Lepton Pair Production Through Two Photon Process in Heavy Ion Collisions

PHYSICAL REVIEW D 102, 094013 (2020)

Lepton pair production through two photon process in heavy ion collisions

Spencer Klein ,1 A. H. Mueller,2 Bo-Wen Xiao,3,4 and Feng Yuan1 1Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 2Department of Physics, Columbia University, New York, New York 10027, USA 3Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China 4School of Science and Engineering, The Chinese University of Hong Kong, 518172, Shenzhen, China

(Received 29 June 2020; accepted 28 October 2020; published 13 November 2020)

This paper investigates the electromagnetic production of lepton pairs with low total transverse momentum in relativistic heavy-ion collisions. We estimate the initial photons’ transverse momentum contributions by employing models where the average transverse momentum squared of the incoming photon can be calculated in the equivalent photon approximation. We further derive an all order QED resummation for the soft photon radiation, which gives an excellent description of the ATLASdata in ultraperipheral collisions at the LHC. For peripheral and central collisions, additional pT -broadening effects from multiple interactions with the medium and the magnetic field contributions from the quark-gluon plasma are also discussed.

DOI: 10.1103/PhysRevD.102.094013

I. INTRODUCTION momentum squared that a parton acquires in the medium of length L. The experimental study of the medium related There have been strong interests in the electromagnetic p -broadening effects of the jet is an important step forward process of dilepton production in two-photon scattering T in heavy-ion collisions in the last few years [1–5]. These to clarify and understand the underlying mechanism for the experiments have made great efforts to select the dilepton jet energy loss in heavy-ion collisions. In the last few years, events through the kinematic constraints where they are there have been a lot of progresses in the understanding of the produced by the purely electromagnetic two-photon reac- pT-broadening effects in dijet, photon-jet, and hadron-jet – tion: γγ → lþl−. These experimental results have attracted productions in heavy-ion collisions [9 14]. It was found quite a lot of attention in the community because they may the vacuum Sudakov effects dominates [9] the transverse provide a potential probe for the electromagnetic property momentum broadening of dijets for the typical dijet kinemat- of the quark-gluon plasma created in heavy-ion collisions. ics measured at the LHC [15,16]. On the other hand, in The medium effects being considered include the medium the RHIC energy regime, the quark-gluon-plasma medium effect is comparable to the Sudakov effects, and the medium induced transverse momentum (pT) broadening from multiple interaction effects and/or the magnetic field effects pT-broadening has been observed in the measurements of from the medium. The confirmation of either effect would hadron-jet correlation [11,17] by the STAR collaboration. We be an important pathway in future explorations of the expect future measurements at both the LHC and RHIC will electromagnetic properties of the quark-gluon plasma. allow us to gain further important and quantitative informa- Furthermore, the physics related to the acoplanarity of the tion on the medium transverse momentum broadening effect. dilepton pair is very similar to that in dijet productions To be able to probe in-medium or magnetic effects in described in QCD. One popular description of the jet medium peripheral collisions, it is important to have a baseline interactions in QCD is the Baier-Dokshitzer-Mueller-Peigne- measurement. Ultraperipheral collisions (UPCs), where the Schiff (BDMPS) approach [6–8]. In addition to jet energy nuclei do not interact hadronically (roughly, with impact parameter b>2R where R is the nuclear radius), can loss, the BDMPS formalism also predicts the pT broadening A A effects, since these two effects are physically related. In the provide this baseline data. UPCs encompass both photo- BDMPS formalism, qLˆ characterizes the typical transverse production and two-photon interactions [18–22] and have a long history. Using the so-called equivalent photon approximation [23], which is also known as the Weizsäcker Williams method [24,25], the photon distribu- Published by the American Physical Society under the terms of tion from a relativistically moving charge particle can be the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to computed from its boosted electromagnetic fields. Photons the author(s) and the published article’s title, journal citation, can strike the oppositely moving nucleus, or they can and DOI. Funded by SCOAP3. interact with the electromagnetic field of the other nucleus,

2470-0010=2020=102(9)=094013(22) 094013-1 Published by the American Physical Society KLEIN, MUELLER, XIAO, and YUAN PHYS. REV. D 102, 094013 (2020) creating a pair of leptons in the two-photon process azimuthal correlations in heavy-ion collisions to di-lepton γγ → μþμ−. In two-photon fusion processes, the final state angular correlations. Because the experimentally measured has a small total pT, so the leptons are nearly back-to-back. lepton pairs [1] have very small pair pT, the associated The first dedicated study of γγ → eþe− was by the STAR physics is a bit different than for dijet correlation. We focus Collaboration in 2004 [26]. STAR studied final states on three important aspects here. The first one is the consisting of the lepton pair, accompanied by neutrons transverse momentum distribution of the incoming photons in each zero degree calorimeter. The neutrons are assumed in the two-photon processes. In particular, these distribu- to come from the mutual Coulomb excitations of the two tions should depend on the impact parameter of the nuclei by two additional photons, not directly associated collisions. However, as far as we know, there is no first- with the pair production [27]. The additional photons principle calculation on the joint transverse momentum and biased the production toward smaller impact parameters impact parameter dependent photon distribution within the (but still with b>2RA) [28]. original EPA framework. On one hand, we first make some Two-photon fusion production of lepton pairs has been approximation and estimate the average transverse momen- calculated with both the equivalent-photon approximation tum for the photons, which turns out a mild dependence (EPA), where the photons are treated as massless [27], and on the impact parameter. On the other hand, we try to via full lowest-order QED calculations, which included generalize the EPA framework by introducing the photon non-zero virtual photon masses [29]. The two calculations Wigner distribution which contains both the transverse agreed well, except that the EPA calculation predicted more momentum and impact parameter information of the pairs with pair pT < 20 MeV=c. The data was in good incoming photon, and find that this generalization indicates agreement with the calculations, favoring the QED calcu- the impact parameter dependence of the transverse momen- lation at a small pair pT. The STAR paper attributed this to tum broadening could be stronger in the central collisions. a breakdown of the EPA approach, but the issue may have We would like to emphasize that further theoretical devel- been that the EPA calculation did not account for the opments are needed to address this issue. change in single-photon transverse momentum (kT) dis- The second one is the QED Sudakov effects in the lepton tribution due to the biasing toward small impact parameters pair production. Much of this study will be similar to the [30]. This can be seen as broadening the pair pT spectrum. previous studies for dijet azimuthal correlations. However, Newer UPC studies of dileptons have mostly focused even beyond the much smaller QED coupling constant, the on J=ψ or heavier vector mesons, with the two-photon QED Sudakov has its own unique features. The formalism is production treated as a background. However, the ATLAS much simpler, and more importantly, the Sudakov contri- collaboration has recently made a high-statistics study of bution has distinguishable behavior compared to the pri- two-photon production of muon pairs in UPCs, finding mordial kT distribution from the two incoming photon mostly good agreement with an EPA calculation, but fluxes. The theory predictions for the UPC events agree accompanied by a tail containing about 1% of the events, very well with the recent measurement from ATLAS [31], of pairs with relatively high acoplanarity [31]. Acoplanarity which, for the first time, clearly demonstrates the importance is closely related to the pair pT. ATLAS focused on of the Sudakov effects in the moderately larger acoplanarity acoplanarity to reduce their sensitivity to resolution effects. region. Third, we discuss the QED medium effects on the At the same time, several experiments have reported on pair pT-broadening due to the leptons. This part is similar to the two-photon production of lepton pairs in peripheral the BDMPS formalism. With this physics included, we collisions, with b<2RA. The ATLAS collaboration stud- compare our calculation to the experimental data and com- ied pair production in lead-lead collisions and observed ment on the implications of the ATLAS measurements. a significant increase in acoplanarity with decreasing It is quite interesting to compare the pT-broadening centrality, consistent with an increase in pair pT [1]. The effects in QED and in QCD, which helps to provide a STAR Collaboration studied pair production at low pT, new perspective of studying the property of quark-gluon pT < 200 MeV/c in gold-gold and uranium-uranium col- plasma created in heavy-ion collisions. The medium lisions [2], also found evidence for momentum broadening, pT-broadening effect of lepton pairs is the probe to the with pT spectra different from calculations without in- electromagnetic constituents of the quark-gluon plasma, – medium effects [32 35]. This broadening could be attrib- whereas the QCD jet pT-broadening effect measures the uted to in-medium scattering of the produced leptons, strong interaction property. The experimental and theoreti- but first, it is necessary to account for possible changes cal investigations of both phenomena will deepen our to the pT spectrum due to changes in the impact-parameter understanding of the hot medium created in these colli- distribution, as induced by either additional photon sions. More importantly, the lepton pT-broadening effects exchange or hadronic interactions. can be clearly seen in the measurements of ATLAS and To better understand these developments from STAR and STAR [1,2]. This is in contrast to the jet pT-broadening ATLAS, we study the acoplanarity of lepton pairs in heavy- effects in the measurement of dijet azimuthal angle corre- ion collisions. We extend our previous studies of dijet lations, due to strong QCD parton showers [9–13].

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A brief summary of our results has been published nucleus momenta (per nucleon), respectively. The leading earlier in Ref. [36]. The rest of the paper is organized as order differential cross section can be written as follows. In Sec. II, we discuss the γγ → lþl− process in σð ½γγ → μþμ−Þ heavy-ion collisions. In Sec. III, we derive the Sudakov d AB ¼ σ ½ γ ð Þ γ ð Þ 2 2 0 xafA xa xbfB xb b⊥ resummation and also compare to recent ATLAS meas- dy1dy2d p1Td p2T urement on the azimuthal correlation of the lepton pair in ð2Þ × δ ðp⃗1 þ p⃗2 Þ; ð3Þ the UPC events and demonstrate the importance of the T T Sudakov contribution to the lepton pair production in these where xa ¼ k1=PA, xb ¼ k2=PB, Q is the invariant mass for QED processes. In Sec. IV, we derive the medium effects the produced lepton pair, and Q2 ¼ M2 ¼ x x s. x on the p -broadening of leptons. In particular, we compare ll a b a;b T represent the momentum fractions of incoming nucleon to the QCD processes in the BDMPS formalism, and carried by the two incoming photons, and s ¼ðP þ P Þ2 argue that the effects observed by the ATLAS collabora- A B is the total hadronic center of mass energy squared per tion are consistent with the parametric estimate of the incoming nucleon pair. The leading order cross section σ0 p -broadening effects for the QED and QCD processes. T is defined as Finally, after discussing possible magnetic effects in Sec. V, we summarize our paper in Sec. VI. ¯ 2 jM0j σ0 ¼ ; ð4Þ 16π2Q4 II. LEADING ORDER PICTURE As shown in Fig. 1(a), the leading order production of with the leading order amplitude squared lepton pairs comes from the photon-photon fusion scatter- 2ð 2 þ 2Þ ing, which is the main ingredient in the STARLIGHT ¯ 2 2 2 t u jM0j ¼ð4πÞ α ; ð5Þ simulation [32,33]. The total cross section for γ þ γ → e tu μþ þ μ− has a so-called t-channel singularity, and one has to include the lepton mass to regulate the divergence. where t and u are usual Mandelstam variables for the 2 → 2 However, we are interested in high mass di-muon produc- process. To simplify the above expression, we have tion with large transverse momentum for each lepton [31], introduced an impact parameter b⊥ dependent photon flux, where the t-channel singularity is absent. which is also known as the joint photon distribution Let us specify the kinematics by setting the momenta function, of outgoing leptons to be p1 and p2 with individual Z γ γ 2 2 ð2Þ transverse momenta p1 and p2 , and rapidities to y1 ½ ð Þ ð Þ ¼ δ ð − − Þ T T fA xa fB xb b⊥ d b1⊥d b2⊥ b1⊥ b2⊥ b⊥ and y2, respectively. The leptons are produced dominantly j⃗ j¼ γ ð ; Þ γ ð ; Þ ð Þ back-to-back in the transverse plane, with pT × fA xa b1⊥ fB xb b2⊥ ; 6 jp⃗1T þ p⃗2Tj ≪ jp1Tj ∼ jp2Tj. Then, neglecting the lepton γ ð ; Þ masses, we find the following longitudinal momenta for the where fA;B xi bi⊥ are individual photon distributions also incoming photon referred as photon flux in the following discussion. These distributions describe the photon distribution at the trans- P pTffiffiffi y1 y2 verse position b ⊥ with respect to the center of the colliding k1 ¼ ðe þ e ÞPA; ð1Þ i s nucleus. The above factorization can be regarded as a semiclassic picture of heavy-ion collisions, where the P T −y1 −y2 k2 ¼ pffiffiffi ðe þ e ÞP ; ð2Þ impact parameter b⊥ is related to the centrality of the s B collisions. For UPC events, b⊥ is normally larger than 2RA where R is the nucleus radius. ¼ 1 j⃗ − ⃗ j ∼ j j ∼ j j A where PT 2 p1T p2T p1T p2T is much greater Incoming photons also carry non-zero transverse than the total momentum pT, PA and PB are the incident momenta, which has to be included because we are interested in the region with a low transverse momentum imbalance for the lepton pair in the final state. In the leading order picture, the total transverse momentum imbalance of the lepton pair equals the total transverse momentum of two incoming photons. In order to under- stand the final state interaction effects of the lepton pair (a) (b) (c) with the hot medium in heavy-ion collisions, such as the FIG. 1. The leading order and next-to-leading order QED medium pT-broadening, we need to have a precise descrip- Feynman graphs for the lepton pair production in two photon tion for the initial state (of the photons) contributions. In the fusion processes. following, we will investigate these contributions.

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A. Impact parameter dependent photon fluxes We start with the analysis of the impact parameter dependent photon flux. In the last few years, there have been much interest in two-photon processes and photo- production processes in peripheral (non-UPC) heavy-ion collisions [1,2,33,34,37]. For these events, we need to understand the photon flux beyond the simple all impact- parameter picture. The impact parameter dependent photon flux can be written as, see, for example Ref. [38] Z 2 ⃗ 2 γ d k k 2 T ik ·b⊥ T 2 xf ðx; b⊥Þ¼4Z α e T F ðk Þ ; ð7Þ A ð2πÞ2 k2 A

2 where FAðk Þ represents the normalized elastic charge form factor for the nucleus. If both the incoming and outgoing nucleus are required to be on-shell, the virtuality of the 2 ¼ð 2 þ 2 2 Þ radiated spacelike photon is then k kT x mp = 2 ð1 − xÞ with mp the nucleon mass. It reduces to k ¼ 2 þ 2 2 ≪ 1 ≡ 1 kT x mp when x . For pointlike particles (FA ), FIG. 2. Impact parameter dependent photon flux as normalized the photon flux becomes to the flux at b⊥ ¼ RA for a typical kinematics at the LHC with −3 −2 x ¼ 10 and RA¼ 7 fm (upper plot) and at RHIC with x ¼ 10 2 γ Z α 2 2 2 (lower plot), respectively. xf ðx; b⊥Þ¼ x m K ðxm b⊥Þ; ð Þ π2 p 1 p 8 process in Ref. [38] to the differential cross section relevant which is the well-known result in classical electrodynam- to the STAR and ATLAS dilepton measurements. In the ics. The above formula has been applied to understand the revised version of Ref. [30] and a recent paper by Li et al. dilepton production in non-UPC events in heavy-ion [39,40], the so-called QED approach [29] has been applied collisions [34], where the contributions from small impact to compute the dilepton production in two-photon proc- parameter (b⊥ nucleons inside the area in UPCs, see, e.g., the STARLIGHT simulation [32]. denoted by b⊥. It is non-trivial to derive the impact-parameter and transverse momentum dependent photon flux. The main B. Transverse momentum dependence difficulty is that the impact parameter b⊥ is a Fourier in the photon fluxes conjugate variable associated with the photon’s transverse Except for Ref. [29], most previous studies ignored the momentum kT. There will be model dependence to com- inter-dependence between the impact parameter b⊥ and the pute the combined distribution from the classic EM fields. photon’s transverse momentum. A recent attempt to In the following, we will estimate the average transverse address this issue was Ref. [30], which extended the momentum squared, and comment on the difficulty to derivation of the total cross section for the two-photon calculate the combined distribution directly.

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Before we get to the details of the models, we would like to emphasize some important features on the transverse momentum distribution for the incoming photons in the nucleus. First, for b⊥ ≪ RA, effective charge contribution is limited to protons inside b⊥ region, and the average 2 transverse momentum squared is proportional to 1=b⊥. This is an important feature which should be satisfied by all model calculations. Similarly, at large b⊥ ≫ RA, because of the uncertainty principle, the average transverse momen- 2 tum squared is also of order 1=b⊥. Around b⊥ ∼ RA, on the other hand, the average transverse momentum may differ FIG. 3. Average transverse momentum squared multiplied by 2 from the above parametric estimates. However, with the b⊥ for the photon distributions as function of the impact −3 −2 constraints from these generic features (b⊥ ≪ RA and parameter b⊥ at x ¼ 10 and x ¼ 10 . Clearly, both cases h 2 i ∼ 1 2 b⊥ ≫ R ), the model calculations of the average transverse predict a generic behavior of kT =b⊥, in particular for small A −2 momentum squared are very much determined. b⊥. At large b⊥ for x ¼ 10 , the above relation breaks down In Eq. (7) the integral variable k is the photon’s because the average transverse momentum is now comparable to T ¼ 10−2 transverse momentum. We may compute the average xmp with x . transverse momentum squared by multiplying the integrand Z by the products of k , 2 2 ð Þ T ð ; Þ¼ 2 b⊥kTJ2 kTb⊥ ð 2Þ ð Þ Z c2 x b⊥ dkT 2 FA k : 15 4 2α 2 2 0 k 2 Z d kT d kT ð − 0 Þ h i¼ i kT kT ·b⊥ kT γ 2 2 e xf ðx; b⊥Þ ð2πÞ ð2πÞ The above results have a very nice feature: the explicit ð 0 Þ2 dependence of average transverse momentum squared kT · kT ð 2Þ ð 02Þ ð Þ × 2 02 FA k FA k ; 11 h 2 i 2 k k kT on the impact parameter b⊥. The overall behavior h 2 i ∝ 1 2 kT =b⊥ is consistent with the above generic discus- where we manipulate the square of the integral in Eq. (7) sions. In addition, we also find that the additional factor with two separate integrals. The average transverse only has a mild dependence on b⊥. As an example, in momentum squared is computed by weighting the integrals ¼ 10−3 0 Fig. 3, we show the typical case for the LHC at x with the scalar product of kT and k . Certainly, there is −2 T and RHIC at x ¼ 10 . For the latter case, because xmp is some model dependence on how we weigh the integrals. comparable to 1=b⊥ at large b⊥, the above relation will be The above formula reproduces the average transverse modified accordingly. momentum squared for the minimum bias case, which provides a cross-check for the above estimate. We can further simplify the above equation by integrat- C. Photon-photon interaction rate ing out the azimuthal angles. For example, Eq. (7) can be in heavy ion collisions written as, With the photon flux for each of the incoming nucleus, we can calculate the two-photon interaction rate following j ð ; Þj2 γð ; Þ¼4 2α c1 x b⊥ ð Þ Eq. (6), xf x b⊥ Z 2 ; 12 b⊥ Z ð Þ¼ 2 2 δð2Þð − þ Þ where c1ðx; b⊥Þ is defined as xaxbfAB b⊥ d b1⊥d b2⊥ b⊥ b1⊥ b2⊥ Z × x f ðb1⊥Þx f ðb2⊥Þ; ð16Þ 2 b⊥kTJ1ðkTb⊥Þ 2 a A b B c1ðx; b⊥Þ¼ dk F ðk Þ; ð13Þ T k2 A where fA;Bðb⊥Þ represents the individual photon fluxes for with J1ðxÞ the J-type generalized Bessel function. A and B nucleus, following the definition in Eq. (7) and xa;b Similarly, we can work out the integrals over the are the photon longitudinal momentum fractions. Figure 4 azimuthal angles in Eq. (11). With that, we arrive at a shows the relative interaction rate as a function of the very simple result for the average transverse momentum impact parameter for the typical kinematics at the LHC squared, and RHIC. The relative interaction rate increases with the     impact parameter and peaks around b⊥ ∼ ð2 − 3ÞRA region 1 ð ; Þ 2 h 2 ið ; Þ¼ 1 þ 1 − c2 x b⊥ ð Þ before decreasing as the impact parameter increases further. kT x b⊥ 2 ; 14 b⊥ c1ðx; b⊥Þ This is essentially consistent with the results shown in Fig. 2, which indicates that the photon distribution is where c2ðx; b⊥Þ is defined as peaked around the edge of each incoming nucleus.

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collisions. This may be due to different treatments of the impact parameter dependence. Therefore, in the following, we try to generalize the EPA approach by introducing the photon Wigner distribution, which con- tains both the impact parameter and transverse momentum information simultaneously.

D. Generalized equivalent photon approximation: Photon Wigner distribution In order to study the joint transverse momentum and impact parameter dependence in the photon flux, we follow the parton Wigner distribution functions intro- duced in Ref. [43] to introduce the photon Wigner distribution,

Z 2 Z − 2 d Δ⊥ Δ dξ d r⊥ þξ−− ð ; Þ¼ i ⊥·b⊥ ixP ikT ·r⊥ xfγ x; kT b⊥ ð2πÞ2 e ð2πÞ3 e FIG. 4. Relative collision rate in collisions between nuclei A    and B as function of the collision impact parameter b⊥ assuming Δ⊥ r⊥ −3 − þ⊥ 0 xa ¼ xb ¼ 10 for the typical kinematics at the LHC (upper × A; F ; ¼ ¼ 10−2 2 2 plot) and xa xb for RHIC (lower plot). We have    b⊥ ¼ 2R Δ normalized the distributions to that at A. þ⊥ ξ− − r⊥ ⊥ ð Þ × F ; 2 A; 2 ; 17

In minimum bias samples, most of the dileptons are where a momentum difference Δ⊥ has been introduced for produced in UPCs. the nucleus states to obtain the impact parameter depend- We can also estimate the average transverse momentum ence. For convenience and simplicity, we also introduce the squared for the collisions, by adding the contributions so-called generalized transverse momentum dependent from both incoming photons’ average transverse momen- (GTMD) photon distributions in the momentum space, tum squared. Figure 5 shows the total transverse momen- −3 Z − 2 tum squared as function of b⊥ for x ¼ x ¼ 10 and dξ d r⊥ þ − a b ij ixP ξ −ik ·r⊥ −2 Γ ðx; k ; Δ⊥Þ¼ e T x ¼ x ¼ 10 , respectively. We see that the average T ð2πÞ3 a b    transverse momentum does not change dramatically Δ⊥ r⊥ depending on the impact parameter. In particular, from − þi 0 × A; 2 F ; 2 peripheral to central collisions, it remains nearly constant.   

It seems that the average transverse momentum squared r⊥ Δ⊥ × Fþj ξ−; − A; ; ð18Þ contribution from the incoming photons depends mildly 2 2 on the collision centrality. Comparing to the results from Ref. [30], we find that our above results predict quite where we keep the transverse indices open to construct different behavior for the average transverse momentum various GTMDs. For our purpose, we deduce the following squared in the two-photon processes in heavy-ion parametrization,

δij Γijð ; Δ Þ¼ ð ; Δ Þ x; kT ⊥ 2 xfγ x; kT ⊥   i j δij þ kþk− − ð ; Δ Þ ð Þ ⃗ ⃗ 2 xhγ x; kT ⊥ ; 19 kþ · k−

where k ¼ kT Δ⊥=2, fγ represents the usual photon distribution, and hγ stands for the so-called linearly polarized photon distribution. For the convenience of the following derivations, we choose a particular decom- position of the above linearly polarized GTMD hγ.We FIG. 5. Average transverse momentum squared for the two- emphasize that different parametrizations can be applied. photon final state in ion-A on ion-B collisions as function of the When we integrate the Wigner distribution over the impact parameter b⊥ for heavy-ion collisions. impact parameter b⊥, the above reproduces the transverse

094013-6 LEPTON PAIR PRODUCTION THROUGH TWO PHOTON … PHYS. REV. D 102, 094013 (2020) momentum dependent (TMD) photon distributions intro- approximation (EPA). In previous EPA models, such as that duced in Refs. [39,40]. implemented in STARLIGHT, the transverse momentum In the classical limit and analogous to the QCD case [44], dependence was introduced as an average over all impact we can find that the above GTMD photon distribution for a parameter. In the following, we extend this approximation heavy nucleus with charge Ze can be written as by applying the photon Wigner distribution introduced above, which we refer to as the generalized EPA (GEPA). xfγðx; kT; Δ⊥Þ¼xhγðx; kT; Δ⊥Þ In our consideration, two nuclei collide at a particular 2 0 impact parameter b⊥, which represents the transverse 4Z α q⊥ · q ¼ ⊥ F ðq2ÞF ðq02Þ; ð Þ ð2πÞ2 2 02 A A 20 distance between the centers of the two nuclei. To produce q q the lepton pair through a short distance process of γ þ γ → lþl−, we assume the interaction point is at the where FA is nucleus form factor, q⊥ ¼ k− ¼ kT − Δ⊥=2, 0 2 2 2 2 transverse place with b1⊥ and b2⊥ with respect to the q ¼ kþ ¼ k þ Δ⊥=2 and q ¼ q þ x m . ⊥ T ⊥ p nuclei centers. In this configuration, the individual photon As an example, we can obtain the Wigner distribution for flux will depend on b1⊥ and b2⊥, respectively. Clearly, the usual photon flux as the Fourier transform of the above the collision impact parameter can be determined by GTMD, ⃗ ⃗ ⃗ b⊥ ¼ b1⊥ − b2⊥. Therefore, we can write down a generic Z 2 d Δ⊥ factorization formula for dilepton production in AA colli- ð ; Þ¼ iΔ⊥·b⊥ ð ; Δ Þ ð Þ γ þ γ → lþl− xfγ x; kT b⊥ ð2πÞ2 e xfγ x; kT ⊥ : 21 sions through as,

þ − As expected from the property of the Wigner distribution, it dσðAB½γγ → μ μ Þ 2 2 2 is straightforward to check that Eq. (21) reduces to the b⊥ dy1dyZ2d p1Td p2TdZb⊥ distribution in Eq. (7) and the kT distribution in Eq. (10) 2 2 2 2 ¼ d k1 d k2 d b1⊥d b2⊥ after integrating over kT and b⊥, respectively. However, the T T above formulas do not have a clear probabilistic interpre- Γijð ÞΓklð Þδð2Þð − þ Þ tation as the impact parameter and transverse momentum × k1T;b1⊥ k2T;b2⊥ b⊥ b1⊥ b2⊥ ð2Þ dependent photon distributions in the factorized cross × HijklðPTÞδ ðpT − k1T − k2TÞ; ð22Þ section calculations. For example, the above formula predicts an oscillation function of kT depending on differ- where ik represent the polarization indices for incoming ent values of b⊥, shown in Fig. 6. That means the two photons in the scattering amplitude and jl for the distribution function calculated from the above equations complex conjugate of the amplitude, Γij and Γkl stand for is not guaranteed to be positive definite. However, when the two photon Wigner distributions. Again, we will focus one applies the photon Wigner distribution or the photon on the kinematic region with the correlation limit where the GTMD to the two-photon fusion processes, one can total transverse momentum of the lepton pair p⃗ is much demonstrate that the corresponding factorized cross section T smaller than the individual transverse momentum p1 is positive definite, which will be discussed in the following T or p2T. Therefore, we can factorize the cross section into subsection. ij kl the hard part Hijkl and soft part Γ and Γ . In particular, the hard part H only depends on the hard momentum E. Generalized equivalent photon approximation ijkl scale PT. In this paper, we focus on the azimuthal angular In order to take into account the transverse momentum average cross sections for individual lepton, i.e., integrating dependence for the incoming photon fluxes in heavy-ion over the azimuthal angle of the lepton (p⃗1T or p⃗2T). In the collisions, we need to generalize the equivalent photon end, one can show that the hard part can be written as

ij kl ik jl il jk Hijkl ¼ σ0½δ δ − δ δ þ δ δ ; ð23Þ

σ ¼ 1 dσ ¼ 2α2 ðuˆ þ tˆÞ where 0 π dt sˆ2 tˆ uˆ . Substituting the Wigner dis- tribution parametrizations in Sec. IIC, we will obtain the following expression for the differential cross section, Z iΔ⊥·b⊥ dσ 2 2 2 e ð2Þ ¼ σ0 d Δ⊥d k1 d k2 δ ðp − k1 − k2 Þ dΩ T T ð2πÞ2 T T T

× ½x1fγðx1;k1T; Δ⊥Þx2fγðx2;k2T; Δ⊥Þ FIG. 6. kT distribution calculated from Eq. (21) for different þ ð ; Δ Þ ð ; Δ ÞH ð Þ impact parameters with arbitrary unit. x1hγ x1;k1T ⊥ x2hγ x2;k2T ⊥ hh ; 24

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2 2 2 where dΩ ¼ dy1dy2d p1Td p2Td b⊥ and Hhh stands for in Ref. [40], the second term in the differential cross section vanishes when we integrate over either b⊥ or p⊥. ⃗ ⃗ ⃗ ⃗ − ⃗ ⃗ ⃗ ⃗ Therefore, for a particular impact parameter b⊥, it leads H ¼ k1þ · k2−k1− · k2þ k1þ · k2þk1− · k2− ð Þ hh ⃗ ⃗ ⃗ ⃗ ; 25 to an oscillation contribution as function of p⊥. In terms of k1þ · k1−k2þ · k2− incoming photon momenta, after averaging over the azi- ⃗ muthal orientation of the lepton pair, the cross section for where k are defined in the previous subsection. the production of lepton pair in AA collisions at fixed With the results for fγ and hγ in the previous subsection, impact parameter b⊥ can be cast into the above reproduces the results in Ref. [40]. As also noted

Z   2 2 2 02 2 02 dσ 2 2 0 1 ð − 0 Þ Z α Fðk1Þ Fðk1 Þ Fðk2Þ Fðk2 Þ ¼ i k1T k1T ·b⊥ 2 2 2 d k1Td k1T 2 e π 2 02 2 02 dy1dy2d p1Td p2Td b⊥ ð2πÞ k1 k1 k2 k2 σ ½ð 0 Þð 0 Þ − ð 0 Þ ð 0 Þ ð Þ × 0 k1T · k1T k2T · k2T k1T × k1T · k2T × k2T ; 26

Z Z Z 2 ¼ 2 2 þ 2 ⃗ ¼ ⃗ − ⃗ ⃗0 ¼ ⃗ − ⃗0 dσ where ki xi M kiT , k2T pT k1T and k2T pT k1T. ¼ð2πÞ 2 dy1dy2 dp1Tp1T dp2Tp2T It is straightforward to check that the contribution from πdb⊥dΔϕ the linearly polarized photons Hhh, i.e., the second term × xWðx; p ;b⊥Þσ0 ð28Þ inside the square brackets in Eq. (26) vanishes when one T integrates over the impact parameter b⊥.Similarly,after where xWðx; pT;b⊥Þ, which encodes the initial momentum integrating over pT with fixed b⊥ and transforming into the impact parameter space, we find that the second term inside imbalance, is defined as Z the square brackets vanishes again, and Eq. (26) reduces to ðj − 0 j Þ 2 2 0 J0 k1T k1T b⊥ the b⊥ dependent expression derived in the conventional xWðx; p ;b⊥Þ¼ d k1 d k T T 1T ð2πÞ2 EPA. Thus, we can reproduce the previous known integrated ijð 0 Þ klð 0 Þ cross section for lepton pair productions. This implies that × x1f x1;k1T;k1T x2f x2;k2T;k2T the calculation for integrated physical quantities (e.g., the ½δijδkl − δikδjl þ δilδjk ð Þ survival factor [45,46]) will not be affected in GEPA. × ; 29 Usually the GTMD or its corresponding Wigner distri- ⃗ ⃗ ⃗ where pT ¼ p1⊥ þ p2⊥. Furthermore, for UPC, we can bution is not positive definite. It is quite interesting to note 2 ∞ that the cross section is in fact positive definite, since it can integrate over b⊥ from RA to , and find be written as σ d UPC ¼ð4π 2 Þ ð Þσ ð Þ 2 2 RA xWUPC x; pT;b⊥ 0; 30 dσ dy1dy2d p1Td p2T 2 2 2 dy1dy2d p1Td p2Td b⊥ ð Þ where xWUPC x; pT;RA can be similarly written as ik jl ij kl ik jl il jk ¼ σ0G G ½δ δ − δ δ þ δ δ 11 22 11 22 xW ðx; p ;R Þ ¼ σ0½ðG − G ÞðG − G Þ UPC ZT A 12 21 12 21 1 2 2 0 þðG þ G ÞðG þ G Þ ð27Þ ¼ d k1 d k 4πR2 T 1T  A  R 2 ð 2Þ ð 2Þ 0 d k1 F k1 F k2 2 ð2 j − jÞ ik ¼ T ik1T ·b⊥ i k ð2Þ 0 RAJ1 RA k1T k1 where G ð2πÞ e k1Tk2T 2 2 . It is also worth δ ð − Þ − T k1 k2 × k1T k1T 0 ð2πÞjk1 − k j mentioning that the above cross section vanishes when T 1T 11 ijð 0 Þ klð 0 Þ b⊥ ¼ 0 and pT ¼ 0. This can be shown by noting that G × x1f x1;k1T;k1T x2f x2;k2T;k2T 22 12 ¼ − 21 becomes the same as G and G G when one sets × ½δijδkl − δikδjl þ δilδjk: ð31Þ b⊥ ¼ 0 and pT ¼ 0. This essentially explains the existence of the so-called displaced peaks in the measurement of the ð Þ 1 Here we normalize xWUPC x; pT;RA by a factor of 4πR2 to momentum imbalance. A make it dimensionless. The numerical evaluation of the above results are shown F. Numerical results in the GEPA framework in Figs. 7 and 8. As anticipated, there is a dip or a displaced To perform the numerical evaluation, let us first average peak in the distribution of the momentum imbalance pT in over the azimuthal angle of the impact parameter b⊥, then the low pT region for the central collisions, since the the corresponding formula reads xWðx; pT;b⊥Þ function vanishes at b⊥ ¼ 0 and pT ¼ 0.

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possible unfolding and correction of the data due to detector responses can help us reach more conclusive phenomeno- logical findings.

III. SOFT PHOTON RADIATION AND SUDAKOV RESUMMATION Higher-order QED corrections as shown in Fig. 1(b,c) can modify the leading order picture described in the previous section. In the following, we focus on the soft photon radiation, which has strong effects on the pT distributions for the lepton pair, especially at moderately large acoplanarity. At one-loop order, soft photon radiations from the final state leptons dominate the small transverse momen- FIG. 7. Transverse momentum imbalance distribution tum for the pair. The soft photon radiation from the xWðx; pT ;b⊥Þ with different values of impact parameters b⊥. lepton propagator is power suppressed for large transverse 2 2 2 The Gaussian form factor Fðk Þ¼exp½−k =Q0 with Q0 ¼ momentum lepton production by order of ks⊥=PT, where 80 MeV and the Wood-Saxon type of form factor yield almost ks⊥ is the radiated photon’s transverse momentum and identical numerical results. PT for the individual lepton’s transverse momentum. Therefore, this type of diagram is discarded in the leading power approximation. We apply the Eikonal approximation to calculate the soft photon radiation contribution, see, e.g., [48],

 μ μ  p p Mð1Þrj ¼ 1 − 2 Mð0Þ ð Þ soft e ; 32 p1 · ks p2 · ks

where Mð0Þ represents the leading order Born amplitude, μ is the polarization index for the soft photon with momen- tum ks, and the minus sign in the bracket comes from the fact that the contributions from the lepton and anti-lepton differ by a minus sign for the soft photon radiation. Therefore, the contribution to the amplitude squared reads as

jϕ −ϕ j 2p1 · p2 α ≡ 1 − 1 2 jMð1Þrj2 ¼ 2 jMð0Þj2 ð Þ FIG. 8. The resulting normalized π distribution. soft e : 33 p1 · ksp2 · ks In terms of the normalized α distribution measured first by This additional soft photon radiation generates additional the ATLAS experiment as shown in Fig. 8, the displaced nonzero transverse momentum for the lepton pair, and the peak becomes less prominent due to the smearing effect after consequence is that the lepton pair will no longer be in the averaging over the individual momenta of the produced back-to-back direction in the transverse momentum plane. leptons. We have also checked that the GEPA calculation In ATLAS, the imbalance between the lepton and antilep- also agrees with the ATLAS data measured in the peripheral ton was measured through the azimuthal angular correla- collisions. This indicates that the transverse momentum tion between them. In order to study the azimuthal angular broadening effect measured in the peripheral collisions distribution of the lepton pair, we calculate the total can be captured by the incoming coherent photon distribu- transverse momentum generated by the radiated photon. tions. In contrast, the transverse momentum broadening Together with the incoming photons’ contributions as due to the coherent incoming photons is not sufficient to discussed in the previous section, this leads to the final accurately describe the data in the central collisions, where total transverse momentum distribution for the lepton pair. additional broadening effects due to the incoherent multiple Alternatively, we also find that we can carry out this scattering and other medium effects as well as Sudakov type calculation by studying the radiated photon contribution to soft photon emissions may become important. We believe the “transverse momentum” imbalance in the lepton frame, that further theoretical developments as well as additional which can be translated into azimuthal angular distribution measurements [47] and experimental efforts including the between the two leptons in the Lab frame. As shown in the

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Appendix, these two frames are consistent in generating the because of the lepton mass. The muon mass mμ ≈ 0.1 GeV azimuthal angular correlation between the two leptons in is relevant because of the total transverse momentum for the the final state. lepton pair is in the same range as the lepton mass. We will show the derivation in the lepton frame, and Therefore, the large logarithms will depend on the relative comment on the lab frame calculations later. In the lepton size between μr (μr ≈ 1=r⊥) and the lepton mass mμ. When frame, the two leptons are moving back-to-back along μr >mμ, it corresponds to the transverse momentum l⊥ zˆ-direction. Soft photon radiation gives the lepton pair a larger than the lepton mass, and we will have a similar l þ − small additional transverse momentum ⊥. Now including double logarithmic term as that in e e → h1h2 þ X μ the lepton mass (assumed to be mass here, for conven- studied in Ref. [51]. This can be understood as the 2 ¼ 2 ¼ 2 ience), we have p1 p2 mμ and the real diagram following cancellation between the real and virtual con- contribution to the soft photon radiation can be written as tributions,

Z 2 α dξ l ðrÞðl Þ¼ e ⊥ ð Þ ðrÞ ðvÞ S ⊥ 2 2 2 2 2 ¯2 2 ; 34 Su ðr⊥ÞþSu ðr⊥Þ π ξ ðl þ ξ mμÞðl þ ξ mμÞ μ >mμ ⊥ ⊥ Z r Z α Q2 l2 2 α μ2 l2 2 ξ ¼ ¼ − d ⊥ Q þ r d ⊥ Q in the transverse momentum space, where ks ·p2=p1 ·p2 2 ln 2 2 ln 2 π 2 l l π 2 l l ¯ mμ ⊥ ⊥ mμ ⊥ ⊥ and ξ ¼ ks · p1=p2 · p1. It is clear that lepton mass plays — α 2 an important role here the lighter the lepton, the more ¼ − 2 Q ð Þ Sudakov radiation. From the kinematics, we know that 2π ln μ2 ; 37 ¯ 2 2 2 2 r ξξ ¼ l⊥=Q . Furthermore, ξ integral is limited by l⊥=Q < −γ ξ < 1. Carrying out ξ integration leads to the following E where μr ¼ c0=r⊥ with c0 ¼ 2e and γE the Euler expression for the soft photon radiation at small transverse constant. In the above calculation, we notice the fact that l momentum of ⊥, the real and virtual contributions completely cancel in the region l⊥ − α 2 Q r⊥ μ α 2l 2 < 2π ln 2 ;mμ < r; ðrÞ d ⊥ Q c0 ð Þ¼ il⊥·r⊥ ð Þ   Su r⊥ 2 2 e ln 2 2 : 36 S ðQ;mμ;r⊥Þ¼ π l l þ mμ u 2 2 2 2 2 ⊥ ⊥ > α Q Q r⊥ mμr⊥ :− 2π ln 2 ln 2 þ ln 2 ;mμ > μr: mμ c0 c0 Compared to the dihadron correlation in eþe− annihilation ð Þ studied in [51], in our case, there is additional complexity 41

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Since soft photon radiation factorizes, all order re- summation can be written as a simple exponential of SuðQ; mμ; r⊥Þ. As discussed above, the derivations can be carried out for the total transverse momentum distribution from soft photon radiation in the Lab frame as well. We will obtain a similar result as that in Eq. (35), where l⊥ is now replaced by p⃗T ¼ p⃗1T þ p⃗2T in the Lab frame. Again, when Fourier transformed into the r⊥-space, we will have the same result as Eq. (41),asr⊥ can be regarded as the Fourier conjugate p variable for p as well. FIG. 9. Compare the T distribution contributions for a typical T lepton pair production kinematics: incoming photon flux, per- To compute the final result for the total transverse turbative [Eq. (35)], and total contribution with the Sudakov momentum distribution of the lepton pair, we convolute resummation. Leptons are produced at midrapidity with invariant the above all order Sudakov contribution with the incoming mass 10 GeV. two photons contributions in the coordinate space to ensure momentum conservation. The transverse momentum dis- tribution for the latter can be formulated from the effective small, and the resummation result is dominated by the first photon approach discussed in the previous section. As an order corrections in this region. On the other hand, between example to illustrate the soft photon radiation effects, we these two regions, the resummation result provides a smooth assume a simple Gaussian form for the initial two photon match between the primordial distribution and one photon contribution, and the total transverse momentum distribu- radiation contribution. tion can be written as It is interesting to compare the distributions of Fig. 9 to the Drell-Yan type lepton pair production, where the lepton Z 2 2 2 r Q pair are produced through quark-antiquark annihilation dN d r⊥ − ⊥ 0 − ð ; Þ ¼ ipT ·r⊥ 4 Su Q;mμ r⊥ ð Þ 2 ð2πÞ2 e e e ; 42 process. In the Drell-Yan process, not only the incoming d pT quark distributions but also the QCD Sudakov effects contribute to a significantly higher transverse momentum. where the first term represents the initial two photon The latter can be seen from the invariant mass dependence contribution as a Gaussian distribution with width of the pT spectrum. In particular, for high mass final states, Q0¼ 40 MeV. The last factor represents an all order like Z-boson production, the low p spectrum is over- resummation of the soft photon radiation contribution. T whelmingly dominated by the Sudakov effects, see, e.g., Although there is no analytic expression for the Fourier the discussions in Ref. [52]. Numerically, for the same transform of the above result in the transverse momentum invariant mass range of lepton pair, the p distribution is space, we find the following solution is a very good T peaked around a few GeV for Drell-Yan process [52], approximation, which is orders of magnitude higher than the spectrum of Z 2 2 the pure QED process shown in Fig. 9. 2 Q r d r⊥ − ð ; Þ − 0 ⊥ eipT ·r⊥ e Su Q;mμ r⊥ e 4 As shown in Eqs. (41) and (42), the Sudakov factor ð2πÞ2     also depends on the mass of the produced lepton (mμ). γ0 2 β 2 Γð1 − βÞe Q0 −2γ p Since the mass of the muon is roughly 200 times larger ≈ E 1 − β 1 − T ð Þ 2 2 e 1F1 ; ; 2 ; 43 πQ0 Q Q0 than the electron mass, one can find that the Sudakov effect is stronger in the eþe− production than in the μþμ− α 2 Q2 production. where 1F1 is a hypergeometric function, γ0 ¼ ln 2, 2π mμ α 2 β ¼ e Q γ π ln 2 þ 2, and E the Euler constant. Numerically, we pT mμ A. Comparison with the UPC data from ATLAS have checked that the above expression gives very close Combining the Sudakov resummation of all order soft result for the Fourier transform of Eq. (42). photon radiation with the incoming photon fluxes contri- In Fig. 9, we compare the contributions from the incoming “ ” bution, we have the following expression for the total photon fluxes ( primordial ) and the perturbative photon transverse momentum of the lepton pair in heavy-ion radiation with the total contribution with Sudakov resum- collisions, mation. First, we notice that the primordial photon flux contribution dies out rapidly around 70 MeV, where the þ − Z 2 perturbative contribution takes over. Second, the Sudakov dσðAB½γγ → μ μ Þ d r⊥ ¼ σ ipT ·r⊥ ð ; Þ resummation result is consistent with the one soft photon 2 2 2 0 2 e W b⊥ r⊥ ; dy1dy2d p1Td p2Td b⊥ ð2πÞ contribution at relatively large transverse momentum. This is ð44Þ understandable, because the electromagnetic coupling αe is

094013-11 KLEIN, MUELLER, XIAO, and YUAN PHYS. REV. D 102, 094013 (2020) where σ0 is defined in Eq. (4). Wðb⊥; r⊥Þ contains the incoming photon flux and all order Sudakov resummation,

−S ðQ;mμ;r⊥Þ Wðb⊥; r⊥Þ¼N γγðb⊥; r⊥Þe u ; ð45Þ where Su is defined in Eq. (41). We have introduced a short notation N for the incoming photon flux contribution. As mentioned in the introduction, in the experiments, the azimuthal angular correlation has been commonly applied to study the low transverse momentum behavior of two final state particles. When the total transverse momentum is relatively small, the two leptons in the final state are FIG. 10. Acoplanarity distribution for lepton pair production almost back-to-back in the transverse plane with a small in UPC events at the LHC [31] for invariant mass from 10 to angle ϕ⊥ ¼ π − ϕ12. We are interested in the small ϕ⊥ ≪ π 100 GeV and rapidity from −0.8 to 0.8. There is a 10% region. For convenience, we take one of the leptons’ normalization uncertainty in the comparison, and the unfolding transverse momentum as reference for −xˆ direction, p1T ∼ effects from the experiments are not taken into account either. ð−jPTj; 0Þ, and p2T is parameterized by p2T ¼ðjPTjcosϕ⊥; j j ϕ Þ PT sin ⊥ . The total transverse momentum is pT. In the In Fig. 10, we show this comparison for the typical ATLAS experiment, the measurements are presented as kinematics of the ATLAS measurement of UPC functions of the so-called acoplanarity α, which is defined events: lepton transverse momentum PT > 4 GeV, rapidity as α ¼jϕ⊥j=π. jYμμj < 0.8, and invariant mass 10 GeV proton in we compare to the ATLAS data. In particular, we have one of the nuclei contributes incoherently to the production compared the invariant mass and rapidity distributions of process. However, the incoherent photon flux is suppressed the lepton pair to the ATLAS measurements for the UPC by 1=Z as compared to the coherent one. Numerically, the events, and found very good agreements. Second, we incoherent contribution is negligible in the low transverse assume a simple Gaussian distribution for the transverse momentum region as compared to the coherent contribu- momentum dependence in the incoming photon flux. The tion. In the relatively large momentum region, the incoher- Gaussian width for each photon flux is computed from ent photon contribution may become important. On the Eq. (10) as function of xa;b. With the above approxima- other hand, its power behavior from the proton form factor tions, we have the following expression for N , ∼1 8 leads to a much steeper declining function (e.g., =qT 2 2 2 ðQ ðx ÞþQ ðx ÞÞr considering a dipole form for the proton form factor) as γ γ − 0 a 0 b ⊥ N ð ; Þ ≈ ½ ð Þ ð Þ 4 compared to that from the soft photon and resummation γγ b⊥ r⊥ xafA xa xbfB xb b⊥ × e ; (∼1=q2 ). It will be interesting to investigate and compare ð46Þ T their contributions with more experimental data available in the future. where the impact parameter b⊥-dependent photon-photon interaction rate can be derived from the GEPA of the previous section. As mentioned above, this can be esti- IV. MEDIUM INTERACTIONS WITH LEPTONS mated following Eqs. (6) and (16) for a typical value of b⊥ In previous sections, we focused on UPCs. In a recent for the UPC events. This is a reasonable assumption when experiment measurement, the ATLAS collaboration has we compare to the experimental data for the normalized extended this idea to central collisions. It was argued that distributions as functions of the acoplanarity α. In the above the two-photon scattering processes produce dilepton ð Þ ð Þ equation, Q0 xa and Q0 xb represent the average trans- with small total transverse momentum (which is similar verse momentum for the photon fluxes of two incoming to UPC events), and these processes probe the electromag- nuclei, respectively. For the typical kinematics of ATLAS netic property of the quark-gluon plasma when they 2ð Þþ measurements [31], we find that the average Q0 xa traverse through the medium in the central collision events. 2 2 −3 Q0ðxbÞ¼ð40 MeVÞ for xa ¼ xb ¼ 10 . This is consis- In particular, the lepton pair has very small transverse tent with the results shown in Fig. 5 as well. momentum from two-photon scattering processes (see the

094013-12 LEPTON PAIR PRODUCTION THROUGH TWO PHOTON … PHYS. REV. D 102, 094013 (2020) discussions in the last section), the medium effects can be rapidity are identical (except for their overall motion) to the evidently measured through the so-called acoplanarity. part of the plasma at rest. If the boost to the CM frame of Before we deal with medium effects in lepton pair the μþμ− is not too large, the approximate boost invariance production in central AA collisions, we would like to of the plasma means that the μþ and μ− can now be viewed comment on the contribution from partonic photon-photon as separating in a plasma at rest, but of course with the scattering. This comes from the photon distribution func- temperature decreasing with time. Clearly the μþ and μ− tions from the nucleons in both nuclei. The differential interact with the plasma independently.1 cross section can be written as those in previous sections, Suppose we view the μþ and μ− in the CM of the ion and the Sudakov resummation can be derived as well. The beams where the μþμ− pair has a finite, but not overly large only difference is that we have to involve the parton rapidity. In this frame the μþ and μ− very slowly separate distributions for the photons from the nucleons [53,54]. from each other. But in this frame the relevant part of the Essentially, the photon distribution function is calculated plasma, the part which was at rest in the CM frame of from the quark distribution. Therefore, the photon PDF the μþμ−, is comoving with the μþμ−. The μþμ− pair and contribution scales with A, whereas the photon flux the comoving part of the plasma are time dilated and the 2 contribution scales with Z . In addition, for the relevant plasma has a long lifetime during which the μþ and μ− kinematics of our study, i.e., low transverse momentum separate and become independent. The key here is to region, we have to apply the nucleon form factor for the recognize that the QGP produced in an AA collision is Λ photon PDFs, which is of order QCD. It is much larger than approximately boost invariant (see e.g., [55]) and has a the typical momentum region of this study, see plots in much longer lifetime than that of the lepton pair dipole. Fig. 9. Therefore, we can safely ignore the photon PDF Therefore, to compute the additional transverse momen- contributions. tum broadening due to the QGP medium effect, we can now treat the leptons independently and go to a frame A. Inclusion of medium effects where each lepton carries zero initial transverse momentum The approach is very similar to that of the dijet azimuthal following the BDMPS formalism. Due to the QED multiple scattering, the typical transverse momentum broadening is angular correlation in heavy-ion collisions, where the PT 2 broadening of energetic jets in hot QCD medium can be then given by the medium saturation momentum Qse.In investigated. However, in the LHC energy range, the this picture, when we square amplitude, a dipole can be dominant broadening effect actually comes from the formed between the lepton in the amplitude and itself in the Sudakov effect [9] for the typical dijet kinematics complex conjugate amplitude. In this case, the correspond- ∼ 1 ∼ 1 [15,16]. From the analysis in the previous section, we find ing dipole has the size r⊥ =q⊥ =Qse. Although the that this is completely opposite for the lepton pair pro- QGP medium created in the middle rapidity region is duction through the two-photon scattering process. The approximately charge neutral, the leptons can still interact QED Sudakov effect is dominated by the incoming photon with the quark constituents of the medium and receive the transverse momentum in a wide range of kinematics. If the additional transverse momentum broadening. medium effect is strong enough, we will be able to probe it Because the lepton only carries an electric charge (we by measuring the azimuthal angular correlations between ignore the weak charge), it only interacts with the electri- the lepton pair. cally charged particles in the medium. As illustrated in It is important to note that each lepton should be treated Fig. 11, the lepton suffers multiple interactions with the as an independent charged particle in this case, since the medium during this process. We can resum the multiple typical distance between the leptons lþ and l− in the QGP photon exchanges between the produced lepton and the medium is much larger than the coherence length (or the electrically charged particles in the medium, and find that lifetime in the natural unit) of the lepton pair ∼1=PT. this results in a QED type time-ordered Wilson line as Intuitively, let us suppose that in the lab frame, which is follows also the center of mass (CM) frame for the AA collision, the pair has a finite rapidity but is not either very far forward or  Z Z U ð Þ¼ − − 2 ð − Þ very far backward moving (the ATLAS collaboration QED x⊥ T exp ie dz d z⊥G x⊥ z⊥ þ − measures μ μ pairs within the range jyj < 2.4). Since  the momentum imbalance pT of the lepton pair is small as − þ − × ρeðz ;z⊥Þ ; ð47Þ compared to PT, one can go to the CM of the μ μ pair simply by a boost along the ions’ beam direction. In this μþμ− μþ μ− CM frame of the pair, the and are moving away 1 from each other at 180° (back-to-back). The medium It is also useful to note that this argument does not apply to the cold nuclear matter case, where the lepton pair should be created in high energy AA collisions is approximately γ ¼ mppffiffi considered coherently. In this case, the time (RA= RA s)of boost invariant, and this is equivalent to say that those the interaction in the pA collision is actually much shorter than parts of the plasma moving with a given (but not too large) the lifetime of the lþl− dipole.

094013-13 KLEIN, MUELLER, XIAO, and YUAN PHYS. REV. D 102, 094013 (2020) Z 4 1 2 ¼ Nc 2 ≡ Ncg −μ2ð −Þ ð Þ Qsg Qsq ln 2 2 dz c z ; 51 CF 4π Λ r⊥ R −μ2ð −Þ¼ A where dz c z 2πR2 is related to the color charge density with R being the size of the target medium [56], and FIG. 11. Multiple interactions of the energetic lepton with the Λ is the QCD Debye mass. The above expression is ⊗ medium, where the symbol represents the scattering center equivalent to the saturation momentum expression for cold from the charged medium. 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 8π αsNc 2 2 2 nuclear matter Qs ¼ 2 ρ R − b xGðx; 1=r⊥Þ [57], Nc−1 α 1 ð 1 2 Þ¼ sCF ρ ð − Þ where xG x; =r⊥ Nc π ln Λ2 2 . where e z ;z⊥ is the random electric charged density of r⊥ the target medium, and the photon propagator Gðx⊥Þ is defined as B. Implication from the ATLAS measurements If we assume the multiple scattering between the Z 2 d q⊥ 1 1 ð Þ¼ iq⊥·x⊥ ¼ ðλ Þ ð Þ produced dilepton and the medium similar to the QCD G x⊥ 2 2 2 e K0 x⊥ ; 48 ð2πÞ q⊥ þ λ 2π case, we can modify the above Wðr⊥Þ of Eq. (45) as,

hˆ i 2 qQEDL r⊥ with λ acting as an infrared (IR) regulator such as the Debye −S ðQ;mμ;r⊥Þ − Wðb⊥; r⊥Þ¼N γγðb⊥; r⊥Þe u e 4 ; ð52Þ mass in QED medium. The value of λ can be estimated according to the range of the electromagnetic interaction in where the last factor comes from the medium contribution the medium, which depends on the QED Debye screening ˆ to the dilepton pT-broadening effects and qQED represents ∼ peTffiffi mass mD 3, where T is the temperature. In our case, the electromagnetic transport coefficient of the quark- λ ∼ ∼ 80 hˆ i simple estimates show that mD MeV which cor- gluon plasma. qQEDL can be identified as the saturation 2 responds to a Debye length a few times less than the typical momentum Qse discussed above. size of the medium created in central heavy-ion collisions. With this additional pT-broadening effects, we can Similar to the QCD qq¯ dipole calculation, the QED calculate the azimuthal angular correlation in heavy-ion incoherent multiple scattering amplitude between the collisions. We assume that the initial distributions remain þ − l l ðqq¯Þ dipole with size r⊥ and the target medium the same as the UPC case and take into account the can be cast into additional broadening effects by parametrizing the qLˆ      for the leptons. 1 1 U þ U† − In Fig. 12, using the UPC as the baseline and assuming QED b⊥ r⊥ QED b⊥ r⊥ 2 2 the mild bperp dependence due to the initial transverse   Q2 r2 momentum broadening, we show this effect by imposing ¼ − se ⊥ ð Þ several different values of the qLˆ . Comparing these curves exp 4 ; 49 to the ATLAS measurements, we find that the effective hˆ i ð100 Þ2 qQEDL range from MeV in most central collisions where the analogR of saturation momentum in QED ð50 Þ2 2 e4 1 − 2 − 2 to MeV in noncentral collisions. In the above- Qse ≡ 4π ln 2 2 dz μeðz Þ. Here, μe is related to the λ r⊥ mentioned GEPA model, we expect that the effective hˆ i expectation of the local charge density fluctuations. As value of qQEDL becomes smaller. The determination of λ hˆ i we can see, only a logarithmic dependence on is left in the qQEDL requires a more sophisticated and detailed com- above dipole amplitude due to cancellation. Normally one parison with the accurate unfolded data. does not have to take into account the multiple scattering for the QED calculation. On the other hand, the dipole size r⊥ ∼ 1=q⊥ is sufficiently large in the soft momentum transfer region where q⊥ is of the order 10 to 100 MeV, 2 2 and this gives Qser⊥ ∼ 1. Similar to the QCD case, the correlation of two charge density follows

− 0− 0 − 0− ð2Þ 0 2 − hρeðz ;z⊥Þρeðz ;z⊥Þi ¼ δðz − z Þδ ðz⊥ − z⊥Þμeðz Þ: ð50Þ

In comparison, we often define the QCD saturation FIG. 12. Medium modifications to the acoplanarity distribu- momentum as follows tion, with different values of the effective qLˆ .

094013-14 LEPTON PAIR PRODUCTION THROUGH TWO PHOTON … PHYS. REV. D 102, 094013 (2020) X In order to extract the p -broadening parameter for the 2 T e2 ¼ N ; ð53Þ leptons, we need to implement the realistic geometry of the q 9 f u;d;s collisions and the photon sources. From that, we shall estimate the average length of the leptons traversing through where we take Nf ¼ 3 with active u, d and s quarks. For the the medium. Compared to the dijet correlation in heavy-ion color factor associated with the quark density, it can be collisions, this is even more important because the photon evaluated as source profile is different from the medium profile created in heavy-ion collisions. In particular, in order to receive the 1 2 ½ a b ½ a b¼ ð Þ additional QGP medium p broadening, the lepton pair 2 Tr T T Tr T T ; 54 T N 9 should propagate through the medium. This indicates that c either the pair is produced inside the medium, or one of the which is an effective color factor by considering the quark- leptons traverses the medium if the pair is produced outside. quark scattering amplitude, where Nc ¼ 3. For gluon This requirement may effectively reduce the overall strength density case, it is of the medium broadening effect. We leave the detailed studies in a future publication. 1 1 ½ c d¼ ð Þ 2 fabcfabdTr T T 2 55 NcðNc − 1Þ C. Comparison with the QCD pT-broadening of jets: Parametric estimate from the quark-gluon scattering amplitude. These are for Assuming that the multiple interactions with the medium the quark jet pT-broadening. Therefore, we can estimate the can be applied to describe both QED for the leptons and the ratio between the QED and QCD saturation scales as QCD for the quarks and gluons, we can compare the sizes of the p -broadening effects from the same formalism, i.e., hqˆ Li α2 21 N 2 α2 7 T QED ¼ e 2 f 9 ¼ e : ð Þ the BDMPS formalism [7,58,59]. hˆ i α2 21 2 1 α2 × 15 56 qQCDL s 2 Nf 9 þ 16 2 s Because the leptons have large transverse momenta, their multiple interactions with the medium can be formulated where again Nf ¼ 3. Here hqLˆ i represents the amount of following the BDMPS [7,58] formalism. In particular, the medium broadening, which is known as the saturation scale pT-broadening can arise from the multiple interaction with in the so-called dipole formalism. The above estimate of the charged particles in the medium. This is quite the same QCD hqLˆ i is for QCD quark jets. For gluon jets, a factor of as that for the quark and gluon jets traversing the quark- CA=CF should be multiplied to the denominator. gluon plasma medium. The quark and gluon also suffer We would like to add a few comments about these pT-broadening because of multiple QCD interactions with observations. One is about the thermal density ratio we the medium. Of course, the major difference is that the applied above. We assume that quark and gluons are lepton interacts with the electric charges while the quark/ thermalized at the same time, which may not be true. gluon jets interact with the color charges. Here, we make There has been concern that the quarks may be thermalized some parametric estimates based on this idea. at a later time, see, for example, the discussions in First, the coupling constant plays an important role. By Ref. [60]. Second, we did not take into account the details definition, the p -broadening depends on two orders of T of the medium property, for example, the associated Debye coupling constants. Since the couplings in QED and QCD masses for QED and QCD. This could introduce additional are dramatically different, thus there is a major difference complexity in the calculations. In addition, for the QCD for the p -broadening effects. T case, there are length dependent double logarithms [61]. Second, the broadening depends on the charge density of Last but not least, the medium path length L can differ the medium. Therefore, we need to estimate the density for the QED and QCD cases, since the electron pair can difference for the electromagnetic QED and strong inter- be created outside the medium. If all these effects are action QCD densities in the quark-gluon plasma. Since taken into account, Eq. (56) may not apply. Nevertheless, only quarks carry electric charges, we can determine the the above equation can serve as a simple formula for a electric charge density from the quark density. For the rough estimate. strong interaction case, both quarks and gluons contribute. The parton density is proportional to the degree of freedom if we assume the thermal distributions of the quarks and D. Comments on the QED energy loss gluons. Therefore, the ratio of quark density vs the gluon In QED medium, the radiative energy loss can be 21 ∶16 density goes as, see, for example, Ref. [60]: 2 Nf , estimated as [58], where Nf is the number of active flavors. There are two sffiffiffiffiffiffiffiffiffi additional differences: one is the color factor in QCD case, pffiffiffiffiffiffiffiffiffiffi α 2 L2E and the other is the charge average between the quarks and Δ ¼ hˆ i ð Þ EQED qL π 3 λ 57 gluons. For electric charge average, we have m

094013-15 KLEIN, MUELLER, XIAO, and YUAN PHYS. REV. D 102, 094013 (2020) in a certain kinematic limit of the induced radiated photon effects. First, we show that the contributions from initial spectrum, where L represents the average path of the lepton electromagnetic fields generated by the colliding nuclei in the medium, λm stands for the mean free path of the cancel out completely in the leading power of the lepton medium. For the kinematics of our interest, we find that the pair production through the two-photon process. Then, we total energy loss is a few percent of the pT-broadening size will discuss how to measure the residual magnetic field (100 MeV). It is so small that we will not be able to observe (due to the created quark-gluon plasma) effects. such effects. The QED case is totally different from QCD jet A. Initial electromagnetic fields contributions: quenching in heavy-ion collisions. First, the energy loss The classical electrodynamics perspective of the leptons is much smaller than the QCD jet energy loss. To estimate the electromagnetic fields of a fast-moving For the leptons going through the medium, there are no nucleus, let us first follow the approach in classical surface bias effects, because the energy loss is completely electrodynamics [62]. Then we provide some discussion negligible. However, we know that the QCD jet energy loss and interpretation in terms of quantum field theory calcu- is important, and surface bias effects are significant in the lations. We can write the electromagnetic fields of a moving realistic simulation of jet physics in heavy-ion collisions. charge propagating along the z direction in the lab frame as

V. MAGNETIC FIELD EFFECTS γqb E ¼ ;B¼ βE ; ð58Þ x ðb2 þ γ2v2t2Þ3=2 y x One measurement might indicate that the pT-broadening effects of the dilepton could come from the magnetic effects where γ ¼ pffiffiffiffiffiffiffiffi1 , β ¼ v=c and we put the impact param- created at the very early time of heavy-ion collisions [2]. 1−β2 The major phenomenological difference between this eter b in the x direction. Let us compute the electromagnetic mechanism and the multiple scattering mechanism dis- field effect for one particular nucleus and choose such a cussed in this paper is that the magnetic effects are strongly frame that the velocities of the back-to-back μþ and μ− can correlated with the event plane. Therefore, if we can be written as vþ ¼ðvx;vy;vzÞ and v− ¼ð−vx; −vy; −vzÞ, measure this correlation, we could distinguish these two respectively. In the eikonal approximation, we can find the mechanisms. force in the y-direction is zero, and the Lorentz forces in the In addition, the contribution from the magnetic effects is x-direction for μþ and μ− can be written as a cross product: B⃗× V⃗, where B⃗ is the magnetic field ⃗ dpþ generated in heavy-ion collisions and V is the lepton x ¼ eE ð1 − β βÞ; ð59Þ velocity. This introduces two important observational dt x z consequences: (1) impact parameter dependence of the dp−x effects is different from the multiple interactions with the ¼ −eExð1 þ βzβÞ; ð60Þ medium discussed in this paper, which will increase with dt decreasing impact parameters. For the magnetic effects, it respectively. Also, we need to include the effect of time increases from UPC to peripheral collisions but will dilation since μþ and μ− are moving along the z direction decrease for more central collisions with decreasing impact with finite velocities. The relative velocities are parameter. (2) The magnetic effects for the additional p -broadening happens in the direction perpendicular to β − β β þ β T β ¼ z β ¼ − z ð Þ both the magnetic field B⃗and the lepton moving direction. þ ; − : 61 1 − βzβ 1 þ βzβ Therefore, if both the lepton and antilepton have the same rapidity and they have the almost same size of transverse Therefore, we find the times that μþ and μ− experience the momentum as in our case, the magnetic effects will cancel electromagnetic shockwave are different and their ratio is out between them. That is, the lepton and antilepton will sffiffiffiffiffiffiffiffiffiffiffiffiffiffi gain the same amount of additional transverse momentum 2 Δtþ 1 − β kick from the magnetic effects but with opposite signs. On ¼ þ ð Þ Δ 1 − β2 : 62 the other hand, if the lepton and antilepton are back-to-back t− − in the zˆ-direction, i.e., their rapidities are opposite to each other, the magnetic fields will have net effects on the It is then straightforward to check that transverse momentum kick and lead to the observed qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 transverse momentum broadening in the lepton pair. Δpþx ¼ eExΔtð1 − βzβÞ 1 − βþ Therefore, we should be able to distinguish these two qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 mechanisms by studying the centrality dependence and ¼ eExΔt 1 þ β βz − β − βz rapidity dependence of the p -broadening effects. In the T ¼ −Δ ð Þ following, we will discuss the details of the magnetic p−x: 63

094013-16 LEPTON PAIR PRODUCTION THROUGH TWO PHOTON … PHYS. REV. D 102, 094013 (2020) Z Z This demonstrates that the momentum broadening exactly ð þ Þ2 αð Þ − αð þ Þ ¼ b⊥ r⊥ e A x dxα e A x r⊥ dxα eq ln 2 : cancel for exact back-to-back lepton pairs due to charge b⊥ neutrality. To get a nonzero transverse momentum broad- ð Þ ening, we need to take the charge dipole moment into 67 account. In coordinate space, the separation of the lepton 1 By taking into account the fact that q ¼ Ze for a relativistic pair is proportional to =Q, where Q is the invariant mass 1 of the produced lepton pair. Therefore, the total transverse heavy-ion, and the 4π difference due to different conven- momentum broadening square must be proportional to tions, we can see that the above result agrees with the h 2 i 1 h 2 i∼ 2 2 2 ∼ Z2e4 ∼ð30 Þ2 Wilson line formalism used below. Pm Q2R2 , where Pm e ExRA R2 MeV , where A A Next, in terms of the final state QED type scattering and ≈ 7 RA fm is the radius of the nucleus. Wilson lines approaches [63–65], we can compute the cross Here we have used a quantum-classical mixed picture to section σγA→μþμ−A as follows. It is straightforward to resum illustrate the broadening of the produced lepton pair due to the multiple photon exchanges between a muon line and the coherent electromagnetic fields of fast-moving nuclei target nuclei, and find that this results in a QED type by assuming that the lepton pair is created before the pair Wilson line passes over the high energy nuclei. This intuitive picture helps to estimate and understand the final state broadening 2 U ðx⊥Þ¼exp ½iZe Gðx⊥Þ; ð68Þ of the lepton pair, while a fully quantum treatment of this QED process is presented in the next subsection. with the photon propagator Gðx⊥Þ defined as

B. Initial electromagnetic fields contributions: 1 ð Þ¼ ðλ Þ ð Þ The quantum field theory perspective G x⊥ 2π K0 x⊥ : 69 Quantitatively, we can also study the p -broadening T λ effect in terms of Wilson lines from the quantum field Here we use as the IR cut off for the moment. When we λ theory perspective. Let us derive the QED Wilson by using study the muon pair production, the dependence will the so-called Lienard-Wiechert potential for a moving point cancel. In this simple model, we assume that the nuclei is a charge particle with charge q ¼ Ze. Again, we can first point particle with charge Ze in order to study the coherent follow the discussion and the convention in Ref. [62], final state effect. The QED multiple scatterings contribute a μþμ− which gives the following four-vector potential phase to the QED type dipole with size r⊥ as follows     qVαðτÞ 1 † 1 αð Þ¼ ð Þ U b⊥ þ r⊥ U b⊥ − r⊥ A x ½ − ðτÞ : 64 QED 2 QED 2 V · x r τ¼τ0  1  jb⊥ þ r⊥j α ¼ 2 α 2 ð Þ Here V ¼ðγc; γcβÞ is the four vector velocity of the exp iZ ln 1 : 70 jb⊥ − 2 r⊥j moving charge. Putting in the light-cone constraint τ ¼ τ0, and using Eq. (14.16) together with the fact dxα ¼ Vαdτ ¼ Furthermore, let us use σγA→μþμ−A as an example to estimate Vαγdt, we can write the order of magnitude of the electromagnetic corrections Z Z ∞ in the final state between the produced muon pair and the α γdt A ðxÞdxα ¼ q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð65Þ target nucleus with charge number Z. The cross section can 2 þ γ2 2 2 −∞ b⊥ v t be written as Z The above integration is divergent, but it becomes finite if 2 2 0 2 2 0 dσγ →μþμ− d b⊥ d b d r⊥ d r we consider the difference of two opposite charges which A A ¼ N ⊥ ⊥ d2p d2P ð2πÞ2 ð2πÞ2 ð2πÞ2 ð2πÞ2 are separated by a distance r⊥. Therefore, the result for a T T 0 0 0 −ip ·ðb⊥−b Þ−iP ·ðr⊥−r Þ lepton pair lepton is × ψðr⊥Þψ ðr⊥Þe T ⊥ T ⊥    Z Z 1 jb⊥ þ 2 r⊥j α α 1 − 2 α e A ðxÞdxα − e A ðx þ r⊥Þdxα × exp iZ ln 1 jb⊥ − 2 r⊥j Z     ∞ 1 jb0 þ 1 r0 j ¼ γ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − −2 α ⊥ 2 ⊥ ð Þ eq dt × 2 2 2 2 × exp iZ ln 0 1 0 ; 71 −∞ b þ γ v t jb⊥ − 2 r⊥j ⊥  1 ϵ·r⊥ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð66Þ where the splitting function ψðr⊥Þ ∼ 2 represents the 2 2 2 2 r⊥ ðb⊥ þ r⊥Þ þ γ v t contribution from the γ → μþμ− splitting. Also if we 1 jb⊥þ2r⊥j Eventually, if we set v ¼ c ¼ 1, we can obtain expand the expression ½1 − exp ð2iZα ln 1 Þ to the jb⊥−2r⊥j

094013-17 KLEIN, MUELLER, XIAO, and YUAN PHYS. REV. D 102, 094013 (2020) lowest order, we should be able to recover the lowest order cancellation of the coherent final state effect is still valid γγ → μþμ− cross section together with the proper normali- even with the formation of QGP. zation N. In the back-to-back limit, it follows that 2 ≫ 2 PT pT. In the case of interest, PT is around several C. Residual magnetic field effects GeV while p is of the order of Λ ¼ 1 ∼ 30 MeV. T m RA in the quark-gluon plasma Let us comment on the above result. First of all, we can In the meantime, as argued in some recent papers compute the Born contribution by expanding the above [66–68], there could be a residual and strong magnetic field Wilson line contribution to the lowest order. In the back-to- in the quark-gluon plasma after the heavy-ion collisions. Due back limit, we can approximately write to the collision symmetry, the magnetic field only contains ⃗  1  the perpendicular component B⊥. As mentioned at the jb⊥ þ r⊥j b⊥ · r⊥ 1 − exp 2iZα ln 2 ≃−i2Zα : ð72Þ beginning of this section, since the Lorentz force vanishes j − 1 j 2 b⊥ 2 r⊥ b⊥ along the direction of the magnetic field, the amount of the pT-broadening from the magnetic effects will have a non- It is then straightforward to estimate that the Born con- trivial correlation with the event plane, which is correlated Z2α2 with the direction of the magnetic field [66–68]. tribution is of the order of N p2 P4 . T T A number of further observations regarding the effects Second, we can estimate the contribution with one more of the residual magnetic fields are as follows. First, the photon exchange in the amplitude level, which means the transverse momentum kick comes from the longitudinal expansion of the Wilson line to the second order. Using the motion vz of the leptons. Therefore, the magnetic effects fact that the lower bound of the b⊥ integral is roughly R , A crucially depend on the rapidity of the leptons. In particular, we can find that the first final state interaction correction is if the lepton and the antilepton move in the same direction, of the order the magnetic effects will cancel out in the total transverse 2α2 2α2 2α2 2α2Λ2 momentum for the pair. Only if they are moving in opposite Z Z ∼ Z Z m ð Þ ˆ N 2 4 2 2 N 2 4 2 : 73 z direction, there will be net effects. In more detail, the net pTPT PTRA pTPT PT transverse momentum kick for the pair is proportional to

m Therefore, the final state interaction is power suppressed by jΔpy jμþμ− ¼ Pm½tanhðyþÞ − tanhðy−Þ; ð74Þ 2α2Λ2 Z m ∼ 10−4 ¼ 1 the factor of P2 for PT GeV. Also, since the T P h 2 i ∼ Λ2 ∼ 1 where m represents the average transverse momentum transverse momentum square average pT m R2 at A kick depending on the geometry of the collisions, yþ and lowest order, we can see that the change due to final state y− are rapidities for the lepton and the antilepton, respec- 2α2Λ2 Δh 2 i ∼ Λ2 Z m interaction is roughly pT m 2 , which is in tively. Therefore, the total pT-broadening effects for the PT agreement with our estimate in previous subsection from pair can be formulated as the perspective of classical electrodynamics. hΔ 2 iB ¼hP2 ð Þi½ ð Þ − ð Þ2 ð Þ In summary, the contributions from the initial electro- pT μþμ− m b⊥ tanh yþ tanh y− ; 75 magnetic fields generated by the colliding nuclei cancel out 2 where hPmðb⊥Þi stands for the average p -broadening with completely in the leading power of pT=PT, while the finite T contribution to the transverse momentum broadening is a centrality dependence. Assuming that a flat distribution power suppressed. This cancellation is also consistent with for the rapidity dependence for the two leptons in the range a factorization argument that the final state interaction of jyμj < 2, and the net magnetic effects can be calculated effect vanishes in this process due to the opposite charges as function of rapidity difference between the lepton pair of the lepton pair. In the case of collisions between two ΔY ¼jyμþ − yμ− j, overlapping nuclei, QGP can be formed in the overlapping 2 B region due to hadronic collisions. Since QGP is mostly hΔp iμþμ− ðΔYÞ T R created by gluons and sea quarks, while the high energy 2 dy1dy2½tanhðy1Þ − tanhðy2Þ dσðy1;y2Þ valance quarks remain largely unaffected and move along ¼hP2 ðb Þi ΔY R m ⊥ σð Þ ; the beam pipe as remnants. Therefore, the electric charge ΔY dy1dy2d y1;y2 of the nuclear remnants after hadronic collisions roughly ð76Þ remains unchanged. In addition, the incoming photons are almost real, and they are typically emitted long before where the rapidity integrals are constrained with the production of the lepton pair and the hadronic collision ΔY ¼jy1 − y2j. In Fig. 13, we plot the normalized dis- take place. Assuming that the incoherent broadening effect tribution as function ΔY for a typical transverse momentum due to the presence of the QGP can be taken into account for the lepton PT ¼ 6 GeV. From this plot, we can clearly separately, we believe that our above discussion about the see that the magnetic effects on the pT-broadening

094013-18 LEPTON PAIR PRODUCTION THROUGH TWO PHOTON … PHYS. REV. D 102, 094013 (2020)

pT-broadening contributions to the acoplanarity of dilep- tons measured in the final state are discussed. First, soft photon radiations are taken into account through the Sudakov resummation; Second, the electromagnetic multiple interactions between the lepton pairs and the electric charges inside the quark-gluon plasma medium are resummed incoherently, since these multiple inter- actions are in general random; Lastly, possible external magnetic fields can be created in heavy-ion collisions and these fields may also bring additional transverse momen- FIG. 13. Normalized magnetic effects on the pT -broadening for the lepton pair as function of their rapidity difference tum broadening. ΔY ¼jyμþ − yμ− j with jyμj < 2.4. As an important baseline, we compare the Sudakov effect to the azimuthal angular correlation of the lepton pair in ultra-peripheral collisions measured by the ATLAS increases with rapidity difference, and approach to the collaboration at the LHC. The comparison indicates that Δ ¼ 3 maximum values around Y . On the other hand, the this effect is crucial to explain the experimental data in the multiple interaction effects with the medium remain a region with moderately larger acoplanarity. relative constant because the medium density does not It appears to us that the pT-broadening effects observed change in this rapidity range. Therefore, the difference by the ATLAS collaboration in the central PbPb collisions between the amount of the pT-broadening effects at differ- may suggest the need for additional medium induced Δ ent Y can serve as an effective measure for the magnetic transverse momentum broadening coming from the multi- effects: ple scattering of the leptons in the medium or other sources. The acoplanarity of dilepton pairs can provide us another ½hΔ 2 i − hΔ 2 i ∝ h ⃗2 i ð Þ pT ΔY¼3 pT ΔY¼0 b⊥ B⊥ b⊥ ; 77 interesting window to study the properties of quark-gluon plasma. which depend on the centrality of heavy-ion collisions. The pT-broadening effects due to the possible magnetic This rapidity dependence can also help to study other fields are considered as well. Through the study on the magnetic effects in heavy-ion collisions, such as the chiral centrality and rapidity dependence, as well as the correla- magnetic effects [66,68]. tion with the orientation of the magnetic field, it is possible Second, the effects, of course, depend on the magnitude to distinguish the transverse momentum broadening effects of the magnetic field. More detailed calculations need to due to the magnetic fields from the multiple scattering formulate the impact dependence of the magnetic fields. effect. We stressed that the broadening coming from the In general, as mentioned above, the magnetic field magnetic fields depends on the rapidity difference between increases from ultra-peripheral to peripheral collisions, the lepton and the antilepton. This rapidity dependence can but starts to decrease toward more central collisions. be used to determine the strength of the magnetic field. Third, the direction of the transverse momentum kick Last but not least, to interpret the pT-broadening is correlated to the direction of the magnetic field. phenomena of dilepton productions observed by STAR Therefore, if this correlation can be measured, we can and ATLAS collaborations as a result of QED interaction further distinguish its contribution from other sources for of the lepton pair with the medium crucially depend on the pT-broadening effects. how precisely we know that the initial state contributions It is interesting to note that a recent measurement by from the incoming photon fluxes of the colliding nuclei. the ATLAS collaboration [4] shows that there is no strong We emphasize that more theoretical developments and rapidity dependence on the medium modification of the experimental measurements are needed to understand this dilepton transverse momentum distribution, which will physics. Only with this being resolved, can we reliably impose a strong constraint on the strength of the magnetic apply this process to study the electromagnetic property of effects discussed above. the quark-gluon plasma created in heavy-ion collisions.

VI. SUMMARY AND DISCUSSIONS ACKNOWLEDGMENTS In this paper, we have investigated the electromagnetic We thank Z. Xu, W. Zha, J. Zhou for discussions and productions of dileptons with very small total transverse comments. This material is based on work partially momentum in heavy-ion collisions, which can provide us a supported by the Natural Science Foundation of China new channel to probe the electromagnetic property of the (NSFC) under Grant No. 11575070, and by the U.S. quark-gluon plasma created in these collisions. In the Department of Energy, Office of Science, Office of above theoretical calculations, besides the initial contribu- Nuclear Physics, under Contracts No. DE-AC02- tions due to the incoming photons, three other important 05CH11231 and No. DE-FG02-92ER40699.

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APPENDIX: AZIMUTHAL ANGULAR In the ATLAS experiment, the measurements are presented DECORRELATION FROM IN THE LAB as functions of the so-called acoplanarity α, which is AND LEPTON FRAMES defined as α ¼jϕ⊥j=π. For the α distribution, we have In experiments, the azimuthal angular correlation Z dv2 between the final state particles has been applied to study dN ¼ 2π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq 2 ð Þ ð Þ α PTf pT : A5 the low total transverse momentum behavior. The two d α>0 2 2 vq − ϕ⊥ leptons in the final state will be back-to-back in the transverse plane with a small angle ϕ⊥ ¼ π − ϕ12.We ϕ ≪ π Let us first assume a simple Gaussian distribution for fðpTÞ are working at the small angle of ⊥ region. For 2 2 −p =Q0 2 convenience, we take one of the leptons’ transverse to check the intuitive α distribution: fðpTÞ¼e T =Q0π. momentum as reference for −xˆ direction, p1T ∼ With that, we find that ð−jPTj;0Þ, and p2T is parametrized by p2T ¼ðjPTj cos ϕ⊥; 2 Z P jP j sin ϕ⊥Þ. In the following, we will show that we can 2 2 T 2 T − 2v dN dvq PT Q q compute the azimuthal angular distribution by taking into ¼ 2π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 0 α π 2 d α>0 2 2 Q0 account additional transverse momentum contributions in vq − ϕ⊥ the Lab frame or the lepton frame. 2bα 2 2 ¼ pffiffiffi −bαα ð Þ π e ; A6 1. Lab frame calculations P π 2 In the lab frame, we calculate the azimuthal angular where bα ¼ T . We can also calculate the average of α as Q0 distribution from the total transverse momentum pT for the lepton pair. This total transverse momentum can come 1 Q2 from initial photons’ contributions or from the final state hα2i¼ 0 ; ð Þ 2 2 π2 A7 medium interaction (pT-broadening). PT We can write down the differential cross section depend- ϕ ing on pT, and in terms of ⊥,wehave

dN 1 dσ 1 2 ¼ ¼ fðp Þ; ðA1Þ 2 Q0 2 σ 2 T hϕ⊥i¼ : ðA8Þ d pT d pT 2 2 PT where fðpTÞ represents a general form of transverse momentum dependence coming from both incoming We have a number of observations. First, it shows that the photon contribution, soft photon radiation, and the average angular broadening is, actually, half of naively expected as the average pT-broadening divided by the pT-broadening effects in the medium. Working out the ’ kinematics, we have lepton s transverse momentum. This will have a significant impact on the interpretation of the experimental measure- 0 ments on the jet p -broadening effects from the azimuthal pT sinðϕ Þ T sinðϕ⊥Þ¼ ; ðA2Þ angular correlations in heavy-ion collisions. Physically, PT the jet pT-broadening spreads toward the perpendicular 0 direction with respect to the jet direction. However, half of where ϕ is the relative azimuthal angle of p⃗T respect to xˆ direction. The differential cross section can be rewritten as that spreading happens in beam direction in the lab frame and will not have observable effects on the azimuthal Z ðϕ Þ angular correlations. Second, the average of hα2i involves dN ¼ PT cos ⊥ ð Þ pTdpT 0 f pT 1 2 dϕ⊥ p cosðϕ Þ an average of =PT, not the average of PT. Z T ϕ In addition, the broadening in azimuthal angular dis- ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos ⊥ 2 ð Þ ð Þ vqdvq PTf pT ; A3 tribution can be model independently related to the addi- v >sin ϕ⊥ 2 2 q vq − sin ϕ⊥ tional pT-broadening in the transverse momentum distribution, where vq ¼ pT=PT. We can further simplify the above 1 Δhl2 i 1 Δh 2 i expression by making approximations in the correlation Δhϕ2 i¼ ⊥ ¼ pT ð Þ ≪ 1 ϕ ≪ 1 ⊥ 2 2 2 2 : A9 limit, i.e., vq and ⊥ , PT PT Z 1 dv2 dN ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq 2 ð Þ ð Þ The pT-broadening can be either formulated in the lab PTf pT : A4 ϕ 2 2 2 frame as p or in the lepton frame as l⊥, see, the d ⊥ v − ϕ T q ⊥ discussions below.

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2. Lepton frame formulation back-to-back with small angular difference ϕ⊥ ≪ 1. It is more convenient to formulate the above results in the From the above kinematics, we will find that lepton frame. It is straightforward to include the medium l ðϕ Þ effects (next section) in this frame, where the leptons are ⊥ sin T sinðϕ⊥Þ¼ moving along the zˆ-direction and additional transverse PT momentum broadening can be directly formulated. E sinðθTÞ sinðϕTÞ sinðθTÞ sinðϕTÞ If the two leptons are moving in the zˆ direction, we ¼ ¼ : ðA11Þ E sin θ0 sin θ0 can parametrize the leading lepton (or reference lepton) ¼ð 0 0 Þ ¼ as p1 E; ; ;E and the away-side lepton as p2 The differential cross section can be rewritten as ðE; l⊥ cos ϕT; l⊥ sin ϕT;Ecos θTÞ in the limit of l⊥ ≪ E, l ¼ θ Z where ⊥ E sin T represents the imbalance between the dN sinðθ0Þ cosðϕ⊥Þ two leptons in the final state. Here, we neglect the masses ¼ l l ðl Þ ð Þ ϕ ⊥d ⊥ ðθ Þ ðϕ Þ f ⊥ : A12 for the leptons since they do not affect the following d ⊥ sin T cos T discussions. The differential cross section can be written as In the following, we take the approximations in the correlation limit, i.e., θ ≪ 1 and ϕ⊥ ≪ 1, and find that dN ¼ dN ¼ ðl Þ ð Þ T 2 2 f ⊥ ; A10 the α distribution can be written as d l⊥ E sin θTd sin θTdϕT Z ϕ 2π θ ðθ Þ where T runs from 0 to . However, T is not directly dN sin 0 2 ¼ 4π θ dθ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E fðl⊥Þ; related to the azimuthal angle between the leptons in the lab α T T 2 2 2 d α>0 θ − sin ðθ0Þϕ⊥ frame. To link to the experimental observables in the lab T ð Þ frame, we need to work out the details. In the lab frame, the A13 leptons have non-zero transverse momentum PT which is same order of E, we will parametrize their momenta as where l⊥ ¼ E θT and α is define above as α ¼jϕ⊥j=π. θ → θ ðθ Þ → p1 ¼ðE; p1T; 0;Ecos θ0Þ and p2 ¼ðE;p2T cosðπ −ϕ⊥Þ; With a simple variable change T T= sin 0 vq,we p2T sinðπ −ϕ⊥Þ;p2zÞ, where p1T ¼ Esinθ0 ∼p2T and can find out that the above equation is identical to that of ϕ⊥ ∼ 0. In the correlation limit, p⃗1T and p⃗2T are Eq. (A5). All discussions above shall also follow.

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