Hadronic matrix elements for new physics

Transcription

1 Hadronic matrix elements for new physics Emanuele Mereghetti Matter over antimatter: the Sakharov conditions after 50 years May 9th, 2017 based on work with V. Cirigliano, W. Dekens, J. de Vries, M. Graesser, C. Y. Seng, A. Walker-Loud

2 Outline 1 Introduction 2 Effective Lagrangians for EDMs and L = 2 3 Hadronic matrix elements for EDMs. Pion-nucleon couplings 4 Hadronic matrix elements for L = 2 5 Conclusion

3 Introduction Sakharov conditions C and CP violation baryon number violation deviation from thermal equilibrium the SM does not explain baryogenesis low-energy experiments crucial for the missing ingredients Electric dipole moments: is there CPV beyond SM? 0νββ decays: what s the nature of neutrinos?

4 Introduction current generation of experiments achieved impressive bounds electron EDM d e e fm ACME collaboration, 14. neutron EDM d n e fm Hg EDM Baker et al, 06. d199 Hg e fm Graner et al, 16.

5 EDMs. Experimental status next generation will improve by one/two orders 1. neutron EDM dn e fm 2. diamagnetic atoms dra e fm 3. proton & deuteron EDM dp, dd e fm

6 Introduction CUORE GERDA NEMO reach of tonne-scale 0νββ KamLAND-Zen coll., similarly for 0νββ decay bounds T 0ν 1/2 on 76 Ge, 130 Te, 136 Xe, 100 Mo can improve by two orders

7 Introduction any non-zero signal in next generation of experiments new physics!... however... to discriminate between new physics scenarios 1. several different orthogonal systems/observables EDMs of leptons, nucleons, light & heavy nuclei electron spectrum in 0νββ 2. systematic connection to flavor and collider physics 3. precise theoretical predictions

8 The Standard Model as an Effective Field Theory Write down all possible operators with SM fields local SU(3) c SU(2) L U(1) Y invariance dimension 4 not much CPV no L or B interactions

9 The Standard Model as an EFT why stop at dim=4? L = L SM + c i, 5 Λ O5 i + c i, 6 Λ 2 O6 i + c i, 7 Λ 3 O7 i +... Λ is the scale of new physics Os are expressed in terms of SM fields Λ v = 246 GeV have the same symmetries as the SM gauge symmetry! but not accidental symmetries as L

10 The Standard Model as an EFT why stop at dim=4? L = L SM + c i, 5 Λ O5 i + c i, 6 Λ 2 O6 i + c i, 7 Λ 3 O7 i +... Λ is the scale of new physics Os are expressed in terms of SM fields Λ v = 246 GeV have the same symmetries as the SM gauge symmetry! but not accidental symmetries as L one dimension 5 operator neutrino masses and mixings Λ GeV

11 The Standard Model as an EFT three/four bosons h self-coupling scalar-gauge Yukawa dipole vector/axial currents four-fermion many dimension 6 1/Λ 2 half of them CPV! Buchmuller & Wyler 86, Weinberg 89, de Rujula et al. 91, Grzadkowski et al and dim. 7, dim. 9 operators ( 1/Λ 3, 1/Λ 5 ) L. Lehman 14, M. Graesser 16,...

12 2. Effective Lagrangians for EDMs and L = 2

13 CP violation at 1 GeV integrate out h, W, Z, t,... match onto SU c(3) U em(1) invariant ops Neglecting small Yukawas one dimension 4 operator: QCD θ term L /T 4 = m θ qiγ5q in principle θ = O(1)... strong CP problem 9 (+ 10 w. strangeness) CPV S = 0 dim. 6 operators 1. 1 pure glue: Weinberg three-gluon operator (gcedm) L /T 6 = gsc G 6v 2 f abc ɛ µναβ G a αβ Gb µρg c ν ρ C G = O ( v 2 Λ 2 )

14 CP violation at 1 GeV 2. 4 (+2) quark bilinears: qedm and qcedm L /T 6 = q m q c (q) γ 2v 2 qiσ µν γ 5 q ef µν mq c (q) g 2v 2 qiσ µν g sg µνγ 5 q 3. 2 (+ 4) LR LR four-quark { L /T 6 = 4G F Σ (ud) 2 1 ( d L u R ū L d R ū L u R dl d R ) + Σ (us) 1 ( s L u R ū L s R s L s R ū L u R ) +Σ (us) 3 ( s L u R ū L s R + s L s R ū L u R ) } + color mixed 4. 2 (+ 4) LL RR four-quark { } L /T 6 = 4G F Ξ (ud) 2 1 dl γ µ u L ū R γ µd R + Ξ (us) 1 s L γ µ u L ū R γ µs R + Ξ (ds) 1 s L γ µ d L dr γ µs R + mix

15 L = 2 Lagrangian at 1 GeV L e=0 L=2 includes ν masses L L=2 = L e=0 L=2 + L e=1 L=2 + L e=2 L=2 ( v L L=2 = (m ν) ij ν T j Cν i 2 ) +... m ν O Λ L e=1 L=2 starts at dim. 6 β decay with the wrong neutrino u d e L (6) L=2 = 2G F 2 {C 6, VL dl γ µ u L ν T L Cγµe R + C 6, VR dr γ µ u R ν T L Cγµe R +C 6, SL dr u L ν T L Ce L + C 6, SR dl u R ν T L Ce L + C 6, T dr σ µν u L ν T L Cσµνe L }

16 L = 2 Lagrangian at 1 GeV L e=2 L=2 starts at dim. 9 [ ] L (9) L=2 = 2G2 F ) (C 9, i e T LC e L + C 9, i e T RC e R O i + e T LCγ µe R C 9, iv O µ i v i=scalar i=vector Scalar operators 1. 1 LL LL four-quark 2. 2 LR LR four-quark O 1 = ū Lγ µ d L ū L γ µd L O 2 = ū Ld R ū L d R, O 3 = ū α L d β R ūβ L dα R 3. 2 LL RR four-quark O 4 = ū Lγ µ d L ū R γ µd R, O 5 = ū α L γ µ d β L ūβ R γµdα R

17 L = 2 Lagrangian at 1 GeV d L u L d L u L d R u R d R u R e e e e C 6, i require a Higgs insertion ( v 3 C 6,i O Λ 3 e.g. ε ijli T Cγ µe dγ µ u H j ( ) v 3 ), C 7,i O (some of the) C 9, i invariant dim. 9 operators, or come from dim. 7 operators ( ) v 3 C 9,i = O Λ, v5 3 Λ 5 Λ 3

18 Connection to models d R d R W R W L W R W L u R u R e.g. left-right (LR) symmetric models 1. W L-W R mixing induces new right-handed W couplings 2 v 2 ξ ij i ϕ D µϕ ū i Rγ µ d j R W LHC, Higgs pheno 2. induce Ξ ud and Ξ us below m W correlations between EDMs, flavor (ɛ /ɛ) and collider physics

19 Connection to models d R d R d R u L W R W L W R W L W L u R u R u R d L e.g. left-right (LR) symmetric models 1. W L-W R mixing induces new right-handed W couplings 2 v 2 ξ ij i ϕ D µϕ ū i Rγ µ d j R W LHC, Higgs pheno 2. induce Ξ ud and Ξ us below m W correlations between EDMs, flavor (ɛ /ɛ) and collider physics

20 Connection to models Ξ (us) 1,2 V usξ us... can be washed out by theory uncertainties need precise ME to be quantitative see Wouter s talk

21 Connection to models m N 1 TeV m WR 2 TeV WARNING crude estimates of ME! S.-F. Ge, M. Linder, S. Patra, 15 LNV at the TeV scale radically changes the picture need precise ME!

22 Strategy new physics Λ v SU L(2) inv. operator at EW scale LHC pheno CPV and L = 2 at 1 GeV C G, c(q) γ, c (q) g, m ν, C 9,1,... Lattice QCD, sum rules NN potentials & currents EDM of heavy systems d Hg, d Ra Chiral Effective Theory Many body d n, EDM of light nuclei T 0ν 1/2 ( )

23 3. Hadronic matrix elements for EDMs. Pion-nucleon couplings

24 CPV at the hadronic level include dim. 4 and dim. 6 CPV in χpt Lagrangian L /T = 2 N ( d0 + d 1τ 3 ) S µ v ν NF µν ḡ0 2F π Nπ τ N ḡ1 2F π π 3 NN + C 1 NN µ ( NS µ N) + C 2 Nτ N µ ( NS µ τ N) at LO, EDMs expressed in terms of a few couplings why π-n couplings? d0, d 1 ḡ 0, ḡ 1 C 1, C 2 neutron & proton EDM, one-body contribs. to A 2 nuclei pion loop to nucleon & proton EDMs leading /T OPE potential short-range /T potential relative size of the coupling depends on chiral/isospin properties of /T source

25 Nucleon EDM. expand in {Q, m π}/λ χ, Λ χ 4πF π 1 GeV d n = d 0 d 1 + a (0) n d p = d 0 + d 1 + a (0) p ḡ 0 2F π ḡ 0 2F π Q Λ 2 χ Q Λ 2 χ + a (1) n + a (1) p ḡ 1 Q 3 2F π Λ 4 χ ḡ 1 Q 2 2F π Λ 3 χ π-n couplings enter at one loop at most as important as d 0, d 1,... but crucial for m 2 π, Q 2 dependence [ ] d n = d 0 d 1 + eg Aḡ 0 (4πF π) 2 ln m2 π µ 2 π m π + ḡ1(κ 0 κ 1 ) m 2 π log m2 π 2 m N 4 µ 2 m 2 N

26 EDMs of Light Nuclei single nucleon EDM corrections to the nuclear wavefunction (CPV NN potential) d A = a (n) ḡ 0 A dn + a(p) A dp + a(0) A ḡ a (1) 1 A F π M NN 2F π M NN M NN 16πF2 π g 2 A m N corrections to wavefunction dominant (in most cases) 300 MeV

27 EDMs of Light Nuclei from EM and U.van Kolck, 15 nuclear matrix elements well under control (... with some exceptions... ) in agreement with power counting C. P. Liu and R. Timmermans, 05; J. de Vries et al, 11; J. Bsaisou et al, 13, J. Bsaisou et al, 15; N. Yamanaka and E. Hiyama, 15

28 Naive estimates of CPV hadronic couplings ḡ 0 /F π ḡ 1 /F π d0,1 Q C 12 Q θ ( ) m 2 O π Λ 2 χ ḡ 0 /F π ḡ 1 /F π d0,1 Q C 12 Q c g,σ (us) ḡ 0 /F π ḡ 1 /F π d0,1 Q C 12 Q Ξ WARNING naive dim. analysis! θ, qcedm, LL RR operators break chiral symmetry ḡ 0 and/or ḡ 1 dominate no isospin breaking = ḡ 0 isospin breaking = ḡ 1

29 Naive estimates of CPV hadronic couplings ḡ 0 /F π ḡ 1 /F π d0,1 Q C 12 Q ḡ 0 /F π ḡ 1 /F π d0,1 Q C 12 Q θ C G,Σ (ud) these hierarchies can be probed in light nuclei! ḡ 0 /F π ḡ 1 /F π d0,1 Q C 12 Q c γ WARNING naive dim. analysis! θ, qcedm, LL RR operators break chiral symmetry ḡ 0 and/or ḡ 1 dominate gcedm and LR LR all couplings important no isospin breaking = ḡ 0 isospin breaking = ḡ 1 qedm only d 0,1

30 CPV π-n couplings. QCD Theta Term can we do better than NDA? ḡ i = ḡ i( θ, c (q) g, Ξ, Σ)? after vacuum alignment L 4 = m qq + r 1 ( θ) ( mε qτ 3q + m sin θ qiγ 5q ) 2 m = mu + m d mε = m d m u m = m um d /(m u + m d ) θ term and mass splitting are chiral partners ( ) ( qiγ5q SU A (2) qα τ q qτ q α qiγ 5q ) Crewther et al, 79 i.e. one spurion enough to construct iso- and T-breaking couplings T violation isospin breaking = 1 ε2 sin 2ε θ ρ θ

31 QCD Theta Term. ḡ δm δ N [MeV] m phys π mπ mπ mπ 241 [MeV] 422 [MeV] 489 [MeV] m π /Λ χ D. Brantley, et al, 16 L = m N NN + 1 [ 2 δmn Nτ 3N 2ρ θ N π τ ] N 2F π δm N = (m n m p) st, strong mass splitting 1 ε 2 ḡ 0 = δm N θ 2ε δm N accessible via existing lattice calculations

32 QCD Theta Term. ḡ 0 Chiral corrections in SU(2) χpt ḡ 0 2F π = ḡ 0 [ 1 + m2 π (4πF π) 2 2F π [ (m n m p) st = δm N 1 + m2 π (4πF π) 2 (( 3g 2 A + 1 ) log µ2 2 m 2 π + g 2 A (( 3g 2 A + 1 ) log µ2 2 m 2 + g 2 A + 1 π 2 )] + δ(3) m N )] + δ (3) m N ρ θ + δḡ 0 2F π 2F π same loop corrections to ḡ 0 and δm N finite LEC δḡ 0 only correct πn coupling T violation isospin breaking = 1 ε2 sin 2ε θ ρ θ

33 QCD Theta Term. ḡ 0 Chiral corrections in SU(2) χpt ḡ 0 2F π = ḡ 0 [ 1 + m2 π (4πF π) 2 2F π [ (m n m p) st = δm N 1 + m2 π (4πF π) 2 (( 3g 2 A + 1 ) log µ2 2 m 2 π + g 2 A (( 3g 2 A + 1 ) log µ2 2 m 2 + g 2 A + 1 π 2 )] + δ(3) m N )] + δ (3) m N ρ θ + δḡ 0 2F π 2F π same loop corrections to ḡ 0 and δm N finite LEC δḡ 0 only correct πn coupling T violation isospin breaking = 1 ε2 sin 2ε θ ρ θ... but... only at NNLO not log enhanced

34 ḡ 0 from QCD Theta Term 2.32(17) [ ] 2.52(29) [ ] 2.28(26) [ ] 2.90(63) [ ] 3.13(57) [ ] 2.51(52) [ ] 2.26(71) [hep-lat/ ] 2.39(12) weighted average δm δ n p [MeV] courtesy of A. Walker-Loud SU(3) analysis shows relation is robust not spoiled by strangeness, or decuplet baryons using lattice average of δm N lattice error ḡ 0 2F π = (15.5 ± 2 ± 1.6) 10 3 θ conservative estimate of theory error

35 Implications F1(q 2,m 2 π)/m 2 π (e fm 3 ) F 1 = F p F n mπ = 150 MeV mπ = 250 MeV mπ = 350 MeV mπ = 450 MeV q (MeV) S 1 = eg Aḡ 0 ( (4πF π) 2 1 5πmπ 4m N ) F n,p(q 2, m 2 π) = d n,p(m 2 π) q2 6m 2 π S n,p(m 2 π) + H(q 2, m 2 π) non-analytic m π dependence of d n,p & momentum dependence of F n,p are predictions compare to LQCD!

36 Quark CEDM can we use the same techniques for higher-dimensional operators? after vacuum alignment L = qiγ 5q m ( θ θ ind) + r qiγ 5 dceq gs 2 qσµν G µν( d CM + d CEiγ 5)q, shift in θ canceled in PQ induced pseudoscalar mass term d CE = diag( d u, d d, d s) d CM = diag( c u, c d, c s) r: ratio of vacuum matrix elements r = 1 qσ µν g sg µνq = dm2 π 2 qq d c 0 d m dm 2 π θind: min. of axion potential in the presence of qcedm ) ( du θ ind = rtr (M 1 dce = r + d d + d ) s m u m d m s

37 Quark CEDM 1. iso-vector (-scalar) qcedm and iso-scalar (-vector) qcmdm are chiral partners 2. qcedm & qcmdm transform as quark masses ( ) ḡ 0 = d d d 1 ε 2 ) 0 + r (m n m p) + δm N ( θ θind, d c 3 d( mε) 2ε ( d ḡ 1 = d 3 r d ) (m n + m p), d c 0 d m exact at NLO, O(m 3 π) to m n + m p corrections drop out see M. Pospelov and A. Ritz, 05, J. de Vries et al, 16

38 Quark CEDM. Chiral corrections to ḡ 0, and ḡ 1 (a) (b) (c) (d) (e) ( ) ḡ 0 = d d d 1 ε 2 ) 0 + r (m n m p) + δm N ( θ θind, d c 3 d( mε) 2ε ( d ḡ 1 = d 3 r d ) (m n + m p) + 2 φ [ ( d 3 d s r d )] (m n + m p), d c 0 d m d c s dm s O(m 2 π/λ 2 χ) SU(2) loops O(m K/Λ χ), O(m 2 K/Λ 2 χ) SU(3) loops broken by counterterms as for θ relations are robust!

39 Quark CEDM. Towards a LQCD implementation To do: 1. bare matrix elements QCD sigma terms qcmdm sigma terms m p qq p = m d d m m mn d3 2 p g s qσ µνg µν q p = d ( ) d 3 d c 0 m N c0 =0 mε p qτ 3q p = mε d d( mε) mε δmn d02 p g s qσ µνg µν τ 3q p = d ( ) d 0 d c 3 δm N, c3 =0 2. non-perturbative renormalization: [ qσ µν G µνq] 1 (Z qq a qq + log(a) m 2 qq ) + Z 2 GG mg µνg µν + Z C qσ µν G µνq power divergence!

40 Quark CEDM. Towards a LQCD implementation but any mixing with qq drops out ( N ( qσ Gq + α qq) N = ( N qσ Gq N need to worry only about log mixing! 0 ( qσ Gq + α qq) 0 0 qq 0 ) 0 qσ Gq 0 N qq N 0 qq 0 3. define renormalization condition that satisfy Ward identities ) N qq N

41 LL RR four-quark operators fixed by meson physics nucleon σ term or δm N LL RR break chiral symmetry and isospin ḡ 0 = ( ) ( ) Im Ξ (us) i + Im Ξ (ds) d i i=1,2 dre Ξ (us) + r i v 2 m d m δm N + δm N d mε mε ( θ θ ind ) i ḡ 1 = 2 ( ) ( ) Im Ξ (us) i Im Ξ (ds) d i i=1,2 dre Ξ (us) r i v 2 m d m N d m i +4 ( ) Im Ξ (ud) d i i=1,2 dre Ξ (ud) r i v 2 m d m N. d m i r i ratio of vacuum ME, r i = F 2 0A i LR/m 2 π holds at NLO, broken at N 2 LO V. Cirigliano, W. Dekens, J de Vries, EM, 16, C. Y. Seng

42 fix the tadpole piece with LQCD input LL RR operators and SU(3) O ab = qγ µ t a P L q qγ µt b P Rq SU L(3) SU R (3) (8 L, 8 R)

43 fix the tadpole piece with LQCD input LL RR operators and SU(3) O ab = qγ µ t a P L q qγ µt b P Rq F4 0 4 Tr ( U t b Ut a) A 1 LR, at LO in χpt, a single mesonic operator ( + 1 for different color contraction)

44 fix the tadpole piece with LQCD input LL RR operators and SU(3) O ab = qγ µ t a P L q qγ µt b P Rq F4 0 4 Tr ( U t b Ut a) A 1 LR, at LO in χpt, a single mesonic operator ( + 1 for different color contraction) S = 0: EDMs (t 4 it 5) (t 4 + it 5) (t 1 it 2) (t 1 + it 2) two LECs to tame them all S = 1: K ππ (t 4 it 5) (t 1 + it 2) BSM contribs. to ɛ /ɛ S = 1: K ( ππ ) (t 6 it 7) t t 8 SM EW penguins S = 2: K 0 K 0 mixing (t 6 it 7) (t 6 it 7) L = 2: π π e e (t 1 it 2) (t 1 it 2) e e

45 Determination of the LECs use K ππ from EW penguins (RBC & UKQCD, 15) A 1 LR(µ = 3 GeV) = (2.2±0.13) GeV 2, A 2 LR(µ = 3 GeV) = (10.2±0.6) GeV 2 LQCD error good agreement with extraction from K 0 K 0 ME

46 Determination of the LECs use K ππ from EW penguins (RBC & UKQCD, 15) A 1 LR(µ = 3 GeV) = (2.2±0.13) GeV 2, A 2 LR(µ = 3 GeV) = (10.2±0.6) GeV 2 LQCD error good agreement with extraction from K 0 K 0 ME Ξ 1,2 generate large ḡ 1 ḡ 1 ( ) 2F = (1.0 ± 0.1) 10 5 Ξ (us) 1 Ξ (ds) 1 + 2Ξ (ud) 1 π tad ( ) (4.7 ± 0.4) 10 5 Ξ (us) 2 Ξ (ds) 2 + 2Ξ (ud) 2 larger than NDA F πλ χ/v ḡ 0 smaller by a factor of 10 larger error from missing direct piece 50% if follows NDA systematically improvable!

47 LR LR operators L /T 6 = 4G F 2 {Σ (ud) 1 ( d Lu Rū Ld R ū Lu R dld R) + Σ (us) 1 ( s Lu Rū Ls R s Ls Rū Lu R) +Σ (us) 3 ( s Lu Rū Ls R + s Ls Rū Lu R) } + color mixed Σ (ud) 1,2 and Σ (us) 1,2 transform like srsl and d Rd L Σ (us) 3,4 belong to ( 6 L, 6 R) no π N couplings for Σ (ud) η N couplings at LO, not very relevant Σ (us) induces ḡ 0 and ḡ 1 similar to qcedm mass - coupling relation survives tadpole piece fixed by K 0 K 0 ME

48 CPV summary. ḡ 0 ḡ 1 d0,1 C 1,2 θ C G, Σ(ud) c g c γ Ξ, Σ (us) : 10% errors : ongoing lattice effort : a long way to go LO non derivative π-n couplings determined by spectrum ḡ 0( θ) well determined, ḡ i(c (q) g ) soon? four-quark might be harder ongoing effort to determine d n( θ), d n(c (q) g ) w. LQCD four-nucleon operators are hopeless?

49 4. Hadronic matrix elements for L = 2

50 Hadronic matrix elements for L = 2 n p e - e - n p 1. standard mechanism leading effects are long distance at LO: nucleon axial and vector form factors at N 2 LO: two-body currents, short-range effects, rel. corrections... well determined hadronic input

51 Hadronic matrix elements for L = 2 n p e - e - n p 2. L (6,7) L=2 u d e still long distance at LO: nucleon axial, vector, scalar, pseudoscalar and tensor form factors well determined hadronic input

52 Hadronic matrix elements for L = 2 n p n p n p e - e - e - e - e - e - n p n p n p L (9) L=2 : new short distance effects ππe c e operators NNπe c e operators NN NNe c e operators G Prezeau, M. Ramsey-Musolf, P. Vogel, 03 relative size of the coupling depends on chiral properties of L = 2 source... but for 4/5 scalar operators ππe c e is enhanced

53 ππ matrix elements d u u d u s d K 0 K 0 s s d chiral symmetry relates π π + to K 0 K 0 oscillation ( 1. LL LL O 1 = ū Lγ µ d L ū L γ µd L (27 L, 1 R) g 27 1F0 4 Lµ32L µ ) Lµ31Lµ F 2. LR LR O 2,3 = ū Ld R ū L d R ( 6 L, 6 R) 0 g Tr(t a U t b U) 3. LL RR O 4,5 = ū Lγ µ d L ū R γ µd R (8 L, 8 R) g 8 8 F Tr(t a U t b U ) ππe c e dominant for O 2,3,4,5, need also πnn and NNNN for O 1 ( 2iπ U = exp F 0 ), L µ = iu µu.

54 ππ matrix elements. Loop corrections at tree level M ππ 6 6 π + O 1+i2,1+i2 6 6 π = K 0 O 6 i7,6 i7 6 6 K 0 M K K 6 6 M ππ 8 8 π + O 1+i2,1+i2 8 8 π = K 0 O 6 i7,6 i7 8 8 K 0 M K K 8 8. different loop corrections for ππ and K K M ππ 8 8 = M K K 8 8 F2 π FK 2 ( ) = M K K 8 8 R 8 8 M ππ = M K K F2 π FK 2 ( ) = M K K 6 6 R 6 6, corrections to decay constants everything else

55 ππ matrix elements. Loop corrections 0.3 Δ Δ8 x 8 Δ6 x μ (MeV) = [ 1 m 2 π (4πF 0 ) 2 4 ( 4 + 5Lπ) m2 K ( 1 + 2L K) + 3 ( ) ] 4 m2 η Lη a 8 8 m 2 K m2 π a 8 8, a 6 6 unknown LECs can be extracted from m s, m dependence of M K K loop corrections are small 8 8 = 0.02 ± 0.30, 6 6 = 0.07 ± 0.20 error from scale variation most of the correction from F π/f K R 8 8 = 0.72 ± 0.21, R 6 6 = 0.76 ± 0.14

56 Extraction of the ππ ME = + + our average for = + + ETM 15 our average for = + RBC/UKQCD 16 = + SWME 15A SWME 14C RBC/UKQCD 12E our average for = = ETM 12D using FLAG averages of K 0 K 0 π + O 2 π = (2.7 ± 0.3 ± 0.5) 10 2 GeV 4 π + O 3 π = (0.9 ± 0.1 ± 0.2) 10 2 GeV 4 π + O 4 π = (2.6 ± 0.8 ± 0.8) 10 2 GeV 4 π + O 5 π = (11 ± 2 ± 3 ) 10 2 GeV 4 benchmark for direct LQCD calc. LQCD error χpt error

57 ππ matrix element for LL LL operators e e K chiral corrections to M K K are large better use K ππ using RBC & UKQCD 15 for π + π O 27 K + Savage, 99 π + O 1 π = (1.0 ± 0.1 ± 0.2) 10 4 GeV 4 quite small... follows chiral counting very well at the same order, need πnn and NN NN operators hyperon decays?

58 Conclusions hadronic uncertainties are important dominant for EDMs of light nuclei d2 H, d3 He, d3 H sizable for heavy system d199 Hg, d225 Ra interplay of LQCD & chiral EFTs can be useful... but more work to do... 10% accuracy on ḡ 0( θ) well defined strategy for ḡ i( c (q) g, Ξ, Σ) 20% - 30% accuracy on L = 2 ππ ME