A path integral for heavy-quarks in a hot plasma
aa r X i v : . [ h e p - ph ] M a y Preprint typeset in JHEP style - PAPER VERSION
A path integral for heavy-quarks in a hot plasma
A. Beraudo , , Centro Studi e Ricerche
Enrico Fermi ,Comprensorio del Viminale, Piazza del Viminale 1, Roma, ITALY; Physics Department, Theory Unit, CERN, CH-1211 Gen`eve 23, Switzerland; Dipartimento di Fisica Teorica dell’Universit`a di Torino andIstituto Nazionale di Fisica Nucleare, Sezione di Torino,Via Pietro Giuria 1, 10154 Torino, ITALYE-mail: [email protected]
J.P. Blaizot
Institut de Physique Theorique, CEA-Saclay, 91191 Gif-sur-YvetteE-mail: [email protected]
P. Faccioli , Dipartimento di Fisica dell’Universit`a di Trento and Istituto Nazionale di Fisica Nucleare, Sezione di Trento (Padova)Via Sommarive 14, I-38100 Povo, TrentoE-mail: [email protected]
G. Garberoglio , , CNISM, Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia; Interdisciplinary Laboratory for Computational Science (LISC), FBK-CMM andUniversity of Trento,via Sommarive 18, I-38123 Povo, TrentoE-mail: [email protected] – 1 – bstract:
We propose a model for the propagation of a heavy-quark in a hot plasma, to beviewed as a first step towards a full description of the dynamics of heavy quark systems in aquark-gluon plasma, including bound state formation. The heavy quark is treated as a nonrelativistic particle interacting with a fluctuating field, whose correlator is determined bya hard thermal loop approximation. This approximation, which concerns only the mediumin which the heavy quark propagates, is the only one that is made, and it can be improved.The dynamics of the heavy quark is given exactly by a quantum mechanical path integralthat is calculated in this paper in the Euclidean space-time using numerical Monte Carlotechniques. The spectral function of the heavy quark in the medium is then reconstructedusing a Maximum Entropy Method. The path integral is also evaluated exactly in the casewhere the mass of the heavy quark is infinite; one then recovers known results concerningthe complex optical potential that controls the long time behavior of the heavy quark. Theheavy quark correlator and its spectral function is also calculated semi-analytically at theone-loop order, which allows for a detailed description of the coupling between the heavyquark and the plasma collective modes.
Keywords:
Thermal Field Theory, Heavy Quark Physics. ontents
1. Introduction 12. A path integral for the heavy-quark propagator 3
3. One-loop calculation 13
4. Numerical results: MC simulations and MEM analysis 21
5. Conclusions 27A. An exactly solvable toy-model 29B. Details on the path integral implementation 36
1. Introduction
Understanding the dynamics of heavy quarks in a quark-gluon plasma, and the fate of theirpossible bound states, has remained a difficult issue, ever since the original proposal of Mat-sui and Satz [1] to view the dissolution of
J/ψ ’s mesons in a quark-gluon plasma as a signalof deconfinement (for a recent review see for instance Ref. [2]). Aside from many studiesbased on the assumption that the dominant effect of the plasma is to screen the interactionpotential, more recently, the problem has been attacked using a “first principle” approach,namely by calculating the QQ spectral functions reconstructed from the corresponding Eu-clidean correlators provided by lattice QCD. The melting of the J/ψ , for instance, is thensignaled by the disappearance of the corresponding peak in its spectral function. The firstresults of such an analysis led to the surprising result that the
J/ψ appears to survive tilltemperatures well above T c [3, 4, 5, 6, 7], in sharp contrast with studies based on screenedpotentials. A comparison between correlators and spectral functions evaluated on the lat-tice and within different potential models was attempted in [8, 9, 10], revealing ambiguities– 1 –n the whole procedure. Another line of first principle calculations was undertaken in anumber of recent papers [11, 12, 13, 14, 15, 16]: in these works, the correlator of a heavyquark pair is calculated directly in real time, revealing that the long time behavior of thedynamics can be encompassed by a Schr¨odinger equation with a complex potential thatdescribes both the effects of screening and, through its imaginary part, of the collisionswith the plasma particles.While it represents an important step forward, this description of the dynamics ofheavy quarks by a Schr¨odinger equation and an effective potential has limitations. Thepotential is calculated, and well defined, only in the limit of infinitely massive quarks.Moreover, a simple potential description emerges only at large times, that is, at time scalesthat are large compared to the typical times characterizing the response of the plasma toperturbations. In the situations which we want eventually to deal with, namely the fateof bound states of heavy quarks in the environment created in ultra-relativistic heavy ioncollisions, all relevant time scales are mixed (see for instance [17]), and a description of thedynamics beyond that provided by a simple Schr¨odinger equation is called for. This paperrepresents an attempt in this direction, building on the approach developed in [13]. Ourstrategy, already sketched in [18], is the following. The heavy quarks are treated as massive,non relativistic, particles coupled to a fluctuating gauge field. The dynamics of the heavyquark is then encoded exactly in a quantum mechanical path integral, while the averageover the gauge field fluctuations is entirely determined by the properties of the medium.If one restricts oneselves to approximations where this average is Gaussian, and hence canbe performed analytically, the gauge fields can be eliminated completely, leaving a pathintegral for a non relativistic particle with a non local (in space and time) self-interactionterm. This path integral is reminiscent of that introduced by Feynman in his treatmentof the “polaron” [19]. An approximation that leads to a Gaussian average (at least in theAbelian case), is the hard thermal loop approximation (HTL) [20]. We shall make useof such an approximation, because of its simplicity, and also because it encompasses thedominant plasma effects that one wants to include: screening effects, collective modes, andcollisions. We emphasize, however, that this approximation, which concerns primarily themedium in which the heavy quark propagates, can be improved without altering the basicstructure of the problem.The present paper has an exploratory character and represents only a first step in thislong-term goal. It focusses on the dynamics of a single heavy quark moving in a plasma oflight charged particles, for which we provide a simple model. We use for the quark-gluonplasma an idealization where only Abelian (in fact, Coulomb) interactions are taken intoaccount. We also assume, for simplicity, that the plasma particles are fermions, i.e., quarks.In short, we model the quark-gluon plasma by an electromagnetic plasma, treated withinthe HTL approximation. This is enough to take into account typical plasma effects, suchas screening, Landau damping of collective excitations, and collisions between the heavyquark and the plasma particles. These phenomena are characterized by a single scale,the Debye screening mass m D , to which, in our numerical studies, we shall give a valuecharacteristic of a quark-gluon plasma at a given temperature (thereby taking effectivelygluons into account). The dynamics of the heavy quarks is then treated exactly within a– 2 –ath integral of the type discussed above, with a non-local self-interaction whose space-timeproperties are controlled by the Debye mass.Our paper is organized as follows. In Sect. 2 we establish the general setting: the basicproperties of the propagator of a heavy particle are recalled, a description of the medium oflight charged particles in which the propagation takes place is given, the path integral forthe heavy quark propagator is constructed. This path integral is calculated exactly in thelimit of an infinitely massive quark, and known results are recovered concerning the longtime behavior of the heavy quark propagator in this limit. Then, in Sect. 3, we calculatethe heavy quark propagator in the one-loop approximation, providing a detailed analysis ofthe coupling of the heavy quark to the collective plasma excitations and of the role of thecollisions. We also calculate the spectral function and the resulting Euclidean correlator. InSect. 4 we present the results of the numerical evaluation of the path integral in Euclideanspace-time, using Monte Carlo (MC) techniques. We use the Maximum Entropy Method(MEM) to reconstruct the spectral density. Within our present implementation of thismethod, we can only reproduce, semi-quantitatively, the main features of the spectraldensity. Finally, Sect. 5 summarizes the conclusions. In Appendix A we give a self-contained presentation of an exactly solvable toy-model which captures general features ofthe heavy quark correlator and its spectral function, and this for any value of the couplingconstant. Appendix B provides details on the numerical evaluation of the path integral.
2. A path integral for the heavy-quark propagator
In this section, we recall general properties of the heavy quark propagator, and establishthe basic path integral that describes the dynamics of the heavy quark coupled to a gaugefield that is integrated out via a Gaussian averaging.
Most of the physical information that we are interested in can be obtained from the studyof the following correlator G > ( t, r | , ) ≡ h ψ ( t, r ) ψ † (0 , ) i , (2.1)where ψ ( t, r ) denotes the heavy quark field. In the following we shall most of the timeuse the simplified notation G > ( t, r ) for G > ( t, r | , ). The expectation value in the aboveformula is a thermal average over the states of light particles (with no heavy quark present)that will be specified later. At this stage, we simply note that the full Hamiltonian H canbe decomposed into three contributions: H = H Q + H med + H int , (2.2)where H Q is the (non relativistic) Hamiltonian describing the heavy quark in vacuum, H med is the Hamiltonian of the medium in which the heavy quark propagates, and H int A somewhat similar model, with however different emphasis, was considered in Ref. [21]. – 3 –epresents the interactions between the medium and the heavy quarks. For the parts thatdepend explicitly on the fermion field, we have H Q = M Z d r ψ † ( r ) ψ ( r ) + Z d r ψ † ( r ) (cid:18) − ∇ M (cid:19) ψ ( r ) , (2.3)and H int = g Z d r ψ † ( r ) ψ ( r ) A ( r ) , (2.4)where A ( r ) represents the local electrostatic potential created by the light particles. Thefull Hamiltonian commutes with the number of heavy quarks N Q :[ H, N Q ] = 0 , N Q = Z d r ψ † ( r ) ψ ( r ) , (2.5)and one can classify its eigenstates according to the number of heavy quarks that theycontain. In particular, one may write a spectral decomposition of the correlator (2.1): G > ( t, r ) = X n, ¯ m e − βE n Z e i ( E n − E ¯ m ) t h n | ψ ( r ) | ¯ m ih ¯ m | ψ † ( ) | n i , (2.6)where the states | n i contain no heavy quark, while the states | ¯ m i contain one heavy quark,i.e., ψ ( r ) | n i = 0 , N Q | ¯ m i = | ¯ m i . (2.7)In eq. (2.6) Z is the partition function of the system without heavy quark. It follows alsofrom Eq. (2.5) that G < ( t, r ) ≡ h ψ † (0 , ψ ( t, r ) i = 0, so that the time-ordered propagator, G ( t ) ≡ i h T ψ ( t ) ψ † (0) i = i θ ( t ) G > ( t ) − i θ ( − t ) G < ( t ) , and the retarded propagator G R ( t ) ≡ i θ ( t ) [ G > ( t ) + G < ( t )] are identical, G ( t ) = G R ( t ) = i θ ( t ) G > ( t ) . Similarly, the spectraldensity is given by the Fourier transform of Eq. (2.1), namely: σ ( ω ) ≡ G > ( ω ) + G < ( ω ) = G > ( ω ) = Z ∞−∞ dt e iωt G > ( t ) , (2.8)with the inverse relation G > ( − iτ ) = Z + ∞−∞ dω π e − ωτ σ ( ω ) . (2.9)In this last equation we have exploited the analyticity of G > ( t ) in the strip − β < Im t < t = − iτ , with 0 < τ < β . Inverting this relation, namely calculating σ ( ω ) fromthe Euclidean correlator G > ( − iτ ) is a difficult (well known) problem that we shall addressbriefly in the last part of this paper.By noting that H med does not depend on ψ , one finds (with all fields in the Heisenbergrepresentation) [ ψ, H ] = [ ψ, H Q + H int ] = (cid:18) M − ∇ M + gA ( t, r ) (cid:19) ψ ( t, r ) , (2.10)so that, from the equation of motion i∂ t ψ ( t, r ) = [ ψ, H ] , we get i∂ t G > ( t, r ) = (cid:18) M − ∇ M (cid:19) G > ( t, r ) + g h A ( t, r ) ψ ( t, r ) ψ † ( ) i . (2.11)– 4 –n the absence of interactions, this equation has the familiar solution G > ( t, r ) = e − iMt (cid:18) M πit (cid:19) / e iM r / t , (2.12)corresponding to the initial condition G > ( t = 0 , r ) = δ ( r ). Note that this initial conditionstill holds in the presence of interactions, i.e., G > ( t = 0 , r ) = δ ( r ), as is easily verified. Afurther exact relation is obtained by considering the equation (2.11) at t = 0: i ∂ t G > ( t, r ) (cid:12)(cid:12) t =0 = (cid:18) M − ∇ M + g h A ( t = 0 , r ) i (cid:19) δ ( r ) . (2.13)Since the thermal average involves only states of the medium which do not contain heavyquarks that could polarize it, we have h A ( r ) i = (1 /Z ) P n e − βE n h n | A ( r ) | n i = 0: theinteractions do not contribute to the leading (linear) order in a small time expansion.Pushing this expansion to second order, one gets − ∂ t G > ( t, r ) (cid:12)(cid:12) t =0 = "(cid:18) M − ∇ M (cid:19) + g h A (0 , r ) i δ ( r ) , (2.14)or, taking a Fourier transform − ∂ t G > ( t, p ) (cid:12)(cid:12) t =0 = (cid:18) M + p M (cid:19) + g h A i , (2.15)where h A i stands for h A ( t = 0 , r = 0) i . Thus at order t , the effect of the interaction isgoverned by the size of the fluctuations of A , an intrinsic property of the medium to bediscussed further later. Note also that the coefficient of t in the expansion of G > ( t, p ) atsmall t is of order g .We may turn these relations for the derivatives of G > ( t, p ) at t = 0 into sum rulesfor the spectral density. From the initial condition (see after Eq. (2.12)), and the relations(2.13) and (2.14) above, one gets, respectively, Z ∞−∞ dω π σ ( ω, p ) = 1 , Z ∞−∞ dω π ωσ ( ω, p ) = M + p M , Z ∞−∞ dω π ω σ ( ω, p ) = (cid:18) M + p M (cid:19) + g h A i . (2.16)The last sum rule assumes that h A i is well defined. However, as we shall see in the nextsubsection, within the approximation used in the present paper h A i is in fact given by adivergent integral, so that this sum rule will not apply. Accordingly the short time behaviorof the correlator will not have a simple Taylor expansion as assumed in the discussion above(Eq. (2.15)).The Fourier transform of G > ( t, r ) used above (see Eq. (2.15)) is of the form G > ( t, p ) = Z d r e − i p · r G > ( t, r ) = h a p ( t ) a † p i , (2.17)– 5 –here a p and a † p are the Fourier transform of the field operators ψ ( r ) and ψ † ( r ), respec-tively, and we have used the translation invariance of the medium in order to implementthe conservation of the total momentum ( h a p a † p ′ i ∼ δ p , p ′ ). For the value t = − iβ , thecorrelator G > ( − iβ, p ) yields the difference of the free energies of the systems with andwithout a heavy quark. To see that, note that this free energy difference is given byexp[ − β ∆ F Q, p ] = 1 Z X n h n | a p e − βH a † p | n i = 1 Z X n e − βE n h n | a p ( β ) a † p (0) | n i = G > ( − iβ, p ) . (2.18)In the first line of Eq. (2.18), the states a † p | n i , while not eigenstates of H , constitute abasis of states with momentum p and containing one heavy quark. Thus, the sum over thestates | n i in the first line of Eq. (2.18) is indeed the partition function for the system withone heavy quark and total momentum p .In preparation for the forthcoming discussion, let us recall that the propagator of theheavy quark, treated as a non relativistic quantum mechanical particle, may be given apath integral representation [19]. With A ( t, x ) considered as a given external potential,we can write ( t > G > ( t, r ) = Z r D z exp (cid:20) i Z t dt ′ (cid:18) M ˙ z − gA ( t, z ) (cid:19)(cid:21) , (2.19)where the symbol R r D z indicates a path integration over paths z ( t ) such that z (0) = 0and z ( t ) = r . The transcription of this expression in imaginary time reads ( τ > G > ( − iτ, r ) = Z r D z exp (cid:20) − Z τ dτ ′ (cid:18) M ˙ z + igA E ( τ, z ) (cid:19)(cid:21) , (2.20)where, aside from making the familiar substitution t → − iτ , we have also introduced theEuclidean field A E ( τ, r ) = − iA ( t = − iτ, r ). The medium is modeled by a plasma of light fermions with Coulomb interactions. Becauseof its large mass, the heavy quark has a small velocity, and consequently its ability to inducemagnetic excitations of the medium is small; accordingly these magnetic excitations willbe ignored. The Hamiltonian reads then H med = Z d r ξ † ( r ) h ξ ( r ) + 12 Z d rd r ′ ˆ ρ ( r ) g π | r − r ′ | ˆ ρ ( r ′ ) , (2.21)where ξ ( r ) and ξ † ( r ) denote the field operators of the light fermions, ˆ ρ ( r ) ≡ ξ † ( r ) ξ ( r ) isthe charge density of the light particles, and h their free Hamiltonian.To the full Hamiltonian of the system corresponds an Euclidean action of the form S = S Q + S int + S med , with S med = Z x ξ ∗ ( x )( ∂ τ + h ) ξ ( x ) + g Z x,x ′ ρ ( x ) K ( x, x ′ ) ρ ( x ′ ) , (2.22)– 6 –nd S int = g Z x,x ′ ρ Q ( x ) K ( x, x ′ ) ρ ( x ′ ) . (2.23)We have set ρ Q ( x ) = ψ ∗ ( τ, r ) ψ ( τ, r ), and Z x ≡ Z d x ≡ Z β dτ Z d r, x ≡ ( τ, r ) . (2.24)The operator K ( x, x ′ ) is given by −∇ r K ( x, x ′ ) = δ ( x − x ′ ) , K ( x, x ′ ) = δ ( τ − τ ′ ) 14 π | r − r ′ | . (2.25)In calculating the partition function of the system, one can proceed in a familiar way, andintegrate over the light fermions after eliminating their density ρ ( x ) in favor of a gaugepotential A E ( x ) ( − igK · ( ρ + ρ Q ) → A E ). One then obtains Z D ( ξ ∗ , ξ ) e − ( S int + S med ) = Z D A E e − S [ A E ] , (2.26)where S [ A E ] = ig Z x A E ( x ) ρ Q ( x ) − Tr ln( ∂ τ + h + igA E ) + 12 Z x,x ′ A E ( x ) K − ( x, x ′ ) A E ( x ′ ) , (2.27)and the field A E obeys periodic boundary conditions in imaginary time, A E (0 , r ) = A E ( β, r ), reflecting the fact that the medium of light particles is in thermal equilibrium attemperature T = 1 /β .At this point we perform the main approximations that will yield a simple model forthe medium. In the expansion of the fermionic determinant (the second term in the r.h.s. ofEq. (2.27)) in powers of A E , we keep only the quadratic term. Furthermore, we keep onlythe leading high temperature approximation to the corresponding 2-point function (theso-called hard-thermal-loop (HTL) approximation [20]). Note that in the case of QED, theHTL approximation automatically truncates the expansion of the determinant at quadraticorder. Further discussion of the validity of this approximation will be made shortly. Atthis point we note that once it is done, we are left with a simple quadratic action: S [ A E ] = ig Z x A E ( x ) ρ Q ( x ) + 12 Z x,x ′ A E ( x ) ˜∆ − ( x, x ′ ) A E ( x ′ ) . (2.28)The propagator ˜∆( x, x ′ ) = h A E ( x ) A E ( x ′ ) i is given in Fourier space by ˜∆ − ( z, q ) = q +Π( z, q ), where Π( z, q ) is the (longitudinal) polarization tensor in the Coulomb gauge:Π( z, q ) = m D (1 − Q ( z/q )) , Q ( x ) ≡ x x + 1 x − , (2.29)with m D = Π( z = 0 , q ) is the Debye mass. The Debye mass is the mass scale thatcharacterizes the response of the medium. – 7 – D ∆ ( τ , r) / T τ/β=0τ/β=0.1τ/β=0.2τ/β=0.3τ/β=0.4τ/β=0.5 Figure 1:
The function ∆( τ, r ) as a function of r/r D = rm D for different values of τ /β (decreasingfrom bottom to top ). Note that as long as τ = 0, ∆( τ, r = 0) is finite. However, ∆(0 , r ) divergeslogarithmically as r → At this point, we note that the equations we have written hold exactly only for thehot electromagnetic plasma. However, at this level of approximation, the main differencewith a quark-gluon plasma lies in the value of the Debye mass that, in a QCD plasma,receives also contributions from gluons. In the numerical studies to be presented below, inorder to get orders of magnitudes that are relevant for the quark-gluon plasma, we shalladjust the Debye mass to the value it would have in a quark-gluon plasma at the consideredtemperature, that is we shall use the QCD HTL expression m D = g s T ( N c / N f / g s / π = α s the strong coupling constant. With α s = 0 . N c = 3 and N f = 2, thisyields a value m D = 713 MeV for T = 300 MeV. Furthermore, the coupling of the heavyquark to the plasma particles involves g s / π multiplied by the Casimir factor C F = 4 / C F into the coupling g , denoting the product g s C F / π by α = g / π . Thus a coupling constant α = 0 . α s = 0 . − ( z, q ) introduced above contains all the information about thescreening phenomena and the response of the medium to the presence of the heavy quark.It differs by a sign from the longitudinal gluon propagator in the HTL approximation(called ∆ L in Ref. [13]). It is convenient to subtract from the latter the instantaneousCoulomb interaction which would contribute here only to the self-interaction of the heavyquark. Thus we define ∆( z, q ) = − (cid:18) q + Π( z, q ) − q (cid:19) . (2.30)This new object ∆( z, q ) is proportional to χ ( z, q ), the density-density correlation functionof the medium: ∆( z, q ) = (1 /q ) χ ( z, q ). One has, in a mixed representation:– 8 –( τ, q ) = Z + ∞−∞ dq π e − q τ ρ L ( q , q )[ θ ( τ ) + N ( q )] , (2.31)where the spectral function ρ L ( ω, q ) reads [13] ρ L ( ω, q ) ≡ π (cid:8) Z L ( q ) [ δ ( ω − ω L ( q )) − δ ( ω + ω L ( q ))] + θ ( q − ω ) β L ( ω, q ) (cid:9) . (2.32)It displays two types of contributions: A continuum term arising from the imaginary partdeveloped by the logarithm in Eq. (2.29) for space-like momenta, and which correspondsphysically to scattering processes, and a pole term, coming from the solution, for time-likemomenta, of q + Π( ω L ( q ) , q ) = 0 , (2.33)which corresponds to an undamped plasma oscillation. Note that the residue Z L ( q ) quicklydies out as q grows beyond m D : Z L ( q ) ∼ qm D exp (cid:18) − q + m D m D (cid:19) , q ≫ m D . (2.34)Collective modes exist only for q < ∼ m D .The approximation that we are using to describe the medium to which the heavyquark is coupled is motivated by its simplicity, and also by the fact that it encompasses theimportant physical phenomena that characterizes weakly coupled plasmas, and that onewants to take into account: polarization and screening effects, collisions with the plasmaparticles. The latter, however, are not treated properly in the HTL approximation, andthis will introduce (small) unphysical features in our results. As a concrete illustration ofthe difficulty we are referring to, consider the function ∆( τ, r ) that will play a central rolein our calculations. This function can be obtained by a Fourier transform of Eq. (2.31) overspatial momentum, and it is displayed in Fig. 1. As indicated in the caption of this figure,∆(0 , r ) is logarithmically divergent as r →
0. This divergence is that of the fluctuation h A i = ∆(0 , h A i = Z d q (2 π ) ρ L ( q , q ) N ( q ) , (2.35)and comes form the continuum part of the spectral function (the contribution of the plas-mon is finite, due to the vanishing of the residue for large wave-numbers; see Eq. (2.34)).As already mentioned, the continuum part of the spectral function describes scatteringprocesses, involving space-like gluons with energy ω , momentum q . In the HTL approxi-mation, the phase space for these processes is given by | ω | ≤ q (see Eq. (2.32)), i.e., it growswithout bound as q increases, leading eventually to a divergence. An analogous divergencealso occurs in the pair correlation function at short distance when this is calculated usingthe Vlasov equation (which is equivalent to the HTL approximation [22]). This is a wellknown difficulty in plasma physics (see e.g. [23]), and it can be cured by a better treatmentof the collisions involving large momentum transfer. Indeed, the HTL approximation isvalid only in the regime where q ≪ p , where p ∼ T is a typical loop momentum (i.e., thetypical momentum of the colliding plasma particles). A proper treatment of the collisions– 9 –ith q ∼ p would lead to a modified phase space and a finite value for h A i . For instance,in a full one-loop calculation, the phase space is given by − q < ω < q for q < ∼ p , but q − p < ω < q for q > p . A possible way to improve the calculation would be to introducea cut-off to separate soft and hard contributions, and apply in each sector appropriate ap-proximations. We shall not do so here, because ∆( t, r ) enters the calculation of the heavyquark correlator only through an integral so that its logarithmic singularity is tamed, andits physical consequences mild. We note however that the divergence of h A i modifies thesmall τ behavior of the heavy quark propagator, and in particular it invalidates the Taylorexpansion dicussed at the end of Sect. 2.1, beyond the linear order. We are now in position to write the propagator of the heavy quark in the form of a pathintegral. Gathering the results of the first two sections, we can write G > ( − iτ, r ) = Z D A E exp (cid:20) − Z x,x ′ A E ( x ) ˜∆ − ( x, x ′ ) A E ( x ′ ) (cid:21) × Z r D x exp (cid:20) − Z τ dτ ′ (cid:18) M ˙ x + ieA E ( τ, x ) (cid:19)(cid:21) , (2.36)This path integral summarizes the model that we use. The dynamics of the heavy quarkin a hot plasma is that of a non relativistic quantum particle moving in a fluctuatingfield A , and this is treated exactly by the path integral. The approximations only enterthe description of the fluctuations of the field A which we assume to be Gaussian and,as we have just discussed, dominated by long wavelengths and low frequencies. Thus,any improvement of the description of the medium will affect only the first part of thefunctional integral (2.36), that is the weight of the integration over the field A , but it willleave intact the second part describing the motion of the heavy quark in the fluctuatingfield. This is an important feature of the present description.As we mentioned earlier, it is convenient to subtract from the correlator ˜∆ the con-tribution of the Coulomb interaction. This is most easily done after having performed theGaussian integral over A E , whence we can just replace ˜∆ by − ∆. One gets G > ( − iτ, r ) = Z r D z e − S [ z ,τ ] , (2.37)where S [ z , τ ] = S [ z , τ ] − ¯ F [ z , τ ], with S [ z , τ ] = Z τ dτ ′ M ˙ z , (2.38)and ¯ F [ z , τ ] = g Z τ dτ ′ Z τ dτ ′′ ∆( τ ′ − τ ′′ , z ( τ ′ ) − z ( τ ′′ )) . (2.39)The real time version of this path integral is obtained by replacing τ by it , and substituting − S [ z , τ ] in Eq. 2.37 by iS [ z , t ] = i ( S + F ) with F [ z , t ] = g Z t dt ′ Z t dt ′′ D ( t ′ − t ′′ , z ( t ′ ) − z ( t ′′ )) , (2.40)– 10 –nd we have used [13] ∆( τ = it, r ) ≡ − iD ( t, r ) . (2.41)The correlator G > ( t, r ), when expressed in terms of the dimensionless variables tT and r T is of the form G > ( t, r ) = T f ( M/T, m D /T, tT, r T ), with f a dimensionless function.When m D → m D /T is fixed, and G > ( t, r ), when t and r are expressed in units of the inverse temperature,depends only on the ratio M/T . We shall refer to this scaling property of the correlatorrepeatedly.The parameter
M/T controls the “diffusion”, described by the correlator (2.12) inimaginary time: the smaller
M/T , the more the heavy quark will move away form theorigin in a given time. Note that this diffusion inhibits the effects of the interaction:because ∆( τ, z ) < ∆( τ,
0) (see Fig. 1), the interaction favors paths for which z remainssmall (their weight in Eq. (2.37) is largest).One may also understand the effect of the interaction in the following way. The heavyquark produces a polarization cloud of light particles around itself. This induced chargescreens that of the heavy quark over a distance scale of order m − D . When the heavy quarkmoves, its polarization cloud tries to adjust and follow its motion, but this takes time(of order m − D ). The heavy quarks sees then a restoring force produced by the laggingpolarization cloud, which limits its excursion.In the limit M/T → ∞ , studied in detail in the next subsection, the heavy quarkis frozen at it initial location: there is then no diffusion, and the effect of interactions ismaximal.
When M → ∞ , the path integral can be calculated exactly. This is because, in this limit,the motion of the heavy quark is frozen and F becomes independent of the coordinates.Thus, in the infinite mass limit, the heavy quark correlator takes the form G > ( t, r ) = δ ( r ) e − iMt e iF ( t ) (2.42)where the function F ( t ) is the functional (2.40) restricted to z = 0: F ( t ) = g Z t dt ′ Z t dt ′′ D ( t ′ − t ′′ , . (2.43)The factor exp ( iF ( t )) mulitplying in Eq. (2.42) the infinite mass limit of the free correlator(2.12), summarizes the effect of the interactions. One can express F ( t ) in terms of theFourier transform D ( ω, q ) of the (time-ordered) gluon propagator [13]: D ( ω, q ) = Z dq π ρ L ( q , q ) q − ( ω + iη ) + iρ L ( ω, q ) N ( ω ) . (2.44)One gets F ( t ) = g Z dω π − cos( ωt ) ω Z d q (2 π ) D ( ω, q ) . (2.45)– 11 –t follows that at short times F ( t ) ≃ g t Z dω π Z d q (2 π ) D ( ω, q ) = g t D ( t = 0 , r = 0) . (2.46)For large time we use lim t →∞ − cos( ωt ) ω = πtδ ( ω ) . (2.47)to obtain F ( t ) ≃ g tD ( ω = 0 , r = 0) ≡ − tV opt . (2.48)An alternative way to obtain this result is to start directly from Eq. (2.43) and to changevariables t ′ − t ′′ → u, ( t ′ + t ′′ ) / → T , and to observe that at large time t , one may integratefreely over u , thereby filtering out the zero frequency part of D ( ω, q ). This yields againEq. (2.48). Thus, the large time ( t ≫ m − D ) behavior of the system is determined by thestatic ( ω = 0) response of the medium. Since, at large times, F is linear in time, Eq. (2.11)for G > ( t, r = 0) is a closed equation that takes the form of a Schr¨odinger equation [13],with an “optical potential” V opt given by V opt ≡ − g Z d q (2 π ) D ( ω = 0 , q )= g Z d q (2 π ) h q + m D − q − i πm D T | q | ( q + m D ) i = − α m D − i αT , (2.49)where we have used the susceptibility sum rule Z ∞−∞ dq π ρ L ( q , q ) q = m D q ( q + m D ) (2.50)in order to perform the q integral needed to calculate D ( ω = 0 , q ). Thus, the opticalpotential involves a real correction to the mass of the heavy quark, and an imaginary partthat takes into account the coupling of the heavy quark to the complex configurations ofthe medium. Alternatively, one may view this imaginary part as reflecting the collisionsof the heavy quark with the particles of the medium. As we shall see in the next section, V opt can be identified with the one-loop on-shell self energy in the infinite mass limit.This imaginary part does not appear in the Euclidean correlator calculated at τ = β ,which exhibits only the mass shift [13]: − T ln G > ( t = − iβ, p ) = M − α m D = M − g iω n = 0 , r = 0) . (2.51)Note that in the present, infinite mass, limit, G > ( t = − iβ, p ) in fact does not depend on p .More generally, the Euclidean correlator has the form (2.42) with iF ( t ) replaced by ¯ F ( τ ),with (see Eq. (2.39)): ¯ F ( τ ) = g Z τ dτ ′ Z τ dτ ′′ ∆( τ ′ − τ ′′ , ) . (2.52)– 12 –ince the dominant effect of the interactions can be characterized by the free energy shift(2.51), it is convenient to separate the corresponding linear growth of ¯ F ( τ ), and write¯ F ( τ ) = ¯ F ( τ ) + ¯ F ( τ ) , (2.53)with ¯ F ( τ ) = g τ Z d q (2 π ) Z dq π ρ L ( q , q ) q = τβ ¯ F ( β ) , (2.54)and ¯ F ( τ ) = g Z d q (2 π ) Z dq π ρ L ( q , q ) q cosh( q ( τ − β/ − cosh( βq / βq / . (2.55)The function ¯ F ( τ ) vanishes at τ = 0 and τ = β . It is symmetric around τ = β/
2, aproperty that follows immediately from the fact that ∆( τ, q ) depends only on | τ | , and isperiodic, ∆( β, q ) = ∆(0 , q ). Note also that the slope of ¯ F ( τ ) at the origin is equal andopposite to that of ¯ F ( τ ), since that of ¯ F ( τ ) vanishes.Many of (but not all) the features of the present M → ∞ limit are shared by the toymodel presented in Appendix A, where one can find a more extended discussion of someof the points addressed in this subsection.
3. One-loop calculation
In this section, we present the results of the one-loop calculation of the heavy quark cor-relator. This provides insight into the dynamics of the heavy quark when the interactionis weak enough for perturbation theory to be applicable. All the numerical results to bepresented are obtained with the value α = 0 . M limit that we have just discussed, will serve asa reference when discussing the results of the Monte Carlo evaluation of the heavy-quarkcorrelator in the next section.The one-loop calculation is easier in momentum space than in coordinate space. Toproceed we consider the analytic propagator G ( z, p ) = − z − E p − Σ( z, p ) , (3.1)where E p = M + p / M , and the one-loop self-energy Σ( z, p ) is given by the diagramdisplayed in Fig. 2. The retarded propagator is obtained as usual by setting z = ω + iη , with ω real. The imaginary part of the retarded propagator yields the heavy-quark spectralfunction σ ( ω, p ) ≡ G R ( ω, p ) = Γ( ω, p )[ ω − E p − Re Σ( ω, p )] + Γ ( ω, p ) / , (3.2)where Γ( ω, p ) ≡ − R ( ω, p ) = − z = ω + iη, p ). Eventually the Euclidean correlatorwill be calculated using Eq. (2.9). Since we shall consider only the case p = 0, we shall usein most of this section the simplified notation Σ( z ) for Σ( z, p = 0), and similarly for otherrelated functions. The relations (3.1) and (3.2) are general, but in the rest of this section, G and Σ will refer to one-loop quantities (unless stated otherwise).– 13 – z Figure 2:
The one-loop self-energy diagram for the heavy quark. The blob on the interaction linereminds that the latter represents a resummed HTL propagator of a longitudinal gluon.
A standard calculation, implementing approximations that are valid when
T /M ≪
1, yieldsthe analytic one-loop self-energyΣ( z ) = g Z d k (2 π ) Z + ∞−∞ dk π ρ L ( k , k ) 1 + N ( k ) z − E k − k . (3.3)Expressing momenta and energies in units of T , one sees that Σ( z ) is a function of theform Σ( z ) = T f ( z/T, M/T, m D /T ). At fixed value of the coupling constant, m D /T isa constant, so that, the only relevant parameter is the ratio T /M , as we have alreadymentioned.By using the explicit expression for the gluon spectral function ρ L ( k , k ) given inEq. (2.32), one can re-write Eq. (3.3) asΣ( z ) = g Z d k (2 π ) (cid:26) Z L ( k ) (cid:20) N ( ω L ( k )) z − E k − ω L ( k ) + N ( ω L ( k )) z − E k + ω L ( k ) (cid:21) ++ Z k dk π π β L ( k , k ) (cid:20) N ( k ) z − E k − k + N ( k ) z − E k + k (cid:21)(cid:27) . (3.4)This expression exhibits two types of contributions that are are illustrated in Fig. 3: a polecontribution whose energy denominators are associated with processes of emission or ab-sorption of collective plasmons by the heavy quark, and a continuum contribution comingfrom the continuum part of the gluon spectral density; the latter contribution representsthe effect of collisions between the heavy quark and the particles of the medium, medi-ated by space-like gluons. It is convenient to evaluate separately these two contributions.Accordingly, we set Γ( ω ) = Γ pole ( ω ) + Γ cont ( ω ).For the pole contribution one gets:Γ pole ( ω ) = g π ( k | E ′ k + ω ′ L ( k ) | Z L ( k ) [1 + N ( ω L ( k ))] ++ X k k | E ′ k − ω ′ L ( k ) | Z L ( k ) N ( ω L ( k )) , (3.5)– 14 – ω ω ω E k E E E k k k kk ω ω LL (k) (k) (a) (b) (c) (d) Figure 3:
The different processes contributing to the imaginary-part of the heavy-quark self-energy: (a)-(b) emission-absorption of a plasmon and (c)-(d) collisions with the plasma particles,mediated by one-gluon exchange. where k and k are implicit functions of ω given by ω = E k + ω L ( k ) , ω = E k − ω L ( k ) , (3.6)and the primes in the denominators of Eq. (3.5) denote derivatives with respect to k .Here ω L ( k ) is the plasmon dispersion relation (see Eq. (2.33)), whose behavior for smallmomenta reads: ω L ( k ) ∼ k ≪ m D ω + 35 k ⇒ ω L ( k ) ∼ k ≪ m D ω pl + 310 k ω pl . (3.7)The solutions of Eqs. (3.6) can be read out from Fig. 4 where the two curves E k ± ω L ( k ) areplotted as a function of k . The first equation (3.6), ω = E k + ω L ( k ), has a single solutionstarting from the plasmon-emission threshold ω = M + ω pl . The number of solutions ofthe second equation depends on the ratio ω pl /M . From Eqs. (3.6) and (3.7) one sees thatfor ω pl > M the dispersion relation ω = E k − ω L ( k ) starts with positive curvature andit contributes to Γ pole with a single solution starting from ω > M − ω pl . In the case ofinterest, T /M ≪
1, we have ω pl < M , and there are two solutions for M/ < ∼ ω < M − ω pl and only one for ω > M − ω pl . Note however that any contribution corresponding to k ∼ M is damped by the plasmon-residue. As a result, there is effectively no pole termcontribution for M − ω pl < ω < M + ω pl , as it can be seen from the left panel of Fig. 5.From Fig. 4 one realizes also that there are values of ω for which the dispersion relations ω = E k ± ω L ( k ) display stationary points which, because of the denominators in Eq. (3.5),lead to singularities in Γ pole ( ω ) ( Van-Hove singularities ), clearly visible in the left panel ofFig. 5. Notice that, as the ratio
T /M gets larger, the one occurring at ω ∼ M/ β L ( k , k ) which has supportfor | k | ≤ k . It follows then from Eq. (3.4) that the continuum contribution to the imaginarypart of Σ comes from values of k, ω such that − k ≤ ω − M − k M ≤ k . The boundaries of– 15 – ω / M Landau-dampingplasmon T=300 MeVplasmon T=600 MeV
Figure 4:
The dashed curves represent, for two different temperatures, the functions E k ± ω L ( k ),with E k = M + k M , and ω L ( k ) the plasma dispersion relation. For k = 0, ω = M ± ω pl , with ω pl = m D / √ M = 1 . ω pl = 412 MeV for T = 300 MeV. The two full lines delineate the support of the continuum partof the gluon spectral function, that is the region − k ≤ ω − M − k M ≤ k . The largest temperature T = 600 MeV corresponds to a plasma frequency ω pl = 824 MeV, very close to the “critical value”3 / M discussed in the text. -1 -0.5 0 0.5 1 1.5 2 ω -M (GeV)00.10.20.30.4 Γ po l e ( ω ) ( G e V ) T=200 MeVT=300 MeVT=500 MeV -1 -0.5 0 0.5 1 1.5 2 2.5 3 ω -M (GeV)00.050.10.150.2 Γ c on t ( ω ) ( G e V ) T=200 MeVT=300 MeVT=500 MeV
Figure 5:
The pole (left) and continuum (right) contribution to Γ( ω ) for a quark mass M = 1 . ω pl = 275, 412 and 687 MeV for the temperatures T = 200, 300 and 500 MeV, respectively.Notice, in the pole contribution, the Van-Hove singularities and the gap, for M − ω pl < ω < M + ω pl ,that increases with temperature. The continuum contribution grows linearly with temperature, andthe threshold at ω = M/ this domain are displayed in Fig. 4, and Γ cont ( ω ) is given byΓ cont ( ω ) = g π ( θ ( ω − M ) Z M + M ( ω ) − M + M ( ω ) k dk β L ( ω − E k , k ) [1+ N ( ω − E k )] ++ θ ( ω − M/ θ ( M − ω ) Z M + M ( ω ) M − M ( ω ) k dk β L ( ω − E k , k ) [1+ N ( ω − E k )] ) , (3.8)– 16 – ω− M (GeV)00.050.10.150.20.250.30.350.40.450.5 Γ ( ω ) ( G e V ) M=1.5 GeVM=4.5GeVM=45 GeVM=infiniteStatic result -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ω -M (GeV)-0.5-0.4-0.3-0.2-0.100.10.2 R e Σ ( ω ) ( G e V ) M=1.5 GeVM=4.5 GeVM=45 GeVM=infinitestatic resulty= ω -M Figure 6:
Imaginary part (left) and real part (right) of the self-energy Σ. The horizontal lines,labelled “static limit”, indicate the values of Γ( M → ∞ ) and ReΣ( M → ∞ ). With the parameters α = 0 . T = 300 Mev, these are respectively 120 MeV and -143 MeV. Within the gap ± ω pl , Γ isan increasing function of M , while ReΣ is a decreasing function of M . Both functions nearly reachthe infinite mass limit when M = 45 GeV. where M ( ω ) ≡ p M +2 M ( ω − M ). Note in particular the lower threshold at ω = M/ k = M of the lower boundary of the support displayedin Fig. 4. This is clearly visible in the plot of Γ cont ( ω ) in the right hand panel of Fig. 5.A similar analysis can be done for the real part of the self-energy. This will not bedetailed here. We just present in Fig. 6 the result of the full calculation of the imaginarypart (left panel) and the real part (right panel) of Σ, for different values of the heavy quarkmass, including the limiting case of infinite mass. Let us recall that in the latter case, wehave analytic results [13] for the on-shell values (corresponding to ω = M ). From Eq. (3.4)one sees that only the continuum part contributes to the on-shell imaginary partImΣ R ( M → ∞ ) = − g k → Z d k (2 π ) N ( k ) ρ L ( k , k ) = − αT ⇒ Γ( M → ∞ ) = αT, (3.9)while the real part receives contribution from both parts of Σ( z ):ReΣ( M → ∞ ) = − g Z d k (2 π ) Z + ∞−∞ dk π ρ L ( k , k ) k = − αm D . (3.10)These values, which coincide with the values obtained for the exact “optical potential”in Eq. (2.49), are indicated by the horizontal lines (labelled as “static result”) in Fig. 6,while the curves representing the full expressions of ReΣ( ω ) and Γ( ω ) in the infinite masslimit are labelled as “M=infinite”. One sees from this figure that the infinite mass limit isnearly attained for M = 45 GeV, and that finite mass effects do not change the qualitativebehavior of the self-energy. To get a quantitative measure of these finite mass effects, wedetermine the shift δM = M ′ − M of the heavy quark mass as given by the solution of theequation M ′ − M = ReΣ( M ′ ) . (3.11)– 17 – ω− M (GeV)0102030405060 σ ( ω ) ( G e V - ) M=1.5 GeVM=4.5 GeVM=45 GeVM=infinite
Figure 7:
One-loop spectral function as a function of ω − M , for various values of the heavy quarkmass, and a fixed tempearture T = 300 MeV. The curve corresponding to M = 45 GeV is hardlydistinguishable from that representing the one-loop infinite mass limit. The smaller the mass M ,the smaller the shift of the main peak. This can be obtained graphically, as the intersection of the line ω − M with Σ( ω ) in Fig. 6.Values of the mass shift δM thus obtained are reported in Table 3.1. T /M δM/T -0.407 -0.4 -0.357 -0.335 -0317 -0.288∆ F Q /T -0.416 -0.409 -0.362 -0.336 -0.318 -0.274ReΣ( M ) /T -0.476 -0.457 -0.41 -0.38 -0.357 -0.326 Table 1:
The mass shift δM obtained from the solution of Eq. (3.11), the one-loop free energyshift ∆ F Q , and the real part of the on-shell self-energy Re Σ( M ) (which equals the exact energyshift in the infinite mass limit), as a function of T /M . One sees from this table that the larger the ratio
T /M , the smaller the mass-shift.This is in line with what one expects from the effects of diffusion that increase as
T /M increases, and inhibits the effect of the interaction. Note also that the mass shift obtainedas the solution of Eq. (3.11) is numerically very close to the free energy shift calculatedfrom the Euclidean correlator G ( − iβ ). It is smaller (in absolute value) than Re Σ( ω = M ),as can be also directly seen in Fig. 6. The spectral density can be readily calculated from the real and imaginary parts of theself-energy (see Eq. (3.2)). It is displayed in Fig. 7. The dominant feature is the existence ofa main peak, approximately located at the value of ω = M ′ , with M ′ given by Eq. (3.11),– 18 – τ/β l n [ G ( τ ) / e xp (- M τ )] M=1.5 GeV (one-loop)M=4.5 GeV (one-loop)M=45 GeV (one-loop)M=infinite (one-loop)M=infinite (exp) τ/β -0.1-0.08-0.06-0.04-0.020 l n [ G ( τ ) / e xp (- M τ )] s y mm M=1.5 GeV (one-loop)M=4.5 GeV (one-loop)M=45 GeV (one-loop)M=infinite (one-loop)M=infinite (exp)
Figure 8:
The quantity ¯ F (1 l ) ( τ ) (see Eq. 3.12)) for various values of the mass and fixed T = 300MeV. For the M = ∞ case we also plot the function ¯ F ( τ ) (the curve labelled “exp”), so as to get ameasure of the accuracy of the one-loop approximation. The value of the free energy shift ∆ F canbe read on the left panel as the value of ¯ F (1 l ) ( β ) and is reported in Table 3.1. In the right panel weplot the same quantities after subtracting the linear τ -dependence driven by the free-energy, thatis the function ¯ F (1 l )2 ( τ ) in Eq. (3.13). as can be expected on general grounds from Eq. (3.2). In addition to the main peak,two secondary bumps appear in the spectrum at values of the energy ω ≈ M ± ω pl , andcome from the energy dependence of the imaginary part of Σ discussed in the previoussubsection. The spectral density satisfies the sum rules (2.16): it is normalized to 1, andits first moment remains equal to M . Note that the infinite mass limit gives an accuratepicture, only mildly modified by finite mass corrections, down to values of the mass of theorder of 1.5 GeV. In particular finite T /M effects seem to be important mainly for theshift and the broadening of the main peak, affecting on the other hand very mildly thesecondary bumps.By using the relation (2.9), one obtains from the one-loop spectral function the corre-sponding Euclidean correlator. This is displayed in Fig. 8, for different values of the heavyquark mass. What is plotted in Fig. 8 is actually the function¯ F (1 l ) ( τ ) = ln G > ( − iτ ) G ( − iτ ) . (3.12)As it can be seen, all the curves, start with zero slope at τ = 0. This is related to thegeneral feature that the interactions do not introduce any corrections linear in τ at small τ , which in turn may be linked to the first two sum rules (2.16)) which are satisfied inthe one-loop approximation. This represents actually an important consistency check ofthe numerical calculation, given the indirect way by which the Euclidean correlator wasobtained. The value of the Euclidean correlator at τ = β measures the free-energy shift ∆ F caused by the addition of the heavy quark, and can be read off Fig. 8. As already obtainedin the case of δM , one finds a smaller shift as the ratio T /M gets larger (see Table 3.1).A different way to plot the Euclidean correlator is offered in the right panel of Fig. 8.There we have separated the linear τ -dependence driven by the free-energy shift, writing– 19 – τ/β G ( τ ) / e xp [- M τ ] M=1.5 GeV (Dyson)M=4.5 GeV (Dyson)M=45 GeV (Dyson)M=1.5 GeV (pert)M=4.5 GeV (pert)M=45 GeV (pert)
Figure 9:
A comparison between the first-order perturbative expansion of the Euclidean correlator G pert ( τ ) ≡ G ( τ ) + G ( τ ) (dashed lines) and the full one-loop correlator G ( τ ) obtained from theresummation of the Dyson series (continuous lines). (see Eq. (2.53)) ¯ F (1 l ) ( τ ) = ¯ F (1 l )1 ( τ ) + ¯ F (1 l )2 ( τ ) , ¯ F (1 l )1 ( τ ) = τβ ¯ F (1 l ) ( β ) . (3.13)The difference of behavior that is observed is quite similar to that obtained in the toymodel presented in Appendix A. Note in particular that the symmetry around β/ M = ∞ limit is lost in the one-loop approximation (also in the infinitemass limit of the latter).Finally it is of interest to study the accuracy of the weak-coupling expansion at shorttime. To that aim, we expand the propagator to order αG ( τ ) = G ( τ ) + G ( τ ) + . . . , (3.14)with G ( τ ) ≡ e − Mτ , and G ( τ ) is given by the one-loop self-energy: G ( τ ) = g e − Mτ Z τ dτ ′ Z τ ′ dτ ′′ Z d k (2 π ) ∆( τ ′ − τ ′′ , k ) e − ( k / M )( τ ′ − τ ′′ ) . (3.15)In order to calculate the time intergral, one may express the gluon propagator in terms ofits spectral density. One gets then: G ( τ ) /e − Mτ = g Z d k (2 π ) Z dk π ρ L ( k , k )[1 + N ( k )] k + k / M τ − g Z d k (2 π ) Z dk π ρ L ( k , k )[1 + N ( k )]( k + k / M ) h − e − ( k + k / M ) τ i . (3.16)– 20 –he result, for the zero-momentum case, is plotted in Fig. 9 and compared with the fullone-loop calculation. As it can be see, for the moderate coupling α = 0 . τ /β . What is perhapssurprising is the dependence on the mass M , which reflects a non analytic behavior atsmall τ . Assume indeed that a Taylor expansion of Eq. (3.15) exists. Then, the leadingterm in this expansion, of order τ , is obtained by setting τ ′ = τ ′′ = 0 in the integrand,leading to the result G ( τ ) /e − Mτ = g τ Z d k (2 π ) ∆(0 , k ) = g τ τ = 0 , r = 0) , (3.17)which would be independent of the mass M . However, as already stressed, ∆( τ = 0 , r = 0)is divergent, so that Eq. (3.15) has no Taylor expansion. The integral over τ ′ and τ ′′ inEq. (3.15) exists however, and because of the exponential factor, it acquires a dependenceon the mass M : it is largest in the limit M → ∞ , and decreases as M/T decreases. Thisis the trend seen in Fig. (9).
4. Numerical results: MC simulations and MEM analysis
In this section we present the results of the numerical evaluation of the path integral forthe heavy-quark correlator. We shall also discuss the spectral density obtained from thelatter through an analysis based on the Maximum Entropy Method (MEM) [24]. Since noambiguity can arise, we use in this section the simplified notation G ( τ, r ) for the Euclideancorrelator in place of G > ( − iτ, r ) used in the rest of the paper. This correlator is obtainedfrom the path integral derived in Sect. 2.3. By taking the ratio of G ( τ, r ) with the freepropagator G ( τ, r ) (see Eq. (2.12)) one obtains G ( τ, r ) G ( τ, r ) = R r D z e − S [ z ] e ¯ F ( z ) R r D z e − S [ z ] = h e ¯ F [ z ,τ ] i , (4.1)with S [ z , τ ] = Z τ dτ ′ M ˙ z , (4.2)and ¯ F [ z , τ ] = g Z τ dτ ′ Z τ dτ ′′ ∆( τ ′ − τ ′′ , z ( τ ′ ) − z ( τ ′′ )) . (4.3)The functional ¯ F [ z , τ ] is a known functional of the path, with ∆( τ, r ) an intrinsic propertyof the plasma, calculated as indicated in Sect. 2.2. The calculation of G ( τ, r ) according toEq. (4.1) amounts to an average that can be performed using Monte Carlo (MC) techniques. In fact, to proceed with the MC calculation, we shall take a slightly different route than thatsuggested by Eq. (4.1). This is because we want to include the effects of the interaction inthe samples of paths used in the averaging. While this may not be necessary in the presentone particle problem, this is essential when dealing with the two particle problem that we– 21 – τ/β l n [ G M C / G fr ee ( τ , r = )] M=infiniteT=0.75T=1T=1.5T=2 0 0.2 0.4 0.6 0.8 1 τ/β l n [ G M C / G fr ee ( τ , r)] r/ β =0.1r/ β =0.5r/ β =1 T = 1 Figure 10:
Left panel: The quantity ¯ F MC ( τ, r = 0) for various temperatures (and M = 7 . T /M decreases the curves move closer to the static result. Right panel: ¯ F MC ( τ, r ) for T = 1 and various values of r . plan to address in the future. Thus, using a standard strategy, we define a propagator G α ( τ, r ) as in Eq. (4.1) but with the action replaced by S α [ z , τ ] = S [ z , τ ] − α ¯ F [ z , τ ] , (4.4)with ¯ F [ z , τ ] given by Eq. (4.3). Clearly, S α [ z , τ ] interpolates between S [ z , τ ], correspond-ing to α = 0, and the full action S [ z , τ ] − ¯ F [ z , τ ] reached for α = 1. By taking thederivative of ln G α with respect to α one obtains1 G α ( τ, r ) ∂G α ( τ, r ) ∂α = R D z ¯ F [ z ] exp [ − S α [ z ]] R D z exp [ − S α [ z ]] = h ¯ F [ z ] i α , (4.5)and G ( τ, r ) is recovered after integration over α :ln G ( τ, r ) G ( τ, r ) = ln (cid:16) h e ¯ F [ z ,τ ] i (cid:17) = Z dα ∂ ln G α ( τ, r ) ∂α = Z dα h ¯ F [ z ] i α . (4.6)The α -dependent average value appearing in the right-hand side of Eq. (4.5) is evaluatedusing a MC algorithm whose details are given in Appendix B. The heavy quark correlator is calculated first in coordinate space, and then at zero spatialmomentum. Calculations have been performed for a fixed mass M = 7 . T = 0 .
75 to T = 2. Recall that all energies in the MC calculation are expressedin units of 197 . T = 1 corresponds to T ≃
200 MeV, and M = 7 . M ≃ . τ /β, r /β , T /M , and m D /T .Actually, since we keep the coupling constant α = 0 . T /M is the only relevant controlparameter. – 22 –or the ease of presentation we define¯ F MC ( τ, r ) ≡ ln G ( τ, r ) G ( τ, r ) . (4.7)The quantity ¯ F MC ( τ, r ) is displayed in Fig. 10 together with its infinite mass limit, thefunction ¯ F ( τ ) (see Eq. (2.52)). Since ∆( τ, z = 0) > ∆( τ, z ), ¯ F [ z , τ ] < ¯ F ( τ ): hence diffusiontends to decrease the magnitude of ¯ F MC ( τ, r ). Thus, the larger the ratio T /M , the largerthe diffusion, and the lower is the corresponding curve in the left panel of Fig. 10. Thepanel on the right hand side of Fig. 10 indicates that the effect of the interactions dependsmildly on r : it attenuates very slowly as r increases.We now consider the correlator projected to zero momentum G ( τ, p = 0) ≡ Z d r G ( τ, r ) . (4.8)It is again convenient to study the ratio e ¯ F MC ( τ, p =0) ≡ G ( τ, p = 0) G ( τ, p = 0) = R d r exp[ − M r / τ ] G ( τ, r ) /G ( τ, r ) R d r exp[ − M r / τ ] . (4.9)This expression lends itself to a convenient numerical evaluation. Indeed, as an outcomeof the MC simulations, for each τ , one knows G/G ( τ, r ) for a discrete, and rotationallysymmetric, set of values { r i } (typically r i < ∼ G ( τ, p = 0) G ( τ, p = 0) = P i r i exp[ − M r i / τ ] G ( τ, r i ) /G ( τ, r i ) P i r i exp[ − M r i / τ ] , (4.10)which is the formula used to obtain G ( τ, p = 0).A further remark is in order. As explained in Appendix B, in the MC calculations, thefunctional ¯ F [ z ] is truncated to the following discrete sum¯ F ′ [ z ] ≡ g N τ X i = j =1 a τ ∆(( i − j ) a τ , z i − z j ) . (4.11)A procedure is introduced then to correct for the missing diagonal ( i = j ) terms, assumingthat these are approximately given by the corresponding terms in the calculation of theknown function ¯ F ( τ ). This amounts to correct the raw data by the quantity h e ¯ F ′ [ z ,τ ] i → h e ¯ F ′ [ z ,τ ] i e C , (4.12)with C = ¯ F ( τ ) − g N τ X i = j =1 a τ ∆(( i − j ) a τ , . (4.13)This correction is linear in τ , and affects for instance the calculation of the free energy,given by the correlator evaluated at τ = β . Table 4.2 summarizes the results. As we see,the correction is small and is no more than a few percent.– 23 – ¯ F MC ( β, p = 0) C0.75 0.382 0.01021 0.366 0.01311.5 0.346 0.01842 0.331 0.0235 Table 2: ¯ F MC ( β, p = 0) for various temperatures obtained with the raw MC data and the correction C in Eq. (4.12). The values of the free-energy shift obtained here for T = 1 and 1 . T /M = 0 .
133 and 0 .
2, respectively. In absolutevalues, the MC (one-loop) free energy shifts are 0.366 (0.336) and 0.346 (0.318), respectively forthe two cases; the one-loop approximation underestimates the free energy shift. τ/β -0.1-0.08-0.06-0.04-0.020 l n [ G ( τ , p = ) / e xp (- M τ )] s y mm M=infinite (exp)T=0.75 (MC)T=1 (MC)T=1.5 (MC)T=2 (MC) τ/β l n [ G ( τ , p = ) / e xp (- M τ )] M=infinite (exp)T=0.75 (MC+renorm)T=1 (MC+renorm)T=1.5 (MC+renorm)T=2 (MC+renorm)
Figure 11:
Left panel: ¯ F MC2 ( τ, p = 0) obtained with the raw MC data, for all the temperaturescovered by our analysis. As usual M = 7 .
5. Right panel: the same quantity after the correctionindicated in Eq. (4.12) (and labelles here as “MC+renorm”).
The quantity ¯ F MC ( τ, p = 0) is shown in Fig. 11 for various temperatures. The left paneldisplays ¯ F MC2 ( τ ) ≡ ¯ F MC ( τ ) − ( τ /β ) ¯ F MC ( β ). The curves are obtained employing directly theraw MC data, which are not affected by the correction (4.12). In the right panel we showthe corrected results. As already found in studying the r = 0 correlator, the curves movecloser to the static result as the ratio T /M decreases, due to the suppression of diffusion.In Fig. 12 we compare the MC results with those of the one-loop calculation presentedin Sect. 3, in which the Euclidean correlator was obtained through the numerical integrationof the corresponding spectral function, according to Eq. (2.9). The MC points start quiteclose to the one-loop curves corresponding to the same value of
T /M , in agreement with theexpectation that the short-time behavior is governed by perturbation theory (as alreadydiscussed in Sect. 3 commenting Figs. 8 and 9). For large values of τ /β the MC resultslie above the one-loop curves. This general behavior is also analyzed within the simpletoy-model presented in Appendix A. – 24 – τ/β -0.1-0.08-0.06-0.04-0.020 l n [ G ( τ , p = ) / e xp (- M τ )] s y mm M=infinite (exp)M=infinite (one-loop)T/M=0.133 (path-integral)T/M=0.133 (one-loop)T/M=0.200 (path-integral)T/M=0.200 (one-loop) τ/β -0.100.10.20.30.40.5 l n [ G ( τ , p = ) / e xp (- M τ )] M=infinite (exp)M=infinite (one-loop)T/M=0.133 (path-integral)T/M=0.133 (one-loop)T/M=0.200 (path-integral - 0.05)T/M=0.200 (one-loop - 0.05)
Figure 12:
A comparison between ¯ F MC ( τ, p = 0) and ¯ F (1 l ) ( τ, p = 0) as a function of the ratio T /M . The one-loop curves are obtained from a numerical integration of the charm ( M = 1 . T = 200 and T = 300 MeV. In the right panel the set of curvescorresponding to T = 300 MeV has been translated downwards by -0.005 in order to make thefigure more readable. We turn now to the reconstruction of the heavy quark spectral density from the Euclideancorrelator obtained with the MC calculation. To do so, we need to invert Eq. (2.9), a wellknown difficult problem. We use here a maximum entropy analysis (MEM), according tothe algorithm described in Ref. [24]. Another exhaustive introduction to the method canbe found in Ref. [25]. In such an approach, one determines the “best” possible spectralfunction, given the information one has about the Euclidean correlator (the “data”), andprior information one has about the spectral density, such as the fact that it is positivedefinite (and hence can be interpreted as a probability density) and that it satisfies somesum rules. The procedure involves the maximization of an entropy function (actually theminimization of a free-energy), which is defined with respect to a default model : in theabsence of data, the spectral density coming out of the entropy maximization is the defaultmodel. There is, of course, a delicate interplay between the effect of the data and that of thedefault model , and the resulting spectral density will in general keep some reminiscenceof the chosen default model. In order to explore such a systematic uncertainty we willconsider two different default models: a constant (within the finite range | ω | < − ( ω − M ) / γ ] / p πγ . In both cases we adjust the parametersof the default model so that the first two sum rules in Eq. (2.16) are fulfilled.Throughout this section the results will be expressed in terms of dimensionless vari-ables, displaying for instance σ ( ω ) T as a function of ¯ ω/T = ( ω − M ) /T , the only parameterleft being the ratio T /M . It is then useful to recall that in the static limit
T /M →
0, thefree-energy shift is − αm D / T ≈ − . ω pl /T ≈ .
373 controls the location of theplasmon absorption/emission peaks .As the first test of the potentiality of the MEM procedure, and of the systematic uncer-tainties attached to the choice of the default model, we reconstruct the (known) one-loop– 25 – ω -M)/T05101520 σ ( ω ) T T/M=0.133 (one-loop)T/M=0.200 (one-loop)T/M=0.333 (one-loop)T/M=0.133 (G one-loop +MEM)T/M=0.200 (G one-loop +MEM)T/M=0.333 (G one-loop +MEM) -4 -3 -2 -1 0 1 2 3 4( ω -M)/T05101520 σ ( ω ) T T/M=0.133 (gauss γ /T=0.1)T/M=0.133 (gauss γ /T=0.2)T/M=0.133 (gauss γ /T=0.3)T/M=0.133 (const)M=infinite (gauss γ /T=0.2)M=infinite (const) Figure 13:
Left panel: a test of the MEM reconstruction of the one-loop spectral function for acharm quark ( M = 1 . G MC ( τ ) in the case T /M = 0 .
133 and to the exactresult for G M = ∞ ( τ ). The constant default model leads systematically to larger shift and a biggerwidth than the Gaussian default model. spectral density from the one-loop Euclidean correlator G (1 l ) ( τ ) obtained in Sect. 3, throughthe integration of the corresponding spectral density. We use a large set of data points( ∼ M = 1 . T = 200 ,
300 and 500MeV. As one can see on the left panel of Fig. 13, the MEM inversion – here performedwith a Gaussian prior – is able to identify the main peak. However this is broader and lessshifted then the exact result: the shift is ∼ − .
15, while it is ∼ − .
35 in the original one-loop spectral density. The method also reconstructs a low-energy secondary bump, thoughless pronounced than the plasmon-absorption peak in the original one-loop spectral func-tion, and it appears also at lower frequency ( ∼ − − . σ (1 l ) ( ω ) is visible in the MEMspectral density. In the right panel of Fig. 13 we illustrate the sensitivity to the choice ofthe default model. There, the MC data at T = 1 are used, as well as the known infinitemass correlator G M = ∞ ( τ ). We consider a Gaussian (with various values of the width) anda constant prior. With a Gaussian prior with width γ/T = 0.1, 0.2 and 0.3, the main peakis shifted respectively by, -0.05, -0.15 and -0.15. The presence of a spectral strength at lowenergy seems to be a quite robust feature of the spectrum, though the broader the defaultmodel, the less pronounced the secondary bump is. In particular for a flat prior one findssimply a very large broadening and negative shift of the main peak.In Fig. 14 we show the results of the MEM inversion of the MC data for G MC ( τ ), forvarious values of T /M , including the exact infinite mass limit corresponding to
T /M = 0.The left panel corresponds to a Gaussian default model with γ = αT /
2. The resultingspectral densities present a broad main peak, slightly shifted with respect to its position inthe vacuum ( M ) by an amount roughly proportional to T (the curves in the dimensionless– 26 – ω -M)/T024681012 σ ( ω ) T M=infinite (gaussian prior)T/M=0.133 (gaussian prior)T/M=0.200 (gaussian prior)T/M=0.267 (gaussian prior) -3 -2 -1 0 1 2 3 4 5( ω -M)/T01234 σ ( ω ) T M=infinite (constant prior)T/M=0.100 (constant prior)T/M=0.133 (constant prior)T/M=0.200 (constant prior)T/M=0.267 (constant prior)
Figure 14:
The MEM spectral densities σ MC ( ω ) for different values of T /M . In the left/rightpanel a Gaussian/constant default model is employed. For comparison the curves obtained from G M = ∞ ( τ ) are also shown. The dashed vertical lines correspond to the static free-energy shift − αm D / T = 0 . ± ω pl , where ω pl /T = 1 .
373 is the plasmafrequency. As clearly seen in the left panel, the Gaussian prior leads to an underestimate of theshift of the main peak (here estimated as the static free energy shift), together with an overestimateof that of the secondary peak at low energy (here estimated by − ω pl ). The dependence on T /M isvery weak. On the right panel one sees that the dependence on
T /M is larger with the constantprior, and in line with what one expects (the curves move gradually towards that corresponding tothe infinite mass limit as
T /M decreases). units employed lie almost on top of each others), but smaller (by a factor ∼
5) than thestatic free-energy shift − αm D / T ≈ . T /M decreases, is also visible. In the right panelthe same data are analyzed using a constant default model. In such a case the spectralfunction exhibits only a broad peak with a sizable negative shift which, as
T /M →
0, results ∼
50% larger than the static free-energy shift. Furthermore the MEM spectral density, withthis choice for the prior, has also a long high-energy tail, at variance with what is foundwith the Gaussian default model.Finally in Fig. 15 a comparison between the MEM and the one-loop spectral functionsis given. The main features discussed above can be seen. In particular, the dependence onthe default model is striking. A Gaussian leads to a very small shift of the main peak. Onthe other hand the constant default model yields a broader and more shifted peak, whosestrength extends to low energy till displaying a partial overlap with the secondary bumpof the curve obtained with the Gaussian prior.
5. Conclusions
In this paper, we have presented an approach to the dynamics of heavy quarks in a hotplasma based on a path integral for non relativistic particles with a non local (in space andtime) self-interaction that summarizes the effects of the medium to which the heavy quarkis coupled. – 27 – ω -M)/T05101520 σ ( ω ) T T/M=0.133 (MC const prior)T/M=0.133 (MC gauss prior)T/M=0.133 (one-loop)T/M=0.200 (MC const prior)T/M=0.200 (MC gauss prior)T/M=0.200 (one-loop)
Figure 15:
The MEM spectral densities σ MC ( ω ) obtained with the two different default models(dotted and dot-dashed curves), compared to σ (1 l ) ( ω ) (continuous curves), for two values of T /M . The path integral providing the heavy-quark Euclidean correlator was evaluated nu-merically using Monte Carlo techniques. The results of this numerical evaluation wereanalyzed and compared to those of the one-loop calculation, and to those of an exactevaluation of the path integral in the infinite mass limit. We showed that the effect of in-teractions is to favor the contribution of straight paths in the path integral, and are indeedmaximum in the infinite mass limit, where the heavy particle stays at rest. Calculationswere done for a value of the coupling constant that would correspond in QCD to a value ofthe strong coupling constant α s ≈ .
3. For such a value the one-loop approximation providesa reasonable first approximation, but deviations with the exact Monte Carlo results wereobtained. The Monte Carlo results move towards those of the infinite mass limit as theratio
M/T increases, as expected.The Monte Carlo calculations of the Euclidean correlator were performed in coordinatespace, but a simple integration over the spatial coordinates gave the correlator for zero-momentum. This allowed us, in particular, to estimate the shift in the free energy of thesystem that is caused by the addition of the heavy quark. We also used the correspondingEuclidean correlator to reconstruct the spectral function, through a MEM analysis. Withinour implementation of this method, we were able only to reproduce the main qualitativefeatures, namely a broad main peak, whose shift is only given semi-quantitatively. Asecondary structure below the main peak, somewhat reminiscent of the plasmon-absorptionpeak of the one-loop spectral function is also seen, but no secondary structure above themain peak is detected, only a long tail at large frequencies is observed (and only with a– 28 –onstant prior). The large sensitivity of the MEM analysis to our choices of default modelsdoes not allow us to draw more robust quantitative conclusions at this stage. On the otherhand, the qualitative features that we were able to reconstruct may be enough to drawconclusions in the two particle problem, which is our ultimate goal.The thorough analysis of the one-particle case that we have presented in this paperpaves the way for several extensions. Clearly the calculation can be improved in severalplaces, and the general setting brought closer to QCD without too much efforts. For in-stance, we have seen that the HTL approximation used in the description of the hot plasmaleads to a somewhat unrealistic description of the effects of collisions. While this affectsonly mildly the heavy quark correlator, and only at small times where the calculation isin complete control (being essentially perturbation theory), this feature can be improvedwithout changing the basic structure of the problem. The calculation of the Euclidean cor-relator of a heavy quark-antiquark pair is within reach. The reconstruction of the spectraldensity of a heavy quark pair from its Euclidean correlator faces the same difficulty as metin lattice QCD: on the one hand, this offers opportunities for more detailed comparisonsbetween the two approaches, on the other hand we note that our path integral for theEuclidean correlator can be calculated with high precision, which could be exploited to de-velop new methods of reconstruction of the spectral density. Finally one may contemplatethe possibility of calculating the path integral directly in real time, perhaps at the cost ofadditional approximations. That would allow us to bypass the problem of the analyticalcontinuation, and would open the possibility of numerous applications.
Acknowledgments
A.B. and J.P.B. gratefully acknowledge ECT* for warm hospitality and financial supportduring the preparation of this work.
A. An exactly solvable toy-model
In this section we present a toy model illustrating some of the features of the calculationsthat are presented in the main text, in particular those features that emerge in the infinitemass limit. The model consists of a fermion of mass M coupled to a single harmonicoscillator that represents the “medium”. The Hamiltonian of the system is written as H = M ψ † ψ + 12 (cid:0) π + m D φ (cid:1) + g ψ † ψ φ, φ ≡ a + a † √ m D , (A.1)where ψ † and ψ are the creation and the annihilation operators of the fermion, { ψ, ψ † } = 1, φ and π are respectively the coordinate of the oscillator and its conjugate momentum,[ φ, π ] = i , and a † , a the associated creation and annihilation operators, [ a, a † ] = 1. Since[ H, ψ † ψ ] = 0, the eigenstates of H can be classified in sectors characterized by the eigen-value of the fermion number operator ψ † ψ . Since the fermion has no internal degree offreedom there are only two sectors to consider, those with ψ † ψ = 0 and with ψ † ψ = 1.– 29 –he first sector corresponds to the medium without the fermion, and the Hamiltonian issimply that of the oscillator H = m D (cid:16) a † a + 1 / (cid:17) . (A.2)The sector with ψ † ψ = 1 mimics the case in which one adds the fermion into the medium.The corresponding Hamiltonian reads H = M + H + gφ ≡ M + H + V, (A.3)and it has the structure of Eq. (2.2). It is easily diagonalized by introducing the shiftedoperators b ≡ a + g q m D and b † ≡ a † + g q m D , with [ b, b † ] = 1 , (A.4)so that H = (cid:18) M − g m D (cid:19) + m D (cid:18) b † b + 12 (cid:19) . (A.5)The spectrum of H is identical to that of H , and the shift in the ground-state energy isgiven by: ∆ E ≡ E − E = M − g m D ≡ M − αm D , α ≡ g m D , (A.6)where we have introduced the dimensionless coupling constant α . The ground state of H is a coherent state characterized by a non-vanishing expectation value of the field φ : h φ i = − gm D = −√ α r m D . (A.7)This expectation value plays the role of the classical field A associated with the polarizationcloud around the heavy quark.One may also consider the non-equilibrium situation that corresponds to adding thefermion into the system in its ground state at t = 0. Following this initial perturbation, thewhole system evolves in time with the Hamiltonian H . It is then not difficult to establishthat the expectation value of φ oscillates around its equilibrium value (A.7) according to h φ i t = h φ i eq (cos m D t − , (A.8)where φ eq is given by Eq. (A.7).This result holds unchanged when the oscillator is in thermal equilibrium at tempera-ture T , that is, h φ i is not affected by thermal fluctuations. Similarly, because the spectra of H and H are identical, the contributions of thermal fluctuations cancel in the differenceof free energies of the systems with and without the fermion, with the result that thisdifference remains equal to the shift in the ground state energy given by Eq. (A.6).Consider now the Euclidean correlator G ( τ ) ≡ G > ( − iτ ) ≡ h ψ ( τ ) ψ † (0) i , (A.9)– 30 –here the expectation value h . . . i ≡ Tr (cid:2) e − βH . . . (cid:3) /Z is taken over states of the mediumwithout the fermion. One has: G > ( − iτ ) = h e Hτ ψe − Hτ ψ † i = h e H τ ψe − H τ ψ † i = e − Mτ D e H τ e − ( H + V ) τ E , (A.10)where, in the last expression, one recognizes the evolution operator in the interactionrepresentation, so that one can write: G > ( − iτ ) = e − Mτ (cid:28) T τ exp (cid:20) − g Z τ dτ ′ φ I ( τ ′ ) (cid:21)(cid:29) . (A.11)A simple calculation yields then the exact result: G > ( − iτ ) = e − Mτ e ¯ F ( τ ) , (A.12)where ¯ F ( τ ) = g Z τ dτ ′ Z τ dτ ′′ ∆( τ ′ − τ ′′ ) . (A.13)Here ∆( τ ) is the Euclidean propagator for the field φ , satisfying periodic boundary condi-tions (∆(0) = ∆( β )):∆( τ ) = h T φ I ( τ ) φ I (0) i = 12 m D h e − m D | τ | (1 + N ) + e m D | τ | N i , (A.14)with N the statistical factor N ≡ βm D − . (A.15)At this point let us note that the model depends on several dimensionful parameters:the mass M , which simply shifts the overall spectrum, and plays no role in the dynamics; theDebye mass m D which characterizes the response of the system to an external perturbation,such as the addition of the fermion; the coupling constant g and the temperature T . Weshall systematically express the coupling between the fermion and the oscillator in termsof the dimensionless coupling α , as in Eq. (A.6). A look at the propagator (A.14) revealsthat m − D appears there as the natural time scale, while the statistical factor depends on m D /T .It is sometimes convenient to write ¯ F ( τ ) as the sum of a term ¯ F ( τ ) linear in τ and aterm ¯ F ( τ ) that is symmetric around β/ F ( τ ) = αm D τ, ¯ F ( τ ) = α (cid:20) cosh( m D ( τ − β/ − cosh( βm D / β ( m D / (cid:21) . (A.16)Clearly, ¯ F ( β ) = ¯ F ( β ) = αm D β = g β ∆( iω n = 0) , (A.17)so that M − (1 /β ) ¯ F ( β ) = ∆ F Q is the difference of free energies of the systems with andwithout the fermion (see Eq. (A.6)). The last equality in Eq. A.17 emphasizes that ¯ F ( β )– 31 –s entirely given by the zero Matsubara frequency part of the oscillator propagator (A.14),with ∆( iω n ) = Z β dτ e iω n τ ∆( τ ) . (A.18)The function ¯ F ( τ ) vanishes at τ = 0 and τ = β , by construction, and has its minimum at τ = β/
2, with value ¯ F ( β/
2) = − α tanh( βm D / τ = 0 is − αm D , so thatthe linear contributions cancel in ¯ F = ¯ F + ¯ F , in accordance with the general result (seeEq. (2.13)). This is also obvious from Eq. (A.13): the small τ behavior starts at order τ .At quadratic order, we have¯ F ( τ ≪ m − D ) ≃ g τ h φ i = 12 αm D τ (1 + 2 N ) , h φ i = 12 m D (1 + 2 N ) = ∆( τ = 0) . (A.19)One recovers the general result between the coefficient of τ and the fluctuation of φ (seeEq. (2.14)). We shall return to the short time behavior of the correlator shortly.We now exploit the analyticity of G > and move to real time. This will allow us inparticular to get the large time behavior of G > ( t ). One gets from Eq. (A.12) G > ( t ) = e − iMt e iF ( t ) , (A.20)with F ( t ) = g Z t ds Z t ds ′ D ( s − s ′ ) , D ( s − s ′ ) = i ∆( τ = is, τ ′ = is ′ ) . (A.21)A simple calculation then yields F ( t ) = g Z ∞−∞ dω π − cos ωtω D ( ω ) , (A.22)with D ( ω ) the Fourier transform of the time-ordered propagator D ( t ) (see Eq. (2.44)). Thelarge time behavior of the correlator follows then from Eq. (2.47): F ( t ≫ m − D ) = g tD ( ω = 0) = αm D t. (A.23)It is entirely determined by the static response of the medium. The comparison withEq. (A.6) reveals that − F ( t ) /t is the interaction contribution to the energy shift causedby the addition of the fermion (see also Eq. (A.17)). This is similar to what happens inthe case of an infinitely massive quark although, in the latter case, a damping term alsoappears next to the free energy shift. No such term appears here because of the discretenature of the spectrum.One can also calculate the spectral function. To do so, it is convenient to start withthe following explicit expression of the propagator (A.20): G > ( t ) = exp [ − α (1+2 N )] exp [ − i ( M − αm D ) t ] ×× exp (cid:2) α (cid:0) N e im D t + (1+ N ) e − im D t (cid:1)(cid:3) , (A.24)– 32 – ω -M)/T05101520253035 σ ( ω ) T α=0.09α=0.36α=1.00α=2.00 m D /T= α γ/ T= α/2 -6 -4 -2 0 2 4 6( ω -M)/T0123456 σ ( ω ) T α=0.36α=1.00α=2.00 m D /T= α γ/ T= α/2 Figure 16:
The spectral function (A.26), for different values of the coupling α . The vertical linesrefer to the position of the “main peak” at ω = M − αm D . The delta functions have been smearedto gaussians with width γ = αT /
2, and the mass m D is adjusted as a function of the temperature, m D = T √ α . As the coupling grows, the individual peaks are smoothed out, leaving a broad,structureless, distribution. and expand the last exponential in powers of α . One gets G > ( t ) = e − α (1+2 N ) e − i ( M − αm D ) t ∞ X n =0 α n n ! n X p =0 (cid:18) np (cid:19) ( N ) p e ipm D t (1+ N ) n − p e − i ( n − p ) m D t . (A.25)The Fourier transform is then obtained immediately and reads σ (¯ ω ) = 2 π e − α (1+2 N ) ∞ X n =0 α n n ! n X p =0 (cid:18) np (cid:19) ( N ) p (1+ N ) n − p δ (¯ ω + αm D − ( n − p ) m D ) , (A.26)where we have set ¯ ω ≡ ω − M . The above spectral density exhibits an infinite numberof peaks in one-to-one correspondence with the transitions between the eigenstates of H .The major peak is located at ¯ ω = − αm D . The expansion of the spectral density to order α K has peaks centered at ¯ ω = − αm D ± km D , with k = 0 , . . . K . Hence, the largerthe coupling, the larger the number of peaks giving a sizable contribution to the spectraldensity.The spectral density is displayed in Fig. 16 for a wide range of values of the coupling α . In order to make contact with the general discussion of a heavy quark in a plasma, wechoose m D = √ αT (this implies among other things that the coupling among the plasmagrows similarly the coupling between the fermion and the plasma particles). Also, for thepurpose of illustrating the global behavior of the spectral function, we smear the deltafunctions by replacing them by gaussians of variance γ ∼ αT . At small coupling, individualpeaks are recognized. For large coupling, the smearing that we have introduced erasesthe individual secondary peaks, leaving a broad distribution which spreads over a larger– 33 –nd larger frequency interval as the coupling grows. Note that the main peak, located at¯ ω = − αm D is shifted to lower frequency as α grows, but the spectral strength remainscentered around ω ∼ M . This behavior may be understood in terms of the sum rulessatisfied by the spectral function.These sum rules are obtained from the derivatives s of G > ( t ) at t = 0: i n ∂ n ∂t n e iMt G > ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z ∞−∞ d ¯ ω π ¯ ω n σ (¯ ω ) . (A.27)These derivatives are easiy calculated and one gets Z ∞−∞ d ¯ ω π σ (¯ ω ) = 1 , Z ∞−∞ d ¯ ω π ¯ ω σ (¯ ω ) = 0 , Z ∞−∞ d ¯ ω π ¯ ω σ (¯ ω ) = αm D (1 + 2 N ) , Z ∞−∞ d ¯ ω π ¯ ω σ (¯ ω ) = αm D . (A.28)These sum rules explain why the spectral weight remains centered around ¯ ω = 0, with awidth increasing with α , while the last sum rule suggest a somewhat larger strength atpositive ¯ ω than at negative ¯ ω . Note that the sum rules that are displayed explicitly hereare at most linear in the coupling α . The first higher order correction, of order α , entersat the level of the ω sum rule.Let us now turn to the one-loop approximation for the time-ordered (or retarded)propagator. The one-loop self-energy of the fermion is easily obtained:Σ(¯ ω + iη ) = αm D (cid:20) N ¯ ω − m D + iη + N ¯ ω + m D + iη (cid:21) . (A.29)The poles of Σ for ¯ ω = ± m D correspond to the energies of the fermion having emitted orabsorbed a quantum of the oscillator, which represent the leading processes that take placeat weak coupling. The inverse retarded propagator reads G − (¯ ω + iη ) = − ¯ ω − iη + Σ(¯ ω + iη ) . (A.30)Thus, the propagator has three poles, at values ¯ ω i solutions of the equation¯ ω − ¯ ω m D [1 + α (1 + 2 N )] − αm D = 0 . (A.31)The general behavior of the solutions may be easily inferred from the graph displayed inFig. 17. The propagator may then be written as G (¯ ω ) = X i z i ¯ ω i − ¯ ω , (A.32)with the residues given by z − i = 1 − ∂ Σ /∂ ¯ ω | i . (A.33)The spectral function takes the form σ (¯ ω ) = 2 π X i z i δ (¯ ω − ¯ ω i ) . (A.34)– 34 – Figure 17:
Graphical solution of the equation ¯ ω = Σ(¯ ω ), with both Σ and ¯ ω expressed in unitsof m D (the curves plotted are Σ(¯ ω/α and ¯ ω/α ). The self-energy (divided by α ) exhibits polesat ¯ ω = ± m D . The intersections with the straight line ¯ ω/α give the locations of the poles of thepropagator. There is always one pole close to ¯ ω = 0. At weak coupling this pole has the largestresidue (the straight line in the figure corresponds to α = 0 . ≈ ± αm D , and their residue saturate the sum rule, leaving very littlespectral weight on the pole at ¯ ω ≈ ω = − m D / (1 + 2 N )). Notethat the intersection of Σ with the vertical axis yields the exact energy shift, Σ(¯ ω = 0) = − αm D . It can be verified that, in the weak coupling limit, this coincides with the general expression(A.26) expanded to order α . The Euclidean correlator is easily obtained from the spectralfunction, and reads G > ( − iτ ) = e − Mτ X i z i e − ¯ ω i τ = e − Mτ e ¯ F (1 l ) ( τ ) , (A.35)which defines the function ¯ F (1 l ) ( τ ). From the correlator calculated for τ = β , one deducesthe one-loop free energy shift ¯ F (1 l ) ( β ) = ln "X i z i e − ¯ ω i β . (A.36)This is to be compared to the exact value ¯ F ( β ) = αm D /T : the one-loop calculationunderestimates the exact result.A comparison between the one-loop and the exact result is offered in Fig. 18. As we didearlier, we may decompose ¯ F (1 l ) ( τ ) = ¯ F (1 l )1 ( τ ) + ¯ F (1 l )2 ( τ ), with ¯ F (1 l )1 ( τ ) = ( τ /β ) ¯ F (1 l ) ( β ).The function ¯ F (1 l )2 ( τ ) is plotted in the right panel of Fig. 18. The agreement of the exactand one-loop correlators may be understood from the fact that the sum rules (A.28) areexactly satisfied at one loop, namely X i z i = 1 , X i z i ¯ ω i = 0 , X i z i ¯ ω i = αm D (1 + 2 N ) , X i z i ¯ ω i = αm D . (A.37)To these we should add the relation P i ¯ ω i = 0, that derives immediately from Eq. (A.31).The sum rules (A.37) are the one-loop transcription of the exact sum rules mentioned– 35 – τ/β l n [ G ( τ ) / e xp (- M τ )] α =1.00 (one-loop) α =1.00 (exact) α =2.00 (one-loop) α =2.00 (exact) τ/β -0.6-0.5-0.4-0.3-0.2-0.10 l n [ G ( τ ) / e xp (- M τ )] s y mm α =1.00 (one-loop) α =1.00 (exact) α =2.00 (one-loop) α =2.00 (exact) Figure 18:
Left: The function ¯ F (1 l ) ( τ ) (continuous curves) compared to the exact result ¯ F ( τ )(dot-dashed curves) as a function of τ /β for large values of the coupling constant α = 1 and α = 2 .Even in this strong coupling regime, the one-loop approximation gives an excellent approximationto the exact result up to values τ < ∼ β/ α . Right: The same for ¯ F (1 l )2 ( τ ) and ¯ F ( τ ). The slope atthe origin is a measure of the free energy shift, which can be also read on the left panel as the valueof ¯ F ( β ). Both plots exhibits clearly that the free-energy shift is underestimated in the one-loopapproximation. Note also the asymmetry, growing with increasing coupling, of the one-loop resultswith respect to τ = β/
2, in contrast to the exact curves. above, Eq. (A.28). They hold exactly at one-loop because the small time behavior of thepropagator involves also a small g expansion and, as it has already been mentioned afterEq. (A.28), up to order τ , the small τ expansion involves terms of the weak couplingexpansion only up to order g . Such terms are taken into account exactly by the one-loopself energy. The fact that the one-loop result is sufficient to describe the small- τ behaviorappears clearly in the left panel of Fig. 18 where, even for large values of the coupling, theone-loop and exact correlators are hardly distinguishable for τ /β small enough.Finally in Fig. 19 we provide a comparison between the one-loop spectral function,given by Eq. (A.34), and the exact result. For weak coupling they look quite similar. Onthe other hand for larger coupling more and more secondary peaks contribute to the exactspectrum, while the one-loop result can display only three peaks. These, having to fulfillthe sum-rules (A.37), result largely distorted. B. Details on the path integral implementation
The path integral that we want to evaluate has the following form (see Eq. (4.5)): h ¯ F [ z , τ ] i α = R r D z ¯ F [ z , τ ] e − S α [ z ,τ ] R r D z e − S α [ z ,τ ] . (B.1)For any chosen value of τ = N τ a τ , where N τ is an integer and a τ a fixed time interval, thepaths are defined by a discrete set of points { z ( τ i ) } , where τ i (0 ≤ τ i ≤ τ ) is a multiple of a τ . We choose natural units ~ = c = k B = 1 and fix the unit of length to be 1 fm, andcorrespondingly the unit of energy (or temperature) to be 197 . ω -M)/T05101520253035 σ ( ω ) T α =0.09 (one-loop) α =0.36 (one-loop) α =0.09 (exact) α =0.36 (exact) -6 -4 -2 0 2 4 6( ω -M)/T0123456 σ ( ω ) T α =0.36 (one-loop) α =1.00 (one-loop) α =2.00 (one-loop) α =0.36 (exact) α =1.00 (exact) α =2.00 (exact) Figure 19:
The one-loop spectral function compared to the exact one. The curves refer to differentvalues of the coupling α , from weak (left panel) to strong (right panel). The one-loop result(continuous curves), characterized by the presence of only three peaks, is compared to the exactone (dot-dashed curves), which has a richer structure. In plotting the curves the delta functionshave been smeared to gaussians in the same way as in Fig. 16. fixed at the value a τ = 0 .
01 fm/c, and the heavy quark mass at M = 7 . W α [ z ] = exp( − S α [ z ]) R [ D z ] e − S α [ z ] . (B.2)Then the average h ¯ F [ z ] i α is evaluated as the arithmetic average over the paths generatedby the equilibrated Markov chain.We used the Metropolis algorithm to generate the Markov chain by a sequence ofelementary moves. A move is defined as follows: starting from a given path z , one selectsat random a time τ i , and displace the corresponding point z ( τ i ) by a quantity δ z uniformlydistributed in a cube of side d centered at z ( τ i ), thus defining a new path z ′ ; the move isaccepted with the probability π = min (cid:20) , exp( − S α [ z ′ ])exp( − S α [ z ]) (cid:21) . (B.3)In the present calculation we start from a straight path connecting the origin (0 ,
0) to thepoint ( τ, r ), and perform at least 10 × N τ moves to reach equilibrium. During this stagethe value of d is adjusted to keep the acceptance ratio of attempted moves between 0 . .
55. Typical values of d were found in the range 0 . − .
08, at the temperature T = 1 .
0. Once the Markov chain has reached equilibrium, one continues generating paths,and the corresponding paths are used in calculating the average values of h ¯ F [ z ] i α . Atleast 10 × N τ paths of the equilibrated chain are used in the calculation of the averagevalue. Finally, the integrand appearing in the right hand side of Eq. (4.6) is evaluated on– 37 –n equispaced array of 10 points in the interval from α = 0 to α = 1. The resulting curveis then interpolated with a cubic spline and integrated using the adaptive Gauss–Kronrodmethod as implemented in the GNU Scientific Library [26].At a given temperature we take tipically between 10 and 20 values of τ to determine G ( τ, r ), with r varying between 0 and 2 fm. Because a large number of paths are used inthe calculation of the average, the statistical errors are negligible: one gets relative errorsof order 10 − for small τ and 10 − for the largest values of τ . Furthermore, since theaverage is taken over a different set of trajectories at each τ , the results at various τ areuncorrelated.There is one issue in this calculation that deserves further comments. It concerns thecalculation of the integral (4.3). The simplest discretized form of this integral reads¯ F [ z , N τ ] ≡ g N τ X i,j =1 a τ ∆(( i − j ) a τ , z i − z j ) . (B.4)This, however, cannot be used as it stands since, as we have seen in Sect. 2.2, ∆(0 ,
0) islogarithmically divergent, so that the diagonal terms i = j in the expression above are illdefined. Before we explain how we have gone around this difficulty, let us examine thecalculation of the same integral in the infinite mass limit, where the paths are frozen atthe origin, i.e., z ( τ i ) = 0 for all i . Then the functional ¯ F [ z , τ ] reduces to the function¯ F ( τ ) (Eq. (2.52)), that we may write, using the same discretization as above but for the“diagonal” terms, as ¯ F ( τ ) ≈ g N τ X i = j =1 a τ ∆(( i − j ) a τ ,
0) + N τ ¯ F ( a τ ) , (B.5)where ¯ F ( a τ ) is the exact value of the integral on a square of side a τ . The comparison of thevalue of ¯ F ( τ ) − N ¯ F ( a τ ) calculated exactly, and from the discretized sum in Eq. (B.5), yieldsan estimate of the discretization error in the evaluation of the integral. As can be seen inFig. 20, this error is of (relative) order 10 − and increases slightly towards small values of τ . We note also that for τ /β > ∼ .
2, the contribution of the diagonal terms amounts to lessthan 10%. We have exploited these features in order to make the following simplifications:i) Only the off-diagonal terms are used in the sampling of paths, that is ¯ F [¯ z, τ ] isreplaced for that purpose by¯ F ′ [ z , N τ ] ≡ g N τ X i = j =1 a τ ∆(( i − j ) a τ , z i − z j ) . (B.6)ii) A correction is applied to compensate for the omission of the diagonal terms, as-suming this correction to be given by N τ ¯ F ( a τ ) (in practice we calculate this correctionfrom the difference between ¯ F ( τ ) and ¯ F ′ [ z , τ ] in Eq. (B.6) estimated for z = 0). Note thatthis correction is presumably an overestimates. Indeed because of diffusion, at time ∼ a τ ,the heavy quark is on the average at a distance ¯ r = p τ / M away from the origin, and– 38 – τ/β τ )-F discr(off-diag) ]/F( τ )[F( τ )-F discr(off-diag) -NF(a t )]/F( τ ) x10 D ∆ ( τ , r) / T τ/β =0 τ/β =0.01 τ/β =0.005 Figure 20:
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