in high-multiplicity pp collisions: lessons from pA
Gluons at small in high-energy nuclei overlap in the longitudinal direction, so the nucleus acts as a single source of gluons, like higher Fock components in a single nucleon, which contribute to inelastic collisions with a high multiplicity of produced hadrons. This similarity helps to make a link between nuclear effects in and high-multiplicity collisions. Such a relation is well confirmed by data for the production rate in high-multiplicity events measured recently in the ALICE experiment. Broadening of transverse momentum is predicted for high-multiplicity collisions.
pacs:11.80.La, 12.40.Nn, 13.85.Hd, 12.38.Qk
Hadron multiplicities larger than the mean value in collisions can be reached due to the contribution of higher Fock states in the proton, containing an increased number of gluons. Correspondingly, the relative rate of production will be also enhanced, because heavy flavors are produced more abundantly in such gluon rich collisions.
We define the ratios of measured multiplicity to average multiplicity in collisions per unit of rapidity as , and differentiate between the general hadron multiplicity ratio and the ratio :
More gluons participating in collisions with explain why rises with increasing . Of course such fluctuations are rare. The absolute value of the production rate in such rare events might be very low. Although qualitatively such a correlation is rather obvious, its quantitative description is far from being trivial. In this paper we outline and employ a close relation between the - correlation in and collisions.
In a boosted high-energy nucleus the longitudinal distances between the bound nucleons are contracting with the Lorentz factor , while the small- glue in each nucleon contracts much less, as kancheli (). For instance, in a collision at the c.m. energy of LHC at the mid rapidity, , where is the mean transverse momentum squared of gluons. So gluons stick out far from the Lorentz contracted nuclear disc. In the nuclear rest frame the same effect is interpreted as a long lifetime of the gluon cloud, which propagates in the longitudinal direction over the distance , four orders of magnitude longer than the mean inter-nucleon spacing in nuclei at LHC energies. Therefore all gluons, which overlap in the transverse plane, also overlap longitudinally. Thus, they can be treated as a single gluon cloud originating from one source with increased density, equivalent to higher Fock states in a single nucleon.
We claim that one can emulate the dependence of on in high-multiplicity pp collisions by the analogous correlation in collisions. In the latter case, one can use as the numerators in the ratios (1) and (2) the mean multiplicities of light hadrons and measured in collisions, while the denominators remain the same as in (1)-(2), so they are also here the mean hadron and the multiplicities in collisions.
Ii High-multiplicity events in and collisions
Multiplicity distributions in and collisions at high energies have been studied ak (); karen1 (); karen2 () within Regge phenomenology, based on the Abramovsky-Gribov-Kancheli (AGK) cutting rules agk (). The simultaneous unitarity cut of elastic amplitude through pomerons corresponds to the production of showers of particles, i.e. to a multiplicity times higher than in one cut pomeron. Weight factors for these graphs are usually estimated either within the eikonal model (see Fig. 1, left), or in the quasi-eikonal approximation karen3 (), taking into account intermediate diffractive excitations.
Particle production in collisions has many similarities to high-multiplicity events. The Glauber model, equipped with the AGK rules, corresponds also to the eikonal graphs in Fig. 1 (left). However, in pA-collisions the Pomerons are attached to different nucleons in the nuclear target. For this reason high-multiplicity events in -collisions are enhanced, because the weight factors are different from , and the graph with cut Pomerons (production of showers) contains the factor . As a result, the inclusive cross section of particle production acquire no nuclear shadowing kancheli70 (); mueller (). The mean number of produced showers, so called number of collisions,
and the mean hadron multiplicity increase as . The nuclear ratio, is defined as a ratio of the mean hadron multiplicities in to collisions. In the Glauber eikonal model it is given by the number of collisions,
However, comparison with data shows that this relation significantly overestimates the hadron multiplicity. The popular parametrization for the nuclear effects,
shows, when fitted to data, that huefner ().
The eikonal Glauber approximation Eq. (4) ignores several corrections. Energy conservation shrinks the allowed energy interval with rising number of cut pomerons froissaron (). The source partons, i.e. the valence and sea quarks, which participate in the multi-pomeron exchange, are distributed in rapidity, and this affects the energy and multiplicity in each cut pomeron. This is taken into account in the quark-gluon string qgsm () or dual parton capella () models, which describe quite well multiple hadron production kaidalov (); kaidalov-karen ().
Another source of reduction of multiplicity are coherence effects. The eikonal description relies on the Bethe-Heitler regime of radiation illustrated in Fig. 1 (left). However, amplitudes of gluon radiation from inelastic interactions on different nucleons interfere, leading to a damping of the radiation spectrum, known as Landau-Pomeranchuk suppression, or gluon shadowing. The related radiation pattern is illustrated in Fig. 1 (right). Maximal suppression occurs when the gluon density glr (); al () saturates, which leads to a modification of the transverse momentum distribution of gluons called color glass condensate mv ().
A comprehensive analysis of data busza () from fixed-target experiments led to . The recent analysis of data in phobos (), at , found good agreement with the simple behavior , where the number of participants for collisions is . This relation is equivalent to Eq. (5) with . While this value of in Eq. (5) underestimates data for by about one standard deviation, a larger value leads to an overestimation by a similar magnitude. For further calculations we treat the interval as a measure of experimental uncertainty. This interval agrees with the multiplicity measured recently at in alice-mult (), , which leads to .
Finally we also want to relate directly to the nuclear mass number . We evaluate the dependence of in Eq. (3), using as a nuclear radius with and equating the nuclear inelastic cross section with the geometrical one. Then we obtain for heavy nuclei
Although can be calculated much more precisely, even including Gribov corrections mine (); kps-heraB () and correlations in nuclei ciofi (), the accuracy of calculation in (6) hardly affects the final result (see below).
Iii Correlation between the multiplicity and rate on nuclei
Both the production rate and the mean multiplicity of light hadrons in collisions rise with . Here we attempt to relate them directly.
The mechanisms of the nuclear effects in production has been debated since the first accurate measurements at SPS in the NA3 experiment na3 (). Besides the usual nuclear enhancement of hard processes by the factor , the production rate of exposes a significant suppression, especially at forward rapidities. Depending on the collision energy, the mechanisms of suppression can include energy loss and break up of the dipoles, higher twist shadowing of charm quarks and leading twist gluon shadowing (see review puzzles ()). No consensus has been reached so far with respect to the mechanisms of production, either in or even in collisions. Therefore we prefer to rely on data here.
where depends on and . The results of the E866 experiment, which have the best accuracy, give at . NA3 data na3 () at lower energy and PHENIX data phenix-psi () at agree with this value. So far the ALICE experiment has measured only at forward rapidities, with a similar nuclear suppression alice-psi ().
Notice that as long as does not vary in a wide energy range, as was discussed above, the relation (8) is nearly energy independent. Indeed, the first factor apparently has no energy dependent ingredients, only the second factor contains the slowly rising , which results in an extremely weak overall energy dependence .
The exponent is known to vary with rapidity. It drops significantly at large Feynman in the fixed-target experiments na3 (); e866 (). Moreover, data agree that scales with . However, data taken so far at RHIC and the LHC correspond to very small and do not show any clear dependence of on . Although theoretical models predict a falling behavior of with , we prefer here to rely on data and provide numerical predictions only at mid-rapidity.
Eq. (8) is the final relation between the multiplicities of and light hadrons in collisions, which we are going to apply to high-multiplicity collisions.
Iv Bridging and high-multiplicity collisions
Before one links multi particle processes in and collisions one may argue that there exists an essential difference. Whereas high-multiplicity collisions necessitate symmetric higher Fock components in both protons, collisions look asymmetric. Although the small- gluon cloud from several longitudinally overlapping nucleons in a high-energy nucleus acts like a higher Fock state in a single nucleon, the proton on the other side may still be in an averaged Fock state. This argument, however, is not correct, because the weight factors for different Fock states in the proton depend on the way they are probed. The scale evolution of the parton distribution function represents a well known example: the gluon density in the proton at small steeply rises with the of the photon probing it. Therefore the mean parton configuration in the proton also drifts to higher Fock components when probed by a collision with a nucleus (see Ref. boosting ()).
This idea is explicitly realized in the quark-gluon string qgsm () or dual capella () models. Multiple inelastic interactions in a collision are not sequential, but occur ”in parallel”, i.e. form a multi-sheet topology. Otherwise, several pomerons could not undergo a simultaneous unitarity cut as is requested by the AGK cutting rules. An example of a double interaction of the proton with two bound nucleons cap-kop () is depicted in Fig. 2.
One can see in Fig. 2 that the proton undergoing multiple interactions has the same number of endpoints of strings as both bound nucleons together, namely two color triplets and two anti-triplets. Therefore the color content inside the proton and the nucleus for a - collision looks symmetric.
Notice that semi-enhanced fan-type reggeon graphs make the rapidity dependence of the multiplicity distribution asymmetric koplik (). However, these graphs are large and the asymmetry is significant only in the nuclear fragmentation region. In the central rapidity region at high energies, the fan diagrams are suppressed by the smallness of the triple-pomeron coupling, which originates from the smallness of gluonic dipoles within the QCD description spots ().
Another source of a possible distinction between high-multiplicity and collisions is the difference in the impact parameter pattern of multiple interactions. It has been known since the early era of Regge theory that multi-shower particle production is characterized by smaller impact parameters of collision than in the production of a single shower, and this is a direct consequence of the AGK cutting rules. Indeed, the mean impact parameter squared for a single pomeron exchange is , where is the standard Regge parametrization for the energy dependent elastic slope bkk (), .
The slope for the elastic amplitude with Pomeron exchange calculated in the eikonal model is times smaller . Thus, the events with multiplicity times higher than the mean value are produced in collisions with impact parameters as small as,
Smallness of the mean impact parameter of a collision means larger transverse momentum of the parton,
where is the transverse momentum gained in a single pomeron interaction.
On the other hand, a parton propagating through a nucleus undergoes multiple interactions with different nucleons, with impact parameters much larger than in (9). Nevertheless, the total transverse momentum gained by the parton is the same as in (10), since the single-Pomeron interaction with every nucleon remains the same, . Moreover, in high-multiplicity events in collisions, where one should convolute multiple-pomeron interactions with separate nucleons with increasing number of collisions, the result (10) remains valid. Indeed, comparison of measured in and collisions at equal hadron multiplicities demonstrate the equality of the mean transverse momenta at not too large multiplicities , which is the range of our further calculations.
However, at higher multiplicities in collisions was found to be considerably higher than in morsch (). This remarkable observation clearly shows an onset of a new dynamics at very high , related to the existence of two scales in the proton. The semihard scale corresponds to the short-range glue-glue correlation radius pisa (), or the small size of instantons shuryak (). Small gluonic spots in the proton kst2 (); spots (), which is a minor effect in the elastic scattering k3p (); totem (), give a significant and rising contribution at high multiplicities. They are characterized by a much higher transverse momenta of gluons and lead to a steep growth of at high multiplicities. This is a much smaller effect in collisions, which gain high multiplicities mainly by increasing .
While these two scales affect light hadron production, they have practically no influence on the production of , which is characterized by an order of magnitude higher scale. For this reason, we can safely apply the results of production in collisions to high-multiplicity events in collisions.
V production in high-multiplicity collisions
Now we are in a position to rely on nuclear effects observed (or calculated) in production in collisions, attempting to predict analogous effects in high-multiplicity events.
v.1 Production rate
We assume that the relation (8), derived for nuclear targets, can be applied to production in collisions. Within the above mentioned uncertainty in the value of , the relation Eq. (8) is plotted in Fig. 3 as the yellow strip. We can now test our hypothesis that the dependence of on in collisions is the same as the dependence of on in collisions, by comparing with data for high-multiplicity collisions alice-psi-mult (). We see that the dependence predicted from data on collisions agrees well with data at mid rapidity.
Notice that although the second factor in (8) was calculated approximately in the black disc limit, the result is rather accurate due to smallness of .
v.2 broadening of
The analogy between high-multiplicity events and collisions can be extended further. It was observed experimentally e866 (), and well understood theoretically jkt (); broad (), that the mean transverse momentum squared of the increases in compared to collisions.
As far as the gluon radiation time exceeds the nuclear size, the radiation process does not resolve between a single and multiple interactions, but is sensitive only to the total kick to the scattering color charge. For instance, if a parton gets the same momentum transfer interacting with a proton or with a nuclear target, the radiation of gluons with large should be the same. This means that multiple interactions, either in high-multiplicity interactions or in collisions, affect the distribution of produced similarly, leading in both case to broadening defined as,
where is the gluon energy in the nuclear rest frame; the nuclear thickness function, is given by the integral of the nuclear density along the parton trajectory.
The coefficient controls the behavior of the universal dipole cross section at small dipole sizes,
This factor steeply rises with energy. For produced at at the mid-rapidity . At this energy and the factor was calculated in broad () at . Inclusion of gluon shadowing corrections broad () substantially reduces this factor . We rely on this value for further evaluations.
The expected broadening of produced in high-multiplicity collisions at and is plotted in Fig. 4 as a solid line, as function of the normalized multiplicity . Calculations are done with . The dashed line shows, for comparison, broadening calculated without shadowing corrections.
High-multiplicity collisions at high energies exhibit features which traditionally have been associated with nuclear effects. Here we observed a close similarity between multiple interactions in and collisions. In order to enhance multiple interactions in the former case one should trigger on high multiplicity of produced hadrons, while in the latter case one can reach the same multiplicity due to the increased number of collisions Eq. (3). We employed the phenomenological description of the mean multiplicity in collisions, Eq. (5), and the observed nuclear effects for production, enabling us to predict the multiplicity dependence of the production rate in collisions. The results agree well with the correlation between and observed in collisions at the LHC alice-psi-mult (). We also predicted broadening for produced in high- compared with mean-multiplicity collisions. We relied on data at mid-rapidity, since the possible rapidity dependence of at the LHC energy is poorly known. If, however, the value of drops at forward rapidities, this will lead to smaller values of at large multiplicities. Although we employed data for collisions integrated over impact parameter, the analysis can be done at different centralities.
This work was supported in part by Fondecyt (Chile) grants 1130543, 1130549, 1100287, and by Conicyt-DFG grant No. RE 3513/1-1.
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