Studying Coulomb correction at EIC and EicC

We study the gauge link contribution to the dipole type transverse momentum dependent distributions of coherent photons, which is conventionally referred to as the Coulomb correction. We further propose to search for the evidence of the Coulomb correction in the Bethe-Heitler process in eA collisions at EIC and EicC.


II. THE COULOMB CORRECTION TO THE PHOTON TMDS
In the Bethe-Heitler process, the incoming electron multiple rescattering off the boosted Coulomb potential can occur either before emitting a photon or after a photon being radiated. At low virtuality, the exchanged photons coherently couple with charged heavy ion as a whole. The multiple coherent Coulomb scattering is much more pronounced in eA collisions than that in ep collisions, because the flux of coherent photons is enhanced by the factor Z 2 . If the calculation is carried out in TMD factorization, the cross section can be expressed as the convolution of the hard part and photon TMD distributions. The imaginary phase accumulated from the final and initial state interactions is summarized into a close loop gauge link in the photon TMD matrix element. In analogy to the gluon TMD distributions [29], the formal operator definition of photon TMDs is given by, (1) where the transverse tensor is commonly defined: δ µν ⊥ = −g µν + p µ n ν + p ν n µ and k 2 ⊥ = δ µν ⊥ k ⊥µ k ⊥ν . Two photon TMDs, f γ 1 and h ⊥γ 1 , are the unpolarized and linearly polarized photon distribution, respectively. U † (0 ⊥ )U (y ⊥ ) and the transverse gauge link which is not explicitly shown here form a close loop gauge link. U (y ⊥ ) is defined as, One should notice that the gauge link here plays the no role in ensuring gauge invariance as photon does't carry charge. As argued above, at low transverse momentum, photons coherently generated by the charge source inside relativistic nuclei dominate the distribution. Both the unpolarized and polarized distributions of coherent photons can be computed with the Weizsäcker-Williams method. If one neglects the gauge link contribution, the photon distributions associated with a boosted Coulomb potential are given by [27,28,30,31], where F is the nuclear charge form factor, and M p is proton mass. The subscript "0" denotes the WW photon distributions. In the small x limit, two photon distributions xf γ 1,0 and xh ⊥γ 1,0 become identical [27,28]. The main purpose of this work is to investigate how the photon distributions are affected by the gauge link. To this end, we first express the gauge potential as, where a photon mass δ is introduced for regulating the infrared divergence. The strength of the field appears in the photon TMD matrix element takes the similar form, where x is the longitudinal momentum fraction carried by photon. The full expression of the photon TMD distributions incorporating the Coulomb correction(the gauge link contribution) is then given by, For a point-like charged particle, the close form solution of the above integration exists. By setting F (q 2 ⊥ ) = 1 and F (q 2 ⊥ + x 2 M 2 p ) = 1, one readily obtains, Inserting these results into Eq.6, One notices that the infrared cutoff scale δ dependence now drops out. Carrying out the integration over y ⊥ and y ′ ⊥ , we arrive at, where 2 F 1 is the hypergeometric function. The unpolarized and the linearly polarized photon TMDs remain the same after taking into account gauge link contribution. The similar relations between the dipole type gluon TMDs have been established in earlier work [32][33][34] for both cases of unpolarized target and transversely polarized target. In the limit Z → 1, the above result is reduced to, which recovers Eq.3 as it should. Furthermore, when k 2 ⊥ ≫ x 2 M 2 p , the photon TMDs are simplified as, This indicates that the gauge link contribution, i.e. the Coulomb correction, to the photon TMDs is vanishing with increasing transverse momentum for the case of point like particle. The photon distributions are altered by the multiple re-scattering effect only in the low transverse momentum region. However, this is no longer true for an extended charge source as shown below. According to Eq. 10, is a function of the single variable |k ⊥ | xMp rather than of two variables |k ⊥ | and x. In Fig. 1, we plot the ratio R = f γ 1 /f γ 1,0 as the function of |k ⊥ | xMp for a point-like particle with the various choices of Z. One sees that the photon TMD is significantly reduced by the Coulomb correction at the low value of |k ⊥ | xMp . In the case of an extended particle, the photon distribution is no longer the function of the single variable |k ⊥ | xMp . The ratio as the function of k ⊥ at different x for a Pb target is displayed in Fig. 1(right). In our numerical estimation, the nuclear charge form factor is taken from the STARlight MC generator [35], where R A = 1.1A 1/3 fm, and a = 0.7fm. This parametrization is very close to the Woods-Saxon distribution. Our numerical results demonstrate that the photon distribution of the charged heavy ion is also suppressed at low k ⊥ due to multiple Coulomb re-scattering. However, in a sharp contrast with the point like particle case, one notices that the ratio exceeds 1 at relatively large k ⊥ . It is also interesting to investigate how the integrated photon distribution is modified by the Coulomb phase. The integration over k ⊥ has to be carried out with extreme caution [11]. For a point like particle, the difference between the integrated dipole type photon distribution(with the Coulomb correction) and the integrated WW photon distribution(without the Coulomb correction) is given by, where f (Zα) ≡ Reψ(1 + iZα) + γ E with ψ(x) = d ln Γ(x)/dx is just the well known universal function derived in the Bethe-Maximon theory [1]. We also numerically test this relation and confirm its validation. This is a quite puzzling result in the sense that photon PDF seems to be process dependent. 1 We will thoroughly explore this issue in a future publication.

III. OBSERVALBES
The photon TMDs with a close loop gauge link can be probed in the Bethe-Heitler process, where xP + k ⊥ is understood as the total momentum transfer via multiple photon exchange. We focus on a specific kinematical region, the so-called correlation limit where the total transverse momentum k ⊥ = p 1⊥ + p 2⊥ of the final state produced particles(γ + e) is much smaller than In such a region, the calculation of the cross section can be formulated either in the CGC framework or in the TMD formalism. The equivalence of the two approaches has been verified for the gluon initiated bremsstrahlung process [32,36]. Obviously, all the analysis can be extended to the corresponding QED process. Since there are two well separated scales in the correlation limit, large logarithm terms arise from unobserved soft photon radiations show up in higher order QED calculations. It is conventional to express the differential cross section in the impact parameter space to facilitate resumming these large logarithms, with µ r = 2e −γE /|r ⊥ |. Here the Sudakov factor e − αe 2π ln 2 P 2 ⊥ µ 2 r takes care of all order soft photon radiation effect up to the double leading logarithm accuracy. The phase space factor is defined as dP.S = dy γ d 2 P ⊥ d 2 q ⊥ , where y γ is the rapidity of the emitted photon. The hard coefficient is given by, where z is the longitudinal momentum fraction of the incoming electron carried by the final state photon. In the above formula, the electron mass has been neglected. This is a very good approximation at EIC and EicC energies. The Fig. 2 displays the two ratios R(q ⊥ ) and R 0 (q ⊥ ) defined as follows, 1 If the differential cross section of forward dilepton production in ultra-peripheral heavy ion collisions is computed in the modern small x formalism, a photon quadruple amplitude shows up. In the correlation limit, the four point function collapses into two point function.
For the Abelian case, Wilson lines and conjugate Wilson lines in the two point function completely cancel out. As a consequence, the gauge link in the corresponding photon TMD is absent. Therefore, photon PDF in this process is free from the Coulomb correction and thus different from the one under consideration. where dσ ep and dσ eA are the exact cross sections of the BH process in ep and eA scatterings respectively. For a comparison, the cross sections dσ 0 ep and dσ 0 eA are computed using the conventional equivalent photon approximation(see Eq.3). The parametrization F (| k|) = 1/(1 + k 2 Q 2 0 ) 2 with Q 2 0 = 0.71 GeV 2 for the proton charge form factor is used for determining the photon distribution of proton. Note that the Coulomb correction to photon distribution of proton is negligible. Therefore, we simply use the WW photon distribution to calculate the cross section in ep collisions. The proton magnetic moment contribution to the BH cross section at low q ⊥ can be neglected. The rapidity y γ in the Fig. 2(right) is defined in the lab frame where electron beam and heavy ion beam energies are 18 GeV and 100 GeV for EIC respectively, while they are 3.5 GeV and 8 GeV for EicC respectively. From Fig. 2, one sees that the ratios R and R 0 are rather different in the most kinematical regions at EIC and EicC energies. If the experimentally measured ratios deviate from these dashed lines(R 0 ) presented in Fig. 2, it would be a clear evidence of the Coulomb correction.
We now turn to study the impact of the Coulomb correction on the polarization dependent observable in the BH process. A cos 2φ azimuthal modulation in the BH cross section is induced by the linearly polarized photons if a virtual photon instead of real one is emitted in the final state. The similar phenomena in QCD has been studied in Ref. [32,36], from which one can readily recover the azimuthal dependent cross section in the QED case, wherek ⊥ = k ⊥ /|k ⊥ | andP ⊥ = P ⊥ /|P ⊥ | are unit transverse vectors. The hard parts take the form, To avoid having to deal with a three scale problem, we restrict to the kinematical region where Q 2 is of the order of P 2 ⊥ . This happens to be the optimal region to observe cos 2φ azimuthal asymmetry as suggested by our numerical estimation. Moreover, as long as Q 2 is sufficiently large, the Coulomb multiple rescattering effect can be neglected for the lepton pair production via the virtual photon decay. 2 We plot the azimuthal asymmetries computed for EIC energy in Fig. 3. Here the azimuthal asymmetries, i.e. the average value of cos 2φ is defined as, cos(2φ) = dσ dP.S. cos 2φ dP.S. dσ dP.S. dP.S. (21) As shown in Fig. 3(right), the asymmetry becomes larger with the increasing photon rapidity. The maximal value of the asymmetry reaches roughly 10%. According to Eq.10, the linearly polarized photon TMD and the unpolarized photon TMD are modified by the multiple Coulomb rescattering effect in the same way, and thus remain identical. If the Sudakov effect were not considered, the azimuthal asymmetry would not be affected by the Coulomb correction. However, the azimuthal averaged cross section and cos 2φ dependent part evolve with the scale P ⊥ following a different pattern. The different initial conditions for the photon distributions would lead to the different cos 2φ asymmetries at higher scale P ⊥ . Fig. 3 displays the cos 2φ asymmetries computed with the photon distributions given in Eq. 6 and the WW photon distributions. One sees that at EIC, the deviation caused by the Coulomb correction is visible though tiny. On the other hand, at EicC, the difference(not shown here) is completely negligible since the evolution effect is much weaker at low energy scale. Therefore, it appears to be not optimistic to observe the Coulomb correction effect via the polarization dependent observable at neither EIC nor EicC.
FIG. 3: The azimuthal asymmetry as the function of q ⊥ (left panel) and yγ (right panel) with and without taking into account the Coulomb corrections for a Pb target at EIC. Q 2 is fixed to be Q 2 = 4 GeV 2 . The asymmetry is averaged over the P ⊥ region [1.5 GeV, 2 GeV]. In the left plot, the emitted photon rapidity yγ is integrated over the region [2, 2.8]. In the right plot, the total transverse momentum q ⊥ is fixed to be 50 MeV.

IV. SUMMARY
In this paper, we performed the detailed analysis of the dipole type photon TMDs associated with a boosted Coulomb potential. Our main focus is on the contribution of the close loop gauge link to photon transverse momentum distributions, which is conventionally refereed to as the Coulomb correction in the study of strong field QED. Due to the large Z enhancement, the Coulomb correction(or gauge link contribution) alters transverse momentum distributions of photons substantially for a charged heavy ion target, as compared to the Weizsäcker-Williams photon distribution. The photon TMDs under consideration can be accessed in the BH process. Our numerical results show that it is promising to observe the Coulomb correction at EIC and EicC. The investigation of the Coulomb correction in the BH process will offer us a clean way to test the TMD formulation of initial/final state multiple re-scattering effects, and would be beneficial for deepening our understanding of the gauge link contribution in QCD processes. Moreover, the accurate account of the Coulomb correction to the BH process is also important for the determination of luminosity at EIC and EicC.