Physics Letters B 732 (2014) 233–240
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Physics Letters B
www.elsevier.com/locate/physletb
High-energy QCD amplitudes at two loops and beyond
Vittorio Del Duca
a
, Giulio Falcioni
b
, Lorenzo Magnea
b
, Leonardo Vernazza
b
a
Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044 Frascati, Italy
b
Dipartimento di Fisica, Università di Torino and INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy
article info abstract
Article history:
Received 8 November 2013
Received in revised form 3 February 2014
Accepted 18 March 2014
Available online 21 March 2014
Editor: B. Grinstein
Recent progress in our understanding of infrared singularities of multi-parton amplitudes has shown
that the simplest form of Regge factorization for high-energy gauge-theory amplitudes fails starting at
next-to-next-to-leading logarithmic accuracy. We provide a framework to organize the calculation of
parton amplitudes at leading power in t
/s, in terms of factorizing and non-factorizing contributions.
This allows us to give explicit expressions for the leading reggeization-breaking terms in two-loop and
three-loop quark and gluon amplitudes in QCD. In particular, using only infrared information, we recover
a known non-factorizing, non-logarithmic double-pole contribution at two-loops, and we compute the
leading non-factorizing single-logarithmic contributions at three loops.
© 2014 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
3
.
1. Introduction
In the high-energy limit, in which the centre-of-mass en-
ergy
√
s is much larger than the typical momentum transfer
√
−t,
so that
|s/t|→∞,witht held fixed, gauge theory scattering am-
plitudes become very simple: they acquire a factorized structure,
where the building blocks are given by a t-channel propagator,
connecting two emission vertices, often called impact factors, char-
acterizing the particles undergoing the scattering. This structure is
often referred to as high-energy factorization: impact factors depend
on the specific scattering process, but they have a simple coupling
to the t-channel propagator, which is process independent.
Going from tree level to loop corrections, the picture remains
the same, but the t-channel propagator gets dressed according to
the schematic form [1],
1
t
→
1
t
s
−t
α(t)
, (1)
where α(t) is a function of the coupling constant, which in the
weak coupling limit becomes a series expansion in the coupling.
Because of the analytic structure of Eq. (1), which is typical of
Regge theory,
α(t) is called Regge trajectory.
Since the amplitude has a t-c
hannel ladder-like structure,
we can assume it to be even under s
↔ u exchange. As a conse-
quence, it must be composed of kinematic and color parts which
E-mail addresses: Vittorio.Del.Duca@cern.ch (V. Del Duca), falcioni@to.infn.it
(G. Falcioni), lorenzo.magnea@unito.it (L. Magnea), vernazza@to.infn.it (L. Vernazza).
are either both even or both odd under s ↔ u exchange. If one
considers t-channel gluon exchange, which is all that is needed at
leading order and at leading logarithmic accuracy in ln
(s/|t|),then
one takes the amplitude to be composed of kinematic and color
parts which are both odd under s
↔ u exchange. To be definite, let
us consider the amplitude for gluon–gluon scattering. In this case,
for the process g
(k
1
) + g(k
2
) → g(k
3
) + g(k
4
),onemaywrite[2]
M
gg→gg
a
1
a
2
a
3
a
4
s
μ
2
,
t
μ
2
, α
s
μ
2
=
4πα
s
μ
2
s
t
T
b
a
1
a
3
C
λ
1
λ
3
(k
1
, k
3
)
×
s
−t
α(t)
+
−
s
−t
α(t)
T
b
a
2
a
4
C
λ
2
λ
4
(k
2
, k
4
)
, (2)
where a
j
and k
j
are the color index and momentum of gluon j,
and T
b
is a color generator in the adjoint representation, so that
(T
a
)
bc
=−i f
abc
. The impact factors, C
λ
i
λ
j
(k
i
, k
j
), depend on the
helicities of the gluons, but, as the notation suggests, carry no s de-
pendence. Both the impact factors and the Regge trajectory, in the
weak coupling limit, can be expanded in powers of the renor-
malized coupling
α
s
(μ
2
): they are then affected by infrared and
collinear divergences, which in Eq. (2) are (implicitly) regularized
by dimensional regularization.
Beyond leading order, one should consider also the exchange of
tw
o or more reggeized gluons. Accordingly, one must include the
contribution to the amplitude in which the kinematic and color
parts are both even under s
↔ u exchange, and in particular the
http://dx.doi.org/10.1016/j.physletb.2014.03.033
0370-2693/
© 2014 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
3
.