Lattice QCD calculation of nucleon charges g A, g S and g T for nedm and beta decay

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1 1 / 49 Lattice QCD calculation of nucleon charges g A, g S and g for nedm and beta decay Boram Yoon Los Alamos National Laboratory with. Bhattacharya, V. Cirigliano, R. Gupta, H. Lin PNDME Collaboration Oct 5, 2015

2 2 / 49 Nucleon Charge g q Γ N q Γ q N = g q,n Γ ψ N Γψ N Can be calculated using lattice QCD Plays important role in understanding SM, probing BSM

3 3 / 49 Neutron Decay SM : V-A Weak decay (e L with ν L ) BSM : Novel S, interactions (e R with ν L )

4 4 / 49 Leading BSM Contribution and Nucleon Charges H eff G F [ε S ūd ē(1 γ 5 )ν e + ε ūσ µν d ēσ µν (1 γ 5 )ν e ] Low energy hadronic part can be calculated from lattice: g u d S p ūd n, g u d p ūσ µν d n Note: p ūγd n = p ūγu p p dγd p g u d Γ in isospin limit Combined with neutron decay experiments, lattice calculation of g u d S and g u d can be used to constrain BSM physics (ε S, ε )

5 Required Precision of g S and g 90% CL contours in [ε S, ε ] plane (Near) future expt. b and B 1 at 10 3 Impact limited by g S and g δg = 2 3 δg S Goal: 10% accuracy in g S and g Bhattacharya, et al., PRD (2012) 5 / 49

6 6 / 49 Neutron Electric Dipole Moment (nedm) Measure for the distribution of positive and negative charge inside the neutron Violates CP Current expt. upper limit : d N < e cm [Baker, et al., PRL, 2006] Standard model estimate : d N e cm [Dar, arxiv:hep-ph/ ] - CPV in SM is not enough to explain observed baryon asymmetry - nedm : good probe of new CP-violating BSM physics

7 7 / 49 Neutron EDM, Quark EDM and ensor Charge Sources of nedm in L CPV eff : θ QCD, quark-edm, chromo-edm, Weinberg 3-gluon,... Quark EDMs L = i 2 d q qσ µν γ 5 qf µν q=u,d,s Neutron EDM from qedms (assuming qedm is the major source) d N = d u g u,n + d d g d,n Hadronic part: nucleon tensor charge + d s g s,n N qσ µν q N = g q,n ψ N σ µν ψ N

8 8 / 49 Neutron EDM, Quark EDM and ensor Charge d q m q in many models d q = y q δ q ; y u y d 1 2, y s y d 20 d N = d u g u,n + d d g d,n = d d [ g d,n d s g s,n δ u g u,n + 20 δ s g s,n δ d δ d ] Precision determination of g s,n is important

9 9 / 49 Lattice QCD Non-perturbative approach to understand QCD Formulated on discretized Euclidean space-time Hypercubic lattice Lattice spacing a Quark fields placed on sites Gauge fields on the links between sites; U µ

10 10 / 49 Systematics for Lattice Calculation Finite Lattice Spacing Simulations at finite lattice spacings a 0.06, 0.09 & 0.12 fm Extrapolate to continuum limit, a = 0 Heavy Quark Mass Smaller quark mass Larger computational cost Simulations at (heavy) pion masses M π 130, 210 & 310 MeV Extrapolate to physical pion mass, M π = M phys π Finite Volume Simulations at finite lattice volume M π L = (L = fm) Extrapolate to infinite volume, M π L = Excited State Contamination

11 11 / 49 MILC HISQ Lattices, n f = ID a (fm) M π (MeV) L 3 M π L a12m (11) 305.3(4) a12m220s (12) 218.1(4) a12m (10) 216.9(2) a12m220l (09) 217.0(2) a09m (08) 312.7(6) a09m (07) 220.3(2) a09m (06) 128.2(1) a06m (04) 319.3(5) a06m (04) 229.2(4) Fermion discretization : Clover (valence) on HISQ (sea) HYP smearing reduce discretization artifact m u = m d

12 12 / 49 hree-point Function Diagrams ME N q i Γq j N Quark-line connected / disconnected diagrams Disconnected diagrams : Complicated and expensive on lattice Canceled in isovector charges g u d Γ Neutron decay needs only connected diagrams (g u d, g u d S ), Quark EDM needs both diagrams (g u, gd, gs )

13 Connected Quark Loop Contribution 13 / 49

14 14 / 49 Nucleon Charge on Lattice Nucleon charges g q Γ (Γ = V,A,S,) defined by N q Γ q N = g q Γ ψ N Γψ N 0 t ins t sep On lattice, g q Γ is extracted from ratio of 3-pt and 2-pt function C 3pt /C 2pt g q Γ C 2pt = 0 χ(t s ) χ(0) 0, C 3pt = 0 χ(t s ) O(t i ) χ(0) 0 χ : interpolating operator of proton χ introduces excited states of proton

15 15 / 49 Removing Excited States Contamination (1) Gaussian smearing on src/snk, (2) Seperation of src/snk in t ins t sep - t ins 0 t ins t sep Separating proton sources far from each other small excited state effect, but weak signal Separating in reasonable range, remove excited state by fitting to C 2pt (t sep ) = A 2 0e M 0t sep + A 2 1e M 1t sep C 3pt (t sep, t ins ) = B 0 e M 0t sep + B 1 e M 1t sep + B 01 [ e M 0t ins e M 1(t sep t ins ) + e M 1t ins e M 0(t sep t ins ) ]

16 16 / 49 Removing Excited States Contamination (a09m130) g u-d, con Extrap t sep =10 t sep =12 t sep =14 a09m130 AMA τ - t sep /2

17 17 / 49 All Mode Averaging (AMA) echnique C imp = 1 N LP N LP C LP (x LP i=1 i ) } {{ } LP estimate + 1 N HP [ N HP j=1 C HP (x HP j ) C LP (x HP j ) } {{ } Correction term All-mode averaging (AMA) [Blum, Izubuchi and Shintani, PRD 88, (2013)] Exploiting translation symmetry & small fluctuation of low-modes LP term is cheap low-precision estimate (inverter with r 10 3 ) HP (high-precision) correction term Systematic error Statistical error N LP N HP brings computational gain (e.g., N LP = 60, N HP = 4) ]

18 18 / 49 MILC HISQ Lattices, n f = ID a (fm) M π (MeV) L 3 M π L a12m (11) 305.3(4) a12m220s (12) 218.1(4) a12m (10) 216.9(2) a12m220l (09) 217.0(2) a09m (08) 312.7(6) a09m (07) 220.3(2) a09m (06) 128.2(1) a06m (04) 319.3(5) a06m (04) 229.2(4)

19 Renormalization of Bilinear Operators q Γ q ZA ZV Lattice results = MS scheme at 2GeV Non-perturbative renormalization using RI-sMOM scheme Calculate ratio Z Γ /Z V : reduce lattice artifact g renorm Γ = Z Γ Z V a12 a09 a q GeV gbare Γ gv bare Z ZV (Use Z V g u d V = 1) a12 a09 a q GeV [PRD 89, (2014)], [arxiv: ] 19 / 49

20 Simultaneous Extrapolation of (a, M π, M π L) g A u-d g Γ (a, M π, L) = c 1 + c 2 a + c 3 M 2 π + c 4 e MπL a (fm) g A u-d M π 2 (GeV 2 ) g A u-d M π L g S u-d a (fm) g S u-d M π 2 (GeV 2 ) g S u-d M π L g u-d a (fm) g u-d M π 2 (GeV 2 ) g u-d M π L 20 / 49

21 Axial Charge g A on the Lattice 21 / 49

22 22 / 49 Axial Charge g A g A u-d a (fm) g A u-d M π 2 (GeV 2 ) Smaller lattice spacing shows smaller value of g A? Excited state contamination!

23 23 / 49 Excited State Effect of g A (bare) g A u-d, con Extrap t sep =8 t sep =9 t sep =10 t sep =11 t sep =12 a12m τ - t sep /2 g A u-d, con a09m310 Extrap t sep =10 t sep =12 t sep = τ - t sep /2 g A u-d, con a06m310 AMA Extrap t sep =16 t sep =20 t sep =22 t sep = τ - t sep /2 g A u-d, con Extrap t sep =8 t sep =10 a12m220l AMA t sep =12 t sep = τ - t sep /2 g A u-d, con a09m220 Extrap t sep =10 t sep =12 t sep = τ - t sep /2 g A u-d, con Extrap t sep =16 t sep =20 t sep =22 t sep =24 a06m220 AMA τ - t sep /2 g A u-d, con a09m130 AMA Extrap t sep =10 t sep =12 t sep = τ - t sep /2

24 24 / 49 Excited State Effect of g A (bare) Source/Sink gaussian smearing radius: 0.66 fm (a12), 0.50 fm (a09) and fm (a06) uned so that it minimizes computation cost and statistical noise but underestimated excited state effect in smaller a ensembles Larger excited state effect observed in smaller a ensembles C 2pt (t sep ) = A 2 0e M 0t sep + A 2 1e M 1t sep A 2 1 /A (a12), 1.2 (a09), and (a06) Excited state contamination makes g A smaller

25 Excited State Effect of g A Fixed source/sink smearing, large src-snk separation 25 / 49

26 26 / 49 Excited State Effect of g A with Different Smearing g A u-d, con Extrap t sep =22 t sep =16 t sep =24 t sep =20 a06m220 AMA σ KG =0.33 fm τ - t sep /2 650 confs g A u-d, con Extrap t sep =18 t sep =20 a06m220 AMA σ KG =0.66 fm t sep =22 t sep = τ - t sep /2 160 confs Aggressive smearing reduces excited state effect, and pushes g A up closer to experimental estimate g A is under investigation

27 27 / 49 Small Excited State Contamination in g S and g a12m a09m a06m310 AMA g S u-d, con Extrap t sep =8 t sep =9 t sep =10 t sep =11 t sep = τ - t sep /2 g S u-d, con Extrap t sep =12 t sep =10 t sep = τ - t sep /2 g S u-d, con Extrap t sep =16 t sep =20 t sep =22 t sep = τ - t sep /2 g S u-d, con Extrap t sep =8 t sep =10 a12m220l AMA t sep =12 t sep = τ - t sep /2 g S u-d, con a09m Extrap t sep =12 t sep =10 t sep = τ - t sep /2 g S u-d, con Extrap t sep =16 t sep =20 t sep =22 t sep =24 a06m220 AMA τ - t sep / Extrap t sep =10 t sep =12 t sep =14 g S u-d, con a09m130 AMA τ - t sep /2

28 28 / 49 Small Excited State Contamination in g S and g g u-d, con Extrap t sep =8 t sep =9 a12m310 t sep =10 t sep =11 t sep = τ - t sep /2 g u-d, con Extrap t sep =10 t sep =12 t sep = a09m τ - t sep /2 g u-d, con Extrap t sep =16 t sep =20 t sep =22 t sep =24 a06m310 AMA τ - t sep /2 g u-d, con Extrap t sep =8 t sep =10 a12m220l AMA t sep =12 t sep = τ - t sep /2 g u-d, con Extrap t sep =10 t sep =12 t sep = a09m τ - t sep /2 g u-d, con Extrap t sep =16 t sep =20 t sep =22 t sep =24 a06m220 AMA τ - t sep /2 g u-d, con Extrap t sep =10 t sep =12 t sep = a09m130 AMA τ - t sep /2

29 Extrapolation to Physical Limit: gs u d, g u d g S u-d g u-d a (fm) a (fm) g S u-d g u-d M π 2 (GeV 2 ) M π 2 (GeV 2 ) Volume extrapolation is not displayed, but included 29 / 49

30 30 / 49 Extrapolation to Physical Limit: g u, gd g u g d a (fm) a (fm) g u g d M π 2 (GeV 2 ) M π 2 (GeV 2 ) g u g d M π L M π L

31 31 / 49 BSM Constraints in β-decay from g u d, gs u d 90% CL contours in [ε S, ε ] plane Nuclear Experiments : transitions, Asym in Gamow-eller β-decay,... Future UCN expt. b and B 1 at 10 3

32 Disconnected Quark Loop Contribution 32 / 49

33 33 / 49 Disconnected Contribution to the Nucleon Charges Disconnected part of the ratio of 3pt func to 2pt func [ ] C 3pt disc = C2pt (t s ) x r[m 1 (t i, x; t i, x) Γ ] C 2pt C 2pt (t s ) M : Dirac operator r[m 1 (t i, x; t i, x) Γ] : disconnected quark loop t ins 0 t sep

34 34 / 49 Difficulties in Disconnected Diagram Calculation [ ] C 3pt disc = C2pt (t s ) x r[m 1 (t i, x; t i, x)σ µν ] C 2pt C 2pt (t s ) Connected calculation needs only point to all propagators Disconnected quark loop needs all x to all propagators Computationally L 3 times more expensive; need new technique Noisy signal Need more statistics t ins 0 t sep

35 35 / 49 Improvement & Error Reduction echniques Multigrid Solver [Osborn, et al., 2010; Babich, et al., 2010] All-Mode Averaging (AMA) for wo-point Correlators [Blum, Izubuchi and Shintani, 2013] Hopping Parameter Expansion (HPE) [hron, et al., 1998; McNeile and Michael, 2001] runcated Solver Method (SM) [Bali, Collins and Schäfer, 2007] Dilution [Bernardson, et al., 1994; Viehoff, et al., 1998]

36 36 / 49 Removing Excited States Contamination t ins t sep - t ins 0 t ins t sep Interpolating operator introduces excited state contamination Remove excited state by fitting to C 2pt (t sep ) = A 1 e M 0t sep + A 2 e M 1t sep C 3pt (t sep, t ins ) = B 1 e M 0t sep + B 2 e M 1t sep + B 12 [ e M 0t ins e M 1(t sep t ins ) + e M 1t ins e M 0(t sep t ins ) ]

37 37 / 49 Removing Excited States Contamination (a12m310, l) g l, disc Extrap t sep = 8 t sep = 9 a12m310 t sep =10 t sep =11 t sep = τ - t sep /2

38 38 / 49 Removing Excited States Contamination (a12m310, s) g s, disc Extrap t sep = 8 t sep = 9 a12m310 t sep =10 t sep =11 t sep = τ - t sep /2

39 Proton ensor Charge : Connected / Disconnected Connected Contribution g u = 0.77(6), Disconnected Contribution g d = 0.20(2) Ens g l g s a12m (23) (19) a12m (40) (27) a09m (22) (21) a09m (54) a06m (65) (55) g l,disc is tiny compared to the connected contributions ake maximum value as systematic error No connected diagrams for g s Extrapolate to physical point Disconnected contributions are non-trivial in g S and g A (O(10%)) [arxiv: ], [arxiv: ] 39 / 49

40 40 / 49 Simultaneous extrapolation of g s in (a, M π) g s g s Π g s = 0.008(9)

41 Results 41 / 49

42 42 / 49 Lattice Results of Nucleon ensor Charge Proton Charges (µ MS = 2 GeV) g u d S = 0.86(18) g u d = 0.99(7) g u = 0.77(6) g d = 0.20(2) g s = 0.008(9) Neutron ensor Charge In isospin limit (m u = m d ), u d from proton g

43 43 / 49 Lattice Calculations of Isovector ensor Charge g u-d PNDME 4f clover/hisq EMC 4f MF LHPC 3f clover LHPC 3f DWF LHPC 3f DWF/asqtad RBC/UKQCD 3f DWF M π 2 (GeV 2 ) EMC 2f MF RQCD 2f clover

44 Disconnected Contribution to ensor Charge his study g l,disc , g s,disc = 0.008(9) Lattice, Abdel-Rehim, et al., 2014, a = fm, M π = 370 MeV, wisted mass g l,disc = (7) Lattice, S. Meinel, et al., 2014, a = 0.11 fm, M π = 317 MeV, Clover 2g l,disc 44 / 49

45 45 / 49 ensor Charge g d g u g s µ his study 0.20(2) 0.77(6) 0.008(9) 2 GeV Quark model 1/3 4/3 QCD Sum Rules (17) 1.4(7)? Dyson-Schwinger (2) 0.55(8) 2 GeV ransversity (33) 0.57(21) 1 GeV ransversity (20) 0.39(15) 1 GeV ransversity GeV 1 Pospelov and Ritz, Pitschmann, et al., Bacchetta, et al., Anselmino, et al., Kang, et al., 2015

46 46 / 49 qedm and ensor Charge Known parameters d N = d u g u,n + d d g d,n + d s g s,n d N < e cm (90% C.L.) [Baker, et al., PRL 2006] g u,n = 0.20(2) g d,n = 0.77(6) g s,n = 0.008(9) Place constraints on d q

47 47 / 49 qedm Constraints du e cm d n e cm 90 CL g s d

48 48 / 49 Constraints in Split SUSY Model Split SUSY: Quark EDM is the dominant BSM source of CPV [Giudice and Romanino, Nucl.Phys. B699, 65 (2004)] [Arkani-Hamed and Dimopoulos, JHEP 0506, 073 (2005)] Bounds of gaugino (M 2 ) and Higgsino (µ) mass params placed Upper limit of d n < e cm in split SUSY placed [arxiv: ]

49 49 / 49 Conclusion g A is under investigation, focusing on the excited state effect Isovector g S and g are obtained within 20% and 7% uncertainty Presented first lattice QCD calculation of nucleon tensor charge including all systematics (a, M π, M π L, disconnected diagrams) Constrained qedms and split SUSY model by using the lattice results combined with experiment