Revisiting unitarity corrections for electromagnetic processes in collisions of relativistic nuclei

Unitarity corrections to several electromagnetic processes in collisions of relativistic heavy nuclei are considered. They are due to the unitarity requirement for the S-matrix and correspond to the exchange of light-by-light scattering block between colliding nuclei. We obtain improved results for the corrections to e+e- and mu+mu- pair production, as well as new results for unitarity corrections to the production of photons via virtual Compton and virtual Delbruck scattering. These corrections can be numerically large; e.g., the mu+mu- pair production cross section is reduced by about 50% and the nuclear bremsstrahlung by about 15 to 20%.


Introduction
The subject of this paper are so-called unitarity corrections, which form a conceptually interesting class of corrections for quantum electrodynamic (QED) processes. The unitarity corrections come from the unitarity requirement for the S-matrix and are relevant for processes in which the lowest order amplitude is large, i.e., in which a large number of photons and/or real electron-positron pairs are typically produced. One class of such processes is bremsstrahlung and lepton pair production in ultrarelativistic heavy-ion collisions.
In order to put the current investigation into perspective, let us briefly recall that heavy-ion collisions definitely concern matter under extreme conditions. The impact parameters ρ relevant for all cross sections under study in the current paper are in the range of the electron Compton wavelength, ρ ∼ λ e = mc = 386 fm .
Now, the typical electric fields, as seen in the laboratory frame, generated by a particle moving at a speed characterized by a relativistic factor γ, are of the order of where E cr is Schwinger's critical field strength. The quantity Zα γ assumes values in the range of ∼ 60 for RHIC a Corresponding author: serbo@math.nsc.ru and ∼ 1800 for the LHC, illustrating that the flash field accompanying the nuclei may well exceed the critical field by several orders of magnitude, for very small spatial regions and very small times. In view of the huge pair production and photoproduction cross sections encountered under these conditions, it is natural that the importance of unitarity corrections has been recognized for the first time within the production of electron-positron pairs in collisions of heavy nuclei (see Ref. [1] and the reviews [2,3]). The unitarity correction for the one electron-positron pair production process has been calculated in Ref. [4] and found to be about 3 ÷ 4 %. Estimates of unitarity corrections for the µ + µ − single-pair production process have been obtained in Ref. [5]; in this case the unitarity correction is found to be large (∼ 50 %). Let us consider these corrections conceptually using the process (3) as an example. In this case the lowest order Feynman diagram is represented by Fig. 1, while the diagrams of the type depicted in Fig. 2 correspond to the unitarity correction. These diagrams include blocks of virtual light-by-light scattering via an electron loop, whose imaginary part corresponds to the production of electronpositron pairs by the Cutkosky rules.
For Z 1 Z 2 α ≫ 1 and γ ≫ 1, it is possible to treat the nuclei as sources of an external field and to calculate the probability of n-pair production P n (ρ) in collisions of two Fig. 1. Feynman diagram for e + e − pair production in heavyion collision (first Born approximation). Bold lines denote nuclei, thin lines denote electrons. nuclei at a fixed impact parameter ρ [6]. The sum over n of the probabilities P n (ρ) for n-pair production must be unity. The unitarity requirement is fulfilled by the Poisson distribution P n (ρ) =n n e n! e −ne , whose sum over n gives one. Heren e ≡n e (ρ) is the averaged number of produced pairs at a given ρ, and the factor exp(−n e ) is the vacuum-to-vacuum transition probability Roughly speaking, the probability for producing one pair, given in perturbation theory byn e , should be modified to readn e exp(−n e ), and this correction is not small for an appreciable value ofn e . This means that also the cross section for the one-pair production σ 1 should be multiplied by an appropriate factor exp(−n e ) in the integral over the impact parameter, which corresponds to the following replacement, where, finally, is the unitarity correction to the one pair production cross section. In Refs. [4] and [5], rather rough approximations were used for the functionn e (ρ). Here, our intention is to use improved approximations for these functions. Thus, the aims of the present paper are: (i) to revise the problem of the impact-parameter dependent pair production probabilityn e (ρ), (ii) to update the unitarity corrections for the previously calculated processes (3) and (4), (iii) to calculate the unitarity correction for photon emission at nuclear collisions In the latter case, the probability for the production of one and only one photon in a heavy-ion collision is modified by the necessity of suppressing the possibility of simultaneous production of electron-positron pairs. In the impact-parameter representation, this implies that the cross section for photo-production σ γ should also be multiplied by the factor exp(−n e ). This corresponds to the replacement where P γ (ρ) is the probability to emit a photon in the collision of two nuclei at a given impact parameter ρ. Therefore, the unitarity correction in this case is given by the expression It should be mentioned that there are two different mechanisms for the photon emission in nuclear collisions: the ordinary bremsstrahlung via virtual Compton scattering (see Fig. 3) and the recently considered (see Refs. [7,8]) emission of photon via virtual Delbrück scattering as illustrated in Fig. 4. Both of the above mentioned corrections to e + e − pair production and to photoproduction correspond to the exchange of virtual light-by-light scattering interactions between the nuclei. Each block of light-by-light scattering brings in an additional factor (Z 1 α Z 2 α) 2 in the amplitude of the corresponding process, and therefore, such corrections can be omitted for the scattering of light ions, for muon-nucleus or electron-nucleus scattering. We definitely need Z to be large for the correction to be appreciable, and consequently this paper is focused on heavy-ion collisions.
We organize our paper as follows. In Sec. 2, we briefly recall relevant physical parameters for modern heavy-ion machines and discuss new approximation for the function n e (ρ). Section 3 is devoted to the actual calculation of the corrections. Specifically, we consider unitarity corrections to e + e − and µ + µ − pair production in Secs. 3

Toward a Revised Representation of the Impact-Parameter Dependent Pair Production Probability
Recently, electromagnetic processes in ultra-relativistic nuclear collisions have found strong and partially renewed interest in numerous papers (see the reviews [2,3] and references therein). Of topical importance are the RHIC collider and the future LHC Pb-Pb option. It is therefore useful to recall the basic physical parameters of these colliders, namely the charge numbers of nuclei Z 1 = Z 2 ≡ Z and their Lorentz factors γ 1 = γ 2 ≡ γ. These are given in Table 1, which is cited here from Ref. [9]. Many of electromagnetic particle-production processes are of imminent importance for two reasons: they are either "dangerous," e.g. in terms of possible beam losses and background, or they are by contrast quite useful for monitoring some experiments at the RHIC and LHC colliders [10]. To fix the conventions used, we mention that natural units with = c = 1 and with the fine-structure constant α ≈ 1/137 are used throughout the text, and we denote the electron and muon masses by m and µ, respectively.
As it was mentioned in the introduction, the importance of unitarity corrections has been recognized for the first time at the process of electron-positron pair production because a large number of real electron-positron pairs are typically produced in ultra-relativistic heavy-ion collisions. In the lowest QED order (Born approximation) this process is described by the Feynman diagram of Fig. 1. Letn e ≡n e (ρ) be the expected (average) number of pairs to be produced in the collision of two nuclei at a given impact parameter ρ. A closed form of the corresponding expression was obtained in Refs. [11,12] although a complete and consistent interpretation of the expressions found was only given later in Ref. [13]. One issue is that n e (ρ) derived in Refs. [11,12] actually requires a further regularization, which was implemented in Refs. [14,15].
As evident from Eqs. (8) and (11), the functionn e (ρ) is a very important quantity for the evaluation of unitarity corrections. However, the obtained close form forn e (ρ) is, in fact, a nine-fold integral and its calculation is very laborious. Therefore, a simpler approximate expression for n e (ρ) is very desirable.
The properties ofn e (ρ) have been studied in detail in Refs. [4,16]. The corresponding functional form for identical heavy nuclei in the region ρ < γ/m reads Here, γ ≫ 1 is the usual relativistic factor (see Table 1). The analytical expressions for the functions F (x, Z) and G(x, Z) have been obtained in Refs. [4,16] only for large values of the impact parameters, 1 ≪ x = m ρ < γ (see the Appendix): is (independent of Z) has been given in Ref. [16] as a rough indicator of the nonlogarithmic (in γ) term in Eq. (12). We improve this approximation by using the results from a first Born approximation, and a different parameterization has been employed altogether, namelȳ  (16), calculated using the approach outlined previously in Ref. [17] x (for a list of numerical values, see Table 2). There is an obvious connection between the above two sets of functions in the limit of low nuclear charge numbers, First of all, it is reassuring to verify, based on numerical data presented in Refs. [16,17], that the equality F (x, 0) = A(x) is valid in the phenomenologically important interval x = 0.01 ÷ 10 with an accuracy better than 5%.
One can now take the data for F (x, 0) as given in Ref. [16] and use a least-squares method in order to fit G(x, 0) to numerical data in Table 2, assuming the functional form (16). A least-squares fit, assuming the dependence gives the best estimates a = 1.4 and b = 1.9, where the prefactor of the logarithm is fixed by the ratio of the two quantities discussed in Eq. (13). The approximation thus obtained differs from B(x)/A(x) in the phenomenologically most important interval x = 0.02 ÷ 5 by less than 2%.
Based on the deviation of F (x, Z = 0) from F (x, Z) by no more than 25% due to Coulomb corrections for heavy nuclei [16], we expect that the same deviation is valid at comparison of G(x, Z = 0) and G(x, Z). Taking into account that the function G(x, Z) is the subleading term of the relative order of 1/L, we can conclude that the approximate expression which involves the function F (x, Z) from Ref. [16], has an accuracy of the order of 5 %. In the calculations reported below, we use this very expression.

Calculation of Unitarity Corrections
3.1 Unitarity corrections for the e + e − pair production Unitarity corrections for the process (3) have been considered in Refs. [4,18]. Based on Eq. (8), we find the unitarity correction is where the integration limits (minimum and maximum impact parameters) are to be specified below. Since we are interested in so-called "silent events" without any "touching" of the nuclei, the physically allowed minimal value of the impact parameter is where R is the nuclear radius. A priori, the upper limit is ρ max = ∞, but due to fast convergence of the integral we can use as well ρ max = 100/m for a quantitative estimate of the unitarity correction. Indeed, for large ρ, the asymptotics (13) allow an expansion of the exponential for the vacuum persistence amplitude in (10) and in analogous expressions used for the unitarity corrections in this article, and thus there is a sufficiently large negative power of ρ characterizing all integrands for large ρ; we can thus neglect of excessively large impact parameters in the evaluation of all unitarity corrections. As a result, the integration region in the variable x can be safely chosen as Here, m is the electron mass. Using the relevant physical parameters, we find x 0 = 0.0361 for Au, x 0 = 0.0368 for Pb, x 0 = 0.0385 for U and x 0 = 0.0213 for Ca. Finally, the relative magnitude of the unitarity corrections for the considered e + e − pair production process (3) is where the known Born cross section reads [19,20] σ Born = 28 27π σ 0 L 3 − 2.198 L 2 + 3.821 L − 1.632 (24) with For light nuclei with not excessive nuclear charge number [(Zα) 4 L ≪ 1], it is possible to calculate in the Born approximation the following integrals where F (x) ≡ F (x, Z = 0) and G(x) ≡ G(x, Z = 0). They can be use to calculate, for light nuclei, the unitarity correction for light nuclei, the total cross section for the production of two e + e − pairs in collisions of light nuclei, and the leading logarithmic asymptotics for the total cross section σ n for n-pair production with n > 2, Using the parameterization (19), we obtain the following numerical results for the coefficients listed in Eq. (26), which enter the formulas listed in Eqs. The results from the previous investigations in Ref. [4] concern only those coefficients which can be defined exclusively in terms of F (x). The previous results read C 2 = 1.33, C 3 = 0.264, and C 4 = 0.066. These differ from the new results listed in (28), because in the previous investigation, a less accurate representation of F (x) was used, which leads to discrepancies especially when higher powers of F (x) enter the integrands as given in (26).
As an example, using the result (28), we found that the cross section for production of two e + e − pair for the Ca-Ca collisions at the LHC collider (γ = 3700) is For heavier nuclei, one cannot use the Born approximation F (x, Z) ≈ F (x, Z = 0) ≡ F (x) any more. In this case, one has to resort to numerical data given in the Table 1 of Ref. [16] for the heavy collision systems Au-Au and Pb-Pb, and employ the relativistic factors as given in Table 1. Indeed, unitarity corrections for the process Z 1 Z 2 → Z 1 Z 2 e + e − have been considered in Ref. [4] and estimated as δ e + e − = −4.1% for Au-Au at RHIC and δ e + e − = −3.3% for Pb-Pb at LHC, where exactly the relativistic factors as given in Table 1 have been employed. We recall that the ratio δ e + e − has been defined in Eq. (23). Using Eq. (19), we are now in the position to give the new values δ e + e − = −5.0% for the RHIC , (30) δ e + e − = −4.0% for the LHC , which differ from those obtained in Ref. [4] by about 20%.

Unitarity corrections for the µ + µ − pair production
Unitarity corrections for the process (4) have been roughly estimated in Ref. [5]. Based on the considerations leading to Eqs. (8) and (11), we can immediately write down the corresponding formula as where P B (ρ) is the probability to produce muon pair in collisions of two nuclei at a given impact parameter ρ in the Born approximation [the Coulomb corrections to this probability, which correspond to multiphoton exchange of the produced e + e − with the nuclei, are parametrically suppressed due to the large muon mass and can be neglected -see Ref. [5] for detail]. For a simple calculation, we can use an expression for P B (ρ) given in the leading logarithmic approximation (LLA) in Ref. [5]. It reads Depending on the value of ρ, the function Φ(ρ) assumes two different asymptotic forms, as shown in Ref. [5], This expression is valid for large values of ln(ρ/R), that is correct for LHC but not for RHIC. Therefore, we consider below the case of the LHC collider only. Using formulae (31)-(33) we obtain where δ µ + µ − is of course defined as the relative magnitude of the unitarity correction in comparison to the Born cross section σ µ + µ − , in analogy with (23). The roughly tenfold increase of the unitarity correction (34) for muon pair production in comparison to (30) for electron-positron pair production demands a qualitative explanation. Indeed, the importance of the unitarity correction is due to the enhanced contribution of the region of small impact parameters in the impact-parameter dependent muon pair production probability (32). Due to the prefactor 1/ρ 2 in (32), the unitarity correction is logarithmically enhanced as it involves an integration proportional to d 2 ρ/ρ 2 over the range of impact parameter 2 R < ρ < 1/m.

Unitarity corrections for Compton-type photoproduction
To a good approximation, tree-level photon emission by nuclear bremsstrahlung is described by the block Feynman diagrams of Fig. 3. Let the cross section dσ a br and dσ b br correspond to the diagrams of Fig. 3(a) and Fig. 3(b), respectively. Roughly speaking, diagram (a) describes the emission of radiation by the Compton scattering of an equivalent photon, generated by nucleus 1, off nucleus 2, whereas for diagram (b), the situation is reversed. The cross section for photoproduction by nuclear bremsstrahlung then is obtained as the sum because the interference term is small and can be safely neglected.
In the LLA, the cross section dσ a br can be calculated using the equivalent photon approximation, in which it is expressed as follows: Here, dn 1 is the number of equivalent photons emitted by nucleus 1 in the energy interval dω and the impact parameter range d 2 ρ, and dσ C (ω, E γ , E 2 , Z 2 ) is the differential cross section for the Compton scattering off nucleus 2, for an energy E γ of the emitted photon, and an energy E 2 of the second nucleus of charge number Z 2 and mass M 2 . The number of equivalent photons reads with the integration region and For the Compton cross section, we can use the following well-known expression, which is valid for a nucleus approximated by a charged point particle. This approach gives a good approximation at least in the region of not too energetic photons, where the nuclear structure can be safely neglected, and it reads Then we integrate Eq. (36) over ω and write the result in the form dσ a br = dP a (ρ) d 2 ρ , where the differential probability dP a (ρ) assumes the form with the Thomson cross section Formally the probability in Eq. (43) is divergent if integrated over all photon energies. However, using reasonable upper and lower bounds for E γ , the probability remains small even at the minimal impact parameter. According to the parameter region relevant for the equivalent-photon approximation (37), this expression is valid in the dominant region 2R ≤ ρ γ 1 /ω min . Integrating Eq. (42) over this region, we obtain dσ a br = 2 where Now the unitarity correction δ γ , expressed as a fraction of the complete nuclear bremsstrahlung cross section, can be obtained by considering diagram (a) alone, and it can be calculated using The main (logarithmically enhanced) contribution to dσ a unit is given by the impact parameter region 2R ≤ ρ m −1 , and therefore, a simple estimate can be given as A more accurate calculation is based on the direct integration of the vacuum persistence amplitude against pair production that involves the number of produced electronpositron pairs according to Eq. (12), and reads where the convergence of the integral is assured by the asymptotics given in Eq. (13) and the upper limit of 100/m for ρ could have been replaced by ∞. An evaluation based on Eq. (50) gives the following result for E γ = 1 GeV, δ γ = −15% for the LHC .

Unitarity correction for Delbrück-type photoproduction
Photon emission in heavy-ion collisions via virtual Delbrück scattering has recently been considered in Refs. [7,8], where a surprisingly large cross sections was found for this case: σ γD = 14 barn for the RHIC , (52) σ γD = 50 barn for the LHC .
The main contribution to these cross section comes from the photon-energy region Below, we estimate the unitarity correction for the discussed process. The probability P γD (ρ), entered the cross section (11) for the photoproduction via the virtual Delbrück scattering, can be easily obtained as a function of the impact parameter ρ in the dominant range 1 ≪ mρ ≪ γ 2 , but the unitarity correction is mainly given by integration in the region mρ ∼ 1. Therefore, while we estimate the unitarity correction here, we stress that a more accurate calculation would require a direct evaluation of the Delbrück-type photoproduction probability P γD (ρ) in the range mρ ∼ 1, which was beyond our scope.
For the sake of simplicity, we consider here only the symmetric case γ = γ 1 = γ 2 with identical nuclei Z = Z 1 = Z 2 . The cross section σ γD can be expressed by the integration of P γD (ρ) over the impact parameter via the relation In LLA we can use the differential cross section in the same form as in (36): where σ D (Z) is a high-energy limit of the Delbrück scattering cross section defined according to Eq. (7) of Ref. [8] and is the number of the equivalent photons. The factor 2 in right-hand-side of dn γ takes into account two possibilities, corresponding two diagrams in Fig. 4. This expression is valid in a parameter range satisfying the two conditions m/γ ω γ/ρ and 1/m ρ γ 2 /m. After integration (56) over ω, we obtain the probability P γD (ρ) in the form Under the restrictions for which the approximation made in Eq. (56) remains valid, the expression (57) is applicable in the dominant region where again the probability is small. We rewrite (11) in the form and now we can use this formula in order to estimate the relative magnitude of the unitarity correction. The functionn e (ρ) is of the order of unity at ρ ∼ 1/m and given by the expression but drops very quickly at larger impact parameters, with an asymptotic behavior ofn e (ρ) ∝ 1/ρ 2 . Since the function P γD (ρ) also drops at large impact parameters [see Eq. (57)], the main contribution to σ unit γD comes from the region ρ ∼ 1/m, and we can estimate the integral (59) as follows, ∼ −P γD (1/m)n e (1/m) π m 2 . Taking into account the result (57), we obtain the estimate P γD (ρ) ∼ 2Z 2 α π 2 σ D (Z) m 2 L at ρ ∼ 1/m . (62) The relative magnitude of the unitarity correction thus is For the nuclear collisions at modern heavy-ion machines with parameters as listed in Table 1, one can estimate unitarity corrections to the photon emission to be on the level of −5%. Let us emphasize, that the form of this correction (63) is different from that for the lepton pair production discussed in Secs. 3.1 and 3.2.

Conclusions
In this article, we have considered unitarity corrections for e + e − and µ + µ − production in heavy-ion collisions, and for the production of photons by nuclear bremsstrahlung and by virtual Delbrück scattering. The main results of the current investigation can be found in Eqs. (27a) for the unitarity correction to e + e − production for collisions of light nuclei, in Eq. (30) for the same process in heavy-ion collisions in modern colliders with parameters as detailed in Table 1, for µ + µ − collisions in modern colliders [see Eq. (34)], where the unitarity correction is numerically large, and in Eqs. (51) and (63) for the unitarity correction to photoproduction in heavy-ion machines, with allowance for both the ordinary nuclear bremsstrahlung and the virtual Delbrück scattering process.
Our results as presented for electron-positron pair production in Eqs. (27a) and (30) are based on a refined treatment of the vacuum persistence amplitude against multipair production implied by Eq. (19), and they represent an update of results previously presented in Ref. [4] for the same corrections. For µ + µ − production, we update the results of Ref. [5]. For ordinary (Compton-type) and Delbrück-type photoproduction, the results for the unitarity corrections are obtained here for the first time to the best of our knowledge.
Finally, we notice that the estimates given here for the coefficients C, D and E in Eq. (28) also enter the total cross section for the production of two e + e − pairs in collisions of light nuclei [see Eq. (27b)] and the leading logarithmic asymptotics for the total cross section σ n for n-pair production with n > 2 [see Eq. (27c)]. From a phenomenological point of view, it is important to remark that all unitarity corrections reduce the one-photon or one-pair production cross sections, and that they can be numerically large [see Eqs. (34) and (51)].