PLB31038S03702693(15)00378010.1016/j.physletb.2015.05.041The AuthorsPhenomenologyTable 1Left: leading chiral order of time (t) and space (x) components of the WIMP and nucleon currents for vector and axialvector interactions, for onebody (1b) and twobody (2b) operators. For the axialvector nucleon operator, terms involving vertices from the NLO chiral Lagrangian (indicated by “2b NLO”) need to be included (see main text for details). The second number (“+2”) refers to the additional suppression originating from the NR expansion of the WIMP spinors, if momentum over WIMP mass is counted in the same way as for the nucleon mass. Right: leading chiral order of the WIMP and nucleon currents for scalar and pseudoscalar interactions.NucleonVA
WIMPtxtx
V1b01+220+2
2b42+224+2
2b NLO––53+2

A1b0+212+20
2b4+222+24
2b NLO––5+23

NucleonSP
WIMP
S1b21
2b35
2b NLO–4

P1b2+21+2
2b3+25+2
2b NLO–4+2
Chiral power counting of one and twobody currents in direct detection of dark matterMartinHoferichterab⁎hoferichter@theorie.ikp.physik.tudarmstadt.dePhilippKlosabAchimSchwenkabaInstitut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, GermanyInstitut für KernphysikTechnische Universität DarmstadtDarmstadt64289GermanybExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, GermanyExtreMe Matter Institute EMMIGSI Helmholtzzentrum für Schwerionenforschung GmbHDarmstadt64291Germany⁎Corresponding author at: Institut für Kernphysik, Technische Universität Darmstadt, 64289, Darmstadt, Germany.Editor: J.P. BlaizotAbstractWe present a common chiral powercounting scheme for vector, axialvector, scalar, and pseudoscalar WIMP–nucleon interactions, and derive all one and twobody currents up to third order in the chiral expansion. Matching our amplitudes to nonrelativistic effective field theory, we find that chiral symmetry predicts a hierarchy amongst the nonrelativistic operators. Moreover, we identify interaction channels where twobody currents that previously have not been accounted for become relevant.KeywordsDark matterWIMPsChiral Lagrangians1IntroductionElucidating the nature of dark matter is one of the most pressing challenges in contemporary particle physics and astrophysics. Still, one of the dominant paradigms rests on a weaklyinteracting massive particle (WIMP), such as the neutralino in supersymmetric extensions of the standard model (SM). A WIMP can be searched for at colliders, in annihilation signals, or in directdetection experiments, where the recoil energy deposited when the WIMP scatters off nuclei is measured. Recent years have witnessed an impressive increase in sensitivity, e.g., from XENON100 [1], LUX [2], and SuperCMDS [3], which will further improve dramatically with the advent of tonscale detectors, XENON1T [4] and LZ [5]. In the absence of a signal, directdetection experiments provide more and more stringent constraints on the parameter space of WIMP candidates. To derive these constraints and to interpret a future signal, it is mandatory that the nucleon matrix elements and the nuclear structure factors, which are required when transitioning from the SM to the nucleon to the nucleus level, be calculated systematically and incorporate what we know about QCD.Effects at the level of the nucleus can be described by an effective field theory (EFT) whose degrees of freedom are nonrelativistic (NR) nucleon and WIMP fields [6,7]. This NREFT has been recently used in an analysis of directdetection experiments [8]. In this approach, scales related to the spontaneous breaking of chiral symmetry of QCD are integrated out, with the corresponding effects subsumed into the coefficients of the EFT. In the context of nuclear forces, such an EFT is called pionless EFT. To derive limits on the WIMP parameter space, information from QCD has then to be included in the analysis in a second step.Alternatively, one can start directly from chiral EFT (ChEFT) to incorporate the QCD constraints from chiral symmetry [9–16], which makes predictions for the hierarchy among one and twobody currents. Based on ChEFT, scalar and axialvector twobody currents were recently considered in [10] and [11,12], respectively. Moreover, lattice QCD can be used to constrain the couplings of twobody currents [17].The goal of this Letter is to combine vector, axialvector, scalar, and pseudoscalar interactions in a common chiral power counting, collect all relevant one and twobody matrix elements, and match the result onto NREFT. This combines our knowledge of QCD at low energies: the onebody matrix elements correspond to the standard decomposition into form factors, while the twobody scalar [9,10], vector [18–20], and axialvector [15,21] currents have been calculated as well, the vector current even at oneloop order. Here, we combine these results for their application in direct detection, extending the axialvector twobody currents to finite momentum transfer and generalizing to the threeflavor case where appropriate. By matching to the NREFT, we find that the chiral symmetry of QCD predicts a hierarchy among the different operators and that twobody currents can be as important as onebody currents in some channels.2Effective Lagrangian and kinematicsWe start from the following dimension6 and 7 effective Lagrangian for the interaction of the WIMP χ, assumed to be a SM singlet, with the SM fields [22](1)Lχ=1Λ3∑q[CqSSχ¯χmqq¯q+CqPSχ¯iγ5χmqq¯q+CqSPχ¯χmqq¯iγ5q+CqPPχ¯iγ5χmqq¯iγ5q]+1Λ2∑q[CqVVχ¯γμχq¯γμq+CqAVχ¯γμγ5χq¯γμq+CqVAχ¯γμχq¯γμγ5q+CqAAχ¯γμγ5χq¯γμγ5q]+1Λ2∑q[CqTTχ¯σμνχq¯σμνq+C˜qTTχ¯σμνiγ5χq¯σμνq]+1Λ3[CgSχ¯χαsGμνaGaμν+CgPχ¯iγ5χαsGμνaGaμν+C˜gSχ¯χαsGμνaG˜aμν+C˜gPχ¯iγ5χαsGμνaG˜aμν], where the Wilson coefficients Ci parameterize the effect of new physics associated with the scale Λ (organizing the interactions in this way assumes Λ to be much larger than the typical QCD scale of 1 GeV). To render the scalar and pseudoscalar matrix elements renormalizationscale invariant we included explicitly the quark masses mq in the definition of the respective operators. We further assumed χ to be a Dirac fermion (in the Majorana case, CqVV=CqVA=CqTT=0), and defined the dual field strength tensor as(2)G˜aμν=12ϵμνλσGλσa, with sign convention ϵ0123=+1. Compared to the operator basis used in [23] we do not include the dimension8 operators related to the traceless part of the QCD energy–momentum tensor. As shown in [23], these operators become relevant for heavy WIMPs and contribute to spinindependent interactions, decreasing significantly the singlenucleon contribution. Finally, we will ignore the tensor operators in (1) and concentrate on the chiral predictions for the V, A, S, P channels.The kinematics for the WIMP–nucleon scattering process are taken as(3)N(p)+χ(k)→N(p′)+χ(k′), the momentum transfer is defined as(4)q=k′−k=p−p′,q2=t, and the pion, η, nucleon, nucleus, and WIMP masses will be denoted by Mπ, Mη, mN, mA, and mχ, respectively (Dirac spinors are normalized to 1). We will also need(5)P=p+p′,K=k+k′.The cross section differential with respect to momentum transfer for the elastic WIMP–nucleus scattering process in the laboratory frame can be expressed as(6)dσdq2=18πv2(2J+1)∑spinsMNR2+O(q0), with nucleus spin J, WIMP velocity v, and NR amplitude MNR defined as(7)M=2mA2mχMNR+O(q2), where M is the relativistic scattering amplitude. In the Majorana case, (6) receives an additional factor of 4.3Chiral power countingWe use the standard chiral power counting [24,25](8)∂=O(p),mq=O(p2),aμ,vμ=O(p), with axialvector and vector sources aμ and vμ. The velocity distribution in dark matter halo models indeed suggests to count the momentum transfer q≲Mπ as O(p) [10]. In the baryon sector we depart from the standard counting in chiral perturbation theory (ChPT) and adopt the more conventional ChEFT assumption (see, e.g., [26–28]) for the scaling of relativistic corrections(9)∂mN=O(p2). This counting is appropriate for a breakdown scale around 500 MeV. As far as the WIMP is concerned, a chiral counting is only required for the NR expansion of the spinors. We assume the same counting as in the nucleon case, but display the corresponding additional powers explicitly. If mχ≳mN, the suppression will be more pronounced, for Mπ≲mχ≲mN the counting should be adapted, and for even smaller mχ the naive counting breaks down.For most of the channels it suffices to consider the leadingorder Lagrangian to determine at which chiral order a given contribution starts. For the onebody matrix elements higher orders are subsumed into the nucleon form factors, which are obtained by their chiral expansion or could be taken from phenomenology. In this work, we consider all contributions up to O(p3). Since the leading twobody terms start at O(p2), this leaves the possibility that the nexttoleadingorder (NLO) pion–nucleon Lagrangian involving the lowenergy constants ci [29] could be required, and this is indeed the case for the spatial component of the axialvector current [11,12] (indicated by “2b NLO” in Table 1). In the same channel, NN contact terms di [30] enter. We define both ci and di in the conventions of [21] (with dimensionless c6 and c7).As a preview of our results, the leading chiral orders of one and twobody currents for time and space components of the axialvector and vector currents, as well as for the scalar and pseudoscalar operators, are listed in Table 1. The suppression by two powers (“+2”) originating from the WIMP spinors is displayed separately. In the following sections, we give results for all one and twobody currents involved in Table 1.4Nuclear matrix elements4.1ScalarAt zero momentum transfer the scalar couplings of the heavy quarks Q=c,b,t can be determined from the trace anomaly of the QCD energy–momentum tensor [31](10)θμμ=∑qmqq¯q+βQCD2gsGμνaGaμν,〈NθμμN〉=mN,βQCD2gs=−(11−2Nf3)αs8π+O(αs2). For Nf=3 active flavors, one obtains(11)〈NmQQ¯QN〉=−αs12π〈NGμνaGaμνN〉=mNfQN, where(12)fQN=227(1−∑q=u,d,sfqN),mNfqN=〈Nmqq¯qN〉. Therefore, at leading order in αs the effect of integrating out the heavy quarks can be absorbed into a redefinition of CgS(13)Cg′S=CgS−112π∑Q=c,b,tCQSS. For the u and dquarks the couplings are intimately related to the pion–nucleon σterm σπN [32](14)fuN=σπN(1−ξ)2mN+ΔfuN,fdN=σπN(1+ξ)2mN+ΔfdN, with ξ=md−mumd+mu=0.36±0.04 [33] and corrections Δfu,dN related to the strong proton–neutron mass difference via the lowenergy constant c5. For the strange quark, the most accurate determination comes from lattice QCD [34]. The above O(αs) analysis may not be accurate enough for the charm quark, see [23,35,36] for a study of higher orders in αs.This analysis generalizes to finite t if one defines(15)mNfqN(t)=〈N(p′)mqq¯qN(p)〉,θ0N(t)=〈N(p′)θμμN(p)〉,fQN(t)=227(θ0N(t)mN−∑q=u,d,sfqN(t)), and replaces fqN→fqN(t), fQN→fQN(t) accordingly.The chiral expansion of σπN starts with(16)σπN=−4c1Mπ2+O(p3), in line with the O(p2) listed in Table 1 for the scalar onebody current. Note, however, that the power 2 does not imply a momentumdependent coupling in this case, but a quarkmass suppression. As far as the tdependence is concerned, the slope of the scalar form factors is dominated by ππ scattering, which is known to not be adequately described by ChPT, but to require a reconstruction based on dispersion relations [37–39]. The tdependence generated by other sources but lightquark scalar form factors was shown to be higher order in the chiral expansion in [10].Defining(17)fN(t)=mNΛ3(∑q=u,d,sCqSSfqN(t)−12πfQN(t)Cg′S), the NR onebody matrix element for the scalar channel becomes11The nucleon spinors include isospin indices according to χs′†fN(t)χs≡12χs′†[(fp(t)+fn(t))1+(fp(t)−fn(t))τ3]χs. The Wilson coefficients match onto the conventions of [40] by means of the identification fN(0)=2GFc0.(18)M1,NRSS=χr′†χrχs′†fN(t)χs, where χr,s (χr′,s′) are NR spinors for the incoming (outgoing) WIMP and nucleon, respectively. M1,NRPS is of higher chiral order since the NR reduction of γ5 produces a term −σ⋅q/(2mχ), which we count as O(p2) for mχ≳mN.4.2VectorThe decomposition of the vector current at the quark level reads(19)〈N(p′)q¯γμqN(p)〉=〈N′γμF1q,N(t)−iσμν2mNqνF2q,N(t)N〉, where the sign of the Pauli term is due to the convention in (4). To obtain a flavor decomposition of the vector current, one usually assumes isospin symmetry (corrections can again be calculated in ChPT [41]):(20)Fiu,p(t)=Fid,n(t),Fid,p(t)=Fiu,n(t),Fis,p(t)=Fis,n(t). In this way, one obtains(21)Fiu,p(t)=Fid,n(t)=2FiEM,p(t)+FiEM,n(t)+Fis,N(t),Fid,p(t)=Fiu,n(t)=FiEM,p(t)+2FiEM,n(t)+Fis,N(t), with electromagnetic form factors FiEM,N(t). At vanishing momentum transfer this defines the vector couplings(22)〈Nq¯γμqN〉=fVqN〈NγμN〉,fVup=fVdn=2fVdp=2fVun=2. Corrections to (22) can be worked out in terms of magnetic moments μN=QN+κN, electric radii 〈rE2〉N, as well as strangeness moments μNs=κNs and radii 〈rE,s2〉N, explicitly(23)F1u,p(t)=2+2(〈rE2〉p6−κp4mp2)t+(〈rE2〉n6−κn4mn2)t+(〈rE,s2〉N6−κs4mN2)t+O(t2),F1d,p(t)=1+(〈rE2〉p6−κp4mp2)t+2(〈rE2〉n6−κn4mn2)t+(〈rE,s2〉N6−κs4mN2)t+O(t2),F1s,N(t)=(〈rE,s2〉N6−κNs4mN2)t+O(t2),F2u,N=κN+O(t),F2d,N=−κN−κNs+O(t),F2s,N=κNs+O(t), with the Sachs form factors(24)GEN(t)=F1N(t)+t4mN2F2N(t)=QN+〈rE2〉N6t+O(t2),GMN(t)=F1N(t)+F2N(t)=μN(1+〈rM2〉N6t)+O(t2).The NR onebody matrix elements involving a nucleon vector current are(25)M1,NRVV=χr′†χrχs′†[f1V,N(t)−q4mN2⋅(q−iσ×P)f2V,N(t)]χs+12mχχr′†[K+iσ×q]χr⋅12mNχs′†iσ×qf2V,N(t)χs,M1,NRAV=12mχχr′†σ⋅Kχrχs′†f1A,N(t)χs−12mNχr′†σχrχs′†[(P−iσ×q)f1A,N(t)−iσ×qf2A,N(t)]χs, where(26)fiV,N(t)=1Λ2∑q=u,d,sCqVVFiq,N(t),fiA,N(t)=1Λ2∑q=u,d,sCqAVFiq,N(t).4.3Axial vectorThe decomposition of the axialvector current at the quark level reads (see, e.g., [42,43])(27)〈N(p′)q¯γμγ5qN(p)〉=〈N′γμγ5GAq,N(t)−γ5qμ2mNGPq,N(t)−iσμν2mNqνγ5GTq,N(t)N〉. GTq,N(t) corresponds to a secondclass current [44], i.e., it violates Gparity, and will be ignored in the following. At vanishing momentum transfer only GAq,N contributes. Its coefficients are conventionally defined as(28)〈N(p)q¯γμγ5qN(p)〉=ΔqN〈Nγμγ5N〉, and isospin symmetry is assumed(29)Δup=Δdn,Δun=Δdp,Δsp=Δsn. The combinations(30)a3p=−a3n=Δup−Δdp=gA,a8N=ΔuN+ΔdN−2ΔsN=3F−D are determined by the axial charge of the nucleon in the case of a3, or can be inferred from semileptonic hyperon decays for a8, yielding D≈0.8, F≈0.46. The third combination(31)ΔΣN=ΔuN+ΔdN+ΔsN is related to the spin structure function of the nucleon, it is not a scaleindependent quantity. At Q2=5 GeV2 and O(αs2) the following values were obtained in [45](32)Δup=0.842±0.012,Δdp=−0.427±0.013,Δsp=−0.085±0.018. Besides the coefficients at zero also the momentum dependence of the flavor combinations(33)Aμ3=Q¯γμγ5λ32Q=12(u¯γμγ5u−d¯γμγ5d),Aμ8=Q¯γμγ5λ82Q=123(u¯γμγ5u+d¯γμγ5d−2s¯γμγ5s) can be analyzed in SU(Nf) ChPT, but due to the anomalously broken U(1)A current this is not the case for the isoscalar component. One obtains(34)〈N(p′)Aμ3N(p)〉=〈N′(γμγ5GA3(t)−γ5qμ2mNGP3(t))τ32N〉,〈N(p′)Aμ8N(p)〉=〈N′(γμγ5GA8(t)−γ5qμ2mNGP8(t))12N〉, with leadingorder results(35)GA3(t)=gA,GA8(t)=3F−D3≡gA8,GP3(t)=−4mN2gAt−Mπ2,GP8(t)=−4mN2gA8t−Mη2. Empirically, the momentum dependence of GA3(t), extracted from neutrino scattering off nucleons and chargedpion electroproduction, follows a dipole fit(36)GA3(t)=gA(1−t/MA2)2, with mass parameter MA around 1 GeV [42,43]. Since for general t the flavor structure cannot be inverted without additional input for the singlet component, we decompose the quark sum according to22At vanishing momentum transfer this equation maps onto the notation of [40] by means of ∑qCqAAΔqN=2GFΛ212(a0+a1τ3).(37)∑qCqAAGA,Pq,N(t)=C0AAGA,P0(t)+C3AAGA,P3(t)τ3+C8AAGA,P8(t), with(38)C0AA=13[CuAA+CdAA+CsAA],C3AA=12[CuAA−CdAA],C8AA=36[CuAA+CdAA−2CsAA], and define(39)gA,PN(t)=1Λ2[C0AAGA,P0(t)+C3AAGA,P3(t)τ3+C8AAGA,P8(t)]. In terms of these quantities, the NR amplitude reads(40)M1,NRAA=−χr′†σχr⋅χs′†[σgAN(t)−q4mN2σ⋅qgPN(t)]χs. Similarly, for the VA channel we define(41)hA,PN(t)=1Λ2[C0VAGA,P0(t)+C3VAGA,P3(t)τ3+C8VAGA,P8(t)] to obtain(42)M1,NRVA=χr′†χr12mNχs′†[σ⋅PhAN(t)−q⋅P4mN2σ⋅qhPN(t)]χs−12mχχr′†[K+iσ×q]χrχs′†[σhAN(t)−q4mN2σ⋅qhPN(t)]χs.4.4PseudoscalarThe pseudoscalar matrix element is usually parameterized as(43)〈N(p′)mqq¯iγ5qN(p)〉=〈N′mNG5q,N(t)iγ5N〉. By means of the Ward identity(44)∑q∂μq¯γμγ5q=∑q2imqq¯γ5q−αsNf4πGμνaG˜aμν, the corresponding form factor G5q,N(t) follows from GAq,N(t) and GPq,N(t), except for the singlet component, where the anomaly does not drop out,(45)G5i(t)=GAi(t)+t4mN2GPi(t),i=3,8. Accordingly, we have(46)M1,NRSP=χr′†χri2χs′†σ⋅qg5N(t)χs,M1,NRPP=12mχχr′†σ⋅qχr12χs′†σ⋅qh5N(t)χs, where(47)g5N(t)=1Λ2[C3SPG53(t)τ3+C8SPG58(t)],h5N(t)=1Λ2[C3PPG53(t)τ3+C8PPG58(t)].5Twobody currents5.1ScalarThe scalar mesonexchange currents, involving both pion and η contributions, have been considered before in [9,10]. The full expression reads(48)M2,NRSS=−χr′†χr(gA2Fπ)2fπMπ2χs1′†χs2′†τ1⋅τ2X12πχs1χs2−χr′†χr(gA2Fπ)2(4α−13)2fηMη2χs1′†χs2′†X12ηχs1χs2, where(49)X12i=σ1⋅q1σ2⋅q2(q12+Mi2)(q22+Mi2),i=π,η, pion decay constant Fπ=92.2 MeV [46], χsi (χsi′) denote NR spinors for the incoming (outgoing) nucleons, with momenta pi (pi′), qi=pi′−pi, α=F/(D+F), and the Wilson coefficients are collected in(50)fπ=1Λ3∑q=u,dCqSSfqπ,fη=1Λ3∑q=u,d,sCqSSfqη, with scalar meson couplings(51)fuπ=mumu+md=0.32±0.03,fdπ=mdmu+md=0.68±0.03, and(52)fuη=13mumu+mdMπ2Mη2=(6.9±0.4)×10−3,fdη=13mdmu+mdMπ2Mη2=(14.7±0.4)×10−3,fsη=23MK02+MK+2−Mπ2Mη2=1.05. One particular feature of the scalar twobody currents is that they cannot be written as a correction to the onebody coupling fN, since the scalar couplings of pions and η mesons probe a different combination of Wilson coefficients [10]. For this reason, even in the isospin limit they cannot be parameterized in terms of a single coupling c0 as conventionally done for the onebody currents, see e.g. [40].5.2VectorThe only twobody vector current up to O(p3) appears in the AV channel(53)M2,NRAV=−2Λ2C3AV(gA2Fπ)2χr′†σχr⋅χs1′†χs2′†i[τ1×τ2]3[σ1⋅q1σ2q12+Mπ2−σ2⋅q2σ1q22+Mπ2+(q1−q2)X12π]χs1χs2. While the nucleon vector current itself has been studied in detail before [18–20], the present application to direct detection is new.In fact, there are neither terms with i=8 nor η contributions to i=3. The reason for this can be traced back to the operator structure of the chiral Lagrangian: the coupling to the vector current occurs via a commutator [vμ,ϕ] of vector source and meson matrix. Expanded in GellMann matrices, this leaves SU(3) structure factors f3ij and f8ij, and the only nontrivial ones, apart from the direct couplings to the nucleon that led to (35), reduce to the SU(2) subset ϵijk.5.3Axial vectorThe axialvector twobody currents are(54)M2,NRAA=1Λ2C3AAχr′†σχr⋅χs1′†χs2′†{[gAFπ2[τ1×τ2]3[c64σ1×q+c4(1−qq2+Mπ2q⋅)σ1×q2]σ2⋅q2q22+Mπ2+2gAFπ2τ23[2c1Mπ2qq2+Mπ2+c3(q2−qq2+Mπ2q⋅q2)]σ2⋅q2q22+Mπ2+2d1τ13(σ1−σ1⋅qqq2+Mπ2)]+(1↔2)+2d2[τ1×τ2]3(σ1×σ2)(1−⋅qqq2+Mπ2)}χs1χs2, where the terms that do not contain an explicit qdependence (q=−q1−q2) and the c6term are taken from [21], while the finiteq pionpole corrections were derived in [15]. The AA twobody current as in [21] has been applied in the calculation of structure factors for spindependent scattering in [11,12], whereas the twobody current in the VA channel,(55)M2,NRVA=−1Λ2C3VAgA2Fπ2χr′†χrχs1′†χs2′†{i[τ1×τ2]3σ2⋅q2q22+Mπ2+(1↔2)}χs1χs2, has not been considered before.For similar reasons as in the vector case there are no i=8 or η contributions from the leadingorder Lagrangian. In principle, one could calculate corrections from the NLO SU(3) Lagrangian, in analogy to the SU(2) result for M2,NRAA. However, there is a large number of poorlyknown lowenergy constants (see [47] or [48] for the matching to SU(2)), which would severely limit the predictive power.Finally, due to the derivative in the Ward identity (44), there are no pseudoscalar twobody currents at O(p3).6Matching to NREFTNext, we express our results in terms of the operator basis from [7](56)O1=1,O2=(v⊥)2,O3=iSN⋅(q×v⊥),O4=Sχ⋅SN,O5=iSχ⋅(q×v⊥),O6=Sχ⋅qSN⋅q,O7=SN⋅v⊥,O8=Sχ⋅v⊥,O9=iSχ⋅(SN×q),O10=iSN⋅q,O11=iSχ⋅q, where S=σ/2 and the velocity is defined as(57)v⊥=K2mχ−P2mN. We find the relations(58)M1,NRSS=χr′†χs′†O1fN(t)χrχs,M1,NRSP=χr′†χs′†O10g5N(t)χrχs,M1,NRPP=1mχχr′†χs′†O6h5N(t)χrχs,M1,NRVV=χr′†χs′†[O1(f1V,N(t)+t4mN2f2V,N(t))+1mNO3f2V,N(t)+1mNmχ(tO4+O6)f2V,N(t)]χrχs,M1,NRAV=χr′†χs′†[2O8f1A,N(t)+2mNO9(f1A,N(t)+f2A,N(t))]χrχs,M1,NRAA=χr′†χs′†[−4O4gAN(t)+1mN2O6gPN(t)]χrχs,M1,NRVA=χr′†χs′†[−2O7+2mχO9]hAN(t)χrχs. This shows that as a result of QCD effects, the operators in the NREFT are not independent. For example, both axial and pseudoscalar operators combine in the nuclear matrix element M1,NRAA. In addition, up to O(p3) only 8 of the 11 operators of (56) are present. However, because M1,NRPS itself enters only at O(p4), they are mapped onto 7 amplitudes, so that the relations cannot be inverted. This is because M1,NRAV and M1,NRVA involve the three operators O7−9. This implies that some operators, e.g. O6, can be isolated by having a particular quarklevel interaction, but this is not possible in general, as demonstrated by the example of O7−9. If we retain subleading corrections in the NR expansion of the spinors, the missing operators appear, accompanied by additional combinations: O11 in terms of M1,NRPS, O2 and O5 in M1,NRVV, O3O8 in M1,NRAV, and O7O8 in M1,NRAA.In the limit where mχ becomes (significantly) larger than the nucleon mass also M1,NRPP should be dropped, as well as the 1/mχ suppressed terms in M1,NRVV and M1,NRVA. In contrast, all twobody currents up to O(p3) are independent of mχ. They appear in the SS, AV, AA, and VA channels.We stress that the above discussion merely pertains to the mapping of operator structures, it does not take into account the evolution of the scale dependence that is required when matching the coefficients of a pionless theory, valid for scales below the pion mass, and ChEFT, defined at chiral scales. This involves also effects related to the limitations of the “Weinberg” counting scheme applied here [49], and would have to be taken into account in the matching relations required for translating NREFT coefficients to the QCD scale. In addition, there may be effects from operator mixing, originating from the interplay between the nucleonspin dependence in the ChEFT WIMP–nucleon scattering operator and that in the highmomentum part of the ChEFT NN potential, which would also have to be considered when evolving NREFT operators to the QCD scale.7Summary and discussionIn this Letter, we have developed the constraints that chiral symmetry of QCD imposes on the nuclear matrix elements that can enter in dark matter direct detection. We provide explicit expressions for one and twobody currents in WIMP–nucleus scattering for vector, axialvector, scalar, and pseudoscalar interactions up to third order in the chiral expansion. The chiral power counting, summarized in Table 1, shows that at this order there are twobody currents that have not been considered and may be of similar or greater importance than some of the onebody operators, see (53) and (55). Moreover, the matching to NREFT shows that not all allowed onebody operators appear at this chiral order and that the operators in the NREFT are not independent.The chiral power counting applies to the one and twonucleon level. In nuclei, the different interactions can lead to a coherent response that scales with the number of nucleons in the nucleus or to a singleparticlelike response. In a next step, we will evaluate the nuclear structure factors, including the contributions from twobody currents, and provide a set of response functions for the analysis of directdetection experiments. This will also allow us to assess how constructive or destructive the interference of operators based on the constraints provided by chiral symmetry proves to be.AcknowledgementsWe thank J. Menéndez for useful discussions, and B. Kubis, U.G. Meißner, and M.J. Savage for comments on the manuscript. 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