On top quark mass effects to $gg\to ZH$ at NLO

We compute next-to-leading order QCD corrections to the process $gg\to ZH$. In the effective-theory approach we confirm the results in the literature. We consider top quark mass corrections via an asymptotic expansion and show that there is a good convergence below the top quark threshold which describes approximately {\num a quarter} of the total cross section. Our corrections are implemented in the publicly available {\tt C++} program {\tt ggzh}.


Introduction
In the upcoming years the general purpose experiments ATLAS and CMS at the CERN LHC will collect a large amount of data which will be used to perform precision studies of various quantities. Among them are certainly the properties of the Higgs boson, in particular its couplings to the other particles of the Standard Model. Important quantities in this context are the production cross sections and partial decay rates of the Higgs boson. The dominant production process is via gluon fusion followed by vector boson fusion and the so-called Higgs-strahlung process pp → V H (V = Z, W ) which is the subject of the current paper. Although pp → V H has a much smaller cross section it is a promising channel to observe, e.g., if the Higgs boson decays to a bb pair once substructure techniques are applied [1].
The leading order (LO) cross section is obtained from the Drell-Yan process for the production of a virtual gauge boson V and its subsequent decay into V H. Next-to-nextto-leading order QCD corrections to this channel have been computed in refs. [2][3][4][5][6] and electroweak corrections have been considered in refs. [7,8]. QCD corrections up to NNLO and electroweak corrections up to NLO for the total cross section have been implemented in the program vh@nnlo [9].
In ref. [10] the loop-induced production channel gg → ZH has been computed at leading order. NLO QCD corrections have been computed in ref. [11] in the heavy top quark limit which significantly simplifies the calculation. They are also implemented in vh@nnlo [9]. Note that the NLO corrections to gg → ZH are formally N 3 LO contributions to pp → ZH. However, due to the numerical importance of the gluon-induced process it is worthwhile to compute gg → ZH to NLO accuracy.
In this paper we study the effect of a finite top quark mass. At LO exact results are available. However, at NLO the occurring integrals are highly nontrivial and their JHEP01(2017)073 evaluation is beyond straightforward application of current multi-loop techniques. We investigate the mass effects by expanding the amplitudes for large m t . This approximation is not valid in all phase space regions. However, it provides an estimate of the numerical size of the power-suppressed terms and thus of the quality of the effective-theory result. Furthermore, it constitutes an important reference for a future exact result since we observe a good convergence of the partonic cross sections below the top quark pair threshold. We only consider the gg channel; similar techniques can also be applied to the loop-induced contributions of the qg and qq channels which are, however, numerically much smaller [11]. In our calculation we do not consider decays of the final-state Z boson.
Similar to gg → ZH also the process gg → HH is mediated by heavy quark loops. NLO and NNLO corrections have been considered in a series of papers [12][13][14][15][16][17][18][19][20][21] applying various approximations. Recently the exact NLO corrections became available [22,23]. The comparison to the approximations shows sizeable differences for the total cross section and the Higgs transverse momentum distribution. However, reasonable agreement between the exact and the in 1/m t -expanded results is found for the Higgs pair invariant mass (m HH ) distribution for not too large values of m HH if the approximated result is re-scaled with the exact LO cross section. Note that the region between the production threshold and the top quark threshold corresponds to about 100 GeV in the case of HH and to about 135 GeV in the case of ZH production which makes the heavy-top expansion more interesting for the latter.
Top quark mass effects have also been computed for the related process gg → ZZ. In ref. [24] 1/m 2 t corrections have been computed at NLO, and interference effects have been considered in [25]. In the latter reference Padé approximation and conformal mapping has been applied to improve the validity of the expansion in 1/m t .
The remainder of the paper is organized as follows: in section 2 we briefly discuss the LO cross section and compare the in 1/m t expanded and exact results. In section 3 we present our findings for the partonic NLO cross section. In particular, we identify the approximation procedure which leads to promising hadronic results, subject of section 4. We summarize our results in section 5. In the heavy-m t approximation the diagrams with internal top quarks reduce to vacuum integrals. The massless triangle diagrams are computed with the help of simple form factor-type integrals which can be expressed in terms on Γ functions (see, e.g., appendix A of ref. [26]). Figure 1. Sample Feynman diagram contributing to gg → ZH at LO and NLO. Solid, wavy, dashed and curly lines denote quarks, Z and Higgs bosons, and gluons, respectively. Internal wavy lines can also represent Goldstone bosons.

JHEP01(2017)073
We perform the calculation for general R ξ gauge and check that the gauge parameter ξ Z present in the Z and Goldstone boson propagators drops out in the result for the cross section. In fact, it cancels between the diagrams with top and bottom quark triangles and a neutral Goldstone boson or a Z boson in the s channel. Note, that for special choices of ξ Z the calculation can be significantly simplified. For example, in Landau gauge the massless triangle contribution with virtual Z boson vanishes [11]. Note that due to Furry's theorem there is no contribution from the vector coupling of the Z. Altogether there are 16 LO Feynman diagrams, all of them are individually finite.
We compute the LO amplitudes both in an expansion for large top quark mass including terms up to order 1/m 8 t , and without applying any approximation and keeping the full top quark mass dependence. In the latter case we have reduced the tensor integrals to scalar three-and four-point integrals which are evaluated using the LoopTools library [27,28]. We want to mention that in the limit m t → ∞ the calculation is significantly simplified. In particular, all top quark triangle contributions with a coupling of the Z boson vanish.
For the numerical results we use the following input values [29] M Z = 91.1876 GeV , where M t is the top quark pole mass. To obtain our numerical results we follow ref. [11] and use the so-called G µ scheme where the electromagnetic coupling constant α and the weak mixing angle (s W ≡ sin θ W ) are defined via Our default PDF set is PDF4LHC15_nlo_100_pdfas [30] which we use to compute both the LO and NLO cross sections. For the strong coupling constant we use the value provided by PDF4LHC15_nlo_100_pdfas which is given by For the implementation of the PDFs we use version 6.1.6 of the LHAPDF library [31] (see https://lhapdf.hepforge.org/) which also provides the running for α s form M Z to the chosen renormalization scale µ R . Our default choice for the latter and for the factorization scale µ F is the invariant mass of the ZH system If not stated otherwise we choose s H = 14 TeV for the hadronic center-of-mass energy.
In figure 2 we compare the partonic cross section of the exact (black solid line) and expanded results (blue dashed lines, see caption for details). One observes a continuous improvement of the large-m t approximations below the top quark pair threshold which is at √ s ≈ 346 GeV. However, the characteristic behaviour at threshold and the drop of In contrast to [25] we apply the Padé approximation at the level of differential cross sections and not at the level of the amplitudes. Furthermore, we refrain from performing a conformal mapping since in our case the gain is marginal.

Partonic NLO corrections
Sample Feynman diagrams contributing to the real and virtual NLO corrections can be found in figure 1. In our calculation we apply standard techniques. In particular, the oneand two-loop integrals are reduced to master integrals using the program FIRE [32]; the resulting master integrals can be found in refs. [33,34]. For the isolation of the soft and collinear infrared divergences we follow ref. [35] which allows to compute differential cross sections. Although we consider top quark mass effects we express our final result in terms of α s defined in the five-flavour theory.
We write the partonic cross section to NLO accuracy in the form where results for the LO cross section have already been discussed in section 2. δσ (virt,red) NLO is the contribution from the reducible diagrams where two quark triangles are connected by a gluon in the t or u channel, see figure 1e for a sample Feynman diagram. In ref. [11] the effective-theory result for the corresponding differential cross section is given, which is obtained by considering the interference with the LO amplitude. We confirm the analytic expression of [11] and add power-suppressed terms up to order 1/m 8 t . Furthermore, we have computed this contribution exactly keeping the full top mass dependence. For the numerical results which we present in section 4 the exact expression is used.
In this section we discuss δσ (approx) NLO . We define the NLO approximation by factoring out the exact LO cross section multiplied by the ratio of the in 1/m t expanded NLO and LO contribution: where "exp-n" means that the corresponding quantity contains expansion terms up to order 1/m n t . In figure 4 we show as (blue) dashed lines the quantities δσ s. We observe a similar behaviour as at LO (cf. figure 2). In particular, it can not be expected that meaningful NLO approximations are obtained for large values of √ s from these expansion terms. However, based on observations at LO we expect that the Padé result provides a reasonable approximation below √ s ≈ 346 GeV. In figure 4 we also show as (yellow) long-and short-dashed curves the quantity δσ (approx) NLO with n = 0 and 8 (the curves for n = 2, 4, 6 lie in between and are not shown for clarity). The shape is now dictated by the LO cross section and has a well-behaved high-energy limit. For curves are close together, however above the top threshold the n = 8 curve is significantly higher.
As an alternative to eq. (3.2) we consider an approach where the exact LO result is factored at the differential level, i.e., before the integration over phase space. Schematically we write where "dPS 2 " indicates that we use this kind of factorization for the two-particle phase space contributions. The contribution from the three-particle phase space (which is numerically small) is added in the infinite top quark mass approximation. The integrand of eq. (3.3) is better behaved than the one for δσ (exp-n) NLO in eq. (3.2), which might lead to better approximations for the total cross section. However, below the top quark pair threshold we only expect small differences between eqs.   2)). This is because the two-particle phase space contributions to the squared matrix elements are proportional to the LO result. Moreover the three-particle contribution is small. As before, the n = 0 and n = 8 curves are close together below the top threshold and significant deviations are observed above.

Numerical results for hadronic cross sections
Numerical results for the LO cross section have already been discussed in section 2. At NLO we write in analogy to eq. H,NLO we consider three possibilities: (i) we either use the in 1/m t expanded partonic results; (ii) we construct an approximation using eq. (3.2) (where the partonic cross sections are replaced by their hadronic counterparts), or (iii) we utilize the differential approach of eq. (3.3). The latter option is only applied to the total cross section. Figure 5 shows the m cut ZH dependence of the NLO contribution δσ   values of m cut ZH one obtains the total cross section which is briefly discussed below. The (blue) dashed curves are obtained from the asymptotically expanded results and the dashdotted (red) curve is obtained from the [2/2]-Padé approximation. The general picture is similar to the one at partonic level. In particular, one observes a good convergence for m cut ZH 350 GeV and one can expect that the Padé result provides a good approximation to the unknown exact result. Note that for m cut ZH = 346 GeV the large-m t approximation gives 13 fb whereas the Padé result leads to 21 fb which corresponds to an increase of more than 50%. The total cross section for m cut ZH = 346 GeV amounts to about a quarter of the total cross section computed in the infinite top quark mass approximation (see also below).
The dashed yellow curve in figure 5 is based on eq. (3.2). It is obtained from the m cut ZHdependence of the exact LO result multiplied by the ratio of the NLO and LO total cross sections taken in the infinite top quark mass approximation. Below m cut ZH 350 GeV this result and the Padé curve lie basically on top of each other. Very similar results are also obtained if the ratio of the m cut ZH -dependent NLO and LO total cross sections are considered in the effective theory limit. For reasons of clarity the corresponding curve is not shown in figure 5.
We refrain from showing the m cut ZH dependence for δσ . However, it is included in the discussion of the total cross section below. Although not visible in the plots we want to remark that the infinite top quark mass approximation of σ (virt,red) H,NLO is off by a factor two. Table 1 shows the values for the total cross section at LO and for four possible approximations at NLO, see caption for details. Note, that in all NLO predictions finite top mass corrections are only considered for √ s < 346 GeV. For higher values of s the infinite top mass limit is applied. The first three approximations treat the top quark as infinitely heavy, whereas the fourth one incorporates the heavy quark effects considered earlier in JHEP01(2017)073 the form of a [2/2]-Padé approximation, which would be our recommendation for the best possible prediction to date. One observes, that the finite top mass corrections shift the total cross section upwards, however, the size is well within the scale uncertainties which are shown for σ [2/2] H,NLO in the last column. Similar uncertainties are also obtained for the other approximations.
The numerical results discussed in this section and in section 3 have been obtained with the help of the program ggzh which can be downloaded from [36]. A brief description of ggzh can be found in the appendix. ggzh can be used to reproduce the numerical results of ref. [11].

Conclusions
The associated production of a Higgs and Z boson is a promising channel in view of the determination of the Higgs boson couplings, in particular the Yukawa coupling to bottom quarks. We compute top quark mass effects to the loop-induced process gg → ZH at NLO in QCD by expanding the Feynman amplitudes in the limit of large top quark mass. Our leading term reproduces the results of ref. [11]. It is not expected that the top quark suppressed terms provide a good approximation for large partonic center-of-mass energies. However, we can show that below the production threshold of two top quarks, say for √ s 350 GeV, the 1/m t -expansion shows a good convergence at NLO. This is strongly supported by the good agreement of the re-scaled NLO approximation using the exact LO cross section and the [2/2]-Padé approximation constructed from expansion terms up to 1/m 8 t . Thus, the corrections computed in this paper provide a good approximation to the m ZH distributions below √ s 350 GeV. This region covers about 25% of the total cross section. Furthermore, the top mass corrections in this region constitute an important cross check once the exact calculation of the NLO corrections to gg → ZH is available. The numerical results presented in this work can be reproduced with the program ggzh which is publicly available from [36].

JHEP01(2017)073
ggzh is written in C++. Before compilation it is necessary to install the libraries CUBA [38], LoopTools [27,28] and gsl [39]. The corresponding paths should be inserted in the file Makefile.local. Afterwards, make starts the compilation.
The input file xsection.cfg defines the channels which shall be considered. Furthermore, one has to decide whether the partonic or hadronic cross section is considered, which pdf set is used and whether the sum of the considered channels is computed or not. Thus, xsection.cfg typically looks as follows Results based on the differential factorization of eq. (3.3) can be obtained via the channels NLO_differential_phase2 and NLO_differential_phase2eta (remember that eq. (3.3) is only applied to two-particle phase space contributions). The parameter diff_order in the input file params.cfg specifies the expansion depth used for the LO and NLO expressions in (3.3).
The second input file params.cfg contains the values for the various input parameters needed for the calculation. It overwrites the default values which are given in params.def together with a brief description of the meaning. The package comes with template files which clarify the syntax.
ggzh is launched by simply calling the executable in the shell

> ./ggzh
All input parameter are repeated in the output and the results for the individual channels is given in the form Besides the total cross section it is also possible to introduce a cut on the invariant mass m ZH which is switched on with use_inv_mass_cutoff: 1 in the file params.cfg.
The numerical values for the cut is specified with inv_mass_cutoff: <m_ZH-value>.
With the help of use_mt_threshold: 1 one switches on the possibility to use the infinite top mass approximation above the value for √ s given by mt_threshold: <mtthr-value>.
ggzh contains the option to vary µ R and µ F independently. Furthermore, it is possible to choose fixed scales (e.g. µ R = M H or µ R = m t ) or identify the scales to the partonic center-of-mass energy.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.