Jets in multiparticle production in and beyond geometry of proton-proton collisions at the LHC

Experimental findings of CMS on properties of jets and underlying events at high multiplicities in proton-proton interactions at 7 TeV are interpreted as an indication of increasing role of central collisions with small impact parameters. We find an indication that the rates of different hard processes observed by CMS and ALICE universally depend on underlying event charged-particles multiplicity until it becomes four times more than average. It is shown that the increase of the overlap area of colliding protons is not sufficient for explanation of the rate of jet production in events with charged-particle multiplicity three times higher than average and some new mechanisms are necessary like interaction of protons in rare configurations of higher than average gluon density. Such mechanisms are not included in the present Monte Carlo event generators. Further studies are proposed


Introduction
The multi-particle production in proton-proton (pp) collisions is governed by several mechanisms of hadron dynamics. The geometry of the collision plays a crucial role in the relative contributions of these mechanisms. Both non-perturbative and perturbative QCD mechanisms contribute to the hadron production.
At the parton level, each event of the inelastic particle production is treated as a combination of hard and soft parton-parton interactions plus partonic remnants of colliding protons. The hard parton-parton interactions result in jets, which appear in final state as well-collimated bunches of hadrons. At large enough transverse momenta of jets, they can be treated within perturbative QCD. The softer components, including also soft ingredients of jets, combine in the so-called underlying event (UE). The interplay between soft and hard contributions certainly depends on the collision energy and on the impact parameter of the collision. The complicated structure of the interaction region was discussed in many papers and, in particular, in [1,2].
Several important features of inelastic pp interactions emerge from the analysis of the data on elastic pp scattering (using s-channel unitarity) and analysis of the transverse parton spread as extracted from hard exclusive processes like γ + p → J/ψ + p. First, one finds that in interactions at the LHC energies protons are completely absorptive at small central area and have a large semi-transparent peripheral zone.(Sec. 2.1). Second, one finds that partons with large fractions of proton energy (x above 10 −3 ) are concentrated in the central absorptive area. A detailed and up-to-date review of the two-scale picture of proton is given in [2].
The goal of the paper is to derive information about dependence of hadron production on the impact parameter using the observables, for which relative importance of hard and soft interactions strongly depends on the impact parameter. We use most recent experimental studies of the processes with a hard trigger and high multiplicities. We determine up to which maximum charged-particle multiplicities impact parameter picture works and where other mechanisms start to dominate. An important tool for our studies is the ratio of the multiplicity of the hard subprocesses for a given range of multiplicity and to the one in bulk of inelastic events. We demonstrate that this ratio exhibits the universality pattern when plotted against N ch / N ch . It is practically the same for jet production with p T > 5 GeV/c and p T > 30 GeV/c, as studied by CMS for associated charged particles detected in |η| < 2.4 range. Moreover, the universality holds, when we compare the CMS ratios with those reported by ALICE for J/ψ, D, B-meson production [3] in a factor of 3 smaller |η| interval. We also argue that a new regime sets in at N ch / N ch ≥ 3, corresponding to very central collisions in which the rise of the hard probe multiplicity exceeds the maximum value allowed by the geometry of the collision. This new regime may correspond to selection of configurations in the colliding nucleons with larger than average relatively-small-x (x ∼ 10 −3 − 10 −2 ) gluon density.
The paper is organized as follows. In Sec. 2 we review the transverse geometry of bulk and hard-probe triggered pp collisions. Next, in section 3, we interpret some recent LHC results [4][5][6][7] as well as more detailed data of CMS collaboration [8] on properties of jets and UE in different charged-particle multiplicity intervals. In section 4, we use jet production as a way to test (calibrate) centrality dependence of events on their multiplicity. We present our conclusions and suggest strategies for further studies in Sec. 5.
2 Geometry of soft and hard pp collisions

Geometry of high-energy pp interactions
The shape of the interaction region can be viewed from the overlap function defined by the unitarity condition for elastic scattering amplitudes. The corresponding formula in the impact parameter representation is where G(s, b) is the overlap function determining the inelastic profile of colliding protons, and is the Fourier-Bessel transform of the elastic scattering amplitude f (s, t) related to the differential cross section as and normalized as σ tot (s) = √ 16πImf (s, 0).
The smallness of the real part of f (s, t) corresponding to small ImΓ(s, b) implies that one can compute G with good precision directly from experimental data obtained by the TOTEM collaboration [9,10] for elastic pp scattering at 7 TeV (up to |t| = 2.5 GeV 2 ) as shown in Fig. 1 (a) by the solid line taken from [11]. Analogously, it can be computed with the Gaussian profile of the diffractive cone Γ(s, b) [2] (the dashed line in Fig. 1a).  Figure 1: Overlap function (a) and probability of inelastic collision with an impact parameter smaller than b (b), according to [11] (solid lines) and [2] (dashed lines).
To illustrate how steeply G(b) drops in transition from black to transparent region we use the fit with a stepwise behavior of the exponential reminding θ-function at λ → 0: It is quite successful. The fit reveals quite strong separation of the two regions at b 0 ≈ 0.3 fm with a width of the transition region λ ≈ 0.1 fm. According to Eq. (5) the exponential becomes three times larger in the narrow strip between the borders of the transition region b 0 − λ and b 0 + λ. The region b < b 0 is completely black while at b > b 0 it becomes more transparent.
When compared to ISR results [11], the overlap function and, consequently, the blackness (opacity) of protons at 7 TeV somewhat increases in the central region approaching complete blackness G(b) = 1. At the same time, much stronger increase by about 40 % is observed in the peripheral region near 1 fm. It could imply, that the b range of inclusive dijet production also increases with increase of energy due to the contribution of smaller and smaller x since for these partons transverse spread is larger.
To illustrate the interplay between the black and gray regions we compute the relative contribution of the impact parameters smaller than b to the inelastic cross section: where σ inel -inelastic cross section of pp collisions. One can see from Fig. 1 (b) that the main contribution to the inelastic cross section originates from the gray area while the dark region (b ≤ b 0 + λ = 0.4 fm) constitutes only about 8 %.

Geometry of dijet production
The transverse distribution of partons in nucleons is given by the generalized parton distributions f j (x, Q 2 , t) which are measured in exclusive hard processes (In the processes we consider Q ∼ p jet T ). Their Fourier transform, f j (x, Q 2 , ρ), determines the geometry of the inclusive hard interactions [12]. Probability that the dijet collision occurs at given b is given by where ρ 1,2 ≡ |ρ 1,2 | are the transverse distances of the two partons from the center of their parent protons [12] and the relation holds. One finds that the transverse spread of F g (x, ρ|Q 2 ) slowly increases with decrease of x at fixed Q 2 and slowly decreases with increase of Q 2 for fixed x.
The distributions of probabilities of hard and soft interactions are compared in Fig.4 in [2]. One can see that hard trigger b distribution is much more narrow than for bulk events. This is reflected in the probability of small b for hard collisions (see plot of b 0 d 2 bP 2 (b) in Fig. 2) to be much higher than for the bulk inelastic collisions (Fig. 1b).

Jet and UE
In this section, we show connection between standard UE studies and recent study of jet properties as a function N ch , exploiting only variables used in these studies, and propose the way to its interpretation. In Sec. 4, we will elaborate the interpretation and substantiate it with more experimental data.
The UE properties are usually studied with reference to the direction of the particle or of the jet with largest p T . Both approaches have advantages and disadvantages. The leading jet is more directly related to the initial parton, however it can be affected by UE contribution. While leading charged particle is less related with the parent parton, it is not affected by UE. Usually, three distinct topological regions in the hadronic final state are thus defined by the azimuthal angle difference ∆φ between the directions, in the plane transverse to the beam, of the leading object (particle or jet) and other hadrons. Hadron production in the nearside region with |∆φ| < 60 • and in the away-side region with |∆φ| > 120 • is expected to be dominated by the hard parton-parton scattering and radiation. The UE structure can be best studied in the transverse region with 60 • < |∆φ| < 120 • .
A number of recent experimental UE studies use the above described techniques [4][5][6][7]. ALICE and ATLAS collaborations use a leading charged particle as a reference, while CMS collaboration uses a leading charged-particle jet. Despite very different pseudorapidity ranges of ALICE and ATLAS experiments (η <0.8 and η <2.5), their results are very close. Of particular interest is particle density in transverse region defined as follows: where N tr ch -charged-particle multiplicity in the transverse region, ∆η -pseudorapidity range studied, ∆(∆φ) -azimuthal width of the transverse region. The transverse charged-particle density as a function of leading object is shown in Fig. 3. The dependence saturates at some p T = p crit T , which is ≈ 4 − 5 GeV/c and ≈ 8 GeV/c for leading charged particle and leading charged-particle jet techniques, respectively. According to the two-scale picture of the proton structure described in Sec. 1 and elaborated in Ref. [2] , the observed plateau can be interpreted as an indication of dominance of central collision.
Since we aim to reveal connection between UE studies [4][5][6][7] and study of jet properties as function of N ch [8], one needs to estimate the N ch as defined in [8], corresponding to the plateau of transverse multiplicity density. The N ch in [8] is defined as a number of charged particles with η < 2.4 and p T > 0.25 GeV/c, while µ tr (shown in Fig. 3) is obtained with charged particles with p T > 0.5 GeV/c. From Fig. 3 we can see that the transverse multiplicity density saturates at µ sat tr ≈ 1.0. The charged-particle multiplicity of UE can be roughly estimated in assumption of flat η-distribution as follows: where δη =4.8, δφ = 2π are the pseudorapidity and azimuthal angle ranges used in [8].
One should also account for different p T cuts of charged particles. The correspondence of UE charged-particle multiplicities for different p T thresholds is obtained using pythia 6 z2* simulation, which describes UE properties quite well: Eq. (10) gives approximately 60 charged particles with p T > 0.25 GeV/c), belonging to UE when it reaches a plateau. According to tables 4 and 5 of [8], a jet contains 5 particles on average, and the jet rate for 50 < N ch ≤ 80 is ≈ 1 jet per event. The second, recoiled jet is usually wider and consists of softer particles, thus may be not found by jet finding algorithm. This is clearly seen in Fig. 2 of Ref. [4]. Therefore, one can conclude that at least 10 charged particles come from jets, and the total N ch , at which collisions become central, equals ≈ 70.
Results on jet production as a function of N ch presented in [8] substantiate the statement. Indeed, we see from table 4 of Ref. [8], that average p T of jets with first threshold lies between 7-8 GeV/c. This value matches to the p crit T at which central pp collisions may occur (see Fig. 3). Therefore, the jet rate at the thresholds 5 GeV/c can serve as a measure of collision centrality. From that table, one can see that events starting from N ch ≈ 60-70 have one jet. This means that the most central collision geometry is reached around these values of N ch . Other mechanisms should be responsible for N ch higher than 70, rather than increasing overlapping area of colliding protons. Indeed, present MC event generators completely fail to describe the charged-particle jet rate with p T thresholds of 30 GeV/c for N ch > 70 (Fig. 7 of [8]). We see that the estimate of N ch of central events obtained from UE studies coincides with the value obtained from the jet studies as a function of N ch .

The impact parameter dependence and beyond
In Sec. 2.2, we have demonstrated that probability of the central collisions is pretty small. It is instructive to compare it with the probability to have multiplicity larger than given N ch , shown in Fig. 4. From the comparison of this plot with the plot for the probability distribution of inelastic events over b, one can make correspondence between impact parameter and N ch (Fig. 5). We see, that N ch (p T > 0.5 GeV/c, |η| < 2.4) ≥ 35 mostly corresponds to b ≤ 0.4 fm. The probability of events with N ch (p T > 0.5 GeV/c, |η| < 2.4) > 35 is about 5 %. However, measured values of N ch reach ≈ 100. Such high values (3 times higher than for b = 0.4 fm) can not be produced even in absolutely head-on collision, if one relies merely on geometric arguments, since increase of overlapping area does not exceed ten-percent level. Figure 4: Fraction of events with N ch > N fixed ch . The N ch is defined as a number of stable charged particles with p T > 0.5 GeV/c and |η| < 2.4. The N ch distribution is taken from [13].
It was shown in Ref. [14], that distribution of the inclusive rates of hard signals over b with respect to the bulk events is given as follows: where σ inel = 55 mb is inelastic cross-section of events with at least 1 charged particle within |η| <2.4 taken from [15]. Such events, having detected particles within a detector acceptance, are further called minimum bias ones and used for most mesurements. If the rate is changed only due to the change of the impact parameter of the collision, the corresponding ratio would reach values of R ∼ 3.8 − 4.2 (Fig. 6). It is worth noticing, that R flattens out already for b ≤ 0.3 ÷ 0.4 fm.
We have extracted R for charged-particle jets for two p T threshold, p ch.jet T > 5 GeV/c and p ch.jet T > 30 GeV/c, from the data taken from Ref. [8] and present them in Figs. 7(a, b). The ratios show a very strong increase beyond N ch ∼ 80. To compare Eq. (11) with the data in Fig. 7 (a, b) we need ideally to plot the rate of jet production as function of N UE ch . Experimentally, the purest way to measure the rate is to select jets produced in one bin of rapidity with multiplicity measured in another bin of rapidity. By doing this, we avoid the contribution of the hadrons produced in the hard trigger component of the event. In practice, with the current data we can only try to correct roughly for this effect by using MC simulations (pythia 6 z2*) to estimate the average multiplicity in the selected jets: ∆N ch = 10 (15) for p ch.jet T > 5 (30) GeV/c. Hence, to correct for the jet contribution we need to reduce the experimental ratios by a factor P (N ch )/P (N UE ch ) and plot them as function of the N UE ch , which is N ch − ∆N ch here. The differential N ch distribution used for computation of the correction is taken from [13] Since the low-N ch events have large fluctuations in |η|, the correction is not reliable for N ch ≤ 50 and the corresponding points are not plotted. One can see that corrected  Figure 6: Geometric probability for two gluons to collide (left y-axis) and inclusive jet production rate with respect to the bulk one (right y-axis). Solid and dashed lines represent two parameterizations of P 2 (b) as given by Eq. (11) in [2].
ratios are approximately the same for two p T cuts which is consistent with the hypothesis that the rates are determined by the initial state of colliding protons.  Figure 7: Ratio of N j at given N ch to N j of bulk events: (a) -for charged-particle jet p ch.jet T > 5 GeV/c, (a) -for charged-particle jet p ch.jet T > 30 GeV/c. The black solid lines represent data sorted according to total N ch . Dashed blue lines represent the ratio if the data would be sorted according to N UE ch (note, that the data points are plotted using total N ch ). To correct the total N ch to the N UE ch , one needs to subtract ≈ 10 (15) particles for p ch.jet T threshold of 5 (30) GeV/c. It is worth noticing, that ALICE has performed studies of similar quantity, R, i.e. the ratio of the J/ψ multiplicity as a function of N ch normalized to minimum bias J/ψ multiplicity [3]. They also reported the same ratio for D and B-meson production. The observed dependences of R on N ch / N are very similar to the one we observe after correcting for the jet contribution (Fig. 8). It is worth emphasizing here, that similarity between R in two measurements is highly non-trivial as the rapidity intervals used for determination of N ch differed by a factor of ∼ 3.
We explained above that the inclusive rate of the jet production at given b as compared to the bulk rate can be calculated using on the information about transverse gluon distributions in nucleons (Eq. (11) of [2]) . Hence, it is convenient to consider relative contributions of different bins in N ch to the total inclusive rate of jet production. The results are presented in Fig. 9  (a,b) for p ch.jet T > 5 GeV/c (30 GeV/c). Since the median of the P 2 distribution corresponds to b ∼ 0.6 fm, we conclude that N ch ∼ 60 should roughly correspond to that value of impact parameter. Also, the corrected value of the ratio, R, for third domain, which corresponds to average b of dijet collisions, has a value of about 2 that is consistent with the expectations of Eq. (11) for average b.
At the same time the highest multiplicity points for both p ch.jet T cuts correspond to R well above 4.0 indicating that new mechanisms play dominant role in this case. One possibility is that the rare high-N ch events are produced in collisions of protons in configuration with gluon fields which are significantly stronger than average. The parameter which determines enhancement in this mechanism is g 1 g 2 /S where g i are the gluon densities in the configurations which generate high multiplicity and S effective area of the overlap [14]. To explain last point of the data this parameter has to be larger than for the average configuration by a factor of 2. This explains much smaller probability of these events than the one given by geometry of the collisions. Another source could be contribution of the higher QCD processes, that are not properly accounted or not accounted at all. However, it is not likely that they would generate the same enhancement for pretty different p ch.jet T cuts. Note also, that measured charged-particle multiplicity reaches values of 170-180 [13]. Then, determining the rate of jet production for these collisions may open a window on the properties of very rare configurations in nucleons.

Summary and conclusions
The role of proton geometry in multiparticle production at LHC energy has been studied using various experimental data. With the help of hard probes we obtain information about the inner regions of the protons. At LHC energies, the core of protons became absolutely absorptive due to high parton density. Collision, involving the proton cores, usually results in high-multiplicity events. Such events contain the large number of high-p T jets. As a tool, we have used the dependence of the jet multiplicity on the charged-particle multiplicity studied in [8]. We compute the ratio, R, of the jet multiplicity in the charged-particle multiplicity intervals (corrected for the hard contribution) to that in minimum bias events. The value of the ratio is determined by geometry of pp collision up to N ch / < N ch >∼ 3.0 corresponding to the average impact parameters of 0.4 fm. We have argued that at higher multiplicities one enters a regime of increase of R, which is not described by geometry of pp collisions. Our analysis strongly suggests that for N ch > 70 ( at 7 TeV), for which the process is dominated by rare central collisions, colliding protons fluctuate into special high-density gluon configurations. This suggests that the events in which ridge was observed prominently also originate from similar collisions. Also, we find an indication that the rates of different hard processes observed by CMS and ALICE universally depend on N ch until it becomes three times more than average, where geometry effects dominate. This is consistent with the hypothesis that their rates are predominantly determined by the initial state of colliding protons. At the same time we observe that it holds even for higher values of N ch / N ch ( up to 4) for x in the 10 −2 ÷ 10 −3 interval. Hence, it would be highly desirable to extend these observations for a wider x-range and also to study jet production for to larger N ch to see whether the jet rate continues to grow, and whether this growth is different for small and moderate x (x > 0.05). Fluctuation of the gluon density may increase relative importance of the multiparton interaction (MPI) mechanism of the jet production. Thus it would be worthwhile to try to determine the rate of MPI in the high multiplicity events.
Further more detailed studies are highly desirable. In particular, it would be preferable to study dependence on multiplicity of the underlying event rather than on total multiplicity. It would be also interesting to study events where multiplicity is significantly smaller than average. One may expect that these events originate from large b collisions, where interaction is dominated by the exchange of a single Pomeron. Since properties of the Pomeron do not depend on b, in this scenario R is practically independent on N ch .
Another possible direction for the experimental studies would be investigation how the production of leading baryons (for example, production of leading neutrons with x F ≥ 0.3) is correlated with N ch / N ch . Indeed, it was argued [16] that the neutron yield in the proton fragmentation region drops with an increase of the centrality of the pp collisions and also decreases faster with increase of x F . If so, one expects that with increase of N ch / N ch the neutron multiplicity for large x F ≥ 0.3 will diminish and it would be very strongly suppressed for N ch / N ch ≥ 4.