Introduction to Direct Dark Matter Detection Experiments

Transcription

1 Introduction to Direct Dark Matter Detection Experiments Chung-Lin Shan Xinjiang Astronomical Observatory Chinese Academy of Sciences School of Physics and Engineering, Sun Yat-Sen University September 17, 2015

2 Evidence and candidates Direct DM detection Evidence and candidates for Dark Matter Evidence for Dark Matter Candidates for Dark Matter Direct Dark Matter detection Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination C.-L. Shan SYSU, September 17, 2015 p. 1

3 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter C.-L. Shan SYSU, September 17, 2015 p. 2

4 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter Rotation curves of spiral galaxies (1970s) [V. C. Rubin and W. K. Ford, Astrophys. J. 159, 379 (1970)] C.-L. Shan SYSU, September 17, 2015 p. 3

5 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter Collision of two clusters of galaxies (Bullet Cluster, 1E ) [ [D. Clowe et al., Astrophys. J. Lett. 648, L109 (2006); Nucl. Phys. Proc. Suppl. 173, 28 (2007)] C.-L. Shan SYSU, September 17, 2015 p. 4

6 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter Collision of two clusters of galaxies (Bullet Cluster, 1E ) [ [D. Clowe et al., Astrophys. J. Lett. 648, L109 (2006); Nucl. Phys. Proc. Suppl. 173, 28 (2007)] C.-L. Shan SYSU, September 17, 2015 p. 5

7 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter Collision of two clusters of galaxies (Bullet Cluster, 1E ) [ [D. Clowe et al., Astrophys. J. Lett. 648, L109 (2006); Nucl. Phys. Proc. Suppl. 173, 28 (2007)] C.-L. Shan SYSU, September 17, 2015 p. 6

8 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter Anisotropy of the cosmic microwave background radiation (CMBR) [NASA/WMAP Science Team (full sky map in 1965)] C.-L. Shan SYSU, September 17, 2015 p. 7

9 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter Anisotropy of the cosmic microwave background radiation (CMBR) [NASA/WMAP Science Team/COBE (1992); M. S. Turner, arxiv:astro-ph/ ] C.-L. Shan SYSU, September 17, 2015 p. 8

10 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter l Anisotropy of the cosmic microwave background radiation (CMBR) [NASA/WMAP Science Team, WMAP 5-year result (2008)] C.-L. Shan SYSU, September 17, 2015 p. 9

11 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter l Anisotropy of the cosmic microwave background radiation (CMBR) Angular scale D`[µK2] Multipole moment, ` [EAS/Planck (2013); Planck Collab., Astron. Astrophys. 571, A1 (2014)] C.-L. Shan SYSU, September 17, 2015 p. 10

12 Evidence and candidates Direct DM detection Evidence Candidates Evidence for Dark Matter A large fraction of the mass/energy in our Universe is dark! Dark Energy: 68.5( )% Dark Matter: 26.5(±1.1)% Baryonic matter: 4.99(±0.22)% Luminous matter: 0.357(±0.006)% Stars: 0.2% 0.3% Neutrinos: < 0.55% CMB photons: (± )% [ : Review of Particle Physics 2014, 2. Astrophysical Constants and Parameters] [ : Review of Particle Physics 2014, 23. Big Bang Nucleosynthesis] [ : Review of Particle Physics 2014, 22. Big Bang Cosmology] C.-L. Shan SYSU, September 17, 2015 p. 11

13 Evidence and candidates Direct DM detection Evidence Candidates Candidates for Dark Matter C.-L. Shan SYSU, September 17, 2015 p. 12

14 Evidence and candidates Direct DM detection Evidence Candidates Candidates for Dark Matter Particles of the Standard Model [ Model of Elementary Particles zh-hant.png] C.-L. Shan SYSU, September 17, 2015 p. 13

15 Evidence and candidates Direct DM detection Evidence Candidates Candidates for Dark Matter Particles of the Standard Model SM particles Name up-quarks down-quarks leptons neutrinos gluons Symbol u, c, t d, s, b e, µ, τ ν e, ν µ, ν τ g photon γ Z boson Z 0 Higgs boson H W bosons W ± C.-L. Shan SYSU, September 17, 2015 p. 14

16 Evidence and candidates Direct DM detection Evidence Candidates Candidates for Dark Matter Particles of typical supersymmetric models Normal particles SUSY partners Name Symbol Name Symbol up-quarks u, c, t up-squarks ũ L, ũ R, c L, c R, t L, t R down-quarks d, s, b down-squarks dl, d R, s L, s R, b L, b R leptons e, µ, τ sleptons ẽ L, ẽ R, µ L, µ R, τ L, τ R neutrinos ν e, ν µ, ν τ sneutrinos ν e, ν µ, ν τ gluons g gluinos g photon γ photino γ Weakly Interacting Massive Particles (WIMPs) Z boson Z 0 Z-ino Z light scalar Higgs h 0 neutralinos χ 0 1, χ0 2, χ0 3, χ0 4 heavy scalar Higgs H 0 neutral higgsinos h 0, H 0 pseudoscalar Higgs A 0 charged Higgs H ± charged higgsinos H ± W bosons W ± W-inos W ± charginos χ ± 1, χ± 2 graviton G gravitino G axion a axino ã C.-L. Shan SYSU, September 17, 2015 p. 15

17 Evidence and candidates Direct DM detection Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Direct Dark Matter detection C.-L. Shan SYSU, September 17, 2015 p. 16

18 Dark Matter searches Evidence and candidates Direct DM detection Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination DM should have small, but non-zero interactions with ordinary matter. = Three different ways to detect DM particles p, e DM DM e, ν µ, γ DM DM (, ) p, e + DM ( ) DM e +, p, D q q Colliders Indirect detection Direct detection C.-L. Shan SYSU, September 17, 2015 p. 17

19 Evidence and candidates Direct DM detection Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Direct Dark Matter detection: elastic WIMP-nucleus scattering WIMPs could scatter elastically off target nuclei and produce nuclear recoils which deposit energy in the detector. The event rate depends on the WIMP density near the Earth ρ 0 the WIMP-nucleus cross sections σ0 SI and σ0 SD the WIMP mass m χ the velocity distribution of incident WIMPs f 1(v) The WIMP-nucleus cross section is about pb, the optimistic expected event rate is 10 4 events/kg-day but could be < 1 event/ton-yr An exponential-like recoil energy spectrum Most events would be with energies less than 50 kev Typical background events due to cosmic rays and ambient radioactivity : signals O(10 6 ) : 1 = bg discrimination 1 : 1.x C.-L. Shan SYSU, September 17, 2015 p. 18

20 Evidence and candidates Direct DM detection WIMP-nucleus cross sections Target material dependence Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Spin-independent (SI) coupling a scalar (and/or vector) interaction the cross section for scalar interaction is approximately proportional to the square of the mass of the target nucleus heavier nuclei, e.g. Ge and Xe, are usually used Spin-dependent (SD) coupling an axial-vector (spin-spin) interaction 19 F, 127 I, and 129/131 Xe are usually used For nuclei with A 30, the SI interaction almost always dominates over the SD interaction. The scattering event rate and the detector sensitivity depend on the mass of target nuclei directly. C.-L. Shan SYSU, September 17, 2015 p. 19

21 Evidence and candidates Direct DM detection WIMP-nucleus cross sections Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Exclusion limits on the (predicted) SI WIMP-nucleon cross section [ C.-L. Shan SYSU, September 17, 2015 p. 20

22 Detection techniques Induced signals Evidence and candidates Direct DM detection Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Ionization (charges) Scintillation (light) Heat (phonons) Quenching factor (nuclear recoil relative efficiency) measured (electron equivalent) recoil energy kev ee true recoil energy kev r Raw/total mass/exposure, fiducial mass/exposure Combinations of two signals a powerful event-by-event rejection method for the background discrimination down to 5 to 10 kev recoil energy C.-L. Shan SYSU, September 17, 2015 p. 21

23 Detection techniques Evidence and candidates Direct DM detection Semiconductor/scintillator detectors Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination ANAIS NaI(Tl), Laboratorio Subterráneo de Canfranc (LSC), Spain. CDEX-TEXONO Ge, China Jin-Ping Laboratory (CJPL), China. CDMS Ge and Si, Soudan Underground Laboratory, Minnesota, USA. CoGeNT Ge, Soudan Underground Laboratory, Minnesota, USA. CRESST Al 2 O 3 /CaWO 4, Laboratori Nazionali del Gran Sasso (LNGS), Italy. DAMA NaI(Tl), LNGS, Italy. DM-Ice NaI(Tl), South Pole. EDELWEISS (EDW) Ge, Laboratoire Souterrain de Modane (LSM), France. HDMS Ge, LNGS, Italy. KIMS CsI(Tl), Yangyang Laboratory (Y2L), South Korea. NaIAD NaI(Tl), Boulby Underground Laboratory, UK. C.-L. Shan SYSU, September 17, 2015 p. 22

24 Detection techniques Evidence and candidates Direct DM detection Liquid noble gas detectors Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination ArDM Dual-phase (gas-liquid) Ar, CERN, Switzerland. DarkSide Dual-phase Ar, LNGS, Italy. DARWIN Dual-phase Ar and Xe, ULISSE, France or LNGS, Italy. DEAP/CLEAN Single-phase Ar and Ne, SNO Underground Laboratory, Canada. LUX Dual-phase Xe, USA. MAX Dual-phase Ar and Xe, USA. PandaX Xe, CJPL, China. WARP Dual-phase Ar, LNGS, Italy. XENON Dual-phase Xe, LNGS, Italy. XMASS Single-phase Xe, SuperKamiokande, Japan. ZEPLIN Single-/dual-phase Xe, Boulby Underground Laboratory, UK. C.-L. Shan SYSU, September 17, 2015 p. 23

25 Detection techniques Evidence and candidates Direct DM detection Superheated droplet/gas detectors COUPP Bubble chamber, CF 3 I, C 3 F 8, and C 4 F 10, USA. D3 Hawaii, USA DAMIC Si, SNO Underground Laboratory, Canada. DMTPC CF 4, USA. DRIFT Xe-CS 2, Boulby Underground Laboratory, UK. Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination MIMAC CF 4, Laboratoire de Physique Subatomique et de Cosmologie (LPSC), France. NEWAGE CF 4, Kamioka mine, Japan. NEWS Laboratoire Souterrain de Modane (LSM), France. Nuclear Emulsion Nagoya, Japan. PICASSO Bubble chamber, C 4 F 10, SNO Underground Laboratory, Canada. SIMPLE C 2 ClF 5, CF 3 I, Laboratoire Souterrain à Bas Bruit (LSBB), France. C.-L. Shan SYSU, September 17, 2015 p. 24

26 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Cosmic muons External natural radioactivity Internal natural radioactivity Fast neutrons Neutron-induced nuclear recoils Multiple-scatter events Electron recoils Surface events Incomplete charge collection C.-L. Shan SYSU, September 17, 2015 p. 25

27 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Cosmic muons Induce fast neutrons O(10 10 ) cosmic muons/m 2 Earth s surface/yr Go deep underground (reduced by a factor of 10 5 to 10 8 ) China Jinping Underground Laboratory (CJPL) ~2400 m CJPL ~8750 m Yalong River 4 [J. Gascon, arxiv:astro-ph/ ; Q. Yue and H. T. Wong, JPCS 375, (2012)] C.-L. Shan SYSU, September 17, 2015 p. 26

28 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination External natural radioactivity Radioactive isotopes in the rock/walls Passive shielding: lead (high-z materials) for MeV γ-ray, (low-z materials) for α-, β-, and low energy γ-rays Internal natural radioactivity Radioactive isotopes contamination in the outer shielding, equipment around the detector, and detector material Radiopure materials Fast neutrons Induced by cosmic-ray in the inner lead shielding Water tank or polyethylene (PE) (materials with high density of hydrogen) C.-L. Shan SYSU, September 17, 2015 p. 27

29 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Multiple-scatter events Mean free path of WIMP-induced events O (light year) Array of detectors CDMS ZIP-detector cryostat and tower [CDMS homepage; P. Cushman, J. Phys. Conf. Ser. 39, 63 (2006)] C.-L. Shan SYSU, September 17, 2015 p. 28

30 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Electron recoils Ionization yield Ionization (S2)/primary scintillation (S1) CDMS-II calibration [CDMS Collab., Z. Ahmed et al., Science 327, 1619 (2010)] C.-L. Shan SYSU, September 17, 2015 p. 29

31 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Surface events Rising time of phonon pulses Self-shielding CDMS-II calibration [CDMS Collab., Z. Ahmed et al., Science 327, 1619 (2010)] C.-L. Shan SYSU, September 17, 2015 p. 30

32 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Surface events Rising time of phonon pulses Self-shielding XENON10 result [XENON10 Collab., J. Angle et al., PRL 100, (2008)] C.-L. Shan SYSU, September 17, 2015 p. 31

33 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Neutron-induced nuclear recoils Mimic WIMP-induced nuclear recoils Neutron veto Boron-loaded liquid scintillator neutron veto [arxiv: ] C.-L. Shan SYSU, September 17, 2015 p. 32

34 Evidence and candidates Direct DM detection Background discrimination Background and background discrimination Elastic WIMP-nucleus scattering WIMP-nucleus cross sections Detection techniques Background discrimination Neutron-induced nuclear recoils Mimic WIMP-induced nuclear recoils Scintillating reflector CRESST-II calibration [CRESST Collab., R. F. Lang et al., Astropart. Phys. 33, 60 (2010)] C.-L. Shan SYSU, September 17, 2015 p. 33

35 Model-Independent Reconstruction of Dark Matter Properties with Direct Detection Experiments Chung-Lin Shan Xinjiang Astronomical Observatory Chinese Academy of Sciences School of Physics and Engineering, Sun Yat-Sen University September 17, 2015

36 Outline Motivation Reconstruction of the 1-D WIMP velocity distribution With measured recoil energies With a non-negligible threshold energy Bayesian reconstruction of the WIMP velocity distribution function Determination of the WIMP mass Determinations of the WIMP-nucleon couplings Estimation of the SI scalar WIMP-nucleon coupling Determinations of ratios of WIMP-nucleon cross sections Summary C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 1

37 Motivation Motivation C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 2

38 Motivation Motivation Differential event rate for elastic WIMP-nucleus scattering dr vmax [ ] f1(v) dq = AF 2 (Q) dv v min (Q) v Here v min (Q) = α Astrophysics Q is the minimal incoming velocity of incident WIMPs that can deposit the recoil energy Q in the detector, A ρ0σ0 mn α m 2m χmr,n 2 2mr,N 2 r,n = mχm N 2 m χ + m N Particle physics ρ 0 : WIMP density near the Earth σ 0 : total cross section ignoring the form factor suppression F (Q): elastic nuclear form factor f 1 (v): one-dimensional velocity distribution of halo WIMPs C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 3

39 Motivation Motivation Differential event rate for elastic WIMP-nucleus scattering dr vmax [ ] f1(v) dq = AF 2 (Q) dv v min (Q) v Here v min (Q) = α Q is the minimal incoming velocity of incident WIMPs that can deposit the recoil energy Q in the detector, ρ0σ0 A 2m χmr,n 2 α mn 2m 2 r,n m r,n = mχm N m χ + m N ρ 0 : WIMP density near the Earth σ 0 : total cross section ignoring the form factor suppression F (Q): elastic nuclear form factor f 1 (v): one-dimensional velocity distribution of halo WIMPs C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 4

40 Reconstruction of the 1-D WIMP velocity distribution Reconstruction of the 1-D WIMP velocity distribution C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 5

41 Reconstruction of the 1-D WIMP velocity distribution With measured recoil energies With measured recoil energies C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 6

42 Reconstruction of the 1-D WIMP velocity distribution With measured recoil energies Reconstruction of the 1-D WIMP velocity distribution Normalized one-dimensional WIMP velocity distribution function { [ ( )]} d 1 dr f 1 (v) = N 2Q dq F 2 (Q) dq N = 2 { [ 1 1 α 0 Q F 2 (Q) ( )] } dr 1 dq dq Q=v 2 /α 2 Moments of the velocity distribution function ( α v n n+1 = N (Q thre ) 2 N (Q thre ) = 2 α [ 2Q 1/2 thre F 2 (Q thre ) [ I n(q thre ) = Q (n 1)/2 Q thre ) [ 2Q (n+1)/2 thre F 2 (Q thre ) ( ) dr dq ( ) dr + I 0 (Q thre ) dq Q=Q thre 1 F 2 (Q) ( )] dr dq dq + (n + 1)I n(q thre ) Q=Q thre ] 1 [M. Drees and CLS, JCAP 0706, 011 (2007)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 7 ]

43 Reconstruction of the 1-D WIMP velocity distribution With measured recoil energies Reconstruction of the 1-D WIMP velocity distribution Ansatz: the measured recoil spectrum in the nth Q-bin ( ) dr r n e kn(q Qs,n) r n Nn dq expt, Q Q n b n Logarithmic slope and shifted point in the nth Q-bin Q Q n n 1 N n ( ) bn (Q n,i Q n) = coth N n 2 i=1 Q s,n = Q n + 1 [ ] sinh(knbn/2) ln k n k nb n/2 ( knb n 2 ) 1 k n Reconstructing the one-dimensional WIMP velocity distribution [ ] [ 2Qs,nrn d ] f 1 (v s,n) = N F 2 (Q s,n) dq ln F 2 (Q) k n Q=Qs,n [ ] N = v s,n = α Q s,n α Qa F 2 (Q a) a [M. Drees and CLS, JCAP 0706, 011 (2007)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 8

44 Reconstruction of the 1-D WIMP velocity distribution With measured recoil energies Reconstruction of the 1-D WIMP velocity distribution Reconstructed f 1,rec (v s,n ) ( 76 Ge, 500 events, 5 bins, up to 3 bins per window) [M. Drees and CLS, JCAP 0706, 011 (2007)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 9

45 Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy With a non-negligible threshold energy C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 10

46 Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Problem! Reconstructed f 1,rec (v s,n ) with a non-negligible threshold energy ( 76 Ge, 2-50 kev, 500 events, m χ = 25 GeV) [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 11

47 Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Modification of the estimator for the normalization constant N Consider the non-zero minimal cut-off velocity where N 2 α [ 2Q 1/2 min F 2 (Q min ) ( ) ] 1 dr + I 0 (Q min, Qmax dq ) expt, Q=Q min ( ) dr = r 1 e k 1(Q min Q s,1) r(q min ) dq expt, Q=Q min I n(q min, Q max ) = Q max Q min [ Q (n 1)/2 1 F 2 (Q) ( Qmax min Q max, Q max,kin = v max 2 ) α 2 ( )] dr dq dq [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 12

48 Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 76 Ge, 2-50 kev, 500 events, m χ = 25 GeV) [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 13

49 Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Modification of the estimator for the normalization constant N Height of the velocity distribution at the non-zero minimal cut-off velocity [ ] [ ] f 1,rec (vmin ) = N 2Q min r(q min ) d F 2 (Q min ) dq ln F 2 (Q) k 1 N f 1,rec (vmin ) Q=Qmin Consider the contribution below the non-zero minimal cut-off velocity N = 2 α [ f 1,rec (vmin ) Q1/2 min + 2Q1/2 min F 2 (Q min ) ( ) ] 1 dr + I 0 (Q min, Qmax dq ) expt, Q=Q min [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 14

50 Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 76 Ge, 2-50 kev, 500 events, m χ = 25 GeV) [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 15

51 Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Modification of the estimator for the normalization constant N Reconstructed f 1,rec (v s,n ) with the input WIMP mass ( 76 Ge, 5-50 kev, 500 events, m χ = 25 GeV) [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 16

52 Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Modification of the estimator for the normalization constant N [ Theoretical bias estimate of v min 0 ] v min f 0 1 (v) dv / v max f 0 1 (v) dv [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 17

53 Reconstruction of the 1-D WIMP velocity distribution Bayesian reconstruction of the WIMP velocity distribution function Bayesian reconstruction of the WIMP velocity distribution function C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 18

54 Reconstruction of the 1-D WIMP velocity distribution Bayesian reconstruction of the WIMP velocity distribution function Formalism Bayesian analysis p(θ data) = p(data Θ) p(θ) p(data) Θ: { } a 1, a 2,, a NBayesian, a specified (combination of the) value(s) of the fitting parameter(s) p(θ): prior probability, our degree of belief about Θ being the true value(s) of fitting parameter(s), often given in form of the (multiplication of the) probability distribution(s) of the fitting parameter(s) p(data): evidence, the total probability of obtaining the particular set of data p(data Θ): the probability of the observed result, once the specified (combination of the) value(s) of the fitting parameter(s) happens, usually be described by the likelihood function of Θ, L(Θ). p(θ data): posterior probability density function for Θ, the probability of that the specified (combination of the) value(s) of the fitting parameter(s) happens, given the observed result C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 19

55 Reconstruction of the 1-D WIMP velocity distribution Bayesian reconstruction of the WIMP velocity distribution function Formalism Probability distribution functions for p(θ) Without prior knowledge about the fitting parameter Flat-distributed p i (a i ) = 1 for a i,min a i a i,max With prior knowledge about the fitting parameter Around a theoretical predicted/estimated or experimental measured value µ a,i With (statistical) uncertainties σ a,i Gaussian-distributed p i (a i ; µ a,i, σ a,i ) = 1 2π σa,i e (a i µ a,i ) 2 /2σ 2 a,i [CLS, JCAP (2014)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 20

56 Reconstruction of the 1-D WIMP velocity distribution Bayesian reconstruction of the WIMP velocity distribution function Formalism Likelihood function for p(data Θ) with Theoretical one-dimensional WIMP velocity distribution function: f 1,th (v; a 1, a 2,, a NBayesian ) Assuming that the reconstructed data points are Gaussian-distributed around the theoretical predictions ) L (f 1,rec(v s,µ), µ = 1, 2,, W ; a i, i = 1, 2,, N Bayesian W ) Gau (v s,µ, f 1,rec(v s,µ), σ f1,s,µ; a 1, a 2,, a NBayesian µ=1 ) Gau (v s,µ, f 1,rec(v s,µ), σ f1,s,µ; a 1, a 2,, a NBayesian [ ] 1 2 / e f 1,rec (v s,µ) f 1,th (v s,µ;a 1,a 2,,a NBayesian ) 2σf 2 1,s,µ 2π σf1,s,µ [CLS, JCAP (2014)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 21

57 Reconstruction of the 1-D WIMP velocity distribution Bayesian reconstruction of the WIMP velocity distribution function Numerical results Input and fitting one-dimensional WIMP velocity distribution functions One-parameter shifted Maxwellian velocity distribution f 1,sh,v0 (v) = 1 ( ) v [ ] e (v ve)2 /v0 2 e (v+ve)2 /v0 2 v e = 1.05 v 0 π v 0 v e Shifted Maxwellian velocity distribution f 1,sh (v) = 1 ( ) v [ ] e (v ve)2 /v0 2 e (v+ve)2 /v0 2 π v 0 v e Variated shifted Maxwellian velocity distribution f 1,sh, v (v) = 1 [ ] { } v e [v (v 0+ v)] 2 /v0 2 e [v+(v 0+ v)] 2 /v0 2 π v 0 (v 0 + v) Simple Maxwellian velocity distribution f 1,Gau (v) = 4 ( v 2 ) π v0 3 e v 2 /v0 2 Modified simple Maxwellian velocity distribution f 1,Gau,k (v) = v 2 ( ) e v 2 /kv0 2 e v max 2 /kv 2 k 0 for v v max N f,k C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 22

58 Reconstruction of the 1-D WIMP velocity distribution Bayesian reconstruction of the WIMP velocity distribution function Numerical results Reconstructed f 1,Bayesian (v) with the input WIMP mass ( 76 Ge, 2-50 kev, 500 events, m χ = 25 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 23

59 Reconstruction of the 1-D WIMP velocity distribution Bayesian reconstruction of the WIMP velocity distribution function Numerical results Reconstructed f 1,Bayesian (v) with the input WIMP mass ( 76 Ge, 5-50 kev, 500 events, m χ = 25 GeV, f 1,sh,v0 (v) f 1,sh,v0 (v), flat-dist.) [CLS, IJMPD 24, (2015)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 24

60 Determination of the WIMP mass Determination of the WIMP mass C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 25

61 Determination of the WIMP mass Determination of the WIMP mass Estimating the moments of the WIMP velocity distribution v n = α n 2Q1/2 min r 1 min F 2 (Q min ) + I 0 2Q(n+1)/2 r min min F 2 + (n + 1)I n (Q min ) I n = a Q (n 1)/2 a F 2 (Q a) Determining the WIMP mass mx m m χ v Y m X R n n = R n m X /m Y R n = ( ) dr r min = = r 1 e k 1 (Q min Q s,1 ) dq expt, Q=Q min [M. Drees and CLS, JCAP 0706, 011 (2007)] 2Q(n+1)/2 min,x r min,x /FX 2 (Q min,x ) + (n + 1)I n,x (X ) 1/n 1 Y 2Q 1/2 (n 0) min,x r min,x /F X 2 (Q min,x ) + I 0,X Assuming a dominant SI scalar WIMP-nucleus interaction m χ σ = (m X /m Y ) 5/2 m Y m X R σ R σ (m X /m Y ) 5/2 R σ = E Y E X [CLS and M. Drees, arxiv: ] 2Q1/2 min,x r min,x /F 2 X (Q min,x ) + I 0,X 2Q 1/2 min,y r min,x /F 2 Y (Q min,y ) + I 0,Y [M. Drees and CLS, JCAP 0806, 012 (2008)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 26

62 Determination of the WIMP mass Determination of the WIMP mass Reconstructed m χ,rec ( 28 Si + 76 Ge, Q max < 100 kev, 2 50 events) [M. Drees and CLS, JCAP 0806, 012 (2008)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 27

63 Determinations of the WIMP-nucleon couplings Determinations of the WIMP-nucleon couplings C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 28

64 Determinations of the WIMP-nucleon couplings Estimation of the SI scalar WIMP-nucleon coupling Estimation of the SI scalar WIMP-nucleon coupling C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 29

65 Determinations of the WIMP-nucleon couplings Estimation of the SI scalar WIMP-nucleon coupling Estimation of the SI scalar WIMP-nucleon coupling Spin-independent (SI) scalar WIMP-nucleus cross section ( ) ( 4 σ0 SI = mr,n 2 [ ] 2 4 Zfp + (A Z)f n π π ) m 2 r,n A2 f p 2 = A 2 ( mr,n m r,p ( ) 4 σχp SI = m 2 π r,p f p 2 f (p,n) : effective SI scalar WIMP-proton/neutron couplings Rewriting the integral over f 1 (v)/v ) 2 σ SI χp ( ) dr = Eρ 2 [( ] 0A 4 )m 2r,p dq expt, Q=Q min 2m χmr,p 2 π fp 2 F 2 2 (Q min ) m 2Q1/2 min r ] min 2 rmin r,n m N F 2 (Q min ) + I 0 1[ F 2 (Q min ) Estimating the SI scalar WIMP-nucleon coupling [ ( )] f p 2 = 1 π ρ E Z A 2 2Q1/2 min,z r min,z Z mz FZ 2(Q + I 0,Z (m χ + m Z ) min,z ) [M. Drees and CLS, PoS IDM2008, 110 (2008)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 30

66 Determinations of the WIMP-nucleon couplings Estimation of the SI scalar WIMP-nucleon coupling Estimation of the SI scalar WIMP-nucleon coupling Reconstructed f p 2 rec ( 76 Ge (+ 28 Si + 76 Ge), Q max < 100 kev, σ SI χp = 10 8 pb, 1(3) 50 events) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 31

67 Determinations of the WIMP-nucleon couplings Estimation of the SI scalar WIMP-nucleon coupling Estimation of the SI scalar WIMP-nucleon coupling Reconstructed f p 2 rec vs. reconstructed m χ,rec ( 76 Ge (+ 28 Si + 76 Ge), Q max < 100 kev, σ SI χp = 10 8 pb, 1(3) 50 events) [CLS, arxiv: ] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 32

68 Determinations of the WIMP-nucleon couplings Determinations of ratios of WIMP-nucleon cross sections Determinations of ratios of WIMP-nucleon cross sections C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 33

69 Determinations of the WIMP-nucleon couplings Determinations of ratios of WIMP-nucleon cross sections Determination of the ratio of SD WIMP-nucleon couplings Spin-dependent (SD) axial-vector WIMP-nucleus cross section ( ) ( ) 32 J + 1 [ Sp a σ0 SD = GF 2 ] 2 π m2 r,n p + S n a n J σ SD χp/n = ( 32 π ) G 2 F m2 r,p/n ( 3 4 ) a 2 p/n J: total nuclear spin S (p,n) : expectation values of the proton/neutron group spin a (p,n) : effective SD axial-vector WIMP-proton/neutron couplings Determining the ratio of two SD axial-vector WIMP-nucleon couplings ( ) SD an a p ±,n [( JX R J,n J X + 1 = Sp X ± S p Y R J,n S n X ± S n Y R J,n ) ( ) JY + 1 Rσ J Y R n ] 1/2 (n 0) [M. Drees and CLS, arxiv: ] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 34

70 Determinations of the WIMP-nucleon couplings Determinations of ratios of WIMP-nucleon cross sections Determination of the ratio of SD WIMP-nucleon couplings Reconstructed (a n /a p ) SD rec,1 ( 73 Ge + 37 Cl and 19 F I, Q min > 5 kev, Q max < 100 kev, 2 50 events, m χ = 100 GeV) [CLS, JCAP 1107, 005 (2011)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 35

71 Determinations of the WIMP-nucleon couplings Determinations of ratios of WIMP-nucleon cross sections Determinations of ratios of WIMP-nucleon cross sections Differential rate for combined SI and SD cross sections ( ) dr = E ρ 0σ SI ( 0 σ [F 2SI SD ) ] dq expt, Q=Q min 2m χm r,n 2 (Q) + χp vmax [ ] σχp SI C pf 2 f1 SD (Q) (v) dv v min v C p 4 3 ( ) [ ] J + 1 Sp + (a 2 n/a p) S n J A Determining the ratio of two WIMP-proton cross sections σχp SD σχp SI = R m,xy F 2 SI,Y (Q min,y )R m,xy F 2 SI,X (Q min,x ) C p,x F 2 SD,X (Q min,x ) C p,y F 2 SD,Y (Q min,y )R m,xy ( ) ( rmin,x EY E X r min,y ) ( ) 2 my m X Determining the ratio of two SD axial-vector WIMP-nucleon couplings ( ) SI+SD an = a p ± ( )[ JX + 1 Sp X c p,x 4 3 J X (c p,x s n/p,x c p,y s n/p,y ) ± c p,x c p,y sn/p,x s n/p,y A X c p,x s n/p,x 2 c p,y s n/p,y 2 ] 2[ F 2 SI,Z (Q min,z )R m,yz F 2 ] SI,Y (Q min,y ) F 2 SD,X (Q min,x ) [M. Drees and CLS, arxiv: ] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 36

72 Determinations of the WIMP-nucleon couplings Determinations of ratios of WIMP-nucleon cross sections Determinations of ratios of WIMP-nucleon cross sections Reconstructed (a n /a p ) SI+SD rec vs. (a n /a p ) SD rec,1 ( 19 F I + 28 Si, Q min > 5 kev, Q max < 100 kev, 3 50 events, σ SI χp = 10 8 /10 10 pb, a p = 0.1, m χ = 100 GeV) [CLS, JCAP 1107, 005 (2011)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 37

73 Determinations of the WIMP-nucleon couplings Determinations of ratios of WIMP-nucleon cross sections Determinations of ratios of WIMP-nucleon cross sections Reconstructed ( ) σχp SD /σχp SI and ( ) σ SD rec χn /σχp SI rec ( 19 F I + 28 Si vs. 23 Na/ 131 Xe + 76 Ge, Q min > 5 kev, Q max < 100 kev, σχp SI = 10 8 pb, a p = 0.1, m χ = 100 GeV, 3/2 50 events) [CLS, JCAP 1107, 005 (2011)] C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 38

74 Summary Summary C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 39

75 Summary Summary Once two or more experiments with different target nuclei observe positive WIMP signals, we could reconstruct WIMP mass m χ 1-D velocity distribution f 1 (v) SI WIMP-proton coupling f p 2 ratio between the SD WIMP-nucleon couplings a n /a p ratios between the SD and SI WIMP-nucleon cross sections σ SD χ(p,n) /σsi χp For these analyses the local density, the velocity distribution, and the mass/couplings on nucleons of halo WIMPs are not required priorly. For a WIMP mass of O(100 GeV), with only O(50) events from one experiment and less than 20% unrejected backgrounds, these quantities could be estimated with statistical uncertainties of 10% - 40%. C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 40

76 Summary Thank you very much for your attention! C.-L. Shan (XAO-CAS) SYSU, September 17, 2015 p. 41