Kaon Weak Matrix Elements in 2+1 Flavor DWF QCD

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1 Kaon Weak Matrix Element in 2 Flavor DWF QCD Renormalization an Phyical Vale Sh Li (for the RBC an UKQCD Collaboration) The XXV International Sympoim on Lattice Fiel Theory Regenbrg 7/27 Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 1 / 13

2 Otline 1 Phyical Parameter 2 Weak Matrix Element for (8,1) Operator Definition of operator Raw matrix element an btraction PQS v. PQN 3 Non-pertrbative Renormalization Mixing with Lower Dimenional Operator Calclation of Fll Mixing Matrix 4 Smmary Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 2 / 13

3 Phyical Parameter Phyical Parameter of the Lattice Configration Gage action Iwaaki, β = 2.13 Fermion action DWF, L = 16 Lattice ize (for NPR: ) a (3) GeV m re.38(4) Sea qark ma.5,.1 Valence qark ma.1,.5,.1,.2,.3,.4 Configration nmber 76, 74 Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 3 / 13

4 Weak Matrix Element for (8,1) Operator Definition of (8,1) Operator Definition of operator Pre (8,1) I = 1/2 Operator Q 3 ( α α ) L,, ( q β q β ) L Q 4 ( α β ) L,, ( q β q α ) L Q 5 ( α α ) L,, ( q β q β ) R Q 6 ( α β ) L,, ( q β q α ) R Operator that have (8,1)(1/2) an (27,1)(1/2) Part Q 1 ( α α ) L (ū β β ) L Q 2 ( α β ) L (ū β α ) L Q ( α α ) L,, e q ( q β q β ) L Q ( α β ) L,, e q ( q β q α ) L Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 4 / 13

5 Weak Matrix Element for (8,1) Operator Platea for (8,1) Operator Raw matrix element an btraction K + π + Platea (Q 6 ) t K = 5, t π = 59 ranom orce t rn = m =.1 m =.1.11 m =.1 m =.2 π + (1/2) O 6 K π + (1/2) O 6 K t t Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 5 / 13

6 Weak Matrix Element for (8,1) Operator Sbtraction for (8,1) Operator Raw matrix element an btraction (8,1) K π weak matrix element have qaratic ivergent part In PQχPT, the power ivergent part can be written a Θ (3, 3) (1 γ5 ) time a momentm-inepenent coefficient Fitting D Θ (8,1) E K D Θ (3, 3) E = 2 α (8,1) 2 K α (3, 3) Fitting mixing coefficient for Q 1 an Q 6 m 2 K m2 π + 2 α(8,1) 1 α (3, 3) m 2 K m2 π (chiral log) + (higher orer) m =.5 m =.5 K / γ 5 K e m =.1 m =.2 m =.3 m average K / γ 5 K m =.1 m =.2 m =.3 m average O 1 (1/2) -.3 O 6 (1/2) m - m m - m Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 6 / 13

7 Weak Matrix Element for (8,1) Operator Raw matrix element an btraction Sbtracte Matrix Element Sbtracte Dπ + Θ + (8,1) E K Dπ = + Θ + (8,1) E K + 2 α(8,1) 2 b α (3, 3) m2 K Dπ + Θ + (3, 3) E K = 4α(8,1) 1 f 2 m K m π + (chiral log) +... Sbtraction for Q 6 with m ea =.5 O nbtracte ivergence term btracte m = m K + b π + O 6 (1/2) m K m π with m re m =.1 m =.5 m =.1 m =.2 m =.3 m no m re m =.1 m =.5 m =.1 m =.2 m =.3 m Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 7 / 13

8 Weak Matrix Element for (8,1) Operator Raw matrix element an btraction Fitting Sbtracte (8,1) Amplite K + b π + O 6 (1/2) m K m π with m re m =.1 m =.5 m =.1 m =.2 m =.3 m no m re m =.1 m =.5 m =.1 m =.2 m =.3 m K + b π + O 6 (1/2) m K m π with m re m =.1 m =.5 m =.1 m =.2 m =.3 m no m re m =.1 m =.5 m =.1 m =.2 m =.3 m Slope of btracte K π matrix element m ea =.5 m ea =.1 with m re no m re with m re no m re Q (12) (12) (11) (11) 1 2 Q (14) (17) (13) (17) 1 1 Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 8 / 13

9 PQS v. PQN Weak Matrix Element for (8,1) Operator PQS v. PQN Smmation part i inglet in fll QCD, non-inglet in partially qenche K pi X Q 3 ( α α) L ` qβ q β L,, Q 4 ` X α β ` qβ q L α L,, X Q 5 ( α α) L ` qβ q β R,, Q 6 ` X α β ` qβ q L α R,, K K q q pi pi Comparion of PQS v. PQN (m ea =.5, btraction with m re) D lope of π + Q i K +E b PQS.9(3.7) (27) (54) (13) 1 1 PQN 4.1(3.8) (35) (56) (14) 1 1 Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 9 / 13

10 Non-pertrbative Renormalization Non-Pertrbative Renormalization Mixing with Lower-Dimenional Operator O cont,ren i = [ j Z ij (µ) Oj lat + k cj k Mixing with: Mixing with Lower Dimenional Operator (µ) Blat k 1 : m/a 2 2 /D + D + m + m : 1/a 2 ] + O (a) Conition: [ Ob ] Tr = amp Tr [i /p ] O b = amp γ 5 γ 5 Fit mixing coefficient to ma epenence an (ap) 2 epenence Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice 27 1 / 13

11 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ 5 γ 5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ 5 γ 5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ 5 γ 5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ 5 γ 5 γ5 γ5 γ 5 γ 5 γ5 γ5 γ5 γ5 γ5 γ5 γ5 γ 5 γ 5 γ5 γ5 γ 5 γ 5 γ5 γ 5 γ 5 γ5 γ 5 γ 5 γ5 γ 5 γ γ γ γ5 γ 5 γ 5 γ 5 γ γ5 γ 5 γ 5 γ 5 γ γ 5 γ 5 γ 5 γ γ 5 γ 5 γ 5 γ 5 Non-pertrbative Renormalization Non-Pertrbative Renormalization Mixing between For-Qark Operator Calclation of Fll Mixing Matrix Work in the bai that ha efinite SU(3) L SU(3) R propertie. Zq 2 Z ki P j { Ob i E j} = F kj To calclate P j { O i b E j} : (eg. Q 1 with Q 1 ) γ 5 γ γ ν Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice / 13 γ 5

12 NPR Coefficient Non-pertrbative Renormalization Calclation of Fll Mixing Matrix Extrapolate to chiral limit Removing mixing with atomatically Manally btract mixing with /D Z 1,1 /Z 2 q at ifferent momentm cale.95.9 Z q Z1,1.9 Z q Z1, nbtrate, linear fit btrate, linear fit nbtrate, qaratic fit btrate, qaratic fit m yn +m re (ap) 2 Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice / 13

13 Smmary Smmary 1 We have analyze K π an K weak matrix element on 2 flavor DWF lattice 2 Sbtraction of power ivergence work for (8,1) operator 3 PQS an PQN metho have cloe relt on 2 partially qenche lattice 4 Non-pertrbative Renormalization finihe 5 Work till in progre: Better chiral fit, Wilon coefficient, phyical meaning... Sh Li (for RBC an UKQCD) () Kaon WME in 2 Flavor DWF (II) Lattice / 13

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