Hadronic matrix elements for Dark Matter and other searches - PowerPoint PPT Presentation

Hadronic matrix elements for Dark Matter and other searches Laurent Lellouch CPT Marseille CNRS & Aix-Marseille U. Budapest-Marseille-Wuppertal collaboration (BMWc) (Phys.Rev.Lett. 116 (2016) 172001 and in preparation) Laurent Lellouch KEK-PH, , 13-16 February 2017

Direct WIMP dark matter detection ↓ χ χ � λ Γ → L N χ = λ Γ N [¯ q [¯ L q χ = q Γ q ][¯ χ Γ χ ] − N Γ N ][¯ χ Γ χ ] q H Quark are confined within nucleons → nonperturbative QCD tool N N Laurent Lellouch KEK-PH, , 13-16 February 2017

WIMP-nucleus spin-independent cross section . . . In low- E limit d σ SI χ A 1 π v 2 [ Zf p + ( A − Z ) f n ] 2 | F X ( q 2 ) | 2 Z X = dq 2 w/ F X ( � q = 0 ) = 1 nuclear FF and χ N couplings ( N = p , n ) f N λ q λ Q � f N � f N = m q + q Q M N m Q q = u , d , s Q = c , b , t p ′ − � p ) � = ( 2 π ) 3 δ ( 3 ) ( � such that ( f = u , . . . , t and � N ( � p ′ ) | N ( � p ) ) uu + ¯ f M N = σ fN = m f � N | ¯ f N f N ud M N = σ π N = m ud � N | ¯ dd | N � , ff | N � For heavy Q = c , b , t (Shifman et al ’78) m Q ¯ QQ � α 2 QQ = − 1 α s s O 6 4 π G 2 + O m Q ¯ � − → 4 m 2 3 Q Laurent Lellouch KEK-PH, , 13-16 February 2017

. . . and relevant hadronic matrix elements Then obtain f N Q in terms of f N q through to M N = � N | θ µ µ | N � , w/    + β ( α s ) θ µ  � � m Q ¯ G 2 m q ¯ µ = ( 1 − γ m ( α s )) qq + QQ 2 α s q = u , d , s Q = c , b , t Integrate out Q = t , b , c and obtain f N c from f N q , q = u , d , s , etc. Will be done to O ( α 3 s ) (Hill et al ’15) , but at LO find   Λ 2 Q ≡ � N | m Q ¯ QQ | N � = 2 QCD f N � f N  + O α s , α 2 � �  1 − q s 4 m 2 M N 27 Q q = u , d , s since 4 πβ ( α s ) = − β 0 α 2 s + O ( α 3 s ) and β 0 = 11 − 2 3 N q − 2 3 N Q For f N q , q = u , d , s , use lattice QCD and Feynman-Hellman theorem � qq | N � = m q ∂ M N f N � q M N = � N | m q ¯ � ∂ m q � m Φ q Laurent Lellouch KEK-PH, , 13-16 February 2017

What is lattice QCD (LQCD)? To describe ordinary matter, QCD requires ≥ 104 numbers at every point of spacetime → ∞ number of numbers in our continuous spacetime → must temporarily “simplify” the theory to be able to calculate (regularization) ⇒ Lattice gauge theory − → mathematically sound definition of NP QCD: U µ ( x ) = e iagA µ ( x ) ψ ( x ) ✻ r r r r r r r r ✻ a ❄ UV (& IR) cutoff → well defined path integral in r r r r r r r r Euclidean spacetime: r r r r r r r r � ¯ � ψ D [ M ] ψ O [ U , ψ, ¯ D U D ¯ ψ D ψ e − S G − r r r r r r r r � O � = ψ ] ✛ T r r r r ❄ r r r r ✲ ✻ � r r r r r r r r D U e − S G det ( D [ M ]) O [ U ] Wick = r r r r r r r r r r r r r r r r D Ue − S G det ( D [ M ]) ≥ 0 & finite # of dofs r r r r r r r r ❄ ✛ r r r r r r r r ✲ → evaluate numerically using stochastic methods L LQCD is QCD but only when N f ≥ 2 + 1, m q → m phys , a → 0, L → ∞ q HUGE conceptual and numerical challenge (integrate over ∼ 10 9 real variables) ⇒ very few calculations control all necessary limits Laurent Lellouch KEK-PH, , 13-16 February 2017

Strategy of calculation Objective: Determine slope of M N wrt m q , q = u , d , s , at physical point Method: Perform many high-statistics simulations with various m q around physical values, various a < ∼ 0 . 1 fm and various L > ∼ 6 fm � 2 M 2 K − M 2 For each compute M π ( → m ud ), M η s = π ( → m s ), M D s ( → m c ) and M N ( → Λ QCD ) Study dependence of m q , q = ud , s , c and M N on M π , M η s , M D s , a and L For each simulation determine a , m Φ q ’s such that M π , . . . take their physical value in a → 0 and L → ∞ limit Compute, at physical point ∂ ln M 2 ∂ ln M N f N � P q = ∂ ln M 2 ∂ ln m q P = π,η s P Laurent Lellouch KEK-PH, , 13-16 February 2017

Lattice details N f = 1 + 1 + 1 + 1 3HEX clover-improved Wilson fermions on tree-level improved Symanzik gluons 33 ensembles w/ total ∼ 169000 trajectories ∼ 500 measurements per configuration 4 a ∈ [ 0 . 064 , 0 . 102 ] fm ; M π ∈ [ 195 , 450 ] MeV w/ LM π > 4 Improvements over BMWc, PRL ’16 ✓ Charm in sea ✓ > ∼ × 100 in statistics ✓ > ∼ × 2 lever arm in m s ✓ Like PRL ’16 FH in terms of quark and not meson masses ✗ No physical m ud , but small enough and know M N from experiment Laurent Lellouch KEK-PH, , 13-16 February 2017

M 2 π ∼ m ud dependence of M N (preliminary) 1150 β =3.2 β =3.2 QCD+QED β =3.3 1100 β =3.4 β =3.5 M N [MeV] 1050 1000 950 900 2 2 2 100 200 300 2 [MeV 2 ] M π Laurent Lellouch KEK-PH, , 13-16 February 2017

M η s ∼ m s dependence of M N (preliminary) β =3.2 β =3.2 QCD+QED 960 β =3.3 β =3.4 β =3.5 940 M N [MeV] 920 900 880 2 2 2 2 2 2 300 400 500 600 700 800 2 [MeV 2 ] M η s Laurent Lellouch KEK-PH, , 13-16 February 2017

Chain rule conversion matrix (preliminary) Have: 0.2 φ m ud /m s ∂ ln M N β =3.2 β =3.3 0.1 ∂ ln M 2 β =3.4 P β =3.5 w/ P = π, η s 0 1.5 φ m s /m s Want: 1 ∂ ln M N 0.5 ∂ ln m q 0 w/ q = u , d , s 2 2 2 2 2 2 2 2 2 100 200 300 300 400 500 600 700 800 2 [MeV 2 2 [MeV 2 M π ] M η ] s  ∂ ln M 2  � 0 . 94 ( 1 )( 1 ) ∂ ln M 2 η s � π 0 . 002 ( 0 )( 1 )  = ∂ ln m ud ∂ ln m ud  ∂ ln M 2 ∂ ln M 2 0 . 06 ( 2 )( 3 ) 1 . 02 ( 0 )( 2 ) η s π ∂ ln m s ∂ ln m s Laurent Lellouch KEK-PH, , 13-16 February 2017

Systematic error assessment (preliminary) Estimated using extended frequentist approach (BMWc, Science ’08, Science ’15) Excited state contamination: 4 time intervals for correlations functions Mass interpolation/extrapolation errors M π ≤ 330 / 360 / 420 MeV different M π/η s dependences (polynomials, Padés, χ PT) continuum extrapolation: O ( α s a ) vs O ( a 2 ) ⇒ 672 analyses which differ by higher order effects AIC weight AIC weight Q weight Q weight flat weight flat weight 1 1 relative weight relative weight 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 20 30 40 50 40 50 60 70 σ udN [MeV] σ sN [MeV] Laurent Lellouch KEK-PH, , 13-16 February 2017

Preliminary results Direct results f N f N ud = 0 . 0430 ( 19 )( 32 ) [ 8 . 6 %] s = 0 . 0564 ( 38 )( 35 ) [ 9 . 2 %] Using SU(2) isospin (BMWc, PRL116) w/ ∆ QCD M N = 2 . 52 ( 17 )( 24 ) MeV (BMWc, Science 347) & m u / m d = 0 . 485 ( 11 )( 16 ) (BMWc, PRL117) f p f p u = 0 . 0153 ( 6 )( 11 ) [ 8 . 1 %] d = 0 . 0264 ( 13 )( 21 ) [ 9 . 4 %] f n f n u = 0 . 0128 ( 6 )( 11 ) [ 9 . 7 %] d = 0 . 0316 ( 13 )( 21 ) [ 7 . 9 %] Using f N ud , s & HQ expansion up to O ( α 4 s , Λ 2 QCD / m 2 c ) corrections (Hill et al ’15) f N f N c = 0 . 0730 ( 5 )( 5 )( ?? ) [ 1 . 0 + ?? %] b = 0 . 0700 ( 4 )( 4 )( ?? ) [ 0 . 9 + ?? %] f N = 0 . 0678 ( 3 )( 3 )( ?? ) [ 0 . 7 + ?? %] t Laurent Lellouch KEK-PH, , 13-16 February 2017

Low-energy effective h - N coupling (preliminary) 8 c b t 6 s p [%] 4 f f d 2 u 0 1 100 10000 m q [MeV] f fN is q contribution to effective coupling of Higgs to nucleon in units of M N ⇒ fraction of M N coming from q contribution to coupling of N to Higgs vev HQ expansion ⇒ Q = c , b , t contributions mainly through their impact on the running of α s f N = 0 . 310 ( 3 )( 3 )( ?? ) [ 1 . 4 + ?? %] Laurent Lellouch KEK-PH, , 13-16 February 2017

Conclusion Scalar quark contents of p & n have been computed with full control over all sources of uncertainties Important for: DM searches; coherent LFV µ → e conversion in nuclei; describing low-energy coupling of N to the Higgs; understanding M N ; π N and KN scattering, etc. f N q , q = u , d , s & N = n , p are now known to better than 10% f N Q , Q = c , b , t and f N to even better precision Correlations between the various quantities will be given Hadronic ME are no longer the dominant source of uncertainty in DM direct detection rate predictions . . . . . . or in the determination of WIMP couplings from possible DM signals Laurent Lellouch KEK-PH, , 13-16 February 2017

BACKUP Laurent Lellouch KEK-PH, , 13-16 February 2017

Comparison N N f f ud s 0.02 0.04 0.06 0.08 0 0.1 0.2 0.3 0.4 0.5 GLS 91 _ GLS 91 _ Pheno. Pavan 02 _ Pavan 02 _ N f =2 Shanahan et al 12 _ Alarcon et al 12 _ Lutz et al 14 _ Shanahan et al 12 _ N f =2+1 Ren et al 14 _ Alvarez et al 13, FH _ An et al 14 _ N f =2+1+1 Lutz et al 14, FH _ Pheno. Alarcon et al 14 _ Ren et al 14, FH _ JLQCD 10, FH N f =2 _ Bali et al 11, ME _ Hoferichter et al 15 _ N f =2+1 ETM 16, ME _ JLQCD 08, FH _ Bali et al 16, ME N f =2+1+1 _ Bali et al 11, FH _ MILC 09, FH _ ETM 16, ME _ BMWc 11, FH _ Bali et al 16, ME _ QCDSF 11, FH _ BMWc 11, FH _ Ohki et al 13, ME _ Junnarkar et al 13, FH _ QCDSF 11,FH _ Gong et al 13, ME _ Yang et al 15, ME _ Yang et al 15, ME _ BMWc 16, FH _ BMWc 16, FH _ _ New dataset, FH New dataset, FH _ 0 0.1 0.2 0.3 0.4 0.5 0.02 0.04 0.06 0.08 Laurent Lellouch KEK-PH, , 13-16 February 2017

Finite-volume effects β =3.2 1000 β =3.2 QCD+QED β =3.3 β =3.4 β =3.5 M N [MeV] 980 960 940 1/10 1/6 1/4 1/3 -1 [fm -1 ] L Fit away leading effects M X ( L ) − M X = cM 1 / 2 L − 3 / 2 e − LM π π M X Compabtible w/ χ PT expectation (Colangelo et al ’10) Laurent Lellouch KEK-PH, , 13-16 February 2017