Renormalization of three-quark operators for baryon distribution amplitudes

Transcription

1 Renormalization of three-quark operators for baryon distribution amplitudes Institut für Theoretische Physik July 25, 2016 Renormalization of three-quark operators for baryon distribution amplitudes 1 / 14

2 Reminder Definition of leading twist octet DAs Φ B Λ ± (x1, x2, x3) = ( 1 2 [V A] B (x 1, x 2, x 3) ± [V A] B (x 3, x 2, x ) 1) Π B Λ (x 1, x 2, x 3) = T B (x 1, x 3, x 2) ( Φ Λ 1 +(x 1, x 2, x 3) = + 6 [V A] Λ (x 1, x 2, x 3) + [V A] Λ (x 3, x 2, x ) 1) ( Φ Λ 3 (x 1, x 2, x 3) = 2 [V A] Λ (x 1, x 2, x 3) [V A] Λ (x 3, x 2, x ) 1) Π Λ (x 1, x 2, x 3) = 6 T Λ (x 1, x 3, x 2) Expansion in terms of shape parameters ( Φ B + = 120x 1x 2x 3 ϕ B 00 P 00 + ϕ B 11P ) ( Φ B = 120x 1x 2x 3 ϕ B 10 P ) f B = ϕ B 00 ( Π B Λ = 120x 1x 2x 3 π B 00 P 00 + π11p B ) ( Π Λ = 120x 1x 2x 3 π Λ 10 P ) f B Λ T = π00 B Renormalization of three-quark operators for baryon distribution amplitudes 2 / 14

3 Renormalization overview Lattice nonpert. ren. RI continuum PT /SMOM MS Bare lattice values have to be renormalized In the end we should be able to give our results in the popular continuum MS scheme This scheme cannot be implemented directly on the lattice We use a nonperturbative RI /SMOM scheme for the lattice renormalization We use continuum perturbation theory to convert from RI /SMOM to MS Renormalization of three-quark operators for baryon distribution amplitudes 3 / 14

4 A tale of two schemes: MS and RI /SMOM MS requires the use of dimensional regularization perform loop integral calculations in 4 2ɛ dimensions expand in ɛ and subtract divergencies order-by-order, usually in the shape of ( G MS (p, µ) = 1 αs(µ) ( )) 1 3π ξ ɛ γe + ln(4π) G bare (p) Renormalization of three-quark operators for baryon distribution amplitudes 4 / 14

5 A tale of two schemes: MS and RI /SMOM MS requires the use of dimensional regularization perform loop integral calculations in 4 2ɛ dimensions expand in ɛ and subtract divergencies order-by-order, usually in the shape of ( G MS (p, µ) = 1 αs(µ) ( )) 1 3π ξ ɛ γe + ln(4π) G bare (p) RI is a class of regularization invariant renormalization schemes defined through a renormalization condition at a specific renormalization point, e.g., 1 =! 1 [ 12p Tr i/p ( ) ] 1(p, G RI µ) 2 p 2 =µ 2 SMOM: Symmetric renormalization point Better behaved than exceptional momentum configurations Renormalization of three-quark operators for baryon distribution amplitudes 4 / 14

6 RI /SMOM renormalization procedure Renormalization point p SMOM : p 2 1 = p 2 2 = p 2 3 = (p 1 + p 2) 2 = (p 1 + p 3) 2 = (p 1 + p 2 + p 3) 2 = µ 2 Amputated four-point vertex function Λ(O p 1, p 2, p 3) fgh αβγ = dx 1dx 2dx 3 e i j p j x j ɛ ijk O(0) f i α (x1)ḡj β (x 2) h k γ (x3) G 1 2 (p 1) α αg 1 2 (p 2) β βg 1 2 (p 3) γ γ Renormalization of three-quark operators for baryon distribution amplitudes 5 / 14

7 RI /SMOM renormalization procedure Renormalization point p SMOM : p 2 1 = p 2 2 = p 2 3 = (p 1 + p 2) 2 = (p 1 + p 3) 2 = (p 1 + p 2 + p 3) 2 = µ 2 Amputated four-point vertex function Λ(O p 1, p 2, p 3) fgh αβγ = dx 1dx 2dx 3 e i j p j x j ɛ ijk O(0) f i α (x1)ḡj β (x 2) h k γ (x3) G 1 2 (p 1) α αg 1 2 (p 2) β βg 1 2 (p 3) γ γ Renormalization condition ( ) L S = Λ S( ) ( O (i) fgh psmom Λ Born( ) O (i) fgh psmom mn i f,g,h α,β,γ Which operators can mix? m αβγ 1! = L RI( L Born) 1 n αβγ ) Renormalization of three-quark operators for baryon distribution amplitudes 5 / 14

8 Operator classification: H(4) Euclidean spacetime has continuous O(4) symmetry For fermions on a lattice this is broken down to the spinorial hypercubic group H(4) (a discrete double cover group with 768 elements) Five irreducible spinorial representations: τ 4 1, τ 4 2, τ 8, τ 12 1, τ 12 2 Classify baryon operators by decomposing direct products E.g. for three quarks: τ 4 1 τ 4 1 τ 4 1 = 5τ 4 1 τ 8 3τ 12 1 Operators belonging to different representations cannot mix Renormalization of three-quark operators for baryon distribution amplitudes 6 / 14

9 Operator classification: H(4) Euclidean spacetime has continuous O(4) symmetry For fermions on a lattice this is broken down to the spinorial hypercubic group H(4) (a discrete double cover group with 768 elements) Five irreducible spinorial representations: τ 4 1, τ 4 2, τ 8, τ 12 1, τ 12 2 Classify baryon operators by decomposing direct products E.g. for three quarks: τ 4 1 τ 4 1 τ 4 1 = 5τ 4 1 τ 8 3τ 12 1 Operators belonging to different representations cannot mix no derivatives 1 derivative 2 derivatives dimension 9/2 dimension 11/2 dimension 13/2 τ 4 1 O 1, O 2, O 3, O 4, O 5... O DD1, O DD2, O DD3,... τ 4 2 O DD4, O DD5, O DD6,... τ 8 O 6 O D1,... O DD7, O DD8, O DD9,... τ 12 1 O 7, O 8, O 9 O D2, O D3, O D4,... O DD10, O DD11, O DD12, O DD13,... τ 12 2 O D5, O D6, O D7, O D8 O DD14, O DD15, O DD16, O DD17, O DD18,... Renormalization of three-quark operators for baryon distribution amplitudes 6 / 14

10 Operator classification: H(4) In the continuum operators with different number of derivatives do not mix On the lattice there is an additional dimensionful quantity: the lattice spacing a Schematically: Dqqq can now mix with 1 a qqq Inverse power of a dependence is very problematic in the continuum limit a 0 We can avoid these problems by only using operators where this cannot happen Are these operators enough to measure all interesting quantities? no derivatives 1 derivative 2 derivatives dimension 9/2 dimension 11/2 dimension 13/2 τ 4 1 O 1, O 2, O 3, O 4, O 5... O DD1, O DD2, O DD3,... τ 4 2 O DD4, O DD5, O DD6,... τ 8 O 6 O D1,... O DD7, O DD8, O DD9,... τ 12 1 O 7, O 8, O 9 O D2, O D3, O D4,... O DD10, O DD11, O DD12, O DD13,... τ 12 2 O D5, O D6, O D7, O D8 O DD14, O DD15, O DD16, O DD17, O DD18,... Renormalization of three-quark operators for baryon distribution amplitudes 7 / 14

11 Operator classification: H(4) no derivatives 1 derivative 2 derivatives dimension 9/2 dimension 11/2 dimension 13/2 τ 4 1 O 1, O 2, O 3, O 4, O 5... O DD1, O DD2, O DD3,... τ 4 2 O DD4, O DD5, O DD6,... τ 8 O 6 O D1,... O DD7, O DD8, O DD9,... τ 12 1 O 7, O 8, O 9 O D2, O D3, O D4,... O DD10, O DD11, O DD12, O DD13,... τ 12 2 O D5, O D6, O D7, O D8 O DD14, O DD15, O DD16, O DD17, O DD18,... O 1, O 2: higher twist, LLL/RRR λ B 2 O 1, O 2: can be combined to form Dosch current O 3, O 4, O 5: higher twist, LLR/RRL λ B 1, λ Λ T O 3, O 4, O 5: can be combined to form Ioffe current O 7, O 8, O 9: leading twist, LLR/RRL f B, f B Λ T O 7, O 8, O 9: correspond to V + A, V A and T, respectively O D5, O D6, O D7: leading twist, LLR/RRL ϕ B 00,(1), ϕb 11, ϕ B 10, π B Λ 00,(1), πb Λ 11, π Λ 10 O D5, O D6, O D7: correspond to first moments of V + A, V A and T, respectively Renormalization of three-quark operators for baryon distribution amplitudes 8 / 14

12 Operator classification: S 3 Under SU(3) quark fields transform according to the fundamental representation 3 For three-quark operators we have the decomposition = The renormalization is defined in the chiral limit and thus in the limit of exact SU(3), where operators belonging to different representations cannot mix Consider a local three-quark operator with separate flavor, spinor and color structures F fgh S αβγ ɛ ijk f i α(0)g j β (0)hk γ(0) Renormalization of three-quark operators for baryon distribution amplitudes 9 / 14

13 Operator classification: S 3 Under SU(3) quark fields transform according to the fundamental representation 3 For three-quark operators we have the decomposition = The renormalization is defined in the chiral limit and thus in the limit of exact SU(3), where operators belonging to different representations cannot mix Consider a local three-quark operator with separate flavor, spinor and color structures F fgh S αβγ ɛ ijk f i α(0)g j β (0)hk γ(0) For the renormalization consider only structures that transform according to S 3 The SU(3) multiplets correspond to the three irreducible representations of S 3: decuplet totally symmetric trivial representation singlet totally antisymmetric signum representation 2 octets mixed symmetric & mixed antisymmetric two-dim. representation Renormalization of three-quark operators for baryon distribution amplitudes 9 / 14

14 Operator classification: S 3 Under SU(3) quark fields transform according to the fundamental representation 3 For three-quark operators we have the decomposition = The renormalization is defined in the chiral limit and thus in the limit of exact SU(3), where operators belonging to different representations cannot mix Consider a local three-quark operator with separate flavor, spinor and color structures F fgh S αβγ ɛ ijk f i α(0)g j β (0)hk γ(0) For the renormalization consider only structures that transform according to S 3 The SU(3) multiplets correspond to the three irreducible representations of S 3: F f π(1) f π(2) f π(3) d F f π(1) f π(2) f π(3) s = F f1f2f3 d = sgn(π)fs f1f2f3 [ ] T (π) F f π(1) f π(2) f π(3) o,t = 2 t =1 F f1f2f3 t t o,t S α π(1) α π(2) α π(3) d S α π(1) α π(2) α π(3) s = S α1α2α3 d = sgn(π)ss α1α2α3 [ ] T (π) S α π(1) α π(2) α π(3) o,t = 2 t =1 t t Sα1α2α3 o,t Renormalization of three-quark operators for baryon distribution amplitudes 9 / 14

15 Renormalization matrix (Nucleon) For the nucleon we have the following renormalization pattern: A simple multiplicative renormalization for f N ( f N ) MS = Z Of ( f N) lat A 2 2 mixing matrix for the higher twist couplings λ N 1 and λ N 2 A 3 3 mixing matrix for the three first moments MS ϕn 00,(1) 2ϕ N 11 = 2ϕ N 10 ZOϕ 11 Z Oϕ 12 Z Oϕ 13 Z Oϕ 21 Z Oϕ 22 Z Oϕ 23 Z Oϕ 31 Z Oϕ 32 Z Oϕ 33 ϕn 00,(1) 2ϕ N 11 2ϕ N 10 lat Renormalization of three-quark operators for baryon distribution amplitudes 10 / 14

16 Renormalization matrix (full beauty) The general case is more complicated due to the additional moments of Π B Renormalization of f B is not necessarily multiplicative Decuplet and singlet renormalization factors are now also relevant for octet baryons ( ( ) ( ) f B Λ Z Of + 2Z Df 2Z Of 2Z Df f B lat Z Of Z Df 2Z Of + Z Df ) MS f B Λ = 1 T 3 ϕ B Λ MS 00,(1) π B Λ 00,(1) 2ϕ B Λ 11 = 1 2π B Λ ϕ B Λ 10 f B T Z Oϕ ZDϕ 11 2Z Oϕ 11 2ZDϕ 11 Z Oϕ ZDϕ 12 2Z Oϕ 12 2ZDϕ 12 3Z Oϕ 13 Z Oϕ 11 ZDϕ 11 2Z Oϕ 11 + ZDϕ 11 Z Oϕ 12 ZDϕ 12 2Z Oϕ 12 + ZDϕ 12 3Z Oϕ 13 Z Oϕ ZDϕ 21 2Z Oϕ 21 2ZDϕ 21 Z Oϕ ZDϕ 22 2Z Oϕ 22 2ZDϕ 22 3Z Oϕ 23 Z Oϕ 21 ZDϕ 21 2Z Oϕ 21 + ZDϕ 21 Z Oϕ 22 ZDϕ 22 2Z Oϕ 22 + ZDϕ 22 3Z Oϕ 23 Z Oϕ 31 2Z Oϕ 31 Z Oϕ 32 2Z Oϕ 32 3Z Oϕ 33 ϕ B Λ 00,(1) π B Λ 00,(1) 2ϕ B Λ 11 2π B Λ 11 2ϕ B Λ 10 lat Structure of renormalization matrix: Renormalization of three-quark operators for baryon distribution amplitudes 11 / 14

17 Renormalization matrix in the SU(3) limit Different renormalization matrices for B Λ and B = Λ ϕ B Λ MS 00,(1) Z Oϕ π B Λ ZDϕ 11 2Z Oϕ 11 2ZDϕ 11 Z Oϕ ZDϕ 12 2Z Oϕ 00,(1) 2ϕ B Λ 11 = 1 Z Oϕ 11 ZDϕ 11 2Z Oϕ 11 + ZDϕ 11 Z Oϕ 12 ZDϕ 12 2Z Oϕ Z Oϕ 21 2π B Λ 3 + 2ZDϕ 21 2Z Oϕ 21 2ZDϕ 21 Z Oϕ ZDϕ 22 2Z Oϕ 11 2ϕ B Λ 10 ϕ Λ MS 00,(1) 2ϕ Λ 11 = 1 3 2ϕ Λ 2π10 Λ ZDϕ 12 3Z Oϕ ZDϕ 12 3Z Oϕ ZDϕ 22 3Z Oϕ 23 Z Oϕ 21 ZDϕ 21 2Z Oϕ 21 + ZDϕ 21 Z Oϕ 22 ZDϕ 22 2Z Oϕ 22 + ZDϕ 22 3Z Oϕ 23 Z Oϕ 31 2Z Oϕ 31 Z Oϕ 32 2Z Oϕ 32 3Z Oϕ 33 lat 3Z Oϕ 11 3Z Oϕ 12 Z Oϕ 13 2Z Oϕ 13 3Z Oϕ 21 3Z Oϕ 22 Z Oϕ 23 2Z Oϕ 23 3Z Oϕ 31 3Z Oϕ 32 Z Oϕ ZS ϕ 2Z Oϕ 3Z Oϕ 31 3Z Oϕ 32 Z Oϕ 33 ZS ϕ 2Z Oϕ 33 + ZS ϕ ϕ Λ 00,(1) 2ϕ Λ 33 2ZS ϕ 2ϕ11 Λ 10 2π Λ 10 ϕ B Λ 00,(1) π B Λ 00,(1) 2ϕ B Λ 11 2π B Λ 11 2ϕ B Λ 10 lat In the SU(3) symmetric limit π 00,(1) ϕ 00,(1), π 11 ϕ 11 and π 10 ϕ 10 The renormalization matrix for all baryons is reduced to MS ϕ 00,(1) ZOϕ 11 Z Oϕ 12 Z Oϕ lat 13 ϕ 00,(1) 2ϕ 11 2ϕ 10 = Z Oϕ 21 Z Oϕ 22 Z Oϕ 23 Z Oϕ 31 Z Oϕ 32 Z Oϕ 33 2ϕ 11 2ϕ 10 Renormalization of three-quark operators for baryon distribution amplitudes 12 / 14

18 Test of the renormalization programme In the continuum DA lmn = On the lattice [dx] x l 1x m 2 x n 3 DA(x 1, x 2, x 3) = = DA (l+1)mn + DA l(m+1)n + DA lm(n+1) ϕ B 00,(1) [V A] B [V A] B [V A] B 001 [dx] (x 1 + x 2 + x 3)x l 1x m 2 x n 3 DA(x 1, x 2, x 3)? = [V A] B 000 f B π B Λ 00,(1) = T 100 B + T010 B + T001 B? = T000 B = ft B Renormalization of three-quark operators for baryon distribution amplitudes 13 / 14

19 Test of the renormalization programme In the continuum DA lmn = On the lattice [dx] x l 1x m 2 x n 3 DA(x 1, x 2, x 3) = = DA (l+1)mn + DA l(m+1)n + DA lm(n+1) ϕ B 00,(1) [V A] B [V A] B [V A] B 001 [dx] (x 1 + x 2 + x 3)x l 1x m 2 x n 3 DA(x 1, x 2, x 3)? = [V A] B 000 f B π B Λ 00,(1) = T 100 B + T010 B + T001 B? = T000 B = ft B [GeV ] [GeV ] ϕ N 00,(1) 4.5 f N ϕ Σ 00,(1) f Σ 4.0 π Σ 00,(1) ϕ Ξ 00,(1) 4.0 ft Σ f Ξ π Ξ 00,(1) f Ξ T ϕ Λ 00,(1) f Λ δm δm Renormalization of three-quark operators for baryon distribution amplitudes 13 / 14

20 Test of the renormalization programme In the continuum DA lmn = On the lattice [dx] x l 1x m 2 x n 3 DA(x 1, x 2, x 3) = = DA (l+1)mn + DA l(m+1)n + DA lm(n+1) ϕ B 00,(1) [V A] B [V A] B [V A] B 001 Numerical results [dx] (x 1 + x 2 + x 3)x l 1x m 2 x n 3 DA(x 1, x 2, x 3)? = [V A] B 000 f B π B Λ 00,(1) = T 100 B + T010 B + T001 B? = T000 B = ft B ϕ N 00,(1) /f N ϕ Σ 00,(1) /f Σ ϕ Ξ 00,(1) /f Ξ ϕ Λ 00,(1) /f Λ π Σ 00,(1) /f Σ T π Ξ 00,(1) /f Ξ T MS 0.988(35) 0.971(17) 0.963(13) 0.971(23) 0.965(17) 0.967(13) bare 0.842(30) 0.827(14) 0.820(11) 0.827(19) 0.822(14) 0.824(11) Renormalization of three-quark operators for baryon distribution amplitudes 13 / 14

21 Summary and outlook We carried out a nonperturbative renormalization in a RI /SMOM scheme, followed by a conversion to MS applying continuum perturbation theory at one-loop accuracy Constructed operators to make full use of the underlying symmetry groups First study to incorporate the full SU(3) baryon octet Renormalization matrices for the normalizations and the first moments of the leading twist DAs (and also for all higher twist normalization constants) Renormalization for the second moments of the leading twist DAs will be available for use in future studies Renormalization of three-quark operators for baryon distribution amplitudes 14 / 14

22 Backup slides Backup slides Renormalization of three-quark operators for baryon distribution amplitudes 1 / 3

23 Backup slides What about O D8? O D8: leading twist, LLL/RRR??? Built from Dirac structures corresponding to twist 4 distribution amplitude Ξ 4 Operator is non-zero but octet baryon matrix elements are zero Verified in the continuum and on the lattice e 11 [O D8 ] N 100 [V A] N 100 T100 N C(t) t/a Renormalization of three-quark operators for baryon distribution amplitudes 2 / 3

24 Backup slides Conversion to MS Renormalization condition ( ) L S = Λ S( ) ( O (i) fgh psmom Λ Born( ) O (i) fgh psmom mn i f,g,h α,β,γ m αβγ 1! = L RI( L Born) 1 n αβγ ) Define conversion matrix Λ MS( ) O m (i) psmom = Cmm Λ RI( ) O (i) psmom m Calculate conversion matrix ( L MS( L Born) ) ( 1 = C mm L RI( L Born) ) 1 = C mm 1 m n = C mn mn m n Renormalization of three-quark operators for baryon distribution amplitudes 3 / 3