JHEP09(2021)161
Published for SISSA by Springer
Received: July 5, 2021
Accepted: September 5, 2021
Published: September 23, 2021
Anomalous couplings in Higgs-boson pair production
at approximate NNLO QCD
Daniel de Florian,
a
Ignacio Fabre,
a
Gudrun Heinrich,
b
Javier Mazzitelli
c
and
Ludovic Scyboz
d
a
International Center for Advanced Studies (ICAS) and ICIFI, ECyT-UNSAM,
Campus Miguelete, 25 de Mayo y Francia, 1650 Buenos Aires, Argentina
b
Institute for Theoretical Physics, Karlsruhe Institute of Technology (KIT),
Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, Germany
c
Max Planck Institute for Physics,
Föhringer Ring 6, 80805 München, Germany
d
Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
Parks Road, Oxford OX1 3PU, U.K.
E-mail: deflo@unsam.edu.ar, ifabre@unsam.edu.ar,
gudrun.heinrich@kit.edu, jmazzi@mpp.mpg.de,
ludovic.scyboz@physics.ox.ac.uk
Abstract: We combine NLO predictions with full top-quark mass dependence with ap-
proximate NNLO predictions for Higgs-boson pair production in gluon fusion, including the
possibility to vary coupling parameters within a non-linear Effective Field Theory frame-
work containing five anomalous couplings for this process. We study the impact of the
anomalous couplings on various observables, and present Higgs-pair invariant-mass dis-
tributions at seven benchmark points characterising different m
hh
shape types. We also
provide numerical coefficients for the approximate NNLO cross section as a function of the
anomalous couplings at
√
s = 14 TeV.
Keywords: NLO Computations, QCD Phenomenology
ArXiv ePrint: 2106.14050
Open Access,
c
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP09(2021)161
JHEP09(2021)161
Contents
1 Introduction 1
2 Higgs-boson pair production within an EFT framework 3
3 Phenomenological results and coupling coefficients 4
3.1 Total cross section 5
3.2 Differential results and heat maps 6
3.3 Fitting procedure and results for the coupling coefficients 9
4 Conclusions 11
A EFT parameter points used for the fit 12
1 Introduction
The Higgs sector is being explored at the LHC with impressive success, having established
the coupling of the Higgs boson to all the electroweak gauge bosons [1], the top [2, 3]
and bottom quarks [4, 5], the tau lepton [6, 7] and recently moving towards indication for
Higgs couplings to muons [8, 9]. In addition to these important results, it is also crucial to
study Higgs boson self-couplings, which provide a way to explore the potential that drives
the electroweak symmetry breaking mechanism. The trilinear Higgs boson self-coupling is
still rather weakly constrained, even though the limits have improved considerably in Run
II [10–13]. The precision of the measurements of Higgs boson couplings to other particles
and itself will increase substantially in the high-luminosity phase of the LHC [14, 15].
Therefore, simulations of the effects of anomalous couplings in the Higgs sector also need
to achieve rather small uncertainties.
Higgs-boson pair production in gluon fusion is a process with direct access to the
trilinear Higgs boson self-coupling c
hhh
. An established deviation from its Standard Model
(SM) value would be a clear sign of new physics. However, if c
hhh
is different from the value
predicted by the SM, it is very likely that other Higgs couplings are also modified, such that
several anomalous couplings, entering the process gg → HH dominantly within an Effective
Field Theory (EFT) framework, should be studied simultaneously. Furthermore, in order
to get reliable predictions, higher-order QCD corrections need to be taken into account,
not only for the SM cross section but also in the case anomalous couplings are included.
The loop-induced process gg →HH at leading order has been calculated in refs. [16–18].
Before the full next-to-leading order (NLO) QCD corrections became available, the
m
t
→∞ limit (“Heavy Top Limit, HTL”), sometimes also called “Higgs Effective Field
– 1 –
JHEP09(2021)161
Theory (heft)” approximation, has been used. In this limit, the NLO corrections were
first calculated in ref. [19] using the so-called “Born-improved HTL”, which involves rescal-
ing the NLO results in the m
t
→ ∞ limit by the LO result in the full theory. In ref. [20] an
approximation called “FT
approx
” was introduced, which contains the real radiation matrix
elements with full top-quark mass dependence, while the virtual part is calculated in the
Born-improved HTL approximation. The NLO QCD corrections with full top-quark mass
dependence became available more recently [21–24]. The NLO results of refs. [21, 22] have
been combined with parton shower Monte Carlo programs in refs. [25–28], where ref. [28]
introduced the possibility of varying five Higgs couplings.
In the m
t
→ ∞ limit, the next-to-next-to-leading order (NNLO) QCD corrections
have been computed in refs. [29–33]. The calculation of ref. [33] has been combined with
results including the top-quark mass dependence as far as available in ref. [34], defining an
NNLO
FTapprox
result which contains the full top-quark mass dependence at NLO as well as
in the double real radiation part. Soft gluon resummation combined with these results has
been presented in ref. [35]. N
3
LO corrections are also available [36, 37], where in ref. [37]
the N
3
LO results in the HTL have been “NLO-improved” using the results of refs. [25, 27].
In ref. [38], NNLO results in the Born-improved heavy top limit including the effect of
anomalous couplings were presented, see also ref. [39] for the NLO case.
The scale uncertainties at NLO are still at the 10% level, while they are decreased
to about 5% when including the NNLO corrections and to about 3% at N
3
LO in the
“NLO-improved” variant. The uncertainties due to the chosen top mass scheme have been
assessed in refs. [23, 24, 40].
In this work we present a study of the anomalous couplings relevant to the process
gg → HH within a non-linear EFT operator expansion, at approximate NNLO in QCD
(which we will denote by NNLO
0
). It builds on the results presented in ref. [38], and extends
them to include the full NLO QCD corrections from ref. [41]. Thus NNLO
0
contains NNLO
results in the heavy top limit, where exact Born expressions have been used whenever the
higher-order corrections in the HTL factorise, as well as NLO corrections with full top-
quark mass dependence.
In particular, we provide coefficients for all the possible combinations of anomalous
couplings that can occur at NNLO for the total cross section, analogous to what has been
provided at NLO in ref. [41]. These coefficients can then be used to reconstruct the cross
section for any combination of coupling values. Furthermore, we show results for seven
benchmark points characteristic of certain shape types of the Higgs-boson pair invariant-
mass distribution m
hh
, which have been identified by an NLO shape analysis presented
in ref. [42].
This paper is organised as follows. In section 2 we describe the theoretical framework
and the definition of the anomalous couplings. Section 3 contains the phenomenological re-
sults, including total cross sections and differential distributions for the benchmark points,
and the description of the fitting procedure for the coefficients of the anomalous couplings,
together with the corresponding results. Finally, we conclude in section 4.
– 2 –
JHEP09(2021)161
Figure 1. Higgs-boson pair production in gluon fusion at leading order in QCD and at chiral
dimension d
χ
= 4. The black dots indicate vertices from anomalous couplings present already at
leading order in the chiral Lagrangian (d
χ
= 2), the black squares denote local operators contribut-
ing at (d
χ
= 4).
2 Higgs-boson pair production within an EFT framework
The calculation builds on the ones presented in refs. [28, 38, 41] and therefore the methods
will be described only briefly here, focusing on the new aspects.
We work in a non-linear EFT framework, sometimes also called Electroweak Chiral
Lagrangian (EWChL) including a light Higgs boson [43, 44] or HEFT (Higgs Effective Field
Theory), not to be confused with the heavy top limit, which is sometimes also called heft.
It relies on counting the chiral dimension of the terms contributing to the Lagrangian [45],
rather than counting the canonical dimension as in the Standard Model Effective Field
Theory (SMEFT). As a consequence, the EWChL is also suitable to describe strong
dynamics in the Higgs sector. Applying this framework to Higgs-boson pair production in
gluon fusion, keeping terms up to, and including, chiral dimension d
χ
= 4, we obtain the
effective Lagrangian relevant to this process as
L ⊃ −m
t
c
t
h
v
+ c
tt
h
2
v
2
!
¯
t t −c
hhh
m
2
h
2v
h
3
+
α
s
8π
c
ggh
h
v
+ c
gghh
h
2
v
2
!
G
a
µν
G
a,µν
. (2.1)
In the EWChL framework there are a priori no relations between the couplings. In general,
all couplings may have arbitrary values of O(1). The conventions are such that in the SM
c
t
= c
hhh
= 1 and c
tt
= c
ggh
= c
gghh
= 0. The EWChL coefficients can be related [39]
to those in the SMEFT at Lagrangian level, however how to treat double insertions of
operators and squared dimension-6 terms at cross section level is less straightforward when
attempting to relate the two EFT frameworks. The diagrams at leading order in QCD and
chiral dimension four are shown in figure 1.
There are different normalisation conventions for the anomalous couplings in the liter-
ature. In table 1 we summarise some conventions commonly used. For the relation to the
corresponding parameters in the SMEFT we refer to ref. [28].
– 3 –
JHEP09(2021)161
Eq. (2.1), i.e. L of ref. [41] Ref. [46] Ref. [39]
c
hhh
κ
λ
c
3
c
t
κ
t
c
t
c
tt
c
2
c
tt
/2
c
ggh
2
3
c
g
8c
g
c
gghh
−
1
3
c
2g
4c
gg
Table 1. Translation between different conventions for the definition of the anomalous couplings.
In ref. [41] the NLO QCD corrections were calculated within this framework. These
corrections have been implemented into the code ggHH [27, 28], which is a public code avail-
able at [47], where the real radiation matrix elements were implemented using the interface
between GoSam [48, 49] and the POWHEG-BOX [50–52] and the virtual two-loop corrections
use the results of the calculations presented in refs. [21, 22]. In turn, approximate NNLO
QCD corrections in a similar EFT framework have been computed in ref. [38], working in
the heavy top limit improved by inserting LO form factors with full top mass dependence.
The results described in this paper are obtained by performing an additive combination of
the ones presented in refs. [38, 41], i.e. keeping the full top mass dependence up to NLO and
adding the approximate results for the genuine NNLO piece. We will denote our results
by NNLO
0
.
We can describe the dependence of the NNLO cross section on the five anomalous
couplings in terms of 25 coefficients a
i
, following refs. [41, 46, 53]:
σ
BSM
/σ
SM
= a
1
c
4
t
+ a
2
c
2
tt
+ a
3
c
2
t
c
2
hhh
+ a
4
c
2
ggh
c
2
hhh
+ a
5
c
2
gghh
+ a
6
c
tt
c
2
t
+ a
7
c
3
t
c
hhh
+ a
8
c
tt
c
t
c
hhh
+ a
9
c
tt
c
ggh
c
hhh
+ a
10
c
tt
c
gghh
+ a
11
c
2
t
c
ggh
c
hhh
+ a
12
c
2
t
c
gghh
+ a
13
c
t
c
2
hhh
c
ggh
+ a
14
c
t
c
hhh
c
gghh
+ a
15
c
ggh
c
hhh
c
gghh
+ a
16
c
3
t
c
ggh
+ a
17
c
t
c
tt
c
ggh
+ a
18
c
t
c
2
ggh
c
hhh
+ a
19
c
t
c
ggh
c
gghh
+ a
20
c
2
t
c
2
ggh
+ a
21
c
tt
c
2
ggh
+ a
22
c
3
ggh
c
hhh
+ a
23
c
2
ggh
c
gghh
+ a
24
c
4
ggh
+ a
25
c
3
ggh
c
t
. (2.2)
While at LO only the first 15 coefficients contribute, at NLO 23 coupling combinations
occur and at NNLO 25 combinations are possible.
It is worth stressing that the functional dependence described in eq. (2.2), which holds
for the exact NNLO cross section, is also valid for the approximate results presented in this
work. To this end, the fact that the combination of the full NLO and approximate NNLO
is done through an additive approach becomes crucial, since e.g. a simple multiplicative
rescaling would not respect the functional form of eq. (2.2) (and, in particular, due to
the new combinations appearing at NNLO it would be ill-defined for certain values of the
anomalous couplings).
3 Phenomenological results and coupling coefficients
Our results are calculated at a centre-of-mass energy of
√
s = 14 TeV. The parton distri-
bution functions PDF4LHC15 [54–57] have been used, interfaced via LHAPDF [58], along
– 4 –
