Nucleon structure near the physical pion mass

Transcription

1 Nucleon structure near the physical pion mass Jeremy Green Center for Theoretical Physics Massachusetts Institute of Technology January 4, 2013

2 Biographical information Undergraduate: , University of Toronto Hon. B. Sc., Math and Physics Summer research: 2004, 2005: knot theory 2006: particle physics (ATLAS) Graduate: (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, / 32

3 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, / 32

4 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, / 32

5 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, / 32

6 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, / 32

7 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 = (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, / 32

8 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, / 32

9 Past calculations 18 (r 1 v ) 2 [GeV -2 ] DWF O(ε 3 ) SSE fit DWF O(p 4 ) CBChPT fit DWF Results, a = fm PDG 2008 Belushkin et al m π [GeV] S. N. Syritsyn et al. (LHPC), Phys. Rev. D 81, (2010), arxiv: 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, / 32

10 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, / 32

11 Challenges at low pion mass Solid curves show expected low-lying N ( ) excited states in a finite box. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

12 BMW action and ensembles N f = 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 = 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, / 32

13 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, / 32

14 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, / 32

15 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, / 32

16 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: (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: , Phys. Rev. D 86, g S, g T Nucleon structure with pion mass down to 149 MeV arxiv: , 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, / 32

17 F v 1 (Q2 ): m π = 254 MeV, , summation Dipole fit F 1 (Q 2 ) F 1 (0) (1+Q 2 /M 2 D )2 to range 0 Q GeV 2. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

18 Isovector Dirac radius (r 2 1 )v Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

19 (r 2 1 )v : chiral extrapolation of summation points 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 ] m π [GeV] Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

20 κ v : chiral extrapolation of summation points κ v = κ p κ n HBChPT+ 32c64 fine 32c96 coarse 24c24 coarse 24c48 coarse 32c48 coarse 48c48 coarse PDG m π [GeV] Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

21 κ v (r 2 2 )v : chiral extrapolation of summation points κ v (r 2 2) v [fm 2 ] HBChPT+ 32c64 fine 32c96 coarse 24c24 coarse 24c48 coarse 32c48 coarse 48c48 coarse PDG m π [GeV] Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

22 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, (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, / 32

23 G v E (Q2 ): lowest pion mass Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

24 G v M (Q2 ): lowest pion mass Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

25 Isovector quark momentum fraction x u d Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

26 x u d : chiral extrapolation of summation points x u d BChPT 32c64 fine 32c96 coarse 24c24 coarse 24c48 coarse 32c48 coarse 48c48 coarse CTEQ m π [GeV] Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

27 Axial charge g A Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

28 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, / 32

29 Axial charge g A ; m π = 200 MeV versus m π = 250 MeV. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

30 Scalar charge g S Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

31 g S : chiral extrapolation g S Mixed Mixed Fine DW Coarse DW Fine Wilson Coarse Wilson Coarse Wilson Coarse Wilson Coarse Wilson 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, / 32

32 Tensor charge g T Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

33 g T : chiral extrapolation Mixed Mixed Fine DW Coarse DW Fine Wilson Coarse Wilson Coarse Wilson Coarse Wilson Coarse Wilson g T 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, / 32

34 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 , , , and ensembles. Agreement with experiment will increase confidence in predictions such as g S = 1.08 ± 0.28 (stat) ± 0.16 (syst) g T = ± (stat) ± (syst) Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

35 Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

36 g A vs. m π L x : summation data Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

37 g A vs. m π L t : summation data Thermal effects? Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

38 F v 2 (Q2 ): m π = 254 MeV, , summation Dipole fit F 2 (Q 2 ) F 2 (0) (1+Q 2 /M 2 D )2 to range 0 < Q GeV 2. Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

39 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, / 32

40 Isovector Pauli radius (r 2 2 )v Jeremy Green (MIT) Nucleon structure near the physical m π JLab, January 4, / 32

41 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, / 32

42 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, / 32

43 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, / 32