ )) ∞ X n =0 G ¯ b b ′ ( θ )2 + (cid:0) b ( θ ) + ¯ b (cid:1) (cid:0) b ( θ ) − ¯ b (cid:1) a ! G ¯ b ! n ! ( θ ′ , θ ) exp ( d ( θ ))where G ¯ b is the Green function of − ddθ (cid:0)(cid:0) a ddθ − ¯ b (cid:1)(cid:1) and the products are products of convolution. For (cid:0) b ( θ ) − ¯ b (cid:1) having periods shorter than a the integrals of ( b ( θ )+¯ b )( b ( θ ) − ¯ b ) a cancel. As a consequence: (cid:18) − ddθ (cid:18) a ddθ − b ( θ ) (cid:19)(cid:19) − ( θ ′ , θ ) ≃ exp ( − d ( θ ′ )) ∞ X n =0 (cid:18) G ¯ b (cid:18)(cid:18) b ′ ( θ )2 (cid:19) G ¯ b (cid:19) n (cid:19) ( θ ′ , θ ) exp ( d ( θ )) (250)We apply this result to the operator arising in our problem. We define ¯ b = ω − (cid:0) ¯ J, θ, Z, G (0 , Z ) + X (cid:1) sothat: G ¯ b ( θ, θ ′ ) = (cid:18) (cid:18) − (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − (cid:0) ¯ J, θ, Z, G (0 , Z ) + X (cid:1)(cid:19)(cid:19) + U ′′ ( X ) (cid:19)(cid:19) − = G ( θ, θ ′ )where ¯ J is the average current. The function G ( θ, θ ′ ) is given by: G ( θ, θ ′ ) = 1 p π exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ | θ − θ ′ | !r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ exp (cid:18) θ − θ ′ σ ¯ X e (cid:19) ≃ p π exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ − σ ¯ X e ! ( θ − θ ′ ) !r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ H ( θ − θ ′ )with: 1¯ X e ( Z ) = ω − (cid:0) ¯ J ( Z ) , θ, Z, G (0 , Z ) + X (cid:1) = arctan (cid:16)(cid:16) X r − X p (cid:17) p ¯ J ( Z ) + G (0 , Z ) + X (cid:17)p ¯ J ( Z ) + G (0 , Z ) + X G ( θ, θ ′ , Z ) = (cid:18) − (cid:18) ∇ θ (cid:18) σ θ ∇ θ − ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) (cid:19)(cid:19) + U ′′ ( X ) (cid:19) − (251)= ∞ X n =0 ( − n (cid:18) G (cid:18)(cid:18) ∇ θ ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X )2 (cid:19) G (cid:19) n (cid:19) ( θ ′ , θ ) × exp R θ ′ θ (cid:0) ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) − ω − (cid:0) ¯ J, θ, Z, G (0 , Z ) + X (cid:1)(cid:1) σ θ ! or in expanded form:˜ G ( θ, θ ′ , Z ) = 1 p π exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ − σ ¯ X e ! ( θ − θ ′ ) !r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ × H ( θ − θ ′ ) ∞ X n =1 ( − n Z θ<θ <...<θ ′ ∇ θ ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X )2 ! × exp R θ ′ θ (cid:0) ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) − ω − (cid:0) ¯ J, θ, Z, G (0 , Z ) + X (cid:1)(cid:1) σ θ ! = 1 p π exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ − σ ¯ X e ! ( θ − θ ′ ) − h ω − ( J ( θ ) ,θ,Z, G (0 ,Z )+ X )2 i θθ ′ !r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ H ( θ − θ ′ ) × exp R θ ′ θ (cid:0) ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) − ω − (cid:0) ¯ J, θ, Z, G (0 , Z ) + X (cid:1)(cid:1) σ θ ! This can be simplified by using that in average: Z θ ′ θ (cid:0) ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) − ω − (cid:0) ¯ J, θ, Z, G (0 , Z ) + X (cid:1)(cid:1) ≃ G ( θ, θ ′ , Z ) = exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ − σ ¯ X e ! ( θ − θ ′ ) − h ω − ( J ( θ ) ,θ,Z, G (0 ,Z )+ X )2 i θθ ′ !p π r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ H ( θ − θ ′ )(252) ≃ exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ − σ ¯ X e ! + (cid:20) ∇ θ ω − ( ¯ J,θ,Z, G (0 ,Z )+ X ) (cid:21)! ( θ − θ ′ ) !p π r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ H ( θ − θ ′ )where the upper bar on a quantity stands for the average computed over the period θ − θ ′ .139 .4.2.2 Green function of ¯ G − ( θ, θ ) Using (249), equation (247) rewrites:0 ≃ ˜ G − δ Ψ ( θ, Z ) (253) − ∇ θ ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) p X − X Z ∇ θ δω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) δ | Ψ ( θ , Z ) | ! δ Ψ ( θ , Z ) ! dθ dZ = ¯ G − δ Ψ ( θ, Z ) − ∇ θ ω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) p X The Green function of the operator ¯ G − defined in (248) is given by a series expansion:¯ G = ∞ X n =0 (cid:18) X (cid:19) n ˜ G ∗ ∇ θ δω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) δ | Ψ ( θ , Z ) | !!! n ˜ G (254)= ∞ X n =0 (cid:18) X (cid:19) n ∇ θ ˜ G ∗ δω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) δ | Ψ ( θ , Z ) | !!! n ˜ G The convolution: ∇ θ ˜ G ∗ δω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) δ | Ψ ( θ , Z ) | !! is computed using (252)and the expression derived previously for the kernel δω − ( J ( θ ) ,θ,Z, G (0 ,Z )+ X ) δ | Ψ( θ ,Z ) | : δω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) δ | Ψ ( θ , Z ) | = C exp (cid:16) − K (cid:12)(cid:12)(cid:12) ( θ i − θ i +1 ) − | Z i − Z i +1 | c (cid:12)(cid:12)(cid:12)(cid:17) p | c ( θ i − θ i +1 ) − | Z i − Z i +1 || where C and K are some parameters depending on the system. The convolution is thus: ∇ θ ˜ G ∗ δω − ( J ( θ ) , θ, Z, G (0 , Z ) + X ) δ | Ψ ( θ , Z ) | !! ( θ i − − θ i +1 ) (255)= Z ∇ θ i − p π exp − r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ − σ ¯ X e ! + (cid:20) ∇ θ ω − ( ¯ J,θ,Z, G (0 ,Z )+ X ) (cid:21)! ( θ i − − θ i ) !r(cid:16) σ ¯ X e (cid:17) + U ′′ ( X ) σ × δ ( Z i − Z i − ) C exp (cid:16) − K (cid:12)(cid:12)(cid:12) ( θ i − θ i +1 ) − | Z i − Z i +1 | c (cid:12)(cid:12)(cid:12)(cid:17) r K (cid:12)(cid:12)(cid:12) ( θ i − θ i +1 ) − | Z i − Z i +1 | c (cid:12)(cid:12)(cid:12) dθ i dZ i = Z ∇ θ i − p π exp (cid:0) − ¯Λ ( θ i − − θ i ) (cid:1) Γ δ ( Z i − Z i − ) C exp (cid:16) − K (cid:12)(cid:12)(cid:12) ( θ i − θ i +1 ) − | Z i − Z i +1 | c (cid:12)(cid:12)(cid:12)(cid:17) r K (cid:12)(cid:12)(cid:12) ( θ i − θ i +1 ) − | Z i − Z i +1 | c (cid:12)(cid:12)(cid:12) dθ i dZ i where: ¯Λ = s(cid:18) σ ¯ X e (cid:19) + 2 U ′′ ( X ) σ − σ ¯ X e + " ∇ θ ω − (cid:0) ¯ J, θ, Z, G (0 , Z ) + X (cid:1) Γ = r π s(cid:18) σ ¯ X e (cid:19) + 2 U ′′ ( X ) σ Z θ i − θ i +1 + | Zi − Zi +1 | c C ¯Λ exp (cid:16) − ¯Λ ( θ i − − θ i ) − K (cid:12)(cid:12)(cid:12) ( θ i − θ i +1 ) − | Z i − − Z i +1 | c (cid:12)(cid:12)(cid:12)(cid:17) r K (cid:12)(cid:12)(cid:12) ( θ i − θ i +1 ) − | Z i − − Z i +1 | c (cid:12)(cid:12)(cid:12) dθ i = Z s θ i − − (cid:18) θ i +1 + | Zi − Zi +1 | c (cid:19) C ¯Λ exp (cid:16) − ¯Λ (cid:16) θ i − − (cid:16) u c + θ i +1 + | Z i − − Z i +1 | c (cid:17)(cid:17) − u (cid:17) K Γ du = Z s θ i − − (cid:18) θ i +1 + | Zi − Zi +1 | c (cid:19) C ¯Λ exp (cid:16) − ¯Λ (cid:16) θ i − − θ i +1 − | Z i − − Z i +1 | c (cid:17) − (cid:16) ¯Λ K (cid:17) u (cid:17) K Γ du = C ¯Λ exp (cid:16) − ¯Λ (cid:16) θ i − − θ i +1 − | Z i − − Z i +1 | c (cid:17)(cid:17) K Γ q ¯Λ K erf r K s θ i − − (cid:18) θ i +1 + | Z i − − Z i +1 | c (cid:19)! and (254) becomes:¯ G ( Z , θ , Z n +1 , θ n +1 )= X n Z n Y l =1 (cid:0) − C ¯Λ (cid:1) n +1 X (cid:16) − ¯Λ (cid:16) θ l − θ l +1 − | Z l − Z l +1 | c (cid:17)(cid:17) K Γ q ¯Λ K erf r K s θ l − (cid:18) θ l +1 + | Z l − Z l +1 | c (cid:19)! × ω ( J ( θ l ) , θ l , Z l , G + X ) ˜ G ( θ n , θ n +1 , Z l ) n Y l =2 dθ l dZ l with: ω − (cid:16) θ ( i ) , Z (cid:17) = G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ( G + X ) dZ (256)Formula (256) can be expanded in terms of current, at the second order of approximation: ω − ( J, θ, Z ) = G [ J ( θ, Z ) , G + X ] (257)+ κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) − ω ( θ, Z ) ω ( θ, Z ) G ′ (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3) ¯ G (0 , Z ) dZ = G (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3) − κN R T ( Z, Z ) (cid:16) ω (cid:16) θ − | Z − Z | c , Z (cid:17) − ω ( θ, Z ) (cid:17) G ′ (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3) ¯ G (0 , Z ) dZ ≃ G (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3) − R κT ( Z,Z ) N (cid:16) G h J (cid:16) θ − | Z − Z | c , Z (cid:17) , ¯ G (0 , Z i ) i − G (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3)(cid:17) G ′ (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3) ¯ G (0 , Z ) dZ and: ω ( J, θ, Z ) (258)= 12 (cid:16) F (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3) + (cid:16)(cid:0) F (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3)(cid:1) +4 (cid:18)Z κN T ( Z, Z ) (cid:18) F (cid:20) J (cid:18) θ − | Z − Z | c , Z (cid:19) , ¯ G (0 , Z i ) (cid:21) − F (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3)(cid:19) ¯ G (0 , Z ) dZ (cid:19) F ′ (cid:2) J ( θ, Z ) , ¯ G (0 , Z i ) (cid:3)(cid:1) (cid:17) where: ¯ G (0 , Z i ) ≃ G (0 , Z i ) + X δ Ψ ( θ, Z ) (259)= X n Z (cid:0) − C ¯Λ (cid:1) n +1 n Y l =1 exp (cid:16) − ¯Λ (cid:16) θ l − θ l +1 − | Z i − Z i +1 | c (cid:17)(cid:17) K Γ q ¯Λ K erf r K s θ l − (cid:18) θ l +1 + | Z i − Z i +1 | c (cid:19)! × ω ( J ( θ l ) , θ l , Z, G + X ) } ˜ G ( θ n , θ n +1 ) (cid:0) ∇ θ ω − ( J ( θ n +1 ) , θ n +1 , Z, G + X ) (cid:1) √ X n +1 Y l =2 dθ l with the convention that θ = θ .For ω ( J ( θ l ) , θ l , Z, G (0 , Z ) + X ) constant in first approximation, the series can be computed. TheFourier transform of (255) is: C ik σ k + ik X e + U ′′ ( X ) p K + √ K + k √ K + k exp (cid:18) − ik (cid:18) | Z i − Z i +1 | c (cid:19)(cid:19) and ¯ G rewrites:¯ G = X n CX ik σ k + ik X e + U ′′ ( X ) p K + √ K + k √ K + k ω (cid:0) ¯ J, G + X (cid:1)! n n Y l =1 exp (cid:18) − ik (cid:18) | Z i − Z i +1 | c (cid:19)(cid:19)! ˜ G The integrals: Z n Y l =1 exp (cid:18) − ik (cid:18) | Z i − Z i +1 | c (cid:19)(cid:19) dZ i with fixed endpoints Z and Z n +1 can be written: Z exp − ik n X l =1 | Z i − Z i +1 | c !! δ Z − Z n +1 − n X l =1 ( Z i − Z i +1 ) ! dZ i = Z exp − ik n X l =1 | Z i − Z i +1 | c !! exp iλ Z − Z n +1 − n X l =1 ( Z i − Z i +1 ) !! dZ i = Z exp − ik n X l =1 | Z i − Z i +1 | c !! exp iλ Z − Z n +1 − n X l =1 ( Z i − Z i +1 ) !! × dλ n Y i =1 (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12) d (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12) dv i where the unit vectors v i are defined such that: Z ( i − − Z ( i ) = v i (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12) We also define: λ. ( Z − Z n +1 ) = | λ | | Z − Z n +1 | cos ( θ ) λ.v i = | λ | cos ( θ i )142he angles θ l are computed in the plane ( λ, Z − Z ) between the projection of v l and Z − Z . π n +1 Z π Z exp − ik n X l =1 | Z i − Z i +1 | c !! exp i | λ | | Z − Z n +1 | cos ( θ ) − n X l =1 | Z i − Z i +1 | cos ( θ i ) !! × d | λ | n Y i =1 (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12) d (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12) sin ( θ i ) dθ i sin ( θ ) dθ = 2 ( π ) n +1 Z sin ( | λ | | Z − Z n +1 | ) | λ | | Z − Z n +1 | n Y i =1 exp (cid:0) i (cid:0) | λ | − kc (cid:1) (cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:1) − exp (cid:0) − i (cid:0) | λ | + kc (cid:1) (cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:1) i | λ |× (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12) d (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12) d | λ | = 2 ( π ) n +1 Z sin ( | λ | | Z − Z n +1 | )2 | λ | n +1 | Z − Z n +1 |× n Y i =1 (cid:18) − dd | λ | Z (cid:18) exp (cid:18) i (cid:18) | λ | − kc (cid:19) (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12)(cid:19) + exp (cid:18) − i (cid:18) | λ | + kc (cid:19) (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12)(cid:19)(cid:19) d (cid:12)(cid:12)(cid:12) Z ( i − − Z ( i ) (cid:12)(cid:12)(cid:12)(cid:19) d | λ |≃ π ) n +1 Z sin ( | λ | | Z − Z n +1 | )2 | λ | n +1 | Z − Z n +1 | n Y i =1 − dd | λ | δ (cid:18) | λ | − kc (cid:19) − iπ (cid:0) | λ | − kc (cid:1) + δ (cid:18) | λ | + kc (cid:19) + iπ (cid:0) | λ | + kc (cid:1) !!! d | λ | = 2 ( π ) n +1 Z sin ( | λ | | Z − Z n +1 | )2 | λ | | Z − Z n +1 | d | λ | d | λ | (cid:0) exp (cid:0) − i (cid:0) | λ | + kc (cid:1) L (cid:1) − (cid:1) i (cid:0) | λ | + kc (cid:1) − (cid:0) exp (cid:0) i (cid:0) | λ | − kc (cid:1) L (cid:1) − (cid:1) i (cid:0) | λ | − kc (cid:1) !! n d | λ | π Z sin ( | λ | | Z − Z ′ | )2 | λ | | Z − Z ′ | X n ikπC σ k + ik X e + U ′′ ( X ) p K + √ K + k √ K + k X ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)! n × d | λ | d | λ | (cid:0) exp (cid:0) − i (cid:0) | λ | + kc (cid:1) L (cid:1) − (cid:1) i (cid:0) | λ | + kc (cid:1) − (cid:0) exp (cid:0) i (cid:0) | λ | − kc (cid:1) L (cid:1) − (cid:1) i (cid:0) | λ | − kc (cid:1) !! n d | λ | ik σ k + ik X e + U ′′ ( X )= 2 π Z sin ( | λ | | Z − Z ′ | ) | λ || Z − Z ′ | − ikπC d | λ | d | λ | ( exp ( − i ( | λ | + kc ) L ) − ) i ( | λ | + kc ) − ( exp ( i ( | λ |− kc ) L ) − ) i ( | λ |− kc ) ! σ k + ik Xe + U ′′ ( X ) √ K + √ K + k √ K + k X ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) d | λ | × ik σ k + ik X e + U ′′ ( X )which can be written: π Z sin ( | λ | | Z − Z ′ | ) | λ || Z − Z ′ | ikd | λ | dk (cid:18) σ k + ik X e + U ′′ ( X ) − ikπC X d | λ | d | λ | (cid:18) ( exp ( − i ( | λ | + kc ) L ) − ) i ( | λ | + kc ) − ( exp ( i ( | λ |− kc ) L ) − ) i ( | λ |− kc ) (cid:19) √ K + √ K + k √ K + k ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)(cid:19) d | λ | d | λ | (cid:0) exp (cid:0) − i (cid:0) | λ | + kc (cid:1) L (cid:1) − (cid:1) i (cid:0) | λ | + kc (cid:1) − (cid:0) exp (cid:0) i (cid:0) | λ | − kc (cid:1) L (cid:1) − (cid:1) i (cid:0) | λ | − kc (cid:1) ! = − exp (cid:0) − i (cid:0) | λ | + kc (cid:1) L (cid:1) | λ | (cid:0) | λ | + kc (cid:1) + exp (cid:0) i (cid:0) | λ | − kc (cid:1) L (cid:1) | λ | (cid:0) | λ | − kc (cid:1) ! L + (cid:0) exp (cid:0) − i (cid:0) | λ | + kc (cid:1) L (cid:1) − (cid:1) i | λ | (cid:0) | λ | + kc (cid:1) − (cid:0) exp (cid:0) i (cid:0) | λ | − kc (cid:1) L (cid:1) − (cid:1) i | λ | (cid:0) | λ | − kc (cid:1) !! π Z sin ( | λ | | Z − Z ′ | ) ik exp ( ik ( θ − θ ′ ))2 | λ | | Z − Z ′ | D d | λ | dk D = σ k + ik X e + U ′′ ( X )+ ikπC X exp (cid:0) − i (cid:0) | λ | + kc (cid:1) L (cid:1) | λ | (cid:0) | λ | + kc (cid:1) + exp (cid:0) i (cid:0) | λ | − kc (cid:1) L (cid:1) | λ | (cid:0) | λ | − kc (cid:1) ! L + (cid:0) exp (cid:0) − i (cid:0) | λ | + kc (cid:1) L (cid:1) − (cid:1) i | λ | (cid:0) | λ | + kc (cid:1) − (cid:0) exp (cid:0) i (cid:0) | λ | − kc (cid:1) L (cid:1) − (cid:1) i | λ | (cid:0) | λ | − kc (cid:1) !! × p K + √ K + k √ K + k ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) The integral over k presents a pole such that ik is real and negative. Due to the exponential in the denomi-nator, | k | << ≃ U ′′ ( X ) + ikπC X exp (cid:0) − i (cid:0) | λ | + kc (cid:1) L (cid:1) | λ | (cid:0) | λ | + kc (cid:1) + exp (cid:0) i (cid:0) | λ | − kc (cid:1) L (cid:1) | λ | (cid:0) | λ | − kc (cid:1) ! L ! r K ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) ≃ U ′′ ( X ) + ikπ CX | λ | cos ( | λ | L ) (cid:18) exp (cid:18) − i kc L (cid:19)(cid:19) L r K ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) so that: ik ≃ − | λ | U ′′ ( X ) L q K CX π cos ( | λ | L ) ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) and the integral over | λ | is such that cos ( | λ | L ) > − Z sin ( | λ | | Z − Z ′ | ) | λ | U ′′ ( X ) L √ K CX πω ( ¯ J, G (0 ,Z )+ X ) cos( | λ | L ) exp (cid:18) − | λ | U ′′ ( X ) L √ K CX πω ( ¯ J, G (0 ,Z )+ X ) cos( | λ | L ) ( θ − θ ′ ) (cid:19) | λ | CX π | λ | | Z − Z ′ | cos ( | λ | L ) L q K ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) d | λ | = − Z sin ( | λ | | Z − Z ′ | ) | λ | U ′′ ( X ) exp (cid:18) − | λ | U ′′ ( X ) L √ K CX πω ( ¯ J, G (0 ,Z )+ X ) cos( | λ | L ) ( θ − θ ′ ) (cid:19) | λ | (cid:16) L q K CX πω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) cos ( | λ | L ) (cid:17) | Z − Z ′ | d | λ | and: δ Ψ ( θ, Z ) = − Z sin ( | λ | | Z − Z ′ | ) | λ | U ′′ ( X ) exp (cid:18) − | λ | U ′′ ( X ) L √ K CX πω ( ¯ J, G (0 ,Z )+ X ) cos( | λ | L ) ( θ − θ ′ ) (cid:19) | λ | (cid:16) L q K CX πω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) cos ( | λ | L ) (cid:17) | Z − Z ′ |× d | λ | (cid:0) ∇ θ ω − ( J ( θ ′ ) , θ ′ , Z ′ , G (0 , Z ) + X ) (cid:1) √ X dθ ′ ≃ Z K sin ( | λ | | Z − Z ′ | ) | λ | U ′′ ( X ) exp (cid:18) − | λ | U ′′ ( X ) L √ K CX πω ( ¯ J, G (0 ,Z )+ X ) cos( | λ | L ) ( θ − θ ′ ) (cid:19) | λ | LCX π cos ( | λ | L )) (cid:0) ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)(cid:1) | Z − Z ′ |× d | λ | ( ∇ θ ω ( J ( θ ′ ) , θ ′ , Z ′ , G (0 , Z ) + X )) p X dθ ′ δ Ψ ( θ, Z ) ≃ Z K sin ( | λ | | Z − Z ′ | ) | λ | U ′′ ( X ) exp (cid:18) − | λ | U ′′ ( X ) L √ K CX πω ( ¯ J, G (0 ,Z )+ X ) cos( | λ | L ) ( θ − θ ′ ) (cid:19) | λ | LCX π cos ( | λ | L )) (cid:0) ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)(cid:1) | Z − Z ′ |× d | λ | ( ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X )) p X dθ ′ − Z K | λ | U ′′ ( X ) exp (cid:18) − | λ | U ′′ ( X ) L √ K CX πω ( ¯ J, G (0 ,Z )+ X ) cos( | λ | L ) ( θ − θ ′ ) (cid:19) ( θ − θ ′ )8 ( LCX π cos ( | λ | L )) (cid:0) ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)(cid:1) × d | λ | dθ ′ (cid:0) ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X ) (cid:1) p X = K sin ( | λ | | Z − Z ′ | ) | λ | U ′′ ( X ) L q K CX πω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) cos ( | λ | L ) | λ | | λ | U ′′ ( X ) ( LCX π cos ( | λ | L )) (cid:0) ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)(cid:1) | Z − Z ′ |×∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X ) p X − Z K | λ | U ′′ ( X ) (cid:16) L q K CX πω (cid:0) ¯ J, G (0 , Z ) + X (cid:1) cos ( | λ | L ) (cid:17) LCX π cos ( | λ | L )) (cid:0) ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)(cid:1) (cid:16) | λ | U ′′ ( X ) (cid:17) d | λ |× (cid:0) ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X ) (cid:1) p X ≃ Z | λ | √ K √ LCX π cos ( | λ | L )) (cid:0) ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)(cid:1) ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X ) p X − Z √ X (cid:0) ω (cid:0) ¯ J, G (0 , Z ) + X (cid:1)(cid:1) U ′′ ( X ) d | λ | ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X ) ≡ N ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X ) − N ∇ θ ω ( J ( θ ) , θ, Z, G (0 , Z ) + X )where we used that the integral over | λ | is constrained on a finite interval. Corrections to (259) due to (245)
We saw in (245), that the backreaction terms, shift | Ψ ( θ, Z ) | by G (0 , Z ).This shift can be absorbed by a redefinition of Ψ ( θ, Z ). Actually, let: δ Ψ ( θ, Z ) = δ ˜Ψ ( θ, Z ) − G (0 , Z ) √ X δ Ψ † ( θ, Z ) = δ ˜Ψ † ( θ, Z ) − G (0 , Z ) √ X For G (0 ,Z ) √ X <<
1, it yields at the first order: | Ψ ( θ, Z ) | + G (0 , Z ) = (cid:16)p X + δ Ψ † ( θ, Z ) (cid:17) (cid:16)p X + δ Ψ ( θ, Z ) (cid:17) + G (0 , Z )= (cid:12)(cid:12)(cid:12) ˜Ψ ( θ, Z ) (cid:12)(cid:12)(cid:12) As a consequence, the computations leading to the minimum are the same as in the previous paragraphs ifwe replace Ψ ( θ, Z ) → ˜Ψ ( θ, Z ). Coming back ultimately to the variable Ψ ( θ, Z ), the minimum (259) for theaction becomes: δ Ψ † ( θ, Z ) = − G (0 , Z ) √ X (260)145 Ψ ( θ, Z ) (261)= X n Z (cid:0) − C ¯Λ (cid:1) n +1 n Y l =1 exp (cid:16) − ¯Λ (cid:16) θ l − θ l +1 − | Z i − Z i +1 | c (cid:17)(cid:17) K Γ q ¯Λ K erf r K s θ l − (cid:18) θ l +1 + | Z i − Z i +1 | c (cid:19)! × ω ( J ( θ l ) , θ l , Z, G (0 , Z ) + X ) } ˜ G ( θ n , θ n +1 ) (cid:0) ∇ θ ω − ( J ( θ n +1 ) , θ n +1 , Z, G (0 , Z ) + X ) (cid:1) √ X n +1 Y l =2 dθ l − G (0 , Z ) √ X and these shifts do not impact the main conclusions of our results. At lowest order in perturbation, the correction to the action is:12
X Z n − † (cid:16) θ ( i ) f , Z i (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( i ) i , Z i (cid:17) (262) × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) × (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − and this shifts the minimum of the potential on the left by a term proportional to (cid:18) ˆΞ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:19) ,so that X → X − ̺ (cid:18) ˆΞ ,n (cid:16) Z i , { Z j } j = i ,θ ( i ) i ,θ ( i ) f (cid:17) θ ( i ) f − θ ( i ) i (cid:19) where ̺ < Appendix 5. Frequency equation
The potential terms (21) along with its corrections (262) evaluated at the background field Ψ yield aconstant U (Ψ ). This constitutes a modification to the scale α which is replaced by α + U ′′ (Ψ ). Thismodifies accordingly the 2 points correlation function. Since U (Ψ ) < α + U (Ψ ) < α . This implies a longer average interaction time.The effective frequencies of the system are obtained from the second derivative of the effective action: δ Γ (Ψ) δ Ψ † (cid:0) θ ( j ) , Z j (cid:1) Ψ (cid:0) θ ( j ) , Z j (cid:1) = G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + "Z θ ( i )1 G − F Θ Ψ θ ( i )1 Ψ † F Θ G − (263) − "Z θ ( i )1 G − F Θ (1 + O , ∞ ) Ψ θ ( i )1 Ψ † (1 − exp ( − x )) F Θ (1 + O , ∞ ) G − − "Z θ ( i )1 G − F Θ (1 + O , ) Ψ θ ( i )1 Ψ † exp ( − x ) F Θ (1 + O , ) G − with F and Θ given by (190). 146e showed that it can be approximated by: δ Γ (Ψ) δ Ψ † (cid:0) θ ( j ) , Z j (cid:1) Ψ (cid:0) θ ( j ) , Z j (cid:1) (264)= G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) + F (cid:0) Ψ † G − Ψ (cid:1) h i ( α ( x ) + F x exp ( − x )) − ¯ ζ + ∇ outθ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) + F (cid:0) Ψ † G − Ψ (cid:1) h i ( β ( x ) − F exp ( − x )) y − ¯ ζ + ∇ outθ ( i ) i ′ Λ ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) + β ( x )( x + exp ( − x ) ( h i − x + ( y − x ))) In the local approximation, the dominant term of (264): G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) = G − − x ) (cid:0) − z + (cid:0) y − x (cid:1)(cid:1)(cid:0) O , ∞ (cid:1) + exp ( − x ) (cid:0) − ¯ O , ∞ + y (cid:0) O , (cid:1) − x (cid:0) O , ∞ (cid:1)(cid:1) (cid:16) θ ( i )1 , θ ( i ) f (cid:17) can be rewritten in its expanded form: G − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n D(cid:0) O , (cid:1) (2) E + P n > n − D(cid:0) O ,n (cid:1) ( n ) E n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) (265)= G − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n (cid:0) O , (cid:1) D(cid:0) O , (cid:1) (1) E + P n > n − (cid:0) O ,n (cid:1) D(cid:0) O ,n (cid:1) ( n − E n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) = G − + h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n (cid:0) O , (cid:1) D(cid:0) O , (cid:1) (1) E + P n > n − (cid:0) O ,n (cid:1) D(cid:0) O ,n (cid:1) ( n − E n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) − (cid:18)(cid:0) O , (cid:1) D(cid:0) O , (cid:1) (1) E + P n > n − (cid:0) O ,n (cid:1) D(cid:0) O ,n (cid:1) ( n − E n − (cid:19) (cid:0) O , (cid:1) D(cid:0) O , (cid:1) (1) E + P n > n − (cid:0) O ,n (cid:1) D(cid:0) O ,n (cid:1) ( n − E n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) = G − h i + D (1 + O , ) (2) E + P n > n ! D (1 + O , ∞ ) ( n ) E n (cid:0) G (cid:0) G − O , G − (cid:1)(cid:1) D(cid:0) O , (cid:1) (1) E + P n > ( G ( G − O ,n G − )) D (
1+ ¯ O ,n ) ( n − E n − ( n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) −G − (cid:0) G (cid:0) G − O , G − (cid:1)(cid:1) D(cid:0) O , (cid:1) (1) E (cid:0) G (cid:0) G − O , G − (cid:1)(cid:1) D(cid:0) O , (cid:1) (1) E + P n > ( G ( G − O ,n G − )) D (
1+ ¯ O ,n ) ( n − E n − ( n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) −G − P n > n − (cid:0) G (cid:0) G − O ,n G − (cid:1)(cid:1) D(cid:0) O ,n (cid:1) ( n − E n − (cid:0) G (cid:0) G − O , G − (cid:1)(cid:1) D(cid:0) O , (cid:1) (1) E + P n > ( G ( G − O ,n G − )) D (
1+ ¯ O ,n ) ( n − E n − ( n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) O ,n = − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) (266) × exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) Λ¯ O , ∞ = − ¯ ζ + ∇ out θ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) exp (cid:16) ˆΞ , ∞ (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) Λand: G − ¯ O ,n (267)= G − O ,n G − = − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) δ (cid:16) θ ( i ) f ′ − θ ( i ) i ′ (cid:17) + − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) f ′ (cid:17)(cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) In the local approximation (267) becomes: G − ¯ O ,n = G − O ,n G − (268)= − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) δ (cid:16) θ ( i ) f ′ − θ ( i ) i ′ (cid:17) + − ¯ ζ n + ∇ out θ ( i ) i ′ Λ ¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i ′ (cid:17) (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) f ′ (cid:17)(cid:17) Moreover, in (265), the sum: O = (cid:0) O , (cid:1) D(cid:0) O , (cid:1) (1) E + X n > n − (cid:0) O ,n (cid:1) D(cid:0) O ,n (cid:1) ( n − E n − is the sum of graphs contributing to the two points correlation, and: S = h i + 12 D (1 + O , ) (2) E + X n > n ! D (1 + O , ∞ ) ( n ) E n computes the sum of all graphs in the background state Ψ. As a consequence, rewriting (265) as: G − + S − O O = G − + ∞ X n =1 ( − n − (cid:0) O n − O n − S (cid:1) The subtraction by the second term removes all the graphs contributing to the inverse two points correlationfunction that can be factored by any n . Moreover, the alternate series removes the graph that can be factoredas a convolution of two graphs through one external leg. As a consequence, the series computes the 1PIgraphs. 148e can thus write: G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) = G −
11 + (cid:0) G (cid:0) G − O , G − (cid:1)(cid:1) D(cid:0) O , (cid:1) (1) E + P n > ( G ( G − O ,n G − )) D (
1+ ¯ O ,n ) ( n − E n − ( n − (cid:16) θ ( i )1 , θ ( i ) f (cid:17) where the upper script recalls that only 1PI part of the series expansion is kept. In an expanded form, wehave: G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) (269)= G − + ∞ X n =0 ( − n G − n(cid:0) G (cid:0) G − O , G − (cid:1)(cid:1) D(cid:0) O , (cid:1) (1) E + X n > (cid:0) G (cid:0) G − O ,n G − (cid:1)(cid:1) D(cid:0) O ,n (cid:1) ( n − E n − ( n − n (cid:16) θ ( i )1 , θ ( i ) f (cid:17) In particular, at the lowest order of the series expansion, a graph: (cid:0) G (cid:0) G − O ,n G − (cid:1)(cid:1) D(cid:0) O ,n (cid:1) ( n − E n − can be replaced by: (cid:0) G (cid:0) G − O ′ ,n G − (cid:1)(cid:1) D(cid:0) O ,n (cid:1) ( n − E n − (270)where:¯ O ′ ,n = − ζ ( n ) + ∇ out θ ( i ) i ′ Λ ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) exp (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17)(cid:17) exp (cid:16) − Λ (cid:16) θ ( i ) f − θ ( i ) i ′ (cid:17)(cid:17) ΛThe term (270) keeps the 1PI part of the initial graph in the state Ψ. The computation of the series (269)149s similar to (243) and gives in the local approximation: G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) (271)= − ∇ θ (cid:18) σ θ ∇ θ − ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17)(cid:19) + α + X ζ ( n ) n ! G (0 , Z ) + Z θ ( i ) − θ ( j ) i | Zi − Zj | c dl j Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n + X − ¯ ζ n + ∇ outθ ( j ) i Λ ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) − ζ ( n ) + ∇ outθ ( i ) i ′ Λ ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17) ( n − × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) (cid:16)(cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) ∇ outθ ( j ) i Λ θ ( j ) f − θ ( j ) i × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − where ω − (cid:0) J, θ ( j ) , ¯ G ( Z ) (cid:1) is solution of: ω − (cid:16) θ ( i ) , Z (cid:17) = G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ¯ G (0 , Z i ) dZ We can now identify the various contributions in (264). First, the expansion (271) of G − F Θ (cid:16) θ ( i )1 , θ ( i ) f (cid:17) includes a potential term which, evaluated at the state Ψ, defines an effective value for α : α + X ζ ( n ) n ! G (0 , Z ) + Z θ ( i ) − θ ( j ) i | Zi − Zj | c dl j Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n + X ζ ( n ) n ! G (0 , Z ) + Z θ ( i ) − θ ( j ) i | Zi − Zj | c dl j Z (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) i − | Z i − Z j | c , Z j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dZ j ! n − X n − (cid:18) ¯ ζ n + ζ ( n ) (cid:19) (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17)(cid:17) × Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) ∇ outθ ( j ) i Λ θ ( j ) f − θ ( j ) i × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − The other terms of (264) also includes a potential term, contributing to the effective value of α : − F (cid:0) Ψ † G − Ψ (cid:1) ( α ( x ) − F x exp ( − x )) ¯ ζ − ( β ( x ) − F exp ( − x )) y ¯ ζ − β ( x )( x + exp ( − x ) (1 − x + ( y − x ))) ! ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) + X n − (cid:16) ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) + ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17)(cid:17) × Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) ∇ outθ ( j ) i Λ θ ( j ) f − θ ( j ) i × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − and a contribution due to the correction terms in (264): F (cid:0) Ψ † G − Ψ (cid:1) ( α ( x ) − F x exp ( − x )) ∇ outθ ( i ) i ′ Λ ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) − ( β ( x ) − F exp ( − x )) y ∇ out θ ( i ) i ′ Λ ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) Gathering these two contributions leads to the effective frequency: ω − e ( J ( θ ) , θ, Z ) = ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) + X n − (cid:16) ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) + ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17)(cid:17) × Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) (cid:16) exp (cid:16) ˆΞ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17)(cid:17) − (cid:17) − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i ∇ out θ ( j ) i Λ × − ¯ ζ n + ¯Ξ ,n (cid:16) Z j , { Z m } m = j , θ ( j ) i , θ ( j ) f (cid:17) θ ( j ) f − θ ( j ) i Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j n − + F (cid:0) Ψ † G − Ψ (cid:1) ( α ( x ) − F x exp ( − x )) (cid:16) ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) − ( β ( x ) − F exp ( − x )) y (cid:16) ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) − e ( J ( θ ) , θ, Z ) ≃ ω − (cid:16) J ( θ ) , θ, Z, G (0 , Z ) + | Ψ | (cid:17) + X n − (cid:16) ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) + ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) × (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17)(cid:17) (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (cid:19) n − + F (cid:0) Ψ † G − Ψ (cid:1) ( α ( x ) − F x exp ( − x )) (cid:16) ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) − F (cid:0) Ψ † G − Ψ (cid:1) ( β ( x ) − F exp ( − x )) y (cid:16) ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) ω − e ( J ( θ ) , θ, Z ) ≃ ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) + ω − (cid:16) J ( θ ) , θ, Z, G + | Ψ | (cid:17) + ω − (cid:16) J ( θ ) , θ, Z, | Ψ | (cid:17) The function ω − is defined by: ω − ( J ( θ ) , θ, Z, G ) = X (cid:16) ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) + ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) n − × (cid:16) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17)(cid:17) × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (cid:19) n − Given (32) and (38):¯Ξ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) ≃ Z θ ( i ) f θ ( i ) i n X l =1 C l − n − δ l − ω − (cid:16) J, θ ( i ) , Z i , | Ψ | (cid:17) Λ l l − Q m =1 δ (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) − | Z i − ¯ Z | c , ¯ Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | Ψ( θ,Z ) | = G (0 ,Z ) dθ ( i ) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) ≃ Z θ ( i ) f θ ( i ) i n − X l =1 C l − n − δ l − ω − (cid:16) J, θ ( i ) , Z i , | Ψ | (cid:17) Λ l l − Q m =1 δ (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) − | Z i − ¯ Z | c , ¯ Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − ζ ( l ) Λ l | Ψ( θ,Z ) | = G (0 ,Z ) dθ ( i ) We can approximate: ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) by the constant − R θ ( i ) f θ ( i ) i P nl =1 C l − n − ζ ( l ) Λ l dθ ( i ) and: (cid:16) ¯Ξ ,n (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17) + ¯Ξ ( n )1 (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ¯Ξ ,n (cid:16) Z i ,θ ( i ) i ′ , { Z j } j = i (cid:17) . Using (32), we can compute the coefficient of δ l − ω − ( J,θ ( i ) ,Z i , | Ψ | ) Λ l l − Q m =1 δ (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ ( i ) − | Zi − ¯ Z | c , ¯ Z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (272). It is152iven by: 12 ∞ X n = l n − C l − n − ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (cid:19) n − l (273)= 12 l ! ∞ X p =0 p ! ˆΞ ,p + l (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f (cid:17) × (cid:18)Z Ψ † (cid:16) θ ( j ) f , Z j (cid:17) Ψ (cid:16) θ ( j ) i , Z j (cid:17) dZ j (cid:19) p ≡ l ! ˇΞ ,l (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f , | Ψ | (cid:17) Thus, ω − ( J ( θ ) , θ, Z, G (0 , Z )) can be defined by its derivatives at G (0 , Z ): δ n ω − ( J ( θ ) , θ, Z, G ) δ n G (0 , Z ) = δ n ω − ( J ( θ ) , θ, Z, G ) δ n G (0 , Z ) ˇΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f , | Ψ | (cid:17) (274)It defines a function similar to ω − (cid:0) θ ( i ) , Z (cid:1) wich satisfies: ω − ( J ( θ ) , θ, Z, G ) = ˆ G J ( θ ) + κN Z T ( Z, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ω ( θ, Z ) W ω ( θ, Z ) ω (cid:16) θ − | Z − Z | c , Z (cid:17) ¯ G (0 , Z i ) dZ for some function ˆ G . The derivatives of ˆ G computed at ¯ J + ¯ G (0 , Z i ) are such that (274) is satisfied. Forweak interactions ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17) ≃ ¯ ζ n <<
1, a solution of (274) is given by:ˆ G ( n ) (cid:0) ¯ J + ¯ G (0 , Z i ) (cid:1) (275) ≃ ω ( J, θ, Z ) ¯Ξ (cid:0) ¯ J, | Z i − Z | (cid:1) κN T ( Z, Z ) ! n × − Z κN ω (cid:18) J, θ − | Z − Z ′ | c , Z ′ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:18) θ − | Z − Z ′ | c , Z ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ ! F ′ [ J, ω, θ, Z, Ψ] ! × (cid:16) ˇΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i , θ ( i ) f , | Ψ | (cid:17)(cid:17) where ω ( J, θ, Z ) is the static solution and denotes the average over time and space of ¯Ξ ( J, θ, l i , | Z i − Z | ). Inthat case, equation (147) computing the successive derivatives δ n ω − ( J ( θ ) ,θ,Z, G ) δ n G (0 ,Z ) is no more valid. Actually,given (275), the successive products of ˆ G ′ (cid:0) ¯ J + ¯ G (0 , Z i ) (cid:1) arising in (146) are negligible. On the contrary, thesuccessive derivatives of ˆ T ( θ, Z, Z , ω, Ψ) that were neglected in the derivation of (147), involve contributionsproportional to ˆ G ( n ) (cid:0) ¯ J + ¯ G (0 , Z i ) (cid:1) that become dominant. The successive derivatives in this approximationare: δ ( n ) ˆ T (cid:16) θ, Z, Z ω, | Ψ | (cid:17) δ G (0 , Z ) ≃ (cid:0) κN T ( Z, Z ) (cid:1) n ˆ F ( n ) h J, ω, θ, Z, | Ψ | i ω ( J, θ, Z ) + (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) F ′ h J, ω, θ, Z, | Ψ | i and: δ n ˆ T ( θ, Z, Z ω, Ψ) δ n G (0 , Z ) ≃ ω ( J, θ, Z ) (cid:0) κN T ( Z, Z ) (cid:1) n ˆ G ( n ) h J, ω, θ, Z, | Ψ | i − (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) G ′ h J, ω, θ, Z, | Ψ | i δ n − ω − ( J ( θ ) , θ, Z, G ) δ n G (0 , Z ) ≃ Z ω (cid:18) J, θ − P nl =1 | Z ( l − − Z ( l ) | c , Z (cid:19) ω ( J, θ, Z ) × (cid:0) κN T ( Z, Z ) (cid:1) n ˆ G ( n ) h J, ω, θ, Z, | Ψ | i − (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) G h J, ω, θ, Z, | Ψ | i ≃ (cid:0) κN T ( Z, Z ) (cid:1) n ˆ G ( n ) h J, ω, θ, Z, | Ψ | i − (cid:18)R κN ω (cid:16) J, θ − | Z − Z ′ | c , Z ′ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (cid:16) θ − | Z − Z ′ | c , Z ′ (cid:17)(cid:12)(cid:12)(cid:12) T ( Z, Z ′ ) dZ ′ (cid:19) G ′ h J, ω, θ, Z, | Ψ | i As a consequence, using (275) and (146): δ n ω − ( J ( θ ) , θ, Z, G ) δ n G (0 , Z ) ≃ (cid:0) ω ( J, θ, Z ) ¯Ξ (cid:0) ¯ J, | Z i − Z | (cid:1)(cid:1) n ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17) ≃ δ n ω − ( J ( θ ) , θ, Z, G ) δ n G (0 , Z ) ˆΞ ,n (cid:16) Z i , { Z j } j = i , θ ( i ) i (cid:17) as needed.The term ω − ( J ( θ ) , θ, Z, Ψ) is defined by: ω − (cid:16) J ( θ ) , θ, Z, | Ψ | (cid:17) = F (cid:0) Ψ † G − Ψ (cid:1) ( α ( x ) − F x exp ( − x )) (cid:16) ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) − F (cid:0) Ψ † G − Ψ (cid:1) ( β ( x ) − F exp ( − x )) y (cid:16) ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) ≃ − F (cid:16) Ψ † U ′′ ( X ) δ Ψ ( θ, Z ) − ∇ θ ω − ( J ( θ ) , θ, Z, G + X ) Ψ † Ψ (cid:17) × ( α ( x ) − F x exp ( − x )) (cid:16) ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) − ( β ( x ) − F exp ( − x )) y (cid:16) ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) = − F (cid:16)p X U ′′ ( X ) δ Ψ ( θ, Z ) − ∇ θ ω − ( J ( θ ) , θ, Z, G + X ) X (cid:17) × ( α ( x ) − F x exp ( − x )) (cid:16) ¯Ξ , ∞ (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) − ( β ( x ) − F exp ( − x )) y (cid:16) ¯Ξ , (cid:16) Z i , θ ( i ) i ′ , { Z j } j = i (cid:17)(cid:17) ( x + exp ( − x ) ( h i − x + ( y − x ))) ppendix 6. Dynamic equations for connectivity functions We adapt the description of ([47]) to our context. The transfer function T from i to j satisfies the followingequation: ∇ θ ( i ) ( n i ) T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17) (276)= − τ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17) + λ (cid:16) ˆ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17)(cid:17) δ (cid:18) θ ( i ) ( n i ) − θ ( j ) ( n j ) − | Z i − Z j | c (cid:19) where ˆ T measures the variation of T due to the signals send from j to i and the signals emitted by i . Itsatisfies the following equation: ∇ θ ( i ) ( n i ) ˆ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17) (277)= ρδ (cid:18) θ ( i ) ( n i ) − θ ( j ) ( n j ) − | Z i − Z j | c (cid:19) × n(cid:16) h ( Z, Z ) − ˆ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17)(cid:17) C (cid:16) θ ( i ) ( n − (cid:17) h C ( ω i ( n i )) − D (cid:16) θ ( i ) ( n − (cid:17) ˆ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17) h D ( ω j ( n j )) o where h C and h D are increasing functions. We depart slightly from ([47]) by the introduction of the function h ( Z, Z ) (they chose h ( Z, Z ) = 1), to implement some loss due to the distance. We may chose for example: h ( Z, Z ) = exp (cid:18) − | Z i − Z j | νc (cid:19) where ν is a parameter. Equation (277) involves two dynamic factors C (cid:0) θ ( i ) ( n − (cid:1) and D ( θ i ( n − C (cid:0) θ ( i ) ( n − (cid:1) describes the accumulation of input spikes. It is solution of the differential equation: ∇ θ ( i ) ( n − C (cid:16) θ ( i ) ( n − (cid:17) = − C (cid:0) θ ( i ) ( n − (cid:1) τ C + α C (cid:16) − C (cid:16) θ ( i ) ( n − (cid:17)(cid:17) ω j (cid:18) Z j , θ ( i ) ( n − − | Z i − Z j | c (cid:19) (278)In the continuous approximation, the solution of (278) is: C (cid:16) θ ( i ) ( n − (cid:17) = Z exp − (cid:0) θ ( i ) ( n − − θ ( i ) ′ (cid:1) τ C + α C Z θ ( i ) ( n − θ ( i ) ′ ω j (cid:18) Z j , θ ′ − | Z i − Z j | c (cid:19) dθ ′ !! × ω j (cid:18) Z j , θ ( i ) ′ − | Z i − Z j | c (cid:19) dθ ( i ) ′ If a static equilibrium ω ( Z j ) exists, expanding around this equilibrium leads to approximate the integral: Z θ i θ ′ i ω j (cid:18) Z j , θ ′ − | Z i − Z j | c (cid:19) dθ ′ by the quantity: ω ( Z j ) (cid:16) θ ( i ) ( n − − θ ( i ) ′ (cid:17) so that: C (cid:16) θ ( i ) ( n − (cid:17) = Z exp (cid:18) − (cid:18) τ C + α C ω ( Z j ) (cid:19) (cid:16) θ ( i ) ( n − − θ ( i ) ′ (cid:17)(cid:19) (cid:18) C + ω j (cid:18) Z j , θ ( i ) ′ − | Z i − Z j | c (cid:19)(cid:19) dθ ′ i (279)155he term D ( θ i ( n − ∇ θ ( i ) ( n − D (cid:16) θ ( i ) ( n − (cid:17) = − D (cid:0) θ ( i ) ( n − (cid:1) τ D + α D (cid:16) − D (cid:16) θ ( i ) ( n − (cid:17)(cid:17) ω i ( Z i ) (280)In the continuous approximation, the solution of (280) is: D (cid:16) θ ( i ) ( n − (cid:17) = Z exp (cid:18) − (cid:18) τ D + α D ω ( Z i ) (cid:19) (cid:16) θ ( i ) ( n − − θ ( i ) ′ (cid:17)(cid:19) (cid:16) D + ω i (cid:16) Z i , θ ( i ) ′ (cid:17)(cid:17) dθ ( i ) ′ (281)As a consequence, the dynamics for transfer functions is a set of two equations: ∇ θ ( i ) ( n i ) T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17) (282)= − τ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17) + λ (cid:16) ˆ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17)(cid:17) δ (cid:18) θ ( i ) ( n i ) − θ ( j ) ( n j ) − | Z i − Z j | c (cid:19) ∇ θ ( i ) ( n i ) ˆ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17) (283)= ρδ (cid:18) θ ( i ) ( n i ) − θ ( j ) ( n j ) − | Z i − Z j | c (cid:19) × n(cid:16) h ( Z, Z ) − ˆ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17)(cid:17) C (cid:16) θ ( i ) ( n − (cid:17) h C ( ω i ( n i )) − D (cid:16) θ ( i ) ( n − (cid:17) ˆ T (cid:16)(cid:16) Z i , θ ( i ) ( n i ) , ω i ( n i ) (cid:17) , (cid:16) Z j , θ ( j ) ( n j ) , ω j ( n j ) (cid:17)(cid:17) h D ( ω j ( n j )) o with C (cid:0) θ ( i ) ( n − (cid:1) and D (cid:0) θ ( i ) ( n − (cid:1) given by (279) and (281).The field translation of (282) and (283) is obtained by including the following potential terms in theaction for the field: Z (cid:18) ∇ θ T (( Z, θ, ω ) , ( Z , θ , ω )) + T (( Z, θ, ω ) , ( Z , θ , ω )) τ (284) − λ (cid:16) ˆ T (( Z, θ, ω ) , ( Z , θ , ω )) (cid:17) δ (cid:18) θ − θ − | Z − Z | c (cid:19)(cid:19) × | Ψ ( θ, Z, ω ) | | Ψ ( θ , Z , ω ) | corresponding to (282) and: Z (cid:18) ∇ θ ˆ T (( Z, θ, ω ) , ( Z , θ , ω )) − ρδ (cid:18) θ ( i ) ( n i ) − θ ( j ) ( n j ) − | Z − Z | c (cid:19) (285) × n(cid:16) h ( Z, Z ) − ˆ T (( Z, θ, ω ) , ( Z , θ , ω )) (cid:17) C ( θ, Z, Z ) h C ( ω ) − D ( θ, Z ) ˆ T (( Z, θ, ω ) , ( Z , θ , ω )) h D ( ω ) o(cid:17) × | Ψ ( θ, Z, ω ) | | Ψ ( θ , Z , ω ) | for (283), with C ( θ, Z, Z ) and D ( θ, Z ) are defined as: C ( θ, Z, Z ) = Z θ exp (cid:18) − (cid:18) τ C + α C ω ( Z ) (cid:19) ( θ − θ ′ ) (cid:19) (cid:18) C + ω (cid:18) Z , θ ′ − | Z − Z | c (cid:19)(cid:19) dθ ′ D ( θ, Z ) = Z θ exp (cid:18) − (cid:18) τ D + α D ω ( Z ) (cid:19) ( θ − θ ′ ) (cid:19) ( D + ω ( Z, θ ′ )) dθ ′ τ C ( Z ) = 1 τ C + α C ω ( Z ) < τ D ( Z ) = 1 τ D + α D ω ( Z ) < ω ( Z, θ ), we can simplify the expressions for C ( θ, Z, Z ) and D ( θ, Z ): C ( θ, Z, Z ) ≃ C ( Z ) = C + ω ( Z ) τ C ( Z ) D ( θ, Z ) ≃ D ( Z ) = D + ω ( Z ) τ D ( Z )After projection on the dependent frequency states the transfer functions become functions T (( Z, θ ) , ( Z , θ ))and ˆ T (( Z, θ ) , ( Z , θ )) respectively. Moreover, we can simplify the action by finding the configurations for T (( Z, θ ) , ( Z , θ )) and ˆ T (( Z, θ ) , ( Z , θ )) that minimize the potential terms (284), (285). It corresponds toset: ∇ θ T (( Z, θ, ω ) , ( Z , θ , ω ))+ T (( Z, θ, ω ) , ( Z , θ , ω )) τ − λ (cid:16) ˆ T (( Z, θ, ω ) , ( Z , θ , ω )) (cid:17) δ (cid:18) θ − θ − | Z − Z | c (cid:19) = 0(286)and:0 = (cid:18) ∇ θ ˆ T (( Z, θ, ω ) , ( Z , θ , ω )) − ρδ (cid:18) θ ( i ) ( n i ) − θ ( j ) ( n j ) − | Z − Z | c (cid:19) (287) × n(cid:16) h ( Z, Z ) − ˆ T (( Z, θ, ω ) , ( Z , θ , ω )) (cid:17) C ( θ, Z, Z ) h C ( ω ) − D ( θ, Z ) ˆ T (( Z, θ, ω ) , ( Z , θ , ω )) h D ( ω ) o We look for solutions of the form: T (cid:18) Z, θ, Z , θ − | Z − Z | c (cid:19) ≡ T ( Z, θ, Z )ˆ T (cid:18) Z, θ, Z , θ − | Z − Z | c (cid:19) ≡ ˆ T ( Z, θ, Z )so that T ( Z, θ, Z ) and ˆ T ( Z, θ, Z ) satisfy: ∇ θ T ( Z, θ, Z ) + (cid:18) T ( Z, θ, Z ) τ − λ ˆ T ( Z, θ, Z ) (cid:19) = 0 (288) ∇ θ ˆ T ( Z, θ, Z ) = ρ (cid:18)(cid:16) h ( Z, Z ) − ˆ T ( Z, θ, Z ) (cid:17) C ( Z ) h C ( ω ( Z, θ )) − ˆ T ( Z, θ, Z ) D ( Z ) h D (cid:18) ω (cid:18) Z , θ − | Z − Z | c (cid:19)(cid:19)(cid:19) (289)Using (288), we replace ˆ T ( Z, θ, Z ) in (289):ˆ T ( Z, θ, Z ) = ∇ θ T ( Z, θ, Z ) λ + T ( Z, θ, Z ) λτ and we arrive to the differential equation satisfied by T ( Z, θ, Z ): ∇ θ T ( Z, θ, Z ) λ + U ( ω ) ∇ θ T ( Z, θ, Z ) + U ( ω ) T ( Z, θ, Z ) = ρC ( Z ) h ( Z, Z ) h C ( ω ( Z, θ )) (290)157here: U ( ω ) = (cid:18) λτ + ρλ (cid:18) C ( Z ) h C ( ω ( Z, θ )) + D ( Z ) h D (cid:18) ω (cid:18) Z , θ − | Z − Z | c (cid:19)(cid:19)(cid:19)(cid:19) U ( ω ) = ρλτ (cid:18) C ( Z ) h C ( ω ( Z, θ )) + D ( Z ) h D (cid:18) ω (cid:18) Z , θ − | Z − Z | c (cid:19)(cid:19)(cid:19) If we consider that the transfer function varies slowly compared to the oscillations of the thread, we canapproximate (290) by a quite static equation: U ( ω ) T ( Z, θ, Z ) = ρC ( Z ) h ( Z, Z ) h C ( ω ( Z, θ ))whose solution is: T ( Z, θ, Z ) = λτ C ( Z ) h ( Z, Z ) h C ( ω ( Z, θ )) C ( Z ) h C ( ω ( Z, θ )) + D ( Z ) h D (cid:16) ω (cid:16) Z , θ − | Z − Z | c (cid:17)(cid:17) (291) ≃ λτ h ( Z, Z )1 + D ( Z ) C ( Z ) h D (cid:16) ω (cid:16) Z ,θ − | Z − Z | c (cid:17)(cid:17) h C ( ω ( Z,θ )) Thus T ( Z, θ, Z ) is a decreasing function of ω (cid:16) Z , θ − | Z − Z | c (cid:17) and an increasing function of ω ( Z, θ ), ashypothesized in the text. The fully static solution associated to (291) is: T ( Z, Z ) = λτ h ( Z, Z )1 + D ( Z ) C ( Z ) h D ( ω ( Z )) h C ( ω ( Z )) We conclude this section by giving the linearized version of (290) around the static solution ( ω ( Z ) , T ( Z, Z )).It is: 0 = ∇ θ ˆ T ( Z, θ, Z ) λ + U ( ω ) ∇ θ ˆ T ( Z, θ, Z ) + U ( ω ) ˆ T ( Z, θ, Z ) (292) − ρC ( Z ) − ˆ T ( Z, Z ) λτ ! h ′ C ( ω ( Z )) Ω ( Z, θ )+ ρ ˆ T ( Z, Z ) λτ (cid:18) D ( Z ) h ′ D ( ω ( Z )) Ω (cid:18) Z , θ − | Z − Z | c (cid:19)(cid:19) where: U ( ω ) = 1 λτ + ρλ ( C ( Z ) h C ( ω ( Z )) + D ( Z ) h D ( ω ( Z ))) U ( ω ) = ρλτ ( C ( Z ) h C ( ω ( Z )) + D ( Z ) h D ( ω ( Z ))) T ( Z, Z ) = λτ h ( Z, Z )1 + D ( Z ) C ( Z ) h D ( ω ( Z )) h C ( ω ( Z )) ˆ T ( Z, Z ) = T ( Z, Z ) h ( Z, Z )ˆ T ( Z, θ, Z ) = T ( Z, θ, Z ) − T ( Z, Z ) h ( Z, Z )Ω ( Z, θ ) = ω ( Z, θ ) − ω ( Z )for ω ( Z ) ≡ ω , this reduces to: ∇ θ ˆ T ( Z, θ, Z ) λ + U ( ω ) ∇ θ ˆ T ( Z, θ, Z ) + U ( ω ) ˆ T ( Z, θ, Z ) (293)= − ρ ˆ T ( Z, Z ) D ( Z ) h ′ D ( ω ) Ω (cid:16) Z , θ − | Z − Z | c (cid:17) λτ + ρC ( Z ) − ˆ T ( Z, Z ) λτ ! h ′ C ( ω ) Ω ( Z, θ )158here: U ( ω ) = 1 λτ + ρλ ( Ch C ( ω ) + Dh D ( ω )) U ( ω ) = ρλτ ( Ch C ( ω ) + Dh D ( ω )) T ( Z, Z ) = λτ h ( Z, Z )1 + DC h D ( ω ) h C ( ω ) C = C + ω τ C D = D + ω τ D which can also be written, up to the second order in derivatives: ∇ θ ˆ T ( Z, θ, Z ) λ + U ( ω ) ∇ θ ˆ T ( Z, θ, Z ) + U ( ω ) ˆ T ( Z, θ, Z ) (294)= ρC ( Z ) h ′ C ( ω ) − ρ ˆ T ( Z, Z ) ( D ( Z ) h ′ D ( ω ) + C ( Z ) h ′ C ( ω )) λτ ! Ω (
Z, θ )+ ρ ˆ T ( Z, Z ) D ( Z ) h ′ D ( ω ) (cid:16) | Z − Z | c ∇ θ Ω (
Z, θ ) − ( Z − Z ) c ∇ θ Ω (
Z, θ ) − ( Z − Z ) ∇ Z Ω (
Z, θ ) (cid:17) λτ Then, to separate the dependences in time and position, we define:ˆ T ( Z, θ ) = Z h ( Z, Z ) ˆ T ( Z, θ, Z ) r π (cid:16) X r (cid:17) + π α ¯ C ( Z ) = 1 r π (cid:16) X r (cid:17) + π α Z h ( Z, Z ) C ( Z )¯ C ( Z ) = 1 r π (cid:16) X r (cid:17) + π α Z h ( Z, Z ) C ( Z ) ˆ T ( Z, Z )ˆ T ( Z ) = 1 r π (cid:16) X r (cid:17) + π α Z h ( Z, Z ) ˆ T ( Z, Z )and ˆ T ( Z, θ ) satisfies: ∇ θ ˆ T ( Z, θ ) λ + U ( ω ) ∇ θ ˆ T ( Z, θ ) + U ( ω ) ˆ T ( Z, θ )= ρ ¯ C ( Z ) h ′ C ( ω ) − ρ (cid:16) D ( Z ) ˆ T ( Z ) h ′ D ( ω ) + ¯ C ( Z ) h ′ C ( ω ) (cid:17) λτ Ω (
Z, θ )+ ρD ( Z ) h ′ D ( ω ) (cid:0) Γ ∇ θ Ω (
Z, θ ) − (cid:0) Γ ∇ θ Ω (
Z, θ ) + c Γ ∇ Z Ω (
Z, θ ) (cid:1)(cid:1) λτ
11 BibliographyReferences11 BibliographyReferences