Isospin breaking corrections to leptonic decay rates on the lattice James Richings RBC/UKQCD Internal Seminar 2018 11/12/2018 James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 1 / 22
RBC/UKQCD Collaboration BNL and BNL/RBRC University of Liverpool Tianle Wang Nicolas Garron Yasumichi Aoki (KEK) Evan Wickenden Mattia Bruno MIT Yidi Zhao Taku Izubuchi David Murphy Yong-Chull Jang University of Connecticut Chulwoo Jung Peking University Christoph Lehner Tom Blum Xu Feng Meifeng Lin Dan Hoying (BNL) Aaron Meyer Luchang Jin (RBRC) University of Southampton Hiroshi Ohki Cheng Tu Jonathan Flynn Shigemi Ohta (KEK) Vera Gülpers Edinburgh University Amarjit Soni James Harrison Peter Boyle Andreas Jüttner Guido Cossu UC Boulder James Richings Luigi Del Debbio Oliver Witzel Chris Sachrajda Tadeusz Janowski Richard Kenway Columbia University Stony Brook University Julia Kettle Jun-Sik Yoo Fionn Ó hÓgáin Ziyuan Bai Sergey Syritsyn (RBRC) Brian Pendleton Norman Christ Antonin Portelli Duo Guo York University (Toronto) Tobias Tsang Christopher Kelly Renwick Hudspith Azusa Yamaguchi Bob Mawhinney Masaaki Tomii KEK Jiqun Tu Bigeng Wang Julien Frison James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 2 / 22
Outline � Lattice QCD introduction � Isospin breaking effects � Isospin breaking corrections to the Pion � Isospin breaking corrections to leptonic decay rates � Sea quark effects James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 3 / 22
Lattice QCD Introduction Lattice QCD � Path integral: �O� = 1 � D [ ¯ ψ, ψ, U ] O e i S Z � Free fermion action: � d 4 x � ¯ ψ f ( i γ µ ∂ µ − m f ) ψ f S F = f � Lattice naive free fermion action: ψ ( n + ˆ µ ) − ψ ( n − ˆ µ ) � � S F = a 4 � � ¯ ψ f ( n ) γ µ − m ψ f ( n ) 2 a n ∈ Λ f � Fermion action: ψ ( n + ˆ µ ) U µ ( n ) − U − µ ( n ) ψ ( n − ˆ µ ) � � S F = a 4 � � ¯ ψ f ( n ) γ µ − m ψ f ( n ) 2 a n ∈ Λ f James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 4 / 22
Lattice QCD Introduction Lattice QCD: gauge fields and the path integral � Gauge links: U µ ( x ) = exp ( iqaA µ ( x )) � Gauge action: U † µ ( x + a ˆ ν ) U µν ( x ) = exp ( − iqa 2 F µν ) U † U ν ( x + a ˆ µ ) ν ( x ) � � S g = 2 1 − 1 � � 2 [ U µν ( x ) + U † µν ( x )] Tr g 2 0 x U µ ( x ) µ ≤ ν x � Euclidean path integral: < O > = 1 � ψ ] e − S Lat [ U ,ψ, ¯ D [ U , ψ, ¯ ψ ] O [ ψ, ¯ ψ ] Z � p [ U ] = 1 � � � e − S g < O > = D [ U ] p ( U ) O ( U ) , det ( D f [ U ]) Z f N 1 O ( U n ) + O ( N − 1 / 2 ) � < O > = lim N N →∞ n = 1 James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 5 / 22
Lattice QCD Introduction Example Calculation: Pion mass � Pion correlator: < 0 |O π + ( x ) O † π + ( y ) | 0 > d ( y ) γ 5 u ( y )] † | 0 � = � 0 | ¯ d ( x ) γ 5 u ( x )[ ¯ γ 5 γ 5 = � 0 | ¯ d ( x ) γ 5 u ( x )¯ u ( y ) γ 5 d ( y ) | 0 � = − tr [ γ 5 S d ( x , y ) γ 5 S u ( y , x )] < 0 |O π ( t ) O π ( 0 ) † | 0 > = 1 � � � e − S [ U ,ψ, ¯ D [ U , ψ, ¯ ψ ] ψ ] tr [ γ 5 S ( x , 0 ) γ 5 S ( 0 , x )] Z < 0 |O π ( t ) O π ( 0 ) † | 0 > = � < 0 |O π ( t ) | n >< n |O π ( 0 ) † | 0 > n | < 0 |O π ( 0 ) | n > | 2 e − E n t = | A π | 2 e − m π t ( 1 + O ( e − ∆ Et )) � = n � Effective mass: � Correlator: C π = | A π | 2 ( e − m π t + e − m π ( T − t ) ) � � C π [ t ] m eff = ln C π [ t + 1 ] James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 6 / 22
Lattice QCD Introduction The propagator � Inverting the Dirac operator: ( / D + m ) ψ = η D ] size = ( 48 3 [ / space × 96 time × 12 spin / colour × 2 complex × 8 double ) 2 The propagator is the sum over paths of link variables. � Low modes: D f + m ] − 1 = [ / ψ λ | ] − 1 + λ − 1 | ψ λ � � ¯ � λ | ψ λ � � ¯ � S f = [ / D f + m − ψ λ | λ λ � Deflation: 3000 m=0.005 m=0.01 2500 Number of CG iterations Using the low modes of the Dirac 2000 operator it is possible to form a better 1500 initial guess of the propagator. 1000 500 0 0 100 200 300 400 500 600 Number of low modes James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 7 / 22
Lattice QCD Introduction Lattice QCD: Calculations u � Spectral quantities: γ 5 γ 5 ¯ d ± / ± = + + FLAG average for + ETM 14E FNAL/MILC 14A u + ETM 13F HPQCD 13A MILC 13A = MILC 11 (stat. err. only) l + ETM 10E (stat. err. only) W + FLAG average for = + BMW 16 RBC/UKQCD 14B RBC/UKQCD 12 Laiho 11 MILC 10 � Leptonic decays: π + JLQCD/TWQCD 10 RBC/UKQCD 10A + PACS-CS 09 BMW 10 = JLQCD/TWQCD 09A (stat. err. only) MILC 09A MILC 09 ν Aubin 08 PACS-CS 08, 08A RBC/UKQCD 08 HPQCD/UKQCD 07 NPLQCD 06 ¯ MILC 04 d FLAG average for = ETM 14D (stat. err. only) ALPHA 13A = BGR 11 ETM 10D (stat. err. only) ETM 09 QCDSF/UKQCD 07 1.14 1.18 1.22 1.26 + ( ) + FLAG average for = + + + ETM 16 FNAL/MILC 13E ν ¯ l − = FNAL/MILC 13C FLAG average for = + JLQCD 16 RBC/UKQCD 15A + RBC/UKQCD 13 FNAL/MILC 12I JLQCD 12 = � Semi-leptonic decays: s u JLQCD 11 RBC/UKQCD 10 RBC/UKQCD 07 FLAG average for = ETM 10D (stat. err. only) ETM 09A = QCDSF 07 (stat. err. only) k 0 π + RBC 06 JLQCD 05 JLQCD 05 non-lattice Kastner 08 Cirigliano 05 Jamin 04 Bijnens 03 Leutwyler 84 ¯ 0.95 0.97 0.99 1.01 d James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 8 / 22
Lattice QCD Introduction CKM matrix: Precision of Lattice QCD Unitarity constraints on the CKM ma- trix are an important bound on the BSM theories. | | | | = + + = + + FLAg average for FLAG average for + ETM 16 ETM 16kpy ETM 14E ETM 14E FNAL/MILC 14A FNAL/MILC 14A FNAL/MILC 13E FNAL/MILC 13E + FNAL/MILC 13C FNAL/MILC 13C ETM 13F ETM 13F HPQCD 13A HPQCD 13A MILC 13A MILC 13A = MILC 11 (stat. err. only) MILC 11 ETM 10E (stat. err. only) ETM 10E = + + = FLAG average for r A L o f e g r v a G F e a BMW 16ulb BMW 16 RBC/UKQCD 15A RBC/UKQCD 14hfa RBC/UKQCD 14B RBC/UKQCD 14B RBC/UKQCD 13 RBC/UKQCD 13 RBC/UKQCD 12 RBC/UKQCD 12 FNAL/MILC 12I FNAL/MILC 12I JLQCD 12 JLQCD 12 Laiho 11 Laiho 11 JLQCD 11 JLQCD 11 MILC 10 MILC 10 JLQCD/TWQCD 10 JLQCD/TWQCD 10 + RBC/UKQCD 10A RBC/UKQCD 10A RBC/UKQCD 10 RBC/UKQCD 10 BMW 10 BMW 10 PACS-CS 09 PACS-CS 09 = JLQCD/TWQCD 09A (stat. err. only) JLQCD/TWQCD 09A MILC 09A MILC 09A MILC 09 MILC 09 Lattice QCD is now at the precision of one Aubin 08 Aubin 08 PACS-CS 08 PACS-CS 08 + RBC/UKQCD 08 RBC/UKQCD 08 RBC/UKQCD 07 RBC/UKQCD 07 HPQCD/UKQCD 07 HPQCD/UKQCD 07 NPLQCD 06 NPLQCD 06 + MILC 04 MILC 04 percent in the light sector: FLAG average for = r A L e g f o a r e v G a = F = ETM 14D (stat. err. only) ETM 14D ALPHA 13A ALPHA 13A BGR 11 BGR 11 + ETM 10D (stat. err. only) ETM 10D ETM 10D (stat. err. only) ETM 10D = ETM 09A ETM 09A ETM 09 ETM 09 = QCDSF/UKQCD 07 QCDSF/UKQCD 07 QCDSF 07 (stat. err. only) QCDSF 07 RBC 06 RBC 06 JLQCD 05 JLQCD 05 HFAG 14 decay F A G 1 4 y d e c a H Hudspith 15 decay and + H h 1 5 u d s p i t 1 5 d e c a y a n d + H Maltman 09 decay and + e + n M 0 9 M a l m a t y d n 0 a a c d 9 = Gamiz 08 decay m G 0 8 a G i e a z c y d 8 0 Hardy 15 nuclear decay a r d y 1 5 n u l e a r d e c c a y H 0.22 0.23 0.973 0.975 James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 9 / 22
Isospin breaking corrections Isospin breaking corrections Two sources of isospin breaking (IB) corrections: � QED isospin breaking corrections: - Difference in the electromagnetic charge on the up and down type quarks. � Strong isospin breaking corrections: - Difference in the up and down quark masses. � How to include these IB effects? - These effects by power counting are of order 1 %: ( m u − m d ) α ≈ 1 / 137 ≈ 1 % , ≈ 1 % Λ QCD - This give us an expansion parameter for each contribution. � What makes this difficult? - Small effect - QED difficult - Expensive James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 10 / 22
Isospin breaking corrections Lattice QFT with Isospin breaking corrections Both of the IB corrections can be calculated in terms of a pertubative expansion: � QED Isospin breaking: - Order alpha correction can be calculated by using a perturbative approach: � �O� = �O� 0 + e 2 ∂ 2 � + O ( α 2 ) ∂ e 2 �O� � 2 � � e = 0 - If the operator O is α independent then the correction has the form: �O� = �O� 0 − e 2 q f q f ′ ν ( y ) � 0 ∆ µν ( x − y ) − ( eq f ) 2 �O V c µ ( x ) V c �O T µ ( x ) � 0 ∆ µµ 2 2 [ G.M.de Divitiis et al.Phys.Rev.D87(2013)114505 ] � Strong Isospin breaking at first order: � � �O� m u � = m d = �O� m u = m d +( m d − m u ) ∂ � � ∂ m �O� = �O� m u = m d +( m d − m u ) �SO� � � � � � � m u = m d m u = m d James Richings (Inernal Seminar 2018) IB Corrections to leptonic decays 11/12/2018 11 / 22
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