Energy dependence of the inelasticity in $pp/p\overline{p}$ collisions from experimental information on charged particle multiplicity distributions

In $p+p(\overline{p})$ collisions at center of mass energy, $\sqrt{s}$, the effective energy left behind by the two leading protons, or correspondingly the inelasticity $K$ Fiete ; Braz_Japn ; YH ; Navarra , is an essential concept because it defines the energy effectively used for producing $n$ new secondary particles. That in turn, determines the dynamics of the interaction in high-energy hadronic and nuclear collisions. The inelasticity varies from event to event, so that one has to introduce an inelasticity distribution $\chi(K,s)$ normalized by Fowler1984

Experimental data on $K$ are very limited and the form of its distribution function, $\chi(K,s)$, has not yet been stablished. It is known as the only experimental information available on $\chi(K,s)$ is from $pp$ interactions at $\sqrt{s}$=16.5 GeV, which exhibits a maximum at $\sim$ $0.5$ Brick1981 . At the ISR energies the mean inelasticity is approximately constant with $<K>$ $\sim$ $0.5$ Kadija .

The energy dependence of the inelasticity is an important problem which has been subject of discussions Fiete ; Wibig ; Navarra2003 ; Navarra1993 ; Navarra1994 ; Musulmanbekov ; BeggioNPA2011 . As example, comparing $p+p(\overline{p})$ with $e^{-}e^{+}$ collisions the $\sqrt{s}$ dependence of the inelasticity in $p+p(\overline{p})$ collisions was calculated in Fiete for three different assumptions on the parameters involved in the analysis and the results were compared with the theoretical study from Kadija .

Although, as mentioned, experimental information on $K(s)$ is limited, the probability for producing $n$ charged particles in final states $P_{n}(s)$, or simply multiplicity distribution, is strictly connected with the inelasticity concept Musulmanbekov ; BeggioNPA2011 . Thus, we can study $P_{n}(s)$ features in order to derive informations on the $K(s)$ behavior, since there are experimental informations available in the full space phase for $P_{n}(s)$ covering the interval of 30 $<$ $\sqrt{s}$ $\leq$ 1800 GeV ABC ; UA51 ; Alexopoulos .

With that in mind, we have studied the relation between $P_{n}(s)$ and $K(s)$ in the framework of a phenomenological procedure related to $P_{n}(s)$ Lam 1982 ; BeggioMV , as well as a formula connecting the inelasticity to the imaginary part of the eikonal function in the impact parameter $b$ space, $\chi_{I}(s,b)$, which was obtained in BeggioNPA2011 . However, in the analysis done in BeggioNPA2011 the $P_{n}(s)$ dataset studied was restrict to collision energies of 52.6, 200, 546 and 900 GeV, and only a limited success was reached in describing the $P_{n}(s)$ data at 200 and 900 GeV. Here, however, we treat the full phase space $P_{n}(s)$ and $K(s)$ at $\sqrt{s}=$ 30.4, 44.5, 52.6, 62.2, 300, 546, 1000 and 1800 GeV) ABC ; UA51 ; Alexopoulos .

Since in our studies $K(s,b)\,\propto\,\chi_{I}(s,b)$, in Beggio Luna we have updated the eikonal formalism of the aforementioned phenomenological procedure in order to describe, in a connected way, $p+p(\overline{p})$ observables in both elastic and inelastic channels through the unitarity condition of the S-Matrix in impact parameter space. All the parameters of the eikonal function, $\chi_{pp}^{\overline{p}p}(s,b)$, were determined carrying out a global fit to all high energy forward $pp$ and $\overline{p}p$ scattering data above $\sqrt{s}$=10 GeV, namely the total cross section, $\sigma_{tot}^{pp,\overline{p}p}$, the ratio of the real to imaginary part of the forward scattering amplitude $\rho^{pp,\overline{p}p}$, the elastic differential scattering cross sections $d\sigma_{el}^{\overline{p}p}/dt$ at $\sqrt{s}$=546 GeV and $\sqrt{s}$=1.8 TeV as well as the TOTEM datum on $\sigma_{tot}^{pp}$ at $\sqrt{s}$=7 TeV. The results obtained in Beggio Luna were compared with the correspondent experimental information and also with the full phase space $P_{n}$ and the $H_{q}$ moments, yielding successful descriptions of all experimental data. In Beggio JPhys G 2017 the phenomenological procedure from Beggio Luna was applied to investigate the $\sqrt{s}$ dependence of the parton-parton inelastic cross sections, parton-parton inelastic overlap functions and the $C_{q}$ moments in proton interactions from $\sqrt{s}$=10 to 14000 GeV, providing also predictions for the ratio $\sigma_{el}(s)/\sigma_{tot}(s)$ as a function of the $\sqrt{s}$, in agreement with the experimental data. Therefore, the success in that global description of elastic and inelastic hadronic observables, over wide interval of $\sqrt{s}$ Beggio Luna ; Beggio JPhys G 2017 , motivated us to investigate the problem of the $\sqrt{s}$ dependence of the $K(s)$ from $P_{n}(s)$ studies.

The main purpose of this paper is to apply the phenomenological procedure formalism in full phase space $P_{n}(s)$ from Beggio Luna , also applied in Beggio JPhys G 2017 , in order to study the energy dependence of the inelasticity based on the experimental information from $p+p(\overline{p})$ multiplicity distributions, since experimental data on $K(s)$ are very limited.

The paper is organized as follows: in the next section we discuss the main ideas associated with the phenomenological procedure as well as their inputs. In Section III, we apply the theoretical formalism computing the inelasticity as a function of $b$ at fixed $\sqrt{s}$, discussing the results. Inelasticity predictions to the LHC energies are made. The concluding remarks are the content of the Section IV.

In $p+p(\overline{p})$ collisions at $\sqrt{s}$ the effective energy for particle production, $E_{eff}$, is the energy left behind by two leading protons and, using four-vector, it may be written Kadija

or

in the case of symmetric events Fiete Kadija and, for quantitative estimation of the inelasticity, we have used the definition Navarra

($0\leq K\leq 1$). We see from Eq. (9) that $\sqrt{s^{\prime}}=E_{eff}$ BeggioNPA2011 , and hence we can rewrite the last Eq. in the form

The factor 2 is due the fact that the $P_{n}(s)$ data are normalized to 2. In turn, the $\beta(s)$ function is related with $\xi(s)$ by Eq. (13), explicitly we have

Using the Eq. (27) we can rewritten the Eq. (26) in the form

With respect to last expression, the DMG eikonal function $\chi_{I}(s,b)$ is completely determined from only elastic channel data analysis (subsection II.B), $\xi(s)$ is determined by the normalized condition given by the Eq. (16), the $\langle N(s)\rangle$ values are obtained from experiments and the $\zeta(s)$ values were obtained by full phase space $P_{n}(s)$ fits Beggio Luna and parametrized by the Eq. (22). Thus, by fixing the value of $\gamma=3.36$, as discussed in the Subsection II.B, we have calculated the $K(s,b)$ as a function of the impact parameter $b$ and the results are displayed in Figs. 1 to 8. The inelasticity behavior is essentially the same at the energies of $\sqrt{s}$ = 62.2 and 44.5 GeV, Fig. 4. The same occurs at 1000 and 546 GeV, Fig. 7. It seems consequence from the fact that at 62.2 and 1000 GeV the theoretical $P_{n}$ does not fits satisfactorily the experimental points in the tail of the distributions.

At the ISR energies the average inelasticity is determined to be about 0.5 Kadija ; Golyak . Interestingly in the present analysis is that the average inelasticity at ISR, when $b$ $\sim$ 0, yields the same value, specifically: $<K>_{ISR}$ = (0.54 + 0.49 + 0.48 + 0.50)/4 $\sim$ 0.5, however, the choice of $b$ $\sim$ 0 is so arbitrary. Based on the results displayed in Fig. 4 and by using the formulae of mathematical expectation of the function $K(s,b)$, we have calculated the average impact parameter, $<b>$, at each ISR energy and the corresponding $K(s,<b>)$ values as well as the new value of $<K>_{ISR}$ $\sim$ 0.16. The results are summarized in Table II and the average inelasticity, thus obtained, do not agree with those from Kadija ; Golyak . However, we recall that the impact parameter dependence of the inelasticity was not analysed in the framework of the both mentioned works, Kadija ; Golyak .

From Fig. 9, where the plots of $K(s)$ versus $b$ at the energies investigated in this work are presented together, is possible see that the particle production processes tend to be more peripheral ($b>1$ $fm$) at the ISR energies of $30.4$, $44.5$, $52.6$ and $62.2$ GeV when compared with the results from other energies investigated. In this interval of $b$ the $K(s)$ values, at the ISR, are rather greater than $K(s)$ values at the others energies at fixed value of $b$. In order to substantiate this statement in Fig. 10 we show the ratios $K(s,b)/K(30.4,0)$ calculated for different collision energies at the impact parameter values of $1.0$, $1.1$, $1.25$ and $1.5$ $fermi$. Based in Fig. 1 we have used $K(30.4,0)$=0.54. In fact, the results presented in Fig. 10 are indicatives that the particle production is more peripheral at the mentioned ISR energies than at other energies studied. This behavior of $K(s)$ is compatible with the minijets production, since semihard processes are more central in the impact parameter than purely soft processes and do not use much collision energy DDeus . The inelasticity $K$ is proportional to the $\chi_{I}(s,b)$, Eq. (28), and in the DGM eikonal model the gluon semihard contribution $\chi_{gg}(s,b)$, Eq. (18), dominates at high energy and the rise of the cross sections with $\sqrt{s}$ is consequence of the increasing number of soft gluons populating the colliding particles, increasing, therefore, the probability of perturbative gluon-gluon collisions at small $x$, which can leads to the appearance of minijets and, as mentioned, do not use much collision energy. This scenario leads to the conclusion that the $K(s)$ decreases as a consequence of the minijet production from semihard soft gluon-gluon interactions when $\sqrt{s}$ increases.

We show in Fig. 11 the energy dependence of the $K(s)$ calculated at $b$ $\approx$ $0$, Eq. (28), and observe a marked decrease in the inelasticity from ISR to LHC, while at the $\sqrt{s}$ $>$ 7 TeV the inelasticity shows a slow decrease. The error bars represent the uncertainties of the parameters $\gamma$ and $\zeta$ propagated to the inelasticity values. The star symbol represents theoretical predictions at the LHC and the solid line is drawn only as guidance for the points. The LHC has measured the multiplicity distributions in a limited pseudorapidity range CMS2011 ; LHCb2012 ; LHCb2014 ; ALICE2016 ; ALICE2017 ; Fiete2017cap , and for this reason we do not compare our results with those from LHC.

We observe that the structure found around the peak in the $P_{n}(s)$ data at higher energies, which appears in the region of low multiplicities, has not been considered in the analysis done in Beggio Luna . However, the $P_{n}(s)$ approach used describes very well the energy dependence of the $F$-moments and reproduces the $H_{q}$ versus $q$ oscillations observed in the experimental data and predicted by QCD Beggio Luna ; BeggioNPA2013 .

With respect to alternative methods of estimating the inelasticity, in Fiete , the coefficient of inelasticity in $p+p(\overline{p})$ collisions and its possible $\sqrt{s}$ dependence was estimated by comparing $p+p(\overline{p})$ with $e^{-}e^{+}$ collisions for three different assumptions on the values for both the parameters involved in the analysis, namely $n_{0}$ and $\Delta m$. The parameter $n_{0}$ corresponds to the contribution from the two leading protons to the total multiplicity, while $\Delta m$ takes the contribution of the masses of the two participating constituent quarks to the centre-of-mass energy into account. Having varied the $n_{0}$ and $\Delta m$ values three different inelasticities were defined. In one of the results the inelasticity decreases from $K$ $\sim$ (0.55 - 0.6) at the ISR energies to 0.4 at $\sqrt{s}$ $=$ 1800 GeV. The two other results indicated the constant value of $K$ $\sim$ 0.35.

Investigating the very high energy $pp$ interactions by cosmic ray data it was shown Wibig that the Feynman scaling violation, in the form proposed by Wdowczyk and Wolfendale, leads to continuous decrease of the inelasticity, which was found be consistent with LHC measurements up to 7 TeV, qualitatively in agreement with our results, Fig. (11).

In another work Navarra2003 and by using methods of information theory approach, calculations of the inelasticity coefficient and its energy dependence were studied, resulting that the inelasticity remains essentially constant in energy, except for a variation around $K$ $\sim$ 0.5 in the range 20 $<$ $\sqrt{s}$ $<$ 1800 GeV to $p+p(\overline{p})$ data.

The Interacting Gluon Model (IGM) was an approach used in studies about the inelasticities and leading particle spectra in hadronic and nuclear collisions YH ; Navarra1993 ; Navarra1994 ; Navarra2003 . In Navarra1993 an extended version of the IGM incorporating the production of minijets was applied and, as a result, it was concluded that the inelasticity slowly increases towards some limited value. The inclusion of minijets reversed the trend of decreasing inelasticities found in previous calculations with the IGM.

In subsequent work Navarra1994 the authors introduced a hadronization mechanism in the IGM concluding that the minijet production leads to inelasticities increasing with $\sqrt{s}$ and that hadronization process does not change this trend.

Based on the above considerations, one can note that the various approaches are largely in conflict with each other in explaining the energy dependence of the inelasticity, reflecting the subtlety of the theme. Hence, we have based the present study on the experimental information on charged particle multiplicity distributions in $p+p(\overline{p})$ collisions. Thus, we provided a new argument in favor of the hypothesis that the $K(s)$ decreases as a function of the center of mass energy.

In the absence of sufficient experimental information on the energy dependence of the inelasticity to test the several existing model predictions, we have based our analysis in the connection between $K(s)$ and the full phase space $P_{n}(s)$ by using a satisfactory modeling to $P_{n}(s)$ adjusted for the experimental reality over a large range of energy, 30 $<$ $\sqrt{s}$ $\leq$ 1800 GeV, which is consistent with several QCD prescriptions Beggio Luna .

In the present approach $K(s,b)$ $\propto$ $\chi_{I}(s,b)$, Eq. (28), and we have adopted the DGM QCD inspired eikonal model luna008 ; luna009 ; Beggio Luna . The only free parameter in the $P_{n}(s)$ formalism adjusted to $p+p(\overline{p})$ experimental data is $\zeta(s)$, (Eq. (22) - Table I), while all the parameters of the eikonal function, $\chi_{pp}^{\overline{p}p}(s,b)$, were determined carrying out a global fit to $\sigma_{tot}^{pp,\overline{p}p}$, $\rho^{pp,\overline{p}p}$ and $d\sigma_{el}^{\overline{p}p}/dt$ data. The results of the fits to $\sigma_{tot}^{pp,\overline{p}p}$, $\rho^{pp,\overline{p}p}$ and $d\sigma_{el}^{\overline{p}p}/dt$ are presented in Beggio Luna . Our results predict the average inelasticity to be $\sim$ 0.5 at the ISR energies if calculated at $b$ $\sim$ 0, in agreement with that from Refs. Brick1981 ; Kadija ; Golyak (see Section III).

The term $\chi_{gg}(s,b)$ in the Eq. (18) gives the main contribution to high multiplicities, being the responsible for the rise of the cross sections with $\sqrt{s}$. Thus, we have concluded that minijets from semihard interactions, arising from scattering of gluons carrying only a very small fraction of the momenta of their parent protons, are the responsible for the decrease of the inelasticity as a function of the $\sqrt{s}$.

Results obtained by using alternative methods to estimate the energy dependence of the inelasticity are in conflict with each other. Thus, based on the experimental information on charged particle multiplicity distributions in $p+p(\overline{p})$ collisions we provided new evidence in favor of the hypothesis that the $K(s)$ decreases when $\sqrt{s}$ is increased.