Entanglement entropy and entropy production in the Color Glass Condensate framework
aa r X i v : . [ h e p - ph ] J un Entanglement entropy and entropy production in the Color Glass Condensateframework.
Alex Kovner and Michael Lublinsky Physics Department, University of Connecticut,2152 Hillside Road, Storrs, CT 06269-3046, USA Department of Physics,Ben-Gurion University of the Negev,Beer Sheva 84105, Israel (Dated: June 18, 2015)We compute the entanglement entropy of soft gluons in the wave function of a fast moving hadronand discuss its basic properties. We also derive the expression for entropy production in a high energyhadronic collision within the Color Glass formalism. We show that long range rapidity correlationsgive negative contribution to the production entropy. We calculate the (naturally defined) temper-ature of the produced system of particles, and show that it is proportional to the average transversemomentum of the produced particles.
I. INTRODUCTION
High energy hadronic scattering is of great current interest. The structure of the hadronic wave function hasbeen the subject of study in the recent years with the aim of better understanding possible manifestations of gluonsaturation[1]. The evolution of this wave function with energy, which drives the evolution of high energy scatteringamplitude is encoded in the so called JIMWLK equation[2–4].The stand out property of this Color Glass Condensate (CGC) wave function is that the soft gluons state“grows”out of that of the valence modes. More specifically the wave function of the soft modes is determined by the colorcharge density of the valence part of the wave function. This suggests a certain “order” in the system, since thesoft gluons are entangled with the valence ones. The measure of (dis)order in a quantum system is entropy. Inparticular entanglement entropy measures the degree of entanglement between different subsets of degrees of freedomin a quantum state. Entanglement entropy measures both how far the system is from a pure state, but also how closeit is to a thermal one. It is therefore interesting to ask what is the entanglement entropy of a CGC wave function.Although entanglement entropy is a characteristic of a hadronic wave function, it also indirectly carries someinformation about the structure of the final state in collision of this hadron with a hadronic target. For examplerecently we have shown that Bose-Einstein correlations present in an incoming CGC state[5] manifest themselvesas ridge type correlations between particles produced in the final state of an energetic collision. The CGC basedcalculation of such correlations[6] provides a possible explanation of the ridge correlations observed by the LHCexperiments in high multiplicity p-p and p-Pb events[7–10].In this paper we calculate the entanglement entropy of the soft modes in the CGC wave function. By soft we meanthe gluon field modes with the longitudinal momentum in the “last” rapidity bin η < ∆ Y ∼ /α s which are stillenergetic enough to participate in scattering at relevant energy. A similar question has been addressed in [11], wherea definition of dynamical entropy has been proposed. In the present paper we do not use an ad hoc definition, butrather directly calculate the standard entanglement entropy of a quantum state.We also consider the entropy of the produced system of soft particles. An earlier attempt in this direction ispresented in[12], while a different approach to the entropy production problem can be found in [13, 14] (see also thereview [15] and references therein). The knowledge of the outgoing wave function in the CGC formalism allows usto directly calculate the entropy of the final state. We show that the correlations between produced particles give awell defined contribution to this entropy. This contribution is negative in accordance with one’s naive expectationthat stronger correlations mean a more ordered state. We define in a natural way the temperature of the producedsystem of particles. We show that in the weak field limit the temperature is given by T = π h k ⊥ i /
2, where h k ⊥ i is theaverage transverse momentum of the produced particles. I. THE DENSITY MATRIX AND RENYI ENTROPY
At high energy a hadronic wave function has a large soft gluon component. These are the softest gluons in the wavefunction which are energetic enough to scatter on a hadronic target. They occupy the rapidity interval 0 < η < ∆ Y with ∆ Y ∼ /α s . In what follows we will calculate the entanglement entropy of this component of the wave function.In the CGC approach the hadronic wave function has the form [16]Ψ[ a, A ] = ψ [ A ] χ [ a, ρ ] (2.1)where a is the soft gluon field modes, A are the valence modes (with rapidities η > ∆ Y ) and ρ a ( x ) is the color chargedensity due to the valence gluons. As long as the color charge density is not too large a good approximation to thesoft part of the wave function is a simple coherent state χ [ a, ρ ] = exp (cid:26) i Z k b ia ( k ) (cid:2) a † ia ( k ) + a ia ( − k ) (cid:3)(cid:27) | i , (2.2)with the Weizs¨acker-Williams field b ia ( k ) = gρ a ( k ) ik i k . The creation and annihilation operators entering the aboveequation are the gluon operators integrated over rapidity, a ai ( k ) ≡ √ ∆ Y Z η< ∆ Y dη π a ai ( η, k ) . (2.3)This structure is typical of the Born-Oppenheimer approximation, where the wave function of the fast degrees offreedom (soft gluons) is determined by the background of slow degrees of freedom (valence gluons). The valence partof the wave function depends on the energy of the process, or in the present context on the total rapidity by whichthe hadron has been boosted from the rest frame, and is subject to JIMWLK evolution[2]. At any fixed energy avalence observable that depends only on the color charge density O [ ρ ] is calculated as h O i = Z D [ ρ ] W Y [ ρ ] O [ ρ ] (2.4)The rapidity dependence of W Y is determined by JIMWLK evolution. In this paper we will not insist that W [ ρ ]solves JIMWLK equation, but instead will consider the (somehwat generalized) McLerran-Venugopalan model[17] W P [ ρ ] = N e − R k µ k ) ρ a ( k ) ρ a ( − k ) (2.5)where N is a normalization factor.Note that the soft wave function eq.(2.2) depends on a single longitudinal degree of freedom, i.e. the gluon fieldmode integrated over rapidity in the interval ∆ Y . Only this mode is relevant to our discussion in this paper. Rapiditydependence becomes significant only when the rapidity interval considered as soft becomes large enough ∆ Y > /α s .For such large rapidity intervals additional rapidity dependence of the wave function appears and our discussion wouldhave to be amended. We will not consider this in the present paper.The reduced density matrix of the soft gluons in the McLerran - Venugopalan model is [5]ˆ ρ = N Z D [ ρ ] e − R k µ k ) ρ a ( k ) ρ a ( − k ) e i R q b ib ( q ) φ ib ( − q ) | ih | e − i R p b jc ( p ) φ jc ( − p ) where we have defined φ ia ( k ) = a ia ( k ) + a † ia ( − k ). The integral over the charge density ρ can be performed with theresult ˆ ρ = X n n ! e − φ i M ij φ j h n Y m =1 M i m j m φ i m | i h | φ j m i e − φ i M ij φ j (2.6)Here we have introduced compact notations: φ i ≡ h a † ai ( x ) + a ai ( x ) i ; M ij ≡ g π Z u,v µ ( u, v ) ( x − u ) i ( x − u ) ( y − v ) j ( y − v ) δ ab (2.7)Here M bears two polarization, two color, and two coordinate indices, collectively denoted as { ij } . In eq.(2.6)summation over discrete and integration over continuous indices is implied.2ur goal is to calculate the Von Neumann entropy of the reduced density matrix eq.(2.6). As a warm-up exercisewe first compute the Renyi entropy. To this end we have to evaluate tr [ˆ ρ ]: tr [ˆ ρ ] = X n,n ′ n ! n ′ ! h | e − φ i M ij φ j h n Y m =1 n ′ Y m ′ =1 M i m j m M i m ′ j m ′ φ j m ′ φ i m | i h | φ j m φ i m ′ i e − φ i M ij φ j | i (2.8)Computation of the matrix elements is straightforward as it involves calculation of two vacuum matrix elements,which can be performed by summing over all possible Wick contractions. The explicit form of the light front vacuumwave function is h φ | i = N e − π φ i φ i (2.9)As this is a Gaussian, it is convenient to introduce G ij ≡ h | e − φ i M ij φ j φ i φ j | i / p det [ π/ ( π + 2 M )] = [ π + 2 M ] − ij (2.10)where we use the shorthand notation π ≡ πδ ij δ ab δ ( x − y ). The matrix elements in eq.(2.8) do not vanish only whenthe total number of φ insertions in each one is even, that is n + n ′ = 2 m (2.11)The result for tr [ˆ ρ ] is a sum of all possible contractions of M G weighted with combinatoric factors. Given that both M and G are diagonal in color, the expansion can be organized in terms of color loops: L ≡ tr [ M G ] ; L ≡ tr [ M GM G ] ; L ≡ tr [ M GM GM GM G ] .... (2.12)Each trace over color gives a factor N c . Performing the averaging over the vacuum state we obtain tr [ˆ ρ ] = det (cid:20) ππ + 2 M (cid:21) X m m X n =0 n !(2 m − n )! h L m (2 m − L m − L m − · · · ] (2.13)Using the identity m X n =0 n !(2 m − n )! (2 m − m X n =0 n !(2 m − n )! (2 m − m − ( m − m m ! (2.14)we find that the summation over m exponentiates: tr [ˆ ρ ] = exp (cid:20) − tr [ln(1 + 2 Mπ )] + 2 L + · · · (cid:21) ≃ exp [ − L + L + · · · ] (2.15)After a more careful examination we find that the other terms exponentiate as well tr [ˆ ρ ] = exp " − tr [ln(1 + 2 Mπ )] + X n n − n L n = exp (cid:20) − tr [ln(1 + 2 Mπ )] − tr [ln(1 − (2 M G ) )] (cid:21) (2.16)Finally using the definition eq.(2.10) we arrive at the closed-form expression tr [ˆ ρ ] = exp (cid:20) − tr (cid:20) ln(1 + 4 Mπ ) (cid:21)(cid:21) (2.17)from which the Renyi entropy is found as σ ≡ − ln tr [ˆ ρ ] = 12 tr (cid:20) ln(1 + 4 Mπ ) (cid:21) (2.18)3 II. VON NEUMANN ENTROPY
We are now ready to compute the Von Neumann entropy of the reduced density matrix defined as σ E = − tr [ˆ ρ ln ˆ ρ ] (3.1)The following identity is very useful for this purposeln ˆ ρ = lim ǫ → ǫ (ˆ ρ ǫ −
1) (3.2)We first compute tr [ˆ ρ N ], for arbitrary N , and then take the limit N → ǫ .Consider the generalization of eq. (2.8) to the calculation of ˆ ρ N . This expression contains a product of N vacuummatrix elements of operators that depend on the field φ . Each one of these matrix elements is calculated independently,and thus the fields entering the calculation of different matrix elements can be considered as independent. We thusdefine the multiplet of replica fields φ αi , α = 1 , ..., N . Thus we can write tr [ˆ ρ N ] = h | e − P Nα =1 φ αi M ij φ αj + P Nα =1 φ αi M ij φ α +1 j | i (3.3)where now | i is the light front vacuum of all the replica fields φ α . Notice the nearest neighbor ”interaction” betweenthe replica fields. The replica fields in eq.(3.3) satisfy periodic boundary conditions φ N +1 = φ .We further rewrite (3.3) tr [ˆ ρ N ] = (cid:18) det[ π ]2 π (cid:19) N/ Z N Y α =1 [ Dφ α ] exp ( − π N X α =1 φ αi φ αi − N X α =1 ( φ αi − φ α +1 i ) M ij ( φ αj − φ α +1 j ) ) (3.4)This is a partition function of a spin chain on a replica space lattice. Eq.(3.4) obviously has the discrete translationalsymmetry in the replica space. The ”action” is diagonalized by Fourier transforming in α :˜ φ n = 1 N N X α =1 e i πN αn φ α ; φ α = N − X n =0 e − i πN αn ˜ φ n (3.5)Notice the periodicity relation ˜ φ N − n = ˜ φ ∗ n . The nearest-neighbor interaction in Fourier space reads( φ αi − φ α +1 i )( φ αj − φ α +1 j ) = X n,m (cid:16) e − i πN n − (cid:17) (cid:16) e − πN m − (cid:17) e − i πN α ( n + m ) ˜ φ ni ˜ φ mj (3.6)Using X α e − i πN α ( n + m ) = N δ n + m,N (3.7)we have X α ( φ αi − φ α +1 i )( φ αj − φ α +1 j ) = N X n (cid:16) e − i πN n − (cid:17) (cid:16) e − i πN n − (cid:17) ˜ φ ni ˜ φ ∗ nj = 4 N X n sin ( πN n ) ˜ φ ni ˜ φ ∗ nj (3.8) tr [ˆ ρ N ] = N N/ (cid:18) det[ π ]2 π (cid:19) N/ Z Y n [ D ˜ φ n ] exp ( − N N − X n =0 ˜ φ ni h π + 4 M sin ( πN n ) i ij ˜ φ ∗ nj ) (3.9)where N N/ is the Jacobian of the transformation (3.5). Using the identity π + 4 M sin ( πN n ) = (cid:20) π + 2 M (cid:18) − cos( 2 πN n ) (cid:19)(cid:21) (3.10)the Gaussian integral in (3.9) is easily computable tr [ˆ ρ N ] = det[ π ] N/ det " N − Y n =0 (cid:20) π + 2 M (cid:18) − cos( 2 πN n ) (cid:19)(cid:21) − / (3.11)4ow we apply (1.396) of Ref. [18], with 1 > M G > N − Y n =0 (cid:20) π + 2 M (cid:18) − cos( 2 πN n ) (cid:19)(cid:21) = (2 M ) N N − Y n =0 (cid:20) − cos( 2 πN n ) + π M (cid:21) = 2 ( M ) N h cosh( N arcCosh[1 + π M ]) − i (3.12)We finally arrive at the following expression for tr [ˆ ρ N ]: tr [ˆ ρ N ] = exp (cid:26) −
12 ln 2 − N tr h ln Mπ i − tr h ln (cid:16) cosh( N arcCosh[1 + π M ]) − (cid:17)i(cid:27) (3.13)The entropy is computed by taking N = 1 + ǫ and keeping the terms linear in ǫ : tr [ˆ ρ ǫ ] ≃ exp ( −
12 ln 2 − ǫ tr (cid:20) ln Mπ (cid:21) − tr " ln π M + ǫ π M r Mπ arcCosh[1 + π M ] ! = 1 + ǫ tr " ln πM − r Mπ arcCosh[1 + π M ] (3.14)Thus we arrive at a closed expression for the entanglement entropy σ E = 12 tr " ln Mπ + r Mπ arcCosh[1 + π M ] = 12 tr ( ln Mπ + r Mπ ln " π M r Mπ ! (3.15)To understand the basic properties of this expression let us consider translationally invariant case. In this case thematrix M is diagonal in momentum space, M abij ( p ) = g µ ( p ) p i p j p δ ab (3.16)In the original MV model µ is a constant and does not depend on momentum, but the momentum dependentGaussian width has been used in recent applications (see e.g. [19]). The contribution to the entropy from largetransverse momentum modes can be calculated by expanding the expression eq.(3.15) to leading order in M , sincefor large momenta ( g µ < p ) the eigenvalues of M are all small. The expression for the entropy in the weak fieldlimit is σ EM ≪ = tr (cid:20) Mπ ln πeM (cid:21) (3.17)Here we have kept the first two terms in the small M expansion, which are not suppressed by powers of M .Thus the dominant ultraviolet contribution is σ EUV ≃ − g π ( N c − S Z d p (2 π ) µ ( p ) p ln g µ ( p ) eπ p θ ( p − g π µ ( p )) (3.18)where S is the total area of the projectile. For the original MV model with momentum independent µ this expressionis logarithmically divergent in the UV. Introducing a UV cutoff Λ we find σ EUV ≃ Q s πg ( N c − S (cid:20) ln g Λ Q s + ln g Λ Q s (cid:21) (3.19)where we have identified the saturation momentum in the standard way as Q s = g µ /π . The UV divergence inthis expression is of course the artifact of the eikonal approximation which in the CGC context is applied for alltransverse momentum modes. In fact the eikonal approximation breaks down when the transverse momentum of thesoft gluons is of the order of their longitudinal momentum. The natural cutoff on the transverse momentum is oforder Λ ∼ P + e − Y ∼ me Y , where P + is the total energy of the hadron and m is a typical hadronic scale and Y is5he rapidity of the soft gluons. This cutoff does not depend on the total energy of the hadron, but rather on someinitial energy at which the eikonal approximation is assume to be valid[38].The contribution from the infrared modes can also be calculated. At small momenta p < Q s /g one can formallyexpand in M − . In this limit the entropy is dominated by the first term in eq.(3.15). Here we also keep the firstsubleading correction: σ EM →∞ ≃ tr [ln e Mπ ] (3.20)and the infrared contribution is σ EIR ≃
12 ( N c − S Z d p (2 π ) ln e g µ ( p ) π p θ ( Q s − g p ) = 38 πg ( N c − SQ s (3.21)Note that we have chosen to separate the integration region into UV and IR at exactly p = Q s /g in eqs.(3.18,3.21).Although the parametric dependence of the separation scale is clear, its exact value is somewhat arbitrary. Ourreason for choosing the above value, is that at this value of p the integrands in eqs.(3.18,3.21) exactly coincide. Thusthis is a unique choice for which the approximation of the total momentum space entropy density by the sum of itsasymptotic expressions is a continuous function of momentum.Combining the two expressions we find approximately for the MV model σ E ≈ σ EUV + σ EIR = SQ s πg ( N c − (cid:20) ln g Λ Q s + ln g Λ Q s + 32 (cid:21) (3.22)A word on the dependence of the entropy on the energy (rapidity) of the hadron. The above calculation is performedat fixed rapidity. The hadronic wave function evolves to higher rapidity via JIMWLK evolution. Since this evolutionis nonlinear, the soft gluon wave function is not given by a Gaussian any more, especially in the saturation regime.Calculating the entropy for a nongaussian wave function is significantly more difficult, although we will describe somesteps toward such a calculation in the last section. However, as has been suggested in the past, approximating thesoft gluon wave function by a gaussian is not a bad approximation for phenomenological purposes as long as theparameters of the Gaussian are taken to evolve with rapidity in a way consistent with JIMWLK evolution[20].In particular one can still use the MV type ansatz but with the function µ Y ( p ) taken to be a solution of theBFKL[21] (or BK[4, 22]) evolution equation. Within this approximation one can derive the evolution of the entropywith the total energy of the hadron. In the weak coupling limit, where the evolution of µ is given by the BFKLequation, differentiating the weak field expression eq.(3.17) we find dσ E dY = − ( N c − S Z d p (2 π ) d k (2 π ) ln[ φ ( p )] K BF KL ( p, k ) φ ( k ) (3.23)where φ ( k ) ≡ g π µ ( k ) k is the gluon unintegrated density, and K BF KL is the kernel of the BFKL equation.In contrast to the MV model, the solution of the BFKL/BK equation at large rapidity exhibits an anomalousdimension which slowly varies with transverse momentum g π µ Y ( p ) ∼ p γ ( p ) ; (3.24)with γ ( p >> Q s ) → γ ( p ∼ Q s ) ≈ .
67 and γ ( p < Q s ) = 0. Note that with such an anomalous dimension the entropydiverges as a power of the cutoff in the ultraviolet as σ UV ∝ Λ γ .Some properties of the entanglement entropy of CGC are worth noting. First, it is proportional to the transversearea of the hadron. This is a natural property of any extensive observable. However σ E is not extensive in thelongitudinal direction. Recall that σ E as calculated above is the entropy on the rapidity slice of width ∆ Y , but σ E is not proportional to ∆ Y . The reason for this is of course clear, since only a single rapidity independent mode onthe interval ∆ Y is entangled with the valence degrees of freedom of the hadron. For the same reason the entropydoes not have any longitudinal UV divergence associated with the fact that we have a sharp boundary between softand valence degrees of freedom in rapidity space. This is very different from a situation one encounters in calculatingentanglement entropy of a finite region of space A in a local field theory when integrating out degrees of freedom in therest of space B . In the latter case the local interaction between the degrees of freedom along the boundary between A and B leads to a UV divergence which depends on the ratio of the correlation length to the (vanishing) width ofthe boundary region. In our case the interaction in the longitudinal direction is nonlocal. The eikonal interactionbetween soft and valence degrees of freedom extends over large distances in rapidity, so that there is no significant6ontribution to the entropy from the (sharp) boundary. In this sense the entanglement entropy of the CGC is akinto the “topological entropy”[23] frequently discussed in the context of condensed matter systems, which is a constantindependent of the boundary or volume of the spatial region under consideration.Recently, the “momentum space” entanglement entropy has been discussed in connection with Wilsonian renor-malization group flow[24]. It would be interesting to explore the relation of these ideas with eq.(3.23) as σ E is indeedcalculated by separating degrees of freedom in momentum space, while the BFKL equation has been interpreted inthe past as RG flow in rapidity variable[25]. IV. ENTROPY PRODUCTION IN COLLISION.
Our next quantity of interest is the entropy production during collision of two hadrons. We calculate the entropyin the rapidity bin of width ∆ Y at rapidity Y away from the valence charges. All the glue at rapidities η < Y + ∆ Y relative to the forward moving particles is considered to be part of the target wave function and as in the standardsaturation approach is represented by the eikonal scattering matrix S ( x ).According to the eikonal paradigm the effect of the collision is to color rotate the valence as well as the soft gluonfield in the wave function, so that at time t = 0 right after the collision ψ [ A ] χ [ a, ρ ] → ˆ Sψ [ A ] χ [ a, ρ ] = ψ [ SA ] χ [ Sa, Sρ ] = ψ [ SA ] exp (cid:26) i Z x ˜ b ia ( x ) S ( x ) (cid:2) a † ia ( x ) + a ia ( x ) (cid:3)(cid:27) | i , (4.1)where ˜ b ai ( x ) is the Weszacker-Williams field produced by the rotated color charge density ˜ ρ b ( z ) = S bc ( z ) ρ c ( z ). Hereˆ S is the second quantized operator of eikonal S -matrix. The action of ˆ S to the right (and ˆ S † to the left) is equivalentto the multiplication of a a ( x ) , a † a ( x ) and ρ a ( x ) by the unitary matrix S ( x ). Since ˆ S is a unitary operator which doesnot mix valence and soft degrees of freedom, it is clear that the initial scattering does not modify the entanglemententropy. To see this explicitly we note that in the basis ˜ ρ ( x ) = S ( x ) ρ ( x ) and ˜ φ ( x ) = S ( x ) φ ( x ) the form of thescattered wave function is identical to that of the incoming one in the original basis. Since the integration measurefor ρ as well as the vacuum wave function for φ ( x ) eq. (2.9) are invariant under this local unitary transformation,changing variables to ˜ ρ and ˜ φ in the calculation of the entropy immediately establishes that the entropy is the same.However the entropy produced in collision is not the same as the entanglement entropy of the final state. Hereone is interested only in characteristics of inelastically produced gluonic state, and not, say in the contribution ofsoft gluons which are part of the wave function of outgoing “bound state” hadrons. Thus the relevant quantity forcalculating the production entropy is not the wave function evolved to t → ∞ , but only its “inelastically produced”part. This is a similar problem to the one we are faced with when calculating any soft gluon observable produced incollision, i.e. single or double gluon inclusive cross section. In that context the solution is well known. Recall, that agluonic observable O ( a, a † ) at t → ∞ is given by the expression [26] h O i = Z dρW [ ρ ] h | e − i R x b ia [ ρ ]( x ) φ ia ( x ) ˆ S † e i R x b ia [ ρ ]( x ) φ ia ( x ) O ( a, a † ) e − i R x b ia [ ρ ]( x ) φ ia ( x ) ˆ Se i R x b ia [ ρ ]( x ) φ ia ( x ) | i (4.2)The multiplication by the extra coherent operator factor accounts for the evolution of the wave function from time t = 0 to time t → ∞ as well as restricting Hilbert space for the calculation of O to that of inelastically producedgluons only.Since this formula is valid for an arbitrary soft observable O ( a, a † ), clearly this means that the density matrix thatdescribes the produced soft gluons has exactly the same form as eq.(2.6) but with the matrix M substituted by M P ≡ g π Z u,v µ ( u, v ) ( x − u ) i ( x − u ) ( y − v ) j ( y − v ) [( S ( u ) − S ( x ))( S † ( v ) − S † ( y ))] ab (4.3)Thus the entropy produced in a hadronic collision, which is the Van Neumann entropy of the density matrix specifiedby eq.(4.3) is given by σ P = 12 tr ( ln M P π + r M P π ln " π M P r M P π ! (4.4)Although the formal expression for σ P is similar to that for σ E , it has a different meaning and its properties aresignificantly different. We stress that σ P is not an entanglement entropy like σ E , but rather the entropy of the state7roduced inelastically in the hadronic collision. As a result σ P is UV finite. This is clear, since the UV divergence inthe calculation of σ P came from the coordinate region x → y → u → v , and M P vanishes in this limit, as opposed to M .Another important property is that σ P as defined in eq.(4.4) is calculated for a single scattering event characterizedby a given eikonal profile S ( x ). Although it certainly makes sense to talk about event by event entropy production, amore global characteristic of scattering would be an event averaged quantity. We will next consider such an averageproduction entropy and will relate it to inclusive gluon production amplitudes.Note, that eq. (4.3) can be written as a product of single inclusive gluon production amplitude Q i defined in [26]averaged over the projectile wave function M Pij = Z D [ ρ ] Q i Q j W P [ ρ ] ≡ h Q i Q j i P (4.5)where Q ai ( x ) = g π Z u ( x − u ) i ( x − u ) [ S ( u ) − S ( x )] ab ρ b ( u ) (4.6)with S being in the adjoint representation.The target average of this expression is directly related to the single inclusive gluon production probability, orrather the phase space density n = dN/dkdydb is n ( k ) = dNd kdyd b = Z xy hh Q ai ( x ) Q ai ( y ) ii P,T e ik ( x − y ) (4.7)where as usual the averaging over the target fields has to be performed with some weight function W [ S ].In order to relate the average entropy to inclusive gluon production amplitudes we expand it in powers of thefluctuation of the matrix M P around its target average M P = ¯ M P + ( M P − ¯ M P ); ¯ M P ≡ Z DS W [ S ] M P [ S ] (4.8)The target average has the form¯ M P = − δ ab g π Z u,v µ ( u, v ) ( x − u ) i ( x − u ) ( y − v ) j ( y − v ) [ P A ( x, y ) + P A ( u, v ) − P A ( x, v ) − P A ( u, y )] (4.9)where the “adjoint Pomeron” is defined as P A ( x, y ) = 1 − N c − h tr [ S † ( x ) S ( y )] i T (4.10)Although ¯ M P is related to the single inclusive gluon production probability, the two are not equal to each other.Assuming translational invariance of µ and P A we have¯ M P = δ ab g Z k,q e − ik ( x − y ) K ij ( k, q ) µ ( k − q ) P A ( q ) (4.11)where K ij ( p, q ) = ( k − q ) i ( k − q ) ( k − q ) j ( k − q ) + k i k k j k − ( k − q ) i ( k − q ) k j k − ( k − q ) j ( k − q ) k i k (4.12)Conveniently assuming parity invariant target, we can decompose h M P i T in momentum space as¯ M P ( k ) = δ ab (cid:20) M l ( k ) k i k j k + M t ( k ) ( δ ij − k i k j k ) (cid:21) (4.13)As opposed to M , the matrix M P has both longitudinal and transverse components. These correspond to gluonsproduced with polarizations parallel and perpendicular to their transverse momentum. Clearly both polarizations ingeneral are produced, since the gluons in question are free gluons rather than a part of the dressing of the wavefunction8f some faster moving valence color charges. The inclusive gluon production cross section [27] is related to the sumof the two polarizations n ( k ) ∝ M l ( k ) + M t ( k ) (4.14)Let us consider the case of small M P , which is relevant for the UV regime. We can then use eq.(3.17)¯ σ P = h tr (cid:20) M P π ln πeM P (cid:21) i T = tr (cid:20) ¯ M P π ln πe ¯ M P (cid:21) − π tr hn h M P M P i T − ( ¯ M P ) o ( ¯ M P ) − i .... (4.15)Using the representation (4.13), the first term in (4.15) can be written as tr (cid:20) ¯ M P π ln πe ¯ M P (cid:21) = − N c − π Z d k (2 π ) h M l ( k ) ln M l ( k ) eπ + M t ( k ) ln M t ( k ) eπ i (4.16)This has the standard form σ = − P i n i ln n i for the system of noninteracting particle species. The index i hererefers to transverse momentum as well as the longitudinal and transverse gluon polarizations. Note, that it cannotbe expressed in terms of the total particle spectrum n ( k ), unless the longitudinal and transverse eigenvalues of M P are proportional to each other. Generically we do not see a reason to expect that this is the case.The second term in (4.15) is related to correlated inclusive two gluon production [6, 28–35]. Pictorially this term canbe represented in Fig.1a. This diagram (apart from the fact that two of the gluon polarization indices do not close) isthe same as the connected “glasma graph” contribution to the two gluon correlation [30]. The “disconnected” diagramin Fig.1b is not included in the expression eq.(4.15). Note that although the diagram in Fig. 1b has a disconnectedtopology, it does in fact contribute to the correlated production in the large N c limit if the target averages do notfactorize[33, 36]. In the language of the Reggeon field theory, this contribution is due to correlated production fromtwo Pomerons, while the diagrams in Fig 1a are due to correlated production from the B-reggeon [37]. µ µ µ µ k kp p p p kkS SS S S SS S(a) (b) FIG. 1:Two contributions to the inclusive two gluon production. Fig.1a - “connected” part that contributes to entropy eq.(4.15).Fig. 1b - “disconnected part that does not contribute to the entropy eq.(4.15).
With this in mind we can write δσ P = 12 π tr hn h M P M P i T − ( ¯ M P ) o ( ¯ M P ) − i (4.17) ∝ Z d kd p h h a † al ( p ) a † bi ( k ) a cj ( k ) a al ( p ) i ( B ) − h a † al ( p ) a al ( p ) ih a † bi ( k ) a cj ( k ) i i ( ¯ M P ) − cbji ( k )9r in a less convoluted way: δσ P = − g π ( N c − Z u,v, ¯ u, ¯ v,y,x, ¯ x,k µ ( u, v ) µ (¯ u, ¯ v ) e ik ( x − ¯ x ) (cid:20) M l ( k ) k i k j k + 1 M t ( k ) ( δ ij − k i k j k ) (cid:21) × ( x − u ) i ( x − u ) ( y − v ) m ( y − v ) (¯ x − ¯ u ) j (¯ x − ¯ u ) ( y − ¯ v ) m ( y − ¯ v ) × tr h (cid:2) ( S ( u ) − S ( x ))( S † ( v ) − S † ( y ))][( S (¯ u ) − S (¯ x ))( S † (¯ v ) − S † ( y )) (cid:3) i T + 12 π Z k n ( k ) (4.18)Again, we see that if the longitudinal and transverse parts of M are proportional to each other at all momenta, theextra contribution to entropy is proportional to the double inclusive gluon correlation function integrated over thegluon momenta. If M l ( k ) = aM t ( k ), the longitudinal and transverse gluon polarizations are separately normalized totheir single inclusive production probability.Interestingly the correlation function in eq.(4.17) is normalized by dividing by the first power of the single inclusivegluon production and not its square. Recall that the correlated gluon production normalized this way remains finite inthe CGC approach in the limit Q s S → ∞ , while the correlation function normalized to the square of single inclusivegluon vanishes in this limit[6, 33].The natural, but nevertheless interesting property of eq.(4.15), is that the contribution of correlations to entropy isnegative. This is in accordance with the intuition that a correlated state contains more order and thus has a smallerentropy.As noted above, the produced entropy is UV finite, since in the UV limit all adjoint Pomerons in eq.(4.9) vanish,and the divergent contribution to the momentum integral in eq.(4.15) disappears. This is even clearer from themomentum space representation of M P eq.(4.11). The perturbative behavior of the adjoint Pomeron at large momentais P ( q ) ∝ Q T /q , which leads to M P ( k → ∞ ) → Q p Q T /g k . Even accounting for the BFKL anomalous dimensionleads to the UV convergent integral in eq.(4.16). The integral in eq.(4.16) is also IR finite, since the expected IRbehavior is P ( q → → /Q T .In general the consequence of eq.(4.9) (neglecting the rotational indices for the moment) is M P ( q ) = π Q P g Q T F (cid:18) Q T q (cid:19) (4.19)where F is a regular function with the limiting behavior F ( x → → const x and F ( x → ∞ ) → const . Usingeq.(4.19) we can estimate the entropy production parametrically σ P ∼ a ( N c − S Q P πg ln bg Q T Q P (4.20)where a and b are constants of order unity. V. DISCUSSION.
In this paper we have calculated the entanglement entropy of the soft gluon modes in the CGC wave function, aswell as entropy produced in hadronic collision at high energy. In these calculations we have relied on the MV modelto represent the valence part of the hadronic wave function.The entanglement entropy is an interesting theoretical quantity, as it is a global characteristic of the soft gluondensity matrix. Its basic properties are determined by the fact that only one longitudinal mode of the soft gluonfield, namely the rapidity independent mode is entangled with the valence part of the wave function. The entropytherefore is not extensive in the length of the rapidity interval, and in this sense is akin to the topological entropydiscussed these days in condensed matter literature[23]. In a certain sense the analogy is closer than it might seem,as the origin of the topological entropy is in the long range quantum entanglement.We find that the entanglement entropy of CGC is formally UV divergent and is dominated by the modes with largetransverse momentum p ⊥ > Q s /g . The contribution from the IR modes is finite and suppressed by a square of thelogarithm of the UV cutoff.We have also derived an expression for the entropy production in an hadronic collision. We have shown that itis related to inclusive gluon production amplitudes, but is not determined exclusively in terms of these amplitudes.In particularly it depends on the production probabilities of longitudinally and transversely (with respect to the10irection of their transverse momentum) polarized gluons separately. We also find that correlated gluon emissiongives a negative contribution to the produced entropy, consistent with the view of entropy as measuring disorder inthe final state. Just like the entanglement entropy, the produced entropy is not extensive in the rapidity interval overwhich it is calculated. The reason is the same: in the CGC approximation all final state particles are produced froma single longitudinal boost invariant mode.Having calculated the entropy, the natural question is whether one can define the corresponding temperature. Thenatural definition would be the usual thermodynamic one T − = dσdE ⊥ (5.1)where E ⊥ is the transverse energy of the system.It is not clear whether this is a sensible quantity for the entanglement entropy. If for E ⊥ we take the obvious E ⊥ ∝ Z d k | k | M ( k ) ∝ Q s Sg Λ (5.2)we obtain for the “entanglement temperature” T E ∝ Λ( N c −
1) ln Λ g Q s → ∞ (5.3)which reflects the fact that the entanglement entropy is dominated by the UV modes and is formally UV divergent.On the other hand for the produced system of particles eq.(4.11) gives E ⊥ ∝ ( N c − S Q P g Q T (5.4)This is just the statement that the average transverse momentum of produced particles is proportional to Q T , whilethe number of particles produced is of the order Q P /g .Since the entropy eq.(4.20) is not only a function of E ⊥ , the definition of temperature is somewhat ambiguous. Todefine the temperature one has to decide which combination of Q P and Q T has to be kept fixed. Choosing whichcombination to keep fixed corresponds to deciding what is the analog of increasing the energy in the same statisticalmechanical system. One reasonable choice is to keep Q P fixed, since this corresponds to increasing the energy of theproduced system by boosting the target, while keeping the projectile unchanged. With this choice using eq.(4.20) wefind T = c Q T = π h k ⊥ i (5.5)where c is a constant of order unity and h k ⊥ i = E ⊥ /N total = R d k | k | M ( k ) R d pM ( p ) is the average transverse momentum perproduced particle. Note, that even though the relation between h k ⊥ i and Q T depends on the details of the function F in eq.(4.19), the relation between h k ⊥ i and temperature is determined exactly. Eq.(5.5) is an intuitively simpleresult showing that the effective temperature is proportional to average momentum per particle. We are neverthelessnot aware of a direct derivation of this result from the density matrix of the system produced in collision.All our calculations were performed in the MV model. The Gaussian weight for the valence charge density averagingmade it possible to express the entropy directly in terms of a single function M P . The actual weight functional evolveswith energy and is determined by solving the JIMWLK equation. Although the calculation of entropy in the case ofnon Gaussian W [ ρ ] is more complicated, one can try to approach it in a similar way.We would have to calculate tr [ˆ ρ N ] = tr Z N Y β =1 D [ ρ β ] W P [ ρ β ] e i R k b β ( k ) φ ( k ) | ih | e − i R k b β ( k ) φ ( k ) (5.6)Integrating over ρ β , and resorting to the same replica trick yields tr [ˆ ρ N ] = (cid:18) det[ π ]2 π (cid:19) N/ Z N Y α =1 [ Dφ α ] exp ( − π N X α =1 φ αi φ αi + 12 N X α =1 Γ[ φ α − φ α +1 ] ) (5.7)11ere as before we have introduced N replicas - one for each vacuum matrix elements. In eq.(5.7) Γ is the effectiveaction. Interestingly, it is directly related to the projectile scattering matrix, since S = Z DρW P [ ρ ] e ig R x,u ρ a ( x ) ( x − u ) i ( x − u )2 [ φ αi ( u ) − φ α +1 i ( u )] ≡ e − Γ[ φ α − φ α +1 ] (5.8)is precisely the scattering matrix element of the projectile on the “target field” φ αi ( u ) − φ α +1 i ( u ).The action Γ can be expanded in terms of connected correlators of the valence color charge densityΓ[ φ ] = Γ ij φ i φ j + Γ ijn φ i φ j φ n + Γ ijnm φ i φ j φ n φ m + ... (5.9)Γ ij = δ Γ[ φ ] δφ i δφ j | φ =0 = − g π Z u,v ( x − u ) i ( x − u ) ( y − v ) j ( y − v ) h ρ a ( u ) ρ b ( v ) i P Γ ijn = δ Γ[ φ ] δφ i δφ j δφ n | φ =0 = − g π Z u,v,w ( x − u ) i ( x − u ) ( y − v ) j ( y − v ) ( z − w ) n ( z − w ) h ρ a ( u ) ρ b ( v ) ρ c ( w ) i P Γ ijnm = − g π Z u,v,w,w ′ ( x − u ) i ( x − u ) ( y − v ) j ( y − v ) ( z − w ) n ( z ′ − w ) ( z ′ − w ′ ) m ( z ′ − w ′ ) h ρ a ( u ) ρ b ( v ) ρ c ( w ) ρ d ( w ′ ) i connP (5.10)where the connected correlators are calculated as h ρ a ( u ) ρ b ( v ) i P = Z D [ ρ ] ρ a ( u ) ρ b ( v ) W P [ ρ ] h ρ a ( u ) ρ b ( v ) ρ c ( w ) i P = Z D [ ρ ] ρ a ( u ) ρ b ( v ) ρ c ( w ) W P [ ρ ] h ρ a ( u ) ρ b ( v ) ρ c ( w ) ρ d ( w ′ ) i connP = Z D [ ρ ] ρ a ( u ) ρ b ( v ) ρ c ( w ) ρ d ( w ′ ) W P [ ρ ] | connected (5.11)If the connected correlators of ρ are small, these formulae can serve as a starting point for perturbative calculation ofthe entropy.An alternative strategy could be to first integrate over φ in eq.(5.6). This leads to the following expression: tr [ˆ ρ N ] = Z N Y β =1 Dρ β exp − N X β =1 (cid:20) − ln W P [ ρ β ] + 12 π Z k,k ′ ( b β +1 ( k ) − b β ( k ))( b β +1 ( k ′ ) − b β ( k ′ )) (cid:21) (5.12)where b N +1 = b . This has the form of discrete lattice model with the same “hopping term” as in Section 2, but witha different on site potential V [ ρ ] = − ln [ W [ ρ ]].To take the limit N → ǫ one would need to perform the integration over the replica fields analytically. If theprojectile is dilute, the potential V [ ρ ] is sharply peaked at small ρ and the integral can be calculated in the “tightbinding approximation”. For the dense projectile on the other hand, the potential is small since large fluctuations ofcharge density are allowed. In this case one can expand around the free “hopping term”. It would be interesting tostudy the effects of non-Gaussianity on the entropy with the help of these approximations for realistic W [ ρ ].In this paper, in order to compute the entanglement entropy (and the entropy production), we have used the densitymatrix of soft gluons derived after tracing over valence gluons. As a further development of the above ideas, one canconsider a further reduction of the density matrix. In particular, since gluons with very small transverse momentaare unobservable, it could be interesting to trace over gluons with momenta below some soft scale and consider theentropy of the remaining modes. Another interesting venue would be to relate two-particle correlations, particularlylong range rapidity correlations, to the concept of mutual information . The latter is defined through entanglemententropy of two different subsets of modes in the hadronic wave-function. Acknowledgments
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