Analysis strategies for general spin-independent WIMP–nucleus scattering

Analysis strategies for general spin-independent WIMP–nucleus scattering

Martin Hoferichter Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA    Philipp Klos Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany    Javier Menéndez Department of Physics, The University of Tokyo, 113-0033 Tokyo, Japan    Achim Schwenk Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

We propose a formalism for the analysis of direct-detection dark-matter searches that covers all coherent responses for scalar and vector interactions and incorporates QCD constraints imposed by chiral symmetry, including all one- and two-body WIMP–nucleon interactions up to third order in chiral effective field theory. One of the free parameters in the WIMP–nucleus cross section corresponds to standard spin-independent searches, but in general different combinations of new-physics couplings are probed. We identify the interference with the isovector counterpart of the standard spin-independent response and two-body currents as the dominant corrections to the leading spin-independent structure factor, and discuss the general consequences for the interpretation of direct-detection experiments, including minimal extensions of the standard spin-independent analysis. Fits for all structure factors required for the scattering off xenon targets are provided based on state-of-the-art nuclear shell-model calculations.

Dark matter, WIMPs, chiral Lagrangians
95.35.+d, 14.80.Ly, 12.39.Fe

I Introduction

Direct searches for the nuclear recoil produced by weakly interacting massive particles (WIMPs) on target nuclei in large-scale detectors provide a prime avenue to unravel the nature of dark matter, complementary to indirect searches for annihilation remnants in astrophysical observations and the production of dark-matter particles in collider experiments Baudis:2016qwx (). However, for the interpretation of current experimental limits, e.g., Aprile:2012nq (); Agnese:2014aze (); Agnese:2015nto (); Akerib:2015rjg (); Xiao:2015psa (); Angloher:2015ewa (); Amole:2016pye (); Agnes:2015ftt (); Armengaud:2016cvl () it is crucial that the nuclear aspects of direct-detection experiments be adequately addressed. This is especially important given the impressive experimental efforts that include future liquid-noble-gas ton-scale experiments already in commissioning such as XENON1T Aprile:2015uzo (), DEAP-3600 Boulay:2012hq (), and ArDM Calvo:2015uln (), or in planning phase, LZ Akerib:2015cja (), XENONnT Aprile:2014zvw (), XMASS Liu:2014hda (), DarkSide-20k DarkSideproject:2016zfx (), and DARWIN Aalbers:2016jon (); but also smaller-scale experiments such as SuperCDMS SNOLAB Brink:2012zza (), DAMIC100 Chavarria:2014ika (), or CRESST Angloher:2015eza () that focus on light WIMPs with masses below .

Standard analyses of WIMP–nucleus scattering are formulated in terms of spin-independent (SI) and spin-dependent (SD) searches Engel:1992bf (), named after the nature of the WIMP–nucleon interactions at low energies. At the same time, SI and SD scattering are characterized by a very different scaling of the corresponding structure factors: while for SI scattering the response is proportional to the total number of nucleons , the scale of SD scattering is set by the spin expectation value of the unpaired nucleon. Due to the coherent enhancement of SI interactions, the corresponding limits on the WIMP–nucleon couplings set by direct-detection experiments are orders of magnitude more stringent than for SD searches, but each type of interaction is sensitive to different operators for the coupling of WIMPs with Standard-Model fields. For instance, while quark–WIMP scalar–scalar and vector–vector terms contribute to the SI response, the SD interaction is generated by axial-vector–axial-vector operators. Additional information on the WIMP nature can be extracted from inelastic scattering off the target nuclei Baudis:2013bba (); McCabe:2015eia ().

Corrections to standard SI and SD responses are conveniently studied in terms of effective field theories (EFTs). In this context, the calculation of nuclear structure factors has been organized in two different ways: first, non-relativistic EFT (NREFT) for nucleon and WIMP fields Fan:2010gt (); Fitzpatrick:2012ix (); Fitzpatrick:2012ib (); Anand:2013yka () allows a study of the nuclear responses as a function of the effective couplings in the EFT, and to extract limits on the coefficients of the NREFT operators Schneck:2015eqa (). Second, in order to translate the NREFT limits to the parameter space of a given new-physics model, the QCD dynamics integrated out in the NREFT approach needs to be included. Particularly important are the consequences of the spontaneous breaking of the chiral symmetry of QCD, which can be explored within the framework of chiral EFT (ChEFT), see Refs. Epelbaum:2008ga (); Machleidt:2011zz (); Hammer:2012id (); Bacca:2014tla () for recent reviews. The analysis within ChEFT establishes relations between different NREFT operators, and provides a counting scheme that indicates at which order contributions beyond the single-nucleon level Prezeau:2003sv (); Cirigliano:2012pq (); Menendez:2012tm (); Hoferichter:2015ipa () need to be included. Recent work in this direction includes ChEFT-based structure factors for the SD response Menendez:2012tm (); Klos:2013rwa (), aspects of SI scattering Cirigliano:2012pq (); Cirigliano:2013zta (); Vietze:2014vsa (), inelastic scattering Baudis:2013bba (), as well as a general ChEFT analysis of one- and two-body currents Hoferichter:2015ipa ().

In the present work we provide a generalization of SI scattering that includes all coherent contributions up to third order in ChEFT Hoferichter:2015ipa (). This involves considering two-body currents, but also momentum corrections to the nucleon form factors predicted at the same ChEFT order. We provide a detailed discussion of the structure factor associated with the scalar two-body current studied before in the literature, and extend the analysis to include the two-body current generated by the coupling of the trace anomaly of the QCD energy-momentum tensor to the pion in flight, which becomes important if the WIMP couples (significantly) via gluonic interactions. In addition, an analysis of the NREFT operators reveals that in general there are six relevant nuclear operators, denoted by , , , , , and  Fitzpatrick:2012ix (); Anand:2013yka (), where corresponds to the standard SI scattering, while a combination of and yields the operator relevant for the SD case. Given that apart from also can be coherently enhanced (which is especially the case in heavy nuclei) and that and interfere, a generalization of the traditional SI analysis should also take the effects from into account Fitzpatrick:2012ix ().

We note that for general SI scattering, new combinations of Wilson coefficients are probed by the two-body currents coupling to the exchanged pion in flight, and also by the corrections to the nucleon form factors and the contributions associated with the operator. This is in contrast to the SD case, where the dominant two-body currents can be absorbed into a redefinition of the one-body structure factors, i.e., the two-body correction is sensitive to the same physics beyond the Standard Model (BSM) as the standard SD interaction Menendez:2012tm (); Klos:2013rwa (). In a similar way to the SI analysis presented here, a more general SD analysis should include the effects of all relevant nuclear operators and two-body currents.

This work is organized as follows. We start with an overview of the main results in Sec. II, where we propose an analysis strategy for direct-detection experiments that generalizes the standard SI case. The general formalism is detailed in Sec. III, where we lay out the decomposition of the WIMP–nucleus scattering rate, collect the relevant nucleon matrix elements, and introduce the Wilson coefficients that parameterize the WIMP–quark and WIMP–gluon interactions. We then formulate a set of generalized structure factors that includes effects from two-body currents, corrections to the nucleon form factors, and the nuclear operator. In Sec. IV we present state-of-the-art nuclear shell-model calculations for the structure factors corresponding to one-body currents in all relevant xenon isotopes, before developing a generalization for the two-body currents in Sec. V. In Sec. VI we discuss the size of the nucleon form-factor corrections as well as the number of independent parameters in generalized SI scattering, and work out in detail the size of the corrections to standard SI scattering for two simple models. We conclude with a short summary in Sec. VII. While our analysis strategy is general, the numerical results presented here are focused on WIMPs scattering off xenon nuclei, leaving the nuclear structure calculations for other targets to future work.

Ii Overview of main results and analysis strategies

Standard analyses of dark-matter direct-detection experiments distinguish between SI and SD scattering based on the nature of the WIMP–nucleon interaction. At the same time, these two cases generate very different nuclear responses, as SI scattering is enhanced by the coherent contribution of all nucleons in the nucleus, whereas the scale of SD scattering is set by a single-nucleon matrix element.

When subleading contributions in EFTs are considered, the classification of the different terms according to the nature of the WIMP–nucleon interaction becomes less useful, given that the coherent enhancement associated with the combined contribution of a significant number of nucleons is also common to NREFT operators that may involve a WIMP or even a nucleon spin operator. Such responses are closer in their experimental signature to the traditional SI interactions in the sense that the associated structure factors are enhanced compared to the single-nucleon case.

Therefore we propose to define generalized SI scattering not by the form of the NREFT operator, but based on whether a coherent enhancement is possible. In this spirit, a general decomposition of the WIMP–nucleus cross section should include the coherently-enhanced corrections generated by

  1. the standard SI isoscalar WIMP–nucleon interaction,

  2. its isovector counterpart,

  3. the interaction of the WIMP with two nucleons via two-body (meson-exchange) currents,

  4. momentum-dependent corrections to the nucleon form factors,

  5. the quasi-coherent response associated with the operator (related to the nucleon spin-orbit operator).

Figure 1: Comparison of the leading structure factors associated with the coherent and quasi-coherent one-body and nuclear responses, the two-body nuclear responses (solid lines) and (dashed lines), and the radius corrections () to the structure factors. The individual contributions are ordered in the legend according to their size at (from top to bottom): the standard SI response corresponding to the isoscalar one-body operator (black), its interference with an isovector contribution (blue) and with the two-body responses and (orange), the purely isovector contribution (green) and the structure factor generated solely by the two-body currents (violet), the momentum-dependent radius correction to (black dashed), and the interference of the standard SI response with the quasi-coherent one-body structure factor (red). The results, representative for all stable xenon isotopes, are shown for the most abundant Xe.

The proposed generalization amounts to the decomposition of the WIMP–nucleus cross section


where is the momentum transfer, the WIMP velocity, and, generically, the nuclear responses are denoted by and the free parameters that include BSM physics by . This cross section includes all coherent contributions mentioned above and all terms up to third order in ChEFT Hoferichter:2015ipa (). First, the standard SI nuclear response, associated with the NREFT operator [see Eq. (III.4) for definitions of the NREFT operators], can be sensitive to protons and neutrons in the same way (isoscalar, ), as considered in standard SI analyses, but also in the opposite way (isovector, ). Given that the heavy nuclei typically used for direct-detection experiments have a substantial neutron excess, the resulting isovector structure factor is coherently enhanced as well. Next, the power counting of ChEFT predicts to this order two-body interactions (parameterized by the nuclear and responses for the coupling to the pion via a scalar current and via the trace anomaly of the QCD energy-momentum tensor , respectively) and momentum-dependent corrections to (represented by ), both of which are coherent. Finally, contributions from subleading NREFT operators can also be significantly coherent, the most relevant being , which is related to the nucleon spin-orbit operator and gives rise to the nuclear response. Here the coherence is also found in both isoscalar and isovector cases.

Equation (II) reflects the different particle, hadronic, and nuclear scales involved in WIMP–nucleus scattering. Within a given new-physics model, WIMPs interact with quark and gluon degrees of freedom, which are then to be embedded into the nucleon sector. In an EFT approach the BSM interaction is encoded in the Wilson coefficients of effective operators, while the nucleon matrix elements are decomposed into nucleon form factors. As a result, the free coefficients , , , , and correspond to a convolution of Wilson coefficients and nucleon matrix elements. In a final step, the nuclear responses , , , and take into account that the scattering occurs in the nucleus, a strongly interacting many-nucleon system. In this work, the relation between the free parameters , , , , and the BSM Wilson coefficients is worked out in Sec. III for the case of a spin- WIMP, see also Eqs. (59)–(62) for the explicit relations. The nuclear responses , , , and are calculated in the framework of the nuclear shell model, with fit functions given for all stable xenon isotopes in Sec. IV for one-body currents and in Sec. V for two-body currents.

The size of the individual terms in Eq. (II) depends on a given new-physics model, which, together with the nucleon matrix elements, fixes the coefficients . Nevertheless the nuclear responses already imply a strong hierarchy by themselves. This is illustrated in Fig. 1, where the different structure factors including interference terms are compared under the assumption that all coefficients are the same. As expected, the dominant correction originates from the interference of isoscalar and isovector responses. Next in the hierarchy is the interference with the two-body responses and . The additional corrections included in Fig. 1 (apart from the pure isovector and pure two-body contributions) vanish at , and are therefore suppressed at small compared to and the two-body structure factors. We have also considered further higher-order NREFT one-body operators, but their contribution is even more suppressed, see Secs. III.4 and IV. Let us emphasize again that the hierarchy of the structure factors in Fig. 1 assumes a common value for the coefficients, but these are not in general independent and relative suppressions or enhancements may occur. In Sec. VI.3 we study the relative size of the isovector and two-body contributions in two simple models, which for instance suggests that the large structure factor tends to be compensated by a large single-nucleon matrix element, leading to a relative two-body effect similar to that of the contribution.

Despite the potential impact of the coefficients on the measured rate, the hierarchy of the nuclear structure factors observed in Fig. 1 is sufficiently pronounced to motivate a minimal extension of the standard SI scattering of the form


with only independent parameters.

Since the nuclear responses can be obtained from nuclear-structure calculations, direct-detection experiments provide constraints on the parameters. Although as discussed above, the limits on the direct-detection rate constrain additional combinations of Wilson coefficients and nucleon matrix elements, so far standard SI analyses have only considered the coefficient , which is then related to the WIMP–nucleon cross section by , with reduced mass . Ideally, to go beyond this approximation a global correlated analysis of direct-detection experiments based on either Eq. (II) or Eq. (II) should be performed in order to determine limits on all parameters at once, which, however, would require the consideration of more than one target nucleus in the analysis.

Barring such a global analysis, one would need to consider slices through the BSM parameter space, e.g., in terms of scans over the Wilson coefficients as in Ref. Cirigliano:2013zta (). Such slices through the parameter space could also be organized in a straightforward extension of present analyses by considering one nuclear response at a time (this is, setting all but one to zero), for instance based on Eq. (II), with parameters [which map onto () Wilson coefficients for a Dirac (Majorana) WIMP]. This would allow one to set limits on different combinations of Wilson coefficients. In particular, due to the role of the two-body responses this kind of analysis would extend the sensitivity of direct-detection experiments to more new-physics couplings than the standard SI single-nucleon cross section studied so far. Depending on the sensitivity of the experiment to the -dependence, the number of relevant structure factors may be reduced, and limits could also be obtained for combinations of the coefficients associated with responses with similar -tail, e.g., and . In that case the one-response-at-a-time analysis could also be performed based on Eq. (II), which originally depends on non-independent coefficients.

In conclusion, we provide a parameterization of the WIMP–nucleus cross section for general SI scattering, which could be applied to generalize the extraction of limits from SI scattering beyond the standard cross section (corresponding to ), e.g., by similar exclusion plots for the additional coefficients in the minimal -parameter extension in Eq. (II), or by more sophisticated scans through the BSM parameter space. For a xenon target, all necessary structure factors are provided in Secs. IV and V.

Iii Formalism

We consider a WIMP scattering off a target nucleus with momenta assigned as


and momentum transfer


as well as


The rate for the detection of a dark-matter particle scattering elastically off a nucleus with mass number , differential in the three-momentum transfer , is then given by


where denotes the (fiducial) mass of the experiment, and the masses of target nucleus and WIMP, respectively, the WIMP–nucleus cross section in the lab frame, the normalized velocity distribution of the WIMP, the WIMP density,  Smith:2006ym () the escape velocity of our galaxy, and


with up to relativistic corrections. The value for the local WIMP density canonically used in the interpretation of direct-detection experiments is , although halo-independent methods have been developed that allow one to eliminate the astrophysical uncertainties in the comparison of different experiments, see, e.g., Refs. Drees:2008bv (); Fox:2010bz (). Alternatively, the detection rate Eq. (6) is often formulated differential in the recoil energy


The WIMP–nucleus cross section itself combines physics from particle, hadronic, and nuclear scales. To separate the nuclear contributions, can be expressed in terms of structure factors Engel:1992bf ()


where refers to the spin of the target nucleus, denotes the Fermi constant, and and are the structure factors for SI and SD scattering, respectively. These structure factors are normalized according to


with proton and neutron numbers and () and proton/neutron spin expectation values . The constants , contain the information about particle and hadronic physics, a relation to be made more precise below. Assuming , the cross section for SI scattering is often represented in the standard form Schumann:2015wfa ()


with nucleon mass and single-nucleon cross section . The nuclear-physics quantity is the only remnant of the structure factor, and is frequently approximated by Lewin:1995rx ()


whose square is known as Helm form factor.

In the following, we revisit this formalism starting from an effective Lagrangian for the interaction of the WIMP with Standard-Model fields presented in Sec. III.1. In Secs. III.2 and III.3 we discuss the relevant nucleon couplings and finally in Sec. III.4 we derive a generalized decomposition for SI scattering that includes two-body currents and the nuclear response.

iii.1 Lagrangian and Wilson coefficients

We consider the following dimension- and - effective Lagrangian for the interaction of a spin- WIMP with quark and gluon fields


where is assumed to be a Standard-Model singlet, the quark masses have been included to make the scalar operator renormalization-group invariant, and the Wilson coefficients parameterize the BSM physics associated with the scale . The effective Lagrangian is defined at the hadronic scale, with the quark sum extending over , after the heavy quarks have been integrated out and their effect has been absorbed into a redefinition of the gluon coefficient , see Eq. (16). In the second formulation of the dimension- Lagrangian the gluon term has been replaced in favor of the trace of the QCD energy-momentum tensor . Equation (III.1) includes the leading operators relevant for coherent WIMP–nucleus scattering, vector and scalar channels, but also retains the axial-vector operator to facilitate the comparison to the SD case. The WIMP could either be a Dirac or Majorana particle, with in the latter case. At dimension , there are spin- operators that can become relevant for the SI scattering of heavy WIMPs Hill:2014yxa (), but their inclusion will be left for future work. Similarly, the operator basis changes for different quantum numbers of the WIMP Goodman:2010ku (); Hill:2014yxa ().

Throughout this work we follow the chiral counting formulated in Refs. Cirigliano:2012pq (); Hoferichter:2015ipa () to organize the calculation. In particular, this implies that momentum corrections to the one-body matrix elements occurring in Eq. (III.1) enter at the same order as the leading two-body contributions, at third order in ChEFT Hoferichter:2015ipa (). The nucleon matrix elements of the operators listed in Eq. (III.1) involve a combination of Wilson coefficients and nucleon couplings. In the next sections, we spell out these combinations, closely following the notation introduced in Ref. Hoferichter:2015ipa ().

iii.2 Scalar couplings

For the scalar channel in Eq. (III.1) we need the following coupling to the nucleon ( or )


where the nucleon scalar form factors are defined as


The form factors for the heavy quarks appear together with the modified gluon Wilson coefficient


after integrating out their effect by means of the trace anomaly of the energy-momentum tensor  Shifman:1978zn (), which also produces


It should be noted that this procedure is accurate at , which may not be sufficient for the quark, see Refs. Kryjevski:2003mh (); Vecchi:2013iza (); Hill:2014yxa () for a study of higher orders in .

We begin with the discussion of Eq. (14) at vanishing momentum transfer, in which case the form factors simply reduce to the scalar couplings of the nucleon. Based on chiral perturbation theory (ChPT), it can be shown that the couplings to and quarks only depend on the value of the pion–nucleon -term , while isospin-breaking corrections are fully determined by the same low-energy constant that governs the strong contribution to the proton–neutron mass difference Crivellin:2013ipa (). Combining dispersive techniques Hoferichter:2015hva () with precision data for the pion–nucleon scattering lengths extracted from pionic atoms Baru:2010xn (); Baru:2011bw () leads to the phenomenological values Hoferichter:2015dsa () for the light-quark couplings quoted in the first line of Table 1. More recently, lattice calculations at physical quark masses have produced significantly lower values for  Durr:2015dna (); Yang:2015uis (); Abdel-Rehim:2016won (); Bali:2016lvx (), which translates to the tension in the scalar couplings shown in Table 1. This tension between phenomenology and lattice Hoferichter:2016ocj () currently constitutes the largest uncertainty in the and couplings.

Hoferichter:2015dsa ()
Durr:2015dna ()
Table 1: Scalar and couplings of the nucleon, in units of .

In contrast to the and quarks, a determination of the scalar coupling to the quark from phenomenology requires the use of relations, whose convergence properties make reliable uncertainty estimates difficult. For this reason, in Table 2 we only quote the values obtained by recent lattice calculations, together with the average from Ref. Junnarkar:2013ac () of previous lattice results. In particular, we assume isospin symmetry for . Finally, Ref. Abdel-Rehim:2016won () also provides a value for the coupling, , to be compared with as extracted from the same reference based on Eq. (III.2) (with ). Within uncertainties, the direct determination from lattice QCD thus agrees with the result extracted by means of the trace anomaly at .

Ref. Durr:2015dna () Yang:2015uis () Abdel-Rehim:2016won () Bali:2016lvx () Junnarkar:2013ac ()
Table 2: Scalar coupling of the nucleon, in units of .

Next, we turn to the finite-momentum-transfer corrections to .111Here and below, couplings without argument are understood to be evaluated at . To the order we are working in ChEFT, it is generally sufficient to keep the radius corrections, i.e., the first order in the expansion around . However, the strong rescattering in the isospin- -wave makes the leading-loop ChPT prediction for the slope of the scalar form factor of the nucleon at  Gasser:1990ap ()


underestimate the true result by nearly a factor of ( and are taken from Ref. Agashe:2014kda ()). For this reason, we make use of the updated dispersive analysis from Refs. Hoferichter:2012wf (); Ditsche:2012fv () and use


Retaining the leading isospin-breaking effect, this correction amounts to the replacement


where we have used  Aoki:2013ldr ().

In analogy to Eq. (18), there is a parameter-free prediction from leading-loop ChPT for the slope of the strangeness radius Cirigliano:2012pq ()


where parameterizes the leading couplings. Numerically, we use as extracted from semileptonic hyperon decays Ratcliffe:1995fk (); Yamanishi:2007zza (), which together with the constraint implies


However, such leading-loop low-energy theorems are known to be sensitive to higher-order corrections Hemmert:1998pi (); Hammer:2002ei (). Therefore, we also considered the coupled-channel dispersive analysis Hoferichter:2012wf (), which in principle provides not only a prediction for but also for . Unfortunately, convergence of the dispersive integrals is much slower for the slope of the strangeness form factor, although the resulting values are not too far from the chiral prediction. All in all, the spread observed in both methods would be covered by a range


leading to


In view of the substantial uncertainties already encountered in the strangeness form factor, we do not make an attempt to quantify radius corrections for the heavy quarks. The leading chiral result, however, can be reconstructed by means of Eq. (III.2) and


Taking everything together, we arrive at the following decomposition of the combination of Wilson coefficients and nucleon form factors relevant for the scalar channel


For the scalar two-body matrix element we also need the couplings to the pion




In Eq. (III.2) we introduced a factor in analogy to the scalar coupling to the nucleon, Eq. (14). The necessity of defining two pion couplings, and in Eq. (III.2), traces back to the fact that the couplings of the scalar current and the trace anomaly of the energy-momentum tensor to the pion differ qualitatively: while the former is constant up to higher-order corrections, the latter becomes momentum dependent and therefore produces a different nuclear structure factor.

iii.3 Vector and axial-vector couplings

In the vector channel there are two sets of couplings to the nucleon


with related to the Dirac and Pauli terms, respectively, in the decomposition of the nucleon form factors of the electromagnetic current. A decomposition analogous to Eq. (26) is given by


Since the matrix element of the Pauli form factor vanishes at zero momentum transfer, the leading term in is sufficient. Assuming isospin symmetry (for corrections see Ref. Kubis:2006cy ()), these couplings expressed in terms of nucleon radii and anomalous magnetic moments become Hoferichter:2015ipa ()


and for the neutron couplings. Numerical values for the nucleon radii and anomalous magnetic moments are collected in Table 3.

Table 3: Nucleon radii and anomalous magnetic moments. The values of , , and are taken from Ref. Agashe:2014kda (), from Ref. Antognini:1900ns (), and as well as from a global analysis of parity-violating asymmetry data Gonzalez-Jimenez:2014bia (). Note that the latter two are strongly correlated, with a correlation coefficient .

For completeness, we also quote the analogous decomposition for the axial-vector channel appearing in Eq. (III.1). In this case, one needs the combinations and with


where the upper/lower sign refers to proton/neutron and the small contribution of the last line above is generally neglected in SD analyses. These relations involve the nucleon spin matrix elements , for which we have assumed isospin symmetry and already used the combinations


Due to the axial anomaly, the singlet combination cannot be analyzed in ChPT, as effects related to the will play a role. However, it can be extracted from the spin structure function of the nucleon, which, at and to order , produces  Airapetian:2006vy (). Further, the dominant radius correction occurring in the isovector contribution in Eq. (III.3) has been included by a dipole ansatz with mass parameter around  Bernard:2001rs (); Schindler:2006it (), while the pseudoscalar poles in prevent a Taylor expansion in .

iii.4 Structure factors

For the definition of the nuclear structure factors we first consider the matching of the one-body operators obtained in ChEFT above onto the NREFT basis of Refs. Fitzpatrick:2012ix (); Anand:2013yka (). This produces the matrix elements


where we have dropped the nucleon and WIMP spinors.222For details see Ref. Hoferichter:2015ipa (). This matching is performed at tree level and hence does not include effects from operator evolution, which could be generated when running the ChEFT operators down to nuclear scales. The NREFT operators are defined by


with spins and velocity


The combination of the different operators in Eq. (III.4) demonstrates how QCD constraints impose relations between the NREFT operators: for the axial-vector channel it is a fixed combination of and that contributes, while the same coefficient that multiplies also appears as a momentum-dependent correction to .

In Eq. (III.4) we only retained those channels that generate coherent or quasi-coherent nuclear responses, compared to the full list studied in Ref. Hoferichter:2015ipa (). These coherent and quasi-coherent responses are denoted as and in Refs. Fitzpatrick:2012ix (); Anand:2013yka (), and are only a subset of the six different nuclear responses generated by the NREFT operators, which also include the , , , and responses. For example, governs standard SI scattering, and it is a combination of and that enters in SD scattering.

Beyond the single-nucleon sector, NREFT operators that involve can be decomposed into two parts Fitzpatrick:2012ix (); Anand:2013yka (). First, there are terms proportional to the relative WIMP velocity with respect to the center-of-mass of the nucleus


where is the sum of the initial and final nucleon momenta. These terms are effectively suppressed by the WIMP velocity with respect to the target and will thus be neglected in the following. Second, also produces contributions involving the velocity operator of the nucleon, which are part of the , , and responses and come with a milder suppression factor . This is the case for the contribution kept in Eq. (III.4), which generates a response. In the end, for coherent SI scattering only scalar and vector interactions remain, and the fact that the response is due only to the vector operator could serve as a tool to discriminate between these two channels.

Figure 2: Diagrams for WIMP–nucleon interactions in ChEFT. Solid (dashed) lines denote nucleons (pions) and crosses the coupling to the WIMP current. Diagram represents a leading one-body term, a radius correction, and a two-body current.

Apart from the one-body operators and the momentum corrections as summarized in Secs. III.2 and III.3, there are two-body currents at the same order in ChEFT, see Fig. 2. The corresponding NR amplitudes take the form


where and are defined in Eq. (III.2), and denote the spin and isospin Pauli matrices of nucleon , respectively, and . Diagrammatically, these amplitudes represent the coupling of the WIMP to the pion in flight via a scalar current and by means of the QCD trace anomaly . The other two-body currents identified in Ref. Hoferichter:2015ipa () in general involve isospin operators as well as spin structures that, after summing over spins, make the diagrams vanish. The only remaining contribution is the exchange diagram from the axial-vector–vector channel, whose isospin structure becomes , only allowing for an isovector coherent enhancement suppressed by with respect to the scalar two-body current. In addition, this two-body current is linear in and does not interfere with  Fitzpatrick:2012ix (). Other contributions such as the vector–vector two-body current also show isovector coherent enhancement only, and are further suppressed in the ChEFT expansion Hoferichter:2015ipa (). For these reasons, we restrict our analysis to the contributions given by Eq. (III.4). It is the presence of the term in the relation between quark-mass and trace-anomaly couplings that necessitates the definition of two structure factors: for a constant term, the contribution could be absorbed into a redefinition of , similarly to in the case of the nucleon coupling [see Eqs. (14) and (III.2)].

In this context, several comments on the role of two-body operators are in order. First, the hierarchy of diagrams shown in Fig. 2 assumes the ChEFT counting originally proposed by Weinberg Weinberg:1990rz (); Weinberg:1991um (). In this counting, the coupling of the scalar current to contact operators is suppressed by two orders in the chiral expansion. Due to the limitations of Weinberg counting, this suppression might be less pronounced in practice, as indicated, e.g., by KSW counting Kaplan:1998tg (); Kaplan:1998we () or by general arguments related to the short-range behavior of nucleon–nucleon wave functions Valderrama:2014vra (). The role of such contact operators at heavy pion masses has been studied in Ref. Beane:2013kca () using lattice QCD, and calculations at or close to the physical pion mass would allow for a check of the ChEFT counting employed here.

Second, while diagram corresponds directly to an NREFT operator from Refs. Fitzpatrick:2012ix (); Anand:2013yka (), the mapping of diagrams and would proceed in an indirect way. The radius corrections are represented by -dependent prefactors of the , see Eq. (III.4). The two-body contributions could be modeled as effective one-body operators, if summed over the second nucleon with respect to a given reference state, symbolically written as , so that the effective one-body operator would become density and state dependent. Such a normal-ordering approximation with respect to a Fermi gas was used in the context of SD scattering Menendez:2012tm (); Klos:2013rwa (). However, in this work we perform a full calculation in harmonic-oscillator basis states, see Sec. V. It is the explicit calculation of all diagrams within ChEFT, instead of a parameterization in terms of effective one-body operators, that allows one to relate the coefficients of the nuclear structure factors to nucleon form factors and new-physics parameters.

For the construction of suitable nuclear structure factors for generalized SI scattering, we first turn to the SD case. Here the result in Eq. (III.3) shows that once the contribution to is neglected, only two independent combinations of Wilson coefficients remain, which can be conveniently identified with the coefficients introduced in Eq. (III)


where for a Dirac (Majorana) spin- WIMP. The structure factor can therefore be decomposed as


or, in terms of so-called proton-only and neutron-only structure factors,


Since both the momentum corrections in Eq. (III.3) and the leading two-body currents Klos:2013rwa () also depend on and only, this implies that the definition of the structure factors Eq. (40) remains applicable even once such corrections are included. In fact, in a normal-ordering approximation the effect from two-body currents amounts to a shift , with predicted from ChEFT. The connection between experimental limits for the direct-detection rate and the Wilson coefficients therefore still proceeds by means of Eq. (III.4).

Our aim is to find a similar decomposition for SI scattering. More precisely, we wish to formulate a set of structure factors that captures the leading corrections, taking into account both the ChEFT expansion and coherence effects in the nucleus, in particular including both one- and two-body operators.

As a first step towards the construction of generalized SI structure factors, we again identify the couplings at vanishing momentum transfer. In this limit we obtain


Indeed, for