Investigating the impact of the gluon saturation effects on the momentum transfer distributions for the exclusive vector meson photoproduction in hadronic collisions
aa r X i v : . [ h e p - ph ] J a n Investigating the impact of the gluon saturation effects on the momentum transferdistributions for the exclusive vector meson photoproduction in hadronic collisions
V. P. Gon¸calves , F. S. Navarra and D. Spiering Instituto de F´ısica e Matem´atica, Universidade Federal de Pelotas,Caixa Postal 354, CEP 96010-900, Pelotas, RS, Brazil and Instituto de F´ısica, Universidade de S˜ao Paulo, CEP 05315-970 S˜ao Paulo, SP, Brazil.
The exclusive vector meson production cross section is one of the most promising observables toprobe the high energy regime of the QCD dynamics. In particular, the squared momentum transfer( t ) distributions are an important source of information about the spatial distribution of the gluonsin the hadron and about fluctuations of the color fields. In this paper we complement previousstudies on exclusive vector meson photoproduction in hadronic collisions presenting a comprehensiveanalysis of the t - spectrum measured in exclusive ρ , φ and J/ Ψ photoproduction in pp and P bP b collisions at the LHC. We compute the differential cross sections taking into account gluon saturationeffects and compare the predictions with those obtained in the linear regime of the QCD dynamics.Our results show that gluon saturation suppresses the magnitude of the cross sections and shifts theposition of the dips towards smaller values of t . PACS numbers: 12.38.-t,13.60.Hb, 24.85.+p
I. INTRODUCTION
Experimental results released in the last years havedemonstrated that photon – induced interactions inhadronic collisions can be used to probe several aspectsof the Standard Model (SM) as well as to test predictionsof Beyond SM Physics (For a recent review see Ref. [1]).In particular, the study of the exclusive vector mesonphotoproduction in hadronic collisions is an importantsource of information about the hadronic structure andalso about QCD dynamics at high energies [2, 3]. Asexclusive processes are driven by the gluon content ofthe target, with the cross sections being proportional tothe square of the scattering amplitude, they are stronglysensitive to the underlying QCD dynamics. Additionally,the squared momentum transfer ( t ) distributions give ac-cess to the spatial distribution of the gluons in the hadronand about fluctuations of the color fields (See e.g. Ref.[4]).In the last years exclusive vector meson photoproduc-tion in hadronic collisions has been discussed by severalauthors considering different assumptions and distinctapproaches (See e.g. Refs. [5–10]). In particular, in Refs.[6, 11] we demonstrated that the experimental LHC Run1 data and the preliminary Run 2 data can be sucessfullydescribed within the color dipole formalism if non - lineareffects in the QCD dynamics are taken into account. Themain advantage of this approach is that the main ingre-dients can be constrained by the very precise HERA dataand hence the predictions for photon – induced interac-tions at the LHC are parameter free. In those previousworks we presented our predictions for the t – integratedobservables – rapidity distributions and total cross sec-tions – which have been measured by the ALICE, CMSand LHCb Collaborations at the LHC in the Run 1. Inprinciple, the t - distributions may be measured in Run2 [1]. This encourages us to extend our previous studies and present the color dipole predictions for the t – spec-trum measured in exclusive vector meson photoproduc-tion in hadronic collisions. In particular, in this paper wewill use the color dipole formalism to describe the photon- hadron interaction, with the scattering amplitude beingexpressed in terms of the impact parameter Color GlassCondensate (bCGC) model, which successfully describesthe t - distributions for the exclusive vector meson pro-duction at HERA. We will compute the t - spectrum forthe exclusive ρ , φ and J/ Ψ photoproduction in pp and P bP b collisions at the LHC energies probed in the Run2. Moreover, in the case of
P bP b collisions, we will con-sider the coherent and incoherent contributions to exclu-sive production, which are associated to processes wherethe nucleus target scatters elastically or breaks up, re-spectively. For a similar analysis considering alternativeapproaches see Ref. [12]. In order to investigate the im-pact of the gluon saturation effects, associated to non -linear contributions for the QCD dynamics at high ener-gies, we will compare our predictions with those obtaineddisregarding these effects, i. e. using a linear model forthe QCD dynamics. As the dipole formalism of exclusiveprocesses has been discussed in detail in our previousworks [6, 11, 13, 14], in the next Section we will only re-view the main elements needed to study exclusive vectormeson photoproduction in hadronic collisions. In SectionIII we will present our predictions for the rapidity and t – distributions and in Section IV we will summarize ourmain conclusions. II. FORMALISM
An ultra relativistic charged hadron (proton or nu-cleus) gives rise to strong electromagnetic fields. In ahadronic collision, the photon stemming from the elec-tromagnetic field of one of the two colliding hadrons caninteract with one photon of the other hadron (photon -photon process) or can interact directly with the otherhadron (photon - hadron process) [15]. In the partic-ular case of exclusive vector meson photoproduction in hadronic collisions, the differential cross section can beexpressed as follows dσ [ h + h → h ⊗ V ⊗ h ] dY dt = (cid:20) ω dNdω | h dσdt ( γh → V ⊗ h ) (cid:21) ω L + (cid:20) ω dNdω | h dσdt ( γh → V ⊗ h ) (cid:21) ω R (1)where the rapidity ( Y ) of the vector meson in the fi-nal state is determined by the photon energy ω in thecollider frame and by the mass M V of the vector me-son [ Y ∝ ln ( ω/M V )]. Moreover, dσ/dt is the differentialcross section for the γh i → V ⊗ h i process, with the sym-bol ⊗ representing the presence of a rapidity gap in the fi-nal state and ω L ( ∝ e − Y ) and ω R ( ∝ e Y ) denoting photonenergies from the h and h hadrons, respectively. Fur-thermore, dNdω denotes the equivalent photon spectrum ofthe relativistic incident hadron, with the flux of a nucleusbeing enhanced by a factor Z in comparison to the pro-ton one. Eq. (1) takes into account the fact that bothincident hadrons can be sources of the photons which willinteract with the other hadron, with the first term on theright-hand side of the Eq. (1) being dominant at positiverapidities while the second term dominating at negativerapidities due to the fact that the photon flux has supportat small values of ω , decreasing exponentially at large ω .As in Refs. [6, 11] we will assume that the photon flux associated to the proton and to the nucleus can be de-scribed by the Dress - Zeppenfeld [16] and the relativisticpoint – like charge [15] models, respectively.In the color dipole formalism, the γh → V h processcan be factorized in terms of the fluctuation of the virtualphoton into a q ¯ q color dipole, the dipole-hadron scatter-ing by a color singlet exchange and the recombinationinto the vector meson V . The final state is characterizedby the presence of a rapidity gap. The differential crosssection for the exclusive vector meson photoproductioncan be expressed as follows dσdt = 116 π |A γh → V h ( x, ∆) | , (2)with the amplitude for producing an exclusive vector me-son diffractively being given in the color dipole formalismby A γh → V h ( x, ∆) = i Z dz d r d b h e − i [ b h − (1 − z ) r ] . ∆ (Ψ V ∗ Ψ) 2 N h ( x, r , b h ) (3)where (Ψ V ∗ Ψ) denotes the wave function overlap betweenthe photon and vector meson wave functions, ∆ = −√ t is the momentum transfer and b h is the impact param-eter of the dipole relative to the hadron target. More-over, the variables r and z are the dipole transverse ra-dius and the momentum fraction of the photon carriedby a quark (an antiquark carries then 1 − z ), respectively. N h ( x, r , b h ) is the forward dipole-target scattering am-plitude (for a dipole at impact parameter b h ) which en-codes all the information about the hadronic scattering, and thus about the non-linear and quantum effects in thehadron wave function. It depends on the γh center - of -mass reaction energy, W = [2 ω √ s ] / , through the vari-able x = M V /W . As in Refs. [6, 11], in what follows wewill consider the Boosted Gaussian model [17, 18] for theoverlap function and the impact parameter Color GlassCondensate (bCGC) model [18] for the dipole – protonscattering amplitude N p . In this model the dipole - pro-ton scattering amplitude is given by [18] N p ( x, r , b p ) = N (cid:16) r Q s ( b p )2 (cid:17) (cid:16) γ s + ln(2 /r Qs ( bp )) κ λ Y (cid:17) rQ s ( b p ) ≤ − e − A ln ( B r Q s ( b p )) rQ s ( b p ) > κ = χ ′′ ( γ s ) /χ ′ ( γ s ), where χ is the LO BFKL char-acteristic function. The coefficients A and B are deter- mined uniquely from the condition that N p ( x, r , b p ), andits derivative with respect to r Q s ( b p ), are continuous at r Q s ( b p ) = 2. The impact parameter dependence of theproton saturation scale Q s ( b p ) is given by: Q s ( b p ) ≡ Q s ( x, b p ) = (cid:16) x x (cid:17) λ (cid:20) exp (cid:18) − b p B CGC (cid:19)(cid:21) γs , (5)with the parameter B CGC being obtained by a fit of the t -dependence of exclusive J/ψ photoproduction. The fac-tors N and γ s were taken to be free. In what follows weconsider the set of parameters obtained in Ref. [19] byfitting the recent HERA data on the reduced ep crosssections: γ s = 0 . κ = 9 . B CGC = 5 . − , N = 0 . x = 0 . λ = 0 . t , with the coherent one being dom-inant when t →
0. The coherent cross section is givenby Eq. (2) in terms of the dipole - nucleus scatteringamplitude N A . As in our previous works [6, 11, 13, 14],we will assume that N A can be expressed as follows N A ( x, r , b A ) = 1 − exp (cid:20) − σ dp ( x, r ) A T A ( b A ) (cid:21) , (6)where T A ( b A ) is the nuclear profile function, which isobtained from a 3-parameter Fermi distribution form ofthe nuclear density normalized to 1, and σ dp is the dipole-proton cross section that is expressed by σ dp = 2 Z d b p N p ( x, r , b p ) (7)with N p given by the bCGC model. For the calculationof the differential cross section dσ/dt for incoherent inter-actions we apply for the vector meson photoproductionthe treatment presented in Ref. [20], which is valid for t = 0. Consequently, we have that dσ inc dt = 116 π Z dzdz ′ d r d r ′ (Ψ V ∗ Ψ)( z, r )(Ψ V ∗ Ψ)( z ′ , r ′ ) h|A| i , (8)with the average of the squared scattering amplitude be-ing approximated by [20] h|A ( r , r ′ , t ) | i = 16 π B p Z d b A e − B p ∆ A N p ( x, r ) N p ( x, r ′ ) A T A ( b A ) × exp n − π ( A − B p T A ( b A ) (cid:2) N p ( x, r ) + N p ( x, r ′ ) (cid:3)o , (9)where N p ( x, r ) is the dipole - proton scattering ampli-tude. The parameter B p is associated to the impact pa-rameter profile function of the proton.In order to investigate the impact of gluon saturationeffects on the exclusive vector meson photoproduction wewill also estimate the differential cross sections assumingthat N p ( x, r , b p ) is given by the linear part of the bCGCmodel, which is N p ( x, r , b p ) = N (cid:18) r Q s ( b p )2 (cid:19) (cid:16) γ s + ln(2 /r Qs ( bp )) κ λ Y (cid:17) , (10) with the same parameters used before in Eq. (4). More-over, in the case of γP b interactions we will assume thatthe dipole - nucleus amplitude can be expressed by N A ( x, r, b A ) = 12 σ dp ( x, r ) AT A ( b A ) (11)with σ dp expressed by Eq. (7) and N p given by Eq.(10). Using Eq. (10) we disregard possible non-lineareffects in the nucleon. On the other hand, using Eq. (11)we disregard the multiple scatterings of the dipole withthe nucleus, which generate non-linear effects in the fullcalculation. d σ / d t d Y [ µ b / G e V ] Non-LinearLinear p + p → ρ + p + p s = 13 TeV t = t min d σ / d t d Y [ µ b / G e V ] Non-LinearLinear p + p → φ + p + p s = 13 TeV t = t min d σ / d t d Y [ nb / G e V ] Non-LinearLinear p + p → J/ ψ + p + p s = 13 TeV t = t min FIG. 1: Rapidity distribution for the exclusive ρ , φ and J/ Ψphotoproduction in pp collisions at √ s = 13 TeV. III. RESULTS
In what follows we will present our predictions for ex-clusive vector meson photoproduction in pp and P bP b collisions at the LHC energies of Run 2. In particular,we will consider pp collisions at √ s = 13 TeV and P bP b at 5.02 TeV. Our main focus will be on the transverse mo-mentum distributions, which are expected to be studiedconsidering the higher statistics of Run 2 [1]. However,firstly let us analyse the impact of the gluon saturationeffects on the rapidity distributions at a fixed value ofthe momentum transfer t . We will estimate Eq. (1) for t = t min , with t min = − m N M V /W . In Fig. 1 wepresent our predictions for the rapidity distributions tobe measured in pp collisions. We observe that the dif-ference between the linear and non - linear predictions d σ / d t d Y [ b / G e V ] Non-LinearLinear
Pb + Pb → ρ + Pb + Pb s = 5.02 TeV t = t min -1 d σ / d t d Y [ b / G e V ] Non-LinearLinear
Pb + Pb → φ + Pb + Pb s = 5.02 TeV t = t min -3 -2 -1 d σ / d t d Y [ b / G e V ] Non-LinearLinear
Pb + Pb → J/ ψ + Pb + Pb s = 5.02 TeV t = t min FIG. 2: Rapidity distribution for the exclusive ρ , φ and J/ Ψphotoproduction in
P bP b collisions at √ s = 5 .
02 TeV. is larger for lighter vector mesons, with the gluon satu-ration effects decreasing the magnitude of the cross sec-tions. In particular, for exclusive ρ photoproduction, thepredictions differ by a factor ≈ Y = 0. On thehand, for the J/ Ψ production, the predictions are simi-lar. These results are expected, since the gluon satura-tion effects are predicted to suppress the contribution ofthe large size dipoles, which are dominant in the ρ case,but contribute less for the J/ Ψ production. Moreover,these results indicate that the analysis of φ production isan important probe of the non - linear QCD dynamics.In Fig. 2, we present our predictions for P bP b collisions.In this case the difference between the linear and non -linear predictions is larger in comparison to the pp one.The difference is a factor of the order of 10 at Y = 0 for ρ production, while for J/ Ψ production it is ≈
2. This re- t | [GeV ]10 -6 -4 -2 d σ / d Y d t [ nb / G e V ] Y = 0, Non-LinearY = 3, Non-LinearY = 0, LinearY = 3, Linear p + p → ρ + p + p s = 13 TeV0 0.5 1 1.5 2 2.5 3 3.5 4| t | [GeV ]10 -6 -4 -2 d σ / d Y d t [ nb / G e V ] Y = 0, Non-LinearY = 3, Non-LinearY = 0, LinearY = 3, Linear p + p → φ + p + p s = 13 TeV0 0.5 1 1.5 2 2.5 3 3.5 4| t | [GeV ]10 -8 -6 -4 -2 d σ / d Y d t [ nb / G e V ] Y = 0, Non-LinearY = 3, Non-LinearY = 0, LinearY = 3, Linear p + p → J/ ψ + p + p s = 13 TeV FIG. 3: Transverse momentum distributions for the exclusive ρ , φ and J/ Ψ photoproduction in pp collisions at √ s = 13 TeVassuming two different values for the vector meson rapidity. sult is also expected, since the saturation scale Q s , whichdefines the onset of the gluon saturation effects, increaseswith the atomic mass number ( Q s ≈ A / ). Our resultsindicate that in exclusive light vector meson photopro-duction in AA collisions we are probing deep in the sat-uration regime. Moreover, we observe that gluon satura-tion effects are non - negligible in the J/ Ψ production. Asverified in pp collisions, the study of φ production can beuseful to understand in more detail the QCD dynamics.Let us now to analyze the impact of the gluon satu-ration effects on the transverse momentum distributions.Initially, let us consider pp collisions at √ s = 13 TeVassuming two different fixed values for the vector mesonrapidity ( Y = 0 and 3). The linear and non - linear t | [GeV ]10 -1 d σ / d Y d t [ b / G e V ] Coh, Non-LinearCoh, LinearInc, Non-LinearInc, LinearY = 0
Pb + Pb → ρ + Pb + Pb s = 5.02 TeV t | [GeV ]10 -2 -1 d σ / d Y d t [ b / G e V ] Coh, Non-LinearCoh, LinearInc, Non-LinearInc, LinearY = 0
Pb + Pb → φ + Pb + Pb s = 5.02 TeV0 0.01 0.02 0.03 0.04| t | [GeV ]10 -4 -3 -2 -1 d σ / d Y d t [ b / G e V ] Coh, Non-LinearCoh, LinearInc, Non-LinearInc, LinearY = 0
Pb + Pb → J/ ψ + Pb + Pb s = 5.02 TeV FIG. 4: Transverse momentum distributions for the exclusive ρ , φ and J/ Ψ photoproduction in
P bP b collisions at √ s = 5 . predictions for exclusive ρ , φ and J/ Ψ photoproductionare presented in Fig. 3. Our results for Y = 0 indicatethat the presence of gluon saturation effects shifts the dippositions to smaller values of the transverse momentum,with the shift being larger for lighter mesons, where thecontribution of these effects is larger. In particular, forthe J/ Ψ production, the shift is small ∆ | t | ≈ . ,while for ρ we have ∆ | t | ≈ . . Moreover, for theproduction of light vector mesons, the number of dips inthe range | t | ≤ is larger when gluon saturationeffects are present. Another important aspect that canbe observed in Fig. 3 is that the position of the dip isnot modified when we increase the rapidity. However, itis not so pronunced as for central rapidities. t | [GeV ]10 -1 d σ / d Y d t [ b / G e V ] Y = 0, Non-LinearY = 3, Non-LinearY = 0, LinearY = 3, Linear
Pb + Pb → ρ + Pb + Pb s = 5.02 TeV0 0.01 0.02 0.03 0.04| t | [GeV ]10 -2 -1 d σ / d Y d t [ b / G e V ] Y = 0, Non-LinearY = 3, Non-LinearY = 0, LinearY = 3, Linear
Pb + Pb → φ + Pb + Pb s = 5.02 TeV0 0.01 0.02 0.03 0.04| t | [GeV ]10 -4 -3 -2 -1 d σ / d Y d t [ b / G e V ] Y = 0, Non-LinearY = 3, Non-LinearY = 0, LinearY = 3, Linear
Pb + Pb → J/ ψ + Pb + Pb s = 5.02 TeV FIG. 5: Transverse momentum distributions for the exclusive ρ , φ and J/ Ψ photoproduction in
P bP b collisions at √ s =5 .
02 TeV. The predictions for the sum of the coherent andincoherent contributions are presented for different values ofthe vector meson rapidity.
In Fig. 4 we present our predictions for
P bP b collisionsat √ s = 5 .
02 TeV. We consider Y = 0 and present sepa-rately the coherent and incoherent contributions. Similarresults are obtained for Y = 3. In the case of the incoher-ent predictions we only present predictions for | t | ≥ . , since the model proposed in Ref. [20] and used inour calculations fails to describe the vanishing of the in-coherent cross section as | t | →
0. As expected, we findthat the coherent cross section clearly exhibits the typi-cal diffractive pattern. Moreover, the coherent processesare characterized by a sharp forward diffraction peak andthe incoherent one by a weak t - dependence. We have verified that the incoherent processes dominate at large- | t | and the coherent ones at small values of the mo-mentum transfer. This is expected, since increasing themomentum kick given to the nucleus the probability thatit breaks up becomes larger. Additionally, the presenceof gluon saturation effects strongly decreases the mag-nitude of the coherent cross sections, in particular forlighter vector mesons, and implies a shift in the positionof the dip to smaller values of t . In the case of the incoher-ent contribution, we have that the linear and non-linearpredictions are similar.Our results indicate that incoherent processes domi-nate at large - | t | and the coherent ones at small valuesof the momentum transfer. Therefore, one can expectthat the analysis of the t dependence can be useful toseparate coherent and incoherent interactions. However,as discussed in detail in Refs. [21, 22], the experimentalseparation of these processes is still a challenge. An al-ternative is the detection of the fragments of the nuclearbreakup produced in the incoherent processes. e.g. thedetection of emitted neutrons by zero - degree calorime-ters. Considering that this separation is not yet possible,in Fig. 5 we present our predictions for the sum of thecoherent and incoherent contributions. We observe thatthe incoherent contribution partially fills the dip in thetranverse momentum distribution. However, it is stillpresent, with its position being affected by gluon satura-tion effects. IV. CONCLUSIONS
The study of exclusive vector meson photoproductionin hadronic collisions is strongly motivated by the expec-tation that this process may allow us to probe the QCDdynamics at high energies, driven by the gluon content ofthe target (proton or nucleus) which is strongly sensitiveto non-linear effects (parton saturation). Our goal in thispaper was to extend and complement previous studiesabout exclusive vector meson photoproduction in pp and P bP b collisions, presenting the color dipole predictionsfor the transverse momentum distributions taking intoaccount gluon saturation effects in the QCD dynamics.In particular, we have used an approach that reproduceswell the available HERA data on vector meson photo andelectroproduction, including data on the t -distributions,as well as the Run 1 LHC data on vector meson photopro-duction. We presented predictions for the t - spectrumof the exclusive ρ , φ and J/ Ψ photoproduction in pp and P bP b collisions, which could be compared with futureexperimental LHC data. In order to estimate the impactof the gluon saturation effects, we also have presenteda comparison with the predictions obtained disregardingthese effects. Our results demonstrate that gluon satu-ration effects reduce the magnitude of the cross sections,with the reduction being larger for lighter vector mesons.Moreover, the gluon saturation effects change the posi-tions of the dips with respect to the linear regime, shiftingthe dips to smaller values of the transverse momentum.Finally, our results indicate that dips predicted by the co-herent contribution in
P bP b collisions should be visible,independently of the fact that this contribution could notbeen easily experimentally separated. These results arerobust predictions of the saturation physics, which canbe used to investigate non-linear QCD dynamics in thekinematical range of the Run 2 of the LHC.
Acknowledgments
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