Mapping color fluctuations in the photon in ultraperipheral heavy ion collisions at the Large Hadron Collider
aa r X i v : . [ h e p - ph ] M a r Mapping color fluctuations in the photon in ultraperipheral heavy ion collisions at theLarge Hadron Collider
M. Alvioli, L. Frankfurt,
2, 3
V. Guzey, M. Strikman, and M. Zhalov Consiglio Nazionale delle Ricerche, Istituto di Ricerca per laProtezione Idrogeologica, via Madonna Alta 126, I-06128 Perugia, Italy Particle Physics Department, School of Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israel Department of Physics, the Pennsylvania State University, State College, PA 16802, USA National Research Center “Kurchatov Institute”,Petersburg Nuclear Physics Institute (PNPI), Gatchina, 188300, Russia
We model effects of color fluctuations (CFs) in the light-cone photon wave function and for thefirst time make predictions for the distribution over the number of wounded nucleons ν in theinelastic photon–nucleus scattering. We show that CFs lead to a dramatic enhancement of thisdistribution at ν = 1 and large ν >
10. We also study the implications of different scales and CFsin the photon wave function on the total transverse energy Σ E T and other observables in inelastic γA scattering with different triggers. Our predictions can be tested in proton–nucleus and nucleus–nucleus ultraperipheral collisions at the LHC and will help to map CFs, whose first indications havealready been observed at the LHC. I. INTRODUCTION
One of key features of high energy processes in the target rest frame is that the wave function of a projectile is thesuperposition of coherent (so-called frozen) configurations [1, 2], which is a consequence of the uncertainty principleand Lorentz slowing down of the interaction time. In the pre-QCD times, coherence of high energy processes hasbeen extensively studied in the photon–nucleon ( γN ) and photon–nucleus ( γA ) collisions, for a review, see [3]. Inparticular, it was established that the resolved photon is dominated by the contribution of the light vector mesoncomponent of the photon wave function, which is responsible for about 70% of σ tot ( γN ). The origin of the photoncomponents, which are responsible for the remaining 30% of the γN cross section, is a matter of debate.In QCD coherence of high energy processes is well understood theoretically and established experimentally, fora review, see, e.g. [4, 5]. A distinctive feature of the QCD dynamics is that the interaction strength of differentconfigurations of quarks and gluons, which are QCD constituents of projectile hadrons, photons, etc., varies. Werefer to this phenomenon as color fluctuations (CFs). In the literature one alternatively uses the term cross sectionfluctuations, which refer predominantly to soft hadron (photon) interactions at high energies.A particular dramatic example of CFs is the phenomenon of color transparency (CT) when, as a consequenceof color screening, the strength of the interaction of a high energy hadron (photon) in a configuration with a smalltransverse size is much smaller than the average interaction strength, for a recent review, see [6]. While CT is a naturalmechanism for the interaction with the strength smaller than the average one, several mechanisms like fluctuationsof transverse size, gluon density, the phenomenon of spontaneously broken chiral symmetry, etc. can contribute tofluctuations with the larger-than-average interaction strength.It has been demonstrated long ago by the direct calculations that the contribution of planar diagrams to the totalcross section of a hadron–hadron collision tends to zero with an increase of the collision energy [7]. Therefore, thecontribution of consecutive multiple rescatterings of a hadron projectile to the total cross section of the hadron–nucleusscattering described by planar Feynman diagrams rapidly decreases with an increase of the invariant collision energy s [8]. Thus, within a quantum field theory multiple interactions of the projectile are dominated at high energiesby the contribution of non-planar diagrams. The Gribov–Glauber approximation [1] has been suggested to resolvethis theoretical puzzle. It accounts for the contribution of non-planar diagrams and employs duality between non-planar diagrams and a sum of the elastic contribution and the diffractive intermediate states (duality between s and t channels) to rewrite formulae in the form rather similar to the Glauber approximation [1]. This theoretical descriptionaccounts for coherence of high energy processes and predicts that the geometry of hA collisions should be rather closeto that expected within the Glauber approach. Hence the Gribov–Glauber approximation is routinely used in theevaluation of geometry of the heavy ion collisions. By virtue of duality the Gribov–Glauber approximation includesdiffractive intermediate states, which allow one to account for energy–momentum conservation, see the discussionin [9]. The presence in the formulae of the contribution of inelastic diffractive states leads to the inelastic shadowingcorrection for σ tot ( pA ) [8]. The inelastic shadowing correction was evaluated in a number of papers and found toagree well with the data, see discussion and references in, e.g., [10].It has been suggested that the interaction matrix of the initial hadron or diffractively produced hadronic states withtarget nucleons, which arises within Gribov–Glauber approach, can be diagonalized [11, 12]. In the particular case,when diffractive intermediate states are resonances, this diagonalization has been performed in [13]. The method ofCFs developed in [14] and discussed below is the further generalization of the Gribov–Glauber approximation, whichallows one to account for the fluctuations of the interaction strength and other implications of QCD.Several effects were observed at collider energies, which naturally emerge in the CF framework. First, the ATLASstudy [15] of the charged-particle pseudorapidity distribution dN ch /dη in proton–lead ( pP b ) collisions at √ s NN = 5 . η showed that CFs affect the collision geometry by broadeningthe distribution of the number of participating nucleons N part for large N part (Fig. 13 of Ref. [15]). It can be interpretedas broadening of the distribution in the number of wounded nucleons naturally emerging in the CF approach [14](note that in these early papers, CFs were called cross section fluctuations). It results in a milder dependence of dN ch /dη ( h N part i /
2) on centrality, especially at large negative rapidities in the Pb-going direction (Figs. 11 and 12 ofRef. [15]) than that expected in the combinatorics of the geometric Gribov–Glauber model [16]. The numerical resultsfor the distribution of CFs in the proton used in the analysis of [15] are consistent with the expectations of [17, 18].The second effect is the observation of a large violation of the Gribov–Glauber approximation for the dependenceof the jet production on the centrality observed in pA collisions at the LHC [19] and in dA collisions at RHIC [20],for which a large- x parton momentum fraction of the proton is involved. The central-to-peripheral R CP ratio issuppressed by as much as 80% at the LHC and 50% at RHIC at the largest measured p T . At the same time thecombinatorics given by the geometric Glauber picture works very well for the collisions with up to eight nucleons, if x is small enough, x ≤ .
1. While CFs only increase the deviation of R CP from unity, this pattern is consistent withthe x -dependence of CFs expected within QCD [21].The third effect is the significant suppression of the rate of ρ meson production in the coherent γA → ρA reactionmeasured in Pb-Pb ultraperipheral collisions (UPCs) at the LHC [22] as compared to the expectations of the vectordominance model combined with the Gribov–Glauber approximation for the photon–nucleus interaction. This wasexplained in [23] by taking into account the effect of CFs in the photon wave function, which reduce the effective ρ –nucleon cross section by suppressing the overlap of the vector meson and photon wave functions and lead to sizableinelastic (Gribov) nuclear shadowing due to the photon inelastic diffraction into large masses.In this paper we argue that one can map CFs in the photon wave function using ultraperipheral collisions (UPCs)of heavy ions at the LHC. Although feasibility of UPC studies was analyzed at length in [24], the studies discussedbelow were not addressed. Primarily this is because such analyses became feasible due to the experience accumulatedin the analysis of pA collisions at the LHC. In a long run studies along these lines at the Electron-Ion Collider [25, 26]would provide a detailed information on CFs in the photon and their dependence on the photon virtuality. The mainchallenge for building a realistic description of the photon–nucleon (nucleus) interactions at collider energies is totake into account the multi-scale structure of the light-cone wave function of the photon associated with presenceof soft and hard intrinsic scales. In particular, the photon wave function contains several types of configurations:large- σ configurations characterized by small transverse momenta k t < . ∼ σ πN ( σ πN is the total pion–nucleon cross section),configurations interacting with σ much larger than σ πN related to the presence of soft large-mass diffraction, andsmall- σ configurations with large k t ≥ W ≤
500 GeV, where hard physics should bewell described within the DGLAP approximation, see [27] and references therein. Thus, a more rapid increase ofparton distributions with energy at extremely small x , which is often discussed in the literature, is beyond the scopeof this paper.We propose a model of CFs in the hadronic component of the photon wave function by combining the informationobtained in the analysis of photoproduction of ρ mesons at the LHC energies [23], which enables us to model thephoton configurations interacting mostly with the strength exceeding the typical ρ –nucleon cross section, with thatobtained in photoproduction of J/ψ mesons [28], which is amenable to the perturbative QCD (pQCD) description ofthe weakly-interacting configurations. Since the information on coherent photoproduction of ρ and J/ψ is availablefor W ∼
100 GeV, we focus in this paper on this energy range. The W dependence of the discussed effects for higher W will be considered elsewhere.We apply the resulting model of the photon CFs to the calculation of the distribution over the number of woundednucleons, ν , involved in the inelastic γA scattering. We show that as a consequence of CFs around the average value,the soft inelastic nuclear shadowing effect is strongly enhanced as compared to pA collisions. We also take into accountan additional effect of the different pattern of the interaction of small dipoles, which leads to the leading twist nuclearshadowing and which is absent in the Gribov–Glauber approximation. This effect leads to the significant probabilityfor small dipoles to interact with several nucleons, which noticeably reduces the distribution over ν for small valuesof ν .This paper is organized as follows. In Sect. II we develop a model for CFs in the photon wave function for thephoton–nucleon interaction. In Sect. III we present and discuss our predictions for the distribution over the numberof wounded nucleons (inelastic interactions) in the inelastic photon–nucleus scattering. In the calculations we use ourmodel for CFs in the photon without and with an additional effect of the leading twist nuclear shadowing for theconfigurations interacting with small cross sections. In Sect. IV, we make a prediction for the transverse energy P E T distribution in γA collisions using as a starting point the model of [15] for the dependence of P E T on ν . Finally, inSect. V we discuss possibilities of special triggers, which would allow one to use γA scattering to map out differentcomponents of the photon wave function. II. COLOR FLUCTUATIONS IN γA SCATTERING: GENERAL FORMALISM
At sufficiently high photon energies E γ in the target rest frame, the coherence length associated with the hadronicfluctuation (component) of the photon wave function of mass M exceeds the target radius R T , l coh = 2 E γ /M > R T .In this case, the forward photon–target amplitude (the total photoabsorption cross section) can be expressed in termsof the dispersion representation over the masses M [29]: σ γN = α e . m . π Z dM M R e + e − → hadrons ( M ) σ MN , (1)where α e . m . is the fine structure constant; R e + e − → hadrons ( M ) = σ ( e + e − → hadrons) /σ ( e + e − → µ + µ − ) is the ratioof the e + e − annihilation cross sections into hadrons (everything) and a muon pair, respectively, of a given invariantmass squared M ; σ MN is the total cross section for the interaction of a given component with the target. It isimportant to emphasize that non-diagonal transitions between different photon components have been neglected inEq. (1), which can be justified at present in the case of a heavy nuclear target [29].In the vector meson dominance model (VMD), 70% of the integral in Eq. (1) is due to the sum of ρ , ω and φ mesons, which interact with hadrons with a strength similar to that of a pion (for ρ and ω ) [3, 30, 31].A straightforward generalization of Eq. (1) to the case of deep inelastic scattering (DIS) leads to a gross violationof the approximate Bjorken scaling, and hence to the contradiction with the leading twist QCD expectations. Inthe framework of the parton model, the qualitative resolution of the paradox was suggested by Bjorken [32] byassuming that the interaction is dominated by the so-called aligned quark–antiquark pairs, where the quarks shareasymmetrically the photon longitudinal momentum and have small transverse momenta p t . Such aligned quark–antiquark pairs configurations are strongly interacting with a nuclear target and correspond to typical vector meson-like (and/or other hadronic) configurations. Their contribution to the total virtual photon–nucleon cross section σ γ ∗ N is suppressed by a factor of µ /M , where µ is a soft QCD scale, which leads to the scaling of σ γ ∗ N .In QCD the situation is somewhat different [33]: in addition to the aligned pairs, configurations with large p t also contribute to σ γ ∗ N ; their noticeable contribution is proportional to α s ( p t ) /p t , where α s is the strong couplingconstant, and grows with an increase of the collision energy.Overall this leads to the following approximate picture of the hadronic component of the wave function of thephoton: the majority of the configurations interact with strengths similar to the one given by CFs in the γ → ρ, ω transitions; they dominate at large and medium σ ≥ σ πN . (They also include the fluctuations in the aligned jetcomponent.) Note that with an increase of collision energies, these configurations are likely to be somewhat morelocalized than those in the elastic vector meson–nucleon scattering [23]. In addition, there is a component whichdominates for small σ and which is described by the perturbative (dipole) wave function interacting with the strengthgiven by perturbative QCD.The formalism of cross section fluctuations was introduced before advent of QCD to explain presence of inelasticdiffraction at small t [11, 12]. Its connection to the Gribov inelastic shadowing for double scattering was pointed outin [34]. The basic idea of this approach is to diagonalize the interaction matrix which arises in the Gribov–Glauberapproach in the basis of elastic and diffractive states. The obtained matrix describes the distribution over the valuesof the cross section. If diffractive states are hadron resonances, this program can be effectively performed [13]. It waspossible to extend this formalism by accounting for the well understood QCD phenomena to reconstruct the form ofthe distribution P γ ( σ, W ) [13, 35], where W is the invariant photon–proton energy. While the form of P γ ( σ, W ) canbe calculated from the first principles only for small σ [36], it can be constrained by the following integral relations: Z dσP γ ( σ, W ) σ ≡ h σ i = σ γp ( W ) , Z dσP γ ( σ, W ) σ ≡ h σ i = 16 π dσ γp → Xp ( W, t = 0) dt , (2)where σ γp ( W ) is the total photon–nucleon cross section; dσ γp → Xp ( W, t = 0) /dt is the cross section of photon diffractivedissociation on the proton including the ρ meson peak, which determines the dispersion of CFs encoded in P γ ( σ, W ).Note that the distribution P γ ( σ, W ) is not normalizable [36], i.e., the integral R dσP γ ( σ, W ) is divergent at the lowerintegration limit due to the infinite renormalization of the photon Green’s function (the vacuum polarization).Therefore, to model CFs in the photon, we build a model interpolating between the regimes of small and large σ .For the former, we use the color dipole model (CDM) of the photon wave function, where the (usually virtual) photonis treated as superposition of quark–antiquark pairs (dipoles). The dipoles interact with the target with cross sectionsgiven by the factorization theorem of perturbative QCD for small dipoles [38]. Note that in the literature there is apopular assumption that the contribution of light vector mesons to the photon–nucleon cross section is dual to theintegral over the small masses of q ¯ q pairs (for example, M ≤ for ρ, ω -mesons). The CDM gives a reasonabledescription of CFs for σ ≪ σ ( πN ). For large σ , σ ≫ σ ( πN ), the CFs are determined by non-perturbative effects bothin terms of the photon configurations involved and the strength of the interaction. Therefore, we use the modifiedVMD (mVMD) approach [23] to model their effects.In our analysis we use the results of the approach developed in [37], which gives a good description of the protonstructure function F p ( x, Q ) down to Q ∼ . In this approach, the dipole cross section σ q ¯ q is built in apiece-wise form. For small dipoles corresponding approximately to d t ≤ . − . σ q ¯ q ( W, d t , m q ) = π d t α s ( Q ) x eff g ( x eff , Q ) , (3)where W is the invariant photon–nucleon center of mass energy, Q = λ/d t for light quarks and Q = m q + λ/d t forheavy quarks; x eff = 4 m q /W + 0 . λ/ ( W d t ); m q = 300 MeV for light u , d and s quarks and m c = 1 . q ¯ q configurations in the photon wave function isclose to that of the pion, d π = 0 .
65 fm, and also leads to a smoother interpolation between small and large σ regimes.The parameter λ = 4 is chosen to best reproduce the HERA data on diffractive J/ψ photoproduction [39]. Note,however, that heavy quarks give a very small contribution to the quantities we discuss below.For large dipole sizes, σ q ¯ q is constrained to be equal to the total pion-nucleon cross section at the appropriate energyat d t = d π = 0 .
65 fm and to slowly grow for d t > .
65 fm. Finally, for the intermediate values of 0 . − . < d t < . σ q ¯ q is modeled as a smooth interpolation between the low- σ q ¯ q (3) and large- σ q ¯ q limits.As a result, one can write the interpolation formula for σ γp ( W ) as σ γp ( W ) = X q e q Z dz d d t σ q ¯ q ( W, d t , m q ) | Ψ γ,T ( z, d t , m q ) | , (4)where z is the fraction of photon momentum carried by the quark in the dipole; d t is the transverse distance betweenthe quark and the antiquark; e q are the quark charge. The photon wave function squared in the mixed momentum–coordinate representation is given in [40].It is worth emphasizing here that the dominant contribution to σ γp in Eq. (4) originates from the nonperturbativeinteractions of large-size multiparton hadron-like configurations in the photon wave function, which do not resemble q ¯ q dipoles. Duality considerations suggest that the contribution of such configurations can be approximated usingthe lightest vector meson. Hence, we first calculate P γ in the model of Eq. (4) and next match it at moderate σ tothe nonperturbative model for CFs for transitions to light mesons.Since σ γp ( W ) = R dσσP γ ( σ, W ), one finds within the model of Eq. (4): P dipole γ ( σ, W ) = X q e q (cid:12)(cid:12)(cid:12)(cid:12) πdd t dσ q ¯ q ( W, d t , m q ) (cid:12)(cid:12)(cid:12)(cid:12) Z dz | Ψ γ,T ( z, d t ( σ q ¯ q ) , m q ) | (cid:12)(cid:12) σ q ¯ q ( W,d t ,m q )= σ . (5)Note that the right-hand side of (5) is expressed in terms of σ q ¯ q ( W, d t , m q ), which is then identified with σ . Theresulting distribution P dipole γ ( σ, W ) as a function of σ for different light quark masses m q and at W = 100 GeV isshown by the green dashed curves. To examine the sensitivity of P dipole γ ( σ, W ) to the choice m q , we varied the lightquark mass in the interval 0 ≤ m q <
350 MeV; the results are shown in Fig. 1, where the dashed curves from theupper to the lower one correspond to m q = 0, m q = 250 MeV, m q = 300, and m q = 350 MeV, respectively.Since in the used model the σ ( q ¯ qN ) cross section does not exceed approximately 40 mb, the resulting distribution P dipole γ ( σ, W ) of Eq. (5) has support only for 0 ≤ σ ≤
40 mb.For large σ , the distribution P γ ( σ, W ) can be well approximated by the distribution P ( σ ) for the γ → ρ transition.Taking the sum of the ρ , ω and φ meson contributions, the resulting distribution reads: P ( ρ + ω + φ ) /γ ( σ, W ) = 119 (cid:18) ef ρ (cid:19) P ( σ, W ) , (6)where P ( σ, W ) is taken from [23]; the coefficient of 11 / ω and φ contributions in the SU(3)approximation (which somewhat overestimates the rather small contribution of φ mesons). The form of P ( σ, W ) ismotivated by P π ( σ, W ) for the pion and takes into account presence of the large-mass diffraction at high energies. Itis also constrained to describe the HERA data on ρ photoproduction on the proton, which requires to account for asuppression of the overlap of the photon and ρ wave function as compared to the diagonal case of the ρ → ρ transition.The resulting P ( ρ + ω + φ ) /γ ( σ ) at W = 100 GeV is shown by the blue dot-dashed curve in Fig. 1. -5 -4 -3 -2 -1
0 10 20 30 40 50 60
W = 100 GeV P γ ( σ ) [ m b - ] σ [mb]P γ dipole , m q = 0 - 350 MeVP ( ρ + ω + φ )/ γ P γ hybrid FIG. 1: The distributions P γ ( σ, W ) for the photon at W = 100 GeV. The red solid curve shows the full result of the hybridmodel, see Eq. (7). The green dashed and blue dot-dashed curves show separately the dipole model and the vector mesoncontributions evaluated using Eqs. (5) and (6), respectively. We build a hybrid model of P γ ( σ, W ) by interpolating between the regime of small σ ≤
10 mb, where perturbativedipole approximation is applicable and there is no dependence on the light quark mass m q , and the regime of large σ , where the soft contribution due to the lightest vector meson dominates (hence we neglect the soft contribution ofconfigurations with the large mass and small k t ). In particular, in our analysis we use the following expression: P γ ( σ, W ) = P dipole γ ( σ, W ) , σ ≤
10 mb ,P int ( σ, W ) ,
10 mb ≤ σ ≤
20 mb ,P ( ρ + ω + φ ) /γ ( σ, W ) , σ ≥
20 mb . (7)where P int ( σ ) is a smooth interpolating function. The resulting P γ ( σ, W ) is shown by the red solid curve in Fig. 1.Our model for P γ ( σ, W ) satisfies the constraints of Eq. (2) and gives the good description of the total and diffractiondissociation photon–proton cross sections at W = 100 GeV. Indeed, for σ γp , we obtain R
100 mb0 dσσP γ ( σ, W ) = 135 µ b, which agrees with the PDG value of σ γp = 146 µ b [41]. For the cross section of diffractive dissociation, we obtain R
100 mb0 dσσ P γ ( σ, W ) / (16 π ) = 240 µ b/GeV . It agrees with our estimate of dσ γp → Xp ( t = 0) /dt ≈ µ b/GeV ,which is obtained by integrating the data of [42] over the produced diffractive masses and extrapolating the resultingcross section to the desired W = 100 GeV.To quantify the width of CFs, one can introduce the dispersion ω σ . For the photon, it can be introduced by thefollowing relation: Z dσσ P γ ( σ, W ) = (1 + ω σ ) (cid:18) ef ρ ˆ σ ρN (cid:19) , (8)where ˆ σ ρN is the ρ meson–nucleon cross section. The use of our P γ ( σ, W ) in Eq. (8) gives ω σ ≈ .
93, which shouldbe compared to ω ρσ ≈ .
54 for the pure ρ meson contribution to P γ ( σ, W ) and to ω πσ ≈ .
45 for CFs in the pion [35].
III. COLOR FLUCTUATIONS AND THE NUMBER OF WOUNDED NUCLEONS IN γA SCATTERING
One of important advantages of the Gribov–Glauber approximation is that it accounts for diffractive processes inthe intermediate states including the photon diffraction into large masses and, therefore, conserves energy–momentumby virtue of duality between the parton model and hadronic descriptions. On the contrary, the Gribov–Glauber model,which accounts for elastic intermediate state only [16], violates energy–momentum conservation for the processes withmultiple multiplicity of wound nucleons; it is proven by direct calculations of the energy released in such processes.A Monte-Carlo procedure including finite size effects in the elementary cross section and short-range correlationsbetween nucleons was developed in [18]. Thus, the formulae for the number of wounded nucleons follow directly fromthe formulae for the CFs but differ from the combinatorics of the Glauber model due to the need to average overvalues of the cross section. For hard processes, nuclear shadowing and its impact on the number of wounded nucleonsis calculated separately through the QCD factorization theorem.It has been understood long ago that the large coherence length prevents cascading of rapid secondary hadronssince they are formed outside of a target. Thus, only low-energy cascades are allowed. Hence, the number of woundednucleons given by the formulae below can be probed by selecting a kinematical region in the rapidity, where thecontribution of cascades is expected to be small, see the discussion in the next section.Previously we used the CF model to calculate the cross section of inelastic interactions with exactly ν nucleons, σ ν ,in pA collisions. The model was found to be consistent with the data at least up to ν ∼
10 [15]. Hence it is natural touse a similar approach to account for the CF in the photon wave function in γA scattering for the interaction strengthcomparable or larger than σ ( πN ) (CF effects due to the contribution of small-size configurations to be discussed later,see Eq. (11)). Then, for the photon–nucleus cross section corresponding to exactly ν inelastic interactions with thetarget nucleons, σ ν , one obtains in the Gribov–Glauber model in the optical model limit: σ ν = Z dσP γ ( σ, W ) (cid:18) Aν (cid:19) Z d ~b (cid:20) σ in ( σ ) T A ( b ) A (cid:21) ν (cid:20) − σ in ( σ ) T A ( b ) A (cid:21) A − ν , (9)where ~b is the impact parameter; σ in is the inelastic, non-diffractive cross section for the configuration characterized bythe total cross section σ ; T A ( b ) = R dzρ A ( b, z ) in the nuclear optical density, where ρ A ( r ) is the density of nucleons.Note that we use σ in = 0 . σ (it is based on our estimate that in the considered range, the elastic cross sectionconstitutes approximately 15% of the total one) and the Wood–Saxon density of nucleons for the Pb target [18] inour analysis. In the derivation of Eq. (9), we employ the discussed above equivalence between the Gribov–Glaubermodel and cross section fluctuations approach. This equivalence becomes trivial, if one uses the approximation ofcompleteness over diffractively produced states. It is worth emphasizing that we consider here soft interaction ofthe multiparton configurations of the hadronic component of the photon wave function. For the interaction of theprojectile consisting exactly of two constituents, only ν = 1 , ν wounded nucleons in γA scattering, P ( ν ), reads: P ( ν, W ) = σ ν P ∞ σ ν , (10)where σ ν are given by Eq. (9). The probability distribution P ( ν, W ) calculated using Eqs. (9) and (10) is shown inFig. 2 by the curve labeled “Color Fluctuations”. For comparison, we also show the results of the calculation, wherethe effect of CFs is neglected and the photon is represented by an effective fluctuation interacting with the total crosssection σ = 25 mb; the corresponding curve is labeled “Glauber”.Equation (9) does not take into account that in QCD, configurations corresponding to a small cross section ofthe interaction with the nucleon at high energies interact with the collective small- x gluon field of the nucleus,which is suppressed compared to the sum of the individual gluon fields of the nucleons due to the phenomenon ofthe leading twist (LT) nuclear shadowing [44]. This is supported by the observation of the large LT shadowing incoherent photoproduction of J/ψ in Pb-Pb UPCs at the LHC [45–47]. This implies that Eq. (9) underestimatesthe probability of the interaction with two and more nucleons for small σ , which is determined by the LT nuclearshadowing. It effectively takes into account the implication of QCD factorization theorem: the presence of themultiparton configurations in a small size q ¯ q configurations which are ignored in the eikonal models and in particularin Eq. (9).To take into the account this effect, we modify Eq. (9) and use the following expression: σ ν = Z ∞ dσP γ ( σ, W ) (cid:18) Aν (cid:19) (cid:20) σ in σ in eff Θ( σ − σ ) + Θ( σ − σ ) (cid:21) Z d ~b (cid:20) σ in eff T A ( b ) A (cid:21) ν (cid:20) − σ in eff T A ( b ) A (cid:21) A − ν , (11)where σ = 20 mb (see details below); σ in /σ in eff ≈ σ/σ eff < σ eff (note that σ in eff = 0 . σ eff ) is a function of σ , which we determine P ( ν ) ν Color FluctuationsGeneralized CFGlauber -6 -5 -4 -3 -2 -1
0 2 4 6 8 10 12 14 16
FIG. 2: The probability distributions P ( ν, W ) of the number of inelastic collisions ν . Predictions of Eqs. (9) and (11) areshown by the curves labeled “Color Fluctuations” and “Generalized CF”, respectively.For comparison, the Gribov-Glaubermodel calculation with σ = 25 mb, which neglects the effect of CFs, is shown by the curve labeled “Glauber”. using the following procedure. For large σ > σ , we set σ eff = σ . For σ < σ , σ eff is defined as the cross sectioncorresponding to the gluon shadowing ratio R g ( x ) [44] calculated in the high-energy eikonal approximation: R g ( x eff , Q ) = xg A ( x eff , Q ) Axg N ( x eff , Q ) = 2 Aσ eff Z d ~b (cid:16) − e − σ eff / T A ( b ) (cid:17) , (12)where x eff and Q are the light-cone momentum fraction and the resolution scale, respectively, which correspondto the dipole cross section for the given cross section σ = σ q ¯ q ( W, d t , m q ) (the transverse size d t ), see Eq. (3). Thisprescription for σ eff is based on the observation that since the non-vector-meson component of P γ ( σ ) is relativelysmall, the gluon shadowing can be considered in a simplified approximation, where CFs for the interaction with N ≥ R g is given by the single effective rescattering cross section σ eff .To estimate the value of σ , we notice that the factor of nuclear suppression of coherent J/ψ photoproduction onnuclei is described very well for the LT nuclear shadowing. In particular, R g ≈ . x = 10 − [47], which accordingto Eq. (12) corresponds to σ eff = 17 mb. Therefore, in our analysis we take σ = 20 mb. Our numerical analysisindicates that the results of our calculation depend weakly on the method of smooth interpolation in Eq. (7) andthe assumption about the value of the ratio σ in /σ in eff . We call the resulting approach to the calculation of photon–nucleus inelastic cross sections σ ν the generalized color fluctuation (GCF) model. The result of the calculation of thedistribution over ν using Eq. (11) is shown in Fig. 2 by the curve labeled “Generalized CF”.The results presented in Fig. 2 deserve a discussion. For one inelastic photon–nucleus interaction ( ν = 1), CFsin the photon lead to an almost a factor of two enhancement of P ( ν ) compared to the calculation neglecting CFs.Thus, an inclusion of the approximately 30% small- σ component of the photon wave function (see the discussion inthe Introduction), leads to a large effect in the inelastic γA scattering. This effect is reduced approximately by afactor of two when we include the LT nuclear shadowing (compare the “Color Fluctuations” and “Generalized CF”curves). As ν increases, the small- σ contribution to the distribution P γ ( σ, W ) becomes progressively less importantand all three models give similar results for 2 < ν <
8, where the contribution of the two terms in the integrand ofEq. (11) approximately compensate each other. For large ν >
10, the two models including the effect of CFs in thephoton predict a much broader distribution P ( ν ) than the model neglecting CFs: the enhancement at large ν comesfrom the contribution of the large-mass inelastic diffractive states implicitly included in Eqs. (9) and (11). IV. COLOR FLUCTUATIONS AND THE DISTRIBUTION OVER TRANSVERSE ENERGY
It is impossible to directly measure the number of inelastic interactions ν for collisions with nuclei. Modeling thedistribution over the hadron multiplicity is also difficult due to the lack of the relevant data from γp scattering andissues with implementing energy–momentum conservation. However, the analysis of [15] suggests that the distributionover the total transverse energy, Σ E T , sufficiently far away from the projectile fragmentation region (at sufficientlylarge negative pseudorapidities) is weakly influenced by energy conservation effects (due to the approximate Feynmanscaling in this region) and is also weakly correlated with the activity in the rapidity-separated forward region. Thisexpectation is validated by a recent measurement of Σ E T as a function of hard scattering kinematics in pp collisionsat the LHC [48].Due to the weak sensitivity to the projectile fragmentation region, we expect that the Σ E T distributions in pA and γA scattering at similar energies should have similar shapes for the same ν . In Ref. [15], a model was developed for thedistribution over Σ E T as a function of centrality in pA scattering at large negative pseudorapidities (in the Pb-goingdirection) and √ s = 5 .
02 TeV. In our discussion below, using the one-to-one correspondence between centrality and ν , we denote this distribution f ν (Σ E T ) = 1 /N evt dN/d Σ E T . In the spirit of the KNO scaling, it is natural to expectthat the distribution over the Σ E T total transverse energy in γA scattering, when normalized to the average energyrelease in pp scattering h Σ E T ( N N ) i , weakly depends on the incident collision energy. That is, the distribution over y = Σ E T ( γN ) / h Σ E T ( γN ) i has approximately the same shape at different energies. Hence we model the distributionover y for photon–nucleus collisions using F ν ( y ) = h Σ E T ( N N ) i f ν ( y ), where the factor of h Σ E T ( N N ) i is a Jacobianto keep normalization of R F ν ( y ) dy = P ( ν ).The results of the calculation of F ν ( y ) are presented in Fig. 3 for the Generalized Color Fluctuations (GCF) modelshowing contributions of events with different ν to the normalized distribution over y . We separately show thecontributions corresponding to ν = 1, 2, 3, and 4, and the total contribution corresponding to the sum over all ν (the curve labeled “Total”). One can see that the net distribution is predicted to be much broader than that for the ν = 1 case corresponding to the γp scattering. Also, our results indicate that for y = Σ E T ( γN ) / h Σ E T ( γN ) i ≤ y in γp scattering can be measured in pA UPCs. A first step would be to test that the y distribution in γp and in the γA process with ν = 1 [for example, in the interaction of the direct photon ( x γ = 1) with a gluon with x A ≥ . y < γp and γA to determine the fraction of the ν = 1and ν > y , a range of ν F G C F ( y ) y = Σ E T ( γ N) / < Σ E T ( γ N) > ν = 1 ν = 2 ν = 4 ν = 10 ν = 15Total -5 -4 -3 -2 -1
10 20 30 40 50
FIG. 3: The probability distributions F ν ( y ) over y = Σ E T ( γN ) / h Σ E T ( γN ) i for different numbers of inelastic interactions ν in the Generalized Color Fluctuations (GCF) model. contributes into the cross section. To a good approximation, h ν i − ∝ y . For y = 10, h ν i reaches 2.8 (2.6, 3.1) forthe GCF (CF, Glauber) model with the variance typically of about ∼ .
15. The resulting smearing over ν for given y does not wipe out the difference between the models for the ν distribution, see Fig. 4. F ( y ) y = Σ E T ( γ N) / < Σ E T ( γ N) >Generalized CFColor FluctuationsGlauber -3 -2 -1
10 20 30 40 50
FIG. 4: The net probability distribution P ν F ν ( y ) as a function of y for different models including (curves labeled “GeneralizedCFs” and “Color Fluctuations”) and neglecting (the curve labeled “Glauber”) CFs in the photon. Since the distribution F ( y ) is predicted to be much broader in γA collisions than in γp scattering, the use ofdifferent forward triggers makes it possible to determine the distribution over ν and use it to determine both h σ i and the variance of the P γ ( σ, W ) distribution for selected configuration. For example, in the CF model of Eq. (9)( cf. [9, 18]), which does not include the LT shadowing effects, one obtains the following relations for the averagenumber of inelastic collisions h ν i , h ν i = Aσ in ( γN ) σ in ( γA ) , (13)and for the variance of the cross section for a specific trigger, (cid:10) σ trig (cid:11) h σ trig i = ( (cid:10) ν (cid:11) / h ν i − A A − R d b T A ( b ) . (14)Obviously similar considerations are applicable for the γA interactions with a special trigger including jet production,production of charm, etc. In the case of forward dijet production, for direct photon for x A ≤ .
01, the leading twistshadowing should set in resulting in a broader distribution over ν as compared to the interactions with x A > . ν = 1), see the discussion in sections 6.3 and 6.4 of [44]. For the resolved photons, the distributionover ν (and hence over Σ E T ) should become broader with an decrease of x γ since hadronic configurations with smaller x γ have a larger transverse size. One also expects that for sufficiently small x γ < .
1, the hard process would selectgeneric configurations in the photon and, hence, the distribution over Σ E T would approach the distribution for generic(without trigger) γA collisions. Note that first studies of diffractive dijet photoproduction in pp , pA and AA UPCs atthe LHC in next-to-leading order (NLO) QCD, where CFs in the photon were used to model the effect of factorizationbreaking, were reported in [49].In the case of production of leading charm, small-size dipoles dominate (the variation of the transverse size isregulated by m c and p t ( charm )), which allows one to study leading twist shadowing effects in the charm channel.For instance, for x ∼ − , one expects h ν i ∼ σ charmin ( γA ) /Aσ charmin ( γp ), seeEq. (13).0 V. CONCLUSIONS
In this paper, we quantify the general property of photon–hadron interactions at high energies that the photon canbe viewed as a superposition of configurations interacting with different cross sections, which we call the phenomenonof color fluctuations (CFs), and propose a model for the distribution P γ ( σ, W ) describing these CFs. Using this modeland also additionally taking into account the effect of leading twist nuclear shadowing for small- σ configurations, wefor the first time give predictions for the distribution over the number of inelastic interactions ν in photon–nucleusscattering. Our results show that CFs lead to a dramatic enhancement of this distribution at the small ν = 1 and thelarge ν >
10 compared to the combinatorics familiar from the Glauber model. We also study the effect of CFs on thetotal transverse energy Σ E T released in inelastic γA scattering with different triggers and point to specific indicationsof the CF effect. Our predictions can be tested in the photon–nucleus ( γA ) interactions in UPCs of ions at the LHC,which are characterized by high-intensity fluxes of quasi-real photons in a wide energy spectrum and which can beviewed as an effective “strengthonometor” of the different components of the photon wave function.It would also allow one to obtain (using central tracking of the LHC detectors) unique information on the centralitydependence of the production of forward hadrons carrying a large fraction of the photon momentum ( x F ≥ . p t and large- x F hadron production. At the same time, very little experimental information is available on suppression of the leadinghadron production at the collider energies and on its W dependence. These and other related topics will be discussedin more detail elsewhere.An alternative approach of [50] assumed the dominance of the single Pomeron exchange in the crossed channel,which branches into many Pomerons interacting with nucleons of nuclei. This approach predicts absence of thedepletion of the yield of leading hadrons in high energy hadron–nucleus collisions. Hence, it is in variance with theexisting data, see e.g. [51]. The physics of the Pomeron branching may become important at the energies significantlyexceeding energies achieved at UPCs and at future eA collider, which are the subject of this paper. Acknowledgments
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