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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 883 (2014) 615–628

www.elsevier.com/locate/nuclphysb

Confronting current NLO parton fragmentation

functions with inclusive charged-particle spectra at

hadron colliders

David d’Enterria

, Kari J. Eskola

b,c

, Ilkka Helenius

b,c

Hannu Paukkunen

b,c,∗

CERN, PH Department, CH-1211 Geneva 23, Switzerland

Department of Physics, University of Jyväskylä, P.O. Box 35, FI-40014 University of Jyväskylä, Finland

Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland

Received 8 November 2013; received in revised form 5 February 2014; accepted 7 April 2014

Available online 13 April 2014

Editor: Tommy Ohlsson

Abstract

The inclusive spectra of charged particles measured at high transverse momenta (p

 2GeV/c)in

proton–proton and proton–antiproton collisions in the range of center-of-mass energies

√

s =200–7000 GeV

are compared with next-to-leading order perturbative QCD calculations using seven recent sets of parton-to-

hadron fragmentation functions (FFs). Accounting for the uncertainties in the scale choices and in the parton

distribution functions, we ﬁnd that most of the theoretical predictions tend to overpredict the measured LHC

and Tevatron cross sections by up to a factor of two. We identify the currently too-hard gluon-to-hadron

FFs as the probable source of the problem, and justify the need to reﬁt the FFs using the available LHC and

Tevatron data in a region of transverse momenta, p

 10 GeV/c, which is supposedly free from additional

non-perturbative contributions and where the scale uncertainty is only modest.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

Corresponding author at: Department of Physics, University of Jyväskylä, P.O. Box 35, FI-40014 University of

Jyväskylä, Finland.

E-mail addresses: dde@cern.ch (D.

d’Enterria), kari.eskola@jyu.ﬁ (K.J. Eskola), ilkka.helenius@jyu.ﬁ

(I. Helenius), hannu.paukkunen@jyu.ﬁ (H. Paukkunen).

http://dx.doi.org/10.1016/j.nuclphysb.2014.04.006

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

616 D. d’Enterria et al. / Nuclear Physics B 883 (2014) 615–628

1. Introduction

The inclusive production of large-transverse-momentum (p

 2GeV/c) hadrons at proton–

proton (p–p) and proton–antiproton (p–

p) colliders provides a ground for testing the factorization

theorem of Quantum Chromodynamics (QCD) [1,2] that predicts the universality and evolution

of the two non-perturbative elements in the theoretical cross-sections: parton distribution func-

tions (PDFs) and parton-to-hadron fragmentation functions (FFs). While the PDFs can nowadays

be tested and constrained by a multitude of different processes in deep-inelastic scattering and

hadronic collisions [3], the variety and kinematic reach of data by which the FFs can be de-

termined is more limited [4,5]. In this context, the inclusive hadron measurements at the LHC

extending to unprecedentedly large values of center-of-mass energy (

√

s) and hadron p

[6],

are particularly useful for studying the FFs and their universality. From an experimental per-

spective, the best precision and widest kinematic reach in the p

-differential cross sections are

achieved when no particle identiﬁcation is performed. The pre-LHC measurements for uniden-

tiﬁed charged-hadron spectra in p–p and p–

p collisions range from ﬁxed-target experiments at

√

s  60 GeV [7,8] to collider experiments covering a wide range of center-of-mass energies

√

s = 200–1960 GeV [9–16]. However, most of these data are restricted to moderate values of

 10 GeV/c, and the accuracy for p

≥10 GeV/c is rather poor. The importance of the new

LHC data to overcome such limitations is underscored by the confusion [17–19] triggered by the

original CDF data [16] which seemed to display deviations from next-to-leading order (NLO)

perturbative QCD (pQCD) calculations up to three orders of magnitude at p

150 GeV/c,but

which was later on identiﬁed as an experimental issue (see the erratum of Ref. [16]).

Interestingly, the NLO pQCD predictions presented along with the recently published

CMS [20,21] and ALICE [22] inclusive charged hadron spectra, appear to overshoot the data

by up to a factor of two in the kinematical region where effects such as e.g. intrinsic transverse

momentum of the colliding partons (intrinsic k

) [23–25], soft-gluon resummation [26,27],or

small-z instabilities of the FFs [28] should not play a major role. Especially since the recent

LHC measurements of the p

-differential cross sections for inclusive jets [29,30] and prompt

photons [31,32] are in perfect agreement with the NLO pQCD expectations [33–35], the data-

vs-theory discrepancies for inclusive charged-hadrons come totally unexpected. Resolving such

inconsistencies is also of relevance for other QCD analyses such as those related to the sup-

pression of high-p

hadrons in ultrarelativistic heavy-ion collisions where the p–p spectra are

required as baseline measurements [36].

In this paper, we present a systematic comparison of the theoretical predictions for unidenti-

ﬁed charged-hadron production to experimental data with a special emphasis on the latest LHC

measurements. Our aim is to demonstrate that such a process in hadron colliders is predominantly

sensitive to the gluon-to-hadron FFs which are presently not well determined and, consequently,

large discrepancies among the modern sets of FFs exist. These differences not only translate into

a signiﬁcant scatter in the corresponding predictions for the cross sections, but none of the current

FF sets can consistently reproduce the current LHC and Tevatron data at p

 10 GeV/c.Asthe

data measured by different experiments at the same collision energies are in mutual agreement,

it seems excluded that such discrepancies are due to an experimental issue. Instead, this hints

to a severe problem in the gluon-to-hadron FFs in most of the existing sets. We conclude that

the gluon FF, which is currently mildly constrained by charged-hadron spectra from hadronic

collisions at RHIC and Sp

pS energies, should be reﬁtted by using the LHC and Tevatron hadron

spectra in the region p

 10 GeV/c, where the theoretical scale uncertainties appear tolerable

and which should be free from additional, non-perturbative hadron production mechanisms.

D. d’Enterria et al. / Nuclear Physics B 883 (2014) 615–628 617

2. The pQCD framework for inclusive hadron production

The cross section for the inclusive production of a single hadron h

with a momentum p

in the collision of two hadrons h

and h

carrying momenta p

and p

respectively, can be

expressed, differentially in transverse momentum p

and (pseudo)rapidity η,as[24,37]

dσ(h

→h

+X)

dη



ij l





,μ

fact





,μ

fact



l→h



z, μ

frag



d ˆσ( ˆp

+ˆp

→ˆp

,μ

ren

,μ

fact

,μ

frag

)

d ˆp

dη



ˆp

. (1)

In this expression, f

,μ

fact

) denote the PDFs of the colliding hadrons evaluated at parton

fractional momenta x

and scale μ

fact

.WewillusetheCT10NLO [38] parametrization through-

out this work. The parton-to-hadron FFs are denoted by D

l→h

(z, μ

frag

) where z is the fraction

of the parton momentum carried out by the outgoing charged hadron. The PDFs and FFs are

convoluted with the partonic coefﬁcient functions d ˆσ for which we use their ﬁxed-order NLO

O(α

) expressions [37,39,40], treating the partons and hadrons as massless particles. In prac-

tice, we evaluate these cross sections employing the INCNLO [37,41] program which we have

modiﬁed to improve the convergence at small values of p

. The ﬁxed-order calculations are

supposed to be adequate for p

 1GeV/c but still sufﬁciently away from the phase-space

boundary p

max

∼

√

s/2 (at midrapidity, η ≈ 0), where soft-gluon resummations [26,27] become

relevant due to large logarithmic contributions from an incomplete cancellation of the infrared

divergences.

Truncating the partonic coefﬁcient functions to O(α

) leads to the well-known scale depen-

dence of the pQCD calculations. For inclusive hadron production, there are three independent

scales: the renormalization scale μ

ren

, factorization scale μ

fact

, and the fragmentation scale

frag

. The sensitivity of the computed cross sections to the variations of these scales is typically

taken as an indication of the size of the missing higher-order corrections. Our default choice is

ren

= μ

fact

= μ

frag

= p

, and we take the scale uncertainty as the envelope enclosed by the

following 16 scale variations [42]



fact

ren

frag



(

), (

, 1), (

, 1,

), (

, 1, 1),

(

, 1, 2), (1,

), (1,

, 1), (1, 1,

(1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1,

(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2).

(2)

We omit the combinations in which μ

ren

and μ

fact

or μ

ren

and μ

frag

are pairwise scaled by a factor

of two in opposite directions due to the appearance of potentially large contributions of the form

log(μ

ren

/μ

fact

) and log(μ

ren

/μ

frag

) in the calculation. The next-to-NLO (NNLO) calculations

are expected to deﬁnitely reduce the scale dependence. However, although the PDF analyses can

be nowadays carried out partly at NNLO level [43–46], the time-like splitting functions needed

in the NNLO evolution of the FFs are not yet fully known [47,48], nor are the NNLO coefﬁcient

functions needed in Eq. (1), although the latter could ﬁnally emerge through the work currently

carried out for jets [49–51].

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