Direct CP violation and the ΔI = 1/2 rule in K ππ decay ...

Direct CP violation and the ΔI = 1/2 rule in K → ππ decay

from the Standard ModelMasaaki Tomii (University of Connecticut)

PRD102,054509 (RBC & UKQCD Collaborations)Co-authors: R. Abbott, T. Blum, P.A. Boyle, M. Bruno, N.H. Christ, D. Hoying, C. Jung,

C. Kelly, C. Lehner, R.D. Mawhinney, D.J. Murphy, C.T. Sachrajda, A. Soni, T. Wang

2

The RBC & UKQCD collaborations

BNL and BNL/RBRC

Ryan AbbotNorman ChristDuo GuoBob Mawhinney

University of Connecticut

Luigi Del DebbioFelix ErbenVera GülpersTadeusz JanowskiJulia KettleMichael MarshallFionn Ó hÓgáinAntonin PortelliTobias TsangAndrew YongAzusa Yamaguchi

Nicolas Garron

Nils AsmussenJonathan FlynnRyan HillAndreas JüttnerJames RichingsChris Sachrajda

Julien Frison

Xu Feng

Bigeng WangTianle WangYidi Zhao

UC Boulder

Yasumichi Aoki (KEK)Peter Boyle (Edinburgh)Taku IzubuchiYong-Chull JangChulwoo JungChristopher KellyMeifeng LinAaron MeyerHiroshi OhkiShigemi Ohta (KEK)Amarjit Soni

Oliver Witzel

Columbia University

Tom BlumDan Hoying (BNL)Luchang Jin (RBRC)Cheng TuMasaaki Tomii

Edinburgh University

University of Southampton

Peking University

University of Liverpool

UAM Madrid

Stony Brook University

Jun-Sik YooSergey Syritsyn (RBRC)

MIT

David Murphy

Mattia BrunoCERN

Christoph Lehner (BNL)

University of Regensburg

Contents

Introduction

‣ CP violation & ΔI = 1/2 rule in K → ππ

‣ Lattice approach

‣ Previous RBC/UKQCD result in 2015

K → ππ matrix elements

Operator renormalization

On going projects

3

CP violation in K → ππKL → ππ invalid in CP limitCP violated in reality

2 important measures ε’ & ε of CP violation‣ Re (ε’/ε)exp = 16.6(2.3) x 10-4‣ Can it be explained by the SM?Key to understanding matter/anti-matter asymmetry

4

|KL> = |K2> + ε|K1>CP odd CP even

|ππ>CP even

ε’ εdirect

CPV

indirect

CPV

I = 0 & I = 2 decay modes

Isospin-definite amplitudes

Convenient decomposition especially for isospin symmetric calculation

A2 precisely calculated (PRL108 (2012) 141601, PRD91 (2015) 074502)

‣ 2 lattice spacings: 2.36 GeV, 1.73 GeV → continuum limit taken

‣ Re A2 = 1.50(4)stat(14)sys x 10-8 GeV, Im A2 = -6.99(20)stat(84)sys x 10-13 GeV

ε’: needs both of A0 & A25

h(⇡⇡)I=0| =p

1/3h⇡0⇡0|+p2/3h⇡+⇡�|

<latexit sha1_base64="8vUhVQMCfyTCqSSqwKHWTgAB2zk=">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</latexit>

h(⇡⇡)I=2| = �p

2/3h⇡0⇡0|+p

1/3h⇡+⇡�|

<latexit sha1_base64="8YHfmAuvd5w+m+cmcYEshxzrxws=">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</latexit>

,

AI = h(⇡⇡)I|HW|Ki

<latexit sha1_base64="YnzHiwrsfc3XzE/GmyCX5f/8q5E=">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</latexit>

cf: (Re A2)exp = 1.479(4) x 10-8 GeV

I3=0

The ΔI = 1/2 rule

Experimental fact

Estimation at LO pQCD: Re A0 = 2 Re A2

Extra factor x10 coming from full QCD or BSM?

6

ReA0

ReA2= 22.45(6)

<latexit sha1_base64="1vRiCpDuNfu1Bs41SgiBQDXyR5g=">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</latexit>

: large suppression of ΔI = 3/2 (A2) mode

Approach to weak processesLarge scale separation

‣ Weak interactions: mW = 80 GeV, mZ = 91 GeV

‣ QCD scale: ΛQCD ≈ 300 MeV

Low-energy effective theory

‣ contributions from heavy particles: effective interactions

74-Fermi interactions

Lattice calculation of MEsEffective Hamiltonian

Lattice calculation of ‣ can involve all contributions from QCD

Renormalization‣ Non-perturbatively

‣ Perturbatively

8

HW =X

i

wRi (µ)O

Ri (µ)

<latexit sha1_base64="hvv/BlaWgINaixVRtNGF7z1I/9g=">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</latexit>

Wilson coefficients Information of t, W, Z, … Calculated by pQCD

Effective operators (e.g. 4-Fermi) Composed of light particles MEs calculated by lattice QCD

Mlati = hout|Olat

i |ini

<latexit sha1_base64="Cn2JoJP8afXLGkIWWI27eiYH7pc=">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</latexit>

Olati ! O

Ri (µ), M

lati ! M

Ri (µ)

<latexit sha1_base64="sG/49oCrU9jexmVahA5+CJoUZYQ=">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</latexit>

wMSi (µ) ! wR

i (µ)

<latexit sha1_base64="BXW8gHEhEOBzATAU2uCVR5vfoWg=">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</latexit>

ΔS = 1 effective operators

9

13 / 37

Operators and Wilson Coe0cients

[Rev.Mod.Phys. 68 (1996) 1125-1144]

[J.Phys.Conf.Ser. 800 (2017) 1, 012008]

● 10 ΔS=1 Weak effective operators:

– Q1-Q

2 “current-current” operators, dominant

contributions to real parts of K→ππ amplitudes.

– Q3-Q

6 “QCD penguin” operators, dominant

contributions to imaginary (CP-violating) parts of amplitudes.

– Q7-Q

10 “EM penguin” operators.

● Wilson coefficients zi, y

i capture high-energy

behavior.

● Involve running from MW down to charm scale

O(1.2 GeV) in QCD PT with some E&M effects (EM penguins).

● Determined to NLO in MSbar scheme.

● NNLO expected in near-future

● Use of PT near charm threshold significant sys. err.

1m30

Current-current operators

QCD penguin operators

EW penguin operators

(sq)V�A(q0q00)V±A = s�µ(1� �5)q

0 · q0�µ(1± �5)q00

<latexit sha1_base64="/UdJtzOrX82MUkc3FFGCKeWYaEo=">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</latexit>

↵,�: color indices

<latexit sha1_base64="Rxc233Y9e57UDATrq9/XR4pVnxY=">AAACCnicbVC5TsNAFFyHK4TLQEmzkCBRoMiOkDiqCBrKIJFDiqPoefOcrLJeW941UhSlpuFXaChAiJYvoONv2BwFJEw1mnlv5+34seBKO863lVlaXlldy67nNja3tnfs3b2aitKEYZVFIkoaPigUXGJVcy2wEScIoS+w7vdvxn79ARPFI3mvBzG2QuhKHnAG2kht+9BTAS14IOIenHo+aihc0cmzlMsOZ6jadt4pOhPQReLOSJ7MUGnbX14nYmmIUjMBSjVdJ9atISSaM4GjnJcqjIH1oYtNQyWEqFrDSeaIHhulQwMTH0RSTy/5vTGEUKlB6JvJEHRPzXtj8T+vmergojXkMk41SjYNClJBdUTHvdAOT5BpMTAEWMLNrZT1IAGmTXs5U4I7/+VFUisV3bPi5V0pX76e1ZElB+SInBCXnJMyuSUVUiWMPJJn8krerCfrxXq3PqajGWu2s0/+wPr8AZm8mZo=</latexit>

Qc1 = (s↵c�)V�A(c�d↵)V�A

<latexit sha1_base64="So221FkP+pOvHmBcGiMJKaH8wjE=">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</latexit>

Qc2 = (sc)V�A(cd)V�A

<latexit sha1_base64="A1Dn3qKVSvK84yxLj1B8SVYdHkA=">AAACGnicbZDLSsNAFIYn9VbjLerSzWAR6sKSlIK6EKpuXLZgL9DEMJlM2qGTCzMToYQ+hxtfxY0LRdyJG9/GaZtFbf1h4Oc753Dm/F7CqJCm+aMVVlbX1jeKm/rW9s7unrF/0BZxyjFp4ZjFvOshQRiNSEtSyUg34QSFHiMdb3g7qXceCRc0ju7lKCFOiPoRDShGUiHXsJpu9QHDK1i2PcShgPjUzWwewvbZ9VifQQz9OegaJbNiTgWXjZWbEsjVcI0v249xGpJIYoaE6FlmIp0McUkxI2PdTgVJEB6iPukpG6GQCCebnjaGJ4r4MIi5epGEUzo/kaFQiFHoqc4QyYFYrE3gf7VeKoMLJ6NRkkoS4dmiIGVQxnCSE/QpJ1iykTIIc6r+CvEAcYSlSlNXIViLJy+bdrVi1SqXzVqpfpPHUQRH4BiUgQXOQR3cgQZoAQyewAt4A+/as/aqfWifs9aCls8cgj/Svn8BsvmdmA==</latexit>

&

enter when nf ≥ 4

sum over q runs for all active quarks

ε’ calculationIsospin-limit formula

10

Re

✓✏0

✏

◆= Re

⇢i!ei(�2��0)

p2✏

ImA2

ReA2� ImA0

ReA0

��= 1.38(5.15)(4.59)⇥ 10�4

<latexit sha1_base64="0bJ65iLinylyryvCwIfccxsV3hs=">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</latexit>

ππ phase shifts

Lellouch-Lüscher finite volume correction

Wilson coefs. pQCD LQCD

AI = FGF

2V⇤usVud

X

i,j

[zi(µ) + ⌧yi(µ)]Zij(µ)h(⇡⇡)I|Qlatj |Ki

<latexit sha1_base64="nvz7RhuFdw+NElWqEiDu2/o2ME8=">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</latexit>

LQCD (+pQCD)

Renormalization matrix

F & δI extracted from ππ scattering study

‣ 2pt function

‣ Lüscher’s method [Commun.Math.Phys. 219 (2001) 31]

(! = ReA2/ReA0)

<latexit sha1_base64="wAGR1dCAhyv7JY74Nwd7U7tggIM=">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</latexit>

hO⇡⇡(~p, t)O⇡⇡(~p, 0)†i

<latexit sha1_base64="FphODoBzCBt/dOpQNLCRd/7/d9I=">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</latexit>

(RBC/UKQCD is preparing a companion paper on this calculation)

Z. Bai et al, (RBC/UKQCD) PRL115(2015) 21, 212001

Simulation parameters‣ 323 x 64 (2+1 Möbius domain-wall fermions)

‣ near physical pion & kaon: mπ = 143.1(2.0) MeV, mK = 490.6(2.2) MeV

‣ lattice cutoff: 1.3784(68) GeV

‣ 216 configurations

Re A0 & Im A0: large statistical & systematic errors

ε’

First result for ε’ in 2015

11Re(✏0/✏)exp = 16.6(2.3)⇥ 10�4

<latexit sha1_base64="G4kHT9Ks+Ed5fiAjxhHWCYmIzOg=">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</latexit>

Re(✏0/✏)SM = 1.38(5.15)stat(4.59)sys ⇥ 10�4

<latexit sha1_base64="7Q80LhUfNmJ2t8GMYOkHznr3LOk=">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</latexit>

2.2σ

disconnected diagrams Truncation of pQCD (small renormalization scale)

This work

Same gauge ensemble but…

‣ 216 → 741 configurations (864 → 5,864 MD time)

‣ Multiple ππ operators → Excited-state contaminations well managed

‣ Renormalization scale nonperturbatively lifted up by step scaling → significant reduction of systematic error

12

13

Computational resources

Main resource‣ Cori @ NERSC (National Energy

Research Scientific Computing Center)‣ 430 M NERSC hours (~core hours)

Supplemental‣ BlueGene/Q (BNL), Hokusai (RIKEN),

Mira (Argonne), KEKSC 1540 (KEK), DiRAC (Edinburgh), Blue Waters (Illinois)

Contents

Introduction

K → ππ matrix elements‣ Extracting on-shell kinematics

‣ ππ scattering phase shift & ππ puzzle

‣ K → ππ


On going projects

14

What’s needed for K → ππ MEsEuclidean correlation function (0-momentum case)

15

Zd3x⇡⇡d

3xKhO⇡⇡(t⇡⇡,~x⇡⇡)HW(t,~0)OK(tK,~xK)†i

<latexit sha1_base64="aGNwFT8MWT+f0dSZY3cey4HRE/Q=">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</latexit>

=X

m,n

h0|O⇡⇡|⇡⇡,mi 1

2E⇡⇡,mh⇡⇡,m|HW|K, ni 1

2EK,nhK, n|O†

K|0ie�m⇡⇡,m(t⇡⇡�t)e�mK,n(t�tK)

<latexit sha1_base64="Kv0DcgzLe46dIIcI3Amhi9+Hva0=">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</latexit>

zero-momentum projection ( )ei~p·~x= 1

<latexit sha1_base64="6QioY0PoUC/mvU0hv1VRExJNChk=">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</latexit>

all possible zero-(total)momentum states that have the same quantum numbers as Oππ/OK

If the lightest state is interesting… look at large tππ - t & t - tK:

! h0|O⇡⇡|⇡⇡, 0i1

2E⇡⇡,0h⇡⇡, 0|HW|K, 0i 1

2EK,0hK, 0|O†

K|0ie�m⇡⇡,0(t⇡⇡�t)e�mK,0(t�tK)

<latexit sha1_base64="xXGaYLy8DlShb729CpMYRC4e0Y0=">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</latexit>

What’s needed for K → ππ MEsEuclidean correlation function (0-momentum case)

15

Zd3x⇡⇡d

3xKhO⇡⇡(t⇡⇡,~x⇡⇡)HW(t,~0)OK(tK,~xK)†i

<latexit sha1_base64="aGNwFT8MWT+f0dSZY3cey4HRE/Q=">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</latexit>

=X

m,n

h0|O⇡⇡|⇡⇡,mi 1

2E⇡⇡,mh⇡⇡,m|HW|K, ni 1

2EK,nhK, n|O†

K|0ie�m⇡⇡,m(t⇡⇡�t)e�mK,n(t�tK)

<latexit sha1_base64="Kv0DcgzLe46dIIcI3Amhi9+Hva0=">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</latexit>

zero-momentum projection ( )ei~p·~x= 1

<latexit sha1_base64="6QioY0PoUC/mvU0hv1VRExJNChk=">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</latexit>

all possible zero-(total)momentum states that have the same quantum numbers as Oππ/OK

If the lightest state is interesting… look at large tππ - t & t - tK:

! h0|O⇡⇡|⇡⇡, 0i1

2E⇡⇡,0h⇡⇡, 0|HW|K, 0i 1

2EK,0hK, 0|O†

K|0ie�m⇡⇡,0(t⇡⇡�t)e�mK,0(t�tK)

<latexit sha1_base64="xXGaYLy8DlShb729CpMYRC4e0Y0=">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</latexit>

Is this what we wanted?

Heavier ππ state neededKaon lightest state is its physical ground state

The lightest ππ state off-shell with 2 stationary pions, Eππ,0 ≈ 270 MeV

‣ need to extract | Eππ = mK ≈ 500 MeV 〉

Possible approaches💡 Finite box → individual pion momenta and spectrum not continuous (2nd lightest

can be of interest)

‣ Analyze correlation functions considering multiple states

‣ Manipulate boundary condition so that the lightest state vanishes (employed)

16

I = 2 calculationImpose anti-periodic boundary conditions (APBC) on d quark in n directions:

‣

‣ Charged pions: anti-periodic on those boundaries:

‣ → lightest state energy:

Isospin rotation (Wigner-Eckart theorem):

‣ A2 can be calculated at on-shell kinematics with APBC d quark17

d(x+ Lex1,...,xn) = �d(x)

<latexit sha1_base64="QVC2U4vAbT588EcnX00vIFXNm0M=">AAACFXicbVDLSsNAFJ34rPFVdelmsAiKtSRSUBdC0Y0LFxVsFZoSJpOJHTqZhJkbaQn9CTf+ihsXirgV3Pk3Th8LXwcGDuecy517glRwDY7zaU1Nz8zOzRcW7MWl5ZXV4tp6UyeZoqxBE5Gom4BoJrhkDeAg2E2qGIkDwa6D7tnQv75jSvNEXkE/Ze2Y3EoecUrASH6xnHs6ssOd3t6F1yGAmZ/3fLfsiTABXe75crCLT/C+CezaA79YcirOCPgvcSekhCao+8UPL0xoFjMJVBCtW66TQjsnCjgVbGB7mWYpoV1yy1qGShIz3c5HVw3wtlFCHCXKPAl4pH6fyEmsdT8OTDIm0NG/vaH4n9fKIDpq51ymGTBJx4uiTGBI8LAiHHLFKIi+IYQqbv6KaYcoQsEUaZsS3N8n/yXNg4pbrRxfVku100kdBbSJttAOctEhqqFzVEcNRNE9ekTP6MV6sJ6sV+ttHJ2yJjMb6Aes9y/qqpzH</latexit>

⇡±(x+ Lex1,...,xn) = �⇡±(x)

<latexit sha1_base64="pMETq9s+ClKDob6Ym6zEAp8NoD0=">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</latexit>

e⇡±(~p, t)|pi=0 = 0, i = 1, . . . , n

<latexit sha1_base64="KMCvCup4Q3oR5Ojmx7wDMcJdldA=">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</latexit>

h⇡+⇡+|

<latexit sha1_base64="w4mRULe/lbwenGA5Pw95XllyP1E=">AAACAHicdVDLSgMxFM34rOOr6sKFm2ARBGHItLW2u6IblxXsAzq1ZNJMG5rJDElGKGM3/oobF4q49TPc+TemD0FFT8jlcM69JPf4MWdKI/RhLSwuLa+sZtbs9Y3Nre3szm5DRYkktE4iHsmWjxXlTNC6ZprTViwpDn1Om/7wYuI3b6lULBLXehTTToj7ggWMYG2kbnbfU4HtcSz6nEIvZjcn03LXzeaQUymXSm4JIsctmHNqCCrkESpD10FT5MActW723etFJAmp0IRjpdouinUnxVIzwunY9hJFY0yGuE/bhgocUtVJpwuM4ZFRejCIpLlCw6n6fSLFoVKj0DedIdYD9dubiH957UQH5U7KRJxoKsjsoSDhUEdwkgbsMUmJ5iNDMJHM/BWSAZaYaJOZbUL42hT+Txp5xy06latirno+jyMDDsAhOAYuOANVcAlqoA4IGIMH8ASerXvr0XqxXmetC9Z8Zg/8gPX2CepLlgI=</latexit>

L (& n) should be tuned so E⇡+⇡+ = mK

<latexit sha1_base64="x+PPaNPKZGQZ5btUvtpTjXru9HM=">AAACAXicdVDLSgMxFM3UV62vqhvBTbAIgjCkD9rOQiiKILipYB/Q1iGTZtrQzIMkI5ShbvwVNy4UcetfuPNvTKcVVPTAvRzOuZfkHifkTCqEPozUwuLS8kp6NbO2vrG5ld3eacogEoQ2SMAD0XawpJz5tKGY4rQdCoo9h9OWMzqb+q1bKiQL/Gs1DmnPwwOfuYxgpSU7u9eVLjy3427Ibo6TNoEn0LMv7WwOmahslapliMyiVa1YU4KKqFyowryJEuTAHHU7+97tByTyqK8Ix1J28ihUvRgLxQink0w3kjTEZIQHtKOpjz0qe3FywQQeaqUP3UDo8hVM1O8bMfakHHuOnvSwGsrf3lT8y+tEyq32YuaHkaI+mT3kRhyqAE7jgH0mKFF8rAkmgum/QjLEAhOlQ8voEL4uhf+TZsHMl0zrqpSrnc7jSIN9cACOQB5UQA1cgDpoAALuwAN4As/GvfFovBivs9GUMd/ZBT9gvH0CCmyWBg==</latexit>

(PRL108 (2012) 141601, PRD91 (2015) 074502)

E2⇡± = m2

⇡+n2(⇡/L)2

<latexit sha1_base64="17QmwP17SSXIxOF3j5MLKWARtZQ=">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</latexit>

h(⇡⇡)I3=1I=2 |H�I3=1/2

�I=3/2 |K+i = 3

2h(⇡⇡)I3=2

I=2 |H�I3=3/2�I=3/2 |K+i

<latexit sha1_base64="0oVBM5RQMhbIcSK0ub3Rj8a8PKI=">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</latexit>

G-parity boundary conditions for I = 0A0 must be calculated with π0π0 final state → APBC useless

G-parity boundary conditions:

‣

‣ All pions: G-parity odd → can extract on-shell kinematics

Dirac matrix mixes u & d → Computationally expensive

Several other challenges confront18

f(x+ Lex1,...,xn) = bGf(x) = bCe�i⇡Iy f(x)

<latexit sha1_base64="sMTRNuIJL3UHEUaK0+Z2u5j5QQk=">AAACPnicbVDPSxtBGJ3Vam38FfXYy9AgWNSwWwT1IIge2oKHFIwGsnGZnf3WDM7OLjPfasKSv8yLf4O3Hr14aBGvHjub5KCxDwYe772Pb74XZlIYdN3fztT0h5nZj3OfKvMLi0vL1ZXVM5PmmkOTpzLVrZAZkEJBEwVKaGUaWBJKOA+vjkv//Bq0Eak6xX4GnYRdKhELztBKQbXpm5jGG73NE7/LkEJQ9AJvy5dRimarF6jB18oB9W9EBKX9vYy+UY7hotgWfiaG4z+D/qBMBNWaW3eHoO+JNyY1MkYjqN77UcrzBBRyyYxpe26GnYJpFFzCoOLnBjLGr9gltC1VLAHTKYbnD+i6VSIap9o+hXSovp4oWGJMPwltMmHYNZNeKf7Pa+cY73UKobIcQfHRojiXFFNadkkjoYGj7FvCuBb2r5R3mWYcbeMVW4I3efJ7cvat7u3U93/t1A6PxnXMkc/kC9kgHtklh+QHaZAm4eSWPJA/5K9z5zw6T87zKDrljGfWyBs4L/8A5gGsmg==</latexit>

(f: isospin representation)

<latexit sha1_base64="FkRieNJGq/kMPjkN68usA9j5sFE=">AAACDnicbVC7TsMwFHXKq5RXgJHFoq1UliqpkHhMFSyMRaKlUhtVjuu0Vh07sh2kKuoXsPArLAwgxMrMxt/gpBmg5Ui2js6599r3+BGjSjvOt1VYWV1b3yhulra2d3b37P2DjhKxxKSNBROy6yNFGOWkralmpBtJgkKfkXt/cp369w9EKir4nZ5GxAvRiNOAYqSNNLCrfRXAWiW9g8olpEqoiHIoiZmiCNdZ2cnALjt1JwNcJm5OyiBHa2B/9YcCx6GZgBlSquc6kfYSJDXFjMxK/ViRCOEJGpGeoRyFRHlJts4MVo0yhIGQ5nANM/V3R4JCpaahbypDpMdq0UvF/7xerINzL6E8ijXheP5QEDOoBUyzgUMqCdZsagjCkpq/QjxGEmFtEiyZENzFlZdJp1F3T+sXt41y8yqPowiOwDGoARecgSa4AS3QBhg8gmfwCt6sJ+vFerc+5qUFK+85BH9gff4AnX+bQg==</latexit>

Charge conjugation 180° isospin rotation

bG✓

ud

◆=

✓�CdT

CuT

◆, bG

✓du

◆=

✓�uTC�1

dTC�1

◆

<latexit sha1_base64="TXMciqdRakvmBIgTfAJ/A1VJins=">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</latexit>

ππ scattering essentialRelations among ππ phase shift δ0, energy Eππ and “momenta” k

‣ ex: 2-boson system in 1+1-dim w/ periodic BC in spatial direction

Relation b/w plane wave at x=L & x=0:

Dispersion relation:

Different relation for 3+1-dim & G-parity BC but same steps:

‣ Extract a few energy levels En from ππ 2pt function (next slide) then corresponding kn

‣ Then δ(kn) can be obtained from Lüscher formula

‣ Derivative δ’(kπ(on-shell)) also needed for Lellouch-Lüscher finite-volume factor F 19

T. Wang, C. Kelly, et al (RBC/UKQCD) 2103.15131

eikL+2i�(k) = 1

<latexit sha1_base64="GT0wOYwmueblJkSEOsCcvxbG+d8=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahIpSkFNSFUHTjwkUF+4Amlsn0ph0yeTAzEUroxo2/4saFIm79B3f+jUmbhVYPXDiccy/33uNEnEllGF9aYWFxaXmluFpaW9/Y3NK3d9oyjAWFFg15KLoOkcBZAC3FFIduJID4DoeO411mfucehGRhcKvGEdg+GQbMZZSoVOrr+5Z0MdwlzLs+rjFrAFyRinc0wefYLPX1slE1psB/iZmTMsrR7Ouf1iCksQ+BopxI2TONSNkJEYpRDpOSFUuICPXIEHopDYgP0k6mX0zwYaoMsBuKtAKFp+rPiYT4Uo59J+30iRrJeS8T//N6sXJP7YQFUawgoLNFbsyxCnEWCR4wAVTxcUoIFSy9FdMREYSqNLgsBHP+5b+kXaua9erZTb3cuMjjKKI9dIAqyEQnqIGuUBO1EEUP6Am9oFftUXvW3rT3WWtBy2d20S9oH990d5af</latexit>

! knL+ 2�(kn) = 2n⇡

<latexit sha1_base64="8C5xTOzRUypup8pa9vP3KpHMryc=">AAACCnicbVDNS8MwHE3n15xfVY9eokOYCKMdA/UgDL148DDBfcBaSpqmW1ialiQVRtnZi/+KFw+KePUv8OZ/Y7b1oJsPAi/v/X4k7/kJo1JZ1rdRWFpeWV0rrpc2Nre2d8zdvbaMU4FJC8csFl0fScIoJy1FFSPdRBAU+Yx0/OH1xO88ECFpzO/VKCFuhPqchhQjpSXPPHRkCB0Vw6HHb09rTkCYQhV9OYGXsMadhHpm2apaU8BFYuekDHI0PfPLCWKcRoQrzJCUPdtKlJshoShmZFxyUkkShIeoT3qachQR6WbTKGN4rJUAhrHQhys4VX9vZCiSchT5ejJCaiDnvYn4n9dLVXjuZpQnqSIczx4KUwZ19EkvMKCCYMVGmiAsqP4rxAMkEFa6vZIuwZ6PvEjatapdr17c1cuNq7yOIjgAR6ACbHAGGuAGNEELYPAInsEreDOejBfj3fiYjRaMfGcf/IHx+QNtppjX</latexit>

(cf: kn = 2n⇡/L in non-interacting case)

<latexit sha1_base64="EWlchTP9wzXJsUKP9fhJCGf+wZ8=">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</latexit>

En = 2p

m2 + k2n

<latexit sha1_base64="8iT7gIypjhj6yyZTFkRy6ccjklM=">AAACBHicbVDLSsNAFJ34rPUVddnNYBEEoSShoC6EogguK9gHNGmYTCft0MkkzkyEErpw46+4caGIWz/CnX/jtM1CWw9cOJxzL/feEySMSmVZ38bS8srq2npho7i5tb2za+7tN2WcCkwaOGaxaAdIEkY5aSiqGGkngqAoYKQVDK8mfuuBCEljfqdGCfEi1Oc0pBgpLflmyZUhvPY5vICOK++FyqKuczL0edcZ+2bZqlhTwEVi56QMctR988vtxTiNCFeYISk7tpUoL0NCUczIuOimkiQID1GfdDTlKCLSy6ZPjOGRVnowjIUuruBU/T2RoUjKURTozgipgZz3JuJ/XidV4ZmXUZ6kinA8WxSmDKoYThKBPSoIVmykCcKC6lshHiCBsNK5FXUI9vzLi6TpVOxq5fy2Wq5d5nEUQAkcgmNgg1NQAzegDhoAg0fwDF7Bm/FkvBjvxsesdcnIZw7AHxifP71tluk=</latexit>

“Lüscher formula” for this toy case

The “ππ puzzle”Large discrepancy b/w lattice & pheno+exp

‣

‣

Lattice analysis was fairly stable, unchanged by…‣ testing single- & 2-state fits

‣ stable w/ varying fit range in the plateau region

How it can be explained?‣ A big reason why we needed to retry I=0 calculation‣ our conclusion: excited states still significant

20

Statistical Improvement

Prediction from dispn theory + expt

216 Configs

Increasing the statistics from 216 to 1438 configurations, the ⇡⇡ correlationfunction is still well described by a single ⇡⇡ state.

It does not solve the �0 puzzle however:

�0 = (23.8 ± 4.9 ± 2.2)� ! �0 = (19.1 ± 2.5 ± 1.2)� (�2/dof = 1.6)

Chris Sachrajda FPCP2020, June 9th 2020 16

Statistical Improvement

(From dispersion theory + expt. data)

1438 cfgs(PRELIMINARY)

216 Configs

Increasing the statistics from 216 to 1438 configurations, the ⇡⇡ correlationfunction is still well described by a single ⇡⇡ state.

It does not solve the �0 puzzle however:

�0 = (23.8 ± 4.9 ± 2.2)� ! �0 = (19.1 ± 2.5 ± 1.2)� (�2/dof = 1.6)


2015 paper

�20150 = 23.8(4.9)(2.2)�, �20200 = 19.1(2.5)(1.2)�

<latexit sha1_base64="aZu1U6yZ6aWpYaW408gIDszJT64=">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</latexit>

�ph+exp0 = 36�

<latexit sha1_base64="rum+Y97yMSx5PzArUjbk58QPr28=">AAACCXicbVDJSgNBEO2JWxy3UY9eGoMgCGFGg8tBCHrxGMEskJmEnk5N0qRnobtHDEOuXvwVLx4U8eofePNv7CwHTXxQ8Hiviqp6fsKZVLb9beQWFpeWV/Kr5tr6xuaWtb1Tk3EqKFRpzGPR8IkEziKoKqY4NBIBJPQ51P3+9civ34OQLI7u1CABLyTdiAWMEqWltoVdGZhuB7gibbuVJb0jeEiG+BKfnLZcygRtWwW7aI+B54kzJQU0RaVtfbmdmKYhRIpyImXTsRPlZUQoRjkMTTeVkBDaJ11oahqREKSXjT8Z4gOtdHAQC12RwmP190RGQikHoa87Q6J6ctYbif95zVQF517GoiRVENHJoiDlWMV4FAvuMAFU8YEmhAqmb8W0RwShSodn6hCc2ZfnSe246JSKF7elQvlqGkce7aF9dIgcdIbK6AZVUBVR9Iie0St6M56MF+Pd+Ji05ozpzC76A+PzB63umRI=</latexit>

G(t) = z0e�E0t & G(t) = z0e

�E0t + z1e�E1t

<latexit sha1_base64="wCatdeffiwyCGypDzzhlwc9oefs=">AAACKnicbVDLSgMxFM3UV62vqks3waJUxDIjBXUhVEV0WcE+oFOHTJppQzMPkjtCHfo9bvwVN10oxa0fYvpYqO2BwLnn3MvNPW4kuALTHBqphcWl5ZX0amZtfWNzK7u9U1VhLCmr0FCEsu4SxQQPWAU4CFaPJCO+K1jN7d6M/Nozk4qHwSP0Itb0STvgHqcEtORkr2zlZe7ycIQv8Ytjsqfk5NYxoW9j+xDbeJ6Fj3VpTUoL+k42ZxbMMfAssaYkh6YoO9mB3Qpp7LMAqCBKNSwzgmZCJHAqWD9jx4pFhHZJmzU0DYjPVDMZn9rHB1ppYS+U+gWAx+rviYT4SvV8V3f6BDrqvzcS53mNGLzzZsKDKAYW0MkiLxYYQjzKDbe4ZBRETxNCJdd/xbRDJKGg083oEKz/J8+S6mnBKhYuHoq50vU0jjTaQ/sojyx0hkroHpVRBVH0it7RB/o03oyBMTS+Jq0pYzqzi/7A+P4BLEyiTg==</latexit>

0

<latexit sha1_base64="7eiRyCb/dgIAVHMNk7xSEePkL7Q=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LBbBU0mkoN6KXjxWsB/YhLLZbtqlu5uwuxFK6L/w4kERr/4bb/4bt20O2vpg4PHeDDPzopQzbTzv2ymtrW9sbpW33Z3dvf2DyuFRWyeZIrRFEp6oboQ15UzSlmGG026qKBYRp51ofDvzO09UaZbIBzNJaSjwULKYEWys9Bjo2A1SxQTtV6pezZsDrRK/IFUo0OxXvoJBQjJBpSEca93zvdSEOVaGEU6nbpBpmmIyxkPas1RiQXWYzy+eojOrDFCcKFvSoLn6eyLHQuuJiGynwGakl72Z+J/Xy0x8FeZMppmhkiwWxRlHJkGz99GAKUoMn1iCiWL2VkRGWGFibEiuDcFffnmVtC9qfr12fV+vNm6KOMpwAqdwDj5cQgPuoAktICDhGV7hzdHOi/PufCxaS04xcwx/4Hz+ADPJkKI=</latexit>

Resolving the ππ puzzleIntroduce multiple ππ operators

‣ In 2015

‣ Additions in 2020

2pt functions

‣ possible to isolate a few lightest states

‣ better way to investigate/manage excited-state contamination 21

O⇡⇡ = ⇡⇡(1, 1, 1)

<latexit sha1_base64="ave28NkuPeJo4hiIsOBt8dwkqU4=">AAACB3icbVDLSgMxFL1TX3V8jboUJFiEClJmpKAuhKIbd1awD+gMJZNm2tDMgyQjlKE7N/6KGxeKuPUX3Pk3pu0stHqSC4dz7iW5x084k8q2v4zCwuLS8kpx1Vxb39jcsrZ3mjJOBaENEvNYtH0sKWcRbSimOG0nguLQ57TlD68mfuueCsni6E6NEuqFuB+xgBGstNS19l0ZmDfdzE2YvmN0gWas7Bzrc9S1SnbFngL9JU5OSpCj3rU+3V5M0pBGinAsZcexE+VlWChGOB2bbippgskQ92lH0wiHVHrZdI8xOtRKDwWx0BUpNFV/TmQ4lHIU+rozxGog572J+J/XSVVw5mUsSlJFIzJ7KEg5UjGahIJ6TFCi+EgTTATTf0VkgAUmSkdn6hCc+ZX/kuZJxalWzm+rpdplHkcR9uAAyuDAKdTgGurQAAIP8AQv8Go8Gs/Gm/E+ay0Y+cwu/ILx8Q3HO5dV</latexit>

ππ

p=(1,1,1)

p=(-1,-1,-1) !! not indicating pion momenta of possible states

� =1p2(uu+ dd)

<latexit sha1_base64="SCIRQSo3LbgHQY1guktVlGuG0gE=">AAACGHicbVDLSsNAFJ3UV42vqEs3g0WoCDUpBXUhFN24rGAf0IQymUzaoZOHMxOhhHyGG3/FjQtF3Hbn3zhNs9DWA5d7OOdeZu5xY0aFNM1vrbSyura+Ud7Ut7Z3dveM/YOOiBKOSRtHLOI9FwnCaEjakkpGejEnKHAZ6brj25nffSJc0Ch8kJOYOAEahtSnGEklDYxzW/i6LegwQPAa2j5HOLWy1BaPXMJ6VrVdxGGSnOXd804HRsWsmTngMrEKUgEFWgNjansRTgISSsyQEH3LjKWTIi4pZiTT7USQGOExGpK+oiEKiHDS/LAMnijFg37EVYUS5urvjRQFQkwCV00GSI7EojcT//P6ifQvnZSGcSJJiOcP+QmDMoKzlKBHOcGSTRRBmFP1V4hHSIUjVZa6CsFaPHmZdOo1q1G7um9UmjdFHGVwBI5BFVjgAjTBHWiBNsDgGbyCd/ChvWhv2qf2NR8tacXOIfgDbfoDhlye1w==</latexit>

⇡⇡(3, 1, 1),

<latexit sha1_base64="Ny0SaDZS77+k0EvrrdzKqdYBH50=">AAAB+3icbVDLSsNAFL2prxpfsS7dDBahQimJFtRd0Y3LCvYBTSiT6aQdOnkwMxFL6K+4caGIW3/EnX/jtM1CWw/3wuGce5k7x084k8q2v43C2vrG5lZx29zZ3ds/sA5LbRmngtAWiXksuj6WlLOIthRTnHYTQXHoc9rxx7czv/NIhWRx9KAmCfVCPIxYwAhWWupbJVcGppswXZWLqlN1zqp9q2zX7DnQKnFyUoYczb715Q5ikoY0UoRjKXuOnSgvw0IxwunUdFNJE0zGeEh7mkY4pNLL5rdP0alWBiiIhe5Iobn6eyPDoZST0NeTIVYjuezNxP+8XqqCKy9jUZIqGpHFQ0HKkYrRLAg0YIISxSeaYCKYvhWRERaYKB2XqUNwlr+8StrnNadeu76vlxs3eRxFOIYTqIADl9CAO2hCCwg8wTO8wpsxNV6Md+NjMVow8p0j+APj8wcD+pJ+</latexit>

Gij(t) = hOi(t)Oj(0)†i =

X

n

Ai,nA†j,ne

�Ent

<latexit sha1_base64="4W+DohUwM0/CriFvfelbhZC6+x0=">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</latexit>

31

FIG. 5: The tmin dependence of fitted ground state energy for the stationary ⇡⇡I=2 channel

(left) with tmax = 25 and the stationary ⇡⇡I=0 channel (right) with tmax = 15. Left: The

circles represent the two-operator, two-state fit and downward pointing triangles the

one-operator, one-state fit. Right: The pentagons represent the one-operator, one-state fit.

The stars and downward pointing triangles show the results from the two two-operator,

two-state fits. Finally the circles show the three-operator, three-state fit for tmin = 3 and 4

while the diamonds show the three-operator, two-state fit for tmin = 5� 8. For the I = 0

channel, including additional operators (especially the �) substantially improves the

determination of the ground state energy.

a total of 9 real fit parameters. Note that, as the lower triangular component of the matrix

is related to the upper triangular component by the time-translational symmetry, we did

not measure these terms in order to reduce the computational cost.

For each case, we perform correlated fits with various choices for tmin and set tmax = 25.

We plot the resulting ground state energy as a function of tmin in the left panel of Figure 5.

As we can see from the plot, the introduction of the second operator does not noticeably

improve the fit result, as the ground state energies given by both fitting strategies are

statistically consistent for all tmin and the statistical errors are also consistent. As we increase

tmin, the ground state energy first decreases, which suggests a non-negligible excited state

contamination for small tmin and then reaches a plateau for tmin ⇡ 10. We adopt the 2-

operator, 2-state fit with the fitting range of 10� 25 for our final result. In Table IV we list

the p-values and the final parameters obtained from that approach. We observe an excellent

p-value indicating a strong consistency between the data and our model.

The fact that Baa is 60� resolved from zero suggests the importance of including these

Effect of multi operators on ππ

22

fit result for lightest(on-shell) ππ energy

in lattice unitsof fit

Result compatible with ph+exp:

�(1,1,1)0 = 19.1(2.5)(1.2)�

<latexit sha1_base64="lkhiMIl8IPSYQBRb9lodZeps1Hk=">AAACFnicbVDLSgMxFM3UV62vqks3wSJMQYdJqWgXQtGNywr2AZ1pyaSZNjTzIMkIZehXuPFX3LhQxK24829MHwttPZcLh3PuJbnHizmTyra/jczK6tr6RnYzt7W9s7uX3z9oyCgRhNZJxCPR8rCknIW0rpjitBULigOP06Y3vJn4zQcqJIvCezWKqRvgfsh8RrDSUjd/5kg/5/QoV7hrd1ITneoqjuEVRBULmSXrvGgiq1TsOIQJ0s0XbMueAi4TNCcFMEetm/9yehFJAhoqwrGUbWTHyk2xUIxwOs45iaQxJkPcp21NQxxQ6abTs8bwRCs96EdCd6jgVP29keJAylHg6ckAq4Fc9Cbif147Uf6lm7IwThQNyewhP+FQRXCSEewxQYniI00wEUz/FZIBFpgonWROh4AWT14mjZKFylblrlyoXs/jyIIjcAxMgMAFqIJbUAN1QMAjeAav4M14Ml6Md+NjNpox5juH4A+Mzx+gkpqo</latexit>

ph+exp

<latexit sha1_base64="nmEnsVpoBatUJaQhtOLhFLo5W8k=">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</latexit>

�20210 (471 MeV) = 32.3(10)(14)�

I=0<ππ| Qi |K> 3pt functions

23

�

K

�

HW

�

K

�

HW

�

K

�

HW �

K

�

HW

type 1 type 2

type 3 type 4

24

A2A propagators, V & W vectors

• V & W vectors

D�1A2A =

NlX

l=1

|�li1

�h�l|+

1

Nh

NhX

h=1

D�1 �

NlX

l=1

|�li1

�h�l|!|⌘hih⌘h|

<latexit sha1_base64="B6eBTOC9EHeqe3Tz5LcWgWifPBI=">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</latexit>

D�1defl

<latexit sha1_base64="r+a0HyzZKbkKdfTTDGegIHAQxGA=">AAAB+nicdVDLSgMxFM3UV62vVpdugkVw45Bpa213RV24rGAf0I5DJs20oZkHSUYp43yKGxeKuPVL3Pk3pg9BRc/lwuGce8nNcSPOpELow8gsLa+srmXXcxubW9s7+cJuW4axILRFQh6Krosl5SygLcUUp91IUOy7nHbc8fnU79xSIVkYXKtJRG0fDwPmMYKVlpx8oS+93IWTDKjH05vk2EqdfBGZ9Vq1alUhMq2yrhNNULmEUA1aJpqhCBZoOvn3/iAksU8DRTiWsmehSNkJFooRTtNcP5Y0wmSMh7SnaYB9Ku1kdnoKD7UygF4odAcKztTvGwn2pZz4rp70sRrJ395U/Mvrxcqr2QkLoljRgMwf8mIOVQinOcABE5QoPtEEE8H0rZCMsMBE6bRyOoSvn8L/SbtkWhWzflUpNs4WcWTBPjgAR8ACp6ABLkETtAABd+ABPIFn4954NF6M1/loxljs7IEfMN4+Ad61k8k=</latexit>

=Nl+NhX

i=1

|ViihWi|

<latexit sha1_base64="Z8XBqOuMrvbzh8DQe5dk1M26hJo=">AAACGnicbVDLSsNAFJ34rPEVdelmsAiCUBIpqItC0Y2rUsE+oIlhMp20QyeTMDMRStrvcOOvuHGhiDtx4984TbvQ1gOXezjnXmbuCRJGpbLtb2NpeWV1bb2wYW5ube/sWnv7TRmnApMGjlks2gGShFFOGooqRtqJICgKGGkFg+uJ33ogQtKY36lhQrwI9TgNKUZKS77luDI0K9CVaeRntOKM77Oaz05rfn88avrUFYj3GHFZ3mDLpyPfKtolOwdcJM6MFMEMdd/6dLsxTiPCFWZIyo5jJ8rLkFAUMzI23VSSBOEB6pGOphxFRHpZftoYHmulC8NY6OIK5urvjQxFUg6jQE9GSPXlvDcR//M6qQovvIzyJFWE4+lDYcqgiuEkJ9ilgmDFhpogLKj+K8R9JBBWOk1Th+DMn7xImmclp1y6vC0Xq1ezOArgEByBE+CAc1AFN6AOGgCDR/AMXsGb8WS8GO/Gx3R0yZjtHIA/ML5+AMh9oMQ=</latexit>

1 i Nl ) |Vii =1

�|�ii, |Wii = |�ii

<latexit sha1_base64="eRNEBRUzpzUWc17X9qRVsXqJk90=">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</latexit>

Nl + 1 i(= Nl + h) Nl + Nh ) |Vii =1

NhD�1

defl|⌘hi, |Wii = |⌘hi

<latexit sha1_base64="aMikqE/6uQ+yD6WuHKVPsUor25s=">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</latexit>

25

Meson fields

• Spin & color contractions leaving mode indices i, j

• Easily summed over time slice → savable data size

• Multiplied with any other meson fields to construct correlation functions

meson field

Γ

i j

i

j

Γ Γ’

<latexit sha1_base64="SAQ+9K3DKYjDSCxHBp2IPFYvn9Y=">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</latexit>

⇧�,ij(t) = hWi|�|Vjit

<latexit sha1_base64="EzWPmyk6jCQpZ0+ShQ0nY0+AwCs=">AAACKHicbVBPS8MwHE3nv1n/TT16CQ7ZhDHaIerNoQc9TnBusI6RZumWLWlLkgqj7ON48at4EVFkVz+J6VbBbT4IvLz3fiS/54aMSmVZEyOzsrq2vpHdNLe2d3b3cvsHjzKIBCZ1HLBANF0kCaM+qSuqGGmGgiDuMtJwhzeJ33giQtLAf1CjkLQ56vnUoxgpLXVyV06NdmJHes4t4hyVIB2MzeReVKdjCM15u1CCA/rrF5JAJ5e3ytYUcJnYKcmDFLVO7t3pBjjixFeYISlbthWqdoyEopiRselEkoQID1GPtDT1ESeyHU8XHcMTrXShFwh9fAWn6t+JGHEpR9zVSY5UXy56ifif14qUd9mOqR9Givh49pAXMagCmLQGu1QQrNhIE4QF1X+FuI8Ewkp3a+oS7MWVl8ljpWyflyv3Z/nqdVpHFhyBY1AENrgAVXAHaqAOMHgGr+ADfBovxpvxZUxm0YyRzhyCORjfP9oKo/Y=</latexit>

⇧�,ij(t)⇧�0,ji(t0)

A2A parameters & index900 low modes from Lanczos algorithm for light quarks (not for strange)

Random noise vectors‣ spin-color and time dilution

‣ Ns x Nc x Nt = 768 noise vectors

# of V & W vectors: 1,668 for light quark, 768 for strange quark

Pion fields: 1,668 x 1,668 matrix; Kaon field: 1,668 x 768 matrix

26

<latexit sha1_base64="8mI2tasfpFP+qyI8x9ZLPUAW7WM=">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</latexit>

⌘h;s,c(~x, t) = ⇠(~x)�h,s+Ns(c+Nct)

Tried with 3 ππ sink operators

Optimal combination of ππ(1,1,1) & σ

‣ r1 & r2 determined from ππ 2pt functions

‣ Orthogonal to 1st excited state

Including ππ(3,1,1) → unstable

‣ 2-state fit with 2 ππ operators

27

Effective MEs

Multiple operators and K ! ⇡⇡ matrix elements

It is convenient to visualise the data by taking “optimal" linear combinations ofoperators which best projects into the ground state.


Multiple operators and K ! ⇡⇡ matrix elements

It is convenient to visualise the data by taking “optimal" linear combinations ofoperators which best projects into the ground state.


Me↵,sink2

<latexit sha1_base64="k6CIh2NF9GI2o2hyci7QFmcHLeg=">AAAB+3icbVBNS8NAEJ34WeNXrEcvwSJ4kJKUgnorevEiVLAf0Maw2W7apZtN2N2IJeSvePGgiFf/iDf/jds2B219MPB4b4aZeUHCqFSO822srK6tb2yWtsztnd29feug3JZxKjBp4ZjFohsgSRjlpKWoYqSbCIKigJFOML6e+p1HIiSN+b2aJMSL0JDTkGKktORb5b4MzVu/9pCRMDyTlI9z36o4VWcGe5m4BalAgaZvffUHMU4jwhVmSMqe6yTKy5BQFDOSm/1UkgThMRqSnqYcRUR62ez23D7RysAOY6GLK3um/p7IUCTlJAp0Z4TUSC56U/E/r5eq8MLLKE9SRTieLwpTZqvYngZhD6ggWLGJJggLqm+18QgJhJWOy9QhuIsvL5N2rerWq5d39UrjqoijBEdwDKfgwjk04Aaa0AIMT/AMr/Bm5MaL8W58zFtXjGLmEP7A+PwBNqqT8A==</latexit>

Me↵,sink6

<latexit sha1_base64="bdY2DaUpLuXfWiOcjLg/OTus2pk=">AAAB+3icbVDLSsNAFJ34rPEV69JNsAgupCRSfOyKbtwIFewD2hgm05t26GQSZiZiCfkVNy4UceuPuPNvnLZZaOuBC4dz7uXee4KEUakc59tYWl5ZXVsvbZibW9s7u9ZeuSXjVBBokpjFohNgCYxyaCqqGHQSATgKGLSD0fXEbz+CkDTm92qcgBfhAachJVhpybfKPRmat/7ZQwZheCIpH+W+VXGqzhT2InELUkEFGr711evHJI2AK8KwlF3XSZSXYaEoYZCbvVRCgskID6CrKccRSC+b3p7bR1rp22EsdHFlT9XfExmOpBxHge6MsBrKeW8i/ud1UxVeeBnlSaqAk9miMGW2iu1JEHafCiCKjTXBRFB9q02GWGCidFymDsGdf3mRtE6rbq16eVer1K+KOEroAB2iY+Sic1RHN6iBmoigJ/SMXtGbkRsvxrvxMWtdMoqZffQHxucPPOaT9A==</latexit>

tsink � t

<latexit sha1_base64="p6H2UAVa+MwseQyZ4cmSVjK9DVE=">AAAB9XicbVBNS8NAEJ3Urxq/qh69BIvgxZJIQb0VvXisYD+gjWWz3bRLN5uwO1FK6P/w4kERr/4Xb/4bt20O2vpg4PHeDDPzgkRwja77bRVWVtfWN4qb9tb2zu5eaf+gqeNUUdagsYhVOyCaCS5ZAzkK1k4UI1EgWCsY3Uz91iNTmsfyHscJ8yMykDzklKCRHro6tLGXaS5HkzPslcpuxZ3BWSZeTsqQo94rfXX7MU0jJpEKonXHcxP0M6KQU8EmdjfVLCF0RAasY6gkEdN+Nrt64pwYpe+EsTIl0ZmpvycyEmk9jgLTGREc6kVvKv7ndVIML/2MyyRFJul8UZgKB2NnGoHT54pRFGNDCFXc3OrQIVGEognKNiF4iy8vk+Z5xatWru6q5dp1HkcRjuAYTsGDC6jBLdShARQUPMMrvFlP1ov1bn3MWwtWPnMIf2B9/gBcnZJy</latexit>

Osink = O⇡⇡(1,1,1), O�, Oopt

<latexit sha1_base64="dvtEYpFpJfULKzZ6aX8aPfb+TLg=">AAACIHicbZBNS8MwGMfT+TbrW9Wjl+AQJozRymB6EIZevDnBvcBaRpqlW1ialiQVRulH8eJX8eJBEb3ppzHbetDNJwn8+P+fhyR/P2ZUKtv+Mgorq2vrG8VNc2t7Z3fP2j9oyygRmLRwxCLR9ZEkjHLSUlQx0o0FQaHPSMcfX0/9zgMRkkb8Xk1i4oVoyGlAMVJa6lt1VwbmbT+VlI8zeAk1ujHVu+xU9DrNKi7UvivpMERzTqNYZX2rZFftWcFlcHIogbyafevTHUQ4CQlXmCEpe44dKy9FQlHMSGa6iSQxwmM0JD2NHIVEeunsgxk80coABpHQhys4U39PpCiUchL6ujNEaiQXvan4n9dLVHDupZTHiSIczy8KEgZVBKdpwQEVBCs20YCwoPqtEI+QQFjpTE0dgrP45WVon1WdWvXirlZqXOVxFMEROAZl4IA6aIAb0AQtgMEjeAav4M14Ml6Md+Nj3low8plD8KeM7x96WaC4</latexit>

Oopt = r1O⇡⇡(1,1,1) + r2O�

<latexit sha1_base64="Ey/avGhQmtt2J4FFSzbCDmU14wQ=">AAACG3icbVBNS8MwGE7n16xfVY9egkOYKKMdA/UgDL14c4L7gLWUNEu3sPSDJBVG6f/w4l/x4kERT4IH/43p1oNuPkngyfO8L8n7eDGjQprmt1ZaWl5ZXSuv6xubW9s7xu5eR0QJx6SNIxbxnocEYTQkbUklI72YExR4jHS98XXudx8IFzQK7+UkJk6AhiH1KUZSSa5Rt4Wv37ppFMsMXkLuWupix1TtqnWq1nEGT5Rcz2VBhwHKXKNi1swp4CKxClIBBVqu8WkPIpwEJJSYISH6lhlLJ0VcUsxIptuJIDHCYzQkfUVDFBDhpNPZMniklAH0I65OKOFU/d2RokCISeCpygDJkZj3cvE/r59I/9xJaRgnkoR49pCfMCgjmAcFB5QTLNlEEYQ5VX+FeIQ4wlLFqasQrPmRF0mnXrMatYu7RqV5VcRRBgfgEFSBBc5AE9yAFmgDDB7BM3gFb9qT9qK9ax+z0pJW9OyDP9C+fgBmf58v</latexit>

t0 =

<latexit sha1_base64="GWt1z9UyANQ64JKqwUya3Ic162Q=">AAAB73icbVDLSgNBEOyNr7i+oh69DAbRU9iVgHoQgl48RjAPSJYwO5lNhsw+nOkVwpKf8OJBEa/+jjf/xkmyB00saCiquunu8hMpNDrOt1VYWV1b3yhu2lvbO7t7pf2Dpo5TxXiDxTJWbZ9qLkXEGyhQ8naiOA19yVv+6Hbqt5640iKOHnCccC+kg0gEglE0UrurAxtPyXWvVHYqzgxkmbg5KUOOeq/01e3HLA15hExSrTuuk6CXUYWCST6xu6nmCWUjOuAdQyMacu1ls3sn5MQofRLEylSEZKb+nshoqPU49E1nSHGoF72p+J/XSTG49DIRJSnyiM0XBakkGJPp86QvFGcox4ZQpoS5lbAhVZShicg2IbiLLy+T5nnFrVau7qvl2k0eRxGO4BjOwIULqMEd1KEBDCQ8wyu8WY/Wi/VufcxbC1Y+cwh/YH3+AJtdjw0=</latexit>

Fit resultsVarious fits‣ : min of (tsink - t) [3-8]‣ tmin: min of (t-tK) [6-8]‣ (# of operators) x (# of states considered)

In 2015, effects of excited states were significantly underestimated

28

K ! ⇡⇡ Fits Results

Examples of results of fits with different numbers of operators, states,tmin = (tH � tK)min and t0min = (t⇡⇡ � tH)min.

Adopt uniform fit with t0min = 5 which is stable for all operators.Evidence that excited state errors significantly underestimated in 2015.






t0min

<latexit sha1_base64="mioWOuEjk7m8691/ifCrlealz/M=">AAAB83icbVBNS8NAEN3Urxq/qh69LBbRU0mkoN6KXjxWsB/QhLLZbtqlu5uwOxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzolRwA5737ZTW1jc2t8rb7s7u3v5B5fCobZJMU9aiiUh0NyKGCa5YCzgI1k01IzISrBON72Z+54lpwxP1CJOUhZIMFY85JWClIDCxC+f9XHI17VeqXs2bA68SvyBVVKDZr3wFg4RmkimgghjT870Uwpxo4FSwqRtkhqWEjsmQ9SxVRDIT5vObp/jMKgMcJ9qWAjxXf0/kRBozkZHtlARGZtmbif95vQzi6zDnKs2AKbpYFGcCQ4JnAeAB14yCmFhCqOb2VkxHRBMKNibXhuAvv7xK2pc1v167eahXG7dFHGV0gk7RBfLRFWqge9RELURRip7RK3pzMufFeXc+Fq0lp5g5Rn/gfP4AovuRcw==</latexit>

Q2

<latexit sha1_base64="CvRbHUg+ZPUVxfGl47gK/dBNB3M=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkpqLeiF48t2A9oQ9lsJ+3SzSbsboQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9djY2t7Z3dgt7xf2Dw6Pj0slpW8epYthisYhVN6AaBZfYMtwI7CYKaRQI7AST+7nfeUKleSwfzTRBP6IjyUPOqLFSp6/DYnNQHZTKbsVdgKwTLydlyNEYlL76w5ilEUrDBNW657mJ8TOqDGcCZ8V+qjGhbEJH2LNU0gi1ny3OnZFLqwxJGCtb0pCF+nsio5HW0yiwnRE1Y73qzcX/vF5qwhs/4zJJDUq2XBSmgpiYzH8nQ66QGTG1hDLF7a2EjamizNiEijYEb/XlddKuVrxa5bZZK9fv8jgKcA4XcAUeXEMdHqABLWAwgWd4hTcncV6cd+dj2brh5DNn8AfO5w9TDY7t</latexit>

Q6

<latexit sha1_base64="0mHzrsFelR0vvt77MTHzUaNOxGE=">AAAB7nicbVDLSgNBEOyNr7i+oh69DAbBU9gV8XELevGYgHlAsoTZyWwyZHZ2mekVQshHePGgiFe/x5t/4yTZgyYWNBRV3XR3hakUBj3v2ymsrW9sbhW33Z3dvf2D0uFR0ySZZrzBEpnodkgNl0LxBgqUvJ1qTuNQ8lY4up/5rSeujUjUI45THsR0oEQkGEUrtbomcuu9q16p7FW8Ocgq8XNShhy1Xumr209YFnOFTFJjOr6XYjChGgWTfOp2M8NTykZ0wDuWKhpzE0zm507JmVX6JEq0LYVkrv6emNDYmHEc2s6Y4tAsezPxP6+TYXQTTIRKM+SKLRZFmSSYkNnvpC80ZyjHllCmhb2VsCHVlKFNyLUh+Msvr5LmRcW/rNzWL8vVuzyOIpzAKZyDD9dQhQeoQQMYjOAZXuHNSZ0X5935WLQWnHzmGP7A+fwBWR2O8Q==</latexit>

t0min


Fit resultsVarious fits‣ : min of (tsink - t) [3-8]‣ tmin: min of (t-tK) [6-8]‣ (# of operators) x (# of states considered)

In 2015, effects of excited states were significantly underestimated

28









t0min


Q2

<latexit sha1_base64="CvRbHUg+ZPUVxfGl47gK/dBNB3M=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KkkpqLeiF48t2A9oQ9lsJ+3SzSbsboQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9djY2t7Z3dgt7xf2Dw6Pj0slpW8epYthisYhVN6AaBZfYMtwI7CYKaRQI7AST+7nfeUKleSwfzTRBP6IjyUPOqLFSp6/DYnNQHZTKbsVdgKwTLydlyNEYlL76w5ilEUrDBNW657mJ8TOqDGcCZ8V+qjGhbEJH2LNU0gi1ny3OnZFLqwxJGCtb0pCF+nsio5HW0yiwnRE1Y73qzcX/vF5qwhs/4zJJDUq2XBSmgpiYzH8nQ66QGTG1hDLF7a2EjamizNiEijYEb/XlddKuVrxa5bZZK9fv8jgKcA4XcAUeXEMdHqABLWAwgWd4hTcncV6cd+dj2brh5DNn8AfO5w9TDY7t</latexit>

Q6

<latexit sha1_base64="0mHzrsFelR0vvt77MTHzUaNOxGE=">AAAB7nicbVDLSgNBEOyNr7i+oh69DAbBU9gV8XELevGYgHlAsoTZyWwyZHZ2mekVQshHePGgiFe/x5t/4yTZgyYWNBRV3XR3hakUBj3v2ymsrW9sbhW33Z3dvf2D0uFR0ySZZrzBEpnodkgNl0LxBgqUvJ1qTuNQ8lY4up/5rSeujUjUI45THsR0oEQkGEUrtbomcuu9q16p7FW8Ocgq8XNShhy1Xumr209YFnOFTFJjOr6XYjChGgWTfOp2M8NTykZ0wDuWKhpzE0zm507JmVX6JEq0LYVkrv6emNDYmHEc2s6Y4tAsezPxP6+TYXQTTIRKM+SKLRZFmSSYkNnvpC80ZyjHllCmhb2VsCHVlKFNyLUh+Msvr5LmRcW/rNzWL8vVuzyOIpzAKZyDD9dQhQeoQQMYjOAZXuHNSZ0X5935WLQWnHzmGP7A+fwBWR2O8Q==</latexit>

t0min


1x1 result used to appear to be on plateau in 2015

clarified w/ more configurations

ε’ calculationIsospin-limit formula

29

Re

✓✏0

✏

◆= Re

⇢i!ei(�2��0)

p2✏

ImA2

ReA2� ImA0

ReA0

��= 1.38(5.15)(4.59)⇥ 10�4


ππ phase shifts

Lellouch-Lüscher finite volume correction

Wilson coefs. pQCD LQCD

AI = FGF

2V⇤usVud

X

i,j

[zi(µ) + ⌧yi(µ)]Zij(µ)h(⇡⇡)I|Qlatj |Ki

<latexit sha1_base64="nvz7RhuFdw+NElWqEiDu2/o2ME8=">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</latexit>

LQCD (+pQCD)

Renormalization matrix

Remaining topic: Renormalization

(! = ReA2/ReA0)

<latexit sha1_base64="wAGR1dCAhyv7JY74Nwd7U7tggIM=">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</latexit>

✔

✔

✔

G-parity BCs are used to extract on-shell kinematicsSignificant ππ excited states are treated better than in 2015

Contents

Introduction


Operator renormalization‣ RI/SMOM scheme & window problem

‣ Step scaling

‣ Our final result

On going projects

30

Power divergenceQuadratic divergence (~ a-2) appears in MEs from

‣ due to mixing 4-quark operators with

‣ Remove by subtraction

31

�

K

�

HW �

K

�

HW

O(m/a2)s�5d

<latexit sha1_base64="OyQiblCkwiEB63EzvataQU7/7fM=">AAACBnicbVDLSgNBEJyNr7i+oh5FGAxCvMTdEFFvQS/ejGAekI1L72Q2GTKzu8zMCiHk5MVf8eJBEa9+gzf/xsnjoNGChqKqm+6uIOFMacf5sjILi0vLK9lVe219Y3Mrt71TV3EqCa2RmMeyGYCinEW0ppnmtJlICiLgtBH0L8d+455KxeLoVg8S2hbQjVjICGgj+bl9T4X2dUEcw13pCHsBSKy8LggB/gnu+Lm8U3QmwH+JOyN5NEPVz316nZikgkaacFCq5TqJbg9BakY4HdleqmgCpA9d2jI0AkFVezh5Y4QPjdLBYSxNRRpP1J8TQxBKDURgOgXonpr3xuJ/XivV4Vl7yKIk1TQi00VhyrGO8TgT3GGSEs0HhgCRzNyKSQ8kEG2Ss00I7vzLf0m9VHTLxfObcr5yMYsji/bQASogF52iCrpCVVRDBD2gJ/SCXq1H69l6s96nrRlrNrOLfsH6+AYWIJcA</latexit>

Condition: hQ0i(t0)K(0)i = 0 at specific t0

<latexit sha1_base64="g86RInwFdv9nbuYRxwW9A3N27CA=">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</latexit>

K → ππ MEs shown earlier are the results after the subtraction

Qi ! Q0i = Qi � ↵is�5d

<latexit sha1_base64="KXjEQTmvD43Fbe5QHrLtPqOMosQ=">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</latexit>

(mixing w/ parity-even operator sd is invalid)

<latexit sha1_base64="vwvfOxWYl+H7b/9i29/x/Ln3s8o=">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</latexit>

�

K

�

HW ０

To remove ln a2 divergence

To construct appropriate Hamiltonian

RI/SMOM scheme (a common nonperturbative scheme)

Renormalization

32

wMSi (µ)

<latexit sha1_base64="jfDPmZX0C64tvGhzBteB4JAKAo8=">AAACBXicbVA7T8MwGHTKq4RXgBGGiAqpLFWCKgFbBQsLUhH0ITUhclynteo4ke2AqigLC3+FhQGEWPkPbPwbnDYDtJxk6XR3n+3v/JgSIS3rWystLC4tr5RX9bX1jc0tY3unLaKEI9xCEY1414cCU8JwSxJJcTfmGIY+xR1/dJH7nXvMBYnYrRzH2A3hgJGAICiV5Bn7jgj0B4/cpU6kcvk16dVNllWdMDnyjIpVsyYw54ldkAoo0PSML6cfoSTETCIKhejZVizdFHJJEMWZ7iQCxxCN4AD3FGUwxMJNJ1tk5qFS+mYQcXWYNCfq74kUhkKMQ18lQyiHYtbLxf+8XiKDUzclLE4kZmj6UJBQU0ZmXonZJxwjSceKQMSJ+quJhpBDJFVxuirBnl15nrSPa3a9dnZdrzTOizrKYA8cgCqwwQlogEvQBC2AwCN4Bq/gTXvSXrR37WMaLWnFzC74A+3zB2r/mIo=</latexit>

ZRij (µ

0)Qlatj

<latexit sha1_base64="sLaH3UNyDAH8AovIQmxs7f6xXKk=">AAACBHicbVDLTsJAFJ3iC+ur6pJNIzHihrSGRN0R3bgEI49ISzMdpjAw0zYzUxPSsHDjr7hxoTFu/Qh3/o0DdKHgSW5ycs69ufceP6ZESMv61nIrq2vrG/lNfWt7Z3fP2D9oiijhCDdQRCPe9qHAlIS4IYmkuB1zDJlPccsfXU/91gPmgkThnRzH2GWwH5KAICiV5BkFRwT6fffWS8lwUnJYcnJa94bdlEI58YyiVbZmMJeJnZEiyFDzjC+nF6GE4VAiCoXo2FYs3RRySRDFE91JBI4hGsE+7igaQoaFm86emJjHSumZQcRVhdKcqb8nUsiEGDNfdTIoB2LRm4r/eZ1EBhduSsI4kThE80VBQk0ZmdNEzB7hGEk6VgQiTtStJhpADpFUuekqBHvx5WXSPCvblfJlvVKsXmVx5EEBHIESsME5qIIbUAMNgMAjeAav4E170l60d+1j3prTsplD8Afa5w/Z6pef</latexit>

Qlati

<latexit sha1_base64="W8eKogfrcNv3ajHlmCqopotDEAc=">AAAB9HicbVDLSgNBEOyNr7i+oh69DAbBU9gVQb0FvXhMwDwgWcPsZDYZMvtwpjcQlnyHFw+KePVjvPk3TpI9aGJBQ1HVTXeXn0ih0XG+rcLa+sbmVnHb3tnd2z8oHR41dZwqxhsslrFq+1RzKSLeQIGStxPFaehL3vJHdzO/NeZKizh6wEnCvZAOIhEIRtFIXlcHdr0nHjNJcdorlZ2KMwdZJW5OypCj1it9dfsxS0MeIZNU647rJOhlVKFgkk/tbqp5QtmIDnjH0IiGXHvZ/OgpOTNKnwSxMhUhmau/JzIaaj0JfdMZUhzqZW8m/ud1UgyuvUxESYo8YotFQSoJxmSWAOkLxRnKiSGUKWFuJWxIFWVocrJNCO7yy6ukeVFxLys39cty9TaPowgncArn4MIVVOEeatAABk/wDK/wZo2tF+vd+li0Fqx85hj+wPr8AYWakfc=</latexit>

Renormalizationperturbative matching

ZRI/SMOM

ij(µ0)

<latexit sha1_base64="KanwFR3tzGvQpWLWPzjyRh/l714=">AAACA3icbVDLSsNAFJ34rPEVdaebYBHrpiZSUHdFN7oo1kcf2MQwmU7asZMHMxOhhIAbf8WNC0Xc+hPu/BunbRbaeuDC4Zx7ufceN6KEC8P4VqamZ2bn5nML6uLS8sqqtrZe52HMEK6hkIas6UKOKQlwTRBBcTNiGPouxQ23dzrwGw+YcRIGN6IfYduHnYB4BEEhJUfbtLin3joJuU/vkqvz/evKRSUtWH68u+doeaNoDKFPEjMjeZCh6mhfVjtEsY8DgSjkvGUakbATyARBFKeqFXMcQdSDHdySNIA+5nYy/CHVd6TS1r2QyQqEPlR/TyTQ57zvu7LTh6LLx72B+J/XioV3ZCckiGKBAzRa5MVUF6E+CERvE4aRoH1JIGJE3qqjLmQQCRmbKkMwx1+eJPWDolkqHl+W8uWTLI4c2ALboABMcAjK4AxUQQ0g8AiewSt4U56UF+Vd+Ri1TinZzAb4A+XzByGRlpA=</latexit>

µ02 = p21 = p22 = (p1 � p2)2

<latexit sha1_base64="aY/0wZbW1oWeA8nGdgZfwLJHfs4=">AAACE3icbVDLSgMxFM3UVx1foy7dBItYBcvMUFAXQtGNywr2AZ1pyaSZNjTzIMkIZeg/uPFX3LhQxK0bd/6N6XQW2nog3JNz7iW5x4sZFdI0v7XC0vLK6lpxXd/Y3NreMXb3miJKOCYNHLGItz0kCKMhaUgqGWnHnKDAY6TljW6mfuuBcEGj8F6OY+IGaBBSn2IkldQzTh3h66kTJMeTrg2vYNyz8mpntayEM3U56do9o2RWzAxwkVg5KYEc9Z7x5fQjnAQklJghITqWGUs3RVxSzMhEdxJBYoRHaEA6ioYoIMJNs50m8EgpfehHXJ1Qwkz9PZGiQIhx4KnOAMmhmPem4n9eJ5H+hZvSME4kCfHsIT9hUEZwGhDsU06wZGNFEOZU/RXiIeIISxWjrkKw5ldeJE27YlUrl3fVUu06j6MIDsAhKAMLnIMauAV10AAYPIJn8AretCftRXvXPmatBS2f2Qd/oH3+AIzOmi4=</latexit>

Qj

<latexit sha1_base64="w5QVQNJ93qNSicsUYZSBSeasbos=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx5bsB/QhrLZbtq1m03YnQgl9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38389hPXRsTqAScJ9yM6VCIUjKKV2j0Tlhr9x3654lbdOcgq8XJSgRz1fvmrN4hZGnGFTFJjup6boJ9RjYJJPi31UsMTysZ0yLuWKhpx42fzc6fkzCoDEsbalkIyV39PZDQyZhIFtjOiODLL3kz8z+umGF77mVBJilyxxaIwlQRjMvudDITmDOXEEsq0sLcSNqKaMrQJlWwI3vLLq6R1UfUuqzeNy0rtNo+jCCdwCufgwRXU4B7q0AQGY3iGV3hzEufFeXc+Fq0FJ585hj9wPn8Ap+2PJQ==</latexit>

p1

<latexit sha1_base64="bGdnzAHiWgzet6b3pLwhLxXT37M=">AAAB7nicbVBNSwMxEJ3Ur7p+VT16CRbBU9kVQb0VvXisYD+gXUo2zbah2WxIskJZ+iO8eFDEq7/Hm//GtN2Dtj4YeLw3w8y8SAlurO9/o9La+sbmVnnb29nd2z+oHB61TJppypo0FanuRMQwwSVrWm4F6yjNSBIJ1o7GdzO//cS04al8tBPFwoQMJY85JdZJ7Z6JPdUP+pWqX/PnwKskKEgVCjT6la/eIKVZwqSlghjTDXxlw5xoy6lgU6+XGaYIHZMh6zoqScJMmM/PneIzpwxwnGpX0uK5+nsiJ4kxkyRynQmxI7PszcT/vG5m4+sw51Jllkm6WBRnAtsUz37HA64ZtWLiCKGau1sxHRFNqHUJeS6EYPnlVdK6qAWXtZuHy2r9toijDCdwCucQwBXU4R4a0AQKY3iGV3hDCr2gd/SxaC2hYuYY/gB9/gCAw48L</latexit>

p2

<latexit sha1_base64="pRNcfLdTsVGY3kP7fcZfGEhI3oA=">AAAB7nicbVBNS8NAEJ3Urxq/qh69LBbBU0lKQb0VvXisYD+gDWWz3bRLN5tldyOU0B/hxYMiXv093vw3btMctPXBwOO9GWbmhZIzbTzv2yltbG5t75R33b39g8OjyvFJRyepIrRNEp6oXog15UzQtmGG055UFMchp91werfwu09UaZaIRzOTNIjxWLCIEWys1B3oyJXD+rBS9WpeDrRO/IJUoUBrWPkajBKSxlQYwrHWfd+TJsiwMoxwOncHqaYSkyke076lAsdUB1l+7hxdWGWEokTZEgbl6u+JDMdaz+LQdsbYTPSqtxD/8/qpia6DjAmZGirIclGUcmQStPgdjZiixPCZJZgoZm9FZIIVJsYm5NoQ/NWX10mnXvMbtZuHRrV5W8RRhjM4h0vw4QqacA8taAOBKTzDK7w50nlx3p2PZWvJKWZO4Q+czx+CR48M</latexit>

p2


p1


p1


p2


p2


p1


Qi

<latexit sha1_base64="KYny2ecrTP2mEEU4fXQKc4oOpbU=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoN6KXjy2YD+gDWWznbRLN5uwuxFK6I/w4kERr/4eb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akm93O/84RK81g+mmmCfkRHkoecUWOlTl+HpeaAD8oVt+ouQNaJl5MK5GgMyl/9YczSCKVhgmrd89zE+BlVhjOBs1I/1ZhQNqEj7FkqaYTazxbnzsiFVYYkjJUtachC/T2R0UjraRTYzoiasV715uJ/Xi814Y2fcZmkBiVbLgpTQUxM5r+TIVfIjJhaQpni9lbCxlRRZmxCJRuCt/ryOmlfVb1a9bZZq9Tv8jiKcAbncAkeXEMdHqABLWAwgWd4hTcncV6cd+dj2Vpw8plT+APn8wemaY8k</latexit>

NP free

Amputated vertex functions

Wilson coefficients

To remove ln a2 divergence

To construct appropriate Hamiltonian

RI/SMOM scheme (a common nonperturbative scheme)

Renormalization

32

wMSi (µ)

<latexit sha1_base64="jfDPmZX0C64tvGhzBteB4JAKAo8=">AAACBXicbVA7T8MwGHTKq4RXgBGGiAqpLFWCKgFbBQsLUhH0ITUhclynteo4ke2AqigLC3+FhQGEWPkPbPwbnDYDtJxk6XR3n+3v/JgSIS3rWystLC4tr5RX9bX1jc0tY3unLaKEI9xCEY1414cCU8JwSxJJcTfmGIY+xR1/dJH7nXvMBYnYrRzH2A3hgJGAICiV5Bn7jgj0B4/cpU6kcvk16dVNllWdMDnyjIpVsyYw54ldkAoo0PSML6cfoSTETCIKhejZVizdFHJJEMWZ7iQCxxCN4AD3FGUwxMJNJ1tk5qFS+mYQcXWYNCfq74kUhkKMQ18lQyiHYtbLxf+8XiKDUzclLE4kZmj6UJBQU0ZmXonZJxwjSceKQMSJ+quJhpBDJFVxuirBnl15nrSPa3a9dnZdrzTOizrKYA8cgCqwwQlogEvQBC2AwCN4Bq/gTXvSXrR37WMaLWnFzC74A+3zB2r/mIo=</latexit>

ZRij (µ

0)Qlatj

<latexit sha1_base64="sLaH3UNyDAH8AovIQmxs7f6xXKk=">AAACBHicbVDLTsJAFJ3iC+ur6pJNIzHihrSGRN0R3bgEI49ISzMdpjAw0zYzUxPSsHDjr7hxoTFu/Qh3/o0DdKHgSW5ycs69ufceP6ZESMv61nIrq2vrG/lNfWt7Z3fP2D9oiijhCDdQRCPe9qHAlIS4IYmkuB1zDJlPccsfXU/91gPmgkThnRzH2GWwH5KAICiV5BkFRwT6fffWS8lwUnJYcnJa94bdlEI58YyiVbZmMJeJnZEiyFDzjC+nF6GE4VAiCoXo2FYs3RRySRDFE91JBI4hGsE+7igaQoaFm86emJjHSumZQcRVhdKcqb8nUsiEGDNfdTIoB2LRm4r/eZ1EBhduSsI4kThE80VBQk0ZmdNEzB7hGEk6VgQiTtStJhpADpFUuekqBHvx5WXSPCvblfJlvVKsXmVx5EEBHIESsME5qIIbUAMNgMAjeAav4E170l60d+1j3prTsplD8Afa5w/Z6pef</latexit>

Qlati

<latexit sha1_base64="W8eKogfrcNv3ajHlmCqopotDEAc=">AAAB9HicbVDLSgNBEOyNr7i+oh69DAbBU9gVQb0FvXhMwDwgWcPsZDYZMvtwpjcQlnyHFw+KePVjvPk3TpI9aGJBQ1HVTXeXn0ih0XG+rcLa+sbmVnHb3tnd2z8oHR41dZwqxhsslrFq+1RzKSLeQIGStxPFaehL3vJHdzO/NeZKizh6wEnCvZAOIhEIRtFIXlcHdr0nHjNJcdorlZ2KMwdZJW5OypCj1it9dfsxS0MeIZNU647rJOhlVKFgkk/tbqp5QtmIDnjH0IiGXHvZ/OgpOTNKnwSxMhUhmau/JzIaaj0JfdMZUhzqZW8m/ud1UgyuvUxESYo8YotFQSoJxmSWAOkLxRnKiSGUKWFuJWxIFWVocrJNCO7yy6ukeVFxLys39cty9TaPowgncArn4MIVVOEeatAABk/wDK/wZo2tF+vd+li0Fqx85hj+wPr8AYWakfc=</latexit>

Renormalizationperturbative matching

ZRI/SMOM

ij(µ0)

<latexit sha1_base64="KanwFR3tzGvQpWLWPzjyRh/l714=">AAACA3icbVDLSsNAFJ34rPEVdaebYBHrpiZSUHdFN7oo1kcf2MQwmU7asZMHMxOhhIAbf8WNC0Xc+hPu/BunbRbaeuDC4Zx7ufceN6KEC8P4VqamZ2bn5nML6uLS8sqqtrZe52HMEK6hkIas6UKOKQlwTRBBcTNiGPouxQ23dzrwGw+YcRIGN6IfYduHnYB4BEEhJUfbtLin3joJuU/vkqvz/evKRSUtWH68u+doeaNoDKFPEjMjeZCh6mhfVjtEsY8DgSjkvGUakbATyARBFKeqFXMcQdSDHdySNIA+5nYy/CHVd6TS1r2QyQqEPlR/TyTQ57zvu7LTh6LLx72B+J/XioV3ZCckiGKBAzRa5MVUF6E+CERvE4aRoH1JIGJE3qqjLmQQCRmbKkMwx1+eJPWDolkqHl+W8uWTLI4c2ALboABMcAjK4AxUQQ0g8AiewSt4U56UF+Vd+Ri1TinZzAb4A+XzByGRlpA=</latexit>

µ02 = p21 = p22 = (p1 � p2)2

<latexit sha1_base64="aY/0wZbW1oWeA8nGdgZfwLJHfs4=">AAACE3icbVDLSgMxFM3UVx1foy7dBItYBcvMUFAXQtGNywr2AZ1pyaSZNjTzIMkIZeg/uPFX3LhQxK0bd/6N6XQW2nog3JNz7iW5x4sZFdI0v7XC0vLK6lpxXd/Y3NreMXb3miJKOCYNHLGItz0kCKMhaUgqGWnHnKDAY6TljW6mfuuBcEGj8F6OY+IGaBBSn2IkldQzTh3h66kTJMeTrg2vYNyz8mpntayEM3U56do9o2RWzAxwkVg5KYEc9Z7x5fQjnAQklJghITqWGUs3RVxSzMhEdxJBYoRHaEA6ioYoIMJNs50m8EgpfehHXJ1Qwkz9PZGiQIhx4KnOAMmhmPem4n9eJ5H+hZvSME4kCfHsIT9hUEZwGhDsU06wZGNFEOZU/RXiIeIISxWjrkKw5ldeJE27YlUrl3fVUu06j6MIDsAhKAMLnIMauAV10AAYPIJn8AretCftRXvXPmatBS2f2Qd/oH3+AIzOmi4=</latexit>

Qj

<latexit sha1_base64="w5QVQNJ93qNSicsUYZSBSeasbos=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx5bsB/QhrLZbtq1m03YnQgl9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38389hPXRsTqAScJ9yM6VCIUjKKV2j0Tlhr9x3654lbdOcgq8XJSgRz1fvmrN4hZGnGFTFJjup6boJ9RjYJJPi31UsMTysZ0yLuWKhpx42fzc6fkzCoDEsbalkIyV39PZDQyZhIFtjOiODLL3kz8z+umGF77mVBJilyxxaIwlQRjMvudDITmDOXEEsq0sLcSNqKaMrQJlWwI3vLLq6R1UfUuqzeNy0rtNo+jCCdwCufgwRXU4B7q0AQGY3iGV3hzEufFeXc+Fq0FJ585hj9wPn8Ap+2PJQ==</latexit>

p1


p2


p2


p1


p1


p2


p2


p1


Qi

<latexit sha1_base64="KYny2ecrTP2mEEU4fXQKc4oOpbU=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoN6KXjy2YD+gDWWznbRLN5uwuxFK6I/w4kERr/4eb/4bt20O2vpg4PHeDDPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTto5TxbDFYhGrbkA1Ci6xZbgR2E0U0igQ2Akm93O/84RK81g+mmmCfkRHkoecUWOlTl+HpeaAD8oVt+ouQNaJl5MK5GgMyl/9YczSCKVhgmrd89zE+BlVhjOBs1I/1ZhQNqEj7FkqaYTazxbnzsiFVYYkjJUtachC/T2R0UjraRTYzoiasV715uJ/Xi814Y2fcZmkBiVbLgpTQUxM5r+TIVfIjJhaQpni9lbCxlRRZmxCJRuCt/ryOmlfVb1a9bZZq9Tv8jiKcAbncAkeXEMdHqABLWAwgWd4hTcncV6cd+dj2Vpw8plT+APn8wemaY8k</latexit>

NP free

Amputated vertex functions

µ0 � ⇤QCD required

<latexit sha1_base64="dfIbbZgoXmBjd4G+76ayckYeOLg=">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</latexit>

: caused large sys. error in 2015µ0 ⌧ ⇡/a required

<latexit sha1_base64="Aw9URyhrx27oTZKHkKuWtQoXNbI=">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</latexit>

Wilson coefficients

Step scalingNonperturbative scale evolution technique

33

Z(µhigh, acoarse) =

✓Z(µhigh, afine)

Z(µlow, afine)

◆Z(µlow, acoarse)

<latexit sha1_base64="xA2cdcuK0Nq8XkdSMx7EXQ5Cv9g=">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</latexit>

fine lattice ensemble created (µhigh ⌧ ⇡/afine)

<latexit sha1_base64="2YlZgG5H6lAe7iZCUMajO0lRmdM=">AAACCnicbVC7TsMwFHXKq4RXgJHFUCGVpSSoErBVsDAWiT6kJooc12ms2klkO0hVlJmFX2FhACFWvoCNv8FtM0DhSFc6Oude+94TpIxKZdtfRmVpeWV1rbpubmxube9Yu3tdmWQCkw5OWCL6AZKE0Zh0FFWM9FNBEA8Y6QXj66nfuydC0iS+U5OUeByNYhpSjJSWfOvQlaFZd3nm5xEdRQV0GYNuSk+Rn4f6zeLEt2p2w54B/iVOSWqgRNu3Pt1hgjNOYoUZknLg2KnyciQUxYwUpptJkiI8RiMy0DRGnEgvn51SwGOtDGGYCF2xgjP150SOuJQTHuhOjlQkF72p+J83yFR44eU0TjNFYjz/KMwYVAmc5gKHVBCs2EQThAXVu0IcIYGw0umZOgRn8eS/pHvWcJqNy9tmrXVVxlEFB+AI1IEDzkEL3IA26AAMHsATeAGvxqPxbLwZ7/PWilHO7INfMD6+AS65mfg=</latexit>

used in 2015

µhigh ' 4.0 GeV

<latexit sha1_base64="u/Shhp58J4FboXh4CrToBnaIpk4=">AAACBnicbVBNS8NAEN34WeNX1KMIi0XwVBIpqLeiBz1WsB/QlLLZTtqlu0nc3Qgl1IsX/4oXD4p49Td489+4bXPQ1gcDj/dmmJkXJJwp7brf1sLi0vLKamHNXt/Y3Np2dnbrKk4lhRqNeSybAVHAWQQ1zTSHZiKBiIBDIxhcjv3GPUjF4uhWDxNoC9KLWMgo0UbqOAe+Cm1fpJ2sz3r9EfYVE3CHyyX34QrqHafoltwJ8DzxclJEOaod58vvxjQVEGnKiVItz010OyNSM8phZPupgoTQAelBy9CICFDtbPLGCB8ZpYvDWJqKNJ6ovycyIpQaisB0CqL7atYbi/95rVSHZ+2MRUmqIaLTRWHKsY7xOBPcZRKo5kNDCJXM3Ippn0hCtUnONiF4sy/Pk/pJySuXzm/KxcpFHkcB7aNDdIw8dIoq6BpVUQ1R9Iie0St6s56sF+vd+pi2Llj5zB76A+vzB9o0mBo=</latexit>

µlow ' 1.5 GeV

<latexit sha1_base64="KxyhCb0u9Wu/w7ssSAs4WEjPQZU=">AAACBXicbVDLSsNAFJ34rPVVdamLwSK4KolU1F3RhS4r2Ac0IUymN+3QmSTOTJQS6sKNv+LGhSJu/Qd3/o3Tx0JbD1w4nHMv994TJJwpbdvf1tz8wuLScm4lv7q2vrFZ2NquqziVFGo05rFsBkQBZxHUNNMcmokEIgIOjaB3MfQbdyAVi6Mb3U/AE6QTsZBRoo3kF/ZcFeZdkfoZj+8H2FVMwC12SscPl1D3C0W7ZI+AZ4kzIUU0QdUvfLntmKYCIk05Uarl2In2MiI1oxwGeTdVkBDaIx1oGRoRAcrLRl8M8IFR2jiMpalI45H6eyIjQqm+CEynILqrpr2h+J/XSnV46mUsSlINER0vClOOdYyHkeA2k0A17xtCqGTmVky7RBKqTXB5E4Iz/fIsqR+VnHLp7LpcrJxP4sihXbSPDpGDTlAFXaEqqiGKHtEzekVv1pP1Yr1bH+PWOWsys4P+wPr8AThsl8Q=</latexit>

a�1fine = 3.148(17) GeV

<latexit sha1_base64="/6dt6eJZbj52Ag5/cLoT+d81Qco=">AAACCnicbVDLSsNAFJ3UV42vqEs3o0WoC0uihdaFUHShywr2AW0Mk+mkHTp5MDMRSohbN/6KGxeKuPUL3Pk3TtsstPXAwOGce7hzjxsxKqRpfmu5hcWl5ZX8qr62vrG5ZWzvNEUYc0waOGQhb7tIEEYD0pBUMtKOOEG+y0jLHV6O/dY94YKGwa0cRcT2UT+gHsVIKskx9rvC05GTeCqf3iXHVgrP4WnJKleLVuXo4Yo0HaNglswJ4DyxMlIAGeqO8dXthTj2SSAxQ0J0LDOSdoK4pJiRVO/GgkQID1GfdBQNkE+EnUxOSeGhUnrQC7l6gYQT9XciQb4QI99Vkz6SAzHrjcX/vE4svaqd0CCKJQnwdJEXMyhDOO4F9ignWLKRIghzqv4K8QBxhKVqT1clWLMnz5PmiSqudHZTLtQusjryYA8cgCKwQAXUwDWogwbA4BE8g1fwpj1pL9q79jEdzWlZZhf8gfb5AydEmAI=</latexit>

a�1coarse = 1.378(7) GeV

<latexit sha1_base64="AqiKv0w6s0cG2UOjfilZ03VgWwg=">AAACC3icbVC7SgNBFJ2Nr7i+opY2Q4IQC8OuBhILIWihZQTzgCSG2cndZMjsg5lZISyxtvFXbCwUsfUH7PwbJ8kWmnjgwuGce7n3HifkTCrL+jZSS8srq2vpdXNjc2t7J7O7V5dBJCjUaMAD0XSIBM58qCmmODRDAcRzODSc4eXEb9yDkCzwb9UohI5H+j5zGSVKS91Mti1dk3RjGhAhYXwXH9tjfI7twmmpnC8dPVxBvZvJWQVrCrxI7ITkUIJqN/PV7gU08sBXlBMpW7YVqk5MhGKUw9hsRxJCQoekDy1NfeKB7MTTX8b4UCs97AZCl6/wVP09ERNPypHn6E6PqIGc9ybif14rUm65EzM/jBT4dLbIjThWAZ4Eg3tMAFV8pAmhgulbMR0QQajS8Zk6BHv+5UVSPynYxcLZTTFXuUjiSKMDlEV5ZKMSqqBrVEU1RNEjekav6M14Ml6Md+Nj1poykpl99AfG5w9kZpi5</latexit>

Final result for ε’

34

Re

✓✏0

✏

◆= Re

⇢i!ei(�2��0)

p2✏

ImA2

ReA2� ImA0

ReA0

��= 1.38(5.15)(4.59)⇥ 10�4


Re(✏0/✏)exp = 16.6(2.3)⇥ 10�4

<latexit sha1_base64="G4kHT9Ks+Ed5fiAjxhHWCYmIzOg=">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</latexit>

SM

<latexit sha1_base64="vYjNbhyeG98sRFAVFqdb2CsjNFA=">AAAB7XicbVBNSwMxEJ2tX3X9qnr0EiyCp7IrBfVW9OJFqGg/oF1KNs22sdlkSbJCWfofvHhQxKv/x5v/xrTdg7Y+GHi8N8PMvDDhTBvP+3YKK6tr6xvFTXdre2d3r7R/0NQyVYQ2iORStUOsKWeCNgwznLYTRXEcctoKR9dTv/VElWZSPJhxQoMYDwSLGMHGSs2ujtz7216p7FW8GdAy8XNShhz1Xumr25ckjakwhGOtO76XmCDDyjDC6cTtppommIzwgHYsFTimOshm107QiVX6KJLKljBopv6eyHCs9TgObWeMzVAvelPxP6+TmugiyJhIUkMFmS+KUo6MRNPXUZ8pSgwfW4KJYvZWRIZYYWJsQK4NwV98eZk0zyp+tXJ5Vy3XrvI4inAEx3AKPpxDDW6gDg0g8AjP8ApvjnRenHfnY95acPKZQ/gD5/MHyBuOoQ==</latexit>

Re(✏0/✏)SM,2015 = 1.38(5.15)stat(4.59)sys ⇥ 10�4

<latexit sha1_base64="G9TxhRdsu1937U8r2j1G4FktbVM=">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</latexit>

ΔI = 1/2 rule also consistent

(ReA0/ReA2)SM = 19.9(2.3)(4.4)

<latexit sha1_base64="o6aK0hnFqahhxnfhys7C1nsINbI=">AAACFHicbVDLSgMxFM34rPU16tJNsAgtyjhTC9qFUHXjRqiPPqBThkyaaUMzD5KMUIZ+hBt/xY0LRdy6cOffmLaz0NYDl3s4516Se9yIUSFN81ubm19YXFrOrGRX19Y3NvWt7boIY45JDYcs5E0XCcJoQGqSSkaaESfIdxlpuP3Lkd94IFzQMLiXg4i0fdQNqEcxkkpy9ANbeNn8LbEPzx3zaNKLBSe5ux7CM2iVjXK+aBwX8iWjVHD0nGmYY8BZYqUkB1JUHf3L7oQ49kkgMUNCtCwzku0EcUkxI8OsHQsSIdxHXdJSNEA+Ee1kfNQQ7iulA72QqwokHKu/NxLkCzHwXTXpI9kT095I/M9rxdI7bSc0iGJJAjx5yIsZlCEcJQQ7lBMs2UARhDlVf4W4hzjCUuWYVSFY0yfPknrRsEpG+aaUq1ykcWTALtgDeWCBE1ABV6AKagCDR/AMXsGb9qS9aO/ax2R0Tkt3dsAfaJ8/U9GZVA==</latexit>

(ReA0/ReA2)exp = 22.45(6)

<latexit sha1_base64="e7IG1buES9MbqpgJFNNrge/oA4U=">AAACD3icbVDJSgNBEO2JW4xb1KOXxqAkIHEmxO0gRL14jGIWSMLQ06lJmvQsdPeIYcgfePFXvHhQxKtXb/6NneWgiQ+KerxXRXc9J+RMKtP8NhJz8wuLS8nl1Mrq2vpGenOrKoNIUKjQgAei7hAJnPlQUUxxqIcCiOdwqDm9q6FfuwchWeDfqX4ILY90fOYySpSW7PR+U7qp7C00Dy5s83DcCzk7hodwgM9xoZAvHmWPc3Y6Y+bNEfAssSYkgyYo2+mvZjugkQe+opxI2bDMULViIhSjHAapZiQhJLRHOtDQ1CceyFY8umeA97TSxm4gdPkKj9TfGzHxpOx7jp70iOrKaW8o/uc1IuWetmLmh5ECn44fciOOVYCH4eA2E0AV72tCqGD6r5h2iSBU6QhTOgRr+uRZUi3krWL+7KaYKV1O4kiiHbSLsshCJ6iErlEZVRBFj+gZvaI348l4Md6Nj/FowpjsbKM/MD5/AMW/mLU=</latexit>

🤝

🤝

= 21.7(2.6)stat(6.2)sys(5.0)EM/IB ⇥ 10�4

<latexit sha1_base64="0T7kba236q8sz0K/SkmsvN+5cqA=">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</latexit>

Breakdown of sys. errors on A0

35

Changes in estimates of systematic errors since 2015 calculation

arXiv:1505.07863

Description 2015 Error 2020 Error

Operator normalisation 15% 5%1

Wilson coefficients 12% unchangedFinite lattice spacing 12% unchanged

Lellouch - Lüscher factor 11% 1.5%2

Residual FV corrections 7% unchangedParametric errors 5% 6%3

Excited state contamination 5% negligible4

Unphysical kinematics 3% 5%Total 27% 21%

1 As a result of step scaling from µ = 1.53 GeV ! 4.00 GeV.2 Better control of ⇡⇡ system due to additional operators.3 Largest uncertainty is due to ⌧ ⇠ 5%.4 Significantly underestimated in 2015.


Breakdown of sys. errors on A0

35

Changes in estimates of systematic errors since 2015 calculation

arXiv:1505.07863

Description 2015 Error 2020 Error

Operator normalisation 15% 5%1

Wilson coefficients 12% unchangedFinite lattice spacing 12% unchanged

Lellouch - Lüscher factor 11% 1.5%2

Residual FV corrections 7% unchangedParametric errors 5% 6%3

Excited state contamination 5% negligible4

Unphysical kinematics 3% 5%Total 27% 21%

1 As a result of step scaling from µ = 1.53 GeV ! 4.00 GeV.2 Better control of ⇡⇡ system due to additional operators.3 Largest uncertainty is due to ⌧ ⇠ 5%.4 Significantly underestimated in 2015.


finer lattice & continuum limit

NP treatment on going

Why changed so much?

36

Re

✓✏0

✏

◆= Re

⇢i!ei(�2��0)

p2✏

ImA2

ReA2� ImA0

ReA0

��= 1.38(5.15)(4.59)⇥ 10�4


2015: 12,3° from real value 2020, exp: mostly real value Doesn’t change too much

2015: accidental cancellation by order 2020: A0 part increased by x3.5

Caused more than 10x difference

= 21.7(2.6)stat(6.2)sys(5.0)EM/IB ⇥ 10�4


Re(✏0/✏)SM = 1.38(5.15)stat(4.59)sys ⇥ 10�4

<latexit sha1_base64="7Q80LhUfNmJ2t8GMYOkHznr3LOk=">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</latexit>

Ultimately misestimation of excited-state contaminations

ContentsIntroduction



On going projects

‣ K → ππ in periodic boundary conditions

‣ New contraction strategy

‣ NP matching of Wilson coefficients from 4 → 3 flavor theory

‣ Finer G-parity lattices

‣ Introducing QED & IB effects37

38

Why periodic BCs?Already have lattice ensembles with physical pion mass• 1 GeV, 243 x 64, 1.4 GeV, 323x64 and …• Continuum limit possible

Hope to introduce QED/IB effects near future • Difficult with G-parity boundary conditions• Periodic BC study valuable

Presence of Eππ = 2mπ state challenging• S/N ratio of Eππ = mK state should be the same as G-parity BC

39

type 4 dominates stats. error

G-parity calculation• types 1,2: averaged over

every 8 time translations• types 3,4: averaged over

every time translation

types 1,2 still expensive but no need of such precision　→ cost reduction?

�

K

�

HW

�

K

�

HW

23

FIG. 3: The contributions of the four Wick contraction topologies type1-type4 to the C2 (left) and

C6 (right) three-point functions with the pp(111) sink operator, plotted as a function of the time

separation between the kaon and the four-quark operator, t, at fixed tK!snksep = 16. For clarity we

plot with an inverted x-axis such that the pp sink operator is on the left-hand side. These

correlation functions include the subtraction of the pseudoscalar operator.

We perform measurements with all three two-pion operators described in Sec. III D. For the

K ! pp matrix elements of the four-quark operators, the full set of Wick contractions for the

pp(111) and pp(311) sink operators can be found in Appendix B.1 and B.2 of Ref. [33], and those

of the s operator in Appendix A of this document. The Wick contractions for the K ! pp matrix

elements of the pseudoscalar operator (with all three sink operators) as well as the K !vacuum

matrix elements of this and the four-quark operators are provided in Appendix B of this document.

In Fig. 3 we plot the contributions of the four classes of Wick contraction illustrated in Fig. 2 to

the three-point functions of the (subtracted) Q2 and Q6 operators with the pp(111) sink operator.

As the individual topologies are not separately interpretable as Green’s functions of the QCD

path integral, their time dependence is not necessarily described by the propagation of physical

eigenstates of the QCD Hamiltonian. As such we cannot combine our data sets with different

tK!snksep when generating such plots, and instead plot with a single, fixed tK!snk

sep = 16. Despite the

inability to interpret the time dependence physically, we can look at the relative contributions of

each topology within the central region of the plot in which the behavior of the combined data is

dominated by the kaon and pp ground-states, i.e. the region in which we perform our fits below.

Our final choices of cut incorporate data from this set in the range 6 t 11 (cf. Sec. IV E 4). In

this window we observe that for both the C2 and C6 correlation functions, the contribution of the

noisy, type4 disconnected diagrams is largely consistent with zero, albeit with much larger errors

�

K

�

HW

type 3

type 1

type 2

�

K

�

HW

type 4

RBC/UKQCD, PRD 102,054509

40

Sparsening HWCost mostly promotional to volume of HW

G-parity calculation: summed HW over whole 3D volume

Plan for this time: reduce the volume of HW (323 → 83: 64x speed up) for types 1 & 2

�

K

�

HW

full vol. sum full vol. sumsparse vol. sum

Summary

41

-10 -5 0 5 10 15 20 25 30 35

RBC/UKQCD 2015RBC/UKQCD 2020

Experiment

More independent calculations desired

Systematic error Isospin breaking effects Truncation error of Wilson coefficients Finite lattice cutoff

= 21.7(2.6)stat(6.2)sys(5.0)EM/IB ⇥ 10�4


3+ times more confs # of ππ operators

1 → 3 excited states well managed

Step scaling in NPR

Re(ε’/ε) (x104)

RBC/UKQCD working hard to figure out these & conclude K → ππ story

Direct CP violation and the ΔI = 1/2 rule in K ππ decay ...

Documents

Transcript of Direct CP violation and the ΔI = 1/2 rule in K ππ decay ...