# Presentation: 911001

## Isospin breaking Study with  Nf=2 domain-wall QCD + Quenched QED Simulation

**Isospin breaking Study with  Nf=2 domain-wall QCD + Quenched QED Simulation**

**Takumi Doi**
- (Univ. of Kentucky / RBRC)

*In collaboration with*

**T.Blum** (Univ. of Connecticut, RBRC)
**M.Hayakawa **(Nagoya Univ.)
**T.Izubuchi **(Kanazawa Univ., RBRC)
**N.Yamada** (KEK)

*for RBC Collaboration*

- Talk @ DWF 10yrs

- 03/15/2007

## Isospin breaking  important physics in QCD/QED.
Mass splitting: 
 = +4.5936(5) MeV
K+ - K0 =  - 3.972(27) MeV
p – n   = - 1.2933317(5) MeV
+ + - - 2 0 = 1.53(11) MeV
- - 0 = 6.48(24) MeV
Light quark mass can be determined by introducing QED
Most fundamental parameters in the standard model
Precise check for the “massless” scenario for strong CP problem
(p-n) : fundamental parameter in nuclear physics
Controls the lifetime of neutron (through the phase space)
Charge symmetry breaking in the N-N interaction

**Mass splitting:**
-  = +4.5936(5) MeV
- K+ - K0 =  - 3.972(27) MeV
- p – n   = - 1.2933317(5) MeV
- + + - - 2 0 = 1.53(11) MeV
- - - 0 = 6.48(24) MeV
_**Light quark mass can be determined by introducing QED**_
- Most fundamental parameters in the standard model
- Precise check for the “massless” scenario for strong CP problem
_**(p-n) : fundamental parameter in nuclear physics**_
- Controls the lifetime of neutron (through the phase space)
- Charge symmetry breaking in the N-N interaction

- Introduction

****** ****dominated by QED**

******                          ****QED**** +****QCD**

**(m****u****-m****d****)**

- Talk @ DWF 10yrs

- 03/15/2007

## Introduction

**Precise theoretical calculation of muon g-2**
- Muon is expected to be sensitive to short-range  New Physics ?
_**Large uncertainty from hadronic contribution**_

**QCD+QED simulation !**

- Talk @ DWF 10yrs

- 03/15/2007

## QED configurations

**Quenched non-compact QED**
**No photon self-coupling**
- ** **_**free theory, coupling does not run**_
**Generating QED configs:**
- Generate *A***(em) in momentum-space
- We must fix the gauge redundancy
-  Coulomb gauge + additional gauge fixing for A0
- Gauge fixing condition can be solved analytically and the action becomes gaussian  simple gaussian random number generation
- No autocorrelation (int=0) even for arbitrary small coupling
- Fourier inversion to x-space
- Wilson line* U****(em)=exp[iA****(em))]                                          *to connect next-neighbor-site

- A.Duncan, E.Eichten, H.Thacker,
-  PRL76(1996)3894

- Talk @ DWF 10yrs

- 03/15/2007

## QCD configurations

**Light quark sector ******** ****chiral symmetry is essential !**
- We employ **the domain wall fermion**
**Nf=2 dynamical domain-wall QCD configs**
- (RBC, PRD72(2005)114505)
- DBW2 gauge action
- a-1 = 1.7GeV (beta=0.8)
- V=163X32, Ls=12  L3 = (2fm)3
- domain-wall height M5=1.8
- sea quark mass=0.02, 0.03, 0.04
-  mq = 1/2 ms –  ms (m = 500-700 MeV), ms=0.0446
- About 200 configs with 25 trajectories separation

***Manifest flavor structure***

- We will use Nf=2+1 confs as well in near future

- Talk @ DWF 10yrs

- 03/15/2007

## Symmetry and SSB with QED on

- Pure QCD
- SU(3)R X SU(3)L X U(1)v  SU(3)V X U(1)V
- 8 NG-bosons for massless quark
- QCD+QED
- Q=diag(+2/3,-1/3,-1/3) = T3+T8/sqrt(3)
- Axial WT identity
- SU(2)Rds X SU(2)Lds X U(1)em X U(1)V  SU(2)Vds X U(1)em X U(1)V
- 3 NG-bosons for massless quark

- 02  etc.

- Talk @ DWF 10yrs

- 03/15/2007

## Meson masses

- QED parametrization + NLO QCD

- NG-boson

- NG-boson

- Non-NG

- Non-NG

- quasi-NG

**The most fundamental LEC with QED on**

- For Iz=0, S=0 channel,                 we ignore the disconnected diagram, we ignore the  mixing of - , –’ (expected to be higher order)

- Talk @ DWF 10yrs

- 03/15/2007

## Extract the mass difference

**We focus on the mass difference directly.**
- (e=0) = A(e=0) exp(-m(e=0) t)
- (e)     = A(e)     exp(-m(e)  t)
**[For visibility]**
- R= (e)/(e=0)
- R (1+ A) – [ m(e)-m(e=0) ] t , (A=(A(e)-A(e=0))/A(e=0))
**The slope of t** is directly related to
-    the **mass difference**
- Statistical fluctuation is expected to be canceled in the ratio, which **improves S/N**
- In the final analysis, we take exp-fit to assure the ground state dominance

- Talk @ DWF 10yrs

- 03/15/2007

## The QED effect on PS-meson

- (msea=0.04)

- (msea=0.03)

- Talk @ DWF 10yrs

- 03/15/2007

## Quark mass determination

- Offset of quark mass in DWF
- Residual quark mass with QED on determined by PCAC
- Fit to the quark mass dependence of neutron mesons and pion mass splittings
-  LECs are determined
- LECs obtained + experimental inputs
- M(0)2  sensitive to (mu+md), insensitive to (mu-md)
-  determine **(mu+md)**
- M(K+)2+M(K0)2  sensitive to ms, (mu+md), insensitive to (mu-md)
-  determine **ms**
- [M(K0)2-M(K+)2] - [M(0)2-M(+)2] sensitive to (mu-md), ms,                     insensitive to (mu+md)
-  determine **(mu-md)**

- Talk @ DWF 10yrs

- 03/15/2007

## Quark masses and splittings

- Masses
- By employing RBC                                                         nonperturbative 1/Zm=0.62
- Systematic error
- neglection of nondegenate mass effect
-  finite V:  estimation  by Cottingham formula                    + vector saturation model  would be negligible
- Splittings

- MILC w/o QED

- Kaon suffer from large systematic error

- Talk @ DWF 10yrs

- 03/15/2007

## Isospin breaking in baryons

- Mass splitting between octets
- p – n   = - 1.2933317(5) MeV
- + + - - 2 0 = 1.53(11) MeV
- - - 0 = 6.48(24) MeV
- Two point correlation function                                     with the operator
- Forward and Backward propagation is averaged to increase statistics

- Talk @ DWF 10yrs

- 03/15/2007

## Plot of (proton)/(neutron)

- The negative slope corresponds to **m(p) > m(n)**   from the QED effect (mu=md)

-  If we rescale to Q=physical,         all the results from different Q are found to agree with each other

- (relative error is smaller for larger Q)

**However,** _S/N is not so enough to extract quantitative results..._

- Talk @ DWF 10yrs

- 03/15/2007

## The idea for the S/N improvement

- Q= +e, -e trick
- Physical observables are expected to
- (Perturbatively, only O(e2n) appear)
-  [ m(+e) + m(-e) ] kill the fluctuation of O(e)
- QED confs: **{A********(em)} ******** {+A********(em), -A********(em)}**

*Very simple idea, but left unaware in the literature...*

**Same Boltzmann Weight !**

- Talk @ DWF 10yrs

- 03/15/2007

## Q= +e, -e trick

**Q= -e only**

**Q=+e only**

**Remarkable Improvement !**

- Talk @ DWF 10yrs

- 03/15/2007

## Proton neutron mass difference from the QED effect

- The lattice result indicates
- M(p) > M(n) (QED) at each msea
-   c.f. Cottingham formula:
-         M(p)-M(n)(QED)= 0.76MeV

**Charge dependence**

**proton-neutron at Q=physical**

- Need more statistics ? Finite V ?

**M(p)-M(n)**

**M(p)-M(n)**

******(em)**

**Physical**

- msea=0.03

- (msea=mval)

- Talk @ DWF 10yrs

- 03/15/2007

## Isospin breaking on triplet

- Insensitive to u/d quark mass difference
- M(+)+M(-) – 2 M(0) * ****Only QED effect !***
- M(+)+M(-) – 2M(0) = O(e^2) + O(mu -- md)
- When isospin symmetry breaks,
-     mixing occurs between 0 , 8 and1
-          c.f. 8(1116) < 0(1193) (<1)
-  Diagonalize 3x3 correlation function matrix (variational method)

- mu md

- +higher order terms

- (uus)

- (dds)

- (uds)

- Talk @ DWF 10yrs

- 03/15/2007

## Isospin breaking in  triplet

- c.f. exp: 1.6MeV

**Charge dependence**

- [Variational method]

- diagonalize

**eigenvectors**

**t**

- (up to n-th excited state)

- 0

- 

**[Q=1.0]**

******(em)**

- msea=0.03

**chiral-extrapolation**

- Talk @ DWF 10yrs

- 03/15/2007

## The QCD part ([md-mu] effect)

- ChPT for baryons (HQchiPT)
- LO  linear in quark mass
- NLO  mq^(3/2) and logs but cancel in splitting
- We perform the simulation with nondegenerate u,d quark masses and extract the linear response to (md-mu)
*Mass difference is again essential !*

- B.C.Tiburzi et al. NPA764(06)274

- (for unquenched case)

- Talk @ DWF 10yrs

- 03/15/2007

## Splittings with various (md-mu)

- (msea=0.03)

**proton-neutron**

**Xi(-)-Xi(0)**

- Talk @ DWF 10yrs

- 03/15/2007

## Splittings with various (md-mu)

- (msea=0.03)

**Sig(+)-Sig(0)**

**Sig(+)+Sig(-) – 2 Sig(0)**

- Talk @ DWF 10yrs

- 03/15/2007

## The isospin breaking from QCD

- p – n                  - 2.55(18)(51) MeV
- Xi(-) - Xi(0)         +3.86(11)(77) MeV
- Sig(+) – Sig(0)    - 3.32(12)(66) MeV
- Sig(-)  – Sig(0)    +3.04(11)(61) MeV
- Sig(+) – Sig(-)     - 6.37(22)(127) MeV
- Inputs: (md-mu)_MS_ = 3.0(6) MeV                                      ( a(md-mu)bare=0.0011(2) ) from meson spectrum

- cf. S.R.Beane, K.Orginos,M.J.Savage hep-lat/0605014
- p – n = - 2.26(57)(42)(10) MeV

- Talk @ DWF 10yrs

- 03/15/2007

## Summary/Outlook

- We have investigated the isospin breaking effect on hadron spectrum using Lattice QCD+QED simulation
-  Determination of the LECs which appear in meson spectrum + experimental input  quark mass
- Further refinement is underway to include the nondegerate              quark mass X QED correction
- In QED effect determination, Q= +e, -e trick gives remarkable improvement, while baryons still need additional work
- The QCD (mu-md) effect on baryons obtained reasonably
- Nf=2+1 (RBC-UKQCD), explicit estimate of finite volume artifact etc. dynamical QED, external EM field NEDM, polarizability

- Talk @ DWF 10yrs

- 03/15/2007

## Residual quark mass

- Because there exists explicit chiral symmetry breaking (Ls ≠∞, in DWF), we must evaluate the residual quark mass _**with QED charge on**_
- One of the largest QED effect in the determination of u, d quark mass

- mres(u)     =  0.001478(40)  mres(d)     =  0.001428(40)  mres(QCD) = 0.001387(39)

- In the chiral limit,

- Talk @ DWF 10yrs

- 03/15/2007

## The determination of LECs

- Fit to the neutral mesons (NG-bosons)

- Leading LEC

- NLO LECs               (L5-2L8), (L4-2L6)

- NLO EM-LECs

- Talk @ DWF 10yrs

- 03/15/2007

## The determination of LECs

- Pion mass splittings

- Using the lattice output for the pion mass splitting,

- NLO EM-LECs

- LO EM-LEC

- Talk @ DWF 10yrs

- 03/15/2007

## Isospin breaking in  doublet

**Charge dependence**

******(em)**

- msea=0.03

**chiral-extrapolation**

- Talk @ DWF 10yrs

- 03/15/2007