Form factors for dark matter capture by the Sun in effective theories

In the effective theory of isoscalar and isovector dark matter-nucleon interactions mediated by a heavy spin-1 or spin-0 particle, 8 isotope-dependent nuclear response functions can be generated in the dark matter scattering by nuclei. We compute the 8 nuclear response functions for the 16 most abundant elements in the Sun, i.e. H, $^{3}$He, $^{4}$He, $^{12}$C, $^{14}$N, $^{16}$O, $^{20}$Ne, $^{23}$Na, $^{24}$Mg, $^{27}$Al, $^{28}$Si, $^{32}$S, $^{40}$Ar, $^{40}$Ca, $^{56}$Fe, and $^{58}$Ni, through numerical shell model calculations. We use our response functions to compute the rate of dark matter capture by the Sun for all isoscalar and isovector dark matter-nucleon effective interactions, including several operators previously considered for dark matter direct detection only. We study in detail the dependence of the capture rate on specific dark matter-nucleon interaction operators, and on the different elements in the Sun. We find that a so far neglected momentum dependent dark matter coupling to the nuclear vector charge gives a larger contribution to the capture rate than the constant spin-dependent interaction commonly included in dark matter searches at neutrino telescopes. Our investigation lays the foundations for model independent analyses of dark matter induced neutrino signals from the Sun. The nuclear response functions obtained in this study are listed in analytic form in an appendix, ready to be used in other projects.


Introduction
The quest for dark matter is at a turning point. Data from direct, indirect and collider searches for dark matter with unprecedented exposure, resolution and extension in energy will finally be available during the next 5-10 years [1][2][3][4][5][6][7]. Efficient strategies to globally interpret these data in terms of dark matter particle mass and interaction properties are of prime importance in astroparticle physics [8][9][10].
Effective theory methods have proven to be a very powerful tool in the analysis of collider data [11][12][13][14], dark matter direct [15][16][17][18][19][20][21][22][23] and indirect [24][25][26] detection experiments, and in combined studies of these different strategies [27,28]. The main advantage of the effective theory approach to dark matter is that it allows for a model independent interpretation of the different observations when all relevant interaction operators are simultaneously explored in multidimensional statistical analyses, as for instance in [16][17][18]. In contrast, comparing a simplistic model for dark matter to observations, important physical properties can be missed, and spurious correlations among physical observables can be enforced by the inappropriately small number of model parameters.
In the context of effective theories for dark matter, the dark matter-nucleus interaction plays a special role, in that its exploration is complicated by non trivial properties related to the internal structure of the nuclei in analysis. The effective theory of dark matter-nucleon interactions [29,30] predicts that 8 independent nuclear response functions -or form factorscan be generated in the dark matter scattering by nuclei. The interpretation of any dark matter experiment probing the dark matter-nucleus interaction is unavoidably affected by the uncertainties within which the 8 nuclear response functions are known. Experiments of this type are dark matter direct detection experiments, and neutrino telescopes searching for neutrinos produced by dark matter annihilations in the Sun. This latter signal depends on the rate of dark matter capture by the Sun, which is in turn determined by the strength of the dark matter interactions with nuclei.
Nuclear response functions needed for model independent analyses of dark matter direct detection experiments have been calculated in [29,31] under the assumption of one-body dark matter-nucleon interactions. Two-body interactions have also been included in [32,33] in an investigation of spin-dependent dark matter-nucleus currents. The nuclear response functions for isotopes of Xe, I, Ge, Na, and F found in these works have been applied to complementary analyses of current direct detection experiments [16,20], and in studies of the prospects for dark matter direct detection [17,18].
In this paper we calculate the 8 nuclear response functions generated in the dark matter scattering by nuclei for the 16 most abundant elements in the Sun. We then use the novel response functions to calculate the rate of dark matter capture by the Sun within the general effective theory of isoscalar and isovector dark matter-nucleon interactions mediated by a heavy spin-1 or spin-0 particle. In the analysis, we comprehensively describe how the capture rate depends on specific dark matter-nucleon interaction operators, and on the elements in the Sun. This study constitutes the first step towards robust model independent analyses of dark matter induced neutrino signals from the Sun.
The paper is organized as follows. In Sec. 2 we provide the equations for computing the rate of dark matter capture by the Sun given an arbitrary dark matter-nucleon interaction. In Sec. 3 we review the effective theory of dark matter-nucleon interactions, while in Sec. 4 we calculate the 8 nuclear response functions predicted by the theory for the most abundant elements in the Sun. We calculate the dark matter capture rate for all isoscalar and isovector dark matter-nucleon interactions in Sec. 5, and we conclude in Sec. 6. The dark matter response functions and the single-particle matrix elements needed in the analysis are listed in the Appendixes A and B, respectively. Finally, in Appendix C we provide the nuclear response functions of this work in analytic form.

Dark matter capture by the Sun
Dark matter particles of the galactic halo with interactions at the electroweak scale can be gravitationally captured by the Sun. For a dark matter particle of mass m χ at a distance R from the center of the Sun, the rate of scattering from a velocity w to a velocity less than the local escape velocity v(R) is given by [34] In Eq. (2.1), E k = m χ w 2 /2, dσ i /dE is the differential cross-section for dark matter scattering by nuclei of mass m i and density n i (R) in the Sun, q is the momentum transfer and E = q 2 /(2m i ) the nuclear recoil energy. The sum in the scattering probability extends over the most abundant elements in the Sun, and the dimensionless parameters µ i and µ ±,i are defined as follows The velocity u in Eq. (2.1) is the velocity of the dark matter particle at R → ∞, where the Sun's gravitational potential is negligible. The relation between u and w is w = u 2 + v(R) 2 , and therefore Ω − v (w) depends on R. In Eq. (2.1), we consider the general case in which the differential scattering-cross section depends both on the momentum transfer q, and on the dark matter-nucleus relative velocity w. We therefore relax the assumption of constant total cross-section, commonly made in this context. This generalization is important in the study of arbitrary dark matter-nucleus interactions, as we will see in the next sections.
Consider now a population of halo dark matter particles with speed distribution at infinity given by f (u). A fraction of them will be captured by the Sun, with a differential capture rate per unit volume given by [34] 3) The total capture rate takes the following form where we integrate over a sphere of radius R , corresponding to the volume of the Sun. The aim of this work is to evaluate Eq. (2.4) within the most general effective theory for dark matter-nucleon interactions mediated by heavy spin-1 or spin-0 particles, using for each element in the Sun the appropriate nuclear response functions. In our calculations we consider the most abundant elements in the Sun, and use the densities n i (R) and the velocity v(R) as implemented in the darksusy code [35]. Accordingly, we include in the analysis the following 16 elements: H, 3 He, 4 He, 12 C, 14 N, 16 O, 20 Ne, 23 Na, 24 Mg, 27 Al, 28 Si, 32 S, 40 Ar, 40 Ca, 56 Fe, and 59 Ni. Finally, we assume the so-called standard halo model [36], with a Maxwell-Boltzmann speed distribution for f (u), and a local standard of rest velocity of 220 km s −1 . We leave an analysis of astrophysical uncertainties [37][38][39] in the evaluation of Eq. (2.4) for future work.

Dark matter-nucleus scattering
In this section we review the theory of dark matter scattering by nucleons and nuclei [29].

Dark matter-nucleon effective interactions
The amplitude for dark matter-nucleon elastic scattering, M, is restricted by energy and momentum conservation, and respects Galilean invariance, i.e. the invariance under constant shifts of the tridimensional particle velocities. These restrictions determine how M depends on the momenta of the incoming and outgoing particles. Let us denote by p (p ) and k (k ) the initial (final) dark matter and nucleon tridimensional momenta, respectively. Momentum conservation implies that only three out of these four momenta are independent in the scattering process. A possible choice of independent momenta is p, k, and q ≡ k − k , where q is the momentum transferred from the nucleon to the dark matter particle. Whereas q is Galilean invariant, p and k are not. Galilean invariance therefore implies that M must depend on the difference v ≡ p/m χ − k/m N , rather than on p and k separately. v is the initial relative velocity between a dark matter particle of mass m χ and a nucleon of mass m N . In addition to particle momenta, M can depend on the dark matter particle and nucleon spins, j χ and j N , respectively. Any non-relativistic quantum mechanical Hamiltonian leading to a scattering amplitude obeying such restrictions can be expressed as a combination of the following five Hermitian operators The five operators in Eq. (3.1) act on the two-particle Hilbert space spanned by tensor products of dark matter and nucleon states, respectively |p, j χ and |k, j N . The operator 1 χN is the identity in this space, whereasŜ χ andŜ N denote the dark matter particle and nucleon spin operators. Finally, iq is the Hermitian transfer momentum operator, andv ⊥ the relative transverse velocity operator. They are Galilean invariant, and characterized by the matrix elements where µ N is the reduced mass of the dark matter-nucleon system, and r is the position vector from the nucleon to the dark matter particle. Notice that energy conservation implies v · q = −q 2 /(2µ N ), and hence v ⊥ · q = 0, with v ⊥ ≡ v + q/(2µ N ). This justifies the use of the notationv ⊥ . In Eqs. (3.2) and (3.3) we adopt a non-relativistic normalization for single-particle states. Only 14 linearly independent quantum mechanical operators can be constructed from (3.1), if we demand that they are at most linear inŜ N ,Ŝ χ andv ⊥ . They are listed in Tab. 1, and labelled as in [40], where the operatorÔ 2 =v ⊥ ·v ⊥ was neglected since it cannot be a leading-order operator in effective theories. They are at most quadratic in the momentum transfer, with the exception ofÔ 15 , that is cubic inq. The restriction on the number of spin/transverse relative velocity operators is a constraint on the spin of the particle mediating the underlying relativistic interaction, that is assumed here to be less than or equal to 1. Tab. 1 also assumes that the mediating particle is heavy compared to the momentum transfer, i.e. long-range interactions are not included.
The most general Hamiltonian density for dark matter-nucleon interactions mediated by a heavy spin-0 or spin-1 particle is hence a linear combination of 14 non-relativistic quantum mechanical operators,Ô k , and is given bŷ In Eq. (3.4), c p k and c n k are the coupling constants for protons and neutrons as implemented in [40], and have dimension mass −2 . By construction, c p 2 = c n 2 = 0. τ 3 is the third Pauli matrix, and 1 denotes the 2 × 2 identity in isospin space. The matrices (1 ± τ 3 )/2 project a nucleon state into states of well defined isospin, i.e. protons and neutrons. As for the building blocks in Eq. (3.1), the operatorsÔ k act on the two-particle Hilbert space spanned by tensor products of dark matter and nucleon states, |p, j χ and |k, j N , respectively. In the calculation of nuclear matrix elements for dark matter scattering by nuclei with well defined isospin quantum numbers, it is convenient to rewrite Eq. (3.4) in terms of isoscalar and isovector coupling constants:Ĥ (3.5) In Eq. (3.5) t 0 = 1, t 1 = τ 3 , and the isoscalar and isovector coupling constants, respectively, c 0 k and c 1 k , are related to c p k = (c 0 k + c 1 k )/2 and c n k = (c 0 k − c 1 k )/2.

Dark matter-nucleus effective interactions
We construct the dark matter-nucleus interaction Hamiltonian density,Ĥ T (r), from Eq. (3.5) under the assumption of one-body dark matter-nucleon interactions. Within this assumption, H T (r) is the sum of A terms of type (3.5), one for each of the A nucleons in the target nucleuŝ The Hermitian and Galilean invariant quantum mechanical operatorsÔ (i) k (r), k = 1, . . . , 15, are listed in Tab. 1. We use the index i to identify the specific nucleon to which dark matter couples in the scattering. Distinct nucleons are characterized by different isospin matrices t τ (i) . In the single-particle state of the ith-nucleon, it is convenient to separate the motion of the nucleus center of mass from the intrinsic motion (relative to the nucleus center of mass) of the nucleon itself. This separation induces the following coordinate space representations forq andv ⊥ :q The operator ∇ x acts on the nucleus center of mass wave function at x, whereas ∇ y acts on the dark matter particle wave function at y. In Eq. (3.9), m T is the target nucleus mass, and µ T the dark matter-nucleus reduced mass. Finally, the operator ∇ r in Eq. (3.10) acts on the constituent nucleon wave function at r, where r denotes the radial coordinate of the dark matter particle in a frame with origin at the nucleus center of mass (notice that in Sec. 3.1, r was the position vector from the single nucleon to the dark matter particle). Separating the center of mass motion from the intrinsic motion of the constituent nucleons, the only operator depending on the position of the nucleons relative to the nucleus centre of mass, r i , isv ⊥ N (x). Operators in Tab. 1 independent ofv ⊥ N (x) act like the identity 1 i on the ith-nucleon position r i . In the coordinate space representation 1 i corresponds to δ(r − r i ).
Combining Eqs. (3.6), (3.8) and (3.10) with Tab. 1, we can finally write the most general Hamiltonian density for dark matter-nucleus interactions mediated by heavy spin-0 or spin-1 particles as a combination of (one-body) charge and nuclear currents [29]: where σ(i) denotes the set of three Pauli matrices representing the spin operator of the ith-nucleon in the target nucleus, and Inspection of Eq. (3.11) shows that dark matter couples to the constituent nucleons through the nuclear vector and axial charges (first line in Eq. (3.11)), the nuclear spin and convection currents (second line in Eq. (3.11)), and the nuclear spin-velocity current (last line in Eq. (3.11)). The 14 dark matter-nucleon interaction operators in Tab. 1 contribute to these couplings in different ways. For instance, the constant spin-independent operator O 1 contributes to the vector charge coupling through the operatorl τ 0 , whereas the constant spindependent operator O 4 contributes to the nuclear spin current coupling through the operator l τ 5 .
The interaction Hamiltonian relevant for nuclear matrix element calculations is finally obtained by integrating the Hamiltonian density (3.6) over space coordinates This latter integration eliminates the delta functions δ(r − r i ) in Eq. (3.11).

Dark matter-nucleus scattering cross-section
From the dark matter-nucleus interaction Hamiltonian (3.13), one can calculate the amplitude for transitions between initial, |i , and final, |f , scattering states. We denote initial nuclear states by |k T , J, M J , T, M T , where J and T are the nuclear spin and isospin, respectively, and M J and M T the associated magnetic quantum numbers. With this notation |i = |k T , J, M J , T, M T ⊗|p, j χ , M χ (M χ is the spin magnetic quantum number of the dark matter particle, omitted in previous equations for simplicity), and an analogous expression applies to |f . We can therefore write In Eqs. (3.14) and (3.15) we use the result Notice that on the right hand side of Eq. (3.16), q and v ⊥ T ≡ v + q/(2µ T ) replace, respectively,q andv ⊥ T , in agreement with Eqs. (3.2) and (3.3). From now on |v| = w denotes the dark matter-nucleus relative velocity in the Sun. Importantly, each line in the transition amplitude (3.15) is equal to the product of a term containing information on the kinematics of the scattering and on the dark matter-nucleon coupling strength, i.e. l , and a term given by a nuclear matrix element.
In order to evaluate the nuclear matrix elements in Eq. (3.15), we perform a multipole expansion of the nuclear charges and currents using a spherical unit vector basis e λ with z-axis along q, and the identities that holds for any vector A, given a spherical unit vector basis e λ . The vector spherical harmonics in Eq. (3.17) are defined in terms of Clebsch-Gordan coefficients and scalar spherical harmonics: The multipole expansion of the nuclear spin current, for instance, leads to Assuming that nuclear ground states are eigenstates of P and CP , only multipoles that transform as eveneven under P and CP contribute to the square modulus of the transition amplitude. With this assumption Σ LM ;τ (q) does not contribute at all, and is therefore not defined here. Expressions similar to Eq. (3.20) can be derived for the remaining charges and currents. Besides the two operators in Eq. (3.21), four additional nuclear response operators contribute to the transition probability, namely Squaring the amplitude (3.15), summing (averaging) the result over final (initial) spin configurations, and demanding that only multipoles transforming as even-even under P and CP contribute, one finally obtains [29] P tot (w 2 , q 2 ) ≡ 1 2j χ + 1 where the dark matter response function are quadratic combinations of the matrix elements l and are defined in Appendix A. They depend on the momentum transfer, the dark matter-nucleus relative velocity, as well as on the dark matter-nucleon interaction strength.
The nuclear response functions in Eq. (3.23) are defined as follows where A and B correspond to pairs of operators in Eqs.
and it involves Wigner 3j-symbols which cancel in Eq. (3.24) after summing over spin configurations because of their orthonormality. In the next section, we will evaluate our nuclear response functions using the Mathematica package of Ref. [40], which assumes the harmonic oscillator basis with length parameter b = [41.467/(45A −1/3 − 25A −2/3 )] fm for the singleparticle states. In this case, the nuclear response functions in Eq. (3.24) only depend on the dimensionless variable y = (bq/2) 2 .
For the ith-element in the Sun, we can finally write the dark matter-nucleus differential cross-section as follows which constitutes the particle physics input in the calculation of the rate of dark matter capture by the Sun.

Nuclear matrix element calculation
In this section we calculate the reduced nuclear matrix elements that appear in Eq. (3.24) for the most abundant elements in the Sun. We list analytic expressions for the associated nuclear response functions in Appendix C. These expressions can be used by the reader in analyses of dark matter induced neutrino signals from the Sun. We perform this calculation using the Mathematica package introduced in [40], which requires as an input the one-body density matrix elements (OBDME) for ground-state to ground-state transitions of the target nuclei in analysis. We compute these OBDME using the Nushell@MSU program [41], which allows for fast nuclear structure calculations based on the nuclear shell model. In order to relate the nuclear matrix elements in Eq. (3.24) to the underlying OBDME, we expand the nuclear operators in Eqs. (3.21) and (3.22), here collectively denoted by A LM ;τ , in a complete set of spherically symmetric single-particles states, |α . Here we assume the nuclear harmonic oscillator model for the radial part of the wave functions associated with the states |α . Within this assumption, single-particle states can be labelled by their principal, angular momentum and spin quantum numbers, respectively n α , l α and s α , and by their total spin and isospin, respectively j α and t α : Here m jα and m tα denote the total spin and isospin magnetic quantum numbers, whereas |α| represents the set of all non magnetic quantum numbers, i.e. |α = ||α|, m jα ; m tα . With this notation, the nuclear operators in Eqs. (3.21) and (3.22) can be expanded as follows The reduced nuclear matrix elements in Eq. (3.24) can be further reduced in nuclear isospin, and hence written as Using the definition of ground-state to ground-state OBDME, namely, we can finally write the reduced nuclear matrix elements in Eq. (3.24) as follows which is the master equation for nuclear matrix element calculations based on the assumption of one-body dark matter-nucleon interactions. Since the nuclear operators A LM ;τ depend on isospin through the matrices t τ (i) only, the doubly reduced matrix elements in Eq. (4.5) can be further simplified as follows where A L is the part of the operator A L;τ acting on nuclear spin and space coordinates. In Appendix B, we provide explicit expressions for the reduced matrix elements on the right hand side of Eq. (4.6), which in the case of the harmonic oscillator single-particle basis are known analytically, and depend on the momentum transfer through the variable y defined above. The Mathematica package in Ref. [40] provides an efficient implementations of these expressions. We now move on to the OBDME calculation. In this computation, the multipole number L is bounded from above, i.e. L ≤ 2J, whereas τ = 0, 1. In contrast, the indexes |α| and |β| in principle span a complete set of infinite single-particle quantum numbers. The nuclear Element 2J 2T P core-orbits valence-orbits Hamiltonian  Table 2. Summary of element specific input parameters needed for the calculation of the OBDME via the Nushell@MSU code. We use the notation of [46] in defining the major shells. For each element in the Sun, we use a model space comprising the core-orbits and valence-orbits reported in this table.
In the "restrictions" column, we list the energetic orbits not included in the calculation in order to make the computation numerically feasible. The interaction Hamiltonians in the next to last column are described in the review [46], and in the corresponding references.
shell model provides a robust framework to restrict the set of relevant |α| and |β| in the OBDME definition (4.4), and to consistently truncate the infinite sums in Eq. (4.5).
In the nuclear shell model nucleons occupy single-particle states degenerate in the total spin magnetic quantum number called sub-shells, or orbits. Sub-shells are solutions of the Schrödinger equation for an empirically determined nuclear potential (e.g. harmonic oscillator potential, Woods-Saxon potential, etc. . . ). Orbits are labeled with conventions similar to those used for atomic orbitals, e.g. the orbit 0p 1/2 has "principal quantum number" 0, orbital angular momentum 1 and total spin 1/2. Groups of energetically close sub-shells form major shells of progressively increasing energy. The set of fully occupied major shells forms the nuclear core. For instance, the orbits 0s 1/2 , 0p 3/2 and 0p 1/2 divide into the s and p major shells, and together form the core of, e.g, 20 Ne, which contains 8 proton/neutron pairs. Analogously, the orbits 1s 1/2 , 0d 3/2 , and 0d 5/2 form the sd major shell, and the orbits 1p 1/2 , 1p 3/2 , 0f 5/2 , and 0f 7/2 the pf major shell. Nucleons that are not in the nuclear core are called valence nucleons. Not all orbits are accessible to valence nucleons since sizable energy gaps separate adjacent major shells. Therefore, valence nucleons tend to only occupy orbits of sufficiently low energy. Restrictions on the number of nucleons allowed in the most energetic orbits are often imposed in order to reduce the computational effort. The set of orbits that are actually accessible in a calculation constitutes the so-called model space. Within this framework, the sums in Eq. (4.5) only extend over orbits in the assumed model space, since the remaining states do not contribute by construction.
The OBDME for orbits corresponding to the nuclear core can be analytically calculated. Only multipoles of nuclear response operators with L = τ = 0 contribute, since in a nuclear  core all orbits are fully occupied. One finds [47] ψ L;τ |α||β| = 2(2J + 1)(2T + 1)(2j α + 1) δ |α||β| δ τ 0 δ L0 . (4.7) The calculation of the OBDME for the remaining orbits in the model space instead requires a numerical approach. We address this problem using the Nushell@MSU program [41,46]. This code mainly relies on three inputs: the target nucleus spin, isospin and parity; the Hamiltonian for valence nucleon interactions (several options are provided with the code); the model space, including restrictions on the number of nucleons in the most energetic orbits. The assumptions made in our calculations are listed in Tab. 2, and closely follow the guidelines provided in Ref. [46], and references therein. Assigned these inputs, the code first calculates the many-body ground-state wave function of the valence nucleon system Ratio of the capture rate of this work for c 0 1 = 0 to the capture rate computed with darksusy for spin-independent dark matter interactions. We report the ratio of total rates (thick black line), and the ratio of partial rates specific to the 16 most abundant elements in the Sun. The two total rates differ by at most 8%. Right panel. Same as in the left panel, but for c 0 4 = 0. In this case the comparison can be performed for the total rates and for H only, since elements heavier than H are not included in darksusy for dark matter spin-dependent interactions.
diagonalizing the assumed interaction Hamiltonian. Then it evaluates the overlap of this wave function with the single-particle states |α according to Eq. (4.4).
In the present analysis, H constitutes a special case, in that it consists of a single valence nucleon system with no-core. Its OBDME can be trivially calculated as follows [47] ψ L;τ |α||β| = δ |α||γ| δ |β||γ| , (4.8) where |γ| corresponds to the 0s 1/2 orbit. Using the OBDME resulting from the methods outlined above, we evaluate the reduced nuclear matrix elements in Eq. (3.24), and hence the dark matter-nucleus scattering crosssection (3.26) for the most abundant elements in the Sun. This cross-section will allow us to calculate the rate of dark matter capture by the Sun (2.4) for all interaction operators in Tab. 1, as we will see next. The nuclear response functions that we obtain in this analysis, i.e. Eq. (3.24), are listed in Appendix C, and can be used by the reader for other projects.

Numerical evaluation of the capture rate
In this section we numerically evaluate the dark matter capture rate by the Sun, Eq. (2.4), using the nuclear response functions derived in the previous section, and collected in analytic form in Appendix C. We study one operator at the time, and for each interaction operator in Tab. 1, we separately consider the corresponding isoscalar and isovector coupling constants. In the figures, we report the dark matter capture rate as a function of the dark matter particle mass, varying m χ in the range 10 -1000 GeV. When a coupling constant is different from zero, it takes the reference value of 10 −3 m −2 v , with m v = 246.2 GeV. Using the same interaction strength in all panels allows for a straightforward comparison between capture    rates associated with different operators. For definiteness, we assume a spin j χ = 1/2 for the dark matter particle.

Constant spin-independent and spin-dependent interactions
We start with the capture rate for the interaction operatorsÔ 1 andÔ 4 , corresponding to constant, i.e. velocity and momentum independent, dark matter-nucleon interactions. Fig. 1 shows the capture rate C as a function of m χ for the two operators. The top panels refer to the couplings constants c 0 1 and c 0 4 , whereas the bottom panels correspond to c 1 1 and c 1 4 . In the plots we report the total capture rate (thick black line), and partial capture rates specific to the 16 most abundant elements in the Sun. Conventions for colors and lines are those in the legends.
In the case c 0 1 = 0 many elements contribute to C in a comparable manner. The leading contributions come from 4 He, 16 Fig. 2 compares the isoscalar rates of Fig. 1 with the spin-independent and spindependent capture rates computed by darksusy. For constant spin-independent interactions, corresponding to theÔ 1 operator, darksusy uses a simplified version of Eq. (2.1), namely where σ i is the total dark matter-nucleus scattering cross-section in the limit of zero momentum transfer, y = (bq/2) 2 , and  which allows to analytically compute Ω − v (w). In the case of constant spin-dependent interactions, corresponding to theÔ 4 operator, darksusy calculates the capture rate for H only, and neglects other elements. Other interaction operators are not included in the program, and cannot be used for comparison.
The left panel of Fig. 2 shows the ratio of the capture rate of this work for c 0 1 = 0 to the capture rate computed with darksusy for spin-independent interactions. We report the ratio of total capture rates, and the ratio of partial rates specific to different elements in the Sun. Whereas for elements like 56 Fe and 59 Ni the two rates differ up to 25% for m χ 1 TeV, the total rate computed with our nuclear response functions and the one obtained from Eq. (5.1) differ by at most 8%. We conclude that for constant spin-independent dark matter-nucleon interactions, the capture rate is only moderately affected by the use of refined nuclear response functions.
The capture rate for constant spin-dependent dark matter interactions computed with darksusy is systematically smaller than the capture rate of this work for c 0 4 = 0. This effect  is however important for dark matter masses larger than 100 GeV only. Neglecting elements heavier than H, and in particular 14 N, induces an error on the total capture rate of about 25% for m χ 1 TeV, as shown in the right panel of Fig. 2.
In summary, the capture rate for the operatorsÔ 1 andÔ 4 found with the nuclear response functions of this work does not dramatically differ from that of previous studies. However, in the future errors at the 10-20% level on the capture rate induced by simplistic form factors might non negligibly alter the interpretation of a hypothetical signal in terms of dark matter particle mass and interaction properties.

Velocity and momentum dependent interactions
We now move on to our results for the capture rate of the operatorsÔ i , i = 3, 5 . . . , 15. We report these results in Figs. 3, 4, 5, 6, 7, and 8, which show total and partial capture rates as a function of the dark matter particle mass. In each panel the thick black line represents the total capture rate, whereas partial rates correspond to colored lines, as explained in the  legends. Inspection of these figures shows that the most important element in the determination of C significantly depends on the dark matter-nucleon interaction operator, on whether the coupling is of isoscalar or isovector type, and on the value of m χ . Elements that contribute the most to the capture rate for at least one interaction operator, and in a specific dark matter particle mass range are H, 4 He, 14 N, 16 O, 27 Al, 56 Fe and 59 Ni. The existence of a variegated sample of elements important in the dark matter capture process shows the significance of detailed nuclear structure calculations. This conclusion is in particular true for interaction operators that favor dark matter couplings to nuclei heavier, and more complex than H or 4 He.
Also the behavior of the capture rate as a function of the dark matter particle mass strongly depends on the nature of the dark matter-nucleon interaction. In the log-log planes of Figs. 3, 4, 5, 6, 7, and 8, we observe steeply decreasing lines, e.g. for c 1 1 = 0, roughly flat lines, e.g. for c 0 11 = 0, bumps, e.g. for c 0 3 = 0, and even more complex behaviors, like in the case of c 1 6 = 0. Different factors intervene in determining the exact dependence of  the capture rate on m χ , including what element dominates the capture process, its nuclear structure and the resulting nuclear response functions, and the intrinsic momentum/relative velocity dependence of the operator in analysis.
Another important result of this work is to observe that the operatorsÔ 1 andÔ 4 do not necessarily dominate the dark matter capture process. We find that the operator O 11 = iŜ χ ·q/m N generates a total dark matter capture rate larger than that associated with the operatorÔ 4 for values of the dark matter particle mass larger than approximately 30 GeV. This result is clearly illustrated in Fig. 9, where we compare the total dark matter capture rate as a function of m χ for the operatorsÔ 1 ,Ô 4 andÔ 11 assuming isoscalar interactions. As in the previous figures, we consider the same value of the coupling constant, i.e. 10 −3 m −2 v , for the three operators. The relative strength of the three interactions in Fig. 9 is hence determined by the matrix elements of the nuclear response operators M LM ;τ (q), Σ LM ;τ (q) and Σ LM ;τ (q) when evaluated for the most abundant elements in the Sun, and by the intrinsic momentum/relative velocity dependence of the three operators. Notice that the response  Figure 9. Total capture rate for the interaction operatorsÔ 1 ,Ô 4 , andÔ 11 . In the three cases we assume the same value for the isoscalar coupling constant, i.e. 10 −3 m −2 v , with m v = 246.2 GeV (we set to zero the isovector coupling constant). The operatorÔ 11 , though never included in experimental analyses, generates a capture rate larger than that associated with the operatorÔ 4 for m χ 30 GeV. operator M LM ;τ (q) affects the cross-sections generated byÔ 1 andÔ 11 , whereas a linear combination of Σ LM ;τ (q) and Σ LM ;τ (q) determines the cross-section associated withÔ 4 .

Conclusions
We have calculated the 8 nuclear response functions generated in the dark matter scattering by nuclei, i.e. Eq. (3.24), for the 16 most abundant elements in the Sun. We have carried out this calculation within the general effective theory of isoscalar and isovector dark matternucleon interactions mediated by a heavy spin-0 or spin-1 particle. This theory predicts 14 isoscalar and 14 isovector dark matter-nucleon interaction operators with a non trivial dependence on velocity and momentum transfer. In contrast, current experimental searches for dark matter focus on 2 constant spin-independent and spin-dependent interaction operators only.
We have used the nuclear response functions found in this work to calculate the rate of dark matter capture by the Sun for the 14 isoscalar and the 14 isovector dark matter-nucleon interactions separately. We find that different elements contribute to the dark matter capture rate in a significant manner. H, 4 He, 14 N, 16 O, 27 Al, 56 Fe and 59 Ni generate the leading contribution for at least one interaction operator, and in a specific dark matter particle mass range. Detailed nuclear structure calculations, like those performed in this work, are hence crucial to accurately compute the rate of dark matter capture by the Sun, in particular for interaction operators that favor dark matter couplings to nuclei heavier, and more complex than H or 4 He.
Another important result found in this work concerns the operatorÔ 11 = iŜ χ ·q/m N , which couples to the nuclear vector charge operator. For m χ 30 GeV, this operator generates a capture rate larger than the rate induced by the operatorÔ 4 =Ŝ χ ·Ŝ N , i.e. the constant spin-dependent operator commonly considered in experimental searches for dark matter. This result was not known previously, and should be kept in mind in the analysis of dark matter induced neutrino signals from the Sun.
Our findings significantly extends previous investigations, where the dark matter capture rate was calculated for constant dark matter-nucleon interactions only (see however [48] for an interesting exception), and using simplistic nuclear form factors. The nuclear response functions obtained in this work are listed in analytic form in Appendix C, and can be used in model independent analyses of dark matter induced neutrino signals from the Sun.

A Dark matter response functions
Below, we list the dark matter response functions that appear in Eq. (3.26). The notation is the same used in the body of the paper. (A.1)

B Single-particle matrix elements of nuclear response operators
Here we list the single-particle matrix elements of the nuclear response operators in Eqs. (3.21) and ( The reduced single-particle matrix elements of the four independent nuclear response operators are given in the following, where to simplify the equations we use the notation α|| · ||β ≡ n α (l α 1/2)j α || · ||n β (l β 1/2)j β , and [λ] = √ 2λ + 1, for any index λ. They read as follows Eqs. (B.2), (B.3), and (B.4) also appear in the calculation of nuclear matrix elements for electroweak lepton-nucleus interactions. The latter expression is instead needed to evaluate the matrix elements of the nuclear response operatorsΦ and Φ , specific to the dark matternucleus scattering. Different combinations of Wigner 3j, 6j and 9j symbols appear in the equations above, which also depend on residual radial matrix elements of spherical Bessel functions and of their derivatives at ρ = qr i . In the case of the harmonic oscillator singleparticle basis, these radial matrix elements can be analytically evaluated as follows n α l α j α |j L (ρ)|n β l β j β = = 2 L (2L + 1)!! y L/2 e −y (n β − 1)!(n α − 1)!Γ(n α + l α + 1 2 )Γ(n β + l β + 1 2 ) ; y] where y = (qb/2) 2 , and 1 F 1 is the confluent hypergeometric function.