Higgs Strahlung at the Large Hadron Collider in the 2-Higgs-doublet model

We present a calculation of all relevant contributions to associated production of a Higgs boson with a weak gauge boson in the 2-Higgs-doublet model (2HDM) at the LHC, pp → ϕ, with ϕ ∈ {h, H0, A} and V ∈ {W, Z}. While for the W ϕ mode, this mostly amounts to a simple rescaling of the Standard Model (SM) cross section, the Zϕ cross section depends on several 2HDM parameters. The ratio σWϕ/σZϕ, for which we present the currently most complete SM prediction, therefore appears to be a sensitive probe of possible New Physics effects. We study its numerical dependence on the top and bottom Yukawa couplings, including their sign. Furthermore, we consider the Wϕ/Zϕ ratio in exemplary 2HDM scenarios and briefly address the effects in the boosted regime. Analogous studies for other 2HDM scenarios will become possible with an upcoming version of the program vh@nnlo which incorporates the 2HDM effects.


Introduction
After the discovery of a Higgs boson at the Large Hadron Collider (LHC) [1,2], one of the top priorities of the experiments is to measure its properties as precisely as possible. Any deviation from the Standard Model (SM) predictions could be valuable information about a possible extended theory. 1 However, precision physics at hadron colliders like the LHC typically suffers from large experimental and theoretical uncertainties. The latter are dominated by the truncation of the perturbative series and insufficient knowledge of the input parameters such as the strong coupling and the parton density functions. Observables where these uncertainties are absent or suppressed are therefore highly desirable.

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Yukawa couplings. In this paper, we thoroughly combine all contributions to the Higgs Strahlung cross section, taking into account 2HDM effects. We will show that modified top and bottom Yukawa couplings mostly affect the Zφ production cross section and thus can lead to significant deviations of the ratio of the Wφ/Zφ cross sections with respect to its SM value. In addition, the final state Zφ can also be produced via s-channel exchange of a (virtual) scalar φ ′ = φ in the 2HDM which is produced through either bb or gg annihilation. This can enhance the Zφ production cross section dramatically. Such effects have been studied for ZA production in refs. [44][45][46][47][48][49][50] and for Zh and ZH 0 production in ref. [50]. Most of these studies were carried out in the framework of the Minimal Supersymmetric SM (MSSM) though, where the parameters of the Higgs sector are tightly constraint, thus restricting the possible effects with respect to a fully general 2HDM. For our numerical analysis, we extended the program vh@nnlo [12,51] for the calculation of associated VH production to Vφ production in the 2HDM. While the main results of the paper are derived from predictions for the total inclusive cross section, we also study the influence of lower cuts on the Higgs' transverse momentum p φ T , motivated by the increased signal-to-background ratio in the so-called boosted regime [52]. We will argue that, while the sensitivity to the Yukawa couplings is generally increased in that case, effects induced by additional (virtual) scalar particles can be strongly reduced. This suggests that events at low-p φ T should not be discarded in experimental analyses.
The remainder of the paper is structured as follows: section 2 reviews Higgs Strahlung in the SM, before discussing the various cross section contributions in the 2HDM. Their implementation within vh@nnlo is described, and the Wφ to Zφ ratio of production cross sections as precision observable is motivated. We provide the most up-to-date numerical value for this ratio in the SM. In section 3, the effect of Yukawa couplings on Higgs Strahlung in the SM is discussed. Section 4 describes different aspects of Higgs Strahlung in the 2HDM and highlights their effects on the Wφ to Zφ ratio. Differences of the observed effects in the boosted regime are discussed in section 5. Conclusions and an outlook for possible future studies are given in section 6. Appendix A summarizes some information on the 2HDM which is relevant for our presentation.
2 Theoretical description of the Higgs Strahlung process

Standard Model
The cross section for Higgs Strahlung in the SM through NNLO QCD can be written as Through NLO QCD, i.e. O(α s ), only the so-called "Drell-Yan-like" (DY) terms σ VH DY contribute. They are given by the production of a virtual gauge boson V * and its subsequent decay into a real gauge boson plus a Higgs; the leading order diagram is shown in figure 1 (a). We exclude contributions from the definition of σ VH DY which involve the coupling of the gauge boson to a closed quark loop, see for example figure 1 (b), which will be JHEP02(2014)023 attributed to σ ggZH . The QCD corrections to σ VH DY are therefore simply given by the QCD corrections to the Drell-Yan process qq → V * , and they are the same for WH and ZH production at any order of perturbation theory. At NNLO, they have been obtained in ref. [12] on the basis of ref. [53].
The other terms of eq. (2.1) only contribute at O(α 2 s ) or higher. They all involve top-or bottom-quark loops. The term σ VH I is very similar for WH and ZH production. Its Feynman diagrams can be obtained by inserting a top-or bottom-quark loop into the gluon lines of (real and virtual) NLO QCD diagrams for pp → V production, and radiating the Higgs boson off this loop. A sample diagram is shown in figure 2 (a); it is their interference with DY-diagrams which leads to σ VH I . The numerical effect of this contribution was evaluated in the heavy-top limit and by neglecting the bottom Yukawa coupling; it was found to be of the order of 1-2% of the total cross section [54].
The terms σ ZH II and σ ggZH collect contributions where the Z boson couples to a closed top-or bottom-quark loop. The analogous terms are absent for WH production as indicated by the Kronecker symbol δ VZ in eq. (2.1). The Higgs boson may then be emitted either from the Z boson, or from the closed quark loop. In σ ZH II , the quark loop is connected via gluons to an external quark line. This results in either two-loop diagrams for qq → ZH, see figure 2 (b), or in one-loop diagrams with two quarks, one gluon, and ZH as external states. The resulting amplitudes have to be interfered with the corresponding lowest order DY-contribution. In ref. [54], σ ZH II was found to be even smaller than σ VH I , typically at the sub-percent level.
Of particular importance for our analysis is the contribution σ ggZH , where the closed top-or bottom-quark loop is connected to two initial state gluons, the Z is radiated off that loop, and the Higgs is emitted from the Z or from the quark loop. The lowest order, i.e. O(α 2 s ), was calculated in refs. [12,18,19]. Due to the two initial state gluons, its numerical contribution to the total cross section can be significantly larger than the one of σ ZH II , although both are formally of the same order of perturbation theory. Its size depends on the center-of-mass energy though: while it was negligibly small for the Tevatron, for example, it amounts to about 4% (6%) at the LHC for 8 TeV (14 TeV). Since its impact on the theoretical uncertainty of the total cross section was quite significant, ref. [55] evaluated it at the next order in perturbation theory, i.e. O(α 3 s ), which is formally an N 3 LO contribution to the Higgs Strahlung process. These QCD corrections where found to be of the order of 100%, similar to what is observed in the gluon fusion process gg → H [56][57][58][59][60][61][62]. While the absolute renormalization scale dependence is almost the same at O(α 2 s ) and O(α 3 s ), the relative variation is smaller at O(α 3 s ) by a factor of two due to the large K-factor.

2-Higgs-doublet model
In the SM, contributions proportional to the bottom Yukawa coupling are typically strongly suppressed with respect to the corresponding top-quark induced terms. This is different in extended Higgs sectors like, for example, the 2HDM. An introduction to the 2HDM including a presentation of all relevant couplings in the neutral Higgs sector is given in appendix A.
In this paper, we consider the production of any of the three neutral Higgs bosons of the JHEP02(2014)023  2HDM, generically denoting them as φ ∈ {h, H 0 , A}, in association with a weak gauge boson V ∈ {W, Z}. The main differences to the SM are as follows: • The couplings g φ V V are different from the SM g H V V coupling.
• The bottom Yukawa coupling can be significantly enhanced and thus cannot be neglected in general.
• In addition to an off-shell gauge boson, also one of the three neutral Higgs bosons can occur as internal particle.
The implications of these changes are as follows. The DY contribution σ Vφ DY is obtained from the SM expression simply by reweighting the Higgs coupling to the gauge bosons: The 2HDM version of the gluon-initiated process, σ ggZφ , on the other hand, depends on g φ ZZ as well as on the relative Yukawa couplings g φ b and g φ t . In addition, instead of the intermediate Z boson in figure 1 (b), a (virtual) CP-odd/even Higgs boson φ ′ can be produced, which decays into a real CP-even/odd Higgs boson and a Z boson, see figure 3 [44][45][46][48][49][50]. As we will see, this contribution is absolutely essential for the correct description of the Higgs Strahlung process in the 2HDM. This is particularly true if the mass of the intermediate particle φ ′ is larger than the threshold, m φ ′ > m Z + m φ , in which case the propagator becomes resonant at the partonic cms energy √ŝ = m φ ′ . We regulate the associated divergence by replacing the propagator as where Γ φ ′ is the total width of φ ′ . Given a specific 2HDM, Γ φ ′ is calculable; we use the value provided by 2HDMC [63,64], which can be linked to the new version of vh@nnlo.
Since the O(α 3 s ) effects for σ ggZφ are only known for vanishing bottom Yukawa coupling [55], we will disregard these corrections in the 2HDM analysis and take into account only the O(α 2 s ) results [12,18,19]. For the 2HDM generalizations σ Vφ I and σ Zφ II of σ VH I and σ ZH II , also Feynman diagrams with closed bottom instead of top quark loops should be taken into account. Due to the fact that one cannot evaluate them in an effective theory analogous to the heavy-top limit, these contributions are extremely difficult to calculate and thus currently unavailable. Considering the smallness of these terms in the SM, however, it should be fair to neglect these terms altogether. Unless stated otherwise, we therefore set σ Vφ I = σ Zφ II = 0 in what follows. Finally, an enhanced bottom Yukawa coupling induces a new contribution σ bbZφ ∼ (y φ b ) 2 with respect to the SM, where the final state Zφ is produced in association with bottom quarks [45,[47][48][49][50]. If the latter are not tagged, the cross section can be calculated in the socalled five-flavour scheme where the perturbative series is re-arranged to resum logarithmic terms that arise from the collinear region of the final state bottom quark momenta (see, e.g., ref. [65] and references therein). The bottom quarks then appear as initial states, bb → Zφ, with proper parton density functions. The relevant Feynman diagrams at leading order are depicted in figure 4. Similar to the process ggZφ, they partly involve intermediate virtual Higgs bosons φ ′ , which we treat in the very same way as described above (see eq. (2.3)).
For the purpose of the current paper, the LO prediction of the bbZφ process shall be sufficient. We will see, however, that this contribution can be quite significant and even numerically dominant, in particular if the intermediate Higgs boson becomes resonant. NLO corrections to this process have been calculated in ref. [49]. They should be included in the analysis once sufficient experimental data are available.
Note that σ bbZφ exclusively refers to that part of the partonic process bb → Zφ which involves a bottom Yukawa coupling. The same process can also be mediated for y b = 0, JHEP02(2014)023 namely as part of σ Zφ DY , which we evaluate with five-flavor PDFs. Assuming massless bottom quarks, there is no interference between these DY-and the bbZφ-diagrams.
To summarize, we write the cross section for Higgs Strahlung in the 2HDM as

Implementation in vh@nnlo
For the numerical evaluation of the individual contributions, we use the program vh@nnlo [51] which we extended to the 2HDM. It turns out convenient to link vh@nnlo to 2HDMC [63,64] to allow for different parametrizations of the 2HDM, consistency checks of the input parameters, and additional information about the decays of the involved scalar particles. The latest version of vh@nnlo includes: • The Drell-Yan-like terms σ Vφ DY through NNLO by integrating the results of zwprod [53] and re-weighting with the proper 2HDM coupling.
• The gluon-initiated terms ggZφ at O(α 2 s ) for finite top and bottom Yukawa coupling, including the full top-and bottom-quark mass dependence, as well as terms with internal Higgs bosons, see figure 3. For the top-and bottom-quark masses, the on-shell values are used throughout the calculation of the ggZφ contribution. The corresponding part of the original vh@nnlo has been fully replaced by an updated code which provides higher flexibility concerning the choice of physical parameters. It was generated with the help of FeynArts/FormCalc [66,67] and requires vh@nnlo to be linked to the LoopTools [67] and CUBA [68] libraries. Regarding the resonant exchange of a Higgs boson φ ′ ∈ {h, H 0 , A} (see figure 3), we introduce the Breit-Wigner function as given in eq. (2.3) and insert the numerical value for the total width of φ ′ provided by 2HDMC [63,64]. The numerical integration of the Breit-Wigner peak is done by suitable Monte Carlo sampling.
• The bottom-quark initiated terms bbZφ at LO. The bottom mass entering the Yukawa coupling is taken to be the MS mass at the energy scale √ŝ of the incoming partons. The resonant contribution from internal (pseudo)scalars (see figures 4 (c,d)) is treated in the very same way as for the ggZφ terms, see above.
The only genuinely new terms with respect to the previous version of vh@nnlo are bbZφ and the s-channel contributions to ggZφ. Corresponding results have been reported on in the literature before [44][45][46][47][48][49][50]. We re-calculated them with the help of FeynArts/FormCalc [66,67], employing an adapted version of the 2HDM model file. Our analytic bbZφ result agrees with the one of ref. [50]. Numerical comparison of our results with the literature is impeded by their rather crucial dependence on the precise value of the parameters of the Higgs sector. Since most of the previous analysis focus on the MSSM, such comparisons could only be done at a rather qualitative level.
For bbZφ (φ ∈ {h, H 0 , A}) and ggZA production, we find satisfactory numerical agreement with ref. [50], but we fail to reproduce the numbers for gg → Zh and gg → ZH 0 . However, while our results for Zh production obey the required decoupling limit by approaching JHEP02(2014)023 the SM result for large M A , for example, we cannot verify this behavior in ref. [50]. The source of this difference is unresolved. 2 In case of the SM, higher order terms as presented in section 2.1 are available and recommended to be included: • Electro-weak contributions by assuming full factorization with the Drell-Yan-like terms, i.e. by replacing [69] σ VH The correction factor δ VH EW is implemented by interpolating a table of numerical results obtained from refs. [13,70].
• The terms σ VH I and σ ZH II , by implementing the results of ref. [54].
• The gluon-initiated terms ggZH at O(α 3 s ) with vanishing bottom Yukawa coupling, by implementing the perturbative correction factor in the heavy-top approximation [55].

Input parameters
Throughout this paper, we use the following numerical values for the SM parameters: where θ C is the Cabbibo mixing angle (mixing with the third quark generation can be safely neglected); the notation for the other quantities should be self-explanatory. Hadronic cross sections are calculated for a cms energy of √ s = 14 TeV. Furthermore, we use a value of α s (m Z ) = 0.119 as input for 2HDMC. For the cross section calculations however, the strong coupling constant is taken according to the corresponding parton density functions. As default set we use MSTW2008NNLO [71], for which α s (m Z ) = 0.11707.

The Wφ/Zφ ratio
From the previous discussion, it is clear that the ratio of the cross sections for Wφ and Zφ production is rather insensitive to radiative corrections in the SM. This feature is shared with the Drell-Yan production of lepton pairs, of course. However, due to the fact that R WZφ is sensitive to Higgs couplings to fermions, it provides a useful handle on possible deviations from the SM as soon as precise measurements of the ratio in eq. (2.7) are available. From

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the experimental point of view, the similarity of the Wφ and Zφ production processes should be reflected in a reduced uncertainty on the measured value of R WZφ as well. Any dependence on the beam luminosity drops out, for example; also b-tagging efficiencies are basically the same in the numerator and the denominator, with a marginal difference from kinematics due to m W = m Z . The theoretical prediction for R WZH within the SM is rather precise, since both the numerator and the denominator are known with NNLO accuracy. Using vh@nnlo, we calculate σ VH DY including electro-weak corrections, σ ggZH at O(α 3 s ), σ VH I , σ ZH II and σ bbZH , using the input parameters from section 2.4. When varying parameters for error estimation, we take the ratio at each point. We use the NNLO PDF sets from MSTW2008 [71], CT10 [72], and NNPDF23 [73], calculate the envelope of the error bands obtained from each set, and take its center and half width as prediction for R WZH and its PDF + α S uncertainty. For the estimation of the scale uncertainty, we vary the renormalization and factorization scale separately in an interval [ 1 3 µ 0 , 3µ 0 ] around our default choice µ 0 = m VH , the invariant mass of the VH system. The total uncertainty is obtained by adding both numbers linearly. Our results for various values of the Higgs mass are listed in table 1.
It is instructive to compare the uncertainty of the ratio R WZH to those of the VH cross sections. For m H = 125 GeV we obtain An uncorrelated error estimate for the ratio, i.e. adding the relative uncertainties of σ WH SM and σ ZH SM in quadrature, would lead to about 5.8% in total compared to 3.4%. Thus we observe that a significant part of the uncertainties indeed cancels in the ratio. The residual uncertainty is dominated by the scale uncertainty of the ggZH contribution to σ ZH , which has no counterpart in σ WH to cancel against.
In the rest of this paper, we will study the influence of possible non-SM physics on the ratio R WZφ for the production of a neutral Higgs boson φ. In a first step, we will simply modify the SM Yukawa couplings, without any particular underlying model in mind. Subsequently, we will study a general 2HDM, taking into account all the effects discussed in section 2.2.
In the following discussion, we use eq. (2.4) as the definition of σ Vφ , without O(α 3 s ) corrections to σ ggZH and electro-weak effects, since they are only known in the SM.

Effect of the Yukawa couplings
In this section, we will not respect any constraints on the third generation Yukawa couplings of the SM, but simply test the sensitivity of the ratio R WZφ on these parameters. For that purpose, we rescale the Yukawa couplings as  The dependence of the ggZH contribution (red/dashed) on the bottom Yukawa coupling is very small. Its y t dependence is non-trivial, though, due to interference terms. Over a large range of y t , the dominant contribution to ggZH originates from terms JHEP02(2014)023   gg → Z * → ZH that do not depend on any Yukawa coupling (see figure 1 (b)). Once y t is larger than about 3 y SM t , the ggZH component rises steeply. It surpasses the DY contribution at about y t = 7 y SM t , exceeding it by more than a factor of two at y t = 10 y SM t . The resulting WH/ZH ratios R WZH , normalized to the SM ratio R SM WZH , are shown as the solid line in figures 5 (d-f). Note that for consistency also the SM result has been obtained according to eq. (2.4) here, with electro-weak corrections and O(α 3 s ) effects to σ ggZH neglected. For κ 4, the deviation from the SM prediction (dashed) does not significantly exceed 10% within the range of κ considered here. An increased top Yukawa coupling, however, leads to a sharp decrease of R WZH : for κ = 10, the value of this ratio is almost three times smaller than its SM prediction.
To conclude this section, let us remark that, apart from the magnitude of the top Yukawa coupling, an anomalous sign of the top Yukawa coupling would be an undoubtful indicator of New Physics. Its measurement clearly requires interference effects; a determination through associated Higgs and single-top production has been suggested in ref. [74,75].
The gg → ZH process offers another, albeit more indirect way to measure the sign of the top Yukawa coupling due to the interference between box-and triangle diagrams (see, JHEP02(2014)023 e.g., figures 1 (b) and (c), respectively) [76]. Since this interference is destructive in the SM, the ratio R WZH decreases when changing the top Yukawa coupling to its negative. For example, at m H = 125 GeV and √ s = 14 TeV, the decrease is about 20% when using the LO result for gg → ZH; we estimate that this decrease can be as large as 30% when higher orders are included. In the boosted regime, this effect should be even more prominent due to the enhanced importance of the gg → ZH process (see section 5).
While the impact of a sign change in the bottom Yukawa coupling relative to its SM value is rather small, it might be worth investigating this issue as well in extended theories. A more detailed analysis is beyond the scope of this paper though.

2-Higgs-doublet model
As discussed in section 2.2, the Higgs Strahlung cross section in the 2HDM is not only affected by the modification of the couplings, but also by the contribution from additional Higgs bosons. In this section we will therefore extend the discussion of the previous section to the full 2HDM. We start with a comment on our choice of the 2HDM parameters, followed by a short note about the influence of internal scalars to Higgs Strahlung at the partonic level of the cross section. Then we focus on the production of the light Higgs and subsequently extend our discussion to the heavy and the pseudoscalar Higgs boson. In each case our main concern is the ratio R WZφ .
As explained before, σ Vφ I and σ Zφ II are not taken into account in the subsequent discussion of production cross sections in the 2HDM. The contribution σ ggZφ is calculated at O(α 2 s ), σ Vφ DY without electro-weak corrections.

Choice of 2HDM parameters
Apart from the SM input, the Higgs Strahlung cross section depends on the 2HDM Yukawa type (see appendix A) as well as the parameters tan β, sin(β − α), and the masses of the neutral Higgs bosons. If an intermediate Higgs state (see, e.g., figure 3) becomes resonant, the total decay width of that state enters the calculation, see eq. (2.3). This introduces a dependence on the parameter m 12 which, however, is weak. We therefore set m 12 = 0 in all our numerical examples. Note that according to stability, perturbativity, and unitarity requirements, this choice is actually allowed only as long as tan β 1. Different choices of m 12 can considerably loosen this constraint though, without affecting our numerical results very much. Identifying the observed Higgs signal at the LHC with the light Higgs of the 2HDM forces the value of sin(β − α) to be close to 1. In this paper, we largely disregard such constraints and always scan the full range of sin(β − α).

ggZφ and bbZφ contributions involving internal (pseudo)scalars
As pointed out in section 2.2, the cross section pp → φZ in the 2HDM includes ggZφ and bbZφ contributions that involve an internal scalar different from the final state Higgs particle φ. Before moving on to the full results, let us look at this contribution in a bit more detail.  Consider the production of the light Higgs h with mass m h = 125 GeV for sin(β − α) = 0.5 and tan β = 2. Figure 6 shows the partonic cross section gg → Zh as a function of the partonic cms energy √ŝ , with (black/solid) and without (red/dashed) the pseudoscalar A as internal particle, for three different values of its mass (a-c). Below the kinematical threshold √ŝ < m h + m Z ≈ 216 GeV, the ggZh contribution vanishes. In figure 6 (a), the mass of the pseudoscalar Higgs is m A = 200 GeV; it is therefore only produced off-shell. Already then, the numerical contribution to the partonic cross section is quite significant. In figure 6 (b) and (c), on the other hand, the pseudoscalar Higgs is assumed heavier and therefore becomes resonant. For m A = 300 GeV ( figure 6 (b)), its total width is Γ A = 2.82 GeV which leads to a sharp peak in the partonic cross section. For m A = 500 GeV (figure 6 (c)), on the other hand, the corresponding width becomes Γ A = 75.0 GeV and the peak is much broader. Independent of m A , the heavy and charged Higgs masses are chosen to be m H 0 = m H ± = 200 GeV.
All cases of figure 6 show the significance of the internal scalar to the ggZh contribution; a similar discussion applies to ggZH 0 and ggZA production, as well as to the bbZφ mode. The numerical effects of these terms on the hadronic cross sections and in particular on the ratio R WZφ will be discussed in the subsequent section. The ggZh terms are displayed as the red/dashed line; their amplitude involves contributions proportional to g h V V and contributions proportional to the Yukawa couplings g h b and g h t . Additionally, the process gg → A * → Zh is relevant, whose amplitude is proportional to g h AZ = cos(β − α) and g A b or g A t . In case of the type I 2HDM, both g h b and g h t (and g A b and g A t ) decrease with increasing tan β, which implies that for larger values of tan β the overall cross section σ ggZh is dominated by g h V V . For smaller values of tan β, the ggZh terms can become dominant even for rather large values of | sin(β − α)|.

Light Higgs
The bbZh contribution (blue/dash-dotted) is of no relevance in the type I 2HDM. Note that for sin(β −α) = ±1, all contributions assume their SM values as then cos α/ sin β = ±1 and therefore y t,b = ±y SM t,b (see appendix A). Figures 7 (d-f) show the corresponding ratios R WZh with respect to the SM ratio R SM WZH for the three different values of tan β as the solid line by taking into account all available contributions. At tan β = 1, the ratio exhibits a remarkably strong dependence on sin(β − α), also close to the edges. Towards larger values of tan β, the curves become flatter for sin(β − α) = ±1 since the Zh-specific contributions become more and more suppressed. At sin(β − α) = 0, Wh production is identical to zero in our approximation (recall that we neglect σ Vh I as well as σ Zh II ). Figures 7 (d-f) also include a dash-dotted curve which, for the parameters of this example, is almost indistinguishable from the solid curve. It shows the ratio R WZh without the σ bbZh contribution; this might be a useful quantity if the bbZφ process can be excluded efficiently by applying a b-tagging veto. In the current example, this is irrelevant, of course, but we will use the same notation in other examples, where the bbZφ process is numerically much more important.
In figure 8 we show the corresponding results for a type II 2HDM. The DY contributions are identical to type I, see figure 7. For tan β = 1, also the ggZh curve is quite similar to the type I case because the b-loop contribution is very small in both cases, and the top Yukawa coupling is identical. Since σ bbZh is again very small, also the total cross section -and therefore also the ratio R WZh , see figure 8 (d) -is almost the same as for type I. In the limit sin(β − α) = ±1, again all curves assume their SM value for the same reason as discussed above. These examples already show that a simple reweighting of the SM cross section by (g h V V ) 2 (which would resemble the behaviour of the DY contributions in the shown figures) is not appropriate in general. Even in regions where g h V V is close (but not equal) to ±1, the Yukawa contributions to the amplitude proportional to g h t and g h b (and g A t and g A b ) can become non-negligible.
If the mass of the pseudoscalar Higgs exceeds the kinematical threshold, m A > m h + m Z , it can be produced on-shell via gg → A → Zh or bb → A → Zh, leading to a substantial increase of these contributions to the cross sections under consideration, see figure 6. The parameters for figure 9 are identical to those of figure 7, except that the JHEP02(2014)023 pseudoscalar mass m A is increased from 200 GeV to 300 GeV. Since now we have to take into account the total width of the pseudoscalar Higgs, which in turn depends on m 12 and the heavy and charged Higgs masses m H 0 and m H ± , we have to specify these parameters as well. Unless stated otherwise, we will set m 12 = 0 and m H 0 = m H ± = m A in what follows.
The DY contributions are obviously not affected by the change in m A . For small tan β and sin(β − α) = ±1, the resonant gg → A → Zh process dramatically increases the ggZh contribution, making it dominant by more than a factor of 10 in most of the sin(β − α) region. The fact that it features a relatively constant behaviour for most values of α can be understood as follows: for small values of tan β, the amplitude is approximately proportional to (g Ah Z ) 2 = cos 2 (β − α). Squaring the propagator of eq. (2.3) results in a Breit-Wigner function. If we write the latter function in the narrow-width approximation the factor cos 2 (β − α) is canceled by 1/Γ A as long as the decay mode A → Zh is dominant. For | sin(β − α)| → 1 the decay mode A → Zh is suppressed. As soon as its partial width drops below the other partial widths, mainly Γ(A → bb), the contribution σ ggZh is instantly turning towards its SM value. In case m A > 2m t the decay channel A → tt opens and lowers the pseudoscalar contributions. For increasing tan β, the suppression of g A t and g A b reduces the impact of the pseudoscalar as internal particle. A similar statement holds for the contribution σ bbZh , but their contribution is down by a factor 10 −2 to 10 −3 .
Similarly to the case m A = 200 GeV, the impact of the ggZh and bbZh terms decreases towards larger tan β and the total cross section is described by the pure DY terms better and better. For tan β 5, however the ratio R WZh significantly deviates from its SM value even in the regions very close to sin(β −α) = ±1 (see figures 9 (d,e)). In fact, for tan β = 1, R WZh < 0.7 (meaning R WZh /R SM WZh < 0.4) for −0.91 < sin(β − α) < 0.99. Figure 10 shows the cross section dependence on sin(β − α) for a type II 2HDM. The parameters are the same as in figure 8, except that the masses m A = m H 0 = m H ± are increased from 200 GeV to 300 GeV. Similarly to the case for m A = 200 GeV, the curve for tan β = 1 very much resembles the one for type I, which is again due to the little impact of the bottom contributions, of course. Towards larger values of tan β, the bbZh fraction becomes dominant very quickly. The behaviour of the ratio R WZh is thus quite remarkable for all values of tan β, see figures 10 (d-f). If the bbZh component is disregarded (for reasons discussed above), the deviation of R WZh from its SM value becomes less drastic towards larger tan β, in particular close to sin(β − α) = ±1, but it still amounts to about 20% for | sin(β − α)| < 0.9 at tan β = 5.
The statements of this section are also valid in case of other choices of the parameters, in particular for other choices of m 12 , since m 12 only affects the decay widths of the internal pseudoscalar, as discussed above. If however the heavy Higgs and/or charged Higgs bosons are light enough to open the decay channels A → ZH 0 and/or A → W ± H ∓ in addition, the decay width of A is enlarged and thus the ggZh and bbZh cross section contributions get reduced. We will discuss this effect in the subsequent section for the heavy Higgs, where A → Zh, A → ZH 0 and A → W ± H ∓ are possible.

Heavy and pseudoscalar Higgs
The heavy Higgs couples to the heavy gauge bosons proportional to g H 0 V V = cos(β − α). Therefore, if the light Higgs resembles the SM-like Higgs, the Drell-Yan-like contributions σ VH 0 DY are strongly suppressed. In contrast, the Yukawa couplings are still sizable and therefore dominate this production process in the region | sin(β − α)| ≈ 1. In figure 11, we show the different contributions to the cross section for pp → ZH 0 in the type II 2HDM when the mass of the pseudoscalar Higgs is below the ZH 0 threshold (m H 0 = m A = 200 GeV). figure 12, on the other hand, demonstrates the presence of a resonance (m H 0 = 200 GeV, m A = 300 GeV). In contrast to the cases discussed in section 4.3, the internal pseudoscalar can then decay via A → Zh, A → ZH 0 and A → W ± H ∓ . Thus, the ggZH 0 and bbZH 0 contributions are not constant, but the proportionality to (g AH 0 Z ) 2 = sin 2 (β −α) dominates. Figure 11 (c-d) and figure 12 (c-d) present the ratios R WZH 0 with respect to the SM ratio. For | sin(β − α)| ≈ 1 the cross section σ WH 0 vanishes, whereas σ ZH 0 is dominated by Yukawa couplings, such that R WZH 0 tends towards zero. The detailed measurement of σ ZH 0 therefore provides important information about the Yukawa couplings of the heavy Higgs in case the light Higgs resembles the SM-like Higgs. The latter effect is less important in case of type I 2HDM with large values of tan β, since all Yukawa couplings are suppressed. The pseudoscalar Higgs of the 2HDM does not couple to the heavy gauge bosons, since g A V V = 0. However, the presence of Yukawa couplings g A t and g A b allows for large ggZA and bbZA cross section contributions. Figure 13 (a) demonstrates the size of both ggZA and bbZA cross section contributions to ZA production in the type II 2HDM with tan β = 20. As expected, the cross sections are hardly dependent on the mixing angle α. However, figure 13 (b) shows the presence of a heavy scalar H 0 , which can decay to ZA. In this case, the cross sections are slightly increased and a dependence on the mixing angle α is induced. Since we omit σ WA I , the production cross section σ(pp → W A) vanishes and a definition of R WZA is not reasonable for the pseudoscalar.

The gluon-induced component in the SM
It is well known that the signal-to-background ratio for VH production is significantly enhanced in the so-called boosted regime [52], where the transverse momentum p H T of the   Higgs boson is large. In this section, we therefore briefly study the influence of a lower cut on p H T on the effects observed in the previous sections. The discussion in this section will be on a rather qualitative level; more quantitative studies are beyond the scope of our paper and will be deferred to a future publication.
First we note that in the SM, the Higgs transverse momentum due to the gg → ZH sub-process is peaked at larger values than when the Higgs is produced through the DY-like process (see, e.g., ref. [76]). The effect of this is that the relative gg → ZH contribution is about twice as large as for the total cross section if one restricts the Higgs transverse momentum to p H T 150 GeV [77] (assuming m H = 125 GeV). Increasing the lower cut on p H T beyond that value leads again to a decrease of the relative gg → ZH portion since the spectrum drops rather sharply towards large p H T . The second step is to study the effect of New Physics on the gg → Zφ and the bb → Zφ sub-processes in the boosted regime. The shape of the p φ T spectrum is clearly unaffected for the DY-like terms in the scenarios considered in sections 3 and 4 of this paper. Figure 14 shows the gg → ZH (red) and the bb → ZH contribution (blue) relative to the SM one for modified top-and bottom-Yukawa couplings (parameters as in figure 5) without a p H T cut (dashed), and when the Higgs transverse momentum is restricted to p H T > 150 GeV (solid). Since σ bbZH is proportional to |y b | 2 in the SM, the corresponding dashed and solid line are identical.

Modified SM Yukawa couplings
For gg → ZH, on the other hand, it is remarkable that almost any modification of the top-Yukawa coupling leads to an increase in the large-p H T fraction with respect to the SM one (solid vs. dashed). A decrease is only observed for y SM t < y t < 2y SM t . Additionally, close to the SM value y t ≈ y SM t , the dependence on κ and thus on the top-Yukawa coupling is increased. Combined with the observations of a generally enhanced SM gg → ZH contribution with respect to the DY-terms, we conclude that the WH/ZH ratio becomes significantly more sensitive to the top-Yukawa coupling in the boosted regime.

2-Higgs-doublet model
Let us now turn from the simple modification of the Yukawa couplings to the full 2HDM. The main new effects here are due to the additional Higgs bosons φ ′ which can occur as virtual particles. Their impact is particularly large when valuesŝ ≈ m 2 φ ′ are kinematically allowed, see section 2.2. Applying a lower cut on p φ T restricts the values ofŝ tô For example, while the lower limit on √ŝ for the inclusive cross section is at m Z + m φ ≈ 216 GeV for m φ = 125 GeV, it moves up to about 370 GeV in the boosted regime with p φ T > 150 GeV. We therefore expect that the effects on the Wφ/Zφ ratio observed in section 4 decrease in the boosted regime.
First we consider the case corresponding to figure 7, i.e. 2HDM type I with m A = 200 GeV. Note that for the total inclusive cross section, the pseudoscalar is already slightly below resonant; a lower cut on p φ T moves the allowed interval forŝ further away from the resonance. Figure 15 shows the contributions of the subprocesses gg → Zh (red) and bb → Zh (blue) in the 2HDM relative to the SM quantities in this case. For the solid lines, a lower cut on p φ T (φ = h for the 2HDM, φ = H for the SM) of 150 GeV was applied, while this cut is absent for the dashed lines. The effect of the cut is non-negligible only for the case tan β = 1, where it leads to a moderate decrease of the gg → Zh fraction relative to the SM. The bb → Zh is completely negligible in this scenario anyway (see figure 7), so the reduction by the p φ T cut observed in figure 15 is irrelevant. For the gg → Zh component, the situation looks similar also in the 2HDM type II with non-resonant pseudoscalar (see figure 16; parameters as in figure 8). For moderate to large tan β, the effect of the p φ T -cut on the bb → Zh component, on the other hand, is quite drastic, leading to a reduction of about an order of magnitude.
The largest impact of a restriction to large p φ T , however, is expected in scenarios where this cut removes effects from resonant intermediate particles.  Quite generally, we conclude from this discussion that a restriction to the boosted regime may not be advantageous when searching for effects of New Physics. A deterioration of the signal-to-background ratio in the inclusive cross section may well be compensated by effects that are otherwise cut away.

Conclusions
We provided the SM prediction for the ratio of the cross sections for Higgs production in association with a W and with a Z boson, including all available and numerically relevant theoretical contributions: the Drell-Yan-like terms including NNLO QCD and NLO electroweak corrections, the so-called top-loop induced terms, as well as gluon-initiated effects through NLO. The residual theoretical uncertainty on this ratio is about 3%, induced by missing higher orders, variations in the PDFs, and the experimental error on α s (m Z ). The gluon-induced terms only affect ZH production, and since they are rather sensitive to New Physics effects, the WH/ZH ratio provides a useful test of the Standard Model, once statistics of the collected data is sufficiently large. As an example, we have considered a general 2HDM where it turns out that intermediate Higgs bosons can give a particularly large contribution to Zφ production. In addition, bottom quark annihilation becomes numerically relevant and in some cases even dominant. The consequence is a significant drop of the Wφ/Zφ ratio R WZφ as compared to its SM value, even close to the value sin(β − α) = 1 which seems to be preferred by the recent LHC results. This is even more so when the intermediate Higgs can become resonant. Restriction to the "boosted Higgs"-regime carries the potential danger of cutting away such resonant contributions; we therefore suggest to apply dedicated analyses also on the inclusive cross sections.
There is a number of ways to extend our study, which we leave for future investigations. Let us name just a few: • In this paper, we focussed on total inclusive Wφ and Zφ production, with only a rather qualitative consideration of the boosted-Higgs regime. Quite generally, restriction to particular kinematical regions or distributions may further improve the experimental and possibly also the theoretical significance of the effects observed here.
• In a more detailed phenomenological analysis, the decay of the final state Higgs boson φ has to be folded in. This does not affect the quantity R WZφ itself, of course, but JHEP02(2014)023 its experimental sensitivity. Note, however, that for m h = 125 GeV, the branching ratio to bb is significant unless the bottom Yukawa coupling becomes exceptionally small. Therefore, this main decay mode should be accessible in most of the 2HDM parameter space.
• Clearly, the Wφ/Zφ ratio should be studied also in other extensions of the SM, for example in supersymmetric models. Due to the tight restrictions on the parameters of the Higgs sector, we expect the effects in the MSSM to be substantially reduced compared to our findings for the 2HDM though.
• It was found that the O(α 3 s ) effects to the gluon fusion contribution σ ggZH in the SM are quite substantial [55]. A precise prediction of the Wφ/Zφ ratio in other models therefore requires the analogous corrections in these models. Similarly, the terms σ Vφ I and σ Zφ II could receive non-negligible contributions from bottom-quark loops if the bottom Yukawa coupling is large. The relevant calculations are non-trivial, however.
Finally, let us point out that a new release of the program vh@nnlo will include all of the 2HDM effects discussed in this paper and will allow for a flexible calculation of the Higgs Strahlung cross section in the 2HDM. Similarly, for the gluon fusion process gg → φ and bottom quark annihilation bb → φ, a link of SusHi [78] to 2HDMC will soon be available. Since ttφ production is typically suppressed in 2HDM models, and a rather good approximation for weak boson fusion is obtained by a simple rescaling of the SM cross section by g φ V V , all relevant cross section for Higgs production at the LHC within a 2HDM will then be available.
Note added. While this paper was in the reviewing process which led us to include section 5, ref. [76] appeared which touches upon similar issues. Obviously, the coupling of the light Higgs h to the gauge bosons equals the SM coupling for sin(β −α) = 1, where also the coupling to the pseudoscalar Higgs vanishes. For convenience of the reader, we add the relative Yukawa couplings in terms of sin(β − α): 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.