Multiparticle correlations in the Schwinger mechanism
aa r X i v : . [ h e p - ph ] A ug Multiparticle correlationsin the Schwinger mechanism
Kenji Fukushima (1) , Fran¸cois Gelis (2) , Tuomas Lappi (2 ,
1. Yukawa Institute for Theoretical Physics, Kyoto University,Oiwake-cho, Kitashirakawa, Sakyo-ku,Kyoto 606-8502, Japan2. Institut de Physique Th´eorique (URA 2306 du CNRS)CEA/DSM/Saclay, Bˆat. 77491191, Gif-sur-Yvette Cedex, France3. Department of Physics,P.O. Box 35, 40014 University of Jyv¨askyl¨a, Finland
Abstract
We discuss the Schwinger mechanism in scalar QED and derive themultiplicity distribution of particles created under an external electricfield using the LSZ reduction formula. Assuming that the electric fieldis spatially homogeneous, we find that the particles of different momentaare produced independently, and that the multiplicity distribution in onemode follows a Bose-Einstein distribution. We confirm the consistencyof our results with an intuitive derivation by means of the Bogoliubovtransformation on creation and annihilation operators. Finally we revisita known solvable example of time-dependent electric fields to present exactand explicit expressions for demonstration.
YITP-09-45IPhT-t09/104
A classic example of a non-perturbative tunneling phenomenon in quantum fieldtheory is the decay of an electric field due to pair creation. The quantum vac-uum is full of virtual particle-antiparticle pairs (i.e. the Dirac sea), which canoccasionally gain enough energy from the external field to become real. Thedecay (or persistence) rate of the QED (quantum electrodynamics) vacuumin the presence of an external electric field was first deduced from the imagi-nary part of the Heisenberg-Euler Lagrangian [1] and formulated in Schwinger’sclassic paper [2]. The phenomenon is commonly referred to as the
Schwinger echanism (see ref. [3] for a comprehensive review.) In the case of QED thecoupling constant e is very small, and it is difficult in practice to achieve largeenough electric fields; the probability of producing an electron-positron pair is,up to a prefactor, ∼ exp[ − πm e / ( eE )], and thus a very strong electric field, E ∼ m e /e ≃ . × V/m, is necessary to observe the phenomenon. Tothe best of our knowledge, the Schwinger mechanism in QED remains to beunambiguously observed experimentally.Pair creation from an electric field became a phenomenologically much morerelevant subject with the realization that the strong nuclear force is describedby a gauge theory called QCD (quantum chromo-dynamics). A popular phe-nomenological view of QCD with confinement is a description in terms of achromoelectric flux tube connecting the color charges of the quarks. If thesequarks are then pulled apart by their momenta, the string formed by the chro-moelectric field can decay via the Schwinger mechanism leading to the decay ofthe system into q ¯ q or color neutral mesons as a result of hadronization. In thiscase, the decay probability is characterized by ∼ exp[ − πm q /σ ], where the QCDstring tension σ ≃ / fm is an energy stored in the chromoelectric flux tubeper unit length. Applications of the particle production by the Schwinger mech-anism range from e + e − annihilation [4, 5] to early models of relativistic heavyion collisions [6, 7, 8, 9, 10, 11, 12]. Recent extensive studies on thermal hadronproduction as a possible manifestation of the Hawking-Unruh effect, that is anequivalent formulation to the Schwinger mechanism in curved space-time, isfound in refs. [13, 14, 15].The QCD coupling constant g , although asymptotically small, is not as smallas the QED one at phenomenologically interesting energies. Even more impor-tant is that the nonlinear dynamics of the gauge fields naturally leads, in somecircumstances, to gauge fields that are parametrically large in the coupling, A µ ∼ /g . A prime example of such a situation is caused by the large occupa-tion numbers of gluonic states in high energy scattering. The transverse gluondensity ∼ Q provides a typical energy scale Q s in such a system. There the non-linear interactions among bremsstrahlung gluons with small Bjorken’s x lead togluon saturation, which is most conveniently described as a coherent color fieldradiated by static (in light cone time) sources. This description is referred to asthe Color Glass Condensate (CGC) (for reviews, see [16, 17, 18, 19]). The colli-sion of two objects whose wavefunction is characterized by Q s in the CGC for-malism achieves a field configuration of longitudinal chromoelectric and chromo-magnetic fields whose strength is also given by Q s . This transient state of mattercontaining strong longitudinal fields is known as the glasma [20, 21, 22]. In thecase of QCD it is of course difficult to achieve the canonical model case of a con-stant electrical field. Generically, a WKB-type non-perturbative evaluation suchas in refs. [13, 14] could be expected to be valid in a case where the field strength gA µ ∼ Q s is much larger than the typical (inverse) time and spatial scales of thefields. A perturbative calculation, on the other hand, is also feasible in the casewhen g is small enough. The particle (gluons and quarks) production associatedwith strong CGC fields has been formulated based on the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula [23, 24, 25, 26, 27, 28, 29, 30, 31] as well2s on the canonical formalism [32]. One of the aims of this paper is to establisha link, by a concrete example, between the formalism of LSZ reduction formulas,which is usually associated with perturbation theory only, and non-perturbativetunneling phenomena of the Schwinger mechanism. More concretely, we want toshow that the LSZ perturbative framework automatically includes the particlesproduced by the Schwinger mechanism, provided the external field is properlyresummed. Thus, this contribution does not need to be added separately byhand.For applying the mechanism of pair creation from a classical field to phe-nomenology one needs, in addition to the vacuum decay rate or spectrum ofpairs, the whole probability distribution of multiparticle production. In manypractical applications of the Schwinger mechanism, there has been a confusion ofterminology, with both the formulas and the concepts of the vacuum decay rate(or persistence probability calculated by Schwinger) and the pair productionrate. The difference between the two was recently nicely discussed in ref. [33],where it is interpreted as a result of temporal correlations between the producedpairs. In fact these two were clearly distinguished already in a classical paperby Nikishov [34]. In the case of the typical QED discussion, the pair productionrate is extremely small, in which case the probability distribution of producedpairs cannot be distinguished from a Poisson distribution. This seems to havebeen the assumption used, without any further justification, also in many QCDphenomenological applications (see e.g. [8] where this is very explicit). As weshall show explicitly in the following, this assumption is not true when the pairproduction is not strongly suppressed, as can typically be the case in e.g. heavy-ion collisions. Instead, the probability distribution of the produced pairs turnsout naturally to be the appropriate (Bose-Einstein or Fermi-Dirac) quantumone . With explicit expressions for the probabilities to produce one, two, etc.particles, the distinction between the vacuum decay rate (related to the proba-bility to produce no pairs) and the pair production rate (the expectation valueof the number of pairs produced), becomes obvious. The fact that there is aquantum statistical (BE or FD) correlation has long ago been realized by someauthors as a requirement that should in principle be built into Monte Carloevent generators [37, 38, 39]. Also, the full computation of the vacuum decayrate should encompass all the multiparticle production processes,–because ofunitarity–, including the quantum statistics. To our knowledge, however, anexplicit derivation of how the BE or FD correlations arise from the Schwingermechanism has been lacking. Besides, the multiparticle distribution has scarcelydrawn attention in the context of the Schwinger mechanism, probably becauseof the absence of experimental access. In fact the spectrum of multiparticleproduction is quite informative and precise data of charged hadron multiplicityfluctuations are already available in p -¯ p [40] and heavy-ion [41] collision experi-ments, where a negative binomial distribution gives a beautiful fit. Of course, toaccount for the high-energy experimental data, a simple treatment of spatially Note that the “inversion of spin statistics” discussed in [35, 36] refers to a formal expres-sion of the vacuum decay rate as an integral over the BE or FD distribution function and notthe actual probability distribution of produced particles.
We here calculate the Schwinger mechanism in terms of the LSZ reductionformulas. To this aim we develop a slightly modified version of the Schwinger-Keldysh formalism to compute the generating functional of the particle andantiparticle spectra.
To avoid encumbering the discussion with unessential details, we consider scalarQED, i.e. a charged scalar field φ coupled to an external vector potential A µ .Moreover, in order to simplify things even further, we neglect any kind of self-interactions among the scalar fields, and the coupling to the external electro-magnetic field enters only via the covariant derivatives, D µ = ∂ µ − i eA µ . Thus,the Lagrangian of this model is: L = ( D µ φ ) ( D µ φ ) ∗ − m φφ ∗ . (1)In most of the considerations of this paper, we need not specify the precise formof the background potential A µ . Only in the final section, we work out the caseof an explicit example of background electric field that leads to exact analyticalresults. We assume that the initial state of the system does not contain any particles orantiparticles. However, because of the background field, transitions to populated4tates are possible. Let us consider the following transition amplitudes, M m,n ( { p i } , { q i } ) ≡ (cid:10) p · · · p m | {z } particles q · · · q n | {z } antiparticles out (cid:12)(cid:12) in (cid:11) , (2)from the vacuum to a populated state. The conservation of electrical chargeimplies that an equal number of particles and antiparticles must be produced,i.e. that this general amplitude is proportional to δ mn . This transition amplitudecan be obtained from the expectation value of time-ordered products of fields: M m,n ( { p i } , { q i } ) = Z m Y i =1 d x i e i p i · x i ( (cid:3) x i + m ) n Y j =1 d y j e i q j · y j ( (cid:3) y j + m ) × (cid:10) out (cid:12)(cid:12) T φ ( x ) · · · φ ( x m ) φ ∗ ( y ) · · · φ ∗ ( y n ) (cid:12)(cid:12) in (cid:11) . (3)Here the on-shell boundary condition is the vacuum one, i.e. p i → p p i + m and q i → p q i + m for particles and antiparticles, respectively. This is ade-quate only if one chooses a gauge in which the background field A µ vanisheswhen time goes to + ∞ . Because the conjugate φ ∗ ( y i ) already takes care of an-tiparticle nature, q i should also be chosen to be positive. Note that, in principle,each field in this formula should be accompanied by a wave-function renormal-ization factor, Z − / . However, since we do not include any self-interactionsamong the fields, these factors are equal to unity here and we can safely ignorethem. All the physical quantities related to particle production in this model can beconstructed from the squared amplitudes |M m,n | . A very useful object thatcontains all this information in a compact form is the generating functionaldefined by [26, 28] F [ z, ¯ z ] ≡ ∞ X m,n =0 m ! n ! Z m Y i =1 d p i z ( p i ) n Y j =1 d q j ¯ z ( q j ) (cid:12)(cid:12)(cid:12) M m,n ( { p i } , { q i } ) (cid:12)(cid:12)(cid:12) . (4)In this functional, z and ¯ z are two functions defined over the 1-particle momen-tum space (unlike what the notation may suggest, they are independent andnot complex conjugates of each other).If one sets the functions z and ¯ z to constants equal to unity, one gets, F [1 ,
1] = ∞ X m,n =0 P m,n , (5) It is always possible to find such a gauge if the electrical field vanishes when time goes toinfinity, a necessary condition to be able to unambiguously define what we mean by “measuringa particle”. If one insists on using a gauge in which A µ is not zero when x → + ∞ , onemust replace the ordinary derivatives by covariant derivatives and the plane waves by gaugetransformed plane waves in eq. (3). The mass-shell condition for p i and q i should also bealtered by the non-zero background gauge field. P m,n is the total probability to have m particles and n antiparticles inthe final state. From unitarity, the sum of all these probabilities must be equalto one, hence F [1 ,
1] = 1 . (6)This is an important constraint on the generating functional F [ z, ¯ z ], that leadsto significant simplification in the computation of inclusive observables.Assuming that this generating functional is known, one can obtain the singleinclusive particle spectrum asd N +1 d p = δ F [ z, ¯ z ] δz ( p ) (cid:12)(cid:12)(cid:12)(cid:12) z =¯ z =1 , (7)the single inclusive antiparticle spectrum asd N − d q = δ F [ z, ¯ z ] δ ¯ z ( q ) (cid:12)(cid:12)(cid:12)(cid:12) z =¯ z =1 , (8)and the double inclusive particle-particle spectrum asd N ++2 d p d p = δ F [ z, ¯ z ] δz ( p ) δz ( p ) (cid:12)(cid:12)(cid:12)(cid:12) z =¯ z =1 . (9)Other combinations of inclusive 2-particle spectra are given byd N −− d q d q = δ F [ z, ¯ z ] δ ¯ z ( q ) δ ¯ z ( q ) (cid:12)(cid:12)(cid:12)(cid:12) z =¯ z =1 , d N + − d p d q = δ F [ z, ¯ z ] δz ( p ) δ ¯ z ( q ) (cid:12)(cid:12)(cid:12)(cid:12) z =¯ z =1 . (10)Note that from their definitions, these two particle spectra are normalized sothat their integrals over p and q are, respectively, Z d p d p d N ++2 d p d p = (cid:10) N + ( N + − (cid:11) , Z d q d q dN −− d q d q = (cid:10) N − ( N − − (cid:11) , Z d p d q d N + − d p d q = (cid:10) N + N − (cid:11) , (11)where N ± denote the number operator for particles and antiparticles in thefinal state respectively. In terms of the total probability introduced in eq. (5)we will see that (cid:10) N + (cid:11) = P m,n mP mn , (cid:10) N − (cid:11) = P m,n nP mn , (cid:10) N + ( N + − (cid:11) = P m,n m ( m − P mn , etc, for which one can find a justification in the appendix A.What these equations mean in the ++ and −− cases is that our 2-particlespectra are defined by summing over all possible pairs of distinct particles inevery event. When summed over all momenta in a given event, this leads to N ± ( N ± −
1) where N ± is the multiplicity of particles (resp. antiparticles) inthat event. Naturally, this requirement of taking distinct particles has no in-cidence on the + − case – since charged particles are always distinct from thecorresponding antiparticles –, which explains the last of eqs. (11).6 .4 Generating functional: computation Let us now proceed to the actual computation of the generating functional F [ z, ¯ z ]. There is usually no closed form answer for this object. Since we areneglecting the self-interactions of the fields φ in our model, however, this be-comes a much simpler calculation. It has been shown before [26, 28] that thegenerating functional F [ z, ¯ z ] is the sum of all the vacuum-vacuum graphs ina slightly modified version of the Schwinger-Keldysh formalism [48], where theoff-diagonal components G − and G − + of the free propagator are altered bythe functions z or ¯ z . Explicitly, the propagators read: G ( p ) = i p − m + i ǫ , G −− ( p ) = − i p − m − i ǫ ,G − ( p ) = 2 πθ ( − p ) z ( p ) δ ( p − m ) ,G − + ( p ) = 2 πθ (+ p ) ¯ z ( − p ) δ ( p − m ) . (12)As one can see, the off-diagonal free propagators are simply multiplied by z ( p )and ¯ z ( p ) respectively. Given these propagators, the rules for calculating F [ z, ¯ z ]are straightforward: i. Draw all the vacuum-vacuum diagrams at the desired order. There aresimply connected and multiply connected graphs. However, one can al-ways limit the calculation to the simply connected ones, and then expo-nentiate the result in order to obtain the full result that also includes themultiply connected ones. ii.
For a given graph, sum over all the possible ways to assign + or − signsto the vertices. iii. A − vertex is the complex conjugate of a + vertex. Let us denote by V ( A ) the value of the coupling of φ, φ ∗ to the background field in a +vertex. Note that this ’potential’ is not simply A µ itself, since there areboth a e ( ∂ µ φ ) φ ∗ A µ and a e φφ ∗ A µ A µ couplings – however, we will notneed its detailed expression in the following. The corresponding − ver-tex is V ∗ ( A ) = −V ( A ) (this identity follows from the hermiticity of theLagrangian). iv. Connect these vertices with the propagators defined in eq. (12).Step i is trivial: there is only one topology of simply connected vacuum-vacuum graph in our model. These are the graphs made of a single closed loop,embedded with the background field A µ , as illustrated in fig. 1. We must sumover the number of insertions of the background potential A µ (from zero toinfinite insertions), and for each of these insertions we must sum over the type+ and − for the corresponding vertex. This double summation can be organizedin blocks, as illustrated in figs. 2 and 3. Vacuum-vacuum graphs are diagrams that have no external legs with respect to φ . ( A ) Figure 1: Topology of the connected vacuum-vacuum diagrams that contributeto ln F . The solid line denotes the free propagator of the charged scalar field φ (the arrow indicates the direction of the flow of positive electric charge). Thewavy line terminated by a circled cross denotes the background gauge potential V ( A ). Note that, in scalar QED, the background ’potential’ is not simply A µ itself, since there are both a e ( ∂ µ φ ) φ ∗ A µ and a e φφ ∗ A µ A µ couplings. Thewavy line represents the sum of these two contributions. = + + + ++ + ... +++ = + - + -- + ... --- Figure 2: Building blocks for the summation of field insertions having a fixedSchwinger-Keldysh vertex assignment. ++--+ + -
Figure 3: Block decomposition of the double summation over the number ofbackground field insertions and the ± assignments at the vertices.8f we denote by T + the sum of graphs in the first line of fig. 2 and by T − thesum of graphs on the second line of the same figure, we can take the remainingsteps ii , iii , and iv and the sum of the vacuum-vacuum diagrams contributingto ln F can be written asln F [ z, ¯ z ] = constant + ∞ X n =1 n tr h T + G − T − G − + i n , (13)where the trace symbol (tr ( · · · )) denotes an integration over the space-timecoordinates (not represented explicitly in the formula) of all the vertices. Thefirst term in this formula, that we simply denoted “constant” but did not writeexplicitly, is independent of z and ¯ z . It is made of all the graphs in which all thevertices are of type + or all of type − (and thus cannot contain z nor ¯ z sincethese come with the G ±∓ propagators). In the second term of this formula,the index n represents the number of the block consisting of one + − and one − + transitions, and the factor 1 /n is a symmetry factor since we can rotate thegraph by one block without altering it. Note that the index n gives the order in z and ¯ z of the corresponding term .It is trivial to perform explicitly the summation in eq. (13) to find,ln F [ z, ¯ z ] = constant − tr ln h − T + G − T − G − + i . (14)The constant term that we did not write explicitly can be determined withoutany calculation so that ln F [1 ,
1] = 0, as required from unitarity (6). Therefore,we have F [ z, ¯ z ] = exp (cid:16) − tr ln h − T + G − T − G − + i(cid:17) exp (cid:16) − tr ln h − T + G − T − G − + i z =¯ z =1 (cid:17) . (15)Although fairly formal, this formula contains all we need to know about theparticle production by an external electromagnetic field in scalar QED.In order to simplify the subsequent discussion, let us restrict ourselves tobackground electric fields that do not depend on the position x . In this caseit is always possible to choose a gauge in which the background vector poten-tial A µ ( x ) is also independent of x , and its Fourier transform is proportionalto a delta function δ ( k ) as far as its dependence on the spatial componentsof the momentum is concerned. In this particular case, the 2-point function T + G − T − G − + has the same entering and outgoing momenta. In order to makemore explicit the z and ¯ z dependence, let us introduce the notation: h T + G − T − G − + i p ≡ z ( p )¯ z ( − p ) L p . (16)The important points here are that L does not contain z and ¯ z , and that thefunctions z and ¯ z carry the same momentum up to a relative sign . With this We see explicitly here that the order in z of a given term is the same as its order in ¯ z ,which reflects the fact that particles and antiparticles can be created only in pairs. Physically, the building block in eq. (16) is the amplitude squared for producing a singleparticle-antiparticle pair. Since the background field is uniform in space, the total momentumof this pair must be zero. Hence the opposite sign for the momentum argument of z and ¯ z . F [ z, ¯ z ] = exp (cid:16) − tr ln h − z ¯ zL i(cid:17) exp (cid:16) − tr ln h − L i(cid:17) . (17)From now on, it is simpler to calculate the trace in momentum space. Indeed,when the background potential is space independent, a unique momentum p runs around the loop.To close this subsection, let us mention a generic property of the trace thatappears in eq. (17). Strictly speaking, when the background electric field isindependent of the location in space, this trace exhibits a factor (2 π ) δ ( ) inmomentum space. This factor should be interpreted as the volume V of the sys-tem , and its presence is an indication that the particle spectra are proportionalto the overall volume. A µ So far, we have not attempted to calculate the object L that appears in thegenerating functional. Since it is built from the T ± , which are Feynman (time-ordered) propagators amputated of their external legs, it is clear that L is re-lated to the propagation of small fluctuations over the background electric field.However, knowing that L is related to the Feynman propagator is inconvenientfor practical calculations because this propagator obeys complicated boundaryconditions. In practice, one should try to rewrite L in terms of propagators thatobey simpler boundary conditions, like the retarded propagator.Let us start from the equation that defines T + to rewrite it into the re-tarded quantities. The resummation that leads to T + can be summarized bythe following Lippmann-Schwinger equation: T + = V + V G T + , (18)where V is the sum of the two couplings to the background field (the derivativecoupling to a single A µ and the non-derivative coupling to A µ A µ ). We do notneed to specify more what V is. Note that we could have written the equationin a slightly different form: T + = V + T + G V . (19)(This just amounts to starting the expansion from the other end-point of thepropagator.) Concerning T − , it is sufficient to note that it is the complexconjugate of T + . In order to check this, one can make the background electric field slightly space dependent,so that it has a compact support in space. One sees now that all the integrals are finite, andthat the single particle spectrum is proportional to the size of the region where the backgroundfield is non-zero. T R is defined from the same equation,but the free Feynman propagator G is replaced by the free retarded propa-gator: T R = V + V G R T R = V + T R G R V , (20)where the free retarded propagator G R ( p ) is defined as G R ( p ) = i p − m + i p ǫ . (21)In order to express T + in terms of T R , the first step is to relate the Feynman andthe retarded propagators. This is done via the following well-known relationship: G = G R + ρ − , (22)where ρ − is a 2-point function whose definition in momentum space is ρ ± ( p ) ≡ πθ ( ± p ) δ ( p − m ) . (23)(Note that ρ − is nothing but G − with z = 1.) From the above equations, wearrive trivially at (1 − V G R − V ρ − ) T + = (1 − V G R ) T R , (24)and subsequently at (1 − (1 − V G R ) − V | {z } T R ρ − ) T + = T R . (25)Thus, we have T + = (1 − T R ρ − ) − T R . (26)Similarly, one can prove, T + = T R (1 − ρ − T R ) − . (27)In fact it is easy to confirm that eqs. (26) and (27) are equivalent by expandingthe inverse quantity in terms of T R ρ − in eq. (26) and ρ − T R in eq. (27). Takingthe complex conjugate of eq. (26), we get, T − = T ∗ + = (1 − T ∗ R ρ − ) − T ∗ R . (28)( ρ − is purely real.) Multiplying eq. (27) by ρ − on the right, we finally obtain, T + ρ − = T R (1 − ρ − T R ) − ρ − = T R ρ − (1 − T R ρ − ) − . (29)Combining everything, we obtain the following expression for z ¯ zL : z ¯ zL = T + zρ − T − ¯ zρ + = T R zρ − (1 − T R ρ − ) − (1 − T ∗ R ρ − ) − T ∗ R ¯ zρ + . (30)11hus we have managed to replace all the Feynman propagators in L by retardedones. The price to pay for this transformation is that we have now an expressionthat is no longer bilinear in the propagators, but has terms at any order ≥ T R is given in eq. (20). For T ∗ R , itreads: T ∗ R = −V − V G ∗ R T ∗ R = −V − T ∗ R G ∗ R V . (31)Here, we have used the fact that, in a unitary theory (like scalar QED with areal background potential), we have V ∗ = −V . Adding up the equations for T R and T ∗ R , we first obtain T R + T ∗ R = −T ∗ R G ∗ R V + V G R T R . (32)Finally, we can eliminate V in the right hand side of this equation, in favor of T R or T ∗ R . This leads easily to: T R + T ∗ R = −T ∗ R h G R + G ∗ R i T R = −T ∗ R h ρ + − ρ − i T R . (33)Note that this relation is a variant of the optical theorem applied to a 2-pointfunction. The left hand side is equal to the discontinuity of the 2-point functionacross the real energy axis, and the right hand side gives the expression of thisdiscontinuity in terms of cut graphs. Thanks to this relationship, it is nowstraightforward to check that(1 − T ∗ R ρ − )(1 − T R ρ − ) = 1 + T ∗ R ρ + T R ρ − . (34)By combining eqs. (30) and (34), we easily arrive at the following simplifi-cation; 1 − z ¯ zL = 1 − T R zρ − (1 + T ∗ R ρ + T R ρ − ) − T ∗ R ¯ zρ + = (1 + T ∗ R ρ + T R ρ − ) − h − ( z ¯ z − T R ρ − T ∗ R ρ + i , (35)which leads to the generating functional: F [ z, ¯ z ] = exp (cid:16) − tr ln h − ( z ¯ z − T R ρ − T ∗ R ρ + i(cid:17) . (36)Here again, thanks to the fact that the background field is uniform, all thefactors inside the logarithm share a single spatial momentum p . The function z has argument p and the ¯ z is evaluated at − p .At this point, we see that all the properties of the distribution of producedparticles are determined by a single quantity, namely the amputated retardedpropagator T R of a scalar particle on top of the background field A µ . Before This is why the relation V ∗ = −V , that is the manifestation of unitarity in this calculation,is crucial in order to obtain eq. (33). z ¯ z T R ρ − T ∗ R ρ + withall the momentum dependence: h z ¯ z T R ρ − T ∗ R ρ + i p,q = ¯ z ( − q ) ρ + ( q ) Z d k (2 π ) z ( k ) T R ( p, k ) ρ − ( k ) T ∗ R ( k, q ) . (37)This formula is completely general. Note that T ∗ R is the same quantity as T R ,in which all the retarded propagators are replaced by advanced ones and V ( A )is replaced by V ∗ ( A ).In eq. (37), the momenta k and q are forced to be on the in-vacuum mass-shell, since they appear inside the distribution ρ ± . However, in the case of k , itturns out to be simpler to have a momentum variable (which we denote here by˜ k ) that obeys the mass-shell condition imposed by the non-zero gauge potentialat x → −∞ . Let us denote G ∞ R the retarded propagator evaluated in thepresence of the (constant) background field A ∞ µ ≡ lim x →−∞ A µ ( x ) . (38)Since there is no electrical field at x → −∞ , the gauge field is a pure gauge inthis limit: A ∞ µ = ∂ µ χ ( x ) , (39)and the propagator G ∞ R is simply obtained by a gauge transformation from thevacuum propagator G R : G ∞ R ( x, y ) = e ieχ ( x ) G R ( x, y ) e − ieχ ( y ) . (40)Let us now rewrite the combination T R ρ − T ∗ R that appears in eq. (37) in termsof the corresponding expressions T ∞ R and ρ ∞− which are naturally functions ofthe modified mass shell momentum ˜ k . This can be done by writing eq. (37) as T R ρ − T ∗ R = T R G R ( G ∞ R ) − | {z } T ∞ R G ∞ R ( G R ) − ρ − (cid:0) ( G R ) − G ∞ R (cid:1) ∗ | {z } ρ ∞− (cid:0) ( G ∞ R ) − G R T R (cid:1) ∗ | {z } ( T ∞ R ) ∗ . (41)Note that because ( D ∞ x + m ) G ∞ R ( x, y ) = δ ( x − y ) , (42)where D ∞ x is the covariant derivative constructed with the asymptotic field A ∞ µ at y → −∞ , and ρ ∞− ( x, y ) is still translationally invariant, it now projectsmomenta to the mass shell in presence of the background field at x → −∞ .Since this gauge field is a pure gauge, it is easy to write the correspondingmass-shell conditions imposed by ρ ∞ + (˜ k ) and ρ ∞− (˜ k ):(˜ k ± eA ∞ ) = m , (43)where the signs ± are to be chosen for the positive and negative energy solutionsrespectively. In a gauge where A = 0, as we shall chose later on, these read:˜ k = E in k and ˜ k = − E in − k , where E in k ≡ p ( k + e A ∞ ) + m . (44)13herefore, eq. (37) can be rewritten as h z ¯ z T R ρ − T ∗ R ρ + i p,q = ¯ z ( − q ) ρ + ( q ) Z d ˜k(2 π ) z ( k ) T ∞ R ( p, ˜ k ) ρ ∞− (˜ k )( T ∞ R (˜ k, q )) ∗ = z ( p )¯ z ( − q ) ρ + ( q ) Z d ˜ k (2 π ) (cid:12)(cid:12)(cid:12) T ∞ R ( p, − ˜ k ) (cid:12)(cid:12)(cid:12) ρ ∞ + (˜ k ) . (45)In the second line, we have changed ˜ k → − ˜ k , and we have exploited the factthat T ∞ R ( p, − ˜ k ) is proportional to δ ( p + k ) in a uniform background field. Notethat thanks to the constraints provided by the ρ + ( q ) and ρ ∞ + (˜ k ) factors, theenergies ˜ k and q are both positive. However, they obey different mass-shellconditions. The outgoing particle energy q follows the in-vacuum dispersionrelation, while ˜ k obeys the dispersion relation in the presence of the backgroundfield A ∞ µ .For practical calculations of T ∞ R ( p, − ˜ k ), it is best to relate this quantity tothe Fourier coefficients of a plane wave propagating on top of the backgroundfield. Since T ∞ R is obtained by amputating the retarded propagator G R with( G R ) − on the right and with ( G ∞ R ) − on the left, we can immediately write: T ∞ R ( p, − ˜ k ) = Z d x e i p · x ( (cid:3) x + m ) Z d y e i˜ k · y ( D ∞ y + m ) G R ( x, y ) | {z } η k ( x ) = lim x → + ∞ Z d x e i p · x ( ∂ x − i E out p ) η k ( x ) . (46)In these formulas E out p is the vacuum on-shell energy E out p ≡ p p + m . Sincethe propagator G R ( x, y ) is a Green’s function of the operator D x + m (now,the covariant derivative D µ is defined with the full background field, not justits asymptotic value in the past), (cid:2) D x + m (cid:3) G R ( x, y ) = δ ( x − y ) , (47)it is easy to check that η k ( x ) obeys the following equation of motion: (cid:2) D x + m (cid:3) η k ( x ) = 0 , (48)provided that ˜ k obeys the negative energy mass-shell condition (43). In orderto find the boundary condition when x → −∞ for η k ( x ), we can replace thefull propagator G R ( x, y ) in eq. (46) by the propagator G ∞ R ( x, y ) that resumsonly the asymptotic field A ∞ , η k ( x ) = x →−∞ Z d y e i˜ k · y ( D ∞ y + m ) G ∞ R ( x, y )= e i˜ k · x . (49) Recall that we will need the trace of h z ¯ z T R ρ − T ∗ R ρ + i and thus q will be equal to themomentum of the produced particle p .
14n order to obtain the final formula, we have used:( D ∞ y + m ) G ∞ R ( x, y ) = δ ( x − y ) . (50)Thus, we see that the initial condition for η k ( x ) is a plane wave, with a momen-tum ˜ k that obeys the mass-shell condition of eq. (43).In a uniform background field, we can simplify a bit the notations by writing T ∞ R ( p, − k ) ≡ − E out p (2 π ) δ ( p + k ) β p , (51)so that h z ¯ z T ∞ R ρ ∞− ( T ∞ R ) ∗ ρ + i p,q = 2 E out p (2 π ) δ ( p − q ) z ( p )¯ z ( − q ) ρ + ( q ) | β p | . (52)The only quantity that we need to determine in order to fully solve the problemis the coefficient β p . This is obtained by solving the equation of motion (48),with a plane wave initial condition when x → −∞ . We note that the initialplane wave is chosen as an antiparticle-like one in eq. (49) and projected intoa particle-like one in eq. (46), the intuitive meaning of which will be clear indiscussions in sec. 3. Let us now use eqs. (37) and (52) in order to obtain results about the spectraof the produced particles. From now on, let us simply denote E out p as E p in thissection, for we have chosen the definition of β p so that E in p will never appear inthe expressions. We shall wait for the next section where the difference between E in,out p and the physical interpretation of β p will be more explicit. The singleinclusive particle spectrum is obtained as the first derivative of the generatingfunctional with respect to z ( p ). We obtaind N +1 d p = δδz ( p ) tr (cid:2) z ¯ zρ + T ∞ R ρ ∞− ( T ∞ R ) ∗ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) z, ¯ z =1 = V (2 π ) | β p | . (53)It should be mentioned that, in accord with the definition (4), the functionaldifferentiations with respect to z ( p ) and ¯ z ( q ) are not accompanied by (2 π ) . Inthe second, we have made the trace explicit. The final δ ( p − k ) comes from thedifferentiation with respect to z ( p ). In the last line, we have performed the d k integration explicitly, and we have interpreted the (infinite) factor (2 π ) δ ( ) asthe volume V of the system. Naturally, the spectrum of antiparticles is identical.For later reference, it will be useful to introduce more compact notations asfollows: n p ≡ d N +1 d p = V (2 π ) | β p | ,f p ≡ (2 π ) V n p = | β p | . (54)15ote that f p has the interpretation of the occupation number for the producedparticles of momentum p . This is clear if we integrate eq. (54) over the momen-tum and write it as (cid:10) N + (cid:11) = Z d p d x (2 π ) f p , (55)where the properly normalized phase space measure d p d x / (2 π ) , or restoringthe Planck constant d p d x /h , is explicit.Let us now turn to the 2-particle spectra. For two particles, we obtaind N ++2 d p d p − d N +1 d p d N +1 d p = δ δz ( p ) δz ( p ) tr (cid:2) z ¯ zρ + T ∞ R ρ ∞− ( T ∞ R ) ∗ z ¯ zρ + T ∞ R ρ ∞− ( T ∞ R ) ∗ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) z =¯ z =1 = δ ( p − p ) n p f p . (56)The uncorrelated part of the 2-particle spectrum shows up naturally in thiscalculation, and we have absorbed it in the left hand side. The right hand siderepresents the correlated component of the 2-particle spectrum. As one can see,particles are correlated only if they have identical momenta. By integrating theprevious equation over p and p , and by using the first of eqs. (11), we obtain: (cid:10) N + ( N + − (cid:11) − (cid:10) N + (cid:11) = Z d p n p f p , (57)or equivalently (cid:10) N + N + (cid:11) − (cid:10) N + (cid:11) = Z d p n p (1 + f p ) = Z d p d x (2 π ) f p (1 + f p ) . (58)The form with an explicit integral over x is the one that would be applied toa system with a phase space density that depends (slowly) on the coordinate.The first term in the right hand side (the 1 in 1 + f p ) is the answer one wouldobtain for a Poisson distribution. That is, if the probability distribution is givenby P mn = δ mn e −h N i h N i m m ! , (59)which defines the Poisson distribution , one would have (cid:10) N + N + (cid:11) − (cid:10) N + (cid:11) = (cid:10) N + (cid:11) (i.e. the variance and the mean are identical). Thus, the deviationsfrom a Poisson distribution are contained in the term proportional to n p f p .Equation (58) indicates that these correlations are Bose-Einstein correlations,i.e. due to stimulated emission of particles in a single quantum state.For one particle and one antiparticle, we get:d N + − d p d q − d N +1 d p d N − d q = δ δz ( p ) δ ¯ z ( q ) h tr (cid:2) z ¯ zρ + T ∞ R ρ ∞− ( T ∞ R ) ∗ (cid:3) + tr (cid:2) z ¯ zρ + T ∞ R ρ ∞− ( T ∞ R ) ∗ z ¯ zρ + T ∞ R ρ ∞− ( T ∞ R ) ∗ (cid:3) i z =¯ z =1 = δ ( p + q ) n p (1 + f p ) . (60)16n this case, the correlation can only exist if the particle and antiparticle haveopposite spatial momenta. Again, the integrated form of this equation reads (cid:10) N + N − (cid:11) − (cid:10) N + (cid:11)(cid:10) N − (cid:11) = Z d p n p (1 + f p ) . (61)One may have wondered why the particle and antiparticle have correlations.This can be understood by the fact that, as we noted repeatedly, the pairof a particle and an antiparticle is created at once so that the particle andantiparticle production preserves the momentum conservation as well as thecharge conservation. Therefore, the correlation (60) reflects the particle-particlecorrelation (56) with p = p and p = − q , which explains the delta function ofspatial momenta in eq. (60). The results of the previous subsection suggest that the distribution of producedparticles obeys the following properties: i. Two particles are correlated only if they have identical momenta. ii.
A particle and an antiparticle are correlated only if they have oppositemomenta. iii.
In a given momentum mode, the distribution of produced particles followsa Bose-Einstein distribution.Let us now present a more general justification of these results. Becausethe background electric field is uniform, a unique momentum p runs around theloop. By using results obtained in the previous subsections, we can rewrite thegenerating functional explicitly as follows: F [ z, ¯ z ] = exp (cid:18) − V Z d k (2 π ) ln h − (cid:0) z ( k )¯ z ( − k ) − (cid:1) f k i(cid:19) . (62)In writing eq. (62), as already discussed previously in this paper, we have as-sumed that the system is placed in a finite volume V . This is indeed necessaryin order to have a finite particle production rate in a constant (in space) externalfield. A consistent quantization of the system in a finite volume requires one tospecify boundary conditions at the edges of V , which leads to the momentum k being a discrete variable. The continuum in k is recovered in the limit V → ∞ .Switching now to a notation which makes this explicit, and remembering thatd k / (2 π ) = P k we can write the generating functional as F [ z, ¯ z ] = Y k
11 + f k − z k ¯ z k f k , (63) Periodic ones being the most convenient. z k ≡ z ( k ) and ¯ z k ≡ ¯ z ( − k ). From thisformula, one sees immediately that the distributions of produced particles in thevarious modes are totally uncorrelated, since the generating functional factorizesas a product of generating functions for single modes: F [ z, ¯ z ] = Y k F k ( z k , ¯ z k ) , F k ( z k , ¯ z k ) ≡
11 + f k − z k ¯ z k f k . (64)By Taylor expanding this formula around z k , ¯ z k = 0, it is easy to obtain theprobability of having m i particles and n i antiparticles in the mode i , P ( { m i } , { n i } ) = Y k δ m k ,n k f k (cid:18) f k f k (cid:19) m k . (65)Given the occupation numbers f k (obtained from β k by solving the equationof motion of a plane wave over the background field), this formula completelyspecifies the distribution of produced particles and antiparticles. Distinct modesare not correlated. In each mode, there must be an equal number of particlesand antiparticles. The distribution of the particle multiplicity in the mode k is a Bose-Einstein distribution of occupation number f k . A Bose-Einsteindistribution is in sharp contrast to a Poisson distribution (which would be theresult in a complete absence of correlations), since its decrease at large m k ismuch slower because of the absence of m k ! in a Poisson distribution (see thedenominator of eq. (59)). As a result, final states with many particles in thesame momentum mode are more likely. We can interpret the results obtained in the LSZ derivation as a Bogoliubovtransformation. To do this explicitly it is useful to switch to canonical quan-tization, which we shall review here shortly as the following manipulations arevery standard ones.From the Lagrange density of eq. (1) one obtains the Hamiltonian of thetheory (apart from the gauge part), H = Z d x h ΠΠ † + i eA ΠΦ − i eA Π † Φ † + ( m − e A )ΦΦ † + ( ~D Φ) · ( ~D Φ) † , i (66)where Φ and Π are operators in the Heisenberg picture satisfying the equal-timecommutation relation, [Φ( x ) , Π( y )] = i δ ( x − y ) . (67)It will be convenient for the following discussion to choose a gauge where A = 0,because in this gauge it is possible to directly associate the time dependence18f the wave function with the physical energy of the particle . We shall thuswork with the following Hamiltonian, H = Z d x h ΠΠ † + m ΦΦ † + ( ~D Φ) · ( ~D Φ) † , i . (68)The whole dynamics of the matter fields is determined by the equations ofmotion ∂ Φ = i [ H, Φ] = Π † ∂ Π = i [ H, Π] = ( ~D − m )Φ † . (69)These can then be expressed as an equation of motion for Φ only, but involvingsecond order time derivatives. The important thing to realize is that becausewe are looking at a theory without self-interactions and coupled to a classicalbackground field , the equations of motion of the field operators are linear;they are in fact the same as the classical equations of motion for the fields. Thesolution to the retarded field equations therefore contains all the informationabout the relation between the field operators Φ and Π at x → −∞ and x → ∞ . In the Heisenberg picture, knowing the relations between the fieldoperators is equivalent to knowing the dynamics of the theory; in particular thewhole probability distribution of the produced particles.We can now introduce the familiar decomposition of the field operators interms of creation and annihilation operators. One can perform this decompo-sition in different bases of operators; in particular the ones that correspond toparticles at x → −∞ (the “in” states) or x → ∞ (the “out” states). Sincethese are just decompositions of the same operator Φ in different bases, one getsthe equality,Φ( x ) = Z d k (2 π ) " a in , k q E in k φ +in , k ( x ) + b † in , k q E in − k φ − in , − k ( x ) = Z d k (2 π ) " a out , k p E out k φ +out , k ( x ) + b † out , k q E out − k φ − out , − k ( x ) (70)Here we have denoted the dispersion relations of particles and antiparticles inthe in-state as E in k and that in the out-state as E out k . We assume that thebackground electric field is turned off adiabatically x → ±∞ , which meansthat A i approaches a constant value A in,out i . The dispersion relation for par-ticles is then E in,out , + k = p m + ( k + e A in,out ) and the one for antiparticles Consider for example a particle at rest in the vacuum: in the gauge A µ = 0 its wave-function is e − i mx . Performing a time-dependent gauge transformation with the function Mx /e will generate a (constant) gauge potential A = − M/e and change the wave-functionto e − i( m − M ) x , which for M > m will seemingly look like a negative energy one. Coupling to a quantum field would induce effective self-interactions through loop correc-tions. in,out , − k = p m + ( k − e A in,out ) . In writing eq. (70) we have used the sym-metry E − k = E + − k to write everything in terms of the particle dispersion relation E in,out k ≡ E in,out , + k . Note that the momentum label k refers to the “canonical”momenta, which are the variables describing the oscillation of the wavefunctionin space. The momentum that is actually measured in a detector is the “ki-netic” one, which in this case is k + e A in,out for particles and k − e A in,out forantiparticles. We are keeping the notations rather general in this section. In thephysical situation we are interested in, the only particles that are measured arethe “out”-ones. A convenient gauge choice, and the one adopted in sec. 4 is thento take A out i = 0, so that one need not distinguish between the canonical andkinematical momenta for particles in the final state–it is enough to rememberthat E in k is different from E out k .The choice of basis in the decomposition eq. (70) of the field operator isdetermined by the boundary conditions for the functions φ ± in , k ( x ) and φ ± out , k ( x ).When we require that they approach plane waves at asymptotic times: φ +in , k ( x ) = e − i E in k x +i k · x for x → −∞ (71) φ − in , k ( x ) = e i E in k x +i k · x for x → −∞ (72) φ +out , k ( x ) = e − i E out k x +i k · x for x → + ∞ (73) φ − out , k ( x ) = e i E out k x +i k · x for x → + ∞ (74)the corresponding operators a in,out , b in,out annihilate the in-state and out-stateparticles and antiparticles, respectively. Note that for further convenience ournotation has been chosen such that the coordinate dependence in both φ + k ( x )and φ − k ( x ) is e +i k · x and thus the usual negative energy plane wave e i k · x corre-sponds to φ −− k ( x ). The canonical commutation relation for Φ and Π is satisfiedif [ a in , k , a † in , p ] = [ b in , k , b † in , p ] = [ a out , k , a † out , p ] = [ b out , k , b † out , p ] = (2 π ) δ ( k − p ) . (75)All the space-time dependence of the field operator Φ is in the coefficient func-tions φ ± in,out , k ( x ); the creation and annihilation operators are time-independent.Because the equation of motion (69) for Φ is linear, the coefficient functions φ ± in,out , k ( x ) must each independently satisfy the same equation. In fact the so-lution to the equation of motion η k ( x ) introduced in eq. (48) is nothing but φ − in , − k ( x ).The relation between the field operators at x → −∞ and at x → + ∞ isencoded in the Bogoliubov coefficients. They are in the transformation matrixbetween the in- and out-basis functions. The solution for φ − in , k ( x ) is again asuperposition of plane waves at x → + ∞ . If the background field dependsonly on time, the modes of different k do not mix and we can introduce theBogoliubov coefficients as the coefficients of this plane wave decomposition by Note that in here these commutation relations do not include a factor 2 E k as is conven-tional. The normalization used here is simpler in the case where E in k differs from E out k . x → + ∞ φ +in , k ( x ) = s E in k E out k (cid:16) α k e − i E out k x +i k · x + β ∗ k e i E out k x +i k · x (cid:17) . (76)By noticing that [ φ +in , k ( x , − x )] ∗ satisfies both the same initial condition as φ − in , k ( x ) and the same equation of motion, one finds the solution at x → ∞ that starts as a negative energy wave aslim x → + ∞ φ − in , k ( x ) = s E in k E out k (cid:16) α ∗ k e i E out k x +i k · x + β k e − i E out k x +i k · x (cid:17) . (77)This can also been seen using e − i k · x φ +in , k ( x ) = [e − i k · x φ − in , k ( x )] ∗ . We can nowdeduce the relation, φ +in , k ( x ) = s E in k E out k (cid:16) α k φ +out , k ( x ) + β ∗ k φ − out , k ( x ) (cid:17) , (78) φ − in , k ( x ) = s E in k E out k (cid:16) α ∗ k φ − out , k ( x ) + β k φ +out , k ( x ) (cid:17) . (79)In the general case of a space dependent background field the Bogoliubovcoefficients are not diagonal in momentum space. However, one can diagonal-ize the transformation matrix from the in- to the out-states, and our followingdiscussion will equally well apply to the eigenstates of the more general transfor-mation instead of individual momentum modes. Inserting the decompositions(78) and (79) into eq. (70) one gets a out , k = α k a in , k + β k b † in , − k ,b † out , k = α ∗− k b † in , k + β ∗− k a in , − k . (80)Consistency with the commutation relations (75) gives the normalization con-dition, | α k | − | β k | = 1 . (81)This normalization condition is a consequence of the charge conservation sym-metry of our Lagrangian . The above relations can be inverted to give a in , k = α ∗ k a out , k − β k b † out , − k , (82) b in , k = α ∗− k b out , k − β − k a † out , − k . (83) One can verify that eq. (81) implies Q out = Q in , where Q ≡ e Z d k (2 π ) ( a † k a k − b † k b k ) . a k | i = b k | i = 0 . (84)A properly normalized state with n particles of momentum k can be constructedas | n k i = (cid:18) a † k √ n ! V (cid:19) n | i . (85)Particles of momentum k are counted with the particle number operator,d ˆ N d k = a † k a k (2 π ) . (86)For example, the expectation value of the number operator on a state with oneparticle of momentum p is h p | d ˆ N d k | p i = h | a p √ V a † k a k (2 π ) a † p √ V | i = δ ( p − k ) . (87)After setting up these conventions let us return to the problem of particleproduction. We consider the situation where there are no particles in at x →−∞ , then the system is in the incoming vacuum state defined by a in , p | in i = b in , p | in i = 0 . (88)We are working in the Heisenberg picture where there is no time evolutionin the states, so the system stays in the | in i state. But at late time x → + ∞ particles are described by the “out” annihilation operators which do notnecessarily annihilate | in i . In order to count the number of outgoing particleswe must count the number of out-particles contained in the state | in i . Forthis it is useful to derive an expression for | in i in terms of the “out” quantities a out , p , b out , p and | out i . That is, we have to solve, a in , p | in i = (cid:0) α ∗ p a out , p − β p b † out , − p (cid:1) | in i = 0 ,b in , p | in i = (cid:0) α ∗− p b out , p − β − p a † out , − p (cid:1) | in i = 0 . (89)This above equation can be easily solved using the following ansatz; | in i = C Y k exp (cid:16) λ k a † out , k b † out , − k (cid:17) | out i , (90)where C is a normalization constant. Applying the condition (89) to theansatz (90) we find the condition, h ∞ X n =1 λ n k n ! α ∗ k nV ( a † out , k ) n − ( b † out , − k ) n − ∞ X n =0 λ n k n ! β k ( a † out , k ) n ( b † out , − k ) n +1 i | out i = 0 . (91)22hifting the summation variable by one in the first term gives ∞ X n =0 λ n k n ! (cid:2) λ k α ∗ k V − β k (cid:3) ( a † out , k ) n ( b † out , − k ) n +1 | out i = 0 , (92)which is satisfied by λ p = V − β p α ∗ p . (93)Now we can fix the normalization constant from h out | exp n λ ∗ p a out , p b out , − p o exp n λ p a † out , p b † out , − p o | out i = 11 − V | λ p | = 1 + | β p | , (94)where we used the normalization condition (81). We then have the expressionfor the initial state in the following form: | in i = Y k (1 + | β k | ) − / exp (cid:20) V − β k α ∗ k a † out , k b † out , − k (cid:21) | out i . (95)This has a clear physical interpretation that the initial vacuum is a superpositionof states with out-state pairs of particles with k and antiparticles with − k .Armed with the explicit expression (95) it is now straightforward to cal-culate, for example, single and double inclusive spectra by taking expectationvalues of the number operator. The spectrum of particles isd N +1 d p = h in | a † out , p a out , p (2 π ) | in i = V (2 π ) (1 + | β p | ) − ∞ X n =1 n (cid:18) | β p | | α p | (cid:19) n = V (2 π ) | β p | . (96)This expression exactly coincides with eq. (53). The two particle spectrum(equal sign) is likewised N ++2 d p d p = h in | a † out , p a out , p (2 π ) a † out , p a out , p (2 π ) − δ ( p − p ) a † out , p a out , p (2 π ) ! | in i = 1(2 π ) h in | a † out , p a † out , p a out , p a out , p | in i = d N +1 d p d N +1 d p + δ ( p − p ) V (2 π ) | β p | . (97)Note that we are explicitly subtracting the delta function contribution to agreewith the definition (11). The variance is (cid:10) N + N + (cid:11) − (cid:10) N + (cid:11)(cid:10) N + (cid:11) = Z d p n p (1 + f p ) , (98)23hich should remind the reader of the particle number fluctuations in a Bose-Einstein system. It is actually easy to see that this is indeed what we have bylooking directly at the probability distribution.In order to see more clearly the structure of the probability distribution wenote that we can decompose the Fock space into a direct product of the Fockspaces for different momenta k . We then write an arbitrary state | Ψ i as a tensorproduct | Ψ i = O k | Ψ i k . (99)It is convenient to group together particles of momentum k and antiparticlesof momentum − k under the same label k . Thus we define the vacuum states | , i k of the subspaces k as a out , k | , i k = b out , − k | , i k = 0 . (100)We can now write | out i as a tensor product of the vacuum states of the differentmodes, | out i = O k | , i k . (101)A state with m particles of momentum k and n antiparticles of momentum − k at x → ∞ is then denoted by | m, n i k = a † out , k √ m ! V ! m b † out , − k √ n ! V ! n | , i k . (102)By taking tensor products of the states | m, n i k for different k one can constructa complete basis for the Fock space.Applying this decomposition to eq. (95) we write | in i = O k (cid:26) (1 + | β k | ) − / exp (cid:20) V − β k α ∗ k a † out , k b † out , − k (cid:21) | , i k (cid:27) , (103)and expanding the exponential we get | in i = O k ( (1 + | β k | ) − / ∞ X m k =0 m k ! (cid:18) V − β k α ∗ k a † out , k b † out , − k (cid:19) m k | , i k ) (104)= O k ( (1 + | β k | ) − / ∞ X m k =0 (cid:18) β k α ∗ k (cid:19) m k | m k , m k i k ) . (105)Thus we see that explicitly the “in” vacuum for a momentum mode k is asuperposition of outgoing particle-antiparticle pair states. The amplitude forbeing in a state with m k pairs is M k = k h m k , m k | P k | in i = 1 p | β k | (cid:18) β k α ∗ k (cid:19) m k , (106)24here we have to introduce P k , the projection operator to the subspace k , totake the inner product . The corresponding probability to have m k pairs inthe mode k and any number of particles in the other momentum modes is P ( m k ) = (cid:12)(cid:12)(cid:12) k h m k , m k | P k | in i (cid:12)(cid:12)(cid:12) = 11 + | β k | (cid:18) | β k | | β k | (cid:19) m k = 11 + f k (cid:18) f k f k (cid:19) m k . (107)This law for the probabilities characterizes the Bose-Einstein, or geometrical,distribution. Since the momentum modes are independent, we can then writethe combined probability distribution as P ( { m k } ) = Y k P ( m k ) = Y k
11 + f k (cid:18) f k f k (cid:19) m k , (108)which is exactly the form at which we already arrived in eq. (65). So far our discussions and formulas are given in a rather general way. In whatfollows, we consider an example which is exactly solvable and demonstrate howthe formulas in the preceding sections lead to the concrete evaluation of spec-trum of produced particle under an external electric field.The LSZ reduction method and the interpretation as a Bogoliubov transfor-mation both require that the asymptotic states at x → ±∞ are well defined.This means that the external fields should be adiabatically vanishing so that wecan define | in i and | out i without ambiguity. Thus, the imposed electric fieldsmust be time-dependent, beginning with zero at x = −∞ , growing finite withincreasing x , and diminishing to zero again as x → + ∞ . It is known that the Klein-Gordon or Dirac equation under the following time-dependent electric field E = (0 , , E ( x )) is exactly solvable [3, 34, 47]; E ( x ) = E (cid:2) cosh( ωx ) (cid:3) . (109)This electric field exponentially goes to zero for | x | ≫ ω − . In the limit of ω → The states | in i and | m k , m k i k live in different Hilbert spaces; one in the whole Fockspace and the other one in its subspace k . In order to take an inner product one thereforehas to project out the k component of the state | in i . Physically this projection means thatwe are not measuring the other momentum modes of the state | in i than k . Thus eq. (106)gives the amplitude to have m k pairs in the mode k and any number of particles in the othermomentum states.
25n which A = 0 and the vector potential associated with the electric field(109) is A = − (0 , , A = R d x E ( x )) . We can immediately recover eq. (109)from E = − ∇ A − ∂ A . After the integration we find the Sauter-type gaugepotential, A ( x ) = Z x d y E ( y ) = Eω (cid:2) tanh( ωx ) − (cid:3) , (110)where we have chosen the integration constant so as to make A ( x ) → x → + ∞ . As we have discussed in sec. 3 this is a very natural choice inthe present case. A constant A amounts to a shift in the third component ofthe momentum p , which is interpreted as a different frame choice. It is mostnatural to sit in the frame in which the particles and antiparticles measured at x = + ∞ are at rest if their p is zero. The gauge potential and the electricfield are sketched in fig. 4. −2 E/ ω E A E x ω x Figure 4: Sketch of the chosen gauge potential and the associated electric fieldas a function of time x . The electric field has a peak at x = 0 whose height is E and width is specified by 1 /ω . The gauge potential has an offset from zero by − E/ω in the in-vacuum at x = −∞ , meaning that the origin of p is shiftedby this offset. An explicit solution for the Sauter-type gauge potential is already known, andso we will simply explain the necessary notation and then jump into the knownexpression of the solution. To make this paper as self-contained as possiblewe supplement the derivation in appendix B in more detail. Introducing thefollowing notation, λ ≡ eEω , (111)the equation of motion in scalar QED is given by the gauged Klein-Gordonequation as h ∂ − (cid:0) ∂ − i λω (cid:2) tanh( ωx ) − (cid:3)(cid:1) − ∂ ⊥ + m i φ ( x ) = 0 , (112) We work with the (+ , − , − , − ) metric convention. x , we can factorize the wavefunction φ ( x ) into the time-dependent part and the spatial plane-wave part, i.e. φ ( x ) = ψ k ( x ) e i k · x . (113)The time dependence is governed by the differential equation, h ∂ + (cid:0) k + λω (cid:2) tanh( ωx ) − (cid:3)(cid:1) + k ⊥ + m i ψ k ( x ) = 0 . (114)We here change the variable x into ξ , defined by ξ ≡ (cid:2) tanh( ωx ) + 1 (cid:3) . (115)With this variable we can eliminate the hyperbolic function from the equationand express it only in terms of meromorphic functions. Moreover, ξ has aconvenient asymptotic behavior. We will later make use of ξ i α = (cid:18) e ωx ωx (cid:19) i α −→ (cid:26) x → + ∞ e αωx for x → −∞ (116)and also(1 − ξ ) i α = (cid:18) e − ωx − ωx (cid:19) i α −→ (cid:26) e − αωx for x → + ∞ x → −∞ (117)to infer the plane-wave boundary conditions.Also, for concise notation we introduce the dimensionless energies in thesame way as in Dunne’s review [3]; µ ≡ E in k ω , ν ≡ E out k ω , (118)where, with the time-dependent gauge potential, the energies in the in- andout-states are, respectively,( E in k ) = ( k − λω ) + k ⊥ + m , ( E out k ) = k + k ⊥ + m . (119)We note that the energy of antiparticles is given by E in − k in the in-vacuum and E out − k in the out-vacuum, respectively. Also we should explain our convention ofthe electric charge e . Our choice is as follows; particles are negatively chargedand antiparticles are positively charged for e >
0. These are reminiscent ofelectrons and positrons in real QED. Therefore, in the above, noticing that k = − k we see that the particle dispersion relation E in k starts with a largerlongitudinal momentum than E out k , which is understood as the deceleration bythe Lorentz force in the direction anti-parallel to the external electric field.27ow we are ready to express the solution of the differential equation. Onecan express two linearly independent solutions in terms of the hypergeometricfunctions as ψ (1) k ( ξ ) = ξ − i µ (1 − ξ ) − i ν F ( − i( λ ′ + µ + ν ) , +i( λ ′ − µ − ν ) ; 1 − µ ; ξ ) ,ψ (2) k ( ξ ) = ξ i µ (1 − ξ ) − i ν F ( − i( λ ′ − µ + ν ) , +i( λ ′ + µ − ν ) ; 1+2i µ ; ξ ) , (120)where λ ′ ≡ p λ − / λ defined in eq. (111). Now we need to take an appropriate linear combination of the two solutionswritten in eq. (120), so that the the boundary condition like eq. (48) can befulfilled. In fact, it will turn out that these solutions already satisfy a simpleplane-wave boundary condition. To make this explicit we should use eqs. (116)and (117) together with the general property of the hypergeometric function; F ( a, b ; c ; ξ → →
1. Therefore, in the limit of x → −∞ (i.e. ξ → ψ (1) k ( ξ → → e − µωx = e − i E in k x , ψ (2) k ( ξ → → e µωx = e i E in k x . (121)In accord with the identification defined by eqs. (71) and (72) we have φ +in , k = ψ (1) k e i k · x , φ − in , k = ψ (2) k e i k · x , (122)The quantities necessary to compute the particle and antiparticle productionare obtained as the coefficient of these solutions in the x → + ∞ limit–whendecomposed in terms of the φ ± out , k . To this end, it is necessary to know thelimit of F ( a, b ; c ; ξ → a , b , and c are complex numbers in this case. Thus, it isconvenient to make a transformation in the argument from ξ to 1 − ξ , which ispossible by means of the following mathematical identity; F ( a, b ; c ; x ) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) F ( a, b ; 1 − c + a + b ; 1 − x )+ Γ( c )Γ( a + b − c )Γ( a )Γ( b ) (1 − x ) c − a − b F ( c − a, c − b ; 1+ c − a − b ; 1 − x ) , (123)which leads to an alternative expression of the solution, ψ (1) k ( ξ ) = ξ − i µ (1 − ξ ) − i ν B ∗ k F ( − i( λ ′ + µ + ν ) , +i( λ ′ − µ − ν ) ; 1 − ν ; 1 − ξ )+ ξ − i µ (1 − ξ ) i ν A k F ( +i( λ ′ − µ + ν ) , − i( λ ′ + µ − ν ) ; 1+2i ν ; 1 − ξ ) , (124)and ψ (2) k ( ξ ) = ξ i µ (1 − ξ ) − i ν A ∗ k F ( − i( λ ′ − µ + ν ) , +i( λ ′ + µ − ν ) ; 1 − ν ; 1 − ξ )+ ξ i µ (1 − ξ ) i ν B k F ( +i( λ ′ + µ + ν ) , − i( λ ′ − µ − ν ) ; 1+2i ν ; 1 − ξ ) , (125)28here we defined, A k ≡ Γ(1 − µ )Γ( − ν )Γ( − i( λ ′ + µ + ν ))Γ( + i( λ ′ − µ − ν )) ,B ∗ k ≡ Γ(1 − µ )Γ(2i ν )Γ( + i( λ ′ − µ + ν ))Γ( − i( λ ′ + µ − ν )) , (126)Here we note that, if we take a naive limit of x → F ( a, b ; c ; 1), it would miss the second term in the identity (123) because ofthe factor (1 − x ) c − a − b which is zero if a , b , c are real but is an oscillatory finitefunction if a , b , c are complex. At this point it is straightforward to deduce theasymptotic plane-wave forms at x → ∞ , that is, ξ → ψ (1) k ( ξ → → A k e − νωx + B ∗ k e νωx = A k e − i E out k x + B ∗ k e i E out k x ,ψ (2) k ( ξ → → A ∗ k e νωx + B k e − νωx = A ∗ k e i E out k x + B k e − i E out k x . (127)Comparing the above behavior with the Bogoliubov transformations (76) and(77), we obtain the Bogoliubov coefficients as α k = s E out k E in k A k , β k = s E out k E in k B k , (128) Before addressing the concrete expressions for the particle spectrum, we list allthe necessary formulas to proceed with the calculations. The Gamma functiongenerally satisfies,Γ(1 + z ) = z Γ( z ) , Γ(1 − z )Γ( z ) = π sin( πz ) , (129)from which we can derive the following useful relations, | Γ(i α ) | = πα sinh( πα ) , | Γ(1 + i α ) | = πα sinh( πα ) , | Γ( + i α ) | = π cosh( πα ) . (130)Then, after some algebra, we reach, | α k | = cosh[ π ( λ ′ + µ + ν )] cosh[ π ( λ ′ − µ − ν )]sinh(2 πµ ) sinh(2 πν ) . (131)and | β k | = cosh[ π ( λ ′ − µ + ν )] cosh[ π ( λ ′ + µ − ν )]sinh(2 πµ ) sinh(2 πν ) . (132)We note here that, using cosh( a + b ) = cosh( a ) cosh( b ) + sinh( a ) sinh( b ) twice,we can easily check that | α k | − | β k | = 1 , (133)29hich is consistent with the condition (81). Now that we have | β k | explicitly,we can get the general probability distribution which is characterized only interms of | β k | . The single inclusive spectrum, for example, isd N +1 d p = V (2 π ) cosh[ π ( λ ′ − µ p + ν p )] cosh[ π ( λ ′ + µ p − ν p )]sinh(2 πµ p ) sinh(2 πν p ) . (134)From this expression we can get the occupation number f p which is obtained byremoving the volume factor V / (2 π ) of the single particle spectrum. Once f p isgiven, the whole probability distribution is known as discussed in the previoussections. We plot f p as a function of p in the unit of m ⊥ ≡ p p ⊥ + m in fig. 5.In drawing fig. 5 we fixed m ⊥ , and set the electric field to the value E = πm ⊥ /e –which is sufficiently strong to create particles in view of the standard expressionof the Schwinger mechanism–, and then we vary the time scale ω . f p p / m ⊥ eE = π m ⊥ ω = m ⊥ ω = 3 m ⊥ ω = 10 m ⊥ Figure 5: Time scale dependence of the produced particle distribution as afunction of p . As ω → f p extendsbetween p ≃ − eE/ω and p ≃ ω = m T in thefigure. In contrast, with increasing ω , the result approaches eq. (136) whichspreads wider than the small- ω case with the distribution center located at p ≃ − eE/ω .Now let us consider two extreme cases. First, we take the constant field limit( ω → N +1 d p → V (2 π ) exp (cid:20) − π ( p ⊥ + m )4 eE (cid:16)
11 + ρ − ρ (cid:17)(cid:21) ( ω → , (135)where ρ ≡ ωp / (2 eE ) taking a value in the range of − eE/ω < p < − < ρ < p > p < − eE/ω , the result is zeroin the ω → p direction corresponds to the momen-tum distribution as a result of the deceleration by the electric field while it isimposed. If ω is small, the electric field lives long, and the produced particlesare pushed down by the Lorentz force along the p direction for longer time.Note that this range bounded from − eE/ω to zero coincides with the rangeof A changing between x = ±∞ . One might expect that the total numberof produced particles diverges in the case of a constant (in time) electric field.This is indeed true; the result of the p integration is nearly proportional to theintegration range 2 eE/ω when ω is small enough, which is divergent as 1 /ω –i.e.as the time during which the external electric field is non-zero.Next, we shall take a look at the opposite limit, i.e. a short-pulse limit, ω → ∞ . In this limit eq. (134) is reduced to,d N +1 d p → V (2 π ) E in p E out p (cid:18) E in p − E out p (cid:19) ( ω → ∞ ) . (136)To have non-zero value the electric field eE should be larger than ωE out p . Eventhough there is no exponential suppression , as compared to the small- ω case,the resulting f p is significantly suppressed by large ω for a fixed maximalstrength of the electric field, as is apparent in fig. 5.Let us finish this subsection with a comment on a related work by Cooper andNayak [49]. In this paper, the authors study the Schwinger mechanism for thepair production of charged scalars in the presence of an arbitrary time-dependentbackground electric field. In particular, they calculate the pair production rate(i.e. the number of pairs created per unit of time) , and conclude that “theresult has the same functional dependence on E as the constant electric field E result with the replacement: E → E ( t )”. However, the comparison of the twolimits in eqs. (135) and (136) indicates that the time-dependent case is unlikelyto be given by a mere replacement E → E ( t ) in the time-independent result. We have calculated the multiplicity of produced particles under a spatially con-stant but time-dependent electric field in scalar QED using a formalism basedon the LSZ reduction formula. We defined and computed the generating func-tional of the particle and antiparticle distribution, from which we determinedthe whole distribution of the production probability. We found that particle When the electric field has a fast time dependence, the perturbative process γ → φφ ∗ canproduce particles. One may question whether this is a well defined concept, since the presence of a timedependent external field makes the definition of proper particle states ambiguous. One mayargue that the only quantities that can be defined unambiguously are those where measure-ments are done only after the external field has died out. From the point of view of theBogoliubov transformation, the transformation coefficients are uniquely determined from theasymptotic plane-wave time-dependence. Of course one may use some working definition ofthe Bogoliubov coefficients at arbitrary intermediate time, but such a treatment implicitlyassumes a quasi-static approximation. | α k | + | β k | = 1then to preserve the anticommutation relation of the transformed operators. Itis easy to show that | β p | gives the occupation number f p , and using the abovenormalization condition we can arrive at the probability distribution, P ( { m s, k } ) = Y s, k (cid:0) − f s, k (cid:1) (cid:18) f s, k − f s, k (cid:19) m s, k , (137)where s refers to the spin and m s, k takes the values 0 or 1. Equation (65) (orequivalently (108)) and the above (137) are our central results. We see that,if we drop higher orders than the quadratic terms in f k for f k ≪
1, all ofthe Bose-Einstein, Fermi-Dirac, and Poisson distributions are trivially reducedto identical answer; 1 − f k for no-particle production and f k for one-particleproduction. The difference emerges at the quadratic order, and it is notablethat the difference remains no matter how small f k is. For instance, the two-particle production probability in the same mode is f k / f k if a Bose-Einstein one, and zero if a Fermi-Dirac one. In otherwords one must properly take account of the quantum statistical nature if themultiparticle correlations are concerned.In the final section of this paper, we have revisited a known exactly solvableexample of time-dependent electric fields. The time-dependence of the Sauter-type potential is actually ideal to think of the Schwinger mechanism; since wecan unambiguously define the asymptotic states in the infinite past and future.This property of adiabatically vanishing external fields is necessary to make thediscussion of particle production meaningful.Using the exact solution we took two extreme limits of constant ( ω → ω → ∞ ) electric fields. In the constant case we found that f p distributes almost uniformly over the range − eE/ω < p <
0. Thereforethe total (integrated) number of produced particles diverges as 1 /ω that isinterpreted as the time duration for which the external electric field is imposed.In the short-pulse case, on the other hand, f p is a double-peak structure and theminimum in-between is located at p = − eE/ω . In practice the latter would beuseful because it is more difficult to sustain larger eE/ω in the laboratory.32 cknowledgments We thank Raju Venugopalan for discussions. K. F. thanks Harmen Warringafor discussions on the time-dependent electric field. K. F. is grateful to Institutde Physique Th´eorique CEA/DSM/Saclay for warm hospitality where this workwas initiated and he is supported by Japanese MEXT grant No. 20740134 andalso supported in part by Yukawa International Program for Quark HadronSciences. T. L. is supported by the Academy of Finland, project 126604. F. G.is supported in part by Agence Nationale de la Recherche via the programmeANR-06-BLAN-0285-01.
A Multiplicity distribution
In some cases, one is interested only in the overall number of particles andantiparticles in the final state, but not in their distribution in momentum space.For such observables, one can define a simpler generating function that does notcontain any information relative to the momentum of the produced particles: G [ u, ¯ u ] ≡ ∞ X m,n =0 u m ¯ u n m ! n ! Z m Y i =1 d p i n Y j =1 d q j (cid:12)(cid:12)(cid:12) M m,n ( { p i } , { q i } ) (cid:12)(cid:12)(cid:12) . (138)Note that this is also equal to: G [ u, ¯ u ] = ∞ X m,n =0 u m ¯ u n P m,n , (139)where P m,n is the probability to have exactly m particles and n antiparticlesin the final state. Obviously, this new generating function can be obtainedfrom the generating functional F [ z, ¯ z ] by setting the functions z ( p ) and ¯ z ( p ) toconstants respectively equal to u and ¯ u : G [ u, ¯ u ] = F [ z ( p ) = u, ¯ z ( p ) = ¯ u ] . (140)Thanks to this relationship, one can obtain quantities such as those defined ineqs. (11) as ordinary derivatives of G [ u, ¯ u ]. For instance, Z d p d p d N ++2 d p d p = ∂ G [ u, ¯ u ] ∂u (cid:12)(cid:12)(cid:12)(cid:12) u, ¯ u =1 = ∞ X m,n =0 m ( m − P m,n . (141)The last equality is obtained from eq. (139), and is the justification for the righthand side in the first of eqs. (11). B Detailed derivation of the solution
With the variable change from x to ξ , simple algebraic procedures lead to ∂ = 2 ωξ (1 − ξ ) ∂ ξ and ∂ = 4 ω ξ (1 − ξ )[ ξ (1 − ξ ) ∂ ξ + (1 − ξ ) ∂ ξ ], from which33e can rewrite the equation of motion (114) in the following form; h ξ (1 − ξ ) ∂ ξ + (1 − ξ ) ∂ ξ + µ ξ − + ν (1 − ξ ) − − λ i ψ k ( ξ ) = 0 . (142)In what follows we will explain how to solve this differential equation. Beforefinding the analytical solution, from this form of the equation we can alreadyconfirm that the asymptotic behavior of the particle solution is e ± i E in k x at x → −∞ and e ± i E out k x at x → + ∞ as it should be. We shall pick upthe most singular terms out of the differential equation (142), which gives (cid:2) ξ∂ ξ + ∂ ξ + µ ξ − (cid:3) ψ in k ( ξ ) = 0 , (143)around ξ = 0 (i.e. x → −∞ ). It is easy to find the solution of this equation as ψ in k ( ξ ) = ξ ± i µ ≃ e ± i E in k x . In the same way we can extract the behavior around ξ → x → + ∞ ) from the singular terms; (cid:2) (1 − ξ ) ∂ ξ − ∂ ξ + ν (1 − ξ ) − (cid:3) ψ out k ( ξ ) = 0 , (144)leading to ψ out k ( ξ ) = (1 − ξ ) ± i ν ≃ e ∓ i E out k x .Now let us return to solving eq. (142). Because we have seen the boundarycondition, it is convenient to factorize the plane-wave pieces as follows; ψ k ( ξ ) = ξ − i µ (1 − ξ ) − i ν ϕ k ( ξ ) , (145)then we can find the equation that ϕ k ( ξ ) should satisfy as n ξ (1 − ξ ) ∂ ξ + (cid:2) − µ − ( − µ − ν + 2) ξ (cid:3) ∂ ξ − ( − i µ − i ν − i λ ′ + 1 / − i µ − i ν + i λ ′ + 1 / o ϕ k ( ξ ) = 0 . (146)Here we recall that the hypergeometric differential equation, n x (1 − x ) ∂ x + (cid:2) c − ( a + b + 1) x (cid:3) ∂ x − ab o f ( x ) = 0 , (147)has two independent solutions given by f (1) ( x ) = F ( a, b ; c ; x ) , f (2) ( x ) = x − c F ( a +1 − c, b +1 − c ; 2 − c ; x ) . (148)Therefore, by the identification of a = 12 − i( λ ′ + µ + ν ) , b = 12 + i( λ ′ − µ − ν ) , c = 1 − µ , (149)we finally arrive at the solution (120). 34 eferences [1] W. Heisenberg and H. Euler, Z. Phys. , 714 (1936),[arXiv:physics/0605038].[2] J. S. Schwinger, Phys. Rev. , 664 (1951).[3] G. V. Dunne, Heisenberg-Euler effective Lagrangians: Basics and exten-sions, in From fields to strings , edited by M. Shifman et al. , pp. 445–522,World Scientific, 2004, arXiv:hep-th/0406216.[4] B. Andersson, G. Gustafson, G. Ingelman and T. Sjostrand,Phys. Rept. , 31 (1983).[5] A. Bialas, W. Czyz, A. Dyrek, W. Florkowski and R. B. Peschanski,Phys. Lett. B229 , 398 (1989).[6] A. Casher, H. Neuberger and S. Nussinov, Phys. Rev.
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