Higgs production in bottom-quark fusion to third order in the strong coupling

We present the inclusive cross section at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD for the production of a Higgs boson via bottom-quark fusion. We employ the five-flavour scheme, treating the bottom quark as a massless parton while retaining a non-vanishing Yukawa coupling to the Higgs boson. We find that the dependence of the hadronic cross section on the renormalisation and factorisation scales is substantially reduced. For judicious choices of the scales the perturbative expansion of the cross section shows a convergent behaviour. We present results for the N$^3$LO cross section at various collider energies. In comparison to the cross section obtained from the Santander-matching of the four and five-flavour schemes we predict a slightly higher cross section, though the two predictions are consistent within theoretical uncertainties.

With the discovery of the Higgs boson by the experiments at the Large Hadron Collider (LHC) at CERN [1,2] particle physics has entered a new era. The last missing degree of freedom of the Standard Model (SM) of particle physics has been established, making the SM a fully predictive theory without any free parameters. For the first time we can now directly measure the properties of a fundamental scalar boson. The remarkable performance of the LHC during Run II brings the Yukawa interactions -the interaction among fundamental fermions and the Higgs boson -within reach of being probed by the experiments. Clearly, exploring the Higgs sector and testing our understanding of the fundamental interactions of nature is one of the main tasks for the LHC.
Since in the SM the coupling strength of the Higgs boson to fermions is proportional to the fermion mass, prospects for coupling measurements are most promising for third-generation matter, i.e., bottom and top quarks and τ leptons. Several models of new physics, including the Minimal Supersymmetric Standard Model, predict enhanced couplings of the Higgs boson to bottom quarks. Consequently, studying the Yukawa coupling to bottom quarks is particularly interesting. Even though the decay of a Higgs boson into bottom quarks benefits from a large branching fraction, it is overwhelmed by QCD background and thus challenging to observe [3,4]. A viable alternative consists in probing the couplings of the Higgs boson to bottom quarks via the production of a Higgs through fusion of a bb pair.
There exist two formally equivalent descriptions of the cross section for the production of a Higgs boson from the fusion of bottom quarks. In the fourflavour scheme (4FS), the bottom quark is treated as a massive quark, which decouples from the evolution of the strong coupling constant and the parton density functions (PDFs) described by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation. At leading order (LO) in perturbative QCD in the 4FS bottom quarks are produced from gluon splitting. Since the bottom quarks are treated as massive, the gluon splitting is free of collinear divergences, but produces logarithms log m 2 b /Q 2 at every order in perturbation theory, where Q is a characteristic scale of the hard process. These logarithms may become large, thereby spoiling the convergence of the perturbative series and requiring resummation. In the five-flavour-scheme (5FS), instead, the bottom quark is treated as a massless parton and is included in the evolution of the strong coupling and the PDFs. The collinear logarithms are then de facto resummed into the PDF evolution, thereby avoiding the appearance of large logarithms order-by-order in perturbation theory. On the other hand, the inclusive cross section in the 5FS neglects power-suppressed terms of order m 2 b /Q 2 , which are automatically included in the 4FS where all mass effects are taken into account. While both schemes are formally equivalent non-perturbatively, the truncation of the perturbative series is necessary for practical calculations and introduces a scheme dependence of the cross section.
Tab. I displays representative Feynman diagrams contributing in the 4FS and 5FS respectively. At LO, the 5FS is described by a two-to-one process. The simple structure of the LO process makes it possible to compute the inclusive cross section to high orders in perturbation theory, and both next-to-leading order (NLO) [5,6] and next-to-next-to-leading order (NNLO) [7] results are available. Computations in the 4FS are technically more involved, due to the higher parton multiplicity and the fact that the mass of the bottom quark is treated exactly. Currently the inclusive cross section in the 4FS is only available through NLO [8][9][10].
It is known that perturbative results may differ substantially between both schemes, and various methods have been proposed to combine the 4FS and 5FS into a single prediction [11][12][13][14][15]. A direct comparison of the   NLO and NNLO results in the 4FS and 5FS respectively is, however, not straightforward, because they correspond to different orders in the perturbative expansion in the strong coupling constant (see Tab. I). A consistent comparison with the 4FS at NLO requires the knowledge of the inclusive cross section in the 5FS at next-to-nextto-next-to-leading order (N 3 LO) in the strong coupling.
In this paper we present for the first time the complete result for the inclusive cross section for Higgs production in bottom-quark fusion at N 3 LO in the 5FS and investigate its phenomenological implications.

THE N 3 LO CROSS SECTION IN THE 5FS
The inclusive hadronic cross section for the production of a Higgs boson can be written as where the sum runs over all parton flavours, f i are parton densities andσ ij are partonic cross sections. The partonic cross sections depend on the ratio z = m 2 H /s, where √ s is the partonic centre-of-mass energy, related to the hadronic centre-of-mass energy √ S by s = x 1 x 2 S through the two Bjorken momentum fractions x 1,2 . µ r and µ f denote the renormalisation and factorisation scales respectively. We work in the 5FS with five massless quark flavours. We assume that the Higgs boson has a non-zero Yukawa coupling y b to the bottom quark, and we neglect couplings of the Higgs boson to all other quark flavours. This implies that we focus on terms in the cross section proportional to y 2 b . The partonic cross sectionsσ ij are expanded through N 3 LO in the strong coupling α s . The complete set of initial state configurations that contribute to the cross section through N 3 LO are shown in Tab. I. The NLO and NNLO corrections to the cross section have been computed in Refs. [5][6][7]. In the remainder of this paper we present for the first time N 3 LO corrections.
In order to compute the partonic cross sections at N 3 LO, we follow the same steps that have been employed in the computation of the N 3 LO corrections to Higgs production in gluon fusion in Refs. [16][17][18]. We have generated all relevant Feynman diagrams with QGraf [19]. Individual Feynamn diagrams are sorted into scalar integral topologies, which are then reduced to a set of master integrals via integration-by-parts identities [20,21] using an in-house code. Finally, the master integrals are computed analytically using the differential equations method [22][23][24][25][26]. All the relevant master integrals are known analytically as a function of z and have been evaluated in the context of the N 3 LO corrections to the gluon-fusion cross section. In particular at N 3 LO three-loop corrections to the Born process contribute, which have been computed for the first time in Ref. [27] using the master integrals computed in Refs. [28][29][30][31][32][33][34] and the results of Ref. [27] agree with our computation. In addition, the N 3 LO cross section receives contributions from partonic subprocesses involving fewer loops but additional real emissions in the final state. Single-real emission contributions from two-loop and squared one-loop diagrams have been considered in Ref. [35][36][37][38][39][40]. The master integrals for double-real virtual and triple-real contributions have been computed in Refs. [16,[41][42][43][44][45] as an expansion around the production threshold of the Higgs boson and exactly as a function of z in Ref. [18]. Here we work exclusively with the master integrals of Ref. [18].
Contributions from different initial states and/or parton multiplicities are individually ultraviolet (UV) and infrared (IR) divergent. We regulate the divergences by working in dimensional regularisation in D = 4 − 2 dimensions. UV divergences can be cancelled by replacing both the bare strong and Yukawa couplings by their renormalised values in the MS-scheme. The UVcounterterm for the strong coupling constant has been de- termined through five loops in Refs. [46][47][48][49][50]. The renormalisation constant for the Yukawa coupling is identical to the quark mass renormalisation constant of QCD in the MS-scheme [7,48,[51][52][53]. IR divergences are absorbed into the definition of the PDFs using mass factorisation at N 3 LO [54][55][56]. The mass factorisation involves convoluting lower-order partonic cross sections with the three-loop splitting functions of Refs. [57][58][59]. We have computed all the convolutions analytically in z space using the PolyLogTools package [60]. We observe that all divergences cancel after UV renormalisation and mass factorisation. We emphasise that this is not only a strong cross check of our result, but, together with the results of Ref. [16] for gluon-initiated processes, this is the first time that the complete set of three-loop splitting functions of Refs. [57,58] has been confirmed by an independent analytic computation. Moreover, this is the first time that the universality of QCD factorisation has been confirmed for hadron collisions for all partonic initial states.
The analytic cancellation of all ultraviolet and infrared singularities provides a strong check of our results. In addition, we have reproduced the soft-virtual N 3 LO cross section of Ref. [61] and the physical kernel constraints of Ref. [62][63][64] for the next-to-soft term of the bottomquark-initiated cross section. We have also checked that all logarithmic terms in the renormalisation and factorisation scales produced from the cancellation of the UV and IR poles satisfy the DGLAP evolution equation. Fi-nally, we have also recomputed the NLO and NNLO cross sections, and we have checked that through NNLO our results are in perfect agreement with the literature results implemented in the code Sushi [65].

BOTTOM-QUARK FUSION AT N 3 LO IN QCD
In this section we present our phenomenological results for inclusive cross section for bottom-quark fusion at N 3 LO in QCD. We assume a Higgs mass of m H = 125.09 GeV. The strong coupling is α s (m 2 Z ) = 0.118 and is evolved to the renormalisation scale µ r using the fourloop QCD beta function in the MS-scheme assuming five massless quark flavours. The Yukawa coupling between the Higgs boson and the bottom quark is proportional to the bottom-quark mass in the MS-scheme, and we evolve it from m b (m b ) = 4.18 GeV [66] to the same renormalisation scale µ r using four-loop running [53]. The band is obtained by varying µ r and µ f independently within the intervals µ r ∈ [m h , 2m h ] and µ f ∈ [m h /8, m h /2] with the restriction that 1/2 ≤ 4µ f /µ r ≤ 2. We observe that cross section predictions based on successive perturbative orders are contained within the bands of the lower order predictions over a wide range of hadronic centre of mass energies. The dependence on the renormalisation and factorisation scales of the hadronic cross section is reduced as the perturbative order is increased. We therefore believe that the residual scale dependence provides a reliable estimate of the missing higher orders beyond N 3 LO. Let us comment on our choice for the relatively small value of the factorisation scale µ f . This choice is motivated by the presence of a small scale m b in the problem and is consistent with previous choices in the literature [7,65,69,70]. We observe that for higher values of µ f , the convergence behaviour of the cross section with the perturbative order deteriorates. In particular, if we choose µ f = µ r = m H , the scale variation bands at NNLO and N 3 LO do not overlap. We therefore conclude that such higher choices of the factorisation scale lead to worse convergence of the cross section, and they should therefore be discarded.
Tab. II presents results for the inclusive cross section for a proton-collider for various hadronic centre-of-mass energies, and we show the main uncertainties affecting the cross section. To obtain central values and PDF uncertainties we use the Monte-Carlo replica method following the PDF4LHC recommendation [67]. All other uncertainties are computed with a fixed PDF set, namely the zeroth member of the PDF4LHC15 nnlo mc set. The QCD scale uncertainty is estimated by varying µ r and µ f as described in the previous paragraph. We also include an uncertainty reflecting the fact that currently there are no N 3 LO PDF sets available. The estimate of this uncertainty was obtained following the recipe introduced in Ref. [17]. The bottom quark mass is affected by an uncertainty of +0.04 −0.03 GeV [66] and we display the resulting uncertainty of our cross section. A more detailed discussion of the uncertainties, including a more comprehensive study of the impact of different PDF sets, will be presented in Ref. [68].

COMMENT ON THRESHOLD EXPANSIONS
Let us comment on the value of the N 3 LO cross section and how it compares to results obtained by approximations based on incomplete calculations at N 3 LO. In particular, the contribution to the cross section when the Higgs boson is produced at threshold and all QCD radiation is soft corresponds to the limit z → 1. In this limit, the cross section develops large logarithms, which can be resummed to all orders. In Ref. [61] the universal structure of the threshold logarithms was combined with the three-loop form factor of Ref. [27] and the constant term for Higgs production at threshold at N 3 LO of Ref. [41] to obtain the analytic result for the partonic cross section at N 3 LO at threshold (at threshold only the bb channel contributes), and predictions for the inclusive cross section were presented. We can use our exact result for cross section at N 3 LO and confront it with the threshold approximation. Fig. 2 shows a comparison of the expansion of the partonic cross sections around the production threshold of the Higgs boson to our exact result after convolution with the PDFs. To this end we performed a series expansion of the partonic cross section around z = 1 and we study the convergence of the threshold expansion in Fig. 2. We observe that, while the series stabilises after several terms, the value of the cross section varies widely if only a few terms are included. While it would be possible to influence the convergence of the threshold expansion by keeping unexpanded some factors that dominate in the limit z → 0, we find it hard to believe that any trustworthy and precise prediction of the N 3 LO cross section can be achieved based solely on the knowledge of a few terms in the threshold expansion, let alone of only the value of the cross section at threshold. This confirms earlier findings in the context of Higgs production in gluon fusion, where a similar convergence behaviour of the threshold expansion at N 3 LO was observed [16,17,71]. Based on these observations, we therefore believe that approximate cross sections at N 3 LO based only on the knowledge of a few terms in the threshold expansion are dangerous and may lead to wrong conclusions about the actual size of the N 3 LO corrections.

COMPARISON TO THE SANTANDER-MATCHING
While the 5FS resums collinear logarithms to all orders, it neglects power-suppressed terms of the order (m b /Q) 2 , where Q is a characteristic hard scale of the process. For the inclusive cross section typically Q ∼ m H , so that (m b /Q) 2 ∼ 10 −3 , and the 5FS is expected to give reliable predictions. Nevertheless, it has been observed that results in the 5FS at NNLO and in the 4FS at NLO may differ substantially. However, this naive comparison is not satisfactory, because results in the 5FS at NNLO and in the 4FS at NLO correspond to different orders in perturbation theory. In particular, the 4FS at NLO includes terms proportional to α 3 s which are not captured by the 5FS at NNLO (see Tab. I). For this reason various methods have been proposed to combine the 4FS and 5FS into a single prediction [11][12][13][14][15]. In this section we present for the first time a comparison of the two schemes that includes partonic cross sections consistently computed through α 3 s . We focus on the comparison to the so-called Santandermatching (S-M) scheme [11], which is a pragmatic way of combining the 4FS and 5FS computations into a single prediction through a weighted average: where the weighting factor is w = log m H m b − 2.  [10,14,65]. We see that the N 3 LO corrections lower the value of the cross section in the 5FS, bringing it closer to the results in the S-M scheme. Moreover, we see that our N 3 LO results and the S-M results agree within scale and PDF uncertainties. Since the S-M scheme is a pragmatic, but very ad hoc, prescription, we do not see any compelling argument to work with this scheme, given that now the 4FS and 5FS are known to the same order in the strong coupling constant and that powersuppressed terms in the 4FS are expected to give small contributions to the inclusive cross section.

CONCLUSIONS
We have presented for the first time the complete computation of the inclusive cross section for the production of a Higgs boson through bottom-quark fusion at N 3 LO. We observe a substantial reduction of the residual scale uncertainty and a good convergence of the perturbative series, provided that the factorisation scale is set to a small value around m H /4, in agreement with previous studies. We have also compared the value of the inclusive cross section at N 3 LO in the 5FS to the S-M of Ref. [72]. We find that the two results agree within uncertainties. Since finite power-suppressed effects due to the bottom-mass are expected to give only small contributions to the cross section, we believe that our result in the 5FS provides the most reliable prediction of the inclusive cross section of Higgs production to date, including consistently all higher-order QCD effects through α 3 s .