Effective theory of dark matter direct detection. Riccardo Catena. Chalmers University of Technology

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1 Effective theory of dark matter direct detection Riccardo Catena Chalmers University of Technology March 16, 216

2 Outline Introduction Dark matter direct detection Effective theory of dark matter-nucleon interactions (Fitzpatrick et. al, 213) Comparison with observations: R. Catena, A. Ibarra, S. Wild, arxiv: R. Catena and P. Gondolo, JCAP 158, 8, 22 (215) R. Catena, JCAP 149, 9, 49 (214) R. Catena and P. Gondolo, JCAP 149, 9, 45 (214) R. Catena, JCAP 147, 7, 55 (214) Direct Detection

3 Evidence for dark matter Dark matter is a dissipation-less fluid that makes up about 5/6 of the total matter in the Universe The most famous evidence: the rotation curves of spiral galaxies rotation curve of M33 dark matter halo!! stellar disk interstellar gas

4 Evidence for dark matter Further evidence: from the motion of stars in the solar neighborhood, to the largest scales we see in the Universe The most compelling evidence: the formation of cosmological structures If only ordinary matter and photons are present in the Universe, galaxies unavoidably form too late The dissipative coupling of electron/protons and photons delays the formation of galaxies Need for a dissipation-less fluid dark matter Yet, the particles forming dark matter have so far escaped detection...

5 Dark matter direct detection / basics Geometry: Basic concept: It searches for nuclear recoil events induced by the non-relativistic scattering of Milky Way dark matter particles in low-background detectors

6 Dark matter direct detection / basics Geometry: Kinematics: - For m χ 1 GeV, one expects a flux of cm 2 s 1 - Expected recoil energy, E R = (2µ 2 T v 2 /m T ) cos 2 θ O(1) kev Modulation: - The inclination of the Earth s orbit induces an annual modulation in the number of recoil events

7 Dark matter direct detection / measurements Example: scintillation plus charge signal (e.g. LUX, XENON, LZ)

8 Dark matter direct detection / DAMA modulation signal DAMA results: data vs cosinusoidal fitting function 2-4 kev Residuals (cpd/kg/kev) DAMA/LIBRA 25 kg (.87 ton yr) Time (day)

9 Dark matter direct detection / physical observables Rate of dark matter-nucleus scattering events: dr = ρ χ ξ T f (v + v e(t)) v dσ T (v 2, q 2 ) d 3 v de R m T m χ de R T v>v min (q) galactic distribution particle nature Modulation amplitude: 1 1 [ ] A(E, E +) = R(E, E +) R(E, E +) E + E 2 June 1st Dec 1st

10 Local dark matter density in 5 steps Assume a mass model for the Milky Way: halo, stellar disk, bulge Calculate the observables: rotation curves, surface density, velocity dispersion of stars, weak lensing optical depth, etc... Compare predictions with astronomical observations: the Bayesian approach has proven to be a powerful tool for this Extract preferred regions in parameter space, e.g. credible regions Translate them into an estimate for the local dark matter density, e.g. posterior PDF

11 Local dark matter density: Bayesian analysis Catena & Ullio 21

12 Local dark matter velocity distribution in steps Simplifying assumption: spherically symmetric galactic gravitational potential Use Eddington s inversion formula to relate the local dark matter velocity distribution to the parameters of the assumed mass model From the posterior PDF of the model parameters, obtain the posterior PDF of local dark matter velocity distribution at sampled velocities

13 Local dark matter velocity distribution: Bayesian analysis Bozorgnia, Catena and Schwetz g χ (u) [(GeV/cm 3 ) (s/km) 2 ] u [km/s]

14 Dark matter-nucleus scattering cross-section Standard paradigm: spin-independent and spin-dependent dark matter-nucleon interactions dσ T = m T 1 A F e iq r i (H de R 2πv 2 SI + H SD) I (2j χ + 1)(2J + 1) spins i=1 nucleus DM state 2 one-body DM-nucleon interaction

15 Spin-independent interaction H SI Scalar/Scalar coupling: L SS = 1 Λ q C SS 3 q χχm q qq S-matrix element: f is i = iū χ(p )u χ(p) d 4 x e i qx N q c q q(x)q(x) N i(2π) 4 δ 4 (q k + k) ξ χ ξ χ ξ N (b + b1τ3)ξ N Underlying non-relativistic Hamiltonian H SI = b τ 1 χ1 N t τ c1 τ 1 χn t τ τ=,1 τ=,1

16 Spin-dependent interaction H SD Axial-Vector/Axial-Vector: L AA = 1 Λ q C AA 2 q χγ µ γ 5χ qγ µγ 5q S-matrix element: f is i = iū χ(p )γ µγ 5u χ(p) d 4 x e i qx N q c q q(x)γ µ γ 5q(x) N i(2π) 4 δ 4 (q k + k) ξ χ σ χξ χ ξ N (a + a1τ3)σ Nξ N Underlying non-relativistic Hamiltonian H SD = a τ σ χ σ N t τ c4 τ Ŝχ Ŝ N t τ τ=,1 τ=,1

17 Dark matter direct detection / physical observables Rate of dark matter-nucleus scattering events: dr = ρ χ ξ T f (v + v e(t)) v dσ T (v 2, q 2 ) d 3 v de R m T m χ de R T v>v min (q) galactic distribution particle nature Modulation amplitude: 1 1 [ ] A(E, E +) = R(E, E +) R(E, E +) E + E 2 June 1st Dec 1st

18 Exclusion limits / Favored regions Billard et al Be Neutrinos CDMSlite (213) DAMIC (212) COHERENT NEUTRIN O SCATTERING 8 B Neutrinos CoGeNT (212) CDMS Si (213) CRESST DAMA COHERENT NEU TRI NO SCATTERING EDELWEISS (211) WIMPMass GeV c 2 SIMPLE (212) COHERENT NEUTRINO SCATTERING Atmospheric and DSNB Neutrinos COUPP (212) ZEPLIN-III (212) CDMS II Ge (29) Xenon1 (212) LUX (213) WIMP nucleon cross section pb

19 Effective Theory (ET) of dark matter direct detection Assumption: q /m V 1, where m V is the mediator mass ET basic symmetries: Galilean and translation invariance ET basic operators: Consider the scattering χ(p) + N(k) χ(p ) + N(k ) Momentum conservation M(p, k, q) Galilean invariance M(v = p/m χ k/m N, q) In general, M = M(v, q, S χ,s N ) We therefore identify five basic operators 1 χn iˆq ˆv = ˆv + ˆq 2µ N Ŝ χ Ŝ N The most general Hamiltonian for χ-n interactions is a power series in ˆq/m V. Each term in the series is a scalar combination of basic operators.

20 Effective Hamiltonian for dark matter-nucleon interactions Only 14 linearly independent operators can be constructed from the basic operators, if we demand that they are at most quadratic in ˆq (and arise from the exchange of a mediator of spin 1) The most general Hamiltonian density is therefore Ĥ(r) = τ=,1 k ck τ Ôk(r) t τ t = 1, t 1 = τ 3 c p k = (c k + c 1 k )/2 and c n k = (c k c 1 k )/2

21 Dark matter-nucleon interaction operators Ô 1 = 1 χn Ô 9 = iŝ χ (ŜN ˆq ( ) Ô 3 = iŝn ˆq m N ˆv Ô 4 = Ŝχ ŜN Ô ( ) Ô 5 = iŝχ ˆq m N ˆv Ô 6 = (Ŝχ ˆq m N ) (ŜN ) ˆq m N Ô 1 = iŝn ˆq m N m N ) 11 = iŝχ ˆq m N ) Ô 12 = (ŜN Ŝχ ˆv Ô 7 = ŜN ˆv Ô 14 = i (Ŝχ Ô 8 = Ŝχ ˆv Ô 15 = (Ŝχ ) ˆq m N ) Ô 13 = i (Ŝχ ˆv (ŜN ) ) ˆq (ŜN m N ˆv ) ) ˆq [(ŜN m N ˆv ] ˆq m N

22 Dark matter-nucleus scattering cross-section In the ET framework, the dark matter-nucleus scattering cross-section is dσ T = m T 1 F de R 2πv 2 (2j χ + 1)(2J + 1) spins A dr e iq r Ĥ i (r) I i=1 2 This expression depends on: - 28 coupling constants - 8 nuclear response functions Available nuclear response functions: - For Xe, Ge, I, Na, F: Anand et al For 16 elements in the Sun: R. Catena & B. Schwabe 215

23 Comparison with null searches I compare theory and observations in global multidimensional statistical analyses varying all model parameters simultaneously Experimental data simultaneously included in the fit: LUX XENON1 XENON1 CDMS-Ge CDMS-LT SuperCDMS CDMSlite PICASSO SIMPLE COUPP

24 Global limits: mass vs interaction strengths R. Catena and P. Gondolo, JCAP 149 (214) log 1 (c 1 m 2 v ) log (c m 2 ) 1 4 v D marginal pdf 1D profile Likelihood 99% CR 95% CL Profile likelihood color code: log (m /GeV) 1 χ log (m /GeV) 1 χ log (m /GeV) 1 χ log (c m 2 ) 1 3 v 2 2 log 1 (c 5 m 2 v ) 2 2 log (c m 2 ) 1 6 v 2 2 log (c m 2 ) 1 7 v log 1 (m χ /GeV) log (m /GeV) 1 χ log (m /GeV) 1 χ log 1 (m χ /GeV) log (c m 2 ) 1 8 v 2 2 log 1 (c 9 m 2 v ) 2 2 log (c m 2 ) 1 1 v 2 2 log (c m 2 ) 1 11 v log 1 (m χ /GeV) log 1 (m χ /GeV) log 1 (m χ /GeV) log 1 (m χ /GeV)

25 Operator interference Pairs of operators, or isoscalar and isovector components of the same operator can interfere For instance, the operators Ô 1 and Ô 3 generate a transition probability proportional to M NR 2 spins = 4π [ 2J + 1 ττ R. Catena and P. Gondolo, JCAP 158, 8, 22 (215) c τ 1 cτ 1 W ττ q 2 M (y) m N 2 ( + q2 q 2 m N 2 4m N 2 c τ 3 cτ 3 W ττ Φ v 2 T cτ 3 cτ 3 W ττ Σ (y) ) ] (y) + cτ 1 cτ 3 W ττ Φ M (y) 1 c 1 c 1 1 (all data) 1 1 c 1 c 1 1 (all data) 1 c 1 c 3 (all data) c 1 (LUX).8 c 1 1 c 3 1 (all data) c 1 1 (LUX).8 log 1 (c 1 m 2 v ) log 1 (c 1 1 m 2 v ) log (m /GeV) 1 χ log (m /GeV) 1 χ

26 DAMA confronts null searches I j Is there a linear combination of Ô k such that DAMA can be reconciled with null searches? j c T 95% -u.l. Aexp-1 (m χ) c < 1 c T 95% -u.l. Aexp-1 (m χ) c < 1 c T 95% -u.l. Aexp-2 (m χ) c < 1 c T 95% -u.l. Aexp-2 (m χ) c < 1 increasing nσ c τ (, ) c τ up to n max σ (, ) c T A nσ-l.l. DAMA[E,E+] (mχ) c > 1 c T A nmax σ -l.l. DAMA[E,E+] (mχ) c > 1 c τ i c τ i R. Catena, A. Ibarra, S. Wild, arxiv:

27 DAMA confronts null searches II 7σ 6σ DD + Super-K (τ + τ /W + W ) DM with non-zero spin 7σ 6σ DD + Super-K (τ + τ /W + W ) DM with zero spin N σ max 5σ Only DD N σ max 5σ Only DD 4σ 4σ 3σ mχ [GeV] Q Na =.3, Q I =.9 3σ mχ [GeV] Q Na =.3, Q I =.9 7σ 6σ DD + Super-K (τ + τ /W + W ) DM with non-zero spin 7σ 6σ DD + Super-K (τ + τ /W + W ) DM with zero spin N σ max 5σ Only DD N σ max 5σ Only DD 4σ 4σ 3σ mχ [GeV] Q Na =.4, Q I =.5 3σ mχ [GeV] Q Na =.4, Q I =.5 R. Catena, A. Ibarra, S. Wild, arxiv:

28 Conclusions A complete classification of all one-body dark matter-nucleon interaction operators has recently been performed Current direct detection data place interesting constraints on dark matter-nucleon interaction operators commonly neglected Destructive interference effects can weaken standard direct detection exclusion limits by up to 1 order of magnitude in the coupling constants Even within this general theoretical framework, it seems to be impossible to reconcile DAMA with null searches