Proton decay matrix elements from chirally symmetric lattice QCD

Transcription

1 , The XXVI International Symposium on Lattice Field Theory 1 / 27

2 Introduction What to Measure Simulation Details Results Non Perturbative Renormalization Summary and Outlook 2 / 27

3 Proton decay is a distinctive signature of many Grand Unified Theories Experiments such as Super Kamiokande are searching for proton decay The current minimum bound on the proton lifetime from Super Kamiokande is years 3 / 27

4 Proton decay is a distinctive signature of many Grand Unified Theories Experiments such as Super Kamiokande are searching for proton decay The current minimum bound on the proton lifetime from Super Kamiokande is years 3 / 27

5 Proton decay is a distinctive signature of many Grand Unified Theories Experiments such as Super Kamiokande are searching for proton decay The current minimum bound on the proton lifetime from Super Kamiokande is years 3 / 27

6 For a generic decay channel, the partial decay width is: [ ( m p Γ(p m + l) = 32π 2 1 ( mm m p ) )] 2 C i W i 0(p m + l) i 2 4 / 27

7 For a generic decay channel, the partial decay width is: [ ( m p Γ(p m + l) = 32π 2 1 ( mm m p ) )] 2 C i W i 0(p m + l) i 2 4 / 27

8 For a generic decay channel, the partial decay width is: [ ( m p Γ(p m + l) = 32π 2 1 ( mm m p ) )] 2 C i W i 0(p m + l) i 2 4 / 27

9 For a generic decay channel, the partial decay width is: [ ( m p Γ(p m + l) = 32π 2 1 ( mm m p ) )] 2 C i W i 0(p m + l) The form factors can be related to a matrix element P L [ W i 0 (q 2 ) i/qw i q(q 2 ) ] u(k, s) = m O i N i 2 4 / 27

10 For a generic decay channel, the partial decay width is: [ ( m p Γ(p m + l) = 32π 2 1 ( mm m p ) )] 2 C i W i 0(p m + l) The form factors can be related to a matrix element P L [ W i 0 (q 2 ) i/qw i q(q 2 ) ] u(k, s) = m O i N i 2 4 / 27

11 For a generic decay channel, the partial decay width is: [ ( m p Γ(p m + l) = 32π 2 1 ( mm m p ) )] 2 C i W i 0(p m + l) The form factors can be related to a matrix element P L [ W i 0 (q 2 ) i/qw i q(q 2 ) ] u(k, s) = m O i N i 2 4 / 27

12 For a generic decay channel, the partial decay width is: [ ( m p Γ(p m + l) = 32π 2 1 ( mm m p ) )] 2 C i W i 0(p m + l) The form factors can be related to a matrix element P L W i 0(q 2 )u(k, s) = m O i N i 2 4 / 27

13 For a generic decay channel, the partial decay width is: [ ( m p Γ(p m + l) = 32π 2 1 ( mm m p ) )] 2 C i W i 0(p m + l) The form factors can be related to a matrix element The operators O i are given by P L W i 0(q 2 )u(k, s) = m O i N O RL = ɛ abc u a (x, t)cp R d b (x, t)p L u c (x, t) O LL = ɛ abc u a (x, t)cp L d b (x, t)p L u c (x, t) i 2 4 / 27

14 Define a general operator of the form O Γ i Γ j = ɛ abc u a (x, t)cγ i d b (x, t)γ j u c (x, t) 5 / 27

15 Define a general operator of the form O Γ i Γ j = ɛ abc u a (x, t)cγ i d b (x, t)γ j u c (x, t) where Γ i are matrices with two spin indices, labelled by, S = 1 P = γ 5 V = γ µ A µ = γ µ γ 5 T = 1 2 {γ µ, γ ν } 1 T = γ 5 2 {γ µ, γ ν } R = P R = 1 2 (1 + γ 5) L = P L = 1 2 (1 γ 5) 5 / 27

16 Define a general operator of the form O Γ i Γ j = ɛ abc u a (x, t)cγ i d b (x, t)γ j u c (x, t) where Γ i are matrices with two spin indices, labelled by, S = 1 P = γ 5 V = γ µ A µ = γ µ γ 5 T = 1 2 {γ µ, γ ν } 1 T = γ 5 2 {γ µ, γ ν } R = P R = 1 2 (1 + γ 5) L = P L = 1 2 (1 γ 5) Operators with this structure are also used later in nucleon correlation functions and in the non-perurbative renormalization 5 / 27

17 We could measure the matrix elements m O i N directly Known as the direct method Three-point functions are required Computationally expensive Alternatively can relate the three-point functions to two-point functions using Chiral Perturbation Theory Known as the indirect method Computationally cheaper Introduces an additional source of error 6 / 27

18 We could measure the matrix elements m O i N directly Known as the direct method Three-point functions are required Computationally expensive Alternatively can relate the three-point functions to two-point functions using Chiral Perturbation Theory Known as the indirect method Computationally cheaper Introduces an additional source of error 6 / 27

19 We could measure the matrix elements m O i N directly Known as the direct method Three-point functions are required Computationally expensive Alternatively can relate the three-point functions to two-point functions using Chiral Perturbation Theory Known as the indirect method Computationally cheaper Introduces an additional source of error 6 / 27

20 We could measure the matrix elements m O i N directly Known as the direct method Three-point functions are required Computationally expensive Alternatively can relate the three-point functions to two-point functions using Chiral Perturbation Theory Known as the indirect method Computationally cheaper Introduces an additional source of error 6 / 27

21 For p π 0 + e +, the chiral perturbation theory gives W RL 0 (p π 0 + e + ) = α(1 + D + F )/ 2f + O(m 2 l /m2 N ) W LL 0 (p π 0 + e + ) = β(1 + D + F )/ 2f + O(m 2 l /m2 N ) 7 / 27

22 For p π 0 + e +, the chiral perturbation theory gives W RL 0 (p π 0 + e + ) = α(1 + D + F )/ 2f + O(m 2 l /m2 N ) W LL 0 (p π 0 + e + ) = β(1 + D + F )/ 2f + O(m 2 l /m2 N ) α and β are low energy constants from the chiral lagrangian 7 / 27

23 For p π 0 + e +, the chiral perturbation theory gives W RL 0 (p π 0 + e + ) = α(1 + D + F )/ 2f + O(m 2 l /m2 N ) W LL 0 (p π 0 + e + ) = β(1 + D + F )/ 2f + O(m 2 l /m2 N ) α and β are low energy constants from the chiral lagrangian They can be calculated from two point functions 0 O RL N = αp L u(k, s) 0 O LL N = βp L u(k, s) 7 / 27

24 Define a class of two point functions f Γ1 Γ 2,Γ 3 Γ 4 (t) = [ ] tr O Γ 1Γ 2 Ō Γ 3Γ 4 P x 8 / 27

25 Define a class of two point functions f Γ1 Γ 2,Γ 3 Γ 4 (t) = [ ] tr O Γ 1Γ 2 Ō Γ 3Γ 4 P x O Γ 1Γ 2 = ɛ abc u a (x, t)cγ i d b (x, t)γ j u c (x, t) 8 / 27

26 Define a class of two point functions f Γ1 Γ 2,Γ 3 Γ 4 (t) = [ ] tr O Γ 1Γ 2 Ō Γ 3Γ 4 P x O Γ 1Γ 2 = ɛ abc u a (x, t)cγ i d b (x, t)γ j u c (x, t) P = 1 2 (1 + γ 4) 8 / 27

27 Define a class of two point functions f Γ1 Γ 2,Γ 3 Γ 4 (t) = [ ] tr O Γ 1Γ 2 Ō Γ 3Γ 4 P x O Γ 1Γ 2 = ɛ abc u a (x, t)cγ i d b (x, t)γ j u c (x, t) P = 1 2 (1 + γ 4) Example: the proton correlation function J p (x, t) J p (0) = f PS,PS (t) x 8 / 27

28 Strategy: First find m N from a correlated fit to the effective mass ( ) fps,ps (t) m eff (t) = log m N t 0 f PS,PS (t + 1) Then find G N from a correlated fit to an effective amplitude G N,eff = 2f PS,PS e m Nt G N t 0 Finally to calculate α and β we use a ratio of two point functions R α (t) = 2G N f RL,PS (t) f PS,PS (t) α R β(t) = 2G N f LL,PS (t) f PS,PS (t) β 9 / 27

29 Strategy: First find m N from a correlated fit to the effective mass ( ) fps,ps (t) m eff (t) = log m N t 0 f PS,PS (t + 1) Then find G N from a correlated fit to an effective amplitude G N,eff = 2f PS,PS e m Nt G N t 0 Finally to calculate α and β we use a ratio of two point functions R α (t) = 2G N f RL,PS (t) f PS,PS (t) α R β(t) = 2G N f LL,PS (t) f PS,PS (t) β 9 / 27

30 Strategy: First find m N from a correlated fit to the effective mass ( ) fps,ps (t) m eff (t) = log m N t 0 f PS,PS (t + 1) Then find G N from a correlated fit to an effective amplitude G N,eff = 2f PS,PS e m Nt G N t 0 Finally to calculate α and β we use a ratio of two point functions R α (t) = 2G N f RL,PS (t) f PS,PS (t) α R β(t) = 2G N f LL,PS (t) f PS,PS (t) β 9 / 27

31 Calculation is carried out on 2+1 flavour Domain Wall Fermion ensembles Iwasaki gauge action (β = 2.13) Fifth dimension size L s = 16 Inverse lattice spacing a 1 = 1.73(3) GeV Two different lattice volumes V = and Two degenerate light quarks with masses am u/d = 0.005*, 0.01, 0.02 or 0.03 One strange quark with mass am s = / 27

32 Calculation is carried out on 2+1 flavour Domain Wall Fermion ensembles Iwasaki gauge action (β = 2.13) Fifth dimension size L s = 16 Inverse lattice spacing a 1 = 1.73(3) GeV Two different lattice volumes V = and Two degenerate light quarks with masses am u/d = 0.005*, 0.01, 0.02 or 0.03 One strange quark with mass am s = / 27

33 Calculation is carried out on 2+1 flavour Domain Wall Fermion ensembles Iwasaki gauge action (β = 2.13) Fifth dimension size L s = 16 Inverse lattice spacing a 1 = 1.73(3) GeV Two different lattice volumes V = and Two degenerate light quarks with masses am u/d = 0.005*, 0.01, 0.02 or 0.03 One strange quark with mass am s = / 27

34 Calculation is carried out on 2+1 flavour Domain Wall Fermion ensembles Iwasaki gauge action (β = 2.13) Fifth dimension size L s = 16 Inverse lattice spacing a 1 = 1.73(3) GeV Two different lattice volumes V = and Two degenerate light quarks with masses am u/d = 0.005*, 0.01, 0.02 or 0.03 One strange quark with mass am s = / 27

35 Improve the signal by: Oversampling and binning of correlation functions Multiple sources per configuration Local Smearing (L), Gaussian Smearing (G) / (G*) and Hydrogen-Like Smearing (H) of operators 11 / 27

36 Improve the signal by: Oversampling and binning of correlation functions Multiple sources per configuration Local Smearing (L), Gaussian Smearing (G) / (G*) and Hydrogen-Like Smearing (H) of operators 11 / 27

37 Improve the signal by: Oversampling and binning of correlation functions Multiple sources per configuration Local Smearing (L), Gaussian Smearing (G) / (G*) and Hydrogen-Like Smearing (H) of operators 11 / 27

38 Fitting Fit by minimising a correlated χ 2 χ 2 (p) = t,t [p eff (t) p] C 1 tt [ peff (t ) p ] With correlation Matrix C tt = 1 N boot N boot n=1 Bootstrap to get central value and errors [ ] [ ] p (n) eff (t) p eff(t) p (n) eff (t ) p eff (t ). 12 / 27

39 Fitting Fit by minimising a correlated χ 2 χ 2 (p) = t,t [p eff (t) p] C 1 tt [ peff (t ) p ] With correlation Matrix C tt = 1 N boot N boot n=1 Bootstrap to get central value and errors [ ] [ ] p (n) eff (t) p eff(t) p (n) eff (t ) p eff (t ). 12 / 27

40 Fitting Fit by minimising a correlated χ 2 χ 2 (p) = t,t [p eff (t) p] C 1 tt [ peff (t ) p ] With correlation Matrix C tt = 1 N boot N boot n=1 Bootstrap to get central value and errors [ ] [ ] p (n) eff (t) p eff(t) p (n) eff (t ) p eff (t ). 12 / 27

41 Nucleon Mass LL LL2 HL HL2 G*L (a) (c) m N,eff 0.7 am N t a(m q +m res ) 13 / 27

42 Nucleon Amplitude LL (b) G N,eff G N t a(m q +m res ) 14 / 27

43 Low energy constant: α LL HL (b) (a) R α a 3 α t a(m q +m res ) 15 / 27

44 Low energy constant: β LL GL (b) R β a 3 β t a(m q +m res ) 16 / 27

45 Statistical error shown previously ( 10%) Finite volume errors Extrapolation errors Errors in renormalisation Still also have an error from using chiral perturbation theory, difficult to quantify this 17 / 27

46 Statistical error shown previously ( 10%) Finite volume errors Extrapolation errors Errors in renormalisation Still also have an error from using chiral perturbation theory, difficult to quantify this 17 / 27

47 Statistical error shown previously ( 10%) Finite volume errors Extrapolation errors Errors in renormalisation Still also have an error from using chiral perturbation theory, difficult to quantify this 17 / 27

48 Statistical error shown previously ( 10%) Finite volume errors Extrapolation errors Errors in renormalisation Still also have an error from using chiral perturbation theory, difficult to quantify this 17 / 27

49 Statistical error shown previously ( 10%) Finite volume errors Extrapolation errors Errors in renormalisation Still also have an error from using chiral perturbation theory, difficult to quantify this 17 / 27

50 Finite Volume Error x x a 3 α No noticeable effect a(m q +m res ) 18 / 27

51 Extrapolation Error points 4 points (b) points 4 points (b) a 3 α a 3 β a(m q +m res ) 18% a(m q +m res ) 17% 19 / 27

52 NPR Non-perturbative MOM scheme renormalisation of the Rome-Southampton group The renormalised operators are O A ren = Z AB O B latt A and B label the spin structure, eg LL Z AB is the mixing matrix O LL and O RL mix with a 3rd operator O A(LV ) Z AB is a 3 3 matrix Exponentially accurate chiral symmetry from Domain Wall Fermions should suppress operator mixing 20 / 27

53 NPR Non-perturbative MOM scheme renormalisation of the Rome-Southampton group The renormalised operators are O A ren = Z AB O B latt A and B label the spin structure, eg LL Z AB is the mixing matrix O LL and O RL mix with a 3rd operator O A(LV ) Z AB is a 3 3 matrix Exponentially accurate chiral symmetry from Domain Wall Fermions should suppress operator mixing 20 / 27

54 NPR Non-perturbative MOM scheme renormalisation of the Rome-Southampton group The renormalised operators are O A ren = Z AB O B latt A and B label the spin structure, eg LL Z AB is the mixing matrix O LL and O RL mix with a 3rd operator O A(LV ) Z AB is a 3 3 matrix Exponentially accurate chiral symmetry from Domain Wall Fermions should suppress operator mixing 20 / 27

55 NPR Non-perturbative MOM scheme renormalisation of the Rome-Southampton group The renormalised operators are O A ren = Z AB O B latt A and B label the spin structure, eg LL Z AB is the mixing matrix O LL and O RL mix with a 3rd operator O A(LV ) Z AB is a 3 3 matrix Exponentially accurate chiral symmetry from Domain Wall Fermions should suppress operator mixing 20 / 27

56 NPR Non-perturbative MOM scheme renormalisation of the Rome-Southampton group The renormalised operators are O A ren = Z AB O B latt A and B label the spin structure, eg LL Z AB is the mixing matrix O LL and O RL mix with a 3rd operator O A(LV ) Z AB is a 3 3 matrix Exponentially accurate chiral symmetry from Domain Wall Fermions should suppress operator mixing 20 / 27

57 NPR Non-perturbative MOM scheme renormalisation of the Rome-Southampton group The renormalised operators are O A ren = Z AB O B latt A and B label the spin structure, eg LL Z AB is the mixing matrix O LL and O RL mix with a 3rd operator O A(LV ) Z AB is a 3 3 matrix Exponentially accurate chiral symmetry from Domain Wall Fermions should suppress operator mixing 20 / 27

58 We define the parity basis of operators SS-SP, PP-PS, AA+AV These are related to the chirality basis of operators we are interested in via LL = 1 4 (SS + PP) 1 (SP + PS) 4 RL = 1 4 (SS PP) 1 (SP PS) 4 A(LV ) = 1 2 AA 1 2 ( AV ) Hence Z C = T Z P T 1, where 1/4 1/4 0 T = 1/4-1/ /2 21 / 27

59 We define the parity basis of operators SS-SP, PP-PS, AA+AV These are related to the chirality basis of operators we are interested in via LL = 1 4 (SS + PP) 1 (SP + PS) 4 RL = 1 4 (SS PP) 1 (SP PS) 4 A(LV ) = 1 2 AA 1 2 ( AV ) Hence Z C = T Z P T 1, where 1/4 1/4 0 T = 1/4-1/ /2 21 / 27

60 We define the parity basis of operators SS-SP, PP-PS, AA+AV These are related to the chirality basis of operators we are interested in via LL = 1 4 (SS + PP) 1 (SP + PS) 4 RL = 1 4 (SS PP) 1 (SP PS) 4 A(LV ) = 1 2 AA 1 2 ( AV ) Hence Z C = T Z P T 1, where 1/4 1/4 0 T = 1/4-1/ /2 21 / 27

61 We define the parity basis of operators SS-SP, PP-PS, AA+AV These are related to the chirality basis of operators we are interested in via LL = 1 4 (SS + PP) 1 (SP + PS) 4 RL = 1 4 (SS PP) 1 (SP PS) 4 A(LV ) = 1 2 AA 1 2 ( AV ) Hence Z C = T Z P T 1, where 1/4 1/4 0 T = 1/4-1/ /2 21 / 27

62 We want to calculate the non-perturbative amputated 3-quark vertex function of these operators where G A abc,αβγδ (p2 ) = ɛ abc (CΓ) α β Γ δγ Qa a α α (p)qb b β β (p)qc c γ γ (p) Q a a α α = S a a α α (p) 1 S a a α α (p) and Γ and Γ are the matrices which appear in O A 22 / 27

63 We want to calculate the non-perturbative amputated 3-quark vertex function of these operators where G A abc,αβγδ (p2 ) = ɛ abc (CΓ) α β Γ δγ Qa a α α (p)qb b β β (p)qc c γ γ (p) Q a a α α = S a a α α (p) 1 S a a α α (p) and Γ and Γ are the matrices which appear in O A 22 / 27

64 We want to calculate the non-perturbative amputated 3-quark vertex function of these operators where G A abc,αβγδ (p2 ) = ɛ abc (CΓ) α β Γ δγ Qa a α α (p)qb b β β (p)qc c γ γ (p) Q a a α α = S a a α α (p) 1 S a a α α (p) and Γ and Γ are the matrices which appear in O A 22 / 27

65 We want to calculate the non-perturbative amputated 3-quark vertex function of these operators where G A abc,αβγδ (p2 ) = ɛ abc (CΓ) α β Γ δγ Qa a α α (p)qb b β β (p)qc c γ γ (p) Q a a α α = S a a α α (p) 1 S a a α α (p) and Γ and Γ are the matrices which appear in O A 22 / 27

66 The renormalization condition in the RI Mom Scheme is Zq 3/2 Z BC M CA = δ BA Where the matrix M is, M AB = G A abc,αβγδ (p2 )P B abc,βαδγ and the projection matrices Pabc,βαδγ A are chosen so that the renormalization condition is satisfied in the free field case where Z q = 1 and Z BC = δ BC. Z AB can then be calculated from M AB using the renormalization condition 23 / 27

67 The renormalization condition in the RI Mom Scheme is Zq 3/2 Z BC M CA = δ BA Where the matrix M is, M AB = G A abc,αβγδ (p2 )P B abc,βαδγ and the projection matrices Pabc,βαδγ A are chosen so that the renormalization condition is satisfied in the free field case where Z q = 1 and Z BC = δ BC. Z AB can then be calculated from M AB using the renormalization condition 23 / 27

68 The renormalization condition in the RI Mom Scheme is Zq 3/2 Z BC M CA = δ BA Where the matrix M is, M AB = G A abc,αβγδ (p2 )P B abc,βαδγ and the projection matrices Pabc,βαδγ A are chosen so that the renormalization condition is satisfied in the free field case where Z q = 1 and Z BC = δ BC. Z AB can then be calculated from M AB using the renormalization condition 23 / 27

69 The renormalization condition in the RI Mom Scheme is Zq 3/2 Z BC M CA = δ BA Where the matrix M is, M AB = G A abc,αβγδ (p2 )P B abc,βαδγ and the projection matrices Pabc,βαδγ A are chosen so that the renormalization condition is satisfied in the free field case where Z q = 1 and Z BC = δ BC. Z AB can then be calculated from M AB using the renormalization condition 23 / 27

70 M AB LL,LL LL,RL LL,A(LV) RL,LL RL,RL RL,A(LV) A(LV),LL A(LV),RL A(LV),A(LV) p 2 24 / 27

71 Rotate the basis Perform a chiral extrapolation We match to the MS scheme at 2GeV This gives U MS latt (2GeV ) LL = 0.662(10) U MS latt (2GeV ) RL = 0.664(8) 25 / 27

72 Rotate the basis Perform a chiral extrapolation We match to the MS scheme at 2GeV This gives U MS latt (2GeV ) LL = 0.662(10) U MS latt (2GeV ) RL = 0.664(8) 25 / 27

73 [1] [2] [1] Donoghue 82 [3] [2] Thomas 83 [5] [4] [3] Meljanac 82 [4] Ioffe 81 [5] Krasnikov 82 [6] [7] [11] [10] [8] [9] [6] Ioffe 84 [7] Tomozawa 81 [8] Brodsky 84 [9] Hara 86 [10] Bowler 88 [11] Gavela 89 [12] JLQCD 00 [13] CP-PACS & JLQCD 04 Putting all these pieces together we get α = (12)(22) [13] [14] [12] [14] RBC 07 [15] RBC 07 [16] This work β = (13)(23) [15] [16] α (GeV 3 ) 26 / 27

74 The direct calculation is currently underway Example: Preliminary results for the W LL 0 (p π+ + ν), on the lattice, with valence quark mass am u = / 27

75 The direct calculation is currently underway Example: Preliminary results for the W LL 0 (p π+ + ν), on the lattice, with valence quark mass am u = Preliminary Data 0.1 W 0 LL t 27 / 27