Dark matter and BSM with nuclei


 Cornelius Strickland
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1 Dark matter and BSM with nuclei Martin Hoferichter Institute for Nuclear Theory University of Washington INT Program on Fundamental Physics with Electroweak Probes of Light Nuclei Seattle, July 11, 2018 PLB 746 (2015) 410, PRD 94 (2016) , PRL 119 (2017) with P. Klos, J. Menéndez, A. Schwenk PRD 97 (2018) with A. Fieguth, P. Klos, J. Menéndez, A. Schwenk, C. Weinheimer M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
2 How to search for dark matter? Search strategies: direct, indirect, collider Assume DM particle is WIMP Direct detection: search for WIMPs scattering off nuclei in the largescale detectors Ingredients for interpretation: DM halo: velocity distribution Nucleon matrix elements: WIMP nucleon couplings Nuclear structure factors: embedding into indirect X X Planck 2013 collider target nucleus direct M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
3 Direct detection of dark matter: schematics Nuclear recoil in WIMP nucleus scattering Flux factor Φ: DM halo and velocity distribution WIMP nucleus cross section XENON1T 2018 Spinindependent: coherent A 2 Spindependent: S p or S n Information on BSM physics encoded in normalization at q = 0 for SI case: σ SI χn M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
4 Direct detection of dark matter: scales q,g X Λ BSM q,g Y X 1 BSM scale Λ BSM : L BSM q,g X 2 Effective Operators: L SM + i,k 1 Λ i BSMO i,k q,g X Λ EW 3 Integrate out EW physics N X π X 4 Hadronic scale: nucleons and pions Λ hadrons N X π X effective interaction Hamiltonian H I 5 Nuclear scale: N H I N Λ nuclei nuclear wave function M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
5 Direct detection of dark matter: scales Λ hadrons N π X 4 Hadronic scale: nucleons and pions N π X effective interaction Hamiltonian H I 5 Nuclear scale: N H I N Λ nuclei Typical WIMP nucleon momentum transfer nuclear wave function q max = 2µ Nχ v rel 200 MeV v rel 10 3 µ Nχ 100 GeV QCD constraints: spontaneous breaking of chiral symmetry Chiral effective field theory for WIMP nucleon scattering Prézeau et al. 2003, Cirigliano et al. 2012, 2013, Menéndez et al. 2012, Klos et al. 2013, MH et al. 2015, Bishara et al In NREFT Fan et al. 2010, Fitzpatrick et al. 2012, Anand et al need to match to QCD to extract information on BSM physics the EFT approach not unique! M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
6 Outline 1 Chiral effective field theory 2 Corrections beyond the standard nuclear responses 3 Calculating nuclear responses 4 Limits on Higgs Portal dark matter 5 Conclusions M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
7 Chiral Perturbation Theory Effective theory of QCD based on chiral symmetry L QCD = q L i /Dq L + q R i /Dq R q L Mq R q R Mq L 1 4 Ga µνg µν a Expansion in momenta p/λ χ and quark masses m q p 2 scale separation Two variants SU(2): u and dquark dynamical, m s fixed at physical value expansion in M π/λ χ, usually nice convergence SU(3): u, d, and squark dynamical expansion in M K /Λ χ, sometimes tricky Λ χ π,k,η M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
8 Chiral EFT: a modern approach to nuclear forces 2N force 3N force 4N force Traditionally: mesonexchange potentials Chiral effective field theory Based on chiral symmetry of QCD Power counting Lowenergy constants Hierarchy of multinucleon forces Consistency of NN and 3N modern theory of nuclear forces Longrange part related to pion nucleon scattering LO NLO 2 N LO 3 N LO Figure taken from M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
9 Nuclear Physics from first principles longrange plan 2015 }{{} }{{} QCD shell model, coupled cluster, inmedium SRG,... }{{} exact manybody: lattice EFT, NCSM, coupledcluster,... }{{} density functional, meanfield theory,... Piecewise overlap of abinitio and various manybody methods match to QCD Consistent NN interactions key at various stages Abinitio not yet up to xenon, but impressive progress M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
10 Chiral EFT: currents Coupling to external sourcesl(v µ, a µ, s, p) Same LECs appear in axial current β decay, neutrino interactions, dark matter Vast literature for v µ and a µ, up to oneloop level With unitary transformations: Kölling et al. 2009, 2011, Krebs et al Without unitary transformations: Pastore et al. 2008, Park et al. 2003, Baroni et al For dark matter further currents: s, p, tensor, spin2, θ µ µ (a) (b) (c) (d) M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
11 Vector current in chiral EFT: deuteron form factors, magnetic moments 1 1 m N /M d G M (q) [µ N ] 0,1 NLO NNLO 0, q [MeV] 10 0 m N /M d G M (q) [µ N ] 0,1 NLO NNLO 0, q [MeV] Kölling, Epelbaum, Phillips 2012 j N3LO /NN(N3LO), Piarulli et al. 3 j N3LO.. p 7 Li 9 /NN(N2LO), Kolling et al. 2 3 H 9 B Li Li 2 H 6 Li 8 B 0 j LO / AV18+IL j N3LO / AV18+IL7 91 EXPT 7 Be 9 C Be n 3 He 2 (b) q [fm 1 ] Bacca, Pastore 2014 Pastore et al m/(m d µ d ) G M µ (µ N ) 4 M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
12 Axialvector current in chiral EFT: νless double β decay GT(1b+2b)/g A bc 1bc p [MeV] Menéndez, Gazit, Schwenk 2011 Normal ordering over Fermi sea effective onebody currents Twobody currents contribute to quenching of g A in Gamov Teller operator g A στ M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
13 Direct detection and chiral EFT (a) (b) (c) (d) Expansion around chiral limit of QCD simultaneous expansion in momenta and quark masses Three classes of corrections: Subleading onebody responses (a) Radius corrections (b) Twobody currents (c),(d) NREFT covers (a), but misses (b) (d) (b): modifies coefficient of O i by momentumdependent factor (c),(d): do not match directly onto NREFT, need normal ordering N N N N O eff i (a)+(b) just ChPT for nucleon form factors, but (c)+(d) genuinely new effects M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
14 Chiral counting Starting point: effective WIMP Lagrangian Goodman et al L χ = 1 [ ] C SS Λ 3 q χχ m q qq + C PS q χiγ 5χ m q qq + C SP q χχ m q qiγ 5q + C PP q χiγ 5χ m q qiγ 5q q + 1 [ Λ 2 C VV q q + 1 [ ] C S Λ 3 g χχαsga µν Gµν a Chiral power counting χγ µ χ qγ µq + C AV q χγ µ γ 5χ qγ µq + C VA q = O ( p ), m q = O ( p 2) = O(M 2 π), aµ, v µ = O(p), ] χγµ χ qγ µγ 5q + C AA q χγ µ γ 5χ qγ µγ 5q construction of effective Lagrangian for nucleon and pion fields organize in terms of chiral order ν, M = O(p ν ) m N = O ( p 2) M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
15 Chiral counting: summary Nucleon V A WIMP t x t x WIMP Nucleon S P 1b V 2b b NLO b A 2b b NLO b 2 1 S 2b 3 5 2b NLO 4 1b P 2b b NLO from NR expansion of WIMP spinors, terms can be dropped if m χ m N Red: all terms up to ν = 3 Twobody currents: AA Menéndez et al. 2012, Klos et al. 2013, SS Prézeau et al. 2003, Cirigliano et al. 2012, but new currents in AV and VA channel Worked out the matching to NREFT and BSM Wilson coefficients for spin1/2 hierarchy predicted from chiral expansion M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
16 Matching to nonrelativistic EFT Operator basis in NREFT Fan et al. 2010, Fitzpatrick et al. 2012, Anand et al O 1 = 1 O 2 = ( v ) 2 O 3 = is N (q v ) O 4 = S χ S N O 5 = is χ (q v ) O 6 = S χ q S N q O 7 = S N v O 8 = S χ v O 9 = is χ (S N q ) O 10 = is N q O 11 = is χ q Matching to chiral EFT (f N,...: Wilson coefficients + nucleon form factors) Conclusions M SS 1,NR = O 1f N (t) M VV ( 1,NR = O 1 f V,N 1 M SP 1,NR = O 10g N 5 (t) (t) + t 4m 2 N f V,N ) (t) M AV 1,NR = 2O 8f V,N (t) + 2 ( O 1 9 f V,N 1 m N M AA 1,NR = 4O 4g N A (t) + 1 m N 2 O 6 g N P (t) MPP 1,NR = 1 O 6 h N 5 m (t) χ O 3 f V,N 2 m N (t) + f V,N ) (t) 2 1 ( ) (t) + to 4 + O 6 f V,N (t) 2 m N m χ { MVA 1,NR = 2O } O 9 h N A m (t) χ O 2, O 5, and O 11 do not appear at ν = 3, not all O i independent 2b operators of similar or even greater importance than some of the 1b operators M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
17 Direct detection of dark matter: scales Λ hadrons N π X 4 Hadronic scale: nucleons and pions N π X effective interaction Hamiltonian H I 5 Nuclear scale: N H I N Λ nuclei nuclear wave function M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
18 Coherence effects Six distinct nuclear responses Fitzpatrick et al. 2012, Anand et al M O 1 SI Σ,Σ O 4,O 6 SD Φ O 3 quasicoherent, spinorbit operator, Φ : not coherent Quasicoherence of Φ Spinorbit splitting Coherence until midshell About 20 coherent nucleons in Xe j = l 1 2 j = l+ 1 2 Interference M Φ O 1 O 3 Further coherent Mresponses from O 5, O 8, O 11, but no interference with O 1 due to sum over S χ M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
19 Spectra and shellmodel calculation Energy (kev) Th 1/2 + 7/23/2 + 7/2 + 3/2 + 5/2 + 9/211/21/2 + 3/ Xe Exp 3/2 + 7/27/2 + (1/2,3/2) + 3/2 + 5/2 + 9/211/21/2 + 3/2 + Excitation energy (kev) Theory Xe Exp & Shellmodel diagonalization for Xe isotopes with 100 Sn core Uncertainty estimates: currently phenomenological shellmodel interaction chiraleftbased interactions in the future? abinitio calculations for light nuclei? M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
20 Consequences for the structure factors ξoi F interference terms e Xe 1e q [GeV] O1 O1 O3 O3 O11 O8 O5 ξoi F 2 + interference terms e Xe 1e q [GeV] O1 O1 O3 O3 O11 O8 O5 ξ Oi kinematic factors for O i, e.g. ξ O1 = 1, ξ O3 = q2 2m 2 N O 11 assumes m χ = 2 GeV much stronger suppressed for heavy WIMPs Structure factors imply hierarchy as long as coefficients do not differ strongly M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
21 Twobody currents ξoi F interference terms Xe Finite at q = 0 q [GeV] O1 O1 O3 O1 2b θ O1 2b scalar 2b θ 2b interference 2b scalar O3 2b θ O3 2b scalar (a) (b) (c) (d) Most important next to IS and IV O 1 Sensitive to new combination of Wilson coefficients, e.g. for scalar channel f N = m ( N ) Λ 3 Cq SS f q N 12πf Q N C S g q=u,d,s f π = Mπ Λ 3 ( q=u,d Cq SS + 8π 9 C S g ) f π q fπ θ = Mπ 8π Λ 3 9 C S g Typically (5 10)% effect, enhanced whenever cancellations occur: blind spots, heavy WIMP limit M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
22 Radius corrections ξoi F interference terms Xe q [GeV] O1 O1 O3 O1 2b θ O1 2b scalar 2b θ 2b interference 2b scalar radius (a) (b) (c) (d) Set scale as q 2 /m 2 N Strong suppression at small q, but potentially relevant later Yet another new combination ḟ N = m ( N ) Λ 3 C SS q ḟ N q 12πḟ N Q C S g q=u,d,s M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
23 Full set of coherent contributions structure factors Xe F+ M 2 [O1] 2 F+ M Fπ θ [O1 2b θ] 2 F+ M F M [O1 IS IV] 2 F+ M Fπ [O1 2b scalar] Fπ θ 2 [2b θ] F M 2 [O1 IV] Fπ 2 [2b scalar] 2q 2 /m 2 N FM + 2 [O1 radius] q 2 /m 2 N FM + F Φ + [O1 O3] q [GeV] Parameterize cross section as dσχn SI dq 2 = 1 4πv 2 (c + M q2 ) F+(q M 2 )+ (c M q2 ) mn 2 ċ M F (q M 2 ) mn 2 ċ+ M + c πf π(q 2 )+cπf θ π(q θ 2 )+ q2 ] [c Φ 2mN 2 + F+ Φ (q 2 )+c Φ F Φ (q 2 2 ) Singlenucleon cross section: σ SI χn = µ 2 N c M + 2 /π c related to Wilson coefficients and nucleon form factors M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
24 Discriminating different response functions structure factors m χ =10GeV 132 Xe F M + 2 F M 2 q 2 /m 2 N FM + 2 q 2 /m 2 N FM 2 Fπ 2 q 4 /m 4 N FM + 2 q 4 /m 4 N FM 2 q 4 /m 4 N FΦ + 2 q 4 /m 4 N FΦ q [GeV] White region accessible to XENONtype experiment Can one tell these curves apart in a realistic experimental setting? Consider XENON1Tlike, XENONnTlike, DARWINlike settings M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
25 Discriminating different response functions discrimination power at 1.28 [%] F M 2 F 2 q 2 /4m 2 F M exposure [10 ton years] DARWINlike setting, m χ = 100 GeV qdependent responses more easily distinguishable If interaction not much weaker than current limits, DARWIN could discriminate most responses from standard SI structure factor M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
26 Twobody currents: SD case Xe S p (u) 1b currents S n (u) 1b currents S p (u) 1b + 2b currents S n (u) 1b + 2b currents S i (u) S i (u) u 131 Xe S p (u) 1b currents S n (u) 1b currents S p (u) 1b + 2b currents S n (u) 1b + 2b currents Nuclear structure factors for spindependent interactions Klos et al Based on chiral EFT currents (1b+2b) Shell model u = q 2 b 2 /2 related to momentum transfer 2b currents absorbed into redefinition of 1b current u M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
27 Twobody currents: SD case ) 2 SD WIMPproton crosssection (cm SuperK (bb) SuperK (τ + τ )  CDMS PICO2L DAMA SuperK (W + W ) LZ Projected PICASSO  KIMS PICO60 ZEPLINIII XENON10 IceCube (bb) IceCube (τ + τ )  XENON100 + IceCube (W  ) W LUX 90% C.L WIMP Mass (GeV/c 3 2 ) Xenon becomes competitive for σ p thanks to twobody currents! M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
28 Higgs Portal dark matter Higgs Portal: WIMP interacts with SM via the Higgs Scalar: H H S 2 Vector: H H V µv µ Fermion: H H f f If m h > 2m χ, should happen at the LHC WIMP nucleon cross section [ cm 2] limits on invisible Higgs decays Scalar Fermion Vector WIMP mass [GeV] PandaXII LUX XENON1T SuperCDMS M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
29 Higgs Portal dark matter Higgs Portal: WIMP interacts with SM via the Higgs Scalar: H H S 2 Vector: H H V µv µ Fermion: H H f f If m h > 2m χ, should happen at the LHC WIMP nucleon cross section [ cm 2] limits on invisible Higgs decays Scalar Fermion Translation requires input for Higgs nucleon coupling Vector WIMP mass [GeV] f N = fq N = fq N 9 +O(αs) m Nfq N = N mq qq N q=u,d,s,c,b,t q=u,d,s Issues: input for f N = outdated, twobody currents missing PandaXII LUX XENON1T SuperCDMS M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
30 Higgs nucleon coupling (a) (b) (c) (d) Onebody contribution f 1b N = 0.307(9) ud (15) s(5) pert = 0.307(18) Limits on WIMP nucleon cross section subsume twobody effects have to be included for meaningful comparison Twobody contribution Need s and θ µ µ currents Treatment of θ µ µ tricky: several illdefined terms combine to Ψ T + V NN Ψ = E b A cancellation makes the final result anomalously small f 2b N = [ 3.2(0.2) A (2.1) ChEFT + 5.0(0.4) A ] 10 3 = 1.8(2.1) 10 3 M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
31 Improved limits for Higgs Portal dark matter WIMP nucleon cross section [ cm 2] Scalar Fermion Vector WIMP mass [GeV] PandaXII LUX XENON1T SuperCDMS M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
32 Improved limits for Higgs Portal dark matter WIMP nucleon cross section [ cm 2] Scalar Fermion Vector WIMP mass [GeV] PandaXII LUX XENON1T SuperCDMS M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
33 Contact terms Scalar source suppressed for (N N) 2 longrange contribution dominant (in Weinberg counting) Typical size (5 10)% reflected by results for structure factors more important in case of cancellations Contact terms do appear for other sources, e.g. θ µ µ related to nuclear binding energy E b Same structure factor in spin2 twobody currents MH, Klos, Menéndez, Schwenk, in preparation M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
34 Outlook: chiral counting for neutrino nucleus scattering Nucleon V A WIMP t x t x WIMP Nucleon S P 1b V 2b b NLO b A 2b b NLO b 2 1 S 2b 3 5 2b NLO 4 1b P 2b b NLO M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
35 Outlook: chiral counting for neutrino nucleus scattering Nucleon V A WIMP t x t x WIMP Nucleon S P 1b V 2b b NLO 5 3 1b A 2b b NLO 5 3 1b 2 1 S 2b 3 5 2b NLO 4 1b 2 1 P 2b 3 5 2b NLO 4 M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
36 Outlook: chiral counting for neutrino nucleus scattering Nucleon V A WIMP t x t x WIMP Nucleon S P 1b V 2b b NLO 5 3 1b A 2b b NLO 5 3 1b 2 1 S 2b 3 5 2b NLO 4 1b 2 1 P 2b 3 5 2b NLO 4 Standard interactions: coherent 2b currents scale with N Z, but Cu VV G ( F = ) 2 cos 2 θ w 3 sin2 θ w proton coupling 2C VV u + C VV d 1b only scales with N 2, not (N + Z) 2 Cd VV G ( F = ) 2 cos 2 θ w 3 sin2 θ w 1 4 sin 2 θ w suppressed Nonstandard interactions: potentially large corrections from scalar 2b currents M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
37 Conclusions (a) (b) (c) (d) Chiral EFT for WIMP nucleon scattering Predicts hierarchy for corrections to leading coupling Connects nuclear and hadronic scales Ingredients: nuclear matrix elements and structure factors Applications: discriminating nuclear responses σ SD p limits from xenon via twobody currents improved limits on Higgs Portal dark matter from LHC searches M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
38 Rate and structure factors Rate dr dq 2 = ρm m A m χ vesc v min d 3 v v f( v ) dσ χn dq 2 Haloindependent methods Drees, Shan 2008, Fox, Liu, Weiner 2010,... Nuclear structure factors Engel, Pittel, Vogel 1992 dσ χn 8G 2 [ ] F dq 2 = (2J + 1)v 2 S A (q)+s S (q) Normalization at q = 0: S S (0) = 2J + 1 c 0 A+c 1 (Z N) 2 4π (2J + 1)(J + 1) S A (0) = (a 0 + a 1 ) S p +(a 0 a 1 ) S n 2 4πJ Assume c 1 = 0 and SI scattering dσ SI χn dq 2 = σsi χn 4v 2 µ 2 FSI 2 (q2 ) N phenomenological Helm form factor F 2 SI(q 2 ) M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
39 GellMann Oakes Renner relation Leading order in SU(2) meson ChPT L ChPT = F 2 π 4 Tr ( d µ U d µu + 2BM ( U + U )) + = ( m u + m d ) BF 2 π 1 2( mu + m d ) B ( π 0 ) 2 ( m u + m d ) Bπ + π + Comparison with QCD Lagrangian LQCD = mu ūu m d dd + BF 2 π = qq qq = ūu = dd GellMann Oakes Renner relation M 2 π = ( m u + m d ) B +O ( m 2 q ) B = qq F 2 π M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
40 GellMann Oakes Renner relation Leading order in SU(2) meson ChPT L ChPT = F 2 π 4 Tr ( d µ U d µu + 2BM ( U + U )) + = ( m u + m d ) BF 2 π 1 2( mu + m d ) B ( π 0 ) 2 ( m u + m d ) Bπ + π + Comparison with QCD Lagrangian LQCD = mu ūu m d dd + BF 2 π = qq qq = ūu = dd Mass difference entirely due to electromagnetism M 2 π ± = M2 π 0 + 2e2 F 2 πz +O ( m d m u ) 2 GellMann Oakes Renner relation M 2 π 0 = 2ˆmB +O( mq 2 ) ˆm = mu + m d 2 B = qq F 2 π M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
41 Example: chiral counting in scalar channel Leading pion nucleon Lagrangian [ L (1) πn = ( Ψ iγ µ µ iv µ) m N + g ( A 2 γµγ 5 2a µ µ π ) ] + Ψ F π no scalar source! WIMP Nucleon S 1b 2 S 2b 3 M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
42 Example: chiral counting in scalar channel Leading pion nucleon Lagrangian [ L (1) πn = ( Ψ iγ µ µ iv µ) m N + g ( A 2 γµγ 5 2a µ µ π ) ] + Ψ F π no scalar source! Scalar coupling f N = m N Λ 3 q=u,d,s C SS q f N q + N mq qq N = f N q m N for q = u, d related to pion nucleonσterm σ πn Chiral expansion Nucleon S WIMP 1b 2 S 2b 3 σ πn = 4c 1 Mπ 2 9g2 A M3 π 64πFπ 2 +O(Mπ 4 ) σ = 5g2 A Mπ 256πFπ 2 +O(Mπ 2 ) slow convergence due to strong ππ rescattering use phenomenology for the full scalar form factor! M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
43 } Γ1s σterm from Roy Steiner analysis of pion nucleon scattering Error analysis σ πn = 59.1± 0.7 }{{} flat directions = 59.1±3.5 MeV ± 0.3 }{{} matching ± 0.5 }{{} systematics ± 1.7 }{{} scattering lengths ± }{{} 3.0 MeV lowenergy theorem Crucial result: relation betweenσ πn and πn scattering lengths σ πn = 59.1 MeV + I s c Is a Is 0+ Pionic atoms: π p/d bound states 3s 3p 3d 2s 2p ] ã + [ 10 3 M 1 π level shift of πh level shift of πd width of πh ǫ1s 1s a [ ] 10 3 Mπ 1 M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
44 A new σterm puzzle Recent lattice calculations of σ πn BMW : σ πn = 38(3)(3) MeV χqcd : σ πn = 45.9(7.4)(2.8) MeV ETMC : σ πn = 37.2(2.6) ( +4.7) 2.9 MeV RQCD : σ πn = 35(6) MeV Similar puzzle in lattice calculation of K ππ RBC/UKQCD , also 3σ level ] ã + [ 10 3 M 1 π level shift of πd level shift of πh BMW ETMC Both puzzles with profound implications for BSM searches: scalar nucleon couplings, CP violation in K 0 K 0 mixing width of πh χqcd a [ ] 10 3 Mπ 1 M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
45 A new σterm puzzle: issues with the pionicatom scattering lengths? Something wrong with pionicatom data? Direct fit to pion nucleon data base , requires careful treatment of Radiative corrections Experimental normalization uncertainties Bottom line: ] a 1/2 [ 10 3 M 1 π 180 π p π p π p π 0 n 165 KH80 πh πd π + p π + p σ πn = 58(5) MeV a [ ] 3/ Mπ 1 independent confirmation of pionicatom data M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
46 c A new σterm puzzle: what could lattice do? a 1/2 c /ā 1/ MeV 4MeV 6MeV 5MeV 8MeV 9MeV 10MeV 3MeV ac 3/2 /āc 3/2 ] ã + [ 10 3 M 1 π level shift of πd level shift of πh BMW ETMC width of πh χqcd a [ ] 10 3 Mπ 1 πn: lattice calculation of a 1/2, a 3/2 test input for πn scattering lengths Possible issues of σterm calculations: Finitevolume corrections Discretization effects Excitedstate contamination M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
47 Status of the phenomenological determination of σ πn Karlsruhe/Helsinki partialwave analysis KH80 Höhler et al. 1980s comprehensive analyticity constraints, old data Formalism for the extraction of σ πn via the Cheng Dashen lowenergy theorem Gasser, Leutwyler, Locher, Sainio 1988, Gasser, Leutwyler, Sainio 1991 canonical value σ πn 45 MeV, based on KH80 input GWU/SAID partialwave analysis Pavan, Strakovsky, Workman, Arndt 2002 much larger value σ πn = (64±8) MeV ChPT fits vary according to PWA input Fettes, Meißner 2000 (same problem in different regularizations (w/ and w/o ) Alarcón et al. 2012) M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
48 Status of the phenomenological determination of σ πn Karlsruhe/Helsinki partialwave analysis KH80 Höhler et al. 1980s comprehensive analyticity constraints, old data Formalism for the extraction of σ πn via the Cheng Dashen lowenergy theorem Gasser, Leutwyler, Locher, Sainio 1988, Gasser, Leutwyler, Sainio 1991 canonical value σ πn 45 MeV, based on KH80 input GWU/SAID partialwave analysis Pavan, Strakovsky, Workman, Arndt 2002 much larger value σ πn = (64±8) MeV ChPT fits vary according to PWA input Fettes, Meißner 2000 (same problem in different regularizations (w/ and w/o ) Alarcón et al. 2012) Our work: two new sources of information on lowenergy πn scattering Precision extraction of πn scattering lengths from hadronic atoms Royequation constraints: analyticity, unitarity, crossing symmetry , M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
49 QCD constraints for subleading nuclear corrections (a) (b) (c) (d) Onebody operators: known nuclear form factors determines radius corrections(b) Axial Ward identity relates g N A,P(t) and fixed combination of O 4,6 in (a) M AA 1,NR = 4O 4gA N (t)+ 1 mn 2 O 6 gp N (t) O 10 only appears in SP channel not coherent and vanishes at q = 0 M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
50 Comparison to NREFT For the leading corrections all O i but O 3 are small not necessary to keep 2 14 parameters in first step But: some new parameters for twobody effects and radius corrections cover coherent responses (+ SD), same order in chiral counting Nucleon operators:1, S N, v, v q, v q = 0 only v can produce new coherent (nuclear) effect Similarly to SD searches: define subleading cross sections pion WIMP scattering NREFT only first step in chain of EFTs need matching to QCD to make connection to BSM, ChEFT one crucial step M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
51 Analysis strategies Parameters (ζ = 1(2) for Dirac (Majorana)): Couplings [ c M ± = ζ 2 ċ± M = ζm2 [ N ḟ p ± 2 f p ± f n + f V,p 1 ḟn + ḟ V,p 1 ± f V,n ] 1 ± ḟ V,n + 1 ( 1 4m N 2 c π = ζf π c θ π = ζfθ π c Φ ± = ζ ( 2 f V,p 2 ± f V,n ) ] 2 f V,p 2 ± f V,n ) 2 f N = m ( N ) Λ 3 C SS q f N q 12πf N Q C S g q=u,d,s f π = Mπ Λ 3 ( q=u,d C SS q + 8π 9 C S g ) f π q f θ π = Mπ 8π Λ 3 9 C S g Conclusions Different c probe different linear combinations of Wilson coefficients Ideally: global analysis of different experiments Oneoperatoratatime strategy: producing limits e.g. on c M and cπ in addition to cm + would provide additional information on BSM parameter space QCD constraints: when considering O 3 should also keep radius corrections M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,
52 Spin2 and coupling to the energymomentum tensor Effective Lagrangian truncated at dim7, but if WIMP heavy m χ/λ = O(1) heavywimp EFT Hill, Solon 2012, 2014 L = 1 Λ 4 { q C (2) q χγ µi νχ 1 2 q ( γ {µ id ν} mq 2 gµν) q + C (2) g ( gµν χγ µi νχ 4 Ga λσ Gλσ ) } a G µλ a G ν aλ leading order: nucleon pdfs similar twobody current as in scalar case, pion pdfs, EMC effect Coupling of trace anomalyθ µ µ to ππ θ µ µ = q m q qq + β QCD G a µν 2g Gµν a π(p ) θ µν π(p) = p µp ν + p µ pν + ( gµν M 2 π p p ) s probes gluon Wilson coefficient C S g M. Hoferichter (Institute for Nuclear Theory) Dark matter and BSM with nuclei Seattle, July 11,