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CP-even scalar boson production via gluon fusion at the LHC

In view of the searches at the LHC for scalar particle resonances in addition to the 125 GeV Higgs boson, we present the cross section for a CP-even scalar produced via gluon fusion at N3LO in perturbative QCD assuming that it couples directly to gluons in an effective theory approach. We refine our prediction by taking into account the possibility that the scalar couples to the top-quark and computing the corresponding contributions through NLO in perturbative QCD. We assess the theoretical uncertainties of the cross section due to missing higher-order QCD effects and we provide the necessary information for obtaining the cross section value and uncertainty from our results in specific scenarios beyond the Standard Model. We also give detailed results for the case of a 750 GeV scalar, which will be the subject of intense experimental studies.


Introduction
The Higgs boson is the only elementary scalar particle of the Standard Model (SM) and the first such particle whose existence has been confirmed experimentally [1,2]. Many well-motivated extensions of the SM predict additional electrically neutral and colourless scalar particles (elementary or not). Experimental searches to hunt for these hypothetical new particles are thus of extreme importance. The LHC experiments have the potential to discover new scalar resonances by using similar techniques and signatures as in the search for the SM Higgs boson. In particular, in this context both the ATLAS an CMS experiments have recently reported an excess in the diphoton mass spectrum at an invariant mass of ∼ 750 GeV [3,4]. Even if it is still premature to speculate if the source of this excess is indeed a new resonance or merely a statistical fluctuation, this excess is exemplary for the kind of signals that may arise from the production of heavy singlet scalars in the current run of the LHC.
While specific models with colour-neutral scalars may differ drastically in the principles which motivate them as well as in their spectra and interactions, the computation of QCD radiative corrections to their production cross section could share many common features that are independent of the underlying model. In particular, the production of hypothetical Higgs-like scalars at the LHC may be favourable in the gluon-fusion process due to the large value of the gluon-gluon luminosity. In this case, the model-dependence of the production cross section may only enter through the coupling describing the (effective) interaction of the Higgs-like scalar with the gluons and quarks.
The purpose of this paper is to provide benchmark cross sections for a colorless CPeven scalar in an effective theory which is similar to the one obtained for the Higgs boson in the Standard Model after integrating out the top-quark. The cross section for the scalar can be obtained from the cross section of the Higgs boson for a wide range of models, by appropriately taking into account the corresponding Wilson coefficient. Our computation is valid through N 3 LO in perturbative QCD and is based on recent results for the SM Higgs boson gluon-fusion cross section to that order [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. We consider mass values for the new scalar particle ranging from 10 GeV to 3 TeV, which is the same range as the one targeted by the searches of ATLAS and CMS in the Run 2 of the LHC [21].
In some extensions of the Standard Model (see, for example, ref. [22] and references therein) the new Higgs-like scalar has a non-negligible coupling to the Standard Model top-quark. For scalar masses below the top-pair threshold the Wilson coefficient in our effective field theory (EFT) can be made to account for a coupling of the scalar to the top-quark. However, for scalar masses around and above threshold, the effective theory approach is not accurate. For this purpose, we extend the effective theory by adding a Yukawa-type interaction of the top-quark with the scalar and compute the additional contributions generated by this coupling.
While we can correct our N 3 LO cross-section in the effective theory for contributions of light Standard Model particles through NLO in QCD, it is not possible to do so in a modelindependent way for contributions from new relatively light particles with a mass smaller than about half the mass of the scalar. The presence of such particles will invalidate our effective-theory computation. To estimate this effect, we consider the difference between the effective theory NLO cross-section and the exact NLO cross-section for a 750 GeV scalar which couples to a new top-like quark, as a function of the mass of the quark. We find that the two cross-sections can differ by more than the QCD scale uncertainty for top-like quark masses as heavy as twice the mass of the scalar. Nevertheless, we observe that one can still make use of the effective theory computation for such a model also for lower values of the quark mass in order to estimate the relative size of the QCD corrections (K− factor).
This article is organised as follows. In section 2 we present the effective theory setup of our calculation and discuss how the results can be adapted to suite a large class of models through the Wilson coefficient. We also discuss the theoretical uncertainty that is associated with the N 3 LO QCD corrections in the effective theory. In section 3, we discuss the inclusion of finite width effects in the calculation and provide a fit of the zero-width cross section to enable the adaptation of our results to models where the scalar resonance is expected to have a non-vanishing but narrow width. In section 4, we investigate the validity of our effective theory approach under the assumption that the coupling between the scalar and the gluons is mediated by a heavy top partner of varying mass. In section 5 we go beyond a purely effective description of the scalar interaction and allow for a nonvanishing coupling between the Standard Model top and the scalar, for which we present the NLO QCD corrections. We conclude in section 6 and provide appendices with tables of cross section values for a wide range of masses of the scalar.

N 3 LO QCD corrections in the effective theory approach
Let us consider a model where the SM is extended by a colourless singlet scalar S of mass m S and width Γ S , which only couples to the SM in a minimal way. For now, we assume that the only SM fields that the new scalar S couples to are the gluons, through an effective operator of dimension five. We will discuss Yukawa couplings to heavy quarks in Section 5. Such a model can be described by a Lagrangian of the form where L S collects the kinetic term and the potential of the scalar S. The vacuum expectation value v is [23], and C S is a Wilson coefficient parametrizing the strength of the coupling of the particle S with the gluons. Except for the value of the Wilson coefficient, this is the same dimensionfive operator as the one coupling the SM Higgs boson to the gluons, after the top-quark has been integrated out.
Since the scalar S couples to gluons, it can be produced at hadron colliders. Its hadronic production cross section can be cast in the form where µ is the mass scale introduced by dimensional regularisation and Λ UV is the scale of new physics, representing collectively the masses of the heavy particles in some ultraviolet (UV) completion of the theory. At all orders in perturbation theory, the cross section is independent of the arbitrary scale µ, with the scale variation of the square of the Wilson coefficient cancelling the one of the matrix-elements η. Since η does not depend on Λ UV , the scale dependence of the Wilson coefficient itself is universal, and it does not depend on the (renormalizable) UV completion of the effective theory. In particular, the renormalization group evolution equation that relates the Wilson coefficient at two different scales reads [24]: with β(µ) the QCD β-function. We see that, as expected, the evolution equation only depends on the low-energy effective theory, and it is independent of the details of the UV completion. As a consequence, if we divide the production cross section by the value of the Wilson coefficient at some reference scale µ 0 , we obtain a universal ratio that does not depend on the UV completion (up to corrections due to differences that the various high-energy theories may induce on the width of the scalar particle S). In particular, since the dimension-five operator in eq. (2.1) is the same as the dimension-five operator mediating Higgs production in gluon fusion in the SM, we can extract the right-hand side of eq. (2.4) from Higgs production in gluon-fusion, Here σ H is the gluon-fusion production cross section of a Higgs boson of mass m S and width Γ S , in a variant of the SM with N f = 5 massless quarks and the top-quark integrated out, m t is the top-quark mass and C(µ 0 , m t ) is the Wilson coefficient multiplying the SM gluonfusion operator evaluated at our reference scale µ 0 . In the MS scheme it reads [25,26] C(µ, m t ) = − with a s = α s (µ)/π, α s being the strong coupling constant in QCD with N f = 5 flavors. We note at this point that eq. (2.5) is valid to all orders in perturbation theory, and so neither σ S nor σ H depend on the arbitrary scales µ 0 and µ. They do, however, acquire a dependence on the scale µ once the perturbative series is truncated at some finite order. It is of course possible to further separate µ into a factorization and a renormalization scale, µ f and µ r , as it is customary, and we will assume in the following that this is done implicitly. In addition, σ S also acquires a dependence on our reference scale µ 0 order by order in perturbation theory, resulting from the truncation of the perturbative expansion of the ratio of Wilson coefficients. Equation (2.5) is the basic formula which allows us to predict the hadronic production cross section of a generic CP-even singlet scalar through gluon-fusion to high order in perturbative QCD. Indeed, the gluon-fusion cross section is known through N 3 LO in QCD in the large-m t limit [19] (for Γ S = 0) as an expansion around the Higgs threshold, and eq. (2.5) easily allows us to convert the result for the Higgs boson into the corresponding N 3 LO cross section for a generic CP-even scalar.
At this point we need to make a small technical comment: both the ratio of Wilson coefficient and the Higgs production cross section on the right-hand side of eq. (2.5) admit a perturbative expansion up to N 3 LO. Multiplying these two perturbative expansions, as suggested by eq. (2.5), introduces additional terms into the perturbative expansion which are of N 4 LO accuracy or higher, thus beyond the reach and the control of our N 3 LO computation. In ref. [19] it was shown that the numerical impact of these terms is captured by the scale variation at N 3 LO if the central scale is chosen as µ = m S /2, and the effects of such terms are therefore accounted for by the scale variation uncertainty assigned to our prediction.
A second comment regards the dependence of the Standard Model Wilson coefficient on the various scales. The SM Wilson coefficient C(µ 0 , m t ) depends on the reference scale µ 0 and the top-quark mass m t . While the value of µ 0 is arbitrary and can be chosen freely, in practise we choose the value of µ 0 to be the mass of the scalar, µ 0 = m S , and we set the value of the mass of the top quark to its physical value. In doing so, we may introduce large logarithms into the computation, order by order in perturbation theory. For example, we see that at NNLO and N 3 LO the Wilson coefficient contains a logarithm of the form log(m t /µ 0 ). Similarly, the Wilson coefficient expression for the scalar C S (µ 0 , Λ UV ) will contain logarithms of the type log(Λ UV /µ 0 ). In order to avoid large logarithms due to widely disparate scales in the ratio of the two Wilson coefficients, it may be preferable to evaluate the SM Wilson coefficient with a top-quark mass value m t ∼ Λ UV rather than its physical value. The value of the top mass entering the SM Higgs cross-section can easily be changed with a simple rescaling, Let us now discuss our predictions for the production cross section σ S . In Figs. 1 -4 we show the effective theory cross section σ H (m S , Γ S = 0, m t ) as a function of m S through N 3 LO, at a proton-proton collider with a center-of-mass energy of 13 TeV. The theory uncertainty bands correspond to a variation of the scale µ = µ r = µ f in the range µ ∈ [m S /4, m S ]. The value of the top mass is set to m t (m t ) = 162.7 GeV and we use the PDF4LHC15 [27] parton distribution function (PDF) set. We observe that for the range of scalar masses between 10 GeV and 3 TeV the N 3 LO scale variation band is always contained inside the NNLO band. This is also true for the lowest values of this mass range, m S ∼ 10 − 50 GeV ( Fig. 1), where one observes especially large corrections at NLO and NNLO. For higher masses, the scale variation is reduced. As an illustration, we show in Fig. 5 the scale dependence for the cross section of a CP-even scalar with a mass of 750 GeV.
Sources of theoretical uncertainty affecting the gluon-fusion cross section at N 3 LO other than scale variation have been considered in detail in ref. [19]. In particular, these sources of uncertainty are due to the lack of N 3 LO parton densities and to the truncation of the threshold expansion for the N 3 LO correction. In order to estimate the uncertainty on our computation, we follow faithfully the uncertainty estimation prescription of ref. [19] with one modification: the uncertainty due to the lack of N 3 LO parton densities is estimated here as the envelope of the corresponding uncertainties formed by using the CT14 [28], NNPDF30 [29] and PDF4LHC sets, which are themselves computed according to the prescription of ref. [19]. The reason for modifying the prescription in this case is that the PDF4LHC set leads to very small uncertainties for scalars in the region 500 − 1000 GeV and, in particular, to a vanishing uncertainty estimate for a scalar with mass around 770 GeV. This feature is not shared with individual PDF sets. We therefore use, conservatively, the envelope of CT14, NNPDF30 and PDF4LHC, which leads to an uncertainty due to the lack of N 3 LO parton densities at the level of 0.9% − 3% for scalars in the range 50 GeV−3 TeV. This uncertainty remains of the order of a few percent also at lower masses, but it increases rapidly to O(10%) for m S 20 GeV. We present the cross section values and uncertainties for this range of scalar masses in Appendix A. In particular, in Tab. 6 we focus on the range between 730 and 770 GeV.

Finite width effects and the line-shape
The results of the previous section hold formally only when the width of the scalar is set to zero. In many beyond the Standard Model (BSM) scenarios, however, finite-width effects cannot be neglected. In this section we present a way to include leading finite-width effects into our results, in the case where the width is not too large compared to the mass. The total cross section for the production of a scalar boson of total width Γ S can be obtained from the cross section in the zero-width approximation via a convolution where Q is the virtuality of the scalar particle. This expression is accurate at leading order in Γ S (m S )/m S . For large values of the width relative to the mass, subleading corrections Let us also note that in order to use eq. (3.1) faithfully, one needs the width as a function of the virtuality of the scalar, which may bear a substantial model dependence. However, it is often the case that the width can be approximated as The invariant mass distribution in this approximation for the production of a CP-even scalar with a mass of m S = 750 GeV and total width from 2 to 10% of the scalar mass is shown in Fig. 6. This result has been obtained from an interpolation of the zerowidth cross section values of Tab. 6 in Appendix A. We caution that if the results of Appendix A are used to derive cross section values with non-zero width effects following the strategy outlined in this section, an additional uncertainty of the order O (Γ S /m S ) should be assigned. In order to facilitate the computation of the line-shape we perform a parametric fit of the production cross section of a zero-width scalar boson as a function of its virtuality (with Λ UV = m S ) for a center-of-mass energy of 13TeV. To guarantee agreement of the fitted cross section with the actual cross section at a level better than 1%, we split the range of interesting scalar boson mass values into three intervals: • in the range m S ∈ [10 GeV, 150 GeV], cf. Tab. 2 in Appendix A, we find • in the range m S ∈ [150 GeV, 500 GeV], cf. Tab. 3, we find

Validity of the EFT approach
So far, we have assumed that the effective theory of eq. (2.1) furnishes an accurate description of the gluon-scalar interaction. However, this assumption may be challenged if the scalar couples to light coloured particles. To investigate this effect, we will consider a scalar of mass m S = 750 GeV which couples to a top-like quark of mass m T . In this scenario, we can compute the cross-section exactly through NLO in perturbative QCD. We can then compare this prediction with the cross-section derived with the effective theory of eq. (2.1).
The red line in Fig. 7 shows the percent difference of the two predictions, as a function of m T for a scalar with a mass of m S = 750 GeV. While in the region m T ∼ m S /2 the effective field theory is inadequate, for larger values of m T the EFT description becomes accurate very quickly. Indeed, for m T ∼ 1.5m S the discrepancy with respect to the exact result is already below the theoretical uncertainty of QCD origin (of about 5%).
Let us now define a rescaled effective theory cross-section, The green line of Fig. 7 shows the relative difference between the rescaled effective theory prediction for the cross-section and the exact NLO cross section. We notice that the rescaled effective theory cross-section reproduces the exact cross section very well (as it has also been the case for the Standard Model Higgs boson [30,31]). We conclude that, while for the study of scalars which couple to relatively light particles one should resort to a calculation within a specific model, also in such situations the effective theory computation can be utilised as a means to compute with a good accuracy the corresponding QCD K− factor with respect to the exact LO cross section.

Top-quark contributions
So far we have considered the case where the interaction between the heavy scalar S and the Standard Model is mediated exclusively by the dimension-five operator in eq. (2.1). In many extensions of the Standard Model new scalars may also couple directly to the quarks of the third generation as well as the W and Z bosons. However, the absence of resonances in top-pair production and four-lepton production suggests that these couplings should be small. It may be nonetheless useful to estimate the contribution to the inclusive S-scalar cross section due to the heavy SM quarks and gauge bosons for the purpose of setting precise experimental constraints on their couplings to S. In this section, we study the effects due to the top quark, which in many scenarios are expected to give the most important contribution.
For light scalar masses, m S < O(m t ), contributions due to the coupling of the top quark can be simply taken into account through N 3 LO in perturbative QCD by using an appropriate Wilson coefficient C S . For heavier scalars with masses around and above the top-pair threshold, however, it is not justified theoretically to integrate out the top-quark. For this reason we study in the following the effect of modifying the Largangian in eq. (2.1) by including a direct Yukawa interaction between the scalar S and the top quark, i.e., we consider the Lagrangian where λ t is the ratio of the Yukawa coupling of the scalar S and the Yukawa coupling of the Higgs boson to the top quark and λ wc = C S C denotes the ratio of the Wilson coefficients in this theory and the SM with the top-quark integrated out.
The production cross section now depends on the values of λ wc and λ t :  It does not take too much effort to show that the NLO cross section can be cast in the form  where is the relative change of the gluon-fusion cross-section in the effective theory from NLO to N 3 LO. Note that in this way make the assumption that the cross-section components that are not known beyond NLO will not have a worse perturbative convergence than in the effective theory. Finally, we enlarge this uncertainty further by

Conclusion
In this paper we have presented the most precise predictions for the production of a CPeven scalar produced in gluon fusion at the LHC. We assume that the coupling of the scalar to the gluons can be described in an effective field theory approach. This enables us to provide precision QCD corrections at N 3 LO to the production cross section at the LHC. We find that for a scalar with a mass of 750 GeV the N 3 LO corrections yield a theoretical uncertainty of ∼ 2% when we choose the central renormalization and factorization scales to be m S /2 and follow the prescriptions layed out in ref. [19]. If we assume that the effective theory Wilson coefficient coupling of the CP-even scalar and the gluons is generated by a heavy top-partner, we can estimate the validity of the effective description as a function of the heavy fermion mass. We find that in the threshold region, when the mass of the top-partner is about half the mass of the scalar, the corrections can be as large as ∼ 60%. However, from direct searches the low mass ranges for toppartners are tightly constrained, so that at least for the discussion of a 750 GeV scalar this configuration is unlikely. For heavier fermion masses the effective theory description performs very well and is already accurate at ∼ 5% when then fermion mass is about 1.5 times the scalar mass.
Additionally, we have considered the case that the new scalar couples directly to the SM top quark. In this case we need to take into account the top corrections to the cross section, which are unfortunately only available through NLO in QCD, and they thus come with uncertainties of up to ∼ 20%. We incorporate the top corrections to the scalar production cross section as a function of the Yukawa coupling of the top quark to the scalar, allowing for a model-dependent choice of the parameters.
In summary, we have provided the ingredients to endow predictions for the production of a CP-even scalar at the LHC with the most precise QCD corrections available. The only assumptions that have been made in the calculation are that the scalar couples to the gluons through an effective theory like operator. By determining the Wilson coefficient for the scalar coupling to the gluons as well as the Yukawa coupling to the top within a concrete model, it is now possible, using the numbers provided here, to easily incorporate QCD corrections through N 3 LO into the cross section predictions for the LHC. While the results presented in this article concern strictly the production of a CP-even scalar, we can use the same techniques to compute the cross-section for resonance production of different CP/spin types which may be phenomenologically relevant (see, for example, ref. [32]). This will be the subject of future works.
A. Reference tables for the production cross section of a CP-even scalar through top-quark loops To it we add linearly the uncertainty from missing N 3 LO PDFs, evaluated following the method of sec. 2, and from the truncation of the threshold expansion. We refer to ref. [19] for an explanation of how the latter is computed. We use the PDF set PDF4LHC14 [27] and derive the combined PDF+α s error following the indications of the PDF4LHC working group.   ±8.5 Table 4: Production cross section for a scalar particle with a mass in the range 500 GeV to 1700 GeV.  ±4.   Table 8: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. (5.4). The theory uncertainty is computed as in eq. (5.5).  Table 9: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. (5.4). The theory uncertainty is computed as in eq. (5.5).  Table 10: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. (5.4). The theory uncertainty is computed as in eq. (5.5).  Table 11: NLO contributions to the production cross section for a scalar particle, as they are defined in eq. (5.4). The theory uncertainty is computed as in eq. (5.5).