# Next-to-leading-order QCD Corrections to Higgs Production in association
with a Jet

Roberto Bonciani<EMAIL_ADDRESS>Dipartimento di Fisica,
Università di Roma “La Sapienza” and INFN Sezione di Roma, Piazzale Aldo Moro
2, 00185 Roma, Italy Vittorio Del Duca<EMAIL_ADDRESS>ETH Zürich,
Institut für theoretische Physik, 8093 Zürich, Switzerland, Physik-Institut,
Universität Zürich, 8057 Zürich, Switzerland INFN, Laboratori Nazionali di
Frascati, 00044 Frascati (RM), Italy Hjalte Frellesvig
<EMAIL_ADDRESS>Niels Bohr International Academy, University of
Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark Martijn Hidding
<EMAIL_ADDRESS>Department of Physics and Astronomy, Uppsala
University, SE-75120 Uppsala, Sweden Valentin Hirschi<EMAIL_ADDRESS>Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland Francesco
Moriello<EMAIL_ADDRESS>DSM AG, Wurmisweg 576, CH-4303
Kaiseraugst, Switzerland Giulio Salvatori<EMAIL_ADDRESS>School
of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, USA
Gábor Somogyi<EMAIL_ADDRESS>Wigner Research Centre for Physics,
Konkoly-Thege Miklós u. 29-33, 1121 Budapest, Hungary Francesco Tramontano
<EMAIL_ADDRESS>Università di Napoli and INFN, Sezione di
Napoli, Complesso Universitario di Monte Sant’Angelo, Via Cintia, 80126
Napoli, Italy

###### Abstract

We compute the next-to-leading-order (NLO) QCD corrections to the Higgs
$p_{T}$ distribution in Higgs production in association with a jet via gluon
fusion at the LHC, with exact dependence on the mass of the quark circulating
in the heavy-quark loops. The NLO corrections are presented including the top-
quark mass, and for the first time, the bottom-quark mass as well. Further,
besides the on-shell mass scheme, we consider for the first time a running
mass renormalisation scheme. The computation is based on amplitudes which are
valid for arbitrary heavy-quark masses.

##  I Introduction

Since the discovery of the Higgs boson [1, 2], one of the main goals of the
Large Hadron Collider (LHC) physics program has been to investigate couplings
and quantum numbers of the Higgs boson as accurately as possible. At the LHC,
the dominant Higgs production mode is via gluon fusion, with the coupling of
the Higgs boson to gluons being mediated by a heavy-quark loop. This gives the
opportunity to test the Standard Model (SM) and to look for possible
deviations from it, which would be footprints of New Physics (NP) [3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

A promising observable to probe possible NP effects is the Higgs $p_{T}$
distribution [20, 21, 22, 23, 24], which allows one to analyse how the
production rate depends on the heavy-quark loop at different $p_{T}$ values.
The Higgs $p_{T}$ distribution is known at leading order in $\alpha_{s}$ for
arbitrary quark masses in the heavy-quark loop [25, 26, 27], and at NLO for
the top-quark mass [28, 29] in the on-shell (OS) mass scheme. Approximate
results are known at NLO [30], which include the bottom-quark mass [31, 32,
33] as well as the top-bottom interference [34], and beyond NLO in the high-
energy limit [35]. Further, the mixed QCD-electroweak contributions to the
Higgs $p_{T}$ distribution are known at leading order [36]. The Higgs
Effective Field Theory (HEFT) approach provides a good approximation to the
Higgs $p_{T}$ distribution when $p_{T}<m_{t}$ [26]. In HEFT, the Higgs $p_{T}$
distribution is known at NNLO in $\alpha_{s}$ [37, 38, 39, 40]. Conversely,
when $p_{T}>m_{t}$ the shape of the Higgs $p_{T}$ distribution in HEFT
overshoots the exact Higgs $p_{T}$ distribution [29].

In this letter, we present the Higgs $p_{T}$ distribution at NLO in
$\alpha_{s}$, with top-quark mass dependence, and for the first time including
also the exact bottom-quark mass contribution. Further, since the top- and
bottom-quark masses are treated as dynamical parameters and not as fixed
numbers, besides the OS mass scheme, we compute the Higgs $p_{T}$ distribution
using for the first time a dynamical mass renormalisation scheme, the
$\overline{\rm{MS}}$ scheme.

##  II Calculation

Our computation of the Higgs $p_{T}$ distribution is based on amplitudes which
are valid for arbitrary quark masses circulating in the heavy-quark loops.

We consider the production of a Higgs boson in association with a jet in
proton-proton collisions, $p+p\to H+j+X$. The corresponding cross section can
be found by convoluting the partonic cross section with the respective parton
distribution functions (PDF), for the partonic channels: $gg\to Hg$,
$q\bar{q}\to Hg$ and $q(\bar{q})g\to Hq(\bar{q})$. At LO in the coupling
constants, this involves the one-loop $2\to 2$ amplitudes. The coupling of the
Higgs boson to light quarks is suppressed by the light-quark mass. Therefore,
only the contributions in which the Higgs boson is coupled to a heavy quark
are important. In this letter we consider the contribution of the top and the
bottom quarks.

At NLO in QCD, we have to consider the $\mathcal{O}(\alpha_{S})$ corrections
to the LO $2\to 2$ amplitudes, together with the one-loop $2\to 3$ amplitudes,
in which the Higgs boson and the jet are produced with an additional parton in
the final state: $gg\to Hgg$, $q\bar{q}\to Hgg$, and $q(\bar{q})g\to
Hq(\bar{q})g$. The calculation of the two-loop $2\to 2$ amplitudes, which has
been performed assembling the relevant master integrals [41, 42, 43], is
briefly described in the following subsection and will be the subject of a
forthcoming publication [44]. The calculation of the one-loop $2\to 3$
amplitudes, which are known analytically [45, 46] and numerically through
public automated tools, has been performed using the results of refs. [47, 46]
reported in the program MCFM-9.1 [48]. We have cross-checked the one-loop
$2\to 3$ amplitudes with automated tools (MG5_aMC$@$NLO [49, 27] and GoSam
[50]) and found perfect agreement for every phase-space point we compared,
including points where an unresolved limit is approached.

After renormalisation of the ultra-violet (UV) divergences, each of the two
sets of amplitudes still contains infra-red (IR) divergences. In order to
regulate such divergences, we used the dipole subtraction scheme [51]. For the
numerical evaluation of the $2\to 3$ amplitudes and the subtraction terms we
have used the program MCFM-9.1 [48]. Although the integration of the $2\to 3$
amplitudes, even when generated with the automated tools, is not time
intensive, we preferred to use the analytic result of refs. [47, 46] which
saved about a factor of a hundred in the integration time of the gluonic one-
loop $2\to 3$ amplitudes.

##  III Two-loop $2\to 2$ amplitudes

The two-loop $2\to 2$ amplitudes for the processes under consideration can be
written in terms of form factors, exploiting their Lorentz and Dirac
structures, where every form factor can be expressed as a combination of
scalar integrals that are divergent in four space-time dimensions. We
regularise both UV and IR divergences in dimensional regularisation. Since the
scalar integrals are not all independent, we use Integration-by-Parts
Identities [52, 53, 54], implemented in the computer programs FIRE [55, 56]
and Kira [57, 58], in order to reduce them to a set of independent integrals,
called Master Integrals (MIs). We calculate the MIs using the differential
equations method [59, 60, 61, 62, 63, 64, 65, 66]. The MIs are expressed as
Laurent series in the dimensional parameter $\epsilon=(4-d)/2$, where $d$ is
the dimension of space-time. The system of first-order linear differential
equations satisfied by the MIs is solved at every order in $\epsilon$ using
DiffExp [67], a Mathematica implementation of the series-expansion method
[68]. The evaluation of all the 447 MIs involved in the calculation is
described in detail in [41, 42, 43]. For a number of points in the physical
phase space region we compared the values of the full set of MIs with results
obtained using the numerical package AMFlow [69, 70], always finding agreement
with the requested full precision (16 digits).

The bare two-loop $2\to 2$ amplitudes need UV renormalisation. We employ two
different schemes. In the first, we renormalise the external fields on-shell,
the strong coupling constant in a mixed scheme in which the light-flavour
contribution is renormalised in $\overline{\text{MS}}$ while the heavy quark
is renormalised at zero momentum, and the Yukawa coupling and the heavy-quark
mass in OS. In the second scheme, we renormalise both the Yukawa coupling and
the heavy-quark mass in $\overline{\text{MS}}$.

Once the UV divergences are renormalised, the two-loop $2\to 2$ amplitudes
still contain residual IR divergences that appear as poles in $1/\epsilon$.
The structure of these IR poles is universal and is described by factorization
formulae [51].

We examined the behaviour of the two-loop $2\to 2$ amplitudes in the soft and
collinear limits of one unresolved parton. To do so, we generated sequences of
phase space points tending to the desired limit and compared the value of the
amplitudes to that predicted by the corresponding one-loop IR factorization
formulae [71, 72, 73, 74, 75, 76]. We found that the amplitudes approach the
limit formulae with the rate expected from the cancellation of the leading
singularity order-by-order in $\epsilon$. We observed the expected behaviour
of the full amplitudes independently of the value of internal quark mass as
well as for the interference of two massive internal quarks.

As a further check, we verified that in the limit of very large transverse
momentum, the full amplitude is in reasonable agreement with the approximated
results computed in ref. [77].

##  IV The Cross Section

The results for the hadronic cross sections have been obtained by combining
the integration of the one-loop $2\to 2$ (Born) and $2\to 3$ (real)
amplitudes, including the subtractions and the initial-state collinear
remnants, with the integration of the two-loop $2\to 2$ (virtual) amplitudes.
To efficiently integrate the virtual part we have first produced a grid using
the virtual correction in the HEFT. To increase the precision in the
computation of the tail of the Higgs $p_{t}$ distribution, we have biased the
weight with an appropriate exponential factor and generated a second grid.
Subsequently we have used these grids to integrate the amplitudes in the full
theory. In particular for every choice of scales we have run about 14k points
on the cross section grid and 16k on the biased one. When we use the
$\overline{\rm{MS}}$ mass renormalisation, the value of the internal quark
masses changes dynamically in each phase space point so that no form factors
or integrals can be recycled from the evaluation of another point. The run
time of the evaluation of the two-loop matrix element for one kinematic
configuration and one choice of masses circulating in the loops varies greatly
depending on the phase-space point and the existing pre-computed points that
DiffExp can transport from. However, we can qualitatively report that it
ranges from 5 minutes to one hour, with a median around 15 minutes.

We implemented our two-loop amplitudes through a MadGraph5_aMC@NLO plugin,
similar to what was done in ref. [36], where the amplitude appeared as an
effective vertex in a UFO [78] model. In that representation, the form factors
are understood as dynamical couplings which will be re-evaluated for each
phase-space point by the tree-level Fortran matrix element code via an
interface to Mathematica. The main advantage of such an implementation is that
it greatly facilitates the distribution and reproducibility of our two-loop
amplitude as it can now readily be generated like any tree-level process in a
version of MadGraph5_aMC@NLO equipped with our plugin. It moreover offers
additional flexibility regarding the selection of the type of contribution the
user is interested in. This includes 1) the possibility of choosing between OS
and $\overline{\text{MS}}$ mass renormalisation schemes, 2) selecting
particular interference terms and which massive quark flavours are included in
each fermion loop, 3) potentially attaching Higgs decay structures to the
production process and finally 4) computing renormalisation scale variations
through a reweighting procedure necessitating only a single evaluation of the
form factors.

For the case of OS top-quark mass renormalisation, we validated this
implementation by aligning our setup with that chosen in refs. [28, 29] and
found great agreement both at the inclusive and differential level.***Note
that in the updated version [29] of ref. [28] on the arXiv, the authors report
a bug fix that affected the results of ref. [28] by about 13% on the total
cross section. We find agreement with the updated prediction [29].

##  V Results

In this section we show a selection of predictions obtained using our
computation. For our simulation we use the value $G_{F}=1.16639\cdot
10^{-5}\,$GeV-2 for the Fermi constant and the
$\rm{NNPDF40\\_nlo\\_as\\_01180}$ [79] PDFs set and $\alpha_{S}$. Jets are
reconstructed using the anti-kt algorithm [80] with resolution variable
$R=0.4$. We select events of proton-proton collisions at $\sqrt{s}=13\,$TeV
where the Higgs boson is produced in association with a jet with transverse
momentum larger than $p^{j_{1}}_{T}>20\,$GeV. The masses of the Higgs boson
and of the internal quarks are set to $m_{H}=125.25\,$GeV, and
$m^{\rm{OS}}_{t}=172.5\,$GeV when considering on-shell top mass
renormalisation, whereas we compute $m^{\overline{\rm{MS}}}_{t}(\mu_{R})$ from
$m^{\overline{\rm{MS}}}_{t}(m^{\overline{\rm{MS}}}_{t})=163.4\,$GeV in the
case of $\overline{\rm{MS}}$ top-quark mass renormalisation, using leading
logarithmic evolution. We also included the effect of having an internal
massive bottom quark attached to the Higgs boson. For a consistent use of the
decoupling scheme with five-flavour evolution of the strong coupling and PDFs
we have chosen to retain the bottom mass only in the loop where it is coupled
to the Higgs boson and set it to zero in the other pure QCD contributions. The
bottom-quark mass and the corresponding Yukawa coupling are computed evolving
$m^{\overline{\rm{MS}}}_{b}(\mu_{R})$ from
$m^{\overline{\rm{MS}}}_{b}(m^{\overline{\rm{MS}}}_{b})=4.18\,$GeV (also with
leading logarithmic accuracy). Our choice for the central renormalisation
($\mu_{R}$) and factorization ($\mu_{F}$) scales is

$\mu_{R}^{0}=\mu_{F}^{0}=\frac{H_{T}}{2}=\frac{1}{2}\left(\sqrt{m_{H}^{2}+p^{2}_{t,H}}+\sum_{i}|p_{t,i}|\right)\\!\,,$
(1)

with the sum running over all partons in the final state. As for the scale
variations, we take the envelope of the seven scale choices obtained by
varying the central scales by a factor of two in both directions excluding the
maximal distance.

Our results for the semi-inclusive Higgs production cross section within our
jet acceptance cut are given in Table 1,

| renormalisation of
---
internal masses
$\sigma_{\mathrm{LO}}$ [pb] |  $\sigma_{\mathrm{NLO}}$ [pb]
top+bottom–($\overline{\rm{MS}}$) | $12.318^{+4.711}_{-3.117}$ | $19.89(8)^{+2.84}_{-3.19}$
top–($\overline{\rm{MS}}$) | $12.538^{+4.822}_{-3.183}$ | $19.90(8)^{+2.66}_{-2.85}$
top–($\rm{OS}$) | $12.551^{+4.933}_{-3.244}$ | $20.22(8)^{+3.06}_{-3.09}$
Table 1: Cross section for Higgs boson production in association with a jet
with transverse momentum larger than $p^{j_{1}}_{T}>20\,$GeV, with top- and
bottom-quarks circulating in the heavy-quark loop, in the $\overline{\rm{MS}}$
scheme (upper line); with top-quark circulating in the loop, in the
$\overline{\rm{MS}}$ scheme (middle line) and in the OS scheme (lower line),
at LO and NLO accuracy.

where we note that the relative scale uncertainty goes down from about 30% at
LO to about 14% at NLO. Further, by comparing the first two lines, we see that
the top-bottom interference yields a negative contribution at LO, and instead
a positive contribution at NLO which cancels the offset between the cross
section with and without top-bottom interference.

The bottom quark is expected to affect the production rate only at small and
intermediate values of the Higgs $p_{T}$ distribution. Approximated results
for the effect of the bottom quark at intermediate $p_{T}$ values have been
previously reported in [34]. In fig. 1, we show the results of our exact
computation for the Higgs $p_{T}$ distribution in 20GeV-wide bins at
intermediate $p_{T}$ values, at LO (left panel) and NLO (right panel)
accuracy, with top- and bottom-quarks circulating in the heavy-quark loops in
$\overline{\rm{MS}}$ (black curve); with top-quark only, in
$\overline{\rm{MS}}$ (blue curve) and in OS (red curve). Vertical bars
represent the statistical error from the Monte Carlo integration. In the left
panel, the first bin is empty, since the LO kinematic contraint,
$p_{T}=p^{j}_{T}$, forces the Higgs $p_{T}$ to be not lower than 20GeV; in the
second bin, the contribution of the top-bottom interference is negative, and
fades away from the third on. In the right panel, the NLO contribution of the
top-bottom interference is negative in the first bin, positive in the second,
negative again, but smaller, in the third, and dies out from the fourth on.
Note that the seven-point scale variation (not shown) provides a much larger
uncertainty than the difference among the bars that are shown in the figure.
Also note that at LO, the impact of the change of top-quark mass
renormalisation scheme is almost indiscernible in the second bin of the left
panel. However, at NLO this impact is far greater, and overall 15 times larger
than at LO in our semi-inclusive cross-section of tab. 1. We leave further
investigation of this enhanced sensitivity to the mass and Yukawa
renormalisation scheme at NLO to future work.

While the scale uncertainty is expected to be reduced in a resummed
calculation of the Higgs $p_{T}$, we anticipate that the difference among the
different predictions will persist. Further investigation of the Higgs $p_{T}$
distribution in the low-energy limit is beyond the aim of the present work and
will be presented elsewhere.

Figure 1: Higgs $p_{T}$ distribution in the intermediate $p_{T}$ range. Figure
2: Higgs $p_{T}$ distribution with top- and bottom-quarks.

In fig. 2, we plot the Higgs $p_{T}$ distribution in 25GeV-wide bins, with
top- and bottom-quarks circulating in the heavy-quark loops in
$\overline{\rm{MS}}$, at LO (green curve) and NLO (red curve) accuracy. The
scale uncertainty bands at LO (yellow band) and NLO (purple band) accuracy are
obtained by taking the envelope of seven-point scale variations. Not to
clutter the plot, we refrain from showing the same distribution with the top-
quark only, either in $\overline{\rm{MS}}$ or in OS, opting for highlighting
their behaviour in the next figures.

Figure 3: NLO/LO ratio of the Higgs $p_{T}$ distribution.

In fig. 3, we plot the ratio of the Higgs $p_{T}$ distribution at NLO over the
same at LO in 50GeV-wide bins, with top- and bottom-quarks circulating in the
heavy-quark loops, in $\overline{\rm{MS}}$ (upper panel); with top-quark only,
in $\overline{\rm{MS}}$ (middle panel) and in OS (lower panel). The scale
uncertainty bands are given by the ratio of the bands at NLO accuracy over the
central value of the Higgs $p_{T}$ distribution at LO. The upper panel
corresponds to the ratio of the red and green curves of fig. 2. We note that
except for the first bin all ratios have a numerical value greater than or
equal 2. In particular, the curves of the upper and middle panels have a
similar, rather flat, shape with a numerical value which is larger than 2 on
most of the $p_{T}$ range, while the curve of the lower panel wiggles about
the value 2.

Figure 4: Ratio of Higgs $p_{T}$ distributions at LO.

In figs. 4 and 5, we plot the ratio of the Higgs $p_{T}$ distribution, with
top- and bottom-quarks circulating in the heavy-quark loops, over the
distribution with the top-quark only, both in $\overline{\rm{MS}}$ (upper
panel); the ratio of the distribution with the top-quark in OS over the one
with top- and bottom-quarks in $\overline{\rm{MS}}$ (middle panel); the ratio
of the distribution with the top-quark in OS over the one with the top-quark
in $\overline{\rm{MS}}$ (lower panel). The scale uncertainty bands as reported
in the y-labels are given by the ratio of the bands of the distributions over
their central values. In fig. 4 the distributions display 25GeV-wide bins and
have LO accuracy; in fig. 5 they display 50GeV-wide bins and have NLO
accuracy. Except for the very first bins, the ratio of the Higgs $p_{T}$
distribution, with top- and bottom-quarks, over the distribution with top-
quark only is flat and equals $1$ (upper panels of figs. 4 and 5). This
emphasises that within the scale uncertainty the contribution of the bottom
quark, and thus of the top-bottom interference, to the Higgs $p_{T}$
distribution is negligible, except at the low end of the $p_{T}$ range.

Since the central values of the ratios of the upper panels of figs. 4 and 5
equal $1$ over almost the whole $p_{T}$ range, the ratios of the middle and
lower panels are basically equal. Focusing on e.g. the lower panels of figs. 4
and 5, we note that the Higgs $p_{T}$ distribution with the top-quark only in
$\overline{\rm{MS}}$ falls off faster than the same distribution in OS, as
$p_{T}$ increases, the more so at LO than at NLO accuracy. This can be
understood by the fact that the top-quark mass has an OS fixed value, while
the running top-quark mass decreases as the $p_{T}$ values, and thus the
renormalisation and factorisation scales, eq. (1), increase. This is at the
origin of the difference between the upper/middle and the lower panels of fig.
3 at high $p_{T}$ values.

Figure 5: Ratio of Higgs $p_{T}$ distributions at NLO.

##  VI Conclusions

Building on two-loop amplitudes for Higgs + three partons [44], which are
valid for arbitrary quark masses circulating in the heavy-quark loops, we have
computed for the first time the NLO QCD corrections to the Higgs $p_{T}$
distribution in Higgs + jet production via gluon fusion, with top and bottom
quarks circulating in the heavy-quark loops. The exact mass dependence on the
top and bottom quarks has been included using, for the first time in this
context, a running mass renormalisation scheme, the $\overline{\rm{MS}}$
scheme. We have also provided predictions for the Higgs $p_{T}$ distribution
with only the top quark, in $\overline{\rm{MS}}$ and OS schemes.

We find that within the scale uncertainty the LO contribution of the bottom
quark, and thus of the top-bottom interference, to the Higgs boson production
is almost erased at inclusive level by the NLO corrections. On the other hand,
at the low end of the $p_{T}$ distribution, the interference induces a non
trivial change of shape. However, for precision studies on the high energy
tail and with the current attainable accuracy, the use of the Higgs $p_{T}$
distribution with only the top quark circulating in the heavy-quark loops is
fully justified. Finally, we find that the Higgs $p_{T}$ distribution with the
top-quark only in $\overline{\rm{MS}}$ falls off faster than the same
distribution in the OS scheme as $p_{T}$ increases. This would have an obvious
impact on any numerical study, requiring then that the choice of mass
renormalisation scheme be done with great care.

###  VI.1 Acknowledgements

We thank S. Catani, G. Luisoni, F. Maltoni and P. Nason for interesting
discussions and A. Schweitzer for his contribution in the earlier development
of the interface of multi-loop amplitudes with MadGraph5_aMC@NLO. H.F. has
received funding from the European Union’s Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie grant agreement No. 847523
‘INTERACTIONS’, and has been partially supported by a Carlsberg Foundation
Reintegration Fellowship. M.H. is supported by the European Research Council
under ERC-STG-804286 UNISCAMP. R.B. is partly supported by the italian
Ministero della Università e della Ricerca (MIUR) under grant PRIN 20172LNEEZ.

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