Origin of Nucleon Mass in Lattice QCD Quark and glue components of hadron mass Decomposition of meson masses πn σ term, strangeness and charmness Decomposition of nucleon mass c QCD Collaboration Trento, Apr. 5, 2017
Motivation Where does this observable 4.6% come from? Where does the proton mass come from, and how? But the mass of the proton is 938.272046(21) MeV. ~100 times of the sum of the quark masses! The Higgs boson make the u/d quark having masses (2GeV MS-bar): mu = 2.08(9) MeV md = 4.73(12) MeV Laiho, Lunghi, & Van de Water, Phys.Rev.D81:034503,2010
Where does the mass of the hadron come from? Canonical Conformal Anomaly M. Chanowitz and J. Ellis, PRD 7, 2490 (1973)
Quark and Glue Components of Hadron Mass Energy momentum tensor P T P P P / M Trace anomaly T mm = -m(1+g m )yy + b(g) 2g G2 Separate into traceless part T and trace part ˆ T qg, 2 1 2 2 2 P T P x q, g ( )( P P P ) / M, x q( ) x g ( ) 1 4 T 3 / 4 M ; Tˆ 1/ 4M 44 44 4
Decomposition of hadron mass Xiangdong Ji, PRL 74, 1071 (1995); PRD 52, 271 (1995) Equation of motion å z (D c Therefore, + m)(x,z) 1 D c + m (z, y) = d x,y áh q ñ - áh E ñ = áh m ñ + O(a 2 ) ì0 for CI Þ í îconstant for DI D c = rd ov 1- D ov / 2 5
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Pseudoscalar meson masses from m π ~ 200 MeV to η c ~3 GeV 7
Decomposition of Pion Mass 50% Quark mass Quark energy Glue Trace anomaly 12% 8% 30% m p 2 m q Feynmann-Hellman theorem: m p m q = m q < p yy p > = 1/ 2m p 8
Vector meson masses from m ρ ~ 800 MeV to J/ψ ~3 GeV 9
ρ Mass 1% 41% Quark mass Quark energy Glue Trace anomaly 23% 35% 10
charmonium Quark mass Quark mass 77% Quark energy Glue Trace anomaly 73% Quark energy Glue Trace anomaly 6% 7% 7% 5% 10% 15% η c J/ψ 11
Hyperfine Splitting 12
2+1 flavor DWF configurations (RBC-UKQCD) La ~ 4.5 fm m π ~ 170 MeV 32^3 x 64, a =0.137 fm La ~ 2.8 fm m π ~ 330 MeV 24^3 x 64, a =0.115 fm La ~ 2.7 fm m π ~ 295 MeV 32^3 x 64, a =0.085 fm (O(a 2 ) extrapolation) La ~ 5.5 fm m π ~ 140 MeV 48^3 x 96, a =0.115 fm La ~ 5.5 fm m π ~ 140 MeV 64^3 x 128, a =0.085 fm
Nucleon Mass mp2 - mp L Empirical form: c 0 + c1mp + c2 mp + c3 mp + c4 mp,sea + c5 e + c6 a 2 L Þ mn = 962.4(7.2) MeV 2 3 2 Simple ChPT form: M 0 + Amp + Bmp + C(mp,vs + a D mix ) 2 Þ mn = 950.4(5.0) MeV 3 2 2 3/2 mp2 - mp L + c4 mp,sea + c5 e + c6 a 2 L 2 mn = 929(16)(7) MeV from BMW15 14
Hadron Structure with Quarks and Glue Quark and Glue Momentum and Angular Momentum in the Nucleon ( u D u d D d)( t) t0 D ( t)( u, d, s) t f t0 t f t0 t f 15
The πnσ term Y. Yang, et al. [ QCD], 1511.09089 u+d The plot shows the data points on three ensembles with different volume and lattice spacing. The data prefers the ruler approximation. The curve shows the quark mass dependence of the sigma term. The darker band for the statistic error and the lighter band for that combined with the systematic ones. σ N (m,a,l) = C 0 m + C 1 m 2 + C 2 a 2 The final prediction is, + C 3 ( m2 L m3 )e m L, 16
The strange σ termy. Yang, et al. [ QCD], 1511.09089 Due the finite lattice spacing, volume and partially quenching effects, biases exist between the data points and curve. It can be cured by the global fit. The final prediction is, σ sn (m,a,l) = C s 0 + C s 1m 2 + C s 2a 2 +C s 3e m L. 17
Yibo Yang (c QCD) et al., PRD94, 054503 (2016) arxiv:1511.15089 s pn = 45.9(7.2)(2.8) MeV s sn = 40.2(11.7)(3.5) MeV H m (u,d,s) / m N = 9(2)% 18
pa Proton mass decomposition YBY, K. Liu, Y. Chen et al, QCD Collaboration, in preparation Different flavors of the Quark Momentum fraction x d = 0.15(3) x u = 0.32(4) x s = 0.03(2) Lattice bare results running to 2GeV
Quark Momentum Fraction -- DI 20
Lattice input to global fitting of PDF <x> s /<x> u/d (DI) = 0.83(7)
pa Proton mass decomposition YBY, K. Liu, Y. Chen et al, QCD Collaboration, in preparation Gluon momentum fraction Lattice bare results running to 2GeV, x g = 0.54(11)
Momenta and Angular Momenta of Quarks and Glue Energy momentum tensor operators decomposed in quark and glue parts gauge invariantly --- Xiangdong Ji (1997) T q mn = i é 4 yg D y + (m «n) ù ë m n û J = q Nucleon form factors Momentum and Angular Momentum ò d é 3 x 1 2 ygg y + x yg (-id)y ù ê 5 4 ú ë û T g mn = F ml F ln - 1 4 d F 2 J mn g = ò d 3 x éë x (E B) ù û p, s T p ' s ' u( p, s)[ T ( q ) p T ( q ) p q / 2m 2 2 1 2 v -it ( q )( q q q ) / m T ( q ) m / 2] u( p ' s ') 2 2 2 3 4 ét Z q,g T 1 (0) q,g éë OPEù û x q/g (m,ms), Z 1 (0) + T 2 (0) ù q,g ê ú ë 2 û q,g J q/g (m,ms)
Renormalization and Quark-Glue Mixing Normalization from Momentum and Angular Momentum Sum Rules áxñ q R = Z q áxñ q L, áxñ g R = Z g áxñ g L, J q R = Z q J q L, J g R = Z g J g L, ìz q áxñ L q + Z g áxñ L g = 1, ï í ïz q J L q + Z g J L g = 1 î 2 Renormalization and Mixing q g ZqT1 (0) Z gt1 (0) 1, q q g g Zq( T1 T2 )(0) Z g( T1 T2 )(0) 1, q g ZqT2 (0) Z gt2 (0) 0 MS R x q ( ) Cqq( ) Cqg ( ) x q MS R x ( ) Cgq( ) Cgg ( ) x g g M. Deka et al., (c QCD) PRD 91, 014505 (2015) No anomalous gravito-magnetic moment S. Brodsky M. Glatzmaier, KFL arxiv:1403.7211 24
Proton mass decomposition Renormalization of the momentum fractions From the lattice bare quantities with the chiral fermion and HYP smeared Iwasaki gluon to that under the MS-bar scheme, at a scale μ=1/a, YBY, et.al. [ QCD], arxiv: 1612.02855 With the joint fit, x q = 50(7)% at MS-bar 2GeV. For the gluon operator renormalization at 1-loop level, the value and the uncertainty (from the estimate of the 4-gluon vertex tadpole contribution) are large and then indicate the convergence problem. The bare value of x g is 54(11)% and that deduced from the momentum fraction sum rule is x g = 50(7)%. pa
Proton mass decomposition pa Comparing the momentum fractions from the experiment YBY, K. Liu, Y. Chen et al, QCD Collaboration, in preparation S. Dulat et al, Phys. Rev. D 93 (2016), 033006
Nucleon Mass Components Quark energy Glue energy Trace anomaly Quark mass x 1/4 27
Summary and Challenges Decomposition of hadron masses into quark and glue components on the lattice is becoming feasible. Normalization, renomalization and mixing to obtain quark and glue momentum fraction in MS scheme at 2 GeV is begin carried out. System errors due to continuum limit at physical pion mass and large lattice volume with chiral fermions are being carried out. Implication on and connection to experiments need to be addressed. 28