Proton Mass and Spin Decomposition • Quark spin – proton spin crisis • Anomalous Ward Identity, Goldberger-Treiman relation and renormalization • Cluster decomposition error reduction • Energy-momentum tensor -- proton spin and mass decomposition • Renormalization of glue operators c QCD Collaboration INT, Oct. 16, 2018 Where does the spin of the proton come from? Quark spins according to the quark model 9 ∧ DS = åDq + Dq ~ 0.3 q 3 Proton Spin Crisis • What’s wrong with the quark model? • Mixture from the glue spin? Anomalous Ward Identity Take 0 ¶m (Am + 2iN f Km ) = i2 å miPi i=u.d.s However, the Chern-Simons current is not gauge invariant. • `Proton spin crisis is the graveyard of all hadronic models’ 4 Lattice Calculations of Quark and Glue Spins • Quark and Glue Momentum and Angular Momentum in the Nucleon (u D u d D d )( t ) ´ Connected insertion (CI) t f Disconnected insertion (DI) ´ D( t )( u , d , s ) t t0 f 5 Quark Spin and Anomalous Ward Identify • Calculation with a local singlet axial-vector current needs a normalization. • AWI needs to be satisfied. • Normalized AWI for overlap fermion for local current 0 k A ¶m Am = i2mP - iN f 2q(x) Renormaliztion and mixing: 0 0 0 Z Ak A ¶m Am = i2ZmmZP P - iN f 2(Zqq(x) + l ¶m Am ) . Overlap fermion --> mP is RGI (ZmZP=1) . Overlap operator for q ( x ) = - 1 / 2 T r g 5 D ov ( x , x ) has no multiplicative renormalization. 2 æ ö 0 a s 3 1 . Espriu and Tarrach (1982) Z A(2-loop)=1-ç ÷ C2(R)N f , è p ø 8 e 2 æ a ö 3 1 l = - s C (R) ç p ÷ 16 2 e 6 è ø Low-mode average for quark loop † fnOfn † -1 Tr loop = Trå + Tr hHOD hH n m + iln 7 Cluster Decomposition Error Reduction Noise Problem with Large Volume • Single falls off exponentially, but noise remains constant sign problem, such as in glueball mass. • nEDM with the θ term is noisy for large volume, because the topological charge fluctuation as (V)1/2. Cluster Decomposition Principle H. Araki, K. Hepp, and D. Ruelle, Helv. Phys. Acta 35, 164 (1962); S. Weinberg, Quantum Theory of Fields,Vol 1, pp. 169 0 |O (x)O (y) | 0 £ Ar-3/2e- Mr , r = x-y (space like), 1 2 s O1 and O2 are color-singlet operators, Asympotic behavior of a boson propagator K1(r) / r. This point-to-point relation should hold for color-singlet operators separated by large Euclidean distance. Variance Reduction via Cluster Decomposition -- Disconnected Insertions • Variance of disconnected insertion KFL, J. Liang, Y.B. Yang, • Vacuum insertion arXiv:1706.06358 • Var(Rmax,t) =1 (indep of t), but Var(Rs,t) = Vs/V. • Gains singal to noise ratio: S/N(R ,t) V S = S/N(L,t) VS • Fast Fourier transform to calculate the truncated sum in relative coordinates ~ V log V operations. 10 Strangeness in the Nucleon 3 32 x 64 (32ID) RBC (4.6 fm, mπ=170 MeV) 11 CP violation angle in the nucleon Tr(C (t)g ) a 1 = 3Q 5 Tr(C2 (t)Ge ) 3 48 x 96 RBC lattice (a = 0.114 fm, mπ = 139 MeV), mπ(valence) = 280 MeV, r (cut off) = 16, S/N increases by ~ 3.6 Quark spin with overlap fermion on domain-wall fermion configurations RBC/UK CD 2+1 flavor DWF Configurations • Overlap fermion is many times more expensive than Wilson fermion to invert • It has chiral symmetry at finite a (Ginsparg-Wilson relation) • Renormalization is easier with Ward identities • Multi-mass inversion with deflation with the same eigenvectors • Typically 5-6 valence quarks for each lattice ensemble 13 gA in DI for the 32ID lattice with cluster decomposition error reduction (CDER) light quark strange quark e-MR C3(R)= C3(¥)+k R M 14 Non-perturbative Renormalization RI/MOM vertex RI to MS-bar matching The same as the conventional flavor irreducible representation due to linearity of the equations, but lattice classification is richer in structure and each components is discernible experimentally (KFL, 1703.04690). 15 DI results -m L g = c +c a2 +c (m2 -m2 )+c (m2 -m2 )+c e p ,v A 0 1 2 p ,v p ,p 3 p ,s p ,p 4 light quark strange quark No statistically significant O(a^2) dependence 16 CI results -m L g = c +c a2 +c (m2 -m2 )+c (m2 -m2 )+c e p ,v A 0 1 2 p ,v p ,p 3 p ,s p ,p 4 u quark d quark No discernable O(a^2) dependence 17 Quark Spin Compoments MS (2 GeV) gA Δ(u+d) Δ(u/d) Δs Δu Δd Δu-Δd 3 DS CI DI (gA ) J. Green -0.0240 0.863 -0.345 1.206 0.494 (45) (7)(14) (6)(9) (20) (11)(15) C. Alexandrou 0.598 -0.077 -0.042 0.830 -0.386 1.216 0.402 (24)(6) (15)(5) (10)(2) (26)(4) (16)(6) (31)(7) (34)(10) 0.580 -0.070 -0.035 0.847 -0.407 1.254 0.405 c QCD (16)(30) (12)(15) (6)(7) (18)(32) (16)(18) (16)(30) (25)(37) -0.10 0.76 -0.41 1.17 0.25 NPPDFpol1.1 (8) (4) (4) (6) (10) (Q2=10 GeV2) DSSV -0.012 0.793 -0.416 1.209 0.366 (Q2=10 GeV2) +(56)-(62) +(28)-(34) +(35)-(25) +(45)-(42) +(62)-(42) J. Green et al., NF=2+1, Clover fermion, mπ = 317 MeV, one lattice C. Alexandrou et al., NF=2, twisted mass fermion, , mπ = 131 MeV, one lattice c QCD, NF=2+1, Overlap fermion, , mπ = 170, 290, 330 MeV, 5 - 6 valence quarks for each of the three lattices 3 3 Expt. gA = 1.2723(23); CalLat: gA = 1.271(13) Quark Spin • Lattice calculation with chiral fermion is getting close in revealing the origin of the smallness of the quark spin – the disconnected insertion is large and negative. • The interplay between the pseudoscalar and topological charge couplings in the anomalous Ward identity is the origin for the negative DI contribution – another example of U(1) anomaly at work. • In the future it would be desirable to use the exponentially local chiral axial-vector current (P. Hazenfratz) to check the 0 calculation where ZA ,norm = 1. 19 Where does the rest of the spin of the proton come from? Glue spin Quark ortibal angular momentum Glue orbital angular momentum Momenta and Angular Momenta of Quarks and Glue . Energy momentum tensor operators decomposed in quark and glue parts gauge invariantly --- Xiangdong Ji (1997) i é1 ù T q = éyg D y + (m « n)ù ® J = d 3x ygg y + x ´yg (-iD)y mn ë m n û q ò ê 5 4 ú 4 ë2 û 1 T g = F F - d F 2 ® J = d 3x éx ´ (E ´ B)ù mn ml ln 4 mn g ò ë û . Nucleon form factors 22 psT,| |'' ps upsTq (,)[()12 pTqp () v q /2 m 2 2 2 -iT34 (qqq )( qmTq ) / ( ) mups / 2] ( ' ') . Momentum and Angular Momentum éT (0) + T (0)ù é ù 1 2 Zq,gT1(0)q,g ëOPEû ® x (m,MS) , Zq,g ê ú ® Jq/g (m,MS) q/g ë 2 û q,g page 21 Normalization, Renormalization and Quark-Glue Mixing Momentum and Angular Momentum Sum Rules RLRL x q Z q x q, x g Z g x g , RLRL JZJJZJq q q, g g g , LL Z x Z x 1, qg q q g g ZTZTqg11(0) (0) 1, 1 ZTTZTT(q q )(0) ( g g )(0) 1, ZJZJLL qg1 2 1 2 q q g g qg 2 ZTZTqg22(0) (0) 0 Mixing x MS( ) C ( ) C( ) x R q qq qg q M. Glatzmaier, KFL MS CC( ) ( ) x R arXiv:1403.7211 x g () gq gg g 22 Quark Spin, Orbital Angular Momentum, and Gule Angular Momentum (M. Deka et al, 1312.4816, PRD) pizza cinque stagioni Dq » 0.25; 2 Lq » 0.47 (0.01(CI)+0.46(DI)); 2 J » 0.28 g These are quenched results so far. Proton Spin Decomposition (2+1 Flavor) Approximate by setting T2 = 0 Motivation Where does this Wobservablehere does t4.6%he p r oton mass come comef frroom?m, and how ? The Higgs boson makes the u/d quark having masses (2GeV MS-bar): But the mass of the proton is mu = 2.08(9) MeV 938.272046(21) MeV. md = 4.73(12) MeV ~100 times of the sum of the quark Laiho, Lunghi, & Van de Water, masses! Phys.Rev.D81:034503,2010 Quark and Glue Components of Hadron Mass Energy momentum tensor PTPPPM| | / Trace anomaly b(g) 2 Tmm = -m(1+g m )yy + G 2g Separate into traceless part and trace part ˆ T T 1 áP |T q,g | Pñ = áxñ (m2 )(P P - d P2 ) / M, áxñ (m2 ) + áxñ (m2 ) = 1 mn q,g m n 4 mn q g ˆ áT44 ñ = -3/ 4M; áTmm ñ = - M 26 Decomposition of hadron mass Xiangdong Ji, PRL 74, 1071 (1995); PRD 52, 271 (1995) – Equation of motion 1 ì0 for CI rDov å(Dc + m)(x,z) (z, y) = d x,y Þ í D c = z Dc + m îconstant for DI 1- Dov / 2 Therefore, 2 áHq ñ - áH E ñ = áHm ñ + O(a ) 27 Overlap fermion on DWF configurations RBC/UK CD 2+1 flavor DWF Configurations Y.B.