Lattice QCD Calculation of Nucleon Tensor Charge T. Bhattacharya, - PowerPoint PPT Presentation

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

Neutron EDM, Quark EDM and Tensor Charge • Quark EDMs at dim=5 L = − i � qσ µν γ 5 qF µν d q ¯ 2 q = u,d,s • Neutron EDM from qEDMs d N = d u g u,N + d d g d,N + d s g s,N T T T • Hadronic part: nucleon tensor charge qσ µν q | N � = g q,N ¯ � N | ¯ ψ N σ µν ψ N T 2 / 36

Neutron EDM, Quark EDM and Tensor Charge • d q ∝ m q in many models y u ≈ 1 y s d q = y q δ q ; 2 , ≈ 20 y d y d d N = d u g u,N + d d g d,N + d s g s,N T T T � � + 1 δ u + 20 δ s g d,N g u,N g s,N = d d T T T 2 δ d δ d ⇒ Precision determination of g s,N is important T 3 / 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 4 / 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 = 3 . 2 ∼ 5 . 4 ( L = 2 . 9 ∼ 5 . 8 fm ) ⇒ Extrapolate to infinite volume, M π L = ∞ 5 / 36

MILC HISQ Lattices, n f = 2 + 1 + 1 L 3 × T ID a (fm) M π (MeV) M π L 24 3 × 64 a12m310 0.1207(11) 305.3(4) 4.54 24 3 × 64 a12m220S 0.1202(12) 218.1(4) 3.22 32 3 × 64 a12m220 0.1184(10) 216.9(2) 4.29 40 3 × 64 a12m220L 0.1189(09) 217.0(2) 5.36 32 3 × 96 a09m310 0.0888(08) 312.7(6) 4.50 48 3 × 96 a09m220 0.0872(07) 220.3(2) 4.71 64 3 × 96 a09m130 0.0871(06) 128.2(1) 3.66 48 3 × 144 a06m310 0.0582(04) 319.3(5) 4.51 64 3 × 144 a06m220 0.0578(04) 229.2(4) 4.25 • Fermion discretization : Clover (valence) on HISQ (sea) • HYP smearing – reduce discretization artifact • m u = m d 6 / 36

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

Connected Quark Loop Contribution 8 / 36

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

Removing Excited States Contamination t ins ¡ ¡ ¡ ¡∞ ¡ t sep -­‑ ¡t ins ¡ ¡ ¡ ¡∞ ¡ t sep ¡ 0 ¡ t ins ¡ • 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 0 t sep + A 2 e − M 1 t sep C 3pt ( t sep , t ins ) = B 1 e − M 0 t sep + B 2 e − M 1 t sep � e − M 0 t ins e − M 1 ( t sep − t ins ) + e − M 1 t ins e − M 0 ( t sep − t ins ) � + B 12 10 / 36

Removing Excited States Contamination (a12m310) 1.24 t sep =10 Extrap t sep =8 t sep =11 1.20 t sep =9 t sep =12 con, u-d 1.16 1.12 g T 1.08 a12m310 1.04 -4 -2 0 2 4 t - t sep /2 • Small excited state contamination (compared to g A , g S ) 11 / 36

Removing Excited States Contamination (a09m310) 1.16 Extrap t sep =10 1.12 t sep =12 con, u-d t sep =14 1.08 g T 1.04 a09m310 1.00 -6 -4 -2 0 2 4 6 t - t sep /2 • Small excited state contamination (compared to g A , g S ) 12 / 36

MILC HISQ Lattices, n f = 2 + 1 + 1 L 3 × T ID a (fm) M π (MeV) M π L 24 3 × 64 a12m310 0.1207(11) 305.3(4) 4.54 24 3 × 64 a12m220S 0.1202(12) 218.1(4) 3.22 32 3 × 64 a12m220 0.1184(10) 216.9(2) 4.29 40 3 × 64 a12m220L 0.1189(09) 217.0(2) 5.36 32 3 × 96 a09m310 0.0888(08) 312.7(6) 4.50 48 3 × 96 a09m220 0.0872(07) 220.3(2) 4.71 64 3 × 96 a09m130 0.0871(06) 128.2(1) 3.66 48 3 × 144 a06m310 0.0582(04) 319.3(5) 4.51 64 3 × 144 a06m220 0.0578(04) 229.2(4) 4.25 13 / 36

Renormalization of Bilinear Operators qσ µν q • Lattice results = ⇒ MS scheme at 2 GeV • Non-perturbative renormalization using RI-sMOM scheme • Calculate ratio Z T /Z V : reduce lattice artifact • Renormalized Tensor Charge : × g bare = Z T g renorm T (Use Z V g u − d = 1 ) T g bare V Z V V a (fm) Z T /Z V 0.12 1.01(3) 0.09 1.05(3) 0.06 1.07(4) 14 / 36

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

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

Disconnected Quark Loop Contribution 17 / 36

Disconnected Contribution to the Nucleon Charges Disconnected part of the ratio of 3pt func to 2pt func � C 3pt � disc = −� C 2pt ( t s ) � x Tr[ M − 1 ( t i , x ; t i , x ) σ µν ] � C 2pt � C 2pt ( t s ) � • M : Dirac operator • Tr[ M − 1 ( t i , x ; t i , x ) σ µν ] : disconnected quark loop t ins ¡ t sep ¡ 0 ¡ 18 / 36

Difficulties in Disconnected Diagram Calculation � disc � C 3pt = −� C 2pt ( t s ) � x Tr[ 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 ¡ t sep ¡ 0 ¡ 19 / 36

Improvement & Error Reduction Techniques • Multigrid Solver [Osborn, et al. , 2010; Babich, et al. , 2010] • All-Mode Averaging (AMA) for Two-point Correlators [Blum, Izubuchi and Shintani, 2013] • Hopping Parameter Expansion (HPE) [Thron, et al. , 1998; McNeile and Michael , 2001] • Truncated Solver Method (TSM) [Bali, Collins and Sch¨ afer, 2007] • Dilution [Bernardson, et al. , 1994; Viehoff, et al. , 1998] 20 / 36

Improved Estimator of Two-point Function N LP N HP C 2pt, imp = 1 1 � � � � C 2pt C 2pt HP ( x j ) − C 2pt LP ( x i ) + LP ( x j ) N LP N HP i =1 j =1 � �� � � �� � LP estimate 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) 21 / 36

Truncated Solver Method (TSM) N LP + N HP N LP 1 1 � � � � M − 1 = | s i � LP � η i | + | s i � HP − | s i � LP � η i | E N LP N HP i =1 i = N LP +1 � �� � � �� � LP estimate Crxn term • Stochastic estimate of M − 1 [Bali, Collins and Sch¨ afer, 2007] – Do calculate exact M − 1 , but estimate with reasonable error 1 1 – Computational cost : 100 ∼ 10000 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 � 22 / 36

Removing Excited States Contamination t ins ¡ ¡ ¡ ¡∞ ¡ t sep -­‑ ¡t ins ¡ ¡ ¡ ¡∞ ¡ t sep ¡ 0 ¡ t ins ¡ • Interpolating operator introduces excited state contamination • Remove excited state by fitting to C 2pt ( t sep ) = A 1 e − M 0 t sep + A 2 e − M 1 t sep C 3pt ( t sep , t ins ) = B 1 e − M 0 t sep + B 2 e − M 1 t sep � e − M 0 t ins e − M 1 ( t sep − t ins ) + e − M 1 t ins e − M 0 ( t sep − t ins ) � + B 12 23 / 36

Removing Excited States Contamination (a12m310, l ) 0.005 t sep = 9 t sep =11 Extrap t sep = 8 t sep =10 t sep =12 0.000 l, disc -0.005 g T -0.010 -0.015 a12m310 -0.020 -3 -2 -1 0 1 2 3 t - t sep /2 24 / 36

Removing Excited States Contamination (a12m310, s ) 0.015 t sep = 9 t sep =11 Extrap t sep = 8 t sep =10 t sep =12 0.010 0.005 s, disc 0.000 g T -0.005 -0.010 a12m310 -0.015 -3 -2 -1 0 1 2 3 t - t sep /2 25 / 36