Phenomenology of the Higgs Effective Lagrangian via FeynRules

The Higgs discovery and the lack of any other hint for new physics favor a description of non-standard Higgs physics in terms of an effective field theory. We present an implementation of a general Higgs effective Lagrangian containing operators up to dimension six in the framework of FeynRules and provide details on the translation between the mass and interaction bases, in particular for three- and four-point interaction vertices involving Higgs and gauge bosons. We illustrate the strengths of this implementation by using the UFO interface of FeynRules capable to generate model files that can be understood by the MadGraph 5 event generator and that have the specificity to contain all interaction vertices, without any restriction on the number of external legs or on the complexity of the Lorentz structures. We then investigate several new physics effects in total rates and differential distributions for different Higgs production modes, including gluon fusion, associated production with a gauge boson and di-Higgs production. We finally study contact interactions of gauge and Higgs bosons to fermions.


Introduction
The discovery of a new state featuring the characteristics of a Higgs boson [1,2] has led to an intense effort to determine whether it consists of the Standard Model (SM) Higgs boson [3,4]. On the one hand, its quantum numbers and properties have been theoretically studied in the context of the Standard Model and many of its extensions , while on the other hand, both Tevatron and LHC experiments have determined that the resonance is likely a scalar particle with properties close to those expected from a Standard Model Higgs boson [47][48][49]. In particular, dedicated studies have tried to unravel new physics by investigating possible deviations of the couplings of the new state to the Standard Model particles .
Alas, no direct evidence of physics beyond the Standard Model has been observed in the light of current data [80,81], up to some anomalous multilepton events recorded by the CMS detector [82] which could find a possible explanation from gauge-mediated supersymmetrybreaking models [83]. With this state of affairs, a practical way for investigating new physics lies in a description based on an effective field theory valid up to a scale Λ lying around the TeV scale. In this context, the dynamics of the elementary particles is described through higher-dimensional operators featuring the Standard Model fields and only constrained by the SM symmetry group. This approach allows to characterize the properties of the newly observed Higgs boson, and also has the advantages to be renormalizable order by order in the E/Λ expansion, E being a typical energy scale for the processes of interest, as well as to exhibit a small number of free parameters as the gauge symmetries and the hierarchy among the operators of different canonical dimensions strongly constrain the set of operators relevant for a given purpose. Consequently, the path of effective field theories for Higgs physics has been widely explored [84][85][86][87][88][89][90][91][92][93][94][95][96][97][98][99], although one of its major drawbacks consists of the loss of unitarity, and thus of predictivity, for energy scales greater than Λ.
In this work, we focus on the dominant dimension-six operators related to Higgs physics, i.e., those involving at least one Higgs or gauge field, and neglect possible fourfermion interactions allowed by the SM gauge symmetries. Whilst limits on the magnitude of the associated Wilson coefficients can be extracted from LEP and Tevatron data, Higgs signal strengths deduced from measurements by both the ATLAS and CMS collaborations also now imply constraints on a subset of the allowed operators [100][101][102][103][104][105]. However, as more information on the Higgs-boson properties is obtained by LHC experiments, investigating new physics effects with a simple fit of the Higgs signal strengths clearly becomes too naive. More statistics indeed allows one for employing various kinematical distributions as powerful handles to look for physics beyond the Standard Model in the Higgs sector. In these perspectives, it is necessary to rely on sophisticated tools that cover the implementation of the Higgs effective operators in Monte Carlo simulation programs, the latter further leading to a possible recasting of the experimental analyses and a subsequent comparison of the theoretical predictions, in the framework of an effective field theory for Higgs physics, with experimental data.
Within the last few years, a framework based on the FeynRules package [106][107][108][109][110][111][112][113] has been developed in order to facilitate the implementation (and validation) of any beyond the Standard Model theory in multi-purpose matrix-element generators [114]. In particular, its virtues have been illustrated in the context of the CalcHep [115][116][117][118], FeynArts/FormCalc [119][120][121][122][123][124], MadGraph [125][126][127][128][129], Sherpa [130,131] and Whizard [132,133] programs. Furthermore, it also offers the possibility to translate any particle physics model in terms of a Python library under the so-called Universal Feyn-Rules Output (UFO) format [134]. Previously designed model formats for Monte Carlo programs are usually imposing restrictions on the color and/or Lorentz structures and the number of external legs that are allowed in the interaction vertices so that the UFO has been conceived to overcome such limitations. It is therefore suitable for effective field theories that often contain vertices with uncommon Lorentz structures.
We present in this work an implementation in FeynRules of a Higgs Effective Field Theory that includes a set of independent dimension-six operators assumed to encompass all possible effects of new physics on the Higgs sector. This set of operators can further be easily complemented by 22 four-fermion operators, irrelevant for Higgs physics (at least at leadingorder and in the context of the LHC phenomenology) and thus omitted here, in order to achieve a full basis of independent dimension-six operators featuring only Standard Model fields. In this way, high-energy physics programs such as Aloha [135], GoSam [136,137], Herwig++ [138,139], MadAnalysis 5 [140,141] and MadGraph 5, capable to fully handle a UFO model, can be directly employed for phenomenological investigations of new physics in the context of the effective field theory model presented in this work. For the sake of the examples, we demonstrate the usefulness of such an implementation by studying various beyond the Standard Model effects in both Higgs production rates and kinematical distributions by means of the FeynRules and MadGraph 5 packages and their interface through the UFO.
Our work is organized as follows. In Section 2, we extensively describe the operators which are included in our implementation. Next, we move onto studying several properties of the Higgs boson as examples of phenomenological studies that can be performed from this implementation. Their thorough study, possibly carried at the most sophisticated level of collider physics simulations, is however beyond the scope of this work, as well as a careful design of experimentally allowed values for the Wilson coefficients of the various effective operators that we briefly comment in Section 3. In Section 4.1, we firstly focus on Higgsboson production by gluon fusion and on the constraints that can be possibly extracted on the custodial symmetry or from measurements of angular distributions in the case the Higgs boson decays via intermediate gauge bosons. Secondly, we consider in Section 4.2 the associated production of a Higgs boson with one weak boson and investigate the ratio of the total production rates at center-of-mass energies of 8 TeV and 14 TeV as well as the invariant-mass spectrum of the Higgs and vector boson system. Thirdly, we assess the effects of some effective operators on the production of a Higgs-boson pair in Section 4.3, focusing on boosted topologies. Finally, we study in Section 4.4 contact interactions of a single Higgs field, a single gauge boson and a fermion pair and compare the LHC reach to limits extracted from LEP data. Our conclusions are given in Section 5.

The most general effective Higgs Lagrangian in the gauge eigenbasis
The Standard Model of particle physics is a quantum field theory which describes the elementary particles and their interactions based on the SU (3) c × SU (2) L × U (1) Y gauge symmetry. The particle content of the model can be classified in terms of a gauge sector, consisting of vector fields responsible for the mediation of the interactions, a chiral sector with the matter building blocks and eventually a Higgs sector related to the breaking of the electroweak symmetry. We now introduce our notations and start with the gauge vector fields lying in the adjoint representation of the relevant gauge subgroup, which we show together with their full representation under the Standard Model gauge group and explicitly indicate the adjoint gauge indices. The chiral content of the theory is defined by three generations of left-handed and right-handed quark (Q L , u R and d R ) and lepton (L L and e R ) fields, which we again present together with their representation under the Standard Model gauge group. Finally, the Higgs sector contains a single SU (2) L doublet of fields, With the first equality, we show the component fields of the doublet after shifting the neutral field h by its vacuum expectation value v. Moreover, we have included the Goldstone bosons G +,0 to be eaten by the weak boson to get their longitudinal degree of freedom.
In the effective field theory-based approach that we adopt, the usual Standard Model Lagrangian L SM is supplemented by higher-dimensional operators that parametrize the possible effects of non-observed states assumed to appear at energies larger than an effective scale identified with the W -boson mass m W or equivalently with the vacuum expectation value of the Higgs field v. Restricting ourselves to operators of dimension less than or equal to six, the most general gauge-invariant Lagrangian L is known for a long time [84][85][86] and can be expressed, in a convenient basis of independent operators O i [93,94], as assuming baryon and lepton number conservation. Moreover, we adopt the decomposition of Ref. [97] and normalize the Wilson coefficientsc i as such. Other choices for the operator basis yield different physics interpretations and care must be taken when comparing various works on the topic. According to the purposes, other bases might be more adequate.
The first piece of this Lagrangian, L SILH , corresponds to the a popular set of CPconserving operators involving the Higgs doublet Φ. Their (conventional) normalization is inspired by scenarios where the Higgs field is part of a strongly interacting sector, where the Wilson coefficientsc are free parameters, λ stands for the Higgs quartic coupling and y u , y d and y are the 3×3 Yukawa coupling matrices in flavor space (all flavor indices are understood for clarity). In this expression, we also denote the U (1) Y , SU (2) L and SU (3) c coupling constants by g , g and g s , whereas the generators of SU (2) in the fundamental representation are given by T 2k = σ k /2, σ k being the Pauli matrices. Additionally, we have introduced the Hermitian derivative operators ← → D µ defined as 6) and the SU (2) invariant products the rank-two antisymmetric tensors being defined by 12 = 1 and 12 = −1. Finally, our conventions for the gauge-covariant derivatives and the gauge field strength tensors are can be supplemented by extra CP -violating operators, where the dual field strength tensors are defined by The third term in Eq. (2.4) contains interactions between two Higgs fields and a pair of quarks or leptons, whilst the fourth term of this Lagrangian addresses the interactions of a quark or lepton pair and one single Higgs field and a gauge boson, In this expression, the matrices T a are the generators of the SU (3) group in the fundamental representation and the γ µν quantities, defined by 13) are the generators of the Lorentz algebra in the (four-component) spinorial representation.
In the most general case, the Wilson coefficientsc i related to the fermionic operators included in the Lagrangians L F i are tensors in flavor space and complex quantities. The last term of Eq. (2.4) refers to operators not directly connected to Higgs physics, but that may be important as affecting the gauge sector and possibly modifying the gauge boson self-energies and self-interactions, with recalling that D µ W νρ k has been defined in Eq. (2.8). Finally, 22 independent baryon and lepton number conserving four-fermion operators are also allowed by gauge invariance. Since they have no effects on Higgs physics, at least at the leading order and in the context of the LHC phenomenology, we omit them from the present manuscript, as already above-mentioned. Moreover, as indicated in Ref. [97], two of the 39 operators that have been introduced are redundant and can be removed through where we sum over the whole chiral content of the theory and θ W stands for the weak mixing angle (see Eq. (2.17) and Eq. (2.18) in Section 2.2). We however include them in our effective field theory description as according to the specific effect of interest, one choice for an operator basis may be more suitable than another. The full set of interactions generated by the 39 operators included in the Lagrangian L of Eq. (2.4) have been implemented in a FeynRules [106,113] model file that can be either downloaded from http://feynrules.irmp.ucl.ac.be/wiki/HEL or obtained from the authors upon request. Employing the various interfaces linking Feyn-Rules to Monte Carlo event generators, the effects of those operators can be further studied at colliders. Care must however be taken with the choice of the Monte Carlo generator to employ for this purpose, as most of them have strong requirements on the allowed Lorentz structures for the vertices 1 . To avoid such limitations and be capable of probing any of the 39 considered operators, we make use, for the few examples of Section 4, of the MadGraph 5 program [129] linked to the UFO [134] version of the model presented above.

The most general effective Higgs Lagrangian in the mass eigenbasis
After electroweak symmetry breaking to electromagnetism, i.e., when the Higgs field acquires its vacuum expectation value, the gauge eigenstates introduced in Eq. (2.1) mix to the physical massive W -boson and Z-boson as well as to the photon A which remains massless, (2.17) In this equation, we have introduced the weak mixing angle at tree-level whose sine and cosine, noted as s W and c W , are defined with respect to the hypercharge (g ), weak (g) and electromagnetic (e) coupling constants, First, it can be observed that some of the new physics operators induce a modification of the kinetic terms of the gauge and Higgs bosons, once the neutral component of the Higgs doublet gets its vacuum expectation value v. For instance, the O G operator of Eq. (2.5) leads to a variation of the gluon field strength tensor squared term given by The interfaces take care of discarding each vertex that is not compliant with the relevant program.
Appropriate field redefinitions (at the first order in the effective couplings) are then required in order to bring back the various kinetic terms to their canonical forms, and read While in our operator basis convention, the mass of the W -boson is agnostic of any new physics effect, there is an impact on both the Z-boson and Higgs-boson masses, In this last relation, we have made use of the minimization condition of the Higgs potential that reads, again in the first order in the effective couplings, (2.23) These considerations and simplicity have motivated us to choose the electroweak input parameters that are used in the FeynRules model file. They consist in the mass of the W -boson m W , the Fermi constant G F , the electromagnetic coupling constant α, as well as the Higgs-boson mass m H . The field redefinitions of Eq. (2.21), the replacement of the internal parameters in terms of the external ones and the removal of any term of higher order that is possibly arising from the procedure sketched above have been fully automated. One could have implemented the effective Lagrangian of Section 2.1 directly in the mass basis. This choice, adopted for instance in Ref. [46], is in general clearer when one aims to properly estimate new physics effects that can be hinted for by a given physical observable. Several of the effective operators introduced in the previous section can indeed give rise to the same interaction in the mass basis, with the same Lorentz structure, so that it is not straightforward, if not impossible, to map a single measurement to a single operator. Moreover, a Lagrangian expressed in the mass basis also offers an easy way to be generalized by including appropriate form factors to model new physics effects. We consequently dedicate this section to a proper comparison of the two approaches, linking the effective Lagrangian of Section 2.1 that is expressed entirely in the gauge basis to its counterpart as read off in terms of mass-eigenstates after electroweak symmetry breaking.
We now turn to the investigation of the interaction terms of the Lagrangian. Focusing on the terms related to the Higgs sector one writes  where L i denotes the set of i-point interactions involving at least one Higgs boson. For the sake of the example, we only work out explicitly the structure of the interactions which are the more relevant for the phenomenology of the Higgs sector at the LHC, namely the three-point and four-point interactions involving at least one Higgs field. Vertices involving a higher number of external legs are in general related to processes associated with smaller cross sections, making them difficult to probe. We therefore refer to the FeynRules implementation for their explicit form in the mass basis.
Eq. (2.25) -Eq. (2.26) Section 2.1 Wc H In the unitarity gauge and in the mass basis, the L 3 Lagrangian reads where the flavor indices of the fermions are understood and where we have introduced the W -boson, Z-boson and photon field strength tensors, W µν , Z µν and F µν . As already mentioned above and in Ref. [46], this form of Lagrangian is sufficient to characterize all Higgs properties in a non-ambiguous way. This contrasts with the initial set of operators in the gauge basis which leads to additional structures that can be removed after several integrations by parts. In addition to all the operators already included in the Lagrangian of Ref. [46], i.e., the Higgs to diboson couplings defined in Table 1 and the Higgs to fermions  interactions of Table 3, we also include the triple Higgs interactions presented in Table 2. Furthermore, the tables also contain a translation dictionary linking our free parameters to the general Lagrangian employed in Ref. [46].  Table 3. Coupling strengths of the interactions of one or several Higgs boson(s) with a fermion pair and possibly an additional gauge boson. We present the relations between the Lagrangian parameters introduced in Eq. (2.25) (first column), where the Lagrangian is expressed in the mass basis, and those associated with the operators of Section 2.1 expressed in the gauge basis (third column). In our notations, V CKM denotes the CKM mixing matrix. Concerning the three-point interactions, we relate our parameters to those of the Lagrangian description of Ref. [46] in the second column of the table.
The set of four-point interactions involving one or several Higgs fields is deduced from where the free parameters are given in the various tables of this section. Hence, Table 3 contains the parameters related to the interactions among one Higgs boson h, a fermionantifermion pair and possibly an additional gauge boson, while Table 2 and Table 4 are respectively dedicated to the Higgs boson self-interactions and to its interactions with vector bosons. We recall that some of the interaction vertices among a single Higgs field and three gluons were already included in the Lagrangian of Eq. (2.25) through the gluon field strength tensor and its dual. Consequently, these are omitted from Eq. (2.26). Along the same line, we include in a gauge-invariant way higher-dimensional Higgs boson and gluon vertices.
Eq. (2.26) hhzz , g hhaz , g hhaz , g hhww , g hzz , g haz , g haz , g hww , g Both Lagrangians L 3 and L 4 not only exhibit Lorentz structures common with the Standard Model interactions, but also some novel ones. For instance, focusing on the Higgs to W -boson trilinear interactions, the complete Feynman rule reads where only the component proportional to the metric is present in the Standard Model. and after splitting the different terms into three-point and four-point contributions, respectively, and neglecting any other vertex containing more than four external legs. We collect the coefficients of the different operators entering those Lagrangians in Table 5 and Table 6 and give their expressions in terms of the Wilson coefficients of the Higgs effective Lagrangian presented in Section 2.1.

Experimental constraints on dimension-six effective operators
The magnitude of the Wilson coefficients associated with the dimension-six operators introduced in Section 2, and thus an estimate of their impact on physical observables, can be g wwww naively computed by a simple power counting [93,97]. In this way, each power of Φ leads to a g N P /M suppression factor, M being the typical mass scale of the new physics sector and g N P the coupling strength of the new states to the Higgs field, while each derivative implies an additional reduction of 1/M . Additionally, in the framework of a given theory, specific operators can be generated at the one-loop level so that additional suppression can be foreseen. Sticking to tree-level, Wilson coefficients such asc H ,c T ,c 6 orc ψ are expected to be of the order ofc and can therefore be quite large for strongly coupled new physics, while in contrast, the coefficients of operators such asc W andc B scale as and are thus expected to be relatively suppressed or enhanced according to the value of the ratio g/g N P .
In addition, the value of the 39 Wilson coefficients introduced in Section 2 is experimentally constrained from various sources. These limits mainly arise from collider data (LEP, Tevatron, LHC), electric dipole moment measurements, rare decay bounds and results of several experiments dedicated to the measurement of the anomalous magnetic moments of the muon and the electron. An exhaustive list of the current bounds on the various coefficients of the considered dimension-six operators can be found in Ref. [97] so that we omit it from the present manuscript.
Some of these constraints however only involve specific combinations of Wilson coefficients instead of a single coefficient. An example lies in the recasting, in the language of the Higgs effective Lagrangian of Section 2.1, of the constraints originating from the electroweak precision parameters derived in Ref. [142], This still leaves open the possibility of a cancellation between two largec W andc B coefficients at the Z-pole. The possibility of such cancellations also holds for other coefficients, such asc HW orc HB , for which there exists no strong bound from the LEP experiments. A more involved example arises from the strong limits on the quantityc W W defined by a combination of several of thec i coefficients, Existing data being too limited to allow for disentangling the individual effects of the different operators, this implies that no constraints can be really inferred in our choice of basis for the dimension-six operators. Those considerations motivate us to avoid a careful design of a benchmark scenario theoretically motivated and not experimentally excluded, a task that can only be performed from a global fit of all data and considering all dimension-six operators, including additional four-fermion interactions, their one-loop mixings as well as field equations of motion linking our basis of operators to an extended one with redundant operators. We instead pick up, in Section 4, several phenomenological examples illustrating possible usages of the Higgs effective field theory implementation achieved in this work, so that it could be possibly used for such global fits in future works.

Interactions between the Higgs boson and the electroweak gauge bosons
As first examples of a possible usage of our implementation, we focus on the restricted set of operators implying modifications of the hV V couplings between one single Higgs boson and a gauge-boson pair. New physics effects possibly arising in such interactions have been largely investigated in pioneering and more recent works prior the discovery of a Higgs boson by the ATLAS and CMS experiments [5-18, 20, 22-24, 26, 87-89, 91, 92, 95, 96, 143-145]. Within the last two years, they have been reassessed in the light of newer LHC data including the Higgs results [28-30, 34, 37, 45, 100-105, 146]. We reproduce and extend, in the rest of this subsection, some of these results by employing the MadGraph 5 package [129], using our FeynRules [106,113] implementation to generate the necessary UFO model files [134]. The events have eventually been analyzed by means of MadAnalysis 5 [140,141].

Probing the custodial symmetry
Among all the Wilson coefficients included in the Lagrangian of Eq. (2.4), thec W parameter has the specificity to imply modifications of both the hZZ and hW W interactions simultaneously. Its value can therefore be probed by independent measurements of the Higgs boson properties in its W W and ZZ decay modes, when considering, e.g., leptonic weak boson decays, Analyzing the full set of LHC data describing collisions at center-of-mass energies of 7 TeV and 8 TeV, considering those two decay patterns and a Higgs-boson production by gluon fusion, the ATLAS collaboration has recently 2 reported a measurement [148] of the ratio of the two corresponding branching fractions estimated relatively to the Standard Model expectations, Assuming a Higgs boson mass of m h = 125.5 GeV, the quantity λ W Z has been found to lie in a range defined by at the 95% confidence level and when keeping the strength of the Higgs boson to two photon coupling as a free parameter. This last assumption allows for beyond the Standard Model effects in the hγγ interactions, the latter being in particular expected from a non-zeroc W parameter too (see Table 1). Employing our framework, we theoretically estimate the two κ parameters and the λ W Z quantity as considering TeV scale new physics [97,[100][101][102][103][104][105] and also agrees with predictions performed by making use of the eHDecay program [97], the extension of the HDecay package [149,150] in the framework of the Higgs effective field theory of Section 2.1.
In the case both thec W andc B coefficients are non-vanishing, as suggested by the LEP limits constraining their difference, the additional effect of thec B parameter on the κ Z quantity can be obtained by replacingc W byc W + tan 2 θ WcB in Eq. (4.5). The corresponding modification in the predictions for the λ W Z parameter is trivially obtained by, e.g., linearizing the new physics effects.

Kinematics of the four-lepton system issued from a h → Z * Z ( * ) decay
In addition to shifting the values of the Higgs signal strengths κ W and κ Z , higher-dimensional operators also affect various kinematical distributions. In particular, several angular distributions related to the leptonic systems arising from decays of the Higgs boson into four fermions such as those introduced in Eq. (4.1) can be drastically modified by new physics effects. Considering the ZZ * channel in the four-lepton mode where the entire final state can be reconstructed, we simulate LHC collisions at a center-of-mass energy of 14 TeV yielding the production of a Higgs boson from a gluon-pair initial state, followed by its decay into four leptons, In our setup, each lepton is required to have a transverse momentum greater than 10 GeV, a pseudorapidity |η| < 2.5 and to be isolated. We define lepton isolation by enforcing a minimum angular distance ∆R of 0.4 between any two of the produced particles. In our notations, ∆R = ∆ϕ 2 + ∆η 2 with ϕ denoting the azimuthal angle with respect to the beam direction. The four-momenta of the four final state leptons can be seen as lying on three distinct planes, one of them being related to the decaying Higgs boson into two (possibly virtual) Z-bosons and the two other being associated with the decay products of each Z-boson. As it is customary in the literature [5-18, 20, 22-24, 28-30, 34, 45], we denote by θ 1,2 the polar angles of the two leptons − 1,2 in the rest frame of the respective decaying Z-bosons, and by ∆φ −  We show in Figure 1 the angular distribution ∆φ 1 2 for the pure Standard model case (green-solid histogram) to which we superimpose predictions obtained for two choices of non-vanishingc W Wilson coefficient. These latter predictions also include Standard Model contributions with which new physics interferes. The black-dashed and blue-solid lines in the figure correspond to values ofc W = 0.3 and -1.5, respectively. The Standard Model spectrum is expected to be a function of both cos(∆φ 1 2 ) and cos(2∆φ 1 2 ) exhibiting a rather gentle slope. This known result [17,151] is recovered by our predictions. In contrast, the structure of the O W dimension-six operators modifies the coefficients of the polynomial in the cosines of the ∆φ 1 2 angle, leading to a steeper dependence. The larger the value of c W , the steeper this function becomes. Consequently, this ∆φ 1 2 distribution can clearly be used, if measured with a decent precision, to constrain the magnitude of the Wilson coefficient associated with the operator O W .

Kinematics of the two-lepton system issued from a h → W * W ( * ) decay
We now consider the W W * Higgs decay mode in the case both weak bosons decay leptonically, in the framework of LHC collisions at a center-of-mass energy of 14 TeV and for similar lepton requirements as in the previous subsection. We study new physics effects arising from non-vanishingc W andc HW parameters in the spectrum of the angular distance ∆R, in the (η, ϕ) plane, among the two produced leptons. We have found that the resulting distribution is barely affected when the two  c HW = 0.5 (blue-solid line) andc W = 0.05 (red-dashed line). In contrast, for larger values of the Wilson coefficients, important differences in the distributions can be expected, as shown on the right panel of the figure for two representative scenarios with respectivelȳ c HW = −1 (blue-solid line) andc W = −1 (red-dashed line). It can be noted that according to the new physics benchmark scenarios the most striking effects can be either expected in the low ∆R region, or in the higher one.

Correlations of new physics effects in the Higgs boson partial widths
As already mentioned in Section 2, one given dimension-six operator may affect several interactions of the Higgs boson to a pair of gauge bosons. This feature is employed in Figure 3 where we compute different Higgs boson partial widths when considering a specific dimension-six operator, allowed to vary in the [−1, 1] range. The results of the different decay channels are then confronted to each other and we study the correlations among the corresponding partial widths. It can be seen that this information could be used in the future to constrain which Wilson coefficients are allowed to be non-negligible by data. It is however clear that this type of investigation consists only of a first step towards a global fit of a complete set of dimension-six operators. In addition, our results are compared in each of the treated cases to the Standard Model predictions represented by an orange dot.
In this way, we present on the upper-left panel of the figure the variations of the h → γ γ and h → Z γ partial widths for non-vanishing CP -conserving and CP -violating parametersc γ andc γ . We observe that those two channels do not offer a clear way to distinguish contributions arising from the two associated O γ andÕ γ operators. The situation is however drastically different when we focus, in the upper-right panel of the figure, on correlations possibly observable in the h → γ γ and h → Z Z channels and that are induced by variations of the samec γ andc γ parameters. In this case, if one assumes precise measurements pointing towards physics beyond the Standard Model in one or the other Higgs decay modes (possibly in collisions at future LHC center-of-mass energies or even at future colliders), it can be observed that these measurements could help to get an handle on the structure of the operators responsible for the assumed observations. More involved cases are treated in the three other panels of the figure, where we confront predictions for several partial widths of the Higgs boson into different gauge-boson pairs for various choices of non-zero Wilson coefficients that are allowed to vary in the [−1, 1] range. For most of the cases, it is found that this method allows for distinguishing among the different operators that could model a specific new physics effect.

Higgs boson production in association with a vector boson
We now turn to the study of the production of a Higgs boson in association with a weak gauge boson pp → V * → h V , (4.10) with V being either a W -boson or a Z-boson. The existing searches are classified according to the lepton multiplicity of the final state, i.e., in dileptonic, singly leptonic and zero-lepton channels. This offers the possibility to make use of the properties of the final state leptons possibly coming accompanied by missing transverse energy to reject the Standard Model background without affecting too much the Higgs signal as the latter could be searched for in its dominant decay mode to two jets originating from the fragmentation of b-quarks.
In this case, the final state kinematics differs from the one expected from gluon fusion which was discussed in the previous subsection [44,152]. In gluon fusion, the Higgs is usually produced with a very little boost as the partonic center-of-mass energy is close to the Higgs-boson mass. Contrary, a two-body system comprised of a Higgs boson and a  W -or Z-boson is issued from an off-shell vector boson that can be seen as radiating a Higgs particle. This off-shellness allows for greater partonic center-of-mass energies, which subsequently increases the sensitivity to non-standard Lorentz structures in the interactions of the Higgs field as modeled by the effective operators of Section 2.1. We exploit these features in the rest of this subsection.

New physics effects in total cross sections for associated Higgs and gauge boson production
Total rates in the associated production of a Higgs boson with a gauge boson [31,[153][154][155] are known to be a powerful handle to obtain information on beyond the Standard Model effects modeled by dimension-six operators. In particular, a class of variables consisting of double ratios of total cross sections has been recently identified as one of the key players for this task [153,156]. More into details, such quantities are defined as ratios of total cross sections at different center-of-mass energies, normalized to the corresponding Standard Model values. Focusing on present and future LHC collision center-of-mass energies ( √ S = 8 TeV and 14 TeV), we investigate in this work the variable where the subscriptc i indicates a computation of the cross section after including the effects of an higher-order operator associated with the Wilson coefficientc i . It is also believed that this type of variables could be useful in the case statistically relevant and separated information for collisions at center-of-mass energies of 13.5 TeV and 14 TeV would be available [157]. We illustrate the use of the variable introduced in Eq. (4.11) in Figure 4 by investigating the process pp → W ± h → ( ν) (bb) . We show the dependence of R on the coefficient ofc HW which turns out to be quite steep whenc HW is of order O(0.1) or smaller, and smoother for larger (absolute) values of this Wilson coefficient. The results however largely depend on the selection requirements (on the final state lepton and missing transverse energy) of the corresponding analysis that could further accentuate the effect of the effective operator.

Invariant mass of a two-body system constituted of a Higgs boson and a gauge boson
The kinematical properties of the system formed by the massive vector boson V and the Higgs boson h may be modified by the presence in the Lagrangian of non-standard operators such as those introduced in Section 2.1. In this context, one interesting observable consists of the invariant-mass distribution of the V h-system [31], as illustrated in Figure 5

Di-Higgs production in vector boson fusion
Recently, the interest for di-Higgs production at the LHC, running at a center-of-mass energy of 14 TeV, has importantly increased. This process indeed allows to get a first grip on the triple-Higgs interaction strength. In the following, we show as an example di-Higgs production in the vector boson fusion mode, where both jets j are forward jets. Several of the effective operators of Section 2.1 can affect such a process and we find that the distribution of the Higgs transverse momentum p T provides information allowing to probe such effects. This is illustrated on Figure 6 where we represent the p T spectrum of both Higgs bosons, including hence one entry for each Higgs boson in the histograms. We compare the Standard Model expectation (green-solid histogram) to new physics results arising fromc W = 0.05 (blue-solid line). We demonstrate in this way how the operator O W (and the associated Wilson coefficient c W ) favors final state configurations with boosted Higgs bosons. This behavior is even more pronounced than the one observed in Section 4.2.2 when the Higgs boson is produced in association with a massive gauge boson. As a consequence, the operator O W is likely to be investigated via new techniques dedicated to Higgs searches in boosted topologies, as presented for example in Refs. [61,[158][159][160].

Associated production of a Higgs and gauge boson from contact interactions with fermions
As a last example of the strength of our implementation, we investigate, in this subsection, possible effects originating from effective operators involving one single Higgs field, one single gauge boson and a fermion-antifermion pair. More especially, we focus on the O HQ operator of Eq. (2.11) which contributes to the associated production of a Higgs boson h together with a massive vector boson V through a contact interaction, (4.14) The kinematical properties due to new physics contributions are thus expected to be largely different from the Standard Model ones where the final state is produced via Higgs radiation from an off-shell gauge boson, as shown in Eq. (4.10). Therefore, observables such as the transverse momentum of the Higgs boson or the invariant mass of the V h system can be foreseen to play key roles in the detection of effects due to a non-vanishingc HQ parameter. We have however found that this operator induces more striking modifications of the total production cross section, turning thus out to be more promising for searches for this type of physics beyond the Standard Model. Considering Higgs boson production in association with a Z-boson, we restrict our analysis to a dileptonic Z-decay into an electron or muon pair, together with a Higgs boson decay into a pair of b-quarks, p p → h Z → (bb) ( + − ) . (4.15) Simulating LHC collisions at a center-of-mass energy at 14 TeV, we impose that the four produced fermions have a transverse momentum larger than 25 GeV, a pseudorapidity satisfying |η| > 2.5 and that the angular separation ∆R between any two objects is greater than 0.4. Normalizing the associated total production rate to the Standard Model one σ SM , one fits the effects of the O HQ operator as κc HQ = σc HQ σ SM = 1.00 − 2.00c HQ + 863c 2 HQ , (4. 16) where σc HQ stands for the cross section as computed when including, in addition to the SM contributions, diagrams involving the O HQ effective operator. This proves that any LHC measurement of this κc HQ quantity that is accurate at the percent level (achievable with the high-luminosity run of the LHC [161]) would surpass the current bounds derived from LEP-I data at the Z-pole, the latter constraining the value ofc HQ to be of order O(10 −3 ) [162].

Conclusions
The Higgs discovery, together with the characterization of its properties and quantum numbers, sets the ground for the approach taken in this paper. With the absence of any evidence for new physics, we have followed the path of effective field theories to describe beyond the Standard Model effects in the Higgs sector. We have first formulated in extensive details the Higgs Effective Lagrangian that we have employed, limiting ourselves to dimension-six operators involving the Higgs and/or gauge bosons and adopting the basis and conventions of Refs. [93,97]. Next, we have implemented this Lagrangian in the framework of FeynRules, a Mathematica package interfaced to several sophisticated Monte Carlo tools. Our implementation allows to study, in particular, various differential distributions related to processes involving one or several Higgs bosons in the context of the LHC collisions. In the high-energy run of the LHC at a center-of-mass energy of 14 TeV, more statistics will be collected, resulting in a better understanding of the Higgs properties. In these perspectives, fits involving the Higgs signal strengths will rapidly fall short to exploit all data, while angular, invariant-mass as well as many other differential distributions will be able to directly probe possible non-standard Lorentz structures of the Higgs interactions. We believe that our implementation will hence allow to unveil physics beyond the Standard Model if related to the Higgs sector.
We have illustrated the strengths of our machinery by making use of the UFO interface of FeynRules to pass the full set of interaction vertices included in our Higgs Effective Lagrangian to the event generator MadGraph 5, without any restriction on the Lorentz structures or on the number of external legs allowed in the vertices. For the sake of the example, we have considered Higgs boson production via gluon fusion and investigated its decay modes to massive vector bosons, Higgs boson associated production with a weak boson, di-Higgs production as well as the effects of contact interactions among a fermionantifermion pair, a Higgs boson and a gauge boson. In all those processes, we have studied either total rates, or several differential distributions, or both, and shown how this has allowed us to investigate how dimension-six operators could manifest themselves in LHC processes. Finally, we have also briefly addressed the way to disentangle the source of a specific effect among the set of effective operators possibly giving rise to it.