Energy evolution of the moments of the hadron distribution in QCD jets including NNLL resummation and NLO running-coupling corrections

Redamy Perez-Ramos 1 David d'Enterria 0 0 CERN, PH Department , CH-1211 Geneva 23, Switzerland 1 Department of Physics, University of Jyvaskyla , P.O. Box 35 (YFL), F-40014 Jyvaskyla, Finland The moments of the single inclusive momentum distribution of hadrons in QCD jets, are studied in the next-to-modified-leading-log approximation (NMLLA) including next-to-leading-order (NLO) corrections to the s strong coupling. The evolution equations are solved using a distorted Gaussian parametrisation, which successfully reproduces the spectrum of charged hadrons of jets measured in e+e collisions. The energy dependencies of the maximum peak, multiplicity, width, kurtosis and skewness of the jet hadron distribution are computed analytically. Comparisons of all the existing jet data measured in e+e collisions in the range s 2-200 GeV to the NMLLA+NLO predictions allow one to extract a value of the QCD parameter QCD , and associated two-loop coupling constant at the Z resonance s(m2Z ) = 0.1195 0.0022, in excellent numerical agreement with the current world average obtained using other methods. Contents 1 Introduction 2 3 4 Evolution equations for the low-x parton fragmentation functions Extraction of s from the evolution of the distribution of hadrons in jets in e+e collisions Conclusions and outlook A Mellin-transformed splitting functions B NMLLA+NLO moments Kn of the distorted Gaussian C Higher-order corrections to the moments of the distorted Gaussian Introduction One of the most ubiquitous manifestations of the fundamental degrees of freedom of Quantum Chromodynamics (QCD), quark and gluons, are the collimated bunches of hadrons (jets) produced in high-energy particle collisions. The evolution of a parton into a final distribution of hadrons is driven by perturbative dynamics dominated by soft and collinear gluon bremsstrahlung [1, 2] followed by the final conversion of the radiated partons into hadrons at non-perturbative scales approaching QCD 0.2 GeV. The quantitative description of the distribution of hadrons of type h in a jet is encoded in a (dimensionless) fragmentation function (FF) which can be experimentally obtained, e.g. in e+e collisions at c.m. energy s, via Dh(ln(1/x), s) = d(ee hX) , tot d ln(1/x) where x = 2 ph/s is the scaled momentum of hadron h, and tot the total e+e hadronic cross section. Its integral over x gives the average hadron multiplicity in jets. Writing the FF as a function of the (log of the) inverse of x, = ln(1/x), emphasises the region of relatively low momenta that dominates the spectrum of hadrons inside a jet. Indeed, the emission of successive gluons inside a jet follows a parton cascade where the emission angles decrease as the jet evolves towards the hadronisation stage, the so-called angular ordering [1, 35]. Thus, due to QCD colour coherence and interference of gluon radiation, not the softest partons but those with intermediate energies (Eh Ej0e.t3) multiply most effectively in QCD cascades [4, 5]. As a result, the energy spectrum of hadrons as a function of takes a typical hump-backed plateau (HBP) shape [46], confirmed by jet measurements at LEP [7] and Tevatron [8] colliders, that can be written to first approximation in a Gaussian form of peak and width : where Q0 is the collinear cut-off parameter of the perturbative expansion which can be pushed down to the value of QCD (the so-called limiting spectrum). Both the HBP peak and width evolve approximately logarithmically with the energy of the jet: the hadron distribution peaks at 2 (5) GeV with a dispersion of 0.7 (1.4) GeV, for a parton with Ejet = 10 GeV (1 TeV). The measured fragmentation function (1) corresponds to the sum of contributions from the fragmentation Dih of different primary partons i = u, d, , g: Dh(ln(1/x), s) = X Z01 dzz Ci(s; z, s) Dih(x/z, s), and, although one cannot compute from perturbation theory the final parton-to-hadron transition encoded in Dh, the evolution of the intermediate functions Dabc describing i the branching of a parton of type a into partons of type b,c can be indeed theoretically predicted. The relevant kinematical variables in the parton splitting process are shown in figure 1 for the splitting a(k) b(k1)+c(k2), such that b and c carry the energy-momentum fractions z and (1 z) of a respectively. The Sudakov parametrisation for k1 and k2, the four-momentum of partons b and c, can be written as 1 z 2n k with the light-like vector n2 = 0, and time-like transverse momentum k2 > 0 such that, k k = n k = 0. Then, the scalar product k1 k2 reads: k2 = 2z(1 z)k1 k2. E, ~k , k1 = zE, ~k1 , k2 = (1 z)E, ~k2 k2 = 2k1 k2 = 2z(1 z)E2(1 cos ) a = (q, g) such that, replacing eq. (1.4) in (1.3), one finally obtains: k = 2z(1 z)E sin 2 . of the MLLA analytical results to the inclusive particle spectra in jets, determining the energy evolution of the HBP peak position was performed in [17]. The solution of the evolution equations for the gluon and quark jets is usually obtained writing the FF in the form where C(s(t)) = 1 + s + s . . . are the coefficient functions, and = 1 + s + s . . . is the so-called anomalous dimension, which in Mellin space at LLA reads, LLA(, s) = 41 + p2 + 8Ncs/ . where is the energy of the radiated gluon and Nc the number of colours. At small or x, the expansion of the FF expression leads to a series of half-powers of s, s + s + s3/2 + . . ., while at larger or x in DGLAP, the expansion yields to a series of integer powers of s, s + s2 + s3 + . . . for FFs and PDFs. In the present work we are mostly concerned with series of half-powers of s generated at small , which can be truncated beyond O (s) in the high-energy limit. In this paper, the set of next-to-MLLA corrections of order O (s) for the single inclusive hadron distribution in jets, which further improve energy conservation [18, 19], including in addition the running of the coupling constant s at two-loop or next-to-leading order (NLO) [20], are computed for the first time. Corrections beyond MLLA were considered first in [21], and more recently in [22], for the calculation of the jet mean multiplicity N and the ratio r = Ng/Nq in gluon and quark jets. We will follow the resummation scheme presented in [21] and apply it not just to the jet multiplicities but extend it to the full properties of parton fragmentation functions using the distorted Gaussian (DG) parametrisation [23] for the HBP which was only used so far to compute the evolution of FFs at MLLA. The approach followed consists in writing the exponential of eq. (1) as a DG with mean peak and width , including higher moments (skewness and kurtosis) that provide an improved shape of the quasi-Gaussian behaviour of the final distribution of hadrons, and compute the energy evolution of all its (normalised) moments at NMLLA+NLO accuracy, which just depend on QCD as a single free parameter. Since the evolution of each moment is independent of the ansatz for the (...truncated)