Two-photon production of dilepton pairs in peripheral heavy ion collisions
TTwo-photon production of dilepton pairs in peripheral heavy ion collisions
Spencer R. Klein
Lawrence Berkeley National Laboratory, Berkeley CA, 94720, USA (Dated: January 16, 2018)The STAR collaboration has observed an excess production of e + e − pairs in relativistic heavy ioncollisions, over the expectations from hadronic production models. The excess pairs have transversemomenta p T <
150 MeV /c and are most prominent in peripheral gold-gold and uranium-uraniumcollisions. The pairs exhibit a peak at the J/ψ mass, but include a wide continuum, with pair invari-ant masses from 400 MeV/c up to 2.6 GeV/c . The ALICE Collaboration observes a similar excessin peripheral lead-lead collisions, but only at the J/ψ mass, without a corresponding continuum.This paper presents a calculation of the cross-section and kinematic for two-photon productionof e + e − pairs, and find general agreement with the STAR data. The calculation is based on theSTARlight simulation code, which is based on the Weizs¨acker-Williams virtual photon approach.The STAR continuum observations are compatible with two-photon production of e + e − pairs. TheALICE analysis required individual muon p T be greater than 1 GeV/c; this eliminated almost allof the pairs from two-photon interactions, while leaving most of the J/ψ decays.
I. INTRODUCTION
Two-photon collisions were extensively studied at e + e − colliders, where each lepton emitted a photon.These reactions were used to study a variety of hadronicfinal states and also four-lepton final states [1, 2]. Morerecently, they have been studied in ultra-peripheral col-lisions (UPCs) of relativistic heavy ions. UPCs are colli-sions where the nuclei physically miss each other (impactparameter b greater than twice the nuclear radius R A ),but interact electromagnetically. The very strong elec-tromagnetic fields which emanate from highly chargedheavy ions lead to large cross-sections for photonuclearand two-photon interactions [3–5]. Two-photon interac-tions of interest include e + e − production in strong fieldsand light-by-light scattering [6]. The reaction γγ → l + l − has been studied in UPCs by the STAR [7], ATLAS [8],ALICE [9] and CMS [10] collaborations, and good agree-ment with lowest order quantum electrodynamics predic-tions was seen.Recently, the STAR experiment observed an excessof e + e − pairs, produced at small transverse momentum( p T <
150 MeV/c) in peripheral gold-gold and uranium-uranium collisions at a center of mass energies of 200GeV/nucleon pair and 193 GeV/nucleon pair respectively[11, 12]. The e + e − invariant mass spectrum of the low p T excess shows a significant peak at the J/ψ , but alsoincludes a continuum component in the mass range from400 MeV to 2.6 GeV. The signal is most prominent inmore peripheral collisions (60% to 80% centrality) thanin those with smaller impact parameters. Here, 0% cen-trality is a head-on collision with impact parameter b = 0,while 100% centrality is a grazing collision with b = 2 R A .ALICE has also reported on an excess of µ + µ − pairsat low p T [13] in very peripheral (70-90% centrality)lead-lead collisions at a center of mass energy of 2.76TeV/nucleon pair. The peak data are consistent with J/ψ photoproduction [14–16], but they do not see signif-icant continuum production.The STAR and ALICE
J/ψ rates and p T spectra are in agreement with expectations from photoproduction,including the cutoff at very low p T due to interferencebetween photoproduction for photons coming from theopposite directions [17, 18]. However, the broad STARmass continuum does not seem compatible with vectormeson photoproduction. The featureless mass distribu-tion and limitation to low p T are both very suggestive oftwo-photon production of e + e − pairs.In this work, we calculate the rates and kinematic dis-tributions for two-photon production of e + e − pairs inperipheral hadronic collisions, and show that it is gener-ally compatible with the STAR observation and ALICEnon-observation of continuum production at low p T . Theresults provide a basis for a more detailed comparison be-tween the data and two-photon theory. II. METHODS
The cross-sections and kinematic distributions for γγ → e + e − in peripheral collisions are calculated us-ing the photon flux predicted by the Weizs¨acker-Williamsmethod, and the lowest order Breit-Wheeler cross-sectionfor γγ → e + e − . For ultra-relativistic particles, the pho-ton flux at a perpendicular distance b from an emittingnucleus with nuclear charge Z is [4, 5, 19] N ( k, b ) = Z απ k ( (cid:126) cγ ) K ( x ) (1)where k is the photon energy, x = kb/γ (cid:126) c , γ is the ionLorentz boost, α ≈ /
137 is the electromagnetic finestructure constant, and K ( x ) is a modified Bessel func-tion.The cross-section depends on the overlap of the photonfluxes, integrated over all possible transverse positions forthe two ions and the location of the photon-photon in-teraction. This can be simplified to a three-dimensionalintegral over the distance from the first ion to the in-teraction site, b , the distance from the second ion site, a r X i v : . [ nu c l - t h ] J a n b , and the angle φ between the two ion-interaction sitevectors [20]. For peripheral collisions, the ion-ion impactparameter range is restricted to match a desired central-ity bin: b min < | b | < b max . The cross-section to producea final state W from photons with energy k and k is σ = (cid:90) ∞ R A πb d b (cid:90) ∞ R A πb db (cid:90) π dφN ( k , b ) N ( k , b )(2) σ ( k k → W ) θ ( b min < | b − b | < b max )where the θ function is 1 when the inequality is satisfied,and 0 otherwise.This approach assumes that the photons come fromthe electromagnetic fields of the entire nucleus, moving atthe full beam velocity. Since the fields at time t are eval-uated based on the nucleus configuration at a retardedtime τ = t − | b i | /γc [21], this assumption should be satis-fied. This approach also assumes that the photon flux iszero within the emitting nucleus ( i. e. for | b | < R A ). Theinclusion of interactions occurring within one of the nu-clei would slightly increase the calculated cross-section,with the size of the increase rising with increasing pairmass.The final state pair mass M ll is given by the two-photon center of mass energy W . The photon energiesmap into W and rapidity y via W = M ll = 2 √ k k and y = 1 / k /k ).The cross-section to produce pairs of leptons with lep-ton mass m is the Breit-Wheeler cross-section [22]: σ ( γγ → l + l − ) =4 πα W (cid:20)(cid:0) m W − m W (cid:1) ln ( W + √ W − m m ) (3) − (cid:114) − m W (cid:0) m W (cid:1)(cid:21) . The angular distribution of the decay electrons also fol-lows Breit-Wheeler, with the leptons preferentially emit-ted in the forward and backward directions: G ( θ ) = 2 + 4 (cid:0) − m W (cid:1) (1 − m W ) sin ( θ ) cos ( θ ) + m W (1 − (1 − m W ) cos ( θ )) . (4)where θ is the angle between the beam direction and oneof the leptons, in the lepton-lepton center of mass frame.The pair p T is the vector sum of the photon k T ; thephoton k T comes from the Weizs¨acker-Williams approach[17, 23]: dNdk T = 2 F ( k /γ + k T ) k T (2 π ) (( k/γ ) + k T ) (5)where F is the nucleon form factor, per Ref. [24]. Theindividual lepton p T includes contributions from this ini-tial γγ p T , plus the transverse kick acquired from thenon-zero θ in Eq. 4. The calculations are done in the framework of theSTARlight Monte Carlo [19, 24]. We modified STARlightto limit the range of integration in impact parameter to auser selectable range, regardless of whether the two nucleioverlap or not, as shown in Eq. 2; this code is publiclyavailable in the trunk of STARlight [25]. STARlight hasbeen extensively compared with UPC data, with goodagreement found for γγ → l + l − , with data from STAR[7], ATLAS [8], ALICE [9] and CMS [10] collaborations.There is a discrepancy at small pair p T where the equiva-lent photon approximation predicts an overabundance ofpairs, compared to both a lowest order Quantum Elec-trodynamics (QED) calculation and data [7]. ATLASalso sees a small tail of events with larger pair p T ; thecollaboration notes that this might be background, or itmight be from the two-photon signal. Also, STARlightassumes that nuclei are spherical. Uranium is aspherical;this introduces an additional uncertainty in the uranium-uranium calculations. III. RESULTS
Most of the γγ → ee cross-section is for near-thresholdpairs, which are not visible in existing detectors. Thesecalculations focus on experimentally accessible inter-actions, so consider only pairs with invariant massesabove 400 MeV/c . Results are presented for five dif-ferent experimental conditions: 60-80% centrality, 40-60% centrality and 10-40% centrality Au-Au collisionsat a center-of-mass (CM) energy of 200 GeV/nucleon,60-80% U-U collisions at a slightly lower CM energy, 193GeV/nucleon (all at RHIC), and 60-80% Pb-Pb collisionsat a center of mass energy of 2.76 TeV/nucleon at theLHC. These calculations are for a central detector, fol-lowing the STAR acceptance [11]. As noted below, theALICE forward muon spectrometer has little acceptancefor two-photon production of dimuons, and an ALICEstudy in the central region would likely have a similaracceptance to STAR.The collaborations report impact parameter range interms of collision centrality. A Monte Carlo Glauber cal-culation [26] is used to convert from the reported cen-tralities into impact parameters. The calculation findscross-sections of 6.8 barns for Au-Au collisions at RHIC,7.8 barns for U-U collisions at RHIC [27], and 7.6 barnsfor Pb-Pb collisions at the LHC at a center of mass en-ergy of 2.76 TeV/nucleon pair, all with errors that arenegligible compared to other uncertainties in the over-all calculation. These cross-sections are 4-8% lower thansome other results [28], likely because the calculation usesslightly lower inelastic proton-proton cross-sections thanother works. We then use a simple black-disk model toconvert from centrality to impact parameter range, so100% centrality corresponds to b max = √ σ had /π . Table Ishows the centrality regions and hadronic cross-sectionsfor those centralities.Figure 1 shows the rapidity distribution dσ/dy for 60- Ion/ Centrality b − range σ had σ ll (restr.) % satisfying visible ll l ± criteria hadronic event60-80% RHIC Au-Au 11.4-13.2 fm 1.36 b 3.7 mb 3.3% 8 . × − . × − . × − . × − . × − TABLE I. Ions and centralities, impact parameters and cross-sections for two-photon production of lepton pairs. The centralitieswere chosen to match the STAR analysis. The table also gives the calculated photoproduction cross-section (within the givenconstraints on pair mass and rapidity), the fraction of those events that pass the individual lepton kinematic cuts, and thefraction of the hadronic events in that centrality that should contain a visible (satisfying the pair and individual leptonconstraints) lepton pair.
Pair Rapidity10 - - - - - / d y ( m b ) s d FIG. 1. dσ/dy for γγ → e + e − with pair mass more than400 MeV/c , for 60-80% centrality gold-gold (solid black his-togram) and uranium-uranium (dashed red histogram) atRHIC and lead-lead collisions at the LHC (dot-dashed blueline).
80% centrality collisions of gold and uranium at RHICand lead at the LHC for M ee > . . Becausethe distribution is almost independent of centrality, onlyone RHIC curve is shown. The integrated cross-sectionsare 6.7 mb, 8.7 mb and 24.2 mb for gold, uranium, andlead respectively. The LHC cross-section is much largerand covers a much wider rapidity range than the RHICcurves, because of the higher beam energy. The uraniumcross-section is about 36% larger than that for gold, lessthan the 54% increase expected from the naive Z scal-ing. Uranium nuclei are larger than gold nuclei, and theper-nucleon collision energy was a bit lower , so the pho-ton flux, Eq. 1 is cut off at about 10% lower energy thanfor gold. The cutoff is also evident in the rapidity distri-bution, which is slightly narrower than for gold-gold.Figure 2 shows the p T spectra for the individual lep-tons for the three systems, along with, for comparison,the lepton p T from photoproduction of J/ψ in gold-goldultra-peripheral collisions at RHIC [17, 24, 29]. The lep-tons from γγ → e + e − are peaked at very low p T ,in sharpcontrast to the leptons from J/ψ decays. This differenceimmediately shows why the ALICE forward muon spec- (GeV/c) T Lepton p0 0.2 0.4 0.6 0.8 1 1.2 1.4 N u m be r / b i n FIG. 2. Individual lepton p T for Au-Au (solid black his-togram) and U-U (dashed green histogram) at RHIC and Pb-Pb collisions at the LHC (dot-dashed red histogram), alongwith the lepton p T from photoproduction of J/ψ in Au-Auultra-peripheral collisions at RHIC (dotted blue line). trometer cut on muon p T > γγ interactions while retaining thepairs from coherent J/ψ photoproduction, even thoughthe two reactions have not dissimilar pair p T spectra.The p T spectra from the three γγ channels are similar,with small changes due to the per-nucleon collision en-ergy and size of the nuclei.We now turn to calculations of cross-sections withinthe STAR acceptance: pairs with pair mass M ee > . and rapidity | y ee | <
1. STAR also requires thatthe individual leptons satisfy p T,e >
200 MeV/c andpseudorapidity | η e | <
1. Table I shows the cross-sectionsfor five different beam/energy/centrality conditions, forhadronic interactions, the cross-sections for γγ → e + e − within the pair rapidity and pair mass range, the prob-ability for thosepaper events to also satisfy the individ-ual lepton pseudorapidity and p T cuts, and, finally, thenumber of pairs within the full STAR acceptance perhadronic collisions. The hadronic cross-section is the to-tal hadronic cross-section times the width of the central-ity bin.For a fixed condition and acceptance, the restricted ) (GeV/c ee M0 0.5 1 1.5 2 2.5 3 3.5 4 N u m be r / b i n
10 ) (GeV/c ee M0 0.5 1 1.5 2 2.5 3 3.5 4 N u m be r / b i n FIG. 3. Pair invariant mass spectra for (top) Au-Au (blacksolid histogram), U-U (green dashed histogram), and Pb-Pb(red dot-dashed histogram) and (bottom) 60-80% centrality(black solid histogram), 40-60 % centrality (red dashed his-togram), and 10-40% centrality (blue dot-dashed histogram).The bottom three histograms are indistinguishable. two-photon cross-section depends mostly on the widthof the centrality bin. The cross-section is higher for Pb-Pb collisions, because of the higher LHC collision energy;the increase in γγ cross-section with energy is much fasterthan the rise in hadronic cross-section. This increase isreflected in the higher number of visible ee pairs for Pb-Pb collisions than for the lower energy RHIC systems.The restricted cross-section is 40% larger for U-U cross-sections than for Au-Au collisions. This is again lessthan the 54% increase expected from the Z scaling, butlarger than the increase in the all-rapidity cross-section,because, for the heavier nucleus, production is more con-centrated at small | y | .The fraction of lepton pairs that pass the individuallepton cuts is about 3.4%, almost independent of the col-lision conditions, with only a small rise for the higher-energy Pb-Pb collisions. This acceptance is so low be-cause, per Eq. 4, the pairs from two-photon interactionsprefer a forward-backward geometry, and so avoid thecentral region.Figure 3 shows the pair invariant mass distributionsfor events within the STAR acceptance. The Pb-Pbdata spectrum is harder than the Au-Au and U-U dis-tributions, because of the higher beam energy. The U-Uspectrum is slightly softer than the Au-Au distribution,because of the slightly larger nuclear size and lower per- (GeV/c) p0 0.005 0.01 0.015 0.02 N u m be r / b i n (GeV/c) p0 0.005 0.01 0.015 0.02 N u m be r / b i n FIG. 4. Pair p T spectra for (top) Au-Au (black solid his-togram), U-U (green dashed histogram), and Pb-Pb (reddot-dashed histogram) and (bottom) for three invariant massranges: 0.4 to 0.76 GeV/c (black solid histogram), 0.76-1.2GeV/c (red dashed histogram), and 1.2-2.6 GeV/c (bluedot-dashed histogram). The bottom three histograms show aclear mass ordering. The lines in the bottom plot are fits toEq. 6 in the displayed region, as discussed in the text. nucleon collision energy. The shape of these distributionsare similar to the STAR data presented in Ref. [11]; Thenumber of events drops by roughly a factor of 10 as thepair mass doubles from 0.5 GeV to 1.0 GeV, in at leastrough agreement with the STAR data.Figure 4 shows the distribution of lepton-pair p T forthe three species (top), and the three Au-Au centralities(bottom). A significant upturn is seen for p T < . , while at higher energies, the distribution looksquasi-exponential. The p T scale is lowest for Pb-Pbbecause, for a fixed photon energy k , the photon k T drops with increasing ion energy. The U-U distributionis slightly softer than the Au-Au spectrum because thelarger nuclear size softens the energy distribution.At low p T , the equivalent photon approach used herediffers from a lowest-order QED calculation, which pre-dicts a drop-off at low p T . Data from γγ → ee in ultra-peripheral collisions also does not show this increase [7].The peak of the pair p T distribution scales roughly as √ . M ee /γ [19]. In Ref. [7], the data diverged from theequivalent photon calculation for p T <
20 MeV/c, fora sample with M ee >
140 MeV/c . If the √ M ee scal-ing holds, the calculated p T spectrum should be OK for p T >
35 MeV/c, or p T > .
001 (GeV/c) .Following the STAR Collaboration [11], these curvesare fit to exponential distributions, dNdp T = a exp( − bp T ) (6)for events in three mass regions, 0.4 to 0.76 GeV/c , 0.76to 1.2 GeV/c and 1.2 to 2.6 GeV/c . The fit region waschosen to avoid the upturn at low p T : 0.002 (GeV/c) < p T < . The fits are shown with thecolored lines in the bottom panel of Fig. 4. The slopesare 430 ± − , 402 ± − and 367 ± − for the low, medium and high mass regionsrespectively. These numbers are near the upper end ofthe uncertainty ranges reported by STAR [11], but thetrend with increasing mass agrees well with the data. Itshould be noted that STAR used a significantly differ-ent fit range, 0.0004 (GeV/c) < p T < .That range overlaps with the low- p T upturn in Fig. 4and would have led to significantly larger fit slopes, indisagreement with the STAR data. Even for the chosenrange, the slope depends slightly on the chosen fit range.These slope differences are not surprising. For k T (cid:29) k/γ , dN/dk T from Eq. 5 scales as F ( k T ) /k T . Thisnaively implies similar slopes, but k scales linearly withpair mass (with some rapidity dependence), so the loca-tion of the transition to the k T (cid:29) k/γ varies with pairmass. The contribution to pair p T from θ , Eq. 3, shouldalso scale with pair mass, with, for the lepton pseudora-pidity cut, a significant dependence on the pair rapidity. Similar fits were made to the Au-Au, U-U and Pb-Pbdistributions, over the range 400 MeV to 4 GeV . Theyyielded slopes of 404 ± − , 502 ± − and 483 ± − , respectively. The Au-Au andU-U points have a similar trend to the STAR fits, albeitwith a larger separation between the two slopes. IV. CONCLUSIONS
Two-photon production of lepton pairs in ultra-relativistic collisions is a well-understood process, stud-ied in many ultra-peripheral collision analyses. In thiswork, we have studies the two-photon production of lep-ton pairs in peripheral collisions, and found that it de-scribes the general characteristics of the STAR observa-tions of a continuum excess of e + e − pairs at low p T quitewell. ALICE did not observe this continuum; this is ex-pected because of the required minimum p T for muons tobe observable in their forward muon spectrometer. How-ever, it should be visible in a future ALICE mid-rapiditystudy if a low enough lepton p T cut can be applied.I thank Jamie Dunlop, Michael Lomnitz, Lijuan Ruan,Shuai Yang, Wangmei Zha and Zhangbu Xu for use-ful discussions. This work was funded by the U.S.Department of Energy under contract number DE-AC-76SF00098. [1] V. M. Budnev, I. F. Ginzburg, G. V. Meledin andV. G. Serbo, Phys. Rept. , 181 (1975).[2] H. Kolanowski, Two-photon physics at e + e − storagerings , Springer-Verlag, 2006.[3] A. J. Baltz et al. , Phys. Rept. , 1 (2008).[4] C. A. Bertulani, S. R. Klein and J. Nystrand, Ann. Rev.Nucl. Part. Sci. , 271 (2005).[5] G. Baur, K. Hencken, D. Trautmann, S. Sadovsky andY. Kharlov, Phys. Rept. , 359 (2002).[6] M. Aaboud et al. [ATLAS Collaboration], Nature Phys. , no. 9, 852 (2017).[7] J. Adams et al. [STAR Collaboration], Phys. Rev. C ,031902 (2004).[8] M. Dyndal [ATLAS Collaboration], Nucl. Phys. A ,281 (2017).[9] E. Abbas et al. [ALICE Collaboration], Eur. Phys. J. C , no. 11, 2617 (2013).[10] V. Khachatryan et al. [CMS Collaboration], Phys. Lett.B , 489 (2017).[11] S. Yang for the STAR Collabora-tion, presented at Quark Matter 2017,https://indico.cern.ch/event/433345/contributions/2373720/[12] W. Zha for the STAR Collabora-tion, presented at Quark Matter 2017,https://indico.cern.ch/event/433345/contributions/2358104/[13] J. Adam et al. [ALICE Collaboration], Phys. Rev. Lett. , no. 22, 222301 (2016).[14] M. Klusek-Gawenda and A. Szczurek, Phys. Rev. C , no. 4, 044912 (2016).[15] W. Zha et al. , arXiv:1705.01460 [nucl-th].[16] W. Shi, W. Zha and B. Chen, arXiv:1710.00332 [nucl-th].[17] S. R. Klein and J. Nystrand, Phys. Rev. Lett. , 2330(2000).[18] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. Lett. , 112301 (2009).[19] A. J. Baltz, Y. Gorbunov, S. R. Klein and J. Nystrand,Phys. Rev. C , 044902 (2009).[20] G. Baur and L. G. Ferreira Filho, Nucl. Phys. A ,786 (1990).[21] J. D. Jackson, Classical Electrodynamics, 2nd edition ,John Wiley & Sons, 1975[22] S. J. Brodsky, T. Kinoshita and H. Terazawa, Phys. Rev.D , 1532 (1971).[23] M. Vidovic, M. Greiner, C. Best and G. Soff, Phys. Rev.C , 2308 (1993).[24] S. R. Klein, J. Nystrand, J. Seger, Y. Gorbunov andJ. Butterworth, Comput. Phys. Commun. , 258(2017).[25] STARlight is available at https://starlight.hepforge.org/.[26] C. Loizides, J. Kamin and D. d’Enterria,arXiv:1710.07098 [nucl-ex].[27] D. d’Enterria, private communication, 2017.[28] W. Fischer et al. , Phys. Rev. C , 014906 (2014).[29] S. R. Klein and J. Nystrand, Phys. Rev. C60