The lepton pair production in heavy ion collisions revisited

Transcription

1 The lepton pair production in heavy ion collisions revisited E. Bartoš,,S.R.Gevorkyan,3,E.A.Kuraev and N. N. Nikolaev 4 Joint Institute of Nuclear Research, 4980 Dubna, Russia Dept. of Theor. Physics, Comenius Univ., Bratislava, Slovak Republic 3 Yerevan Physics Institute,375036,Yerevan,Armenia 4 Institut f. Kernphysik, Forshchungszentrum Jülich, D-545 Jülich, Germany and L. D. Landau Institute for Theoretical Physics, 443 Chernogolovka, Russia Abstract We derive the first terms up to the fourth order in the fine structure constant in the amplitude of lepton pair production in the Coulomb field of two relativistic heavy ions. We investigate the lowest (Born) term in detail and obtained the differential cross section in this approximation for any momentum transfer and final state momenta. Using the same technique we calculate the next orders of perturbation series (up to fourth order in fine structure constant) and investigate the limiting cases. It has been shown that the terms of perturbation series are gauge invariant and finite. We discussed also the Coulomb interaction between the colliding nuclei and its influence on the lepton pair yield. Explicit expressions for the process of multiple pairs creation have been obtained in the framework of the eikonal model. Introduction The process of lepton pairs production in collision of relativistic heavy ions attracted a rising interest in the past years which is connected mainly with operation of RHIC (Lorentz factor γ = E/M = 00) and LHC (γ = 3000). The total cross section of the process A + B e + e + A + B ()

2 in its lowest order (Born approximation, Fig. a,b) has been known long ago (see [] and references there in). The modern problem is to take into account the Coulomb corrections (CC) to the amplitude of this process some of which are depicted on Fig. a-d. In spite of the fact that CC are determined by the powers of fine structure constant α =,this 37 quantity entered the amplitude of the process () in combination (αz ) n (αz ) n (αz Z ) n 3, where Z,Z are the nucleus charge numbers, n + n is the number of exchanged photons between nuclei and the pair, n 3 0 is the number of photons exchanged between the colliding nuclei. For heavy ion collisions the parameter αz is of the order of unity and the problem arise to take into account the effect of CC. The exact knowledge of total yield of lepton pairs in heavy ion collisions is important, because the pair production consist a huge phone in experiments with relativistic heavy ions. For example the total cross section for the process () at RHIC energies is tens kilobarns for the heavy ion collisions. At LHC energies this quantity becomes a hundreds kilobarns if one estimate it according to the Racah formula []. Moreover, the intensive pair production can destroy the ion beams circulating in the accelerator as a result of electron capture by heavy ion (see for example [3]). Recently the problem of CC in process () was investigated in the number of papers [4, 5, 6, 7] with the unexpected result. The authors [5, 6] solved the Dirac equation for an electron in the Coulomb field of two highly relativistic nucleus. Using the crossing symmetry they connected the amplitude for electron scattering in the Coulomb fields of two colliding nucleus with amplitude for reaction (). As the result the total cross section turn out to coincide with the Born approximation for the process () so that CC haven t impact on the total lepton pairs yield. The same result have been obtained in the work [7] in which the authors use the eikonal approximation for electron scattering at high energies. They took advantage of the common wisdom that the interaction of lepton pair with Coulomb fields of two highly relativistic nucleus can be represented as a product of eikonal amplitudes for interaction of electron and positron with nucleus A and B separately. Such approach allows them to take into account the contribution of CC in amplitude of process () with the quoted above result: the total cross section exactly coincide with the Born term. This result induced the series of critical papers [8, 9], where it was discussed, why this result is so unexpected and how assumptions which have been done in its derivation can affect this result. As was observed in [8] the main contribution to the CC in process () comes from the case when one of the colliding ions radiate a single photon and the produced pair interact with another nucleus by means of arbitrary number of photons (see Fig. a d) In fact the contributions of such type can be obtained with the use of common Weizsäcker-Williams (WW) approximation, which leads to appearance of large logarithms in energy for such terms, which can t be cancelled by the next order of CC where such logarithms are absent. From the other case it has been shown [9] that the change in the order of integration in regularize integrals (the Coulomb phase for electron scattering diverges, so it needs regularization) completely change the final result. In the recent publication [6] the two approaches employed in investigation of this problem were analyzed in detail, with the conclusion that the perturbation approach in fact correspond to the exclusive process of the coherent production of lepton pairs in the collisions of relativistic ions (i.e. reaction ()). As to the approach based on retarded solutions of the Dirac equation it gives the inclusive cross section for production of arbitrary number of the lepton pairs, i.e. in this way one obtain the contribution of inclusive pairs production. Nevertheless the issue of calculation of the CC contribution in correct way in process () is an open task up to now and it turn out to be more complex problem that it seems from the first glance. All this

3 stimulated us to investigate the first order terms contribution in the amplitude for exclusive process under discussion in more details, with the aim to understand how one can takes into account the contribution of CC in heavy ion collision in the correct way. Our paper is organized as follows. In Sec. we deduce the amplitude for the lepton pair creation in the collision of relativistic particles in the lowest order of perturbation theory (Born approximation). We are restricted by kinematics of pionization region so the created pair doesn t influence the kinematics of the colliding particles. We supposed that momenta transfer between the produced pair and initial particles are much smaller in comparison with the energy of initial particles, e.g. we consider the case of their quasielastic scattering. The most general analytical expressions have been obtained for differential cross section in compact form and analyzed using the well known WW approximation. For this limiting case we cited the full expression for the total cross section of the process under consideration. In the Sec. 3 we consider all possibilities for pair production by three photon exchange, so called amplitudes of the type M (). Here we present the general formulae for its contribution in the differential cross section and explicitly show its contribution in the main logarithmic approximation. Our result for the amplitude appears to be the generalization of the known result for the amplitude of the process of pair photoproduction in the odd charge state gained in the paper [0]. We have done the numerical estimate for the relevant contribution to the total cross section in the main (double log) approximation. Here we give the results of the amplitude computation and corresponding contributions in the cross section from processes of two photon exchange between each ion and pair and the processes when one of the ions exchange the single photon to the pair and another one is connected with a pair by three photons in all possible ways. In the last case, our result appears to be generalization of the known result in the case of photoproduction of the pair in even charge state []. In the Sec. 4 we analyze how the Coulomb interaction between ions influence on the pair creation process. Analyzing the low order diagrams we show that the effect of the multiphoton exchanges leads to the well known phase factor e iαz Z ln q λ for every amplitude obtained above. Nevertheless as we show it is not the only effect of the multiphoton exchanges and we suggest some ansatz for general form of the amplitude. In the Sec. 5 we consider the amplitudes and the cross sections of several pairs production process. Some explicit formulae for the case when one neglects the exchanges between properly ions (n 3 = 0) are presented. The accuracy of our approximation is estimated. In Appendix A we give the expressions which allows us to do the loop momenta integrations. In Appendix B we give the result of computation amplitudes corresponding to first CC M (),M() (3),M. In Appendix C we cite some details of construction of the amplitudes and the cross sections. The Born term We are interested in the exclusive process of lepton pair production in the collision of two relativistic nuclei with charge numbers Z,Z A(p )+B(p ) e + (q + )+e (q )+A(p )+B(p ). We define the usual c.m.s. total energy squared of colliding nuclei s =(p + p ) =4γ γ m m (3) 3

4 where m, and γ, are the mass and relativistic Lorenz factors (γ = E ) of the colliding m nuclei. Keeping in mind the fast decrease of γ γ e + e cross sections with the momentum transfer squared, i.e. the photons virtuality and with the invariant mass squared of the pair s p =(q + + q ), we will work in the kinematics s =(p + p ) q, q,s p. (4) Later on we calculate the amplitudes neglecting the pieces of the order m, so our results are s valid for relativistic particles with power accuracy. The total cross section of the reaction in the lowest order in fine structure constant (Born approximation) is known long, we still spend some time on it for the sake of illustration of the technique and approximations which we use later on in the CC derivation. We follow the technique developed in []. The lowest order Feynman diagrams describing the process are depicted on the Fig. a,b. The corresponding amplitude read M () () = Z Z i(4α) ū(p qq )γ µu(p )ū(p )γ νu(p )g µα g νβ ū(q )T αβ v(q + ) (5) ˆq ˆq + + m T αβ = γ β (q q + ) m γ ˆq ˆq + + m α + γ α (q q + ) m γ β Using the Gribov s decomposition of metric tensor g µν = gµν + s (pµ p ν + pν pµ )itiseasyto show that at high energies the main contribution to (5) is given by its longitudinal part ( ) g µα g νβ p ν s p µ p α p β. As a result the expression (5) can be cast in the form M () () = Z Z is(8α) ū(q qq )R () () v(q +)N N (6) where R () () = s pµ p ν T µν, N = sū(p )ˆp u(p ),N = sū(p )ˆp u(p ). For spinless projectile and target as well as for the colliding fermions at definite chirality state N = N =. In what follows we will use the standard Sudakov expansion of all the momenta in the two almost light-light vectors p, p and the two-dimensional transverse component (q p = q p =0) q = α p + β p + q,q = α p + β p + q,q ± = α ± p + β ± p + q ± (7) p p = p p s, p p = p p s, p = p = O( m6 s ), s p p p p From the on mass shell condition, for instance p =(p q ) = m,p =(p q ) = m, one can easily obtain β α m s,α β q s. (8) 4

5 Later on we will neglect (where it is possible) the α,β which greatly simplify calculations so that q β p + q,q α p + q, q = q + β m, q = q + α m. (9) β α Hereafter q i always stand for q i etc. With the above parameterization the invariant mass of the pair is s p =(q + + q ) =(q + q ) α β s ( q + q ). (0) The conservation laws provides the following relations among the introduced variables thus the limits of β, α variation are β = β + β +,α = α + α +, q + q = q + q +, () s p +( q + q ) <β, α <. () s As we shall see, the production amplitudes take a simple form in terms of the variables z ± = β ± /β,z = z + and k± = q ± z ± q. In what follows we will work in the kinematics of pionization region (similar approach was used in [3]), which give the main contribution to the total cross cross section of the process under consideration. For such kinematics the following limitations on β,α take place s α β. Using the above approximation the expression (6) can be represented as m M () () = is N N ( q + β M )( q + α M ) 6 Z Z ū(q )R () () v(q +). (3) It is convenient to use the gauge invariance constraint q µ T µν = 0, which in view of (9) entails After some algebra one finds p ν T νµ = qν β T νµ q β e ν T νµ, e ν i = q i q i. (4) R () () = q sβ [ mê R +ê ˆQ +z + Q e + q z + z R ] ˆp (5) where R = S ( k ) S ( k + ), Q = k S ( k )+ k + S ( k + ),S ( k)= k + z + z q + m, 5

6 k± = q ± z ± q,z ± = β ± /β,z + + z =. This representation of the Born term is explicitly gauge invariant and has transparent physical meaning. The factor q i q i + m i βi (αi ) is the familiar bremsstrahlung amplitude for the photon with transverse momenta q i. The square of this amplitude give the well known luminosity factors [3]. To take into account the heavy ions dimensions it is enough to replace the factors N,N in the expression (3) by the colliding nuclei formfactors F (qi ). A straightforward calculation gives ū(q )R () 4s () v(q +) = z {( ) +z q m +4z + z q R +(z+ + z ) Q +4z } +z (z + z )R q Q. (6) The different terms in (6) have a simple meaning in terms of the transverse and scalar (often called longitudinal) polarizations of the virtual photon γ and there is one-to-one correspondence with the discussion of helicity structure function in diffractive DIS [4] and diffractive production of vector mesons [3]. Namely, in the expression (5) the first term corresponds to excitation of the pair with the sum of the q q helicities λ + λ = ± by the transverse photons, the second and the third terms correspond to excitation of the pair with λ + λ =0 by the transverse photons, the last term describes excitation of the pair with λ + λ =0by scalar photons. Correspondingly, the terms 4z + z q R describe the pair production by scalar photons, whereas the last term describes the ST (LT ) interference. Evidently, the LT interference vanishes upon the phase space integrations. Because of the exact conservation of the helicity of relativistic electrons, the above structure of the amplitude (3) and its square (6) will be retained also for higher order production processes. We put here the alternative form of R () () R () () = q [ ] mê R +ê ˆQ sα +y + Q e + q y + y R ˆp, R = S ( l ) S ( l + ), Q = l + S( l + )+ l S( l ), S ( l)= m + q y + y + l, l ± = q ± y ± q,y ± = α ±,y + + y =. (7) α The differential cross section of the process is connected with the full amplitude by the relation dσ = 8s M dγ 4. (8) The summation is carried on by final particles polarization. The phase space volume for four particles in the final state can be expressed through the variables introduced above in the following way dγ 4 = d 3 p d 3 p d 3 q d 3 q + δ 4 (p () 8 E E + p p p q + q ) E E + 6

7 = ds p dβ d q 4() 8 d q dγ ±, (9) s( α )( β ) β dγ ± = d3 q + E + d 3 q E δ 4 (q + q q + q ). In obtaining this expression we used the following relations for all final momenta d 3 q E =d4 qδ(q m )=sdαdβd qδ(sαβ q m ). The expressions (6), (6) allows one to calculate the differential cross section of the process under consideration in its lowest order (by fine structure constant) in most general way. In order to illustrate the main features of the process we cite here the principal steps of the calculation of the total cross section in the widely exploited Weizsäcker Williams (WW) approximation. We focus on the double WW region of []. The total cross section of pair production γ γ e+ e drops rapidly as soon as one of the photons is off mass shell, which imposes the upper limit of integration over q, < m, the upper limits of integration β < β max = m/m, and α < α max = m/m,,theα and β are related by (9) and the lower limit of integration over β β min = s p = s pm sα max sm. (0) Because of the virtualities of photons are small q, < m e 6 R ch A 0 R A 0.5 A /3 GeV, where R A.A /3 fm is the nuclear radius, the effect of form factors can be neglected. It is not the case for production of muon pairs or hadronic states, though. Notice that because β max,α max, the spin of the nucleus is not important. Then one can extend to nuclear collisions the results of [] for point like lepton beams and the contribution from double WW kinematics to the total cross section equals σ tot = Z Z α 4 4m β max ds p dβ σ γγ (s p ) s p β β min m 0 q dq ( q + m β ) m 0 q dq ( q + m α ), () where for the on mass shell photons [( σ γγ (s p )= 4α + 8m s p s p ) ( ) 6m4 sp ln s p m + sp 4m ( ) + 4m 4m. s p s p As a matter of fact, the dependence on the large masses m, of colliding nuclei is spurious and can be eliminated by the rescaling β = xβ max and α = yα max,sos p =4m e γ γ xy and x min = () s p 4m e γ γ 0, x max =. (3) 7

8 Then the contribution of the double WW region is obtained from eq. (9) of [] by the substitution sσ m 4γ γ, so that σ = α4 Z Z 8 m 7 ( = α4 ZZ 8 m 7 34 ) 7 ln + ln ρ 8 ζ(3) ln + 34 )] 7 ln 8ln 3 [ ρ 3 8ρ +30.3ρ ],ρ=ln(γ γ ). (4) [ ( ) ( 7 33 ρ 3 4 6ln ρ + ( ) 66 Because the γ γ e + e cross section starts vanishing for large virtuality of photons well before the effect of nuclear form factor becomes substantial, the evaluation of the contribution from q i > m can be performed in the pointlike nucleus approximation. Then for the total cross section one can use the Racah formula [] (in [5] this formula is cited with the misprint in the third coefficient) σ = α4 Z Z 8 [ ρ 3.98ρ +3.84ρ.636 ]. (5) m 7 Thus the total cross section of the process under consideration in the Born approximation is the growing function of the invariant energy (as the third power of logarithm). This is the direct consequence of the fact that both photons are quasi real, which led to second power of logarithm in (5) and one power come from the common for all cases integration by rapidity β. Before going to calculation of properly Coulomb corrections let us do the following remark. The lepton pair created by two photons (Born approximation) produced in charge even state. The exchange of the odd number of photons between the Coulomb fields of heavy ions and produced leptons will let to a pairs in charge odd state which don t interfere with Born amplitude. Remembering that we are restricted by fourth order of α in the amplitude of the process, later on we have to take into account only the interference of Born amplitude with the amplitude where the four photons exchange take place between leptons and colliding particles (see Fig. 3). Now we can go to the main topic of present work, calculations of Coulomb corrections(cc). We begin from the simplest case which are provided by three photons exchange between the produced lepton pair and colliding particles. 3 The Coulomb corrections; the triplet (C-odd pair state) and singlet (C-even pair state) production We start with the discussion of the triplet pair production. The amplitude which correspond to first CC consist of two parts: M () = M () + M () (), where the amplitude M corresponds to the case when the lepton pair is attached to colliding particle A by one photon and connects with B through two photons in all possible ways. The case when two photons are attached to A (one to B) is described by the term M (). As would be shown later this terms can be obtained one from another by simple substitution, so it is enough to consider one of them. 8

9 The amplitude M () is described by six Feynman diagrams some of which are depicted at Fig. a c. The Sudakov parameterization momentum of the second exchanged photon is k = α p + β p + k. The integrations in α and β can be done by the standard methods as described in Appendix B. Because of the helicity conservation, the result for the two photon exchange contribution to the production amplitude M () = s (4α)3 Z Z () ( q + m β )ū(q ) ˆR v(q +)N N, (6) is very similar to the Born amplitude, with = q sβ d k ˆR () = q [ mê R () sβ +ê () ˆQ +z + Q () e + q z + z R ] () ˆp [ k ( q mê R (k)+ê ˆQ (k)+z + Q (k) e + q z + z R (k) ] ˆp (7) k) where in close analogy to amplitudes of diffractive DIS R (k) =S ( k )+S ( k + ) S ( k k) S ( k + k), Q = k S ( k ) k + S ( k + ) ( k k)s ( k k) ( k + k)s ( k + k). (8) For the calculation of square of an amplitude we can use directly the result (7), i.e., () M z {( ) +z q m +4z + z q () (R ) In general the expressions for R () and Q () for them for the case q 0 is given in the Appendix A. The amplitude M () A is readily obtained from M () +(z + + z )( Q () ) +4z + z (z + z )R () q Q () }. (9) are rather complex quantities. An explicit form with the two-photon exchange between the lepton pair and the nucleus via substitutions β α,z ± y ± q q, e e, ˆp ˆp,Z Z,S S. (30) The contribution to the total cross section from M () and z-integration is straightforward. The discussion of Sec. on the vanishing of the cross section with the virtuality of the photon is fully applicable to σtot odd and the limits of integration in q and β are the same as in the calculation of the Born cross section. In terms of the rescaled variable x we need to evaluate ρ = x min dx x so that to the leading logarithmical accuracy Numerical estimation for P = d q m d q q ( q + m x ) σ tot odd = α6 Z Z (Z + Z )ρ m P. (3) d q [ m 4 (R () ) + 3 m ( Q () ) ] 9 q =0 (3)

10 with the results of Appendix A gives P 3, 6. We want to emphasize, that the general expression contains the leading logarithms contribution as well as the nonleading ones, omitting only terms whose contribution is suppressed by factor m and which is vanished in the high energy limit. s The approximation q 0 corresponds to Weizsäcker-Williams approximation for the case when the scattered proton (muon) is not detected and will scatter at very small angles. We see that for the case with semi inclusive setup (when the scattered proton (muon) is fixed) we have more complicate picture compared with the case of photoproduction. In Appendix A we give the explicit expressions for -loop momentum integrals for amplitudes M () for the case of small transfer momentum q. We put here also the contribution to the total cross section from the interference of amplitudes M () and M (). It is enhanced only by the first power of large logarithm ρ due to bust effect σodd tot = α6 ZZ 3 ρ 3 d m q d q q q ds p dγ ± 4 Sp(ˆq + m)r () (ˆq () + m) R. (33) In Appendix C we put the explicit expression for the relevant trace. Using the amplitudes M () and M (3) (see Fig. 3) obtained in Appendix B, we put here the leading contributions to the C-even state pair creation cross section, one from the interference of Born (,) and (,) exchange mechanisms σ = α6 (Z Z ) 3 ρ dz d q d q d q m 0 z ( z ) q q and the one from the interference of (,) and (,3), (3,) mechanisms where m 4 Sp(q + m)r (q () + m) R () (34) σ (3) () + σ() (3) = α6 (Z Z ) (Z + Z )ρ m P, (35) P = d q d q m [m R q R ] Q Q3. (36) q =0 Here we make a point that P = P. Indeed, the product R R 3 contains altogether 6 terms, and making the proper shifts of the integration variables, one can establish one-to-one correspondence to the 6 terms in R. This can be regarded as a manifestation of the well known AGK rules, [6]. Indeed, the two photon-exchange production of lepton pairs can be viewed as a diffraction excitation of the photon, whereas the 3-photon exchange can be viewed as an absorption correction to the one-photon exchange. Furthermore, a comparison with the Davis-Bethe-Maximon result [7] for pair production cross section in the Coulomb field σ γz e+ e Z (s) = 8Z α 3 [ ln s 9m m 09 ] 4 (Zα) (37) n= n(n +(Zα) ) and using the WW-approximation gives P = P = 8 ζ(3) which was verified by numerical calculations as was mentioned above. 0

11 4 Eikonal type Feynman Diagrams The third class of Feynman diagrams (FD) contains in the lowest order (-loop) approximation FD with three exchanged photons, one of them connects the projectiles,. There present 8 FD of such form, which combine to 4 FD with one of exchanged photons having the pair production gauge-invariant insertion. Separating the longitudinal and the transverse integrals the contribution of Born amplitude and the one arising from this set of FD results in the replacement R () () (q,q,α)=r () () (q,q )+iαz Z J I (38) with ( dα J = i α + i0 ) ( dβ α i0 i β + i0 ) (39) β i0 and d k I = ( k + λ ) R() () (k + q,k+ q ). (40) The quantity J is equal to unity. As for the quantity I it have the infrared singularities. Let us show that only one region k 0 produce the infrared divergent contribution. Really, the two additional regions q + k 0, q + k 0 do not produce the infrared divergent contributions due the gauge property of Born amplitude R () ()(q, r) 0, q 0, r 0. So we have I =ln q () (q,q )+B, (4) λ R() where we introduce some auxiliary vector q. The quantity B do not contain the infrared singularities. Let us see what happened in higher orders of PT. In two loop level there present 8 FD of such classes of FD, whose contribution can be written in the form! (iαz Z ) I J. (4) Again one see J = (this property is valid in any order of PT: J n =)and I = d k ( k + λ ) d k ( k + λ ) R() () (k + k + q,k + k + q ). Again one can see that the main infrared singularity corresponds to the expansion of the known eikonal phase factor R () () (q,q )e iαz Z ln q λ. (43) Following the logics of the paper [8] we can arrange the probable form of amplitudes considered above with taking into account the arbitrary number of exchanged photons between projectiles R (i) (j) (q,q ) e iαz Z ln q [ λ (i) R (j) (q,q )+ib (i,j) () + B (i,j) +... ]. (44) Let consider the problem of taking into account of eikonal photons from the spatial point of description. For this aim let use another form of Born amplitude. The obtained above

12 results can be written as M = M 0 + M + M = i s s [ d 6 k B(q + k, q k) Z Z N N B(q q ) iαz Z k + λ d! (αz Z ) k d ] k B(q + k + k,q k k ) ( k + λ )(, (45) k + λ ) with B(q,q )= q q e α e β ū(q )T αβ v(q + )/(qq ). Introducing the additional momentum integration and the delta function corresponding to the conservation low δ (k + k +... q )= () d be i b( k + k +... q ), (46) we see that amplitude M can be read M = i s s (α) Z Z N N d b e i q b iαz Z Φ( b) Φ B ( b q ), (47) with Φ( d k e i k ( b b)= k + λ = C +ln ), Φ B ( b, q )= b λ d k ei k b B( k, q k). (48) where C =0.577 is the Euler constant. This result confirms the general ansatz given above that all the dependence on photon mass λ can be represented as a phase factor. We see from these expressions that eikonal photons exchange amplitudes can not be expressed in terms of Born amplitude with some phase factor. The corresponding contributions excluding the Born term to the total cross section will be enhanced by only the first power of ρ. 5 Production of any number of pairs As we learned from the previous section restricting ourselves by consideration of terms containing 3rd and nd powers of large logarithms we can omit the class of eikonal type FD as well as ones of the form M (i) (j),i,j>considering the process of the single pair creation. There present several mechanisms of many pairs production.the chain mechanism corresponds to emission of all pairs from one exchanged photon line.in lowest order approximation it has large logarithmical enhancement but is not optimal in terms of Z Z α parameter. It turns out that the optimal series of FD consists from eikonal type FD with some of exchanged photons containing the pair emission insertions (see Fig. 4a). The matrix element of such FD with n + exchanged photons, k from them having the single pair creation insertion, has a form ( Z Z α ) k (8) k+ ( iz Z α) n+ k M n,k = sn N k! 4s s...s k (n + k)! d b Φn+ k ( k b) Φ B ( b, r i ), (49) i=

13 with r i = q +i +q i is the 4-momentum of i-th created pair, s i = ri is its invariant mass square. When summing on the number of exchanged between nuclei virtual photons one obtains ( Z Z α ) k M k = M n,k = sn N (8) k+ e iαz Z ln q n=k 4Π k λ j=s j k! d b e i q b e iz Z α ln( k b q) Φ B ( b, r i ). (50) i= Let us now consider the phase volume dγ k of the final state dγ k =() 4 3k d3 p d 3 p E E k i= d 3 q +i E +i d 3 q i E i δ 4 (p +p p p r... r k ),r i = q i +q +i (5) where we use Sudakov parameterization for pair component momenta q ±i = α ±i p + β ±i p + q ±i, r i = α r i p + β r i p + r i, besides that we consider the kinematical situation when β r >β r >...>β r k. For this case we introduce the auxiliary 4-vectors q = p p = r + q,q = r + q 3,..., q k = r k + q k+,q k+ = p p. (5) Expanding these vectors as well in Sudakov s form q i = α i p + β i p + qi, we can conclude that β > β >... > β k > (γ γ ) for this situation. Really in contradiction to the chain kinematics when only one such kind of kinematics works in general case all possible k! rearrangements take place. Phase volume can be put in the form dγ k = k+ s 7k 3 5k i= d q i k j= dβ j β j β j+ dφ j,β k+ =0, dφ j = ds j dγ j,dγ j = d3 q j d 3 q +j δ 4 (q j r j q j+ ). (53) E j E +j Taking into account all possible kinematical situations we can see that the integration of the phase volume on the bust variables β,..., β k provides the enhancement factor ρ k. Let us now consider the case k = n +, which corresponds to the case when all the exchanged photons have a pair creation insertion and there is no pure photons connecting ions lines without such an insertion (see Fig. 4b). Really this case can be considered as a toy model. The matrix element of such gauge invariant set of FD has a form with M n+ = s (iα Z Z ) n+ (8) n+ N N (n +)! n i= d k i n+ j= k j r j k j k j (r j k j ) Q j s j (54) n k n+ = q k j,r j = q +j + q j,q j = e µ j ẽ ν j ū(q )T j µνv(q j +), j j= e j = k j k j, ẽ j = (q j q j+ k j ) q j q j+ k j ),s j =(q j+ + q j ) +( q j q j+ ). (55) 3

14 Using the property of fast decreasing the quantity Q j dγ j with the increasing virtualities of photons q j, q j+ and the relation 4m ds j s j Qj dγ j = 8 9m, (56) the total cross section of k pairs production (as usually we suppose that all the pairs are different or omit the identity effects) will have a form σ n+ = m ((n +)!) R n = ( 4α 4 Z Z ρ ) n+ C n+, C n+ = n j= d k j 9 n+ l= n+ i= d q i k l q l q l+ k l k l (q l q l+ k l ),k n+ = q R n, (57) mn n k j. The region of integration on q i must be restricted by qi <m due to fast decreasing of off mass shell matrix element squared of production of pair by two virtual photons. The reasonable choice C k = leads to the sum of cross sections of producing two or more pairs σ n = ( n= m ) I 0 (y) y 4 j=,y= α Z Z 4ρ. (58) 3 We see in the frames of this model that the probability of n pairs creation have an Poisson form P n = σ n yn σj (n)! e y. (59) The sum on the number of pairs created behaves at large ρ =ln(γ γ )asσ = σ n e c ρ, which is more weak than the unitarity violating behavior σ (γ γ ) ε, which is typical for QCD for any small positive ε, but nevertheless violate the Froissart bound. This result must be corrected by at least two reasons. One consist in taking into account the exchanged between nuclei photons. Another possibility is related to screening effects which manifest itself in FD containing the light light scattering blobs as a virtual corrections to the production amplitudes (see Fig. 4c). One may expect that screening effect will results in relaxation of high energy behavior of (58). Concluding this section let us estimate the accuracy of our calculations. For this aim let us compare the cross sections σ chain of producing two pairs by chain mechanism (when both pairs are emitted by the single exchange photon) [9] and the cross section σ of two pairs creation by mechanism of emission from two exchanged photon lines (see Fig. 4d). Their ratio is σ chain ( ) ρ. (60) σ Z Z α this ratio is of the order 0. for ρ 0 and Z Z 70. This quantity determine the accuracy of our approach in the case of multiple pairs production. We are grateful to E. Levin for pointing out this possibility. 4

15 6 Conclusions In conclusion we list shortly the main results of our work:. For the terms with one photon attach to one of the colliding ion (with any number k of photons exchanged between lepton pair and another ion) the leading contribution (ρ ) to the cross section can be obtained by means of Weizsäcker-Williams approximation using the cross-section of lepton pair photoproduction process. Nonleading contributions cannot be obtained by WW approximation nevertheless as was shown above the relevant expressions can be obtained.. We explicitly calculate the logarithmically enhanced contributions to the cross sections from the interferences of kind M () () M, M () F (Z n Zm M (), which cannot be expressed in the form ) as was claimed in the papers [8, ]. 3. We obtain the formulas for the total cross sections of creation of any number of pairs in leading logarithm approximation and estimate its accuracy at the level 0%. We would like to note that the theoretical support of our ansatz for amplitudes of inelastic processes write down in the form generalized eikonal representation needs further theoretical support. Besides we want to note the problem of Z, Z dependence of M(Z,Z, q ± )in the case q ± m. The factorization (i.e. the representation of matrix element in the form M f(q ± )f(z )f(z )) is presumably violated by M types of contributions. Acknowledgements We are grateful to Stanislav Dubnicka for the participation in the initial stage of this work and to Valery Serbo for useful discussions. The work of S. G. and E. K. are supported by INTAS and E. B. and E. K. have support from SR-000 grant. E. K. is grateful of participants of DESY Theory seminar for interesting discussions. References [] V. M. Budnev, I. F. Ginzburg, G. V. Meledin and V. G. Serbo, Phys. Rep. C5 (975) 83; V. Baier, V. Fadin, V. Khoze and E. Kuraev, Phys. Rep. 78 (98) 93. [] G. Racah, Nuovo Cimento 4 (937) 93; L. Landau and E. Lifshits, Sow. Phys. 5 (934) 06. [3] C. A. Bertulani, G. Baur, Phys. Rep. 63 (988) 99. [4] B. Segev,J. C. Wells, Phys. Rev. C59, (999) 753. [5] A. J. Baltz, L. McLerran, Phys. Rev. C58, (998) 679. [6] A. J. Baltz, F. Gelis, L. McLerran, A. Peshier, nucl-th/0004 (00); BNL-NT-0/. [7] U. Eichmann, J. Reinhardt and W. Greiner, Phys. Rev. A6 (000) [8] D. Yu. Ivanov, A. Schiller, V. G. Serbo, Phys. Lett. B454, (999) 55. 5

16 [9] R. N. Lee and A. I. Milstein, Phys.Rev. A6 (000) [0] I. F. Ginzburg, S. L. Panfil and V. G. Serbo, Nucl. Phys. B84 (987) 685. [] D. Ivanov and K. Melnikov, Phys. Rev. D57 (998) 405. [] E. Kuraev, L. Lipatov, Yad. Fiz. 6 (97) 060. [3] E. Kuraev, N. Nikolaev and B. Zakharov, JETP Letters 68 (998) 696. [4] N. N. Nikolaev, A. V. Pronyaev and B. G. Zakharov, Phys. Rev. D59, (999) [5] R. Lee, A. Milstein and V. Serbo, hep-ph/00804 (00). [6] V. Abramovski, V. Gribov and O. Kancheli, Yad. Phys. 8 (973) 595. [7] H. Davies, H. A. Bethe and L. Maximon, Phys. Rev. 93 (954) 788. [8] D. Yennie, S. Frautchi and H. Suura, Ann. Phys. 96 v [9] V. Serbo, JETP Lett. (970) 39; L. Lipatov and G. Frolov, Sov. J. Nucl. Phys. 3 (97) 333. [0] D. Ivanov, E. Kuraev, A. Schiller and V. Serbo, Phys. Lett. B44 (998) 453. [] I. F. Ginzburg and D. Yu. Ivanov, Nucl. Phys. B388 (99)

17 Appendix A We put here the result of two-dimensional integration on loop momenta scalar and vector ones d k I( q ) = ( k + λ )(( q k) + λ )(( q (A.) k) + m ) = [ q q + ] ln q q + λ + [ ] q q + q + q + ln m q + A ln q m + A ln q + m, A = q q q A = q q + q [ ] + + [ ] q q q + q + q + q + m q q +, q q + [ ] + + [ ] q q q + q + q + q + m q q, q q + I( q ) = = d k q q q + ln q + λ q + k ( k + λ )(( q k) + λ )(( q k) + m ) [ ln q +ln q ] + q m ] + [ q q + q + ln q + m q q ln q q q whereweusethenotation a = a + m and besides = q q + m q, q = q + + q., (A.) Using these integrals we obtain for the case q = q + q + d k R (k) R = ( k + λ )(( q k) + λ ) = = ( ) q q q + ln q q + + ( q + m q q q + ) ln q q q +, (A.3) R (k) = q k q + q + k q +, for scalar combination which enters into M () for the case of small q and d Q k Q (k) = ( k + λ )(( q k) + λ ) = (A.4) [ ( m =( q q + ) q + ) ln q q + + ( q + m q q q ) ln q ] q + q + + q ( [( q q + q ) ln q ) + q m ln m q q + q q ( + q + ln q + m q ln q )], m 7

18 Q (k) = q q + k q k q q + q + k q + k q +. Appendix B It can be shown [],[0], that matrix elements M (), M () (3) can be written in form of matrix element M () () given above with the replacements R, Q R, Q for matrix element M () and R, Q R 3, Q 3 for matrix element M () (3) respectively. The explicit forms of R i, Q i are R = S( k )+S( k + )+S( k k) S( k + k), Q = k S( k ) k + S( k + )+( k k)s( k k)+( k + k)s( k + k) (B.) and R 3 = S( k )+S( k k )+S( k k )+S( k q + k + k ) ( k k + ), Q 3 = k S( k )+( k k )S( k k )+( k k )S( k k ) ( k q + k + k )S( k q + k + k )+( k k + ). (B.) At q 0, β these expressions agree with the results [0], [] obtained for the case of pair photoproduction. The matrix element M has a form M = isn N α 4 4 (Z Z ) ū(q )R v(q +), (B.3) with R d = k d k k ( q k ) k ( q k ) ( + P ud)m. (B.4) In the limiting case q 0, β these expressions agree with the ones obtained in [0], [] for the case of pairs photoproduction. The quantity R has so called up down symmetry, which is decoded in the permutation operator P ud, which acts as follows P ud F (p,p,α,β,q,k,q,k )=F (p,p,β,α,q,k,q,k ) (B.5) and the quantity m is with m = R 43 + R 34 + R 43 + R 34 + R 34 + R 43, (B.6) R 43 = β sa p (q q + m)p, R 34 = p (q k + m)p (q k k + m)p ( q + + q k + m)p s β +, β q q k k + q k q + + q k R 43 = p (q q + k + m)p ( q + + k + m)p, (B.7) s α q + + k + α + q q + k 8

19 R 34 = p (q k + m)p ( q + q k + m)p, s β + q + k + β q q k R 43 = p (q k + m)p ( q + + q + k k + m)p ( q + + k + m)p s β +, β q q + + q + k k + q k q + + k R 43 = p (q q + k + m)p ( q + + k + k + m)p ( q + + k + m)p s β +. β q q + + k + k + q + + k q q + + k The symmetry property caused by the charge-conjugation symmetry of amplitude. There are 4 FDs contributing to M. Instead of them it is convenient to consider 4 = 96 FDs which take as well the permutations of emission and absorption points of exchanged photons to the nuclei. The result must be divided by (!). This trick provides the convergence of α,β integrals. All the 4 FDs, describing the interaction of 4 virtual photons with the pair componenta are relevant: providing the convergence of β,α integrations. When closing the contour if integrations say in upper planes only from them become relevant. We do not succeeded in trying to write down the result in such elegant form as M () (,3). We note as well that besides the charge conjugation symmetry the matrix element M () (3) possess the additional one Bose-symmetry, which manifest itself in the symmetry of the integrand m () (3) under k,k permutation. Appendix C Here we put down the explicit form of the trace which appears in calculation of interference term of amplitudes M () and M () : s Sp(ˆq + m )[ mr ê +ê ˆQ + L ]ˆp (ˆq+ m )[ mr ê + ˆQ ] ê + L = m [ m R R (e e )+(e Q Q e )+L L + L (e Q )+L (e Q ) ] +m R R (q e q + e )+(q e Q q + Q e )+L L (q + q ) +L (q q + Q e )+L (q e Q q + ), (C.) whereweusethenotations (ab) = 4 Spâ ˆb = a b, (abcdef) = 4 Spâ ˆb ĉ ˆd ê ˆf. 9

20 p p 0 p p 0 q q q + p q q q + p 0 p q q p 0 (a) (b) Fig. : Feynman diagrams for the two photon pair production in the process A + B A + B + e + e p p 0 k q q q + q k (b) Λ(c) p p 0 (a) Fig. : Some Feynman diagrams for the three photon pair production in the process A+B A + B + e + e 0

21 p p 0 k q k k q k q q + Λ p p 0 (a) (b) (c) Fig. 3: Some Feynman diagrams for the four photon pair production in the process A + B A + B + e + e (a) ::: (b) Λ (c) (d) + :::Λ+ ::: Fig. 4: Feynman diagrams for multiple pair production