11institutetext: Institute for Theoretical Physics, Karlsruhe Institute of
Technology, 76131 Karlsruhe, Germany 22institutetext: Institute for
Astroparticle Physics, Karlsruhe Institute of Technology, 76344 Eggenstein-
Leopoldshafen, Germany33institutetext: Physik-Institut, Universität Zürich,
CH-8057 Zürich, Switzerland 44institutetext: Theoretical Physics Department,
CERN, 1211 Geneva 23, Switzerland 55institutetext: Institute for Particle
Physics Phenomenology, Durham University, Durham, DH1 3LE, UK 66institutetext:
Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
UK 77institutetext: Institut für Theoretische Teilchenphysik, Karlsruhe
Institute of Technology, 76128 Karlsruhe, Germany

# Top-quark mass effects in $H$+jet and $H$+2 jets production

X. Chen 4 A. Huss 5 S. P. Jones 1,2,3 M. Kerner 3 J.-N. Lang 6 J. M. Lindert
3,7 H. Zhang

###### Abstract

We present calculations of Higgs boson production via gluon-gluon fusion in
association with one or two additional jets at next-to-leading order in QCD.
The calculation of $H$+jet is exact in the treatment of the top-quark mass,
whereas for the $H$+2 jets calculation the two-loop virtual amplitudes are
approximated via a reweighting with leading-order mass effects, while keeping
all top-quark mass effects in the real radiation contributions. For $H$+jet
production, this study extends a previous calculation, revealing an error in
the previous results. For total and differential cross sections, we present
new results and compare the QCD corrections with the infinite top-mass limit,
for which we find a strikingly good agreement if all amplitudes are rescaled
by the leading-order mass dependence.

††preprint:  KA-TP-21-2021 TTP21-035 CERN-TH-2021-153 P3H-21-070 ZU-TH 48/21
IPPP/21/39

##  1 Introduction

The current and upcoming runs of the Large Hadron Collider (LHC) are stress-
testing the Standard Model (SM) of particle physics at an unprecedented level.
In this respect one of the main objectives of Run 3 and the high-luminosity
phase of the LHC (HL-LHC) will be a further detailed investigation of the
Higgs sector. The abundant future data samples will allow the range of Higgs
analyses to be extended to multi-dimensional measurements and high-energy
tails of kinematic distributions. A key observable in this regard is the
transverse momentum distribution of the Higgs boson, $p_{\rm{T},H}$, which
serves as a unique probe of physics Beyond the Standard Model (BSM)
Arnesen:2008fb ; Banfi:2013yoa ; Azatov:2013xha ; Grojean:2013nya ;
Harlander:2013oja ; Dawson:2014ora ; Langenegger:2015lra ; Grazzini:2016paz ;
Deutschmann:2017qum ; Battaglia:2021nys . Given the expected experimental data
sets, this distribution will be measured both inclusively and in association
with jets up to several hundreds of $\mathrm{GeV}$ in the Higgs transverse
momentum ATLAS:2018ibe ; CMS:2018qgz . Already now experimental measurements
by ATLAS and CMS yield sensitivity up to few hundred $\mathrm{GeV}$
ATLAS:2021nsx ; Sirunyan:2020hwz .

Both in the inclusive case and for the production in association with jets,
the dominant Higgs production mode in the SM originates via a top-quark loop
in gluon-gluon fusion, however, at large transverse momentum eventually also
vector-boson fusion (VBF), Higgsstrahlung (VH) and top-pair associated Higgs
production contribute significantly Becker:2020rjp . For the production in
association with jets, additional constraints on jet invariant masses and/or
rapidities allow the relative fraction of the VBF events to be enhanced
Campanario:2018ppz ; Buckley:2021gfw . Precise VBF measurements will allow us
to constrain on the one hand the electroweak (EW) couplings of the Higgs, and
on the other hand when restricting to large Higgs transverse momentum they
will allow for complementary constraints on models of new physics
Artoisenet:2013puc ; Maltoni:2013sma ; Greljo:2015sla ; Degrande:2016dqg ;
Greljo:2017spw ; Araz:2020zyh . In this regard, one of the dominant
uncertainties in VBF measurements originates from the background modelling of
the gluon-induced Higgs production mode.

The loop-induced nature of the gluon-gluon fusion Higgs production process
makes higher-order corrections notoriously difficult to calculate. In QCD
fixed-order perturbation theory including mass effects, inclusive Higgs
production at next-to-next-to-leading order (NNLO) was calculated only very
recently Czakon:2021yub , Higgs plus jet production is known at next-to-
leading order (NLO) Jones:2018hbb ; Lindert:2018iug ; Neumann:2018bsx ;
Kerner:2019qgb , while Higgs plus dijet production (and beyond) is only known
at leading order (LO) DelDuca:2001eu ; DelDuca:2001fn ; Neumann:2016dny ;
Andersen:2018kjg ; Budge:2020oyl . In the NLO computations of Higgs plus jet
production the crucial two-loop virtual contributions have been obtained
numerically Borowka:2015mxa ; Borowka:2017idc  in Refs. Jones:2018hbb ;
Kerner:2019qgb  respectively via a suitable high-energy expansion
Kudashkin:2017skd  in Refs. Lindert:2018iug ; Neumann:2018bsx .

Formally, below the top-quark threshold, higher precision can be achieved via
the heavy top loop (HTL) approximation, effectively integrating out the top-
quark loop Wilczek:1977zn . In the HTL approximation, inclusive Higgs
production is known at N3LO Anastasiou:2015vya ; Dulat:2017prg ; Cieri:2018oms
; Mistlberger:2018etf ; Chen:2021isd , Higgs plus jet production at NNLO
Boughezal:2013uia ; Chen:2014gva ; Boughezal:2015dra ; Boughezal:2015aha ;
Chen:2016zka , and Higgs plus dijet production at NLO Campbell:2006xx ;
vanDeurzen:2013rv  (for Higgs plus trijet production see Cullen:2013saa ). For
inclusive Higgs production and for $p_{\rm{T},H}<m_{t}$ (where $m_{t}$ is the
top-quark mass) predictions in the full SM and in the HTL agree at the percent
level. For $p_{\rm{T},H}\ll m_{t}$ eventually fixed-order perturbation theory
becomes unreliable and a matching to higher logarithmic accuracy becomes
mandatory Bozzi:2005wk ; deFlorian:2011xf ; Becher:2012yn ; Bizon:2017rah ;
Chen:2018pzu ; Bizon:2018foh ; Re:2021con , and also bottom-quark effects have
to be considered Mantler:2012bj ; Grazzini:2013mca ; Banfi:2013eda ;
Melnikov:2016qoc ; Lindert:2017pky ; Melnikov:2017pgf ; Caola:2018zye .

At the other end of the spectrum, for $p_{\rm{T},H}>m_{t}$, the accuracy of
the HTL quickly deteriorates due to a different high-energy scaling compared
to the full theory Caola:2016upw . For $p_{\rm{T},H}=500(1000)\,\mathrm{GeV}$
the two differ by a factor of about 4(10). In this high-energy regime in order
to improve with respect to the HTL additional $\mathcal{O}(1/m_{t})$
corrections have been investigated Harlander:2012hf ; Neumann:2014nha ;
Neumann:2016dny . Overall, the above cited explicit fixed-order computations
have shown that higher-order corrections computed in the HTL rescaled with
lower-order predictions with explicit mass dependence yield remarkably good
approximations of the full result despite the fact that the HTL is not valid
in this energy regime. Therefore, it appears to be justified to tentatively
apply this very same procedure also at the highest perturbative orders, where
validation of the approximation is not yet possible. An example of such an
approximation at the currently highest available perturbative order is
presented in Ref. Becker:2020rjp  for Higgs plus jet production where NNLO
corrections in the HTL are combined with NLO corrections in the full SM.
However, defining a reliable uncertainty on such approximations remains
crucial. Such approximations of reweighting higher-order computations in the
HTL with exact lower order results are also at the basis of all currently
available NLO Monte Carlo predictions matched to parton showers for Higgs plus
(multi-)jet production Campbell:2012am ; Hamilton:2012rf ; Buschmann:2014sia ;
Frederix:2016cnl ; Greiner:2016awe .

In this paper we present fixed-order NLO QCD computations for $pp\rightarrow
H+j$ and also $pp\rightarrow H+jj$ including top-quark mass effects. The
computation for $pp\rightarrow H+j$ continues the study of Ref. Jones:2018hbb
, i.e. two-loop virtual corrections in the full SM are evaluated numerically
via SecDec-3 Borowka:2015mxa ; Borowka:2017idc . Here we present additional
kinematic observables besides the $p_{\rm{T},H}$ distribution already shown in
Ref. Jones:2018hbb  and compare the relative higher-order corrections in the
full SM against the HTL, and also an alternative approximation known as
$\rm{FT}_{\rm approx}$, which has been introduced in Ref. Maltoni:2014eza  in
the context of calculations for multi-Higgs production. In the $\rm{FT}_{\rm
approx}$ all ingredients of the NLO computation are computed exactly, except
for the two-loop virtual contributions, which are approximated in the HTL and
reweighted with LO mass dependence. In the case of $pp\rightarrow H+jj$
production we compare results in the HTL and in the $\rm{FT}_{\rm approx}$, as
the exact five-point two-loop virtual amplitudes remain beyond current
technology. The main aim of this study is three-fold: firstly, we would like
to offer complementary kinematic information for the $pp\rightarrow H+j$
process, while also offering a (partial) validation of the results already
presented in Ref. Jones:2018hbb . In this respect, the present study uncovered
an issue affecting the real corrections included in Ref. Jones:2018hbb , which
has subsequently been rectified. Secondly, we would like to investigate the
$pp\rightarrow H+jj$ process in a kinematic regime relevant for VBF analyses.
The comparison of NLO/LO ratios (usually known as K-factors) among HTL,
$\rm{FT}_{\rm approx}$ and the full SM will help to put results obtained in a
reweighted HTL on a more solid footing. Thirdly, results presented in this
study can be seen as an intermediate step towards an NNLO computation of
$pp\rightarrow H+j$ including exact mass effects wherever possible.

Technically, the computations of the NLO corrections to $pp\rightarrow H+j$
and $pp\rightarrow H+jj$ production are performed within the NNLOJET fixed-
order Monte Carlo framework, which employs antenna subtraction for the
handling of infrared (IR) singularities GehrmannDeRidder:2005cm ;
GehrmannDeRidder:2005aw ; Daleo:2006xa ; Daleo:2009yj ; Glover:2010im ;
GehrmannDeRidder:2011aa ; GehrmannDeRidder:2012ja ; Ridder:2012dg ;
Currie:2013vh . All loop-squared amplitudes are evaluated via a new interface
between NNLOJET and OpenLoops2 Cascioli:2011va ; Buccioni:2017yxi ;
Buccioni:2019sur  based on the latest (soon to be released) version
OpenLoops2.2 which in turn implements a new reduction method called Otter
otter:2021 , which ensures excellent numerical stability in particular of the
loop-induced real radiation amplitudes deep into the unresolved regime. We
will investigate and discuss this numerical stability issue explicitly.

The structure of the paper is as follows. In Section 2 we describe the
computational setup and employed tools. Numerical results for $H$+jet and
$H$+2 jets production will be presented in Section 3. We will conclude in
Section 4.

##  2 Analysis framework and tools

In this paper, we present predictions for the production of a boosted Higgs
boson in the gluon-fusion channel. We consider Higgs bosons produced in
association with one ($pp\rightarrow H+j$) or two ($pp\rightarrow H+jj$) jets
with NLO QCD corrections and include the effects of a finite top-quark mass
either fully or via a suitable approximation. For $pp\rightarrow H+j$ we
compute the transverse momentum distribution of the Higgs boson including a
finite top-quark mass, which has appeared previously in the literature
Jones:2018hbb ; Lindert:2018iug ; Neumann:2018bsx ; Kerner:2019qgb , as well
as the Higgs boson plus jet invariant mass distribution. For $pp\rightarrow
H+jj$, the virtual corrections involve two-loop amplitudes for $2\rightarrow
3$ scattering. The mathematical complexity of the virtual corrections makes
their computation currently intractable using either numerical or analytical
methods. We therefore adopt an approximation scheme to the full theory
($\mathrm{FT}_{\mathrm{approx}}$) Frederix:2014hta ; Maltoni:2014eza  for the
NLO QCD corrections to Higgs boson plus two jet production. Specifically, we
include the exact top-quark mass dependence (SM) in the real corrections and
infrared singular subtraction terms while using the virtual corrections in the
heavy top-quark limit (HTL) re-weighted by the full Born level contribution on
an event-by-event basis. In fact, although the full matrix elements relevant
to the virtual contributions of $\rm{H}+2\,$jets production are currently not
available, nevertheless, their explicit infrared divergence at NLO can be
predicted by the Catani dipole structure Catani:1998bh :

$\displaystyle\text{Pole}\\{|\mathcal{M}^{2}_{4}(m_{t},\mu_{R}^{2};\\{p\\})|^{2}\\}=\sum\textbf{\emph{I}}^{(1)}(\epsilon,\mu_{R}^{2};\\{p\\})|\mathcal{M}^{1}_{4}(m_{t};\\{p\\})|^{2},$
(1)

where $m_{t}$ is the top-quark mass, $\mu_{R}^{2}$ is the renormalisation
scale, $\\{p\\}$ is the momentum set regarding all external particles,
$|\mathcal{M}^{m}_{n}|^{2}$ is the matrix element with $n$ legs and $m$ loops
and $\textbf{\emph{I}}^{(1)}(\epsilon,\mu_{R}^{2};\\{p\\})$ is the dipole
operator containing all explicit IR divergences in $d$ space-time dimensions.
The explicit expressions for dipole operators at squared matrix element level
can be found in GehrmannDeRidder:2005cm . We estimate the finite contribution
of $|\mathcal{M}^{2}_{4}(m_{t},\mu_{R}^{2};\\{p\\})|^{2}$ by re-weighting the
corresponding matrix element in the HTL approximation
($m_{t}\rightarrow\infty$) using:

$\displaystyle|\mathcal{M}^{2}_{4}(m_{t},\mu_{R}^{2};\\{p\\})|^{2}\rightarrow|\mathcal{M}^{1}_{4}(\infty,\mu_{R}^{2};\\{p\\})|^{2}\frac{|\mathcal{M}^{1}_{4}(m_{t};\\{p\\})|^{2}}{|\mathcal{M}^{0}_{4}(\infty;\\{p\\})|^{2}}.$
(2)

Consequently, Eq.(2) also recovers the explicit pole structure in Eq.(1) and
the explicit pole cancellation in the second bracket of Eq.(3) is
automatically retained.

The $\mathrm{FT}_{\mathrm{approx}}$ scheme has proved to be remarkably
reliable for Higgs plus one jet Chen:2016zka  production and, to a lesser
extent, di-Higgs Grazzini:2018bsd  production (however in the latter case, it
is much less reliable for differential distributions) at the LHC. We implement
and present the first application of this approximation to Higgs boson plus
two jet production at NLO in QCD.

In the following sections we document the detailed implementation of our
calculations. The NNLOJET program is used as a parton level event generator
and all Born and real radiation one-loop contributions are computed using
OpenLoops2.2. The two-loop matrix elements involving a finite top-quark mass
for $pp\rightarrow H+j$ are computed exactly using SecDec-3.

###  2.1 Parton level event generator: NNLOJET

NNLOJET is a parton-level event generator equipped with flexible histogram
analysis tools and scattering matrix elements evaluated in both the HTL
approximation and in the full SM. It implements the antenna subtraction method
to cancel infrared singularities from higher order QCD corrections
GehrmannDeRidder:2005cm ; GehrmannDeRidder:2005aw ; Daleo:2006xa ;
Daleo:2009yj ; Glover:2010im ; GehrmannDeRidder:2011aa ;
GehrmannDeRidder:2012ja ; Ridder:2012dg ; Currie:2013vh  while retaining the
fully differential information of final state particles.

In this study, we combine NNLOJET with loop induced matrix elements provided
by OpenLoops2 and SecDec-3 to study finite top-quark mass corrections for
$\rm{H}+$jet and $\rm{H}+2\,$jets production with NLO QCD corrections. The HTL
and SM have the same infrared singular behaviour for both real emissions and
virtual corrections, the antenna subtraction method can therefore be readily
applied to regulate infrared singularities at NLO for loop-induced processes.
Schematically, the fully differential NLO contribution takes the form:

$\displaystyle{\rm d}\sigma_{H+n{\rm jet}}^{\rm{NLO}}$ $\displaystyle=$
$\displaystyle\int_{{\rm
d}\Phi^{H}_{n+1}}\bigg{[}|\mathcal{M}^{1}_{n+3}(m_{t};\\{p\\})|^{2}-\sum_{\\{p\\}}X^{0}_{3}|\mathcal{M}^{1}_{n+2}(m_{t};\\{\tilde{p}\\})|^{2}\bigg{]}$
(3) $\displaystyle+$ $\displaystyle\int_{{\rm
d}\Phi^{H}_{n}}\bigg{[}|\mathcal{M}^{2}_{n+2}(m_{t},\mu_{R}^{2};\\{p\\})|^{2}+\sum_{\\{p\\}}\mathcal{X}^{0}_{3}|\mathcal{M}^{1}_{n+2}(m_{t};\\{p\\})|^{2}\bigg{]},$

where $\Phi^{H}_{n}$ is the final state phase space of one Higgs plus $n$
partons, $X^{0}_{3}$ represents the tree-level three-parton antenna functions,
$\mathcal{X}^{0}_{3}$ represents the corresponding integrated antenna
functions in $d$-dimensions and $\\{\tilde{p}\\}$ is the momentum set after
antenna mapping with one less parton compared to the $\\{p\\}$ momentum set.
We use $X^{0}_{3}$ and the corresponding reduced matrix elements
($|\mathcal{M}^{1}_{n+2}|^{2}$) to capture the infrared singular behaviour of
real radiations in $|\mathcal{M}^{1}_{n+3}|^{2}$, leading to a infrared finite
contribution of the first bracket in Eq.(3). By adding back the integrated
antenna functions $\mathcal{X}^{0}_{3}$ in the second bracket of Eq.(3), we
render the integrand over the $\Phi^{H}_{n}$ phase space IR finite. The
explicit IR divergences (which appear as poles in the regulator $\epsilon$)
cancel analytically with the explicit IR poles from the virtual contribution
($|\mathcal{M}^{2}_{n+2}|^{2}$).

###  2.2 One-loop contributions: OpenLoops2.2

All one-loop amplitudes contributing at the Born and real radiation level are
provided by the OpenLoops2.2 package, an upcoming improved version of the
OpenLoops2 program which implements a new reduction method called Otter
otter:2021 . Compared to the original algorithm Cascioli:2011va , OpenLoops2
includes significant improvements in numerical stability and performance for
the computation of tree-loop interference amplitudes. These improvements were
achieved by a combination of the so-called on-the-fly reduction algorithm
Buccioni:2017yxi  and an automated stability system. However, the current
implementation, which was designed for tree-loop interferences, cannot be
directly applied to loop-induced amplitudes, such as those required for the
present computation. In OpenLoops2.2 a new tensor integral reduction method
has been developed, based on the on-the-fly reduction algorithm
Buccioni:2017yxi , that can also be used for loop-squared amplitudes. It
profits from recent improvements in performance and methods developed for
handling numerical instabilities. More specifically, numerical instabilities
are avoided by using certain freedoms in the selection of the reduction
identities, and analytical any-order expansions of three-point tensor
integrals in the limit of small Gram determinant Buccioni:2019sur .
Furthermore, within OpenLoops2.2 residual instabilities are captured by a
rescaling test and tensor integrals are recomputed in quadruple precision in
case the given accuracy is not reached. This upgrade to quadruple precision is
efficient, and also important for computations in deep infrared regions.
OpenLoops2.2 depends on Collier Denner:2016kdg  only for double precision
scalar integrals and on OneLOop vanHameren:2010cp  for quadruple precision
scalar integrals.

#### Numerical stability

The numerical stability of OpenLoops2.2 is crucial for the calculations of
$\rm{H}+$jet and $\rm{H}+2\,$jets productions presented in this paper,
especially to contributions from infrared kinematical regions, which are
numerically challenging. In Fig. 1 we illustrate as a benchmark the stability
of the critical $\mathrm{gg\to Hgg}$ and $\mathrm{gg\to Hggg}$ amplitudes
subject to single soft or collinear radiation. The degree of softness and
collinearity are defined as

$\displaystyle\xi_{\mathrm{soft}}=E_{\mathrm{soft}}/\sqrt{s},\quad\xi_{\mathrm{coll}}=\theta_{ij}^{2},$
(4)

where $E_{\mathrm{soft}}$ is the energy of the soft particle, and
$\theta_{ij}$ denotes the angle of the collinear branching. The numerical
stability is defined as

$\displaystyle\mathcal{A}=\log_{10}\Big{(}\frac{\mathcal{W}-\mathcal{W}^{(0)}}{\mathcal{W}^{(0)}}\Big{)},$
(5)

where $\mathcal{W}$ denotes the one-loop-squared matrix element,
$\mathcal{W}^{(0)}$ is the benchmark result, and $\mathcal{A}$ corresponds to
the number of stable digits up to a minus sign. As can be seen from these
plots, the numerical accuracy remains very high all the way down to the deep
infrared regime.

Figure 1: Stability plots in IR regions for one-loop-squared matrix elements
in $\mathrm{gg\to Hgg}$ and $\mathrm{gg\to Hggg}$ versus the degree of
collinear $\xi_{\text{coll}}$ or soft $\xi_{\text{soft}}$ singularity obtained
with OpenLoops2.2. For each value of $\xi_{\text{soft/coll}}$, the numerical
accuracy is calculated with a sample of $10^{3}$ randomly distributed infrared
events. Unstable points are detected by a rescaling test and rescued if the
relative accuracy of $10^{-6}$ is not reached. The rescue step reevaluates the
tensor integrals to quad precision. The accuracy of the so-obtained value is
determined by comparing it to a quadruple precision benchmark whose accuracy
is also assessed by a rescaling test. The plotted central points and variation
bands correspond, respectively, to the average and 100% confidence interval of
$\mathcal{A}$.

#### Numerical performance

In Table 1 we present results for the average evaluation time of samples of
random phase-space points using various different modes/versions of OpenLoops.
In particular, here certain parts of the amplitude are evaluated in double or
in quadruple precision, or a realistic error estimate of the amplitudes is
performed. In summary, performance is greatly improved in OpenLoops2.2 which
in particular makes a tensor integral based rescaling test cheap. Moreover,
since OpenLoops2.2 operates also in quadruple precision with very high
numerical efficiency, numerically unstable points can be rescued in a reliable
way, which has largely been prohibitive for loop-squared amplitudes in
OpenLoops2. In fact, OpenLoops2.2 allows for rescue of unstable points in a
new hybrid mode, where only the tensor integrals are evaluated in higher
numerical precision resulting in a 8-fold and 3-fold increase in runtime
compared to pure double precision for $gg\rightarrow Hgg$ and $gg\rightarrow
Hggg$ respectively, compared to a roughly 80-fold increase in runtime for full
quadruple precision. In practice, and as used for the present computation of
this paper in OpenLoops2.2 a combination of pure double precision with this
new hybrid mode is used. Based on a pure double precision evaluation the
stability for every phase-space point is estimated based on a rescaling test
of only the tensor integrals. Then, only for critical points the tensor
integrals are reevaluated in quadruple precision based on the hybrid mode.

Mode |  $gg\rightarrow Hgg$ (time/psp) |  $gg\rightarrow Hggg$ (time/psp)
---|---|---
OL2.1+Collier DP | 13ms | 0.56s
OL2.1+Collier DP + error estimation | 19ms | 0.89s
OL2.1+CutTools QP | 43000ms | 2300s
OL2.2+Otter DP | 8.9ms | 0.29s
OL2.2+Otter DP + error estimation | 11ms | 0.32s
OL2.2+Otter DP+QP tensor integrals | 68ms | 0.87s
OL2.2+Otter QP | 740ms | 23s
Table 1: Runtimes for loop-squared amplitudes for $gg\to Hgg$ and $gg\to Hggg$
in OpenLoops. All numbers have been produced on an Intel(R) Core(TM) i7-7700
CPU @ 3.60GHz. The first three rows correspond to the modes so far available
in OpenLoops2 Buccioni:2019sur . These employ double precision evaluation with
Collier (first and second row), where in the second row tensor integrals are
computed twice using the COLI and DD branches of Collier in order to obtain an
error estimate. The third row employs CutTools and the entire amplitude is
evaluated in quadruple precision. For the amplitudes at hand the resulting
runtimes in quadruple precision are prohibitive to be used as rescue system.
The lower four rows represent evaluation based on the new Otter method in
OpenLoops2.2. In this case the error estimation is performed via a rescaling
test where all tensor integrals are recomputed with rescaled kinematics. The
sixth row corresponds to a new hybrid mode where only the tensor integrals are
evaluated in quadruple precision, and everything else in double precision. The
last row shows the performance for a full quadruple precision evaluation
within OpenLoops2.2.

###  2.3 Two-loop contributions: SecDec-3

For $pp\rightarrow H+j$ production, we evaluate the two-loop virtual
contributions with exact top-quark mass dependence as presented in Refs.
Jones:2018hbb ; Kerner:2019qgb . Briefly, the two-loop amplitudes, which
depend on four mass scales (the Mandelstam invariants $s$ and $t$ as well as
the two masses $m_{t}$ and $m_{h}$), are expressed in terms of a basis of
master integrals using the program Reduze2 vonManteuffel:2012np . In order to
obtain the integral reduction in a reasonable time and to reduce the size of
the resulting amplitude, the ratio of the Higgs boson mass to the top-quark
mass is fixed according to $m_{H}^{2}/m_{t}^{2}=12/23$. The master integrals
are then sector decomposed using the program SecDec-3 Borowka:2015mxa ;
Borowka:2017idc  and numerically integrated on Graphics Processing Unit (GPUs)
using the Quasi-Monte Carlo method Li:2015foa ; Borowka:2018goh . To improve
the stability of the amplitude we select a quasi-finite basis of master
integrals as outlined in Ref. Kerner:2019qgb , this differs from the basis
originally used in Ref. Jones:2018hbb . We observe that the new choice of
master integrals also significantly reduces the complexity of the coefficients
of the master integrals appearing in the amplitude and thus the size of the
code.

The results presented here are produced using a total of 6497 phase-space
points for the two-loop virtual contribution. In Ref Jones:2018hbb , a
fraction of the phase-space points were distributed such that they provide a
good estimate of the total cross section (assuming a jet cut of
$p_{\rm{T},j}>30\,\mathrm{GeV}$) and additional phase-space points were
generated to sample the tail of the $p_{\rm{T},H}$ distribution. We reuse
these existing phase-space points and also compute an additional 1007 points
to populate the large invariant mass region $m_{Hj}$ for
$p_{\rm{T},j}>300\,\mathrm{GeV}$.

##  3 Numerical results

###  3.1 Setup

As an extension of the study of Higgs plus one jet production at NLO
Jones:2018hbb , we adopt the same input parameters and numerical setup in the
current calculation. To quantify the impact of increasing the number of final
state jets, we keep the input parameters consistent between $\rm{H}+$jet and
$\rm{H}+2\,$jets production. The counting of the number of jets in this study
is inclusive. There is no difference between inclusive and exclusive jet
counting for LO while results at NLO accuracy receive contributions from real
emissions including events classified with one addtional jet. For the Higgs
and top-quark mass we use $m_{H}=125\,\mathrm{GeV}$ and
$m_{t}={173.055}\,\mathrm{GeV}$. The top-quark Yukawa coupling
$\lambda_{t}=\sqrt{2}m_{t}/v$ is determined by the vacuum expectation value
(vev) of the Higgs $v=\frac{M_{W}\sin\theta_{W}}{\sqrt{\pi}\alpha_{\rm
QED}}=246.219\,$GeV and the top-quark mass. We use the five flavour scheme
assuming light quarks are massless in both inital and final states.

Throughout our calculation, the top-quark mass is renormalised using the on-
shell (or pole mass) scheme. It has been pointed out in the literature that
several Higgs boson production processes (including Higgs boson production in
association with jets) are sensitive to the scheme and scale used to
renormalise quark masses Baglio:2018lrj ; Baglio:2020ini ; Baglio:2020wgt ;
Amoroso:2020lgh . For example, at LO, the difference between the pole mass
scheme and the $\overline{\mathrm{MS}}$ scheme at scale $m_{H,j}/2$ was found
to be around $12\%$ for $m_{H,j}=700\,\mathrm{GeV}$ and
$p_{T,j}>300\,\mathrm{GeV}$. At larger $p_{T,H}$ the difference between the
two schemes grows and can reach $\approx 25\%$ for $p_{T,H}=1\,\mathrm{TeV}$.
In off-shell Higgs boson production, off-shell Higgs boson decay to photons
and in Higgs pair production, the NLO corrections reduce the mass scheme
uncertainty to approximately half that of the LO Baglio:2018lrj ;
Amoroso:2020lgh . By analogy, we may expect that the mass scheme uncertainty,
which we do not assess, is similar in size to our NLO scale uncertainties (see
below).

We employ the PDF4LHC15_nlo_30_pdfas PDF set Butterworth:2015oua  throughout
and all of our predictions are at a center of mass energy of $\sqrt{S}=13$
TeV. Renormalisation and factorisation scales are chosen as

$\displaystyle\mu_{\rm R,F}=\xi_{\rm R,F}\cdot H_{T}/2,\quad\mathrm{with}\quad
H_{T}=\sqrt{m_{H}^{2}+p_{\rm{T},H}^{2}}+\sum\limits_{j}|p_{\rm{T},j}|\,,$ (6)

where the sum includes all final state partons. Our central scale corresponds
to $\xi_{\rm R,F}=(1,1)$ and we determine scale uncertainties via the standard
7-point factor-2 variations $\xi_{\rm R,F}=(2,2),(2,1),(1,2),$
$(1,1),(1,\frac{1}{2}),$ $(\frac{1}{2},1),(\frac{1}{2},\frac{1}{2}))$. Any
reconstructed jets are clustered via the anti-kt Cacciari:2008gp  algorithm
with $R=0.4$. We apply the following cuts:

$\displaystyle H+\mathrm{jet}:$ $\displaystyle\quad
p_{\rm{T},j}>30\,\mathrm{GeV},$ (7) $\displaystyle H+2\,\mathrm{jets}:$
$\displaystyle\quad p_{\rm{T},j_{1}}>40\,\mathrm{GeV}\ \mathrm{and}\
p_{\rm{T},j_{2}}>30\,\mathrm{GeV}\,.$ (8)

The latter asymmetric jet cuts avoid a perturbative instability in the limit
$p_{\rm{T},H}\to 0$ $\mathrm{GeV}$.

###  3.2 Fiducial total cross sections

$\sigma$[pb] |  | Inclusive $p_{T,H}$ |  $p_{T,H}>300$ GeV
---|---|---|---
| LO | NLO | K | LO | NLO | K
$\rm{H}+$jet | HTL | $8.22^{+3.17}_{-2.15}$ | $13.57^{+2.11}_{-2.09}$ | $1.65$ | $0.086^{+0.038}_{-0.024}$ | $0.160^{+0.033}_{-0.030}$ | $1.86$
$\rm{FT}_{\rm approx}$ | $8.56^{+3.30}_{-2.24}$ | $14.06(1)^{+2.17}_{-2.16}$ | $1.64$ | $0.046^{+0.020}_{-0.013}$ | $0.088^{+0.019}_{-0.017}$ | $1.91$
SM | $8.56^{+3.30}_{-2.24}$ | $14.15(7)^{+2.29}_{-2.21}$ | $1.65$ | $0.046^{+0.020}_{-0.013}$ | $0.089(3)^{+0.020}_{-0.017}$ | $1.93$
$\rm{H}+2\,$jets | HTL | $2.87^{+1.67}_{-0.99}$ | $4.33^{+0.59}_{-0.80}$ | $1.51$ | $0.120^{+0.071}_{-0.042}$ | $0.160^{+0.012}_{-0.025}$ | $1.33$
$\rm{FT}_{\rm approx}$ | $2.92^{+1.70}_{-1.01}$ | $4.45(1)^{+0.63}_{-0.83}$ | $1.52$ | $0.068^{+0.040}_{-0.024}$ | $0.092^{+0.008}_{-0.015}$ | $1.35$
SM | $2.92^{+1.70}_{-1.01}$ | $-$ | $-$ | $0.068^{+0.040}_{-0.024}$ | $-$ | $-$
Table 2: Integrated cross sections at LO and NLO in the HTL and $\rm{FT}_{\rm
approx}$ approximations and with full top-quark mass dependence (SM) for
$\rm{H}+$jet and $\rm{H}+2\,$jets production together with corresponding
K-factors. Uncertainties correspond to the envelope of 7-point scale
variations. For $\rm{H}+$jet production we require
$p_{\rm{T},j}>30\,\mathrm{GeV}$, while for $\rm{H}+2\,$jets production we
require
$p_{\rm{T},j_{1}}>40\,\mathrm{GeV},\,p_{\rm{T},j_{2}}>30\,\mathrm{GeV}$. On
the left no further phase-space restrictions are considered, while on the
right we additionally require $p_{T,H}>300\,\mathrm{GeV}$. Numerical
integration errors larger than permil level are indicated in brackets.

Applying the computational setup in Section 3.1, we document the fiducial
total cross section for $\rm{H}+$jet and $\rm{H}+2\,$jets produciton in Tab.
2. For reference we present results for the HTL, $\rm{FT}_{\rm approx}$ and SM
predictions. We also include the fiducial total cross section for boosted
Higgs with $p_{T,H}>300$ GeV. Further fiducial cross sections with varying
$p_{T,H}$ cuts are listed in Appendix A. The numbers for $\rm{H}+$jet reported
here agree, within the statistical uncertainty, with the updated version of
Ref. Jones:2018hbb .

Together with predictions obtained with the central scale defined in Eq. (6)
we show the upper and lower values obtained by the envelope of 7-point scale
variations. For $\rm{H}+$jet production without the fiducial constraint of
$p_{T,H}$, the top-quark mass effects lead to an increase of 4.3% (4.6%) at LO
(NLO) comparing to HTL. There is an increase of about 1% in the total NLO
cross section when comparing the $\rm{FT}_{\rm approx}$ result with the full
top-quark mass dependence. The NLO/LO K-factor is consistent among HTL,
$\rm{FT}_{\rm approx}$ and SM at about 1.65.

With the fiducial constraint of $p_{T,H}$ larger than 300 GeV, we observe a
similar amount (2.3%) of relative increase from the $\rm{FT}_{\rm approx}$
prediction to the result with full top-quark mass effects at NLO. In contrast,
the HTL prediction is 78% (81.8%) larger than the full theory ($\rm{FT}_{\rm
approx}$) result due to large logarithmic corrections from the disparity in
scales between $m_{t}^{2}$ and $p_{T,H}^{2}$. The NLO/LO K-factor for
$p_{T,H}>300$ GeV is however again almost universal among HTL, $\rm{FT}_{\rm
approx}$ and SM at about 1.9. To be precise, the K-factor in the full SM is
$5.3\%$ resp. $2.6\%$ larger compared to the HTL and $\rm{FT}_{\rm approx}$
predictions.

For $\rm{H}+2\,$jets production without the fiducial constraint on $p_{T,H}$,
the $\rm{FT}_{\rm approx}$ prediction induces a 1.7% (2.8%) increase compared
the HTL approximation at LO (NLO). Considering the fiducial constraint of
$p_{T,H}$ larger than 300 GeV, the HTL fiducial total cross section is about
1.76 (1.74) times the $\rm{FT}_{\rm approx}$ predictions at LO (NLO). However,
again in both selections NLO K-factors are universal with mass corrections in
the $\rm{FT}_{\rm approx}$ below the $2\%$ level.

For the absolute cross sections the extra jet emission in $\rm{H}+2\,$jets
production decreases the total $\rm{H}+$jet cross section by more than a
factor of 3 from $14.06$ pb ($\rm{H}+$jet) to $4.45$ pb considering the
$\rm{FT}_{\rm approx}$ predictions at NLO in both cases. In contrast, in the
boosted Higgs boson regime, the total NLO cross sections are comparable, while
the NLO K-factor is about $46\%$ larger in the $\rm{H}+$jet computation
compared to the $\rm{H}+2\,$jets one. This can be understood from the
restriction to the back-to-back configuration in the $\rm{H}+$jet computation
at LO. Multi-jet configurations recoiling against the hard Higgs only open up
at NLO, where they are effectively described at LO. In contrast in the
$\rm{H}+2\,$jets computation such configurations already contribute at LO.

In order to quantify top-quark mass effects in various fiducial regions and to
explore the possibility to extrapolate the mass effect to higher order
corrections, in the following sections we present in-depth comparisons of
differential cross sections for $\rm{H}+$jet and $\rm{H}+2\,$jets production.

###  3.3 Fiducial differential cross sections for $\rm{H}+$jet production

Figure 2: Transverse momentum distribution of the Higgs (left) and the hardest
jet (right) in $\rm{H}+$jet production. We show LO predictions in the full SM
(LOSM, magenta) and the HTL (LOHTL, green) as well as NLO predictions in the
$\rm{HTL}$ (NLOHTL, blue), the $\rm{FT}_{\rm approx}$ (NLOFTapprox, red) and
the full SM (NLOSM, orange). The upper panel shows absolute predictions. The
first ratio plot shows corrections with respect to LOSM, while the second
ratio plot shows NLO corrections normalised to the respective LO prediction,
i.e. NLOHTL/LOHTL, NLOSM/LOSM, and NLOFTapprox/LOSM. Shaded bands correspond
to scale variations. Error bars indicate integration uncertainties.

In Figs. 2-3 we present numerical results for $\rm{H}+$jet production at the
LHC with $\sqrt{S}=13$ TeV. We compare NLO corrections in the $\rm{HTL}$ with
the $\rm{FT}_{\rm approx}$ approximation, and with the full SM.

In Fig. 2 we consider the transverse momentum distribution of the Higgs on the
left and of the jet on the right. These distributions are identical at LO, and
are also highly correlated at NLO. As is well known, for $p_{T,X}>m_{t}\
(X=H,j)$ the HTL approximation breaks down and predicts a very different high-
energy scaling compared to the SM. However, for NLO/LO ratios in the bottom
panels of Fig. 2, at least at the $O(10\%)$ level, higher-order QCD
corrections in the $\rm{HTL}$ agree well with the $\rm{FT}_{\rm approx}$ and
the full SM. Further improvement of agreement is observed between the
$\rm{FT}_{\rm approx}$ and the full SM at the $O(5\%)$ level within numerial
uncertainties. The scale variation bands in the ratio plots are obtained by
fixing the observable at the central scale choice in the denominator while
taking the envelope of scale choices in the numerator by the 7-point factor-2
variations. We observe consistent agreement for the size of scale variations
also for the ratio plots. In $p_{\rm{T},H}$ the corrections are at the level
of $90-110\%$, while in $p_{\rm{T},j}$ they are at the level of $60-80\%$, for
the entire considered range, i.e shape corrections are mild. Examining the
corrections in more detail an overall increase of the corrections of $\approx
5-10\%$ can be appreciated for the full SM compared to the HTL, with a mild
relative increase at large transverse momenta slightly larger for
$p_{\rm{T},j}$ than for $p_{\rm{T},H}$. Corrections in the $\rm{FT}_{\rm
approx}$ and the HTL agree exactly up to $p_{\rm{T},H}/p_{\rm{T},j}\sim
m_{t}$. Beyond that, corrections in the $\rm{FT}_{\rm approx}$ are a few
percent larger compared to the HTL, with again a very slight increase at large
transverse momenta. The remaining scale uncertainties at NLO are at the
$20-25\%$ level throughout.

In Fig. 3 we turn to a different kinematic regime in $\rm{H}+$jet production
by considering invariant mass distributions in the Higgs-jet system on the
left for inclusive $\rm{H}+$jet production and on the right considering
$p_{\rm{T},H}>300\,\mathrm{GeV}$. For the inclusive $m_{Hj}$ distribution the
HTL NLO result increases mildly by $10-20\%$ in the tail of the distribution
compared to the $\rm{FT}_{\rm approx}$ and the full SM. However, the NLO
K-factor is universal in all three predictions decreasing from about $1.7$ at
small $m_{Hj}$ to about $1.5$ at large $m_{Hj}$ with similar relative size of
scale variation. Finally, in the exclusive $p_{\rm{T},H}>300\,\mathrm{GeV}$
phase-space absolute predictions in the HTL and $\rm{FT}_{\rm approx}$ diverge
for increasing $m_{Hj}$. However, also here the relative NLO corrections are
found to be largely identical for the HTL, $\rm{FT}_{\rm approx}$ and full
top-quark mass results. This holds at large $m_{Hj}$ as well as at small
$m_{Hj}$, which for $m_{Hj}<600$ GeV is kinematically inaccessible at LO due
to the $p_{\rm{T},H}>300\,\mathrm{GeV}$ requirement.

From the detailed comparison of $p_{\rm{T},H}$, $p_{\rm{T},j}$ and $m_{Hj}$
distributions of $\rm{H}+$jet production, we observe excellent agreement of
differential NLO/LO K-factors (for the central scale and scale variations)
among theory predictions using HTL, $\rm{FT}_{\rm approx}$ and the exact top-
quark mass dependence. This observation validates the multiplicative
reweighting procedure introduced in Chen:2016zka  at histogram level and
further strengthens the reweighed predictions for $\rm{H}+$jet production at
NNLO accuracy Becker:2020rjp .

Figure 3: Invariant mass distribution of the Higgs and the jet for inclusive
$\rm{H}+$jet production (left) and with $p_{\rm{T},H}>300\,\mathrm{GeV}$
(right). Colour coding and labelling as in Fig. 2.

###  3.4 Fiducial differential cross sections for $\rm{H}+2\,$jets production

In Figs. 4-7 we turn to the numerical results for $\rm{H}+2\,$jets production.
We compare NLO corrections in the $\rm{HTL}$ with the $\rm{FT}_{\rm approx}$
approximation, focussing on multi-jet observables and jet correlations with
and without boosted Higgs kinematics (i.e. inclusive and with an additional
$p_{\rm{T},H}>300\,\mathrm{GeV}$ requirement). As discussed in the
introduction these are phenomenologically highly relevant for the modelling of
$\rm{H}+2\,$jets backgrounds in analyses for VBF Higgs production.

Figure 4: Transverse momentum distribution of the Higgs (left) and the hardest
jet (right) in $\rm{H}+2\,$jets production. Colour coding and labelling as in
Fig. 2.

In Fig. 4 we consider the transverse momentum of the Higgs (left) and of the
hardest jet (right). These plots can directly be compared with the
corresponding ones for $\rm{H}+$jet production in Fig. 2. Again we observe
very large deviations of the nominal predictions at large transverse momenta.
At the same time also the QCD corrections are sizeable: around $60-70\%$ at
small $p_{\rm{T}}$ and around $30\%$ for $p_{\rm{T},H}/p_{\rm{T},j}=1$ TeV.
However the relative NLO corrections normalised to the respective LO show a
universal behaviour, i.e. they are identical in the HTL and the $\rm{FT}_{\rm
approx}$.

A similar picture as for the transverse momentum distributions emerges when
looking at the distribution in the invariant mass of the Higgs and the hardest
jet, as depicted in Fig. 5. Both, for the inclusive selection (as shown on the
left), and for the boosted Higgs selection with an additional
$p_{\rm{T},H}>300\,\mathrm{GeV}$ requirement, the QCD corrections in the
$\rm{FT}_{\rm approx}$ identically track the corresponding corrections in the
HTL, while the nominal predictions substantially diverge.

Figure 5: Invariant mass distribution of the Higgs and the hardest jet system
in $\rm{H}+2\,$jets production. Colour coding and labelling as in Fig. 2.

Next, in Fig. 6 we turn to the phenomenologically important dijet invariant
mass distribution. Again we consider an inclusive selection (left) and a
boosted selection requiring $p_{\rm{T},H}>300\,\mathrm{GeV}$. In the inclusive
phase-space the NLO corrections are about $50\%$ in the $\rm{FT}_{\rm approx}$
with hardly any variations over the considered $m_{j_{1}j_{2}}$ range.
Corrections in the HTL are identical to the $\rm{FT}_{\rm approx}$ at small
$m_{j_{1}j_{2}}$ and slightly reduce to about $40\%$ in the multi TeV range,
i.e. up to $10\%$ smaller than in the $\rm{FT}_{\rm approx}$. With the boosted
selection, corrections in both the $\rm{FT}_{\rm approx}$ and the HTL are at
the level of $30-40\%$ and marginally reduce in the tail of the
$m_{j_{1}j_{2}}$ distribution.

Finally in Fig. 7 we plot the rapidity difference between the two jets in
$\rm{H}+2\,$jets production, again with an inclusive selection on the left and
a boosted selection on the right. In the inclusive case there is hardly any
variation in the NLO corrections over the considered rapidity range with a
K-factor at the 1.5 level. For the boosted selection, the K-factor decreases
slightly from about 1.35 to 1.25 from small to high rapidity differences. For
both selections and over the entire rapidity range corrections in the
$\rm{FT}_{\rm approx}$ and the HTL agree at the percent level. We observed
very similar findings also in other angular correlation observables including
e.g. the rapidity difference between the Higgs and the hardest jet.

Overall in all considered observables we find a remarkable agreement of the
relative corrections computed in the HTL and the $\rm{FT}_{\rm approx}$–
despite up to several order of magnitude variations in nominal predictions.
This clearly points towards a factorisation of QCD higher-order corrections
from the heavy fermion loop mediating the coupling of the Higgs boson.

Figure 6: Invariant mass distribution of the di-jet system in $\rm{H}+2\,$jets
production. Colour coding and labelling as in Fig. 2.

Figure 7: Rapidity difference between the two hardest jets in $\rm{H}+2\,$jets
production. Colour coding and labelling as in Fig. 2.

##  4 Conclusions

In this paper we have presented precise differential predictions for
$\rm{H}+$jet and $\rm{H}+2\,$jets production at the LHC at NLO including top-
quark mass effects. For the former process our prediction incorporates the
exact top-quark mass dependence. Instead, in our study of $\rm{H}+2\,$jets
production, the two-loop virtual matrix elements are computed in the HTL
approximation (infinite top-quark mass) and reweighed by the full LO result,
while the exact top-quark mass dependence is retained in the Born and real
radiation contributions. Our results are produced using the NNLOJET event
generator with one-loop amplitudes provided by OpenLoops2.2 (to be released
soon) which implements a novel tensor reduction method based on the on-the-fly
reduction algorithm of OpenLoops. The two-loop virtual matrix elements
including top-quark mass effects contributing to $\rm{H}+$jet production are
evaluated using SecDec-3.

We find that the inclusion of the exact top-quark mass dependence in the two-
loop virtual matrix elements enhances the cross section for $\rm{H}+$jet
production at NLO by about $0.6\%$ with respect to the
$\mathrm{FT}_{\mathrm{approx}}$ prediction, and by about $4.3\%$ with respect
to the HTL prediction. However, the NLO/LO K-factor is found to be universal
amongst the three predictions at about $1.7$. Noteworthy, this universality is
broadly also found examining the corrections to the $p_{T,H}$ and $p_{T,j}$
distributions. Although above the top-quark threshold the HTL approximation
becomes formally invalid, relative NLO corrections agree at the 10% (5%) level
between HTL ($\rm{FT}_{\rm approx}$) respectively and the full theory.

Regarding the $\rm{H}+2\,$jets production process we have produced
distributions for the $p_{T}$ of the Higgs boson and leading jet as well as
invariant mass distributions for the Higgs + leading jet system and the
leading dijet system. In particular the latter is crucial for gluon-fusion
Higgs production backgrounds in VBF Higgs production analyses. At the
inclusive level the $\mathrm{FT}_{\mathrm{approx}}$ cross section is $2.8\%$
larger than the HTL prediction. Differentially, we observe that the
approximate inclusion of the top-quark mass has the largest, though still
rather mild, impact on the $p_{T}$ distributions above the top-quark
threshold. Relative higher-order corrections in the HTL and the
$\mathrm{FT}_{\mathrm{approx}}$ always agree at below the $10\%$ level.

Overall, the top-quark mass effects at NLO are observed to be rather mild. In
particular they are expected to be at the same level or even smaller than the
current uncertainties due to the scheme dependence of the top-quark mass.

The computations presented are relevant for analyses of Higgs production at
large transverse momentum and also for Higgs plus multi-jet backgrounds in
Higgs production via vector boson fusion. Through detailed comparisons, this
study proves the reliability of higher-order corrections computed in the HTL
rescaled with lower-order predictions with explicit mass dependence. The
multiplicative reweighting procedures are of similar theoretical
uncertainties, within $10\%$, for both event-by-event and bin-by-bin
rescaling. This paves the way towards computations of $pp\rightarrow H+j$
including approximate mass effects at NNLO and higher QCD accuracy.

## Acknowledgments

We are grateful to Alexander Karlberg for contributions at the early stage of
this work. We thank Thomas Gehrmann, Gudrun Heinrich, Joey Huston and Stefano
Pozzorini for useful discussions and comments on the manuscript. We thank the
University of Zurich S3IT (http://www.s3it.uzh.ch) and Swiss National
Supercomputing Centre (CSCS) for providing support and computational
resources. This research is supported in part by the Swiss National Science
Foundation (SNF) under contract 200020-175595 and by CSCS under project ID
UZH10. This research was supported in parts by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762
- TRR 257. J.L. is supported by the Science and Technology Research Council
(STFC) under the Consolidated Grant ST/T00102X/1 and the STFC Ernest
Rutherford Fellowship ST/S005048/1. S.P.J. is supported by a Royal Society
University Research Fellowship (Grant URF/R1/201268). J.-N.L. and H.Z. are
supported by the Swiss National Science Foundation (SNF) under contract
BSCGI0-157722.

##  Appendix A Inclusive cross sections

In Tab. 3 and Tab. 4 we list as reference integrated cross sections
$\sigma(p_{T,H}>p_{T,H}^{\text{cut}})$ in function of $p_{T,H}^{\text{cut}}$
for $pp\rightarrow H+j$ and $pp\rightarrow H+jj$ respectively. Shown are cross
sections at LO and NLO and related K-factors considering the HTL and
$\rm{FT}_{\rm approx}$ approximations, and for $pp\rightarrow H+j$ also the
results with exact top-mass dependence. The setup is given in Section 3.1.

$p_{T,H}^{\text{cut}}$ | LO${}_{\text{HTL}}$ | NLO${}_{\text{HTL}}$ | K${}^{\text{NLO}}_{\text{HTL}}$ | LO${}_{\text{SM}}$ | NLO${}_{\rm{FT}_{\rm approx}}$ | K${}^{\text{NLO}}_{\rm{FT}_{\rm approx}}$ | NLO${}_{\text{SM}}$ | K${}^{\text{NLO}}_{\text{SM}}$
---|---|---|---|---|---|---|---|---
50 | $4453^{+1755}_{-1181}$ | $8482^{+1755}_{-1522}$ | $1.90$ | $4566^{+1800}_{-1212}$ | $8682^{+1793}_{-1557}$ | $1.90$ | $8732^{+1858}_{-1585}$ | $1.91$
100 | $1430^{+585}_{-389}$ | $2732^{+578}_{-502}$ | $1.91$ | $1391^{+570}_{-379}$ | $2645^{+557}_{-485}$ | $1.90$ | $2669^{+575}_{-494}$ | $1.92$
150 | $593^{+249}_{-164}$ | $1121^{+235}_{-207}$ | $1.89$ | $528^{+222}_{-146}$ | $989^{+205}_{-182}$ | $1.87$ | $996^{+216}_{-186}$ | $1.89$
200 | $284^{+121}_{-79.6}$ | $533^{+111}_{-98.9}$ | $1.88$ | $219^{+94.2}_{-61.7}$ | $411^{+85.9}_{-76.5}$ | $1.88$ | $417^{+90.0}_{-78.4}$ | $1.90$
250 | $151^{+65.2}_{-42.6}$ | $281^{+58.2}_{-52.2}$ | $1.87$ | $97.4^{+42.5}_{-27.7}$ | $184^{+39.3}_{-34.8}$ | $1.89$ | $189^{+41.2}_{-36.1}$ | $1.94$
300 | $85.9^{+37.6}_{-24.5}$ | $160^{+32.9}_{-29.7}$ | $1.86$ | $45.9^{+20.3}_{-13.2}$ | $87.8^{+19.2}_{-16.9}$ | $1.91$ | $90.1^{+19.7}_{-17.4}$ | $1.96$
350 | $51.8^{+22.9}_{-14.8}$ | $95.9^{+19.7}_{-17.9}$ | $1.85$ | $22.9^{+10.2}_{-6.62}$ | $44.0^{+9.71}_{-8.53}$ | $1.92$ | $45.1^{+10.2}_{-8.85}$ | $1.97$
400 | $32.5^{+14.5}_{-9.38}$ | $60.1^{+12.3}_{-11.2}$ | $1.85$ | $12.0^{+5.39}_{-3.48}$ | $23.0^{+5.10}_{-4.49}$ | $1.92$ | $23.6^{+5.38}_{-4.66}$ | $1.98$
450 | $21.2^{+9.49}_{-6.14}$ | $39.1^{+8.00}_{-7.33}$ | $1.84$ | $6.52^{+2.96}_{-1.91}$ | $12.6^{+2.80}_{-2.46}$ | $1.93$ | $12.9^{+2.97}_{-2.56}$ | $1.98$
500 | $14.2^{+6.39}_{-4.13}$ | $26.2^{+5.36}_{-4.92}$ | $1.84$ | $3.67^{+1.68}_{-1.08}$ | $7.09^{+1.59}_{-1.40}$ | $1.93$ | $7.25^{+1.69}_{-1.45}$ | $1.97$
550 | $9.71^{+4.40}_{-2.84}$ | $17.9^{+3.66}_{-3.37}$ | $1.84$ | $2.14^{+0.98}_{-0.63}$ | $4.11^{+0.92}_{-0.81}$ | $1.92$ | $4.18^{+0.96}_{-0.83}$ | $1.96$
600 | $6.79^{+3.09}_{-1.99}$ | $12.5^{+2.56}_{-2.36}$ | $1.84$ | $1.28^{+0.59}_{-0.38}$ | $2.47^{+0.56}_{-0.49}$ | $1.93$ | $2.50^{+0.59}_{-0.51}$ | $1.96$
650 | $4.82^{+2.21}_{-1.42}$ | $8.88^{+1.83}_{-1.69}$ | $1.84$ | $0.78^{+0.36}_{-0.23}$ | $1.50^{+0.34}_{-0.30}$ | $1.92$ | $1.53^{+0.36}_{-0.31}$ | $1.96$
700 | $3.48^{+1.60}_{-1.03}$ | $6.42^{+1.33}_{-1.22}$ | $1.85$ | $0.49^{+0.23}_{-0.15}$ | $0.94^{+0.21}_{-0.19}$ | $1.92$ | $0.96^{+0.23}_{-0.19}$ | $1.96$
750 | $2.54^{+1.17}_{-0.75}$ | $4.68^{+0.97}_{-0.89}$ | $1.84$ | $0.31^{+0.15}_{-0.09}$ | $0.60^{+0.14}_{-0.12}$ | $1.94$ | $0.62^{+0.14}_{-0.13}$ | $1.97$
800 | $1.87^{+0.87}_{-0.56}$ | $3.45^{+0.71}_{-0.66}$ | $1.84$ | $0.20^{+0.1}_{-0.06}$ | $0.39^{+0.09}_{-0.08}$ | $1.92$ | $0.39^{+0.09}_{-0.08}$ | $1.94$
Table 3: Integrated cross sections in fb depending on $p_{T,H}^{\text{cut}}$ at LO and NLO in the HTL, $\rm{FT}_{\rm approx}$, and with full top-quark mass dependence (SM) for $\rm{H}+$jet production at the LHC with $\sqrt{S}=13$ TeV together with corresponding K-factors. Uncertainties correspond to the envelope of 7-point scale variations. We require $p_{\rm{T},j}>30\,\mathrm{GeV}$.  $p_{T,H}^{\text{cut}}$ | LO${}_{\text{HTL}}$ | NLO${}_{\text{HTL}}$ | K${}^{\text{NLO}}_{\text{HTL}}$ | LO${}_{\text{SM}}$ | NLO${}_{\rm{FT}_{\rm approx}}$ | K${}^{\text{NLO}}_{\rm{FT}_{\rm approx}}$
---|---|---|---|---|---|---
50 | $2365^{+1377}_{-813}$ | $3511^{+446}_{-631}$ | $1.48$ | $2387^{+1391}_{-821}$ | $3574^{+477}_{-651}$ | $1.50$
100 | $1355^{+790}_{-466}$ | $2025^{+267}_{-368}$ | $1.50$ | $1317^{+769}_{-454}$ | $1991^{+276}_{-368}$ | $1.51$
150 | $671^{+392}_{-232}$ | $969^{+110}_{-169}$ | $1.44$ | $601^{+353}_{-208}$ | $881^{+107}_{-157}$ | $1.47$
200 | $355^{+208}_{-123}$ | $497^{+47.9}_{-83.4}$ | $1.40$ | $281^{+166}_{-97.5}$ | $399^{+42.1}_{-68.7}$ | $1.42$
250 | $201^{+118}_{-69.6}$ | $274^{+22.7}_{-44.4}$ | $1.36$ | $136^{+80.2}_{-47.2}$ | $188^{+17.8}_{-31.6}$ | $1.39$
300 | $120^{+70.5}_{-41.6}$ | $160^{+11.8}_{-25.3}$ | $1.34$ | $68.1^{+40.3}_{-23.7}$ | $92.4^{+7.73}_{-15.2}$ | $1.36$
350 | $74.8^{+44.0}_{-26.0}$ | $98.5^{+6.53}_{-15.2}$ | $1.32$ | $35.5^{+21.1}_{-12.4}$ | $47.7^{+3.70}_{-7.73}$ | $1.34$
400 | $48.4^{+28.5}_{-16.8}$ | $62.9^{+3.82}_{-9.56}$ | $1.30$ | $19.2^{+11.4}_{-6.71}$ | $25.4^{+1.82}_{-4.05}$ | $1.33$
450 | $32.2^{+19.0}_{-11.2}$ | $41.5^{+2.36}_{-6.23}$ | $1.29$ | $10.7^{+6.38}_{-3.75}$ | $14.1^{+0.96}_{-2.23}$ | $1.32$
500 | $21.9^{+12.9}_{-7.62}$ | $28.0^{+1.51}_{-4.18}$ | $1.28$ | $6.16^{+3.68}_{-2.16}$ | $8.06^{+0.52}_{-1.27}$ | $1.31$
550 | $15.2^{+8.99}_{-5.30}$ | $19.4^{+1.01}_{-2.87}$ | $1.27$ | $3.64^{+2.18}_{-1.28}$ | $4.74^{+0.30}_{-0.74}$ | $1.30$
600 | $10.8^{+6.37}_{-3.75}$ | $13.7^{+0.69}_{-2.02}$ | $1.27$ | $2.20^{+1.32}_{-0.77}$ | $2.85^{+0.18}_{-0.44}$ | $1.29$
650 | $7.72^{+4.57}_{-2.70}$ | $9.79^{+0.49}_{-1.44}$ | $1.27$ | $1.36^{+0.82}_{-0.48}$ | $1.76^{+0.11}_{-0.27}$ | $1.29$
700 | $5.61^{+3.33}_{-1.96}$ | $7.10^{+0.35}_{-1.05}$ | $1.27$ | $0.85^{+0.51}_{-0.30}$ | $1.10^{+0.07}_{-0.17}$ | $1.29$
750 | $4.12^{+2.45}_{-1.44}$ | $5.22^{+0.25}_{-0.77}$ | $1.26$ | $0.55^{+0.33}_{-0.19}$ | $0.70^{+0.04}_{-0.11}$ | $1.29$
800 | $3.06^{+1.82}_{-1.07}$ | $3.88^{+0.19}_{-0.57}$ | $1.27$ | $0.36^{+0.22}_{-0.13}$ | $0.46^{+0.03}_{-0.07}$ | $1.28$
Table 4: Integrated cross sections in fb depending on $p_{T,H}^{\text{cut}}$
at LO and NLO in the HTL and $\rm{FT}_{\rm approx}$ for $\rm{H}+2\,$jets
production at the LHC with $\sqrt{S}=13$ TeV together with corresponding
K-factors. Uncertainties correspond to the envelope of 7-point scale
variations. We require $p_{\rm{T},j_{1}}>40\,\mathrm{GeV},\,$
$p_{\rm{T},j_{2}}>30\,\mathrm{GeV}$.

## References

  * (1) C. Arnesen, I. Z. Rothstein, and J. Zupan, Smoking Guns for On-Shell New Physics at the LHC, Phys. Rev. Lett. 103 (2009) 151801, [arXiv:0809.1429].
  * (2) A. Banfi, A. Martin, and V. Sanz, Probing top-partners in Higgs+jets, JHEP 08 (2014) 053, [arXiv:1308.4771].
  * (3) A. Azatov and A. Paul, Probing Higgs couplings with high $p_{T}$ Higgs production, JHEP 01 (2014) 014, [arXiv:1309.5273].
  * (4) C. Grojean, E. Salvioni, M. Schlaffer, and A. Weiler, Very boosted Higgs in gluon fusion, JHEP 05 (2014) 022, [arXiv:1312.3317].
  * (5) R. V. Harlander and T. Neumann, Probing the nature of the Higgs-gluon coupling, Phys. Rev. D 88 (2013) 074015, [arXiv:1308.2225].
  * (6) S. Dawson, I. M. Lewis, and M. Zeng, Effective field theory for Higgs boson plus jet production, Phys. Rev. D 90 (2014), no. 9 093007, [arXiv:1409.6299].
  * (7) U. Langenegger, M. Spira, and I. Strebel, Testing the Higgs Boson Coupling to Gluons, arXiv:1507.01373.
  * (8) M. Grazzini, A. Ilnicka, M. Spira, and M. Wiesemann, Modeling BSM effects on the Higgs transverse-momentum spectrum in an EFT approach, JHEP 03 (2017) 115, [arXiv:1612.00283].
  * (9) N. Deutschmann, C. Duhr, F. Maltoni, and E. Vryonidou, Gluon-fusion Higgs production in the Standard Model Effective Field Theory, JHEP 12 (2017) 063, [arXiv:1708.00460]. [Erratum: JHEP 02, 159 (2018)].
  * (10) M. Battaglia, M. Grazzini, M. Spira, and M. Wiesemann, Sensitivity to BSM effects in the Higgs $p_{T}$ spectrum within SMEFT, arXiv:2109.02987.
  * (11) ATLAS Collaboration, Prospects for differential cross-section measurements of Higgs boson production measured in decays to ZZ and $\gamma\gamma$ with the ATLAS experiment at the High-Luminosity LHC, ATL-PHYS-PUB-2018-040 (2018).
  * (12) CMS Collaboration, Sensitivity projections for Higgs boson properties measurements at the HL-LHC, CMS-PAS-FTR-18-011 (2018).
  * (13) ATLAS Collaboration, Study of Higgs-boson production with large transverse momentum using the $H\rightarrow b\bar{b}$ decay with the ATLAS detector, ATLAS-CONF-2021-010 (3, 2021).
  * (14) CMS Collaboration, A. M. Sirunyan et al., Inclusive search for highly boosted Higgs bosons decaying to bottom quark-antiquark pairs in proton-proton collisions at $\sqrt{s}=$ 13 TeV, JHEP 12 (2020) 085, [arXiv:2006.13251].
  * (15) K. Becker et al., Precise predictions for boosted Higgs production, arXiv:2005.07762.
  * (16) F. Campanario, T. M. Figy, S. Plätzer, M. Rauch, P. Schichtel, and M. Sjödahl, Stress testing the vector-boson-fusion approximation in multijet final states, Phys. Rev. D 98 (2018), no. 3 033003, [arXiv:1802.09955].
  * (17) A. Buckley et al., A comparative study of Higgs boson production from vector-boson fusion, arXiv:2105.11399.
  * (18) P. Artoisenet et al., A framework for Higgs characterisation, JHEP 11 (2013) 043, [arXiv:1306.6464].
  * (19) F. Maltoni, K. Mawatari, and M. Zaro, Higgs characterisation via vector-boson fusion and associated production: NLO and parton-shower effects, Eur. Phys. J. C 74 (2014), no. 1 2710, [arXiv:1311.1829].
  * (20) A. Greljo, G. Isidori, J. M. Lindert, and D. Marzocca, Pseudo-observables in electroweak Higgs production, Eur. Phys. J. C 76 (2016), no. 3 158, [arXiv:1512.06135].
  * (21) C. Degrande, B. Fuks, K. Mawatari, K. Mimasu, and V. Sanz, Electroweak Higgs boson production in the standard model effective field theory beyond leading order in QCD, Eur. Phys. J. C 77 (2017), no. 4 262, [arXiv:1609.04833].
  * (22) A. Greljo, G. Isidori, J. M. Lindert, D. Marzocca, and H. Zhang, Electroweak Higgs production with HiggsPO at NLO QCD, Eur. Phys. J. C 77 (2017), no. 12 838, [arXiv:1710.04143].
  * (23) J. Y. Araz, S. Banerjee, R. S. Gupta, and M. Spannowsky, Precision SMEFT bounds from the VBF Higgs at high transverse momentum, JHEP 04 (2021) 125, [arXiv:2011.03555].
  * (24) M. Czakon, R. V. Harlander, J. Klappert, and M. Niggetiedt, Exact top-quark mass dependence in hadronic Higgs production, arXiv:2105.04436.
  * (25) S. P. Jones, M. Kerner, and G. Luisoni, Next-to-Leading-Order QCD Corrections to Higgs Boson Plus Jet Production with Full Top-Quark Mass Dependence, Phys. Rev. Lett. 120 (2018), no. 16 162001, [arXiv:1802.00349].
  * (26) J. M. Lindert, K. Kudashkin, K. Melnikov, and C. Wever, Higgs bosons with large transverse momentum at the LHC, Phys. Lett. B 782 (2018) 210–214, [arXiv:1801.08226].
  * (27) T. Neumann, NLO Higgs+jet production at large transverse momenta including top quark mass effects, J. Phys. Comm. 2 (2018), no. 9 095017, [arXiv:1802.02981].
  * (28) M. Kerner, Top mass effects in HJ and HH production, PoS RADCOR2019 (2019) 020.
  * (29) V. Del Duca, W. Kilgore, C. Oleari, C. Schmidt, and D. Zeppenfeld, Higgs \+ 2 jets via gluon fusion, Phys. Rev. Lett. 87 (2001) 122001, [hep-ph/0105129].
  * (30) V. Del Duca, W. Kilgore, C. Oleari, C. Schmidt, and D. Zeppenfeld, Gluon fusion contributions to H + 2 jet production, Nucl. Phys. B 616 (2001) 367–399, [hep-ph/0108030].
  * (31) T. Neumann and C. Williams, The Higgs boson at high $p_{T}$, Phys. Rev. D 95 (2017), no. 1 014004, [arXiv:1609.00367].
  * (32) J. R. Andersen, J. D. Cockburn, M. Heil, A. Maier, and J. M. Smillie, Finite Quark-Mass Effects in Higgs Boson Production with Dijets at Large Energies, JHEP 04 (2019) 127, [arXiv:1812.08072].
  * (33) L. Budge, J. M. Campbell, G. De Laurentis, R. K. Ellis, and S. Seth, The one-loop amplitudes for Higgs + 4 partons with full mass effects, JHEP 05 (2020) 079, [arXiv:2002.04018].
  * (34) S. Borowka, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun. 196 (2015) 470–491, [arXiv:1502.06595].
  * (35) S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222 (2018) 313–326, [arXiv:1703.09692].
  * (36) K. Kudashkin, K. Melnikov, and C. Wever, Two-loop amplitudes for processes $gg\to Hg,qg\to Hq$ and $q\bar{q}\to Hg$ at large Higgs transverse momentum, JHEP 02 (2018) 135, [arXiv:1712.06549].
  * (37) F. Wilczek, Decays of Heavy Vector Mesons Into Higgs Particles, Phys. Rev. Lett. 39 (1977) 1304.
  * (38) C. Anastasiou, C. Duhr, F. Dulat, F. Herzog, and B. Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops, Phys. Rev. Lett. 114 (2015) 212001, [arXiv:1503.06056].
  * (39) F. Dulat, B. Mistlberger, and A. Pelloni, Differential Higgs production at N3LO beyond threshold, JHEP 01 (2018) 145, [arXiv:1710.03016].
  * (40) L. Cieri, X. Chen, T. Gehrmann, E. W. N. Glover, and A. Huss, Higgs boson production at the LHC using the $q_{T}$ subtraction formalism at N3LO QCD, JHEP 02 (2019) 096, [arXiv:1807.11501].
  * (41) B. Mistlberger, Higgs boson production at hadron colliders at N3LO in QCD, JHEP 05 (2018) 028, [arXiv:1802.00833].
  * (42) X. Chen, T. Gehrmann, E. W. N. Glover, A. Huss, B. Mistlberger, and A. Pelloni, Fully Differential Higgs Boson Production to Third Order in QCD, Phys. Rev. Lett. 127 (2021), no. 7 072002, [arXiv:2102.07607].
  * (43) R. Boughezal, F. Caola, K. Melnikov, F. Petriello, and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, JHEP 06 (2013) 072, [arXiv:1302.6216].
  * (44) X. Chen, T. Gehrmann, E. W. N. Glover, and M. Jaquier, Precise QCD predictions for the production of Higgs + jet final states, Phys. Lett. B 740 (2015) 147–150, [arXiv:1408.5325].
  * (45) R. Boughezal, F. Caola, K. Melnikov, F. Petriello, and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order, Phys. Rev. Lett. 115 (2015), no. 8 082003, [arXiv:1504.07922].
  * (46) R. Boughezal, C. Focke, W. Giele, X. Liu, and F. Petriello, Higgs boson production in association with a jet at NNLO using jettiness subtraction, Phys. Lett. B 748 (2015) 5–8, [arXiv:1505.03893].
  * (47) X. Chen, J. Cruz-Martinez, T. Gehrmann, E. W. N. Glover, and M. Jaquier, NNLO QCD corrections to Higgs boson production at large transverse momentum, JHEP 10 (2016) 066, [arXiv:1607.08817].
  * (48) J. M. Campbell, R. K. Ellis, and G. Zanderighi, Next-to-Leading order Higgs + 2 jet production via gluon fusion, JHEP 10 (2006) 028, [hep-ph/0608194].
  * (49) H. van Deurzen, N. Greiner, G. Luisoni, P. Mastrolia, E. Mirabella, G. Ossola, T. Peraro, J. F. von Soden-Fraunhofen, and F. Tramontano, NLO QCD corrections to the production of Higgs plus two jets at the LHC, Phys. Lett. B 721 (2013) 74–81, [arXiv:1301.0493].
  * (50) G. Cullen, H. van Deurzen, N. Greiner, G. Luisoni, P. Mastrolia, E. Mirabella, G. Ossola, T. Peraro, and F. Tramontano, Next-to-Leading-Order QCD Corrections to Higgs Boson Production Plus Three Jets in Gluon Fusion, Phys. Rev. Lett. 111 (2013), no. 13 131801, [arXiv:1307.4737].
  * (51) G. Bozzi, S. Catani, D. de Florian, and M. Grazzini, Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC, Nucl. Phys. B 737 (2006) 73–120, [hep-ph/0508068].
  * (52) D. de Florian, G. Ferrera, M. Grazzini, and D. Tommasini, Transverse-momentum resummation: Higgs boson production at the Tevatron and the LHC, JHEP 11 (2011) 064, [arXiv:1109.2109].
  * (53) T. Becher, M. Neubert, and D. Wilhelm, Higgs-Boson Production at Small Transverse Momentum, JHEP 05 (2013) 110, [arXiv:1212.2621].
  * (54) W. Bizon, P. F. Monni, E. Re, L. Rottoli, and P. Torrielli, Momentum-space resummation for transverse observables and the Higgs p⟂ at N3LL+NNLO, JHEP 02 (2018) 108, [arXiv:1705.09127].
  * (55) X. Chen, T. Gehrmann, E. W. N. Glover, A. Huss, Y. Li, D. Neill, M. Schulze, I. W. Stewart, and H. X. Zhu, Precise QCD Description of the Higgs Boson Transverse Momentum Spectrum, Phys. Lett. B 788 (2019) 425–430, [arXiv:1805.00736].
  * (56) W. Bizoń, X. Chen, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, A. Huss, P. F. Monni, E. Re, L. Rottoli, and P. Torrielli, Fiducial distributions in Higgs and Drell-Yan production at N3LL+NNLO, JHEP 12 (2018) 132, [arXiv:1805.05916].
  * (57) E. Re, L. Rottoli, and P. Torrielli, Fiducial Higgs and Drell-Yan distributions at N3LL′+NNLO with RadISH, arXiv:2104.07509.
  * (58) H. Mantler and M. Wiesemann, Top- and bottom-mass effects in hadronic Higgs production at small transverse momenta through LO+NLL, Eur. Phys. J. C 73 (2013), no. 6 2467, [arXiv:1210.8263].
  * (59) M. Grazzini and H. Sargsyan, Heavy-quark mass effects in Higgs boson production at the LHC, JHEP 09 (2013) 129, [arXiv:1306.4581].
  * (60) A. Banfi, P. F. Monni, and G. Zanderighi, Quark masses in Higgs production with a jet veto, JHEP 01 (2014) 097, [arXiv:1308.4634].
  * (61) K. Melnikov, L. Tancredi, and C. Wever, Two-loop $gg\to Hg$ amplitude mediated by a nearly massless quark, JHEP 11 (2016) 104, [arXiv:1610.03747].
  * (62) J. M. Lindert, K. Melnikov, L. Tancredi, and C. Wever, Top-bottom interference effects in Higgs plus jet production at the LHC, Phys. Rev. Lett. 118 (2017), no. 25 252002, [arXiv:1703.03886].
  * (63) K. Melnikov, L. Tancredi, and C. Wever, Two-loop amplitudes for $qg\to Hq$ and $q\bar{q}\to Hg$ mediated by a nearly massless quark, Phys. Rev. D 95 (2017), no. 5 054012, [arXiv:1702.00426].
  * (64) F. Caola, J. M. Lindert, K. Melnikov, P. F. Monni, L. Tancredi, and C. Wever, Bottom-quark effects in Higgs production at intermediate transverse momentum, JHEP 09 (2018) 035, [arXiv:1804.07632].
  * (65) F. Caola, S. Forte, S. Marzani, C. Muselli, and G. Vita, The Higgs transverse momentum spectrum with finite quark masses beyond leading order, JHEP 08 (2016) 150, [arXiv:1606.04100].
  * (66) R. V. Harlander, T. Neumann, K. J. Ozeren, and M. Wiesemann, Top-mass effects in differential Higgs production through gluon fusion at order $\mathcal{O}(\alpha^{4}_{s}$), JHEP 08 (2012) 139, [arXiv:1206.0157].
  * (67) T. Neumann and M. Wiesemann, Finite top-mass effects in gluon-induced Higgs production with a jet-veto at NNLO, JHEP 11 (2014) 150, [arXiv:1408.6836].
  * (68) J. M. Campbell, R. K. Ellis, R. Frederix, P. Nason, C. Oleari, and C. Williams, NLO Higgs Boson Production Plus One and Two Jets Using the POWHEG BOX, MadGraph4 and MCFM, JHEP 07 (2012) 092, [arXiv:1202.5475].
  * (69) K. Hamilton, P. Nason, C. Oleari, and G. Zanderighi, Merging H/W/Z + 0 and 1 jet at NLO with no merging scale: a path to parton shower + NNLO matching, JHEP 05 (2013) 082, [arXiv:1212.4504].
  * (70) M. Buschmann, D. Goncalves, S. Kuttimalai, M. Schonherr, F. Krauss, and T. Plehn, Mass Effects in the Higgs-Gluon Coupling: Boosted vs Off-Shell Production, JHEP 02 (2015) 038, [arXiv:1410.5806].
  * (71) R. Frederix, S. Frixione, E. Vryonidou, and M. Wiesemann, Heavy-quark mass effects in Higgs plus jets production, JHEP 08 (2016) 006, [arXiv:1604.03017].
  * (72) N. Greiner, S. Höche, G. Luisoni, M. Schönherr, and J.-C. Winter, Full mass dependence in Higgs boson production in association with jets at the LHC and FCC, JHEP 01 (2017) 091, [arXiv:1608.01195].
  * (73) F. Maltoni, E. Vryonidou, and M. Zaro, Top-quark mass effects in double and triple Higgs production in gluon-gluon fusion at NLO, JHEP 11 (2014) 079, [arXiv:1408.6542].
  * (74) A. Gehrmann-De Ridder, T. Gehrmann, and E. W. N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056, [hep-ph/0505111].
  * (75) A. Gehrmann-De Ridder, T. Gehrmann, and E. Glover, Gluon-gluon antenna functions from Higgs boson decay, Phys. Lett. B 612 (2005) 49–60, [hep-ph/0502110].
  * (76) A. Daleo, T. Gehrmann, and D. Maître, Antenna subtraction with hadronic initial states, JHEP 04 (2007) 016, [hep-ph/0612257].
  * (77) A. Daleo, A. Gehrmann-De Ridder, T. Gehrmann, and G. Luisoni, Antenna subtraction at NNLO with hadronic initial states: initial-final configurations, JHEP 01 (2010) 118, [arXiv:0912.0374].
  * (78) E. W. Nigel Glover and J. Pires, Antenna subtraction for gluon scattering at NNLO, JHEP 06 (2010) 096, [arXiv:1003.2824].
  * (79) A. Gehrmann-De Ridder, E. W. N. Glover, and J. Pires, Real-Virtual corrections for gluon scattering at NNLO, JHEP 02 (2012) 141, [arXiv:1112.3613].
  * (80) A. Gehrmann-De Ridder, T. Gehrmann, and M. Ritzmann, Antenna subtraction at NNLO with hadronic initial states: double real initial-initial configurations, JHEP 1210 (2012) 047, [arXiv:1207.5779].
  * (81) A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover, and J. Pires, Double Virtual corrections for gluon scattering at NNLO, JHEP 02 (2013) 026, [arXiv:1211.2710].
  * (82) J. Currie, E. Glover, and S. Wells, Infrared Structure at NNLO Using Antenna Subtraction, JHEP 04 (2013) 066, [arXiv:1301.4693].
  * (83) F. Cascioli, P. Maierhofer, and S. Pozzorini, Scattering Amplitudes with Open Loops, Phys. Rev. Lett. 108 (2012) 111601, [arXiv:1111.5206].
  * (84) F. Buccioni, S. Pozzorini, and M. Zoller, On-the-fly reduction of open loops, Eur. Phys. J. C 78 (2018), no. 1 70, [arXiv:1710.11452].
  * (85) F. Buccioni, J.-N. Lang, J. M. Lindert, P. Maierhöfer, S. Pozzorini, H. Zhang, and M. F. Zoller, OpenLoops 2, Eur. Phys. J. C 79 (2019), no. 10 866, [arXiv:1907.13071].
  * (86) J.-N. Lang et al, OTTER: On-The-fly Tensor Reduction, In preparation.
  * (87) R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, P. Torrielli, E. Vryonidou, and M. Zaro, Higgs pair production at the LHC with NLO and parton-shower effects, Phys. Lett. B 732 (2014) 142–149, [arXiv:1401.7340].
  * (88) S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161–171, [hep-ph/9802439].
  * (89) M. Grazzini, G. Heinrich, S. Jones, S. Kallweit, M. Kerner, J. M. Lindert, and J. Mazzitelli, Higgs boson pair production at NNLO with top quark mass effects, JHEP 05 (2018) 059, [arXiv:1803.02463].
  * (90) A. Denner, S. Dittmaier, and L. Hofer, Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations, Comput. Phys. Commun. 212 (2017) 220–238, [arXiv:1604.06792].
  * (91) A. van Hameren, OneLOop: For the evaluation of one-loop scalar functions, Comput. Phys. Commun. 182 (2011) 2427–2438, [arXiv:1007.4716].
  * (92) A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction, arXiv:1201.4330.
  * (93) Z. Li, J. Wang, Q.-S. Yan, and X. Zhao, Efficient numerical evaluation of Feynman integrals, Chin. Phys. C 40 (2016), no. 3 033103, [arXiv:1508.02512].
  * (94) S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, and J. Schlenk, A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec, Comput. Phys. Commun. 240 (2019) 120–137, [arXiv:1811.11720].
  * (95) J. Baglio, F. Campanario, S. Glaus, M. Mühlleitner, M. Spira, and J. Streicher, Gluon fusion into Higgs pairs at NLO QCD and the top mass scheme, Eur. Phys. J. C 79 (2019), no. 6 459, [arXiv:1811.05692].
  * (96) J. Baglio, F. Campanario, S. Glaus, M. Mühlleitner, J. Ronca, M. Spira, and J. Streicher, Higgs-Pair Production via Gluon Fusion at Hadron Colliders: NLO QCD Corrections, JHEP 04 (2020) 181, [arXiv:2003.03227].
  * (97) J. Baglio, F. Campanario, S. Glaus, M. Mühlleitner, J. Ronca, and M. Spira, $gg\to HH$ : Combined uncertainties, Phys. Rev. D 103 (2021), no. 5 056002, [arXiv:2008.11626].
  * (98) S. Amoroso et al., Les Houches 2019: Physics at TeV Colliders: Standard Model Working Group Report, in 11th Les Houches Workshop on Physics at TeV Colliders: PhysTeV Les Houches, 3, 2020.  arXiv:2003.01700.
  * (99) J. Butterworth et al., PDF4LHC recommendations for LHC Run II, J. Phys. G 43 (2016) 023001, [arXiv:1510.03865].
  * (100) M. Cacciari, G. P. Salam, and G. Soyez, The anti-$k_{t}$ jet clustering algorithm, JHEP 04 (2008) 063, [arXiv:0802.1189].