Nucleon structure near the physical pion mass Jeremy Green Center for Theoretical Physics Massachusetts Institute of Technology January 4, 2013
Biographical information Undergraduate: 2003 2007, University of Toronto Hon. B. Sc., Math and Physics Summer research: 2004, 2005: knot theory 2006: particle physics (ATLAS) Graduate: 2007 2013 (expected), MIT Ph. D, Department of Physics Research (advisor: John Negele): Lattice QCD spatial diquark correlations nucleon structure Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 2 / 32
Outline Introduction Isovector nucleon observables State of past lattice calculations Lattice methodology Challenges BMW action; ensembles Systematic errors Results Vector form factors; radii and magnetic moment Momentum fraction Axial charge Tensor and scalar charges Conclusions Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 3 / 32
Nucleon observables Electromagnetic form factors: p(p, s ) qγ µ q p(p, s) = u(p, s ) where = P P, Q 2 = 2. Quark momentum fraction: ( ) γ µ F q 1 (Q2 ) + iσ µν ν 2m p F q 2 (Q2 ) u(p, s), p(p, s ) qγ {µ D ν } q p(p, s ) = x q u(p, s )γ {µ P ν } u(p, s) Axial, tensor, and scalar charges: p(p, s ) uγ µ γ 5 d n(p, s) = g A u p (P, s )γ µ γ 5 u n (P, s) p(p, s ) ud n(p, s) = g S u p (P, s )u n (P, s) p(p, s ) uσ µν d n(p, s) = g T u p (P, s )σ µν u n (P, s) Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 4 / 32
Vector form factors Dirac and Pauli form factors: p(p, s ) qγ µ q p(p, s) = u(p, s ) where = P P, Q 2 = 2. Isovector combination: ( ) γ µ F q 1 (Q2 ) + iσ µν ν 2m p F q 2 (Q2 ) u(p, s), F v 1,2 = F u 1,2 F d 1,2 = F p 1,2 F n 1,2, where F p,n 1,2 are form factors of the electromagnetic current in a proton and in a neutron. Dirac and Pauli radii defined via slope at Q 2 = 0: F 1,2 (Q 2 ) = F 1,2 (0)(1 1 6 r2 1,2 Q2 + O(Q 4 )); F 2 (0) = κ, the anomalous magnetic moment. Proton charge radius, (re 2)p = (r1 2)p + 3κp, has 7σ discrepancy between 2m 2 p measurements from e p scattering and from Lamb shift in muonic hydrogen. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 5 / 32
Quark momentum fraction Measure forward matrix element of quark energy-momentum operator: p(p, s ) qγ {µ D ν } q p(p, s) = x q u(p, s )γ {µ P ν } u(p, s). x q is the average momentum fraction carried by quarks q and q; focus on isovector combination x u d. Phemonenological values come from measurements of parton distribution functions in deep inelastic scattering experiments. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 6 / 32
Axial charge Benchmark observable: p(p, s ) uγ µ γ 5 d n(p, s) = g A u p (P, s )γ µ γ 5 u n (P, s). Naturally isovector observable; PDG: g A /g V = 1.2701(25) from β decay of polarized neutrons. Is also the difference between contributions from the spin of u and d quarks to the total proton spin. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 7 / 32
Scalar and tensor charges p(p, s ) ud n(p, s) = g S u p (P, s )u n (P, s) p(p, s ) uσ µν d n(p, s) = g T u p (P, s )σ µν u n (P, s) Not measured experimentally. Recent interest because they are needed to know leading contributions to neutron β decay from BSM physics. There are plans to measure the tensor charge at JLab using the upcoming 12 GeV upgrade. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 8 / 32
Past calculations 18 (r 1 v ) 2 [GeV -2 ] 16 14 12 10 8 DWF O(ε 3 ) SSE fit DWF O(p 4 ) CBChPT fit DWF Results, a = 0.084 fm PDG 2008 Belushkin et al 2007 6 4 0.1 0.2 0.3 0.4 0.5 m π [GeV] S. N. Syritsyn et al. (LHPC), Phys. Rev. D 81, 034507 (2010), arxiv:0907.4194 Past calculations required large m π m phys π extrapolations; chiral perturbation theory allows for rapid changes, e.g. log-divergence in Dirac radius as m π 0. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 9 / 32
Lattice calculations Two kinds of quark contractions for N (t s )O(t)N (0) : Computationally-expensive disconnected diagrams cancel for isovector quantities, so we are focusing on those for now. Use sequential propagator through the sink: propagators from fixed source and sink allow for measurement at all intermediate times. Cost grows linearly with number of source-sink separations T; past calculations typically used just one. Excited-state contamination exp( ET ). Signal-to-noise exp( (m N 3 2 m π )T ). Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 10 / 32
Challenges at low pion mass Solid curves show expected low-lying N ( 1 2 + ) excited states in a finite box. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 11 / 32
BMW action and ensembles N f = 2 + 1 tree-level O(a)-improved Wilson fermions coupled to double-hex-smeared gauge fields. Pion mass ranging from 149 MeV to 356 MeV. Nine coarse lattices with a = 0.116 fm; one fine lattice with a = 0.09 fm. No disconnected diagrams, so we focus on isovector observables. Three source-sink separations for controlling excited-state contributions: T {0.9, 1.2, 1.4} fm. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 12 / 32
Ensembles Areas of circles scale with number of measurements: largest is 10,000. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 13 / 32
Contributions to systematic error From best-controlled to worst-controlled: Quark masses: smallest pion mass is 149 MeV, so m π 135 MeV chiral extrapolation is under control. Excited states: use of multiple source-sink separations allows for clear identification of observables where excited states are a problem; the summation method seems to work well for reducing excited-state contamination. Finite volume: effects expected to be small with m π L 4, and two volumes at m π = 254 MeV allow for fully-controlled test of finite-size effects. Finite temperature: at small m π, L t is smaller than the typically used L t 2L x, but three different time extents at m π 250 MeV are useful for identifying possible problems. Discretization: one finer lattice at m π = 317 MeV for consistency check, but no a 0 extrapolation. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 14 / 32
Systematic error: excited states Usual approach for extracting matrix elements (forward case): C 2pt (t) = N (t)n (0) C 3pt (T, τ ) = N (T )O(τ )N (0) Ratio-plateau method: compute ratio R(T, τ ) = C 3pt (T, τ )/C 2pt (T ) = c 00 + c 10 e Eτ + c 01 e E (T τ ) + c 11 e ET +..., where c 00 is the desired ground-state matrix element. Then average a fixed number of points around τ = T/2, yielding asymptotic errors that fall off as e ET /2. Summation method: compute sums S(T ) = R(T, τ ) = b + c 00 T + dte ET +..., τ then find their slope, which gives c 00 with errors that fall off as Te ET. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 15 / 32
Results Collaborators: JRG, J. W. Negele, A. V. Pochinsky (MIT) S. N. Syritsyn (LBNL) M. Engelhardt (New Mexico State U.) S. Krieg (FZ Jülich & U. Wuppertal) Nucleon structure from Lattice QCD using a nearly physical pion mass arxiv:1209.1687 (r1 2)v, κ v, (r2 2)v, x u d, g A Nucleon scalar and tensor charges from Lattice QCD with light Wilson quarks arxiv:1206.4527, Phys. Rev. D 86, 114509 g S, g T Nucleon structure with pion mass down to 149 MeV arxiv:1211.0253, PoS(Lattice 2012)170 (r 2 1 )v, g A, g S, g T Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 16 / 32
F v 1 (Q2 ): m π = 254 MeV, 32 3 48, summation Dipole fit F 1 (Q 2 ) F 1 (0) (1+Q 2 /M 2 D )2 to range 0 Q 2 0.5 GeV 2. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 17 / 32
Isovector Dirac radius (r 2 1 )v Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 18 / 32
(r 2 1 )v : chiral extrapolation of summation points 0.8 0.7 0.6 HBChPT HBChPT+ 32c64 fine 32c96 coarse 24c24 coarse 24c48 coarse 32c48 coarse 48c48 coarse µp + PDG PDG 2012 (r 2 1) u d [fm 2 ] 0.5 0.4 0.3 0.2 0.1 0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m π [GeV] Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 19 / 32
κ v : chiral extrapolation of summation points 6 5 4 κ v = κ p κ n 3 2 1 HBChPT+ 32c64 fine 32c96 coarse 24c24 coarse 24c48 coarse 32c48 coarse 48c48 coarse PDG 2012 0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m π [GeV] Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 20 / 32
κ v (r 2 2 )v : chiral extrapolation of summation points κ v (r 2 2) v [fm 2 ] 6 5 4 3 2 HBChPT+ 32c64 fine 32c96 coarse 24c24 coarse 24c48 coarse 32c48 coarse 48c48 coarse PDG 2012 1 0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m π [GeV] Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 21 / 32
Sachs form factors G E (Q 2 ) = F 1 (Q 2 ) Q2 2m N F 2 (Q 2 ) G M (Q 2 ) = F 1 (Q 2 ) + F 2 (Q 2 ) Slopes at Q 2 = 0 give rms charge and magnetic radii. Compare: 1. Isovector m π = 149 MeV, summation data from lattice calculation. 2. Parameterization of experimental data: J. J. Kelly, Simple parametrization of nucleon form factors, Phys. Rev. C 70, 068202 (2004). 4 parameters for each of G Ep, G Mp, G Mn ; 2 parameters for G En : determined from fit to experiment. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 22 / 32
G v E (Q2 ): lowest pion mass Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 23 / 32
G v M (Q2 ): lowest pion mass Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 24 / 32
Isovector quark momentum fraction x u d Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 25 / 32
x u d : chiral extrapolation of summation points 0.25 0.20 x u d 0.15 0.10 0.05 BChPT 32c64 fine 32c96 coarse 24c24 coarse 24c48 coarse 32c48 coarse 48c48 coarse CTEQ6 0.00 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m π [GeV] Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 26 / 32
Axial charge g A Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 27 / 32
Axial charge g A m π 250 MeV, L x = 24; L t = 48 versus L t = 24. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 27 / 32
Axial charge g A 32 3 48; m π = 200 MeV versus m π = 250 MeV. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 27 / 32
Scalar charge g S Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 28 / 32
g S : chiral extrapolation g S 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 Mixed 20 3 64 Mixed 28 3 64 Fine DW 32 3 64 Coarse DW 24 3 64 Fine Wilson 32 3 64 Coarse Wilson 48 3 48 Coarse Wilson 32 3 48 Coarse Wilson 24 3 48 Coarse Wilson 32 3 96 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m 2 π [GeV 2 ] Using middle source-sink separation; also analyzed data from earlier studies. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 29 / 32
Tensor charge g T Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 30 / 32
g T : chiral extrapolation 1.20 1.15 Mixed 20 3 64 Mixed 28 3 64 Fine DW 32 3 64 Coarse DW 24 3 64 Fine Wilson 32 3 64 Coarse Wilson 48 3 48 Coarse Wilson 32 3 48 Coarse Wilson 24 3 48 Coarse Wilson 32 3 96 1.10 g T 1.05 1.00 0.95 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m 2 π [GeV 2 ] Using middle source-sink separation; also analyzed data from earlier studies. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 31 / 32
Conclusions Major new advance for calculations of nucleon structure observables: combination of both 1. m π close to physical 2. serious effort toward control over excited-state contamination Calculated isovector Dirac and Pauli radii, anomalous magnetic moment, and momentum fraction are consistent with experiment. Benchmark axial charge still inconsistent with experiment, but the influence of thermal states has been identified as a serious candidate for being the culprit. We are planning to do a controlled study of finite volume and time effects using 32 3 48, 24 3 48, 32 3 24, and 24 3 24 ensembles. Agreement with experiment will increase confidence in predictions such as g S = 1.08 ± 0.28 (stat) ± 0.16 (syst) g T = 1.038 ± 0.011 (stat) ± 0.012 (syst) Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 32 / 32
Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 33 / 32
g A vs. m π L x : summation data Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 34 / 32
g A vs. m π L t : summation data Thermal effects? Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 35 / 32
F v 2 (Q2 ): m π = 254 MeV, 32 3 48, summation Dipole fit F 2 (Q 2 ) F 2 (0) (1+Q 2 /M 2 D )2 to range 0 < Q 2 0.5 GeV 2. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 36 / 32
Isovector anomalous magnetic moment κ v Normalized relative to the physical magneton: κ norm = mphys N m lat F lat 2 (0). N Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 37 / 32
Isovector Pauli radius (r 2 2 )v Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 38 / 32
g S renormalization: using Z S plotted: Z S g bare S Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 39 / 32
g S renormalization: using m s m ud plotted: (m s m ud )g S m phys s m phys ud m2 K,phys m2 π,phys m 2 K m2 π Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 40 / 32
g S renormalization: using m ud + m res plotted: (m ud+m res )g S m phys ud m2 π,phys m 2 π Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, 2013 41 / 32