JHEP03(2018)048
are available [7, 8]. The NNLO result was completed in ref. [9] in which the three-loop
matching coefficient of the effective operator for two Higgs bosons and two, three or four
gluons was computed. Note that it differs from that of single Higgs boson producti on.
The re sul t of [
9] has been complemented by power-suppressed terms i n the top quark
mass in ref. [
10], where the soft-virtual approximation was constructed. The resummation
of threshold-enhanced logarithms to next-to-next-to-leading logarithmic (NNLL) accur acy
has been perfor me d in refs. [
11, 12] and differential distributions through NNLO for various
observables were c ompu te d in ref. [
13] in the heavy-top li mi t . Finally, exact NLO results
became available in refs. [
14, 15] using a numerical approach for the computation of t he
two-loop v i r t ual corrections. Bas ed on these results t he transverse momentum resumma-
tion has been considered in ref. [
16]. More recently the results of refs. [14, 15] have been
matched to parton shower s in refs. [
17, 18].
In this paper we study a class of massive two-loop four-point functions with massless
external particles. We describe in detail the methods used for the computation of the
amplitudes and in particular the evaluation of the master integrals. We aim to study
double Higgs boson production via the process gg → HH. Numerical NLO results are
available [
14, 15], however the calculation of cross sections is computationally expe nsi ve
and we want to provide an independent cross check in th e high-ene r gy region. We wish to
provide results in terms of compac t analytic expressions which can be used to construct
simple approximations or can be used directly in the kinematic region in which they are
valid. In thi s paper we provide the first step towards this goal by considering the part of
the amplitude whi ch is expressed in terms of planar master integrals.
We perform our calculation in the limit of vanishing Higgs boson mass which provides,
as we will demonstrate in section
3, a good approximation to the general case where
m
H
6= 0. Furthermore, finite Higgs-mass effec ts can be incorporated by a simple Taylor
expansion. Recently the amplitudes for single-Higgs boson plus jet production have been
considered in the limit of lar ge Higgs transverse momentum [
19]. In this reference an
expansion for small Higgs boson mass has been performed and thus the underlying integrals
are the same as those of our calculation, so part of our findings can be cross checked
against ref. [
19].
In the recent literature one can find se veral calculations where two-lo op box integrals
are also involved. However, the underlying integral families and/or the kinematics of the
external and internal masses are different. For example, in ref. [
20] the amplitude of a
Higgs boson and three partons has been considered. In the limit m
H
→ 0 their integral s
are also the same as ours. However, this limit cannot been taken since the calculation
is perf or med in the Euclidean region with the assumption m
2
H
< s < 0 an d the results
are expres se d in terms of multiple polylogarithms, which can not easily be analytically
continued into ot her regions. Si mi l ar arguments apply to other recent calculations such
as [
21] or [22]; analytic results have been ob tai ne d in terms of multiple polylogarithms
which can in principle be evaluated numerically, but are very unwieldy. This is a another
reason why we have decided to perform an expansion in the high ener gy limit. Our final
results have a simple structure in terms of harmonic polyl ogari t hms and can be evaluated
numerically in a fast and rel i abl e manner.
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