The effective theory of nuclear scattering for a WIMP of arbitrary spin

We introduce a systematic approach to characterize the most general non-relativistic WIMP-nucleus interaction allowed by Galilean invariance for a WIMP of arbitrary spin $j_\chi$ in the approximation of one-nucleon currents. Five nucleon currents arise from the nonrelativistic limit of the free nucleon Dirac bilinears. Our procedure consists in (1) organizing the WIMP currents according to the rank of the $2 j_\chi+1$ irreducible operator products of up to $2 j_\chi$ WIMP spin vectors, and (2) coupling each of the WIMP currents to each of the five nucleon currents. The transferred momentum $q$ appears to a power fixed by rotational invariance. For a WIMP of spin $j_\chi$ we find a basis of 4+20$j_\chi$ independent operators that exhaust all the possible operators that drive elastic WIMP-nucleus scattering in the approximation of one-nucleon currents. By comparing our operator basis, which is complete, to the operators already introduced in the literature we show that some of the latter for $j_\chi=1$ were not independent and some were missing. We provide explicit formulas for the squared scattering amplitudes in terms of the nuclear response functions, which are available in the literature for most of the targets used in WIMP direct detection experiments.


Introduction
In one of its most popular scenarios dark matter (DM) is believed to be composed of Weakly Interacting Massive Particles (WIMPs) with a mass in the GeV-TeV range and weak-type interactions with ordinary matter. Such small but non vanishing interactions can drive WIMP scattering off nuclear targets, and the measurement of the ensuing nuclear recoils in low-background detectors (direct detection, DD) represents the most straightforward way to detect them.
The most popular WIMP candidates are provided by extensions of the Standard Model such as Supersymmetry or Large Extra Dimensions which are in growing tension with the constraints from the Large Hadron Collider (LHC). As a consequence, model-independent approaches have become increasingly popular to interpret DM search experiments .
In particular, since the DD process is non-relativistic (NR), on general grounds the WIMP-nucleon interaction can be parameterized with an effective Hamiltonian H that complies with Galilean symmetry. The effective Hamiltonian H to zero-th order in the WIMPnucleon relative velocity v and momentum transfer q has been known since at least Ref. [24], and consists of the usual spin-dependent (SD) and spin-independent (SI) terms. To first order in v, the effective Hamiltonian H has been systematically described in [25,26] for WIMPs of spin 0 and 1/2, and less systematically described in [27,28] for WIMPs of spin 1 and in [29] for WIMPs of spin 3/2. An extension to spin-1/2 inelastic DM to first-order approximation in the WIMP mass difference can be found in [30].
In this paper we systematically extend the WIMP-nucleon effective interaction approach to the case of a WIMP with arbitrary spin j χ . As in [25][26][27][28][29], we focus on elastic WIMPnucleus scattering and include one-nucleon currents only [25,26]. The effective Hamiltonian H is a sum of WIMP-nucleon operators O j t τ , each multiplied by a coefficient c τ j , Here τ is an isospin index (0 for isoscalar and 1 for isovector), t 0 = 1, t 1 = τ 3 are nucleon isospin operators (the 2 × 2 identity and the third Pauli matrix, respectively), and the O j 's (j = 1, N ) are operators in the space of WIMP-nucleon states. Alternatively, the sum over the isospin index τ can be replaced by a sum over protons and neutrons using the following relations between the isoscalar and isovector coupling constants c 0 j and c 1 j and the proton and neutron coupling constants c p j and c n j , (1. 2) The O j operators introduced in [26][27][28] are listed in the first and second columns of Table 1 (the third column shows their expressions in terms of the operators O X,s,l that we introduce systematically in Section 3). The symbol 1 χN denotes the identity operator, q is the momentum transferred from the WIMP to the nucleus, 1 a tilde over q denotesq = q/m N (and q= q/m N ), where m N is the nucleon mass, S χ and S N are the WIMP and nucleon spins, respectively, and S ij = δ ij − 1 2 (S χi S χj + S χj S χi ) is a DM spin-1 operator (see Section 6.1 for its identification with the symbol S used in [27,28]). Moreover, is the WIMP-nucleon relative velocity, and where µ χN is the reduced WIMP-nucleon mass. The operators listed in Table 1 are invariant under Galilean transformations. The operators O 1 and O 4 are the only two operators to zero-th order in v χN and q. If terms up to first order in v χN are included, H in [25,26] contains 4 terms for a WIMP of spin 0 Table 1. Non-relativistic Galilean invariant operators discussed in the literature ( [26][27][28]) for a dark matter particle of spin 0, 1/2 and 1, and their relation with the WIMP-nucleon operators O X,s,l defined in Eqs. (3.22). Notice that the sign convention for the momentum transfer q used in this table and throughout the paper is opposite to that of Refs. [26][27][28].
(O 1,3,7,10 ) and 15 terms for a WIMP of spin 1/2 (O 1,3,..., 16 ). Earlier work on effective WIMPnucleon interactions beyond the usual SI and SD considered only operators independent of v χN [3]. Later work to include WIMPs of spin 1 enlarged the effective Hamiltonian to a total of 18 terms in [27] (O 1,..., 18 ) and eventually 24 terms in [28] (O 1,...,24 ). Beyond spin 1, Ref. [29] shows a particular example for a WIMP of spin 3/2. Our systematic treatment shows that some of these operators are not independent. Specifically, a look at the third column in Table 1  The expected DD scattering rate is obtained by evaluating the effective Hamiltonian H between initial and final nuclear states. The expected differential rate for WIMP-nucleus elastic scattering off a nuclear target T , differential in the energy deposited E R , is given by where M is the mass of the detector, N T is the number of target nuclei per unit detector mass, ρ χ is the mass density of dark matter in the neighborhood of the Sun, m χ is the WIMP mass, f (v) is the WIMP speed distribution in the reference frame of the Earth, and v T,min (E R ) is the minimal speed an incoming WIMP needs to have in the target reference frame to deposit energy E R . For elastic WIMP scattering, where m T is equal to the nuclear target mass and µ χT is equal to the WIMP-nucleus reduced mass. As shown in [26], the differential cross section dσ T /dE R in Eq. (1.5) can be put into the form where the sums contain products of WIMP and nuclear response functions R τ τ ′ k (v, q) and F τ τ ′ T k (q) (the latter are the nuclear response functions W τ τ ′ T k (q) in [25,26] apart from a multiplying factor). In the expression above, WIMP and nuclear physics are factorized in the product of the nuclear response functions F τ τ ′ T k , which depend on q 2 , and the WIMP response functions R τ τ ′ k , which depend on c τ j , q 2 , and ( v + where v χT is the WIMP-nucleus relative velocity and µ χT is the WIMP-nucleus reduced mass. The index k runs over combinations of nucleon currents. This factorization holds if two-nucleon effects [21,31,32] are neglected. To generalize the expressions above for a WIMP of arbitrary spin, the crucial observation is that, thanks to the factorization between the WIMP and the nucleon currents, the latter are unchanged and completely fixed irrespective of the WIMP spin. This has two consequences: (i) the effective operators O j t τ for a WIMP of arbitrary spin can be obtained in a systematic way by saturating the nucleon current with increasing powers of the vectors q, v + χN and S χ ; (ii) the shell model determinations of the nuclear response functions F τ τ ′ T k (q) available in the literature [26,33] can also be used for WIMPs of spin higher that 1.
The only new ingredients required to upgrade the cross section of Eq. (1.7) to a WIMP of arbitrary spin are WIMP response functions R τ τ ′ k that include the WIMP-nucleon operators for WIMPs of any spin. We compute their explicit expressions and give them in Eqs. (5.43). To obtain such expressions we find it convenient to define WIMP-nucleon interaction Hamiltonian operators in terms of tensors irreducible under the rotation group. The corresponding operator basis O X,s,l is given in Eqs. (3.22) or, alternatively, in Eqs. (3.23), and differs from that of the O j operators of Eq. (1.1). The third column in Table 1 gives the dictionary between the two operator bases, from which an analogous dictionary among the corresponding Wilson coefficients c τ j can be obtained in a straightforward way. This paper is organized as follows. In Section 2 we review the nuclear currents that arise in the non-relativistic limit of nucleon Dirac bilinears. In Section 3 we introduce a basis O j t τ of WIMP-nucleon interaction operators for the Hamiltonian of Eq. (1.1) and a WIMP of arbitrary spin. In Section 4 we "put the nucleons inside the nucleus" and present the ensuing effective WIMP-nucleus Hamiltonian. In Section 5 we derive the squared WIMP-nucleus scattering amplitude, resulting in Eqs. (5.43), which are the main result of this paper. We discuss our findings in Section 6 and conclude in Section 7.

Non-relativistic nucleon currents
There is a standard procedure to find all possible non-relativistic one-nucleon current operators in a nucleus. First one finds the free-nucleon operators that appear in the non-relativistic limit of the free nucleon currents (the Dirac bilinears). Then one sums the corresponding density operators over the A nucleons in the nucleus. 2 In the non-relativistic limit, the nucleon Dirac bilinears ψ f Γψ i , where Γ is any combination of Dirac γ matrices and ψ is the Dirac spinor for a relativistic free nucleon, reduce to linear combinations of five non-relativistic bilinears χ † f O X t τ N χ i , where χ is a non-relativistic Pauli spinor for the nucleon, t τ N is the isospin operator (τ = 0, 1 for the isoscalar and isovector parts, respectively), and O X is one of the free-nucleon operators Here σ N is the vector of Pauli spin matrices acting on the spin states of the nucleon N , and v + N is the operator (in the position representation), where r N and m N are the position vector and the mass of the nucleon N . The operator v + N is defined so that its matrix elements between free nucleon states are where v N,i and v N,f are the initial and final velocities of the nucleon. By contrast, the nucleon velocity operator is For a nucleon in a nucleus, one introduces one-nucleon current densities where a volumedensity version of the operator v + N appears. Each of the five free-nucleon operators O X t τ N (X = M, Σ, ∆, Φ, Ω) has a corresponding one-nucleon current density defined by Here the index N refers to the nucleon on which the operator acts (we abuse the notation N by using it to refer to a particular free nucleon and also as a summation index over nucleons bound in a nucleus). Moreover, the symbol · · · sym stands for the symmetrization operation This symmetrized operator is Hermitian and is the volume density version of the operator v + N , in the sense that its free-nucleon matrix elements between nucleon wave functions ψ 1 ( r N ) and ψ 2 ( r N ) obey the relation This justifies the replacement of v + N with [δ( r − r N ) v N ] sym in passing from a free-nucleon operator to a one-nucleon current in a nucleus.
Hence the following correspondence applies between free-nucleon operators and onenucleon currents, In problems involving the transfer of a momentum q to the nucleus (such as the problem we are interested in, namely the scattering of WIMPs off nuclei), another variant of the onenucleon currents appears. These are currents defined in the Breit frame of the nucleus, namely the reference frame in which the nucleus momentum changes sign when the momentum q is transferred. 3 The velocity of the Breit frame is where v T,i and v T,f are the initial and final velocities of the nucleus. Since the velocity of the nucleus v T equals the velocity of the center of mass of the system of nucleons, For elastic scattering, the energy transferred to the nucleus in the Breit frame is zero. The use of the Breit frame in the definition of form factors for particles of any spin has been discussed in [34]. The Breit frame is particularly relevant for nucleon form factors (see, e.g., [35][36][37]). 4 In the notation of [25,26], v ⊥ T is used in place of our v + χT in Eq. (4.1), and there is no v + T . Moreover, v ⊥ is used in place of our v + χN in Eq. (3.4). To err in the direction of clarity, we have chosen to maintain the particle labels as subscripts and to use the different symbol + in place of ⊥ to distinguish v ⊥ T in [25,26] from our v + T in Eq. (2.9). Let be the nucleon velocity in the nucleus Breit frame corresponding to momentum transfer q. The Breit-frame currents are defined as the symmetrized currents with v N replaced by v N T , The Breit-frame currents are related to the symmetrized currents via (2.14) One also defines the non-symmetrized currents in the Breit frame When we later consider the scattering of WIMPs in the Born approximation, the plane wave WIMP wave functions contribute a factor e i q· r to the amplitude, and the Fourier transform of the one-nucleon Breit-frame currents appears, Substituting Eqs. (2.13) into Eqs. (2.16), and using the relation one finds the following identities between the Fourier-transformed symmetrized and nonsymmetrized one-nucleon currents in the Breit frame,

WIMP-nucleon operators
In this section we describe the effective interaction Hamiltonian of a WIMP with a free nucleon. The five free-nucleon operators O X (X = M, Ω, Σ, ∆, Φ) in Eq. (2.1) depend on the nucleon velocity, which is not invariant under Galilean boosts. Indeed, to comply with Galilean invariance one must introduce five corresponding WIMP-nucleon operators O X (X = M, Ω, Σ, ∆, Φ) that depend on the relative WIMP-nucleon velocity instead (in the following we drop the hat on top of operators, unless it is needed for clarity) However from the non-relativistic limit of the nucleon Dirac bilinears one knows that v N appears in the combination v + N of Eq. (2.17). If the WIMP has spin-1/2 the same argument implies that the analogous combination appears also from the non-relativistic limit of the WIMP Dirac bilinear. Then combining Eqs. (3.1) and (3.2) one concludes that the WIMP-nucleon operators consistent to Eq. (2.1) must be: We now show that this conclusion holds also for a WIMP of arbitrary spin. In order to do so one writes the non-relativistic Hamiltonian H χN for an interacting system made of a WIMP χ and a nucleon N , The most general interaction Hamiltonian V χN depends on the WIMP spin operator S χ , the nucleon spin operator S N , and, imposing Galilean invariance, on the relative WIMPnucleon position operator r χN and its conjugate relative momentum operator p χN . Moreover, Eqs. (2.1) imply that the interaction Hamiltonian is either independent of p χN or linear in p χN . In the latter case, since V χN must be Hermitian and it depends on the non-commuting operators r χN and p χN , a prescription needs to be set up on the order in which these two operators appear. Any combination of the form f 1 ( r χN ) p χN f 2 ( r χN ), where f 1 ( r χN ) and f 2 ( r χN ) are arbitrary functions, can be rearranged with the r χN dependence on the left of the operator p χN by commuting p χN and f 2 ( r χN ) and regarding their commutator as an extra term in the Hamiltonian. Thus there is no loss of generality in assuming that the dependence on r χN is on the left of p χN , as in f ( r χN ) p χN . Then an Hermitian term in the Hamiltonian is obtained by constructing the symmetric combination Since the nucleon has spin 1/2, the interaction Hamiltonian V χN can be split into terms independent of the nucleon spin operator S N and terms linear in S N (notice that the nonrelativistic limit of the nucleon Dirac bilinears in Section 2 shows that symmetric tensor terms of the form p χN,i S N,j + p χN,j S N,i do not appear). So the interaction Hamiltonian V χN must have the form Here we have introduced the relative WIMP-nucleon velocity operator v χN defined by The interaction amplitude for the WIMP-nucleon scattering process (in the Born approximation) is then given by where p χ,i , p χ,f , p N,i , p N,f are the initial and final momenta of the WIMP and the nucleon, and in the integral we have explicitly separated the motion of the center of mass with coordinates ( R, p tot ). The integral appearing in Eq. (3.10) is a function of q = p χN,i − p χN,f and v + χN = ( p χN,i + p χN,f )/(2µ χN ). The dependence on v + χN gives the operators in Eq. (3.3) multiplied by functions of q, namely the Fourier transforms V τ X q, S χ = d 3 r χN e i q· r χN V τ X r χN , S χ (3.11) of the potentials in Eq. (3.8). As a way of example, the explicit contribution to the amplitude from V ∆ is Analogous steps show that also the contributions from V Φ and V Ω are proportional to the v + χN operator. This shows that the effective operators of Eq. (3.3) written in terms of v + χN must drive the WIMP-nucleon interaction also for WIMPs of spin higher that 1/2. Notice that for elastic WIMP-nucleon scattering, The WIMP-nucleon operators O X are related to the free-nucleon operators O X by means of the relations, obtained by using v + The On the other hand, the effective interaction term for a WIMP of spin j χ must contain up to the product of 2j χ WIMP spin vectors, in order to mediate transitions where the third component of the WIMP spin changes from ±j χ to ∓j χ . Using index notation S i for the i-th component of the vector S χ (we drop the subscript χ in S χ,i for more readability), there are interaction terms containing no S i or a product of s factors S i up to s = 2j χ , In other words, there are 2j χ + 1 possible products of the WIMP spin operator for a WIMP of spin j χ . Each product can be labeled by the number s of WIMP spin factors S i . An alternative way to reach the same conclusion is to show that the 2j χ +1 products in Eq. (3.15) are a basis in the space of spin operators for spin j χ . Once the number of WIMP spin factors is fixed to s, and the scalar or vector nature of the free-nucleon operator O X is considered, the number of q factors is constrained by rotational invariance. In particular, in the case of a scalar nucleon operator O X (X = M, Ω), the WIMP operator o must be a scalar, and all the indices i 1 · · · i s in S i 1 · · · S is must be saturated by terms q i 1 · · · q is . The resulting WIMP operator is S i 1 · · · S is q i 1 · · · q is . On the other hand, in the case of a vector nucleon operator O X (X = Σ, ∆, Φ), a vector WIMP operator o is needed, and the s indices in S i 1 · · · S is must be saturated by an appropriate number of q factors in order to obtain a vector. This can be achieved in three ways: (1) by using s − 1 factors of q to produce A further consideration informs our choice of basis interaction terms. In the calculation of the cross section for WIMP-nucleus scattering, traces of the S i 1 · · · S is operators are needed. The latter are greatly simplified if for the products of WIMP spin operators one uses irreducible tensors (i.e., belonging to irreducible representations of the rotation group). Irreducible tensors are completely symmetric under exchange of any two of their indices and have zero trace under contraction of any number of pairs of indices (they are symmetric traceless tensors). In addition, an irreducible tensor of rank s has 2s + 1 independent components, and belongs to the irreducible representation of the rotation group of spin s. Irreducible tensor operators of different rank are independent, in the sense that the trace of their product is zero. As a consequence, there are no interference terms in the cross section between irreducible operators of different spin. Therefore we use the following 2j χ + 1 irreducible spin tensors as a basis in the spin space of a WIMP of spin j χ , Here, borrowing the notation of [38], we use an overbracket over an expression containing a set of indices to indicate that the free indices under the bracket are completely symmetrized and all of their contractions are subtracted. For example, Notice that 1 = 1 and A i = A i . More details are given in Appendices D.2 and D.3. When the potentials V X ( r χN , S χ ) in Eq. (3.8) are expanded onto the basis (3.16), the coefficients of the expansion are tensor functions of ranks from 0 to 2j χ + 1 of the magnitude r χN = | r χN |, These tensor functions can be written as derivatives of scalar functions of r χN . For instance, introducing a factor (−1) s for our later convenience, When the same procedure is applied to the Fourier transforms V τ X ( q, S χ ) in Eq. (3.11), the coefficient functions are tensor products of the form iq i 1 iq i 2 · · · iq is multiplied by scalar functions of the magnitude q = | q|. For example, The scalar functions V τ X,s,l (q) will give the q dependence of the coefficients c τ X,s,l (q) in Eq. (3.24) below.
Using the irreducible spin products in Eq. (3.16) in place of those in Eq. (3.15), we are lead to introduce the scalar WIMP operators 20) and the vector WIMP operators The three vector operators correspond to the three possible combinations of angular momenta s (the number of S factors) and l (the number of q factors) with total angular momentum 1.
Following the procedure outlined above we define the following basis of WIMP-nucleon operators O X,s,l , all of which are irreducible in WIMP spin space and Hermitian, Each operator of Eqs. (3.22) is to be multiplied by the isoscalar or isovector operator t 0 or t 1 = τ 3 to form O τ X,s,l = O X,s,l t τ . The basis operators in Eqs. (3.22) can also be written in vector notation as follows, where the overbrackets amount to taking the symmetric traceless part of the product of WIMP spin matrices (in the following equation and in Tables 2-6 we use the notation S N and S χ for the nucleon and WIMP spins, respectively) The indices in the symbol of the operator O X,s,l follow the following scheme. The first index X is the nucleon current (X = M , Ω, Σ, ∆, and Φ for the nucleon currents 1, v + χN · σ N , σ N , v + χN , and v + χN × σ N , respectively). The second index s is the number of WIMP spin operators S χ appearing in O X,s,l . This can be considered as the spin of the operator. It ranges from s = 0 to twice the WIMP spin s = 2j χ . The third index l is the power of the momentum exchange vector q i in the operator O X,s,l . This can be considered as the angular momentum of the operator. A factor of i is introduced for every power of q. We include the operator O ∆,s,s+1 in our list of basis operators even if it is zero for elastic scattering because v + χN · q = 0; it may appear in inelastic scattering in which the nucleus transitions to another energy level.
The relation between our operators and those defined in [25,26] and [27] is listed in Table 1 (see Section 6.1 for the case of WIMP spin 1). Notice that following common usage in the WIMP dark matter community we define q as the momentum transferred to the nucleus, whereas [25,26] use q for the momentum lost by the nucleus; thus our q and that in [25,26] have opposite signs. Tables 2-6 summarize the explicit forms of the effective operators for WIMPs of spin 0, 1/2, 1, 3/2, and 2.
A general WIMP-nucleon operator O χN is a linear combination of the basis WIMPnucleon operators in Eqs. (3.22), The coefficients c τ X,s,l (q) are in principle functions of the magnitude q of the momentum transfer, determined by the Fourier transforms of the potentials in Eq. (3.8) as c τ X,s,l (q) = m l N V τ X,s,l (q). In some phenomenological studies they have been taken as constants.
We can group the basis operators according to the five nucleon currents X = M, Ω, Σ, ∆, Φ Eq. (3.25) applies to WIMP interactions with a free nucleon.

Effective WIMP-nucleus Hamiltonian
We now pass from the Hamiltonian describing the interaction of a WIMP with a free nucleon to the effective Hamiltonian that describes the interaction of the WIMP with the whole nucleus. Under the approximation that the WIMP interacts only with one nucleon at a time (the one-nucleon approximation), what we need to do is to "put the nucleon inside the nucleus" and use the relative velocity of the WIMP with respect to the nucleus (i.e., the center of mass of the system of nucleons).
Let v χT be the WIMP velocity in the reference frame of the nucleus center of mass. (4.1) In the notation of [25,26], For elastic WIMP-nucleus scattering, and (4.4) The recipe to "put the nucleon inside the nucleus" is to replace the free-nucleon operators O X t τ by their respective symmetrized nucleon current densities j τ X . In more detail, Eqs. (2.14) and (3.14) imply the following replacements A Fourier transform (which applies for WIMP wave functions that are plane waves) leads to the WIMP-nucleus effective Hamiltonian where In this Section we outline the procedure to calculate the square of the amplitude for the scattering process driven by the effective Hamiltonian of Eq. (4.7). As already pointed out, the factorization between the nuclear currents j τ X , j τ X and the WIMP currents l τ X , l τ X implies that, compared to the results in the literature for a WIMP of spin ≤1 [25][26][27] , the nuclear part of the calculation will not change when the currents (3.26) are used to describe the interaction of a WIMP with arbitrary spin. As a consequence, part of the procedure has already been described elsewhere [25,26]. Nevertheless, for completeness, in this Section we review the full calculation, albeit focusing on how to obtain the WIMP spin averages from the currents of Eqs. (3.26). In the latter derivation the convenience of assuming irreducible representations of the rotation group for the basis WIMP-nucleon operators introduced in Section 3 becomes apparent, as all the results are obtained by using the two master equations (5.19-5.20) for traces of products of irreducible spin operators. The proof of some of the derivations used in this Section, including those of Eqs. (5.19-5.20), are provided in the Appendices.

Sum/average over nuclear spins
Nuclear targets in direct dark matter detection experiments are usually unpolarized, thus the cross section is summed over final nuclear spins and averaged over initial nuclear spins. Let H fi = f| H|i indicate the transition matrix element of the effective Hamiltonian between an initial WIMP-nucleus state |i and a final WIMP-nucleus state |f . The sum/average over nuclear polarizations is defined as a sum over final nuclear azimuthal quantum numbers M f and an average over initial nuclear azimuthal quantum numbers M i , Here J i and J f denote the initial and final total angular momentum of the nucleus. As far as the nuclear part is concerned, the calculation requires to expand the nuclear currents j τ X , j τ X in spherical and vector spherical harmonics, and to obtain the sums over initial and final nuclear spins for each nuclear current multipole operator making use of the Wigner-Eckart theorem.
When the Fourier transform of the non-symmetrized nucleon currents in Eqs. (2.15) is expanded into multipoles one obtains for the scalar currents, and for the vector currents. In the expression above, which is obtained using the multipole expansion of the scalar and vector plane waves provided in Appendix A, the one-nucleon operators X τ JM , X ′τ JM and X ′′τ JM (with X=M, Σ, ∆, Φ, Ω) arise [39][40][41]. We provide them explicitly in Eq. (C.3). For the vector operators X=Σ, ∆, Φ, we follow the standard notation that double-primed quantities indicate a longitudinal multipole (L), single-primed quantities correspond to a transverse-electric multipole (TE) and unprimed quantities indicates a transverse-magnetic multipole (TM). Moreover, in the expressions above Y The operators j τ ∆ , j τ Φ and j τ Ω in Eq. (C.3) correspond to the non-symmetrized nuclear currents of Eqs. (2.15). As explained in Section 2 the WIMP-nucleus scattering process is driven by the symmetrized currents in Eqs. (2.18). So after symmetrization one obtains with We provide the details of the rest of the calculation of the sum/average over nuclear spins in Appendix B. The result is Table 7. Parity of the nucleon currents under space reflection P and time reversal T . Columns P J and T J list the parities of their J-th multipole moments (the notation L, TE, and TM stands for longitudinal, transverse electric, and transverse magnetic multipole, respectively). The last column lists the allowed J's in a ground state that is P and T (or CP ) invariant.
Here we use the notation of [25], where the nuclear response functions F τ τ ′ X are defined by with J f || X τ J (q)||J i being the reduced matrix elements of the one-nucleon multipole operator X τ JM defined in Eq. (C.3). Ref. [26] uses the notation We write F τ τ ′ X (q) for F τ τ ′ XX (q). In Eq. (5.7) only the multipole operators X=M , Σ ′ , Σ ′′ , ∆, Φ ′′ and Φ ′ appear, which correspond to P and T invariant nuclear ground states. These are the only allowed responses under the assumption that the nuclear ground state is an eigenstate of P and CP. The parity of the nucleon currents and their multipoles under space-reflection P and time-reversal T are collected in Table 7.

Sum/average over WIMP spins
The sum/averages over the nuclear spins Eqs. (5.7) contain products of the WIMP currents ℓ τ X and ℓ τ X . The average of these products over the initial WIMP spins and their sum over the final WIMP spins defines the unpolarized WIMP response functions R τ τ ′ XY , apart from conventional factors. We indicate the sum/average over WIMP spins with an overline over the product of WIMP currents. (The context makes it clear if the overline denotes a sum/average over nuclear spins or WIMP spins; a double overline denotes a sum/average over both.) Thinking of the WIMP currents ℓ τ X and ℓ τ X as matrices in WIMP spin space, and thus of ℓ τ * X as the Hermitian conjugate of the matrix ℓ τ X , we have and similar relations for the vector WIMP currents. In particular, taking the average over nuclear and WIMP spins of Eq. (5.7) yields where, matching the notation of [26], We now use Eqs. (4.8) and the fact that the ℓ X ℓ * Y are functions of the vector q only. Thus, for example, ℓ τ M ℓ τ ′ * X,i is proportional toq i , with coefficient given by On the other hand ℓ τ X,i ℓ τ ′ * Y,j is the sum of a term in δ ij −q iqj , a term inq iqj , and a term in ǫ ijkqk , with respective coefficients given by We can express the WIMP response functions R τ τ ′ XY in terms of the coefficients L τ τ ′ XY . Writing L τ τ ′ XX = L τ τ ′ X and introducing we obtain The last step is the calculation of the traces of the WIMP currents contained in the coefficients L τ τ ′ XY . In Section 3 we chose to write the effective Hamiltonian in terms of irreducible tensors S i 1 · · · S is of products of WIMP spin operators. As a consequence, all the traces can be calculated by making use of the two following master equations Here B jχ,s = s! (2s + 1)!! s! (2s − 1)!! K jχ,0 · · · K jχ,s−1 , The first few values of B jχ,s are A proof of the equations above is provided in Appendix D.3. Let us start with the scalar currents, which are readily obtained. For example, The vector currents 2ℓ τ Σ,i , ℓ τ ∆,i , and 2ℓ τ Φ,i have similar expressions, and we give details about the calculation of L τ τ ′ Σ only. We need 1 2j χ + 1 tr S i 1 · · · S is q i 1 · · · q i s−1 a τ Σ,isi S j 1 · · · S js q j 1 · · · q j s−1 a τ ′ * Σ,jsj , Split a τ Σ,isi into a part parallel toq is and a part perpendicular toq is , tr S i 1 · · · S is q i 1 · · · q i s−1 a τ Σ,isi S j 1 · · · S js q j 1 · · · q j s−1 a τ ′ * Here we used The quantities L τ τ ′ ΦM and L τ τ ′ Σ∆ are obtained as follows Finally, inserting the expressions for L τ τ ′ XY into (5.18), the explicit expressions in the next subsection are obtained for the eight response functions R τ τ ′ XY with X=M , Φ ′′ , Φ ′′ M , Φ ′ , Σ ′′ , Σ ′ , ∆ and ∆Σ ′ .

Results
The unpolarized differential cross section for WIMP-nucleus scattering is given by the expression (our v +2 χT ≡ ( v + χT ) 2 is equal to v ⊥2 T in the notation of [25]) where the sum is over are given in terms of the nuclear response functions in Eq. (5.8) and available in the literature by the expressions The functions R τ τ ′ k v +2 χT , q 2 are the WIMP response functions, given for WIMPs of any spin by (see Eq. 5.21). The equations above are valid for a WIMP of arbitrary spin j χ and are the main result of the present paper. In particular, the adoption of the irreducible tensors in Eq. (3.16) implies that for a given value of s=2j χ a different set of WIMP response functions R τ τ ′ X arises for each set of the operators O X,s,l introduced in Section 3. For a WIMP of spin j χ all the operators O X,s,l with s ≤ 2j χ contribute to the cross section.

Discussion
In this Section we discuss some of the consequences of the results obtained in the previous Sections.

The case of spin 1
In Section 3 we expressed the WIMP-nucleon interaction Hamiltonian operators in terms of tensors irreducible under the rotation group. The case j χ = 1 has already been discussed in the literature in terms of reducible operators [27,28], so it is instructive to compare the two approaches.
The authors of Ref. [27] introduce a symbol S in expressions of the kind a · S · b, where a and b are vectors (see, e.g., their Eq. (4)). They call it the symmetric combination of polarization vectors ǫ i . In their Appendix they give the expression We want to identify the symbol S with an operator S in WIMP spin space (in this section we keep the hat over WIMP spin operators). We find the definitions of S and S ij as operators in Ref. [27] a little obscure. We interpret them as definitions in a particular basis, and then translate them to basis-independent definition in terms of the WIMP spin operators S i (where i = 1, 2, 3). In particular, we identify the quantities ǫ s i in [27] with the components of the WIMP spin eigenstate |1, s in the linear polarization basis |e i , i.e., As standard, the linear polarization states in the x, y, and z directions |e i (with i = 1, 2, 3) are given in terms of the angular momentum eigenstates |1, m (with m = +1, 0, −1) by The latter expression matches the formula iS k = ǫ ijk ǫ † i ǫ j after Eq. (B4) in [27] if it is interpreted as i S k = ǫ ijk |e j e i |, i.e., if the following identifications are made: ǫ i → |e i and ǫ † i → e i |. This motivates our interpretation of the definition of S ij in the Appendix of Ref. [27], namely S ij = 1 2 (ǫ † i ǫ j + ǫ † j ǫ i ), as Our goal is to write the operator S ij so identified in terms of products of the spin operators S i (where i = 1, 2, 3). In the |e i basis, from Eq. (6.6), Also, Using the symmetrization symbol {..} and j χ = 1 in the relation S ij can also be written as The substitutions S ij → − S i S j + 1 3 δ ij 1 and q → − q produce the relations in Table 1 between the spin-1 operators O 17,...,20 and the operators O X,s,l introduced in Section 3.
Similarly, we find the definition of the S ij in Ref. [28] as operator also a little confusing. The definition in their equation (3.4) is consistent with the operator S ij that we identify in Eq. (6.10) if their equation (3.4) is interpreted as the transition amplitude of the operator S ij between initial and final helicity eigenstates. Let the initial and final helicity eigenstates for a spin-1 particle be |h, s , |h ′ , s ′ , (6.13) respectively. We identify the quantities e si and e ′ s ′ i in [28] with Then from Eq. (6.6) we have which equals S s ′ s ij in [28] and reproduces their equation (3.4). This clarifies that the symbols S in Dent et al. [27] and Catena et al. [28] can be identified with the operators We now show that the additional operators O 21,...,24 of order q introduced in [28] are not independent in the one-nucleon approximation. These operators do not arise from the non-relativistic limit of a high energy amplitude. They are obtained by combining S in rotationally invariant combinations with S N , q and v + χN . Consider for example the operator O 21 = v + χN · S · S N . Using Eq. (6.12) one obtains Since S i S j v + χN,i S N,j = S i S j v + χN,i S N,j , and in one-nucleon approximation v + χN,i S N,j does not contribute to the scattering process, i.e., it is not included among the currents in Eq. (2.1), the first term in the right hand side of Eq. (6.17) vanishes. Thus in the one-nucleon-scattering approximation, In general any interaction term depending on v + χN and S N must be projected onto the currents of Eq. (2.1) using the decomposition In this way, for the additional operators O 22,...,24 defined in [28], we obtain

The counting of independent operators
The procedure outlined in Section 3 consists in coupling one of the five nucleon currents of Eq. (2.1) to WIMP currents ordered according to the rank of the irreducible operators S is · · · S is (s = 0, 1, 2, ...). The power l of the transferred momentum q descends from rotational invariance. . This implies that each value of s > 0 contributes 2+ 3× 3 = 11 new operators (10 for elastic scattering). Since s ranges from 0 to 2j χ , the total number of independent operators for a WIMP of spin j χ is 4 + 10 × 2j χ = 4 + 20j χ for elastic scattering (5 + 11 × 2j χ = 5 + 22j χ for inelastic scattering). If we restrict the counting to operators that are independent of the WIMP-nucleon relative velocity, we keep only X = M, Σ, and find that at s = 0 there are two operators and that each s > 0 contributes 4 operators (one with X = M and three with X = Σ). This gives a total of 2 + 8j χ velocity-independent basis operators. The number of linearly-independent operators for WIMPs of spin 0, 1/2, 1, 3/2, and 2 are collected in Table 8. The number of operators introduced so far in the literature for WIMP spin j χ ≤ 1 is 24, as shown in Table 1. This number coincides with our counting of 24 basis operators for elastic scattering of WIMPs of spin j χ ≤ 1. This is only a coincidence. The total number of independent operators that have appeared in the literature so far is actually 19, as 1 of those in Table 1   (see their absence from Table 1 and their presence in Table 4). In addition, for inelastic scattering, one should add the linearly independent operators, Ref. [26] introduced 14 independent operators for j χ ≤ 1/2, in agreement to our counting for elastic scattering: the 16 operators O 1,...,16 , minus the two operators O 2 and O 16 , the former being quadratic in v + and the latter being a linear combination of O 12 and O 15 . Ref. [27] introduced two additional operators for j χ = 1, O 17 and O 18 , accounting for 16 of the 24 independent operators for j χ = 1. Ref. [28] introduced six additional operators O 19,...,24 , but only three of them are linearly independent, bringing the number of independent operators for j χ = 1 to 19 out of 24. Our addition of the operators in Eq. (6.21) completes the 24 linearly independent operators for elastic scattering of WIMPs of spin j χ = 1.

Examples of differential scattering rates
In Figs. 1-4 we provide a few examples of the expected spectrum of the differential rate in Eq (1.5) as driven by some of the irreducible effective operators introduced in Eqs. (3.22). In particular Fig. 1 shows the differential rate for a 10 GeV mass WIMP on xenon and for the 10 irreducible effective operators O X,2,l that arise for a WIMP with j χ ≥1. Fig. 2 shows the differential rate for the operators O X,3,l arising for a WIMP with j χ ≥3/2. Figs. 3 and 4 show the analogous cases for a fluorine nuclear target. All the spectra are normalized to 1 event.
For the WIMP velocity distribution f (v), a truncated Maxwellian with escape velocity 550 km/s and rms velocity 270 km/s in the Galactic rest frame is adopted. In these plots one can observe how the spectra shift to larger recoil energies E R for growing j χ due to the correlation between E R and the power of q/m N in the squared amplitude. Such correlation implies also a suppression of the contribution of higher-rank operators compared to lower-rank operators when their couplings are of the same order of magnitude. It must be remarked that from the point of view of a non-relativistic effective theory, one cannot rule out the possibility that the scattering rate of a WIMP with spin j χ is driven by one of the higher-rank operators.
We expect this to lead to non-standard phenomenological consequences.

Conclusions
In the present paper we have introduced a systematic approach that, in the one-nucleon approximation, describes the most general non-relativistic WIMP-nucleus interaction allowed        [26,33].
In particular, we have expressed the WIMP-nucleon interaction Hamiltonian operators in terms of tensors irreducible under the rotation group. This has several advantages: • it includes all the operators allowed by symmetry, including those that do not arise as the low-energy limit of standard point-like particle interactions with spin ≤1 mediators; • it avoids double counting, allowing to show that some of the operators introduced in the literature for the spin-1 WIMP case are not independent (see Section 6.1); • it greatly simplifies the calculation of the cross section, that was obtained from the two master equations (5.19-5.20) for the traces of WIMP spin operators; • for a given WIMP spin j χ the scattering cross section is given by a sum of cleanly separated contributions from irreducible operators of ranks 0, 1, 2, 3, . . . , up to 2j χ , without interference terms (since irreducible operators of different rank do not interfere).
All the Wilson coefficients c X,s,l are defined up to arbitrary functions of the transferred momentum q 2 . Moreover, as shown in Table 1, in some cases the change of basis from reducible to irreducible operators involves momentum-dependent coefficients.
From the phenomenological point of view, contributions from irreducible operators of higher rank are shifted to larger recoil energies compared with contributions from operators of lower rank. It may happen that lower rank operators vanish and the WIMP scattering rate is dominated by a higher rank operator. We expect this to lead to non-standard phenomenological consequences.

A Multipole expansion of a vector plane wave
It is well known that a plane wave e i q· r can be expanded into spherical harmonics according to the equation Here L, TE, and TM stand for longitudinal, transverse electric and transverse magnetic, respectively; the TE and TM terms start at J = 1; are the longitudinal, transverse electric and transverse magnetic spherical harmonics, defined in terms of the vector spherical harmonics where C JM Lα1β is the Clebsch-Gordan coefficient for coupling angular momenta Lα and 1β into JM , andê β is the standard spherical basiŝ The orthogonality relation for the vector spherical harmonics, then gives The sums over M involving derivatives of the Y JM are evaluated by differentiating the addition theorem of spherical harmonics where P J (µ) is the Legendre polynomial of order J with µ = cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos(φ 1 − φ 2 ). (D.9) For example, with the → indicating the limit (θ 1 , φ 1 ) → (θ 2 , φ 2 ),

D.2 Some relations between symmetric and symmetric traceless tensors
By definition, the symmetric traceless part A i 1 ·is of an rank-s tensor A i 1 ···is is obtained by first symmetrizing A i 1 ···is completely with respect to all of its indices, and then subtracting all the possible traces, i.e., contractions of pairs of indices, double pairs of indices, . . . , s/2tuple pairs of indices. There is a general formula for the resulting expression (cfr. Eq. (2.2) in [42] and [43], and (2.44) in [44], where the connection with Legendre polynomials is also explained), where the sum is over the number p of traces (or of Kronecker δ's) in the right hand side, ⌊s/2⌋ is the largest integer smaller than or equal to s/2, is the coefficient of x s−2p in the Legendre polynomial P s (x) of order s (in the standard normalization P s (1) = 1), 16) and curly brackets indicate complete symmetrization with respect to the free indices inside the brackets, with the sum over the permutations π of 12 · · · s. For products of spin operators S and a vector q, Eq. (D. 13) gives Recall that for a particle of spin j χ , The quantity c n,k is the coefficient of x n−2k in the monic Legendre polynomial P n (x) of degree n (in a monic polynomial, the coefficient of the term of highest degree is equal to 1), The first few cases, relevant for WIMPs of spin up to 2, are The coefficients can be compared to those appearing in the Legendre polynomials x , (D.28) The reason for the equality of these coefficients is that Legendre polynomials are the expressions in polar angles of the symmetric traceless tensors that define electrostatic multipoles. The identities that connect these quantities are where θ is the angle between q and r.