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We present a computation of the next-to-next-to-leading-order (NNLO) QCD corrections to the production of a Higgs boson in association with a

Production of the Higgs boson in association with the

The importance of associated

Fully differential NNLO QCD results for associated

The purpose of this paper is to repeat the computation of Ref.

Second, the type of distributions for which large QCD corrections were found in Ref.

Finally, we perform the computation using the local subtraction scheme described recently in Ref.

The remainder of the paper is organized as follows. In Sec.

The goal of this section is to review the subtraction scheme for NNLO QCD calculations

We note that the NNLO QCD corrections to inclusive

At next-to-leading order, an understanding of how to do this in full generality using both slicing and subtraction methods was achieved more than twenty years ago

One of these methods, the so-called sector improved residue-subtraction scheme, was developed in Refs.

We will illustrate the main idea of Ref.

Our goal is to extract singularities from Eq.

We need to extract all these singularities in an unambiguous way. We begin with soft singularities. We write

Here, we define the double-soft limit as

We deal with the two terms on the right-hand side of Eq.

The procedure continues with collinear subtractions that are again applied to the terms on the right-hand side in Eq.

There are two major ingredients to this phase space splitting. First, we partition the phase space into two double-collinear partitions and two triple-collinear partitions. In the two double-collinear partitions, the gluons can only have singularities if

The contributions of the double-collinear partitions can be computed right away since all singular limits are uniquely established. The situation is more complex for the triple-collinear partitioning where this is not the case. Indeed, in triple-collinear configurations we need to consider the two cases of the gluons being either close or well-separated in rapidity. To this end, we further partition the phase space into four sectors. Taking as an example the

A detailed discussion of this approach can be found in Ref.

We extend the computation of Ref.

We compute all the channels with an additional quark-antiquark pair in the final state

We compute the contribution of the

We include the NNLO contributions to the associated production

In this section, we present results of the computation of the NNLO QCD corrections to the process

The Higgs and the

We employ dynamic renormalization and factorization scales that we take to be proportional to the invariant mass of the

We report our results for

Results for

As the next step, we study the NNLO corrections in more detail, focusing on the case of

Results for

This is achieved with a runtime of approximately 500 CPU hours. This degree of precision is unnecessary when considering

NNLO QCD corrections to kinematic distributions can also be computed with a high degree of numerical stability. In Fig.

Results for the rapidity and the transverse momentum distributions of the Higgs boson. Upper panes—results in consecutive orders of perturbation theory. Lower panes—ratios of NLO to LO and NNLO to NLO. The renormalization and factorization scales are set to

Results for rapidity and transverse momentum distributions of the charged lepton from the decay of a

In this section, we discuss a fully exclusive computation of NNLO QCD corrections to Higgs boson decay to a

We consider the Standard Model Lagrangian, integrate out the top quark and neglect the interaction of the Higgs boson with quarks of the first two generations. Interactions of the Higgs boson with hadronic constituents are then described by an effective Lagrangian

The two constants

The computation of NNLO QCD corrections to Higgs boson decay to two

In this subsection, we compute NNLO QCD corrections to the decay

First, as we already mentioned, we work in the approximation of massless

Second, integrals of the double-soft eikonal factors are identical to the production case and can be re-used in the

They are functions of a momentum fraction in the production case and just numbers in the decay case.

Third, it turns out that the calculation of

The last point concerns the contribution of the

We continue by presenting some numerical results of the calculation. Again, our goal in this section is not to discuss phenomenology of the Higgs boson decay to a

The numerical computation yields the following result for the decay rate of the Higgs boson to a

It is instructive to compare Eq.

It is also interesting to compute jet rates in

Following Ref.

We mentioned above that a nonvanishing Wilson coefficient

Some of these contributions are shown in Fig.

Illustrative interference diagrams that contribute to the

These contributions are soft and collinear finite for

However, since the calculation in the previous subsection was performed with massless

The validity of the massless approximation assumes that the logarithmic dependence on the

It is clear that a proper description of the interference contributions requires a computation with fully massive

We are now in a position to discuss the physical process

To define an expansion of Eq.

Using this notation, we define the physical cross sections computed through different orders in QCD perturbation theory

We are now in a position to discuss the results of the computation. To define the

We are grateful to G. Salam for providing us with his private implementation of the algorithm

Finally, following the experimental analyses, we may impose an additional requirement that the vector boson has a transverse momentum

We begin by presenting the fiducial volume cross sections for the process

The results for the fiducial cross sections in Eqs.

We now turn to differential distributions. We begin by identifying the

The invariant mass of a

To understand what causes these large effects, we split the difference between approximate NNLO and full NNLO into two terms—NNLO radiation in the decay (

The invariant mass of a

Next, we consider the transverse momentum of the

Same as Fig.

To understand the relative impact of different contributions, we again split the full NNLO into two different parts,

The different contributions to the distribution of the sum of transverse momenta of the

It is also interesting to study the angular separation

The

Another distribution that is subject to large modifications if the cut on the vector boson transverse momentum is applied is the transverse momentum distribution of the hardest

The transverse momentum of the hardest

As the last example, we show in Fig.

The transverse momentum distribution of the charged lepton. Left pane—without the

The goal of this section is to compare fixed-order QCD predictions for

As we have seen in the previous section, radiative corrections to kinematic distributions in the

Moreover, some of these regions, e.g.

As in the previous section, we study the

Comparison of fixed-order and parton shower predictions for the normalized invariant mass distribution of the two

Same as Fig.

Same as Fig.

For the

Turning to the

Next, in Fig.

Given the different jet algorithms used in the fixed-order and parton shower calculations, it is interesting to investigate to what extent the details of the jet definition affect these results. In Figs.

The invariant mass of a

Same as Fig.

In this paper, we presented a computation of the NNLO QCD corrections to the associated production of the Higgs boson

We pointed out an interesting contribution to Higgs decay to

We found a number of kinematic distributions in the

We compared fixed-order predictions for the

We would like to thank the Munich Institute for Astronomy and Particle Physics (MIAPP) for hospitality and partial support during the programs