Non-perturbative renormalization of Tensor currents in SF schemes

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1 Non-perturbative renormalization of ensor currents in SF schemes David Preti in collaboration with P. Fritzsch, C. Pena 33rd International Symposium on Lattice Field heory Kobe, 4 July 25

2 Outline Motivation ensor operator in SF formalism and O(a) improvement Perturbative -loop computation:. Renormalization constants and cutoff effects 2. Scheme matching and 2-loop SF Anomalous Dimension Non-Perturbative renormalization and running: 3. Nf= 4. Nf=2 5. Nf=3 (ongoing) Kobe, Lattice 25

3 Motivation. he only bilinear missing in the ALPHA collaboration NP renormalization program!!!! 2. ensor currents enter in some interesting physic processes like rare heavy meson decay processes. eg: B! K l + l B! l + l... [ Atoui et al. Eur.Phys.J. C74 (24) 5, 286 ] [ Horgan et al. Phys.Rev. D89 (24) 9, 945] [ Fermilab arxiv:57.68 [hep-ph] ] [ Horgan et al. Phys.Rev.Lett. 2 (24) 223 ] [ alk by Straub at Moriond ] Kobe, Lattice 25

4 ensor Operator O(a) improvement O(a) operator improvement is achieved by considering I µ = µ + V µ Where the improvement coefficient computed at -loop within the SF c =.896()C F g 2 + O(g 4 ) I k = k + V k [Sint et al. Nucl.Phys. B52 (997) ] Focusing on the electric part of the operator the improvement reads V since we restrict ourselves to zero momentum the second term in the parenthesis vanishes. Kobe, Lattice 25

5 SF correlation functions In the SF scheme, contracting the tensor operator with a boundary source we define the tensor correlator k (x )= a6 2 X h k (x )( (y) k (z))i y,z k I (x )=k (x k V (x) x and the boundary-to-boundary correlators k = f = a2 6L 6 a2 2L 6 X h( (y ) k (z )( (y) k (z))i y,z,y,z X h( (y ) 5 (z )( (y) 5 (z))i y,z,y,z Kobe, Lattice 25

6 SF Renormalization Scheme We impose mass independent renormalization conditions of the form Z ( ) (g, L/a) ki (L/2) f /2 k = k (L/2) f /2 k m =m c,g = defining a class of renormalization schemes that depend on the free parameter α and the angle θ entering spatial boundary conditions Following the standard SF iterative renormalization procedure, we define the SSF (s) he continuum SSF is then defined as (g 2 )=exp log(2) = ( Z p (g 2 ) g Z p (g 2 ) g Z(s) (u, a/l) = (g,a/2l) Z (s) (g, a/l) ḡ2 (L)=u ) dg (g ) (g ) dg (g ) (s) (u) =lim Kobe, Lattice 25 a! (s) (u, a/l) (u) =ḡ 2 (2L), u=ḡ 2 (L) composite op [Capitani et al. Nucl.Phys. B544 (999) ] coupling [ Della Morte et al. Nucl.Phys. B73 (25) ]

7 SFP Renormalization We impose mass independent renormalization conditions of the form Z ( ) (g, L/a) ki (L/2) f /2 k = k (L/2) f /2 k m =m c,g = Expanding in powers of g 2 Z (g, L/a) =+ X n= g 2n Z (n) (L/a) and k (x )= X n= g 2n k (n) (x ) at -loop the improvement for the tensor correlator reads k I (x )=k () (x )+g 2 k () (x )+ag 2 c k () V (x) x + O(ag 4 ) and the -loop renormalization constant for ( k() (x ) f () k () (x + ac () ) 2f () Z () (x, L/a) = = k () V (x) x k () (x ), F () = F () + F () bi + m c F [Sint and Weisz Nucl.Phys. B545 (999) ] Kobe, Lattice 25

8 -loop SSF cutoff effects In order to study the approach to the continuum limit we define the relative deviation (g, 2 L/a) k(g, 2 u=g 2 L/a) = SF (L) (u) = k g 2 + O(g 4 (u) ) [ Sint et al. JHEP 63 (26) 89 ] 2 k = Z() (2L/a) Z() (L/a) log(2) [ Sint et al. Nucl.Phys. B545 (999) ] k for O(a) improved fermions theta= theta=.5, f theta=.5, k theta=., f theta=., k a/l Kobe, Lattice 25

9 -loop SSF cutoff effects In order to study the approach to the continuum limit we define the relative deviation (g, 2 L/a) k(g, 2 u=g 2 L/a) = SF (L) (u) = k g 2 + O(g 4 (u) ) k = Z() [ Sint et al. JHEP 63 (26) 89 ] (2L/a) Z() (L/a) log(2) [ Sint et al. Nucl.Phys. B545 (999) ].5.5 k, f, csw=, k, csw= k, f, csw= k, k, csw= k k at = a/l Kobe, Lattice 25

10 SFP Renormalization Considering the asymptotic expansion in powers of a/l Z () (L/a) ' X a L r + s log L a = and focusing on = coefficients: [Bode et al. Nucl.Phys. B68 (2) 48] For the two schemes and for the three values of theta we checked s = (g) = g 2 ( () + g 2 () + g 4 (2) + O(g 6 )) (g) = g 3 (b () + g 2 b () + g 4 b (2) + O(g 6 )) () = 2C F (4 ) 2.5 /s f /s k.().().(7).(7).97(3).97(5) Kobe, Lattice 25

11 SFP Renormalization Considering the asymptotic expansion in powers of a/l Z () (L/a) ' X a L r + s log L a = [Bode et al. Nucl.Phys. B68 (2) 48].5.5. Z (), theta=. Z () log(l/a), theta=. Z () f, theta=.5 Z () f log(l/a), theta=.5 Z () k, theta=.5 Z () k log(l/a), theta=.5 Z () f, theta=. Z () f log(l/a), theta=. Z () k, theta=. Z () k log(l/a), theta= a/l Kobe, Lattice 25

12 SF anomalous dimensions are given by Scheme Matching where () = () SF,ref = () SF,lat () SF = () ref +2b () ref,lat and () () () g () g =2b log(µl) 4 (c, + c, N f ) [Sint and Sommer Nucl.Phys. B465 (996) 7-98] and the matching coefficient SF-LA are the finite parts of the -loop renormalization coefficients considering two ref schemes () SF,lat = r ( MS ref = RI [Skouroupathis et al. Phys.Rev. D79 (29) 9458 ] [Capitani et al. Nucl.Phys. B593 (2) ] [ Gracey Nucl.Phys. B667 (23) ].5 (),f SF.433(3)-.6764(42)N f.6945(55)-.224(6)n f.2693(28)-.3364(84)n f (),k SF.433(3)-.6764(42)N f.636(65)-.886(6)n f.956(6)-.783(5)n f Kobe, Lattice 25

13 SF anomalous dimensions are given by Scheme Matching where () = () SF,ref = () SF,lat () SF = () ref +2b () ref,lat and () () () g () g =2b log(µl) 4 (c, + c, N f ) [Sint and Sommer Nucl.Phys. B465 (996) 7-98] and the matching coefficient SF-LA are the finite parts of the -loop renormalization coefficients considering two ref schemes.5 () SF,lat = r ( MS ref = RI (),f SF / (),k SF /.8486(8) (25)N f.43(33)-.32()n f.594(6)-.9(5)n f.8486(8) (25)n f.4376(33)-.()n f.58()-.46(3)n f Kobe, Lattice 25 [Skouroupathis et al (29)] [hep-lat] [Capitani et al. (2)] hep-lat/74 [Gracey (23)] () MS /.9.9N f

14 NP computation of SSF Z ( ) (g, L/a) ki (L/2) f /2 k = k (L/2) f /2 k g = Reminding the definition of the SSF (s) Z(s) (u, a/l) = (g,a/2l) Z (s) (g, a/l) ḡ2 (L)=u We have performed NP calculation for several values of Nf N f # coup status 4.5 preliminary preliminary 3 8(SF).5 ongoing-preliminary Kobe, Lattice 25 * talks by S. Sint & P. Fritzsch

15 Continuum Extrapolation N f = (s) (u) =lim a! (s) (u, a/l) UV f f a/l a/l IR Kobe, Lattice 25

16 Continuum Extrapolation N f = (s) (u) =lim a! (s) (u, a/l) UV k k a/l a/l IR Kobe, Lattice 25

17 Continuum Extrapolation N f =2 (s) (u) =lim a! (s) (u, a/l) UV f.5 k (a/l) (a/l) 2 IR Kobe, Lattice 25

18 Continuum Extrapolation N f =3 (s) (u) =lim a! (s) (u, a/l) () (u, a/l) = (u, a/l) + k (a/l)u.25 (u,a/l) () (u,a/l).2 (.,a/l) (a/l) 2 Kobe, Lattice 25

19 Continuum Extrapolation N f =3 (s) (u) =lim a! (s) (u, a/l) () (u, a/l) = (u, a/l) + k (a/l)u.35 (u,a/l) () (u,a/l).3 (.488,a/L) (a/l) 2 Kobe, Lattice 25

20 (s) Continuum Extrapolation N f =3 (u) =lim a! (s) (u, a/l) () (u, a/l) = (u, a/l) + k (a/l)u.55 (u,a/l) () (u,a/l).5.45 (2.2,a/L) (a/l) 2 Kobe, Lattice 25

21 SSF N f = (s) () (u) =+ s u + s (2) u 2 + s (3) u 3 + O(u 4 ) apple 2 = () log(2) + 2 ( () ) 2 () + b (log(2)) 2 = () log(2) f k u u Kobe, Lattice 25

22 SSF N f =2 (s) () (u) =+ s u + s (2) u 2 + s (3) u 3 + O(u 4 ) apple 2 = () log(2) + 2 ( () ) 2 () + b (log(2)) 2 = () log(2) f k u u Kobe, Lattice 25

23 SSF N f =3 (s) () (u) =+ s u + s (2) u 2 + s (3) u 3 + O(u 4 ) apple 2 = () log(2) + 2 ( () ) 2 () + b (log(2)) 2 = () log(2) u Kobe, Lattice 25

24 Running he RG evolution between two scales is defined as ( Z ) ḡ(µ2 ) U(µ 2,µ )=exp dg (g) Z(g,aµ 2 ) =lim (g) a! Z(g,aµ ) ḡ(µ ) Once defined the perturbative evolution coefficient as ĉ(µ) = O RGI O(µ) = appleḡ2 (µ) 4 2b exp ( Z ḡ(µ) dg (g) (g) ) b g We can match it to the NP computations to an high energy µ P =2 n µ HAD Where U is given by products of SSFs ĉ(µ HAD )=ĉ(µ P )U(µ P,µ HAD ) ny [ (u i )] i= Kobe, Lattice 25

25 Running N f = f k ORGI/O(µ). ORGI/O(µ)..9.9 /2.8 2/2 2/ µ/ /2.8 2/2 2/ µ/ f k O RGI O(µ had ).94(7).4(6) µ P / = MeV GeV [ S. Capitani et al., Nucl. Phys. B544 (999) 669 ] Kobe, Lattice 25

26 Running N f =2 f k ORGI/O(µ).2. ORGI/O(µ).2..9 /2.8 2/2 2/ µ/.9 /2.8 2/2 2/ µ/ f k O RGI O(µ had ).4(9).45(7) µ P = MeV 49.GeV Kobe, Lattice 25

27 P running systematics N f = f k ORGI/O(µ) /2 2/2 2/3 3/3 2/3 MSbar ORGI/O(µ) /2 2/2 2/3 3/3 2/3 MSbar µ/ µ/ If () = (2) () O f RGI O(µ P ) =.327(6) O k RGI O(µ P ) =.333(4) P systematics is not negligible! Kobe, Lattice 25

28 Conclusions Perturbative Side: We have computed -loop renormalization coefficients and studied cutoff effects as a function of the angle theta From a matching with perturbative schemes we extracted the NLO anomalous dimensions in SF schemes Non-Perturbative Side: We have computed renormalization constants for Nf=,2,3 (ongoing) NP running over various orders of magnitude via recursive procedure through SSF RGI matching factors have been computed. Nf=3 in progress in parallel with the mass (Patrick Fritzsch s alk)! Kobe, Lattice 25

29 Backup