Lepton pair production in ultra-peripheral collisions

The lepton pair production in ultra-peripheral collisions (UPC) is studied in the classical field approximation. We derive a general form of the cross section in terms of photon distributions which depend on the transverse momentum and coordinate based on the wave packet form of nuclear wave functions. Such a general form of the cross section in the classical field approximation contains the results of the generalized equivalent photon approximation (EPA) as well as the corrections beyond EPA in the Born approximation. By rewriting the general form of the cross section in light-cone coordinates, we find a good connection with the transverse momentum dependent distribution (TMD) factorization formalism in the Born approximation. Our numerical results are consistent with current experimental data.

Another type of effects is related to the non-perturbative production of lepton pairs in strong electric fields through Schwinger mechanism [54]. Recent developments along this line include the lepton pair production in strong magnetic fields by Schwinger mechanism [55][56][57][58] and the vacuum birefringence [59][60][61].
Recently the lepton pair production through strong electromagnetic fields in ultraperipheral collisions (UPC) has drawn broad interest. Back to 1930s, Weizsacker and Williams considered the electromagnetic field produced by a fast moving particle as an equivalent flux of quasi-real photons [62,63]. This approximation is called Weizsacker-Williams method or equivalent photon approximation (EPA) [64]. A related process is the lepton pair production through collisions of two real photons and was studied by Breit and Wheeler [65] under the condition that the total energy of two photons should be greater than the mass of the lepton pair. The STAR Collaboration at Relativistic Heavy Ion Collider (RHIC) has measured the lepton pair (ll) production process in UPC [66]. There are also several measurements related to nonlinear effects of quantum electrodynamics (QED) such as the vacuum birefringence [67] and the light-by-light scattering [68]. The transverse momentum spectra of the lepton pair in peripheral collisions of heavy ions are found to be significantly broader than the ones in UPC by STAR [69] and by the ATLAS collaboration at the Large Hadron Collider (LHC) [70]. Such broadenings may arise from medium effects in peripheral collisions, therefore, UPC may provide a baseline for future studies of medium effects.
Several theoretical methods are available to describe the lepton pair production in UPC.
A widely used method is the EPA or the generalized EPA (gEPA). The total cross section of γγ → ll has been calculated in EPA in the classical field approximation of QED [71]. The original EPA calculations predict that the position (transverse momentum) of the peak in the transverse momentum spectrum is less than 20 MeV, inconsistent with the experimental data [72][73][74]. Therefore the generalized EPA was proposed [71,[74][75][76][77][78][79] to give the correct position of the peak in the transverse momentum spectrum [77][78][79].
The azimuthal asymmetry in the lepton pair from linearly polarized photons in UPC has been studied in the TMD factorization formalism [80,81] similar to polarized gluons [82][83][84].
In this formalism, the photon Wigner functions are introduced into the cross section which depend on the transverse momentum and coordinate [85,86]. The broadening of transverse momentum has also been studied in the TMD formalism [87].
With all these different formulations in different perspectives of the process, it is natural to ask if there is a unified description. In this paper, we will derive a general form of the cross section based on wave-packet nuclear wave functions, which incorporates photon distributions with the dependence on the transverse momentum and coordinate. The cross sections in (g)EPA and the TMD formalism can be derived from the general form. The numerical results of the cross section in the general form are in a good agreement with experimental data. This paper is organized as follows. In Sec. II, we derive a general form of the cross section in terms of transverse momentum and coordinate dependent photon distributions. In Sec.
III, we implement the classical field approximation. In Sec. IV we take the ultra-relativistic limit to reproduce the results of (g)EPA. In Sec. V, we rewrite the general form of the cross section in light-cone coordinates and make a connection with results of the TMD formalism.
The numerical results of the cross section in the general form are given in Sec. VI and a comparison is made with experimental data as well as the results of (g)EPA and the TMD formalism. The main results of this work are summarized in Sec. VII.
Throughout this paper, we choose the metric g µν = diag{+, −, −, −} for ordinary co- 2 and x T = (x 1 , x 2 ) are also used. A vector a µ can be written as a µ = a + n µ + + a − n µ − + a µ T where a ± = (a 0 ± a 3 )/ √ 2 and n µ ± are light-like vector satisfying n 2 + = n 2 − = 0 and n + · n − = 1. The inner product of two vectors in light-cone coordinates is

II. GENERAL FORM OF CROSS SECTIONS FOR LEPTON PAIRS
In this section, we give a general form for the differential cross section of lepton pairs in UPC, with the detailed derivation being given in Appendix A.
As shown in Fig. 2, we consider collisions of two nuclei A 1 and A 2 moving alone ±z direction which are displaced by an impact parameter b T and generate a pair of leptons l and l along with other particles X 1 , · · · , X f , · · ·, where four-momenta of particles are given in parentheses. Here P µ A1 = (E A1 , P A1 ) and P µ A2 = (E A2 , P A2 ) with E A1 = P 2 A1 + M 2 1 and E A2 = P 2 A2 + M 2 2 are on-shell momenta of two nuclei with masses M 1 and M 2 respectively. The three-momenta of two nuclei are P A1 = (0, 0, P z A1 ) and P A2 = (0, 0, −P z A1 ) in the center of mass frame of the collision. In order to describe collisions at fixed impact parameters, we need to assume the wave functions of colliding nuclei to be wave packets. We follow the standard way in quantum field theory to obtain the cross section at the impact parameter b T in Eq. (A5). Making the ansatz (A9) for the longtitudinal momentum amplitudes of wave-packets and completing the integrals over longitudinal momenta of wave-packets we arrive at Here P 1 , P 2 , P 1 and P 2 are on-shell momenta of nuclear wave-packets given by P 1 = (E P 1 , P 1T , 2P z A1 − P z 1 ), P 2 = (E P 2 , P 2T , −2P z A1 + P z 1 ), P 1 = (E P 1 , P 1T , P z 1 ) and P 2 = (E P 2 , P 2T , −P z 1 ). As the solution to the energy conservation P z 1 is a function of transverse momenta P 1T , P 2T , P 1T and P 2T . We see that the z components of P 1 + P 2 and P 1 + P 2 are vanishing. The function G(x 2 ) is defined in Eq. (A9), which is a positive and decreasing function of x 2 satisfying G(0) = 1. We have used the shifts of transverse momenta ∆ 1T ≡ P 1T − P 1T and ∆ 2T ≡ P 2T − P 2T , and relative velocity of two nuclei v given in Eq.
(A10) as a function of transverse momenta.
In the tree level of Feynman diagrams (sometimes called Born approximation) as shown in Fig. 1, the lepton pair is produced in the photon fusion process (3) Here, we assume the photon γ(p 1 ) or γ(p 2 ) comes from the nuclei A 1 (P 1 ) or A 2 (P 2 ), respectively. Note that, each photon does not have to come from the nuclear center. We can identify b iT in Eq. (2) as the transverse distance between γ(p i ) and which are related to the impact parameter b T = b 1T − b 2T as shown in Fig. 2.
The invariant amplitude M can be obtained through the matrix element of the operator T or the T-matrix element. The results are given in Eqs. (A16,A17). By inserting these results for invariant amplitudes into Eq. (2), the differential cross section can be put into the form where P z 1 is a function of nulcear transverse momenta which approaches P z A1 if all nulcear transverse momenta are vanishing, L µν;σρ is the lepton tensor in Eq. (A21), and we have used variables ∆ 1T and ∆ 2T with to replace P 1T , P 2T , P 1T and P 2T . We have used the Wigner functions for photons which are similar to those in Ref. [85,86]. We will discuss them carefully in the TMD factorization formalism in Sec. V.
We can also integrate Eq. (4) over b 1T and b 2T and express the differential cross section ×L µν;σρ (p 1 , p 2 ; p 1 − P 1 + P 1 , p 2 − P 2 + P 2 ; k 1 , k 2 ) We can also integrate Eq. (7) over b T to obtain where we have used P z 1 = P z A1 as the solution to the energy conservation equation with the conditions P 1T = P 1T and P 2T = P 2T , so the on-shell momenta of nuclear wave-packets now become P 1 = (E P 1 , P 1T , P z A1 ) and P 2 = (E P 2 , P 2T , −P z A1 ). We have also defined as the TMD correlation function for unpolarized nuclei.
In this section, we have given a general form for the impact parameter dependent cross section. The cross sections in Eqs. (7,8) are similar to the TMD factorization for photons.
We see that S σµ (P, p) appear in Eqs. (7,8) as non-perturbative soft correlation functions for photons. In the next section, we will show how to handle soft correlation functions in the classical field approximation.

III. CLASSICAL FIELD APPROXIMATION
In this section, we take the classical field approximation for the photon field A µ (x). We follow the idea in early works [71,75,76] and extend their original formalism by including impact parameters b 1T and b 2T .
We consider collisions of two nuclei in the center of mass frame as shown in Fig. 2.
The matrix elements of photon operators in Eq. (A18) are assumed to take the form with i = 1, 2 labeling two nuclei where denote the classical photon fields in (11), and a µ i denote the components of a µ i for a = p, p , P, P that are perpendicular to u µ i , a i · u i = 0, which are given by with u µ i ≡ γ(v, 0, 0, 1), g µν T ≡ diag(0, −1, −1, 0), and ∆ µν u i ≡ g µν − u µ i u ν i . Note that a µ i are actually the spatial components of a µ i in the comoving frame of the nucleus labeled by i. So a µ i can be decomposed as a µ i = (a i · u i )u µ i + a µ i . The explicit forms of a µ i for a = p, p are given by a µ i under the conditions a i · u i = 0 Now we can derive the cross section from Eq. (A18) for collisions of two equal nuclei (Z 1 = Z 2 = Z and v 1 = v 2 = v) by using Eqs. (11,12) in Eq. (A18) and completing the integrals over p 1 , p 2 , p 1 · u 1 and p 2 · u 2 . The result is where p 1 , p 2 , p 1 and p 2 are given by Eq. (14), and p i = p i − P i + P i for i = 1, 2. Note that p i , p i , P i and P i are momenta perpendicular to the four-velocity u µ i which have three independent components, while quantities with index T are transverse ones perpendicular to the beam direction. We denote the 0-components of photon momenta p i and p i as ω i and ω i for i = 1, 2, respectively, then the conditions In the derivation of (15), the four-momentum integrals are

Completing the inetgrals over
where we have used P z 1 = P z A1 as the solution to the energy conservation equation with the conditions P 1T = P 1T and P 2T = P 2T . The result of (16) is consistent with the results in Ref. [71,75].
On the other hand, we can carry out the integrals over where Note that p 1T and p 2T are now free variables. We see in the above formula that the couplings of impact parameters with nuclear momenta have been converted to those with photon momenta by integration over nuclear transverse momenta P 1T and P 2T . Completing the integrals over b 1T and b 2T will gives a term in the inetgrand.
If we make a similar ansatz to (A9) for transverse momentum amplitudes we can simplify Eq. (17) significantly, where G T (x 2 ) may differ from G(x 2 ) for longitudinal momenta but with similar behavior: it is a positive function with G T (0) = 1 and decreases rapidly with growing x 2 . Then the integrals over P 1T and P 2T in Eq. (17) can be completed and we obtain where we have used P z 1 = P z A1 as the solution to the energy conservation equation with the conditions P 1T = −P 1T and P 2T = −P 2T , and ω i = ω i for i = 1, 2.
We emphasize that one of our main results in this work is the cross section (20) with impact parameter dependence which encodes the information of photons in transverse phase space. This is the basis for the derivation of EPA results in Sect. IV at the relativistic limit.
We will show that Eq. (20) contains all high order contributions of γγ → ll at the tree level.
In Sect. VI, we will compute cross sections based on Eq. (20) and compare our results with experimental data.
We will also parametrize the TMD correlation function S µν in Eq. (9) and implement the TMD factorization formalism in Sect. V.

IV. CONNECTION TO EPA
In this section, we will derive the EPA result from Eq. (20). Now we evaluate (17). In order to simply notations, from now on, we resume the use of p 1 , p 2 , p 1 and p 2 in L µν and L σρ * for p 1 , p 2 , p 1 and p 2 respectively if there is no ambiguity. It is convenient to rewrite u 1µ u 2ν Γ µν (p 1 , p 2 ; k 1 k 2 ) in light-cone coordinates where i, j = x, y stand for transverse directions, we have used Eq. (A23), and p ± 1,2 are given by At the relativistic limit v → 1 and γ → ∞, the following power counting rules hold in light-cone coordinate It implies that photons are almost on-shell [71], Following (23), the first, second and third term of (21) tively. Therefore the leading order contribution comes from the first term. The expansion in (21) can also be interpreted as the photon virtuality expansion. Details can be found in Appendix B.
At the relativistic limit, the cross section (20) can be put into a compact form where σ 0 is the leading order contribution from the first term of Eq. (21) and δσ represents the corrections from other terms. In the above formula, i, j, k, l are indices of two transverse directions, and p i 1,2 and p i 1,2 denote the directions (unit vectors) of p 1T,2T and p 1T,2T respectively which play the role of polarization vectors of photons so the summation over i, j, k, l represents that over photon polarizations. Therefore we can define the last two lines of Eq. (20) as a kind of cross section of the photon fusion to produce dileptons σ γγ→ll (p 1 , p 2 ; p 1 , p 2 ) = 1 8ω 1 ω 2ˆd When p 1 = p 1 and p 2 = p 2 , σ γγ→ll (p 1 p 2 , p 1 p 2 ) becomes the ordinary cross section for production of lepton pairs by two photons The leading order cross secction from Eq. (20) is put into the form If we approximate p iT ≈ p iT for i = 1, 2, we have Then Eq. (28) can be written in a compact way where the photon flux spectra from A 1 and A 2 are for i = 1, 2. The impact parameter structure in Eq. (30) is similar to the space-dependent photon flux defined in Ref. [85,86,89]. The approximation that leads to the result of (30) is called the generalized EPA or gEPA [71].
We can complete the integrals over b T , b 1T , and b 2T in Eq. (30) to obtain where n A1 and n A2 are the photon fluxes generated by A 1 and A 2 respectively and are given by for i = 1, 2. Note that the photons are very close to real ones as shown in Eq. (24). The approximation that leads to the result of (32) is called EPA.
About δσ in (25), the cross section beyond the EPA, with the same approximation as for Eq. (30), we obtain where I is defined through where L µν ≡ L µν (p 1 , p 2 ; k 1 , k 2 ) and L σρ * ≡ (p 1 , p 2 ; k 1 , k 2 ) which are given by (A22). Note that the first term inside the square brackets gives σ 0 . The explicit form of I is given by So we see that the cross section (17) contains not only the result of EPA, but also the result beyond the EPA.

V. CONNECTION TO TMD FACTORIZATION FORMALISM
In this section, we will discuss the connection of our results to those from the TMD factorization formalism. Here we just give a concise comparison between the results from our approach and those from the TMD factorization formalism. The details of the TMD factorization formalism applied to UPC can be found in Refs. [80,81,87,90]. The recent developments in the impact-parameter dependent cross sections in terms of photon Wigner functions are given in Ref. [85,86].
We work in the light-front formalism and discuss photon Wigner functions based on Ref. [85,86]. The gauge invariant TMD correlation function of EM fields [91] for an unpolarized nucleus moving in +z direction is defined as Similarly one can define the TMD correlation function W µν;ρσ (P 2 , p 2 ) for the nucleus moving in −z direction. In Sect. III we use the Lorentz gauge. Both W µν;ρσ and the cross section are gauge invariant, therefore, in this section we follow the standard TMD factorization formalism to choose the light-cone gauge, A ± = 0, for nuclei 1 and 2 moving in ±z direction.
In the light-cone gauge the classical solutions to EM fields read where the upper/lower sign corresponds to nuclei 1 and 2 moving in ±z direction, as an comparison, the result in the Lorentz gauge is given in Eq. (11). One can verify that the TMD correlation functions are related to the correlation functions in Eq. (9) as W +ν;+σ (P 1 , p 1 ) = (p + 1 ) 2 S νσ (P 1 , p 1 ), Note that there is a different sign from Ref. [81].
In the order of twist-2, we can parametrize correlation functions for nuclei 1 and 2 of the same species asˆd where x 1 = p + 1 /P + 1 and x 2 = p − 2 /P − 2 denote light-cone momentum fractions. The TMD distribution f γ (x 1 , p 2 1T ) represents the usual unpolarized photon distribution, while h γ T (x 1 , p 2 1T ) represents the distribution function of linearly-polarized photons in an unpolarized nucleus [91]. The cross section (8) up to twist-2 can be expressed by where we have assumed E P 1 ≈ E A1 and E P 2 ≈ E A2 and performed the integrals over P 1T and P 2T .
The TMD distributions f γ (x, p 2 T ) and h γ T (x, p 2 T ) are non-perturbative in nature and cannot be derived directly from the perturbation theory. In the classical field approximation based on the classical solution (38) we obtain which are consistent with the photon flux n A (ω, p T ) in Eqs. (33). The details for the derivation of the above result can be found in Appendix C. Inserting Eq. (42) into Eq. (41) yields, It is straightforward to prove that the above result is equivalent to Eq. (30).
In the classical field approximation, the twist-3 correlation functions are given bŷ dp − 1 W +i;+− (P 1 , and the twist-4 correlation function iŝ No higher order (twist-3 and twist-4) correlation functions contribute to the tree level diagram of γγ → ll.
The total cross section in the TMD factorization formalism reads, where with I given by Eq. (36). It is remarkably that σ twist n is exactly δσ in Eq. (34). Therefore, since we have already shown that σ twist 2 = σ 0 in Eq. (43), the total cross section (46) in the TMD factorization formalism is consistent with the EPA result (25).
Thus we have shown the equivalence of EPA and TMD factorization formalism. We would like to emphasis that our main result (20) includes corrections beyond the EPA or twist-2. Note that, in principle, one also needs to consider the Sudakov factor in the TMD factorization theorem [80,81,86,87]. It is still a chanllenge to add the effective Sudakov factor into our framework, which we reserve for a future study.

VI. NUMERICAL RESULTS
In this section, we will present numerical results for the cross section and compare them with STAR measurements at 200 GeV Au+Au collisions at RHIC [67,69]. The cross section can be expressed as a multi-dimension integral over independent variables The derivation of the above formula as well as the explanation of all variables are given in Appendix D. Here we have introduced a factor P 2 (b T ) which will be explained shortly. The high-dimension integral of the cross section (48) is a challenge in the numerical calculation.
In this paper we use the ZMCintegral package [99,100] which has been applied to calculation of high dimensional integrals in heavy ion collisions [101].
For the numerical calculation we choose the nucleus form factor [73,80,81,102] as where a = 0.7 fm, R A = 1.2A 1/3 fm is the nucleus radius with A being the number of nucleons, and ρ 0 = 3A/(4πR 3 A ) is the nucleon number density to make F (q = 0) = 1 [102]. One can also choose R A = 1.1A 1/3 fm as in Ref. [80,81].
Another factor that should be taken into account is that two nuclei may undergo mutual Coulomb excitation and emit neutrons [81]. To consider such effects, we need to introduce an extra factor P 2 (b T ) in the calculation of the cross section, where P(b T ) is the probability of emitting a single neutron from an excited nucleus and can be parametrized as [103], where N γ denotes the number of photons that can be absorbed by a nucleus, and w is defined Note that the probability of emitting more than one neutron is highly suppressed, so multiple neutron emission is usually neglected. The formula (50) has been widely used [72,76,78,[103][104][105]. Other choices are P(b T ) w [76] and P(b T ) = w exp(−w) [76,78,103,104,106].
In this work, we choose (50) in our numerical calculation.
We follow STAR experiments [67,69] to choose that the transverse momentum of the electron and that of the positron is greater than 200 MeV but the transverse momentum of the electron-positron pair is less than 100 MeV.
In Table I are the results from Ref. [74] (times a factor 9.1), XnXn [67] and STARLight [73], respectively. The data points are from STAR measurements [67], while the shaded areas stand for the experimental uncertainty.
In Fig. 3, we plot differential cross sections as functions of the invariant mass of the lepton pair M ee . The STAR data [67] as well as the results of some models [73,74]  The differential cross sections as functions of lepton pair's transverse momenta are presented in Fig. 4. We compare our results with STAR data [67], the results of Ref. [74] and STARLight [73]. Our results using Eq. (20) are in a good agreement with the data in the low transverse momentum region, while they give a little larger values than the data for transverse momenta larger than 0.02 GeV. The difference may come from possible higher order corrections [79,107].
In with and without contributions of magnetic fields, respectively. The data points are from STAR measurements [67].
of M ee . Our results are consistent with experimental data, so we do not need to consider extra contributions from magnetic fields at the early stage. According to Refs. [85,87,90], the Sudakov form factor may play a role to the broadening of the transverse momentum.
That difference between our results and experimental data may come from the choice of parameters. In Fig. 6, we show the results from different P(b T ). For the invariant mass spectrum of the lepton pair, the result with P(b T ) w [76] matches the data, but the transverse momentum spectrum of the lepton pair is much higher than the data. The result with P(b T ) of Ref. [78] under-estimate both invariant mass and transverse momentum spectra. These results imply that we need to study the parameter dependence systematically, which we leave for a future study.  The numerical results for the differential cross sections with respect to invariant mass and transverse momentum of the lepton pair are in a good agreement with STAR data. One approximation we have made in our results is that wave packets are assumed to be narrow so that they are similar to plane waves. In the future this approximation will be relaxed so that the effects from broader wave packets can be studied. Other effects such as Sudakov factor as well as the dependence on various parameters are expected to be investigated in the future.

Appendix A: Derivation of impact-parameter dependent cross section
In this appendix, we will derive the cross section which depends on the impact parameter b T . We assume the wave functions of two colliding nuclei moving in ±z direction (see Fig.   2) can be written in the form of wave packets following the argument of Ref. [108], where P 1 = (E P 1 , P 1 ) and P 2 = (E P 2 , P 2 ) with E P 1 = P 2 1 + M 2 1 and E P 2 = P 2 2 + M 2 2 are on-shell momenta of two nuclei with masses M 1 and M 2 , φ(P 1 ) ≡ φ(P 1 − P A1 ) and φ(P 2 ) ≡ φ(P 2 − P A2 ) denote the amplitudes of momentum states which are centered at the nuclear momenta P A1 = (0, 0, P z A1 ) and P A2 = (0, 0, −P z A1 ) respectively. They satisfy the normalization condition (2π) −3´d3 k|φ(k)| 2 = 1. Here we have introduced the impact parameter b µ = (0, b T , 0) into the state of A 1 , which is the transverse distance between the centers of two nuclei.
The initial state of nuclei can be written as while the final state is assumed to be in momentum states instead of wave packets where the indices 'in' and 'out' stand for the in-and out-states at t → ±∞, respectively, and X f with momenta K f denote all other particles that are produced in collisions.
The cross section reads (A4) Using Eqs. (A2,A12) and completing the integration over P z 1 and P z 2 in the above formula, we obtain where ∆ 1T ≡ P 1T − P 1T , M denotes the invariant amplitude defined through the T-matrix element. By rewriting the delta function for transverse momenta as and adding an integralˆd The cross section (A5) can now be put into the form where we have used ∆ 2T ≡ P 2T − P 2T . Now we look at the momentum amplitudes φ(P 1 ) and φ(P 2 ) in nuclear wave functions as wave packets. Normally the total cross section does not depend on the form of φ(P 1 ) and φ(P 2 ) [108]. A natural choice for φ(P 1 ) and φ(P 2 ) is that they can be factorized as φ(P i ) where φ z (P z i ) are distributions of longitudinal momenta that P z 1 and P z 2 are centered at P z A1 and −P z A1 respectively, and φ T (P iT ) are distributions of transverse momenta that P iT are centered at zero. As a simple ansatz we assume the longitudinal part takes the form where G(x 2 ) is a positive function with G(0) = 1 and decreases rapidly with growing x 2 , and the factor 1/4 inside G-functions is by convention. So we can carry out the integrals over P z 1 and P z 2 to remove two delta-functions and setting P z 1 = 2P z A1 − P z 1 and P z 2 = −2P z A1 − P z 2 in the integrand. Then we complete the integrals over P z 1 and P z 2 to remove two delta-functions for longitudinal momenta and energieŝ dP 1z dP 2z δ(P z where we have used P z 1 = 2P z A1 − P z 1 and P z 2 = −2P z A1 − P z 2 , and P z 1 = −P z 2 are solved as functions of transverse momenta from the energy conservation. Furthermore the condition as a function of transverse momenta. Completing the integrals over P z 1 , P z 2 , P z 1 and P z 2 in Eq. (A8) by using (A9) and (A10), we obtain Eq. (2) for the cross section. Now we deal with the invariant amplitude M through the matrix element of the operator T (T-matrix element) as with the T-matrix element being parametrized as where M denotes the invariant amplitude of the process. By definition, the T-matrix element can be written in the form where T denotes the time-ordered operator. The lepton part is evaluated as Note that in Eqs. (A16,A17) we have suppressed the spin indices of lepton spinors.
Using Eqs. (A16,A17) in Eq. (2), the cross section is put into the form where we have used the identity and the relation We have also used the lepton part as where L µν and L σρ * are defined by A useful property of L µν (p 1 , p 2 , k 1 , k 2 ) is the following identity [71] p 1µ L µν = p 2ν L µν = 0.
In (A18) we have also reduced the photon matrix element as This is because each term in the second approximate equality can be proved to be identical after change of variables, giving a symmetry factor 4. So one can just take the first term and multiply it by 4 to obtain the photon matrix element. We can rewrite the integrals over p 1 , p 2 , p 1 and p 2 in (A18) as In deriving the above formula, we have converted all photon fields in momentum space to coordinate space as and changed all coordinate variables to X i = (x i + x i )/2 and y i = x i − x i for i = 1, 2, then we carried out the integration over X i after shifting x 1 and x 2 for the photon fields in Inserting (A25) into Eq. (A18), the cross section is rewritten as Equations (A18,A27) are our starting point.
Appendix B: Some useful formula in light-cone coordinates A four-vector or a four-tensor can be written in light-cone variables. For example, the coordinate is written as Minkowski metric tensor becomes with µ = +, −, 1, 2. So we have x µ = g µν x ν = (x − , x + , −x T ). It is convenient to decompose an arbitrary four-vector a µ by light-like Sudakov vectors n µ + and n µ − , satisfying n 2 + = n 2 − = 0 and n + · n − = 1, where a ± = (a 0 ± a 3 )/ √ 2 and a µ T = g µν T a ν with the space-like transverse projector The inner product of two vectors is a · b = a − b + + a + b − − a T · b T . The explicit forms of light-like Sudakov vectors are The four-momenta of colliding nuclei can be written as where M 1,2 and u µ 1,2 are masses and four-velocities of two nuclei respectively and u µ 1,2 are given by with γ 1,2 = 1/ 1 − v 2 1,2 being the Lorentz factor. Then the four-momenta of two nuclei can be written in the form where light-cone variables are For high energy nuclei with v 1,2 ∼ 1, we have P + From the classical photon field (11), we have p i · u i = p i · u i = 0 for i = 1, 2. Then p 1 and p 2 can be decomposed as where At high energies, we have v 1,2 → 1, so it is easy to verify On the other hand, in the rest frame of a nucleus, it is assumed therefore, we have which means photons are almost on-shell.
We assume a collision of two identical nuclei, so we have v 1 = v 2 . To simplify u 1µ u 2ν L µν in Eq. (17), we use Eq. (A23) and obtain where i = x, y denotes two transverse directions. From Eq. (B10) or p z 1,2 /ω 1,2 = ±1/v (the upper/lower sign corresponds to nucleus 1 and 2 respectively), u 1µ u 2ν L µν can be rewritten as where i, j = x, y denote two transverse directions. The result of (B15) is consistent with Ref. [71]. We can also use the light-cone coordinate and obtain (B16) So we evaluate u 1µ u 2ν L µν as It seems that there is an extra factor v 2 in the first term of Eq. (B17) in comparison with Eq.
as the solution to the Maxwell equation. Here u µ = γ(1, 0, 0, v) is the four-velocity of the nucleus satisfying P µ = M u µ . We use the same ansatz as (12) for the matrix element in Eq. (C9) where u µ ≡ γ(v, 0, 0, 1) satisfying u · u = 0. Using the above formula in (C9) we obtain the transverse component where we have used d 4 p = d(p · u)d(−p · u)d 2 p T in carrying out the integral over p , and the constraints by the delta-functions give p µ = p µ .
Then we take an integral over p − for W +i;+j (P, p) and obtain dp − W +i;+j (P, p) = 1 4π 3 M Z 2 e 2 u + F (−p 2 ) −p 2 where p denotes the momentum that satisfies p · u = 0, in Sect. III we denote it as p as defined in Eq. (14) but here we suppressed the bar to simplify the notation. Comparing with Eq. (C5) we can extract × [u 1σ u 2ρ L σρ * (p 1 , p 2 ; k 1 , k 2 )] (2π) 4 δ (4) ( where p 2T = p 1T + p 2T − p 1T , ω 1 = ω 1 and ω 2 = ω 2 . Note that the photon momenta p 1 , p 2 , p 1 , p 2 are momenta that satisfy p i · u = 0 and p i · u = 0 for i = 1, 2, in Sect. III we denote them as p 1 , p 2 , p 1 , p 2 as defined in Eq. (14) but here we suppressed the bars to simplify the notation. Then we can complete the integrals over ω 1 , ω 2 and p 2T to remove the delta functions, fixing ω 1 and ω 2 as and fixing p 2T and p 2T as p 2T = k 1T + k 2T − p 1T and p 2T = k 1T + k 2T − p 1T . This gives the cross section of the form σ = Z 4 α 2 64π 4 γ 4 v 2ˆd y 1 d 2 k 1T dy 2 d 2 k 2T where we have used the rapidity and transverse momentum as independent variables for lepton momenta so that the lepton momenta k 1 and k 2 can be expressed as Here the rapidity and transverse mass are defined as Then we have d 3 k/E k = dyd 2 k T . Now we choose independent integration variables of the cross section. The impact parameter is written as The transverse momentum shift is defined as so the integral over p 1T can be replaced by ∆ T . Since the invariant mass and transverse momentum spectra of lepton pairs were measured in experiments, we should define the four-momentum of the lepton pair P µ ee = k µ 1 + k µ 2 = (P 0 ee , P T ee , P z ee ) = M 2 ee + (P T ee ) 2 cosh Y ee , P T ee cos φ ee , P T ee sin φ ee , where Y ee is the rapidity of the lepton pair, and M ee is the invariant mass of the lepton pair given by M 2 ee = P 2 ee = 2m 2 e + 2k 1 · k 2 = 2P ee · k 2 .
Instead of k µ 1 and k µ 2 , we choose P µ ee and two variables of k µ 2 as independent variables since k µ 2 satisfies the constraint (D9). Two variables of k µ 2 are chosen to be the pseudo-rapidity η k2 and the azimuthal angle φ k2 of the transverse momentum, so k µ 2 can be expressed by k µ 2 = m 2 e + k 2 2T cosh 2 η k2 , k 2T cos φ k2 , k 2T sin φ k2 , k 2T sinh η k2 .
In summary, we can choose following integration variables (Y ee , P T ee , φ ee , M ee ), (η k2 , φ k2 ), (p 1T , φ p1 ), (∆ T , φ ∆ ), (b T , φ b ), where we used the parenthesis to enclose variables from the same source. In terms of these independent variables, the cross section (D3) can be rewritten in the form of Eq. (48).