Baryon semi-leptonic decay from lattice QCD with domain wall fermions

Size: px

Start display at page:

Download "Baryon semi-leptonic decay from lattice QCD with domain wall fermions"

Octavia Gaines
5 years ago
Views:

1 4th ILFTN Workshop at Hayama, Mar. 8-11, 2006 Baryon semi-leptonic decay from lattice QCD with domain wall fermions Shoichi Sasaki (RBRC/U. of Tokyo) in collabration with Takeshi Yamazaki (RBRC)

2 Baryon semi-leptonic decay The octet baryons (p,n,λ,σ,ξ) admit various β-type decays. neutron beta decay B B + l ± + ν l ( ν l ) n p + e + ν e CVC (conserved vector current) hypothesis: p V + µ A + µ n =2 p V 3 µ A 3 µ p Weak matrix element Iso-vector nucleon matrix element way to access the nucleon structure

3 Baryon semi-leptonic decay The octet baryons (p,n,λ,σ,ξ) admit various β-type decays. hyperon beta decay B B + l ± + ν l ( ν l ) Alternative way to determine V us other than K l3 decays Δs=1 decay W ν e e T. Kaneko s talk Σ (sdd) n(udd) the weak mixing element (CKM) Vus

4 Baryon semi-leptonic decay The octet baryons (p,n,λ,σ,ξ) admit various β-type decays. hyperon beta decay B B + l ± + ν l ( ν l ) Alternative way to determine V us other than K l3 decays Vital input to analysis of strange quark spin fraction Σ(= u + d + s) Expt. = ± (g A /g V ) np = u d (g A /g V ) Λp = (2 u d s)/3 (g A /g V ) ΞΣ = ( u + d 2 s)/3 (g A /g V ) Σn = d s Assumption : SU(3) symmetry

5 Baryon semi-leptonic decay The octet baryons (p,n,λ,σ,ξ) admit various β-type decays. hyperon beta decay B B + l ± + ν l ( ν l ) Alternative way to determine V us other than K l3 decays Vital input to analysis of strange quark spin fraction s = ± 46 The hidden uncertainty of Δs coming from unknown SU(3) breaking in hyperon beta decays.

6 Baryon semi-leptonic decay The octet baryons (p,n,λ,σ,ξ) admit various β-type decays. hyperon beta decay B B + l ± + ν l ( ν l ) Alternative way to determine V us other than K l3 decays Vital input to analysis of strange quark spin fraction SU(3) breaking in hyperon beta decays

7 Baryon semi-leptonic decay B B + l ± + ν l ( ν l ) These decays are described by 6 form factors B V α A α B =ū B (p)[f 1 (q 2 )γ α + f 2(q 2 ) 2M B +g 1 (q 2 )γ α γ 5 + g 2(q 2 ) 2M B σ αβ q β + f 3(q 2 ) 2M B q α σ αβ γ 5 q β + g 3(q 2 ) 2M B q α γ 5 ]u B (p ) 1st class: f 1 (q 2 ), f 2 (1 2 ), g 1 (q 2 ), g 3 (q 2 ) 2nd class: f 3 (q 2 ), g 2 (q 2 )

8 Baryon semi-leptonic decay B B + l ± + ν l ( ν l ) These decays are described by 6 form factors B V α A α B =ū B (p)[f 1 (q 2 )γ α + f 2(q 2 ) 2M B +g 1 (q 2 )γ α γ 5 + g 2(q 2 ) 2M B σ αβ q β + f 3(q 2 ) 2M B q α σ αβ γ 5 q β + g 3(q 2 ) 2M B q α γ 5 ]u B (p ) 1st class: f 1 (q 2 ), f 2 (1 2 ), g 1 (q 2 ), g 3 (q 2 ) Sach s form factors G E (q 2 )=f 1 (q 2 )+ q2 f 2 (q 2 ) 2M B G M (q 2 )=f 1 (q 2 )+f 2 (q 2 )

9 Baryon semi-leptonic decay B B + l ± + ν l ( ν l ) These decays are described by 6 form factors B V α A α B =ū B (p)[f 1 (q 2 )γ α + f 2(q 2 ) 2M B +g 1 (q 2 )γ α γ 5 + g 2(q 2 ) 2M B σ αβ q β + f 3(q 2 ) 2M B q α σ αβ γ 5 q β + g 3(q 2 ) 2M B q α γ 5 ]u B (p ) 1st class: f 1 (q 2 ), f 2 (1 2 ), g 1 (q 2 ), g 3 (q 2 ) g V = lim q 2 0 f 1(q 2 ) g A = lim q 2 0 g 1(q 2 ) forward limit

10 Baryon semi-leptonic decay B B + l ± + ν l ( ν l ) These decays are described by 6 form factors B V α A α B =ū B (p)[f 1 (q 2 )γ α + f 2(q 2 ) 2M B +g 1 (q 2 )γ α γ 5 + g 2(q 2 ) 2M B σ αβ q β + f 3(q 2 ) 2M B q α σ αβ γ 5 q β + g 3(q 2 ) 2M B q α γ 5 ]u B (p ) 2nd class: f 3 (q 2 ), g 2 (q 2 ) SU(3) limit (neutron beta decay) f 3 (q 2 )=0, g 2 (q 2 )=0

11 Contents Nucleon axial charge Finite size effect O(a 2 ) correction Sasaki, Orginos, Ohta, Blum, Phys. Rev. D68 (03) Nucleon form factors Finite size effect Nf=2 & Nf=0 DWF results (preliminary) Hyperon beta decay SU(3) breaking effect An exploratory study in quench (Nf=0) DWF calculation

12 Nucleon axial charge

13 Nucleon axial charge ga Well measured quantity in experiment Iso-symmetry gives rise to the relation p V + µ A + µ n =2 p V 3 µ A 3 µ p from neutron beta decay; ga/gv= (35). The simplest nucleon matrix elements lowest moment (no covariant derivative) zero momentum transfer no disconnected diagram a benchmark calculation = a gold plated test

14 But expt (30) n p + e + ν e 1.2 g A = < 1 >!u-!d % discrepancy ( ga g A = lim q 2 0 g 1(q 2 ) g V ) expt. =1.2670(30) Quenched QCD (MIT) Full QCD (SESAM) Full QCD (SCRI-LHPC) long-standing problem M PS 2 [GeV 2 ]

15 Lattice calculation of ga (before 2002) type group fermion lattice β volume configs m π L g A quench KEK 1) Wilson 16 3 x (2.2 fm) (25) Kentuchy 2) Wilson 16 3 x (1.5 fm) (10) DESY 3) Wilson 16 3 x (1.5 fm) (90) LHPC-SESAM 7) Wilson 16 3 x (1.5 fm) (98) QCDSF 4) Wilson 16 3 x x (1.5 fm) 3 (1.6 fm) 3 O(500) O(300) 1.14(3) 32 3 x (1.6 fm) 3 O(100) QCDSF-UKQCD 5) Clover 16 3 x x (1.5 fm) 3 (1.6 fm) 3 O(500) O(300) 1.135(34) 32 3 x (1.6 fm) 3 O(100) full LHPC-SESAM 7) Wilson 16 3 x (1.7 fm) (106) SESAM 6) Wilson 16 3 x (1.5 fm) (20) 1. M. Fukugita et al., Phys. Rev. Lett. 75 (1995) K.F. Liu et al., Phys. Rev. D49 (1994) M. Göckeler et al., Phys. Rev. D53 (1996) S. Capitani et al., Nucl. Phys. B (Proc. Suppl.) 79 (1999) R. Horsley et al., Nucl. Phys. B (Proc. Suppl.) 94 (2001) S. Güsken et al., Phys. Rev. D59 (1999) D. Dolgov et al., Phys. ReV.D66 (2002)

16 Low value of ga in lattice QCD Possible systematic errors Quenching! g Full A < g Quench! ~ A at a~0.1 fm ~ 5-10 % Finite lattice spacing! (g A ) at a 0 > (g A ) at a~0.1 fm! ~ 5 % Determination of Z A! Z Non-pert A < Z Pert A (Clover )! ~ 10 % Finite volume No estimation?

17 Lattice calculation of ga (before 2002) type group fermion lattice β volume configs m π L g A quench KEK 1) Wilson 16 3 x (2.2 fm) (25) Kentuchy 2) Wilson 16 3 x (1.5 fm) (10) DESY 3) Wilson 16 3 x (1.5 fm) (90) LHPC-SESAM 7) Wilson 16 3 x (1.5 fm) (98) QCDSF 4) Wilson 16 3 x x (1.5 fm) 3 (1.6 fm) 3 O(500) O(300) 1.14(3) 32 3 x (1.6 fm) 3 O(100) QCDSF-UKQCD 5) Clover 16 3 x x (1.5 fm) 3 (1.6 fm) 3 O(500) O(300) 1.135(34) 32 3 x (1.6 fm) 3 O(100) full LHPC-SESAM 7) Wilson 16 3 x (1.7 fm) (106) SESAM 6) Wilson 16 3 x (1.5 fm) (20) 1. M. Fukugita et al., Phys. Rev. Lett. 75 (1995) K.F. Liu et al., Phys. Rev. D49 (1994) M. Göckeler et al., Phys. Rev. D53 (1996) S. Capitani et al., Nucl. Phys. B (Proc. Suppl.) 79 (1999) R. Horsley et al., Nucl. Phys. B (Proc. Suppl.) 94 (2001) S. Güsken et al., Phys. Rev. D59 (1999) D. Dolgov et al., Phys. ReV.D66 (2002)

18 DWF calculation of ga Big advantage in dealing with the axial symmetry - g A are supposed to respect the axial WT identity - Empirically known as Goldberger-Treiman relation: M N g A = f π g πn Excellent chiral properties of DWF lighter pion mass Especially, a significant relation ZV=ZA for local lattice currents - up to O(a 2 ) T. Blum et al, PRD66 (02) A ratio ga latt / gv latt directly yields the renormalized value of ga Just require to calculate the ratio of three-point functions!

19 CVC hypothesis The vector weak current is the iso-spin rotation of the electromagnetic current, j em µ = 2 3 V u µ 1 3 V d µ + [I +,j em µ ]= dγ µ u p dγ µ u n = p [I +,j em µ ] n = p j em µ p n j em µ n g V = lim q 2 0 p dγ µ u n = lim q 2 0 p jem µ p = 1 way to calculate the renormalization factor Z V = gv ren /gv lat =1/g lat V

20 CVC hypothesis (cont d) Further consideration of iso-spin symmetry provides lim q 2 0 p jem µ p = lim q 2 0 p V d µ p = lim q 2 0 p V u µ V d µ p =1 For the local lattice currents in the chiral limit, Three different determination of ZV can expose an O(a 2 ) lattice artifact. V f µ = Z V V f µ + O(a 2 ) in the continuum

21 Check of the relation ZA=ZV Nf0: DWF-DBW2 at β=0.87 (a -1 =1.3GeV) coarse lattice x 32 x 16 with M5=1.8 Sasaki-Orginos-Ohta-Blum, PRD68 (03) Z V = 1 g latt V N(t ) N(0) N(t )V 4 (t) N(0) Z V am f

22 Check of the relation ZA=ZV Nf0: DWF-DBW2 at β=0.87 (a -1 =1.3GeV) coarse lattice x 32 x 16 with M5=1.8 Sasaki-Orginos-Ohta-Blum, PRD68 (03) Z V = 1 g latt V N(t ) N(0) N(t )V 4 (t) N(0) Z V Z A Z A Acon 0 (t)π(0) A 0 (t)π(0) Y. Aoki et al., PRD69 (04) am f

23 Check of the relation ZA=ZV Nf0: DWF-DBW2 at β=0.87 (a -1 =1.3GeV) coarse lattice x 32 x 16 with M5=1.8 Sasaki-Orginos-Ohta-Blum, PRD68 (03) Z V = 1 g latt V N(t ) N(0) N(t )V 4 (t) N(0) Z V (u-d) Z V (j) Z V (d) Z A 0.12 Violation of lim q 2 0 p jem µ p = lim q 2 0 p V d µ p = lim q 2 0 p V u µ V d µ p implies an O(a 2 ) lattice artifact am f

24 Check of the relation ZA=ZV Nf0: DWF-DBW2 at β=0.87 (a -1 =1.3GeV) coarse lattice x 32 x 16 with M5=1.8 Sasaki-Orginos-Ohta-Blum, PRD68 (03) a -1 =1.3GeV Z V (u-d) Z V (j) Z V (d) a -1 =2.0GeV Z V (u-d) Z V (j) Z V (d) Z A am f am f

25 Check of the relation ZA=ZV Nf0: DWF-DBW2 at β=0.87 (a -1 =1.3GeV) coarse lattice x 32 x 16 with M5=1.8 Sasaki-Orginos-Ohta-Blum, PRD68 (03) In the chiral limit Z V =0.796(3) 0.78 Z A =0.7776(5) am f 8 Z V Z A % systematic error stemming from determination of Z-factor mainly due to O(a 2 ) corrections

26 Quench DWF calculation of ga 1.4 ga (exp.) = (3) Sasaki-Orginos-Ohta-Blum, PRD68 (03) g A 1.2 g ren A = glat A gv lat = N(t )A 3 (t)n(0) N(t )V 4 (t)n(0) the lightest pion mass, Mπ ~ 0.39 GeV relatively large volume, V ~ (2.4 fm) 3 large statistics, 416 configs mild quark mass dependence 0.8 Linear extrapolation yields L=2.4 fm (DWF/DBW2) ga = (27) M PS 2 [GeV 2 ]

27 Quench DWF calculation of ga 1.4 ga (exp.) = (3) Sasaki-Orginos-Ohta-Blum, PRD68 (03) g A 1.2 the lightest pion mass, Mπ ~ 0.39 GeV relatively large volume, V ~ (2.4 fm) 3 large statistics, 416 configs mild quark mass dependence 0.8 L=2.4 fm (DWF/DBW2) L=1.2 fm (DWF/DBW2) L=1.5 fm (DWF/Wilson) clear finite volume dependence M PS 2 [GeV 2 ] a 20 % increase from 1.2 fm to 2.4 fm resolve the long-standing problem!

28 Nf2: DWF-DBW2 at beta=0.80 (a-1 =1.7GeV) 163 x32x12 (L=1.9 fm): O(5000) MC trajectories m sea=2, 3, 4 (Mπ=0.49, 0.61, 0.70 GeV) m res=0137(5) f π= 134.0(42), fk=157.4(38) MeV B K MS (2 GeV)=0.495(18) RBC collaboration, Phys. Rev. D72, (05)

29 Nf2: DWF-DBW2 at beta=0.80 (a-1 =1.7GeV) 163 x32x12 (L=1.9 fm): 220 statistics m sea=3, 4 m sea=2 (underway) Nucleon structure function Blum, Lin, Ohta, Orginos, Sasaki

30 Nf=2 DWF calculation of ga g A L=2.4 fm (DWF/DBW2/Nf=0) L=1.9 fm (DWF/DBW2/Nf=2) M PS 2 [GeV 2 ]

31 Nucleon form factors

32 Neutron beta decay n p + e + ν e B V α A α B =ū B (p)[f 1 (q 2 )γ α + f 2(q 2 ) 2M B σ αβ q β +g 1 (q 2 )γ α γ 5 + g 3(q 2 ) 2M B q α γ 5 ]u B (p ) Under CVC hypothesis (rigid isospin symmetry) f 3 (q 2 )=0 g 2 (q 2 )=0 p V + µ A + µ n =2 p V 3 µ A 3 µ p weak matrix element isovector nucleon matrix element

33 Nf2: DWF-DBW2 at beta=0.80 (a-1 =1.7GeV) 163 x32x12 (L=1.9 fm): 220 statistics m sea=3, 4 (Mπ=0.61, 0.70 GeV)

34 Vector (Dirac form factor; f1=gv) 0.8 m sea =4 (DWF/DBW2/Nf=2) g V (Q 2 ) / g V (0) Dipole (exp.) with M V =0.86 GeV g V (Q 2 g V (0) )= (1 Q 2 /MV 2 )2 M V = 0(6) GeV Q 2 (GeV 2 )

35 Vector (Dirac form factor; f1=gv) 0.8 m sea =3 (DWF/DBW2/Nf=2) m sea =4 (DWF/DBW2/Nf=2) g V (Q 2 ) / g V (0) Dipole (exp.) with M V =0.86 GeV g V (Q 2 g V (0) )= (1 Q 2 /MV 2 )2 M V = 0(6) GeV Q 2 (GeV 2 )

36 Axial-vector (g1=ga) m sea =4 (DWF/DBW2/Nf=2) 0.8 g A (Q 2 ) / g A (0) M A = 1.85(1) GeV ra~0.38fm 0.2 g A (Q 2 g A (0) )= (1 Q 2 /MA 2 )2 Dipole (exp.) with M A =8(8) GeV large discrepancy ra~0.63fm r 2 A = 12 M 2 A Q 2 (GeV 2 )

37 Axial-vector (g1=ga) 0.8 m sea =3 (DWF/DBW2/Nf=2) m sea =4 (DWF/DBW2/Nf=2) g A (Q 2 ) / g A (0) M A = 1.85(1) GeV 0.2 Dipole (exp.) with M A =8(8) GeV r 2 A = 12 M 2 A Q 2 (GeV 2 ) No sea-quark mass dependence!

38 Nf0: DWF-DBW2 at beta=0.87 (a-1 =1.3GeV) 163 x32x16 (L=2.4 fm): 119 statistics m f=4, 5, 6, 8 (Mπ= GeV)

39 Axial-vector (g1=ga) 0.8 L=1.9 fm (DWF/DBW2/Nf=2) L=2.4 fm (DWF/DBW2/Nf=0) g A (Q 2 ) / g A (0) M A = 1.85(1) GeV M A = 1.39(1) GeV ra~0.38fm ra~0.50fm 0.2 Dipole (exp.) with M A =8(8) GeV ra~0.63fm r 2 A = 12 M 2 A Q 2 (GeV 2 ) Finite volume effect?

40 No quark mass dependence! 0.8 m f =4 (DWF/DBW2/Nf=0) m f =5 (DWF/DBW2/Nf=0) m f =6 (DWF/DBW2/Nf=0) g A (Q 2 ) / g A (0) M A = 1.39(1) GeV 0.2 Dipole (exp.) with M A =8(8) GeV Q 2 (GeV 2 )

41 Nf0: DWF-DBW2 at beta=0.87 (a-1 =1.3GeV) 123 x32x16 (L=1.8 fm): 400 statistics

42 1.6 mπ ~ 0.76 GeV Finite volume effect on ga an gv g lat A Nf0: DWF-DBW2 at beta=0.87 (a -1 =1.3GeV) Large finite volume effect on nucleon axial charge less volume dependence for nucleon vector charge g lat V 2.4 fm g lat A g lat V = N(t )A 3 (t)n(0) N(t )N(0) = N(t )V 4 (t)n(0) N(t )N(0) L

43 Axial-vector Vector Nf0/L=2.4 fm Nf0/L=1.8 fm Dipole (exp.) with M A =0.86 GeV g A (Q 2 )/g A (0) g V (Q 2 )/g V (0) Nf0/L=2.4 fm Nf0/L=1.8 fm Dipole (exp.) with M A =8 GeV Q 2 Q 2 Finite volume effect is observed!

44 Axial-vector Vector Nf0/L=2.4 fm Nf0/L=1.8 fm Nf2/L=1.9 fm Dipole (exp.) with M A =0.86 GeV g A (Q 2 )/g A (0) g V (Q 2 )/g V (0) Nf0/L=2.4 fm Nf0/L=1.8 fm Nf2/L=1.9 fm Dipole (exp.) with M A =8 GeV Q 2 Q 2

45 Hyperon beta decay

46 Baryon semi-leptonic decay According to DWF study of neutron beta decay Better control of determination of Z factor, thanks to excellent chiral properties of DWF ga/gv in neutron beta decay is well reproduced within about 5 % accuracy even in quenched calculation The lightest baryon (nucleon) seems to be fitted in 2.4 fm lattice size (quench) In the next stage, we explore the hyperon beta decay in lattice QCD

47 Hyperon Beta Decay (Expt.) B Blν f SU(3) 1 g 1 /f 1 (Exp.) (g 1 /f 1 ) SU(3) n p ± 030 F + D Λ p ± 15 F + 1D 2 3 Ξ Λ ± 5 F 1D 2 3 Σ n ± 17 F D Ξ 0 Σ ± 0.21 F + D Ξ Ξ 0-1 N/A F D g V = lim q 2 0 f 1(q 2 ) g A = lim q 2 0 g 1(q 2 )

48 Hyperon Beta Decay (Expt.) B Blν f SU(3) 1 g 1 /f 1 (Exp.) (g 1 /f 1 ) SU(3) n p ± 030 F + D Λ p ± 15 F + 1D 2 3 Ξ Λ ± 5 F 1D 2 3 Σ n ± 17 F D Ξ 0 Σ ± 0.21 F + D Ξ Ξ 0-1 N/A F D g V = lim q 2 0 f 1(q 2 ) g A = lim q 2 0 g 1(q 2 ) Ξ Σ is the direct analogue of n p under d s Ξ Ξ is the direct analogue of Σ n under d s

49 Hyperon Beta Decay (Ξ ) Ξ Σ is the direct analogue of n p under d s W ν e e Ξ 0 (ssu) V us Σ + (suu) highly sensitive to SU(3) breaking Center of mass correction approach (Ratcliffe) " (ga/gv)np > (ga/gv)ξσ" 8-10% 1/Nc expansion approach (Flores-Mendieta-Jenkins-Manohar) " (ga/gv)np > (ga/gv)ξσ" 20-30%

50 Summary on Ξ (exp.) First and Single experiment at - g1 / f1 = 1.17 ± 0.28(stat) ± 5(syst) - g2 / f1 = 1.7 ± 2.0(stat) ± 0.5 (syst) no evidence for a nonzero value of the g2 form factor (2nd-class) Assumming g2 / f1 = 0 n p: g1 / f1= (35) - g1 / f1 = 1.32 ± 0.21(stat) ± 5(syst) no indication of flavor SU(3) breaking effects

51 2nd-class current B V α A α B =ū B (p)[f 1 (q 2 )γ α + f 2(q 2 ) 2M B +g 1 (q 2 )γ α γ 5 + g 2(q 2 ) 2M B σ αβ q β + f 3(q 2 ) q α 2M B σ αβ γ 5 q β + g 3(q 2 ) q α γ 5 ]u B (p ) 2M B Time reversal invariance requires all 6 form factors to be real With respect to transformation under (extended) G-parity, 1st class Gf 1,2 (Q 2 )G 1 =+f 1,2 (Q 2 ) Gg 1,3 (Q 2 )G 1 = g 1,3 (Q 2 ) 2nd class Gf 3 (Q 2 )G 1 = f 3 (Q 2 ) (extended) G-parity invariance requires f 3 (Q 2 )=0 g 2 (Q 2 )=0 Gg 2 (Q 2 )G 1 =+g 2 (Q 2 ) G = Ce iπt 2,5,7

52 2nd-class current B V α A α B =ū B (p)[f 1 (q 2 )γ α + f 2(q 2 ) 2M B +g 1 (q 2 )γ α γ 5 + g 2(q 2 ) 2M B σ αβ q β + f 3(q 2 ) q α 2M B σ αβ γ 5 q β + g 3(q 2 ) q α γ 5 ]u B (p ) 2M B Time reversal invariance requires all 6 form factors to be real With respect to transformation under (extended) G-parity, 1st class Gf 1,2 (Q 2 )G 1 =+f 1,2 (Q 2 ) Gg 1,3 (Q 2 )G 1 = g 1,3 (Q 2 ) 2nd class Gf 3 (Q 2 )G 1 = f 3 (Q 2 ) (extended) G-parity invariance requires SU(3) breaking f 3 (Q 2 ) 0 g 2 (Q 2 ) 0 Gg 2 (Q 2 )G 1 =+g 2 (Q 2 ) G = Ce iπt 2,5,7

53 An exploratory study Extract the 2nd-class form factors (g2 and f3) SU(3) breaking = existence of non-zero g2 and f3 Quantify the SU(3) breaking effect on g1 / f1 double ratio: Σ(t )A 3 (t)ξ(0) Σ(t )V 4 (t)ξ(0) N(t )V 4 (t)n(0) N(t )A 3 (t)n(0) = g 1 (q 2 max) f 1 (q 2 max) δf 3 (q 2 max) ( ) f1 (0) g 1 (0) SU(3) ( )( ) g1 (0) f1 (0) f 1 (0) g 1 (0) SU(3) δ = M Ξ M Σ M Ξ + M Σ q 2 max = (M Ξ M Σ ) 2

54 Nf0: DWF-DBW2 at beta=0.87 (a-1 =1.3GeV) 163 x32x16 (L=2.4 fm): 119 statistics m l=4, 5, 6 (Mπ=0.53, 0.60, 0.65 GeV) fixed strange quark mass at m s=8

55 2nd-class form factors ml=5, ms=8 1.5 preliminary preliminary 0.5 G 2 (Q 2 ) / F 1 (Q 2 ) 0.5 F 3 (Q 2 ) / F 1 (Q 2 ) δ = M Ξ M Σ M Ξ + M Σ =14(1) Q 2 (GeV 2 ) Q 2 (GeV 2 ) g 2 (0.25 GeV 2 ) f 1 (0.25 GeV 2 ) =0.24(18) f 3 (0.25 GeV 2 ) f 1 (0.25 GeV 2 ) =0.25(8)

56 2nd-class form factors ml=5, ms=8 1.5 Hyperon beta decay Neutron beta decay 0.5 Hyperon beta decay Neutron beta decay G 2 (Q 2 ) / F 1 (Q 2 ) 0.5 F 3 (Q 2 ) / F 1 (Q 2 ) preliminary preliminary Q 2 (GeV 2 ) Q 2 (GeV 2 ) g 2 (0.25 GeV 2 ) f 1 (0.25 GeV 2 ) =0.24(18)(13) f 3 (0.25 GeV 2 ) f 1 (0.25 GeV 2 ) =0.25(8)(13)

57 2nd-class form factors Preliminary result (quenched lattice QCD) - at δ=14(1) (ml=5, ms=8) g 2 (0.25 GeV 2 ) f 1 (0.25 GeV 2 ) =0.24 ± 0.18 ± 0.13 KTeV@FNAL - δ=5 (phys.) g 2 (0) f 1 (0) =1.7 ± 2.0 ± 0.5

58 Other form factors ml=5, ms= F 1 (Q 2 ) / F 1 (0) G 1 (Q 2 ) / G 1 (0) Q 2 (GeV 2 ) Q 2 (GeV 2 ) F 2 (Q 2 ) / F 1 (0) G 3 (Q 2 ) / G 1 (0) Q 2 (GeV 2 ) Q 2 (GeV 2 )

59 Other form factors ml=5, ms=8 0.8 Hyperon beta decay Neutron beta decay 0.8 F 1 (Q 2 ) / F 1 (0) G 1 (Q 2 ) / G 1 (0) Q 2 (GeV 2 ) Q 2 (GeV 2 ) F 2 (Q 2 ) / F 1 (0) G 3 (Q 2 ) / G 1 (0) Q 2 (GeV 2 ) Q 2 (GeV 2 )

60 the SU(3) breaking effect on g1 / f1 Consider the following double ratio at the rest frame (p,k=0) Σ(t )A 3 (t)ξ(0) Σ(t )V 4 (t)ξ(0) N(t )V 4 (t)n(0) N(t )A 3 (t)n(0) = g 1 (q 2 max) f 1 (q 2 max) δf 3 (q 2 max) ( ) f1 (0) g 1 (0) SU(3) = ( )( ) g1 (0) f1 (0) f 1 (0) g 1 (0) SU(3) + O(δ 2 ) where δ = M Ξ M Σ M Ξ + M Σ q 2 max = (M Ξ M Σ ) 2

61 0.10 exp: -4(17) δ exp =50 = (g A/gV )np (ga/gv )ΞΣ SU(3) breaking (ga/gv )np ~ 4 ± 2% breaking (lower bound) (4, 8) (ml, ms) =(5, 8) (6, 8) Theoretical expectation δ δ δ = M Ξ M Σ M Ξ + M Σ

62 Summary/Outlook The computation of weak matrix elements in lattice QCD is now progressing with steadily increasing accuracy by utilizing domain wall fermions (DWF). - DWF has a big advantage in dealing with the axial symmetry - It is easy to determine Z-factors for V(A) local currents Neutron beta decay corresponds to a gold plated test The axial-vector channel significantly suffers from the finite volume effect, while it is hardly observed in the vector channel.

63 Summary/Outlook The computation of weak matrix elements in lattice QCD is now progressing with steadily increasing accuracy by utilizing domain wall fermions (DWF). - DWF has a big advantage in dealing with the axial symmetry - It is easy to determine Z-factors for V(A) local currents Neutron beta decay corresponds to a gold plated test There is a challenging issue to explore the SU(3) breaking effect in hyperon beta decay through a first principles calculation. Our exploratory study in quenched DWF calculation: Succeeded in evaluating 2nd-class form factors from lattice QCD Observed the SU(3) breaking effect with higher accuracy than expt.

64 Generating 2+1 flavor DWF configurations Large scale production run DWF + Iwasaki gauge action Lattice cutoff: a -1 ~ 1.7 GeV (β 2.13, c1= 0.331) Box size: V=24 3 x 64 x 16 L ~ 2.8 fm mlight = 3/4, 1/2, 1/4 of mstrange QCDOC with RBRC & BNL Lattice theorists Mπ ~ 350, 500, 750 MeV in collaboration with Columbia, UKQCD

65 Summary/Outlook The computation of weak matrix elements in lattice QCD is now progressing with steadily increasing accuracy by utilizing domain wall fermions (DWF). - DWF has a big advantage in dealing with the axial symmetry - It is easy to determine Z-factors for V(A) local currents Neutron beta decay corresponds to a gold plated test There is a challenging issue to explore the SU(3) breaking effect in hyperon beta decay through a first principles calculation. Future: 2+1 flavors DWF calculation (RBC+UKQCD) is promissing for theoretical research on the SU(3) breaking effect in baryon semileptonic decays.

Hadron Structure from Lattice QCD

Hadron Structure from Lattice QCD Huey-Wen Lin University of Washington 1 Outline Lattice QCD Overview Nucleon Structure PDF, form factors, GPDs Hyperons Axial coupling constants, charge radii... Summary