Progress of the AMIDAS Package for Reconstructing WIMP Properties

Progress of the AMIDAS Package for Reconstructing WIMP Properties Chung-Lin Shan Xinjiang Astronomical Observatory Chinese Academy of Sciences 4th International Workshop on Dark Matter, Dark Energy, and Matter-Antimatter Asymmetry December 31, 2016

Outline AMIDAS package Motivation Reconstruction of the 1-D WIMP velocity distribution With measured recoil energies With a non-negligible threshold energy With a model of the WIMP velocity distribution 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) December 31, 2016, NTHU p. 1

AMIDAS package AMIDAS package A Model-Independent Data Analysis System C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 2

AMIDAS package AMIDAS package AMIDAS: A Model-Independent Data Analysis System for direct Dark Matter detection experiments and phenomenology C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 3

AMIDAS package AMIDAS package AMIDAS: A Model-Independent Data Analysis System for direct Dark Matter detection experiments and phenomenology DAMNED Dark Matter Web Tool (ILIAS Project) http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ [CLS, AIP Conf. Proc. 1200, 1031; arxiv:0910.1971; Phys. Dark Univ. 5-6, 240 (2014)] TiResearch (Taiwan interactive Research) http://www.tir.tw/phys/hep/dm/amidas/ C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 3

AMIDAS package AMIDAS package AMIDAS: A Model-Independent Data Analysis System for direct Dark Matter detection experiments and phenomenology C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 4

AMIDAS package Motivation Motivation C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 5

AMIDAS package Motivation Motivation Differential event rate for elastic WIMP-nucleus scattering dr vmax [ ] f1(v) dq = AF 2 (Q) dv v Here v min (Q) = α Q v min (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) December 31, 2016, NTHU p. 6

AMIDAS package Motivation Motivation Differential event rate for elastic WIMP-nucleus scattering dr vmax [ ] f1(v) dq = AF 2 (Q) dv v Here v min (Q) = α Q v min (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) December 31, 2016, NTHU p. 6

AMIDAS package 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 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) December 31, 2016, NTHU p. 6

AMIDAS package 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) December 31, 2016, NTHU p. 6

Reconstruction of the 1-D WIMP velocity distribution Reconstruction of the 1-D WIMP velocity distribution C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 7

Reconstruction of the 1-D WIMP velocity distribution With measured recoil energies With measured recoil energies C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 8

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) December 31, 2016, NTHU p. 9 ]

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 = 2 1 1 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) December 31, 2016, NTHU p. 10

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) December 31, 2016, NTHU p. 11

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) December 31, 2016, NTHU p. 12

Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Reconstruction of f 1 (v) with a non-negligible threshold energy 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, 1550090 (2015)] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 13

Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Reconstruction of f 1 (v) with a non-negligible threshold energy Modification of the renormalization constant N = 2 α [ f 1,rec (vmin ) Q1/2 min + 2Q1/2 min F 2 (Q min ) ( ) ] 1 dr + I 0 (Q min, Q dq max) expt, Q=Q min where [ ] [ ] f 1,rec (vmin ) 2Q min r(q min ) d F 2 (Q min ) dq ln F 2 (Q) k 1 Q=Qmin ( ) 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) ( )] dr dq dq a Q (n 1)/2 a F 2 (Q a) ( Qmax min Q max, Q max,kin = v max 2 ) α 2 [CLS, IJMPD 24, 1550090 (2015)] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 14

Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Reconstruction of f 1 (v) with a non-negligible threshold energy 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, 1550090 (2015)] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 15

Reconstruction of the 1-D WIMP velocity distribution With a non-negligible threshold energy Reconstruction of f 1 (v) with a non-negligible threshold energy Theoretical bias estimate of [ v min 0 ] vmin f 0 1(v) dv / v max f 0 1(v) dv [CLS, IJMPD 24, 1550090 (2015)] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 16

Reconstruction of the 1-D WIMP velocity distribution With a model of the WIMP velocity distribution With a model of the WIMP velocity distribution Bayesian analysis C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 17

Reconstruction of the 1-D WIMP velocity distribution With a model of the WIMP velocity distribution Bayesian reconstruction of f 1 (v) 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 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(data): evidence, the total probability of obtaining the particular set of data 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) December 31, 2016, NTHU p. 18

Reconstruction of the 1-D WIMP velocity distribution With a model of the WIMP velocity distribution Bayesian reconstruction of f 1 (v) 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 1408, 009 (2014)] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 19

Reconstruction of the 1-D WIMP velocity distribution With a model of the WIMP velocity distribution Bayesian reconstruction of f 1 (v) 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 1408, 009 (2014)] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 20

Reconstruction of the 1-D WIMP velocity distribution With a model of the WIMP velocity distribution Bayesian reconstruction of f 1 (v) 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) C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 23

Reconstruction of the 1-D WIMP velocity distribution With a model of the WIMP velocity distribution Bayesian reconstruction of f 1 (v) 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, 1550090 (2015)] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 24

Determination of the WIMP mass Determination of the WIMP mass C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 25

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) ( ) dr r min = dq = r 1 e k 1 (Q min Q s,1 ) expt, Q=Q min [M. Drees and CLS, JCAP 0706, 011 (2007)] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 26

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 [CLS and M. Drees, arxiv:0710.4296] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 26

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:0710.4296] 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) December 31, 2016, NTHU p. 26

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) December 31, 2016, NTHU p. 27

Determinations of the WIMP-nucleon couplings Determinations of the WIMP-nucleon couplings C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 28

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) December 31, 2016, NTHU p. 29

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 ) C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 30

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:1103.0481] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 31

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 C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 32

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 + 127 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) December 31, 2016, NTHU p. 33

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:0903.3300] C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 34

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 + 127 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) December 31, 2016, NTHU p. 35

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 + 127 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) December 31, 2016, NTHU p. 36

Summary Summary C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 37

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 (with an assumed ρ 0 ) 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 With an assumed f 1,th (v), one can fit f 1 (v) by using Bayesian analysis. For these analyses the local density, the velocity distribution, and the mass/couplings on nucleons of halo WIMPs are not required priorly. C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 38

Summary Thank you very much for your attention! C.-L. Shan (XAO-CAS) December 31, 2016, NTHU p. 39