Pair creation in boost-invariantly expanding electric fields and two-particle correlations
aa r X i v : . [ h e p - ph ] F e b Pair creation in boost-invariantly expanding electric fieldsand two-particle correlations
Naoto Tanji ∗ Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan
Abstract
Pair creation of scalar particles in a boost-invariant electric field which is confined in theforward light cone is studied. We present the proper-time evolution of momentum distribu-tions of created particles, which preserve the boost invariance of the background field. Thetwo-particle correlation of the created particles is also calculated. We find that long-rangerapidity correlations may arise from the Schwinger mechanism in the boost-invariant electricfield.
UT-Komaba/10-8
Boost invariance in a longitudinal beam direction is a key ingredient in theoretical descriptionsof ultrarelativistic heavy-ion collisions. In 1983, Bjorken [1] introduced a boost-invariant initialcondition for Landau’s hydrodynamic model [2]. Because the hydrodynamic equations as wellrespect the boost symmetry, the following space-time evolution of the system is also boost-invariant, which leads to the central plateau in the particle rapidity distributions. Owing tothis symmetry, the fluid evolution is described by the fluid proper time τ = √ t − z and thetransverse coordinates x ⊥ and does not depend on the space-time rapidity η = ln t + zt − z , wherethe longitudinal beam direction is taken along the z axis.Before the hydrodynamic evolution, the system of ultrarelativistic heavy-ion collisions is alsoassumed to hold the boost symmetry, because shortly after the collision, the nuclei are highlyLorentz-contracted pancakes receding from the collision point at nearly the speed of light, sothat there is no scale in the longitudinal direction. For example the flux-tube model (the Low–Nussinov model [3, 4]), which assumes the generation of longitudinal color-electric fields justafter high-energy particle collisions, comprehends the boost invariance. Also in the frameworkof the color glass condensate, the formation of boost-invariant longitudinal electric fields andthe longitudinal magnetic fields as well is predicted, of which the state is called glasma [5]. Suchelectric fields would depend on τ and x ⊥ but do not depend on the space-time rapidity η .Starting from the boost-invariant classical fields, how does the system evolve into a boost-invariant and locally thermalized quark-gluon plasma? One possible approach to this problemis solving the classical Yang–Mills equations and investigating the dynamics of the classicalelectromagnetic fields [5, 6, 7]. Because of the expansion of the system, the electromagneticfields decay in time. To understand the formation process of the quark-gluon plasma, however,the notion of particle or quantized fields is necessary. In particular, the classical Yang–Millsequations cannot describe the quark production. If one introduces charged quantum fields ∗ E-mail address : [email protected] τ line instead of an equal-time line inorder to give a description preserving the boost invariance. The proper quantization procedure inthe boost-invariant curvilinear coordinates [14] provides a boost-invariant description of the paircreation. To concentrate on revealing the boost-invariant dynamics of the particle production,we treat the scalar QED under constant electric fields which is independent of η and neglect theback-reaction of pair creation to the background field. We drop the dependence of the electricfields on τ and x ⊥ for simplicity. Under the constant electric field, we can derive the analyticsolution for the distribution of created particles. Extension to the more general case, quark paircreation under a color-electric field and taking the back-reaction into account, is possible in asimilar way as that in Ref. [13].Pair production in electric fields confined to a bounded region has been studied by Martinand Vautherin [15]. They calculated the pair creation rate by using the Balian-Bloch expansionof Green’s functions and concluded that if the boundaries move with the speed of light, finite-sizecorrelations vanish and the pair creation rate is the same as Schwinger’s original one. In theircalculation, information from outside the light cone is not used. However, it should be usedas an initial condition on the forward light cone because inside and outside the light cone arecausally connected. Furthermore, calculating the pair creation rate is not sufficient to revealthe time evolution of the system [12]. To properly describe the system evolution, we must setan initial condition and investigate the time evolution of the quantum fields. A study on thisline has been carried out by Cooper et al. [16, 17]. Because the electric field is absent outsidethe forward light cone, there is no particle created. Therefore, the initial state on the light coneshould be a vacuum. In Ref. [16], the initial vacuum is defined so that a charge current, whichis regularized by the adiabatic regularization, vanishes at an initial time.However, in quantum field theory, there is an ambiguity in identifying a vacuum. Requiringthe absence of a particle or current at an initial time is not sufficient to determine one initialvacuum. Because we work in the flat Minkowski space, the vacuum must hold the Poincar´einvariance, even if we use the curvilinear coordinates. Hence, the vacuum we should employ forthe particle mode defined in the τ - η coordinates must be the same as that associated with theusual quantization of free fields in the Cartesian coordinates. In other words, because particlescaptured by detectors are considered as the particle modes associated with the usual plane waveexpansion of the field operator in the Cartesian coordinates, the state before the collision should2e the vacuum for that particle mode.Because the configuration of the electric field is not symmetric under translation along thebeam direction ( z axis) but symmetric under translation of the space-time rapidity η , particlescreated from the electric field are eigenstates of the momentum conjugate to η (which will bedenoted as λ in this paper), instead of the usual longitudinal momentum p z . In Ref. [16],the relation between these two kinds of momentum variables is given based on a coordinatetransformation in classical mechanics, and the rapidity independence of particle spectra is shown.In contrast, we will derive a relation between the two eigenstates of these momenta as an operatorrelation in a field-theoretical sense and show that the eigenstate of the momentum conjugateto η is a quantum-mechanical superposition of the eigenstate of the longitudinal momentum p z . This prescription enables us to study multiparticle correlations. Because particles arecreated as a coherent superposition of several momentum modes, they are correlated in themomentum (rapidity) space. Two-particle correlations between particles produced in the earlystage of heavy-ion collisions [18] attract interests in the context of the near-side ridge phenomenaobserved in heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) [19, 20, 21] andrecently in proton-proton collisions at the LHC [22]. Multiparticle correlations arising from theSchwinger mechanism in spatially homogeneous electric fields have been studied by Fukushima,Gelis and Lappi [23]. We will investigate how that correlation is modified if an electric fieldspans only inside the forward light cone.This paper is organized as follows. In Sec. 2, we review the field quantization in termsof the τ - η coordinates. Discussions in this section are to a large degree based on Ref. [14].Because of the translational invariance to the η direction, particle modes are parametrized bythe momentum conjugate to η . In Sec. 3, the physical meaning of the momentum conjugate to η is investigated. After these provisions, in Sec. 4, we will study particle production in the boost-invariant electric field which is confined in the forward light cone. The proper-time evolutionof the momentum distribution functions is displayed there. Finally, two-particle correlationsbetween particles created in the boost-invariant electric field are investigated in Sec. 5. τ - η coordinates In this section, we will review the canonical quantization in terms of the curvilinear τ - η coordi-nates, as a basis to study the pair creation in a boost-invariant electric field which spans onlybetween two charged plates receding from each other at the speed of light.The τ - η coordinates ( τ, η, x ⊥ ) are related with the Cartesian coordinates as τ = p t − z , η = 12 ln (cid:18) t + zt − z (cid:19) , x ⊥ = ( x, y ) . (1)In terms of the τ - η coordinates, the Klein–Gordon equation is written as (cid:18) τ ∂ ∂τ + τ ∂∂τ − ∂ ∂η − τ ∂ ∂x ⊥ + m τ (cid:19) φ ( τ, η, x ⊥ ) = 0 . (2) φ is a charged scalar field with a mass m ( = 0) and a charge e . We quantize the field φ on anequal τ -line instead of an equal t -line, imposing the canonical commutation relation (cid:2) φ ( τ, η, x ⊥ ) , π ( τ, η ′ , x ′⊥ ) (cid:3) = iδ ( η − η ′ ) δ ( x ⊥ − x ′⊥ ) , (3)where π ( τ, η, x ⊥ ) = τ ∂∂τ φ † ( τ, η, x ⊥ ) is canonical conjugate momentum.3ecause of the translational invariance to the x ⊥ and η directions, a solution of Eq. (2) canbe expanded by Fourier modes as φ ( τ, η, x ⊥ ) = Z dλd p ⊥ ˜ φ p ⊥ ,λ ( τ ) 1 p (2 π ) e i p ⊥ · x ⊥ e iλη . (4)This equation tells us that a particle mode defined on an equal τ line can be parametrized byquantum numbers p ⊥ and λ . To define this particle mode, let us decompose the field operatorinto a “positive frequency” part and a “negative frequency” part: φ ( τ, η, x ⊥ ) = Z dλd p ⊥ h χ + p ⊥ ,λ ( τ ) a p ⊥ ,λ + χ − p ⊥ ,λ ( τ ) b †− p ⊥ , − λ i p (2 π ) e i p ⊥ · x ⊥ e iλη = Z dλd p ⊥ h φ + p ⊥ ,λ ( τ, η, x ⊥ ) a p ⊥ ,λ + φ − p ⊥ ,λ ( τ, η, x ⊥ ) b †− p ⊥ , − λ i , (5)where φ ± p ⊥ ,λ ( τ, η, x ⊥ ) = χ ± p ⊥ ,λ ( τ ) 1 p (2 π ) e i p ⊥ · x ⊥ e iλη (6)are c-number solutions of the field equation (2). The mode functions φ ± p ⊥ ,λ ( τ, η, x ⊥ ) are set toobey the orthonormal conditions( φ + p ⊥ ,λ , φ + p ′⊥ ,λ ′ ) τ = δ ( λ − λ ′ ) δ ( p ⊥ − p ′⊥ ) , ( φ − p ⊥ ,λ , φ − p ′⊥ ,λ ′ ) τ = − δ ( λ − λ ′ ) δ ( p ⊥ − p ′⊥ ) , (7)( φ + p ⊥ ,λ , φ − p ′⊥ ,λ ′ ) τ = 0 . The inner product ( φ , φ ) τ is defined as( φ , φ ) τ ≡ i Z τ =const dηd x ⊥ τ φ † ←→ ddτ φ ! , (8)where φ † ←→ ddτ φ ≡ φ † · ddτ φ − ddτ φ † · φ . (9)Owing to the orthonormal conditions (7), the commutation relations[ a p ⊥ ,λ , a † p ′⊥ ,λ ′ ] = δ ( λ − λ ′ ) δ ( p ⊥ − p ′⊥ ) , [ b p ⊥ ,λ , b † p ′⊥ ,λ ′ ] = δ ( λ − λ ′ ) δ ( p ⊥ − p ′⊥ ) , (10)[ a p ⊥ ,λ , b p ′⊥ ,λ ′ ] = [ a † p ⊥ ,λ , b † p ′⊥ ,λ ′ ] = 0 , are deduced from the canonical commutation relation (3), and the operators a † p ⊥ ,λ and b † p ⊥ ,λ acquire the role of the creation operator of a particle with quantum numbers p ⊥ and λ . Fur-thermore, the expression of the charge operator in terms of a p ⊥ ,λ and b p ⊥ ,λ ˆ Q = e ( φ, φ ) τ = e Z dλd p ⊥ h a † p ⊥ ,λ a p ⊥ ,λ − b p ⊥ ,λ b † p ⊥ ,λ i (11)indicates that a † p ⊥ ,λ is the creation operator of a particle with charge + e and b † p ⊥ ,λ is the creationoperator of an antiparticle with charge − e . 4n order that Eq. (5) is a solution of the Klein–Gordon equation (2), χ ± p ⊥ ,λ ( τ ) must satisfy (cid:18) τ ∂ ∂τ + τ ∂∂τ + λ + m ⊥ τ (cid:19) χ ± p ⊥ ,λ ( τ ) = 0 , (12)where m ⊥ is the transverse mass defined by m ⊥ = m + p ⊥ . Note that the role of transversedegrees of freedom is only shifting the mass. Solutions of Eq. (12) can be expressed by the Besselfunctions. For example, we may use the Bessel function of the first kind: χ + p ⊥ ,λ ( τ ) = r π π | λ | J − i | λ | ( m ⊥ τ ) χ − p ⊥ ,λ ( τ ) = r π π | λ | J i | λ | ( m ⊥ τ ) . (13)The normalization factor has been determined by the condition (7) or equivalently iτ χ + † p ⊥ ,λ ←→ ddτ χ + p ⊥ ,λ ! = 1 ,iτ χ − † p ⊥ ,λ ←→ ddτ χ − p ⊥ ,λ ! = − , (14) iτ χ + † p ⊥ ,λ ←→ ddτ χ − p ⊥ ,λ ! = 0 . One can construct a Fock space by using the creation and annihilation operator associated withthe solutions (13). However, the choice of solutions χ ± p ⊥ ,λ is not unique. There remains thefreedom of the Bogoliubov transformation:˜ χ + p ⊥ ,λ ( τ ) = α p ⊥ ,λ χ + p ⊥ ,λ ( τ ) + β ∗ p ⊥ ,λ χ − p ⊥ ,λ ( τ )˜ χ − p ⊥ ,λ ( τ ) = α ∗ p ⊥ ,λ χ − p ⊥ ,λ ( τ ) + β p ⊥ ,λ χ + p ⊥ ,λ ( τ ) , (15)where the coefficients α p ⊥ ,λ and β p ⊥ ,λ satisfy the normalization condition | α p ⊥ ,λ | − | β p ⊥ ,λ | = 1 . (16)For any set of such coefficients, ˜ χ ± p ⊥ ,λ are also solutions of Eq. (12) fulfilling the orthonormalcondition (14). Therefore, there are infinite numbers of ways to decompose the field operator φ into a “positive frequency” part and a “negative frequency” part, and each decomposition givesdifferent sets of creation and annihilation operators: φ ( τ, η, x ⊥ ) = Z dλd p ⊥ h χ + p ⊥ ,λ ( τ ) a p ⊥ ,λ + χ − p ⊥ ,λ ( τ ) b †− p ⊥ , − λ i p (2 π ) e i p ⊥ · x ⊥ e iλη = Z dλd p ⊥ h ˜ χ + p ⊥ ,λ ( τ )˜ a p ⊥ ,λ + ˜ χ − p ⊥ ,λ ( τ )˜ b †− p ⊥ , − λ i p (2 π ) e i p ⊥ · x ⊥ e iλη (17)The two kinds of creation and annihilation operators introduced above are related by the fol-lowing Bogoliubov transformation: a p ⊥ ,λ = α p ⊥ ,λ ˜ a p ⊥ ,λ + β p ⊥ ,λ ˜ b † p ⊥ ,λ b † p ⊥ ,λ = α ∗ p ⊥ ,λ ˜ b † p ⊥ ,λ + β ∗ p ⊥ ,λ ˜ a p ⊥ ,λ . (18)5ecause creation and annihilation operators are mixed by the Bogoliubov transformation, thevacuum | i defined by a p ⊥ ,λ | i = b p ⊥ ,λ | i = 0 and the vacuum | ˜0 i defined by ˜ a p ⊥ ,λ | ˜0 i =˜ b p ⊥ ,λ | ˜0 i = 0 are inequivalent: | i 6 = | ˜0 i . (19)That is, there are many inequivalent vacua in quantum field theory. This problem is not owingto the use of the special coordinates, the τ - η coordinates, but is inherent in quantum field theory.In the Cartesian coordinates, however, positive and negative frequency solutions are selected as φ ± p ( t, x ) = 1 p ω p (2 π ) e ∓ iω p t + i p · x (20)according to the symmetry of the space and time. Now, ω p is a one-particle energy: ω p = p p + m . In the τ - η coordinates, such a selection of positive and negative frequency solutionsis not obvious because the metric depends on τ and there is no translational symmetry on τ .A proper set of positive and negative frequency solutions, in other words, a proper definitionof particle in the τ - η coordinates can be found in the following way [14]. The use of the τ - η coordinates is only a matter of description and should not change the physics. Therefore, evenif one uses the τ - η coordinates, a physical vacuum must be the same as that defined in theCartesian coordinates. That is, a “positive frequency” solution in the τ - η coordinates must be asuperposition of only the positive frequency solutions defined in the Cartesian coordinates; thenegative frequency cannot be mixed. We can construct the solutions satisfying such a conditionby using the Hankel functions as follows: φ ± p ⊥ ,λ ( τ, η, x ⊥ ) = χ ± p ⊥ ,λ ( τ ) e i p ⊥ · x ⊥ + iλη p (2 π ) , (21)where χ + p ⊥ ,λ ( τ ) = √ π i e π λ H (2) iλ ( m ⊥ τ ) (22) χ − p ⊥ ,λ ( τ ) = h χ + p ⊥ ,λ ( τ ) i ∗ = − √ π i e π λ H (1) − iλ ( m ⊥ τ ) . (23)Using the integral representation of the Hankel function [24] H (2) ν ( z ) = ie πiν/ π Z ∞−∞ dt e − iz cosh t + νt [ | Re ν | < , Im z < , (24)one can rewrite φ ± p ⊥ ,λ ( τ, η, x ⊥ ) as follows: φ + p ⊥ ,λ ( τ, η, x ⊥ ) = 1 √ π Z ∞−∞ dp z √ ω p φ + p ( t, x ) e iλy p (25) φ − p ⊥ ,λ ( τ, η, x ⊥ ) = 1 √ π Z ∞−∞ dp z √ ω p φ − p ( t, x ) e − iλy p , (26)where y p denotes rapidity corresponding to momentum p : y p = tanh − ( p z /ω p ). The conditionIm z < m − iǫ ,which is a usual prescription in quantum field theory. An important point is that the positivefrequency and the negative frequency solutions are not mixed with each other. In the forward6ight cone, one can decompose the field operator in two ways by using the mode solutions in theCartesian coordinates φ ± p ( t, x ) [Eq. (20)] and in the τ - η coordinates φ ± p ⊥ ,λ ( τ, η, x ⊥ ) [Eq. (21)]: φ ( t, x ) = Z d p h φ + p ( t, x ) a p + φ − p ( t, x ) b †− p i (27)= Z d p ⊥ dλ h φ + p ⊥ ,λ ( τ, η, x ⊥ ) a p ⊥ ,λ + φ − p ⊥ ,λ ( τ, η, x ⊥ ) b †− p ⊥ , − λ i . (28)Then, the relations between the creation or annihilation operator associated with φ ± p ( t, x ) andthat with φ ± p ⊥ ,λ ( τ, η, x ⊥ ) are obtained as a p = ( φ + p , φ ) t = 1 p πω p Z ∞−∞ dλ e iλy p a p ⊥ ,λ ,b † p = − ( φ −− p , φ ) t = 1 p πω p Z ∞−∞ dλ e iλy p b † p ⊥ ,λ . (29)Their inverse relations are a p ⊥ ,λ = 1 √ π Z ∞−∞ dp z √ ω p e − iλy p a p , b † p ⊥ ,λ = 1 √ π Z ∞−∞ dp z √ ω p e − iλy p b † p . (30)From these equations, we see that the particle mode respecting the boost invariance is a super-position of various momentum modes. Because the particle modes and the antiparticle modesare not mixed with each other, a vacuum defined in the Cartesian coordinates a p | i = b p | i = 0is also a vacuum for the particle mode defined in the τ - η coordinates: a p ⊥ ,λ | i = b p ⊥ ,λ | i = 0 . (31)Using these particle modes defined by Eq. (29), we can properly calculate field-theoretical quan-tities in terms of the τ - η coordinates. In the previous section, we have formulated the field quantization in terms of the τ - η coordinatesand derived the relation between the particle modes respecting the boost invariance and thoseassociated with the usual plane wave [Eq. (29)]. In that process, the quantum number λ hasbeen introduced as a conjugate variable to η . However, the physical meaning of λ is not obviousfrom the discussions in the previous section. We will investigate it in this section. First, we study the meaning of λ in the framework of the classical mechanics in the τ - η coor-dinates. For simplicity, we analyze a (1+1)-dimensional system. The action of a free particleis S = − m Z dt p − ˙ z = − m Z dτ p − τ ˙ η , (32)7here ˙ z ≡ dzdt (33)˙ η ≡ dηdτ . (34)Hence, the Lagrangian in terms of the τ - η coordinates is L τ = − m p − τ ˙ η . (35)Motion of the particle is expressed as z = z ( t ) or η = η ( τ ), both of which give the same motionif the particle is in the forward light cone. The momentum conjugate to η is λ = ∂L τ ∂ ˙ η = mτ ˙ η p − τ ˙ η . (36)Using the relation ˙ η = 1 τ t ˙ z − zt − z ˙ z , (37)one can rewrite Eq. (36) as λ = m ( ˙ zt − z ) √ − ˙ z = p z t − ω p z, (38)where p z is the momentum conjugate to z and ω p is the energy of the particle: p z = m ˙ z √ − ˙ z (39) ω p = m √ − ˙ z . (40)To make the meaning of λ clearer, let us boost this system by the velocity v z = z ( t ) /t inthe z direction, where z ( t ) is the position of the particle at time t . By this boost, t and z aretransformed as t → t ′ = τ (41) z → z ′ = 0 , (42)where the prime denotes a boosted quantity. Because λ is a boost-invariant quantity, λ = p z t − ω p z = p ′ z t ′ − ω ′ p z ′ = p ′ z τ . (43)This equation clearly tells us the meaning of λ : λ/τ is the longitudinal momentum of theparticle in the frame which moves with the longitudinal velocity v z = z ( t ) /t . Note that thevelocity v z = z ( t ) /t is time-dependent unless the motion of the particle is a straight line fromthe origin z ( t ) ∝ t . That is why p ′ z = λ/τ depends on τ even if the particle is free and λ isconstant.Under the gauge field A τ = 0 , A η = Eτ giving a constant electric field, the action of acharged particle is S = − m Z dτ p − τ ˙ η − e Z dτ A η ˙ η , (44)which leads the equation of motion ddτ (cid:18) λ − eEτ (cid:19) = 0 . (45)If λ = 0 at τ = 0, this equation implies that p ′ z = λ/τ = eEτ . It seems that the accelerationthe particle feels is half compared with the description in terms of the Cartesian coordinates: p z = eEt . This is because v z = z ( t ) /t varies under the electric field and thereby the frame wherethe longitudinal momentum p ′ z = λ/τ is defined is also changing in time.8 .2 Quantum field theory In this subsection, we investigate the physical meaning of λ in the frame work of quantum fieldtheory. The key of this analysis is Eq. (29), which relate the particle modes in terms of the τ - η coordinates and those in the Cartesian coordinates.Let us define a one-particle state with quantum number λ and q ⊥ as | q ⊥ , λ i = s τ (2 π ) V η a † q ⊥ ,λ | i , (46)where the normalization factor is introduced so that h q ⊥ , λ | q ⊥ , λ i = 1, and V η is a space volumeon the τ -constant hypersurface, introduced as V η = L η L = Z τ dη Z d x ⊥ = τ Z dη e i ( λ − λ ) η Z d x ⊥ e i ( p ⊥ − p ⊥ ) · x ⊥ = τ (2 π ) δ ( λ − λ ) δ ( p ⊥ − p ⊥ ) . (47)By calculating expectation values of several quantities with this one-particle state, we shallexamine the meaning of λ .First, we study the expectations of particle number operators. The expectation of the numberoperator in the a p ⊥ ,λ basis is dNdλ ′ d p ⊥ = h q ⊥ , λ | a † p ⊥ ,λ ′ a p ⊥ ,λ ′ | q ⊥ , λ i = τ (2 π ) V η (cid:2) δ ( λ − λ ′ ) δ ( p ⊥ − q ⊥ ) (cid:3) = δ ( λ − λ ′ ) δ ( p ⊥ − q ⊥ ) . (48)This is a trivial result since the expectation is taken by the eigenstate of p ⊥ and λ , and doesnot tell us anything about the physical meaning of the quantum number λ .Next, we study the expectation of the number operator in the a q -basis: dNd p = h q ⊥ , λ | a † p a p | q ⊥ , λ i = 12 πω p Z dλ ′ e − iλ ′ y p Z dλ ′′ e iλ ′′ y p h q ⊥ , λ | a † p ⊥ ,λ ′ a p ⊥ ,λ ′′ | q ⊥ , λ i = 1 ω p τ (2 π ) V η (cid:2) δ ( p ⊥ − q ⊥ ) (cid:3) = 1 ω p τL η δ ( p ⊥ − q ⊥ ) . (49)Converting the longitudinal momentum p z to the rapidity y p , we can get the rapidity distribution dNdy p d p ⊥ = ω p dNd p = τL η δ ( p ⊥ − q ⊥ ) . (50)This is independent of rapidity y p . The one-particle state | q ⊥ , λ i contains all rapidity modeswith equal weight. This result is consistent with the boost invariance of the state | q ⊥ , λ i in thelongitudinal direction. 9urthermore, because several momentum modes are condensed in the state | q ⊥ , λ i , h q ⊥ , λ | a † p a p ′ | q ⊥ , λ i = 1 √ ω p ω p ′ e − iλ ( y p − y p ′ ) τ (2 π ) V η δ ( p ⊥ − q ⊥ ) δ ( p ′⊥ − q ⊥ ) (51)is nonzero even if p z = p ′ z . This implies that nontrivial correlations in the longitudinal momen-tum space may arise from the particle production in the boost-invariant electric field. We willinvestigate it in Sec. 5.Because the rapidity distribution (50) is also independent of λ , the meaning of λ is notobtained from it. To further investigate the meaning of λ , we study the energy-momentumtensor. In the Cartesian coordinates, the energy-momentum tensor for free complex scalar fieldsis T µν = √− g h ∂ µ φ † ∂ ν φ + ∂ ν φ † ∂ µ φ − g µν (cid:16) ∂ ρ φ † ∂ ρ φ − m φ † φ (cid:17)i (52)( µ, ν = 0 , , , g µν = diag(1 , − , − , −
1) is the metric of the flat Cartesian coordinatessystem and √− g = p | det g µν | = 1. Also in the τ - η coordinates, it has the same form: T αβ = √− g h ∂ α φ † ∂ β φ + ∂ β φ † ∂ α φ − g αβ (cid:16) ∂ γ φ † ∂ γ φ − m φ † φ (cid:17)i (53)( α, β = τ, , , η ), where g αβ = diag(1 , − , − , − /τ ) is the metric of the τ - η coordinatessystem, and √− g = p | det g αβ | = τ .The relations between the differentials in the Cartesian coordinates and those in the τ - η coordinates are ∂ τ ∂ ∂ τ ∂ η = cosh η − sinh η − sinh η η ∂ ∂ ∂ ∂ . (54)Note that the matrix in the right hand side accords with the matrix representing the Lorentzboost along the z axis with the velocity v z = z/t = tanh η :Λ µν = cosh η − sinh η − sinh η η . (55)Therefore, ˜ T µν ≡ √− g T ττ T τ T τ τ T τη T τ T T τ T η T τ T T τ T η τ T ητ τ T η τ T η τ T ηη (56)equals the energy-momentum tensor transformed by the Lorentz boost (55):˜ T µν = Λ µσ Λ νρ T σρ . (57)For later reference, let us define the η -frame as the frame which is boosted by (55). Then,˜ T µν ( τ, η ) is the energy-momentum tensor observed in the η -frame.The expectation value of ˜ T ( τ, η ) = T τη ( τ, η ), which represents the momentum density inthe z direction observed in the η -frame, with the one-particle state | p ⊥ , λ i is h p ⊥ , λ | T τη ( τ, η ) | p ⊥ , λ i = λτ V η . (58)10his means that the momentum the state | p ⊥ , λ i contains is λ/τ . This result is consistent withthe result of the classical mechanics in the previous subsection. In the present case, however, η is a parameter indicating a space point, while in the classical mechanics η is a mechanicalvariable representing a point where a particle exists. Therefore, Eq. (58) means that momentum λ/τ is distributed at any space point of η . In this section, we study the pair creation in a constant electric field which exists only insidethe forward light cone.The metric of the τ - η coordinates has a singularity in τ = 0. To avoid this singularity, wesuppose that the electric field is zero when 0 ≤ τ < τ and is switched on at τ = τ >
0. Suchan electric field is given by the gauge A η ( τ ) = ( Eτ ( τ < τ ) Eτ ( τ ≥ τ ) (59)and A τ = 0. The pure gauge A η = Eτ at τ < τ is introduced so that the gauge field iscontinuous. It will be confirmed later that if τ is sufficiently small, results of the calculationsare insensitive to the values of τ .The Klein–Gordon equation under this gauge field is (cid:2) τ ∂ τ + τ ∂ τ − D η − τ ∂ ⊥ + m τ (cid:3) φ ( τ, η, x ⊥ ) = 0 , (60)where D η = ∂ η + ieA η . A solution of this equation can be expanded in the same way as the freefield case: φ ( τ, η, x ⊥ ) = Z d p ⊥ dλ h φ + in p ⊥ ,λ ( τ, η, x ⊥ ) a in p ⊥ ,λ + φ − in p ⊥ ,λ ( τ, η, x ⊥ ) b in †− p ⊥ , − λ i . (61) φ ± in p ⊥ ,λ ( τ, η, x ⊥ ) is a positive or negative frequency mode function satisfying Eq. (60). The su-perscripts ‘in’ specify the initial condition for the field: At τ < τ , there is no electric field sothat the mode functions are free ones. The free solutions at τ < τ are given by the gaugetransformation ( A η = 0 → Eτ /
2) of Eq. (21): φ + in p ⊥ ,λ ( τ, η, x ⊥ ) = √ π i e π λ H (2) iλ ( m ⊥ τ ) 1 p (2 π ) e i p ⊥ · x ⊥ + i ( λ − Eτ ) η ,φ − in p ⊥ ,λ ( τ, η, x ⊥ ) = − √ π i e π λ H (1) − iλ ( m ⊥ τ ) 1 p (2 π ) e i p ⊥ · x ⊥ + i ( λ − Eτ ) η . (62)The mode functions φ ± in p ⊥ ,λ at τ ≥ τ are constructed so that φ ± in p ⊥ ,λ and their derivative withrespect to τ are continuous, respectively, at τ = τ . Their explicit forms can be found in theappendix A. By expanding the field operator with these mode functions φ ± in p ⊥ ,λ as Eq. (61), wecan obtain the annihilation operators a in p ⊥ ,λ and b in p ⊥ ,λ and the associated vacuum state | , in i ,which is defined as a in p ⊥ ,λ | , in i = b in p ⊥ ,λ | , in i = 0 . (63)We choose this vacuum as the state of this system. This choice is a manifestation of the initialcondition that there is no particle before the electric field is turned on. An important pointis that this vacuum is also the vacuum for particles defined in the Cartesian coordinates as11iscussed in Sec. 2. Therefore, although the τ - η coordinates can cover only inside the forwardlight cone, the absence of particles outside the forward light cone is guaranteed.Because we now use the Heisenberg representation, the state | , in i does not change duringtime evolution. What evolves in time is a definition of particles or, in other words, creation andannihilation operators of particles. Such a time-dependent definition of particles is introducedby decomposing the field operator into positive and negative frequency instantaneously [12]: φ ( τ, η, x ⊥ ) = Z d p ⊥ dλ h φ + ( τ ) p ⊥ ,λ ( τ, η, x ⊥ ) a p ⊥ ,λ ( τ ) + φ − ( τ ) p ⊥ ,λ ( τ, η, x ⊥ ) b †− p ⊥ , − λ ( τ ) i , (64)where φ ± ( τ ) p ⊥ ,λ ( τ, η, x ⊥ ) is a positive or negative frequency solution of the equation of motion (60)under the pure gauge A η = A η ( τ = τ ): φ + ( τ ) p ⊥ ,λ ( τ, η, x ⊥ ) = √ π i e π λ H (2) iλ ( m ⊥ τ ) 1 p (2 π ) e i p ⊥ · x ⊥ e i [ λ − eA η ( τ )] η ,φ − ( τ ) p ⊥ ,λ ( τ, η, x ⊥ ) = − √ π i e π λ H (1) − iλ ( m ⊥ τ ) 1 p (2 π ) e i p ⊥ · x ⊥ e i [ λ − eA η ( τ )] η . (65)The operators a p ⊥ ,λ ( τ ) and b p ⊥ ,λ ( τ ) give a time-dependent particle definition. The instantaneousmode functions φ ± ( τ ) p ⊥ ,λ ( τ, η, x ⊥ ) are related with the in mode functions φ ± in p ⊥ ,λ ( τ, η, x ⊥ ) as follows: φ + in p ⊥ ,λ ( τ, η, x ⊥ ) = Z d p ′⊥ dλ ′ h(cid:16) φ + ( τ ) p ′⊥ ,λ ′ , φ + in p ⊥ ,λ (cid:17) τ φ + ( τ ) p ′⊥ ,λ ′ ( τ, η, x ⊥ ) − (cid:16) φ − ( τ ) p ′⊥ ,λ ′ , φ + in p ⊥ ,λ (cid:17) τ φ − ( τ ) p ′⊥ ,λ ′ ( τ, η, x ⊥ ) i = α p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) φ + ( τ ) p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ, η, x ⊥ )+ β ∗ p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) φ − ( τ ) p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ, η, x ⊥ ) (66) φ − in p ⊥ ,λ ( τ, η, x ⊥ ) = Z d p ′⊥ dλ ′ h(cid:16) φ + ( τ ) p ′⊥ ,λ ′ , φ − in p ⊥ ,λ (cid:17) τ φ + ( τ ) p ′⊥ ,λ ′ ( τ, η, x ⊥ ) − (cid:16) φ − ( τ ) p ′⊥ ,λ ′ , φ − in p ⊥ ,λ (cid:17) τ φ − ( τ ) p ′⊥ ,λ ′ ( τ, η, x ⊥ ) i = β p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) φ + ( τ ) p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ, η, x ⊥ )+ α ∗ p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) φ − ( τ ) p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ, η, x ⊥ ) , (67)where the Bogoliubov coefficients α p ⊥ ,λ ( τ ) and β p ⊥ ,λ ( τ ) are introduced by ∗ (cid:16) φ + ( τ ) p ′⊥ ,λ ′ , φ + in p ⊥ ,λ (cid:17) τ = α p ′⊥ ,λ ′ ( τ ) δ ( p ⊥ − p ′⊥ ) δ ( λ − λ ′ + eA η ( τ ) − eA η ( τ )) , (cid:16) φ + ( τ ) p ′⊥ ,λ ′ , φ − in p ⊥ ,λ (cid:17) τ = β p ′⊥ ,λ ′ ( τ ) δ ( p ⊥ − p ′⊥ ) δ ( λ − λ ′ + eA η ( τ ) − eA η ( τ )) . (68)For the explicit forms of the Bogoliubov coefficients, see Eqs. (A18) and (A19). These coefficientssatisfy the condition | α p ⊥ ,λ | −| β p ⊥ ,λ | = 1. By the definition of χ ± in p ⊥ ,λ ( τ ), β p ⊥ ,λ ( τ ) = 0 for τ < τ .Inserting Eqs. (66) and (67) into Eq. (61) and comparing it with Eq. (64), we get the rela-tion between the particle definition associated with in-solutions and the instantaneous particledefinition: a p ⊥ ,λ ( τ ) = α p ⊥ ,λ ( τ ) a in p ⊥ ,λ − eA η ( τ )+ eA η ( τ ) + β p ⊥ ,λ ( τ ) b in †− p ⊥ , − λ + eA η ( τ ) − eA η ( τ ) , b †− p ⊥ , − λ ( τ ) = α ∗ p ⊥ ,λ ( τ ) b in †− p ⊥ , − λ + eA η ( τ ) − eA η ( τ ) + β ∗ p ⊥ ,λ ( τ ) a in p ⊥ ,λ − eA η ( τ )+ eA η ( τ ) . (69) ∗ Actually, we should write α p ⊥ ,λ ( τ, τ ) instead of α p ⊥ ,λ ( τ ) in Eq. (68). However, we omit one time-argumentfor simplicity, because the instantaneous decomposition of the field [Eq. (64)] is physically relevant only at τ = τ . a = m ⊥ eE = 0 . a = m ⊥ eE = 0 . √ eEτ = 0 . λ/ √ eEτ = 1 ( m / eE = 0 . √ eEτ = 0 . dNd p ⊥ dλ = h , in | a † p ⊥ ,λ ( τ ) a p ⊥ ,λ ( τ ) | , in i = | β p ⊥ ,λ ( τ ) | V η τ (2 π ) . (70)This means that particles are created from the electric field. Because of the charge and themomentum conservation, antiparticles have always opposite momentum to particles: h , in | b † p ⊥ ,λ ( τ ) b p ⊥ ,λ ( τ ) | , in i = | β − p ⊥ , − λ ( τ ) | V η τ (2 π ) . (71)In Figs. 1,2, the proper-time evolution of the distribution function f p ⊥ ,λ ( τ ) = (2 π ) dNd x ⊥ dηd p ⊥ dλ = | β p ⊥ ,λ ( τ ) | (72)is plotted. Figure 1 shows the longitudinal momentum distributions with fixed transverse mo-mentum p ⊥ = 0 for m / eE = 0 . a = m ⊥ eE and √ eEτ =0 . λ/τ instead of λ because λ/τ denotes a momentum in the η -frame asexplained in Sec. 3. Figure 2 exhibits the transverse momentum distribution with fixed lon-gitudinal momentum λ/ √ eEτ = 1 for m / eE = 0 .
1. Hereafter, all figures are shown in adimensionless unit scaled by √ eE .These momentum distributions are very similar to those in the uniform electric fields (seeRef. [12]): • Particles are created with approximately zero longitudinal momentum. • In the transverse direction, the distributions are nearly Gaussian exp (cid:16) − πm ⊥ eE (cid:17) . • After being created, they are accelerated to the longitudinal direction by the electric fieldaccording to the classical equation of motion.The similarity of the transverse momentum distributions is quite reasonable, because the fieldsare free in the transverse direction also in the present case. In contrast, the similarity of thelongitudinal momentum distributions is only a superficial one. The physical contents are quitedifferent between the uniform electric field case and the present case, because the physicalmeanings of the longitudinal momentum are distinct. In the uniform case, the longitudinalmomentum is defined in the center of mass frame. On the other hand, in the present case, thelongitudinal momentum λ/τ denotes the momentum observed in the η -frame, which is the frameboosted by (55) from the center of mass frame. Therefore, Fig. 1 does not mean particles arecreated with zero longitudinal momentum in the center of mass frame, but it means particlesare created with the scaling velocity distributions. That is, a particle created at the point( t, z ) has the velocity z/t from the first instance when it is created. This result supports theassumption in Refs. [9, 10] that a source term in a kinetic equation contains the factor δ ( y − η ).In our field-theoretical treatment, however, the velocity is not exactly v z = z/t because ofquantum fluctuation. After being created, particles undergo the acceleration by the field andtheir velocity distribution deviates from the scaling one, of which processes are expressed by λ = eEτ + const. in a classical level.The momentum distribution with respect to momentum defined in the Cartesian coordinates,in other words, momentum in the center of mass frame, is calculated as follows: dNd p = h , in | a † p a p | , in i = 12 πω p Z dλ Z dλ ′ e − i ( λ − λ ′ ) y p h , in | a † p ⊥ ,λ ( τ ) a p ⊥ ,λ ′ ( τ ) | , in i = 12 πω p Z dλ f p ⊥ ,λ ( τ ) L (2 π ) , (73)14a) Longitudinal momentum distribution f p ⊥ ,λ ( τ ) at fixed time √ eEτ = 10 (b) Time evolution of the particle numberdensityFigure 4: (color online). Dependence on the time of the switch-on τ ( a = 0 . dNd x ⊥ d p ⊥ dy p = ω p L dNd p = 1(2 π ) Z dλ f p ⊥ ,λ ( τ ) (74)is independent of rapidity. Because we have assumed the perfect boost invariance, the centralplateau in the rapidity distribution extends to infinity. In reality, the speed of nuclei aftera collision is not exactly the speed of light, so that the configuration of the electric field isnot perfectly boost-invariant, and there should be some cutoff in the plateau. Note that thisindependence of rapidity is a direct consequence of Eq. (29) and is not affected by the explicitform of f p ⊥ ,λ ( τ ). Therefore, even if the electric field has proper-time dependence or the back-reaction is taken into account, the rapidity distribution (74) is always independent of rapidityas long as the boost invariance of the electric field is assumed.In Fig. 3, the time evolution of the rapidity distributions (74) divided by τ is plotted. Theyshow linear increase at later time. That is, the particle number density increases quadratically inproper time. This is because (i) pair creation happens constantly some time after the switch-onof the field and (ii) the space volume the electric field spans increases linearly in τ .The dependence on the time of the switch-on τ is shown in Fig. 4. Figure 4(a) representsthe longitudinal momentum distribution at fixed time √ eEτ = 10, and Fig. 4(b) does the timeevolution of the particle number density. Three lines corresponding to τ = 0 . , . τ forsufficiently small τ ( √ eEτ . . Because particles are created as a coherent superposition of several momentum modes in theelectric field which exists only inside the forward light cone, they are correlated in the momentumspace. In this section, we study the two-particle correlations in the momentum space betweenparticles created from the boost-invariant field.The two-particle spectrum is defined as dN d pd q = h , in | a † p a † q a p a q | , in i . (75)In the case of the pair creation in a spatially uniform electric field [23], the two-particle spectrumis dN d pd q = dNd p dNd q + { f p ( t ) } V (2 π ) δ ( p − q ) , (76)15here f p ( t ) is the momentum distribution function of particles created from the field. Thefirst term in the right-hand side is an uncorrelated part, which is a product of one-particledistribution dNd p = h , in | a † p a p | , in i = f p ( t ) V (2 π ) , and the second term is a correlated part.Particles are correlated only if they have identical momenta. This correlation is due to theBose–Einstein statistics.Let us see how this two-particle spectrum is modified if the electric field localizes in theforward light cone. Using Eqs. (29) and (69), one can derive dN d pd q = dNd p dNd q + 1 ω p ω q (cid:12)(cid:12)(cid:12)(cid:12)Z dλ π e − iλ ( y p − y q ) f p ⊥ ,λ ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) L (2 π ) δ ( p ⊥ − q ⊥ ) , (77)instead of Eq. (76). Converting the longitudinal momenta to rapidities gives dN d p ⊥ dy p d q ⊥ dy q = dNd p ⊥ dy p dNd q ⊥ dy q + (cid:12)(cid:12)(cid:12)(cid:12)Z dλ π e − iλ ( y p − y q ) f p ⊥ ,λ ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) L (2 π ) δ ( p ⊥ − q ⊥ ) . (78)For later reference, we introduce the correlation function and its longitudinal part as C ( y p , p ⊥ ; y q , q ⊥ ) ≡ dN d p ⊥ dy p d q ⊥ dy q − dNd p ⊥ dy p dNd q ⊥ dy q dNd p ⊥ dy p dNd q ⊥ dy q = (cid:12)(cid:12)R dλ π e − iλ ( y p − y q ) f p ⊥ ,λ ( τ ) (cid:12)(cid:12) (cid:8)R dλ π f p ⊥ ,λ ( τ ) (cid:9) δ ( p ⊥ − q ⊥ ) L / (2 π ) , (79) C L (∆ y = y p − y q , p ⊥ ) ≡ (cid:12)(cid:12)R dλ π e − iλ ∆ y f p ⊥ ,λ ( τ ) (cid:12)(cid:12) (cid:8)R dλ π f p ⊥ ,λ ( τ ) (cid:9) . (80)From Eq. (77) we can see that the correlation in the transverse direction is the same as that inEq. (76): L (2 π ) δ ( p ⊥ − q ⊥ ). This correspondence is reasonable because in the transverse directionthe field is free in both cases. This correlation is short-range in momentum space. What isremarkable is the correlation in the longitudinal direction. The correlated part no longer containsthe delta function δ ( p z − q z ) and is given by the Fourier transform of the momentum distribution: R dλ π e − iλ ( y p − y q ) f p ⊥ ,λ ( τ ). Therefore, particles are correlated even if their longitudinal momentaare different, while in Eq. (76) particles are correlated only if they have an identical momentum.Because the rapidity correlation is given by the Fourier transform of the momentum distribution,its width is approximately the inverse of the width of momentum distribution f p ⊥ ,λ ( τ ) in λ space.In particular, if the distribution is proportional to δ ( λ ), in other words, if the particles’ velocitydistribution is the scaling one, C L (∆ y, p ⊥ ) = 1 for any ∆ y and the rapidity-correlation range isinfinite.Figure 5 exhibits the time evolution of (a) the longitudinal correlation C L under the constantelectric field and (b) the corresponding momentum distributions f p ⊥ ,λ , whose Fourier transfor-mations give the correlation functions through Eq. (80). Because of the pair creation andsubsequent acceleration, the width of the longitudinal momentum distribution grows in time.As a result, the longitudinal correlation fades away as time goes on. This can be understoodas follows: Because particles which are created at the same time are strongly correlated, thecorrelation is long-range at first. As particle production continues to happen, many particleswhich are created at different times begin to coexist, so that the correlations are attenuated atlater times.In Fig. 6, the a = m ⊥ eE dependence of (a) the longitudinal correlations at fixed time √ eEτ = 1and (b) the corresponding momentum distributions are shown. The two peak structures seen16a) the longitudinal rapidity correlations C L (∆ y ) (b) the longitudinal distributions f p ⊥ ,λ ( τ )Figure 5: (color online). Time dependence of the correlations and the corresponding momentumdistributions ( m ⊥ / eE = 0 . √ eEτ = 0 . C L (∆ y ) (b) the momentum distributions f p ⊥ ,λ ( τ )Figure 6: (color online). a = m ⊥ eE dependence of the correlations and the corresponding momen-tum distributions ( √ eEτ = 1 and √ eEτ = 0 . a . This is because the height of the momentumdistributions depends on a exponentially ( ∼ e − πa ) [12] and their width scaled by √ eE is nearlyindependent of a , so that the relative width of the momentum distributions increases withincreasing a . This a dependence means that the heavier the transverse mass or the weaker theelectric field, the shorter the rapidity correlation.Before closing this section, let us note that although we have dealt with the constant electricfield which is independent of τ , the formula (77) is valid for electric fields having arbitrary τ dependence. Thus, we can use Eq. (77) even if the back-reaction of the particle productionis taken into account, within the approximation that interaction between produced particlesthrough quantum gauge fields is neglected. In this paper, we have studied the pair creation of scalar particles in electric fields which spanonly inside the forward light cone as a model for plasma formation in heavy-ion collisions. Wehave assumed the electric field is symmetric under the Lorentz boost along the longitudinal beamdirection and investigated how particle production happens maintaining the boost invariance ofthe field.To give a description which holds the boost invariance, we have developed the field quanti-17ation in terms of the curvilinear τ - η coordinates. This formalism may be useful for not onlythe study of pair creation but also for other studies that quantum fields in a boost-invariantlyexpanding system are involved. Although we have treated scalar fields for simplicity, general-ization to Dirac fields or vector fields is straightforward. A detailed formulation of Dirac fieldsin the τ - η coordinates has been presented in Ref. [25].We have shown that under the boost-invariant field, particles are created as a coherentsuperposition of several eigenmodes of the longitudinal momentum. As a result, the rapiditydistribution of the particles is independent of rapidity, and the particles have the scaling velocitydistribution v z = z/t from the first instance they are created. A significant point is that thisflow of particles consists of quantum-mechanical states. Each particle spreads in the forwardlight cone and forms the scaling velocity distribution as a quantum state. It is not the case thata cluster of classical particles forms this flow like in the Bjorken flow in classical hydrodynamics.This fact brings nontrivial rapidity correlations among the particles.We have calculated the two-particle correlations between particles created from the boost-invariant field, and found that the correlation is short-range with respect to the transversemomentum, which originates in the Bose–Einstein correlation, and is long-range with respectto the longitudinal rapidity. These features may remind us of the near-side ridge phenomenaobserved in nucleus-nucleus collisions at the RHIC [19, 20, 21] and recently in proton-protoncollisions at the LHC [22]. However, to make quantitative predictions, further improvements ofthis study, such as introducing a proper-time dependence of the electric field due to expansionof the system and taking the back reaction of the pair creation into account, are necessary. Theback reaction can be introduced by the Maxwell equation1 τ ddτ (cid:18) τ ddτ A η ( τ ) (cid:19) = −h , in | j η | , in i , (81)where j η denotes the charge current operator of the produced particles. In our calculation, thelongitudinal correlation fades away as proper time goes on, because under the constant field theparticle production continues to happen and many particles which are created at different timescoexist at later times. Hence, we can conclude that the life time of the electric field should beshort to obtain a long-range rapidity correlation. Furthermore, considering a tubelike structureof the electric field and effects of radial flow may be important for the transverse correlation [18].It is also interesting to study whether the correlation can survive under a thermalization process,which is not included in our treatment, or under the subsequent hydrodynamic evolution. Acknowledgments
The author thanks Professor T. Matsui for enlightening discussions and careful reading of themanuscript and Professor H. Fujii for helpful discussions and comments on the manuscript.Comments by Professor T. Hirano are greatly appreciated. The author also thanks YoshiakiOnishi for his kind comments and discussions. The author is supported by the Japan Societyfor the Promotion of Science for Young Scientists.
Appendix A Solutions of the Klein–Gordon equation underthe boost-invariant field
In this appendix, we show a set of solutions of the Klein–Gordon equation [Eq. (60)] (cid:2) τ ∂ τ + τ ∂ τ − D η − τ ∂ ⊥ + m τ (cid:3) φ ± p ⊥ ,λ ( τ, η, x ⊥ ) = 0 (A1)18nder the gauge A η = Eτ with the initial condition such that they are continuous with φ ± in p ⊥ ,λ ( τ, η, x ⊥ ) = χ ± (0) p ⊥ ,λ ( τ ) 1 p (2 π ) e i p ⊥ · x ⊥ + i ( λ − Eτ ) η , (A2)respectively, at τ = τ , where χ + (0) p ⊥ ,λ ( τ ) = √ π i e π λ H (2) iλ ( m ⊥ τ ) (A3) χ − (0) p ⊥ ,λ ( τ ) = h χ + (0) p ⊥ ,λ ( τ ) i ∗ . (A4)Under the gauge potential A η = Eτ , a set of normalized solutions of Eq. (A1) can beconstructed by the confluent hypergeometric functions U ( a, b, z ) [24] as follows:˜ φ ± p ⊥ ,λ ( τ, η, x ⊥ ) = ˜ χ ± p ⊥ ,λ ( τ ) 1 p (2 π ) e i p ⊥ · x ⊥ e iλη , (A5)where ˜ χ + p ⊥ ,λ ( τ ) = 1 √ e − π ( a + λ ) (cid:18) eE τ (cid:19) i λ e − i eEτ U (1 / ia + iλ, iλ, ieEτ /
2) (A6)˜ χ − p ⊥ ,λ ( τ ) = h ˜ χ + p ⊥ ,λ ( τ ) i ∗ . (A7)These solutions ˜ φ ± p ⊥ ,λ are not continuous with Eq. (A2) at τ = τ . To find solutions which areconnected smoothly to Eq. (A2) at τ = τ , we decompose the solutions ˜ φ ± p ⊥ ,λ by φ ± ( τ ) p ⊥ ,λ , whichare defined by Eq. (65):˜ φ + p ⊥ ,λ ( τ, η, x ⊥ ) = Z d p ′⊥ dλ ′ h(cid:16) φ + ( τ ) p ′⊥ ,λ ′ , ˜ φ + p ⊥ ,λ (cid:17) τ φ + ( τ ) p ′⊥ ,λ ′ ( τ, η, x ⊥ ) − (cid:16) φ − ( τ ) p ′⊥ ,λ ′ , ˜ φ + p ⊥ ,λ (cid:17) τ φ − ( τ ) p ′⊥ ,λ ′ ( τ, η, x ⊥ ) i = A p ⊥ ,λ + eA η ( τ ) ( τ, τ ) φ + ( τ ) p ⊥ ,λ + eA η ( τ ) ( τ, η, x ⊥ )+ B ∗ p ⊥ ,λ + eA η ( τ ) ( τ, τ ) φ − ( τ ) p ⊥ ,λ + eA η ( τ ) ( τ, η, x ⊥ ) (A8)˜ φ − p ⊥ ,λ ( τ, η, x ⊥ ) = Z d p ′⊥ dλ ′ h(cid:16) φ + ( τ ) p ′⊥ ,λ ′ , ˜ φ − p ⊥ ,λ (cid:17) τ φ + ( τ ) p ′⊥ ,λ ′ ( τ, η, x ⊥ ) − (cid:16) φ − ( τ ) p ′⊥ ,λ ′ , ˜ φ − p ⊥ ,λ (cid:17) τ φ − ( τ ) p ′⊥ ,λ ′ ( τ, η, x ⊥ ) i = B p ⊥ ,λ + eA η ( τ ) ( τ, τ ) φ + ( τ ) p ⊥ ,λ + eA η ( τ ) ( τ, η, x ⊥ )+ A ∗ p ⊥ ,λ + eA η ( τ ) ( τ, τ ) φ − ( τ ) p ⊥ ,λ + eA η ( τ ) ( τ, η, x ⊥ ) , (A9)where the Bogoliubov coefficients A p ⊥ ,λ ( τ, τ ) and B p ⊥ ,λ ( τ, τ ) are defined, respectively, by A p ⊥ ,λ ( τ, τ ) = iτ (h χ + (0) p ⊥ ,λ ( τ ) i ∗ ←→ ddτ ˜ χ + p ⊥ ,λ − eA η ( τ ) ( τ ) ) (A10) B p ⊥ ,λ ( τ, τ ) = iτ (h χ + (0) p ⊥ ,λ ( τ ) i ∗ ←→ ddτ ˜ χ − p ⊥ ,λ − eA η ( τ ) ( τ ) ) . (A11)Substituting Eqs. (A3), (A4) and (A6) into these equations, one can derive the explicit forms ofthe Bogoliubov coefficients. Albeit Eqs. (A8) and (A9) are mathematically valid for any positive19 and τ , the expansion by the mode functions φ ± ( τ ) p ⊥ ,λ ( τ, η, x ⊥ ) is physically meaningful only at τ = τ . Therefore, it is sufficient to know the Bogoliubov coefficients having the same timeargument τ = τ : A p ⊥ ,λ ( τ ) ≡ A p ⊥ ,λ ( τ, τ ) and B p ⊥ ,λ ( τ ) ≡ B p ⊥ ,λ ( τ, τ ). After completing thedifferentiation in Eqs. (A10) and (A11), we can set τ = τ and obtain the following equations: A p ⊥ ,λ ( τ ) = √ π e − π ( a − eEτ ) (cid:18) eEτ (cid:19) i ( λ − eEτ ) e − i eEτ × (cid:26) (1 + 2 ia ) H (1) − iλ ( m ⊥ τ ) U ( 12 + ia + i ( λ − eEτ ) , i ( λ − eEτ ) , i eEτ ) − m ⊥ τ H (1)1 − iλ ( m ⊥ τ ) U ( 12 + ia + i ( λ − eEτ ) , i ( λ − eEτ ) , i eEτ ) (cid:27) (A12) B ∗ p ⊥ ,λ ( τ ) = √ π e − π ( a − eEτ ) (cid:18) eEτ (cid:19) i ( λ − eEτ ) e − i eEτ × (cid:26) (1 + 2 ia ) H (2) iλ ( m ⊥ τ ) U ( 12 + ia + i ( λ − eEτ ) , i ( λ − eEτ ) , i eEτ )+ m ⊥ τ H (2)1+ iλ ( m ⊥ τ ) U ( 12 + ia + i ( λ − eEτ ) , i ( λ − eEτ ) , i eEτ ) (cid:27) . (A13)Taking the relations (A8) and (A9), we can construct the mode functions φ ± p ⊥ ,λ which arecontinuous with Eq. (A2) (which is equivalent to φ ± ( τ ) p ⊥ ,λ ) at τ = τ as follows: φ + p ⊥ ,λ ( τ, η, x ⊥ ) = A ∗ p ⊥ ,λ ( τ ) ˜ φ + p ⊥ ,λ − eA ( τ ) ( τ, η, x ⊥ ) − B ∗ p ⊥ ,λ ( τ ) ˜ φ − p ⊥ ,λ − eA ( τ ) ( τ, η, x ⊥ ) (A14) φ − p ⊥ ,λ ( τ, η, x ⊥ ) = A p ⊥ ,λ ( τ ) ˜ φ − p ⊥ ,λ − eA ( τ ) ( τ, η, x ⊥ ) − B p ⊥ ,λ ( τ ) ˜ φ + p ⊥ ,λ − eA ( τ ) ( τ, η, x ⊥ ) . (A15)Further using Eqs. (A8) and (A9), we can expand the mode functions φ ± p ⊥ ,λ by the instantaneousfree solutions at τ = τ : φ + p ⊥ ,λ ( τ, η, x ⊥ ) = (cid:8) A ∗ p ⊥ ,λ ( τ ) A p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) − B ∗ p ⊥ ,λ ( τ ) B p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) (cid:9) × φ + ( τ ) p ⊥ ,λ + eA η ( τ ) − eA ( τ ) ( τ, η, x ⊥ )+ n A ∗ p ⊥ ,λ ( τ ) B ∗ p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) − B ∗ p ⊥ ,λ ( τ ) A ∗ p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) o × φ − ( τ ) p ⊥ ,λ + eA η ( τ ) − eA ( τ ) ( τ, η, x ⊥ )= α p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) φ + ( τ ) p ⊥ ,λ + eA η ( τ ) − eA ( τ ) ( τ, η, x ⊥ )+ β ∗ p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) φ − ( τ ) p ⊥ ,λ + eA η ( τ ) − eA ( τ ) ( τ, η, x ⊥ ) (A16) φ − p ⊥ ,λ ( τ, η, x ⊥ ) = α ∗ p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) φ − ( τ ) p ⊥ ,λ + eA η ( τ ) − eA ( τ ) ( τ, η, x ⊥ )+ β p ⊥ ,λ + eA η ( τ ) − eA η ( τ ) ( τ ) φ + ( τ ) p ⊥ ,λ + eA η ( τ ) − eA ( τ ) ( τ, η, x ⊥ ) , (A17)where the Bogoliubov coefficients α p ⊥ ,λ ( τ ) = A ∗ p ⊥ ,λ − eA η ( τ )+ eA η ( τ ) ( τ ) A p ⊥ ,λ ( τ ) − B ∗ p ⊥ ,λ − eA η ( τ )+ eA η ( τ ) ( τ ) B p ⊥ ,λ ( τ ) (A18) β p ⊥ ,λ ( τ ) = A p ⊥ ,λ − eA η ( τ )+ eA η ( τ ) ( τ ) B p ⊥ ,λ ( τ ) − B p ⊥ ,λ − eA η ( τ )+ eA η ( τ ) ( τ ) A p ⊥ ,λ ( τ ) (A19)have been introduced. 20 eferences [1] J. D. Bjorken, Phys. Rev. D27 (1983) 140[2] L. D. Landau, Izv. Akad. Nauk SSSR, 17 (1953) 51 [English translation by Ter Haar, in Collected Papers of L. D. Landau (Pergamon, New York, 1965), p.569][3] F. Low, Phys. Rev. D12 (1975) 163[4] S. Nussinov, Phys. Rev. Lett. 34 (1975) 1286[5] T. Lappi and L. McLerran, Nucl. Phys. A772 (2006) 200 [arXiv:hep-ph/0602189][6] G. Gatoff, A. K. Kerman and D. Vautherin, Phys. Rev. D38 (1988) 96[7] H. Fujii and K. Itakura, Nucl. Phys. A809 (2008) 88 [arXiv:0803.0410][8] J. Schwinger, Phys. Rev. 82 (1951) 664[9] K. Kajantie and T. Matsui, Phys. Lett. 164B (1985) 373[10] G. Gatoff, A. K. Kerman and T. Matsui, Phys. Rev. D36 (1987) 114[11] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper and E. Mottola, Phys. Rev. Lett. 67(1991) 2427; Phys. Rev. D45 (1992) 4659[12] N. Tanji, Ann. Phys. 324 (2009) 1691 [arXiv:0810.4429][13] N. Tanji, Ann. Phys. 325 (2010) 2018 [arXiv:1002.3143][14] C. Sommerfield, Ann. Phys. 84 (1974) 285[15] C. Martin and D. Vautherin, Phys. Rev. D40 (1989) 1667[16] F. Cooper, J. M. Eisenberg, Y. Kluger, E. Mottola and B. Svetitsky, Phys. Rev. D48 (1993)190 [hep-ph/9212206][17] B. Mihaila, J. F. Dawson and F. Cooper, Phys. Rev. D78 (2008) 116017 [arXiv:0811.1353][18] A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan, Nucl. Phys. A810 (2008) 91[arXiv:0804.3858][19] J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 95 (2005) 152301[arXiv:nucl-ex/0501016][20] A. Adare et al. (PHENIX Collaboration), Phys. Rev. C78 (2008) 014901 [arXiv:0801.4545][21] B. Alver et al. (PHOBOS Collaboration), Phys. Rev. Lett. 104 (2010) 062301[arXiv:0903.2811][22] V. Khachatryan et al.et al.