Lattice QCD Calculation of Nucleon Tensor Charge

Transcription

1 1 / 36 Lattice QCD Calculation of Nucleon ensor Charge. Bhattacharya, V. Cirigliano, R. Gupta, H. Lin, B. Yoon PNDME Collaboration Los Alamos National Laboratory Jan 22, 2015

2 2 / 36 Neutron EDM, Quark EDM and ensor Charge Quark EDMs at dim=5 L = i 2 Neutron EDM from qedms d N = d u g u,n q=u,d,s d q qσ µν γ 5 qf µν + d d g d,n + d s g s,n Hadronic part: nucleon tensor charge N qσ µν q N = g q,n ψ N σ µν ψ N

3 3 / 36 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

4 4 / 36 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

5 5 / 36 Physical Results from Unphysical Simulations Finite Lattice Spacing Simulations at finite lattice spacings a 0.06, 0.09 & 0.12 fm Extrapolate to continuum limit, a = 0 Heavy Pion Mass Lattice simulation: 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 =

6 6 / 36 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

7 7 / 36 hree-point Function Diagrams ME N q i σ µν q j N Quark-line connected / disconnected diagrams Disconnected diagrams : complicated and expensive on lattice

8 Connected Quark Loop Contribution 8 / 36

9 9 / 36 Nucleon Charge on Lattice Nucleon tensor charge g q is defined by N qσ µν q N = g q ψ N σ µν ψ N On lattice, g q 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 0 t ins t sep

10 10 / 36 Removing Excited States Contamination 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 Put operator reasonable range, 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 ) ]

11 11 / 36 Removing Excited States Contamination (a12m310) g con, u-d a12m310 Extrap t sep =8 t sep =9 t sep =10 t sep =11 t sep = t - t sep /2 Small excited state contamination (compared to g A, g S )

12 12 / 36 Removing Excited States Contamination (a09m310) g con, u-d a09m310 Extrap t sep =10 t sep =12 t sep = t - t sep /2 Small excited state contamination (compared to g A, g S )

13 13 / 36 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)

14 14 / 36 Renormalization of Bilinear Operators qσ µν q Lattice results = MS scheme at 2GeV Non-perturbative renormalization using RI-sMOM scheme Calculate ratio Z /Z V : reduce lattice artifact Renormalized ensor Charge : g renorm = Z Z V gbare gv bare (Use Z V g u d V = 1) a (fm) Z /Z V (3) (3) (4)

15 15 / 36 Simultaneous extrapolation of (a, M π, M π L) g (a, M π, L) = c 1 + c 2 a + c 3 M 2 π + c 4 e MπL La#ce Spacing a 0 Pion Mass M π M π phys La#ce Volume M π L g u g d Preliminary!

16 16 / 36 Simultaneous extrapolation of (a, M π, M π L) g (a, M π, L) = c 1 + c 2 a + c 3 M 2 π + c 4 e MπL La#ce Spacing a 0 Pion Mass M π M π phys La#ce Volume M π L g u d g u+d Preliminary!

17 Disconnected Quark Loop Contribution 17 / 36

18 18 / 36 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)σ µν ] C 2pt C 2pt (t s ) M : Dirac operator r[m 1 (t i, x; t i, x)σ µν ] : disconnected quark loop t ins 0 t sep

19 19 / 36 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

20 20 / 36 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]

21 21 / 36 Improved Estimator of wo-point Function C 2pt, imp = 1 N LP C 2pt N LP i=1 LP (x i) + 1 } {{ } LP estimate N HP N HP j=1 [ ] C 2pt HP (x j) C 2pt LP (x j) } {{ } Crxn term All-mode averaging (AMA) [Blum, Izubuchi and Shintani, 2013] with Multigrid solver for Clover in Chroma [Osborn, et al., 2010] Exploiting translation symmetry & small fluctuation of low-modes LP term is cheap low-precision estimate HP (high-precision) correction term Systematic error Statistical error N LP N HP brings computational gain (e.g., N LP = 60, N HP = 4)

22 22 / 36 runcated Solver Method (SM) M 1 E = 1 N LP N LP s i LP η i + 1 i=1 } {{ } LP estimate N LP +N HP N HP i=n LP +1 ( s i HP s i LP ) η i } {{ } Crxn term Stochastic estimate of M 1 [Bali, Collins and Schäfer, 2007] Do calculate exact M 1, but estimate with reasonable error 1 Computational cost : of exact calculation Same form as AMA - C 2pt M 1 - Sum over source positions Sum over random noise sources η i : complex random noise vector s i : solution vector; M s i = η i

23 23 / 36 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 ) ]

24 24 / 36 Removing Excited States Contamination (a12m310, l) Extrap t sep = 8 t sep = 9 t sep =10 t sep =11 t sep =12 g l, disc a12m t - t sep /2

25 25 / 36 Removing Excited States Contamination (a12m310, s) Extrap t sep = 8 t sep = 9 t sep =10 t sep =11 t sep =12 g s, disc a12m t - t sep /2

26 Proton ensor Charge : Connected / Disconnected Connected Contribution g u g d g u d g u+d 0.788(64) 0.223(25) 1.020(75) 0.567(62) Disconnected Contribution Ens g l g s a12m (24) (24) a12m (46) (32) a09m (19) (25) a09m (69) a06m (94) (60) 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 26 / 36

27 27 / 36 Simultaneous extrapolation of g s in (a, M π) g s, disc g s, disc Π g s = 0.002(11)

28 Results 28 / 36

29 Lattice Results of Nucleon ensor Charge Preliminary! Proton ensor Charge (µ MS = 2 GeV) g u = 0.79(7) g d = 0.22(3) g u d = 1.02(8) g u+d = 0.57(6) g s = 0.002(11) Neutron ensor Charge In isospin limit (m u = m d ), u d from proton g 29 / 36

30 Proton ensor Charge his study (µ MS = 2 GeV) g u = 0.79(7), g d = 0.22(3) g u d = 1.02(8) g u+d = 0.57(6) Lattice QCD estimates for g u d [LHPC, EMC, RQCD, PNDME] 30 / 36

31 Proton ensor Charge his study g l,disc , g s,disc = 0.002(11) 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 31 / 36

32 32 / 36 Proton ensor Charge his study g u = 0.79(7), g d = 0.22(3) (µ MS = 2 GeV) Quark model g u = 4 3, gd = 1 3 Dyson-Schwinger [Pitschmann, et al., 2014] g u = 0.55(8), g d = 0.11(2) (ζ 2 = 2 GeV) Experiments (HERMES and COMPASS) g u = 0.57(21), g d = 0.18(33) (Q 2 = 1.0 GeV 2 ) [Bacchetta, et al., JHEP 2013] g u = , gd = (Q 2 = 0.8 GeV 2 ) [Anselmino, et al., PRD 2013]

33 33 / 36 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 = 0.22(3) g d,n = 0.79(7) g s,n = 0.002(11) Place constraints on d q

34 34 / 36 qedm Constraints 90% C.L. parameter space of d u and d d, assuming g s = 0

35 35 / 36 qedm Constraints d N = d u g u,n + d d g d,n + d s g s,n g u,n = 0.223(28), g d,n = 0.788(68), g s,n = 0.002(11) Since g s = 0 within error, cannot give constraints on d s

36 36 / 36 Conclusion Presented first lattice QCD calculation of nucleon tensor charge including all systematics (a, M π, M π L, disconnected diagrams) Constrained qedms by the results combined with experiment Need more study on g s to constrain d s