On the sensitivity of present direct detection experiments to WIMP-quark and WIMP-gluon effective interactions: a systematic assessment and new model-independent approaches
aa r X i v : . [ h e p - ph ] A ug On the sensitivity of present direct detection experiments toWIMP–quark and WIMP–gluon e ff ective interactions: asystematic assessment and new model–independent approaches. Sunghyun Kang, Stefano Scopel, Gaurav Tomar, Jong–Hyun Yoon
Department of Physics, Sogang University, Seoul, Korea, 121-742
Abstract
Assuming for Weakly Interacting Massive Particles (WIMPs) a Maxwellian velocity distributionin the Galaxy we provide an assessment of the sensitivity of existing Dark Matter (DM) directdetection (DD) experiments to operators up to dimension 7 of the relativistic e ff ective field theorydescribing dark matter interactions with quarks and gluons . In particular we focus on a system-atic approach, including an extensive set of experiments and large number of couplings, bothexceeding for completeness similar analyses in the literature. The relativistic e ff ective theoryrequires to fix one coupling for each quark flavor, so in principle for each di ff erent combinationthe bounds should be recalculated starting from direct detection experimental data. To addressthis problem we propose an approximate model–independent procedure that allows to directlycalculate the bounds for any combination of couplings in terms of model–independent limits onthe Wilson coe ffi cients of the non–relativistic theory expressed in terms of the WIMP mass andof the neutron–to–proton coupling ratio c n / c p . We test the result of the approximate procedureagainst that of a full calculation, and discuss its possible pitfalls and limitations. We also providea simple interpolating interface in Python that allows to apply our method quantitatively. Keywords:
Dark Matter, Weakly Interacting Massive Particles, Direct detection, E ff ectivetheories PACS: + d,
1. Introduction
One of the most popular scenarios for the Dark Matter (DM) which is believed to contributeto up to 27% of the total mass density of the Universe [1] and to more than 90% of the haloof our Galaxy is provided by Weakly Interacting Massive Particles (WIMPs) with a mass in theGeV-TeV range and weak–type interactions with ordinary matter. Such small but non–vanishinginteractions can drive WIMP scattering events o ff nuclear targets, and the measurement of theensuing nuclear recoils in low–background detectors (direct detection, DD) represents the moststraightforward way to detect them. Indeed, a large worldwide e ff ort is currently under way to Email addresses: [email protected] (Sunghyun Kang), [email protected] (Stefano Scopel), [email protected] (Gaurav Tomar), [email protected] (Jong–Hyun Yoon)
Preprint submitted to Astroparticle Physics August 7, 2019 bserve WIMP-nuclear scatterings, but, with the exception of the DAMA collaboration [2, 3,4, 5] that has been observing for a long time an excess compatible to the annual modulation ofa DM signal, many other experiments using di ff erent nuclear targets and various background–subtraction techniques [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] have failed to observe anyWIMP signal so far.The calculation of DD expected rates is a ff ected by large uncertainties, of both astrophysicaland particle–physics nature. For instance, most of the explicit ultraviolet completions of theStandard Model that stabilize the Higgs vacuum contain WIMP exotic states that are viable DMcandidates and for which detailed predictions for WIMP–nuclear scattering can be worked out,leading in most cases to either a Spin Independent (SI) cross section proportional to the square ofthe target mass number, or to a Spin–Dependent (SD) cross section proportional to the productof the WIMP and the nucleon spins. Crucially, this allows to determine how the WIMP interactswith di ff erent targets, and to compare in this way the sensitivity of di ff erent detectors to a givenWIMP candidate, with the goal of choosing the most e ff ective detection strategy. However, thenon–observation of new physics at the Large Hadron Collider (LHC) has prompted the needto go beyond such top–down approach and to use either “e ff ective” or “simplified” models toanalyze the data [19], implying a much larger range of possible scaling laws of the WIMP–nucleon cross section on di ff erent targets. Moreover, the expected WIMP–induced scatteringspectrum depends on a convolution on the velocity distribution f ( ~ v ) of the incoming WIMPs,usually described by a thermalized non–relativistic gas described by a Maxwellian distributionwhose root–mean–square velocity v rms ≃
270 km / s is determined from the galactic rotationalvelocity by assuming equilibrium between gravitational attraction and WIMP pressure. Indeed,such model, usually referred to as Isothermal Sphere, is confirmed by numerical simulations [20],although the detailed merger history of the Milky Way is not known, allowing for the possibilityof the presence of sizable non–thermal components for which the density, direction and speed ofWIMPs are hard to predict [21].As far as the latter issue is concerned, for definiteness, in the following we will adopt forthe velocity distribution f ( ~ v ) of the incoming WIMPs a standard thermalized non–relativistic gasdescribed by a Maxwellian distribution.On the other hand, in the present paper we wish to focus on the former issue of the scal-ing law in direct detection, and in particular on how to compare the sensitivities of di ff erentexperimental set–ups on WIMP–quark and WIMP–gluon e ff ective interactions by making useof model–independent bounds obtained independently at a lower scale on WIMP–nucleon non–relativistic operators. In particular, since the DD process is non–relativistic (NR) it has beenunderstood some time ago [22, 23] that the most general interaction besides the SI and the SDcross sections can be parameterized with an e ff ective Hamiltonian that complies with Galileansymmetry, containing at most 15 terms in the case of a spin–1 / H ( r ) = X τ = , X j = c τ j O j ( r ) t τ , (1)where the O j operators are listed in [23] and t = t = τ denote the 2 × c j and c j , are related to those to protons and neutrons c pj and c nj by c pj = c j + c j and c nj = c j − c j .Indeed, the NR couplings c τ j represent the building blocks of the low–energy limit of anyultraviolet theory, so that an understanding of the behaviour of such couplings is crucial for the2nterpretation of more general scenarios. As a consequence, the NR e ff ective theory (NREFT) ofEq. (1) has been extensively used in the literature to analyze direct detection data [24, 25, 26, 27,28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].In light of this, in Ref. [39] we provided an assessment of the overall present and futuresensitivity of an extensive list of both present and future WIMP direct detection experimentsassuming systematically dominance of one of the possible terms of the NR e ff ective Hamilto-nian in the calculation of the WIMP–nucleon cross section. In particular, compared to previousanalyses adopting the same approach, in Ref. [39] the bounds on the NREFT are presented in anovel model–independent way: for each of the couplings of Eq. (1) a contour plot of the moststringent 90% C.L. bound on the WIMP–nucleon cross section among a comprehensive set of14 existing experiments is provided as a function the WIMP mass m χ and of the ratio of theWIMP–neutron and WIMP–proton couplings c n / c p (along with a color code showing the experi-ment providing it). This approach allows to make the best constraints on the WIMP–proton crosssection available in a model–independent way (in Appendix A a simple code is introduced thatallows to interpolate the m χ – c n / c p planes of Ref. [39] to get the corresponding numerical val-ues) so that, with the exception of cancellations among di ff erent NR operators, such bounds canbe directly used to get constraints for a given relativistic e ff ective DM scenario when taking itsnon–relativistic limit, without the need to go through the calculation of the experimental boundstarting from the data and to apply the standard machinery used in [39]. The latter includes morerefined treatments beside a simple comparison between theoretical predictions and upper bounds,such as background subtraction or the optimal-interval method [40], and may not be trivial formodel builders, who have only access from experimental papers to the bounds on the standardisoscalar spin–independent or WIMP–proton / WIMP–neutron spin–dependent cross sections.However, in spite of its generality, such approach presents some drawbacks: in particular, theinterference of di ff erent NR operators and especially the sensitivity of such e ff ect to the runningof the couplings from the energy scale of the ultraviolet theory to the nucleon scale [41, 42, 43]are di ffi cult to include in a model–independent way, as well as a possible momentum dependenceof the Wilson coe ffi cients of the NR theory. In particular, the latter can arise in the case of along–range interaction such as for electric–dipole or magnetic–dipole DM [44, 45]. Moreoveran additional momentum dependence arises when one needs to include the light-meson poles inthe case the DM couples to the axial quark current [46, 47].For the reasons listed above, the use of the limits on the NR couplings of Ref. [39] to calculatethe bounds for an e ff ective relativistic model defined at a much larger scale needs to be tested.This is the first main goal of the present paper, where we wish to use the results of Ref. [39]to calculate the bounds on a specific example of relativistic e ff ective theory, and compare theoutcome to the full calculation. In particular, we will assess the sensitivity of present DD exper-iments to a set of operators up to dimension 7 describing dark matter interactions with quarks q and gluons L χ = X q X a , d C ( d ) a , q Q ( d ) a , q + X b , d C ( d ) b Q ( d ) b , (2)where the C ( d ) a , q , C ( d ) b are dimensional Wilson coe ffi cients. The sums run over the dimensionsof the operators, d = , , a and b . If not specified otherwise, weconventionally fix the Wilson parameters at the Electroweak (EW) scale, that we identify withthe Z boson mass. The operators Q ( d ) a , q , Q ( d ) b that we will analyze are listed in Eqs.(3,4,5), and arethe same analyzed in [48]. Analyses on similar sets of relativistic e ff ective operators can also befound in [49, 50, 51, 43]. 3n particular, once, for each of the relativistic models we consider, the NR Wilson coe ffi -cients c τ j at the nucleon scale are obtained from the C ( d ) a , q ’s or C ( d ) b ’s, the expected DD rates onlydepend on the non–relativistic response functions [52, 53]. In our analysis we will follow closelyRef. [39], so that we address the reader to that paper for the formulas that we use to calculate theexpected rates for WIMP–nucleus scattering.In Ref. [39] we found that 9 experiments out of a total of 14 present Dark Matter searchescan provide the most stringent bound on some of the e ff ective couplings for a given choice of( m χ , c n / c p ). We include the same experiments in the present analysis: XENON1T [6], CDM-Slite [9], SuperCDMS [10], PICASSO [12], PICO–60 (using a CF I target [13] and a C F one[14]), CRESST-II [15, 54], DAMA (average count rate [55]), DarkSide–50 [18]. The details ofhow each experimental limit has been obtained can be found in the Appendix A of Ref. [39] withthe exception of the PICO–60 result with a C F target of Ref. [56] that was recently updatedwith its final result [14]. In particular, in [14] an additional exposure of 1404 kg days at thresh-old 2.45 keVnr was included in the analysis, lower than that of Ref. [56] where an exposure of1167 kg days with threshold 3.3 keVnr was used. So, compared to Ref. [39], for PICO–60 wehave added the additional run at 2.45 keVnr and updated the e ffi ciency of both runs using theresult from Fig.3 of [14]. We stress that in the literature only the bounds from a few experiments(typically XENON1T and PICO–60) are discussed, and only for a few of the e ff ective models ofEqs.(3,4,5) [50, 41, 42, 48, 57, 37, 38, 58]. So the second main goal of the present paper is tofocus on a systematic approach, including a number of experiments and of e ff ective couplingsthat both exceed for completeness previous analyses.The approach of the present analysis is complementary to that of Ref. [39], but itself not de-void from drawbacks. In particular, the matching of the Wilson coe ffi cients C ( d ) a , q ’s of the WIMP–quark relativistic interaction into the c τ j ’s of the NR WIMP–nucleon Hamiltonian is highly de-generate, since, in principle, the relativistic e ff ective theory requires to fix one coupling for eachquark flavor q , while the NR theory contains only protons and neutrons. In other words, someassumptions must be made on how the C ( d ) a , q ’s scale with the flavor q . A frequent approach inthe literature is to parameterize the theory in terms of a single coupling C ( d ) a , q common to allquarks [59, 60], and in our analysis we will do the same. However it is worth pointing out thatthis assumption would not be applicable to the case of the supersymmetric neutralino, for which,for instance, C (6)4 , q scales as the Z –boson coupling in the case of a Higgsino, or C (7)5 , q depends onthe mass and the weak isospin of the quark for a Gaugino–Higgsino mixing. This has importantphenomenological consequences: for instance, the ratio of the two vacuum expectation valuesin supersymmetry, traditionally parameterized as tan β , selects through the Yukawa couplingswhether the neutralino couples preferentially to up–type or down–type quarks through Higgs ex-change, and it is well known in the literature that large and low tan β values imply very di ff erentphenomenological scenarios. Indeed, in the case of a generic scaling of the WIMP–quark cou-plings the only possible way to obtain a consistent limit without reanalyzing the experimentaldata is to calculate the ratio c n / c p from the C ( d ) a , q ’s and directly use the NR bounds of [39]. Thelimit obtained in this way is only valid if one NR coupling dominates the predicted rate and thereare no cancellations among the contributions of di ff erent NR couplings. So a specific goal ofour analysis is also to assess the validity of such a procedure and to discuss the impact of suchcancellations in the di ff erent relativistic models we consider.The paper is organized as follows. In Section 2 we list the relativistic E ff ective Field Theory(EFT) terms that we consider in our analysis and we summarize how we calculate the NR Wilsoncoe ffi cients starting from each of them; Section 3 is devoted to our quantitative analysis, where4e will provide updated exclusion plots for each relativistic model assuming a common coupling C ( d ) a , q for all quarks; in Section 4 we discuss the impact of interferences among di ff erent NRcouplings, showing that in most cases only one non–relativistic operator dominates the expectedrate and the bounds. We will provide our conclusions in Section 5. Finally, in Appendix A weprovide a simple interpolation code written in Python that, based on the conclusions of Section 4,allows to reproduce most of the results of Section 3 and to generalize them to other choices ofthe C ( d ) a , q couplings assuming that one non–relativistic operator dominates the expected rate.
2. Relativistic e ff ective models In this Section we outline the procedure that we follow to obtained the numerical resultsof Section 3. We use the code DirectDM [48, 47] to calculate the nonperturbative matchingof the e ff ective field theory describing dark matter interactions with quarks and gluons at theEW scale to the e ff ective theory of nonrelativistic dark matter interacting with nonrelativisticnucleons (alternative analyses based on chiral e ff ective field theory can be found for instancein [61, 62, 63, 64]). For this reason we follow closely the notation of Ref.[48, 65] and considerthe same relativistic operators.In particular, we consider the two dimension-five operators: Q (5)1 = e π ( ¯ χσ µν χ ) F µν , Q (5)2 = e π ( ¯ χσ µν i γ χ ) F µν , (3)where F µν is the electromagnetic field strength tensor and χ is the DM field, assumed here to be aDirac particle. Such operators correspond, respectively, to magnetic–dipole and electric–dipoleDM and imply a long–range interaction [66] . The dimension-six operators are Q (6)1 , q = ( ¯ χγ µ χ )( ¯ q γ µ q ) , Q (6)2 , q = ( ¯ χγ µ γ χ )( ¯ q γ µ q ) , Q (6)3 , q = ( ¯ χγ µ χ )( ¯ q γ µ γ q ) , Q (6)4 , q = ( ¯ χγ µ γ χ )( ¯ q γ µ γ q ) , (4)and we also include the following dimension-seven operators: namely: Q (7)1 = α s π ( ¯ χχ ) G a µν G a µν , Q (7)2 = α s π ( ¯ χ i γ χ ) G a µν G a µν , Q (7)3 = α s π ( ¯ χχ ) G a µν e G a µν , Q (7)4 = α s π ( ¯ χ i γ χ ) G a µν e G a µν , Q (7)5 , q = m q ( ¯ χχ )( ¯ qq ) , Q (7)6 , q = m q ( ¯ χ i γ χ )( ¯ qq ) , Q (7)7 , q = m q ( ¯ χχ )( ¯ qi γ q ) , Q (7)8 , q = m q ( ¯ χ i γ χ )( ¯ qi γ q ) , Q (7)9 , q = m q ( ¯ χσ µν χ )( ¯ q σ µν q ) , Q (7)10 , q = m q ( ¯ χ i σ µν γ χ )( ¯ q σ µν q ) . (5)In the equations above q = u , d , s denote the light quarks, G a µν is the QCD field strength tensor,while e G µν = ε µνρσ G ρσ is its dual, and a = , . . . , ff ect among the vector and axial–vector currents of Eq.(4) isknown to be induced by the running of the couplings above the EW scale [41, 42]. In particular, The anapole coupling ( ¯ χγ µ γ χ ) ∂ ν F µν leads instead to an e ff ective contact interaction. A recent discussion is providedin [67]. ff ectivetheory contains only an axial coupling at the high scale, changing dramatically the DD crosssection scaling with the nuclear target and the ensuing DD constrains. For this reason, in the caseof the operators of Eq. (4) with a vector–axial quark current, besides the results valid for a givene ff ective operator at the EW scale we also show the corresponding ones when the same operatoris defined at the scale µ scale = µ scale to m Z = . We then use the output of runDM as an input for DirectDM toperform the remaining running from m Z to the nucleon scale, where the hadronization of theoperators Q ( d ) a , q in eqs. (3,4,5) leads at leading order in the chiral expansion only to single-nucleon(N = p,n) currents, i.e., schematically: < N | ¯ q Γ q | N > = X Γ ′ Ω Γ ′ N ¯ Ψ N Γ ′ Ψ N ,< N | G a µν G a ,µν | N > = Ω ′ N ¯ Ψ N Ψ N ,< N | G a µν ˜ G a ,µν | N > = Ω ′′ N ¯ Ψ N γ Ψ N , (6)with Γ , Γ ′ = , γ µ , γ µ γ , γ , σ µν and Ψ N the nucleon field. Also for the quantities Ω , Ω ′ and Ω ′′ (which in general can depend on external momenta) we rely on the output of DirectDM (seeappendix A of [48]). In particular, the matching of the axial-axial partonic level operator, as wellas that of the coupling between the DM particle to the QCD anomaly term leads to pion and etapoles that can be numerically important, and that we include in our analysis. Specifically [48]: h N | ¯ q γ µ γ q | N i = ¯ Ψ N (cid:20) F q / NA ( q ) γ µ γ + m N F q / NP ′ ( q ) γ q µ (cid:21) Ψ N , (7) h N | m q ¯ qi γ q | N i = F q / NP ( q ) ¯ Ψ N i γ Ψ N , (8) h N | α s π G a µν ˜ G a µν | N i = F N ˜ G ( q ) ¯ Ψ N i γ Ψ N , (9)with: F q / NP , P ′ ( q ) = m N m π − q a q / N π + m N m η − q a q / N η + b q / N , (10) F N ˜ G ( q ) = q m π − q a N ˜ G ,π + q m η − q a N ˜ G ,η + b N ˜ G , (11)where we use DirectDM and runDM when applicable to calculate the coe ffi cients a and b fromthe high–energy couplings, and q represents here the squared four–momentum transfer. Finally,taking the non–relativistic limit, we obtain the coe ffi cients c τ i of the e ff ective Hamiltonian ofEq. (1), which turn out to be proportional to the initial relativistic dimensional coupling C ( d ) a , q ,and, in general, depend on the WIMP mass m χ and on the exchanged momentum q (the latter In particular, we assume the benchmark “QuarksAxial” in [68], with a vanishing DM-Higgs coupling. ff erential cross section is proportional to thesquared amplitude: d σ T dE R = m T π v T " j χ + j T + |M T | , (12)with v T ≡ | ~ v T | the WIMP speed in the reference frame of the nuclear center of mass, m T thenuclear mass, j T , j χ are the spins of target nucleus and WIMP, and [23]:12 j χ + j T + |M T | = π j T + X τ = , X τ ′ = , X k R ττ ′ k c τ i , c τ ′ j , ( v ⊥ T ) , q m N W ττ ′ Tk ( y ) . (13)In the above expression the squared amplitude |M T | is summed over initial and final spins, the R ττ ′ k ’s are WIMP response functions which depend on the couplings c τ j as well as the transferredmomentum ~ q , while: ( v ⊥ T ) = v T − v min , (14)and: v min = q µ T = m T E R µ T , (15)represents the minimal incoming WIMP speed required to impart the nuclear recoil energy E R .Moreover, in equation (13) the W ττ ′ Tk ( y )’s are nuclear response functions and the index k representsdi ff erent e ff ective nuclear operators, which, under the assumption that the nuclear ground state isan approximate eigenstate of P and CP , can be at most eight: following the notation in [22, 23], k = M , Φ ′′ , Φ ′′ M , ˜ Φ ′ , Σ ′′ , Σ ′ , ∆ , ∆Σ ′ . The W ττ ′ Tk ( y )’s are function of y ≡ ( qb / , where b isthe size of the nucleus. For the target nuclei T used in most direct detection experiments thefunctions W ττ ′ Tk ( y ), calculated using nuclear shell models, have been provided in Refs. [23, 69].Details about the definitions of both the functions R ττ ′ k ’s and W ττ ′ Tk ( y )’s can be found in [23]. Inparticular, using the decomposition: R ττ ′ k = R ττ ′ k + R ττ ′ k ( v ⊥ T ) = R ττ ′ k + R ττ ′ k (cid:16) v T − v min (cid:17) , (16)the correspondence between each term of the NR e ff ective interaction in (1) and the W ττ ′ Tk ( y )nuclear response functions is summarized in Table 1. Notice that W M corresponds to the standardSI interaction, while W Σ ′′ + W Σ ′ (with W Σ ′ ≃ W Σ ′′ ) to the standard SD one.Finally, for the WIMP local density we take ρ loc = / cm and for the velocity distribu-tion we assume a standard isotropic Maxwellian at rest in the Galactic rest frame boosted to theLab frame by the velocity of the Sun, v ⊙ =
232 km / s, with root–mean–square velocity v rms = / s and truncated at the escape velocity u esc =
550 km / s.
3. Analysis
In this Section for each of the models Q ( d ) a , q , Q ( d ) b listed in Eqs. (3–5) we show the present con-straints on the correspondent dimensional coupling C ( d ) a , q (assumed to be the same for all flavors)7oupling R ττ ′ k R ττ ′ k coupling R ττ ′ k R ττ ′ k M ( q ) - 3 Φ ′′ ( q ) Σ ′ ( q )4 Σ ′′ ( q ), Σ ′ ( q ) - 5 ∆ ( q ) M ( q )6 Σ ′′ ( q ) - 7 - Σ ′ ( q )8 ∆ ( q ) M ( q ) 9 Σ ′ ( q ) -10 Σ ′′ ( q ) - 11 M ( q ) -12 Φ ′′ ( q ), ˜ Φ ′ ( q ) Σ ′′ ( q ), Σ ′ ( q ) 13 ˜ Φ ′ ( q ) Σ ′′ ( q )14 - Σ ′ ( q ) 15 Φ ′′ ( q ) Σ ′ ( q ) Table 1: Nuclear response functions corresponding to each coupling, for the velocity–independent and the velocity–dependent components parts of the WIMP response function, decomposed as in Eq.(16). In parenthesis is the power of q in the WIMP response function. and C ( d ) b from the list of experiments summarized in the Introduction (XENON1T [6], CDM-Slite [9], SuperCDMS [10], PICASSO [12], PICO–60 (using a CF I target [13] and a C F one [14]), CRESST-II [15, 54], DAMA (average count rate) [55], DarkSide–50 [18]) in terms oflower bounds on the e ff ective scale ˜ Λ defined through: C ( d ) a , q , C ( d ) b ≡ Λ d − . (17)As a default choice in all cases we fix C ( d ) a , q , C ( d ) b at the EW scale, identified as the Z –boson mass, µ scale = m Z . Only for the 6–dimensional interaction terms Q (6)3 , q and Q (6)4 , q we also show in Fig. 4the result obtained when the C ( d ) a , q coupling is fixed at the scale µ scale = Q (6)3 , q passing from µ scale = m Z to µ scale = ff ect [41, 42] due to the mixing between Q (6)3 , q and Q (6)1 , q induced by the running from 2TeV to m Z . In particular, without such mixing Q (6)3 , q gives rise to two NR operators that have botha spin–dependent type scaling with the target, O and O , the latter also velocity suppressed [39],while the mixing due to running induces a Q (6)1 , q component leading to the SI O operator (seeTables 1 and 2) that overwhelms the other contributions in spite of the loop–suppressed Wilsoncoe ffi cient. Such e ff ect is also present for the Q (6)4 , q operator due to the mixing with Q (6)2 , q , althoughin this case the e ff ect on the exclusion plot is less sizable. It is worth pointing out here thatthe mixing between vector and vector–axial currents is driven by the coupling between the DMparticle and the quarks of the third family, so it is not present if the latter is assumed to vanish.In such case the results of Fig. 4 would coincide to those of Fig. 3.In all the plots the data are analyzed in the same way of the experimental collaborations toobtain lower bounds on the e ff ective scale ˜ Λ defined in Eq. (17) as a function of the WIMP mass m χ assuming a single flavor–independent coupling C ( d ) a , q common to all quarks. In particular, forall experiments with the exception of SuperCDMS and DarkSide–50 we compare the expectedrate to the 90% C.L. upper bound on the count rate in each energy bin assuming zero background.Namely, for XENON1T we have assumed 7 WIMP candidate events in the range of 3 PE ≤ S ≤
70 PE, as shown in Fig. 3 of Ref. [6] for the primary scintillation signal S1 (directly in PhotoElectrons, PE), with an exposure of 278.8 days, fiducial volume of 1.3 ton and the e ffi ciency takenfrom Fig. 1 of Ref. [6]. In the analysis of DarkSide–50, we subtract the estimated background8y fitting the data at fixed m χ to the sum S i ( ˜ Λ ) + λ b i in terms of the two free parameters ˜ Λ and λ [39], with S i the expected WIMP signal in each energy bin i and b i taken from Fig. 3 of [18]using the exposure of 6786.0 kg days. This latter procedure is particularly e ff ective when thespectral shapes of the signal and of the background are di ff erent. For DarkSide–50 the estimatedspectrum of the background is rising with the recoil energy, so it yields a weaker constraint forinteractions types with an explicit momentum dependence that lead to a signal rising with energyin a way similar to the background. This loss of constraining power is the reason of the peculiarshapes of some of the exclusion plots for DarkSide–50 in Figs. 1–9. The latest SuperCDMSanalysis [10] observed 1 event between 4 and 100 keVnr with an exposure of 1690 kg days. Toanalyze the observed spectrum we apply the the maximum–gap method [40] with the e ffi ciencytaken from Fig. 1 of [10] and the energy resolution σ = p . + . E ee / keVee keVeefrom [70]. In the case of CDMSlite, we consider the energy bin 0.056 keV < E ′ < ± − (Full Run 2 rate, Table II of Ref. [9]). We havetaken the e ffi ciency from Fig. 4 of [9] and the energy resolution σ = q σ E + BE R + ( AE R ) , with σ E = A = × − and B = E th = E th = CF I target and in this case we adopt an energythreshold of 13.6 keV and an exposure of 1335 kg days [13]. For DAMA we consider the upperbound from the average count rate (DAMA0) which has been taken from [55] (rebinned from0.25-keVee- to 0.5-keVee-width bins). We assume constant quenching factors q = q = σ = ee / keVee) + √ E ee / keVeein keV. For the CRESST-II experiment, we considered the Lise module analysis from [15] withenergy resolution σ = ffi ciency from Fig. 4 of [71]. In our analysis,we have selected 15 events for 0.3 keVnr < E R < Q (5)1 , q , Q (5)2 , q , Q (6)1 , q , Q (6)2 , q , Q (7)1 , Q (7)2 , Q (7)5 , q , Q (7)6 , q and Q (7)10 , q the most constraining experiments areDarkSide–50 at low WIMP mass and XENON1T at larger m χ . As can be seen by combining Ta-ble 2 (that allows to see the correspondence between each Q ( d ) a , q , Q ( d ) b term and NR operators) andTable 1 (where the correspondence between each NR operator O i and the nuclear response func-tions W ττ ′ Tk ( y )’s is shown) one can see that all such interactions take contributions from the W M nuclear response function, leading to a SI scaling of the cross section (possibly combined withexplicit dependence from the exchanged momentum q and from the WIMP incoming speed ).Indeed, due to its very low threshold DarkSide–50 drives the exclusion plot at low mass, but onlyfor interactions that do not require a nuclear spin (its target is Ar), while at larger masses theSI coupling enhances the sensitivity for scatterings o ff xenon in XENON1T. The second class ofexclusion plots is represented by the models Q (6)3 , q , Q (6)4 , q , Q (7)3 , Q (7)4 , Q (7)7 , q , Q (7)8 , q , and Q (7)9 , q for whichthe exclusion plot is driven by PICASSO and PICO60 (and, sometimes, by CDMSLite) at lowWIMP mass, and by XENON1T at larger WIMP masses. In such cases, as can again be seen As far as the O and O NR operators are concerned, the SI part of the nuclear response function usually dominatesin spite of the fact that it is velocity suppressed [39]. m χ (GeV) − Λ ( G e V ) (5)1 = e8π ( ̄χσ μν χ)F μν μ scale = m Z EFTmax NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50 m χ (GeV) − Λ ( G e V ) (5)2 = e8π ( ̄χσ μν iγ χ)F μν μ scale = m Z EFTmax NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50
Figure 1: Lower bound on the e ff ective scale ˜ Λ defined in Eq. (17) for the operators Q (5)1 , q (left) and Q (5)2 , q (right) . In bothcases the dimensional couplings C (5)1 , q and C (5)2 , q are fixed at the EW scale µ scale = m Z . In the region below the solid cyanline the limits are inconsistent with the validity of the EFT based on the simple criterion introduced in Section 3. −1 m χ (GeV) ̃ Λ ( G e V ) (6)1,q = ( ̄χγ μ χ)( ̄qγ μ q)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50 −1 m χ (GeV) ̃ Λ ( G e V ) (6)2,q = ( ̄χγ μ γ χ)( ̄qγ μ q)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50
Figure 2: The same as in Fig. 1 for Q (6)1 , q (left) and Q (6)2 , q (right) . −1 m χ (GeV) −1 ̃ Λ ( G e V ) (6)3,q = ( ̄χγ μ χ)( ̄qγ μ γ q)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50 −1 m χ (GeV) −1 ̃ Λ ( G e V ) (6)4,q = ( ̄χγ μ γ χ)( ̄qγ μ γ q)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I) Figure 3: The same as in Fig. 2 for Q (6)3 , q (left) and Q (6)4 , q (right) . −1 m χ (GeV) ̃ ) ( G e V ) (6)3̃q = ( (χγ μ χ)( (qγ μ γ q)μ scale = 2TeV EFTXENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50 −1 m χ (GeV) ̃ ) ( G e V ) (6)4̃q = ( (χγ μ γ χ)( (qγ μ γ q)μ scale = 2TeV EFTXENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50
Figure 4: The same as in Fig. 3 for µ scale = −1 m χ (G V) −1 ̃ − ( G V ) (7)1 = α s ( ̄χχ)G aμν G aμν μ scale = m Z EFTma( NR op.XENON1TCDMSLiteDAMA0PICO60(μ F )PICASSOSuperCDMSCRESST-IIPICO60(μF I)DS50 −1 m χ (G V) −1 ̃ − ( G V ) (7)2 = α s ( ̄χiγ χ)G aμν G aμν μ scale = m Z EFTma( NR op.XENON1TCDMSLiteDAMA0PICO60(μ F )PICASSOSuperCDMSCRESST-IIPICO60(μF I)DS50
Figure 5: The same as in Fig. 2 for Q (7)1 (left) and Q (7)2 (right) . −1 m χ (GeV) −2 −1 ̃ Λ ( G e V ) (7)3 = α s ( ̄χχ)G aμν ̃ G aμν μ scale = m Z EFTmax NR op̄XENON1TCDMSLiteDAMA0PICO60(μ F )PICASSOSuperCDMSCRESST-IIPICO60(μF I) −1 m χ (GeV) −2 −1 ̃ Λ ( G e V ) (7)4 = α s ( −χiγ χ)G aμν ̃ G aμν μ scale = m Z EFTm ) NR op.XENON1TCDMSLiteDAMA0PICO60(μ F )PICASSOSuperCDMSCRESST-IIPICO60(μF I) Figure 6: The same as in Fig. 2 for Q (7)3 (left) and Q (7)4 (right) . −1 m χ (GeV) −1 ̃ Λ ( G e V ) (7)5,q = m q ( ̄χχ)( ̄qq)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50 −1 m χ (GeV) −1 ̃ Λ ( G e V ) (7)6,q = m q ( ̄χiγ χ)( ̄qq)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50
Figure 7: The same as in Fig. 2 for Q (7)5 (left) and Q (7)6 (right) . −1 m χ (GeV) −2 −1 ̃ Λ ( G e V ) (7)7,q = m q ( ̄χχ)( ̄qiγ q)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I) −1 m χ (GeV) −2 −1 ̃ Λ ( G e V ) (7)8,q = m q ( ̄χiγ χ)( ̄qiγ q)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I) Figure 8: The same as in Fig. 2 for Q (7)7 (left) and Q (7)8 (right) . −1 m χ (GeV) −1 ̃ Λ ( G e V ) (7)9,q = m q ( ̄χσ μν χ)( ̄qσ μν q)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I) −1 m χ (GeV) −1 ̃ Λ ( G e V ) (7)10,q = m q ( ̄χiσ μν γ χ)( ̄qσ μν q)μ scale = m Z EFTma) NR op.XENON1TCDMSLiteDAMA0PICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I)DS50
Figure 9: The same as in Fig. 2 for Q (7)9 (left) and Q (7)10 (right) . from Tables 2 and 1, the operator M is always missing in the NR limit, while the response func-tions Σ ′ and / or Σ ′′ are always present, leading to a SD–type scaling of the cross section for whichlarge detectors containing fluorine are competitive with xenon.Some of the limits shown in Figs. 1–9 may be so weak that they put bounds on values ofthe ˜ Λ scale which are inconsistent with the validity of the e ff ective theory. In such case onecan simply conclude that the present experimental sensitivity of direct detection experiments isnot able to put bounds on the corresponding e ff ective operator. A criterion for the validity ofthe EFT is to interpret the scale ˜ Λ in terms of a propagator g / M ∗ with g < √ π and M ∗ >µ scale , since in our analysis we fixed the boundary conditions of the EFT at the scale µ scale .This is straightforward for dimension–6 operators, while in the case of operators whose e ff ectivecoupling has dimension di ff erent from -2 only matching the EFT with the full theory wouldallow to draw robust conclusions. In particular, in this case ˜ Λ can be interpreted in terms ofthe same propagator times the appropriate power of a typical scale of the problem µ ′ scale , whichdepends on the ultraviolet completion of the EFT. For instance, in the operator Q (7)5 , q = m q ( ¯ χχ )( ¯ qq )the quark mass may originate from a Yukawa coupling, so the missing scale is an Electroweakvacuum expectation value in the denominator. To fix an order of magnitude we choose to fix µ ′ scale = µ scale , so that the bound ˜ Λ > µ scale / (4 π ) / ( d − can be derived. Such limit is shown as ahorizontal solid line in Figs. 1–9. In particular, for models Q (7)2 , Q (7)3 , Q (7)4 , Q (7)6 , Q (7)7 , Q (7)8 and Q (7)9 the bound on the ˜ Λ scale lies above such curve in all the WIMP mass range, implying thatthe sensitivities of present direct detection experiments to such couplings may not be su ffi cient toput meaningful bounds. However we stress again that this can only be assessed when a specificultraviolet completion of the e ff ective theory is assumed.In Appendix A we introduce NRDD_constraints , a simple interpolating code written inPython that can reproduce most of the results of this Section by assuming that one NR couplingdominates in the low–energy limit of the interactions of Eqs.(3)–(5). In Figs. 1–3 and Figs. 5–9the output of such code is indicated by “max NR op.” and represented by the red–dashed curve.One can see that, with the exception of models Q (7)7 , q and Q (7)8 , q , the “max NR op.” matches thelower edge of the excluded region obtained through a full calculation.13 . Interference and momentum e ff ects in the NR theory No matter which among the relativistic interactions listed in Eqs.(3,4,5) is generated at ahigher scale by some beyond–the–standard–model scenario, Dark Matter DD scattering is a low–energy process completely described by the NR e ff ective theory of (Eq. 1). This implies that thelimits discussed in the previous Section can be expressed in terms of NR operators only. Inparticular, in the case of interactions between the DM particle and the quark current, this wouldhave the advantage to present the limits from existing experiments in a way independent fromthe choice of the C ( d ) a , q couplings for each flavor q , since the NR e ff ective theory depends onlyon WIMP mass and on the ratio between the WIMP–neutron and the WIMP–proton couplings r ≡ c n / c p . Indeed, in ref. [39] we obtained updated upper bounds on the e ff ective cross section: σ N i = max( σ pi , σ ni ) , (18)with: σ p , ni = ( c p , ni ) µ χ N π , (19)( µ χ N is the WIMP–nucleon reduced mass) assuming constant couplings c p , ni and, systematically,dominance of one of the possible NR interaction terms O i of Eq. (1), providing for each of thema two–dimensional plot where the contours of the most stringent 90% C.L. upper bounds to σ N i were shown as a function of the two parameters m χ (WIMP mass), and c n / c p . One possibledrawback of this approach is however that, in general, a given relativistic coupling leads to morethan one NR operator. In addition to that, as explained in Section 2, the NR coe ffi cients c τ i maydepend explicitly on the exchanged momentum, leading, in practice, to contributions which areequivalent to including additional NR operators of the type F α i ( q ) O i (where, for each operator O i di ff erent momentum dependences are possible, as for instance in Eqs.(10,11)). In fact, setting: c τ i ( m χ , q ) ≡ ˆ c τ i ,α ( m χ ) F α i ( q ) , (20) R ττ ′ k ≡ c τ i c τ ′ j ˆ R ττ ′ k , i j = ˆ c τ i ,α ˆ c τ ′ j ,β ˆ R ττ ′ k , i j F α i ( q ) F β j ( q ) , (21)the squared amplitude (13) can be rewritten as:12 j χ + j T + |M T | = π j T + X i j X αβ X τ,τ ′ ˆ c τ i ,α ˆ c τ ′ j ,β X k ˆ R ττ ′ k , i j W ττ ′ k ( q ) F α i ( q ) F β j ( q ) , so that the expected rate R can be expressed as a sum over all possible interferences among thecontributions from each generalized NR term F α i ( q ) O i : R = X i j X αβ X τ,τ ′ ˆ c τ i ,α ˆ c τ ′ j ,β hO i O j F α i ( q ) F β j ( q ) i ττ ′ . (22)In the equation above each term hO i O j F α i ( q ) F β j ( q ) i ττ ′ simply represents the factor that mul-tiplies ˆ c τ i ,α ˆ c τ ′ j ,β at fixed i , j , α , β , τ , τ ′ in the expected rate. The terms contributing to the sums over i , j , α , β for each of the interactions discussed in Section 3 are listed in Table 2.To discuss whether it is correct to assume dominance of one e ff ective operator F α i ( q ) O i at atime we introduce the parameters: 14 −1 m χ (GeV) ε (7)7,q = m q ( ̄χχ)( ̄qiγ q)μ scale = m Z XENON1TCDMSL tePICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I) m χ (GeV) −3−2−10123 ε α β ij XENON1T ̂c τ10 ̂c τ ′ ⟨ τ10 τ ′ ⟩̂c τ10 ̂c τ ′ ⟨ τ10 τ ′
10 m (m −q ) ⟩̂c τ10 ̂c τ ′ ⟨ τ10 τ ′
10 m (m −q )(m −q ) ⟩̂c τ10 ̂c τ ′ ⟨ τ10 τ ′
10 m (m −q ) ⟩̂c τ10 ̂c τ ′ ⟨ τ10 τ ′
10 m (m −q ) ⟩̂c τ10 ̂c τ ′ ⟨ τ10 τ ′
10 m (m −q ) ⟩ Figure 10: Parameter ǫ defined in Eq. (23) as a function of the WIMP mass m χ for model Q (7)7 , q . (left) Parameter ǫ as afunction of the WIMP mass m χ for all the experiments and energy bins considered in Section 3. (right) Contributions ǫ αβ ij (each arising from one of the terms listed in Table 2) for the specific example of the XENON1T experiment. −1 m χ (GeV) ε (7)8,q = m q ( ̄χiγ χ)( ̄qiγ q)μ scale = m Z XENON1TCDMSL tePICO60(C F )PICASSOSuperCDMSCRESST-IIPICO60(CF I) m χ (GeV) −3−2−10123 ε α β ij XENON1T ̂c τ6 ̂c τ ′ ⟨ τ6 τ ′ ⟩̂c τ6 ̂c τ ′ ⟨ τ6 τ ′ (m −q ) ⟩̂c τ6 ̂c τ ′ ⟨ τ6 τ ′ (m −q )(m −q ) ⟩̂c τ6 ̂c τ ′ ⟨ τ6 τ ′ (m −q ) ⟩̂c τ6 ̂c τ ′ ⟨ τ6 τ ′ (m −q ) ⟩̂c τ6 ̂c τ ′ ⟨ τ6 τ ′ (m −q ) ⟩ Figure 11: The same as in Fig. 10 for model Q (7)8 , q . αβ i j = P ττ ′ ˆ c τ i ,α ˆ c τ ′ j ,β hO i O j F α i ( q ) F β j ( q ) i ττ ′ P ττ ′ P lm P ρσ ˆ c τ l ,ρ ˆ c τ ′ m ,σ hO l O m F ρ l ( q ) F σ m ( q ) i ττ ′ , ǫ ≡ max i , j ,α,β ( | ǫ αβ i j | ) . (23)By numerical inspection we find that, with the exception of the operators Q (7)7 , q and Q (7)8 , q , the ǫ parameter in all the energy bins of all the experiments included in our analysis never exceeds1.7. Actually, we have checked that such extreme value (indicating destructive interference)corresponds to the highest energy bins of DAMA0 where the rate is exponentially suppressedby the velocity distribution and irrelevant for the constraint. In all other cases ǫ < ∼ F α i ( q ) O i in the calculation of theexpected rate in the determination of the exclusion plot implies an inaccuracy within ± ≃ ǫ > ǫ <
1, but in most cases much smaller.In Figs. 10 and 11 we plot ǫ as a function of the WIMP mass m χ for the two operators Q (7)7 , q and Q (7)8 , q . In particular, while the dominant contribution for Q (7)3 and Q (7)4 corresponds tothe constant term in Eq.(11), as shown in Figs. 10 and 11 the situation is di ff erent for Q (7)7 and Q (7)8 , where each of the terms O n , O n m N m π − q , O n m N m η − q (with n = ǫ ≫ ǫ αβ i j contributionsfor the specific example of the XENON1T experiment. Indeed for the interaction terms Q (7)7 and Q (7)8 such e ff ect is natural since in this particular case the momentum–independent term O n isnext-to-leading order in chiral counting [48] compared to the terms O n m N m π − q and O n m N m η − q .Indeed, our conclusions on the ǫ parameter are not unexpected, since the scaling of the ratefor di ff erent NR operators is very di ff erent, and depends also on experimental inputs, so barringaccidental cancellations or clear–cut situations, like the one of axial operators Q (7)7 , q and Q (7)8 , q ,dominance of one NR operator appears natural. The numerical tests in this Section confirm this.In Appendix A we introduce NRDD_constraints , a simple code that exploits this feature tocalculate approximate bounds on the couplings C ( d ) a , q and C ( d ) b . In particular, while the resultsof Section 3 have been obtained by assuming a single coupling C ( d ) a , q common to all quarks,using NRDD_constraints such constraints can be generalized to a generic dependence of suchcouplings on the flavor q .
5. Conclusions
Assuming for WIMPs a Maxwellian velocity distribution in the Galaxy we have exploredin a systematic way the relative sensitivity of an extensive set of existing DM direct detectionexperiments to each of the operators Q ( d ) a , q , Q ( d ) b listed in Eqs. (3–5) up to dimension 7 describingdark matter e ff ective interactions with quarks and gluons. In particular we have focused on asystematic approach, including an extensive set of experiments and large number of couplings,both exceeding for completeness similar analyses in the literature. For all the operators we havefixed the corresponding dimensional coupling C ( d ) a , q at the scale µ scale = m Z and used the code Di-rectDM [47] to perform the running from m Z to the nucleon scale and the hadronization to single-nucleon (N = p,n) currents, including QCD e ff ects and pion poles that arise in the nonperturbativematching of the e ff ective field theory to the low–energy Galilean–invariant nonrelativistic e ff ec-tive theory describing DM–nucleon interactions. For operators Q (6)3 , q and Q (6)4 , q we have also used16he runDM code [68] to discuss the mixing e ff ect among the vector and axial–vector currents in-duced by the running of the couplings above the EW scale, when the DM vector–axial couplingis assumed to be the same to all quarks.We find that operators Q (5)1 , q , Q (5)2 , q , Q (6)1 , q , Q (6)2 , q , Q (7)1 , Q (7)2 , Q (7)5 , q , Q (7)6 , q and Q (7)10 , q take contributionswhich correspond to a Spin Independent scaling of the cross section (possibly combined withexplicit dependence from the exchanged momentum q and from the WIMP incoming speed)leading to an exclusion plot driven by DarkSide–50 at low WIMP mass and XENON1T at larger m χ . On the other hand for models Q (6)3 , q , Q (6)4 , q , Q (7)3 , Q (7)4 , Q (7)7 , q , Q (7)8 , q , and Q (7)9 , q the cross sectionscaling law is of the Spin–Dependent type, leading to an exclusion plot driven by PICASSOand PICO60 (and, sometimes, by CDMSLite) at low WIMP mass, and by XENON1T at largerWIMP masses.We also find that for models Q (7)2 , Q (7)3 , Q (7)4 , Q (7)6 , Q (7)7 , Q (7)8 and Q (7)9 the present experimentalsensitivity of direct detection experiments appears not to be able to put bounds consistent to thevalidity of the EFT, although only matching the EFT with the full theory would allow to drawrobust conclusions.The matching between the relativistic e ff ective theory to the NR one implies a redundancy ofthe parameters C ( d ) a , q , implying that in many cases the DD constraints, that only depend on the ratio c ni / c pi between the WIMP–neutron and the WIMP–proton couplings, can only be discussed forspecific benchmarks. In particular in our exclusion plots we have assumed a flavor–independentcoupling, C ( d ) a , q = C ( d ) a . However, we have shown how, once the WIMP mass m χ and the c ni / c pi ratio are fixed, for all the Q ( d ) a , q models with the exception of Q (7)7 , q and Q (7)8 , q the expected rate isnaturally driven by a dominant contribution from one of the NR operators O i (possibly modifiedby a momentum–dependent Wilson coe ffi cient, O i → O i F α i ( q )) without large cancellations.This implies that the bounds directly obtained within the context of the NR theory by assumingdominance of one of the F α i ( q ) O i can be used as discussed in Ref. [39] to obtain approximateconstraints valid for any choice of the Q ( d ) a , q parameters, with an inaccuracy within a factor oftwo, but usually smaller. To perform such task in Appendix A we provide a simple interpolatinginterface in Python. On the other hand, in the case of Q (7)7 , q and Q (7)8 , q the terms with a momentum–independent coe ffi cient O and O are next-to-leading order compared to the terms F i ( q ) O and F i ( q ) O which depend on the pion and eta propagators F i ( q ) = / ( m π − q ), 1 / ( m η − q ), sothat they cannot be assumed to dominate. Indeed, in this case all the terms F n ( q ) O n are naturallyof the same order with large cancellations among them. Appendix A. The program
The
NRDD_constraints code provides a simple interpolating function written in Pythonthat for a given generalized NR diagonal term ( F α i ) O i O i among those listed in Table 2 (with theexception of those proportional to a meson pole) calculates the most constraining limit amongthe experiments analyzed in [39] on the e ff ective cross section: σ N i ,α = max( σ pi ,α , σ ni ,α ) , (A.1)(for F α i = q ) − ) with: σ p , ni ,α = (ˆ c p , ni ,α ) µ χ N π , (A.2)17s a function of the WIMP mass m χ and of the ratio r i = ˆ c ni ,α / ˆ c pi ,α . The ˆ c p , ni ,α coe ffi cients aredefined in Eq. (21). The code requires the SciPy package and contains only four files, the code
NRDD_constraints.py , two data files
NRDD_data1.npy and
NRDD_data2.npy , and a drivertemplate
NRDD_constraints-example.py . The module can be downloaded from https://github.com/NRDD-constraints/NRDD or cloned by git clone https://github.com/NRDD-constraints/NRDD
By typing: import NRDD_constraints as NR two functions are defined. The function sigma_nucleon_bound(inter,mchi,r) returns theupper bound ( σ N i ,α ) lim on the e ff ective cross section of Eq.(A.1) in cm as a function of the WIMPmass mchi and of the ratio r = r in the ranges 0 . < m χ < − < r < .The inter parameter is a string that selects the interaction term and can be chosen in the listprovided by the second function print_interactions() : NR.print_interactions()[ ' O1_O1 ' , ' O3_O3 ' , ' O4_O4 ' , ' O5_O5 ' , ' O6_O6 ' , ' O7_O7 ' , ' O8_O8 ' , ' O9_O9 ' , ' O10_O10 ' , ' O11_O11 ' , ' O12_O12 ' , ' O13_O13 ' , ' O14_O14 ' , ' O15_O15 '' O5_O5_qm4 ' , ' O6_O6_qm4 ' , ' O11_O11_qm4 ' ] The list above includes all the O i O i F i ( q ) terms in Table 2 with the exception of those depend-ing on pion and eta poles which, as explained in Section 4, are either subdominant in the caseof models Q (7)3 and Q (7)4 , or, in the case of Q (7)7 and Q (7)8 , imply absence of a dominant term alto-gether (see Figs. 10 and 11) so that the procedure described in this Section leads to an inaccurateestimation of the constraints. This can be seen explicitly in Fig. 8, where the red–dashed curveapproximates poorly the bound obtained with a full calculation. The upper bound returned by sigma_nucleon_bound(inter,mchi,r) corresponds to the results of [39] with the exceptionof the interaction terms with momentum dependence in the Wilson coe ffi cient, that have beenadded to include the long–range interactions of Eq.(3).The driver NRDD_constraints-example.py calculates the exclusion plot on the referencecross section: σ relre f = C µ χ N π , (A.3)assuming for the interaction between the DM particle χ and the nucleon Ψ N the relativistic e ff ec-tive Lagrangian: L = C ¯ χγ µ χ Ψ N γ µ γ Ψ N → C X N = p , n (cid:16) c N O N + c N O N (cid:17) , (A.4)18ith c p = -2, c p = m N / m χ and r i = c ni / c pi . Assuming dominance of one operator O i at a time thedriver implements a function coeff(inter,mchi,r) that calculates the largest Wilson coef-ficient between proton and neutron in absolute value max ( | c pi ( m χ , r ) | , | c ni ( m χ , r ) | ) and plots thebound ( σ relre f ) lim on σ relre f as [39]:( σ relre f ) lim = min i = , ( σ N i ) lim ( m χ , r i ) max ( c pi ( m χ , r i ) , c ni ( m χ , r i ) ) , (A.5)for the specific choice r = r =
1. In particular, for a given value of the WIMP mass mchi : sigma_lim_rel_min=large_numberfor inter in [ ' O7_O7 ' , ' O9_O9 ' ]:c=coeff(inter,mchi,r)sigma_lim_nucleon_NR=NR.sigma_nucleon_bound(inter,mchi,r)sigma_lim_rel=sigma_lim_nucleon_NR/c**2sigma_lim_rel_min=min(sigma_lim_rel_min,sigma_lim_rel) Such limit is also converted into an upper bound ( ˜ Λ ) lim in GeV on ˜ Λ using:( ˜ Λ ) lim = µ χ N ( ~ c ) q ( σ relre f ) lim π d − , (A.6)with ~ c = × − cm GeV and d = Acknowledgements
This research was supported by the Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry of Education, grant number 2016R1D1A1A09917964.
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