Low Energy Effective Field Theory Operator Basis at $d \le 9$

We obtain the complete operator bases at mass dimensions 5, 6, 7, 8, 9 for the low energy effective field theory (LEFT), which parametrize various physics effects between the QCD scale and the electroweak scale. The independence of the operator basis regarding the equation of motion, integration by parts and flavor relations, is guaranteed by our algorithm, whose validity for the LEFT with massive fermions involved is proved by a generalization of the amplitude-operator correspondence. At dimension 8 and 9, we list the 35058 (756) and 704584 (3686) operators for three (one) generations of fermions categorized by their baryon and lepton number violations $(\Delta B, \Delta L)$, as these operators are of most phenomenological relevance.

1 Introduction The standard model (SM) has been proved to be a success according to current experiment data at the LHC, yet several unsolved basic problems about our universe motivate both theorists and experimentalists to search for new physics beyond the SM, which have yielded a null result up to Λ NP ∼ TeV scale. Under the circumstances, the effective field theory (EFT) provides a model-independent way to parameterize new physics above Λ NP using interactions involving higher dimensional operators at energy scale < Λ NP .
While the physics above the electroweak scale Λ EW is described by the standard model effective field theory (SMEFT) [1][2][3][4][5][6][7], we also need the low-energy effective field theory (LEFT) below Λ EW , which has gauge symmetry SU (3) C × U (1) EM and all the SM fermions but the top quark. Part of the LEFT operators can be derived from the SM by integrating out the massive vectors W ± , Z, the top quark t and the Higgs boson h. For example, the historical Fermi interactions generated by integrating out the W ± boson is essentially a part of the LEFT operators at the dimension-6 level, and had been applied to various processes like the β decay, the muon decay, and the lepton flavor violation [8][9][10][11].
Furthermore, flavor physics such as B-physics and K-physics utilizes these LEFT operators to parameterize new physics contributions in the B and K meson decay processes, e.g. [12,13]. However, generic LEFT operators are only subject to Lorentz invariance and gauge symmetry, which is far more rich than those derived in the SM.
Therefore, it is necessary to write down the complete set of the higher dimensional LEFT operators. Recently [26][27][28], the LEFT operators have been written up to dimension 7.
The traditional way to obtain the operator basis is to consider all possible operators, and then eliminate the operator redundancies caused by equation of motion (EOM), integration by parts (IBP), repeated flavor structure and Fierz identities, e.g. [4], which is quite involved when the number of possible structures blows up at higher dimensions. So far the traditional way has been applied to dimension 5, 6 in Ref. [26] and dimension 7 in Ref. [28], which present the result in the form of a set of operators with complicated flavor relations. On the other hand, the systematical procedure proposed in Ref. [1,2] is capable of generating independent structures of operators automatically, and presents the flavor-specified operators without extra redundancies. However, the procedure makes use of the amplitude-operator correspondence [29], and was designed only for massless amplitudes, which seems to be not compatible with LEFT. In this work, we present the massive amplitudes in the form introduced in Ref. [30,31], which have a one-to-one correspondence to a set of massless amplitudes when the spin S ≤ 1/2, while the correspondence between massive amplitudes and operators remains valid. Hence we argue that the procedure should work as well for EFTs with massive fermions, such as LEFT.
We sketch the procedure as follows. We first use the amplitude operator correspondence to convert the problem of finding the operator basis into finding the amplitude basis, which is further divided into separate tasks as to find: 1.
a basis of Lorentz factors expressed as functions of spinor-helicity variables; 2. a basis of gauge factors expressed in terms of invariant group tensors. The basis of Lorentz factors is generated via the method in [2,32], translated from semi-standard Young tableau (SSYT), which after converting to the operators are automatically free from the IBP and EOM redundancy. Meanwhile, using the Littlewood-Richardson rules, we construct a set of singlet Young tableau of the gauge group indices from the constituting particles, which induces the basis of gauge group factors. A complete and independent flavor-blind operator basis are obtained by direct product of these two factors. Afterwards, we symmetrize the flavor indices of the repeated fields to obtain the so-called p-basis operators, where the ones with permutation symmetries allowed by the flavor number make up our final results as an independent basis of flavor-specified operators.
As a key feature, our form of operators include a flavor Young tableau, which indicates the independent components in the Wilson coefficient tensor, hence no more flavor relations are necessary.
In this work, we apply the procedure above to the LEFT. The dimension 5, 6, 7 operator bases are reproduced with new form of flavor structures as indicated above, compared to the previous literature [26,28]. More importantly, we also generate the dimension 8, 9 operator bases for the first time, which constitutes the main part of this work. We adopt the chiral fermion notation in order to be consistent with other literature [26], as well as to keep the correspondence with the spinor helicity amplitudes explicit.
The paper is organized as follows. In section 2, We introduce the notation and discuss the validity of amplitudeoperator correspondence in the massive spinor case. In particular, we describe the main ideas of how to deal with the amplitude bases, group structures and permutation symmetries in section 2.3. In the section 3, We list the complete operator bases in order of dimension from 5 to 9. Our conclusion is presented in section 4.

General Framework
In this section, we describe the framework of writing down the operator basis of the LEFT. We start by reviewing the massive spinor helicity formalism [30,31], and then illustrate a one to one correspondence between the massless and massive amplitude basis for particles with spin S ≤ 1/2. Together with the massive version of the amplitude operator correspondence that we formulate later, we extend the framework for massless EFTs introduced in [1,2] to include massive fermions, and summarize the procedure of enumerating the complete and independent operator basis of LEFT.

Amplitude Basis Including Massive Fermions
Let us first review the massive spinor helicity formalism developed recently in [31], and build the connection between massive spinor and amplitude basis in this subsection. In spinor formalism, a momentum vector can always be decomposed into two spinors, i.e. the spinor helicity variables, where α,α are the SU (2) L × SU (2) R Lorentz indices, I is the little group indices, which is SU (2) for massive particle, and U (1) for massless particle (I can be omitted). Note that det p αα = p µ p µ = m 2 . Therefore p αα has rank 1 in massless case and rank 2 for massive particles. These indices are raised and lowered by the 2-index Levi-Civita symbols defined as, ǫ 12 = −ǫ 21 = ǫ 21 = −ǫ 12 = 1, λ αI = ǫ αβ λ I β , λ αI = ǫ IJ λ J α ,λα I = ǫαβλ Iβ ,λα I = ǫ IJλ J α . (2.2) We normalize the spinors as λ Iα λ J α = −mǫ IJ ,λ IαλJα = mǫ IJ , λ I α λ βI = mǫ αβ ,λ Iαλβ I = mǫαβ (2.3) In the standard polar coordinates where p µ = (E, p sin θ cos φ, p sin θ sin φ, p cos θ), the general solutions of the spinors are the following, where ζ ± satisfy ǫ IJ ζ −I ζ +J = 1 and abbreviation c θ ≡ cos θ 2 , s θ ≡ sin θ 2 e iφ . ζ ± are determined by the direction along which we take the spin components, for instance ζ + = (1, 0), ζ − = (0, 1) mean that we take the spin components along the momentum direction. Because we work in the in-coming convention in this paper, the out-going physical momenta are thus written as in-coming momentum with E < 0. For such momentum, we define λ I α (−p) = −λ I α (p) andλ Iα (−p) =λ Iα (p). In the massless limit, E − p, η,η → 0, the SU (2) little group reduces to the transverse U (1) rotation, while I can be omitted. Only two of spinors in eq. (2.4,2.5) are still valid, The resulting Lorentz scalar can be represented by angle spinor brackets and square spinor brackets By this rule, we have a well-defined dimensionality of the amplitudes as the highest power of spinor brackets in it. Under this definition, the complete amplitude basis at a certain dimension is defined modulo lower dimensional amplitudes.
It will be clear shortly that the EOM redundancy of the corresponding operators is eliminated through this removal.
Also due to the total symmetry of the spinor indices, the r pairs of little group indices in eq. (2.10) must be contracted between λ I andλ I , because otherwise the amplitude would vanish. Therefore the integer r stands for the number of spinor pairs (λ (I) ,λ (I) ) in both cases that constitute factors of momentum due to eq. (2.1). The 2S + 1 choices of n represent independent forms 2 of contribution the spin-S massive particle could have.
It is easy to observe the correspondence between the massless factor and the massive factor when S = |h| and n = 0, 2S, which maps momenta to momenta, and the rest of the massless spinors to massive spinors with totally symmetric little group indices 3 . The mapping covers all the possible massive amplitudes only when S ≤ 1 2 , in that n = 0, 2S are the only two choices. Therefore, we have the explicit form of the mapping as For higher spin massive particles, the map could not be established, and our framework would fail. However, it is already enough for us to enumerate the amplitude basis involving massive scalars and fermions by enumerating the massless ones and then simply adding the little group indices according to eq. (2.12).

Amplitude-operator correspondence including massive fermions
With the massive local amplitude basis in hand, we still need the massive version of the amplitude operator correspondence to finally obtain the operator basis. First we convert the Lorentz indices µ in the operators into the spinor indices α andα and we have the following definitions of various notations used in constructing operator basis: 14) where ξ, χ, ζ are the Weyl fermions, Ψ and Ψ M denote the Dirac and Majorana fermion respectively, F L/R = 1 2 (F ±iF ) are the chiral basis of the gauge bosons. Note that in terms of Weyl components, we need a pair of them to represent the independent chiral components of the Dirac fermion, but only one for a Majorana fermion. Nevertheless, all the Weyl degrees of freedom share the same form of spinor wave functions u I = λ I α λα I ,ū I = (−λ α I ,λα I ), v I = −λ αĨ λα I ,v I = λ αI ,λ Iα , (2.16) 2 It is true that λ I andλ I are not independent due to the Dirac equations. However, given the standard form in eq. (2.10), by removing the terms with factor of masses, the amplitude with various n are indeed independent. 3 The mapping is referred to as the "first reduction" in [33], which is claimed to be valid only for the "maximal helicity categories", namely n = 0, 2S. Our formulation is from a different perspective, which better fits in our framework.
which solve the Dirac equations (p / − m)u I = 0,ū I (p / − m) = 0,ū I u J = 2mδ J I , u Iū I = p / + m, (2.17) (p / + m)v I = 0,v I (p / + m) = 0,v I v J = 2mδ I J , v Iv I = p / − m. (2.18) We focus on u I andv I associated with incoming fermions and anti-fermions, since all momenta are incoming in our convention. The amplitudes generated by fermion bilinears are given by the usual Feynman rules. 20) Therefore the amplitude operator correspondence between spinor variables with free little group indices and fourcomponent spinor fields can be read from the second and the last expressions in above equations: The pair of spinor helicity variables λ J iλ iJ with contracted little group indices can be translated into derivative acting on particle i yielding following correspondences: where for completeness we also include the massless case denoted by spinor variables without I or J indices. One may question about do the presence of the covariant derivatives generate other local amplitudes with more gauge bosons, which exist in ordinary Feynman rules. However, these vertices are not gauge invariant, and the final gauge invariant amplitudes with contributions from these operators are non-local, and our amplitude operator correspondence applies to local amplitudes only.
Let us take a closer look at the derivatives in eq. (2.24) and (2.25). The total symmetries among the spinor indices in amplitudes are very clear. Otherwise, the resulting amplitude must vanish or reduce the dimension due to the on-shell condition λ I i[α λ J iβ] = m i ǫ IJ ǫ αβ . Therefore the derivatives acting on each fields in the operators also take the totally symmetric spinor indices. It is easy to see that any pair of anti-symmetric spinor indices if associated with derivatives can be always converted to other types or reduced dimesion with EOM and the relation i[D µ , D ν ] = F µν , where we are supposed to have obtained a complete basis: (2.26) Therefore, the building block in constructing the operator basis is in the following form: ∈ (j l + r i − |h i |, j r + r i − |h i |) (2.27) where the powers of the indices α i andα i indicate that they are already totally symmetrized, thus the whole object transforms as (j l + r i − |h i |2, j r + r i − |h i |) under sl(2, C) = su(2) l × su(2) r , given that Φ transforms as (j l , j r ).
The IBP redundancy of operators is solved by the manifest momentum conservation in the amplitude, which can be illustrated by the following example: corresponds to the operator equivalence where terms that convert to other type steming from 1 I 1 J = m 1 ǫ IJ s by EOM are omitted. Hence, taking momentum conservation into account, the amplitude basis corresponds to an IBP non-redundant basis of operators.
where B and T determine the Lorentz structure and gauge structure of the operators respectively, and our goal to find complete and independent operator basis is equivalent to find independent T 's and B's given the type of the operators.
Note that we have suppressed the flavor indices on the right hand side of eq. (2.28) for the moment , and thus call them as "flavor-blind" operators, where we effectively treat all the particles as distinguishable ones, the redundancy related to the repeated fields or equivalently constraints from spin-statistics among the identical particles are tackled in the next section.

The Operator Basis
Having established the correspondence, the next step is to construct concrete amplitude basis satisfying momentum conservation and spin-statistics constraint. Such amplitude basis could be translated into operator basis following eq. (2.28).
In this subsection, we summarize the method of constructing the amplitude basis, which we elaborate in Ref. [1,2].
In general, the Lorentz factor of the amplitude, determined only by the helicities of the constituting particles, can be expressed as where n andñ denote the numbers of λ andλ pairs, as shown in eq. (2.29).
To find the complete and independent amplitude basis, we use the method introduced by Ref. [32] and further developed by Ref. [1]. It is proved in Ref. [32] that the amplitudes modulo total momenta form irreducible representations of SU (N ) group are denoted by the primary Young diagrams,    2.31) where N is the number of particles in the amplitudes. The base vectors of the irreducible representations are given by SU (N ) semi-standard Young tableaus (SSYTs). The number of indices i to fill in the Young diagram, denoted by #i, is determined by #i =ñ − 2h i , where {h i } denote the set of helicity of the ith particle in the class and are sorted in the order h i ≤ h i+1 , i = 1, · · · , N − 1. Going through all possible ways to fill in the numbers and obtain a SSYT will give us the complete and non-redundant amplitude basis expressed by SSYTs. In fact, the Fock's condition of Young tableaus corresponds to IBP and Fierz/Schouten identities of operators. Thus choosing SSYTs means picking out all independent Young tableaus, which corresponds to picking out all independent Lorentz structures of operators.
The SSYTs can be translated to amplitudes using following relations for each columns, where E is the Levi-Civita tensor of the SU (N ) group. These amplitudes can be further translated to Lorentz structures of operators using amplitude-operator correspondence.
The gauge group factors are expressed by Levi-Civita tensors that contract with the fundamental indices of the fields.
Anti-fundamental representation and adjoint representation of fields can be converted to fundamental representation by (2.33) Following the procedures in Ref. [1], the gauge group factors can be obtained by constructing the singlet Young tableau using the modified Littlewood-Richardson rule with the corresponding indices of each field filled in.
Finally, we need to find the flavor structure in the presence of repeated fields. To achieve that, we utilize the basis b [λ] x=1,...,d λ in the irreducible left ideal of the group algebraS m , which is a set of independent symmetrizers spanning an irreducible representation space of the permutation group S m . By acting them on the flavor indices of an operator, we obtain either zero or an irreducible flavor tensor, whose independent components are given by the SU (n f ) SSYT's.
We thus express the result as such irreducible tensors in the form Y 1 is the famous Young symmetrizer 4 and O is some monomial operator. Take S 3 group for example, we show the action of Young symmetrizer as following To get all the independent irreducible representation spaces, one must find a collection of operators {O ζ } such that Y   [2]. It is usually highly non-trivial considering the complexity of redundancy relations of the operators. But with the Young tableau basis obtained above, we could find the unique coordinates of any Y In practice, the coordinates of Y [λ] 1 • O ζ can be obtained by finding the matrix representation of the group elements D T (π) and D B (π) on the complete and independent bases of T i and B j obtained by the Young tableau approaches 4 In our prescription, Y  Thus for any operator O = C ij T i B j , the permutation yields a transformation of the coordinate C ij : Alternatively, one could obtain the p-basis by inner product decomposition as in Ref. [1,2], where one first finds the irreducible Lorentz factors B [λ] x and gauge factors T G x and then combine them with the CG coefficients of the permutation group as 1 . It is then straightforward to select a set {O ζ } with independent projections, which we refer to as the "desymmetrization" process in [2].
When we contract the flavor tensor with the Wilson coefficient tensor C prs , the Young symmetrizer could be interpreted as acting on C prs with inverse permutations while all the other components in the tensor C prs are either 0 or related to the above ones by the tensor symmetry [λ].

Lists of Operators in LEFT
In this section, we list all the independent dimension 5, 6, 7, 8 and 9 operators in LEFT, and the number of operators with various lepton number violation |∆L|, are summarized for each dimension in table 2 and table 3. Although the twocomponent Weyl spinors are used to construct the operator basis, we changed them into four-component Dirac spinors

Lists of the Dim-5 Operators
Class F L ψ 2 : 6 types

Lists of the Dim-7 Operators
Class F L 2 ψ 2 : 10 types

Classes involving Two-fermions
Class F L 2 ψψ † D: 8 types

Classes involving Four-fermions
Class

Classes involving Six-fermions: ψ 6
There are 58 complex types in this class.