Effective Field Theory for Dark Matter Direct Detection up to Dimension Seven

We present the full basis of effective operators relevant for dark matter direct detection, up to and including operators of mass dimension seven. We treat the cases where dark matter is either a Dirac fermion, a Majorana fermion, a complex scalar, or a real scalar, allowing for dark matter to furnish a general representation of the electroweak gauge group. We describe the algorithmic procedure used to obtain the minimal set of effective operators and provide the tree-level matching conditions onto the effective theory valid below the electroweak scale.


I. INTRODUCTION
In order to compare results of different Dark Matter (DM) direct detection experiments one needs a theoretical framework in which to interpret the data. Since the typical momentum exchanges between DM and nuclei are small, q max ≲ 200 MeV, the interactions are well described within Effective Field Theory (EFT) without recourse to detailed particle physics models . The interactions are ordered according to the dimensionality of the operators, giving an effective interaction Lagrangian Here, C a are dimensionless Wilson coefficients, Λ is the NP scale, generically of the order of the mediator mass, and the sum runs over the operator labels, a, and the mass dimension of the operators, d. The EFT description is valid as long as the mediators are heavier than the momenta exchanges, q max ≪ Λ.
The UV theories of DM interactions with the SM generically give a nonzero contribution to at least one of the operators with d ≤ 7. This set of operators already covers, for instance, all the different chiral structures one can form out of two DM and two SM fermion fields.
Since the operators with low mass dimensions typically give the dominant contributions to the scattering rates, it suffices for most purposes to truncate the sum in (1) at d = 7.
Indeed, the commonly used DM EFTs include operators up to mass dimension seven [24,25].
However, the sets used in the literature are not complete -they do not contain the full basis of independent dimension-seven operators. In the present manuscript we rectify this situation.
We construct a complete EFT basis for operators up to and including dimension seven, coupling fermionic and scalar DM to the visible sector. We provide two sets of operators: a basis valid at low energies, µ ∼ 2 GeV, as well as a basis in the unbroken electroweak phase, valid at or above the electroweak scale µ = µ EW ∼ m Z . There are a number of operators at dimension seven that were not considered before. We show that the complete basis can be chosen in such a way that these additional operators contain derivatives acting only on the DM currents. In the case of fermionic DM such operators are likely to be only a subleading correction to the operators already considered previously. The situation is qualitatively different for scalar DM. Here, the operators with derivatives acting on DM fields enter already at mass dimension six and can form the leading contribution (for instance, if DM is a pseudo-Nambu-Goldstone boson). Throughout this paper we restrict our consideration to operators bilinear in the DM fields. To obtain a basis that is closed under renormalization group (RG) evolution, one would have to include operators with a different number of DM fields.
The formulation of the EFT at scale µ depends on which degrees of freedom are relevant at that particular scale. As a starting point we formulate in Section II A the EFT for fermionic DM that is valid at µ ∼ 2 GeV and describes the couplings of DM to the u, d, s quarks, leptons, gluons, and photons. In Section II C we then perform a nonperturbative matching, at µ ∼ 2 GeV, to an EFT that describes the couplings of DM to nonrelativistic protons and neutrons. Section II B contains the details about the renormalization group running from the electroweak scale down to µ ∼ 2 GeV. In Section III we provide the necessary ingredients to relate to the EFT above the electroweak symmetry-breaking scale. In Section III A we first present the complete basis of dimension-seven operators above the electroweak scale, while in Section III B we perform the matching at the electroweak scale onto the five-flavor EFT by integrating out the Higgs, W , and Z bosons and the top quark. In Section IV we extend the above formalism to the case of scalar DM. Section V contains our conclusions.
Finally, in Appendix A we spell out the details on the construction of the operator bases, while in Appendix B we collect the relevant results for the nonrelativistic reduction of the DM and nucleon fields.

II. LOW-SCALE INTERACTIONS OF FERMIONIC DM
We start by considering Dirac fermion DM, with field χ, and write down the complete basis of dimension-five, dimension-six, and dimension-seven operators at µ ∼ 2 GeV, for interactions between DM and quarks, gluons, photons, and leptons, Eq. (1). When writing the basis we assume that there is a conserved global dark U (1) D quantum number, which forbids currents of the formχ c Γχ, where χ c is the charge-conjugated DM field, and Γ denotes a generic string of Dirac matrices. The obtained basis is valid at all scales below the electroweak scale, µ ≲ M W , if matching corrections at the charm-and bottom-quark thresholds and RG running are taken into account, as detailed in Section II B. The nonperturbative matching onto an EFT coupling DM to nonrelativistic nucleons is presented in Section II C.
We also comment on the modifications needed when considering Majorana DM.

A. Fermionic DM coupling to quarks, gluons and photons
The basis for dimension-five and dimension-six operators coincides with the one in Refs. [19,25]. The two dimension-five operators are the magnetic and electric dipole oper-ators, where F µν ≡ ∂ µ A ν − ∂ ν A µ is the electromagnetic field-strength tensor, and σ µν = i 2 [γ µ , γ ν ]. The four dimension-six operators are formed from products of vector and axial-vector currents, The subscript f runs over the quark and lepton fields with masses m f < µ, so that f ∈ {u, d, s, e, µ, τ, ν e , ν µ , ν τ }. (For SM neutrinos only the left-handed combination appears.) The above basis is easily extended to all scales up to the electroweak breaking scales. For scales m c ≲ µ ≲ m b , in addition the charm quark is a propagating degree of freedom, so the operators with f = c need to be included, and for m b ≲ µ ≲ µ EW also the operators involving the bottom-quark field, f = b.
At dimension seven, there are four operators involving products of gluon field strength whereG a µν = 1 2 ε µνρη G ρη a . There are also six operators involving a scalar or tensor fermion current. As they naturally arise from operators which, above the electroweak scale, involve a Higgs field, we include an explicit factor of m f and count them as dimension seven (see Section III): The operators in Eqs. (5)-(9) were already used in Ref. [19]. The full basis of dimensionseven operators includes, in addition, the Rayleigh operators coupling DM to two photon field strength tensors, Q and four four-fermion operators with derivatives acting on the DM currents, 1 This completes the EFT basis of dimension-seven operators for fermionic DM that couple DM bilinears to the fields in the visible sector. We have checked in two different ways that this set of operators forms a complete basis: first, using an algorithmic procedure that explicitly implements the equations of motion and various algebraic identities, outlined in Appendix A, and second, with a procedure based on the conformal Hilbert series [26][27][28].

B. RG running
If the initial conditions for Wilson coefficients are set in the four-flavor EFT, m c ≲ µ ≲ m b , or in the five-flavor EFT, m b ≲ µ ≲ µ EW , the Wilson coefficients C 10,f in Eqs. (2)- (9) can be read off from Ref. [8]. The effect of weak mixing below the weak scale, potentially important for dimension-six operators, Eq. (3), was presented in Ref. [29], including the leading logarithmic QCD corrections. Note that the additional dimension-seven operators with derivatives on DM fields, Q 15,f , . . . , Q 18,f in Eqs. (12) and (13), have vanishing one-loop anomalous dimension, and there are no matching corrections at the quark thresholds.
The CP-even Rayleigh operators Q 11 , Q 12 in Eq. (10) mix into the scalar operators Q 5,f , Q 6,f in Eq. (7) with anomalous dimension 1 There are four more independent structures closely related to the operators in Eqs. (12) and (13), namely, These involve a chirality-flipping fermion current. Therefore, following our counting rules, they get multiplied by the corresponding fermion masses, rendering them dimension eight.
The CP-odd Rayleigh operators Q 13 , Q 14 in Eq. (11) do not mix into the pseudo-scalar operators in Eq. (8), contrary to their QCD counterparts, due to the absence of a QED anomaly. The RG evolution of the operators in Eqs. (2)- (13) is implemented in the DirectDM code [25].

C. Couplings to nonrelativistic nucleons and nuclear response
The scattering of DM on nuclei is dominated by DM scattering on single nucleons. To leading order in the chiral expansion the scattering is therefore described by the interactions of DM with nonrelativistic nucleons and protons for which the EFT Lagrangian is [5,25], with two momentum-independent nonrelativistic operators, and a set of momentum-dependent operators (displaying only the ones needed below), where N = p, n. The momentum exchanges are defined as where p 1(2) is the momentum of incoming (outgoing) DM particle, and similarly k 1(2) for incoming (outgoing) nucleon.
The chirally leading nonrelativistic reduction of the operators (2)-(9) was already given in Refs. [1,19]. The nonrelativistic reduction for the remaining dimension-seven operators with quark currents is The vector and axialvector form factors are defined through where our notation suppresses the dependence of nucleon states on their momenta, i.e., . The contributions of the operators Q 15,q , . . . , Q 18,q to the coefficients c N i in the nonrelativistic Lagrangian are thus 15,q + · · · , 15,q + · · · , 16,q + · · · , 18,q .
The ellipsis denotes the contribution of the operators in Eqs.
(2)-(9), with the results given in Eqs. (17)-(26) of Ref. [25]. The coefficient c N 14 receives only contributions from dimensionseven operators. In the above expressions we do not show the contributions of the Rayleigh operators, since their hadronic matrix elements are poorly known [30]. The coefficient c N 1 , for instance, receives a nonperturbative contribution from the CP-even Rayleigh operator, Q 11 ∼ (χχ)F F in Eq. (10). In the EFT with nonrelativistic nucleons one would need to keep both this contribution to O N 1 , as well as the Rayleigh operator Q 11 itself, with appropriately modified Wilson coefficient due to the matching. The operators Q (7) 12 ∼ (χīγ 5 χ)F F , Q 13 ∼ (χχ)FF , and Q (7) 14 ∼ (χīγ 5 χ)FF similarly give contributions in the nonperturbative matching to the non-relativistic operators respectively. The matching also modifies the Wilson coefficients of these Rayleigh operators.
At present only NDA estimates for the effects of the matching are available [30][31][32][33].
This completes the matching of the most general EFT Lagrangian for fermionic DM coupling to SM through operators up to and including dimension seven. The cross section for DM scattering on nuclei, due to DM scattering on a single nucleon, is given by Eqs. (40) and (46)  Here, we allow DM to be part of a multiplet of the electroweak SU (2), with a possibly nonzero hypercharge, Y χ . We denote the multiplet field by χ and its electromagnetically neutral component 2 (the DM) by χ 0 . We assume the DM interactions to be invariant under a global dark U (1) D symmetry, with the DM multiplet charged under U (1) D .
The size of the EFT operator basis depends on the dimensionality of DM SU(2) multiplet, d χ . One needs to distinguish three different cases. For a DM multiplet with d χ ≥ 3 the basis 2 In Section II we did not display the superscript "0" in order to shorten the notation.
is the most general one, and is given below. If DM is a doublet, d χ = 2, one can use the completeness relation for Pauli matrices, to reduce the basis to a smaller set. Below we comment on which of the operators should be dropped for the case of d χ = 2 in order to obtain a complete set of operators without redundant members. For singlet DM, d χ = 1, further reductions of the basis occur: because the SU(2) generators in that representation are zero, several operators vanish trivially.

A. EFT basis above the electroweak scale
We start the construction of EFT for DM interactions above the electroweak symmetry breaking scale by discussing the renormalizable interactions, i.e., the kinetic term in the If DM is an electroweak singlet, For such cases a phenomenologically interesting possibility is that the leading interactions between DM and the visible sector, relevant for direct detection experiments, are due to higher dimension operators. We collect them in the effective Lagrangian Here, the label a runs over different operators of dimension d, the C The complete basis for dimension-five and dimension-six operators of DM interacting with the SM can be found in Ref. [36], and we repeat it here for completeness. We present the complete basis of the dimension-seven operators here for the first time.

Dimension-five operator basis
The CP-conserving dimension-five operators are while the CP-odd operators have an extra insertion of γ 5 , Here and below, H is the SM Higgs doublet. All SU (2) (and, below, also color) indices are not displayed explicitly and are assumed to be contracted within the brackets. Note that if χ is a SU (2) singlet, the operators Q 6 , and Q 8 are absent. In a perturbative UV theory the operators Q

Dimension-six operator basis
At dimension six there are many more operators. We do not consider flavor violating operators, keeping our discussion minimal. For each SM fermion generation, i = 1, 2, 3, there are then eight operators that are products of DM currents and quark currents, Here Q L denotes the left-handed quark doublet, and u R , d R the right-handed up-and downtype quark singlets, respectively. The corresponding operators involving lepton currents are Q where L L denotes the left-handed lepton doublet, and ℓ R the right-handed down-type lepton singlet. Finally, there are four dimension-six operators involving Higgs currents, The Higgs currents are defined in terms of hermitian combinations of the covariant derivatives, Additional operators with covariant derivatives acting on the DM fields vanish via the DM equations of motion, up to total derivatives.
As in the case of dimension-five operators, the basis simplifies if DM is a SU (2) singlet. In this case, the operators Q 12,i , Q 15 , and Q (6) 17 vanish and should be dropped from the basis.

Dimension-seven operator basis
We group the dimension-seven operators into several classes, depending on how many derivatives they contain and to which SM fields they couple. Our convention for the covariant whereτ {ab} =τ aτ b +τ bτ a . Among those, the eleven operators Q kV with k = 1, 4,5,8,9,12,13,15,18,19,21 are CP even, while the remaining eleven are CP odd. In the definitions of the operators we included loop factors with appropriate gauge couplings, since these operators arise in perturbative theories at one loop or higher.
For d χ < 3 not all of the above operators are linearly independent. For DM that is an electroweak doublet, d χ = 2, the operators Q 21V , Q 22V should be dropped from the basis, since they are equivalent to Q 15V , Q 16V . Similarly, for DM that is an electroweak singlet, d χ = 1, the operators Q 9V , . . . , Q 14V and Q (7) 19V , . . . , Q 22V should be dropped from the basis. b. Gauge-Higgs operators. For d χ ≥ 3 there are 12 operators that couple DM currents to a set of mixed Higgs and gauge-boson operators, The definitions of the operators include an appropriate gauge coupling and a loop factor, because they are expected to arise at one loop when matching from a UV theory. For DM that is part of an electroweak doublet, d χ = 2, the operators Q 11V ′ and Q 12V ′ should be dropped from the basis, while for singlet DM, d χ = 1, in addition the operators Q 3V ′ ,4V ′ , and Q 7V ′ ,...,10V ′ should be dropped. c. Four-Higgs operators. For d χ ≥ 3 there are six dimension-seven operators that involve only DM and Higgs fields, The operators Q 1H , Q 3H , and Q 5H are CP even, while Q 2H , Q 4H , and Q 6H are CP odd. For DM that is an electroweak singlet, d χ = 1, the operators Q 3H , . . . , Q 6H are redundant and should be dropped, while for DM that is a doublet, d χ = 2, only the operators Q 5H and Q For DM that is an electroweak doublet, d χ = 2, the operators Q 9H ′ and Q 10H ′ are redundant and should be dropped from the basis. Similarly, for electroweak singlet DM, d χ = 1, the operators Q 3H ′ , Q 4H ′ , Q 7H ′ ,...,10H ′ should be dropped. e. Yukawa-like operators. The operators with chirality-flipping currents on both the DM and the SM side are whereH = iσ 2 H * is the charge-conjugated Higgs field. The above operators are not Hermitian so that they enter the effective Lagrangian, Eq. (37), with complex coefficients. Their Hermitian conjugates have to be included, modifying (37) to where the "h.c." is present in the sum only for the operators Q 1Y,i , . . . , Q 18Y,i . Alternatively, one could work with a set of 36 Hermitian operators with real coefficients, by defining Q (7) i− = iQ (7) i −i Q kY,i is part of a complete basis for both d χ ≥ 3 and for d χ = 2, while if DM is an electroweak singlet, d χ = 1, the operators with k = 3, 4, 6, 9, 10, 12, 15, 16, 18, should be dropped from the basis.

B. Tree-level matching at the electroweak scale
Next, we perform the tree-level matching of the EFT for DM interactions above the electroweak scale onto the EFT valid below the electroweak scale. In the matching at µ ∼ µ EW we integrate out the heavy states with masses of the order of the electroweak scale -the Higgs boson, the top quark, and the W and Z bosons -resulting in the effective Lagrangian (1) with five active quark flavors. The results for the gauge-invariant EFT containing dimension-five and dimension-six operators can be found in Ref. [36]. Here, we present the additional contributions due to the presence of dimension-seven operators.
We use the conventions of Ref. [37]. The gauge fields in the broken phase are with the sine of the weak mixing angle s w ≡ g 1 / g 2 1 + g 2 2 . Accordingly, we choose the following explicit form of the SU (2) generators with I χ = (d χ − 1)/2, and the indices k, l running over the values −I χ , . . . , I χ − 1, I χ . The

Higgs doublet field is
where h denotes the physical Higgs field, and G ± , G 0 the pseudo-Goldstone fields, eaten by the longitudinal components of W ± and Z.
where the χ field now only denotes the DM field, i.e., the neutral component of the DM multiplet, which in addition has been transformed by a simple chiral rotation, χ ′ = exp i 2 γ 5 ϕ χ, where (see also Ref. [38]) while the new mass term is 5H cos ϕ .
The corrections from dimension-five and dimension-six operators were already derived in Ref. [36].
From now on we will assume that the above field and mass redefinitions have been performed, and drop the primes on the Wilson coefficients, the χ fields, and the DM mass m χ .
Next we give the contributions from dimension-seven operators above the electroweak scale, when matching onto the EFT below the electroweak scale (the contributions from dimension-five and dimension-six operators are given in Ref. [36]). The gauge-Higgs operators, Eqs. (63)-(67), contribute to the dimension-five dipole operators after the Higgs obtains its vacuum expectation value, leading tô The ellipsis denotes the tree-level contributions from the dimension-five UV operators, as well as the one-loop contributions from renormalizable gauge interactions. The explicit expressions for both of these contributions can be found in Ref. [36]. Note that there are no tree-level contributions from the operators Q 9V ′ and Q 10V ′ . The matching of dimension-seven electroweak operators (51), (52) onto the gluonic operators (5), (6) is given byĈ The ellipsis denotes the one-loop contribution of the dimension-five scalar operators, given in Ref. [36].
The tree-level matching of the operators (53)-(61) onto the Rayleigh operators (10)- (11) is given bŷ Note that there are no tree-level contributions to the interactions of the neutral component χ 0 from the operators Q 13V , Q 14V , Q 19V , and Q 20V . The Yukawa-like operators, Eq. (77)-(85), match onto the dimension-seven scalar-current four-fermion operators, (7)-(9), by replacing the Higgs with its vacuum expectation value.

IV. SCALAR DM
Next we turn our attention to complex scalar DM, again allowing for DM to be part of an electroweak multiplet (we comment on the changes required for scalar DM below). There are only two renormalizable couplings between scalar DM and the SM, If these Higgs-portal couplings are absent or are very small, the higher dimension operators can become important. This can happen, for instance, if φ is a pseudo-Goldstone boson in which case the above two operators are forbidden by a shift symmetry.
The nonrenormalizable interactions of scalar DM with the visible sector start at dimension six, Note that, in order to simplify the notation, we denote the operators and Wilson coefficients with the same symbols as we did for fermionic DM in the previous two sections. The Hermitian conjugate in the sum is present only when the operators are not Hermitian. This is the case for the operators in Eqs. (135)-(137), whose Wilson coefficients can be complex.
where we included a loop factor with appropriate gauge couplings, see Eq. (62). For DM that is an electroweak doublet, d χ = 2, the operators Q 9 and Q 10 should be dropped from the basis as they are not linearly independent, while for DM that is an electroweak singlet, d χ = 1, in addition the operators Q The left-right covariant derivatives have been defined below Eq. (50), with the appropriate representation of the τ a matrices understood. For DM that is an electroweak doublet, d χ = 2, the operator Q 17 should be dropped from the basis. For singlet DM also the operators Q and Q 16 should be dropped. The Yukawa-like operators are, for both d χ ≥ 3 and d χ = 2, For singlet DM, d χ = 1, the operators Q 19,i , Q 21,i , Q 23,i do not appear in the basis. Note that the above operators, Q Finally, the operators with fermion currents are, both for d χ ≥ 3 and d χ = 2, For singlet DM, d χ = 1, the operators Q 25,i and Q 29,i , should be dropped from the basis. For completeness, we also display the dimension-seven operators for scalar DM.
For DM that is an electroweak doublet, d χ = 2, the operator, Q 3,i is not linearly independent and should be dropped in order to obtain a basis. For singlet DM both Q (7) 2,i and Q (7) 3,i are zero.
This completes the construction of the EFT basis for operators with mass dimension seven, for complex scalar DM. We have checked that the basis is complete both with the algorithmic procedure outlined in Appendix A, as well as with the conformal Hilbert series method [26][27][28].
For real scalar DM, the operators Q Note that the operators Q 1,f and Q 2,f vanish for real DM. In the remainder of this section we present the explicit tree-level matching conditions from the EFT above the electroweak scale to the five-flavor EFT. The resulting Wilson coefficients can then be evolved, using the RG equations, down to the hadronic scale µ ∼ 2 GeV, where the matching onto the nuclear EFT is performed. The details for the RG running and the subsequent matching to the nuclear EFT for the above operators has been given in Ref. [25], with the exception of the Rayleigh operators Q 8 . Their nonperturbative matching is directly analogous to the case of the fermionic operators Q 11 ∼ and Q 11 , for which, however, only NDA estimates are available at present (for the details see the previous section).
Tree-level matching at the electroweak scale gives, for the Wilson coefficients of the operators with the fermionic vector and axial currents in Eq. (145), 29,i + C The operators involving scalar currents, Eq. (146), obtain the following contributions: for up-type quarks, for down-type quarks, and for charged leptons. The matching for the gluonic operators, Eq. (147), giveŝ while for the Rayleigh operators we havê C 9 (10) . Eq. (7), while t-channel scalar mediator exchanges lead to a linear combination of scalar and tensor operators, Eqs. (7) and (9), after Fierz transformations.
The EFT we presented is valid as long as the mediators are heavier than the momenta exchanges in the physical process one is considering. For direct detection the typical momentum exchange is below 200 MeV. This means that the direct detection processes for a large class of the UV theories of DM would be described by the EFT discussed in this paper.
Still, one can think of ways to extend the framework further. For instance, one could extend the basis to include operators with several DM multiplets, or to relax the assumption of global dark U (1) D . We leave these for future work.
Permutations of identical fields and index symmetries. We explicitly check the disparity of operators under simultaneous permutations of (anti-)symmetric indices and index sets of identical fields, keeping track of fermionic signs. This also allows us to eliminate operators that vanish already via antisymmetry of indices.
Chirality of standard model fermions. We eliminate all operators that vanish via the chirality of the respective fields.
De-facto symmetry of covariant derivatives. Pairs of covariant derivatives acting on the same field can be decomposed into parts that are symmetric and antisymmetric, respectively.
Since the antisymmetric part is proportional to the commutator [D µ , D ν ], which is in turn given in terms of gauge field strength tensors, it is already covered by operators with a lower count of covariant derivatives.
Integration by parts. Since total derivatives leave the action invariant, expanding them using the Leibniz rule yields additional relations between our operators. Schouten identities. These follow from the fact that there is no totally antisymmetric tensor with more indices than the vector space dimension, other than the zero tensor. In our case, antisymmetrizing an object with more than four Lorentz indices yields zero. For instance, antisymmetrizing the indices of the Levi-Civita symbol together with µ in and dropping terms that vanish by symmetry, implies that the whole operator vanishes.
Moreover, Schouten identities can also yield relations likē which follows from antisymmetrizing the Levi-Civita symbol, together with the index ν, on the left-hand side.
Bianchi identity. The Bianchi identity for any gauge field strength tensor G a µν is given by In particular, this implies that the covariant derivative of the dual tensor vanishes, Additional reductions for special representations of SU (2). Our operator basis is valid for a general irreducible representation of SU (2) that the dark matter field χ furnishes. Additional relations, however, can make some operators redundant for specific representations.
For the fundamental representation, d χ = 2, some of the operators in the general basis are linearly dependent due to the completeness relation, Eq. (35). The reduced set of operators is given explicitly in the main text. For the trivial representation of SU (2), d χ = 1, one simply drops all the operators with an SU (2) generator in the DM current.
Additional reduction for the Majorana / real scalar case. For every Majorana DM bilinear we can use the Majorana condition to replace χ with its charge conjugate (see, e.g., [44]).
A dark matter bilinear that is odd under charge symmetry can then be dropped from the operator basis in the case of Majorana DM, eliminating DM vector and tensor currents (the covariant derivatives acting within the bilinear can change this). The list of these operators is given explicitly at the end of Section III A. In complete analogy, the operators listed at the end of Section IV A vanish for real scalar DM.
This procedure, together with the previously discussed treatment of partial integration and index symmetries, captures all relations implied by equations of motion. To see this, consider a completely general EOM-vanishing operator where Ψ is any Dirac field, Γ is any sum of products of Dirac matrices, possibly contracted with other expressions in the remainder of the operator, which is denoted by X. This operator, when expressed in our basis, does not induce any relations beyond the ones constructed above. Eqs. (A9) and (A10) can be repeatedly applied within each term of Γ to reduce it to a sum of terms with a maximum of two gamma matrices (and possibly one γ 5 , which can always be anticommuted to the rightmost position and eliminated if an even number of them exists). We have thus expressed the above operator as a linear combination of the vanishing operators already constructed. The case where the covariant derivative acts on the left, or if additional derivatives appear, can be treated analogously.
Finally, as a consistency check of the constructed basis, we compared it with the operator counts for a given field content that were derived using the conformal Hilbert series method [26][27][28]. This approach allows for a systematic group-theoretic treatment of equations of motion and inclusion of integration-by-parts identities. It calculates an object called the Hilbert series, given by H(D, ϕ 1 , . . . , ϕ N ) = k,r 1 ,...,r N c k,r 1 ,...,r N ϕ r 1 1 . . . ϕ r N N D k , where ϕ i and D are complex numbers that stand in for the fields of the theory and the covariant derivative. The coefficients c k,r 1 ,...,r N are the desired operator counts.
The Mathematica package provided in Ref. [28], with the addition of the fermionic or scalar DM field in an SU (2) multiplet, gives the total number of independent, Lorentz-and gauge-invariant operators of a certain mass dimension, as well as their field content. We have verified that the number of operators within each operator class obtained in this way exactly matches the number of operators constructed using our algorithmic procedure.
The ellipses denote higher orders in the expansion in 1/m χ . The remaining nonrelativistic currents, as well as the corresponding products with the expanded nuclear form factors, have already been presented in Ref. [19].