# NNLO+PS predictions for Higgs production via bottom annihilation

Christian Biello  Max-Planck Institute for Physics,
Boltzmannstr. 8, 85748 Garching, Germany

###### Abstract

We report on the implementation of a new NNLO+PS event generator for the Higgs
production via bottom annihilation [1] , using the MiNNLOPS method in the
Powheg framework. The calculation has been carried out in the five flavour
scheme (5FS), where the bottom mass is neglected. We compare our results
against fixed-order predictions at NNLO as well as resummed predictions at
next-to-next-to-leading-logarithmic (NNLL) accuracy. We also present a
preliminary study in the four flavour scheme (4FS) setup, achieving a new
level of precision in the massive scheme.

MPP-2024-101

##  1 Introduction

Accurate simulation of $b\bar{b}H$ production at the LHC is crucial for
identifying deviations from the Standard Model predictions and distinguishing
between new physics signals and background noise. This mode directly probes
the bottom Yukawa coupling and serves as the primary background for studying
double Higgs production, enhancing the LHC’s potential to uncover new
phenomena and deepen our understanding of particle physics.

In particular, Higgs production via bottom fusion is theoretically intriguing
for exploring different flavor schemes. In the five-flavor scheme (5FS), the
bottom quark is treated as massless, allowing for the resummation of collinear
logarithmic terms through DGLAP evolution. However, this approach is not able
to capture power corrections related to the bottom mass, leading to less
accurate descriptions of bottom kinematic distributions. Alternatively, the
decoupling scheme treats bottoms as massive quarks within a four-active-flavor
setup (4FS), explicitly considering mass effects but without resumming large
logarithms. Total cross-sections are currently known from fixed-order
calculations: N3LO QCD in 5FS [2] and NLO QCD in 4FS [3].

Combining NNLO QCD calculations with parton showers (NNLO+PS) is a key
challenge in collider theory. It’s crucial for linking precise theory
predictions with accurate experimental measurements. For $b\bar{b}H$
production, the first matching was performed for NLO predictions in 4FS using
the MC@NLO [4] and Powheg [5] method. Here we focus on the first fully-
differential calculation of NNLO QCD matched to parton shower using the
MiNNLOPS method.

##  2 The method

Powheg [6] is a fully tested method for the matching of NLO predictions with
parton shower which avoids the introduction of an unphysical matching scales.
Powheg generates the hardest emission which alone gives the correct NLO result
with an appropriate Sudakov form factor and interfaces the events with a
Shower Monte Carlo for the subsequent radiations. The starting point of our
calculation is the Higgs plus one jet production described by the Powheg
formula,

$\mathrm{d}\sigma_{\text{HJ}}=\mathrm{d}\Phi_{\text{HJ}}\,\bar{B}^{\text{{Powheg}}}\,\left\\{\Delta_{\text{pwg}}(\Lambda_{\text{pwg}})+\mathrm{d}\Phi_{\text{rad}}\Delta_{\text{pwg}}(p_{\text{T,rad}})\frac{R_{\text{HJ}}}{B_{\text{HJ}}}\right\\},$
(1)

where $\mathrm{d}\Phi_{\text{HJ}}$ is the HJ phase space,
$\Delta_{\text{pwg}}$ is the Powheg Sudakov form factor with a cutoff
$\Lambda_{\text{pwg}}=0.89$ GeV. In addition, $\mathrm{d}\Phi_{\text{rad}}$
and $p_{\text{T,rad}}$ are the phase space measure and the transverse momentum
of the radiation with respect to HJ production.

The MiNLO′ procedure [7] improved NLO multijet calculations with the
appropriate choice of scales and with the inclusion of a Sudakov form factor:
this avoids an unphysical merging scale. MiNNLOPS empowers the MiNLO′ master
formula with other ingredients from the color singlet transverse momentum
resummation. The $\bar{B}$ function thus reads:

$\bar{B}^{\text{{{MiNNLO${}_{\rm
PS}$}}}}=e^{-\tilde{S}({p_{\text{T,$H$}}})}\left\\{B\left(1+\tilde{S}^{(1)}\right)+V+\int\mathrm{d}\Phi_{\text{rad}}R+D^{(\geq
3)}({p_{\text{T,$H$}}})F^{\text{corr}}\right\\},$ (2)

where the Born $B$, virtual $V$ and real $R$ contributions are evaluated using
${p_{\text{T,$H$}}}$ as the renormalisation scale for the strong couplings.
Here $\tilde{S}({p_{\text{T,$H$}}})$ is the MiNLO′ Sudakov form factor and
$\tilde{S}^{(1)}$ is its first order coefficient. The extension of MiNNLOPS
compared to MiNLO′ is encoded in the $D^{(\geq 3)}$ terms multiplied by a
spreading function $F^{\text{corr}}$. The $D^{(\geq 3)}$ terms contain all the
required ingredients to obtain the NNLO accuracy for inclusive observables.

##  3 Results in 5FS

We provide numerical predictions for Higgs production via bottom-quark
annihilation at the LHC, for 13 TeV center-of-mass energy. The mass of the
Higgs boson is set to 125 GeV and its width to zero. Using the 5FS with
massless bottom quarks, we employ a non-zero Yukawa coupling in
${\overline{\rm MS}}$ scheme, derived from $m_{b}(m_{b})=4.18$ GeV and evolved
via four-loop running. Our default setup uses the NNLO set of NNPDF40 parton
densities with 5 active flavors and $\alpha_{s}(m_{Z})=0.118$ [9]. Theory
scale uncertainty is assessed with a standard 7-point scale variation. For
predictions matched to a parton shower (PY8), we use Pythia8 with the A14
tune. We maintain Higgs boson stability and disable effects such as
hadronization, multi-parton interactions (MPI), and QED radiation.

Using the MiNNLOPS method, we obtain a total inclusive cross section of
$0.509(8)_{-5.3\%}^{+2.9\%}$ pb. Such value is in agreement with the fixed-
order result of $0.518(2)_{-7.5\%}^{+7.2\%}$ pb from SusHi [10]. MiNLO′
predicts a higher value of $0.571(1)_{-22.7\%}^{+17.4\%}$ pb for the fully
inclusive total cross-section, while it is compatible with the MiNNLOPS result
for high values of ${p_{\text{T,$H$}}}$ (see Figure 2), where both generators
effectively provide NLO accuracy. We validate our MiNNLOPS generator against
the fixed-order calculation [11]. From Figure 2, it is clear that our
predictions are agreement with the fixed-order predictions in the high
$p_{\text{T,H}}$ regime. However, the fixed-order calculation produces a
divergence for vanishing $p_{\text{T,H}}$, while the MiNNLOPS prediction
remains finite.

Figure 1: Higgs transverse spectrum from MiNLO′ and MiNNLOPS predictions with
the Shower.

Figure 2: Comparison of the fixed-order NNLO prediction with MiNNLOPS at LHE
level.

We further compare the MiNNLOPS results with the analytic resummation at
NNLO+NNLL [12] in Figure 3. Here we provide the MiNNLOPS predictions at the
LHE level in the left plot while the right one shows the results after
showering with Pythia8. The agreement is acceptable for small value of the
transverse momentum. In the high $p_{\text{T}}$ region, the resummed results
closely match with NNLO+NNLL before the shower. This agreement surpass the
fixed-order comparison (Figure 2), highlighting the impact of resummation. As
observed in the second plot of Figure 3, the inclusion of the parton shower
clearly affects high transverse momenta, leading to an approximate 10%
increase in the MiNNLOPS prediction. However, this modification remains
contained within the scale-uncertainty bands.

Figure 3: Predictions for the Higgs transverse spectrum from the analytic
resummation at NNLO+NNLL compared with our MiNNLOPS generator before (left)
and after (right) the Parton Shower.

##  4 Preliminary studies in 4FS

The 4FS implementation of the MiNNLOPS generator [13] is more complex due to
the higher multiplicity and the presence of massive colored final states. We
present for the first time some preliminary results at the integrated level.
We adapted the MiNNLOPS method for heavy-quark pair production plus a color
singlet for $b\bar{b}H$ production by taking into account the Yukawa scale
dependence. The two-loop amplitude for massive bottom quarks is not known, but
we can perform an approximation in order to capture the amplitude up to power
corrections in the bottom mass. Starting from the amplitude with massless
bottom quarks [14], we apply the massification procedure [15]. We use a
generalised approach which includes the leading mass effects of the bottom
quarks in the loops [16].

By choosing the Higgs mass for the Yukawa scale and a quarter of the
transverse invariant mass for the Born strong couplings, the NLO+PS generator
gives a cross-section of $0.381(2)_{-16\%}^{+20\%}$ pb. This value is about
$55\%$ smaller than the NLO 5FS prediction. The theoretical tension between
the two schemes shows the importance of the collinear logarithmic
contributions. Conversely, MiNNLOPS predicts $0.464(9)_{-13\%}^{+14\%}$ pb
which is closer to the NNLO 5FS values. Indeed, for the first time, we are
observing an agreement of the predictions in the two schemes within the scale
uncertainty.

##  5 Conclusion

We presented the first matching of NNLO predictions with parton shower for the
Higgs production via bottom fusion using the MiNNLOPS method. We discussed the
predictions in the 5FS and introduced the current studies in the 4FS with
remarkable preliminary results.

## Acknowledgments

I am indebted to Chiara Savoini and Simone Zoia for useful conversations and
checks during the conference regarding the 4FS calculation. I express my
gratitude to the organisers of the Rencontres de Moriond for giving me the
opportunity to partecipate with a grant.

## References

## References

  * [1] C. Biello, A. Sankar, M. Wiesemann, G. Zanderighi, Eur.Phys.J.C 84 (2024) 5, 479
  * [2] C. Duhr, F. Dulat, B. Mistlberger, Phys.Rev.Lett. 125 (2020) 5, 051804
  * [3] S. Dittmaier, M. Krämer, M. Spira, Phys.Rev.D 70 (2004) 074010
  * [4] M. Wiesemann, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, P. Torrielli, JHEP 02 (2015) 132
  * [5] B. Jager, L. Reina, D. Wackeroth, Phys.Rev.D 93 (2016) 1, 014030
  * [6] P. Nason, JHEP 11 (2004) 040
  * [7] K. Hamilton, P. Nason, C. Oleari, G. Zanderighi, JHEP 05 (2013) 082
  * [8] P.F. Monni, P. Nason, E. Re, M. Wiesemann, G. Zanderighi, JHEP 05 (2020) 143
  * [9] NNPDF collaboration, Eur. Phys. J. C 82 (2022) 428
  * [10] R.V. Harlander, S. Liebler and H. Mantler, Comput. Phys. Commun. 184 (2013) 1605
  * [11] R.V. Harlander, A. Tripathi, M. Wiesemann, Phys.Lett.B 693 (2010) 269-273
  * [12] R.V. Harlander, A. Tripathi, M. Wiesemann, Phys.Rev.D 90 (2014) 1, 015017
  * [13] C. Biello, J. Mazzitelli, A. Sankar, M. Wiesemann, G. Zanderighi, in preparation
  * [14] S. Badger, H.B. Hartanto, J. Krys and S. Zoia, JHEP 11 (2021) 012
  * [15] A. Mitov, S. Moch, JHEP 05 (2007) 001
  * [16] G. Wang, T. Xia, L.L. Yang, X. Ye, JHEP 05 (2024) 082