Lattice Measurement of the Delta I=1/2 Contribution to Standard Model Direct CP-Violation in K → ππ Decays at Physical Kinematics: Part II Daiqian Zhang Dept. of Physics, Columbia University 23 June 2014 @ Columbia University, New York.
RBC & UKQCD Collaboration ( K → ππ subgroup) ◮ BNL ◮ Taku Izubuchi ◮ Tata Institute of ◮ Chulwoo Jung Fundamental Research ◮ Christoph Lehner ◮ Andrew Lytle ◮ Amarjit Soni ◮ Trinity College ◮ Columbia ◮ Nicholas Garron ◮ Ziyuan Bai ◮ University of Southampton ◮ Norman Christ ◮ Chris Sachrajda ◮ Christopher Kelly ◮ Tadeusz Janowski ◮ Robert Mawhinney ◮ University of Edinburgh ◮ Jianglei Yu ◮ Daiqian Zhang ◮ Peter Boyle ◮ Julien Frison ◮ Connecticut ◮ Tom Blum
UKQCD RBC Rudy Arthur (Odense) Ziyuan Bai (Columbia) Peter Boyle (Edinburgh) Thomas Blum (U Conn/RBRC) Luigi Del Debbio (Edinburgh) Norman Christ (Columbia) Shane Drury (Southampton) Xu Feng (Columbia) Jonathan Flynn (Southampton) Tomomi Ishikawa (RBRC) Julien Frison (Edinburgh) Taku Izubuchi (RBRC/BNL) Nicolas Garron (Dublin) Luchang Jin (Columbia) Jamie Hudspith (Toronto) Chulwoo Jung (BNL) Tadeusz Janowski Taichi Kawanai (RBRC) (Southampton) Chris Kelly (RBRC) Andreas Juettner (Southampton) Hyung-Jin Kim (BNL) Ava Kamseh (Edinburgh) Christoph Lehner (BNL) Richard Kenway (Edinburgh) Jasper Lin (Columbia) Andrew Lytle (TIFR) Meifeng Lin (BNL) Marina Marinkovic Robert Mawhinney (Columbia) (Southampton) Greg McGlynn (Columbia) Brian Pendleton (Edinburgh) David Murphy (Columbia) Antonin Portelli (Southampton) Shigemi Ohta (KEK) Thomas Rae (Mainz) Eigo Shintani (Mainz) Chris Sachrajda (Southampton) Amarjit Soni (BNL) Francesco Sanfilippo Sergey Syritsyn (RBRC) (Southampton) Oliver Witzel (BU) Matthew Spraggs Hantao Yin (Columbia) (Southampton) Jianglei Yu (Columbia) Tobias Tsang (Southampton) Daiqian Zhang (Columbia)
Outline 1. Motivation 2. Method ◮ Weak matrix elements. ◮ Decay amplitude. 3. Current results. ◮ ππ phase shift. ◮ K → ππ ( I = 0) weak matrix elements, decay amplitude A 0 . 4. Conclusion
Motivation ◮ First ab initio calculation of direct CP-violation (in K → ππ ). current experiment result: Re ( ǫ ′ /ǫ ) = 1 . 65(26) × 10 − 3 ǫ ′ = ie i ( δ 2 − δ 0 ) ReA 2 [ ImA 2 − ImA 0 √ ] (1) ReA 0 ReA 2 ReA 0 2 current lattice result: Only has Re ( A 2 ) and Im ( A 2 ), both with < 10% error. (mainly from stat and Wilson coefficients) Once we obtain A 0 with ≈ 20% error, could compare ǫ ′ with experiments.
Weak matrix elements � ππ | Q i | K � ◮ G-parity Boundary introduces even larger numbers of contractions. − γ 4 γ 2 ¯ d T u ¯ � ππ � = ¯ u γ 5 d d γ 5 u u T γ 4 γ 2 ¯ d ¯ + ¯ u γ 5 d d γ 5 u Not like single pion, the 10 matrix elements � ππ | Q i | K � each contains 256 possible contractions. One has to figure out the linear combination: 256 � � ππ | Q i | K � = c ij [ Contraction j ] j =1
Weak matrix elements � ππ | Q i | K � ◮ G-parity boundary introduces subtlety in momentum directions. γ 4 γ 2 ¯ − γ 4 γ 2 ¯ d T − u d T Y Under G-parity boundary con- u γ 4 γ 2 ¯ d T − u dition, the degrees of freedom doubles in momentum space. γ 4 γ 2 ¯ d T − γ 4 γ 2 ¯ d T u Allowed quark momentum are in ’diagonal’ direction. X p y p y p y p x p x p x No G-parity twist 1 G-parity twist 2 G-parity twists
Weak matrix elements � ππ | Q i | K � ◮ Reducing errors from ’disconnected’ diagrams. T K T π Since the ππ ( I = 0) state cou- ples with vacuum, the ampli- Q i tude doesn’t decay as separa- tion increases, small fluctua- tion could result in huge error. ⇓ In order to reduce the ππ ( I = 0) to vacuum coupling, we π chose to use the localized me- K Q i son source, and separate the π two pions in time direction.
Weak matrix elements � ππ | Q i | K � ◮ Using localized source (all-to-all propagators). π π Shaded boxes are where the K K random sources have been π π used. � Tr { γ µ (1 − γ 5 ) L ( � x op , t op ; t π ) γ 5 L w ( t π ; t π ′ ) γ 5 L w ( t π ′ ; t K ) γ 5 � x op x op , t op ) } · Tr { γ µ (1 − γ 5 ) L ( � S ( t K ; � x op , t op ; � x op , t op ) } † γ µ (1 − γ 5 ) v i † γ µ (1 − γ 5 ) v j { w ′ m � x op } · { w j x op } · π ik t π π kl π K lm = x op x op t ′ t K � x op The complexity is ( Mode Number ) 2 × ( Volume ) × ( T size ) × 144 Mode number for light quark is 2436, volume is 32 3 × 64, T is 64.
From M i = � ππ | Q i | K � to decay amplitude Bare M i on Lattice Finite volume correction [1] ⇓ M i in infinite volume Lat → RI/SMOM matching at 1.52GeV [2] ⇓ M i in RI/SMOM scheme RI/SMOM → MS matching at 1.52GeV [3] ⇓ M i in MS scheme times MS Wilson coefficients at 1.52GeV [4] ⇓ Decay amplitude A 0 [1] Laurent Lellouch et al. HEP-LAT/0003023; [2] C.Sturm et al. ARXIV:0901.2599 [3] Christoph Lehner et al. ARXIV:1104.4948; [4] Buchalla et al. HEP-PH/9512380;
Lattice setup and measurement time ◮ Used 32 3 × 64 lattice, DWF+IDSDR action, a − 1 ≈ 1 . 38 GeV , (4 . 6 fm ) 3 box, physical pion and kaon. With G-parity boundary in X,Y,Z directions. ◮ Measurement time on IBM BG/Q 512-node machine: time flops Generating eigenmodes 3.6h 22 Gflops/Node Quark propagator (CG) 7.5h 38 Gflops/Node Meson field contraction 5h ∼ 20 Gflops/Node Total ∼ 17h
Result: Meson spectrum � E 2 π − p 2 E π m K E ππ ( I =0) π Lat 0.19834(67) 0.1021(12) 0.35490(32) 0.3888(86) MeV 273.71(92) 140.9(17) 489.76(44) 537(12) 2 m π 0.5 I=2, 3 G.B.C. I=0, 3 G.B.C. 0.4 Phase Shift (radian) 0.3 0.2 0.1 0 -0.1 300 400 500 600 700 800 ππ center-of-mass energy (MeV) Figure : Phase shift
Result: Weak matrix elements and decay amplitude � ππ | Q 2 | K � = (1 . 30 ± 0 . 96) × 10 − 3 , using 50 configurations, fitting from 4 to 8: K π π 0.02 � ππ | Q 2 | K � 0.015 0.01 0.005 0 -0.005 -0.01 0 2 4 6 8 10 12 14 16 T
Result: Weak matrix elements and decay amplitude � ππ | Q 6 | K � = ( − 1 . 35 ± 0 . 37) × 10 − 2 , using 50 configurations, fitting from 4 to 8: K π π � ππ | Q 6 | K � 0.04 0.02 0 -0.02 -0.04 0 2 4 6 8 10 12 14 16 T
Conclusion ◮ K → ππ ( I = 0) decay amplitude is underway, with physical π , K , and physical kinematics. Estimate 100 more measurements in order to get 50% error for A 0 . The measurement will take a few months. ◮ Future work: ◮ estimate lattice artefacts / do the same computation on a finer lattice. ◮ Match at higher scale in MS scheme / use dynamic charm quark.
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