Higgs production in heavy quark annihilation through next-to-next-to-leading order QCD

The total inclusive cross section for charged and neutral Higgs production in heavy-quark annihilation is presented through NNLO QCD. It is shown that, aside from an overall factor, the partonic cross section is independent of the initial-state quark flavors, and that any interference terms involving two different Yukawa couplings vanish. A simple criterion for defining the central renormalization and factorization scale is proposed. Its application to the $b\bar bH$ process yields results which are compatible with the values usually adopted for this process. Remarkably, we find little variation in these values for the other initial-state quark flavors. Finally, we disentangle the impact of the different parton luminosities from genuine hard NNLO effects and find that, for the central scales, a naive rescaling by the parton luminosities approximates the full result remarkably well.


Introduction
Models with an extended Higgs sector typically predict a spectrum of Higgs bosons with very diverse properties (see, e.g., Ref. [1]). This means that the relative importance of individual processes for the total production cross section can be very different compared to the Standard Model (SM) Higgs boson H, where the main contribution to the total cross section at the Large Hadron Collider (LHC) is given by gluon fusion, gg → H (see, e.g., Refs. [2][3][4]). In particular, quark-associated production can be much more important than for SM Higgs production. For example, in supersymmetric theories, it it can naturally occur that at least one of the neutral Higgs bosons φ would be predominantly produced at the LHC in association with bottom-quarks, pp → φbb. Also an enhanced coupling to charm quarks can occur in many beyond-the-SM (BSM) scenarios, leading to non-negligible contributions of associated Higgs-charm production [5]. Similarly, the cross section for charged Higgs bosons φ ± may receive contributions from associated csφ − /csφ + or cbφ − /cbφ + production, and we could even imagine flavor-violating contributions of the form (bs +bs)φ to neutral Higgs production [6].
The proper theoretical description of associated bbφ production has a long and still ongoing history. The main argument has been centered around the question whether the so-called 4-or 5-flavor scheme (referred to as 4FS or 5FS in what follows) is more approprate to obtain the best approximation of the total inclusive cross section. In the 4FS, bottomquark parton densities are neglected, so that the dominant leading-order (LO) partonic process for bbφ production is gg → bbφ (the cross section for qq → bbφ is about a factor of ten smaller at the LHC). Integration over the final state bottom quark momenta leads to logarithms of the form ln(m b /m φ ) in the total inclusive Higgs production cross section, where m b and m φ is the bottom-quark and the Higgs-boson mass, respectively. The 5FS resums these terms to all orders in the strong coupling α s by introducing bottom-quark parton densities, and describing the LO partonic cross section as bb → φ. In the partonic calculation, the bottom-quark mass is set to zero (except where it occurs in the Yukawa coupling), and all collinear divergences are absorbed into the parton density functions (PDFs) through mass factorization. Concerning the sub-process gg → φbb, there is a potential mismatch of this approach with the treatment of the bottom-quark threshold in the parton densities. However, by comparing the massless with a massive calculation in this sub-channel [7], such effects could be shown to be negligible w.r.t. the overall theoretical accuracy.
The current experimental analyses are based on a combination of results from both approaches through the so-called Santander-matching formula [8], where the 4FS and 5FS results-the former at next-to-leading order (NLO) [9,10], the latter at next-to-NLO (NNLO) QCD [11]-enter with Higgs-mass dependent weights. For larger Higgs mass, the logarithms discussed above become more important, so the 5FS is expected to provide the more reliable result, and thus receives a larger weight. This is indeed confirmed by approaches aiming at a theoretically better-founded matching of the underlying processes [12][13][14]. 1 Due to the small value of the charm-quark mass m c ∼ 1 GeV, a charm-initiated approach for the calculation of the total inclusive cross section, cc → φ, is preferable over a 3-flavor scheme (3FS) description (LO process gg → ccφ) already for much smaller values of the Higgs boson mass. It can be evaluated both in the 4FS and the 5FS, where in the latter case the bottom quark plays the role of a spectator. Since, as we will show below, interference effects involving the bottom and the charm Yukawa coupling are absent, the only technical difference in evaluating the 4FS and the 5FS result for σ(cc → φ) is a change of the PDF set. All results in this paper are obtained in the 5FS.
Analogous considerations apply to other quark-associated production modes. As we will show in this paper, the corresponding NNLO partonic cross sections differ only by an obvious overall factor, given by the ratio of the respective Yukawa couplings, as long as the dynamical quark masses (as opposed to the Yukawa couplings) are neglected. The latter condition is anyway necessary in a partonic formulation of these scattering processes.
We can therefore use the known partonic NNLO results for the process bb → φ [11], and translate them into hadronic cross sections for arbitrary initial-state quarks. This will be explained in more detail in the next section. Section 3 uses these results to determine the central renormalization and factorization scales for all heavy-quark initiated Higgs production processes, and provides theoretical predictions through NNLO. In addition, the impact of hard radiation is disentangled from the purely PDF-induced effects. Section 4 contains our conclusions.

Calculation
We denote by Q Qφ the process for the associated production of a Higgs boson φ with a Q Q pair in the 5FS, whose LO Feynman diagram is given by Fig. 1. Depending on the specific flavors of Q and Q , φ can be electrically neutral or charged. Within QCD, renormalization of the Q Qφ coupling, and thus also its anomalous dimension, is independent of the quark flavors Q and Q . Since we work in the massless-quark limit throughout this paper, the underlying theory is chirally symmetric, which means that all our results apply to scalar as well as pseudo-scalar Higgs bosons φ (see also Ref. [11]); scalar/pseudo-scalar interference terms vanish.
At NLO QCD, aside from the virtual corrections to the LO process, the real radiation processes QQ → gφ, gQ → Q φ, and gQ →Qφ need to be taken into account in the calculation of the total cross section. Similarly, at NNLO QCD, there are the two-loop virtual corrections to the LO process, and the one-loop virtual corrections to the NLO real-emissions processes. In addition, double-real emission processes occur. Those with two external gluons are: QQ → ggφ, Qg → Qgφ,Q g →Q gφ, gg →QQ φ. The squared amplitude composed of these processes contains only a single fermionic trace.
This may be different for processes with four external quarks. Their amplitudes are given by C: Qq → Q qφ,Q q →Qqφ, Qq → Q qφ,Q q →Qqφ and correspond to the Feynman diagrams shown in Fig. 2. Here, q denotes a quark of arbitrary flavor, andq the corresponding anti-quark. The square of each of these processes involves two fermionic traces, one of which contains both Higgs couplings.
Let us now look at potential interference terms. If q ∈ {Q, Q }, the initial and final states of A, B, and C are different, and they obviously cannot interfere. If q ∈ {Q, Q }, there are AC and BC interference terms, which involve a single fermionic trace.
All contributions above are independent of the specific quark flavors Q and Q . For Q = Q = q, however, it seems that also A and B interfere with each other, leading to a term with two fermionic traces, each of which contains one Higgs coupling. However, in the limit of zero quark masses, the traces are over an odd number of Dirac matrices and vanish.
In conclusion, aside from an overall constant Yukawa factor, 2 the NNLO partonic cross section for the process Q Qφ is independent of the quark flavors Q and Q , as long as quark masses are neglected. Along the same lines, one observes that, for Q = Q , any interference terms between QQ-and Q Q -initiated Higgs production vanishes through NNLO.
Let us remark that in the analogous case of Drell-Yan production, i.e. φ = V ∈ {W, Z}, the AB interference term, which exists only for Z-production, is not zero. The doublequark emission corrections for W -production are therefore different from those of Zproduction [16]. 3 It follows that the hadronic Q Qφ cross section for the collision of hadrons h 1 and h 2 can be obtained by simply convolving the 5FS partonic cross section for bbφ production with the appropriate PDFs. For example, we may define whereσ bb =σ bb (m 2 H /ŝ) is the partonic cross section for the SM process bb → H + X, which can be found in Ref. [11],ŝ is the partonic center-of-mass energy, and ⊗ denotes the convolution Furthermore, f j (x) and f j (x) are the parton densities in the hadron h j , with f, f ∈ {q,q, g} and q ∈ {d, u, s, c, b}. The component of the hadronic Q Qφ cross section which is induced by the partonic QQ initial state can then be written as where β QQ is the squared ratio of the QQ φ and the SM bbH coupling. In particular, we Similarly, we can define where and the sum runs over all quark flavors q. The partonic cross sectionsσ ij on the right hand side are ij-initiated components of the partonic SM bbH cross section; explicit expressions can be found in Ref. [11].
In this way, we can calculate where q may be any (anti-)quark except Q orQ . The total inclusive hadronic cross section is then given by the sum of all the terms in Eqs. (1) and (6).
The implementation of this result in bbh@nnlo [11] (which is now part of SusHi [20]) is straightforward and will be publically available in the next version of SusHi. 4 3 Numerical results

Determination of the central scales
As a reference, the upper two plots of Fig. 3 show the first three perturbative orders for the bbφ cross section for m φ = 125 GeV and β bb = 1 as a function of the factorization scale µ F (left), and the renormalization scale µ R (right). These results are well-known [11]; they corroborate the choice (μ R ,μ F ) = (1, 1/4) as the central values for the scales [21][22][23], where we have introduced the normalized scaleŝ We may formalize the justification of this choice by considering the variation ∆ F of the NNLO hadronic cross section σ within the intervalμ R ∈ [1/10, 10], while fixingμ F : The central factorization scaleμ F can then be defined as the value ofμ F where ∆ F is minimal. The analogous procedure (with R↔F) can be used to define the central renormalization scaleμ We performed this study for all heavy-quark initiated processes by calculating σ on an equidistant 21 × 21 logarithmic grid in the (μ R ,μ F ) plane, i.e., using the valuesμ R ,μ F ∈ {10 n/10 , n = −10, −9, . . . , 9, 10}. When quoting numbers, we will round these values to two significant digits; e.g., we will refer toμ F = 10 −3/5 = 0.2512 . . . simply asμ F = 0.25, orμ F = 1/4 for that matter.
For m φ = 125 GeV, we findμ (0) F = 1/4 in this way, independent of the quark flavors Q and Q . This is an interesting observation, because this value has been derived specifically for Q = Q = b using kinematical considerations [21][22][23]; the fact that all other quark-initiated processes seem to favor the sameμ (0) F is not at all obvious from these discussions. Following the above procedure, the central renormalization scale turns out to beμ   However, in all cases, the minima are sufficiently shallow to justify also the choice (μ  1/4). Exemplary plots for the bbφ, ccφ, and bsφ processes are shown in Fig. 3 for m φ = 125 GeV, and in Fig. 4 for m φ = 600 GeV. All cross sections correspond to β QQ = 1, i.e., the Yukawa coupling is assumed identical to the one for SM bbH production. For the N n LO curve, it is evaluated from m b (m b ) = 4.18 GeV by (n + 1)-loop evolution with n f = 5 active flavors to 5 m b (µ R ) ≡ m    [24]. In the SM, the ccφ cross section is therefore about 6-7 times smaller than the bbφ cross section. All plots have been produced with the MSTW2008 PDF sets [25] as implemented in the LHAPDF library [26,27], and the associated value of α s (M Z ) = 0.139/0.120/0.117 at LO/NLO/NNLO.
Recall that the role of the central values is to determine the position of a "reasonable" interval forμ F andμ R ; the variation of the cross section within this interval should then give a clue of the associated theoretical error induced by missing higher-order effects. Due to the unphysical nature of the renormalization and factorization scale, any procedure to "determine" their central values is formally arbitrary, though not necessarily sensible. The fact that, at (μ R ,μ F ) = (μ (0) R ,μ (0) F ), the NNLO corrections are significantly smaller than the NLO ones in all cases studied here (see Figs. 3 and 4), confirms that the procedure defined above is indeed sensible. Other observations concerning the choice of the central scale in the case of the Q Qφ processes will be recalled in Section 3.4. 5 The notation m (n f ) q indicates that mq is renormalized in n f -flavor QCD.

Cross sections and uncertainties
Numerical values for the cross sections at the LHC with 13 TeV are shown in the form σ ± ∆ µ ± ∆ PDF in Table 1, where ∆ µ and ∆ PDF are the scale and PDF uncertainty, respectively, to be defined below. Again, for all processes, we assumed the Yukawa coupling to be equal to the SM Table 1. The PDF uncertainty ∆ PDF is determined from the 25 eigenvector sets of MMHT2014 while setting (μ R ,μ F ) = (1, 1/4); the central value σ is the mid-point of the PDF error interval.
We checked that we obtain comparable results when using the CT14 PDF set [29], while the default NNPDF3.0 set [30] typically leads to larger bbφ and bcφ + cross sections (15% and 11% at m φ = 125 GeV, respectively), which is most likely due to the significantly smaller bottom-quark mass assumed in that set (m b | NNPDF3.0 = 4.18 GeV). This motivates a study of the sensitivity of the results to the heavy-quark masses m b,PDF , m c,PDF used in the PDF fits. To that aim, we use the mbrange nf5 and mcrange nf5 versions of MMHT2014, whose member sets correspond to different values of the bottom and charm quark masses.  Table 1: Cross sections for the processes Q Qφ at 13 TeV. The Q Qφ coupling is assumed equal to the SM bbH coupling in all cases. The cross sections hold for both scalar and pseudo-scalar Higgs bosons. The first uncertainty is due to scale variation, the second one denotes the PDF error (see main text for more details). observe that the cross sections and the PDF quark masses are anti-correlated, and that a change of m b,PDF by 1% changes the bbφ cross section by about 1.5% at m φ = 125 GeV, and a little less at m φ = 600 GeV; the bcφ + and bsφ cross sections change by about 0.7%. On the other hand, the ccφ cross section for m φ = 125 GeV changes by about 0.9% when varying m c,PDF by 1%, and a little more at m φ = 600 GeV. Note that such a variation is largely compensated by a corresponding change of the quark mass in the Yukawa coupling (see Ref. [31]).
Note that, for the bbφ process, a much more thorough account of the quark-mass effects has been obtained by a consistent matching between the various energy regimes involved in this process [13,14]. A similar analysis could be performed for the general Q Qφ processes discussed here; however, we expect the corresponding effects to be much smaller than other theoretical and expected experimental uncertainties for these processes.

Effects due to the hard process
At LO, the partonic Q Qφ cross section iŝ and thus the LO hadronic cross section for Q Qφ production is proportional to the LO bbφ cross section, where is the f f parton luminosity in h 1 h 2 collisions. To a first approximation, one might be tempted to apply the rescaling of Eq. (11) also at higher orders. In order to see to what extent such an approximation is valid, let us study the double ratio at NLO and NNLO QCD, where at each order it is understood to use the appropriate set of PDFs, e.g., MSTW2008nlo at NLO, and MSTW2008nnlo at NNLO.
Any deviation from R Q Q φ = 1 is due to the hard scattering process. These effects depend on the choice ofμ R andμ F . Fig. 6 shows theμ F dependence of R Q Q φ atμ R = 1 for m φ = 125 GeV; the effects typically decrease/increase towards larger/smallerμ R . Interestingly, one observes that also here the scaleμ F = 1/4 plays a special role: in all cases, the hard-scattering effects become minimal around this value. Moreover, we have checked that this observation is virtually independent ofμ R (at least withinμ R ∈ [1/10, 10]). For m φ = 600 GeV, one observes an analogous behavior, albeit again at slightly lowerμ F , see Fig. 7.

Remarks on the central scale
Despite the fact that the "determination" of a central scale is not an unambiguous concept, it may be worth recalling the special role of this quantity for the Q Qφ processes. Let us focus on the case m φ = 125 GeV for the sake of simplicity. On the one hand, our studies of Section 3.1 have singled out the values (μ R ,μ F ) = (1, 1/4) as preferable from a perturbative point of view, see also Ref. [11]. As noted above, this is compatible with kinematic considerations based on the behavior of the bottom-quark parton densities [21][22][23]. In addition, it turns out that, for the bbφ process, the LO perturbative predictions within the 4FS and the 5FS agree at the 5-10% level for this choice of scales, while they differ by more than a factor of four for (μ R ,μ F ) = (1, 1), for example [32]. Furthermore, it was found that the total cross section is almost completely exhausted by the partonic bb channel at (μ R ,μ F ) = (1, 1/4), while all other channels are very small [33]. Specifically, for m φ = 125 GeV, one finds σ(bbφ) = σ(bbφ) [1.04 − 0.06 + 0.02] at (μ R ,μ F ) = (1, 1/4) , where the first number in square brackets denotes the bb, the second the (b +b)g, and the third the sum of the remaining channels. For (μ R ,μ F ) = (1, 1), on the other hand, we find a large cancellation between the bb and the (b +b)g channel: σ(bbφ) = σ(bbφ) [2.39 − 1.56 + 0.17] at (μ R ,μ F ) = (1, 1) .
That latter observation seems closely related to the one of Section 3.3 which shows that the differences between different initial state partons due to genuinely hard effects practically vanish at (μ R ,μ F ) = (1, 1/4). For the sake of completeness, let us note that analogous observations hold at larger Higgs masses, albeit at slightly different values for the central scales.
The author is not aware of any solid theoretical arguments that would explain why all of these observations occur at this one particular choice for the unphysical scales; one may simply characterize them as a mere couriosity. On the other hand, if there is an explanation, it would certainly be useful for the reduction of the theoretical uncertainty in perturbative calculations.

Conclusions
Cross sections for the production of neutral and charged scalar and pseudo-scalar particles via quark annihilation have been calculated through NNLO QCD. The results should be useful for studying models with an extended Higgs sector at hadron colliders. Explicit predictions for cross sections in exemplary cases of a SM-like and a heavy Higgs boson have been provided. For more general analyses, the next version of the public program SusHi [20] will provide easy access to these cross sections.
As a final remark, we note that the considerations of Section 2 can be extended to higher orders of perturbation theory, and are not restricted to the total inclusive cross section. The N 3 LO results of Refs. [34,35] as well as the differential results of Refs. [36][37][38][39] can therefore be generalized to Q Qφ processes by a simple replacement of the parton densities as well.