Proton Mass and Decomposition • spin – spin crisis • Anomalous Ward Identity, Goldberger-Treiman relation and • 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 in the

Nucleon ()(uD ) udD dt ´ Connected insertion (CI)

t f Disconnected insertion (DI) ´  Dtu( 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: Z 0k ¶ A0 = i2Z mZ P - iN 2(Z q(x) + l ¶ A0 ) A A m m m P f q m m

. 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 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 )/2 v q m 222 -iT34 (q )() qqq /(  )/ 2  mTqm ups ] ( ' ') . 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 Zqg TZ11(0)(0) T  1,  1 Z()(0)()(0) TTZqqgg TT 1, ZJZJLL  qg1212 q q g g  qg 2 Zqg TZ22(0)(0) T  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 áT ñ = -3/ 4M; áTˆ ñ = - M 44 mm

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, áH ñ - áH ñ = áH ñ + O(a2 ) q E m 27 Overlap fermion on DWF configurations RBC/UK CD 2+1 flavor DWF Configurations Y.B. Yang et al., 1808.08677

SU(4|2) mixed action HBCPT

2 MN = 960(13) MeV, c /dof = 0.52 Fix g =1.2723, M = 931(8) MeV, c 2 /dof = 1.5 A N 28 Non-perturbative Renormalization

• Renormalized q and g in MS-bar at μ

• Renormalization of glue operator in propagator is very noisy.

29 Glue Renormalization

• Off-diagonal and traceless renormalization in RI/MOM

• CDER

30 Cluster Decomposition Error Reduction (CDER)

Y.B. Yang et al., 1805.00531

21,166 243 x 64 N =2 Clover configs, 70,834 243 x 64 quenched configs, f m = 450 MeV, a =0.117 fm, use 10% a=0.098 fm, use 1% for CDER π for CDER 31 -1 ZGG and g with CDER

R = 0.9 fm, R =1.4 R the 1 2 1, Perturbative mixing with quark error is reduced by ~ 300 times on the 48I lattice with L = 5.5 fm. Y.B. Yang et al., 1805.00531

(V/VR1)½ (V/VR2)½ ~ 300 32 q, g and Normalization

Znorm( å < x > f + < x >g )=1 f =u,d ,s

Y.B. Yang et al., 1808.08677

33 Comparison with Global Fitting of MS-bar at 2 GeV

34 Proton Mass Decomposition

35 < x >u-d = 0.151(28)(29) CT14 --- 0.158(6)(23) 36 Summary and Challenges

• Lattice calculations of the physical 2+1 flavor dynamical fermions at the physical pion point and with extrapolations to continuum and infinite volume limits are becoming available even with chiral fermions.

• Decomposition of proton spin and hadron masses into quark and glue components on the lattice is feasible, pending reasonalbe statistics of non-perturbative renormalization. Large momentum frame for the proton to calculate glue helicity remains a challenge.

• Together with evolution, factorization, perturbative QCD, lattice QCD results with small enough statistical and systematic errors can compare directly with experiments and have an impact in advancing our understanding of the underline physics of the hadron structure (form factors, PDF, neutron electric dipole moment, g-2, etc).

37 38 Status of Proton Spin Problem

• The crisis is over and most of the puzzles are solved. However, challenges still remain in experiments and lattice calculations to totally understand the proton spin decomposition.

39 Glueball Masses on 483 x 96 lattice (L= 5.5 fm)

Scalar: cutoff at R=9 reduces Pseudoscalar: cutoff at R=11 reduces error by ~ 4 which is ≈ (25/9)3/2 error by ~ 3 which is ≈ (25/11)3/2 41 AWI and Goldberger-Treiman Relation

• Generalized Goldberger-Treiman relation is violated badly at small momentum transfer g (q2)+q2 /2m h (q2)¹ m /m g (q2)+2g (q2) A N A q N P Q • Consider

42 43 Orbital Angular Momentum

skyrmion Trinacria, Erice

# AWI vs Goldberger-Treiman Relation • AWI is an operator relation which should be satisfied in any state. • GTR is based on applying the derivative on the nucleon state which is susceptible to excited state contamination. 0 3 • ZA ,norm the same as ZA ,norm in CI.

45