SMEFTsim 3.0 -- a practical guide

The SMEFTsim package is designed to enable automated computations in the Standard Model Effective Field Theory (SMEFT), where the SM Lagrangian is extended with a complete basis of dimension six operators. It contains a set of models written in FeynRules and pre-exported to the UFO format, for usage within Monte Carlo event generators. The models differ in the flavor assumptions and in the input parameters chosen for the electroweak sector. The present document provides a self-contained, pedagogical reference that collects all the theoretical and technical aspects relevant to the use of SMEFTsim and it documents the release of version 3.0. Compared to the previous release, the description of Higgs production via gluon-fusion in the SM has been significantly improved, two flavor assumptions for studies in the top quark sector have been added, and a new feature has been implemented, that allows the treatment of linearized SMEFT corrections to the propagators of unstable particles.


Introduction
LHC physics is about to enter a precision era that will span over the next two decades. During this time, new opportunities to hunt for new physics will arise: direct searches of new particles will be complemented by indirect searches, that target possible deviations from the predictions of the Standard Model (SM). While the isolation of this kind of signatures is not without challenges, indirect searches present some very attractive features. Most notably, they do not rely on specific assumptions about the nature of the new physics under scrutiny and, at the same time, their sensitivity in terms of new physics scales can potentially extend beyond the energy reach of the collider.
The Standard Model Effective Field Theory (SMEFT) is the best established theory framework to describe such effects. Its formulation employs the degrees of freedom and gauge symmetries of the SM and it is structured as an infinite series of operators sorted by canonical dimension. At the observables level, it reproduces a series expansion in (E/Λ), being E the typical energy exchanged in a process and Λ the mass scale that characterizes the beyond-SM (BSM) dynamics. The condition (E/Λ) 1, indicating the near decoupling of the new physics sector, is necessarily assumed.
The notation is fixed in Section 1.1. Section 2 focuses on the bosonic sector and it reviews the field and parameter redefinitions required to ensure a canonical parameterization of the kinetic terms and scalar potential. Section 3 is devoted to the flavor structure of the SMEFT and it defines the five scenarios implemented in SMEFTsim. Some significant changes have been made compared to version 2, that are documented in detail, and two new flavor options have been introduced (top, topU3l) that comply with the recommendations for studies of top quark observables [33]. Section 4 provides a general discussion of how the extraction of numerical values for the SM parameters is affected in the presence of higher-dimensional operators, and illustrates the treatment of these effects in SMEFTsim.
Section 5 documents the implementation of Higgs interactions that are purely loopgenerated in the SM, namely hγγ, hZγ, hgg. Compared to version 2, the description of Higgs-gluon vertices has been substantially improved, such that it can now model one-loop SM interactions with up to 4 gluons. Section 6 focuses on SMEFT effects in the propagators of unstable particles, that arise due to modifications of their pole masses or decay widths. A new feature has been introduced in version 3.0, that enables the inclusion of such effects, linearized in the EFT parameters, in Monte Carlo simulations. To our knowledge SMEFTsim is the first publicly available UFO model to implement such a tool. Sections 7, 8 provide recommendations for the use of SMEFTsim in Mathematica and in MadGraph5_aMC@NLO respectively, and in Section 9 we conclude.
Additional useful material is provided in the Appendices: analytic expressions of the decay widths implemented in the propagator corrections (App. A), a list of changes made in version 3.0 (App. B), tables to facilitate the conversion between flavor assumptions (App. C), between theory and code notation (App. D) and between SMEFTsim and dim6top or SMEFT@NLO (App. E). Finally, App. F documents the validation of the UFO models, that followed the procedure recommended in [44].

Basics and notation
We consider the SMEFT Lagrangian truncated at the dimension-6 level: (1.1) We neglect all lepton-and baryon-number violating terms, which includes the dimension-5 Weinberg operator that generates a Majorana mass term for neutrinos. For future convenience, the SM Lagrangian is split into four terms: where q, l represent the left-handed quark and lepton doublets respectively, and u, d, e the righthanded quarks and leptons. H is the Higgs doublet andH = εH * , with ε = iσ 2 the 2dimensional Levi-Civita tensor. Y d , Y u , Y l are the 3 × 3 Yukawa matrices of the down and up quarks and of the charged leptons. Covariant derivatives are defined with a plus sign, ie.
T a ≡ λ a /2, a = {1, . . . , 8} are the SU (3) c generators, with λ a the Gell-Mann matrices, and σ i , i = {1, 2, 3} are the Pauli matrices. y q = 1/6 denotes the hypercharge of the q field and g s , g W , g 1 are the SU (3) c ×SU (2) L ×U (1) Y coupling constants. As a general rule, color indices are denoted by a, b, c, d, SU (2) L indices by i, j, k and flavor indices by p, r, s, t. Summation over identical indices is always understood, unless otherwise specified. The Lagrangian L 6 contains a complete and non-redundant basis of dimension-6 operators Q α constructed with the SM fields and invariant under the SU (3) c × U (2) L × U (1) Y gauge symmetry. SMEFTsim implements the Warsaw basis [39], whose operators are collected in 8 groups, following the classification of Ref. [45]. Class 8 is further split into 4 subgroups: 2 L 6 = L with the sum running over the class-n operators {Q α } defined in Table 1 and C α denoting the associated Wilson coefficients. Both Q α and C α generally carry flavor indices, that are implicitly contracted in Eq. (1.10). In this basis, explicit CP violation is carried by the real coefficients C G , C W , C H G , C H W , C H B , C H W B and by the imaginary parts of the Wilson coefficients associated to non-hermitian fermionic operators, namely those in L The operators definitions use the following notation: (1.12) 2 Note that L implicitly contains (Q Hud + h.c.), as this operator is not Hermitian. Q G f abc G aν µ G bρ ν G cµ ρ Q eW (l p σ µν e r )σ i HW i µν Q ee (ē p γ µ e r )(ē s γ µ e t ) (q p γ µ T a q r )(ū s γ µ T a u t ) Q eH (H † H)(l p e r H) Q ll (l p γ µ l r )(l s γ µ l t ) Q ledq (l j p e r )(d s q tj ) (1) lq (l p γ µ l r )(q s γ µ q t ) Q (1) lequ (l j p e r )ε jk (q k s u t ) Q (3) lq (l p γ µ σ i l r )(q s γ µ σ i q t ) Q (3) lequ (l j p σ µν e r )ε jk (q k s σ µν u t ) Table 1. L 6 operators in the Warsaw basis [39], categorized into eight classes L (n) 6 as in [45]. Only baryon number-conserving invariants are retained. The flavor indices p, r, s, t are suppressed in the operators' names.

Higgs sector
The operator Q H introduces a perturbation of the Higgs potential: The true minimum of the potential, that triggers the electroweak symmetry breaking, is We have introduced the "bar" notation for Wilson coefficients: Note that because v = v T + O(Λ −2 ) and O(Λ −4 ) contributions are entirely neglected, the two quantities v, v T are interchangeable whenever they multiply a Wilson coefficent. 3 The Higgs field H is expanded around its vacuum expectation value (vev) as with G + , G 0 the charged and neutral Goldstone bosons and h the physical Higgs boson. In the broken phase, the kinetic terms of the scalar fields receive corrections from the operators Q H , Q HD . As the scope of SMEFTsim is limited to tree-level calculations, we choose to work in unitary gauge and neglect EFT effects in the Goldstone sector, both in the present discussion and in the code implementations. The Goldstone bosons case and the generalization of the gauge fixing procedure in the SMEFT were addressed in Refs. [3,29,[55][56][57][58][59].
Using integration by parts, the kinetic term of the physical Higgs boson takes the form This replacement is formally operated on the entire L SMEFT . However, when applied to L 6 , its net effect is of O(Λ −4 ). As we work at O(Λ −2 ), the replacement only needs to be performed on L SM . This holds for all field and parameter redefinitions introduced in the following, unless otherwise specified. For the same reason, all quantities in a Wilson coefficient's prefactor are understood to be defined in the SM limit. The main consequence of (2.7) is that the Wilson coefficients C H , C HD are recast into an overall rescaling of all SM Higgs couplings. The resulting Higgs potential is (2.8) In the FeynRules implementation, the redefinitions of the physical Higgs field, Eq. (2.7), and of the vev, Eq. (2.2), are embedded in the definition of the Higgs doublet.

Gauge sector
Upon EWSB, the operators Q HG , Q HW , Q HB , Q HW B induce corrections to the kinetic terms of the gauge bosons. The first three lead to overall rescalings: G a µν G aµν 1 − 2C HG +. . . (2.9) The canonical normalization is easily restored at O(Λ −2 ), via the field redefinitions G a µ → G a µ (1 +C HG ) , W i µ → W i µ (1 +C HW ) , B µ → B µ (1 +C HB ) .
(2. 10) In order to leave the covariant derivatives unchanged, the coupling constants need to be redefined at the same time. Neglecting O(Λ −4 ) corrections: The operator Q HW B introduces a kinetic mixing between the B and W 3 fields of the form (2.12) The rotation [45] -7 -removes this residual mixing and leads to fully canonical and diagonal kinetic terms. Once Eqs. (2.10), (2.11), (2.13) have been applied, the electric-charge eigenstates W ± are obtained via the usual rotation W 1 (2.14) while the mass term of the neutral bosons is diagonalized by The rightmost rotation is unitary up to O(Λ −4 ) corrections, and therefore does not reintroduce kinetic mixing at d = 6. Equivalently, with a shifted weak mixing angle θ defined as (2.17) After all the coupling and field redefinitions have been applied, a generic covariant derivative has the form D µ = ∂ µ + iQ g 1 g W g 2 1 + g 2 W A µ 1 −C HW B g 1 g W g 2 W + g 2 1 (2.18) where t 3 denotes the eigenvalue of the 3 rd SU (2) L generator (t 3 = ±1/2 for left-handed fields and t 3 = 0 for right-handed ones) and Q = t 3 + y is the electric charge. We have also introduced the shorthand notation s θ = sin θ, c θ = cos θ, with θ defined as in Eq. (2.17). The dots stand for potential gluon and W ± terms, for which there are no residual L 6 corrections. Eq. (2.18) shows that the contributions from Q HW , Q HB , Q HG are fully reabsorbed in the definition of the fields and gauge couplings. As a consequence, these operators have no physical impact in the pure gauge sector, and they only contribute to Higgs-gauge interactions [60]. On the other hand, the operator Q HW B introduces net modifications of all γ and Z couplings.
- 8 -In the former case the correction is a universal rescaling of the electromagnetic constant, while in the latter case the corrections depend on the field's charges. In particular, in the Higgs case (t 3 = −1/2, Q = 0) this implies a correction ∝C HW B to the Z mass term. The physical interpretation of these contributions requires defining a set of input observables and is deferred to Section 4.
In SMEFTsim, the redefinitions described in this subsection are applied simultaneously at the Lagrangian level in the FeynRules model. The coupling constants' rescaling in Eq. (2.11) is implemented in the replacement list redefConst. The field redefinitions are operated in the mass and charge eigenstate basis: the replacement list rotateGaugeB implements the net mismatch between the series of rotations (2.10), (2.13), (2.15) and the usual SM rotations, ie.  is the correction to the mixing angle stemming from Eq. (2.17).

Flavor assumptions
The SMEFT Lagrangian defined in Sec. 1 is not invariant under flavor rotations of the fermion fields, so the latter should always be defined in order to avoid ambiguities. In SMEFTsim, the fields q, l, u, d, e are defined in the mass basis of the charged leptons and of the up-type quarks, in which the Yukawa matrices in Eq. (1.5) take the form The superscript (d) denotes diagonal matrices and V is the CKM matrix. This basis choice is consistently employed in the definition of both L SM and L 6 , and for all the flavor assumptions implemented in SMEFTsim. The only special case are the top and topU3l models, where quark mixing is entirely neglected setting V ≡ 1.
-9 -Upon EWSB, the Lagrangian can be written in terms of the fermionic mass eigenstates. By definition the relation between the SU (2) L and mass bases is trivial for all fermion fields, except the left-handed quark doublet 4 : In unitary gauge, the relevant terms in the SM Lagrangian are therefore The CKM matrix is implemented in the Wolfenstein parameterization [61]: (3.5) The numerical values employed for the parameters are listed in Table 17.
SMEFTsim implements five alternative flavor scenarios: one with fully arbitrary indices, and four based on the implementation of different global symmetries. Three of these scenarios have been present since the first release, and two have been newly introduced in version 3.0. The following sub-sections review in detail the properties of the L 6 operators within each setup and provide the corresponding parameter counting. A dictionary between the different flavor assumptions is provided in Appendix C.

general: general flavor structure
Without further assumptions on the flavor structure of the SMEFT, L 6 contains the operators in Table 1, summed over all possible flavor combinations: Not all flavor combinations included in this way are independent, due to intrinsic symmetry properties of the effective operators. SMEFTsim does not remove redundant terms from the 4 For economy of notation, we use the same letters u, d, e for the right-handed fields and for the mass eigenstates, both of them carrying flavor indices. To avoid ambiguities, the latter always carry L, R subscripts, while the former don't.
-10 -sums in Eqs. (3.6), (3.7). Instead, the symmetry relations are enforced in the definition of the tensor Wilson coefficients C α,pr (st) : only a minimum number of independent parameters  is defined for each operator, as reported in Appendix D, Tables 19, 20, and all the entries of C α,pr(st) are functions of these parameters, consistent with the relations described below.

Classes 5 and 6
The operators in L (5,6) 6 are not hermitian. Therefore each Wilson coefficient has 9 independent complex entries. In total, this gives 198 real parameters (counting independently real and imaginary parts).

Class 7
All operators in L 6 , except Q Hud , are hermitian. In this case, the diagonal entries of the Wilson coefficients are real, and the off-diagonal ones are related by In total, this class depends on 81 real parameters.

Class 8 a
All operators in L (8a) 6 are hermitian. Moreover, each of the two currents that compose them is itself hermitian. Therefore the following relation holds: In the operators Q ll , Q qq , Q qq , the two currents contain the same fields, which leads to an additional exchange symmetry C prst = C stpr . (3.10) Each of these three operators has then 15 real entries (C pppp , C pprr , C prrp ) and only 9 are independent, and 66 complex entries, 18 independent. Operators Q lq have each 9 real entries, all independent, and 72 complex ones, only 36 independent.
In total, this class depends on 297 real parameters.

Class 8 b
All operators in L (8b) 6 are hermitian and composed of two hermitian currents, so relation (3.9) holds for all Wilson coefficients in this class. Eq. (3.10) is valid in addition for C uu , C dd , C ee .
The operator Q ee is peculiar: because the e field is a singlet under both SU (2) L and SU (3) c , this term is invariant under Fierz rearranging. This leads to the additional constraint C prst = C ptsr . (3.11) The coefficient C ee has then 15 real entries, 6 independent, and 66 complex entries, only 15 independent. The counting for the other operators is the same as for the invariants in class 8a, so L (8b) 6 has a total of 450 real parameters.

Class 8 c
All operators in L (8c) 6 are hermitian and composed of two hermitian currents, but no other symmetry is present. Therefore only relation (3.9) holds for all Wilson coefficients, leaving a total of 648 parameters.

Class 8 d
Finally, all operators in L 8d 6 are non-hermitian. No symmetry relation is present and this class has 810 independent real parameters.
The number of independent parameters is considerably reduced if a flavor symmetry is assumed. The maximal symmetry available for the SM fermion fields is the symmetry of the kinetic terms [62]: Each field is assigned to a 3 representation of the associated group: denoting a generic U (3) ψ transformation by Ω ψ , the transformation rules are [45] q Vector currentsψ p γ µ ψ r are trivially made invariant under U (3) 5 by imposing a δ pr contraction, that corresponds to the singlet composition of a3 and 3 representations. This is immediate to see applying the field transformations and using Ω ψ Ω † Chirality-flipping currents, with either scalar or tensor Lorentz structure, violate the flavor symmetry. To allow the introduction of fermion masses, it is customary to promote the Yukawa couplings to spurions of the flavor symmetry, by assigning them transformation properties (3.14) In this way the structuresd are formally invariant. When the U (3) 5 symmetry is imposed, the flavor structure of each operator can be factored out of the Wilson coefficient, that becomes a scalar quantity: In the construction of the U (3) 5 symmetric Lagrangian, we do not define a power counting for insertions of the Yukawa couplings. Instead, we simply choose to retain the leading invariant structures for each operator, corresponding to no Yukawa insertions in L (7,8a,8b,8c) 6 , one insertion in L (5,6) 6 and two insertions in C Hud and L (8d) 6 .

Classes 5 and 6
All the operators in L (5,6) 6 require the insertion of a Yukawa coupling: where the last equality in each line holds in the up-quarks mass basis, Eq. (3.1). Note that no net mixing among down-type quarks is induced in the mass basis, as the V in the spurion cancels against V † in theq field, Eq. (3.2). In fact, by construction all the operators in L (5,6) 6 have the same flavor structure as the SM Yukawas. Because the operators are non-hermitian, the associated C α are complex. These classes therefore introduce 22 independent real parameters.

Class 7
All the currents appearing in the operators of class 7, except Q Hud , are invariant under U (3) with X α = 1. This implies that the flavor structure of this class is exactly the same as in the SM kinetic terms. For instance, the charged quark current induced by the operator Q Hq is aligned with the SM one (Eq. (3.3)), that contains CKM mixing. 5 In order to make Q Hud invariant, it is necessary to insert the spurion product The number of independent real parameters is 9, as 7 out of 8 operators are hermitian.

Class 8 a
Containing only vector currents, all the operators in L (8a) 6 are U (3) 5 invariant with the trivial flavor contraction X α = δ pr δ st .
The operators Q ll , Q qq additionally allow the "crossed" contraction X α = δ pt δ sr . This is an independent structure that cannot be arbitrarily rearranged into X α : applying Fierz transformations in this case would introduce additional operators with SU (2) L triplet and SU (3) c octet contractions, see Sec. 3.4. Therefore these operators are split into two invariants each, weighted by independent Wilson coefficients: qq,prst + . . . (3.20) As all operators are hermitian, L (8a) 6 contains 8 real parameters. 5 Note that this implies that, even though all Wilson coefficients are real, SM-sourced CP violation, due to the CKM phase in charged left-handed currents, is generally present in L 6 .

Class 8 b
All operators in L (8b) 6 are invariant with X α = δ pr δ st . The operators Q uu , Q dd additionally admit independent crossed contractions X α = δ pt δ sr , and are treated analogously to Q ll , Q . This is not the case for Q ee that, as mentioned above, is invariant under Fierz rearrangements: in this particular case the two flavor contractions are equivalent. In total, there are 9 real parameters in L (8b) 6 .

Class 8 c
All the operators in L (8c) 6 admit the invariant contraction is X α = δ pr δ st , leading to 8 independent real parameters.

Class 8 d
Finally, operators in L (8d) 6 require one Yukawa coupling insertion for each current. As they are not invariant under Fierz transformations, the operators Q (1), (8) quqd admit two independent contractions, mapped to one another by interchanging the twoq fields. 6 Because the operators are non-hermitian, there are 14 real parameters in L (8d) 6 .

MFV: linear Minimal Flavor Violation
The Minimal Flavor Violation ansatz [62,64,65] assumes that the only sources of flavor and CP violation in L SMEFT are those already present in the SM, namely the Yukawa couplings and the CKM phase. The requirement on CP violation implies that the Wilson coefficients of CP-odd bosonic operators scale with the Jarlskog invariant J [66,67]: As the J suppression is stronger, for instance, than a loop factor, these coefficients can be safely neglected within the scope of SMEFTsim. The corresponding operators are therefore not implemented in the MFV version. An analogous argument applies to sources of explicit CP 6 The two X α structures (4 real parameters) for Q (1), (8) quqd were not included in previous versions of SMEFTsim. I thank the authors of Ref. [63] for pointing this out.
-14 -violation in the fermion sector. In the Warsaw basis, these are the imaginary parts of the Wilson coefficients in L 6 , that are not defined either in the SMEFTsim MFV models.
The requirement on flavor violation is realized imposing a U (3) 5 symmetry on the fermion fields and allowing for arbitrary U (3) 5 -invariant spurion insertions in the currents, that generate flavor violating effects. Such insertions are organized in an expansion in powers of the Yukawa couplings, that can be either resummed (obtaining a non-linear MFV formulation [68]) or treated as a truncated series. SMEFTsim adopts the latter option and retains contributions up to one power of Y l and up to 3 powers of Y u , Y d .
The relevant spurion structures at this order are The first column indicates the spurions' representation under the while the second provides the corresponding transformation rules. All of them are hermitian and they satisfy In the mass basis of the up quarks (Eq. (3.1)) the spurions take the form Additional relevant structures in this basis are (3.33)

Classes 5 and 6
With the power counting chosen, L (5,6) 6 take the form -15 - where the parameters C are also allowed for operators Q uX , Q dX respectively, but they are not independent due to Eqs. (3.28), (3.29). These two classes contain a total of 27 real parameters.

Class 7
For the operators in class 7 we have Hl,pr + C He δ pr Q He,pr The total number of independent parameters in this class is 14.

Class 8 a
The operators of class 8a are composed of the same currents as those of class 7. MFV corrections have therefore an analogous structure. With the power counting chosen, the -16 -independent contractions are C ll δ pr δ st + C ll δ pt δ sr Q ll,prst lq,prst .
In the case of operators Q ll , Q

Class 8 b
The MFV Lagrangian in class 8b has the form ud,prst . (3.37) As in the U (3) 5 symmetric case, Q uu , Q dd admit two independent flavor contractions, and spurion insertions in only one of their currents is required, by symmetry. Class 8b therefore contains a total of 19 independent parameters.

Class 8 c
For class 8c we have a total of 28 independent parameters: 38) The total number of parameters is 13.

top, topU3l: U (2) 3 symmetry in the quark sector
Two new sets of models have been introduced in version 3.0, that implement a flavor structure consistent with the recommendations of Ref. [33] for the SMEFT interpretation of top quark measurements. The formalism builds upon [69][70][71] and is defined by the following assumptions: • quarks of the first two generations and quarks of the 3rd are described by independent fields. We denote them respectively by (q p , u p , d p ) with p = {1, 2} and by (Q, t, b).
• a symmetry U (2) 3 = U (2) q × U (2) u × U (2) d is imposed on the Lagrangian, under which only the light quarks transform: • mixing effects in the quark sector are neglected and V CKM ≡ 1 is assumed.
This choice greatly simplifies the structure of the Lagrangian, as mixing between the light and heavy quarks can only be introduced through extra U (2) spurions [63,69].
With this notation, the SM Lagrangian is with the Yukawas of the light quarks while y t , y b do not transform under any symmetry. As a consequence, only (LR), (RL) currents with light quarks need to be weighted by Yukawa insertions.
It is convenient to construct a U (2) 3 invariant basis mapping the fermionic operators of Table 1 to the notation with 6 quark fields. We choose the set given in Table 2, where, analogously to the U (3) 5 case, we retain the least Yukawa-suppressed U (2) 3 -invariant contractions for each operator in the Warsaw basis.
In the lepton sector we consider two alternative ansätze: (a) a U (1) 3 l+e = U (1) e × U (1) µ × U (1) τ symmetry under which the fields transform as This matches the "baseline" scenario in Ref. [33] and corresponds to simple flavordiagonality. It is implemented in the top models.
In the lepton sector, this setup matches exactly the structure of the U35 and MFV models. It is more restrictive compared to U (1) 3 l+e and contains fewer free parameters. It is implemented in the topU3l models.
In the U (1) 3 l+e symmetric case, no transformation rule needs to be assigned to Y l , as leftand right-handed leptons transform under the same symmetry. This implies that (LR), (RL) lepton currents are weighted by Y l in the topU3l models but not in the top ones.

Classes 5 and 6
The basis of quark operators for L (5) 6 and L (6) 6 in Table 2 is easily constructed splitting the quark currents for the first 2 and the 3rd generations. Insertions of Y † u , Y † d in light quark currents, that are required for U (2) 3 invariance, are embedded in the operator definitions. L (5,6) 6 contain in total 32 real parameters (16 complex) coming from quark invariants. When U (1) 3 l+e is imposed (top models) on the lepton fields, Q eH,pr , Q eW,pr , Q eB,pr admit 3 independent contractions each, one per generation. When the more restrictive U (3) 2 is imposed (topU3l models), each operator is associated to only one complex Wilson coefficient.
The total number of real independent parameters in L (5,6) 6 is therefore 50 in the top case and 38 in the topU3l case. The Lagrangian is

Class 7
Class 7 depends on 12 real parameters from quark operators, plus 9 (3) real parameters from lepton operators in the top (topU3l) case. Q Hud is defined with a Y u Y † d insertion to preserve U (2) 3 , while Q Htb is independent of the Yukawas.
Hl,pp + (C Hl,pp + (C He ) pp Q He,pp , Hl,pr + C Hl δ pr Q Hl,pr + C He δ pr Q He,pr ,

Class 8 a
Class 8a contains 2 operators with 4 quarks. Mapping them to the formalism with 6 quark fields, each of them admits 5 independent U (2) 3 invariant contractions, that can be written and analogously for Q qq . In practice, for analyses involving top quark processes it is convenient to trade "crossed" flavor contractions, as well as Q QQ , for operators with a color octet structure. This is motivated by top processes being largely dominated by QCD interactions in the SM. The rotation is done using Fierz rearrangements and the completeness relations for SU (2) and SU (3) Consistent with the recommendations in Ref. [33], SMEFTsim implements the invariants in Table 2, that are related to those in Eqs. (3.48)-(3.50) and their Q qq counterparts as: Eqs. (3.53)-(3.57) can be written compactly as Q Warsaw = R Q top , with Q Warsaw , Q top the two "operator vectors" and R a rotation matrix. The relation among the Wilson coefficients is then derived equating the Lagrangian written in the two bases: with C Warsaw , C top the coefficients vectors. The solution is  64) and the inverse The operator Q ll admits 2 independent contractions in the U Note that the allowed flavor contractions in the U (1) 3 l+r and U (3) 2 cases are the same, but the different symmetry properties generally lead to different relative normalizations. For instance, -24 -considering the (1111) and (1122) entries, one has where the relative 2 between the C ll contributions to Q 1122 and Q 1111 is due to U (3) 2 requiring to sum over both the 1122 and 2211 contractions, that are equivalent for this particular operator. In total, L (8a) 6 contains 31 independent real parameters in the top case and 16 in the topU3l case.

Class 8 b
A basis rotation analogous to the one performed in L (8a) 6 is applied to Q uu , Q ud in L (8b) 6 . No modification is needed for Q (1), (8) ud as in this case the color octet contraction is already manifest. The set of 5 independent U (2) 3 -invariant contractions in the Warsaw basis is in this case and analogously for the Q dd counterparts. Using Fierz transformations and Eqs. (3.51),(3.52): where the operators on the right-hand side of the equations are defined in Table 2. The relations among Wilson coefficients are -25 -and the inverse The operator Q ee admits 6 independent contractions in the top case, with indices that we choose in the set P ee = {1111, 2222, 3333, 1122, 1133, 2233} . is mapped into 6 (2) independent invariants in the top (topU3l) case. The Lagrangian for class 8a has the form and it depends on 40 (27) real independent parameters in the top (topU3l) case.

Class 8 c
No basis rotation is required in L (8c) 6 , and the quark currents are mapped directly. In the lepton sector, Q le admits 1 independent contraction in the U (3) 2 case (neglecting the subleading contribution ∝ Y 2 l ) and 12 in the U (1) 3 l+e case. We choose those with indices prst in the set are not hermitian and therefore the associated Wilson coefficients are complex.
-26 - The Lagrangian reads and it depends on 54 (31) independent real parameters in the top (topU3l) case.

Class 8 d
Finally, the operators in L (8d) 6 are also mapped directly to the notation with 6 quark fields. U (2) 3 invariance requires an insertion of a light Yukawa couplings for each (qu) or (qd) current and an insertion of Y l for each (le) current, as indicated in Table 2.
This class includes a total of 64 (40) real parameters in the top (topU3l) case:

Comparison with the literature
We conclude this section with a comparison of the parameterizations presented in this section with other recent results in the literature. As a quantitative reference, Table 3 summarizes the number of independent real parameters for each class of L 6 operators and flavor setup.   Table 3. Number of independent real parameters in each class of dimension 6 operators, for the 5 flavor structures implemented in SMEFTsim.
Compared to previous versions of SMEFTsim [1], the following changes were made: • the dependence on the CKM matrix in currents involving left-handed down quarks was neglected in the effective operators defined in the general and MFV versions. It has been restored in version 3.0.
• four parameters corresponding to the real and imaginary parts of C quqd were missing in the U35 and MFV models, and have now been included.
• the MFV models have been modified: all Yukawas are now retained in the spurions, instead of only y t , y b . Moreover, the Lagrangian is now organized according to a power counting in the quark Yukawas, that led to some flavor-violating terms (eg. quqd . . . ) being dropped, and others (eg. ∆ d C uH , C The U35 and MFV models can be compared, for instance, to the U (3) 5 spurion analyses presented in Refs. [63,72]. Ref. [72] contains an exhaustive classification of all the flavor spurions associated with SM fermion currents in the presence of a U (3) 5 symmetry. In their notation, S u , S d correspond to ∆ U , ∆ D respectively, while both S qu , S qd are mapped to ∆ Q . The structure Y u Y † d corresponds to ∆ U D and, since we only retain linear insertions of Y l , ∆ L = ∆ E = 0 in SMEFTsim. Any other spurion leads to baryon and/or lepton number nonconservation, and therefore does not have an equivalent in the Lagrangian considered here.
Ref. [63] presented a detailed classification of all the U (3) 5 and U (2) 5 invariant structures in the Warsaw basis. In the U (3) 5 case, their results can be directly compared with the parameterizations of the U35 and MFV models in SMEFTsim, while the U (2) 3 case can be compared (in the quark sector) to the top and topU3l models. We find complete agreement in the characterization of the structures, and the operator countings are consistent once a few differences in the organization of the invariants are taken into account: • the U (3) 5 and U (2) 5 Lagrangians in Ref. [63] are organized according to a power counting in the Yukawas, while for the U35, top and topU3l models in SMEFTsim we simply choose to retain the leading invariant for each operator in the Warsaw basis.
• in the MFV models we retain terms up to order . This choice is different from the power counting in Ref. [63], that truncates at ( • the Lagrangian of the MFV models includes spurions ∝ Y 2 d , that were neglected in Ref. [63]. • in the U (2) case, different symmetries were chosen for the lepton sector: U (2) 3 in Ref. [63] vs U (1) 3 l+e and U (3) 2 in the top and topU3l models.
The structure of the top and topU3l versions builds upon those of Refs. [7,9,33]. The main difference compared to these works is that in SMEFTsim the parameterization has been systematically extended to all operators of the Warsaw basis, including at the same time CP violating terms, interactions that do not involve the top quark, and spurion insertions of the light quark Yukawas.

Input parameters
Once the kinetic terms have been canonically normalized and the flavor structure has been fixed, the Lagrangian parameters can be assigned numerical values, with a procedure that is sometimes referred to as "fixing an input parameter scheme" or "finite renormalization". This section revisits this procedure in the SM and in the SMEFT, using a general formalism that accounts for terms up to arbitrary EFT order. They can be applied to both tree level and loop calculations but, in the latter case, this procedure needs to be combined with the usual renormalization to reabsorb UV divergences. In Sections 4.2, 4.3 these formulas are applied to the Warsaw basis case, to recover the known tree-level results, see eg. [1,2,45,49,60,73,74]. Aspects specific to the NLO case have been discussed in [48,50,51,53,75,76].
The Lagrangian parameters are fixed imposing a set of defining conditions that relate them to (pseudo-)observables: for a Lagrangian with N independent parameters g = {g i . .
n denotes a function of the parameters g. If M = N and (4.1) is an invertible system of equations, the solution The numerical values of the parameters g are then univocally determined by measurements of O.
In the SM case, one has 19 independent parameters, that we can classify as α s ,θ, QCD The procedure outlined above is most often employed to determine the value ofθ and of the EW+Higgs and Yukawa parameters. On the other hand, the determination of the CKM parameters and of α s usually relies on a large number of observables: in these cases, the system (4.1) is not invertible and the parameters' values are extracted via a global fit. When transitioning from the SM to the SMEFT, a large number of additional parameters enters the Lagrangian, namely the cutoff Λ and the Wilson coefficients C α . Fixing their numerical values in terms of measured observables is obviously still an open challenge (and indeed the ultimate goal of the present work), so these quantities are necessarily left free in the Lagrangian. Nevertheless, they play a role in the finite renormalization procedure, because the observables O employed to fix the SM quantities generically receive contributions from higher dimensional operators. Working order by order in the EFT expansion, the relations in (4.1) are modified into 7 where C here generically represents the set of relevant Wilson coefficients, that can be associated to operators of any dimension. In cases where the system of Eq. (4.1) can be inverted, (4.4) can also be solved expanding around the SM solution. The result has the general form: is the SM solution and the following K terms are SMEFT corrections that depend on the Wilson coefficients. The leading term in the solution (4.5) is defined imposing that the SM relation holds: The explicit form of the remaining K terms is found inserting Eq. (4.5) into (4.4), expanding in Λ and requiring that SMEFT corrections cancel order by order in the resulting expression. Iteratively, one finds . . .
n is the inverse of the Jacobian matrix and in Eq. (4.9) the sum runs over all possible terms with m < d and such that m + d 1 + · · · + d D = d. All functions and derivatives appearing explicitly in Eqs (4.7)-(4.9) are evaluated at the SM solution for the parameters g ≡ K (0) (O) and the indices n, k, j, i 1 . . . i D are implicitly contracted internally and summed over. A generic predicted observable P inherits a dependence on the corrections F (d≥2) n to the input quantities. Analogous to O, P will have the generic form where P (0) is the SM expression and P (d≥2) encode direct EFT contributions to P, induced by effective operators entering the relevant Feynman diagrams. Calculating P in the SMEFT starting from input quantities O means inserting the expressions of g in Eq. (4.5) into Eq. (4.11). This operation introduces "indirect" EFT contributions, that are a direct consequence of the F (d≥2) n terms in Eq. (4.4). The dependence on the latter quantities can be made explicit: where the m, n indices are summed over and, as above, all functions are implicitly evaluated The coefficients A, B are found via chain differentiation: Here A n and the first term in A n account for linear K (2) corrections to g in the P (0) and P (2) function respectively. The first term in B mn contains double K (2) insertions 8 in P (0) , while the second terms of A (2) n and B mn both stem from K (4) contributions in P (0) . The net effect of the finite renormalization procedure is that all the EFT corrections to input measurements are recast into corrections to predicted quantities: if P is an input observable P ≡ O q , all EFT corrections in Eq. (4.12) cancel order by order in the EFT. This happens by construction and follows trivially from the defining conditions imposed. It can be checked explicitly: in this case and assuming that O is a set of independent quantities, also ∂F Eq. (4.12) provides a dictionary between different input parameter schemes: comparing sets O and O , the difference in the predicted P is which is easily evaluated via Eqs. (4.13)-(4.15). This result is consistent with those in the Appendix of Ref. [60] and in Ref. [77].

Implementation in SMEFTsim
SMEFTsim implements the finite renormalization procedure via replacements of the form 9 Here "double insertions" refers to any contribution quadratic in the L6 coefficients. This includes contributions from the square of a diagram with one EFT insertion, as well as from the interference between SM and EFT diagrams with two EFT vertices, or EFT diagrams with a single interaction ∝ C 2 . The latter generally stem from field or parameter redefinitions in the Lagrangian. 9 Ref. [60] used the notationḡi →ĝi + δgi from Ref. [49]. This is completely equivalent to the one used here, dropping the bars.
-32 - and δg i encodes all the dependence on the Wilson coefficients. In the FeynRules models, these replacements are operated at the Lagrangian level via the lists redefConst (applied simultaneously to the redefinitions in Eq. (2.11)) and redefVev, and the hats are subsequently dropped in the notation. In this way, all the SM parameters appearing explicitly in the final L SMEFT are hatted quantities, ie. they are conveniently defined in the exact same way as in the SM and their numerical value is directly defined by the input observables chosen.
The shifts δg appear explicitly in the interaction terms, and they are responsible for propagating input shifts corrections to the computed processes. By construction, the dependence on δg themselves is universal, while their expressions in terms of Wilson coefficients are fixed by the input scheme choice: As noted in Sec. 2, because we work at order Λ −2 , the replacements of Eq. (4.19) only need to be performed on L SM and only linear terms in δg need to be retained. Moreover, one can replace v T →v in theC α notation, Eq. (2.4).
This procedure is implemented for parameters listed in the Higgs, EW and Yukawa sectors in (4.3), as described below. Eq. (4.4) makes manifest that the extraction of SM parameters from global fits can become problematic when generalized to the SMEFT. Whenever this set of equations is not invertible, it is not possible to find a simple form for g i that expands around the SM solution. A consistent treatment of EFT corrections to such input observables would require to extract simultaneously g i and C α , which can be very unpractical or even unfeasible, in the presence of blind directions.
In the case of the CKM parameters, this issue has been overcome in Ref. [78], where an optimal set of 4 input measurements was proposed, that allows a treatment of the CKM angles and phase analogous to that of EW parameters. Its implementation is left for future versions of SMEFTsim.
The case of α s poses a bigger challenge. The strong coupling constant can be determined from a particularly vast range of processes [79], and its extraction is often correlated to that of other physical quantities, such as parton distribution functions (PDFs). A proof-of-concept analysis of SMEFT effects on the PDFs determination was presented in Ref. [80], that explored the consequences of including four-fermion operators in a fit to deep-inelastic scattering data. Further studies are needed in order to define an optimal strategy for the treatment of SMEFT contributions in this context. For the time being, input shift corrections associated to the determination of α s are omitted in SMEFTsim.

Higgs and EW sectors
The electroweak sector of the SM contains 4 independent quantities, that can be chosen as g = {g 1 , g W , v, λ}. The 4 (pseudo-)observables needed to fix their values are usually taken in Hl,11 +C  Table 4. Expressions of input parameter shifts and the kinetic correction ∆κ H (defined in (2.6)) in terms of Wilson coefficients. The left column is common to all flavor versions, while ∆G F varies as indicated in the right column. We use the notationC While m h always needs to be retained in order to fix λ , the choice of the 3 remaining inputs is free, and several combinations have been adopted in the literature. SMEFTsim implements the two alternative schemes {α em , m Z , G F } and {m W , m Z , G F }, providing independent UFO models for both. The fine structure constant α em (0) is taken to be measured in Thomson scattering 10 , the Fermi constant G F measured in muon decays µ − → e − ν µνe , and m W , m Z , m h are defined as the bosons' pole masses, see Ref. [1] and references therein for further details. With these definitions, at tree level: 11 The ∆ quantities are dimensionless and defined in Table 4: ∆G F is inferred computing the muon decay width at tree level in the SMEFT, while the remaining shifts can be read from the relevant Lagrangian terms. In particular, the contributions inC HW B to ∆α em , ∆m 2 Z follow directly from Eq. (2.18) and ∆m 2 h follows from Eq. (2.8).
10 As we work at tree level, only direct SMEFT corrections to Thomson scattering (i.e. to the determination of αem(0)) are included here. The determination of αem(mZ ) at one loop in the SMEFT is another major open problem, as potential EFT contributions in the running have not been estimated to date. The main challenge in this task is posed by non-perturbative effects, particularly those arising as αem runs through the hadronic resonances region. 11 The normalization of ∆GF has been modified compared to previous SMEFTsim versions in order to homogenize the notation with the remaining shifts.
having defined the weak angleθ as The Jacobian J = ∂O (α) /∂g defined in Eq. (4.10) takes the form Taking the inverse and plugging it in Eq. (4.7), one obtains explicit expressions for the parameter shifts defined as in (4.20): It can be convenient, as a shorthand notation, to define a shift for sin 2 θ. In the input schemes considered here, this is always a predicted quantity, that can be expressed as where we defined the shift The second line was evaluated with generic flavor indices for ∆G F , and it can be easily mapped to other flavor structures with the dictionary in App. C. Finally, it is worth noting that electromagnetic interactions do not receive any corrections in this scheme: consistent with α em being an input quantity.
with the weak angle defined by (4.37) The Jacobian J = ∂O (m W ) /∂g takes the form -36 -and from Eq. (4.7) one has With this input scheme choice, α em is now a predicted quantity. From Eq. (4.12): (4.43) It can be instructive to write the final form of the Higgs potential, once the input shifts are applied onto Eq. (2.8). For both input schemes considered here, the result is (4.44)

Yukawa sector
To fix the SM Yukawa couplings, we take fermion masses as input quantities. From the propagators' poles, at tree level, we have In the other flavor setups ψ = {l, u, d} and all quantities are 3 × 3 matrices. The SMEFT corrections ∆M ψ are given in Table 5 for each flavor assumption. The SM solutions areŶ We use the notationC α = C α (v 2 /Λ 2 ) and the results are given in the mass basis of the up-quarks and charged leptons. and the shifts δY ψ have the form where ∆G F enters via Eq. (4.41) and ∆M ψ is non-diagonal and non-hermitian in general. The expressions (4.46), (4.47) can be easily generalized to setups where M ψ is not diagonal, by applying the appropriate flavor rotations to both sides of the equations. The net effect of the finite renormalization procedure is that ∆M ψ corrections to the fermion mass terms are recast into corrections to the hψψ couplings. In unitary gauge, the Lagrangian resulting from the replacements (2.7), (4.47) is In the top, topU3l models analogous terms with t, b quarks are also present.
In the FeynRules and UFO implementations, the common shifts δv, δλ, δY ψ are automatically replaced with the corresponding expressions in terms of Wilson coefficients. On the other hand, the dependence on the EW shifts δg 1 , δg W is left explicit in the Lagrangian, as it is identical for all EW input schemes. Once an inputs set is selected, these shifts can be traded for Wilson coefficients expressions: in Mathematica this is done via the replacement lists alphaShifts or MwShifts. In the UFO models the relations are embedded in the definitions of dg1, dgw as internal parameters.

SM loop-generated Higgs interactions
Because SMEFTsim is designed as a tree-level model, it cannot reproduce processes that only occur at 1-loop. In fact, estimating SMEFT corrections to observables that are genuinely loopgenerated both in the SM and at d = 6 level is beyond the scope of SMEFTsim. Nevertheless, there are cases where a 1-loop SM processes receives tree L 6 corrections. This notably happens in a few relevant Higgs production and decay channels.
In order to allow an estimate of interference terms between L 6 and SM diagrams for the processes gg → h, h → γγ, h → Zγ, SMEFTsim implements effective SM interactions obtained in the large m t limit. This formally corresponds to matching the SM onto an EFT (we will refer to this as "top-EFT") where the top quark has been integrated out. The advantage of this approach is that the top loops are effectively reduced to point vertices that can be inserted in tree diagrams. The obvious caveat is that the top-EFT is only valid in a limited kinematic region, as discussed below. Hgg where The corresponding g coefficients are fixed via a 1-loop matching procedure of the SM onto the top-EFT. For the hγγ and hZγ interactions we use the results from Refs. [81][82][83][84], that -39 -include loops of both top quarks and W bosons: The loop functions I f , I w , I Z w are evaluated in the limit where the Higgs boson is on-shell and higher order corrections are simply obtained via Taylor-expansion, retaining terms up to O(m −2 t m −6 W ): Note that the d = 7 operators produce interactions with one Higgs and up to 6 gluon legs. While the full gauge-invariant Lagrangian is implemented in the FeynRules models, 12 There is a sign difference in the definition of O3 compared to Refs. [85,86]. The sign of C3 is also affected by the sign in the covariant derivative definition, that was taken with the opposite convention in Ref. [87].
-40 -only vertices with up to 4 gluon legs (hgg, hggg, hgggg) were exported to the UFOs. The Feynman rules of the hggggg and hgggggg vertices are extremely complex both in the color and Lorentz structures, to the point that their inclusion makes the Monte Carlo event generation computationally challenging. They are available upon request.

Validity of the approximations used
The Higgs interactions described in this section are implemented to the specific purpose of enabling the simulation of Higgs production and decay processes. In general, these vertices should not be inserted into other arbitrary processes. In MadGraph5_aMC@NLO, the insertions can be controlled at the diagram generation level via the interaction order SMHLOOP = 1 that is assigned to all the g couplings in the Lagrangian (5.1), see also Sec. 8.2.
The following limitations should also be kept in mind: • The implementation relies on the top-EFT formalism, that is only valid when the momentum q flowing through the effective vertex is q < m t . This condition is always fulfilled for gg → h with no extra jets, for which the top-EFT reproduces the 1-loop SM cross-section within an accuracy of a few permille. With more complex final states, a validity threshold is present and it can translate differently in terms of measured observables, depending on the process. For both pp → hj and pp → hjj, the total cross section is dominated by gg-and qg-initiated channel, for which the m t → ∞ approximation breaks down roughly at p T (h) 250 GeV [86,87,[89][90][91][92]. Within the top-EFT validity regime, the d = 7 implementation reproduces the SM 1-loop result within an accuracy of few %, see also Ref. [87]. The large m t approximation fails most significantly in qq-initiated processes that, nevertheless, give a negligible contribution to the total cross section. This behavior is due to the quarks' PDFs preferring significantly larger x compared to the gluon one, which leads to largeŝ contributions being suppressed for gg and qg initial states, but not for qq [91].   Processes gg → h + nj with n ≥ 3 cannot be fully reproduced with SMEFTsim, even with the inclusion of hggggg, hgggggg vertices, because a complete matching to O(m −2 t ) onto these vertices would require d = 9 top-EFT operators.
• In addition to the validity of the large m t approximation, the implementation of hγγ, hZγ assumes an on-shell Higgs in the parameterization of the loop function.

Comparison to previous versions of SMEFTsim
Previous SMEFTsim versions only implemented the hgg, hγγ and hZγ vertices, while interactions with higher numbers of gluons were omitted. In version 3.0, all the vertices induced by the operator O Hgg and vertices with up to 5 legs (4 gluons) from O (2,3,4,5) Hgg are included. Moreover, the hgg interaction was previously parameterized in the on-shell Higgs limit, analogously to hγγ, hZγ, via a coupling [83,84,94] Hgg + with p 1 , p 2 the momenta of G µ , G ν respectively. In the limit p 2 1,2 = 0, Hgg .

Propagator corrections
Mass terms and decay widths of the SM particles generally receive corrections from L 6 operators. In order to compute amplitudes consistently at O(Λ −2 ), these corrections need to be included in the propagators. In unitary gauge the propagator of a generic unstable vector V , scalar S or fermion ψ has the form In the SMEFT we can write, for each particle, m = m SM + δm, Γ = Γ SM + δΓ , (6.4) where the shifts δm, δΓ collect all the contributions from d ≥ 6 operators. The corresponding propagator expressions be expanded to linear order in the shifts [95] P µν V = P µν,SM where the expressions for P SM are given by Eqs.
The corrections read with the shorthand notation Note that, since ∆P ∝ D(q 2 ) −1 , propagator corrections are expected to be relevant in the on-shell kinematic region and suppressed when the particle is largely off-shell. The dominant contributions are therefore approximated by the on-shell expressions: 13 (6.13)

Implementation in SMEFTsim
SMEFTsim 3.0 implements propagator corrections for the Z, W, h bosons and for the top quark. The user has two alternative options for including them in SMEFT predictions: 13 Longitudinal contributions for vector bosons were neglected here.
-44 -(a) using the linearized propagator expressions of Eqs. (6.5)-(6.7). In this case the pole of the propagator remains located at m SM , and the dependence on the Wilson coefficients, stemming both from δm and δΓ, is linear at the amplitude level. This option is selected fixing linearPropCorrections = 1 (or any value = 0) in the param_card.
(b) using the propagator expressions in Eqs. (6.1),(6.2), with shifted masses. In this case the pole of the propagator is located at m = m SM + δm while width corrections are entirely dropped. 14 The dependence on the Wilson coefficients is generally non-linear, as contributions ∝ 1/C α are induced in the amplitude. This is the default option and it's selected with linearPropCorrections = 0.
While option (a) is recommended for consistency of the EFT expansion, we caution the user that the linearization can be problematic, particularly in the presence of mass corrections. Formally, expanding around the complex pole of the propagator is not a gauge-invariant operation [96][97][98]. Numerically, significantly large discrepancies between methods (a) and (b) can occur, as illustrated in Figure 2  In this way, for instance, linearized Z-propagator corrections to pp → µ + µ − can be estimated computing the pp → Z → µ + µ − and pp → Z → µ + µ − amplitudes, and using the interaction order NPprop to isolate the pure SM/interference/quadratic contributions as detailed in Sec. 8.3. Note that linearized propagator corrections are available only in the UFO models, as the propagators are modified directly in the propagators.py file and not in FeynRules. 16

Mass and width corrections implemented
All the mass and width shifts implemented in SMEFTsim are computed to O(Λ −2 ), ie. linearly in the Wilson coefficients. Because m Z , m h and m t are taken as input parameters, 14) The  [79,102]. These are free parameters in the models, that can be modified by the user. The quantities δΓ/Γ SM tree are calculated at tree level (both numerator and denominator) using the width computation tools in FeynRules. They include all 2-body decays and are extracted in the limit V CKM = 1, with all fermion masses set to zero, except those of the b and t quarks. Analytic expressions are given in Appendix A.
The correction to the total Higgs width is computed using individual K-factors for each decay channel, as in Ref. [ with f running over the set {γγ, Zγ, gg, bb, cc, τ + τ − } plus the allowed 4-fermion channels. In the SMEFTsim implementation, only 4-fermion decays proceeding via charged currents (h → W W * → 4f ) are retained, in order to simplify the analytic expressions. Channels mediated by neutral bosons (h → ZZ * , Zγ * , γ * γ * , g * g * → 4f ) give subdominant corrections, that are estimated in a 3 − 5% change to the dependence onC HW ,C HB ,C HD and a change 1% for the other Wilson coefficients [54]. Γ SM,best h is a free parameter in the models and can be modified by the user. The bestfit branching ratios, instead, are embedded numerically in the δΓ h expressions and cannot be changed. The values employed are reported in Table 6. The relative deviations δΓ h→f /Γ SM h→f tree for 2-body decays are computed with the FeynRules tools, retaining the full dependence on all the relevant fermion masses and Yukawa couplings. If a given Yukawa coupling y f is set to zero in the param_card, all contributions to δΓ h originating from the h → ff decay channel are dropped. For the h → 4f channels we take the analytic results of Ref. [54], that neglect all fermion masses and quark mixings. Note that the results in Ref. [54] were given for the U (3) 5 flavor symmetric case, and they have been generalized to the other flavor assumptions in SMEFTsim. Full analytic results for SMEFT corrections are reported in Appendix A. = 0.24161 [103]. They include only charged current contributions and are summed over all allowed flavor combinations.

Usage in Mathematica
The FeynRules files in SMEFTsim can be imported in Mathematica The FeynRules code is split over different files that contain the required operators, parameters and Lagrangian definitions. The implementation is such that only the objects matching the selected flavor structure and EW input scheme are defined upon loading. In all models, the following Lagrangians are defined (definitions were given in Section 1): • LGauge = L gauge . Contains the SM terms plus the linearized SMEFT corrections due to field redefinitions and input parameter shifts. • LGaugeP. Same as LGauge, but with at least one W or Z boson replaced with the corresponding dummy field W1, Z1.
-48 - • LHiggs = L Higgs . Contains the SM terms plus the linearized SMEFT corrections due to field redefinitions and input parameter shifts.
• LHiggsP. Same as LHiggs, but with at least one W , Z or Higgs boson replaced with the corresponding dummy field W1, Z1, H1.
• LFermions = L fermions . Contains the SM terms plus the linearized SMEFT corrections due to field redefinitions and input parameter shifts.
• LFermionsP. Same as LFermions, but with at least one top quark or W, Z boson replaced with the corresponding dummy field t1, W1, Z1.
• LYukawa = L Yukawa . Contains the SM terms plus the linearized SMEFT corrections due to field redefinitions and input parameter shifts.
• LYukawaP. Same as LYukawa, but with at least one top quark or Higgs boson replaced with the corresponding dummy field t1, H1.
• LSM = L SM . The SM Lagrangian without any SMEFT correction.
• LSMlinear. The SM lagrangian plus the linearized SMEFT corrections due to field redefinitions and input parameters shifts.
• LSMloopP. Same as LSMloop, but with at least one Higgs or Z boson replaced with the corresponding dummy field H1, Z1.
• LSMincl = LSMlinear + LSMloop The last 3 Lagrangians contain extremely long expressions. It is strongly recommended to use them with care and avoid calling these variables unless strictly necessary. The parameters notation in the code is provided in Appendix D. In addition, the following parameters lists are defined in all models: • WC6. The list of all Wilson coefficients.
In the general model the list WC6indices is defined in addition. In this case WC6 contains eg. cHuIm11, cHuIm33, while WC6indices contains cHu[ff1_,ff2_] with blank flavor indices.
• shifts. The list of all shift parameters, such as dGf, dMZ2, dgw, dg1 etc. The complete list is given in Table 18.
• d6pars. List of all SMEFT quantities, including Wilson coefficients with and without free indices, and shifts.
Two handy functions are also defined: •   Finally, it is not recommended to export the UFO models independently, unless only a small subset of operators is included. The UFOs provided in the GitHub repository have been exported in a specific, optimized way and the python files have been manipulated a posteriori in order to enable some of the new features. The notebook with the original export procedure is available upon request.

Usage in MadGraph5_aMC@NLO
This section provides recommendations for the use of SMEFTsim in MadGraph5_aMC@NLO. It is in no way meant as a manual for the functionalities of MadGraph5_aMC@NLO itself, for which we defer the reader to the appropriate references, see eg. [43,106,107].
The SMEFTsim package provides 10 pre-exported UFO models, one for each flavor setup and input parameter scheme. Each of them contains the full L SMEFT defined in Sec. 1 and 3 and L SMhloop defined in Sec. 5. The manipulations and redefinitions described in Sec. 2 and 4 have been consistently applied. The vertices contained in the models are derived in unitary gauge and the ghost fields have been removed: SMEFTsim is designed for LO event generation and does not support the NLO syntax. A list of the SMEFT parameters defined in the codes is provided in Appendix D, with a mapping to the notation used in this notes. All the UFO models have been validated following the recommendations in Ref. [44], as detailed in Appendix F.
-53 -Although the selection of an appropriate model is of course up to the taste of the user, each flavor setup is meant to optimize the parameterization of a certain class of effects in the SMEFT. For instance, the top and topU3l models are designed to single out the couplings of the top and bottom quarks [33], and they only differ in that the top case provides more freedom to distinguish the lepton flavors. The U35 models allow to work with a minimal number of parameters and are recommended for flavor-blind processes or whenever the flavor structure can be assumed to be strictly SM-like. At the other side of the spectrum, the general models provide maximal freedom and can be used to study flavor-violating processes or to realize arbitrary flavor structures beyond those implemented. An operator by operator comparison of the different flavor structures is provided in Appendix C.
As discussed in Sec. 6, the use of the {m W , m Z , G F } input scheme is particularly recommended for processes involving W bosons, as it avoids the problematic introduction of SMEFT corrections to the W pole mass. The {α em , m Z , G F } and {m W , m Z , G F } input scheme implementations are expected to give results that differ most significantly in the dependence on the Wilson coefficients C HW B , C HD , C

Parameter cards and restrictions
The model parameters are grouped in blocks, that are explicitly shown in the parameter cards. This block is absent in the MFV models.
SMEFTFV -the (∆C α ) parameters of the MFV setup, with default value 0. This block is only present in MFV models.
SWITCHES -the parameter linearPropCorrections, that can be used to switch ON/OFF the linearization of SMEFT corrections in the propagators. The default value is 0 (OFF).
The SMINPUTS block contains G F , α s and either α em or m W depending on the input scheme.
The use of restriction cards allows to reduce the number of diagrams generated for a given process. Two restriction cards are provided by default with each UFO: • restrict_massless.dat. The masses and Yukawa couplings of all fermions, except the top and bottom quarks, are set to 0. The CKM matrix is set to the identity. The Wilson coefficients are set to arbitrary numerical values.
• restrict_SMlimit_massless.dat. As in restrict_massless.dat, but with all Wilson coefficients set to 0.
The restrictions should be applied at the stage where the model is imported, eg.: § ¤ import model SMEFTsim_A_U35_MwScheme_UFO_v3_0 -massless ¦ ¥ In this way, all the parameters that are set to either 0 or 1 in the restriction are fixed to their value and cannot be edited further. Sets of parameters that are assigned an identical value in the restriction are fixed to be identical: while their numerical value can still be edited, they cannot be disentangled from one another. Diagrams that are proportional to a vanishing parameter will not be generated. The use of one of the massless restrictions is recommended for LHC studies, because it simplifies significantly the calculations. There are of course several possible strategies for the use of these restrictions in MadGraph5_aMC@NLO: for instance, one can create a modified version of restrict_SMlimit_massless.dat turning on one Wilson coefficient with some arbitrary value = 0. Importing the model with this modified restriction allows to generate events with the chosen coefficient only, while all the other operators are forbidden. Alternatively, if the model is imported with restrict_massless.dat, all the Wilson coefficients are retained: all the allowed SMEFT diagrams will be generated and all the parameters can be freely edited at the event generation stage. Note that, to achieve this, all the Wilson coefficients in restrict_massless.dat are assigned different non-vanishing and non-unitary values 18 , that will need to be changed prior to the event generation. To simplify this operation, a "restricted" parameter card param_card_massless.dat is provided in the UFO, where all the Wilson coefficients are set to 0. This card can be directly copied in the PROC/Cards/ directory of the exported process and modified at will.

Interaction orders
A standard feature of UFO models is that every coupling parameter is assigned an interaction order, ie. a "flag" that allows to control the number of coupling insertions in generated Feynman diagrams. Each parameter carries an arbitrary number of interaction orders.

Definitions
In the SMEFTsim UFO models the interaction orders are assigned as reported in Table 7. 18 In previous versions of SMEFTsim, the Wilson coefficients in the restriction cards were all set to the special value 9.999999e-01, that in principle allows to set the parameters to 1 without fixing their value. However, this syntax is not fully supported by MadGraph5_aMC@NLO, and is occasionally source of unexpected numerical behavior in UFO models with a very large number of parameters, such as SMEFTsim general or MFV.

Order
Parameters assigned Hgg . . . g  The orders QED and QCD are assigned as customary in the standard SM UFO implementations, with the exception of the SMEFT cutoff Λ, that has been assigned QED=-1 such that the combination (v/Λ) is order-less. This causes the C H correction to the h 3 interaction, that is proportional tov 3 C H /Λ 2 (see Eq. (4.44)), to have overall order QED = -1. This is the only vertex with an inevitably negative interaction order. Special care is therefore recommended when generating processes sensitive to this contribution.
The interaction order SMHLOOP labels the SM loop-generated Higgs interactions introduced in Sec. 5. Since by definition they are proportional to the SM gauge couplings, the g (k) Hgg parameters additionally carry QCD=2 and the g Hγγ , g HZγ parameters carry QED=2.
The interaction order NP (New Physics) is assigned to all the Wilson coefficients and shifts indistinctly. In addition, starting from version 3.0, individual interaction orders have been introduced for each effective operators. The same order NPc[a] is assigned to all the associated CP-conserving and violating parameters, irrespective of the flavor indices carried. For instance, in the top models, the parameters Re(C eH ) pp , Im(C eH ) pp for p = {1, 2, 3} all have order NPceH=1. In the U35, MFV, top and topU3l models, distinct interaction orders are assigned to independent flavor contractions. For instance C ll and C ll have orders NPcll and NPcll1 respectively. In the top models, the parameters (C ll ) pprr have order NPcll, while the (C ll ) prrp contractions have order NPcll1, etc. In most cases the label [a] coincides with the name root of the associated Wilson coefficient, that can be read off from the tables in Appendix D. If in doubt, the user can resort to the .fr source files or check explicitly the couplings.py file to identify the exact orders assigned to a given parameter or coupling.
All the CP-violating parameters, that belong to the SMEFTCPV block, have an order NPcpv=1. Analogously, all the (∆C α ) quantities in the MFV models, that belong to the SMEFTFV block, have an order NPfv=1.
The shift quantities introduced in Sec. 4.2 are assigned the orders reported in Tab. 8. Finally, the order NPprop labels the interactions of the dummy fields W , Z , h , t carrying linearized propagator corrections, see Sec. 6.1. It is carried by a dummy internal parameter propCorr that only takes values 0/1, when the linearPropCorrections switch is set to 0/a nonzero value. Its application is discussed in the next subsection. By default, the interaction order NPprop is "switched off", as it is assigned an upper limit of 0 interactions, that can be lifted as shown below. No upper limit is set for the other orders.
The interaction orders SMHLOOP, NP, NPshifts, NPprop, NPcpv, NPfv have been assigned hierarchy 99. MadGraph5_aMC@NLO will therefore generally avoid insertions of the associated vertices, unless these orders are specified.

Recommended use
Interaction orders are specified at the stage of process generation in MadGraph5_aMC@NLO, eg:  -57 -where = is equivalent to <=, while == selects uniquely the order specified. The syntax XX=n acts at the amplitude level, ie. it specifies the total number of couplings with order XX to be inserted in each Feynman diagram. The syntax XXˆ2 acts instead at the squared amplitude level. This functionality works very nicely for EFT studies, as it allows to disentangle contributions at different orders in the expansion. Although a priori SMEFTsim can be used for computations to any allowed order in Λ, it implements the SMEFT Lagrangian consistently expanded only up to O(Λ −2 ). This means that any SMEFTsim prediction beyond this order is necessarily incomplete in the Effective Theory. It is worth noting that this statement does not concern only higher dimensional operators in L SMEFT , but also affects the dependence on some of the Wilson coefficients in L 6 . For instance, it was stressed at multiple stages in Sections 2 and 4 that terms of order Λ −4 or higher were neglected in the field and parameter redefinitions performed, as well as in the treatment of input parameters. The impact of these L 6 contributions has been discussed in Ref. [108] for the case of O(Λ −4 ) corrections to 1 → 2 decays, using the geoSMEFT formalism [59].
Complete results truncated at O(Λ −2 ) can be obtained with the syntax NP<=1 NPˆ2<=1, that retains only SM plus SM-L 6 interference contributions. Contributions of order Λ −4 stemming from the square of an O(Λ −2 ) amplitude, although incomplete, are also commonly included in the SMEFT calculations: they are selected with NP=1 NPˆ2==2 or just NP==1. For the reasons above, it is generally recommended to use the specification NP<=1 (or NP=1) for any process, to limit the number of EFT insertions to one per Feynman diagram. A generic observable computed in this way will have the form where σ SM , σ α , σ αβ intuitively denote the SM, interference and quadratic contributions respectively. σ SM and σ αα are always positive quantities, while σ α , σ αβ with α = β can take negative values. Table 9 shows examples of how the interaction order syntax can be used to disentangle these contributions, for a simple case with two Wilson coefficients. The expressions directly generalize if three or more parameters are present. The interaction order NPshifts is meant to provide more control on the Wilson coefficients that enter field/parameter redefinitions and input parameter corrections. For instance, setting NPshifts=0, only "direct" contributions from these operators will be retained. Setting NP==1 NPshifts==1 isolates the pure shifts contributions instead and forbids the "direct" ones.
Finally, as discussed in Sec. 5, the loop-generated SM Higgs couplings implemented in SMEFTsim are defined in the m t → ∞ limit, and their use should be limited to (on-shell) Higgs production and decay processes. Outside of this regime, it is strongly recommended to use SMHLOOP=0.  Table 9. Examples of interaction-order syntax that select different EFT contributions to a generic observable with the dependence given in Eq. (8.1), for the case of 2 Wilson coefficients C α , C β .

Propagator corrections and decay widths
As discussed in Sec. 6, SMEFT corrections are generally present in the propagators of unstable particles, due to d = 6 operators modifying their masses and/or decay widths. Sec. 6.1 outlined two alternative methods for estimating these contributions in a given process. In the following we illustrate how they can be implemented in MadGraph5_aMC@NLO. Due to the absence of additional interaction orders, there is unfortunately no equivalent to Table 9 in this case: propagator corrections from different operators and EFT orders cannot be disentangled at this level. Important: in order to avoid unwanted insertions of the dummy fields in standard process generations, the functionality described here has to be activated in 2 steps: (i) in the file coupling_order.py, the expansion_order option for the order NPprop has to be set to a number ≥ 2 (recommended: 99). The default is 0, which forbids dummy interactions completely. (ii) The parameter linearPropCorrections in the param_card has to be set to a non-zero value. If this is not the case, dummy vertices will be included in the diagrams, but they will be idle, as they are proportional to propCorr = 0/1 for linearPropCorrections = 0/non-zero.

Method (b): full corrections
As an alternative to linearization, propagator corrections can be estimated following more canonical procedures. This generally means computing processes with the propagator forms in Eqs. (6.1)-(6.3), with mass and decay parameters that depend explicitly on the Wilson coefficients, to either linear or quadratic order. The resulting process will thus exhibit a non-polynomial dependence on the SMEFT parameters.
The most relevant caveat here is that the implementation of the Wilson coefficient dependence is necessarily different for masses and widths. In the former case, it is possible to define mass parameters as internal and assign them an analytic expression, eg MW = MWsm + dMW, with dMW defined as in (4.33). In SMEFTsim the dMW term is only included when linearPropCorrections = 0, and switched off otherwise. Note also that the expression of a generic δm is extracted at the Lagrangian level and is purely of O(Λ −2 ).
On the other hand, due to how Monte Carlo generators and their interface to parton shower or decay modules are structured, decay widths cannot be defined as internal parameters in UFO models. Therefore the only way their SMEFT expressions can be inserted in the calculation is by letting MadGraph5_aMC@NLO compute them, by setting the relevant widths to Auto in the param_card. The on-the-fly calculation will include all allowed 2-body decays and rely on the pre-computed decay results collected in the file decays.py, which include O(Λ −4 ) terms. 20 Note that, with this procedure, the decay widths will need to be re-evaluated every time the value of a relevant Wilson coefficient is modified.
With both mass and width corrections evaluated as above, the denominator of a generic propagator has the form 2) 19 Remember that NPprop counts the number of dummy vertices, so, in this case, the order specified is twice the number of dummy propagators. 20 Since only 1 → 2 decays are included, these results consistently stem from the square of O(Λ −2 ) amplitudes.
Dummy fields are not included in the pre-computed decay widths.
-60 -where δm and δ (1) Γ are of O(Λ −2 ) and δ (2) Γ is of O(Λ −4 ). Eq. (8.2) contains therefore terms up to O(Λ −6 ). An observable computed for a process with k internal lines corrected in this way, will contain terms up to O(Λ −12 k ) at the denominator. In principle this functional dependence can be reconstructed fitting the appropriate rational function to a sufficient number of benchmark points. However, it is recommended to reduce the proliferation of higher-order terms in the propagators, by evaluating mass and width corrections separately and by treating propagator corrections to different internal states individually, whenever possible. This is achieved avoiding to switch on at the same time the linearPropCorrections flag (that turns on δm W ) and the Auto computation of a decay width, or of two decay widths simultaneously.

Example: Higgs production and decay including W, Z propagator corrections
As a practical example for the use of SMEFTsim in MadGraph5_aMC@NLO, we compute SMEFT corrections to Higgs production and decay processes that are mediated by W, Z exchange, which allow to illustrate the propagator corrections feature.

STXS forqq → hqq
We consider two bins of the stage 1.1 Simplified Template Cross Section (STXS) parameterization [109][110][111][112], for the EWqq → hqq production channel at low Higgs p T . They are defined by the cuts [111]: with y h the rapidity of the Higgs boson. In each bin, the Higgs production cross section in the SMEFT can be parameterized as: Table 10 reports the values of σ α /σ SM for the relevant fermionic operators, computed at parton level using SMEFTsim in the U35 flavor-symmetric, {m W , m Z , G F } input scheme version. The following procedure was followed: 1. 50000 events are generated for each bin in MadGraph5_aMC@NLO, for where the -vbf flag indicates that the model is imported with a custom restriction card restrict_vbf.dat, that in this case sets to zero the masses and Yukawa couplings of all fermions except the bottom and top quarks, as well as all the Wilson coefficients that -61 -are known not to contribute to the process. The remaining ones are set to a random non-zero value in this card.
Applying the STXS defining cuts to these events gives the tree-level SM cross sections The events are reweighted using the reweight module in MadGraph5_aMC@NLO [113]. Individual weights are computed for each Wilson coefficient, splitting contributions from operator insertions in the vertices (labeled as "direct", inclusive of the shift terms) and from insertions in the W, Z propagators (labeled as "propagator"). This is done setting each coefficient to 1 and the SMEFT cutoff scale LambdaSMEFT to 1 TeV. For instance, for the C ll parameter, the reweight_card.dat for the direct contributions is § ¤ change process q q > h q q QCD =0 NP =1 NP^2==1 NPprop =0 SMHLOOP =0 launch --rwgt_name = SMEFTsim -cll1 -direct set cll1 1 set cHl1 0 set cHl3 0 set cHe 0 set cHq1 0 set cHq3 0 set cHu 0 set cHd 0 done ¦ ¥ For estimating the pure propagator contributions the first two lines are replaced with § ¤ change process q q > h q q QCD =0 NP =0 NPprop =2 NPprop^2==2 SMHLOOP =0 launch --rwgt_name = SMEFTsim -cll1 -propagator ¦ ¥ Analyzing the reweighted events gives σ αv 2 Λ 2 for each C α . The numbers in Table 10 are finally obtained dividing by σ SM and normalizing toC α .
The results show that propagator corrections are negligible in the VBF regime, where the relative SMEFT corrections to the cross section is  Table 10. Values of σ a /σ SM for the relevant fermionic Wilson coefficientsC α contributing toqq → hqq and h → e + e − µ + µ − . For the first two columns σ is the cross section in the VBF-like and VH-like STXS bins defined in the text, while in the third the numbers refer to the partial decay width. The results are given for {m W , m Z , G F } inputs with a U (3) 5 flavor symmetry, and neglecting all fermion masses. The "direct" contributions stem from operator insertions in vertices, including parameter shifts, while the "propagator" ones stem from the corrections to the W, Z decay width in internal propagators. The lines highlighted in color are those for which the latter are most relevant.
propagator effects. In this case, the relative SMEFT correction is where the numerical prefactors reflect the proportions of W and Z bosons produced. In fact, the largest numerical effects in Tab. 10 are observed in the operators entering δΓ W .
An analysis of the Z-mediated Higgs decay h → e + e − µ + µ − was performed following a procedure analogous to the one described forqq → hqq. In the decay case, one Z boson is always on-shell, leading to significant contributions from the intermediate Z propagator. The relative SMEFT correction to the decay width is found to be The breakdown into fermionic Wilson coefficients is given in Table 10 and it agrees with the analytic results of Ref. [54].
-63 - The SMEFTsim package contains models in FeynRules and in the UFO format, that implement the complete Warsaw basis of dimension six operators, under different flavor assumptions and with different choices of the input quantities for the EW sector. Its main scope is the Monte Carlo simulation of LHC processes in the SMEFT, but it can also be employed for simple analytic calculations, exploiting the FeynRules interface in Mathematica. This work reviewed the theoretical elements that are implemented in SMEFTsim and presented the improvements in version 3.0. The most significant changes compared to previous releases are the addition of two new flavor structures for top quark physics, the implementation of a brand new tool for the inclusion of SMEFT corrections in the propagator of unstable particles and the general improvement of the code, particularly of the parameterization of Higgs-gluon interactions in the SM.
As in previous versions, SMEFTsim 3.0 supports the WCxf exchange format [19]. The corresponding interface will be updated shortly after the code release. Finally, support for the translation of the UFO models to python3 will be provided in the near future.
where T ψ 3 = ±1/2 is the isospin eigenvalue and Q ψ is the electric charge of the fermion ψ. We also define: At tree level in the SM, the partial decay width of the Z boson into aψ p ψ p pair, with ψ = {ν, l − , u, d} and flavor p, is where N ψ c is the number of colors of the fermion species ψ. The relative SMEFT correction to a partial width can be inferred differentiating in the g ψL , g ψR couplings and inserting the expressions of their SMEFT shifts: Using the expression of δs 2 θ provided in Eq. (4.32), one obtains (A.10) Flavor violating decays are absent at O(Λ −2 ). As m b = 0 is retained, the Z →bb result contains additional terms. The partial width expression in the SM is where the first term stands for the contributions in Eq. (A.10). The relative SMEFT correction to the total decay width is finally obtained as with f running over all the allowed fermion pairs and Br SM Z→f , Γ SM Z computed directly from the tree level expressions.

A.2 W boson
At tree level in the SM, the partial decay width of the W + boson into a fermion pair with N f C = {1, 3} the number of colors. Only flavor conserving decays are considered here, as CKM mixing is neglected. The relative SMEFT correction for each channel is The total W + decay width in the SM is Since in this case the branching ratios are simple rational numbers, the relative SMEFT correction simplifies into

A.3 Higgs boson
The SM partial widths for two-body Higgs decays are: Hgg /x 2 t , (A. 27) with the hγγ, hZγ, hgg couplings defined in Sec. 5. The SMEFTsim implementation retains the masses of the tau lepton, charm and bottom quarks in Eq. (A.24). Four-body decays were included neglecting neutral current contributions, CKM mixing and all fermion masses. The analytic expressions were taken from Ref. [54] and generalized to all flavor setups. For each individual decay channel, the partial decay width in the SM is the appropriate color multiplicities. The numerical factor comes from the phase space integration, that is performed takinĝ m W = 80.387 GeV.

A.4 Top quark
To a very good approximation, the top quark decays exclusively to W b. The SM width is Decaying the W in final state does not lead to any additional contribution to δΓ t /Γ SM t . This happens because the W is always on-shell, so its decay essentially factorizes out: using the narrow width approximation, one trivially has Note that this conclusion only holds at the SM-L 6 interference level, while at O(Λ −4 ) additional SMEFT corrections arise through contact vertices (tb)(νl), (tb)(du).

B What's new in version 3.0
Here we briefly summarize the most significant updates and features introduced in SMEFTsim 3.0, compared to previous versions: • The flavor assumptions top and topU3l described in Sec. 3.4 have been added.
• The flavor structure of all models has been generally improved. See Sec. 3.5 for details.
• The treatment of propagator corrections described in Sec. 6 has been implemented, enabling the estimate of linearized EFT corrections to the W, Z, H, t widths and to the W mass.
• The treatment of SM loop-generated Higgs interactions has been improved, particularly in the Higgs-gluon case. See Sec. 5 for the general treatment and Sec. 5.2 for a detailed comparison with previous implementations.
• All interaction vertices with up to 6 legs are now included in all UFO models. In the previous version, only 4-point functions were retained.
• The numerical values of the SM parameters have been updated, see Table 17. The default value has been set to 0 for all Wilson coefficients.
• All complex Wilson coefficients are expressed in terms of their real and imaginary parts, rather than absolute values and phases.
• Individual interaction orders have been defined for each operator. Additionally, interaction orders NPcpv, NPshifts, NPprop, NPfv have been added, to provide more control on each class of EFT contributions. See Sec. 8.2 for further details.
• In the UFO models, the SMEFT parameters have been organized in parameters blocks: SMEFTcutoff contains only Λ and SMEFT (SMEFTcpv) contain CP conserving (violating) Wilson coefficients. In the MFV models the flavor-violating ∆C α parameters are contained in the additional SMEFTFV block. See Sec. 8.1.
• The normalization of ∆G F (dGf) has been modified. This is explicit in the FeynRules Lagrangian but does not have any consequence for the UFO models.

C Conversion tables between flavor assumptions
This Appendix collects the results of Sec. 3 and compares the flavor structure of the fermionic operators across the five setups considered. Tables 11 -16 provide a dictionary between the different models: in order to translate between two flavor assumptions it is sufficient to exchange the corresponding expressions within each table block. All structures are given explicitly in terms of diagonal Yukawa matrices and of the CKM matrix V . In the top and topU3l cases, Y u and Y d are 2 × 2 matrices and V = 1 is assumed.
general (C eW ) pr Same as (C eH ) pr .
general (C eB ) pr Same as (C eH ) pr .
general (C uW ) pr Same as (C uH ) pr .
general (C uB ) pr Same as (C uH ) pr .
general (C uG ) pr Same as (C uH ) pr .

D Parameter definitions in the code implementation
This Appendix provides tables to facilitate the interpretation of the FeynRules and UFO implementations in terms of the theory discussion in the main text. Table 17 lists the external parameters that are are defined in all SMEFTsim models, specifying the corresponding code name and default numerical value. Table 18 shows the nomenclature used for the Wilson coefficients of the bosonic operators and for the shift quantities defined in Sec. 4.1. Tables 19 -24 do the same for the Wilson coefficients of fermionic operators, for each flavor assumption. As a common rationale, primes are replaced by 1 in the code name and real and imaginary parts are specified by with Re, Im suffixes. If needed, flavor indices are fully specified and appended at the very end of the code names. In the MFV models, the coefficients ∆ q n (C α ) are denoted Delta[n][q]c[a]. Although the correspondence between parameters names is most often direct, some notational changes were necessary, particularly in the top and topU3l implementations. Most notably the lowercase q has been replaced with j in all the parameters' and operators' names, as the q/Q distinction between light and heavy quark fields is problematic for non-case-sensitive interfaces. Analogously, the coefficient C Hb is denoted as cHbq to avoid conflict with cHB, while C bB is denoted as cbBB, distinct from cbb. The internal parameter C tH is denoted as ctHH to avoid conflict with the cosine of the weak angle cth.

E Comparison to other SMEFT UFO models
In this section we compare SMEFTsim with other UFO models dedicated to SMEFT studies, and provide a mapping of the common parameters. For the time being, the comparison is restricted to dim6top [33,117] and SMEFT@NLO [34,118], that are both based on the Warsaw basis.
For each model we summarize the main features and provide conversion tables with the parameters defined in SMEFTsim. To our knowledge, SMEFTsim is currently the only publicly available UFO model that implements linearized SMEFT corrections to propagators. [33,117] contains LO UFO models dedicated to EFT studies in the top sector. Here we refer specifically to dim6top_LO_UFO and dim6top_LO_UFO_each_coupling_order in the version published in May 2020.
Flavor structure. dim6top is based on the recommendations provided in Ref. [33], and it assumes a U (2) 3 flavor symmetry in the quark sector and a (U (1) l+e ) 3 in the lepton sector. U (2) 3 breaking terms are also available and they are implemented explicitly, ie. without promoting the quark Yukawas to spurions of the flavor symmetry. Contractions inducing both flavor-conserving and violating neutral currents are included.
All fermion masses and Yukawa couplings are neglected, except those of the top and bottom quarks of the tau lepton. The CKM is taken to be the unit matrix.
Operators implemented. dim6top contains only operators that modify the interactions of the top quark, and CP violating terms are included. Most operator definitions are identical to those in SMEFTsim top, topU3l. In a few cases, the invariants implemented differ by a Fierz rotation, as detailed in Ref. [33].
Input parameters. Both input schemes {α em , m Z , G F } and {m W , m Z , G F } are supported in dim6top. Since purely bosonic and leptonic operators are omitted, this only affects the numerical values assigned to the SM parameters and not the dependence on the Wilson coefficients.
SM loop-generated Higgs couplings. Not implemented.
dim6top matches very closely the top and topU3l versions of SMEFTsim and it can also be mapped to the general one. A correspondence with other flavor versions of SMEFTsim can only be established partially, due to incompatibilities in the assumed flavor structure.
The mapping between Wilson coefficients defined in dim6top and in the top, topU3l versions of SMEFTsim is provided in Tables 25,26. The mapping to the general version of SMEFTsim is provided in Tables 27, 28. In both cases, the first table contains parameters with a one-to-one correspondence, while the second contains parameters that require a basis -85 -rotation. For example, the point cQlM1=1, cQl31=3 in dim6top corresponds to cQl111=4, cQl311=3 (or clq1Re1133=4, clq3Re1133=3) in SMEFTsim.
Overall minus signs in the mapping are due to the fact that dim6top and SMEFTsim use opposite sign conventions for the definition of covariant derivatives. Although the operator definitions are identical, the relative sign between the L 6 contribution and the corresponding SM coupling is flipped in a few cases. The physics results are identical in both models once this is accounted for. The presence of explicit Yukawa couplings in the Tables is due to the different treatment of flavor symmetry breaking terms.
Wilson coefficients inducing flavor-changing neutral currents can be mapped to parameters in SMEFTsim general, and the corresponding tables are available upon request.
Flavor structure. SMEFT@NLO assumes a flavor symmetry U (2) q × U (3) d × U (2) u in the quark sector and U (1) 3 in the lepton sector, which is the same as in SMEFTsim top and in dim6top, except for the treatment of down quarks.
All fermion masses and Yukawa couplings are neglected, except those of the top quark.
Operators implemented. SMEFT@NLO contains all the operators in classes (1)-(7) and those in class (8) that contain a top quark. Terms that violate the flavor symmetry have been consistently dropped. CP violating terms are omitted.
Input parameters. SMEFT@NLO implements the {m W , m Z , G F } input scheme.
SM loop-generated Higgs couplings. Higgs couplings in the m t → ∞ limit are not implemented, but Higgs-gluon interactions can be fully reproduced at 1-loop in QCD.
Given its flavor structure, SMEFT@NLO can be directly mapped to SMEFTsim in the top, topU3l and general versions. The mapping of Wilson coefficients between SMEFT@NLO and the top, topU3l versions of SMEFTsim is provided in Tables 29,30. The mapping to the general version is provided in Tables 31, 32. In both cases, the first table contains the mapping of parameters with a one-to-one correspondence, while the second contains parameters that require a basis rotation. As for dim6top, the sign convention used in SMEFT@NLO is the opposite compared to SMEFTsim, which leads to some minus signs in the conversion. class

F Validation of the UFO models
The 10 UFO models contained in the SMEFTsim package have been validated following the recommendations of Ref. [44]: the procedure relies on pairwise comparisons between models, based on the values returned for a set of squared amplitudes. Each comparison is performed with the dedicated MadGraph5_aMC@NLO plugin [119]: given a list of 2 → n processes and of points in parameter space, the SM squared amplitude |A SM | 2 , the pure SM-L 6 interference 2 Re A SM A * 6 and the quadratic L 6 contribution |A 6 | 2 are calculated at one random phasespace point for each process and parameter point. The validation is considered successful if the squared amplitudes evaluated with each model pair agree within a permille. Larger discrepancies are ignored if they do not show a consistent pattern across different processes and the squared amplitude is < 10 −16 for both models. Figure 3 illustrates diagrammatically the set of comparisons performed: the top, topU3l and general versions of SMEFTsim have been compared to dim6top (version of May 2020) and SMEFT@NLO (both versions of August 2019 and September 2020, only for models with {m W , m Z , G F } scheme). An internal validation was also carried out, comparing models with different flavor assumptions and same input scheme, and vice versa. The arrows in the figure indicate that, in the comparison, the parameters of the first model were mapped onto those of the latter: the flow generally goes towards more restrictive flavor assumptions.
The validation was performed on the processes listed in Tab. 33, that were chosen so as to probe most effective operators independently. All Wilson coefficients have been included in the comparison, with the exception of those inducing flavor-changing neutral currents.
All fermion masses and Yukawa couplings were retained for internal validation, while only those implemented in dim6top or SMEFT@NLO were included when comparing to these models. CKM mixing has been neglected in all cases. Comparisons between models with different input schemes were done requiring NPshifts=0 in order to filter out the expected discrepancies.
Linearized propagator corrections have been validated with an analogous procedure, using