Direct detection of WIMPs.pdf

Direct detection of WIMPs David G.Cerdeno",Anne M.Green 10 a Departamento de Fisica Teorica C-XI,and Instituto de Fisica Teorica UAM-CSIC, Universidad Autonoma de Madrid,Cantoblanco,E-28049 Madrid,Spain bSchool of Physics and Astronomy,University of Nottingham University Park,Nottingham,NG7 2RD,UK October 13,2021 Abstract A generic weakly interacting massive particle(WIMP)is one of the most attractive candidates 'yd-onst to account for the cold dark matter in our Universe,since it would be thermally produced with the correct abundance to account for the observed dark matter density.WIMPs can be searched for directly through their elastic scattering with a target material,and a variety of experiments are currently operating or planned with this aim.In these notes we overview the theoretical calculation of the direct detection rate of WIMPs as well as the different detection signals.We discuss the various ingredients(from particle physics and astrophysics)that enter the calculation and review the theoretical predictions for the direct detection of WIMPs in particle physics models. 6 1 Introduction If the Milky Way's DM halo is composed of WIMPs,then the WIMP flux on the Earth is of order 105(100 GeV/mx)cm-2s-1.This flux is sufficiently large that,even though the WIMPs are weakly interacting,a small but potentially measurable fraction will elastically scatter off nuclei.Direct detection experiments aim to detect WIMPs via the nuclear recoils,caused by WIMP elastic scattering,in dedicated low background detectors [1].More specifically they aim to measure the rate,R,and energies,Er,of the nuclear recoils (and in directional experiments the directions as well). In this chapter we overview the theoretical calculation of the direct detection event rate and the potential direct detection signals.Sec.2 outlines the calculation of the event rate,in- cluding the spin independent and dependent contributions and the hadronic matrix elements. Sec.3 discusses the astrophysical input into the event rate calculation,including the local WIMP velocity distribution and density.In Sec.4 we describe the direction detection signals, specifically the energy,time and direction dependence of the event rate.Finally in Sec.5 we discuss the predicted ranges for the WIMP mass and cross-sections in various particle physics models. "This contribution appeared as chapter 17,pp.347-369,of "Particle Dark Matter:Observations,Models and Searches"edited by Gianfranco Bertone,Copyright 2010 Cambridge University Press.Hardback ISBN 9780521763684,http://cambridge.org/us/catalogue/catalogue.asp?isbn=9780521763684

arXiv:1002.1912v1 [astro-ph.CO] 9 Feb 2010 Direct detection of WIMPs ∗ David G. Cerde˜noa , Anne M. Greenb a Departamento de F´ısica Te´orica C-XI, and Instituto de F´ısica Te´orica UAM-CSIC, Universidad Aut´onoma de Madrid, Cantoblanco, E-28049 Madrid, Spain b School of Physics and Astronomy, University of Nottingham University Park, Nottingham, NG7 2RD, UK October 13, 2021 Abstract A generic weakly interacting massive particle (WIMP) is one of the most attractive candidates to account for the cold dark matter in our Universe, since it would be thermally produced with the correct abundance to account for the observed dark matter density. WIMPs can be searched for directly through their elastic scattering with a target material, and a variety of experiments are currently operating or planned with this aim. In these notes we overview the theoretical calculation of the direct detection rate of WIMPs as well as the different detection signals. We discuss the various ingredients (from particle physics and astrophysics) that enter the calculation and review the theoretical predictions for the direct detection of WIMPs in particle physics models. 1 Introduction If the Milky Way’s DM halo is composed of WIMPs, then the WIMP flux on the Earth is of order 105 (100 GeV/mχ) cm−2 s −1 . This flux is sufficiently large that, even though the WIMPs are weakly interacting, a small but potentially measurable fraction will elastically scatter off nuclei. Direct detection experiments aim to detect WIMPs via the nuclear recoils, caused by WIMP elastic scattering, in dedicated low background detectors [1]. More specifically they aim to measure the rate, R, and energies, ER, of the nuclear recoils (and in directional experiments the directions as well). In this chapter we overview the theoretical calculation of the direct detection event rate and the potential direct detection signals. Sec. 2 outlines the calculation of the event rate, including the spin independent and dependent contributions and the hadronic matrix elements. Sec. 3 discusses the astrophysical input into the event rate calculation, including the local WIMP velocity distribution and density. In Sec. 4 we describe the direction detection signals, specifically the energy, time and direction dependence of the event rate. Finally in Sec. 5 we discuss the predicted ranges for the WIMP mass and cross-sections in various particle physics models. ∗This contribution appeared as chapter 17, pp. 347-369, of “Particle Dark Matter: Observations, Models and Searches” edited by Gianfranco Bertone, Copyright 2010 Cambridge University Press. Hardback ISBN 9780521763684, http://cambridge.org/us/catalogue/catalogue.asp?isbn=9780521763684 1

2 Event rate The differential event rate,usually expressed in terms of counts/kg/day/kev (a quantity referred to as a differential rate unit or dru)for a WIMP with mass mx and a nucleus with mass mN is given by dR =P0 dowN(v,ER)dv, vf(v)dER (1) dER mN mx Jumin where po is the local WIMP density, (v,ER)is the differential cross-section for the ER WIMP-nucleus elastic scattering and f(v)is the WIMP speed distribution in the detector frame normalized to unity. Since the WIMP-nucleon relative speed is of order 100kms-1 the elastic scattering occurs in the extreme non-relativistic limit,and the recoil energy of the nucleon is easily calculated in terms of the scattering angle in the center of mass frame,0* En =2(1-cos0) (2) mN where uN mxmN/(mx+mN)is the WIMP-nucleus reduced mass. The lower limit of the integration over WIMP speeds is given by the minimum WIMP speed which can cause a recoil of energy ER:Umin=V(mN ER)/(2u).The upper limit is formally infinite,however the local escape speed vesc (see Sec.3.2),is the maximum speed in the Galactic rest frame for WIMPs which are gravitationally bound to the Milky Way. The total event rate(per kilogram per day)is found by integrating the differential event rate over all the possible recoil energies: o器e,E (3) mN mx Jvmin where Er is the threshold energy,the smallest recoil energy which the detector is capable of measuring. The WIMP-nucleus differential cross section encodes the particle physics inputs (and as- sociated uncertainties)including the WIMP interaction properties.It depends fundamentally on the WIMP-quark interaction strength,which is calculated from the microscopic description of the model,in terms of an effective Lagrangian describing the interaction of the particular WIMP candidate with quarks and gluons.The resulting cross section is then promoted to a WIMP-nucleon cross section.This entails the use of hadronic matrix elements,which de- scribe the nucleon content in quarks and gluons,and are subject to large uncertainties.In general,the WIMP-nucleus cross section can be separated into a spin-independent(scalar) and a spin-dependent contribution, dowN dowN dowN dER dER (4) SI dER /SD Finally,the total WIMP-nucleus cross section is calculated by adding coherently the above spin and scalar components,using nuclear wave functions.The form factor,F(ER),encodes the dependence on the momentum transfer,q=v2mNER,and accounts for the coherence 2

2 Event rate The differential event rate, usually expressed in terms of counts/kg/day/keV (a quantity referred to as a differential rate unit or dru) for a WIMP with mass mχ and a nucleus with mass mN is given by dR dER = ρ0 mN mχ Z ∞ vmin vf(v) dσW N dER (v, ER) dv , (1) where ρ0 is the local WIMP density, dσWN dER (v, ER) is the differential cross-section for the WIMP-nucleus elastic scattering and f(v) is the WIMP speed distribution in the detector frame normalized to unity. Since the WIMP-nucleon relative speed is of order 100 km−1 s −1 the elastic scattering occurs in the extreme non-relativistic limit, and the recoil energy of the nucleon is easily calculated in terms of the scattering angle in the center of mass frame, θ ∗ ER = µ 2 N v 2 (1 − cos θ ∗ ) mN , (2) where µN = mχmN /(mχ + mN ) is the WIMP-nucleus reduced mass. The lower limit of the integration over WIMP speeds is given by the minimum WIMP speed which can cause a recoil of energy ER: vmin = q (mN ER)/(2µ 2 N ). The upper limit is formally infinite, however the local escape speed vesc (see Sec. 3.2), is the maximum speed in the Galactic rest frame for WIMPs which are gravitationally bound to the Milky Way. The total event rate (per kilogram per day) is found by integrating the differential event rate over all the possible recoil energies: R = Z ∞ ET dER ρ0 mN mχ Z ∞ vmin vf(v) dσW N dER (v, ER) dv , (3) where ET is the threshold energy, the smallest recoil energy which the detector is capable of measuring. The WIMP-nucleus differential cross section encodes the particle physics inputs (and associated uncertainties) including the WIMP interaction properties. It depends fundamentally on the WIMP-quark interaction strength, which is calculated from the microscopic description of the model, in terms of an effective Lagrangian describing the interaction of the particular WIMP candidate with quarks and gluons. The resulting cross section is then promoted to a WIMP-nucleon cross section. This entails the use of hadronic matrix elements, which describe the nucleon content in quarks and gluons, and are subject to large uncertainties. In general, the WIMP-nucleus cross section can be separated into a spin-independent (scalar) and a spin-dependent contribution, dσW N dER = dσW N dER SI + dσW N dER SD . (4) Finally, the total WIMP-nucleus cross section is calculated by adding coherently the above spin and scalar components, using nuclear wave functions. The form factor, F(ER), encodes the dependence on the momentum transfer, q = √ 2mN ER, and accounts for the coherence 2

loss which leads to a suppression in the event rate for heavy WIMPs or nucleons.In general, we can express the differential cross section as dowN dER (oF,(ER)+iDr(En）, mN (5) where oo SI.SD are the spin-independent and -dependent cross sections at zero momentum transfer. The origin of the different contributions is best understood at the microscopic level,by analysing the Lagrangian which describes the WIMP interactions with quarks.The contribu- tions to the spin-independent cross section arise from scalar and vector couplings to quarks, whereas the spin-dependent part of the cross section originates from axial-vector couplings These contributions are characteristic of the particular WIMP candidate (see,e.g.,[2])and can be potentially useful for their discrimination in direct detection experiments. 2.1 Spin-dependent contribution The contributions to the spin-dependent (SD)part of the WIMP-nucleus scattering cross section arise from couplings of the WIMP field to the quark axial current,5q.For example,if the WIMP is a (Dirac or Majorana)fermion,such as the lightest neutralino in supersymmetric models,the Lagrangian can contain the term Ca4(6X)(@nu59). (6) If the WIMP is a spin 1 field,such as in the case of LKP and LTP,the interaction term is slightly different, Ladeupa(Bp8B)(@°sq). (7) In both cases,the nucleus,N,matrix element reads (NIGYYsqIN)=2X (NIJNIN), (8) where the coefficients relate the quark spin matrix elements to the angular momentum of the nucleons.They can be parametrized as y≈AS+△"S (9) where J is the total angular momentum of the nucleus,the quantities Ag"are related to the matrix element of the axial-vector current in a nucleon,(n)=2s,and (Sp.n)=(NISp.nN)is the expectation value of the spin content of the proton or neutron group in the nucleusl.Adding the contributions from the different quarks,it is customary to define ap=∑ △g:an=】 g-△ (10) q=,asV②G IThese quantities can be determined from simple nuclear models.For example,the single-particle shell model assumes the nuclear spin is solely due to the spin of the single unpaired proton or neutron,and therefore vanishes for even nuclei.More accurate results can be obtained by using detailed nuclear calculations. 3

loss which leads to a suppression in the event rate for heavy WIMPs or nucleons. In general, we can express the differential cross section as dσW N dER = mN 2µ 2 N v 2 σ SI 0 F 2 SI (ER) + σ SD 0 F 2 SD(ER) , (5) where σ SI, SD 0 are the spin-independent and -dependent cross sections at zero momentum transfer. The origin of the different contributions is best understood at the microscopic level, by analysing the Lagrangian which describes the WIMP interactions with quarks. The contributions to the spin-independent cross section arise from scalar and vector couplings to quarks, whereas the spin-dependent part of the cross section originates from axial-vector couplings. These contributions are characteristic of the particular WIMP candidate (see, e.g., [2]) and can be potentially useful for their discrimination in direct detection experiments. 2.1 Spin-dependent contribution The contributions to the spin-dependent (SD) part of the WIMP-nucleus scattering cross section arise from couplings of the WIMP field to the quark axial current, ¯qγµγ5q. For example, if the WIMP is a (Dirac or Majorana) fermion, such as the lightest neutralino in supersymmetric models, the Lagrangian can contain the term L ⊃ α A q ( ¯χγµ γ5χ)(¯qγµγ5q). (6) If the WIMP is a spin 1 field, such as in the case of LKP and LTP, the interaction term is slightly different, L ⊃ α A q ǫ µνρσ(Bρ ↔ ∂µ Bν )(¯qγσ γ5q). (7) In both cases, the nucleus, N, matrix element reads hN|qγ¯ µγ5q|Ni = 2λ N q hN|JN |Ni, (8) where the coefficients λ N q relate the quark spin matrix elements to the angular momentum of the nucleons. They can be parametrized as λ N q ≃ ∆ (p) q hSpi + ∆(n) q hSni J , (9) where J is the total angular momentum of the nucleus, the quantities ∆q n are related to the matrix element of the axial-vector current in a nucleon, hn|qγ¯ µγ5q|ni = 2s (n) µ ∆ (n) q , and hSp,ni = hN|Sp,n|Ni is the expectation value of the spin content of the proton or neutron group in the nucleus1 . Adding the contributions from the different quarks, it is customary to define ap = X q=u,d,s α A q √ 2GF ∆ p q ; an = X q=u,d,s α A q √ 2GF ∆ n q , (10) 1These quantities can be determined from simple nuclear models. For example, the single-particle shell model assumes the nuclear spin is solely due to the spin of the single unpaired proton or neutron, and therefore vanishes for even nuclei. More accurate results can be obtained by using detailed nuclear calculations. 3

interaction of the WIMP and the gluon scalar density in the nucleon,with fc =1- They are determined experimentally, fu=0.020±0.004，fa=0.026±0.005，fs=0.118±0.062， (19) with fru=frd,fra=fr,and frs=frs The uncertainties in these quantities,among which the most important is that on frs,mainly stem from the determination of the m-nucleon sigma term. The vector coupling (which is present,for example,in the case of a Dirac fermion but vanishes for Majorana particles)gives rise to an extra contribution.Interestingly,the sea quarks and gluons do not contribute to the vector current.Only valence quarks contribute, leading to the following expression 00= B路 (20) 64π with BN=av(A+Z)+a(2A-Z) (21) Thus,for a general WIMP with both scalar and vector interactions,the spin-independent contribution to the scattering cross section would read dowN 2mN dER sI [Zf+(A-Z)f")2+ B 256 F2(ER) (22) In most cases the WIMP coupling to neutrons and protons is very similar,fp fn,and therefore the scalar contribution can be approximated by dowN 、dEr/sI 2mN A((ER). (23) TU2 The spin-independent contribution basically scales as the square of the number of nucleons (A2),whereas the spin-dependent one is proportional to a function of the nuclear angular momentum,(J+1)/J.Although in general both have to be taken into account,the scalar component dominates for heavy targets (A 20),which is the case for most experiments (usually based on targets with heavy nuclei such as Silicon,Germanium,Iodine or Xenon). Nevertheless,dedicated experiments exist that are also sensitive to the SD WIMP coupling through the choice of targets with a large nuclear angular momentum. As we have seen,the WIMP direct detection rate depends on both astrophysical input (the local DM density and velocity distribution,in the lab frame)and particle physics input (nuclear form factors and interaction cross-sections,which depend on the theoretical frame- work in which the WIMP candidate arises).We will discuss these inputs in more detail in Secs.3 and 5 respectively. 2.3 Hadronic Matrix Elements The effect of uncertainties in the hadronic matrix elements has been studied in detail for the specific case of neutralino dark matter [5-9].Concerning the SI cross section,the quantities f in Eq.(19)can be parametrized in terms of the nucleon sigma term,N,(see in this respect,e.g.,Refs.7;9])which,in terms of the u and d quark masses reads N(m+m)(Nlu+d), (24) 5

interaction of the WIMP and the gluon scalar density in the nucleon, with f p T G = 1 − P q=u,d,s f p T q. They are determined experimentally, f p T u = 0.020 ± 0.004, f p T d = 0.026 ± 0.005, f p T s = 0.118 ± 0.062, (19) with f n T u = f p T d, f n T d = f p T u, and f n T s = f p T s. The uncertainties in these quantities, among which the most important is that on fT s, mainly stem from the determination of the π-nucleon sigma term. The vector coupling (which is present, for example, in the case of a Dirac fermion but vanishes for Majorana particles) gives rise to an extra contribution. Interestingly, the sea quarks and gluons do not contribute to the vector current. Only valence quarks contribute, leading to the following expression σ0 = µ 2 N B2 N 64π , (20) with BN ≡ α V u (A + Z) + α V d (2A − Z). (21) Thus, for a general WIMP with both scalar and vector interactions, the spin-independent contribution to the scattering cross section would read dσW N dER SI = 2 mN πv2 " [Zf p + (A − Z)f n ] 2 + B2 N 256# F 2 (ER). (22) In most cases the WIMP coupling to neutrons and protons is very similar, f p ≈ f n , and therefore the scalar contribution can be approximated by dσW N dER SI = 2 mN A2 (f p ) 2 πv2 F 2 (ER). (23) The spin-independent contribution basically scales as the square of the number of nucleons (A2 ), whereas the spin-dependent one is proportional to a function of the nuclear angular momentum, (J + 1)/J. Although in general both have to be taken into account, the scalar component dominates for heavy targets (A > 20), which is the case for most experiments (usually based on targets with heavy nuclei such as Silicon, Germanium, Iodine or Xenon). Nevertheless, dedicated experiments exist that are also sensitive to the SD WIMP coupling through the choice of targets with a large nuclear angular momentum. As we have seen, the WIMP direct detection rate depends on both astrophysical input (the local DM density and velocity distribution, in the lab frame) and particle physics input (nuclear form factors and interaction cross-sections, which depend on the theoretical framework in which the WIMP candidate arises). We will discuss these inputs in more detail in Secs. 3 and 5 respectively. 2.3 Hadronic Matrix Elements The effect of uncertainties in the hadronic matrix elements has been studied in detail for the specific case of neutralino dark matter [5–9]. Concerning the SI cross section, the quantities f p T q in Eq.(19) can be parametrized in terms of the π nucleon sigma term, ΣπN , (see in this respect, e.g., Refs.[7; 9]) which, in terms of the u and d quark masses reads ΣπN = 1 2 (mu + md)hN|uu¯ + ¯dd|Ni , (24) 5