Determining WIMP Properties with the AMIDAS Package Chung-Lin Shan Xinjiang Astronomical Observatory Chinese Academy of Sciences Center for Future High Energy Physics, Chinese Academy of Sciences August 12, 2016
Outline AMIDAS package Motivation 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) CFHEP-CAS, August 12, 2016 p. 1
Outline Dark Matter searches DM should have small, but non-zero interactions with SM 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 (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 2
AMIDAS package AMIDAS package A Model-Independent Data Analysis System C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 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/ Online interactive simulation/data analysis system Full Monte Carlo simulations Theoretical estimations Real/pseudo- data analyses C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 4
AMIDAS package AMIDAS package AMIDAS: A Model-Independent Data Analysis System for direct Dark Matter detection experiments and phenomenology C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 5
AMIDAS package Motivation Motivation C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 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 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) CFHEP-CAS, August 12, 2016 p. 7
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) CFHEP-CAS, August 12, 2016 p. 8
C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 9
With measured recoil energies With measured recoil energies C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 10
With measured recoil energies 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) CFHEP-CAS, August 12, 2016 p. 11 ]
With measured recoil energies 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) CFHEP-CAS, August 12, 2016 p. 12
With measured recoil energies 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) CFHEP-CAS, August 12, 2016 p. 13
With a non-negligible threshold energy With a non-negligible threshold energy C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 14
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) CFHEP-CAS, August 12, 2016 p. 15
With a non-negligible threshold energy Reconstruction of f 1 (v) with a non-negligible threshold energy 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) ( )] 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) CFHEP-CAS, August 12, 2016 p. 16
With a non-negligible threshold energy Reconstruction of f 1 (v) with a non-negligible threshold energy 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, 1550090 (2015)] C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 17
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) CFHEP-CAS, August 12, 2016 p. 18
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, 5-50 kev, 500 events, m χ = 25 GeV) [CLS, IJMPD 24, 1550090 (2015)] C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 19
With a non-negligible threshold energy Reconstruction of f 1 (v) with a non-negligible threshold energy [ Theoretical bias estimate of v min 0 ] v min f 0 1 (v) dv / v max f 0 1 (v) dv [CLS, IJMPD 24, 1550090 (2015)] C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 20
With a model of the WIMP velocity distribution With a model of the WIMP velocity distribution Bayesian analysis C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 21
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(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) CFHEP-CAS, August 12, 2016 p. 22
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) CFHEP-CAS, August 12, 2016 p. 23
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) CFHEP-CAS, August 12, 2016 p. 24
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) 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) CFHEP-CAS, August 12, 2016 p. 25
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) CFHEP-CAS, August 12, 2016 p. 26
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, 5-50 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) CFHEP-CAS, August 12, 2016 p. 27
Determination of the WIMP mass Determination of the WIMP mass C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 28
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) CFHEP-CAS, August 12, 2016 p. 29
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) CFHEP-CAS, August 12, 2016 p. 30
Determinations of the WIMP-nucleon couplings Determinations of the WIMP-nucleon couplings C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 31
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) CFHEP-CAS, August 12, 2016 p. 32
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 0 4 1 2 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) CFHEP-CAS, August 12, 2016 p. 33
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) CFHEP-CAS, August 12, 2016 p. 34
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) CFHEP-CAS, August 12, 2016 p. 35
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:0903.3300] C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 36
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) CFHEP-CAS, August 12, 2016 p. 37
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) CFHEP-CAS, August 12, 2016 p. 38
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) CFHEP-CAS, August 12, 2016 p. 39
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) CFHEP-CAS, August 12, 2016 p. 40
Summary Summary C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 41
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. 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) CFHEP-CAS, August 12, 2016 p. 42
Summary Thank you very much for your attention! C.-L. Shan (XAO-CAS) CFHEP-CAS, August 12, 2016 p. 43