Asymptotic Analysis of the parton branching equation at LHC Energies

An asymptotic solution to the QCD parton branching equation is derived using the method of Laplace transformation and saddle point approximation. The distribution is applied to charged particle multiplicity distributions in proton-proton collisions at √ s = 0.9, 2.36, and 7 TeV for |η| < 0.5, 1.0, 1.5, 2.0, 2.4, and 8 TeV for |η| < 0.5, 1.0, 1.5, as well as 13 TeV data for |η| < 0.8 and 2.5.


Introduction
The charged particle multiplicity distribution is a global observable of the final state of high energy collisions.The shape of the distribution offers information on the dynamics of particle production processes as well as any correlation in particle production.Photon counting distributions in optical experiment and theory has shown that the independent production of photons lead to a Poisson distributed counting distribution.In the context of high energy proton-proton (pp) and electron-positron (e + e − ) collisions, the charged-particle multiplicity distributions are most often described using the Negative Binomial Distribution (NBD) [1] and even the double NBD [2].The NBD describes cascades (or showers) of particles originating from an initial number of ancestor particles produced after the collision [3].
The total multiplicity distribution of partons within a jet [4] can be expressed as Markov branching processes of quarks (q) and gluons (g) [5].Considering the processes g → g + g, q → q + g, g → q + q, and g → g + g + g with average branching probabilities of A, Ã, B, and C respectively, the QCD parton branching differential equation for producing m quarks and n gluons after branching [6][7][8] is given as: e-mail: w.y.wang@nus.edu.sg and Q is the initial parton invariant mass, Q 0 is the hadronisation mass, µ is a QCD mass scale of a few GeV, N c is the number of colours, and N f is the number of flavours.The above equation yields the Generalised Multiplicity Distribution (GMD) [6][7][8] as a possible exact solution in the case where B = C = 0. Given an initial m and k number of quarks (q) and gluons (g), the probability of obtaining n final state particles is given by the GMD: where and n is the average number of observed final state particles.Special cases of the GMD are the NBD when k = 0, and Fury-Yule Distribution [9] when k = 0. Furthermore, the NBD converges to the PD for k → ∞.
However, an exact solution including all the processes in equation 1 has never been derived.In this paper, an asymptotic solution is obtained using the method of Laplace transformation and the saddle point approximation.The result corrects an error by A. H. Chan and C. K. Chew [10] in the use of the Stirling's formula.The resulting asymptotic multiplicity distribution is then applied to measurements of charged particle multiplicity distributions in pp collisions by the ALICE [11], ATLAS [12], and CMS [13] collaborations at the LHC.

Asymptotic solution of the QCD parton branching equation
Using a continuous-variable approximation, where n is replaced by x ∈ R and performing a Laplace transform with initial condition , equation 1 becomes When x > k , the initial condition ensures that P(x > k , t = 0) = 0.In the case where x = k , To further simplify equation 4 when x > k , let ln(P * (x, s)) = L(x) and approximate L(x+1) ∼ L(x) and L(x − 2) ∼ L(x − 1) as we expect large number of gluons to be created at high energies.Equation 4 then becomes where a Taylor expansion has been made for all the L(x − 1) terms about L(x) up to the first differential coefficient.
To simplify the analysis, let Integrating equation 8, and letting L 1 (k , s) = ln P * (x = k , s) (Eq.6), A better approximation can then be obtained by expanding the Taylor series up to the second term, Differentiating equation 8, and substituting into equation 11, which on integrating gives The required asymptotic solution is obtained by the inverse Laplace transform where and The integral of equation 14 is expected to be dominated by the highest saddle point at s = s 0 for large t.Using the saddle point approximation, the asymptotic solution is given by Substituting the corresponding expressions for g(s 0 ), f (s 0 ), and f (s 0 ), equation 17 becomes which can be further evaluated by looking for the stationary function of the exponential term in the above equation This gives the expression which by letting k = γ β+1 , we obtain For large values of x, the Stirling's approximation is given by x! = x x+1/2 e x (2π) 1/2 .Taking note that x! = Γ(x + 1) and replacing x by n, the final form of the asymptotic solution to the QCD parton branching equation is obtained as Replacing α, β, γ, and D with their corresponding expressions in equation 9 and letting A = At, A † = A † t, B = Bt, and C = Ct, With the last substitution, the branching probabilities A, A † , B, C are now reinterpreted as the branching probabilities scaled by the QCD evolution parameter.The ratios between the branching probabilities (e.g.A † /A) remain fixed regardless of the scale.Applied to charged particle multiplicity data, equation 23 represents the probability of obtaining n charged particles from the hadronisation of gluons in QCD jets, via the branching processes associated with the parameters A, A † , B, and C.

Results
Figure 1 shows charged particle multiplicity distributions in pp collisions for five pseudorapidity windows |η| < 0.5, 1.0, 1.5, 2.0, and 2.4 as measured by the CMS collboration at √ s = 0.9, 2.36, and 7 TeV.Comparison is made with the best fit asymptotic multiplicity distribution.The corresponding parameter values are given in table 1.
Similarly, figure 2 (top) shows charged particle multiplicity distributions for |η| < 0.5, 1.0, and 1.5 at √ s = 8 TeV measured by the ALICE collaboration. Figure 2 (bottom) shows the same distributions for |η| < 0.8 and 2.5 at √ s = 13 TeV measured by the ATLAS collaboration.The asymptotic multiplicity distribution is found to describe data from CMS collaboration well.However it is noted that there is increasing difficulty in describing the tail of the distribution as the pseudorapidity range increases.This suggests that the asymptotic multiplicity distribution is insufficient to describe all the processes involved in pp collisions and the subsequent fragmentation.At √ s = 8 TeV the asymptotic multiplicity distribution appears to describe the tail of the distribution well.However, it poorly models the data at mid-multiplicity (n ∼ 70) and low-multiplicity (n ∼ 10) regimes.At √ s = 13 TeV, the data is well modelled for |η| < 0.8 but described poorly for |η| < 2.5, where the tail of the distribution is overestimated from n ∼ 70.
Referring to table 1, the values of k , B and C generally tend to 0 or some small value for centreof-mass energies of up to 8 TeV.This implies that quark pair creation and the four-gluon vertex occur at a relatively insignificant rate during the branching process.Quark bremsstrahlung and gluon fission dominate the branching process, and the entire process occurs with no initial gluons.Hence at the above centre-of-mass energies, charged particle multiplicities are still better described by the NBD (k = 0).However the non-zero k values at √ s = 13 TeV seem to suggest that more initial gluons are produced at higher centre-of-mass energies.

Conclusion
From the QCD parton branching equation, the asymptotic multiplicity distribution is derived using the method of Laplace transform and saddle point approximation.Revision of the derivation from previous work is done with the correct Stirling's formula.Starting with an initial number of gluons k , the distribution describes the probability of producing n gluons via four processes that contribute to the overall quark and gluon distribution inside QCD jets, with each process occuring with a different probability.The asymptotic multiplicity distribution is applied to charged particle multiplicity distribution data at √ s = 0.9, 2.36, 7, 8, and 13 TeV from the ALICE, ATLAS, and CMS collaborations.The parameters k , A † , A, B, and C are estimated and the result is found to describe the observed multiplicity well at √ s = 0.9, 2.36, and 7 TeV well except for the tail of the distribution.The parameter k reduces to 0 (yielding NBD), suggesting that branching processes up to 7 TeV are dominated by gluon fission and quark bremsstrahlung with negligible initial gluon densities.
The asymptotic multiplicity distribution is found to be inadequate in modelling observed multiplicities at √ s = 8 TeV and more so at 13 TeV, where it fails to describe both the mid-and highmultiplicity regions at larger pseudorapidity intervals.This hints at the presence of additional processes which should be accounted for.

Table 1 .
Parameter values for the asymptotic multiplicity distribution.