NLO QCD corrections to SM-EFT dilepton and electroweak Higgs boson production, matched to parton shower in POWHEG

We discuss the Standard Model Effective Field Theory (SM-EFT) contributions to neutral- and charge-current Drell-Yan production, associated production of the Higgs and a vector boson, and Higgs boson production via vector boson fusion. We consider all the dimension-six SM-EFT operators that contribute to these processes at leading order, include next-to-leading order QCD corrections, and interface them with parton showering and hadronization in Pythia8 according to the POWHEG method. We discuss existing constraints on the coefficients of dimension-six operators and identify differential and angular distributions that can differentiate between different effective operators, pointing to specific features of Beyond-the-Standard-Model physics.


Introduction
The Standard Model (SM) of particle physics is an extremely successful theory. It is in good agreement with the precise measurements at the Z pole [1], with numerous flavor observables [2][3][4], the observed CP violation in the kaon and B meson systems [4], and the non-observation of electric dipole moments [5][6][7][8][9]. Even the hints for deviations from the SM, observed in the muon anomalous magnetic moment [10] and in direct CP violation in kaon decays [11], seem to indicate that the scale of new physics is high, in the 10 to 100 TeV range. In this scenario, the manifestations of new physics at the Large Hadron Collider (LHC) can be described by non-renormalizable higher-dimensional operators, which are invariant under the SM gauge group and whose effects are suppressed by powers of the scale Λ that characterizes the new physics. Extending the SM with these higher-dimensional operators gives rise to an effective field theory (EFT) in which the lowest-dimensional operators provide the dominant effects as long as Λ is well above the electroweak scale. If, in addition, we assume there are no new light fields and that the Higgs boson observed at the LHC belongs to an SU(2) L doublet, this EFT is referred to as the SM-EFT [12][13][14][15][16].
In this paper we study the complete set of dimension-six operators, in the basis of ref. [13], that gives tree-level contributions to the neutral-current (NC) and charged-current (CC) Drell-Yan processes, to the associated production of the Higgs and a W or Z boson (W H and ZH production), and to Higgs boson production via vector boson fusion (VBF). We include next-to-leading order (NLO) QCD corrections to the partonic processes, and interface with the parton shower according to the POWHEG method [54][55][56], extending the original works of refs. [57][58][59].
SM-EFT analyses of NC and CC Drell-Yan processes have mainly considered contributions at leading order in QCD [26,28,29,32,60,61] or included higher-order QCD corrections via a form-factor rescaling [62,63]. Here we overcome this limitation by including generic NLO QCD corrections, and interfacing to the parton showers. This allows us to reliably estimate the theoretical error on the SM-EFT cross section. It also provides a tool for the study of generic observables, more differential than the dilepton invariant or transverse mass. Some of these observables, such as the lepton angular distributions, can be useful to disentangle the contributions from different effective operators and could therefore provide important clues on the nature of the new physics, were deviations from the SM to be observed in Drell-Yan production.
For Higgs boson production, NLO QCD corrections to the contributions to W H, ZH, and VBF of a subset of SM-EFT operators have been studied in refs. [34,[36][37][38][39]42]. Here we consider a slightly larger set of operators, with less restrictive flavor assumptions for the couplings to quarks. Also in this case, including the effective operators in POWHEG allows for the study of differential and angular distributions, which could help disentangle the effects of different dimension-six operators.
The paper is organized as follows. In section 2 we introduce the operator basis, and establish our notation for the coefficients of the dimension-six operators we implemented in POWHEG. We discuss the NC and CC Drell-Yan processes in section 3, while in sections 4 and 5 we describe, respectively, the associated production of a Higgs boson and a W or Z boson, and Higgs production via vector boson fusion, before concluding in section 6. We relegate several SM-EFT corrections to the Higgs boson decay to appendix A.

JHEP08(2018)205 2 The operator basis
Before discussing dimension-six operators, we recall a few SM ingredients needed to establish our conventions. The SM Lagrangian is completely determined by the invariance under the Lorentz group, the gauge group SU(3) c × SU(2) L × U(1) Y , and by the matter content. We consider here the SM in its minimal version, with three families of leptons and quarks, and one scalar doublet. The left-handed quarks and leptons transform as doublets under SU (2) while the right-handed quarks, u R and d R , and charged leptons, e R , are singlets under SU(2) L . We do not include sterile right-handed neutrinos, but their effects on e.g. W production can be straightforwardly included [26]. The scalar field ϕ is a doublet under SU(2) L . In the unitary gauge we have where v = 246 GeV is the scalar vacuum expectation value (vev), h is the physical Higgs field and U (x) is a unitary matrix that encodes the Goldstone bosons. Byφ we denotẽ ϕ = iτ 2 ϕ * . The gauge interactions are determined by the covariant derivative where B µ , W I µ and G a µ are the U(1) Y , SU(2) L and SU(3) c gauge fields, respectively, and g , g, and g s are their gauge couplings. Furthermore, τ /2 and t a are the SU(2) L and SU(3) c generators, in the representation of the field on which the derivative acts. In the SM, the gauge couplings g and g are related to the electric charge and the Weinberg angle by gs w = g c w = e, where e > 0 is the charge of the positron and s w = sin θ W , c w = cos θ W . We will shortly discuss how these relations are affected in the presence of dimension-six operators. The hypercharge assignments under the group are 1/6, 2/3, −1/3, −1/2, −1, and 1/2 for q L , u R , d R , L , e R , and ϕ, respectively. The SM Lagrangian then consists of the Lorentz-and gauge-invariant terms with dimension d ≤ 4 that can be constructed from the above fields.
The processes we aim to study, Drell-Yan, W H, ZH, and VBF, are affected by many dimension-six operators. Following the notation of ref. [13], we classify the relevant operators according to their content of gauge (denoted by X), fermion (ψ), and scalar fields (ϕ). The operators that contribute at tree level fall in the following five classes L = L X 2 ϕ 2 + L ψ 2 Xϕ + L ψ 2 ϕ 2 D + L ψ 2 ϕ 3 + L ψ 4 . (2.4) Here L X 2 ϕ 2 contains operators with two scalars and two gauge bosons. At the order we are working and for the processes we are considering, the only relevant operators are the JHEP08(2018)205 ones involving SU(2) L and U(1) Y gauge bosons: where W I µν and B µν denote the SU(2) L and U(1) Y field strengths, andX µν = ε µναβ X αβ /2. The operators in the first line of eq. (2.5) are CP-even, while those on the second line violate CP. Here, and in what follows, we define the coefficients of dimension-six operators to be dimensionless. Thus, C ϕX and C ϕX scale as v 2 /Λ 2 , where Λ is the new physics scale. The remaining operators in the class L X 2 ϕ 2 involve the gluon field strength G µν . While these are very interesting for Higgs boson production via gluon fusion, they contribute to the processes we are considering -all of which are (anti-)quark-initiated -only at higher order, and we neglect them.
The classes ψ 2 Xϕ and ψ 2 ϕ 2 D both contain fermion bilinears. The former class consists of dipole operators, of which we focus on the dipole couplings of quarks and leptons to the SU(2) L and U(1) Y gauge bosons: (2.7) L ψ 2 ϕ 2 D contains corrections to the SM couplings of the quarks and leptons to the Z and W bosons, which, in gauge invariant form, appear as Lϕ , c eϕ are hermitian, 3 × 3 matrices, while ξ is a generic 3 × 3 matrix. The operators c Lϕ , c eϕ couple lepton bilinears to the weak bosons, thus affecting Z and W production at tree level. However, these operators are strongly constrained by LEP measurements of the physics JHEP08(2018)205 at the Z pole [1]. Furthermore, the Z and W production cross sections induced by these operators have the same shape as the SM, making it hard to identify them at collider. For these reasons, we neglect them in what follows.
The remaining operators involving quark bilinears in the dimension-six SM-EFT Lagrangian are corrections to the quark and lepton Yukawa couplings These couplings are mainly constrained by Higgs boson production via quark-antiquark annihilation and by the Higgs boson branching ratios, but we will see that the quark couplings Y u,d also give interesting contributions to W H and ZH. Finally, we consider the following semileptonic four-fermion operators, Of these operators, only a few affect charged currents, introducing new Lorentz structures, such as scalar-scalar and tensor-tensor interactions. All of the above operators modify neutral currents and the couplings are, in general, four-indices tensors.
Having introduced the dimension-six operators of interest, we will first discuss how these new interactions affect the SM couplings. The assumptions on the flavor structure of the introduced couplings are discussed in section 2.2.

Corrections to SM couplings
In the SM the gauge couplings g and g , the electric charge e, the Fermi constant G F , the Weinberg angle sin θ W , the gauge boson masses, and the Higgs boson mass and vev are not independent, but are connected by several relations, which hold at tree level, and are modified in a testable way by radiative corrections. Including the higher dimensional operators of eqs. (2.5), (2.6), (2.8) and (2.10) modifies these relations, making it important to specify the input that is taken from the experiments. Here we use as experimental input the values of the weak boson and Higgs boson masses m Z , m W , m H [64], G F , the electric charge, and sin θ W as extracted from the lepton asymmetries at LEP [64]. In the presence of new physics, these experimental values would include contributions from dimension-six operators, which we now describe.
The operators in eq. (2.5) affect the normalization of the gauge bosons, and their masses, inducing, in particular, mass and kinetic mixing between W 3 µ and B µ . Here we follow ref. [12], and diagonalize the gauge boson mass terms up to corrections of O(v 4 /Λ 4 ), by rotating the W 3 and B µ fields according to

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where we have introduced The subscript 0 has been introduced to stress that eqs. (2.13)-(2.15) involve the Weinberg angle in the SM c 0w = cos θ 0 In a similar way, we redefine W 1,2 µ to ensure the kinetic term of the W boson is canonically normalized. The weak boson masses are then given by ref. [12]  where G F is the measured Fermi constant, extracted from muon decay, G F = 1.166 · 10 −5 GeV −2 . Within the general SM-EFT the above relation receives corrections from additional operators that we do not consider here [12]. For example, at dimension-six eq. (2.19) is altered by the operator c Lϕ in eq. (2.8), which modifies the leptonic couplings of the W . Using the physical masses and G F , we define the effective couplings (2.20) which are useful in expressing the couplings of the quarks to the W and Z bosons. The electric charge is given by ref. [64] Finally, we use the lepton asymmetry parameter A e , as measured at LEP [64], to determine the Weinberg angle. A e is related to the ratio of the electron vector and axial coupling, and, since we are not introducing newēeZ interactions, this ratio is only affected by the definition of the Z field,

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Using g e V /g e A = 0.151 ± 0.002 [64], we define an effective s w With the above definitions, the SM couplings of the Z and W boson to quarks can be written as where T 3 is the third component of isospin, The couplings of the Z and W boson to the Higgs boson are given by We will compute the effects of dimension-six operators with respect to these definitions. Lastly, the Yukawa operators in eq. (2.9) affect the definition of the quark masses. We define the Yukawa Lagrangian in the presence of dimension-four and dimension-six operators as and Y 0 u,d denote the Yukawa couplings in the SM. The matrices Y u and Y d can always be made diagonal and real by appropriate field redefinitions, and are determined by the observed quark masses, M u,d = vY u,d . In the mass basis, Y u,d are generic 3 × 3 complex matrices. In what follows we will neglect the contribution of Y u,d , which, for processes involving light quarks, is always negligible, and we will focus on Y u,d , which, being unrelated to the light quark masses, need to be independently constrained.

Flavor assumptions
In the code we try to keep the flavor structure of the couplings in eqs. (2.6), (2.8) and (2.10) as generic as possible. In the lepton sector, we work under the assumption of lepton universality, which implies that the four-fermion and leptonic dipole operators in eqs. (2.10) and (2.6) have a trivial leptonic structure, with the same couplings to the electron, muon, and tau. This assumption can easily be relaxed by setting the vdecaymode flag in the powheg.input file to separately only select electron, muon or tau final states. At the moment the code does not allow for lepton-flavor-violating couplings. In this paper we neglect neutrino masses and define the operators in the neutrino flavor basis.

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To define the couplings to quarks, we first rotate to the quark mass basis, by performing the following transformation where U u,d L,R are unitary matrices that diagonalize the mass matrices (2.29) and the CKM matrix is For most operators, the effects of the transformation can be trivially absorbed by a redefinition of the couplings. For example, for the right-handed W current ξ, the net effect of switching to the mass basis is to replace the Since ξ is an arbitrary matrix, the change has no practical effect. c U ϕ , c Dϕ , Y u,d , C eu , C ed , C Lu , C Ld only affect neutral currents, and we simply define them to be 3 × 3 matrices in the mass basis. Such a redefinition does not work for C Qe , which induces new neutral-current couplings for both u-type and d-type quarks. The Lagrangian in the mass basis is In this case we define the coupling for d-type quarks, C Qe ≡ (U d L ) † C Qe U d L , resulting in the appearance of factors of the CKM matrix for u-type couplings. Dropping the prime, we get As a consequence, we label the elements of C Qe with d-type indices in table 1.
The rotation (2.28) also has nontrivial consequences for the operators c LeQu , which affect both neutral and charged currents. In these cases, we will absorb the rotation matrices that appear in neutral-current interactions in the definition of the coefficients of effective operators, resulting in the appearance of explicit factors of the CKM matrix in charged-current interactions.
For example, for c (1) Qϕ and c Qϕ we find, in the unitary gauge and the weak basis, Transforming to the mass basis, the left-handed couplings of the Z boson to u-type and d-type quarks are These Z couplings in the mass basis, c Qϕ, U and c Qϕ, D , are 3 × 3 hermitian matrices. As a consequence, the W couplings are given by

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and are in general non-diagonal, even if one assumes that there are no tree-level flavorchanging neutral currents. In a similar way, we define the Γ u,d W,γ matrices to be the couplings of the Z and γ dipoles in the mass basis, which results in the following couplings for the W dipoles Finally, the charged-current interactions mediated by semileptonic four-fermion operators are are the neutral-current couplings of these operators in the mass basis. With these definitions of the purely left-handed semileptonic operators, additional factors of V CKM appear in the pp → νν channel. Hermiticity of the Lagrangian in eqs. (2.8) and (2.10) forces the couplings c U ϕ , c Dϕ , c Qϕ, U , c Qϕ, D , C eu , C ed , C Lu , C Ld and C Qe to be 3 × 3 hermitian matrices. This implies that CP-violating phases can only appear in off-diagonal elements. Since we do not expect collider probes to be competitive with low-energy precision experiments, at present these couplings are set to be real. CP-violating phases can be straightforwardly incorporated, if necessary. The dipole couplings Γ u,d,e W,γ , the right-handed charge-current ξ, the Yukawa couplings Y u,d , and the scalar and tensor interactions C LedQ , C LeQu are generic 3 × 3 matrices. For these couplings, both the real and imaginary parts can be turned on. The notation for the coefficients of the operators that can be set in the input file is listed in table 1.
As an alternative to considering generic flavor structures, the code also provides the option to assume the Minimal Flavor Violation (MFV) [65] framework. In this scenario, the allowed forms of the couplings are constrained by flavor symmetries, which significantly decreases the number of free parameters. The implications for the flavor structures of the couplings, as well as the implementation of this scenario in the code, are discussed in appendix B.
In addition to the couplings of dimension-six operators, the user can input the elements of the CKM matrix by setting CKM Vij, with i ∈ {u,c,t} and j ∈ {d,s,b}, in the input file. By default the CKM matrix is assumed to be real, and terms of O(λ 2 ), where λ ∼ |V us |, and higher are neglected.

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Operator Operator C ϕW, ϕB, ϕW B , CC ww, CC bb, CC wb C ϕW ,ϕB,ϕW B CC wwt, CC bbt, CC wbt ImLeQu ik ImLedQ jl Table 1. Notation for the coefficients of the dimension-six operators that can be set in POWHEG. The flavor structures reflect the assumptions discussed in section 2.2. The indices i, k run on u-type quark flavors i, k ∈ {u,c,t}, while j, l on d-type quark flavors j, l ∈ {d,s,b}. With the assumptions of section 2.2, c Qϕ, U , c Qϕ, D , c U ϕ , c Dϕ , C LQ,U , C LQ,D , C eu , C ed , C Lu , C Ld and C Qe are real symmetric matrices. The notation i ≥ k and j ≥ l indicates that the elements {uu,uc,ut,cc,ct,tt} and {dd,ds,db,ss,sb,bb} can be set by the user, while the remaining elements are not independent.

Renormalization
The coefficients of most operators in eq. (2.4) do not run in QCD at one loop. The exceptions are the quark Yukawa couplings in eq. (2.9) and the scalar operators in eq. (2.10), whose coefficients Y u,d , C LeQu and C LedQ obey the same renormalization group equation (RGE) as the quark masses, the dipole operators in eq. (2.6), and the tensor operator in eq. (2.10). The scalar and tensor operators satisfy LeQu , (2.37) to NC and CC Drell-Yan, associated production of a Higgs and a weak boson, and Higgs boson production via vector boson fusion. For each process, a indicates that the contribution has been implemented in POWHEG, a × that the operator contributes at tree level, but has been neglected because of the reasons explained in the text, while a − indicates that the operator does not contribute at leading order.

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where the two loop anomalous dimensions are [66][67][68] γ (0) Here C F = 4/3, N C = 3 and n f = 5 is the number of light flavors. All the other coefficients do not run in QCD, and we neglect electroweak loops. By default, all dimension-six corrections are switched off in the POWHEG input card. To investigate the effect of one or more dimension-six operator, the user needs to set the flag dim6 to 1 and specify the values of the dimensionless coefficients defined in eqs. (2.5), (2.6), (2.8) and (2.10) in the input file. The notation for the coefficients of the operators that can be set in the input file is listed in table 1, while table 2 summarizes the processes we considered, and which operator can be turned on and off in each process. For all processes except VBF, the user can set flavor off-diagonal coefficients, which induce tree-level processes that are absent in the SM, such as ds → e + e − or cū → he + e − . To generate these processes, the flag fcnc should be set to 1 in the POWHEG input card.

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The new physics scale Λ, at which the coefficients are defined, can be specified by setting the flag LambdaNP to the desired value. Eqs. (2.37) and (2.38) are then used to run the coefficients from Λ to µ R , the renormalization scale of the process of interest. By default, LambdaNP = 1 TeV. For the coefficients that do not have QCD evolution, the flag LambdaNP is irrelevant.

Neutral-and charged-current Drell-Yan production
We start by analyzing the contributions of the dimension-six SM-EFT operators to the processes pp → + ν , −ν , + − and νν. The SM background to these processes is known with very high accuracy, including fixed next-to-next-LO (NNLO) QCD corrections [69][70][71] and NLO electroweak (EW) corrections [72][73][74][75][76]. More recently, the interface of the NNLO predictions with the parton shower has been presented in refs. [77][78][79], and a quantitative assessment of the size of different QCD and EW corrections has been discussed in ref. [80]. Contributions from SM-EFT operators, at LO in QCD, have been considered, for example, in refs. [26,32,60,61] and rescaled by the SM NNLO K-factor in refs. [62,63]. In this work, we have calculated the NLO QCD corrections to the partonic processes mediated by SM-EFT operators in table 2 and interfaced with the parton shower according to the POWHEG method, extending the original work of ref. [57].
If the invariant mass of the leptons is close to the weak boson masses, NC and CC Drell-Yan production is dominated by the production of a W or Z boson, which subsequently decays into leptons. In this region, these processes are in principle sensitive to modifications of the SM W and Z couplings to quarks, represented by the operators c Qϕ,U , c Qϕ,D , c U ϕ and c Dϕ , or to new interactions of the W and Z boson: the W coupling to right-handed quarks, mediated by ξ, and the chiral-symmetry-breaking dipole couplings Γ u,d,e W,γ . For W production, the contributions of c Qϕ,U and c Qϕ,D amount to a shift in the values of the CKM mixing matrix, which, in the presence of these operators, is no longer unitary. Instead, ξ induces couplings to quarks with opposite chirality to the SM. If one considers observables that are symmetric under the exchange of the charged lepton and the neutrino momenta, the contributions of these operators are identical to those of the SM, making it difficult to obtain strong constraints [81]. The right-handed nature of ξ would manifest, for example, in a larger fraction of right-handed polarized W bosons. Appreciable deviations, however, require large couplings that are already ruled out by other collider and low-energy observables, such as associated production of a W and a Higgs boson [81]. Similarly, in Z production, the cross section induced by the operators c Qϕ,U , c Qϕ,D , c U ϕ and c Dϕ is qualitatively very similar to the SM, and thus other processes, such as e + e − → qq at the Z pole, or pp → HZ, provide stronger bounds. In the case of the dipole couplings Γ u,d,e W,γ , the cross section has a different shape, being enhanced for large values of W boson transverse mass or the dilepton invariant mass. Furthermore, the different chiral structure of the vertex could be identified by looking at the W and Z boson polarization.
When the dilepton invariant mass is much larger than the W or Z boson mass, NC and CC Drell-Yan production receive important contributions from the four-fermion operators in eq. (2.10). C LQ,U , C LQ,D , C LedQ and C (1,3) LeQu contribute to the neutral and charged-JHEP08(2018)205 current processes, while the remaining operators in eq. (2.10) only contribute to neutral currents. Most of the semi-leptonic interactions, namely C LQ,U , C LQ,D , C eu , C ed , C Lu , C Ld and C Qe , couple left-and right-handed quarks to left-or right-handed leptons, and therefore modify the helicity structures which already exist in the SM. On the other hand, C LedQ and C (1) LeQu are scalar couplings, while C LeQu has the form of a tensor coupling, all of which give rise to new helicity structures.
We computed the W and Z production cross sections in the presence of the dimensionsix operators in table 2, including NLO QCD corrections, and implemented them in the POWHEG BOX V2. The cross section has the schematic structure where σ indicates a generic (differential) cross section in the ν, + − or νν channels. If one neglects light-quark mass effects, only the semileptonic four-fermion operator C LQ,U and C LQ,D and the vertex corrections c Qϕ,U and c Qϕ,D interfere with the SM amplitude for pp → ν, due to helicity considerations. In the case of Z production, the index i in the interference term in eq. (3.1) runs over the operators We included dimension-eight effects to guarantee the positivity of the cross section even for arbitrary large values of the couplings. For completeness, we also included interferences between different dimension-six operators. The chiral structures of the effective operators strongly limit the number of such interference terms and in practice one needs only to consider the interference of the photon and Z dipoles in pp → + − , and of the u-type scalar and tensor operators, C LeQu , in pp → + − and pp → ν . When extracting bounds on the coefficients of effective operators, one should make sure to be working in the regime of validity of the EFT. For the operators that interfere with the SM one can check that the bounds are not dominated by terms quadratic in the new physics couplings. For a clear discussion of this point, we refer to refs. [32,82]. For operators with different chiral structures than the SM, the leading contribution to the W and Z production cross sections is quadratic in the BSM coupling, and goes as Λ −4 . One might then worry that the constraints on these operators are not reliable because of missing Λ −4 contributions from the interference of genuine dimension-eight operators with the SM. This concern is justified for bounds obtained from the total cross section, or from differential distributions that are not sensitive to the quark/lepton chiralities. However, dimension-six operators with different chiral structures than the SM leave clear signatures in observables such as the W and Z polarization fractions, that cannot be mimicked by dimension-eight operators. Thus, in the presence of deviations from the SM, it will be possible to disentangle dimension-six operators with different chiral structure than the SM from genuine dimension-eight effects by studying more differential observables.
We show the resulting pp → e + ν e + e −ν e cross section at √ S = 13 TeV as a function of the W transverse mass, m W T , in figure 1, where m W T is defined as  Here p T and p ν T are the charged-lepton and the neutrino transverse momenta, respectively, and ∆φ ν is their azimuthal separation. In blue we depict the SM cross section, while the remaining lines depict the contributions of the four-quark operators C LQ,u − C LQ,d (green), C LeQu , respectively. In figure 2 we show pp → e + e − , as a function of the dilepton invariant mass. In addition to the operators shown in figure 1, we also included the coupling of left-handed quarks to right-handed leptons, C Qe , which does not contribute to pp → ν. The photon dipole Γ u γ gives a correction to the cross section, which is roughly 70% of the weak dipole at large m e + e − . The remaining u-type couplings, C eu and C Lu , give rise to corrections to the cross sections that have similar shape as C LQ,u and C Qe . Not shown in the figure are d-type couplings, Γ d W , Γ d γ , C LQ,d , C ed , C Ld , C LedQ , which are qualitatively very similar to the corresponding u-type operators, with some suppression from the d-quark PDF.
The cross sections were evaluated using the PDF4LHC15 nlo 30 PDF set [83], with the factorization and renormalization scales set to µ F = µ R = m , where m is the invariant JHEP08(2018)205  For both W and Z production, the uncertainties of the NLO SM cross section are about 2-3% at low m W T or m e + e − , and increase to about 10% at m W T , m e + e − ∼ 1-2 TeV, where they are dominated by PDF uncertainties. We find that the cross sections induced by the dimension-six operators that couple to the light quarks are affected by similar errors. In particular, the PDF uncertainties for both the SM and the dimension-six cross sections dominate at large m W T or m e + e − , where they are about 10-15%. The scale variations for operators with a similar chiral structure as the SM, such as C L,Qu or C Qe , as well as the dipole operators and the semileptonic tensor operators are all very similar, being at most around 5%. The scalar operators C LedQ and C (1) LeQu , on the other hand, have larger scale uncertainties, close to 10% at high invariant mass.
The cross section induced by the four-fermion and dipole operators, as a function of m W T or m l + l − , falls more slowly than in the SM, and thus the effects are more visible for large invariant mass. This is evident from the middle panels of figures 1 and 2, which show the ratio of the differential cross sections in the presence of dimension-six operators and in JHEP08(2018)205 the SM. For the four-fermion operators that interfere with the SM, namely C LQ,u , C LQ,d , C eu , C ed , C Lu , C Ld and C Qe , the ratio scales as (m W T /v) 2 or (m l + l − /v) 2 , whenever the interference term dominates. The dipole operators Γ u,d,e W,γ do not interfere with the SM, and contribute at dimension-eight. In this case, the amplitude contains a W /Z propagator, but it has an additional factor of momentum with respect to the SM, so that the squared amplitude is also enhanced by a factor of (m W T /v) 2 . Finally, the scalar and tensor operators, C

Bounds on effective operators
We can take advantage of the enhancement at high m W T and m + − by interpreting the ATLAS and CMS searches for new high-mass phenomena in the dilepton or lepton and missing energy final states [84][85][86][87][88][89][90][91][92] as bounds on the coefficients of the effective operators. As an example of the kind of constraints one can obtain, we consider the analyses of refs. [92] and [91], which used data collected by the ATLAS collaboration during the LHC Run II at 13 TeV, with integrated luminosity of 36.1 fb −1 . We calculated the pp → e + ν e , pp → e −ν e and pp → e + e − differential cross sections, interfacing to Pythia8.219 [93,94] parton shower and hadronization, using the POWHEG method. We used the same binning as the ATLAS collaboration, which considers 7 bins in m W T , between 130 GeV and 7 TeV [92], and 10 bins in m e + e − , from m Z up to 6 TeV [91]. We applied the same cuts as the experimental collaboration on the leptons and missing energy, but did not perform a detailed detector simulation to determine acceptance and efficiency. In addition, we did not simulate additional SM backgrounds, such as tt and diboson production, but used the expected background events listed in refs. [92] and [91]. For these reasons our bounds have to be interpreted as a crude estimate of the reach of the LHC.
The resulting 90% confidence limits are shown in table 3. We turn on all operators at the new physics scale µ 0 = 1 TeV, assuming that only the couplings to the first generation of quarks are non-zero, and use the electron channels of refs. [92] and [91] Table 3. 90% CL bounds on the coefficients of SM-EFT operators that contribute to CC and NC Drell-Yan production, and the corresponding estimates of the scale Λ, assuming The bounds on the first and second columns use, respectively, SM-EFT cross sections at LO and NLO in QCD. Λ is extracted from the NLO bounds, and, for asymmetric limits, the weaker limit is used. on the second column include NLO QCD corrections. We show the limits in two cases. The first bound in each column includes all the bins of refs. [92] and [91]. In this case, the limits are dominated by the last bin in m W T , m W T ∈ [3, 7] TeV, and m e + e − , m e + e − ∈ [3, 6] TeV. The second bound excludes the highest bin, probing m W T and m e + e − up to 3 TeV. From table 3, one can see that the processes considered in this section are not very sensitive to the dipole couplings Γ u,d W,γ . These coefficients are constrained at the 10% level. If we translate these bounds to an estimate of the new physics scale Λ, using Γ u,d W,γ ≡ v 2 /Λ 2 , we see that the effective scale would be 1 TeV. This is smaller than the scales directly accessed by the experiment, implying the EFT framework might not be applicable in this region and the resulting limits should be interpreted with some caution. On the other hand, the semileptonic four-fermion operators are strongly constrained, in several cases at better than the permil level, corresponding to Λ ∼ 10 TeV. For scalar and tensor operators, the strong bounds suggest that the last two bins are in the regime of validity of the EFT, Λ m W T , m e + e − . For the remaining operators we checked that for the values of the couplings in table 3 the interference term, linear in v 2 /Λ 2 , is larger than the quadratic piece.

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Comparing the first and second column of table 3 we see that including NLO QCD corrections impacts the coefficients of dimension-six operators at the 10%-20% level. As expected from figures 1 and 2, the effect is larger for the analysis that uses all bins of refs. [92] and [91], which is dominated by the bin of highest invariant mass, and the scalar operators C LedQ and C (1) LeQu are the most affected by NLO corrections. It is interesting to compare the constraints of table 3 with complementary constraints that can be extracted from low-energy observables. First, we point out that the imaginary parts of several operators can be stringently constrained by electric-dipole-moment (EDM) measurements. For example, the imaginary parts of C (1) LeQu,LedQ and C (3) LeQu are probed by measurements of T -violation in paramagnetic systems and in diamagnetic atoms, respectively. Below the QCD scale, the scalar and tensor couplings induce spin-independent and spin-dependent electron-nucleon interactions, often referred to asC S andC T , respectively [24,95,96]. The scalar coupling is most sensitively probed in measurements of T -violation in ThO, while the EDM of 199 Hg is sensitive toC T . The current experimental constraints [8,97] roughly imply Im C Similarly, the imaginary parts of the dipole couplings also induce EDMs at low energies. The resulting limits are especially stringent for the couplings to the photon, Γ u,d γ , which directly contribute to the neutron EDM [98,99]. Instead, the Γ u,d W couplings induce the neutron EDM at the one-loop level [16]. The current experimental constraints [5,8,100] give Im Γ u,d W 10 −6 and Im Γ u,d γ 10 −9 . The low-energy constraints that can be set on the real parts of the couplings are weaker, and comparable to the direct limits in table 3. Focusing on the four-fermion operators C (1,3) LeQu and C LedQ , we can map them into the scalar, pseudoscalar, and tensor couplings introduced in refs. [26,27] LeQu − C LedQ and * T = C LeQu . The pseudoscalar coupling P is strongly bound by the leptonic decay of the pion, and, in particular, by the ratio Γ(π → eν)/Γ(π → µν), which is suppressed by m 2 e /m 2 µ in the SM. S is constrained by superallowed β decays [101], while the tensor coupling T affects radiative pion decays, such as π → eνγ and the neutron decay parameters [25][26][27]. The analysis of refs. [25][26][27]30] shows that the bounds on the pseudoscalar coupling are at the level | P | 5 · 10 −4 , while for the scalar and tensor | T,S | 10 −3 , thus making collider observables very competitive with low-energy probes for the real parts of these couplings.

Disentangling dimension-six operators
In light of the comparable sensitivities of low-and high-energy observables, it is very important to construct collider observables that, in the presence of deviations from the SM, are able to differentiate between the various dimension-six operators. Examples are angular distributions. For pp → ν we work in the W helicity frame, that is the frame in which the ν system is at rest, with the z-axis chosen to be in the direction of the sum of the momenta of the charged lepton and neutrino in the laboratory frame. We consider the differential distribution with respect to cos θ * , where θ * is the polar angle of the charged lepton in this frame. When the charged lepton and the neutrino emerge from the decay of a W boson, the differential cos θ * distribution is related to the W boson polarization  fractions, F 0 , F L , and F R [102,103]. We can write the differential distribution as where the dots denote higher powers of cos θ * , which can arise if the ν system has total angular momentum J > 1. σ denotes any differential cross section that does not depend on the kinematics of individual leptons. If the process is mediated by a W boson, the W polarization fractions are given by for W ± , respectively. For electron-neutrino invariant mass far from the W mass, we take eq. (3.4) as the definition of F 0,L,R . In figure 3 we show the values of F 0 and F L − F R = −A 4 /2, for m W > m W cut , where m W denotes the invariant mass of the lepton and the neutrino. The error bands include scale and PDF variations. For illustration sake, we chose larger coefficients compared to figures 1 and 2, but we stress that even for values closer to the limits discussed in section 3.1 the effect on F 0,L,R is significant. In the SM, for small values of p W T , A 0 ∼ 0 and A 4 is negative, reflecting the fact that the W boson is mostly produced with left-handed polarization [102]. The values of A 0 and A 4 are not modified by the operators C LQ,u and C LQ,d , which have the same chiral structure as the SM, and couple left-handed quark to left-handed lepton fields. On the other hand, the dipole operators, and the scalar and tensor four-fermion operators have a different chiral structure. In the case of the dipoles, the interaction of the W to the quarks flips chirality. As a result the W boson is mostly  produced with longitudinal polarization, and, if the dipole operators dominate the cross section, F 0 approaches 1, while A 4 should go to zero. This can explicitly be seen from the tree-level amplitude, that goes as sin 2 θ * . Indeed figure 3 shows that, in the presence of a non-vanishing Γ u W , as the W invariant mass grows and the contribution of Γ u W becomes more important, F 0 increases and F L − F R decreases. Since, for Γ u W = 0.05, the total cross section is at most 50% larger than in the SM, the polarization fractions shown in figure 3 do not reach the values expected when the dipole couplings dominate. The cross section induced by a scalar interaction, such as C (1) LeQu and C LedQ , is, at LO, independent of cos θ * . In this case, when the cross section is dominated by the scalar contribution, F L − F R goes to zero and F 0 ∼ 1/3, which is in agreement with figure 3. Finally, for tensor interactions, the tree-level cross section goes as cos 2 θ * , suggesting that F 0 ∼ −1 and A 4 ∼ 0, which is once again confirmed by figure 3.
While measurement of angular distributions in the W rest frame are experimentally challenging, because of the incomplete knowledge of the neutrino momentum, we hope that the discriminating power shown in figure 3 motivates the study of variables correlated with cos θ * and m W [104], which might reveal similar information.
For Z production, we work in the Collins-Soper frame [105] and show F 0 and A 4 as a function of a cut on the leptons invariant mass in figure 4. In the SM F 0 is very close to zero, and does not significantly depend on the Z invariant mass. A 4 is also close to zero at the Z peak, and its dependence on m Z is determined by Z/γ * interference. As the dilepton invariant mass grows, the SM cross section becomes dominated by the coupling to the SU(2) L gauge boson, resulting in a larger A 4 . C LQ,u and C LQ,d do not change this picture, while new dipole, scalar, and tensor operators would dramatically change the value of F 0 and A 4 , in the same way as in W production. The operator C Qe , which induces new JHEP08(2018)205 couplings of left-handed quarks to right-handed leptons, does not change the longitudinal polarization, but, when it becomes dominant with respect to the SM, causes A 4 to flip sign. Additional information can be extracted from the remaining angular coefficients, which encode the dependence on the azimuthal angle φ * . This brief discussion shows that angular distributions can play a very important role in understanding the origin of new physics, were a deviation from the SM to be observed.
Finally, we note that it is straightforward to include the contributions of gaugeinvariant dimension-six operators involving light sterile neutrinos ν R [26], or lepton-number violating (LNV) dimension-seven operators [18,21,106]. For example, in the presence of ν R it is possible to construct two more scalar (O quν and O lq in the notation of ref. [26]) and one more tensor operator (O lq ). It is easy to see that, for charged-current processes, O quν , O lq and O lq give rise to corrections to the squared amplitude that are identical to LeQu , respectively. Thus, were the cross section of figure 1 and the angular distributions of figure 3 to show evidence of scalar or tensor operators, the effect can be attributed to non-standard operators with either left-or right-handed neutrinos (or even to LNV charged-current scalar and tensor operators). In the first case, however, one would expect to see a correlated deviation in pp → + − . On the other hand, if the scalar or tensor operators involve ν R , or are LNV, SU(2) L gauge-invariance relates the corrections to pp → ν to pp → νν, while pp → + − is unaffected.

Associated production of a Higgs boson and a W or Z boson
Associated production of a Higgs boson and a W or Z boson is the third most important Higgs boson production mechanism at the LHC. Theoretical predictions for the SM background are available at NNLO in QCD [107][108][109][110] and are matched to the parton shower up to the NNLO+PS level [111]. Contributions from sets of SM-EFT operators, including NLO QCD corrections, have been studied in refs. [34,[36][37][38]42]. Refs. [34,[36][37][38] focused on couplings to the gauge bosons, such as C ϕW , C ϕB and C ϕW B . Ref. [42] included also the quark operators c Qϕ,U , c Qϕ,D , c U ϕ and c Dϕ , with the well-motivated flavor assumptions of universal couplings to the first two generations. Compared to ref. [42], the SM-EFT operators of table 2 include the right-handed current operator ξ, the Yukawa couplings Y u,d and the dipole operators Γ u,d W,γ . In addition, we allow c Qϕ,U , c Qϕ,D , c U ϕ and c Dϕ to have quite general flavor structures, the only restriction being that the off-diagonal elements of these matrices are required to be real. For all the operators in table 2 we include NLO QCD corrections, and interface with the parton shower, building upon the original NLO+PS POWHEG BOX code in ref. [59].
W H and ZH are very sensitive to the modifications of the W and Z boson couplings to quarks, written in a gauge invariant way in eq. (2.8). These operators require two scalar fields, which induce local quark-Higgs-gauge boson interactions that lead to a significant enhancement of the cross section. Other operators that enter these processes are the dipole operators, since once again SU(2) L -invariance forces the presence of the Higgs field, and operators of the form ϕ † ϕX µν X µν and ϕ † ϕX µνX µν . The contributions of these three classes of operators to the cross section are greatly enhanced at large p T of the Higgs boson

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Single  Table 4. 90% CL level bounds on the coefficients of effective operators that contribute to W H and ZH production. For operators involving quarks, we turned on couplings to the u and d quark.
The table on the left shows the bounds obtained assuming that a single operator is turned on at a scale Λ = 1 TeV. On the right, marginalized bounds, from W H and ZH production and Higgs boson decays to γγ, γZ, W W * and ZZ * .
or when the Higgs-weak boson invariant mass is large. W H and ZH production also receive tree-level corrections from the non-standard Yukawa couplings Y u,d . Even though these couplings are effectively dimension-four, they induce a correction to the cross section that is not proportional to the mass of the vector boson, and thus is also enhanced by We can appreciate the sensitivity of W H and ZH production to non-standard quark and gauge boson couplings by considering the W H and ZH signal strengths, defined as We computed the signal strength at √ S = 8 TeV using the PDF4LHC15 nlo 30 PDF sets [83], and extracted bounds using the combined results from the ATLAS and CMS collaborations [112] µ W H (8TeV) = 0.89 +0. 40 −0.38 , µ ZH (8TeV) = 0.79 +0.38 −0.36 .

(4.2)
We estimated the scale uncertainties on the signal strength by varying the renormalization and factorization scales µ F and µ R between (m V + m H )/2 and 2(m V + m H ), where m V denotes the mass of the vector boson. The PDF uncertainties were estimated by evaluating the cross section using the 30 members of the PDF4LHC15 nlo 30 PDF sets. For the operators we considered, the theoretical uncertainties are approximately 10%. These uncertainties were taken into account in the extraction of the constraints by following the Rfit approach [113]. The left panel in table 4 shows the 90% CL bounds on the coefficients on effective operators that contribute to W H and ZH production, under the assumption that only one operator is turned on at the new physics scale Λ = 1 TeV. For simplicity we assumed that only the couplings to the u and d quark are affected. We see that the 8 TeV data, despite JHEP08(2018)205 the large uncertainties, constrain non-standard couplings to light quarks at the 10% level or better. The only exceptions are the photon dipole operators Γ u,d γ , which give a small contribution to ZH and are better constrained by NC Drell-Yan production. Constraints on the couplings to gauge bosons C ϕW , C ϕB and C ϕW B , and their CP-odd counterparts, C ϕW , C ϕB and C ϕW B , are also weaker, ranging from 10% to 40%.
If we simultaneously turn on all the operators, the limits on the quark operators c Qϕ,U , c Qϕ,D , c U ϕ , c Dϕ , ξ, Y u,d and Γ u,d W are weakened by a factor of 2. The bounds on the gauge operators, in particular C ϕB and C ϕW B , are more affected, becoming significantly weaker. On the other hand, these operators give tree level corrections to the Higgs boson decays, both to h → W W * and h → ZZ * as well as h → γγ and h → γZ. Since the former two decays are generated at tree level in the SM while the latter two are loop-suppressed, the gauge operators are relatively more important for h → γγ and h → γZ. The expressions for the tree level corrections to h → γγ, γZ, W W * and ZZ * induced by all the operators introduced in section 2 are given in appendix A.
In the right panel of table 4 we show the limits on the coefficients of dimensionsix operators from W H, ZH, and the Higgs boson branching ratios, obtained under the assumption that all couplings are present at the scale of new physics. We used the combined Run-I ATLAS and CMS results for µ h→γγ , µ h→W W * and µ h→ZZ * [112], and the ATLAS bound on µ h→γZ [114] µ h→γγ = 1.14 +0. 19 −0.18 , µ h→W W * = 1.09 +0.18 −0.16 −0.23 µ h→γZ < 6.6. (4.3) We see that the Higgs boson branching ratios provide better constraints on C ϕB(B) and C ϕW B(W B) , while they have no significant effect on the remaining operators, for which the W H and ZH cross sections are more sensitive observables. Additional constraints on the Higgs-gauge operators come from electroweak precision observables [16], while strong constraints on non-standard Yukawas, at the ∼ 1% level, can be obtained from the total Higgs boson production cross section and from the Higgs boson decays [115][116][117].
We have so far only made use of the total W H and ZH cross sections. More differential information, such as the p T spectrum of the Higgs boson or the distribution in invariant mass of the Higgs and vector boson, will allow to further tighten the constraints on the effective operators. In addition, in the presence of deviations from the SM expectations, such information would help to disentangle the possible new physics mechanisms. In ref. [81] we discussed in detail W H production, and suggested that the angular distributions of the charged lepton in the W rest frame can be used to disentangle the different dimension-six operators. We further identified two angular coefficients, A 3 and A 5 , which would pinpoint, respectively, the coupling of the W to right-handed quarks of the first generation, ξ ud , or the CP-odd operator ϕ † ϕW µνW µν . The same discussion can be extended to ZH production.
In figure 5 we show the cumulative ZH cross section, as a function of a cut on the Z transverse momentum, for √ S = 13 TeV. The blue line denotes the SM cross section, while we set the couplings of dimension-six operators so that the total cross section with no cut on p Z T is 20% larger than the SM. The bands denote scale and PDF uncertainties, estimated JHEP08(2018)205 using the PDF4LHC15 nlo 30 PDF sets. For the operators we considered the theoretical uncertainties are around 5%-10%, and roughly independent on the value of p Z T cut . The middle panel of figure 5 shows that for all dimension-six operators the contribution to the cross section increases at large p T . The increase is most marked for the dipole operators Γ u,d W . In this case, if one neglects small SM Yukawas, there is no interference with the SM, and the contribution of the dipole operators is dominated by diagrams in which the Z and Higgs boson are created directly in the qq annihilation. These diagrams are enhanced by (s/v 2 ) 3 with respect to the SM, where √ s is the partonic center of mass energy. Similarly, the interference of C ϕW with the SM does not contribute to the cumulative cross section. The correction is thus quadratic in C ϕW , and, for large s, is enhanced by (s/v 2 ) 2 with respect to the SM. In the case of C ϕW and of the vertex corrections c Qϕ,U , c Qϕ,D , c U ϕ and c Dϕ , there is competition between the interference and the quadratic pieces, which are enhanced by s/v 2 or (s/v 2 ) 2 compared to the SM, respectively. Finally, the non-standard Yukawa cross section is also enhanced by p 2 T /m 2 Z,W at large p T . Data on differential distributions from the LHC Run II would therefore be extremely helpful in constraining the effective operators in table 4 at the few percent level.
The bottom panel of figure 5 shows the ratio of the NLO and LO cross sections, for the SM and dimension-six operators. In most cases the NLO corrections are about 10%-20% across the values of p Z T cut considered in figure 5. The exceptions are non-standard Yukawa couplings, for which the NLO corrections decrease at higher p Z T cut .

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As in the case of W H production, angular distributions of the leptons emitted in the decay of the Z boson can help disentangle the effects of different dimension-six operators. We work in the Z-boson rest frame, with the direction of the z-axis along the momentum of the Z boson in the lab frame. θ * is the polar angle of the electron in this frame. The x-axis is in the direction orthogonal to the Higgs boson and Z momentax ∼ ( p Z × p H ). In this frame, we define the azimuthal angle φ * as the angle between the plane containing the Z and the Higgs bosons, and the plane containing the Z and the electron. That is and we note that φ * is invariant under boosts along the Z momentum p Z . The angular distribution of the Z boson in this frame is parameterized by 8 coefficients The coefficients A 0 and A 4 are related to the Z-boson helicity fractions [77,103,111].
The difference between left-and right-handed polarization depends on A 4 and on the ratio , where c L,R are the couplings of the Z boson to left-and righthanded leptons. The operators we study do not modify these couplings at tree level, and α is fixed by the Weinberg angle, sin θ W .
In figure 6 we show F 0 as a function of a cut on the Z transverse momentum. The figure shows that the Z boson is produced with a high degree of longitudinal polarization in the SM, and if the SM is modified by the couplings c Qϕ,U , c Qϕ,D , c U ϕ ( and c Dϕ , which is not shown in the figure). These couplings have the same helicity structure as the SM Z couplings, and thus it is not surprising that they do not affect the polarization much. The situation is quite different for dipole operators, that prefer the Z boson to have transverse polarization, pushing F 0 to zero at large p T . In the case of W H production, the operators Γ u W and Γ d W produce the W boson in an almost complete left-handed and right-handed polarized state, respectively [81]. In the case of ZH production, the difference between leftand right-polarization, captured by the coefficient A 4 , is always small, so that both Γ u W and Γ d W produce a right-polarized Z boson approximately half of the times. The operator C ϕW also prefers the Z to be transversely polarized. The theoretical uncertainties from PDF and scale variations, shown by the bands in figure 6, are small.
The remaining coefficients in eq. (4.5) also carry important information. For example, A 5 is sensitive to CP violation. A 5 vanishes in the SM, at NLO in QCD, but it receives a relatively large contribution from the interference of the operator C ϕW with the SM, as shown in figure 7. The remaining dimension-six operators, at least with the assumptions  of section 2 on their flavor structure, do not contribute to A 5 , thus making it a clean observables to pinpoint C ϕW . Notice however that A 5 is affected by a larger theoretical error, which is dominated by scale variations.

JHEP08(2018)205 5 Vector boson fusion
Finally we discuss the corrections to Higgs boson production via vector boson fusion. The total cross section for Higgs boson production through VBF has been recently computed in the SM at N 3 LO in QCD [118]. Fully-differential distributions are available up to NNLO [119] and the interface with parton showering is available at NLO+PS accuracy [58]. For this study, we computed the NLO QCD corrections to both the SM and the dimension-six SM-EFT contributions to the VBF cross sections, building upon the POWHEG implementation presented in ref. [58]. The contribution of SM-EFT operators, including NLO QCD corrections, has been considered in refs. [34,36,42]. Here we study the operators in table 2. In addition to the flavor assumptions discussed in section 2.2, in VBF we impose the further restriction of no tree-level flavor-changing neutral currents, so that only the diagonal components of Γ u,d W,γ , c Qϕ,U , c Qϕ,D , c U ϕ and c Dϕ can be non-zero. This additional assumption arises naturally in well motivated flavor models, such as minimal flavor violation [65], where off-diagonal elements are suppressed by small CKM elements or Yukawa couplings, and are irrelevant for collider observables. This assumption has the practical advantage of reducing the channels to be considered.
As in ref. [58], we neglect contributions from V H production, with the vector boson decaying hadronically, and, for channels with two identical quarks in the final state, we neglect effects due to the interchange of identical quarks. These effects are suppressed in the experimentally interesting region with two widely separated jets of large invariant mass. In the implementation of ref. [58], POWHEG generates events with the CKM matrix set to the identity, thus reducing the number of Feynman diagrams and singular regions. The final state quarks are then reweighted by the SM CKM matrix before they are showered by a Monte Carlo program. One could use the same approach for dimension-six operators that do not couple to quarks, such as C ϕW . On the other hand, if the SM-EFT operators couples to quarks, such reweighting is not possible, and one needs to generate events with the proper flavor configuration. Some of the operators described in section 2, such as Γ u,d γ or C ϕB , induce contributions to VBF mediated by the exchange of a photon. These contributions can generate collinear divergences, which are cut-off by the p T and rapidity cuts in the experimentally relevant region, but can make the generation of events extremely inefficient. In order to avoid these singular regions, if the coefficients of operators involving photons are turned on, the generation of events should be performed setting the bornsuppfact flag to an appropriate value in the powheg.input file.
In figure 8 we plot the differential cross section with respect to the dijet invariant mass, at √ S = 8 TeV. We applied the standard VBF cuts, requiring the invariant mass of the two jets to be m jj > 600 GeV, the rapidity separation |y j 1 − y j 2 | > 4.2, and that all jets have p T j > 20 GeV and |y j | < 5. We computed the cross section at the renormalization scale µ = 2m Z , and used the first member of PDF4LHC15 nlo 30 PDF set.  Qϕ , including VBF does not improve the constraints discussed in section 4.
The lower sensitivity of VBF compared to ZH and W H production can be appreciated from the bottom panel of figure 8, which shows that the VBF cross section induced by SM-EFT operators grows slowly as the dijet invariant mass increases. One advantage of VBF is the possibility to flavor tag the jets in the final states, providing an additional handle on the flavor structure of SM-EFT operators.
Finally, the bottom panel of figure 8 shows the ratio of the NLO and LO predictions. Also in this case NLO corrections range between 10% and 30%.

Conclusion
The SM-EFT is a powerful tool to characterize deviations from the SM, under the assumption that new physics arises at scales heavier than directly accessible at the LHC. In this paper, we have discussed the corrections of SM-EFT dimension-six operators to Drell-Yan, associated production of the Higgs boson and a weak boson, and Higgs boson production via vector boson fusion. We have included NLO QCD corrections, and interfaced to the parton shower using the POWHEG method. The resulting POWHEG BOX V2 computer code will be made publicly available on the powhegbox.mib.infn.it webpage. As possible applications, we have discussed the constraints on SM-EFT operators from the NC and CC Drell-Yan cross sections at high dilepton or transverse mass, and the ZH, W H and VBF signal strengths. We find that high-mass NC and CC Drell-Yan put strong constraints JHEP08(2018)205 on new vector, axial, scalar or tensor semileptonic contact operators, which are now competitive with the low-energy constraints from pion and kaon decays and nuclear β decay. ZH, W H and VBF are on the other hand sensitive to modifications of the Higgs couplings and to new couplings of the Higgs and vector bosons to quarks. The Higgs boson signal strengths at the LHC Run I, notwithstanding their relatively large uncertainties, already suggest that the scale of these new interactions has to be larger than 1 TeV. The constraints can be significantly improved by measurements of the signal strength and, more importantly, of differential distributions as the Higgs boson p T spectrum at the LHC Run II. We have pointed out that differential and angular distributions will play an important role in removing the degeneracy between different SM-EFT operators, thus providing clues on the form of physics beyond the SM.

A Formulae for the Higgs boson decay
The operators in table 2 that contribute to W H, ZH and VBF also give tree level corrections to the Higgs boson decays to γγ, γZ, W W * and ZZ * . In this appendix, we give the relevant formulae.

A.1 h → γγ and h → γZ
In the SM the decays h → γγ and h → γZ are mediated by loops involving Higgs boson couplings to the top quark and to the weak bosons. In the SM-EFT the decay widths are modified, at tree level, by the operators C ϕW , C ϕB and C ϕW B , and by their CP-odd counterparts, C ϕW , C ϕB and C ϕW B . We find N c = 3 denotes the number of colors, Q t = 2/3 the top charge, and we are neglecting contributions from quarks lighter than the top. The variables τ and λ are τ t = 4m 2 t /m 2 H , τ w = 4m 2 W /m 2 H , λ t = 4m 2 t /m 2 Z and λ w = 4m 2 W /m 2 Z . The loop functions A H t and A H W are given in ref. [120]: The decay rates for h → W W * and h → ZZ * are induced at tree level in the SM, and are modified by all the operators in table 2 that affect ZH and W H production. Including terms quadratic in the coefficient of effective operators, we find JHEP08(2018)205 The functions are defined as We do not consider Yukawa corrections to h → W W * and h → ZZ * , since non-standard Yukawa would mainly affect the decay of a Higgs boson into jets. Expressions for Γ(h → qq) can be found in ref. [120].

B Minimal Flavor Violation
In this appendix we briefly discuss the Minimal Flavor Violation framework, its implications for the couplings of the operators in the SM-EFT, and its implementation into the code.
MFV is based on the flavor-symmetry group SU(3) Q × SU(3) u × SU(3) d acting on the quark fields, and SU(3) L × SU(3) e acting on the leptons [65]. Within the SM the above symmetry group is only broken by the Yukawa interactions. MFV makes the assumption that this remains the case when going beyond the SM, such that BSM interactions only break the above symmetries through insertions of the Yukawa matrices which now act as spurions. Given the transformation properties of the Yukawa matrices one can then derive all possible structures for a given dimension-6 coupling. Here we take into account all possible structures, while neglecting the small Yukawa couplings, i.e. we take y b,c,s,d,u → 0 and y τ,µ,e → 0.
With these assumptions we find the following allowed structures for the corrections to the W and Z boson couplings c Qϕ,U , c Qϕ,D , c Dϕ and c U ϕ c Dϕ = a Dϕ 1,

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Operator Table 5. Notation for the coefficients of the different flavor structures in the MFV framework, discussed in appendix B, which can be set in POWHEG.
Similarly, the allowed structures for semileptonic four-fermion operators are 2) and (B.5) lead to operators involving explicit top quarks, which do not have any effects in the processes considered in this paper. As a result, we do not explicitly include these pieces in the code. Note that in the MFV framework, the assumption of lepton-flavor universality, made in section 2.2, simply follows from the flavor symmetries.
Finally, the couplings that do not appear in eqs. (B.1)-(B.6) can be set to zero given our assumptions. The reason is that most of these operators mix left-and right-handed quark fields, such as the ψ 2 ϕ 2 X and ψ 2 ϕ 3 classes as well as C (1,3) LedQ,LeQu , or couple right-handed u-type quarks to right-handed down-type quarks in the case of ξ. All of these couplings therefore come with at least one insertion of a Yukawa matrix, unlike the couplings in eqs. (B.1)-(B.6). Since we neglect small Yukawa couplings this means we do not have to consider most of these interactions. The only exceptions would be the u-type dipole operators and the corrections to the u-type Yukawa couplings, for which a term proportional to the top Yukawa survives. Nevertheless, as these terms always involve explicit top quarks, we neglect them in the code.
In order to apply the MFV framework as described above the user has to set the mfv flag to 1. The couplings of the dimension-six operators are then constrained to the forms of eqs. (B.1)-(B.6). The notation for the a i and b i coefficients that can be set in the input file are listed in table 5.