The E&M Corrections to K → ππ

Joe Karpie(Columbia University)

Norman Christ and Tuan Nguyen(Columbia University)

Xu Feng(Peking University)

DWF25,December 17, 2021

The RBC & UKQCD collaborationsUC Berkeley/LBNLAaron Meyer

BNL and BNL/RBRCYasumichi Aoki (KEK)Peter Boyle (Edinburgh)Taku IzubuchiYong-Chull JangChulwoo JungChristopher KellyMeifeng LinHiroshi OhkiShigemi Ohta (KEK)Amarjit Soni

CERNAndreas Jüttner (Southampton)

Columbia UniversityNorman ChristDuo GuoYikai HuoYong-Chull JangJoseph KarpieBob MawhinneyAhmed ShetaBigeng WangTianle WangYidi Zhao

University of ConnecticutTom BlumLuchang Jin (RBRC)Michael RiberdyMasaaki Tomii

Edinburgh UniversityMatteo Di CarloLuigi Del DebbioFelix ErbenVera GülpersTim HarrisRaoul HodgsonNelson LachiniMichael MarshallFionn Ó hÓgáinAntonin PortelliJames RichingsAzusa YamaguchiAndrew Z.N. Yong

KEKJulien Frison

University of LiverpoolNicolas Garron

Michigan State UniversityDan Hoying

Milano BicoccaMattia Bruno

Peking UniversityXu Feng

University of RegensburgDavide GiustiChristoph Lehner (BNL)

University of SiegenMatthew BlackOliver Witzel

University of SouthamptonNils AsmussenAlessandro BaroneJonathan FlynnRyan HillRajnandini MukherjeeChris Sachrajda

University of Southern DenmarkTobias Tsang

Stony Brook UniversityJun-Sik YooSergey Syritsyn (RBRC)

Introduction

• K → ππ decay and direct CP violation [R. Abbott et al,arXiv:2004.09440]

ε′ =ieδ2−δ0√

2

ReA2

ReA0

(ImA2

ReA2− ImA0

ReA0

)(1)

• ∆I = 1/2 rule, where ReA0ReA2

∼ 20, amplifies possiblecorrections

• Naıvely 1% isospin breaking, such as QED and light quarkmass difference, can be ∼ 20% corrections

• In order to study how to include QED in a finite box with twohadrons, begin with the simpler system of π − π scattering

• For K → ππ, see C. Kelly’s and M. Tomii’s talks from thismorning

• For ππ scattering, see T. Wang’s talk this afternoon

QEDL in a finite box

• Luscher quantization works for interactions with a finiterange, such as QCD unlike QED• QEDL is a popular way of including QED into a finite box by

removing the zero modes. VQEDL(x) =

∑|k|6=0

e ik·x

k2

• Adds new 1/L power law corrections to be determined

• Relation between finite volume energies with QEDL andscattering phase shift, in non-relativistic regime, derived byBeane and Savage [S. Beane and M. Savage, arXiv:1407.4846]and implemented by NPLQCD and QCDSF [S. Beane et al,arXiv:2003.12130] at heavy pion masses for understandingnucleon scattering

• Correction for Lellouch-Luscher for K → ππ discussed inproceedings [Y. Cai and Z. Davoudi, arXiv:1812.11015]

Splitting up QED

• First, choose the Coulomb gauge• Separates transverse radiation and instantaneous Coulomb

interaction• The non-Lorentz covariance of the gauge is not issue. Use rest

frame of the Kaon

• Second, truncate the Coulomb interaction to finite range R• The truncated interaction works perfectly with Luscher

quantization in a finite volume• Gives residual power law corrections in R−1

• The remainder of the Coulomb interaction, and the power lawcorrections in R, can be fixed after the fact

• Three pieces

• Truncated Coulomb potential VTC = e2

r θ(R − r) [Numerically]• Complement of the truncated Coulomb potential

VTC = e2

r θ(r − R) [Analytically]• Transverse Radiation [Neglected for now]

Truncated Coulomb potential

• Truncated Coulomb potential works perfectly well withLuscher quantization given• R < L/2 so that the potential remains within the finite volume• R >∼ Λ−1

QCD so that VTC can be corrected with out needingquark dynamics

• Limits need to be studied realistic calculations

• Unlike QEDL, the only 1/L power law corrections are thosefrom neglected higher partial waves, instead have 1/R powerlaw corrections• 1/R can be studied without changing the ensemble!• 1/R can also be corrected analytically

• All that remains are exponentially suppressed L and Rcorrections

Long Distance Calculation[N. Christ, X. Feng, JK, T. Nguyen, arXiv:2111.04668 ]

Calculating Corrections

• Want to remove all power corrections in R from thetruncation, neglecting exponentially suppressed terms

• With sufficiently large R, the interacting pions can be treatedas elementary particles in infinite volume

• Include VTC into the Lippmann-Schwinger series

M =

K

(a)

(b) (c)

K +

+ K K + ...K

K K

−k + P2

k + P2k′ + P

2

−k′ + P2

K K

−k + P2−k′′ + P

2

k + P2

k′′ + P2

Calculating Corrections

• (a) Self Energy Correction [Suppressed]

• (b) 2-PI Kernel Correction [Suppressed]

• (c) Exchange Diagrams Correction [Dominant]

M =

K

(a)

(b) (c)

K +

+ K K + ...K

K K

−k + P2

k + P2k′ + P

2

−k′ + P2

K K

−k + P2−k′′ + P

2

k + P2

k′′ + P2

Exponentially Suppressed Corrections

• When the two pion energy is below the 4 pion threshold, theself energy and 2-PI Kernel Correction can be calculated inEuclidean space and analytically continued to Minkowski space

• These diagrams will be dominated by short range interactionsdue to exponential decay of the propagators

• When VTC is included, at least 2 propagators must travel thelarge distance R giving exponential suppression of the diagram

• [N. Christ, X. Feng, JK, T. Nguyen, arXiv:2111.04668 ]

The Exchange Diagrams Correction

(a)

= K +

+ K K + ...K

K K

M M

(d)(c)

M M

p + P2

−p + P2

−p′ + P2

p′ + P2

k + P2

−k + P2

−k′ + P2

k′ + P2

k − k′

(a) (b)

• Generally diagrams involve off-shell scattering kernel andoff-shell pion EM vertex

• VTC restricts to long distance regions where they go on shell

• On shell quantities are obtainable from LQCD or combinationof experiment and phenomenology

The Exchange Diagrams Correction

(a)

= K +

+ K K + ...K

K K

M M

(d)(c)

M M

p + P2

−p + P2

−p′ + P2

p′ + P2

k + P2

−k + P2

−k′ + P2

k′ + P2

k − k′

(a) (b)

MTC ,l =

1

2l + 1

l∑m=−l

∫ ∫d4k ′d4kΨout

lm (k ′,P)∗KTC (k ′, k,P)Ψinlm(k,P) (2)

Ψin/outlm (k ,P) = ψ0

lm(k ,P) + ψin/outlm (k ,P)

The Exchange Diagrams Correction

(a)

= K +

+ K K + ...K

K K

M M

(d)(c)

M M

p + P2

−p + P2

−p′ + P2

p′ + P2

k + P2

−k + P2

−k′ + P2

k′ + P2

k − k′

(a) (b)

δTCl =ωp

2p

1

2l + 1

l∑m=−l

∞∑l ′=0

l ′∑m′=−l ′

∫ ∞0

dz

∫ ∫dΩp′dΩpVTC

(|~z |)

·[Ylm(p′)∗Yl ′m′(p

′)∗Ylm(p)Yl ′m′(p)

]F (|~p − ~p ′|)2 sin2

(p|~z | − πl

2+ δl

)

Short Distance Calculation

Reminder of Finite Volume Quantization

• Relationship between finite volume energy levels and infinitevolume phase shifts

Valid when interaction is exponentially suppressed atdistance smaller than volume

Truncated Coulomb potential naturally works in finitevolume for R < L/2

• Quantization condition (q = Lp/2π)

δ0(p) + φ(q) = nπ (3)

tanφ(q) = − π3/2q

Z00(1, q)(4)

Perturbation Theory

• Energy and phase shifts can be expanded in the couplingE = E (0) + αE (1) + . . .δ0 = δ

(0)0 + αδ

(1)0 + . . .

• Perturbative corrections to the energy calculated by

E (1) =〈Oππ(tf ) 1

2

∫d3r1d

3r2ρ(r2, tV )VTC(|r2 − r1|L)ρ(r1, tV )Oππ(ti )〉〈Oππ(tf )Oππ(ti )〉

(5)

• Perturbative corrections to Luscher quantization equation

δ(1)0 (p(0)) = −

dδ0(p)

dp+

dφ(q)

dq

L

2π

p=p(0)

E (0)

4p(0)E (1) , (6)

p(0) =√

(E (0)/2)2 −m2

Renormalization of the quark mass

• Ensemble used was generated at physical pion mass withisospin symmetry

Wish to maintain pion mass in isospin breaking calculation

• The quark mass must be adjusted perturbatively to keep thephysical pion mass fixed when QED is added

Tune αm(1)q correction to counteract QED pion mass shift

• Uses same matrix element as the light quark mass splitting

• Apply same quark mass shift from single pion to thescattering pions case

Lattice Setup

• Diagrams for single pion

• Currents meant at fixed time for Coulomb interaction

• Analogous diagrams needed for two pions

Exchange Diagram

Self Energy Diagram

Scalar Current

Lattice Setup

• RBC/UKQCD’s 24ID ensemble, a−1 = 1 GeV, mπ = 140MeV, 243 × 64

• Form correlation functions with Wall source and Wall sinkinterpolators

• Coulomb photons inserted stochastically

• Self-Energy diagram created with a sequential propagator(Still being calculated)

• Other diagrams by tying together source and sink propagators

• Use R/a = 2.0, 4.0, 6.0, 8.0, 10.0, 11.5 to test dependence

• Matrix elements obtained through summation method

R(T ) =

∑t C3(T , t)

C2(T )= MT + b (7)

Exchange diagram for single pion

• Linear behavior in 1/R from truncating the interaction whichis 1/r

• R = 11.5, which has interaction filling almost entire volume,may be seeing edge effects from finite volume

• Preliminary to study in detail without self energy diagram

5.0 7.5 10.0 12.5 15.0 17.5T

0.2

0.3

0.4

0.5

0.6

R = 2.0

R = 4.0

R = 6.0

R = 8.0

R = 10.0

R = 11.5

0.1 0.2 0.3 0.4 0.51/R

0.012

0.014

0.016

0.018

0.020

Exchange diagram for two pions

• Linear behavior in 1/R from truncating the interaction whichis 1/r

• R = 11.5, which has interaction filling almost entire volume,may be seeing edge effects from finite volume

• Preliminary to study in detail without self energy diagram

5 10 15 20T

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 2.0

R = 4.0

R = 6.0

R = 8.0

R = 10.0

R = 11.5

0.1 0.2 0.3 0.4 0.51/R

0.0225

0.0250

0.0275

0.0300

0.0325

0.0350

0.0375

Scalar current insertion• Scalar current insertion for light quark mass splitting

• Appears noisier than the insertion of photons for exchangediagram

6 8 10 12 14 16 18T

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

6 8 10 12 14 16 18T

7.0

7.5

8.0

8.5

9.0

9.5

10.0

Figure: The scalar current diagram for single pion with M = 0.077(4)(Left) and two pions with M = 0.154(9)(Right)

What needs doing to get K → ππ

• Finish diagram for π+π+ scattering and study R dependencein phase shifts

• Study with twisted boundary conditions to move away fromthreshold

• Formalism for transverse radiation in ππ scattering andK → ππ

• Generalize truncated Coulomb interaction formalism toneutral ππ scattering with two channels

• Expand truncated Coulomb interaction formalism to K → ππdecays

• Work on methods for efficient calculation of QED insertionsto K → ππ decay amplitudes

Conclusions

• Can add Coulomb interactions to Luscher finite volumequantization without new 1/L power law corrections, at costof new 1/R corrections

• If R is within appropriate limits, 1/R corrections can becalculated analytically in the infinite volume

• Need to study R dependence in realistic lattice calculation tostudy the limits and R independence of final phase shifts

• Need to add back in the transverse radiation to solve fullrelativistic problem