The SMEFTsim package, theory and tools

We report codes for the Standard Model Effective Field Theory (SMEFT) in FeynRules — the SMEFTsim package. The codes enable theoretical predictions for dimension six operator corrections to the Standard Model using numerical tools, where predictions can be made based on either the electroweak input parameter set α^ewm^ZG^F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{{\widehat{\alpha}}_{ew},{\widehat{m}}_Z,{\widehat{G}}_F\right\} $$\end{document} or m^Wm^ZG^F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{{\widehat{m}}_W,{\widehat{m}}_Z,{\widehat{G}}_F\right\} $$\end{document}. All of the baryon and lepton number conserving operators present in the SMEFT dimension six Lagrangian, defined in the Warsaw basis, are included. A flavour symmetric U(3)5 version with possible non-SM CP violating phases, a (linear) minimal flavour violating version neglecting such phases, and the fully general flavour case are each implemented. The SMEFTsim package allows global constraints to be determined on the full Wilson coefficient space of the SMEFT. As the number of parameters present is large, it is important to develop global analyses on reduced sets of parameters minimizing any UV assumptions and relying on IR kinematics of scattering events and symmetries. We simultaneously develop the theoretical framework of a “W-Higgs-Z pole parameter” physics program that can be pursued at the LHC using this approach and the SMEFTsim package. We illustrate this methodology with several numerical examples interfacing SMEFTsim with MadGraph5. The SMEFTsim package can be downloaded at https://feynrules.irmp.ucl.ac.be/wiki/SMEFT.


Introduction
When physics beyond the Standard Model (SM) resides at scales larger than the electroweak scale (Λ v T ), one can utilise an expansion in this ratio of scales to construct an Effective Field Theory (EFT). 1 Such an EFT can capture the low energy, or infrared (IR), limit of physics beyond the SM so long as no light hidden states are in the particle spectrum andv T /Λ < 1 is assumed/experimentally indicated. When these conditions are satisfied, and a SU L (2) scalar doublet with hypercharge y H = 1/2 is assumed to be present in the IR limit of the underlying sector, the theory that results from expanding in (currently) experimentally accessible scales divided by the heavy scales of new physics is the Standard Model Effective Field Theory (SMEFT).
The SMEFT is well defined and has been studied with increased theoretical sophistication in recent years. This theory can capture the IR limit of a wide range of possible extensions of the SM, consistent with the stated assumptions. Such SM extensions can address the strong evidence for dark matter and neutrino masses in addition to the theoretical issue of the hierarchy problem motivating ∼ TeV scale new physics. The interest in the SMEFT is due to the significant growth in available experimental data due to the continued operation of the LHC, and is also due to the theoretical developments reviewed in ref. [1]. In recent years, it has has become more widely understood that to gain the most out of studying the current LHC data set, and the high luminosity LHC data set, utilizing the SMEFT is valuable. Many LHC measurements will effectively be made below the scale Λ, even if new states are discovered with masses m ∼ Λ. This enables a practically useful, and simplifying, expansion in ∼v T /Λ when considering physics beyond the SM.
Putting in place this theoretical framework allows a general constraint program to be systematically developed over the long term, and also enables the combination of 1 HerevT is the vev defined as the gauge independent vacuum expectation value of the Higgs field including the effect of dimension six operators, H † H ≡v 2 T /2.

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LHC data with the extensive amount of lower energy data in a consistent field theory setting. Efforts in this direction have been hampered by the lack of a general coding of the leading SMEFT corrections to the SM in FeynRules [2][3][4] to date (using the theoretical approach of sections [2][3][4][5], to enable numerical studies. A major result of this paper is to address this issue by reporting a series of novel SMEFT implementations into FeynRules that have been developed and are now released for public use at http://feynrules.irmp.ucl.ac.be/wiki/SMEFT. The codes supplied define the SMEFTsim package, and they cover two different approaches to how the SM Lagrangian parameters are extracted from experimental measurements; i.e. the two electroweak input parameter schemes {α ew ,m Z ,Ĝ F } and {m W ,m Z ,Ĝ F }. SMEFTsim also consists of three different symmetry assumptions for the SMEFT operator basis, a fully flavour general SMEFT, a U(3) 5 -SMEFT with non SM phases, and a Minimal Flavour Violating (MFV) [5][6][7][8][9] version. We discuss these theories in section 4 and we present details of the structure of the coding of the SMEFTsim package in section 6. Using SMEFTsim, LHC studies using EFT methods are still challenging as the number of real parameters present is very large in the leading lepton and baryon number conserving corrections to the SM: there are 76 parameters in the case the number of generations n f = 1 and 2499 parameters in the case n f = 3. Even before this precise counting of parameters was determined [10], the understanding that the number of parameters was large led to pessimism that a general EFT approach could be pursued with collider data. As is well known, to ensure the key point of the EFT approach is not lost due to theoretical inconsistency, a full non-redundant set of operators (allowed by the assumed symmetries) must be retained in a consistent EFT at each order in the power counting of the theory.
Retaining all operators in the SMEFT does not imply that global fits to interesting experimental data necessarily involve the full set of 2499 parameters. Rather remarkably, a SMEFT physics program, although challenging, can be practically carried out at the LHC. The second main result of this paper is to develop the theoretical support for leading order (LO) EFT studies in a collider environment with subsets of parameters. We lay out the theoretical foundation of this approach in section 10 and define a 'WHZ pole parameter' program using this reasoning.
Reduced parameter sets can be adopted, despite neglecting terms the same order in the power counting of the EFT, for two main reasons. First, flavour symmetry assumptions, well motivated out of low scale experimental constraints, can be used. This leads to consistent alternate theories in addition to the general SMEFT: a U(3) 5 -SMEFT and a MFV-SMEFT. A simple corollary that we also systematically exploit defining the 'WHZ pole parameters' is that terms that violate symmetries preserved, or approximately preserved, in the SM interfere in a numerically suppressed fashion. Second, the number of relevant parameters is dramatically reduced in a global study of processes involving near on-shell intermediate narrow states of the SM. Exploiting such kinematics is already generic in well measured processes that are distinguished from large non-resonant backgrounds in a hadron collider environment and we advocate pushing this approach to its logical, and experimentally attainable limit.

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Case CP even CP odd WHZ Pole parameters General SMEFT (n f = 1) 53 [10] 23 [10] ∼ 23 General SMEFT (n f = 3) 1350 [10] 1149 [10] ∼ 46 U(3) 5 SMEFT ∼ 52 ∼ 17 ∼ 24 MFV SMEFT ∼ 108 -∼ 30 Table 1. Parameter counts in the general SMEFT flavour cases for n f generations, and the approximate number of parameters that feed into a W-Higgs-Z pole parameter program in the Warsaw basis, as discussed in section 10. Also shown are the parameter counts in the U(3) 5 limit and the MFV SMEFT case. The symbol ∼ indicates that these latter results are approximate counts for a leading order analysis, with leading flavour breaking spurion insertions, as discussed in section 10.
Taking all of this into account, the feasibility of a 'WHZ pole parameter' approach is illustrated in table 1 -which shows a manageable set of parameters to simultaneously study and constrain considering the global data set. We advocate dedicated experimental analyses be developed along these lines taking advantage of this dramatic simplification of the SMEFT approach at LHC. We demonstrate this numerically using SMEFTsim and MadGraph5 [11] in section 10.5.
So long as an appropriate theoretical error is assigned for this reduction in parameters, this approach can be adopted without introducing undue UV bias or blocking the possibility of building an inverse map to new physics sectors through the SMEFT. The is because these are IR assumptions and simplifications of the SMEFT projected into well measured LHC observables. Although the number of parameters is still 23 in the case of one generation and 46 in the case of three generations, we note that the number of models considered and experimentally constrained in the past decades in dedicated particle physics studies is substantially larger. As soon as a decoupling limitv T /Λ < 1 is present, vast arrays of possible extensions to the SM project into the compact and well defined SMEFT formalism. LHC results indicate that at this time it is reasonable to systematically consider and use the assumption thatv T /Λ < 1 via the SMEFT. In this case, it is much more efficient to project experimental results into the SMEFT using SMEFTsim, rather than into a endless series of models based on yet more assumptions. We encourage the LHC experimental collaborations to develop and study this approach using the tool provided.

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with the SM Lagrangian [20][21][22], defined as where H is an SU L (2) scalar doublet andH j = jk H † k with kj = − jk and 12 = 1, j, k = {1, 2} and topological Lagrangian terms are neglected. The gauge covariant derivative is defined with the convention D µ = ∂ µ + ig 3 T A A A µ + ig 2 t I W I µ + ig 1 yB µ , where T A are the SU c (3) generators, t I = τ I /2 are the SU L (2) generators, and y is the U Y (1) hypercharge generator. 2 The Yukawa matrices are complex with the relation Y i = √ 2M i /v to the complex mass matrices M i , whose real eigenvalues are the fermion masses. The chiral projectors are defined as P R/L = (1 ± γ 5 )/2. The fields {q, } are left handed and the fields {e, u, d} are right handed. We use the definition σ µ ν = i [γ µ , γ ν ]/2 and at times the short hand notation ψ for a general fermion field, and X µ for a general gauge field is used.
We use notation where the parameters of the canonically normalized Lagrangian (i.e. couplings, masses) carry bar superscripts. 4 The canonically normalized fields are generally indicated with a script font: {G, W, B}. The procedure for canonically normalizing is the same in both input parameter schemes, and we follow the approach laid out in ref. [10]. In unitary gauge, the Higgs doublet is expanded as This results in a canonically normalized h field when the Lagrangian is written in mass eigenstate fields. Note that the distinction betweenv T andv is at dimension eight whenv multiplies a Wilson coefficient C i . As such we can trade C iv JHEP12(2017)070 are working. The gauge fields are redefined as to take them to canonical form. The modified coupling constants are also redefined so that the products g 3 G A µ = g 3 G A µ , etc. are unchanged (at O(1/Λ 2 )) when canonically normalizing the theory.
Furthermore, the mass eigenstate basis for {W 3 µ , B µ } in the SMEFT is given by [10,12] W 3 while the mass eigenstate fields of the SM {Z µ , A µ } are defined via The relation between the mass eigenstate fields in the two theories are . (2.10) These expressions hold in both input parameter schemes. The notationθ denotes the weak angle defined in terms of input parameters (see next section). The three angles θ,θ,θ differ by quantities proportional to the L (6) Wilson coefficients. When such factors multiply a Wilson coefficient C i , the three notations are equivalent up to neglected dimension eight corrections. The SMEFTsim codes use unitary gauge and several simplifications that have taken place are dependent on this gauge choice. Nevertheless, the approach laid out here does not have any intrinsic theoretical assumption that blocks it being extended to other gauges to enable next to leading order (NLO) SMEFT results. 5 This approach to canonically normalizing the mass eigenstate Lagrangian in the SMEFT has been shown to be extendable to R ξ gauge fixing in ref. [37]. Gauging the SMEFT is subtle and will not be discussed at JHEP12(2017)070 length here but we note that utilizing unitary gauge for one loop calculations is well known to be best avoided. Even R ξ gauge requires a careful treatment of novel ghost interactions introduced in the gauge fixing terms rotating between the gauge and mass eigenstates, as discussed in ref. [37][38][39]. It has been shown that the related issues involving ghosts in one loop calculations can be overcome when the leading order (LO -i.e. only retaining L (6) corrections) approach of this work is adopted.

Treatment of inputs
We use notation where the input parameters used to define the numerical values of Lagrangian parameters, and quantities derived from input parameters carry hat superscripts. To determine the numerical value of the SM Lagrangian parameters from the {α ew ,m Z ,Ĝ F } EW inputs, the L SM Lagrangian parameters are fixed by the following definitions at tree level (with The Lagrangian parameters in the SMEFT differ from the SM Lagrangian terms due to L (6) local operator corrections. A generic parameter κ receives a shift from its SM value due to L (6) operators given by δκ =κ −κ , (3.2) and in the SM limit (C i → 0) one has δκ → 0. We define the short hand notation 6 and using this notation, related results are [1,10,[13][14][15][16]

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Using the {α ew ,m Z ,Ĝ F } EW input parameters one has {ᾱ ew ,m Z } = {α ew ,m Z }. We still define a parameter δm 2 Z for latter convenience. Note thatḠ F for EW applications is defined asḠ F = 1/ √ 2v 2 T . The input parameters are measured at different experimental scales p 2 {0,m 2 Z ,m 2 µ } and are defined as follows.

Extraction ofα ew
The extraction ofα ew occurs in the measurement of the Coulomb potential of a charged particle in the Thomson limit (p 2 → 0). The low scale extraction ofα ew in the SMEFT is given by In this expression, Σ AA is the two point function of the canonically normalized photon field in the SMEFT at a fixed order in perturbation theory. Formally unknown finite terms in the low scale extraction are present due to the vacuum polarization of the photon in the q 2 → 0 limit, and in addition further related uncertainties are introduced using this input parameter in running this low scale parameter up through the Hadronic resonance region (due to π ± loops etc.). The expression for Σ AA is generally rearranged into the form where the last quantity in square brackets dominantly leads to an uncertainty that is far larger than the low scale measurement uncertainty in the limit q 2 → 0. This introduces a significant numerical uncertainty as these nonperturbative contributions must be estimated. This is the dominant effect, leading to a parameterα ew (m 2 Z ) defined incorporating this correction and leading to the reduced theoretical precision given by [41,42] 1/α ew (m Z ) = 127.950 ± 0.014, while 1/α ew (0) = 137.035999139 (31). (3.11) We use as a code input the value ofα ew (m Z ) =α ew (m Z ) given in refs. [41,43] which includes an estimate of this numerical uncertainty. Whenα ew is used as an input parameter, the mapping of this expression to Lagrangian parameters is given in eq. (3.1).

Extraction ofĜ F
The extraction ofĜ F defined by the Effective Lagrangian generated in the p 2 m 2 W limit of the SM interactions (see ref. [44]) is defined at the scale µ 2 ∼ m 2 with the measurement of µ − → e − +ν e + ν µ defining the numerical value through the measurement of the total muon lifetime (τ µ ). The extraction ofĜ F in this manner is subject to the condition that the ν flavours are summed over experimentally. This is JHEP12(2017)070 required so that unitarity allows the neglect of contributions to the corresponding decay rate due to the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [45,46] relating the weak and mass eigenstates of the Neutrino's, which is suppressed in eq. (3.12). WhenĜ F is used as an input parameter in the SMEFT, the introduction of the shift δG F is still required for EW studies at the LHC. This is because the predictions of observables for LHC do not follow from the lower scale (µ 2 ∼ m 2 µ v 2 T ) Effective Lagrangian given in eq. (3.12).

Extraction ofm Z
The extraction ofm Z is defined in the simultaneous fit to the LEPI pseudo-observables {m Z ,Γ Z ,σ 0 had ,R 0 } as defined in ref. [47] that occurs in the pole scan through the Z mass that was preformed at LEP. 7 The extraction ofm Z occurs with the subtraction of soft initial and final state QED radiation (captured in a QED radiator function denoted R QED ) from the peak cross-section σ 0 as [47] σ The inputm Z is defined to be extracted from the simultaneous fit to these pseudoobservables. Whenm Z is used as an input the introduction of the shift δm 2 Z given in eq. (3.3) still appears in some cases, when experimental predictions depend on the inferred values ofĝ 1 ,ĝ 2 .
Using the {m W ,m Z ,Ĝ F } input parameter scheme offers several theoretical advantages: • The use ofm W has been shown to have a subdominant measurement bias to the quoted experimental error in the SMEFT [48]. This has not been established for α ew . Furthermore, the impressive intrinsic precision of the low scale measurements ofα ew is not relevant for the comparison of the two schemes, due to the large error introduced runningα ew up through the hadronic resonance region being dominant. Due to this, the percentage errors of each input parameter are within a factor of two as far as global constraint studies at EW scales are concerned.
• The use ofm W as an input allows SMEFT studies to expand around the physical poles defining scattering amplitudes such asψ ψ →ψ ψψ ψ through charged currents in a double pole expansion, leading to more consistent global constraint studies, see refs. [1,15] for more discussion.
• The use of anm W input parameter has some advantages when developing one loop results in the SMEFT, see the discussion in ref. [39,[49][50][51]. In addition, the scales of the input parameters are closer together using {m W ,m Z ,Ĝ F } reducing logarithmic enhancements present in runningα ew for LHC predictions, and related mixing effects with L (6) operators.

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The precise extraction ofm W at the Tevatron occurred historically after LEPI-II operations. This fact largely explains the current dominance of the {α ew ,m Z ,Ĝ F } input scheme. Initial investigations of the input scheme dependence of the global constraint conclusions in the SMEFT framework indicate that this scheme dependence is below experimental uncertainties [16]. The advantages of the {m W ,m Z ,Ĝ F } input scheme are substantial enough that transitioning to this approach is theoretically favoured. We provide codes utilizing each input parameter scheme to aid in this transition. Using the {m W ,m Z ,Ĝ F } input scheme, the SM Lagrangian parameters are defined aŝ (3.14) In this scheme {δG F , δm 2 h } are unchanged and

Extraction ofm W
An input parameterm W can be extracted using a fit to the transverse mass m T W at hadron colliders. Recall that when utilizing transverse variables (defined in the plane orthogonal to the collision axis) one defines a missing E T vector summing over all visible final state particles i. E miss T is so reconstructed in the case of W → ν decays and this defines | p ν T |. Combined with a measured p T of an identified (the momentum of the lepton in the plane transverse to the collision axis), these variables are used to construct where φ ν is the angle between the leptons in the plane perpendicular to the collision axis.
In the limit of | p W T | → 0 one has m 2 T = s sin 2 θ where θ is defined as the angle between JHEP12(2017)070  the W boson decay products and the beam axis in the W boson rest frame and s is the partonic energy of the produced W . The Jacobian of transforming between the variable θ and m T is given in the p W T = 0 case by m T /(s − m 2 T ). This introduces a sharp Jacobian peak in the m T spectrum that allows an extraction of the W mass from the shape of the spectra and its endpoint. Precise extractions ofm W are strongly impacted by detector resolution effects and p W T = 0 requiring template fits to the derived spectra to fit form W . See refs. [48,[52][53][54][55] for more details and the mapping tom W from such spectra.

Numerical values of inputs
The numerical values used to define the mass and coupling input parameters in each scheme are given in table 2. In addition, when including flavour violating effects, the Cabibbo Kobayashi Maskawa (CKM) matrix [59][60][61] is defined through the Wolfenstein parameterization [62] as  where the numerical parameters are input as [41]: Here "name" refers to the label of the parameter in the FeynRules codes. Note that the Pontecorvo Maki Nakagawa Sakata (PMNS) [46,63] matrix is not implemented in SMEFTsim as neutrino masses are neglected, but it can be directly incorporated in an extension.
In the context of the SMEFT, experimental extractions and fits to the CKM matrix elements get corrections due to L (6) operators. Such corrections define a difference between "bar" and "hatted" CKM quantities that are neglected here. The reason we have neglected these effects on the CKM inputs is that to our knowledge, no complete analysis in the SMEFT defining such corrections to the global fit to Wolfenstein parameters exists in the literature. Analyses that build up results central to the effort to determine such corrections include refs. [40,44,[64][65][66]. When such results are available they will be included in the SMEFTsim package as an update.

Fermion mass inputs
The fermion mass inputs are given in table 2. The relation of these measured quantities and SMEFT Lagrangian parameters is more subtle than in the SM. Following ref. [10] L Y = −h ψ R,r [Y ψ ] rs ψ L,s + . . . (3.24) and the relation between the complex fermion mass matrix [M ψ ] rs and effective complex Yukawa coupling in the SMEFT is given by These matrices are not simultaneously diagonalizable. Nevertheless the rotational freedom of the fermion kinetic terms (see eq. (4.1)) that leaves the kinetic terms invariant allows the mass matrix to be taken to diagonal form so that The effective Yukawa matrices are then off diagonal in general and complex in the mass eigenstate basis

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In taking [M ψ ] rs to diagonal form, non-SM phases that are present in the SMEFT are shifted into the general 3 × 3 V CKM matrix. The rephasing freedom of the SM fermion field kinetic terms still reduces V CKM to a unitary matrix with the same number of parameters as in the SM (i.e. three real parameters and one phase). New non-SM relative phases do persist in the effective Yukawa couplings in general. In the case of U(3) 5 symmetry where C * ψH rs → C * ψH [Y ψ ] rs with C * ψH ∈ C non-SM phases remain. In the case of MFV being assumed C * ψH rs → C * ψH [Y ψ ] rs and C * ψH ∈ R, and non-SM phases are absent.

Remaining inputs
The remaining inputs in either scheme are the Higgs mass and the strong coupling given directly bym The Higgs mass is directly extracted from the fitted reconstructed peaks in h → γ γ and h → Z Z → + − + − , see ref. [58] for a discussion. The extraction ofĝ 3 is performed by multiple methods. One of the most prominent in recent years is a joint fit to e + e − event shapes and leading non-perturbative corrections, for a review see ref. [67]. The Lagrangian parameterλ is a derived quantity from the input ofm h andĜ F which defineŝ 4 Flavour symmetries in the operator basis

Flavour symmetry assumptions
A general SMEFT contains a large number of real parameters in L (6) , as listed in table 1. There are 1350/53 CP-even parameters and 1149/23 CP-odd parameters in L (6) for three/one generations [10]. Most of the parameters in the SMEFT are in the ψ 4 operators due to flavour indices. This makes clear the importance of flavour symmetry assumptions and carefully utilizing numerical suppressions of ψ 4 operators contributing to cross-sections. We present codes that span several different flavour symmetry assumptions in the operator basis: a flavour symmetric U(3) 5 case that allows CP violating phases, a (linear) minimal flavour violating version where flavour change follows the SM pattern and new CP violating phases are neglected, and the general L (6) case. In this section, we summarize the required theoretical results for each case.

U(3) 5 limit
The U(3) 5 limit refers to the limit of unbroken global flavour symmetry in the SM Lagrangian, restored in the limit Y u,d,e → 0. To define this global symmetry group we define JHEP12(2017)070 the relation between the weak (unprimed) basis and the mass (primed) basis as Each U rotation defines a U(3) flavour group. The U(3) 5 group of the SM is defined as The relative U rotations between components of the lepton and quark SU L (2) doublet fields define the PMNS and CKM matrices as At times, it is useful to have defined the unbroken flavour groups of the SM (with the U(1) global flavour number groups removed) as the quark and lepton subgroups and a Yukawa matrix transforms as Y i ∼ {G q , G } for these groups. Yukawa spurion transformations are defined as so that one can restore the full {G q , G } flavour symmetry by inserting the Yukawa matrices in a manner that makes flavour singlet structures manifest. Furthermore, retaining only the top and bottom quark Yukawa coupling, defines a phenomenologically interesting breaking of the U(3) 5 limit where y i = √ 2m i /v for a quark of mass m i in the SM. We supply a FeynRules code implementing full diagonal Yukawa matrices that directly simplify numerically to this U(3) 5 breaking limit. Below we demonstrate the breaking of this limit that occurs numerically in the U(3) 5 codes, and only retain the leading breaking terms linear in y b , y t .
The L (6) operators are broken down to the Classes given in table 10. The Wilson coefficients of the operators in Classes 1,2,3 and 4 are unchanged going to the U(3) 5 flavour symmetric limit and allowing complex Wilson coefficients. The following Wilson coefficients are then defined in the U(3) 5 limit.

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In the last step we have further neglected all Yukawa's other than the top and bottom Yukawa's in the leading breaking of the U(3) 5 limit while rotating to the mass eigenstate basis. The operators where the fermion fields are taken to the mass eigenstate basis are indicated with a prime superscript. Note that this is an IR limit defined in the SMEFT and {C bH , C tH } ∈ C in general. The breaking of the U(3) 5 limit also follows from inserting In these terms, the expansion in the y 2 t ∼ 1 dependence can be considered to be implicitly absorbed into an effective Wilson coefficient parameter for the Class 5 and 6 operators.
Class 6 (ψ 2 XH) operators: in the U(3) 5 limit the Wilson coefficients of the ψ 2 H 3 and ψ 2 XH operators ∈ C as Y e,u,d are 3 × 3 complex matrices in general. In the last approximation, again, all Yukawa's other than the top and bottom are neglected in the leading breaking of the U(3) 5 limit while rotating to the mass eigenstate basis.
the leading breaking of the U(3) 5 limit. Rotating to the mass eigenstate basis has formally led to a redefinition of the initial Wilson coefficient of the form C → U (L/R) † C U (L/R) and these rotations have been absorbed into a redefinition of the Wilson coefficients on the right hand side of the . An implicit sum over flavour indices rr = {11, 22, 33} has been used here. As the operators {Q Hq , C He , Q Hu , C Hd } ∈ R. The U(3) 5 limit of the four fermion operators is more subtle. The (LL)(LL) operators simplify to [10,40] lq } ∈ R due to the operators being self-Hermitian and not transforming under any external group. Furthermore, The use of C and C denote the two different flavour contractions and the operators with all fermion fields identical also satisfy Q ijkl = Q klij due to relabeling freedom of dummy indices in all flavour symmetry cases, and in the flavour general case. The (RR)(RR) operators simplify in the U(3) 5 limit to where the C ee operator only allows one term due to the fact that the e fields are singlets , and a Fierz identity [10] reduces the number of effective parameters.
ud } ∈ R due to Hermitian operators that are not transforming under an external flavour group.
The (LL)(RR) operators are trivial and have one real Wilson coefficient for each operator. The chirality flipping ψ 4 operators have the U(3) 5 limits lequ } ∈ C. This limit does not forbid CP violation beyond the SM due to the presence of complex Wilson coefficients. Again, in the last approximation all Yukawa's other than the top and bottom are neglected.

MFV breaking
Assuming that a CP violating phase only appears in L (6) due to the SM source of CP violation present in the CKM matrix, and that the breaking of flavour symmetry in the SMEFT follows the breaking pattern in the SM, defines the MFV paradigm [7] (see also refs. [5,6,8,9]).
The reasons to adopt these strong symmetry assumptions are basically twofold. First, the set of experimental constraints derived in the flavour physics program push a naive flavour violating suppression scale in the SMEFT expansion Λ TeV, rendering SMEFT studies of LHC in this case unlikely to extract evidence of physics beyond the SM. 8 The converse point also holds in that if SMEFT studies do uncover deviations in LHC data, then an underlying sector must be consistent with strong lower energy flavour constraints. If this occurs due to a MFV symmetry breaking pattern then this symmetry assumption acts to reduce tuning of parameters. Second, and equally important is that the introduction of such (IR) symmetry assumptions render systematic SMEFT constraint studies feasible to practically carry out. 9 We introduce the Jarlskog invariant [61,62,70,71] as in the MFV limit (at leading order in the MFV expansion) {CG, CW , C HG , C HW , C HB , C HW B } ∝ J, (4.37) which renders {CG, CW , C HG , C HW , C HB , C HW B } and neglectable in studies that also neglect next to leading order corrections, such as the codes reported here, as loop suppressions are ∝ g 2 SM /16π 2 J are not systematically included.

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MFV does not preserve flavour as in the U(3) 5 limit, but dictates that the flavour breaking pattern follows the SM. Following ref. [7] an MFV expansion can be constructed by expanding in flavour invariants determined using the spurion transformation properties in eq. (4.6). The Class X 3 , H 6 , D 2 H 4 , X 2 H 2 Lagrangian terms are unchanged from the U(3) 5 limit results given above. The leading MFV breaking spurion's are given by Here and below ∆ indicates a relative normalization of terms resulting from the spurion insertions and the first line is simplified with the implicit understanding that the y 2 t /y 2 b spurion breakings leading to flavour change will be inserted for the d L /u L fields respectively expanding out the Q L doublet field. The spurions transform as for S u pr , S d pr , S q pr under the global favour symmetries G q . The S q spurion that dictates flavour changing neutral currents, acts to absorb the rotation matrices between the weak and mass eigenstate bases for the fermion fields. Similarly the S u , S d spurions absorb the rotation matrices going to the mass eigenstate basis fields. Incorporating the effects of all of these spurion breakings leads to the following extra parameters in the supplied codes constructed in unitary gauge.
For the Class 5,6 (ψ 2 H 3 ,ψ 2 XH) operators, the effects of the spurions S u,d can be absorbed into a redefinition of C uH ,C dH noted above, as only the third generation entry is retained. The same point also holds for the (LR)(RL) and (LR)(LR) operators. The Class 7 (ψ 2 H 2 D) operators have the following extra parameters in the linear MFV breaking limit: The (LL)(LL) operators introduce the extra parameters (4.50)

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where the ∆C i , ∆C i parameters are normalizations that can differ from the one multiplying the spurion insertions. The (RR)(RR) operators have the extra parameters The (LL)(RR) operators have the extra parameters The remaining operators follow the pattern of the U(3) 5 limit.

Operator normalizations
The normalization used in the SMEFTsim codes also differs from other codes, which should be noted in comparing results. The HEL implementation [72], eHDECAY [73], Higgs Characterization [74] and ROSETTA [75] use a varying suppression scale 1/m 2 W or 1/v 2 for operators. Furthermore, these codes normalize a subset of operators by powers of gauge couplings.
Following Weinberg [76] we take a different approach that conforms with a traditional EFT construction. We retain the general EFT with the most general interaction terms consistent with the assumed symmetries without extra UV specific dynamical content or assumptions. The L (6) operators are normalized in the SMEFTsim codes to a naive mass dimension suppression scale Λ 2 . Operators with field strengths are not normalized to be proportional to a corresponding SM gauge coupling, or suppressed by 16π 2 . The former normalization is not required to respect SU C (3) × SU L (2) × U Y (1) symmetry and the latter is not model independent 10 . No assignment of UV specific coupling factors can be made 10 A historically widespread approach of suppressing operators containing field strengths by loop factors was shown to not be a model independent EFT statement in ref. [77]. See also the discussion in ref. [78] agreeing with these developments.

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in L (6) without introducing further UV assumptions, so we do not include such factors in the FeynRules codes.
Such normalizations can introduce a very problematic non-commutation with the equations of motion when interested in EFT studies that seek to obtain basis independent results. Furthermore, unusual arguments that imply some SMEFT operator bases are preferred have also appeared in the literature related to this challenge. These problems can be avoided if the corresponding Wilson coefficients of the operators normalized differently are then varied sufficiently widely in experimental studies to cancel a chosen normalization. By using a normalization by the naive mass dimension suppression scale 1/Λ 2 we avoid placing this serious burden on a user of the SMEFTsim codes. We note that this standard EFT approach is also used in DsixTools [79] and in Hto4l [80]. This makes it easy to interface with these two programs in the future. We caution that it does not follow, when using a 1/Λ 2 normalization, that scan procedures assuming a homogeneous size for the Wilson coefficients is sufficient to cover all possible UV scenarios.
When comparing results with other codes, we caution that to our knowledge, the SMEFTsim codes, and the implementations of ref. [37,79], are the only example of complete (public) codings of the L (6) SMEFT available to date. 11 Note also that (of these complete codes) only SMEFTsim incorporates input parameter corrections. Missing operators can have non-intuitive consequences on the interpretation of Wilson coefficients that are retained comparing two SMEFT codes, and make comparing complete operator basis results to incomplete results (that are also at times ill-defined) challenging. This is due to the equations of motion being extensively used to define the SMEFT in a minimal basis at L (6) , so that the resulting Wilson coefficients in the reduced basis reflect many removed operator forms not retained. In short, when comparing SMEFTsim results to other codes caveat emptor.

One loop functions
The codes supplied are designed to enable numerical studies of the LO (tree-level) interference of the SMEFT with the SM, while neglecting NLO corrections. This approach is phenomenologically insufficient if universally applied to all SM interactions.
The processes h → gg, h → γ γ, h → γ Z only occur at one loop in the SM due to renormalizability. To obtain a non-zero interference for these processes as a leading numerical correction, we implement the one loop functions for these processes in the SM following the results in refs. [81][82][83][84][85]. An explicit SM Lagrangian term LHSMloop that is defined as

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has been included with The loop functions are In the codes supplied they have been defined in a Taylor expansion up to cubic terms in the arguments

SMEFTsim FeynRules packages
The SMEFTsim package is designed based on the theoretical outline of the previous sections and consists of several model files for the tree-level analysis of the L (6) SMEFT corrections. It contains both model files for FeynRules [4] and pre-exported UFO files [3] to be interfaced e.g. with MadGraph5 aMC@NLO [11]. Two independent models sets are supplied, called "Models set A" and "Models set B": each contains three different theories: a fully flavour general SMEFT, a U(3) 5 − SMEFT with non-SM complex phases and MFV − SMEFT. In addition, each case has two different input schemes available {α em ,m Z ,Ĝ F } and {m W ,m Z ,Ĝ F }. The two models sets differ JHEP12 (2017)070 in the structure and in the technical implementation of L (6) , but they produce equivalent results: the use of both sets is recommended for debugging and validation of the numerical results.
All the models are built upon the default SM implementation in FeynRules [86], from which they inherit the SM fields, parameters and Lagrangian definitions. The original file has been extended and modified to include the complete set of L (6) baryon and lepton number conserving operators of the Warsaw basis [19] and the input numbers have been updated according to table 2. The SM loop-induced effective couplings of the Higgs to gg, γγ and Zγ have also been included, as detailed in the previous section. At this stage, the ghost Lagrangian has been left in its SM form. As a consequence the models give valid results only in unitary gauge, so $FeynmanGauge = False has been enforced in all cases.
The main purpose of the SMEFTsim package is to provide a complete tool for the analysis of the tree level interference terms between the L (6) dependent amplitude and the SM amplitudes in a measured process. The implementation of the entire parameter space of the SMEFT and the automatic inclusion of the shifts due to the choice of an input parameters set is a key feature. In this spirit, the models are not meant to be employed for the extraction of accurate SM predictions and they are not equipped for NLO calculations in MadGraph5 aMC@NLO. The results obtained with SMEFTsim have a finite theoretical uncertainty O(%) for the interference term predicted due to neglected higher orders in the SMEFT effective expansion (L (8) + · · · ) and radiative corrections that are not included.
In this section we provide further details about the implementation of the package in FeynRules and MadGraph5.

Definition of the Wilson coefficients
All the Wilson coefficients are assigned a specific interaction order called NP = 1. See refs. [4,11] for a definition of interaction order and other options relevant to the Mad-Graph5 implementation. They are defined to be dimensionless, as the cutoff scale of the EFT has been defined as an independent external parameter called LambdaSMEFT with a default value of 1 TeV, that can be modified by the user. LambdaSMEFT is defined with an interaction order QED = -1, so that the ratiov 2 /Λ 2 has overall QED = 0. The dim-6 operators of the ClassN = 1, 2, . . . 8 classified as in table 10 L6 The full dim-6 operators, N =1,...,8 L6clN. LagSMEFT LSMincl + L6. Table 3. Lagrangian terms defined in the SMEFTsim code.

Definition of the Lagrangian
All the models contain the Lagrangian terms listed in table 3. In particular, L (6) has been split into 8 terms, one for each Class defined in table 4. The Lagrangians are by definition Hermitian, while the individual L (6) operators are not in general.
The Lagrangian is entirely written in the fermion mass eigenbasis, in which the Yukawa matrices are real and diagonal and the CKM matrix is consistently inserted in charged quark currents. By default all the Yukawa matrices have 3 non-zero diagonal entries and all fermion masses (except those of neutrinos) are non-vanishing. Restriction files are supplied for both the FeynRules and UFO models, that set to zero all the fermions' masses and Yukawas except those of the t and b quarks. Analogously, the CKM matrix is defined as a 3 × 3 unitary matrix in the Wolfenstein parameterization [62], but it can be restricted to the 2 × 2 Cabibbo rotation in FeynRules.

Field redefinitions and shifts
The field redefinitions required to have canonically normalized kinetic terms and the parameter shifts induced by the choice of a set of input parameters are automatically performed in the code, consistent with sections 2 and 3. This means that all the parameters appearing in the output Lagrangian are "hatted" quantities.
The shift in m W induced in the alpha scheme is peculiar in that it does not suffice to have the shift reproduced correctly in the Lagrangian, but it is also necessary to embed it in the definition of the W field for it to be read properly by MadGraph5. This is done defining MW as an internal parameter that includes the shift dMW. This solution is ineffective for the FeynArts/FormCalc/FeynCalc interface [87][88][89] that defines mass parameters independently. When employing the {α ewmz ,Ĝ F }-scheme models within either of these frameworks, the user needs to apply manually the replacement in all the analytic expressions.
qq , Q  In the flavour general models, the L (6) operators constructed out of the fermion fields have free flavour indices that are contracted with those of the associated Wilson coefficients. The latter ∈ C in the flavour space, and are therefore defined as internal tensorial parameters in FeynRules, with norms and phases given independently for all the complex entries. Hermiticity and symmetry constraints require some entries to be real and enforce relations among different entries of a Wilson coefficient matrix, reducing the number of free parameters as detailed in section 2. This has been taken into account in the codes. are therefore retained in the FeynRules model. The restriction cards for massless light fermions consistently set to zero the flavour-diagonal spurions, as they become redundant in this limit. The replacement of the Wilson coefficients in terms of the spurions is done explicitly in the Lagrangian (via the Definitions attribute), so as to make the number of independent contractions manifest and to allow the reduction of symbolic CKM insertions (unitarity enforces cancellations in the product of CKM insertions stemming from field and spurions definitions). The Jarlskog invariant is neglected and CP violating operators expected to be proportional to it are not implemented as they are significantly numerically suppressed.

Code structure
This models set contains one main file called SMEFTsim A main.fr that imports fields definitions from SMEFTsim A fields.fr and parameters definitions from SMEFTsim A parameters.fr. The latter contains switch commands that select the appropriate parameter definitions depending on the flavour framework and input parameters scheme selected: before importing the model in Mathematica, the user should define the two flags Flavor and Scheme that take the values {general, U35, MFV } and {alphaScheme, MwScheme} respectively. The definitions of α ew , m W , sθ, δg 1 , δg 2 depend on the input scheme choice, while the flavour specification determines which Wilson coefficients set is imported among d6 parameters general.fr, d6 parameters U35.fr, d6 parameters MFV.fr (see below) and, consequently, the form of δG F and of the redefinition of the Yukawa couplings. The three files differ mainly in the implementation of the coefficients for fermionic operators, as described in section 6.4.
The definitions of the L (6) operators are also dependent on the flavour assumption adopted, and they are imported from one among the files SMEFTsim A operators general.fr, SMEFTsim A operators U35.fr, SMEFTsim A operators MFV.fr.

Inputs and shifts
The redefinitions of the Higgs and gauge fields required to bring the kinetic terms to their canonical form (see section 2) are applied automatically in the code. For the Higgs field, this takes place in the doublet field definition, while for the gauge bosons a field redefinition called rotateGaugeB is applied on the SM Lagrangian at the mass eigenstates level. This choice avoids performing unnecessary rotations on the gauge fields appearing in L (6) . The redefinition of the vev and of the coupling constants due to fixing the input parameters set (see section 3) is done applying the replacements redefConst and redefVev on the SM Lagrangian terms.
The shifts δG F , δm 2 Z , δm 2 h , δg 1 , δg 2 , δs 2 θ , δm W are left explicitly in the Lagrangian and they are defined in FeynRules as internal parameters depending on the Wilson coefficients, so that they are automatically assigned the correct numerical value in MadGraph5. the Z coupling to a pair of neutrinos: It is worth noting that the Lagrangian expressions containing these quantities have the same form irrespectively of the input scheme chosen. What distinguishes the {α ew ,m Z ,Ĝ F } from the {m W ,m Z ,Ĝ F } choice is the exact dependence of the shifts on of Wilson coefficients. The models contain a replacement list called either alphaShifts or MwShifts that allows one to make explicit the Wilson coefficient dependence in algebraic expressions. The replacements should be applied via ReplaceRepeated in Mathematica.
Finally, all the models contain the definition of the functions LinearWC and SMlimit. The former linearizes analytical expressions in the L (6) corrections, while the latter sets them to zero, recovering the unshifted SM expression.

Comments on the implementation for different flavour structures
The definition of the L (6) operators and associated Wilson coefficients has been optimized for each of the three flavour setups considered. In the flavour general model the fermionic operators have free flavour indices and the corresponding Wilson coefficients are defined as tensorial parameters. In the U(3) 5 symmetric models the flavour contractions for all the fermionic operators are incorporated in the definition of the operators themselves. This allows a reduction in the number of diagrams in the UFO model and consequently the computation time. All the Wilson coefficients are therefore scalar numbers (∈ R for Hermitian operators).
In the MFV case, only the Wilson coefficients of quark operators carry flavour indices and they are defined as the appropriate combination of flavour invariants. Because the model is written in the fermion mass basis and the Yukawa matrices are real and diagonal, it is sufficient to define three spurions to implement the spurion breaking given in eq. (4.41). In this way, for instance, C Hq can be introduced as where the two components of the S q spurion have been assigned independent coefficients ∆ u C (1) Hq and q = (u L , V CKM d L ). One can immediately verify that expanding the SU L (2) components gives the correct expression where CKM insertions accompany S d in the (ū L u L ) current and S u in the (d L d L ) current. Four fermion operators and their coefficients are defined in an analogous way.
The following notation is adopted for spurion insertions: the coefficients of the identity contractions are denoted with a final 0 (e.g. ceW0 ). The coefficients accompanying spurion insertions have names starting with Delta: for operators that allow only one spurion insertion the associated coefficient is called DeltacXX (where XX stands for the operator JHEP12(2017)070 name). Wherever both the S u and S d spurions are allowed we assign them coefficients called DeltaucXX and DeltadcXX respectively. For four fermion operators that admit spurion insertions in both currents, those in the first current have a coefficient Delta1cXX (or Delta1ucXX , Delta1dcXX ) and those in the second one have coefficients Delta2cXX (or Delta2ucXX , Delta2dcXX ). All the parameters appearing here are real, as the only phases allowed from the MFV ansatz are those stemming from the CKM matrix. In the master code the InteractionOrderHierarchy is defined but the InteractionOrderLimit is not specified. 13 In addition to the QCD and QED, we specify the NP orders for the interactions that arise from dimension-6 operators.

SM inputs
The SMdefs.fr is a universal subroutine consisting of the definition of the gauge groups (U1Y, SU2L, SU3C) and the indices associated with these groups. As the gauge group is not enlarged in the SMEFT, this subroutine are retained the same as the SM default implementation. The description of the SM fields is contained in a separate subroutine. Even if no new field is introduced in the SMEFT, we make modifications for this subroutine, offering two versions for {α ew ,m Z ,Ĝ F } or {m W ,m Z ,Ĝ F } input schemes separately, as explained in section 3. The difference occurs on the W ± boson mass MW. It is set Internal in the SMfields alphascheme.fr file, while externally given a numerical value in the SMfields mWscheme.fr file.
The second part of the model file is the definition of the model parameters, which include the SM parameters and the Wilson coefficients of L (6)  SMEFT .
fr Figure 1. Illustrative structure of the SMEFT model set B files.
symmetric limits, the Yukawas and CKM must be defined before the Wilson coefficients. For this purpose, we divide the SM parameters into two blocks: YUKAWA and SMINPUTS.
In the subroutine SMYukawas.fr, the mass of all fermion (mass eigenstates) belonging to YUKAWA block are externally given numerical values.
The remaining SM parameters are exclusively defined in parms alphascheme.fr and in parms mWscheme.fr for the {α ew ,m Z ,Ĝ F } or {m W ,m Z ,Ĝ F } input schemes, respectively. The other block SMINPUTS includes three external parameters: aEWM1, Gf and aS in the parms alphascheme.fr, and with the substitution of aEWM1 by MW in the subroutine labeled as parms mWscheme.fr.

Wilson coefficients
In general, the full list of Wilson coefficients contain two types of parameters: scalar parameters for flavour-singlet Wilson coefficients and tensorial parameters when the Wilson coefficients carry flavour indices. The dependence on the flavour space can be reduced as shown in section 4. In the model file, there are three versions of subroutine SMEFTparms.fr, SMEFTparms MFV.fr and SMEFTparms FLU.fr provided corresponding to the cases of flavour general SMEFT, MFV-SMEFT and U(3) 5 − SMEFT, respectively. In each file, all the flavour-singlet Wilson coefficient parameters are stored in the NEWCOUP block. The coefficients associated with spurion breaking in the MFV limit are individually stored in a block named MFVCOUP. We stress that the attributes BlockName and OrderBlock cannot be specified for tensorial parameters. By default, their block name are taken as FRBlock#X in sequence. In addition, the shifts on the Higgs vev and coupling, gauge boson mass and gauge couplings dGf, dvev, dlam, dW, dM2Z, dgw, dg1, dsw2 due to the L (6) contribution are constructed in the subroutine parms alphascheme.fr and parms mWscheme.fr.

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Among them, the W mass shift dW presented in the {α ew ,m Z ,Ĝ F } input scheme follows eq. (6.1). and the analytic forms for the remaining ones are summarized in section 3.

Lagrangian construction
The SMEFT Lagrangian is constructed in the subroutine Lag.fr. The SM Lagrangian LSM is implemented by default. As already shown, in the presence of L (6) a shift at the leading NP order is induced on the Higgs vev and SM gauge couplings compared to the SM values, and meanwhile the field redefinition for the SM fields are also demanded, leading to a conversion from the initial SM Lagrangian LSM to a new defined SM Lagrangian LSMshift. Here a series of substitution rules (redefXXX) employed encode the above mentioned shifts and redefinitions. It is clear that the terms of higher order in Wilson coefficients are included in the LSMshift. This not only brings the inconsistency in the perturbative expansion but also increases the difficulty in the numerical computation. In fact, for the purpose of the phenomenological study, one may be interested in a SMEFT Lagrangian linearized in Wilson coefficients. To this end, we practically introduce an auxiliary variable For completeness, we add to the the inclusive SM Lagrangian the dimension-5 effective couplings of the SM Higgs to gg, γγ and Zγ which is defined in the LhSMloop function given in section 5.1.

LSMincl := LSMlinear + LhSMloop;
On the other hand, the effect of shifts and redefinitions on the dimension-six terms L (6) are higher NP order and can be safely dropped in the SMEFT. As a result, the SMEFT Lagrangian is L SM + L (6) denoted in the code as LagSMEFT := LSMincl+L6.

General recommendations for the use of the UFO models in Mad-Graph5
When generating a process in the SMEFT it is always necessary to specify the order NP=1 to make sure that all and only the diagrams giving linear L (6) contributions are included. Notice that the InteractionOrderLimit is not specified by default in the model

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files. In order to extract the tree level interference contribution between L (6) and the SM amplitudes, we suggest generating the process with the syntax NP^2==1 in MadGraph5. In general, due to the fact that SM Lagrangian parameters (g i , Y i ,v . . .) can multiply the Wilson coefficients in the Lagrangian, a given interaction vertex can have multiple interaction orders. For instance, the Zēe coupling stemming from Q He is proportional tô v 2 g 1 C He /Λ 2 and has therefore interaction order {NP=1, QED=1}. There is one coupling that has negative QED order, namely the contribution to the trilinear Higgs coupling h 3 stemming from Q H , which is proportional to C Hv 3 /Λ 2 and has therefore order {NP=1, QED=-1}. One should be careful when generating processes that include this coupling, as this may alter the intuitive interaction order hierarchies among diagrams. Before generating events or calling a survey in MadEvent it is preferable to set all the relevant widths to auto in the param card. This is because the value of the particle width is used to compute some cross-sections in a narrow-width approximation in MadGraph5. The values assigned by default to the widths in the model are those computed in the SM (sometimes including radiative corrections) and they are often inconsistent with a treelevel SMEFT prediction. 14 Note for the general SMEFT, it takes a few minutes to load the model, and at most two hours to generate the UFO files.

A pole parameter global SMEFT fit
The SMEFTsim codes enable LHC SMEFT studies to be carried out combined with lower energy data reported at LEP and other experiments, while all parameters in L (6) are retained. Such a global SMEFT physics program is of interest long term, due to its importance for the development of model independent constraints. This approach enables hints of new physics that could emerge in the data in time to be understood and decoded systematically by combining measurements of deviations in a well defined field theory setting. This approach is also valuable as it is a way to record the data in a field theory interpretation that allows the SM to break down at higher energies for the long term.
Developing simplified fits as an intermediate step towards the general fit case is also important. This can be done minimizing UV assumptions and exploiting the kinematics of the relevant collider scattering events, in addition to the SMEFT power counting and flavour symmetries. This approach can be followed when defining a 'WHZ pole parameter' program to constrain an interesting subset of SMEFT parameters. We consider one of the main applications of the SMEFTsim package is to directly enable this effort to be undertaken in the LHC experimental collaborations. The idea is to use the fact that L (6) operator forms interfere with a SM process for the numerically leading correction, to supplement the power counting of the SMEFT systematically by using:

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• Resonance domination of signal events enforced with optimized phase space cuts to further suppress (primarily) Class 8 ψ 4 operators. This is discussed in section 10.1.
• Numerical suppressions in interference due to the presence of small symmetry breaking effects in the SM due to light quark masses, helicity configurations, CP violation, CKM suppressions and the GIM mechanism [90] in SM amplitudes.
Exploiting these IR physics effects in the SMEFT, in addition to the usual power counting of the theory, significantly reduces the number of WHZ pole parameters to a manageable set. Many studies have been performed that utilize one or more of these IR effects, but we believe the systematic approach laid out here goes beyond past literature. It is important to note that the parameter reduction that makes a LO SMEFT effort feasible using such IR physics occurs if flavour symmetry assumptions are explicitly adopted or not, see

Resonance domination and numerically suppressed interference
The interference with the SM leads to a relative kinematic suppression of ψ 4 operators, compared to the parameters retained in a 'WHZ pole parameter' program in several experimental signals of interest. 15 This occurs so long as scattering events in a measurement are dominantly proceeding through a near on-shell phase space pole (i.e. To understand the relevance of the 'WHZ pole parameters' for hadron collider measurements, consider a general scattering amplitude depicted in figure 2. The total amplitude can be decomposed around the physical poles of the narrow bosons of the SM in the process These arguments are the logical extension of the reasoning used to justify the neglect of ψ 4 operators in studying LEPI data, as discussed in refs. [13,91].

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' Contents -1 - Figure 2. Decomposing a general scattering amplitude in the SMEFT into pole and non-pole parameters. Here the black dot indicates a possible insertion of L (6) and shifts are only shown on the final states as an illustrative choice, but also appear on the initial states in the suppressed + · · · contributions. A WHZ pole parameter is shown in case a). figure 2 illustrates that L (6) corrections that are the same order in the power counting can modify a resonant process in the SM (as in A a and figure 2 a), lead to a contribution to the scattering amplitude with fewer poles than in the SM process (as in A b , · · · A j−1 and figure 2 b), or lead to a contribution to the process with no internal SM poles (as in A j ) from narrow SM bosons. Here the p 2 i factors stand for general Lorentz invariants of dimension two. Assume that selection cuts are made so that the process is numerically dominated by a set of leading pole contributions of ≤ N narrow B bosons. Then the leading SMEFT cross-section corrections in this phase space volume Ω are expected to be dσ δΩ pole The differential cross-sections (dσ SM /δΩ) 1,2 are distinct in each case and C i correspond to WHZ pole parameters that are of the form H|L SM |H , the C j correspond to scheme dependent corrections to the intermediate propagators 16 and the C k correspond to a subset of operators that lead to three point interactions with more than one derivative. Additional corrections to the measured processes and relevant backgrounds also exist in the SMEFT, but they can be relatively numerically suppressed in a SMEFT oriented experimental analysis. Consider the interference with a complex Wilson coefficient in L (6) , denoted C, that occurs when a resonance exchange is not present compared to the leading resonant SM signal result (shown in figure 2 d)). The interference terms in the corresponding observable then scale as In the near on-shell region of phase space (  SMEFT then has the additional numerically subleading corrections For this reason, the numerical effect of the non-pole parameters are relatively suppressed by a factor of compared to a Wilson coefficient that is a (scheme independent) pole parameter. Such a suppression factor appears for each missing resonance selected for with selection cuts. This relative numerical suppression occurs in addition to the power counting in the SMEFT. It is the combination of these two suppressions that is experimentally and theoretically relevant.
As experimental selection cuts for narrow SM B bosons do not isolate all of the poles in a process in general, in some cases ψ 4 operators can be classified as leading parameters numerically, and should be retained in a global SMEFT analysis examining such a process. For example, considerψψ →ψψψψψψ which can occur through many Feynman diagrams, including the diagrams in figure 3. In the left two diagrams of figure 3, isolation cuts to identify and reconstruct only the two bosons that decay into pairs of final state fermions do not suppress these contributions. A ψ 4 operator can then be classified as a leading contribution to be retained in an alternate interesting subset of SMEFT parameters for this process, if the remaining phase space selection cuts did not further suppress these contributions. 17 We use the nomenclature 'WHZ pole parameters' which is intended to signal that we exclude such cases by definition in this parameter set.
Other parameters neglected from the 'WHZ pole parameters' set are numerically suppressed due to the lack of tree level flavour changing neutral currents in the SM, see section 10.2 for details. We reiterate, these additional numerical suppressions come about due to IR kinematics and symmetries in near on-shell regions of phase space for the narrow B bosons selected for, not UV assumptions. As NLO corrections are neglected in the SMEFTsim package anyway, these numerically suppressed effects can be consistently neglected -so long as an appropriate theoretical error is assigned for this approximation.  Pole parameter SMEFT studies are an important step towards a global SMEFT analysis. This approach has some similarity to a pseudo-observable (PO) approach to LHC data [92][93][94][95][96][97][98][99], and can be considered to extend and improve it by embedding this approach it in a well defined field theory. Some minor differences in the approaches do exist, and follow from the use of a consistent L SM EF T construction, as we discuss below.

Retaining a general flavour conserving anomalous dipole interaction
due to the arguments of the appendix on the experimental constraints on dipole operators, leads to off-shell interference with the SM Lagrangian in figure 4 a) and c) that give Here a general chiral interaction in the SM is parameterized by g R/L . As the interference is suppressed by m ψ , it follows that C ψX insertions can be initially neglected in LO global SMEFT studies involving light fermions (ψ = {t, b}), not due to experimental constraints, or a UV model assumption, but as a numerical suppression due to the IR physics of the SMEFT. This can be done until experimental precision advances to overcome such additional numerical suppressions. This is in agreement with ref. [95,[97][98][99] argued in the context of a PO framework. Nevertheless, the results of the appendix argue for retaining dipole operators when ψ = {t, b}, and we note that the inclusion of top dipole interactions has been shown to have an important effect on Higgs phenomenology in many works, including refs. [100][101][102][103]. Note that if such L (6) terms are not arbitrarily neglected, then contributions such as shown in figure 2 c) require a deconvolution of possible non-SM soft emissions in the LHC collider environment to extract model independent PO. For more discussion see refs. [1,92,98]. IR SM-L (6) interference effects can also justify the neglect of flavour off diagonal three point interactions. Such interactions can be present due to Class 5, 6 and 7 operators.  (10.8) due to the presence of a GIM mechanism [90], with m j the mass of the internal quarks summed over. Similarly the amplitude following form the effective one loop coupling hψ i ψ k also has a GIM suppression [108][109][110] A h ij 3v Tḡ

b) scales as [104-107]
Interference with flavour off diagonal corrections then experiences an additional numerical suppression of this form in both cases. This can be used to justify neglecting such effects in LO SMEFT analyses when neutral currents are present. No such extra GIM suppression is present in the case of charged currents. 18 For this reason, flavour off diagonal entries in the Wilson coefficient matrices C Hl , C Hq can be practically retained in a LO analysis, while neglecting the remaining flavour off diagonal Wilson coefficients of the Class 5 and Class 7 operators. Even so, the numerical size of the first-third generation charged currents in the SM is smaller than neglected one loop corrections as a result of the CKM parameter suppression, and such SMEFT parameters are thus neglected in the WHZ pole parameter counts.
A similar argument holds for the Wilson coefficient matrix C Hud , however in this case, the neglect of this set of parameters follows from the corresponding right handed currents not interfering at leading order with the SM interactions. Such interference first comes about proportional to two insertions of light quark masses [112].
The interference of CP violating phase stemming from operators of Classes 1,4,5,6,7 with the corresponding effective operators generated in the SMEFT are also numerically suppressed to the level of neglected loop corrections. These operators are neglected in the pole parameter set but we note that we provide a fully general SMEFT code, and a U(3) 5 -SMEFT code with all phases so that CP violating effects can be studied as desired using the SMEFTsim package.
Neglecting numerically suppressed contributions from L (6) operators in a LO 'WHZ pole parameter' program can be justified in this manner. Developing the SMEFT in time to the level of NLO corrections is required for the interpretation of the most precise experimental data, see refs. [1,17,18,39,[49][50][51] for discussion and results developing this effort. In the mean time LO fits in a pole program can and should be pursued. A theory error metric must be chosen in this effort to make such simplifying LO approximations. 18 Note that a one loop result for a flavour changing neutral current in the SM can be compared to the one loop improvement of an effective three point interaction in the SMEFT due to the insertion of L (6) in the loop diagram. The latter does not in general experience the extra suppression from the GIM mechanism on top of the one loop suppression, and can introduce a number of L (6) parameters not present in a tree level analysis. This is another reason that one loop SMEFT results are of interest when incorporating precise experimental constraints such as LEP data into a global SMEFT fit, see refs. [13,14,17,18,51,111].

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Class Parameters Total Count 23 46 Table 5. LO parameter counts in the general SMEFT flavour cases for n f generations for a 'WHZ pole parameter' program. The parameters retained are those that lead to contributions to near on-shell regions of phase space, do not experience suppressions by light quark mases or GIM suppression when interfering with the SM, or violate CP and carry a resonant enhancement in this region of phase space.
Some theory error metrics were proposed in refs. [13,14,17,18]. In addition, we note the exact number of parameters in a pole constraint program is weakly basis dependent as exchanging ψ 4 operators for operators without fermion fields is largely blocked as the latter do not carry sufficient flavour indices. As the vast majority of the parameters of the SMEFT reside in the operators with a maximal set of flavour indices the simplification in the number of parameters present is dramatic. The SMEFTsim codes do not contain anomalous one loop flavour changing neutral current interactions for the SM. The loop level generation of CP odd operators in the SM proportional to the Jarlskog invariant are also absent. As such, a restricted set of parameters is a natural result of numerical simulations using SMEFTsim. The restricted set of parameters comes about when the leading order interference terms with the SM are calculated, which is the purpose of the SMEFTsim package. These arguments on numerically suppressed interference also lead to the corresponding second order terms in a constructed χ 2 to fit experimental data also being suppressed. As such retaining such parameters in a fit is subject to large theoretical uncertainties. In addition L (8) is of the same order as such terms in a constructed χ 2 and neglected and only retaining a subset of O(1/Λ 4 ) corrections is not basis independent.
Restricting to a 'WHZ pole parameter' program the number of parameters in a LO global constraint program in the Warsaw basis are estimated in table 1.

'WHZ pole parameter' counts
The parameter counts in the case of the n f = 1, 3 SMEFT are given in table 5. The only subtlety is in the counting of the (LL)(LL) operators. The expression for the shift in the JHEP12(2017)070 Class Parameters Hl , C Hq , C Hq , C He , C Hu , C Hd } ∈ R, ∼ 7 Total Count ∼ 24 Table 6. Parameter counts in the U(3) 5 SMEFT for a 'WHZ pole parameter' program. The parameter counts that are approximate also rely on expanding numerically in the Yukawa eigenvalues.
Class Parameters Hl , C Hq , C Hq , C He , C Hu , C Hd } ∈ R, ∼ 13

{∆C
(1) Hq , ∆C Hu , ∆C Hd } ∈ R 8 (LL)(LL) {C ll , C ll } ∈ R 2 Total Count ∼ 30 Table 7. Parameter counts in the MFV SMEFT for a 'WHZ pole parameter' program. The parameter counts that are approximate rely on expanding numerically in the Yukawa eigenvalues. For ∆C (1) Hq , ∆C Hq the notation corresponds to two spurion insertions as defined in eq. (4.41).
extracted value of the Fermi constant in the general SMEFT is Further,the diagonal entries of the self Hermitian operator C Hl ∈ R. This leads to the parameter counts given in table 5. The parameter counts for the U(3) 5 -SMEFT and MFV-SMEFT in each operator Class are given in table 6 and table 7 respectively.

Selection/identification cuts in a pole program
A detailed study of efficient experimental cuts to pursue a WHZ pole parameter program is beyond the scope of this work. In this section, we discuss the plausibility of enforcing stronger narrow width selection cuts at LHC to enable this effort. We then illustrate numerically the effect of such selections on L (6) parameter dependence. Our purpose is to demonstrate an application of the Model files, and to numerically illustrate how the scaling rules in eq. (10.5) translate into simulated results.

Narrow width phase space selection cuts for Z
The general prospects for isolating the near on-shell region of phase space for the W, Z to enable a SMEFT program are strong at LHC, and consistent with standard particle identification strategies. For example, enforcing a quasi-narrow Γ Z selection cut using + − or J J final states is standard in many LHC searches with an identified Z. Historically Z identification used a dilepton invariant mass cut [113] of ∼ 10 Γ Z in ATLAS [113] and ∼ 12 Γ Z in CMS [114]. Recent studies at higher operating energies have used tighter cuts ∼ ±{3 Γ Z , (3 + 0.01 p T )} for selecting {e + e − , µ + µ − } final states and ∼ ±7 Γ Z selection for JJ final states in a search at √ s = 13 TeV for heavy resonances decaying into diboson pairs reported by ATLAS [115]. Similar results from CMS [116] use comparable criteria.
Although wider selection criteria have also been utilized at ATLAS and CMS, for example in refs. [117][118][119][120] we strongly encourage the development of analyses to enable cleaner SMEFT interpretations of LHC data in a pole parameter program, through optimized narrow width selection cuts. The effect of further select cuts is non-trivial and requires dedicated experimental studies. For example, high p T selection cuts on the width of a reconstructed Z is important, with a narrow peak persisting for a boosted on-shell Z and a broader peak from off-shell production of a Z dictated by the p T cut selected. Detailed numerical studies are called for with realistic detector simulations and all selection criteria imposed. We note that preliminary studies indicate that the narrow boosted on-shell peak is a subdominant, but non-negligible, source of ∼ 20% of total events in Z → e + e − . when p T (e) > 100 GeV is enforced on each electron.
The near resonance numerical suppression of interest in collider SMEFT studies degrades linearly with the width of this cut and we advocate enforcing even stronger selection criteria when a narrow resonance is present. This is an important alternative experimental strategy to systematically develop that is currently understudied. To our knowledge, intrinsic detector energy resolution is not a barrier to significantly tighter selection cuts of this form, as it is < Γ Z for identified leptons.

Narrow width phase space selection cuts for W
Enforcing selection criteria for a narrow width region of phase space for W →q q decays is also feasible. An example is the use of ∼ ±8 Γ Z selection for JJ final states at √ s = 13 TeV in ref. [115]. For {W, Z} using JJ selection cuts of ∼ ±7 Γ B is chosen due to the degradation of the reconstructed boson mass as a function of jet p T [121]. Tighter selection criteria

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for JJ invariant mass cuts can be imposed at the cost of rejecting high p T jet events. Inspection of ref. [121] (figure 7) indicates that a tighter cut of ∼ ±5 Γ B is feasible when vetoing high p T jet events > 1 TeV. More subtle is W → ν where transverse variables are used. The scaling that follows from eq. (10.5) on resonance domination of the populated phase space still applies when transverse variables are used, even though no resonance peak is reconstructed experimentally. Enforcing a narrow width condition using transverse variables 19 considering the uncertainties involved in E miss T reconstruction and pileup is a challenge. Define the final state and ν in the boosted W frame in a plane orthogonal to the z collision axis as where {θ, φ} are defined relative to the z collision axis and the boost factor is defined as It follows that a narrow width region of phase space Typical selection criteria for W leptonic decays are |p ν T | > 25 GeV and m T > 40 GeV [113][114][115][116]. The lower bound on m T is limited by QCD backgrounds. As such, to enforce a near on-shell region of phase space |p T | 40 GeV leptons with a minimized |p ν T | is preferred. The uncertainty on the missing energy reconstruction ∆|p ν T | ∼ 15 GeV ∼ 7 Γ W is the basic limiting factor. Reducing this uncertainty is limited by pile up at √ s = 13 TeV energies in the RunII collision environment, see ref. [122][123][124]. Dedicated studies to optimize selecting for near on-shell W boson decays in W → ν decays are warranted, but pessimism on strongly enforcing these selection cuts is reasonable. Due to these challenges, the numerical illustrations below utilize JJ final states for isolating the W boson resonance region of phase space.

Numerical illustration
This section provides a simple numerical demonstration of the arguments in section 10, employing the SMEFTsim package and MadGraph5. We present a basic analysis of the impact of pole vs. non-pole parameters for three LHC processes that receive significant resonant contributions in the SM when narrow width regions of phase space are selected for: Contents -1 -  Figure 5 shows a sample of the diagrams that contribute to these processes in our study: process (i) is Drell-Yan production that is resonant for m( + − ) m Z . In process (ii) the EW production of the quark pair is subdominant compared to the gluon-mediated diagrams and, for the sake of cleanliness in the analysis, it is forbidden at the generation level, setting the interaction order limit QED<=2 in MadGraph5. As a consequence, this process has only one resonant structure corresponding to the Z peak in the invariant mass distribution of the lepton pair. Finally, process (iii) serves an illustration for processes with two resonances, around m W in the (q uqd ) mass spectrum and around m Z in the dilepton. The QCD contributions are conveniently removed in this case when generating events in MadGraph5, requiring QCD=0.
We choose 3 representative operators of the Warsaw basis: Q Hl , whose Wilson coefficient belongs to the category of pole parameters and the two four-fermion operators Q qe , Q (1) qq , that give (qq)(qq) and (qq)(¯ ) contact interactions respectively. We adopt the U(3) 5 -SMEFT (for Q (1) qq so that only the flavour contraction Q For each process we generate one event sample for the SM production and one for each interference term with one of the three effective operators considered, included one by one. 20 The operator Q (1) qq is included only in process (iii). 20 The authors are well aware that one at a time operator analyses are generally not representative of consistent IR limits in the SMEFT [125]. For our numerical illustration, we reluctantly consider a one at a time operator analysis acceptable to examine the numerical result of the scaling arguments underlying the pole parameter program.

Specifications in MadGraph5
SM int. order wz,ww,wh = auto mmll = 40 mmllmax = 120 Table 8. Details of the options specified in MadGraph5 for the generation of events for each of the processes considered (see text for details).
The processes are generated in MadGraph5 using the UFO model from set A for the U(3) 5 symmetric case and with the {α em ,m Z ,Ĝ F } scheme. Restriction cards are employed to set all the fermion masses and Yukawa couplings to zero (with the exception of those of the top and bottom quarks) and to fix the values of the Wilson coefficients. In the SM limit all of them are vanished, while for the operator insertions the corresponding coefficient is set to C i = 1 with Λ = 1 TeV. Interaction order limitations are used both to produce cleaner signals and to isolate the interference term in the case of L (6) insertions. Specifications of the first class are QED<=2 for process (ii) and QCD=0 for process (iii), that reduce the number of relevant SM diagrams as explained above. The interference is instead isolated 21 using the recently introduced MadGraph5 syntax NP^2==1.
Each event sample contains 10 4 events and it is produced setting the widths of the W, Z, h bosons for automatic evaluation in MadGraph5 and restricting the phase space with broad kinematic cuts on the invariant masses of fermion pairs, as summarized in table 8. The invariant mass of the lepton pair in the final state is always required to be in the region between 40 and 120 GeV and the invariant mass of the (q uqd ) pair is required to be in the same window for process (iii). This selection allows a more efficient scan of the near-resonant regions of the parameter space.
The invariant mass spectra of the relevant fermion pairs are extracted analyzing the event samples with ROOT [126]. In the case of the SM-L (6) interference terms, the histograms are further rescaled by |σ(C i , int.)|/σ(SM) so that their bin content can be directly compared to that of the SM distributions. In this way all the histograms have the same (arbitrary) normalization, which is such that the SM production has 10 4 events in the kinematic region for which the events were generated.
The resulting distributions for the interference terms are shown in the lower panels of figures 6-8 (colored lines) and can be easily compared. The figures also show, for reference, the error band on the complete distribution for the SM case (grey band). The latter is estimated bin-by-bin as 22 ∆N k = √ N k where N k is the number of events in the 21 The estimate of the interference term obtained with this procedure is more accurate and numerically stable than the estimate obtained e.g. generating the full process with LSM + L (6) and subsequently subtracting the pure SM contribution. 22 In our analysis the SM distribution has two sources of uncertainty: a statistical uncertainty due to analyzing a finite event sample and an error on the overall normalization of the distribution, stemming from the uncertainty in the determination of the total cross-section in MadGraph5. The latter, however, is an effect of order ∼ 8 ‰ and can be safely neglected.  Figure 6. Left: invariant mass distribution of the + − pair from Drell-Yan production (process (i)). Right: invariant mass distribution of the + − pair from process (ii). The top panels show the complete spectra obtained in the SM limit and in the presence of one effective operator with C i /Λ 2 = 1 TeV −2 (only the interference term is retained in these cases). The black and blue lines are overlapping in the right figure. The lower panels show the absolute size of the pure SM-L (6) interference term for each operator (colored lines) compared to the statistical uncertainty on the SM distribution (grey band). bin k. Finally, the top panels show the total predictions for the SM and for the SM + one operator, obtained as the sum of the SM and interference histograms. Note that, while in the lower panel we plot the absolute size of the interference terms, in the upper panel its sign is kept into account. In particular, the interference is always negative for C qe and positive in all the other cases. The normalization of the histograms is arbitrary and such that the SM production has 10 4 events in the kinematic region for which the events were generated.

Results for processes (i) and (ii)
Processes (i) and (ii) are both singly-resonant in the SM, and they show a particularly clean enhancement/suppression effect of the L (6) contributions in the distributions in near the resonant region of phase space.
As shown in figure 6, in both cases the impact of the "pole operator" Q Hl is enhanced around the Z peak. On the other hand, the four-fermion operator Q qe has a very small impact overall, which, with the statistics presented here, is always smaller than the statistical uncertainty on the SM production alone. It is worth noting that, around the Z resonance, it undergoes a further suppression, that can be appreciated as a dip in the curve in the low panels of figure 6.
The relative suppression of the Q qe vs. Q Hl operator emerging around a B boson resonance can be quantified as N (C qe )/N (C  Figure 7. Invariant mass spectrum of the q uqd pair in process (iii) before (left) and after (right) applying a narrow selection cut on m( + − ) to isolate the Z peak. The top panels show the complete distributions obtained in the SM limit and in the presence of one effective operator with C i /Λ 2 = 1 TeV −2 (only the interference term is retained in these cases). The black, yellow and blue lines are overlapping in both figures. The lower panels show the absolute size of the pure SM-L (6) interference term for each operator (colored lines) compared to the statistical uncertainty on the SM distribution (grey band). case, N (C i ) is given by the number of entries for the bins between 89 and 93 GeV of the interference histograms. The numbers obtained in this way are summarized in table 9. This gives ∼ 1/620 and ∼ 1/570 for processes (i) and (ii) respectively (see table 9), which is consistent with the estimate of eq. (10.5).

Results for process (iii)
Because process (iii) can present two resonances, both the invariant mass spectra of the q uqd (figure 7) and dilepton (figure 8) pairs are analyzed. For each pair we show the results obtained directly from the generated sample vs. after applying an additional narrow cut on the invariant mass of the complementary fermion pair: in one case (figure 7, right) we select 89 GeV < m( + − ) < 93 GeV to isolate the Z peak and observe its impact on the (q uqd ) spectrum, while in the other (figure 8, right) we select 78 GeV < m(q uqd ) < 93 GeV to isolate the W + peak and observe its impact on the m( + − ) distribution.
While these cuts do not have a visible impact on the total spectra, they significantly reduce the size of the pure interference term for one four-fermion operator: selecting resonant + − pairs suppresses the contribution of Q qe , while selecting resonant q uqd pairs removes that of Q qq . This effect is clearly visible in the lower panel of figure 7 (8), comparing the blue (yellow) curves in the left and right plots.
As for processes (i) and (ii), the enhancement of the interference term for C  Figure 8. Invariant mass spectrum of the + − pair in process (iii) before (left) and after (right) applying a narrow selection cut on m(q uqd ) to isolate the W + peak. The top panels show the complete distributions obtained in the SM limit and in the presence of one effective operator with C i /Λ 2 = 1 TeV −2 (only the interference term is retained in these cases). The black, yellow and blue lines are overlapping in both figures. The lower panels show the absolute size of the pure SM-L (6) interference term for each operator (colored lines) compared to the statistical uncertainty on the SM distribution (grey band).

Process Resonant region
N (C qe )/N (C  Table 9. Approximate ratios of the number of events contained in the central peak regions of the relevant invariant mass spectra determined by the pure SM-L d=6 interference of non-pole (C qe , C suppression of the four-fermion operators, instead, is less evident and can only be seen as a tiny dip in the central bins of the m( + − ) distributions for C qe (figure 8, lower panels).
Finally, the relative suppression of the Q qe and Q Hl ratio with cuts on m(q uqd ) essentially reflect the ratio of the cross-sections obtained for the interference terms, as the selection is ineffective in these cases. Restricting to the relevant resonant region induces an additional suppression of a factor 3-4. This effect is also observed in processes (i) and (ii).
In this paper we have advanced the SMEFT physics program on multiple fronts. We have developed and reported the SMEFTsim package, a set of FeynRules implementations of the general SMEFT, the U(3) 5 -SMEFT and the MFV-SMEFT theories as defined in sections 2, 3, 4. We have provided these results in two input parameter schemes, {α ew ,m Z ,Ĝ F } or {m W ,m Z ,Ĝ F } in all three cases. We have supplied two code sets based on this theoretical outline for validation purposes.
We have also systematically developed a theoretical framework of a WHZ pole parameter program in the SMEFT as a key application to pursue with the SMEFTsim package. The idea is to maximally exploit numerical suppressions in the scattering and interference of the SMEFT corrections to the SM in addition to the power counting of the EFT. These IR effects come about due to approximate symmetries and the use of near on shell regions of phase space enforced with selection cuts at hadron colliders. Rather ironically, a key complaint against EFT methods at hadron colliders -an excess of parameters, can be arguably overcome using stronger versions of the very selection cuts that underlie standard particle identification of narrow bosons of the SM in such environments. We have advanced this argument and numerically demonstrated the impact of the pole parameter scaling using the SMEFTsim package interfaced with MadGraph5. Although our results do not rise to the level of realistic selection cuts at LHC with full detector effects included, we believe they are sufficient and promising enough to strongly motivate the initiation of a systematic pole parameter approach to data analysis at LHC.
The enormous data rate at LHC in RunII and in the high luminosity run is such that stronger selection cuts sacrificing pure rate in favour of cleaner SMEFT motivated measurements, at lower energies, are of interest and reasonable to consider and develop. Such cuts can enable a systematic program of constraining physics beyond the SM using powerful EFT techniques, that are already argued to be relevant by the lack of beyond the SM resonances discovered to date at LHC. The number of parameters present in such WHZ pole parameter efforts is manageable, and the SMEFTsim package allows the systematic study and optimization of selection cuts to develop this program, examining quantitatively how neglected terms are suppressed with tight or weak selection cuts. We strongly encourage the LHC experimental collaborations to study and develop a pole parameter SMEFT approach to LHC measurements using the tools provided, in addition to standard searches already in place. A Parameter tuning and experimental constraints on Class 5, 6 operators

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Operators of Class 6 (ψ 2 XH) in the Warsaw basis can also contribute anomalous three point interactions, in addition to operators of the form H|L SM |H . In this appendix we examine the constraints on these operators due to naturalness concerns and some experimental considerations. In section 4 the insertion of a SM Yukawa matrix in this operator Class is made to formally restore U(3) 5 flavour symmetry. One can utilize one loop corrections in the L SM EF T to examine the theoretical support of this approach and establish if flavour diagonal interactions due to ψ 2 XH operators with C ψX ∝ y i , with y i a Yukawa coupling, introduce a significant tuning of parameters in the SMEFT. Consider the one loop diagrams shown in figure 9 calculated in dimensional regularization with d = 4 − 2 in MS. The divergence structure of these diagrams for the operator C eB is given by The counterterms are introduced as The contribution to the chiral wavefunction renormalization factors cancels the divergences in the first line of eq. (A.1), while the counterterm for C eH reported in ref. [10] (see figure 9 c) exactly cancels the divergences in the second line once the SMEFT wavefunction renormalization is taken into account. The finite terms follow the same pattern and a shift to the light fermion mass is present that is not proportional to y i as a correction to the effective Yukawa coupling. Numerically as y e (µ < TeV) 3× 10 −6 , y µ (µ < TeV) 6 × 10 −4 , y τ (µ < TeV) 0.01, (A.6) y u (µ < TeV) 6 × 10 −4 , y c (µ < TeV) 4 × 10 −5 , y t (µ < TeV) 1, (A.7) y d (µ < TeV) 3 × 10 −5 , y s (µ < TeV) 6 × 10 −4 , y b (µ < TeV) 0.02, (A.8) dipole operators do not have to be ∝ y i to avoid this tuning of parameters.
Operators of Class 5 directly lead to a tree level contribution to an effective Yukawa interaction for Λ ∼ TeV that are numerically which implies expanding around a U(3) 5 symmetric limit in the SMEFT by inserting a SM Yukawa matrix for Class 5 operators reduces parameter tuning. This then requires a U(3) 5 limit of the SMEFT be taken consistently in the counterterm matrices, which requires the insertion of a SM Yukawa matrix for Class 6 operators in the U(3) 5 -SMEFT and MFV-SMEFT. Finally, utilizing the SMEFT Lagrangian formalism, dipole operators can be directly related to a shift to a measured anomalous magnetic moment The muon anomalous magnetic moment is highly constrained [41] δa µ 288(63)(49)×10 −11 which argues for the neglect of the flavour symmetric component of this dipole interaction operator Wilson coefficient when i = µ, but this conclusion does not hold for all L (6) dipole interaction terms. Experimental constraints on C ψX rs , with r = s are significant in some cases, but not for all possible flavour transitions, see the recent discussion in refs. [127][128][129][130]. Furthermore, the separation of flavour diagonal and flavour off diagonal interactions of the dipole operators is a scale dependent distinction, flavour mixing is extensive in the SMEFT RGE (see refs. [10,35]), and the scales of the experimental constraints are separated from the scales used to probe these interactions at the LHC. The general neglect of all Class 6 operators does not seem to be supported due to experimental constraints in the SMEFT at this time.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.