Meson wave functions from the lattice. Wolfram Schroers

Transcription

1 Meson wave functions from the lattice Wolfram Schroers

2 QCDSF/UKQCD Collaboration V.M. Braun, M. Göckeler, R. Horsley, H. Perlt, D. Pleiter, P.E.L. Rakow, G. Schierholz, A. Schiller, W. Schroers, H. Stüben, J.M. Zanotti

3 Outline Basics of lattice calculations Renormalization Chiral perturbation theory QCDSF results (hep-ph/ ) Summary & Outlook

4

5 Lattice QCD Goal: Qualitative & quantitative insight into hadronic functions from first principles Comparison experiment theory Credibility for predictions Advantage: Vary parameters, e.g., m q, N C, N F

6 Quark masses Heavy quark regime: confinement, flux tubes, adiabatic potential Light quark regime: chiral symmetry breaking, instantons, chiral perturbation theory

7 Getting observables In principle, three extrapolations Infinite-volume extrapolation Continuum extrapolation Chiral extra- (or inter-) polation

8 Fermion discretizations Unimproved Wilson fermions Practically very important question! Improved Wilson Improved staggered fermions Ginsparg-Wilson fermions Domain-wall Overlap

9 Simple, well understood Fermion discretizations Unimproved Wilson fermions Improved Wilson Improved staggered fermions Ginsparg-Wilson fermions Domain-wall Overlap

10 Simple, well understood Fermion discretizations Unimproved Wilson fermions Improved Wilson Improved staggered fermions Ginsparg-Wilson fermions Domain-wall Overlap Cheap

11 Simple, well understood Fermion discretizations Unimproved Wilson fermions Improved Wilson Improved staggered fermions Ginsparg-Wilson fermions Cheap Domain-wall Chiral, O(a 2 ), but expensive Overlap

12 The challenge Light valence fermions are possible today Light sea quarks remain major issue Possible solutions: Hybrid calculations Full GW: Either full DWF or Overlap Full Wilson-type fermions

13 Renormalization Matching between lattice regularization at scale a -1 and continuum MS-bar scheme Perturbative Non-perturbative: RI-MOM scheme Non-perturbative: Schrödinger functional

14 Perturbative Analytical calculation possible No lattice-systematic errors, no statistical uncertainties Lattice pert.-theory restricted to leading order Uncontrolled error from higher orders Only feasible if Z is O(1)

15 NP-renorm: RI-MOM Possible for any lattice action Requires gauge-fixing Lattice Gribov copies May fail to yield result, but can be diagnozed Applicable to arbitrary Z-factors Does not require resampling of gauge fields But needs chiral extrapolation

16 NP-renorm: SF Applicable to any lattice action No gauge fixing and no chiral extrapolation required (exactly chiral) Requires resampling of gauge fields at several volumes (operator-dependend) Matching to MS-bar must be done on your own Very complicated to be used

17 Chiral extrapolations Unresolved problem: <x> u-d Success story: g A

18 Open issue: <x> u-d Phys.Rev. D71: (2005) (M. Göckeler et al)

19 m K (u-d) RGI v 2b (r 0 m PS ) (a / r 0 ) 2 Nucl.Phys.Proc.Suppl. 140: (2005) (J. Zanotti et al)

20 0.4 "=8.0 "= m 2! (GeV)2 PoS LAT2005:363 (2005) (T. Streuer et al)

21 FIG. 9: The ratio of the flavor non-singlet momentum fraction to the helicity distribution (octagons). The experimental expectation is marked by the burst symbol. hep-lat/ (K. Orginos et al)

22 Momentum Fraction: x u d 0.25 <x> u-d m! 2 / f! 2 D.B. Renner, talk at Lattice 2006

23 LHPC: Hybrid calculations Hybrid approach: Asqtad & DWF Achievement: 5% acc. at m π =354 MeV Lattice sizes (2.5fm) 3 and (3.5fm) 3 Six constants: f π, m Δ -m N, g NΔ, g A, g ΔΔ, C First three: physical values, others are fit Total error from constr. parameters: <1%

24 Phys.Rev.Lett. 96: (2006) Combined results full QCD g A LHPC/MILC LHPC/SESAM RBCK QCDSF/UKQCD Experiment m! (GeV )

25 Phys.Rev.Lett. 96: (2006) Combined results full QCD g A See also: Phys.Lett.B639: LHPC/MILC LHPC/SESAM RBCK QCDSF/UKQCD Experiment m! (GeV )

26 Results for g A Experimental (neutron β decay) g A = (29) PRL 96: (2006) g A = 1.226(84) hep-lat/ g A = 1.31(9)(7)

27 Leading twist meson DAs q xp Π(p) q (1 x)p φ(x, µ 2 )

28 Scale dependence φ(x, µ 2 ) k 2 <µ2 d 2 k φ(x, k ) Process-independent, carries info on meson Scale dependence: ERBL-evolution Bethe-Salpeter equation Eigenfunctions are Gegenbauer polynomials C 3/2 n (2x 1)

29 φ(x, µ 2 ) = 6x(1 x) For π, ρ, η, η and Φ: n=0 a n (µ 2 )C 3/2 n (2x 1) G parity => a odd = 0 => mirror symmetry K, K * : a odd 0

30 φ(x, µ 2 ) = 6x(1 x) n=0 Goal: For π, ρ, η, η and Φ: Compute lowest moments a 1 (µ 2 ), a 2 (µ 2 ) G parity => a odd = for 0 kaon and a 2 (µ 2 ) => mirror symmetry for pion a n (µ 2 )C 3/2 n (2x 1) K, K * : a odd 0

31 Lattice calculations Ω O {µ0...µ n } Π( p) = i f Π p µ0... p µn ξ n Local operator via light-cone OPE Yields moments of DAs w.r.t. x x = 1 (1 + ξ) 2 Matrix elements from ratios of two-point functions

32 Choice of operators O a 41 = O {41}, p = (2π/L, 0, 0) O b 44 = (O 44 1/3(O ii )), p = 0 O a 412 = O {412}, p = (2π/L, 2π/L, 0) O b 411 = (O 411 1/2(O O 433 )), p = (2π/L, 0, 0)

33 Renormalization ξ 2 = ZS 412 Z O4 ξ 2 bare + ZS mix Z O4

34 Changing schemes

35 Working points Dynamical Clover, n f =2

36 Mass-degenerate quarks <" 2 > m! 2 / GeV 2 χpt: Chen et.al., PRL92:202001(2004)

37 0.3 <! 2 > a 2 / fm 2

38 Our result: ξ 2 π (µ 2 = 4 GeV 2 ) = 0.269(39) a π 2 (4 GeV 2 ) = 0.201(114) Compare to Del Debbio et.al., NPPS119:416(2003): ξ 2 π (µ 2 = 4 GeV 2 ) = 0.286(49) Larger than asymptotic value: ξ 2 π (µ 2 ) = 0.2

39 F π0 γ*γ at leading twist Diehl et.al., EPJC22:439(2001)

40 F π0 γ*γ various models Figure from Bakulev et.al., PLB578:91(2004)

41 F π0 γ*γ various models 0.1 a a 2 Figure from Bakulev et.al., PLB578:91(2004)

42 F π0 γ*γ various models 0.1 a We CAN distinguish different models! a 2 Figure from Bakulev et.al., PLB578:91(2004)

43 Mass non-deg. quarks <! 2 > Only β=5.29, est. syst. 2 error 2 m from cont. extrap. K/ GeV

44 Averaging over 4 values of κ sea : ξ 2 K (µ 2 = 4 GeV 2 ) = 0.260(6) ξ 2 K / ξ 2 π 1 Chernyak&Zhitnisky: Ball et.al. Khodjamirian et.al.: 0.59(4) 1

45 <"> m K 2 - m! 2 / GeV 2

46 Averaging over 4 values of κ sea : ξ K (µ 2 = 4 GeV 2 ) = (5) a K 1 (4 GeV 2 ) = (9)(29) Recent controversy in literature, see Ball et.al., hep-lat/ : a K 1 (4 GeV 2 ) = 0.05(25) Compatible with hep-lat/ (Next talk of A. Jüttner) a K 1 (4 GeV 2 ) = 0.055(5)

47 Summary a π 2 (4 GeV 2 ) = 0.201(114) : larger than asymptotic values, can distinguish models a K 2 (4 GeV 2 ) = 0.175(18)(47) : about the same as, also distinguishes models a π 2 a K 1 (4 GeV 2 ) = (9)(29) : compatible with sum-rule estimate

48 Outlook Lower pion masses (300 MeV and below) Improved chiral perturbation theory (J.W. Chen, private communication) Higher twist contributions Other mesons Nucleon - N. Regensburg U.