Determining the nonperturbative Collins-Soper kernel from lattice QCD
Markus A. Ebert,
*
Iain W. Stewart,
†
and Yong Zhao
‡
Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
(Received 12 November 2018; published 19 February 2019)
At small transverse momentum q
T
, transverse-momentum dependent parton distribution functions
(TMDPDFs) arise as genuinely nonperturbative objects that describe Drell-Yan like processes in hadron
collisions as well as semi-inclusive deep-inelastic scattering. TMDPDFs naturally depend on the hadron
momentum, and the associated evolution is determined by the Collins-Soper equation. For q
T
∼ Λ
QCD
the
corresponding evolution kernel (or anomalous dimension) is nonperturbative and must be determined as an
independent ingredient in order to relate TMDPDFs at different scales. We propose a method to extract this
kernel using lattice QCD and the large-momentum effective theory, where the physical TMD correlatio n
involving light-like paths is approximated by a quasi-TMDPDF, defined using equal-time correlation
functions with a large-momentum hadron state. The kernel is determined from a ratio of quasi-TMDPDFs
extracted at different hadron momenta.
DOI: 10.1103/PhysRevD.99.034505
I. INTRODUCTION
In the past decades, advances in theory and experiment
have made it possible to explore the structure of the proton
beyond the simplest longitudinal momentum distributions.
Key observables are transverse momentum distributions
(TMDs), which measure the intrinsic transverse momentum
q
T
of partons in the proton, as well as describing the
probability to produce particles at larger q
T
in high energy
collisions. These TMDs are probed directly by experiments
on Drell-Yan, semi-inclusive deep inelastic scattering
(SIDIS), and other processes. Recently, progress has been
made in determining transverse-momentum dependent
parton distribution functions (TMDPDFs) by using lattice
QCD [1–6] to study equal-time correlators. Such correla-
tors are a key ingredient in the large-momentum effective
theory (LaMET) [7,8], where one computes a lightcone
correlator using an equal-time correlator in a boosted
proton state. For TMDPDFs the first theoretical studies
in LaMET have been carried out in Refs. [9–11].
The TMDPDF f
TMD
i
ðx;
b
T
; μ; ζÞ for a parton of flavor i
depends on x, the fraction of the hadron momentum carried
by the struck parton,
b
T
, the Fourier-conjugate of the
transverse momentum
q
T
, and on a virtuality scale μ.
In addition, it depends on a scale ζ, which is related to the
momentum of the hadron or equivalently the hard scale
of the scattering process. Measuring nonperturbative
TMDPDFs, whether from experiment or lattice, thus
requires to specify the scales ðμ
0
; ζ
0
Þ where the
TMDPDF is extracted. For instance, lattice calculations
are restricted to μ
2
0
∼ ζ
0
∼ Oð4 GeV
2
Þ due to finite lattice
spacing, while for example the application to Drell-Yan
production uses μ
2
∼ζ ∼m
2
Z
≈ð91 GeVÞ
2
. The TMDPDFs
thus need be evolved from ðμ
0
; ζ
0
Þ to the phenomenologi-
cally relevant scales ðμ; ζÞ, using
f
TMD
i
ðx;
b
T
; μ; ζÞ¼f
TMD
i
ðx;
b
T
; μ
0
; ζ
0
Þ
× exp
Z
μ
μ
0
dμ
0
μ
0
γ
i
μ
ðμ
0
; ζ
0
Þ
× exp
1
2
γ
i
ζ
ðμ;b
T
Þln
ζ
ζ
0
: ð1Þ
The first exponential in Eq. (1) is the μ evolution and
the second exponential is the Collins-Soper evolution in ζ
[12–14] in the formulation of Ref. [15], with γ
i
μ
and γ
i
ζ
being the associated anomalous dimensions. Here γ
i
ζ
is the
Collins-Soper kernel, often denoted by
˜
K.
The μ evolution in Eq. (1) is perturbative as long as both
μ
0
, μ ≫ Λ
QCD
, analogous to the perturbative DGLAP
evolution for collinear PDFs. In contrast, the Collins-
Soper kernel involves the b
T
-dependent anomalous
dimension γ
i
ζ
ðμ;b
T
Þ, which becomes nonperturbative in
the region b
−1
T
∼ q
T
∼ Λ
QCD
, even if μ ≫ Λ
QCD
. Relating
*
ebert@mit.edu
†
iains@mit.edu
‡
yzhaoqcd@mit.edu
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, jou rnal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D 99, 034505 (2019)
2470-0010=2019=99(3)=034505(10) 034505-1 Published by the American Physical Society