Standard Model Effective Field Theory up to Mass Dimension 12

We present a complete and non-redundant basis of effective operators for the Standard Model Effective Field Theory up to mass dimension 12 with three generations of fermions. We also include operators coupling to gravity via the Weyl tensor. The results are obtained by implementing the algorithm of Li et al., and provided in the form of ancillary files.


Introduction
Owing to the lack of discoveries of particles beyond the Standard Model (SM), one of the main research directions over the past decade has been Effective Field Theories (EFTs). Given a renormalizable low-energy theory L like the SM, EFTs parametrize possible new heavy physics beyond that theory in terms of operators of mass dimension larger than four. The Wilson coefficients compensate for the surplus mass dimension through inverse powers of a generic new-physics scale Λ which is assumed to be much larger than the energy at which this physics is probed. The operators are composed of the fields of L and obey the same gauge symmetries as L.
In order to uniquely constrain the Wilson coefficients by experiment, a number of possible redundancies among the operators need to be taken into account. On the one hand, these arise because operators which vanish by equations-of-motion (EoMs) can be eliminated through a re-definition of the field variables in the path integral [4,5]. Moreover, operators differing by a total derivative can be identified. Finally, operators which are related by algebraic identities associated with the group structure of the underlying internal or Lorentz symmetry can have dependencies.
The EFT constructed from the SM is usually referred to as Standard Model Effective Field Theory (SMEFT). At mass dimension five, it contains only the lepton-number violating Weinberg operator (up to generation multiplicities). Currently, the non-redundant basis of operators is known up to mass dimension nine [1,2,[6][7][8][9][10][11]. Beyond that, only the number of independent operators that form a basis 1 is currently known, albeit broken up into operators with a specific field content. This number can be obtained from a suitable Hilbert series [12,13] or by explicitly considering the field quantum numbers and permutation symmetries [14]. The fact that it grows roughly exponentially with the mass dimension implies that, beyond a certain order in 1/Λ, it is necessary to give the task of explicitly constructing the basis to a computer.
In fact, in Refs. [1,2], partly building on concepts developed in Ref. [15], a fully algorithmic approach was used to determine the SMEFT basis up to mass dimensions eight and nine. Utilizing various functions of Refs. [16,17], 2 its implementation has been published as a Mathematica package [3], but its application currently appears to be restricted to mass dimensions equal to or less than nine.
An automated approach to constructing EFT bases is also desirable from the point of view that as-of-yet undiscovered light particles might still exist which could couple to the SM via effective operators. Examples for this are sterile neutrinos [18] or axion-like particles [19]. In addition, one should expect that the low-energy limit of some theory that includes both the SM interactions as well as gravity will be described by an effective Lagrangian which extends SMEFT by gravitational fields, resulting in General Relativity⊕SMEFT (GRSMEFT) [20].
Finally, an efficient automated approach to constructing EFTs will allow one to study operator bases at higher mass dimension, and thus operators with a richer structure. For example, it is only starting at mass dimension ten that B − L can be violated at ∆(B − L) = 4, where B and L are the baryon and lepton number [21,22]. Furthermore, studying higher-dimensional operators, or specific sub-sets thereof, may give insight on a possible all-order structure of such operators; see Ref. [23], for example.
In this paper, we report on a re-implementation of the algorithm of Ref. [1][2][3] into SageMath [24] which is a free open-source mathematics software system, and the symbolic manipulation system FORM [25,26], both licensed under the GNU General Public License (GPL) [27]. Using this code, named AutoEFT, we have evaluated the (on-shell) SMEFT basis as well as the GRSMEFT basis up to mass dimension 12.
A description of AutoEFT, its usage, and the program itself can be found in Ref. [28]. The main purpose of the present paper is to make the SMEFT and GRSMEFT operators of mass dimensions 10, 11, and 12 accessible to the public. 3 However, the number of operators is too large for listing them in this paper. Instead, we provide them in electronic form, using a notation which is both rather compact, but also straightforward to interpret.
The remainder of the paper is structured as follows. In Section 2 we briefly review the algorithm of Refs. [1][2][3] on which the results of this paper are based. This includes the treatment of scalar fields, chiral fermions, vector bosons, as well as gravitons. The majority of Section 3 is devoted to describing the format in which we encode the operators. The results themselves are provided in the form of ancillary files which accompany this paper. Section 4 contains our conclusions and an outlook.

Constructing EFT Operator Bases
In this section, the concepts introduced in Refs. [1,2] for the systematic construction of an EFT operator basis are briefly reviewed. Contrary to the formal representation in Refs. [1,2], only the central ideas are illustrated.

Field Representations
In a standard construction of SMEFT operators, as it was done for the Warsaw basis at mass dimension six [7], for example, the fundamental building blocks are taken to be complex scalars ϕ, Dirac spinors Ψ L/R , field strength tensors F µν , and the derivative 4 D µ . It turns out that for the operator construction at higher mass dimensions, it is more convenient to characterize the fields and derivatives by the irreducible representation (j l , j r ) in which they transform under the Lorentz group SL(2, C) ≃ SU(2) l × SU(2) r . This corresponds to adopting a chiral basis, where the SM fields are scalars, two-component Weyl spinors, or chiral field-strength tensors: ϕ ∈(0, 0) , Here, α, β, . . . andα,β, . . . denote indices of the fundamental representation of SU(2) l and SU(2) r , respectively. Note that each of the fields in relation (1) has a unique helicity value The derivative transforms non-trivially under both SU(2) l and SU(2) r , Thus, the derivative of a field has the same helicity as the field itself. This definition of the fields is equivalent to the conventional representation of the SM fields. The translation between the two notations is given by: where ψ C denotes a charge conjugated spinor, and the σ-matrices are given by Here, I is the 2 × 2 identity matrix and ⃗ σ = σ 1 , σ 2 , σ 3 denotes the Pauli matrices.

Construction of the Operators
The operators can be classified into families 5 , characterized by the tuple where n Φ equals the number of fields with helicity h Φ , and n D is the number of derivatives. It is further useful to define which correspond to the sum of j l/r of each field and derivative as defined in relation (1). Lorentz invariance and a given mass dimension d of the operators put constraints on the n i such as N + n l + n r = d , where N is the total number of fields, and thus restrict the set of allowed families.
Since Hermitian conjugation exchanges the two SU(2) representations of the Lorentz group, taking the conjugate of the operators of a particular family generates the conjugate family Hence, one can identify real families that satisfy (n F L , n ψ , n ϕ , n ψ † , n F R ; n D ) † = (n F L , n ψ , n ϕ , n ψ † , n F R ; n D ) .
5 In Ref. [1] the term subclass is used instead.
All operators in a real family are either Hermitian, or their conjugate operator is part of the same family. The remaining families are all complex, such that any operator features a distinct Hermitian conjugate version, which is part of the conjugate family.
All operators in a specific family can be further characterized by their type, which corresponds to a specific multiset of fields from a given model. Types of the same field content are identified by ordering the fields by increasing helicity and sorting fields of the same helicity alphanumerically. For example, the dimension-five Weinberg operator, consisting of two lepton doublets and two Higgs doublets, belongs to the family (0, 2, 2, 0, 0; 0) and the type L 2 H 2 .

Lorentz Structure
In this section, only the Lorentz structure of an operator will be considered, while all internal symmetry and generation indices of the fields will be neglected. All operators of a particular family can thus be identified for the purpose of this discussion.
Since any operator O of the EFT will be constructed from the objects of relations (1) and (3), it will be of the form where ) are multi-indices with m i = n i + 2j l,i andm i = n i + 2j r,i , where n i is the number of derivatives acting on the field Φ i , and (j l,i , j r,i ) defines its representation according to relation (1). The indices α are to be associated with the derivatives acting on Φ i , while the remaining indices are associated with the field itself.

Redundancies
One of the most challenging aspects when constructing EFTs is the elimination of redundancies, i.e. operators that are related to other operators by certain identities. As stated in Ref. [1], for the chiral convention of the fields, the structure of T Lorentz must be a polynomial in ϵ αβ and ϵαβ, where ϵ is totally anti-symmetric, and ϵ 12 = ϵ 21 = ϵ12 = ϵ21 = 1. This already eliminates all redundancies from Fierz identities (up to Schouten identities). Furthermore, redundancies due to EoMs or the identity i[D µ , D ν ] = F µν , where the sum is over all gauge groups, can be eliminated by substituting in Eq. (11), where (α i ) = (α ) sym denote totally symmetric multi-indices. In the following, we refer to the combined object on the r.h.s. of relation (12) as building block.
The remaining redundancies to be considered are due to integration-by-parts (IbP) and Schouten identities. These can be avoided by introducing an auxiliary SU(N ) group, where N equals the total number of fields (cf. Eq. (8)). Under this group, the (dotted) spinor indices of T Lorentz transform as the (anti-)fundamental representation, i.e.
with α i ,α i defined in Eq. (11), and U (U † ) group elements of SU (N ) in the (anti-)fundamental representation. For a given family, T Lorentz contains n l ϵ-tensors with undotted and n r ϵ-tensors with dotted indices. Consequently, T Lorentz must transform in the representation under the auxiliary SU (N ) group. According to the Littlewood-Richardson (LR) rule, the two factors can be decomposed as where dropping all diagrams represented by the dots eliminates the Schouten identities.
plus diagrams that contain at least one column with N − 1 entries which will be discussed below. A basis of tensors that transform under this irreducible representation can be constructed from Semi-Standard Young Tableaux (SSYTx) of shape λ and content 6 µ = [n r − 2h 1 , . . . , n r − 2h N ] (cf. Eqs. (2), (7), and (8)) as follows: Replace the first n r columns through the SU(N ) relation (no summation implied) 6 A Young tableau with content (or weight) µ = [i, j, k, . . . ] has i entries of the number 1, j entries of the number 2, k entries of the number 3, and so on. This requirement ensures that the resulting tensors have the correct indices corresponding to the fields specified in the family. A Young tableau is called semi-standard if the entries weakly increase along each row and strictly increase down each column.
where E denotes the N -dimensional Levi-Civita symbol, and a, b, . . . , x, y, z is some permutation of 1, 2, . . . , N . Subsequently identify each column with an ϵ-tensor according to Since the set of all SSYTx forms a basis in the vector space of a particular representation, performing these steps for every such SSYT leads to a set of Lorentz tensors T Lorentz which is non-redundant. To see that it is also complete, consider a Young diagram other than Eq. (16) in the decomposition of Eq. (15). As pointed out above, it contains at least one column of length N − 1. Such columns correspond to tensors proportional to N j=1 ϵ α i α j ϵα jαk . The index pair (α j ,α j ) implies a derivative acting on the j-th field, so that the sum corresponds to a total derivative.

Internal Symmetries
After identifying all independent Lorentz structures, one can continue in a similar manner with the remaining symmetries of the underlying low-energy theory. In the following, the expression internal symmetry refers to a global or a local U(1) or SU(n) symmetry. In analogy to Section 2.3, all Lorentz and generation indices will be neglected in this section, and only the transformation properties of the fields under the internal symmetry group are considered.
Concerning Abelian internal symmetries, it is required that the total charge of an operator under this symmetry vanishes. In the SM, for example, this is achieved by only considering combinations of fields such that the sum of their hypercharges equals zero, hence forming a U(1) invariant operator. However, one may also allow for the breaking of a certain U(1) symmetry, specified by the amount the associated charge of the operators may deviate from zero. This can be useful, for example, if one is only interested in operators that violate baryon number conservation up to a certain degree ∆B.
For each non-Abelian symmetry, the modified LR rule [1] is used to construct all sets of independent tensors. To apply this method, the fields of the low-energy theory must be characterized in terms of fundamental SU(n) indices only. Consider, for example, the non-Abelian part of the SM, i.e., SU(3) ⊗ SU(2). In terms of Young diagrams, the fundamental representations are identified as 3 = for SU(3), and 2 = for SU (2). For the SM fields that transform under the fundamental representations, the symmetry of their indices can be identified with the states where Q and L denote a left-handed quark and lepton doublet, respectively; q the righthanded quark fields; and H the Higgs doublet. Here, a, b, c, . . . and i, j, k, . . . denote fundamental indices of SU(3) and SU(2), respectively.
The fields that do not transform under the fundamental representation of the internal symmetry group can nevertheless be written in terms of quantities with fundamental indices only. In particular, for the SM, where A and I are adjoint indices of SU(3) and SU (2). The λ A are the Gell-Mann matrices for SU (3), and the τ I are the SU(2) Pauli matrices.
Considering the Young diagrams for the adjoint representation of SU(3) and SU(2), given by 8 = and 3 = , one can identify the symmetry of the fundamental symmetry group indices of G abc and W ij with the states meaning that G abc is symmetrized in a ↔ b and subsequently anti-symmetrized in a ↔ c, while W ij is totally symmetric in i ↔ j. Equivalently, the anti-fundamental representations of SU(3) and SU(2) are given by3 = = and2 = = , respectively. The symmetry of the fundamental symmetry group indices can be identified with where the only non-trivial symmetry is given by Q † ab i and q † ab which are anti-symmetric under the exchange a ↔ b.
From the discussion above, it is clear that one can write a generic operator as where I i is a multi-index containing SU(n) fundamental indices only. The independent set of tensors T SU(n) is constructed by combining the Young tableaux of each field in the operator according to the modified LR rule, and subsequently identifying each column of the resulting tableau with an SU(n) ϵ-tensor.

Permutation Symmetries
It is important to note that, up to this point, the algorithm considers all fields that occur in an operator as distinct. But if an operator contains several copies of the same field, new redundancies can occur. This happens because in expressions like Eqs. (11) and (23), the sum over indices is not explicitly carried out. Therefore, the Lorentz and internal symmetry groups can no longer be treated independently as soon as the operator contains identical fields.
As a generalization of this, a theory may contain several copies of fields that transform identically under the Lorentz and internal symmetry groups and are thus indistinguishable for our considerations. In the spirit of the SM, we will refer to these copies as generations 7 .
Owing to the Lorentz and internal symmetry of the operator, not all combinations of generations are independent of one another, in general.
Both of these problems can be treated in the same way when introducing generation indices for all fields, even for those that occur only in a single generation. The expression repeated fields denotes the product of fields which at most differ by their generation index. The general strategy is to decompose any operator into a sum of terms with specific permutation symmetry λ of the generation indices for repeated fields [15]. 8 Following the philosophy of Ref. [1], the permutation symmetry is not inscribed on the operator by symmetrizing the generation indices themselves, but at the level of the Lorentz and internal symmetry tensors they are multiplied with. Consider a specific type, i.e. the set of operators which contain a certain set of fields. Up to now, the algorithm has generated a set of tensors T Lorentz = {T Lorentz 1 , . . . , T Lorentz l } for the Lorentz symmetry, and } for each internal symmetry; see Sections 2.3 and 2.5. These tensors are now combined in such a way that they reflect the permutation symmetries, labeled by λ: where the K are obtained from the plethysm technique and inner-product decomposition (for details, see Ref. [1]). The redundancies due to the permutation symmetries are reflected in the fact that K is an n × m matrix with n ≤ m = l · l 1 · · · l k in general. The set of independent operators is then given by contracting the fields with T λ j for each j = 1, . . . , n and each permutation symmetry λ, and enumerating the generation indices according to the set of SSYTx. Explicit examples will be given below.

Including Gravity
The algorithm described above can be extended to theories which include fields with higher spin [3,20,29]. In particular, gravitational interactions can be taken into account by considering the Weyl tensor C µνρσ as an additional building block in the EFT construction [20]: where R µνρσ denotes the Riemann tensor, R µν = R λ µλν the Ricci tensor, R = R µ µ the Ricci scalar, and g µν the metric tensor, and the indices between square brackets are to be anti-symmetrized. In analogy to Section 2.1, we write the Weyl tensor in terms of leftand right-handed components that transform under irreducible representations (j l , j r ) of the Lorentz group: where Denoting by n C L/R the number of left-/right-handed Weyl tensors in the operators, we extend the families (6) to While all previously considered fields satisfy the relation with [·] denoting the mass dimension in four space-time dimensions and h Φ the helicity, this relation is violated for the Weyl tensor, which satisfies instead. Therefore, the algorithm needs to account for the actual mass dimension of the Weyl tensor. For example, instead of Eq. (8), we get where n GR ≡ n C L + n C R can take values from 0, . . . , min(d/2, N ), and the definitions in Eq. (7) are modified, such that An example for an operator type of dimension-12 in GRSMEFT will be given below.

Application to SMEFT and GRSMEFT
According to the algorithm described in Section 2, we construct on-shell operator bases for SMEFT and GRSMEFT up to mass dimension 12, using our implementation AutoEFT.
To be consistent with the existing literature, we adopt the all-left chirality convention for the SM fields. The EFT field content is given in Table 1. These fields are related to the conventional notation by Eqs. (4), (20), (26), and (27). For example, the SM Dirac spinors are given by where all internal symmetry and generation indices are suppressed.
For a given mass dimension, the operators are grouped into families (6), and in each family, the operators are classified by their type (i.e., the field content). For each of these types, an output file is generated in YAML format. 9 It contains the relevant information to construct all operators belonging to this type. We found this way of presenting the results a reasonable compromise which allows one to encode the large number of operators in a rather compact form without leaving too much calculational effort to the user.
In the following, four representative cases are displayed to show how the output files can be interpreted in terms of Section 2.

No Repeated Fields
As a first example, let us consider the dimension-10 operators of type encoded in the file operators/5/1FL_2psi_1phi_1FR_2D/1WL_1Q_1uC_1H_1WL+_2D.yml included in smeft_10.tar.xz. They contain a weak gauge field and its conjugate (1WL, 1WL+), a quark doublet (1Q), a charge-conjugate up-type quark singlet (1uC), a Higgs field (1H), and two derivatives (2D). This also illustrates the naming scheme of the output files of AutoEFT. The content of this file is displayed in Listing 1. We deliberately include redundant information in the output files in order to facilitate their interpretation and to allow for consistency checks when constructing the explicit operators.  Table 1: SM field content in the all-left chiral convention (cf. Section 2.1). Spinor indices are denoted by α and β, while a, b, c and i, j denote fundamental indices of the gauge groups SU(3) and SU(2), respectively. Each fermion carries an additional generation index g. The Young tableaux of the SU(n) groups shown in columns three and four have been introduced in Section 2.5. The symbol denotes the characters that represent the corresponding field in the ancillary files. Hermitian conjugated fields are denoted by a trailing "+". The covariant derivative is represented by the character "D". The file is structured by certain keywords which we explain in the following: version: The version of AutoEFT which was used to produce the output file.
type: A list 10 of two elements. The first element (line 4) specifies the multiplicities of 10 In YAML, this is called a sequence. Its elements are marked by the leading dashes in lines 4 and 5 of Listing 1.
the fields and derivatives in the operator. It is equivalent to the specification in Eq. (34). The second element (line 5) states that the operator type is complex, (i.e. the operators are not Hermitian). The Hermitian conjugate type is contained in a different file. In the present case, this would be 1WL_1H+_1Q+_1uC+_1WL+_2D.yml. For Hermitian operators, the second entry of type is real.
generations: Provides the number of generations for each field. In this case, it specifies that there is only a single generation of WL, WL+, and H, but three generations of quarks Q and uC.
n_terms: The total number of operators with independent Lorentz and SU(n) index contractions. It does not take into account the number of generations though. In Listing 1, there are five independent ways of contracting the Lorentz indices of the operators while there is only one option for SU(3), and two for SU (2), so the number of terms n_terms is 5 · 1 · 2 = 10.
n_operators: The total number of independent operators, taking into account the independent values the generation indices can take. For Eq. (34), there are 3 · 3 = 9 independent combinations of the Q and u C generation indices for each of the ten terms, hence n_operators is 10 · 9 = 90. The indices are denoted by <i>_<j>, where <i> is the position of the field that carries this index, and <j> is the position of the index on the field. Per invariant contraction, each index appears exactly twice, and summation is implied. For the Lorentz group, <i>_<j>~denotes dotted indices. Furthermore, the indices are associated with the building blocks of relation (12), rather than with the fields. Note that the dotted and undotted indices of the building blocks are understood to be (separately) symmetrized (cf. Section 2.4).
The symbol eps denotes the ϵ-tensor with eps(1,2)=eps(2~,1~)=1 for the Lorentz group, and eps(1,2,...,n)=1 for any internal SU(n) group. All indices not associated with the symmetry group in question are suppressed on the fields. 11 permutation_symmetries: The list of permutation symmetries according to Section 2.6. Since the operator type of Eq. (34) does not involve repeated fields, this entry (lines 22-36) is redundant and is included only for consistency. Details will be discussed in Section 3.2 for a non-trivial example.
In Listing 1, the invariant contractions are given by (cf. Eqs. (11) and (23)) with Q NRF from Eq. (34). The operation T • Q implies that all indices other than those of T are suppressed in Q. Since only contractions with the Lorentz tensors determine the positions of the derivatives, we omit the latter in the contractions with tensors of the internal symmetries.
The complete set of independent operators T · Q NRF is thus given by all possible combi-nations of the Lorentz and SU(n) tensors: Three examples for the ten independent terms (cf. line 7 in Listing 1) are given by Up to this point, we have suppressed the generation indices of the quarks. Since quark and anti-quark transform differently under the symmetry groups, we can simply attach a generation index to each of the two quark fields in Eq. (34).

Repeated Fields
The operator Eq. (34) treated in Section 3.1 is special in the sense that it does not contain repeated fields, i.e. all fields in this type transform differently under the symmetry group of the theory. As mentioned in Section 2.6, if repeated fields are present, new redundancies can arise. The output file thus requires information on the permutation symmetry. As an example, we consider the dimension-11 operators of type contained in the file operators/8/6psi_2phi/2L_1dC_1eC_2uC_2H.yml that is included in smeft_11.tar.xz. Its content is displayed in Listing 2. Listing 2: Content of file 2L_1dC_1eC_2uC_2H.yml, which encodes the operators of Eq. (38).
The meaning of the first 20 lines in this file was explained in the previous section. If all fields in Eq. (38) were different, one would arrive at 10 terms, according to the 5 · 1 · 2 = 10 independent Lorentz and SU(n) tensors: Attaching generation indices to the fields, the number of operators would be 10·3 6 = 7290, because there are six fields with three generations each. This, however, neglects the permutation symmetries arising from the repeated fields. In this example, there are three sets of repeated fields, each of which occurs twice, as can be seen from the first element of type (line 4). This reduces the actual number of terms and operators to those given in lines 7 and 8. As described in Section 2.6, this is because, on the one hand, the permutation symmetries induce dependencies between the terms in Eq. (39), reducing the number of independent terms. On the other hand, a particular permutation symmetry restricts the number of independent combinations of generation indices in the field monomials.
The fillings are the generation indices, i.e. i, j, k, l, m, n ∈ { 1, 2, 3 }, and r = s = 1, because there are three generations of leptons and quarks, and only one of Higgs bosons.
For this permutation symmetry, only the contractions with two combinations of the ten tensors in Eq. (39) are independent. They are given by (cf. Eq. (24)) where S (ng) λ Φ is the set of independent generation indices for the repeated field Φ with permutation symmetry λ Φ and n g generations. The generation indices of fields that appear only once are still independent and are therefore not restricted further, i.e. k, l ∈ { 1, 2, 3 }. In total, out of the 3 6 combinations of the generation indices, only 3 · 6 · 1 · 3 2 = 162 need to be considered for this particular representation of the permutation symmetry. There are thus 2 · (3 · 6 · 1 · 3 2 ) = 324 independent operators of this type, in agreement with line 25.

SMEFT at Mass Dimension 12
The complete basis of operators up to mass dimension 12 for SMEFT is provided, in the format described in Sections 3.1 and 3.2, in the ancillary files together with this paper. It took a few seconds to generate the operator basis at mass dimensions 5, 6, and 7, a few minutes for mass dimension 8 and 9, and a few hours for mass dimension 10 and 11. At mass dimension 12, the largest amount of time was spent by AutoEFT on incorporating the permutation symmetry of the operators with six gluonic field strength tensors, which takes of the order of 10 4 CPU hours. For example, for the type G 6 L , the 2175 contractions of the field strength tensors with the 15 independent Lorentz and 145 independent SU(3) tensors is reduced to only eight independent operators. Their explicit expressions are too long to display them in this paper. They are encoded in the file operators/6/6FL/6GL.yml which is about 140 KB in size. Instead, as an example of a dimension-12 operator, we display the following type of eight-fermion operators in Listing 3: contained in the file operators/8/4psi_4psi+/3L_1eC_1L+_2dC+_1uC+.yml that is included in smeft_12.tar.xz.
The corresponding sets of independent generation indices are thus given by containing 1, 8, and 10 elements, respectively. The permutation symmetries of the generation indices m, n of the two down-type quarks are given by and thus the corresponding sets of generation indices are S 1,1 with 6 and 3 elements, respectively (see Eq. (43)).
In Listing 3, all combinations of the symmetries Eqs. (48) and (50) are present, and each combination is given by exactly one term. Therefore, including the 3 3 generation multiplicities of e C , L † , and u † C , the total number of independent operators is given by in agreement with line 8 of Listing 3.

GRSMEFT at Mass Dimension 12
The complete basis of operators up to mass dimension 12 for GRSMEFT is provided, in the format described in Sections 3.1 and 3.2, in the ancillary files together with this paper.
As an example, we present in Listing 4 the operators of type  Listing 4: Content of the file 1CL_2L_1eC_2dC+_1uC+_1D.yml, which encodes the operators of the type (52). Table 1, the symbol CL denotes the left-handed Weyl tensor C L defined in Eq. (27). From the examples above, the reader should by now be able to reconstruct all operators from this listing.

Ancillary Files
The complete set of SMEFT and GRSMEFT operators up to mass dimension 12 with three generations of fermions are published as ancillary files together with this paper. In addition, we provide a table with the numbers of types, terms, and operators contained in a given family (see Tables 2 and 3 as an example) as well as the associated Hilbert series for each mass dimension. This information can be extracted solely from the attached operator files.
The operators are provided in the format described in Sections 3.1 and 3.2. The files are included in the archives smeft_<ndim>.tar.xz and grsmeft_<ndim>.tar.xz, where <ndim> is the mass dimension. 14 They are collected in directories as For SMEFT up to mass dimension 12, the size of the files amounts to about 442 MB. This number seems to grow roughly with the number of operators, from which we infer that it will reach 1 TB at mass dimension 20, and 1 PB at mass dimension 26.
The Hilbert series provides the number of operators per type and thus constitutes a helpful check on our results. We have compared our numbers for SMEFT and GRSMEFT up to dimension 12 to the results for the Hilbert series as obtained by ECO [13] and found full agreement. We have further performed a number of consistency checks on our results, for example that no two K matrices for a particular operator type contain linearly dependent rows, etc. An immediate comparison of the results at lower mass dimension to the existing literature is highly non-trivial though, due to different representations of the final basis [1,2,10]. 15

Conclusions
We have evaluated the SMEFT and GRSMEFT operator basis with three generations of fermions up to mass dimension 12. They were obtained by re-implementing the algorithm of Refs. [1][2][3] into a non-commercial software. Aside from the results obtained in this paper, we also confirm the completeness of the algorithm up to mass dimension 12. The operators are provided in the form of searchable and compact ancillary files.
In future work, we plan to extend the capabilities of AutoEFT in various respects, for example to keep operators that vanish by equations-of-motion [31], as they are needed for the renormalization of the operators. Furthermore, we plan to implement general basis transformations in the spirit of Ref. [32,33], which would allow the fully automated matching of EFTs to an ultraviolet complete theory, for example by combining AutoEFT with methods like UOLEA [34][35][36][37] (see also Ref. [38]).