# Pseudo-observables in electroweak Higgs production

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## Abstract

We discuss how the leading electroweak Higgs production processes at the LHC, namely vector-boson fusion and Higgs\(+W/Z\) associated production, can be characterized in generic extensions of the Standard Model by a proper set of pseudo-observables (PO). We analyze the symmetry properties of these PO and their relation with the PO set appearing in Higgs decays. We discuss in detail the kinematical studies necessary to extract the production PO from data, and present a first estimate of the LHC sensitivity on these observables in the high-luminosity phase. The impact of QCD corrections and the kinematical studies necessary to test the validity of the momentum expansion at the basis of the PO decomposition are also discussed.

## Keywords

Higgs Boson Contact Term Custodial Symmetry Momentum Expansion Lead Order Prediction## 1 Introduction

Characterizing the properties of the Higgs boson, both in production and in decay processes, with high precision and minimum theoretical bias, is one of the main goal of future experimental efforts in high-energy physics and a promising avenue to shed light on physics beyond the Standard Model (SM). In this context, a useful tool is provided by the so-called Higgs pseudo-observables (PO) [1, 2, 3, 4, 5]. The latter constitute a finite set of parameters that are experimentally accessible, are well defined from the point of view of quantum field theory (QFT), and characterize possible deviations from the SM in processes involving the Higgs boson in great generality. More precisely, the Higgs PO are defined from a general decomposition of on-shell amplitudes involving the Higgs boson – based on analyticity, unitarity, and crossing symmetry – and a momentum expansion following from the dynamical assumption of no new light particles (hence no unknown physical poles in the amplitudes) in the kinematical regime where the decomposition is assumed to be valid.

The idea of PO has been formalized the first time in the context of electroweak observables around the *Z* pole [6, 7], while the generalization relevant to analyze Higgs decays has been presented in Ref. [1]. In this paper we further generalize the PO approach to describe electroweak Higgs-production processes, namely vector-boson fusion (VBF) and associated production with a massive SM gauge boson (VH).

The interest of such production processes is twofold. On the one hand, they are closely connected to the \(h\rightarrow 4\ell ,2\ell 2\nu \) decay processes by crossing symmetry, and by the exchange of lepton currents into quark currents. As a result, some of the Higgs PO necessary to describe the \(h\rightarrow 4\ell ,2\ell 2\nu \) decay kinematics appear also in the description of the VBF and VH cross sections (independently of the Higgs decay mode). This fact opens the possibility of combined analyses of production cross sections and differential decay distributions, with a significant reduction on the experimental error on the extraction of the PO. On the other hand, studying the production cross sections allows us to explore different kinematical regimes compared to the decays. By construction, the momentum transfer appearing in the Higgs decay amplitudes is limited by the Higgs mass, while such a limitation is not present in the production amplitudes. This fact allows us to test the momentum expansion that is intrinsic in the PO decomposition, as well as in any effective field theory approach to physics beyond the SM.

Despite the similarities at the fundamental level, the phenomenological description of VBF and VH in terms of PO is significantly more challenging compared to that of Higgs decays. On the one hand, QCD corrections play a non-negligible role in the production processes. Although technically challenging, this fact does not represent a conceptual problem for the PO approach: the leading QCD corrections factorize in VBF and VH, similarly to the factorization of QED corrections in \(h\rightarrow 4\ell \) [8]. As we will show, this implies that NLO QCD corrections can be incorporated in general terms with suitable modifications of the existing Monte Carlo tools. On the other hand, the relation between the kinematical variables at the basis of the PO decomposition (i.e. the momentum transfer of the partonic currents, \(q^2\)) and the kinematical variables accessible in *pp* collisions is not straightforward, especially in the VBF case. As we will show, this problem finds a natural solution in the VBF case due to strong correlation between \(q^2\) and the \(p_\mathrm {T} \) of the VBF-tagged jets.

The paper is organized as follows: in Sect. 2 we present the decomposition in terms of PO of the electroweak amplitudes relevant to VBF and VH, analyzing the relation with the decay PO already introduced in Ref. [1]. In Sect. 3 we present a phenomenological analysis of the VBF process, discussing in detail the implementation of QCD corrections, and the key role of the jet \(p_\mathrm {T} \) for the identification of the PO. An estimate of the statistical error expected on the PO extracted from VBF in the high-luminosity phase at the LHC is also presented. A similar discussion for the VH processes is presented in Sect. 4. A detailed discussion as regards the validity of the momentum expansion, and how to test it from data, is presented in Sect. 5. The results are summarized in Sect. 6.

## 2 Amplitude decomposition

*s*-channel as in the previous cases, but in the

*t*-channel. Strictly speaking, in VH and VBF the quark states are not on-shell; however, their off-shellness can be neglected compared to the electroweak scale characterizing the hard process (both within and beyond the SM).

Following Ref. [1], we expand the correlation function in Eq. (1) around the known physical poles due to the propagation of intermediate SM electroweak gauge bosons. The PO are then defined by the residues on the poles and by the non-resonant terms in this expansion. By construction, terms corresponding to a double pole structure are independent from the nature of the fermion current involved. As a result, the corresponding PO are universal and can be extracted from any of the processes mentioned above, both in production and in decays.

### 2.1 Vector-boson fusion Higgs production

Higgs production via vector-boson fusion (VBF) receives contribution both from neutral- and charged-current channels. Also, depending on the specific partonic process, there might be two different ways to construct the two currents, and these two terms interfere with each other. For example, in \(u u \rightarrow u u h\) two neutral-current processes interfere, while in \(u d \rightarrow u d h\) there is an interference between neutral and charged currents. In this case it is clear that one should sum the two amplitudes with the proper symmetrization, as done in the case of \(h\rightarrow 4e\) [1].

*Z*, and \(W^{\pm }\)), and to define the PO (i.e. the set \(\{\kappa _i, \epsilon _i\}\)) from the residues of such poles. We stop this expansion neglecting terms which can be generated only by local operators with dimension higher than six. A discussion as regards limitations and consistency checks of this procedure is presented in Sect. 5. The explicit form of the expansion of all the form factors in terms of PO can be found in Ref. [1]

^{1}and will not be repeated here. We report here explicitly only expressions for the longitudinal form factors, which are the only ones containing PO not present also in the leptonic decay amplitudes:

*Z*and

*W*boson to a pair of fermions: within the SM \(g_Z^f = \frac{g}{c_{\theta _W}} (T_3^f - Q_f s_{\theta _W}^2)\) and \(g^{ik}_{W} = \frac{g}{\sqrt{2}} V_{ik}\), where

*V*is the CKM mixing matrix and \(s_{\theta _W}\) (\(c_{\theta _W}\)) is the sine (cosine) of the Weinberg angle.

^{2}The functions \(\Delta ^\mathrm{SM}_{L, n.c. (c.c.)} (q_1^2, q_2^2)\) denote non-local contributions generated at the one-loop level (and encoding multi-particle cuts) that cannot be re-absorbed into the definition of \(\kappa _i\) and \(\epsilon _i\). At the level of precision we are working, taking into account also the high-luminosity phase of the LHC, these contributions can safely be fixed to their SM values.

As anticipated, the crossing symmetry between \(h\rightarrow 4 f\) and \(2 f \rightarrow h \, 2 f\) amplitudes ensures that the PO are the same in production and decay (if the same fermions species are involved). The amplitudes are explored in different kinematical regimes in the two type of processes (in particular the momentum transfer, \(q_{1,2}^2\), are space-like in VBF and time-like in \(h\rightarrow 4f\)). However, this does not affect the definition of the PO. This implies that the fermion-independent PO associated to a double pole structure, such as \(\kappa _{ZZ}\) and \(\kappa _{WW}\) in Eq. (6), are expected to be measured with higher accuracy in \(h\rightarrow 4 \ell \) and \(h\rightarrow 2\ell 2\nu \) rather than in VBF. On the contrary, VBF is particularly useful to constrain the fermion-dependent contact terms \(\epsilon _{Z q_i}\) and \(\epsilon _{Wu_i d_j}\), which appear only in the longitudinal form factors. For this reason, in the following phenomenological analysis we focus our attention mainly on the LHC reach on these parameters. Still, we stress that the PO framework is well suited to perform a global fit including production and decay observables at the same time.

### 2.2 Associated vector-boson plus Higgs production

By VH we denote the production of the Higgs boson with a nearly on-shell massive vector boson (*W* or *Z*), starting from and initial \(q\bar{q}\) state. For simplicity, in the following we will assume that the vector boson is on-shell and that the interference with the VBF amplitude can be neglected. However, we stress that the PO formalism clearly allows one to describe both these effects (off-shell *V* and interference with VBF in the case of \(V\rightarrow \bar{q} q\) decay) simply applying the general decomposition of neutral- and charged-current amplitudes as outlined above.

*V*, the form factors can only depend on \(q^2\). Already from this decomposition of the amplitude it should be clear that differential measurements of the VH cross sections as a function of \(q^2\) [9], as well as in terms of angular variables that allow one to disentangle different tensor structures, are an important input to constrain the PO.

### 2.3 Parameter counting, symmetry limits, and dynamical assumptions on the PO

We now want to analyze the number of free parameters and the symmetry limits for the newly introduced PO appearing in VBF and VH production, compared to the decay PO introduced in Ref. [1]. The additional set of PO (the “production PO”) is represented by the contact terms for the light quarks. In a four-flavor scheme, in absence of any symmetry assumption, the number of independent parameters for the neutral-current contact terms is 16 (\(\epsilon _{Z q^{i j}}\), where \(q=u_L, u_R, d_L, d_R\), and \(i,j=1,2\)): eight real parameters for flavor diagonal terms and four complex flavor-violating parameters. Similarly, there are 16 independent parameters in charged currents, namely the eight complex terms \(\epsilon _{W u^{i}_{L} d^{j}_{L}}\) and \(\epsilon _{W u^{i}_{R} d^{j}_{R}}\).

^{3}

*H*, which contribute also to non-Higgs observables. As a result, it is possible to derive relations between the Higgs PO and electroweak precision observables, as well as relations among Higgs PO which reduce the number of independent parameters. The matching to the SMEFT at the dimension-6 level in various bases, and the explicit relations among Higgs PO which follow, can be found in Refs. [1, 2]. Limiting the attention to the (presumably dominant) tree-level contributions, generated by dimension-6 operators, the following relations can be derived [2]:

*Z*- and

*W*-couplings to SM fermions, \(\delta g_{1,z}\) and \(\delta \kappa _\gamma \) are the anomalous triple gauge couplings (aTGC), and \(T^3_f\) and \(Y_f\) are the isospin and hypercharge quantum numbers of the fermion

*f*. Moreover, the custodial-symmetry relation (13) is automatically enforced at the dimension-6 level.

*Z*- and

*W*-pole observables within the SMEFT, with a generic flavor structure, can be found in Refs. [15, 16]. A combined fit to LEP-II WW and LHC Higgs signal strengths data, which removes all the flat directions in the determination of aTGC within the SMEFT has been presented in Ref. [17]. Combining some of these recent fits (in particular

*Z*- and

*W*-pole couplings from Ref. [15] and aTGC from Ref. [17]) we find the following numerical constraints on the quark contact terms (within the SMEFT):

*h*

*VV*vertices belong to this category, e.g. in Refs. [18, 19, 20, 21]. A specific example of this scenario are the parametric expressions of the Higgs PO in terms of the so-called “Higgs characterization framework” introduced in Refs. [18, 19]

^{4}:

Using FeynRules [10] we implemented a general UFO model [11] containing all the Higgs PO (including also decays [1]). The model itself will promptly be made available online [22] and allows for comprehensive phenomenological Monte Carlo studies at the LHC. A detailed implementation of the Higgs PO framework in a Monte Carlo tool including NLO QCD corrections will be presented in a subsequent publication.

## 3 Higgs PO in VBF production

### 3.1 VBF kinematics

Vector-boson fusion Higgs production is the largest of all electroweak Higgs-production mechanisms in the SM at the LHC. It is highly relevant in the context of experimental Higgs searches due to its striking signature, i.e. two highly energetic forward jets in opposite detector hemispheres, which allows an effective separation from the backgrounds. In this chapter we study the phenomenology of VBF production in the PO framework. We mainly concentrate our discussion on measuring the quark contact term PO, \(\epsilon _{Z q_i}\) and \(\epsilon _{Wu_i d_j}\), namely the residues of the single pole terms in the expansion of the longitudinal form factors in Eq. (6).

There is a potential caveat to the above argument: the color flow approximation ignores the interference terms that are of higher order in \(1/N_C\). Let us consider a process with two interfering amplitudes with the final-state quarks exchanged, for example in \(u u \rightarrow u u h\). The differential cross section receives three contributions proportional to \(|F^{f f^\prime }_L (t_{13},t_{24})|^2\), \(|F^{f f^\prime }_L (t_{13},t_{24}) F^{f f^\prime }_L (t_{14},t_{23})|\) and \(|F^{f f^\prime }_L (t_{14},t_{23})|^2\), where \(t_{ij}=(p_i-p_j)^2=-2E_i E_j (1-\cos \theta _{i j})\). For the validity of the momentum expansion it is important that the momentum transfers (\(t_{i j}\)) remain smaller than the hypothesized scale of new physics. On the other hand, imposing the VBF cuts, the interference terms turn out to depend on one small and one large momentum transfer. However, thanks to the pole structure of the form factors, they give a very small contribution.

Even though in some experimental analyses, after reconstructing the momenta of the two VBF-tagged jets and the Higgs boson, one could in principle compute the relevant momentum transfers \(q_1\) and \(q_2\), adopting the pairing based on the opening angle, in an hadron collider environment like the LHC this is unfeasible. Furthermore, for other Higgs decays modes, such as \(h\rightarrow 2\ell 2\nu \), it is not possible to reconstruct the Higgs boson momentum. Therefore, we want to advocate the use of the \(p_\mathrm {T} \) of the VBF jets as a proxy for the momentum transfers \(q^2_{1,2}\).

*z*axis) give

In order to confirm the above conclusion, in Fig. 2 we show a density histogram in two variables: the (observable) \(p_\mathrm {T} \) of the outgoing jet and the (unobservable) momentum transfer \(\sqrt{-q^2}\) obtained from the correct color flow pairing (the left and the right plots are for the SM and for a specific NP benchmark, respectively). These plots indicate a very strong correlation of the jet \(p_\mathrm {T} \) with the momentum transfer \(\sqrt{-q^2}\) associated with the correct color pairing. We stress that this conclusion holds both within and beyond the SM. Therefore, we encourage the experimental collaborations to report the unfolded measurement of the double differential distributions in the two VBF-tagged jet \(p_\mathrm {T} \): \(\tilde{F}(p_{T j_1}, p_{T j_2})\). This measurable distribution is indeed closely related to the form factor entering the amplitude decomposition, \(F_L(q_1^2, q_2^2)\), and encodes (in a model-independent way) the dynamical information as regards the high-energy behavior of the process. Moreover, as we will discuss in Sect. 3.3, the extraction of the PO in VBF must be done preserving the validity of the momentum expansion: the latter can be checked and enforced setting appropriate upper cuts on the \(p_\mathrm {T} \) distribution. As an example of the strong sensitivity of the (normalized) \(\tilde{F}(p_{T j_1}, p_{T j_2})\) distribution to NP effects, in Fig. 3, we show the corresponding prediction in the SM (left plot) and for a specific NP benchmark (right plot).

### 3.2 NLO QCD corrections in VBF

In the following we will illustrate that the perturbative convergence for exclusive VBF observables can be improved when using a dynamical scale \(\mu _0=H_\mathrm {T}/2 \) (with \(H_{\mathrm {T}}\) being the scalar sum of the \(p_\mathrm {T} \) of all final-state particles) with respect to a fixed scale \(\mu _0=m_W\). In particular, here we will focus on the \(p_\mathrm {T} \) spectra of the VBF jets – as inputs for a fit of the Higgs PO. To this end we employ the fully automated Sherpa+OpenLoops framework [33, 34, 35, 36, 37, 38] for the simulation of EW production of \(pp\rightarrow hjj\) at LO and NLO QCD in the SM. Before applying the VBF selection cuts defined in Eq. (17) we cluster all final-state partons into anti-\(k_{\mathrm {T}}\) jets with \(R=0.4\) and additionally require a rapidity separation of the two hardest jets of \(\Delta \eta _{j_1j_2} > 3\). This additional requirement, could slightly reduce the capability of differentiating different tensor structures [19], however, such a cut is, on the one hand, experimentally required in order to suppress QCD backgrounds.^{5} On the other hand, without such a cut NLO predictions for the \(p_\mathrm {T} \) spectra of the jets become highly unstable when the VBF jet selection is just based on the hardness of the jets, i.e. a bremsstrahlung jet is easily amongst the two hardest jets and spoils the correlation between the \(p_\mathrm {T}\) of the jets and the momentum transfer, as discussed in Sect. 3.1.

In Fig. 4 we plot the \(p_\mathrm {T}\) distributions of the hardest and the second hardest jet using a dynamical scale \(\mu _0=H_\mathrm {T}/2 \). On the left one-dimensional \(p_\mathrm {T}\) spectra are plotted, while on the right we show the corresponding two-dimensional NLO correction factors \(K^{\text {NLO}}=\sigma ^{\text {NLO}}/\sigma ^{\text {LO}}\).

Here CT10nlo PDFs [39] are used both at LO and NLO and uncertainty bands correspond to 7-point renormalization (only relevant at NLO) and factorization scale variations \(\mu _{\mathrm {R},\mathrm {F}}=\xi _{R,F}\mu _0\) with \((\xi _\mathrm {R},\xi _\mathrm {F})=(2,2)\), (2, 1), (1, 2), (1, 1), (1, 0.5), (0.5, 1), (0.5, 0.5).

Thanks to the dynamical scale choice NLO corrections to the one-dimensional distributions are almost flat and amount to about \(-15~\%\), while the dependence in the two-dimensional distribution remains moderate with largest corrections for \(p_\mathrm {T,j_1} \approx p_\mathrm {T,j_2} \).

In the following section we will detail a fit of Higgs PO based on LO predictions of VBF using the scale choice and setup developed in this chapter. Here we already note that this fit is hardly affected by the overall normalization of the predictions. Thus, with respect to possible small deviations from the SM due to effective form factor contributions we expect a very limited sensitivity to QCD effects assuming a similar stabilization of higher-order corrections as observed for the SM employing the scale choice \(\mu _0=H_\mathrm {T}/2 \).

In order to verify this assumption and to improve on the Higgs PO fit, we are currently extending the simulations within the Higgs PO framework to the NLO QCD level. To this end, the framework has been implemented in the OpenLoops one-loop amplitude generator in a process independent way. Here, the \(\mathcal {O}(\alpha _S)\) rational terms of \(R_2\)-type required in the numerical calculation of the one-loop amplitudes in OpenLoops have been obtained generalising the corresponding SM expressions [40].

The implementation of the dipole subtraction and parton-shower matching in the Sherpa Monte Carlo framework is based on the model independent UFO interface of Sherpa [41] and is currently being validated.

### 3.3 Prospects for the Higgs PO in VBF at the HL-LHC

The extraction of the PO from the double differential distribution \(\tilde{F}(p_{T j_1}, p_{T j_2})\) has to be done with care. Here we make an attempt to perform such an analysis. In the following we estimate the sensitivity of the HL-LHC, operated at 13 TeV with 3000 fb\(^{-1}\) of data, on measuring the PO assuming maximal flavor symmetry in a seven-dimensional fit to \(\kappa _{ZZ}\), \(\kappa _{WW}\), \(\epsilon _{Z u_L}\), \(\epsilon _{Z u_R}\), \(\epsilon _{Z d_L}\), \(\epsilon _{Z d_R}\), and \(\epsilon _{W u_L}\). The ATLAS search for \(h\rightarrow WW^*\) reported in Ref. [42] considers the VBF-enriched category in which the detection of two jets consistent with VBF kinematics is required. The expected yields in this category are reported in Table VII of Ref. [42]. After the final selection cuts at 8 TeV with 20.3 fb\(^{-1}\) of integrated luminosity, the expected number of Higgs VBF events in the SM is 4.7 (compared to 5.5 background events) in the \(e\mu \) sample. Rescaling the number of expected events with the expected HL-LHC luminosity (3000 fb\(^{-1}\)) and cross section, we expect about 2000 SM Higgs VBF events to be collected by each experiment. In the following, we make a brave approximation and neglect any background events in the fit and assume that the HL-LHC will observe a total of 2000 events compatible with the SM expectations.

In our analysis we choose the binning in the double differential distributions in the two VBF-tagged jet \(p_\mathrm {T} \) as \(\{30-100-200-300-400-600\}\) GeV. We use the UFO implementation of the Higgs PO in the Sherpa Monte Carlo generator [34, 41] to simulate VBF Higgs events over the relevant PO parameter space in proton-proton collisions at 13 TeV c.m. energy. Here we employ the VBF selection cuts as listed in Eq. (17) with the additional requirement \(\Delta \eta _{j_1j_2} > 3\). We verified that the results of the fit are independent on the precise value of this last cut. Renormalization and factorization scales are set to \(\mu _{\mathrm {R}/\mathrm {F}}=H_\mathrm {T}/2 \), as discussed in Sect. 3.2.

*a*is a label for each bin. Assuming that the HL-LHC “would-be-measured” distribution is SM-like and describing the number of events in each bin with a Poisson distribution, we construct a global likelihood

*L*and evaluate the best-fit point from the maximum of the likelihood. We then define the test statistic, \(\Delta \chi ^2 = - 2 \log (L/L_{\mathrm {max}})\), as a function of the seven PO. For more details of the statistical analysis see “Appendix”.

In Fig. 5, we show in red the \(1\sigma \) (\(\Delta \chi ^2 \le 1\)) and \(2\sigma \) (\(\Delta \chi ^2 \le 4\)) bounds for each PO, while profiling over all the others. The expected uncertainty on the \(\kappa _{ZZ,WW}\) is rather large (with a loosely bounded direction: \(\delta \kappa _{ZZ} \approx -3 \delta \kappa _{WW}\)), however, in a global fit to all Higgs data, these PO are expected to be much more precisely constrained from \(h\rightarrow 4\ell , 2\ell 2\nu \) decays. The most important conclusion of this analysis is that at the HL-LHC all five production PO can be constrained at the percent level. In the following we test the robustness of this conclusion.

^{6}As a result, \(\kappa _{ZZ}\) and \(\kappa _{WW}\) become unconstrained but the constraints on the contact terms do not change qualitatively. We thus conclude that their bounds do come from the shape information, i.e. the normalized distribution \(\tilde{F}(p_{T j_1}, p_{T j_2})\).

Furthermore, we have checked that the uncertainties on the entries of the \(X^a\) matrices, due to the finite statistics of our Monte Carlo simulations, do not impact the fit results. Details of this analysis are reported in “Appendix”. The approach sketched there can also be used to estimate the uncertainty of our result caused by missing higher-order theory corrections, most notably NLO electroweak effects. As anticipated, the latter can exceed the 10 % level in VBF [31, 32]; however, the largest contributions are due to factorizable corrections (EW Sudakov logarithms and soft QED radiation) that can be re-absorbed by a redefinition of the PO. From the results in Ref. [43] for the related process \(e^+ e^- \rightarrow \nu \bar{\nu }h\) we estimate non-factorizable NLO electroweak corrections to barely reach \(10~\%\) in some dedicated corners of the phase space (being typically well below such values in most of the phase space). To be conservative, we assign uncorrelated relative errors of \(10~\%\) in each element of the matrices \(X^a\), by introducing appropriate nuisance parameters, and redo the fit. Profiling over these nuisance parameters, in the Gaussian approximation, we find the following \(1\sigma \) uncertainties for the PO: \(\Delta \kappa _{ZZ} = 0.94\), \(\Delta \kappa _{WW} = 0.31\), \(\Delta \epsilon _{Z u_L} = 0.022\), \(\Delta \epsilon _{Z u_R} = 0.027\), \(\Delta \epsilon _{Z d_L} = 0.033\), \(\Delta \epsilon _{Z d_R} = 0.055\), and \(\Delta \epsilon _{W u_L} = 0.009\). Interestingly, comparing these with the Gaussian errors shown above, we conclude that the estimated sensitivity does not worsen significantly, indicating that statistical errors will still dominate. It is worth noting that the theoretical uncertainties are more relevant for the determination of \(\kappa _{ZZ}\) and \(\kappa _{WW}\) and less relevant for the contact terms PO.

Now that we have obtained the constraint on the PO, we can a posteriori check the consistency condition of the analysis, namely, that we are in the regime of small deviations from the SM prediction. In Fig. 6, we show the envelope of the allowed deviations in the leading-jet \(p_\mathrm {T} \) distribution, obtained by varying the PO inside the \(2\sigma \) region. As can be seen, the size of the distribution is well constrained up to \(400~\text {GeV}\). Equivalently, using \(| \epsilon _{X_f} | \lesssim 0.01\) to check the consistency condition (20), we find \(0.01 \times (600~\text {GeV})^2/m_Z^2 \lesssim 1\), suggesting that we have performed an analysis in a kinematical region where the momentum expansion is indeed reliable.

## 4 Higgs PO in VH production

### 4.1 VH kinematics

*W*or

*Z*boson are respectively the third and fourth most important Higgs-production processes in the SM, by total cross section. Combined with VBF studies, they offer complementary handles to limit and disentangle the various Higgs PO. Due the lower cross sections, so far these processes are mainly studied in the highest-rate Higgs decay channels, such as \(h\rightarrow b\bar{b}\) [44, 45, 46, 47] and \(h\rightarrow WW^*\) [48, 49, 50, 51]. The drawback of these channels are large backgrounds, which are overwhelming in the \(b\bar{b}\) case and of the same order as the signal in the \(WW^*\) channels. In the following we skip over the challenges and the difficulties due to the presence of large backgrounds in these dominant modes, focusing only on \(V+h\) decay channels with a good S/B ratio (which should become accessible at the HL-LHC). In those channels we analyze the prospects for the extraction of the corresponding production PO.

In the *Wh* process, for a leptonic *W* boson decay, the \(p_\mathrm {T,W} \) can not be reconstructed independently of the Higgs decay channel. It is tempting to consider the \(p_\mathrm {T} \) of the charged lepton from the *W* decay as correlated with the *Wh* invariant mass. However, we checked explicitly that any correlation is washed out by the decay.

### 4.2 NLO QCD corrections in VH

At the inclusive and exclusive level QCD corrections to VH processes are well under control [26, 27, 54]. The dominant QCD corrections of Drell–Yan-like type are known fully differentially up to NNLO [55, 56, 57] and on the inclusive level amount to about \(30~\%\) with respect to the LO predictions for both *Wh* and *Zh*. Remaining scale uncertainties are at the level of a few percent.

*Zh*in the SM looking at differential distributions in \(p_{\text {T},Z}\) and \(m_{Zh}\), while the qualitative picture is very similar for

*Wh*. The employed setup is as detailed already in Sect. 3.2, while here we do not apply any phase-space cuts. Although the natural scale choice for VH clearly is \(\mu _0=Q= \sqrt{(p_{h}+p_{Z})^2}\), here we employ a scale \(\mu _0=H_\mathrm {T}/2 \). With this scale choice the resulting differential distributions (to be utilized in the Higgs PO fit) are almost free of shape effects due to higher-order QCD corrections. A study of a similar stabilization including deformations in the Higgs PO framework will be performed in the near future.

In the case of *Zh* besides Drell–Yan-like production there are loop-induced contributions in \(g g \rightarrow Z h\) mediated by heavy quark loops, which in particular become important in the boosted regime with \(p_\mathrm {T,H} >200\) GeV [58, 59].

Besides QCD corrections also EW corrections give relevant contributions and shape effects to VH processes due to Sudakov logarithms at large energies. They are known at NLO EW [60, 61] and decrease the LO predictions by about \(10~\%\) for \(p_{\text {T},Z}=300\) GeV and by about \(15~\%\) for \(p_{\text {T},W}=300\) GeV. We stress that, as in the VBF case, the dominant NLO EW effects are factorizable corrections which can be re-absorbed into a redefinition of the PO.

### 4.3 Prospects for the Higgs PO in \(\varvec{Zh}\) at the HL-LHC

In order to estimate the reach of the HL-LHC, at 13 TeV and \(3000~ \text {fb}^{-1}\) of integrated luminosity, for measuring the Higgs PO in *Zh* production, we consider the all-leptonic channel \(Z \rightarrow 2\ell \), \(h \rightarrow 2\ell 2\nu \). The 8 TeV ATLAS search in this channel [51] estimated 0.43 signal events with \(20.3~\text {fb}^{-1}\) (Table X of [51]). By rescaling the production cross section and the luminosity up to the HL-LHC we estimate approximately \(\sim \)130 signal events at the SM rate.

*Z*boson. In order to control the validity of the momentum expansion we apply an upper cut of \(p_\mathrm {T} ^\mathrm{max} = 280~ \text {GeV}\), which corresponds approximately to \(q^2 \approx 600~ \text {GeV}\) (see Fig. 7). We bin the \(p_\mathrm {T,Z}\) distribution as \(\{0-20-40-60-80-100-120-160-200-240-280\}~\text {GeV}\). Using the UFO implementation of the PO within Sherpa we generate \(p p \rightarrow Zh\) events at \(13~\text {TeV}\) of c.o.m. energy. As in the VBF case, in each bin we have obtained the expression of the number of events as a quadratic function in the PO:

*a*denotes again the label of each bin. We assume the number of events for each bin to follow a Poisson distribution and we build the likelihood \(L(\kappa )\) as a function of the five PO listed above. The best-fit point is defined by \(L_{\text {m}ax}\) and we determine \(\Delta \chi ^2 = -2\log L / L_\mathrm{max}\). In Fig. 5 we show the resulting \(1\sigma \) (\(2\sigma \)) intervals for each PO with solid (dashed) blue lines, when all other PO are profiled. The expected bounds obtained in the

*Zh*channel are comparable in strength with the ones obtained in the VBF channel. In Fig. 9 we illustrate the \(2\sigma \) allowed deviation of the \(p_\mathrm {T,Z} \) distribution

*Zh*invariant mass spectrum provides very similar errors as those shown in Fig. 5. Again Gaussian errors obtained by expanding the likelihood as a quadratic function around the minimum overestimates the errors compared to the ones shown in Fig. 5, although here not as badly as in the VBF case:

### 4.4 Prospects for the Higgs PO in \(\varvec{Wh}\) at the HL-LHC

In the case of *Wh* production, in all the channels used for the Run-1 analysis, the signal manifests itself as a small excess over a large (dominating) background; see e.g. Ref. [51]. A detailed analysis for such processes should be performed evaluating carefully the backgrounds, which is beyond the scope of this work. However, given the high luminosity we are looking at, the golden channel \(h\rightarrow 4\ell \), \(W\rightarrow \ell \nu \) becomes an interesting viable possibility. It has been estimated by ATLAS that 67 signal SM events will be present with 3000 fb\(^{-1}\) of integrated luminosity [62]. We have thus decided to analyze the prospects of this clean channel only, to constrain the \((\kappa _{WW}, \epsilon _{W u_L})\) PO, with an analogous likelihood analysis as those performed for the *Zh* and VBF channels.

*Zh*analysis discussed above. In Fig. 5 we show the resulting \(1\sigma \) (\(2\sigma \)) intervals for each PO with solid (dashed) green lines, when the other PO is profiled. In this case the Gaussian approximation works well and provides the following \(1\sigma \) errors:

We conclude the last two phenomenological sections stressing that we have performed simplified estimates of the HL-LHC sensitivity on the contact term PO by separately considering a limited set of collider signatures. It is reasonable to expect that, including all possible signatures and performing a global fit, the sensitivity can significantly improve. However, such a global analysis should also consider the effect of backgrounds, neglected in this study.

## 5 Validity of the momentum expansion

The momentum expansion at the basis of the PO decomposition is an expansion on kinematical variables that are experimentally accessible. As such, the radius of convergence of this expansion can be checked, a posteriori, by means of experimental data. In particular, as pointed out in Sect. 3.3, a crucial check is represented by the consistency condition (20), where \(q^2_\mathrm{{max}}\) is controlled by \((p_\mathrm {T,j})^\mathrm{max}\) in VBF and \(m_{Vh}\) in VH (or, less efficiently, by \(p_\mathrm {T,Z} \) and \(p_\mathrm {T,H} \) in VH). More generally, the high-momentum behaviors of \(\mathrm{d}^2\sigma /\mathrm{d}p_\mathrm {T,j_1} \mathrm{d}p_\mathrm {T,j_2} \) (VBF) and \(\mathrm{d}\sigma /\mathrm{d} m_{Vh}\) (VH) provide a direct probe of the validity of the momentum expansion, or the absence of nearby NP poles.

Besides these direct probes of the high-momentum behavior of the cross sections, a further check to assess the validity of the momentum expansion is obtained comparing the fit performed including the full quadratic dependence of \(N^\mathrm{{ev}}_a\) on the PO, with a fit in which the \(N^\mathrm{{ev}}_a\) are linearized in \(\delta \kappa _X \equiv \kappa _X-\kappa _X^\mathrm{{SM}}\) and \(\epsilon _X\). The idea behind this procedure is that the quadratic corrections to physical observable in \(\delta \kappa _X\) and \(\epsilon _X\) are formally of the same order as the interference of the first neglected term in Eq. (6) with the leading SM contribution.

If the two fits (linear vs. quadratic) provide similar results, one can safely conclude that the terms neglected in the PO decomposition are indeed subleading. In principle, if the two fits yield significantly different results, the difference might be used to estimate the uncertainty due to the neglected higher-order terms in the momentum expansion. In practice, as will be illustrated below, this estimate turns out to be rather pessimistic and often an overestimate of the uncertainty on the PO.

*Zh*analysis, while only in the

*Wh*case the two fits give comparable results:

*Zh*, could explain the loose constraints obtained in the linear fit. If this was true, we should find that in simple models with less parameters the linear and quadratic fit should agree.

To check if the constraints obtained on the contact terms can, in fact, be used to bound explicit new physics scenarios, we employ a simple toy model. To this end, we extend the SM with a new neutral vector boson, \(Z'\), coupled to specific fermion currents (to be defined below) and to the Higgs, such that it contributes to VBF and VH (or better *Zh*) production. Since the goal of this section is to examine the validity of the momentum expansion with an explicit new physics example, we ignore all other phenomenological constraints on such a model (for example, electroweak precision tests, direct searches, etc.).^{7}

One the one hand, we compute the bounds on the mass and couplings of this new state from the analysis of the double differential \(p_\mathrm {T} \) distribution in VBF Higgs production (and the \(p_\mathrm {T} ^Z\) distribution in *Zh*). On the other hand, we integrate out the heavy \(Z'\) and match to the Higgs PO framework. Finally, we compare the bounds in the full model with the ones obtained from the Higgs PO fit.

### 5.1 Effect of the \({Z'}\) in VBF

We consider the case where the \(Z'\) couples to both the down and the up right-handed quarks, with two independent couplings, \(g_{Z'}^{d_R}\) and \(g_{Z'}^{u_R}\). In addition, we fix the \(Z'\) mass to two benchmarks values: (a) 700 GeV and (b) 2000 GeV. The main results of the analysis are shown in Fig. 10.

On the one hand, we perform a fit to the Higgs PO \(\epsilon _{Z u_R}\) and \(\epsilon _{Z d_R}\), while fixing all other PO to zero, and translate this bound on the relevant parameter space of the \(Z'\) model, namely the \(\{g_{d_R}, g_{u_R}\}\) plane. We report the results of the fit obtained with full quadratic dependence on the PO, as well as the results in which \(N_\mathrm{{ev}}\) is linearized in \(\delta \kappa _X\) and \(\epsilon _X\). In both cases, \(95~\%\) CL bounds are obtained by requiring \(-2\log L / L_{\max } \le 5.99\). On the other hand, using exactly the same binning and statistical treatment, we directly fit the \(Z'\) model parameters.

*t*-channel, and therefore its main effect is to reduce the amplitude for high values of \(q^2\).

### 5.2 Effect of the \({Z'}\) in *Zh*

In order to assess the validity of the momentum expansion in associated production, it is convenient to look first at the underlying partonic cross section. In Fig. 11 we show the partonic cross section \(d \bar{d} \rightarrow Z h\), as a function of the *Zh* invariant mass, for the two benchmark points of \(Z'\) model introduced above.

Both benchmark points have been chosen such that they generate the same contact term when the \(Z'\) is integrated out, \(\epsilon _{Z d_R} = 1.68 \times 10^{-2}\), which is within the \(2\sigma \) bound of our PO fit. The width of the \(Z'\) has been fixed to \(100~\text {GeV}\) and \(200~\text {GeV}\) for the light and heavy scenario, respectively. Using Eq. (31) and assuming no other decay mode is present, this corresponds to \(g_H \simeq 0.097 ~ (3.0)\) in the light (heavy) scenario. We have checked that our conclusions do no change by varying the total width, as long as the condition \(\Gamma _{Z'} \ll M_{Z'}\) is satisfied.

As expected, in the light scenario the cross section in the full model strongly deviates from the PO one well before the \(600~\text {GeV}\) cut-off imposed in the fit, implying that our PO fit is not reliable in this case. On the other hand, the scenario with a heavy and strongly coupled \(Z'\) shows a very good agreement with the full PO analysis up to \(\sim \)1 TeV, i.e. well above the UV cut-off of our analysis, implying that the analysis can be safely applied to such scenarios, and that it could be even improved by setting a slightly higher cut-off. In both cases, from Fig. 11 is clear that the linearized dependence on the PO is not sufficient to describe the cross section, even for energies much smaller than the \(Z'\) mass.

From this analysis we can anticipate the results of a comparison of various fits of *Zh* data, i.e. full model fit vs. PO fits using quadratic and linear dependence, as already done in the VBF case. In Fig. 12 we show the results of such fits. We stress that in all cases the analysis was exactly the same: we have analyzed the \(p_\mathrm {T} ^Z\) distribution up to \(280~\text {GeV}\), employing always the same binning (as discussed in Sect. 4.3). The solid red line represents the 95 % CL bound in the full model while the solid (dashed) blue line shows the bound obtained from the PO fit with quadratic (linear) dependence.

The distributions in Fig. 11 allow a straightforward interpretation of these results. In the heavy-\(Z^\prime \) case, the full quadratic expansion in the Higgs PO describes very well the \(m_{Zh}\) distribution before the cut-off of \(600~\text {GeV}\), while keeping only the linear dependence underestimates the new physics contribution. It is thus expected that in this case the bound will be much worse. In the light-\(Z^\prime \) case, both expansions with Higgs PO underestimate the cross section, thus providing a worse bound than in the full model. Still, the quadratic dependence does a significantly better job in approximating the complete model than the linear one, as in the VBF case.

From this illustrative toy-model example we can draw the following general conclusion with respect to the validity of the PO expansion: for underlying models that respect the momentum expansion, hence for models where the PO extracted from data satisfy, a posteriori, the consistency condition (20), the quadratic fit provides more reliable and thus more useful constraint on the PO. In such models the difference between quadratic and linear fit represents a large overestimate of the errors.

Summary of the “production PO”, namely the PO appearing in VBF and VH in addition to those already present in Higgs decays (classified in Ref. [1]). In the second column we show the independent PO needed for a given set of amplitudes, assuming both CP invariance and \(U(2)^3\) flavor symmetry. The additional variables needed if we relax these symmetry hypotheses are reported in the third and fourth columns. In the bottom row we show the independent PO needed for a combined description of VBF and VH under the hypothesis of custodial symmetry. The number of independent PO range from 12 (sum of the first two lines) to 4 (bottom row, second column)

Amplitudes/processes | \(U(2)^3\) flavor symm | Flavor non universality | CPV |
---|---|---|---|

Neutral currents | \(\epsilon _{Z u_L}, \epsilon _{Z u_R}\) | \(\epsilon _{Z c_L}, \epsilon _{Z c_R}\) | |

(VBF\(_{n.c.}\)+ | \(\epsilon _{Z d_L}, \epsilon _{Z d_R}\) | \(\epsilon _{Z s_L}, \epsilon _{Z s_R}\) | |

Charged currents | Re(\(\epsilon _{Wu_L}\)) | Re(\(\epsilon _{Wc_L}\)) | Im(\(\epsilon _{W u_L}\)) |

(VBF\(_{c.c.}\)+ | Im(\(\epsilon _{W c_L}\)) | ||

VBF and VH | \(\epsilon _{Z u_L}, \epsilon _{Z u_R}\) | \(\epsilon _{Z c_L}, \epsilon _{Z c_R}\) | |

(with custodial symm) | \(\epsilon _{Z d_L}, \epsilon _{Z d_R}\) | \(\epsilon _{Z s_L}, \epsilon _{Z s_R}\) |

In view of these arguments, we encourage the experimental collaborations to report the results of both linear and quadratic fits, as well as to perform such fits using different \(p_\mathrm {T} \) cuts.

## 6 Conclusions

Higgs physics is entering the era of precision measurements: future high-statistics data will allow us not only to determine the overall signal strengths of production and decay processes relative to the SM, but also to perform detailed kinematical studies. In this perspective, an accurate and sufficiently general parameterization of possible NP effects in such distributions is needed. In this paper we have shown how this goal can be achieved in the case of VBF and VH production, generalizing the concept of Higgs PO already introduced in Higgs decays.

As summarized in Table 1, the number of additional PO appearing in all VBF and VH production amplitudes is manageable. In particular, assuming CP invariance, flavor and custodial symmetry, only four new PO should be added to the set of seven PO appearing in \(h\rightarrow 4\ell , 2\ell 2\nu , 2 \ell \gamma , 2\gamma \) in the same symmetry limit [1]. This opens the possibility of precise global determinations of the PO from combined analyses of production and decay modes, already starting from the next LHC runs.

As extensively illustrated in Sects. 3 and 4, the key aspects of VBF and VH is the possibility of exploring sizable momentum transfers in the Green functions of Eq. (1). On the one hand, this maximizes the sensitivity of such processes to PO that are hardly accessible in Higgs decays. On the other hand, it allows us to test the momentum expansion that is intrinsic in the PO decomposition as well as in any EFT approach to physics beyond the SM. Key ingredients to reach both of these goals are precise differential measurements of \(\mathrm{d}^2\sigma /\mathrm{d}p_\mathrm {T,j_1} \mathrm{d}p_\mathrm {T,j_2} \) in VBF and \(\mathrm{d}\sigma /\mathrm{d} m_{Vh}\) in VH (or appropriate proxies such as \(p_\mathrm {T,H} \) and \(p_\mathrm {T,Z} \)). We thus encourage the experimental collaborations to directly report such differential distributions, especially in the kinematical regions corresponding to high momentum transfer.

As far as the PO fits in VBF and VH are concerned, we suggest to perform them setting a maximal cut on \(p_\mathrm {T,j} \) and \(m_{Vh}\), to ensure (and verify a posteriori) the validity of the momentum expansion. As illustrated by matching the PO framework to simplified dynamical NP models, it is also important to report the results of fits using both linearized and quadratic expressions for the cross sections in terms of PO. According to our preliminary estimates, the production PO could be measured at the percent level at the HL-LHC (in the case of maximal flavor symmetry, without the need of imposing custodial symmetry). This level would be sufficient to constrain (or find evidence of) a wide class of explicit NP models and, among other things, to perform non-trivial tests of the relations between electroweak observables and Higgs PO expected in the SMEFT.

## Footnotes

- 1.
With respect to [1] we modified the labels of the form factors: \(F_1 \rightarrow F_L\), \(F_3 \rightarrow F_T\), and \(F_4 \rightarrow F_{CP}\), and analogously for the \(G_i\).

- 2.
More precisely, \((g^{ik}_{W})_\mathrm{SM} =\frac{g}{\sqrt{2}} V_{ik}\) if

*i*and*k*refers to left-handed quarks, otherwise \((g^{ik}_{W})_\mathrm{SM}=0\). - 3.
Strictly speaking, having defined the quark doublets in the basis where down quarks are diagonal, the \(\epsilon _{Z u_L^{i j}}\) have a non-vanishing off-diagonal component [1]. However, this can be neglected for all practical purposes.

- 4.
- 5.
In fact, in most VBF analyses an even tighter selection of \(\Delta \eta _{j_1j_2} > 4.5\) is imposed.

- 6.
In order to stabilize the fit we assign a Gaussian distribution for \(\mu \) centered around 1 with \(\sigma = 10\).

- 7.

## Notes

### Acknowledgments

We would like to thank S. Höche and S. Kuttimalai for help with the UFO interface of Sherpa. Also we would like thank M. Duehrssen-Debling, S. Pozzorini, and A. Tinoco Mendes for useful discussions. This research was supported in part by the Swiss National Science Foundation (SNF) under contract 200021-159720.

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