Xue-Min Jiang(江學(xué)敏) and Yun-De Li(李云德)

Department of Physics,Yunnan University,Kunming 650091,China

AbstractWith a photon spectrum of high energy charged partons,both differential and total cross sections of J/ψ and Υ(1s)inclusive photoproduction in ultra-peripheral p-Pb Collisions(UPCs)at =5.5 TeV are given.A direct photon process and a resolved photon process including fragment and non-fragment contributions are considered.The total cross section is compared to the inelastic production results in coherent p-Pb collisions at the same center-of-mass energy.Our results seem non-negligible.

Key words:ultra-peripheral collisions,Υ(1s),J/ψ,photoproduction

1 Introduction

The photoproduction of heavy quarkonium is an interesting and important topic in the nuclear collisions.It helps to probe the parton distributions and the process of strong interactions.Furthermore,the photoproduction of charmonium is even observable by experiments in the semi-central nucleus-nucleus collisions and even larger than the hadronic contributions in extremely small transverse momentum.[1?3]More studies about photoproduction from Ultra-peripheral to central collisions have been performed.[4?15]Therefore,the detailed studies of subprocess about photoproduction seems urgent and valuable.

Ultra-peripheral collisions(UPCs)are such reactions occurring at large impact parameters more than twice of the colliding nuclear radii.In UPCs,an accelerated charge can be considered as a source of quasi-real photons(the method of Fermi,Weizs?cker and Williams[16])and one of these quasi-real photons may interact either with the opposite ion(photonuclear interaction)or with one of its quasi-real photons(two photon interaction).The physics of ultra-peripheral collisions is reviewed in Refs.[17–21].

In these two interactions mentioned above,most recent studies mainly considered the quasi-real photons as radiated by the whole incident nucleus.[22?24]However,under ultra-relativistic condition,a hadron can also be regarded as a beam of freely moving elementary constituents.[25]Therefore the photons can also be radiated by partons in the incident nucleus.The photon spectrum for a quark can be described as:[25?26]

where α is the electromagnetic coupling parameter,E(E?m),Qqand m are the energy,the charge and the mass of the charged parton correspondingly,x denotes the photon energy in units of E.

By using this photon spectrum of high energy charged partons,the photoproduction processes in ultra relativistic heavy ion collisions have been investigated.[27?28]Besides,the cross section for J/ψ production in ultra peripheral pp collisions at=7 TeV via two photon interaction using this photon spectrum has been calculated in Ref.[29].In this current work,we calculate the cross section of both Υ(1s)and J/ψ production in ultra peripheral p-Pb collisions with this photon spectrum of high energy charged partons.

In p-Pb collisions the photon can be emitted by either proton or nucleus.The cross section is

E(dσγp/dp3)denotes the differential cross section of the process where the photon is emitted by the Pb nucleus,and E(dσγPb/dp3)denotes the differential cross section of the process where the photon is emitted by the proton.In each of them a direct photon process and a resolved photon process are considered,and similar to the two-photon process in Ref.[29],each sub processes can be further divided into a fragment and a non-fragment one.

For a direct γH process,where H stands for proton or the Pb nucleus,the photon can be emitted by nucleus(or nucleon)HB,and then interacts with the partons of nucleon(or nucleus)HAthrough a non-fragment or a fragment process.For the non-fragment process,the differential cross section for each subprocess is

where xa,xbdescribe the parton’s momentum fraction,is the momentum fraction carried by photon andx1and x2are defined asandwhereandBesides,y is the rapidity,M is the mass of the quark pair andis the energy in the center-of mass system.

fa/A(xa)and fb/B(xb)are the parton distributions of nucleus,[30]which satisfy[28]

R(x,A)is the nuclear modification factor,[31]which equals 1 for proton.fa/p(x)is the parton distribution of protons and fa/n(x)is the parton distributions of neutrons.[30]A is the total nucleons number and Z is the total proton number of the nucleus.For Pb there is A=208 and Z=82,while for proton there is A=Z=1.

For a fragment direct photon process,after the interaction of the parton and photon,the heavy quark pair is produced via the fragmentation of an intermediate particle,which can be photon,quark or gluon,and then the heavy quark pair evolves into the final J/ψ or Υ(1s).The differential cross section for each subprocess is

where x=xbx1/(xaxbz?xax2)and xamin=x1/(1?x2),xbmin=xax2/(xa?x1),zmin=x2/xb+x1/xa.The partonic subprocesses γb→ a′b′considered includes γg → gg,γg → γγ,γg → γg,γq → gq,γq → γq.We use the differential cross section in Ref.[35].D(z)is the fragmentation function.[36?37]

For a resolved γH process,the photon from nucleus(or nucleon)HBfluctuates into a quark-antiquark pair and then the quark or antiquark interacts with the partons of nucleon(or nucleus)HAthrough a non-fragment or a fragment process.The differential cross section of the non-fragment process for each subprocess is

In a fragment resolved photon process,after the parton-parton interaction,heavy quark pair is produced via the fragmentation of an intermediate particle which can be photon,quark or gluon,and then the heavy quark pair evolves into the final J/ψ or Υ(1s).The differential cross section for each subprocess is

The cross section changing with respect to the transconsidered are gg→qq,gq→gq,gq→γq,qq→qq′→ qq′.We use the differential cross section in Ref.[35].verse momentum pTin=5.5 TeV p-Pb collisions are plotted in Figs.1–4.The upper limit of the transverse momentum is set as pT=20.0 GeV for both Υ(1s)and J/ψ.Considering the validity of NRQCD,[28]the lower limit of the transverse momentum is set as pT=9.5 GeV for Υ(1s)and pT=3.5 GeV for J/ψ.The strong coupling constant is

with nf=5,Λ=0.2 GeV andwhere m denotes the mass of J/ψ or Υ(1s).The value of the mass are taken from Ref.[39]for mJ/ψ=3.096 GeV and mΥ=9.46 GeV.

Fig.1 (Color online)Subprocesses for J/ψ production in p-Pb collision at center-of-mass energy

Fig.2 (Color online)γp(blue dashed),γPb(blue dotted),total direct photon(red dashed),total resolved photon processes(red dotted)and the total contribution(black solid)for J/ψ production in p-Pb collision at center-of-mass energy

Figure 1 presents the J/ψ production from both nonfragment and fragment process of direct photon and resolved photon process in p-Pb collision at center-of-mass energy=5.5 TeV.It can be seen that the major contribution comes from the non-fragment resolved photon process.

Figure 2 presents the J/ψ production from γp, γPb,total direct photon,total resolved photon process and the sum of the γp and γPb process(or the sum of the total direct photon and total resolved photon process)in p-Pb collision at center-of-mass energy=5.5 TeV.It can be seen that the contribution from γp process and γPb process are almost the same and the contribution from the resolved photon process is slightly larger than the direct photon process in the region around pT<16 GeV and smaller than the direct photon process in the region around pT>16 GeV.

Figure 3 presents the Υ(1s)production from the same subprocesses as in Fig.1,which are non-fragment and fragment process of direct photon and resolved photon process.It can be seen that the major contribution comes from the non-fragment direct photon process in the region around pT<10.5 GeV and in the region around pT>10.5 GeV,the major contribution comes from the fragment resolved photon process.

Fig.3 (Color online)The same as Fig.1 for Υ production.

Fig.4 (Color online)The same as Fig.2 for Υ production.

Figure 4 presents the Υ(1s)production from γp,γPb,total direct photon,total resolved photon process and the sum of the γp and γPb process(or the sum of the total direct photon and total resolved photon process)as in Fig.2.It can be seen that the contribution from γp process and γPb process are almost the same,and the contribution from total resolved photon process is larger than the total direct photon process.

To compare our results with other theoretical results,we calculate the differential cross sections dσ/dy at y=0.For J/ψ the cross sections are

and for Υ(1s)the cross sections are

By comparing with the results of the inelastic production of J/ψ and Υ(1s)in coherent p-Pb collisions in Ref.[24],our result dσJ/ψ/dy=491.89 nb is about 30%of the result in Fig.6 in Ref.[24]at y=0 for J/ψ and dσΥ/dy=3.51 nb is about 12%of the result in Fig.7 in Ref.[24]at y=0 for Υ(1s).It seems that our results are non-negligible.It also can be noticed that the differences between the γp and γPb process is larger than ours for J/ψ production at y=0 in Ref.[24].

In coherent photoproduction the colliding nucleus emit photons as a whole,but in our calculations the nucleus are treated as a bunch of freely moving constituents so the interactions are incoherent,[28]which lead to an additional parton distribution function in the cross sections Eq.(3)compared to the cross sections in Ref.[24].There are other differences between the calculations of Ref.[24]and ours.Firstly,different parton distributions are being used.Secondly,the long-distance matrix elements in Ref.[24]are based on Color Singlet Model[40]while in our calculations the color octet states are also considered.Thirdly,K-factors are included in Ref.[24],which takes into account higher-order corrections[24]while our results only include leading order calculations.Finally,only direct photon production are considered in Ref.[4]and in our calculations the resolved ones are also considered.

In summary,by using a photon spectrum of high energy charged partons we investigate the cross section for J/ψ and Υ(1s)photoproduction in a p-Pb ultra peripheral collisions at=5.5 TeV.The photoproduction interaction is divided into a direct photon process and a resolved photon process.All the subprocesses are further divided into fragment and non-fragment process.Our results of the total cross section seem non-negligible compared to the inelastic production results in coherent p-Pb collisions at the same center-of-mass energy in Ref.[24].