Physics Letters B 772 (2017) 446–451
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Soft-gluon resolution scale in QCD evolution equations
F. Hautmann
a,b,c
, H. Jung
d
, A. Lelek
d
, V. Radescu
e
, R. Žleb
ˇ
cík
d
a
Rutherford Appleton Laboratory, Chilton OX11 0QX, United Kingdom
b
Theoretical Physics Department, University of Oxford, Oxford OX1 3NP, United Kingdom
c
Elementaire Deeltjes Fysica, Universiteit Antwerpen, B 2020 Antwerpen, Belgium
d
Deutsches Elektronen Synchrotron, D 22603 Hamburg, Germany
e
CERN, CH-1211 Geneva 23, Switzerland
a r t i c l e i n f o a b s t r a c t
Article history:
Received
7 April 2017
Received
in revised form 29 June 2017
Accepted
4 July 2017
Available
online 10 July 2017
Editor:
G.F. Giudice
QCD evolution equations can be recast in terms of parton branching processes. We present a new
numerical solution of the equations. We show that this parton-branching solution can be applied to
analyze infrared contributions to evolution, order-by-order in the strong coupling α
s
, as a function of
the soft-gluon resolution scale parameter. We examine the cases of transverse-momentum ordering and
angular ordering. We illustrate that this approach can be used to treat distributions which depend both
on longitudinal and on transverse momenta.
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
The evolution of QCD parton cascades is an essential element of
theoretical predictions for production processes with high momen-
tum
transfer at high-energy colliders. It has been realized since
long that realistic predictions for collider processes require tak-
ing
into account contributions to QCD evolution not only from
collinear parton radiation, associated with the renormalization
group behavior at high momenta, but also from soft gluon radi-
ation,
including its color coherence properties [1–5]. Infrared radi-
ation
contributions are controlled by a finite resolution scale δ of
order δ ∼ O(
QCD
/μ), where μ is the hard scattering scale and
QCD
≈ 1fm
−1
is the natural scale of strong interactions.
In
this paper we study the effects of soft-gluon emission
and the soft-gluon resolution scale in cases in which not only
longitudinal-momentum degrees of freedom but also transverse-
momentum
degrees of freedom are necessary for reliable theo-
retical
predictions. We address the issue of taking into account
simultaneously soft gluon radiation, with light-cone momentum
fraction z → 1, and transverse momentum q
⊥
recoils in the par-
ton
branchings along the QCD cascade. This is relevant for instance
in multiple-scale problems, such as the high invariant-mass region
of heavy particle spectra and the high-energy limit of hadropro-
duction
processes, where transverse-momentum dependent (TMD)
factorization theorems apply (see e.g. [6] for a recent review).
While
analytic resummation methods exist, based on these the-
orems,
for sufficiently inclusive variables such as, e.g., heavy-boson
E-mail address: francesco.hautmann@desy.de (F. Hautmann).
transverse spectra, the viewpoint in this work is to aim at a for-
mulation
by which one could also treat exclusive components of
the final states. Parton-shower algorithms do provide such a for-
mulation,
and are widely used as an effective alternative (though
limited in accuracy) to analytic resummation methods. While great
progress has been achieved in the last decade on matching and
merging methods [7] to combine parton showers with perturbative
calculations through next-to-leading order, several open questions
still remain, both conceptual and technical, on the appropriate use
of parton distribution functions in parton showers [8] and on the
treatment of the shower’s transverse momentum kinematics [9].
In
the context of analytic methods, the behavior of parton
distributions near the endpoint z → 1motivates the use of in-
frared
subtractive techniques which lead to a generalization of the
“plus” distribution [10] including transverse degrees of freedom.
In this paper we describe a new calculation, based on the unitar-
ity
method [1] to recast evolution equations in terms of Sudakov
form factors and real emission kernels, and present a study of the
soft-gluon resolution scale in the cases of inclusive parton distri-
butions
and of transverse-momentum dependent parton distribu-
tions.
We analyze different ordering variables, including transverse
momentum ordering and angular ordering. The method set up
in this paper can be applied systematically order-by-order in the
strong coupling α
s
, at leading order as well as at next-to-leading
and higher orders. In this article we present the basic results and
numerical leading-order applications. Details of the method and
applications including next-to-leading order will be presented else-
where [11].
http://dx.doi.org/10.1016/j.physletb.2017.07.005
0370-2693/
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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