Collinear Parton Dynamics Beyond DGLAP Hao Chen1Max Jaarsma2 3Yibei Li1Ian Moult4Wouter J. Waalewijn2 3and Hua Xing Zhu1 1Zhejiang Institute of Modern Physics Department of Physics Zhejiang University Hangzhou 310027 China

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Collinear Parton Dynamics Beyond DGLAP

Hao Chen,1Max Jaarsma,2, 3 Yibei Li,1Ian Moult,4Wouter J. Waalewijn,2, 3 and Hua Xing Zhu1

1Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou, 310027, China

2Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands

3Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics,

University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

4Department of Physics, Yale University, New Haven, CT 06511

Renormalization group evolution equations describing the scale dependence of quantities in quan-

tum chromodynamics (QCD) play a central role in the interpretation of experimental data. Arguably

the most important evolution equations for collider physics applications are the Dokshitzer-Gribov-

Lipatov-Altarelli-Parisi (DGLAP) equations, which describe the evolution of a quark or gluon frag-

menting into hadrons, with only a single hadron identiﬁed at a time. In recent years, the study of

the correlations of energy ﬂow within jets has come to play a central role at collider experiments,

necessitating an understanding of correlations, going beyond the standard DGLAP paradigm. In

this Letter we derive a general renormalization group equation describing the collinear dynamics

that account for correlations in the fragmentation. We compute the kernel of this evolution equa-

tion at next-to-leading order (NLO), where it involves the 1 →3 splitting functions, and develop

techniques to solve it numerically. We show that our equation encompasses all previously-known

collinear evolution equations, namely DGLAP and the evolution of multi-hadron fragmentation

functions. As an application of our results, we consider the phenomenologically-relevant example

of energy ﬂow on charged particles, computing the energy fraction in charged particles in e+e−→

hadrons at NNLO. Our results are an important step towards improving the understanding of the

collinear dynamics of jets, with broad applications in jet substructure, ranging from the study of

multi-hadron correlations, to the description of inclusive (sub)jet production, and the advancement

of modern parton showers.

Introduction.—Jets and their substructure play a cen-

tral role in modern collider experiments, both in searches

for beyond the Standard Model physics, as well as for

studying quantum chromodynamics (QCD) [1–3]. Due

to the conﬁnement process, jets are complicated multi-

scale objects, formed by the fragmentation of an initial

energetic quark or gluon at short times, into a collimated

spray of hadrons at long times. Because of this multi-

scale nature, renormalization group equations (RGE)

that describe the scale evolution of jets play a crucial

role in the interpretation of nearly all experimental data.

The most celebrated evolution equation is the

Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [4–

6] equation, describing the evolution of a quark or gluon

fragmenting into hadrons, with a single hadron identiﬁed

at a time and all hadrons summed over. This equation

plays a central role in the description of all aspects of

jets, from perturbative calculations, to parton shower al-

gorithms, to the evolution of fragmentation functions.

Because of this, it has received signiﬁcant theoretical at-

tention, and been computed to high orders [7–9].

Driven by the high energies, and exceptional resolution

of the detectors at the Large Hadron Collider (LHC),

there has been signiﬁcant recent interest in understand-

ing the correlations in energy ﬂow within jets, a ﬁeld

known as jet substructure [1–3]. The theoretical descrip-

tion of such correlations requires an understanding of

the scale evolution of correlations in the fragmentation

process, giving rise to non-linear RGEs, and going be-

yond the standard DGLAP evolution equations. While

FIG. 1. A parton with momentum kfragments into an identi-

ﬁed set of hadrons with momentum fraction PR, distinguished

by a speciﬁed quantum number (e.g. electric charge). The

scale evolution of this process is described by a non-linear

renormalization group evolution.

non-linear evolution equations for soft correlations have

existed for quite some time [10–15], similar non-linear

evolution equations incorporating correlations between

collinear partons are not known. Much like their soft

analogs, such non-linear collinear evolution equations will

also be essential for testing higher order collinear correc-

tions to next-generation parton showers [16–24].

In this Letter we derive a general evolution equation for

the fragmentation of collinear partons at next-to-leading

order (NLO), accounting for all correlations, which in-

volves for the ﬁrst time the complete structure of the

1→3 splitting functions. We also show that our evolu-

tion equation can be reduced to the DGLAP equation,

arXiv:2210.10061v1 [hep-ph] 18 Oct 2022

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im1

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···

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Ti1(x1)

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Ti2(x2)

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Tim1(xm1)

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Tim(xm)

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Ki!i1i2...im

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FIG. 2. Perturbative evolution is described by the splitting of

a single parton, i, into mpartons, each of whose momentum

fractions are tracked.

as well as the N-hadron fragmentation functions.

There are many applications of our evolution equa-

tions for jet substructure at the LHC. A key applica-

tion is the study of observables on tracks (charged parti-

cles) [25, 26], which will enable the measurement of more

sophisticated jet substructure observables. In particu-

lar, the use of tracks has played an important role in the

study of the collinear limit of energy correlators [27–29]

using CMS Open Data [30, 31]. This same equation can

be used to describe (sub)jet production for small values

of the radius, e.g. to describe the leading (sub)jet re-

quires knowledge about the other (sub)jets [32, 33]. We

therefore develop algorithms to eﬃciently solve our equa-

tions numerically for phenomenological applications, and

illustrate their use in a physical observable, the energy

fraction in charged hadrons in an e+e−collision.

The Master Equation for Collinear Evolution.—We de-

rive the general collinear evolution equation by studying

the renormalization of a universal object referred to as a

“track function” [25, 26]. In light-cone gauge, it is deﬁned

for quarks as [25, 26]

Tq(x) =Zdy+dd−2y⊥eik−y+/2X

δx−P−

k−

2Nc

trγ−

2h0|ψ(y+,0, y⊥)|XihX|¯

ψ(0)|0i,(1)

with a similar deﬁnition for gluons. Here PRdenotes the

momentum of a subset R⊂Xof hadrons, as illustrated

in Fig. 1. The RGE of these universal non-perturbative

quantities tracks the energy fractions of all partons in

a splitting, as is shown schematically in Fig. 2, leading

to a complicated non-linear evolution equation, whereas

DGLAP considers the momentum fraction of one parton

produced in a splitting at the time, summing over them.

In [34, 35], it was shown how to derive RG equations

for the ﬁrst six moments of the track functions. In this

Letter, we will extend this to derive the complete RGE

at NLO.

Considering the perturbative splitting illustrated in

Fig. 2, at NLO, we can have a splitting into at most

three-partons. The general form of the evolution equa-

tion for the track functions is therefore

d ln µ2Ti(x) = ashK(0)

i→iTi(x) + K(0)

i→i1i2⊗Ti1Ti2(x)i

+a2

shK(1)

i→iTi(x) + K(1)

i→i1i2⊗Ti1Ti2(x)

+K(1)

i→i1i2i3⊗Ti1Ti2Ti3(x)i+O(a3

s),

(2)

where as=αs/(4π) is the coupling and the convolutions

involving the evolution kernels Kare written explicitly

below in Eq. (3).

Since track functions are scaleless in dimensional regu-

larization, to derive their evolution equations, we follow

the approach of [34, 35] of computing an auxiliary ob-

servable which introduces a scale. We take this to be

a jet function diﬀerential in both the energy fraction of

charged particles, and the mass of all particles [36]. Af-

ter renormalization in the mass, the remaining poles are

associated with the track function renormalization.

To derive the evolution equations, one must integrate

out the angular variables appearing in the 1 →3 split-

ting function [37]. The derivation of IR ﬁnite evolution

equations then requires the cancellation poles between

the one-loop 1 →2 [38, 39] and tree-level 1 →3 [38, 39]

splitting functions. This is non-trivial as they can be

overlapping so that standard plus distributions cannot

be used. To overcome this, we use sector decomposition

[40–42], and identify a convenient set of variables which

disentangle the divergences.

We have computed the evolution kernels in both N=

4 super-Yang-Mills (SYM) and for all partonic channels

in QCD. The QCD kernels are provided in an attached

notebook. We will use the simpler N= 4 SYM kernels

to illustrate the form of the track function evolution,

d ln µ2T(x) = −25ζ3a2T(x)+Z1

dx1Z1

dx2Z1

dzT(x1)T(x2)δx−x1

1 + z−x2

1 + z(3)

×4a1

z+

+ 16a2ζ21

z+

+2 ln2(1+z)−ln zln(1+z)

z

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CollinearPartonDynamicsBeyondDGLAPHaoChen,1MaxJaarsma,2,3YibeiLi,1IanMoult,4WouterJ.Waalewijn,2,3andHuaXingZhu11ZhejiangInstituteofModernPhysics,DepartmentofPhysics,ZhejiangUniversity,Hangzhou,310027,China2Nikhef,TheoryGroup,SciencePark105,1098XG,Amsterdam,TheNetherlands3InstituteforTheoreticalPhysi...

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Collinear Parton Dynamics Beyond DGLAP Hao Chen1Max Jaarsma2 3Yibei Li1Ian Moult4Wouter J. Waalewijn2 3and Hua Xing Zhu1 1Zhejiang Institute of Modern Physics Department of Physics Zhejiang University Hangzhou 310027 China

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