# Top-bottom interference effects in Higgs plus jet production at the LHC

###### Abstract

We compute next-to-leading order QCD corrections to the top-bottom interference contribution to production at the LHC. To achieve this, we combine the recent computation of the two-loop amplitudes for and , performed in the approximation of a small -quark mass, and the numerical calculation of the squared one-loop amplitudes for and performed within OpenLoops. We find that QCD corrections to the interference are large and similar to the QCD corrections to the top-mediated Higgs production cross section. We also observe a significant reduction in the mass-renormalization scheme uncertainty, once the NLO QCD prediction for the interference is employed.

^{†}

^{†}preprint: IPPP/17/20, TTP17-012

Detailed exploration of the Higgs boson properties is a major part of the physics program at the Large Hadron Collider (LHC). It is hoped that studies of the Higgs couplings will reveal possible physics beyond the Standard Model, especially if it mostly manifests itself through interactions with the Higgs bosons. The goal, therefore, is to precisely measure Higgs boson couplings to various particles in the Standard Model and to search for small deviations. For example, assuming that the energy scale of New Physics is close to , generic modifications of the Higgs couplings are expected at the level of , where we used for the Higgs field vacuum expectation value. A variety of explicit BSM scenarios conforms with these expectations Gupta:2013zza (), suggesting that achieving a few percent precision in studies of the Higgs couplings may indeed provide interesting information about physics beyond the Standard Model.

Compared to these theoretical goals, existing measurements of the Higgs couplings leave much to be desired Khachatryan:2016vau (). Currently, Higgs couplings to electroweak gauge bosons are known to a precision between ten and twenty percent and Higgs couplings to third generation fermions to about hundred percent. The Higgs boson couplings to first and second generation fermions are practically unconstrained. It is expected that the situation will dramatically improve with the continued operation of the LHC. For example, it is estimated CMS:2013xfa () that, by the end of the high-luminosity phase, the Higgs couplings, that can be extracted from its major production and decay channels, will be determined with a few percent precision. There are several unknowns that may affect the validity of these projections, including progress in reducing the uncertainties in theoretical predictions and the ability of experimentalists to come up with new ideas but, barring revolutionary breakthroughs, these estimates give us a ballpark of what can be expected.

Determination of Higgs couplings at the LHC requires theoretical predictions for relevant processes, including both signal and background. A case in point is the Higgs boson transverse momentum distribution, whose theoretical understanding is important to properly describe the kinematics of the Higgs decay products, but may also give us access to physics beyond the Standard Model Grazzini:2016paz ().

Higgs bosons at the LHC are mostly produced in gluon collisions. If additional
gluons are radiated, a Higgs boson recoils against them; this mechanisms
leads to a non-trivial Higgs -spectrum whose theoretical
description requires good understanding of QCD dynamics.
In the case of a point-like coupling of the Higgs to gluons,
perturbative QCD (pQCD) provides an established framework to describe the Higgs
-spectrum, including
fixed order QCD computations recently extended to
next-to-next-to-leading order (NNLO)
nnlohjet () and the resummation computations known in the
next-to-next-to-leading logarithmic (NNLL) approximation
resum1 ().^{1}^{1}1 A related
topic of jet-veto resummation in Higgs production is discussed
in Refs. resum2 ()
However, the Higgs coupling to gluons in the
Standard Model is the result of a quantum process where gluons fluctuate into a quark-antiquark
pair that annihilates into a Higgs boson.
Because of the differences in fermion Yukawa couplings,
the largest contribution to the coupling in the Standard Model comes
from top quark loops, followed by contributions of bottom and charm quarks.
For values of the Higgs transverse momentum , the top loop
contribution to the coupling can be considered point-like to a very good approximation
and we can apply the full
power of pQCD to describe it with high precision.
However, the bottom and charm loops are not point-like for moderate values
of the transverse momentum
and both the perturbative behavior and
the possibility to perform resummations are much less understood for
these contributions to the effective coupling.

Moreover, it is known that the bottom and charm quark contributions to amplitudes
develop a peculiar, Sudakov-like dependence on the Higgs boson transverse momentum Ellis:1987xu (); bcontrib (). Taking
the bottom quark contribution as an example, we find
Banfi:2013eda (). These double logarithms are not accounted
for in the standard resummation framework^{2}^{2}2See Refs.logs ()
for recent attempts to understand the origin of these logarithms and
the possibility to resum them. Grazzini:2013mca () and they significantly
enhance the contribution of bottom loops to the Higgs production cross section in gluon
fusion, compared to naive expectations. In fact, the bottom loop contribution
to Higgs production in the Standard Model is estimated to be close to minus five percent Anastasiou:2016cez ()
and, therefore, significant on the scale of precision goal discussed
earlier.

It is interesting to remark that the “substructure” of the coupling is precisely what makes the Higgs transverse momentum distribution an interesting observable from the point of view of physics beyond the Standard Model. For example, current constraints on the charm Yukawa coupling are weak but, if the charm Yukawa coupling deviates significantly from its Standard Model value, the charm contribution to increases, and the relevance of the annihilation channel for Higgs production grows. These modifications may result in observable effects in the Higgs transverse momentum distribution. It was pointed out in Ref. Bishara:2016jga () that studies of the Higgs boson transverse momentum distribution lead to very competitive constraints on the charm Yukawa coupling; for example, it is expected Bishara:2016jga () that at high-luminosity LHC, the charm Yukawa coupling can be constrained to lie in the interval at the confidence level. Although not quite relevant for this paper, we also note that at very high values of the transverse momentum , the contribution of top quark loops can be resolved; this allows to probe for a point-like component of the coupling that may originate from physics beyond the Standard Model.

This discussion suggests that the shape of the Higgs boson transverse momentum distribution, from moderate to high -values, is important for a proper description of the kinematic features of Higgs bosons produced at the LHC and, also, may provide important information about physics beyond the Standard Model. Accurate Standard Model predictions for this observable are key for achieving these goals. As we already mentioned, the pQCD description of the Higgs boson transverse momentum distribution, in the approximation of the point-like coupling, is rather advanced, see Refs. nnlohjet (); Monni:2016ktx (), but there is very little understanding of how its not-point-like component is affected by QCD radiative corrections. To clarify this issue, we report on the computation of QCD radiative corrections to top-bottom interference contribution to Higgs boson production at the LHC in this Letter.

The calculation of the NLO QCD corrections to the top-bottom interference is non-trivial and we briefly summarize its salient details. The leading order production of the Higgs boson with non-vanishing transverse momentum occurs in different partonic channels, namely , , and . At leading order these processes are mediated by top or bottom loops (the charm contribution in the SM is negligible). The one-loop amplitudes are known exactly as functions of external kinematic variables and the quark masses Ellis:1987xu ().

At NLO, the production cross section receives contributions from real and virtual corrections. Since the leading order process only occurs at one-loop, the virtual corrections require two-loop computations that include planar and non-planar box diagrams with internal masses. The computation of such Feynman diagrams is a matter of active current research that includes attempts to develop efficient numerical methods that can be used in physical kinematics nummass () and to extend existing analytic methods to make them applicable to two-loop Feynman diagrams with internal masses masses ().

However, if we focus on the top-bottom interference and its impact on Higgs production at the LHC, we can simplify the calculation by using the fact that the mass of the -quark, , is numerically small. Indeed, since , where is a typical Higgs boson transverse momentum, Feynman diagrams that describe Higgs production can be expanded in series in for the purposes of LHC phenomenology. We have checked at leading order that the use of scattering amplitudes either exact or expanded in leads to at most few percent differences in the interference contribution to the Higgs distribution, down to . Since the interference contribution changes the Higgs boson transverse momentum spectrum by at leading order, the percent difference between expanded and not expanded results is irrelevant for phenomenology.

Unfortunately, the expansion in is non-trivial since the Higgs boson production cross section depends logarithmically on the -quark mass. Therefore, we need to devise a procedure to expand scattering amplitudes in and extract the non-analytic terms. This can be done by deriving differential equations for master integrals that are needed to describe the two-loop corrections to and then solving them in the limit Melnikov:2016qoc (). Indeed, since we can derive differential equations to describe the dependence of the master integrals on the mass parameter and on the Mandelstam kinematic variables, and since all the information about singular points of a particular Feynman integral is contained in the differential equations that this Feynman integral satisfies, we can systematically solve the differential equation in series of and extract the non-analytic behavior. We note that a similar method was used to compute the top-bottom interference contribution to the inclusive Higgs production cross section in Ref. Mueller:2015lrx ().

We have used this method to calculate all the relevant two-loop scattering amplitudes to describe the production of a Higgs boson in association with a jet Melnikov:2016qoc (); Melnikov:2017pgf (). In our computation, all quarks in the initial and final states are massless, so that -initiated processes are not included. The two-loop amplitudes mediated by top quark loops, required to describe the interference, are computed in the approximation of an infinitely heavy top quark schmidt ().

To produce physical results for production, we need to combine the virtual corrections discussed above with the real corrections that describe inelastic processes, e.g. , etc. Computation of one-loop scattering amplitudes for these inelastic processes is non-trivial; it requires the evaluation of five-point Feynman integrals with massive internal particles. Nevertheless, such amplitudes are known analytically since long ago DelDuca () and were recently re-evaluated in Ref. Neumann:2016dny ().

In this Letter we follow a different approach, based on the automated numerical computation of one-loop scattering amplitudes developed in recent years. One such approach, known as OpenLoops Cascioli:2011va (), employs a hybrid tree-loop recursion. Its implementation is publicly available openloops () and has been applied to compute one-loop QCD and electroweak corrections to multi-leg scattering amplitudes for a variety of complicated processes (see e.g. Refs. olrefs (); Jezo:2016ujg ()) and as an input for the real-virtual contributions in NNLO computations (see e.g. Ref. Cascioli:2014yka ()).

For applications in NNLO calculations the corresponding real-virtual one-loop contributions need to be computed in kinematic regions where one of the external partons becomes soft or collinear to other partons. We face a similar situation for the loop-induced process discussed in this Letter. Indeed, the loop-squared real contribution has to be evaluated in phase-space regions where a final-state parton becomes unresolved. Although the singular contribution of the real emission graphs is easily identified and subtracted, it is important to control the approach of the singular region of the squared one-loop amplitudes. A reliable computation in such kinematic regions is non-trivial, but the OpenLoops approach appears to be perfectly capable of coping with this challenge thanks to the numerical stability of the employed algorithms. An important element of this stability is the program COLLIER collier () that is used to perform the tensor integral reduction in a clever way via expansions in small Gram determinants.

We have implemented all virtual and real amplitudes in the POWHEG-BOX powheg (), where infra-red singularities are regularized via FKS subtraction Frixione:1995ms (). All OpenLoops amplitudes are accessible via a process-independent interface developed in Ref. Jezo:2016ujg (). The implementation within the POWHEG-BOX will allow for an easy matching of the fixed-order results presented here with parton showers at NLO. At leading order this has been done in Ref. mctb ().

Using the methods described above, we calculated the NLO QCD corrections to the top-bottom interference contribution to production in hadron collisions. We identify the interference contribution through its dependence on top-bottom Yukawa couplings. For the Higgs production cross section, we write

(1) |

where individual contributions to the differential cross section scale as , , . Given the hierarchy of the Yukawa couplings, , the last term in Eq.(3) can be safely neglected. Note, however, that if one focuses on Higgs-related observables that are inclusive with respect to the QCD radiation, receives contributions from Higgs boson production in association with -quarks, e.g. . These processes change inclusive Higgs boson observables at below a permille level which makes them irrelevant unless b-jets in the final state are tagged.

Our main focus is the top-bottom interference contribution . Considering the virtual corrections, we write

(2) |

The leading order (one-loop) term in this formula is known, including full mass dependence. The NLO (two-loop) amplitudes with the top quark are only known in the limit and we use as an approximation for . In principle, one can improve on this by computing corrections to , see Ref. Harlander:2012hf (), but it is not expected that such power corrections will have significant impact on the results for the interference at moderate, , values of the Higgs transverse momentum. The real emission contributions are computed with exact top- and bottom-mass dependence throughout the paper.

In what follows, we present the QCD corrections to the top-bottom interference contribution to the Higgs boson transverse momentum distribution and to the Higgs rapidity distribution in production. We consider proton collisions at the LHC and take the mass of the Higgs boson to be .

We work within a fixed flavor-number scheme and do not consider bottom quarks as partons in the proton. We use the NNPDF30 set of parton distribution functions Ball:2014uwa (). We also use the strong coupling constant that is provided with this PDF set. We renormalize the -quark mass in the on-shell scheme and use as its numerical value. We choose renormalization and factorization scales to be equal and take, as the central value where the sum runs over all partons in the final state.

To quantify the impact of the top-bottom interference on an observable , it is convenient to define the following quantity

(3) |

where is a set of phase-space variables. Note that we do not expand the cross section in the denominator in Eq.(3) in powers of . Therefore, any change in in consecutive orders in perturbation theory would reflect differences in QCD corrections to the interference and the point-like contribution to production. In what follows we present as a function of the Higgs boson transverse momentum and the (pseudo-)rapidity .

The impact of the top-bottom interference on the Higgs boson transverse momentum distribution is shown in Fig. 1. We observe that the leading order interference changes the Higgs boson transverse momentum distribution by at and at . Since the QCD corrections to color-singlet production in gluon annihilation are large and since it is not clear a priori if the QCD corrections to the interference are similar to the QCD corrections to the point-like cross section, large modifications of these LO results can not be excluded. The NLO computation, illustrated in Fig. 1, clarifies this point. There, filled bands in blue for the leading and red for the next-to-leading order predictions show the result for computed in the pole mass renormalization scheme. The widths of the bands indicate changes in the predictions caused by variations of renormalization and factorization scales by a factor of two around the central value . In fact, we observe that differences between leading and next-to-leading order are very small. For example, appears to be smaller than by less than a percent at and, practically, coincides with it at higher values of . We emphasise that these small changes in imply sizable, , corrections to the interference proper that, however, appear to be very similar to NLO QCD corrections to the point-like cross section . The scale variation bands are very narrow (at leading-order hardly visible) due to a cancellation of large scale variation changes between numerator and denominator in Eq.(3). Similar results for the Higgs boson rapidity distribution for events with are shown in Fig. 2.

The above result for the scale variation suggests that the uncertainties in predicting
the size of top-bottom interference effects in production are small
since both the size
of corrections and the scale variation bands are similar to the corrections to the point-like
cross section. Such a conclusion, nevertheless, misses an important source of uncertainties related
to a possible choice of a different mass-renormalization scheme.
Indeed, since the leading order interference contribution is proportional to the square of the bottom
mass and since at leading order a change in the mass renormalization scheme simply
amounts to the use of different numerical values for in calculating , it is
easy to see that this ambiguity is very significant. Indeed, suppose that we choose to renormalize the bottom
mass in the scheme and we take as
input parameter.^{3}^{3}3We calculated this value using the program RunDec rundec ()
with the input value .
Since ,
this implies that is reduced by more than a factor of two, practically
independent
of the value. This large leading order variation is shown as a hashed blue band in Figs. 1,2,
where we have taken and
as the two boundary values.

This large ambiguity in the leading order value of is somewhat reduced at next-to-leading order where the effect of the mass renormalization scheme change is less dramatic but, nevertheless significant. The scheme dependence at NLO, for the setup explained in the previous paragraph, is shown as a hashed red band. We observe that for , the mass renormalization scheme uncertainty is reduced by almost a factor of two, whereas the reduction of uncertainty is only marginal at higher . This happens because in the final result for the interference at high transverse momenta there is a significant cancellation between and , c.f. Eq.(2). Since the first term involves leading order -quark contributions, it experiences large variations when the -quark mass renormalization scheme is changed and this causes large variations in at high . The interference contribution to the Higgs rapidity distribution in Fig. 2 shows similar features. The mass variation band at NLO is smaller than the LO variation band at large absolute values of the pseudo-rapidity (dominated by small ) and practically indistinguishable from it at the central rapidity values (dominated by large ).

In summary, we computed the NLO QCD corrections to the top-bottom interference contribution to Higgs boson production in association with a jet at the LHC. This is the first computation of QCD radiative corrections to Higgs production at this order in perturbation theory that goes beyond the point-like approximation for the coupling. Our results show that corrections to the interference are large yet they appear to track very well corrections to the point-like component of the cross section. The strong dependence of the LO interference on the mass-renormalization scheme is reduced at NLO but at high values of the Higgs transverse momentum or at central rapidity, the remaining ambiguities are significant. It is not clear how the situation at high and/or small absolute can be further improved. However, we want to emphasize that in these kinematic regions the interference is numerically small compared to the contribution. Nevertheless, with this result at hand, one can try to provide the best possible theoretical predictions for the Higgs transverse momentum distribution that combine the known results for the -resummation, NNLO corrections to in the point-like approximation with the top-bottom interference. All the ingredients are now available. We plan to return to this problem before long.

Acknowledgments We thank Tomas Jezo for valuable help with the POWHEG-BOX and Fabrizio Caola for useful conversations. The research of K.M. was supported by the German Federal Ministry for Education and Research (BMBF) under grant 05H15VKCCA.

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