Physics Letters B 735 (2014) 266–271
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The role of Glauber exchange in soft collinear effective theory and the
Balitsky–Fadin–Kuraev–Lipatov Equation
Sean Fleming
University of Arizona, Tucson, AZ 85721, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
25 April 2014
Received
in revised form 10 June 2014
Accepted
16 June 2014
Available
online 20 June 2014
Editor:
B. Grinstein
In soft collinear effective theory (SCET) the interaction between high energy quarks moving in opposite
directions involving momentum transfer much smaller than the center-of-mass energy is described by
the Glauber interaction operator which has two-dimensional Coulomb-like behavior. Here, we determine
this n–
¯
n collinear Glauber interaction operator and consider its renormalization properties at one loop. At
this order a rapidity divergence appears which gives rise to an infrared divergent (IR) rapidity anomalous
dimension commonly called the gluon Regge trajectory. We then go on to consider the forward quark
scattering cross section in SCET. The emission of real soft gluons from the Glauber interaction gives rise
to the Lipatov vertex. Squaring and adding the real and virtual amplitudes results in a cancelation of IR
divergences, however the rapidity divergence remains. We introduce a rapidity counter-term to cancel the
rapidity divergence, and derive a rapidity renormalization group equation which is the Balitsky–Fadin–
Kuraev–Lipatov
Equation. This connects Glauber interactions with the emergence of Regge behavior in
SCET.
© 2014 The Author. 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
3
.
Factorization of high-energy interactions in QCD is the sys-
tematic
separation of different momentum regions into universal
factors to all orders in the strong coupling constant α
s
. All-order
proofs of factorization, which were first carried out by Collins,
Soper and Sterman [1–4] rely on a set of powerful theoretical tools.
Among these are: power counting, pinch analysis via the Landau
equations [5], and the Coleman–Norton Theorem [6]. The Landau
equations allow for the isolation of pinch singularities which, via
the Coleman–Norton Theorem can be identified with long-distance
(infrared) physics. Generically pinch singularities can be identified
with one of three momentum regions: collinear, soft, or Glauber.
In the collinear region internal propagators become collinear with
external particles, and in the soft region they become soft relative
to external particles. In either of these limits particles can approx-
imately
stay on their mass shell. The Glauber region, however, is
special as it corresponds to off-shell modes (Glauber modes) with
k
⊥
k
+
, k
−
, which leads to a two-dimensional Coulomb-like in-
teraction
between and amongst collinear and soft particles [1,7].
The presence of Glauber interactions is problematic because they
can destroy factorization [7,8]. Fortunately, it has been shown that
for sufficiently inclusive quantities the sum over final-state cuts
cancels unwanted pinches, and thereby eliminates Glauber contri-
butions
[9–11].
E-mail address: fleming@physics.arizona.edu.
An alternative approach in deriving factorization is to use effec-
tive
field theory (EFT). The EFT that describes the soft and collinear
degrees of freedom which arise in factorization is soft collinear ef-
fective
theory (SCET) [12–15], and in Ref. [16] it was shown how
the perturbative factorization theorems of QCD are reproduced in
SCET. However, SCET as it was originally formulated did not include
Glauber type interactions. An attempt to include Glauber interac-
tions
between collinear quarks moving in opposite directions in
SCET was made in Ref. [17] where factorization of the Drell–Yan
cross section was reconsidered. Unfortunately, this attempt did not
account for the overlap between different moment regions and
failed as a result. The analysis was taken up in Ref. [18] where
it was concluded that “for the exclusive Drell–Yan amplitude the
correct effective theory would require Glauber modes.” Though the
authors did not consider under which circumstances the contri-
bution
from Glauber interactions cancels. In addition, a number
of groups have considered the role of Glauber interactions be-
tween
collinear and soft degrees of freedom in dense QCD matter
[19–21]. More recently, an attempt to include a Glauber interac-
tion
between two collinear particles moving in opposite directions
has been presented [22,23].
A
second, seemingly unrelated issue concerning the formula-
tion
of SCET was raised in Refs. [24,25], where it was pointed
out that Regge behavior appears to fall outside of the usual or-
ganizing
scheme of SCET. Specifically, Regge behavior refers to
the emergence of power-law behavior for scattering amplitudes.
http://dx.doi.org/10.1016/j.physletb.2014.06.045
0370-2693/
© 2014 The Author. 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
3
.
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