Flavor Twisting for Isovector Form Factors

Transcription

1 Flavor Twisting for Isovector Form Factors Brian C. Tiburzi Duke University LHP2006, August 1 st TwBCs & Form Factors p.1

2 Outline Flavor Twisted Boundary Conditions and Isovector Form Factors Quantized momentum and form factors TwBCs & Form Factors p.2

3 Outline Flavor Twisted Boundary Conditions and Isovector Form Factors Quantized momentum and form factors Twisted boundary conditions TwBCs & Form Factors p.2

4 Outline Flavor Twisted Boundary Conditions and Isovector Form Factors Quantized momentum and form factors Twisted boundary conditions Kinematical effects TwBCs & Form Factors p.2

5 Outline Flavor Twisted Boundary Conditions and Isovector Form Factors Quantized momentum and form factors Twisted boundary conditions Kinematical effects Dynamical effects TwBCs & Form Factors p.2

6 Limitations near q = 0 Operator insertion method H (p ) O H(p) = j O j f j (q 2 ) TwBCs & Form Factors p.3

7 Limitations near q = 0 Operator insertion method H (p ) O H(p) = j O j f j (q 2 ) f j (q 2 ) q 2 [GeV 2 ] O j q TwBCs & Form Factors p.3

8 Limitations near q = 0 Operator insertion method H (p ) O H(p) = j O j f j (q 2 ) f j (q 2 ) q 2 [GeV 2 ] O j q L = 24a, a = 2 GeV 1 : q min = 2π/L 500 MeV TwBCs & Form Factors p.3

9 Nucleon isovector form factor Definition & Chiral expansion F 2 (q 2 ) = G p M (q2 ) G n M (q2 ) = µ iso + f(q 2 /4m 2 π) TwBCs & Form Factors p.4

10 Nucleon isovector form factor Definition & Chiral expansion F 2 (q 2 ) = G p M (q2 ) G n M (q2 ) = µ iso + f(q 2 /4m 2 π) F 2 (q 2 ) m π = 0.14 GeV m π = 0.25 GeV m π = 0.35 GeV q 2 [ GeV 2 ] Deviation F 2 (q 2 ) = F 2(q 2 ) F 2 (0) q 2 F 2 (0) TwBCs & Form Factors p.4

11 Nucleon isovector form factor Definition & Chiral expansion F 2 (q 2 ) = G p M (q2 ) G n M (q2 ) = µ iso + f(q 2 /4m 2 π) F 2 (q 2 ) m π = 0.14 GeV m π = 0.25 GeV m π = 0.35 GeV q 2 [ GeV 2 ] Deviation F 2 (q 2 ) = F 2(q 2 ) F 2 (0) q 2 F 2 (0) Chiral corrections, recoil corrections, lattice point TwBCs & Form Factors p.4

12 Twisted boundary conditions U U = 1 global symmetry of action, e.g. U = exp i θ a i T a C TwBCs ψ(x i + L) = Uψ(x i ) TwBCs & Form Factors p.5

13 Twisted boundary conditions U U = 1 global symmetry of action, e.g. U = exp i θ a i T a C TwBCs ψ(x i + L) = Uψ(x i ) x ψ(d/ + m Q)ψ single valued TwBCs & Form Factors p.5

14 Twisted boundary conditions U U = 1 global symmetry of action, e.g. U = exp i θ a i T a C TwBCs ψ(x i + L) = Uψ(x i ) x ψ(d/ + m Q)ψ single valued Momentum modes p i = 2πn i /L + θ i /L TwBCs & Form Factors p.5

15 Twisted boundary conditions U U = 1 global symmetry of action, e.g. U = exp i θ a i T a C TwBCs ψ(x i + L) = Uψ(x i ) x ψ(d/ + m Q)ψ single valued Momentum modes p i = 2πn i /L + θ i /L Rewrite ψ(x) = exp( i θa x TC a /L)ψ(x) periodic TwBCs & Form Factors p.5

16 Twisted boundary conditions U U = 1 global symmetry of action, e.g. U = exp i θ a i T a C TwBCs ψ(x i + L) = Uψ(x i ) x ψ(d/ + m Q)ψ single valued Momentum modes p i = 2πn i /L + θ i /L Rewrite ψ(x) = exp( i θa x TC a /L)ψ(x) periodic ψ( D/ x + m Q ) ψ TwBCs & Form Factors p.5

17 Twisted boundary conditions U U = 1 global symmetry of action, e.g. U = exp i θ a i T a C TwBCs ψ(x i + L) = Uψ(x i ) x ψ(d/ + m Q)ψ single valued Momentum modes p i = 2πn i /L + θ i /L Rewrite ψ(x) = exp( i θa x TC a /L)ψ(x) periodic ψ( D/ x + m Q ) ψ D µ = D µ + ib µ uniform gauge field TwBCs & Form Factors p.5

18 Twisted boundary conditions U U = 1 global symmetry of action, e.g. U = exp i θ a i T a C TwBCs ψ(x i + L) = Uψ(x i ) x ψ(d/ + m Q)ψ single valued Momentum modes p i = 2πn i /L + θ i /L Rewrite ψ(x) = exp( i θa x TC a /L)ψ(x) periodic ψ( D/ x + m Q ) ψ D µ = D µ + ib µ uniform gauge field B µ = (θ a i T a /L, 0) flavor charges TwBCs & Form Factors p.5

19 Twisted boundary conditions U U = 1 global symmetry of action, e.g. U = exp i θ a i T a C TwBCs ψ(x i + L) = Uψ(x i ) x ψ(d/ + m Q)ψ single valued Momentum modes p i = 2πn i /L + θ i /L Rewrite ψ(x) = exp( i θa x TC a /L)ψ(x) periodic ψ( D/ x + m Q ) ψ D µ = D µ + ib µ uniform gauge field B µ = (θ a i T a /L, 0) Partial twisting flavor charges T a C SU(N N) val SU(N + M N) TwBCs & Form Factors p.5

20 Field momenta of hadrons Hadrons pointlike in chiral EFTs TwBCs & Form Factors p.6

21 Field momenta of hadrons Hadrons pointlike in chiral EFTs Gauge B µ field for partial twisting TwBCs & Form Factors p.6

22 Field momenta of hadrons Hadrons pointlike in chiral EFTs Gauge B µ field for partial twisting Mesons D µ φ = D µ φ + i[b µ, φ] E π + = m 2 π + B 2 π +, B π + = B u B d Sachrajda & Villadoro PLB 609 TwBCs & Form Factors p.6

23 Field momenta of hadrons Hadrons pointlike in chiral EFTs Gauge B µ field for partial twisting Mesons D µ φ = D µ φ + i[b µ, φ] E π + = m 2 π + B 2 π +, B π + = B u B d Sachrajda & Villadoro PLB 609 Baryons D µ B ijk = D µ B ijk + i(b i µ + B j µ + B k µ)b ijk E p = M p + B2 p 2M p, B p = 2B u + B d Tiburzi PLB 617 TwBCs & Form Factors p.6

24 Numerical investigations Meson dispersion relations Quenched de Divitiis, Petronzio & Tantalo PLB (r 0 E) (r 0 θ /L) 2 Dynamical partially twisted Flynn, Jüttner & Sachrajda PLB632 TwBCs & Form Factors p.7

25 Flavor changing operators Twist flavors differently get momentum transfer! TwBCs & Form Factors p.8

26 Flavor changing operators Twist flavors differently get momentum transfer! Nucleon axial current J + 5µ = uγ µγ 5 d TwBCs & Form Factors p.8

27 Flavor changing operators Twist flavors differently get momentum transfer! Nucleon axial current J + 5µ = uγ µγ 5 d Lattice correlator C(t, t ) = 0 P(x, t) J 5µ + (x, t )Ñ (0, 0) 0 x,x TwBCs & Form Factors p.8

28 Flavor changing operators Twist flavors differently get momentum transfer! Nucleon axial current J + 5µ = uγ µγ 5 d Lattice correlator C(t, t ) = 0 P(x, t) J 5µ + (x, t )Ñ (0, 0) 0 x,x C(t, t ) = P Bp (t)j + 5µ (t )N Bn (0) TwBCs & Form Factors p.8

29 Flavor changing operators Twist flavors differently get momentum transfer! Nucleon axial current J + 5µ = uγ µγ 5 d Lattice correlator C(t, t ) = 0 P(x, t) J 5µ + (x, t )Ñ (0, 0) 0 x,x C(t, t ) = P Bp (t)j + 5µ (t )N Bn (0) Momentum transfer q = B p B n = B π + Tiburzi PLB 617 TwBCs & Form Factors p.8

30 Further numerical investigations Meson decay constants Flynn, Jüttner & Sachrajda PLB632 π + (0) J = if πb π + f π reliably extracted S = 1 currents Guadagnoli, Mescia and Simula PRD73 K π form factors Systematics? π (p ) sγ µ u K 0 (p) TwBCs & Form Factors p.9

31 Isospin relations Flavor non-changing currents? TwBCs & Form Factors p.10

32 Isospin relations Flavor non-changing currents? Isospin rotation isovector currents Tiburzi hep-lat/ p u Γ d n = p u Γ u p n u Γ u n TwBCs & Form Factors p.10

33 Isospin relations Flavor non-changing currents? Isospin rotation isovector currents Tiburzi hep-lat/ p u Γ d n = p u Γ u p n u Γ u n Calculate isospin change with TwBCs relate to isovector observables TwBCs & Form Factors p.10

34 Isospin relations Flavor non-changing currents? Isospin rotation isovector currents Tiburzi hep-lat/ p u Γ d n = p u Γ u p n u Γ u n Calculate isospin change with TwBCs relate to isovector observables Arbitrary bilinears, e.g. Γ = γ 5 γ {µ D µ1... D µn } TwBCs & Form Factors p.10

35 Isospin relations Flavor non-changing currents? Isospin rotation isovector currents Tiburzi hep-lat/ p u Γ d n = p u Γ u p n u Γ u n Calculate isospin change with TwBCs relate to isovector observables Arbitrary bilinears, e.g. Γ = γ 5 γ {µ D µ1... D µn } Vector current electromagnetic current p u γ µ d n = p J em µ p n J em µ n TwBCs & Form Factors p.10

36 Dynamical effects due to twisting Exploited isospin breaking B u B d for kinematical gain TwBCs & Form Factors p.11

37 Dynamical effects due to twisting Exploited isospin breaking B u B d for kinematical gain Boundary conditions affect long-distance physics: VOLUME EFFECTS TwBCs & Form Factors p.11

38 Dynamical effects due to twisting Exploited isospin breaking B u B d for kinematical gain Boundary conditions affect long-distance physics: VOLUME EFFECTS Exactly what EFTs address: long-range dynamical effects, model independent TwBCs & Form Factors p.11

39 Dynamical effects due to twisting Exploited isospin breaking B u B d for kinematical gain Boundary conditions affect long-distance physics: VOLUME EFFECTS Exactly what EFTs address: long-range dynamical effects, model independent Isospin splittings with m u = m d m 2 π ± m 2 π 0 = m2 π f 2 π [I(0, m 2 π) I(B π +, m 2 π)] TwBCs & Form Factors p.11

40 Dynamical effects due to twisting Exploited isospin breaking B u B d for kinematical gain Boundary conditions affect long-distance physics: VOLUME EFFECTS Exactly what EFTs address: long-range dynamical effects, model independent Isospin splittings with m u = m d m 2 π ± m 2 π 0 = m2 π f 2 π [I(0, m 2 π) I(B π +, m 2 π)] Similar for nucleon M p M n 0 TwBCs & Form Factors p.11

41 Dynamical effects due to twisting Exploited isospin breaking B u B d for kinematical gain Boundary conditions affect long-distance physics: VOLUME EFFECTS Exactly what EFTs address: long-range dynamical effects, model independent Isospin splittings with m u = m d m 2 π ± m 2 π 0 = m2 π f 2 π [I(0, m 2 π) I(B π +, m 2 π)] Similar for nucleon M p M n 0 Shifts 1% for m π 300 MeV, 2.5 fm TwBCs & Form Factors p.11

42 Axial form factors in finite volume Special: besides tree-level, q 2 dependence arises from recoil terms that are competitive with two-loops TwBCs & Form Factors p.12

43 Axial form factors in finite volume Special: besides tree-level, q 2 dependence arises from recoil terms that are competitive with two-loops Axial radius and pseudoscalar form factor have negligible volume dependence [ p(0) J 5 + ñ(0) = σ G A (0) < r2 A > ] Bπ [ g A +B π + (B π + σ) Bπ 2 + m 2 + π + < r2 A > 3 ] TwBCs & Form Factors p.12

44 Axial form factors in finite volume Special: besides tree-level, q 2 dependence arises from recoil terms that are competitive with two-loops Axial radius and pseudoscalar form factor have negligible volume dependence [ p(0) J 5 + ñ(0) = σ G A (0) < r2 A > ] Bπ [ g A +B π + (B π + σ) Bπ 2 + m 2 + π + < r2 A > 3 ] Ideal testing ground TwBCs Chiral physics TwBCs & Form Factors p.12

45 Isovector magnetic moment Take B u 2 = B 0 and J + µ=3 p(0) J + 3 ñ(0) = ib 2M F 2(B 2 ) + L FV + K FV TwBCs & Form Factors p.13

46 Isovector magnetic moment Take B u 2 = B 0 and J + µ=3 p(0) J + 3 ñ(0) = ib 2M F 2(B 2 ) + L FV + K FV L FV ordinary FV modified with TwBCs Beane PRD70 TwBCs & Form Factors p.13

47 Isovector magnetic moment Take B u 2 = B 0 and J + µ=3 p(0) J + 3 ñ(0) = ib 2M F 2(B 2 ) + L FV + K FV L FV ordinary FV modified with TwBCs Beane PRD70 K FV new effect due to TwBCs TwBCs & Form Factors p.13

48 Isovector magnetic moment Take B u 2 = B 0 and J + µ=3 p(0) J + 3 ñ(0) = ib 2M F 2(B 2 ) + L FV + K FV L FV ordinary FV modified with TwBCs Beane PRD70 K FV new effect due to TwBCs δ L [µ] K, m π = 0.14 GeV L+K, m π = 0.14 GeV K, m π = 0.25 GeV L+K, m π = 0.25 GeV K, m π = 0.35 GeV K+L, m π = 0.35 GeV L [ fm ] TwBCs & Form Factors p.13

49 Isovector magnetic moment Extracting F 2 (B 2 ): Volume Effects, B 2 resolving power -0.1 δ L [ F 2 (B 2 )] m π = 0.25 GeV m π = 0.30 GeV m π = 0.35 GeV 0 π/4 π/2 3π/4 π θ q 500 MeV 25 MeV in fixed volume TwBCs & Form Factors p.14

50 Comparison with background fields Study dependence of observables with continuous parameter: TwBCs θ, BFs A µ TwBCs & Form Factors p.15

51 Comparison with background fields Study dependence of observables with continuous parameter: TwBCs θ, BFs A µ BFs multivalued action, FV? or Dirichlet BCs, FV? TwBCs single valued, FV effects calculable TwBCs & Form Factors p.15

52 Comparison with background fields Study dependence of observables with continuous parameter: TwBCs θ, BFs A µ BFs multivalued action, FV? or Dirichlet BCs, FV? TwBCs single valued, FV effects calculable BFs Each observable, vector, axial, twist-two new fields TwBCs different operator insertion, some propagators can be recycled TwBCs & Form Factors p.15

53 Comparison with background fields Study dependence of observables with continuous parameter: TwBCs θ, BFs A µ BFs multivalued action, FV? or Dirichlet BCs, FV? TwBCs single valued, FV effects calculable BFs Each observable, vector, axial, twist-two new fields TwBCs different operator insertion, some propagators can be recycled BFs radii, curvature calculable with increasingly complicated fields TwBCs access form factors directly TwBCs & Form Factors p.15

54 Comparison with background fields Study dependence of observables with continuous parameter: TwBCs θ, BFs A µ BFs multivalued action, FV? or Dirichlet BCs, FV? TwBCs single valued, FV effects calculable BFs Each observable, vector, axial, twist-two new fields TwBCs different operator insertion, some propagators can be recycled BFs radii, curvature calculable with increasingly complicated fields TwBCs access form factors directly BFs isoscalar can be done time consumingly TwBCs??? TwBCs & Form Factors p.15

55 Summary TwBCs produce continuous hadron momentum TwBCs & Form Factors p.16

56 Summary TwBCs produce continuous hadron momentum Flavor changing currents TwBCs & Form Factors p.16

57 Summary TwBCs produce continuous hadron momentum Flavor changing currents Isospin symmetry & isovector currents TwBCs & Form Factors p.16

58 Summary TwBCs produce continuous hadron momentum Flavor changing currents Isospin symmetry & isovector currents VOLUME EFFECTS TwBCs & Form Factors p.16

59 Summary TwBCs produce continuous hadron momentum Flavor changing currents Isospin symmetry & isovector currents VOLUME EFFECTS Nucleon axial current TwBCs & Form Factors p.16

60 Summary TwBCs produce continuous hadron momentum Flavor changing currents Isospin symmetry & isovector currents VOLUME EFFECTS Nucleon axial current Nucleon isovector magnetic moment TwBCs & Form Factors p.16

61 Summary TwBCs produce continuous hadron momentum Flavor changing currents Isospin symmetry & isovector currents VOLUME EFFECTS Nucleon axial current Nucleon isovector magnetic moment Further studies needed: EFT & Lattice TwBCs & Form Factors p.16

62 Summary TwBCs produce continuous hadron momentum Flavor changing currents Isospin symmetry & isovector currents VOLUME EFFECTS Nucleon axial current Nucleon isovector magnetic moment Further studies needed: EFT & Lattice Isoscalar??? TwBCs & Form Factors p.16