Lattice QED and QCD - 2009 Taipei Workshop on Lattice QCD

<strong>Lattice</strong> <strong>QED</strong> <strong>and</strong> <strong>QCD</strong> 

Taku Izubuchi 

for Riken-BNL-Columbia/UK<strong>QCD</strong> collaboration 

RIKEN BNL Reserch Center 

Taku Izubuchi, <strong>Taipei</strong>, TW<strong>QCD</strong>09, Decmber 15, <strong>2009</strong> 1

Electromagnetic Splittings 

<strong>QED</strong> + <strong>QCD</strong> simulations 

[T.Blum, T.Doi, M.Hayakawa, T.Izubuchi, S.Uno, N.Yamada <strong>and</strong> R.Zhou] , in preparation 

[R. Zhou, S. Uno] , “Isospin symmetry breaking in 2+1 flavor <strong>QCD</strong>+<strong>QED</strong>” PoS(LAT<strong>2009</strong>) 182. 

[T.Izubuchi] , “Studies of the <strong>QCD</strong> <strong>and</strong> <strong>QED</strong> effects on Isospin breaking” PoS(KAON09) 034. 

[R. Zhou, T.Blum, T.Doi, M.Hayakawa, T.Izubuchi, <strong>and</strong> N.Yamada] , 

“Isospin symmetry breaking effects in the pion <strong>and</strong> nucleon masses” PoS(LAT2008) 131. 

[T. Blum, T. Doi, M. Hayakawa, T.Izubuchi, N. Yamada] , 

“Determination of light quark masses from the electromagnetic splitting of psedoscalar meson 

masses computed with two flavors of domain wall fermions” Phys. Rev.D76 (2007) 114508 

“The isospin breaking effect on baryons with Nf=2 domain wall fermions” PoS(LAT2006) 174 

“Electromagnetic properties of hadrons with two flavors of dynamical domain wall fermions” 

PoS(LAT2005) 092 

“Hadronic light-by light scattering contribution to the muon g-2 from lattice <strong>QCD</strong>: Methodology” 

PoS(LAT2005) 353 


Isospin Breaking Effects 

• The first principle calculations of isospin breaking effects 

due to electromagnetic (EM) <strong>and</strong> the up, down quark mass 

difference are necessary for accurate hadron spectrum, 

quark mass determination. 

• Isospin breaking’s are measured very accurately : 

m π ± − m π 0 = 4.5936(5)MeV, 

m K ± − m K 0 = −3.9272(27)MeV 

mN − mP = 1.2933317(5)MeV 

cf. splitting for vector meson is consistent with zero 

experimentally. 

u 

2/3 e 

q 

Q e 

¼ + 

(repulsive) 

¼ 0 

(attractive) 

• Positive mass difference between Neutron (udd) <strong>and</strong> Proton (uud) stabilizes proton 

thus make our world as it is. mN − mP = 1.2933317(5)MeV 

Taku Izubuchi, <strong>Taipei</strong>, TW<strong>QCD</strong>09, Decmber 15, <strong>2009</strong> 3 

d 

1/3e 

q 

-Q e

Isospin <strong>and</strong> SU(3)F Breaking Effects on spectrum 

• PS meson spectrum <strong>and</strong> quark masses. 

• Asymmetry due to Quark mass differences : 

mu = md = ms 

• Asymmetry due to <strong>QED</strong> interactions : 

Qu = 2/3e, Qd = Qs = −1/3e 

• <strong>QCD</strong> axial anomaly makes m ′ 

η heavy. 

• A few % effect: O(mu − md), O(α) 

¼ ¡ 

K 0 K + 

d¹u 

d¹s u¹s 

s¹u 

¼ 0 ´ ´ 0 

s ¹ d 

K ¡ ¹ K 0 

• Could mu 0, which would explain the very small Neutron EDM ? (Strong CP problem) 

[D.Nelson,G.Fleming, G.Kilcup,PRL90:021601,2003. ] 

• m + 

ρ 

− m0 

ρ , Γ ρ +, Γ ρ 0 are related to the conversion of Γ(τ → Hadrons) to Γ(e + e − → 

Hadrons) to determine leading <strong>QCD</strong> correction to muon g − 2. 


u ¹ d 

¼ 0

ChPT with EM 

• Axial WT identity with EM for massless quarks (NF = 3), 

Lem = eAem µ(x)¯qQemγµq(x), Q = diag(2/3, −1/3, −1/3) 

∂ µ A a 

µ = ieAem µ q [T a , Q] γ µ γ5q − α 

2π tr (QT a ) F µν 

em e Fem µν , 

neutral currents, four A a 

µ (x), are conserved (ignoring O(α2 ) effects): 

π 0 , (K 0 , K 0 , η8) are still a NG bosons. 

• NG field U(x) = e iΦ(x)/F 0 of SU(NF )L × SU(NF )R/SU(NF )V 

Lχ = 1 2 

FEM 4 

1 D 

2 

+ F0 DµU 

4 † E 

DµU 

+ 

D 

χU † + χ † E 

U 

DµU = ∂µU − iQReAem,µU + iUQLeAem,µ . 

symmetric under 

U → gRUg † 

L , 

χ = 2B0diag(mu, md, · · · ) → gRχg † 

L , 

QL → gLQLg † 

L , QR → gRQRg † 

R , 

+ C 

D 

QRUQLU †E 

, 


• Using the half field u = √ U 

u → gRuh † = hug † 

L 

with the SU(3)V transformation h = h[gL, gR, U]. 

• List basic building blocks in the convinient form, e O = u † Ou † 

eO → h e Oh under O → gROg † 

L . 

• <strong>QCD</strong> Building blocks 

uµ = i g DµU = {u † (∂µu − iRµu) − u(∂µu † − iLµu † )}, 

χ± = eχ ± eχ † 

• <strong>QED</strong> Building blocks 

eQL = uQLu † , e QR = u † QLu, 

• For SU(2)+Kaon ChPT, Kaon multiplet K = (K + , K 0 ) T is transformed as K → hK 

<strong>and</strong> Kaon building blocks are 

KK † , DµKK † ∓ KDµK † 


• For SU(3)+ EM PQChPT, there are one O(e 2 ) term (Dashen’s term C), <strong>and</strong> 14(unitary)+2(PQ) 

O(e 2 m) terms. 

• PS mass formula at O(p 4 , p 2 e 2 ) [Bijnens Danielsson, PRD75 (07)] 

M 2 

π ± = 2mB0 + 2e 2 C 

f 2 0 

+O(m 2 log m, m 2 ) + I0e 2 m log m + K0e 2 m 

M 2 

π0 = 2mB0 

+O(m 2 log m, m 2 ) + I±e 2 m log m + K±e 2 m 

• Dashen’s theorem : 

The difference of squared pion mass is independent of quark mass up to O(e 2 m), 

∆M 2 

π 

≡ M 2 

π ± − M 2 

π 0 = 2e 2 C 

f 2 0 

+ (I± − I0)e 2 m log m + (K± − K0)e 2 m 

C, K±, K0 is a new low energy constant. I±, I0 is known in terms of them. 

• We will also use preliminary mass formula from SU(2)+Kaon+EM PQChPT, which would 

have a better convergence for our NF = 2 + 1 simulation region treating Kaon as a 

heavy Iso-doublet: [M.Hayakawa & S. Uno, T.Blum & TI] 

K(x) = e iMv·x k(x), 

L0 = ∂µK † ∂ µ K − M 2 K † K → (−2iMv µ k † ∂µk + ∂µk † ∂ µ k) 


<strong>QCD</strong>+<strong>QED</strong> lattice simulation 

• In 1996, Duncan, Eichten, Thacker carried out SU(3)×U(1) simulation to do the EM 

splittings for the hadron spectroscopy using quenched Wilson fermion on a −1 ∼ 1.15 

GeV, 12 3 × 24 lattice. [Duncan, Eichten, Thacker PRL76(96) 3894, PLB409(97) 387] 

• Using NF = 2 + 1 Dynamical DWF ensemble (RBC/UK<strong>QCD</strong>) would have benefits of 

chiral symmetry, such as better scaling <strong>and</strong> smaller quenching errors. 

• Especially smaller systematic errors due to the the quark massless limits, 

mf → −mres(Qi), has smaller Qi dependence than that of Wilson fermion, κ → 

κc(Qi) (PCAC). 

• Generate Coulomb gauge fixed (quenched) non-compact U(1) gauge action with 

= exp[−iAem µ(x)]. 

β<strong>QED</strong> = 1. U EM 

µ 

• Quark propagator, Sq i (x) with EM charge Qi = qie with Coulomb gauge fixed wall 

source 

D ˆ (U EM 

µ 

) Qi SU(3) ˜ 

× Uµ Sq (x) = bsrc, (i = up,down) 

i 

qup = 2/3, qdown = −1/3 


photon field on lattice 

• non-compact U(1) gauge is generated by using Fast Fourier Transformation (FFT). 

Coulomb gauge ∂jAem j(x) = 0, Ãem µ=0(p0, 0) = 0 with eliminating zero modes. 

(NF = 2 + 1: Feynman gauge) 

• static lepton potential on 16 3 × 32 lattice (β<strong>QED</strong> = 100, 4,000 confs) vs lattice 

Coulomb potential. 

• L=16 has significant finite volume effect for ra > 6 ∼ 1.5r0 ∼ 0.75 fm. It would be 

worth considering for generation of U(1) on a larger lattice <strong>and</strong> cutting it off. 

0 wilson_vs_r.dat_shift 

V_t13.dat 

V_t14.dat 

V_t15.dat 

-0.2 

-0.4 

-0.6 

-0.8 

Coulomb potential V(r)-V(1) 

ncU(1) simulation vs FFT prediction at beta=100 

0 1 2 3 4 5 

0 L=16 

L=32 

L=64 

L=128 

-0.2 

-0.4 

-0.6 

-0.8 

-1 

Finite size effect 

0 5 10 15 20 


simulation parameters 

• NF = 2 + 1 Dynamical DWF configuration for <strong>QCD</strong> 

• a −1 = 1.784(44) GeV. 

• Quark masses: 

aml,sea = 0.005, 0.01, 0.02, 0.03 

amval = 0.001, 0.005, 0.01, 0.02, 0.03, 0.04 

∼ 12(valence only, mπ ∼ 240 MeV), 25, 40, 70, 100, 130 MeV. 

• One sea strange quark point, amh = 0.04 

(∼ 20% heavier than the physical). 

• 16 3 × 32 (1.8 fm) 3 <strong>and</strong> 24 3 × 64 (2.7 fm) 3 . 

• Ls = 16, mresa = 0.00321 or a couple of MeV. 

• EM charge: e = ±0.3028 = p 4π/137 

• ∼ 200 configurations for each m with 20 (40 for ml = 0.005) traj separation. 

• one or two <strong>QED</strong> configuration per a <strong>QCD</strong> configuration. 

• All 16 meson connected correlators + Neutron, Proton, +Baryons. 


EM spectrum on lattice 

• By neglecting O(α 2 ) <strong>and</strong> O((mu − md) 2 ), we approximate π 0 mass squared by that 

of π 3 , which doesn’t have the noisy disconnected diagram. 

• We will not use π 0 mass to determine quark masses. 

• The correlator for π 3 , ρ 3 meson is calculated using the interpolation field of the a = 3 

component of isospin: 

C X 0(t) = 1 

2 

hD 

J uu 

X 

uu† 

(t)JX (0) 

E 

conn 

+ 

D 

J dd 

X 

dd† 

(t)JX (0) 

E 

conn 

i 

, X = π, ρ 

• massless limit of DWF D is defined through Axial Ward identity of degenerate quarks, 

mf = −mres = − J a 

5q (t)P a E 

(0) / 〈P a (t)P a (0)〉 

O(α) effect is parametrized in the generic form 

mres(α) = mres(0) + C2(Q 2 

1 

+ Q2 

2 ) 

for currents made of quarks of charges q1 <strong>and</strong> q3 (No q1q3 term). 

mres(0), C2 → 0 at Ls → ∞. 


Effect of the residual chiral symmetry breaking’s in 

NF = 2 + 1 <strong>QCD</strong>+<strong>QED</strong> simulations 

• mres for Ls = 16 <strong>and</strong> 32 for V=16 3 lattice. 

• Also fit to the charge splittings δm 2 of neutral meson 

δm 2 = M 2 

PS 0(e = 0) − M 2 

PS 0(e = 0) = δmres(q 2 

1 

+ q2 

3 ) 

• Ls = 16 <strong>and</strong> 32 consistent with quark mass shift mres(e) = mres(0) + C2(q 2 

1 

consistent with PCAC : residual chiral symmetry breaking is under a control. 

U(L) U(R) 

q(L) q(R) 

0 2 ... Ls/2-1 ... Ls-1 

mf 

Ω 

δm 2 (<strong>Lattice</strong> Unit) 

0.0003 

0.00025 

0.0002 

0.00015 

0.0001 

5e-05 

0 

16 3 Ls=16 <strong>and</strong> 32 result, fit range 0.01-0.02 

Ls=16 

Ls=32 

B0*C2*(...) 

dmres*(...) 

0 0.05 0.1 0.15 0.2 

m 2 ps 

+ q2 

3 ) 


Analysis methods 

• Analysis method I (main method) : 

Fit correlator for each charge combination separately, 

then calculate the mass splittings under the jackknife. 

X = π, ρ, N :∆MX = M X ± − M X 0, 

• Analysis method II (Illustration purpose)): 

Subtract charged correlator by neutral correlator, 

<strong>and</strong> fit it by a linear function in t: 

CX(t) = A(e 2 )e −M X (e2 )t 

C X ±(t) − C X 0(t) 

C X 0(t) 

G(t; q1, q2) = 

1 

2 

1 

2 

e 2 : q1q2 q 2 1 q 2 2 

= ∆MX × t + Const 

1 

2 

1 

2 

e 4 : q2 1q 2 2 


1 

2 

q 2 1 q2 2 

1 

2

propagator ratio 

• G(t) = 〈J5(0)J5(t)〉 at m = 0.04 <strong>and</strong> 0.03. 

0.03 

0.02 

0.01 

0 

-0.01 

-0.02 

-0.03 

∆ G (t)/ G(t) ps 

m sea =m val =0.04 

∆M=2.5MeV 

∆M=10MeV 

∆M = 5 MeV 

down-down 

up-down 

up-up 

-0.04 

0 5 10 15 

0.03 

0.02 

0.01 

0 

-0.01 

-0.02 

-0.03 

∆ G (t)/ G(t) ps 

m sea =m val =0.03 

∆M=2.5MeV 

∆M=10MeV 

∆M = 5 MeV 

down-down 

up-down 

up-up 

-0.04 

0 5 10 15 

• Fluctuations due to SU(3) are comparable to that from U(1): by double the <strong>QED</strong> 

statistics: ∆Mπ reduces by ∼ 4, 10, (30) % for A4, J5, (N) resp. at m = 0.04. 

σ 2 

<strong>QCD</strong> 

+ 0.5σ2 

<strong>QED</strong> 

σ 2 <strong>QCD</strong> + σ2 <strong>QED</strong> 

= (0.9) 2 =⇒ σ<strong>QED</strong>/σ<strong>QCD</strong> ∼ 0.85 


O(e) error reduction 

• On the infinitely large statistical ensemble, 

term proportional to odd powers of 

e vanishes. But for finite statistics, 

〈O〉 e = 〈C0〉 + 〈C1〉 e + 〈C2〉 e 2 + · · · 

〈C2n−1〉 could be finite <strong>and</strong> source of 

large statistical error as e 2n−1 vs e 2n . 

• By averaging +e <strong>and</strong> −e measurement 

on the same set of <strong>QCD</strong>+<strong>QED</strong> configuration, 

1 

2 [〈O〉 e +〈O〉 −e ] = 〈C0〉+〈C2〉 e 2 +· · · 

O(e) is exactly canceled. 

2 2 

-mdd 

m ud 

0.0014 

0.0012 

0.001 

0.0008 

0.0006 

0 0.01 0.02 0.03 0.04 0.05 

m l 


ChPT+EM at NLO 

• Double expansion of M 2 

PS (m1, q1; m3, q3) in O(α), O(mq). 

<strong>QCD</strong> LO: 

M 2 

PS = χ13 = B0(m1 + m3) 

<strong>QCD</strong> NLO: (1/F 2 

0 ×) 

(2L6 − L4)χ 2 

13 + (2L5 − L8)χ13 ¯χ1 + χ13 

<strong>QED</strong> LO: (Dashen’s term) 

2C 

(q1 − q3) 2 

F 2 0 

<strong>QED</strong> NLO: ( ¯ Q2 = P q 2 

sea−i , no ¯ Q1 in SU(3)N F ) 

X 

I=1,3,π,η 

RIχI log(χI/Λ 2 

χ ), 

−Y1 ¯ Q2χ13 + Y2(q 2 

1 χ1 + q 2 

3 χ3) + Y3q 2 

13 χ13 − Y4q1q3χ13 + Y5q 2 

13 ¯χ1 

+χ13 log(χ13/Λ 2 

χ )q2 

13 + ¯ B(χγ, χ13, χ13)q 2 

13 χ13 − ¯ B1(χγ, χ13, χ13)q 2 

13 χ13 + · · · 

• <strong>QED</strong> LO adds mass to π ± at mq = 0, <strong>QED</strong> NLO changes slope,B0, in mq. 

• Partially quenched formula (msea = mval) SU(3)N F [Bijnens Danielsson, PRD75 (07)] 

SU(2)N F +heavy Kaon+FiniteV [Hayakawa Uno, PTP 120(08) 413] [RBC/UK<strong>QCD</strong>; 

T.Blum’s talk] (also [ C. Haefeli, M. A. Ivanov <strong>and</strong> M. Schmid, EPJ C53(08)549] ) 


mass-squared difference (lattice units) 

SU(3) PQChPT fit in NF = 2 + 1 <strong>QCD</strong>+<strong>QED</strong> simulations 

0.0015 

0.001 

0.0005 

• SU(3) PQChPT fit. 

0 

m sea = 0.02 

ud meson 

dd meson 

uu meson 

ds meson 

us meson 

ss meson 

0 0.02 0.04 0.06 0.08 0.1 

(m 1 +m 2 ) 

• a −1 ∼ 1.8 GeV from Ω − baryon 

mass (no log in NLO). 

• Five degenerate up/down quark 

masses in the simulation: 

∼ 12(valence only), 25, 40, 70, 100 

MeV. 

• One strange quark point 

(∼ 20% heavier than the physical). 

• Two volumes: 

(1.8 fm) 3 <strong>and</strong> (2.7 fm) 3 

• Determine 3 <strong>QCD</strong> LEC + 5 <strong>QED</strong> LEC (also 3 <strong>QCD</strong> LEC for fπ) 

• In total about 240 charge,quark mass combinations are 

measured. 

MPS(m1, q1; m2, q2; ml) 


• By fitting charge splitting 

SU(3)+EM ChPT LEC [R. Zhou] 

δM 2 = M 2 

PS (m1, q1; m2, q2; ml) − M 2 

PS (m1, 0; m2, 0; ml) 

by SU(3) ChPT+EM formula at NLO, 3 <strong>QCD</strong> LECs (1 LO + 2 NLO), 5 <strong>QED</strong> LECs (1 LO + 4 

NLO) are determined. 

• Requiring m1, m3, ml ≤ 40 MeV (70 MeV), 48 (120) partially quenched data for 

MPS(m1, q1; m2, q2; ml) are used in the fit (to see NNLO effects). 

• Finite volume effects are observed by repeating the fit on (1.8 fm) 3 <strong>and</strong> 

<strong>and</strong> (2.7 fm) 3 : δm 2 

π0 


0.0016 

0.0014 

0.0012 

0.001 

0.0008 

0.0006 

0.0004 

0.0002 

unitary point 

wlog 0.001-0.02 

wlog 0.001-0.01 

24 3 lat. 

0 

0 0.05 0.1 

m 

0.15 0.2 

2 

ps 

has negligible FV, δm2 

π ± has ∼ 10 % increase. 


0.0016 

0.0014 

0.0012 

0.001 

0.0008 

0.0006 

0.0004 

0.0002 

unitary point 

wlog 0.01-0.03 

wlog 0.01-0.02 

16 3 lat. fit range:0.01-0.03 

0 

0 0.05 0.1 

m 

0.15 0.2 

2 

ps 


Finite Volume Effect 

SU(3) LEC 

10 6 C 10 2 Y2 10 3 Y3 10 3 Y4 10 2 Y5 10 3 δmres χ 2 /dof 

ours 0.27(19) 1.59(10) -10.6(7) 9.8(16) 2.00(68) 5.08(9) 2.11(73) 

BD 7.3 0.38 1.58 2.83 -0.953 N/A N/A 

• Ours are fit for (2.7 fm) 3 , mq = 12 − 40 MeV. 

• BD: A parameter set chosen for illustration, (55) of [J. Bijnens <strong>and</strong> N. Danielsson Phys. 

Rev. D. 75, (2007) 014505] 

• The Dashen’s term, C, is very small in our fit. This may indicate (1) FV (2) sea strange 

term behaves as a constant in our fit. (3) poor convergence of SU(3). 


SU(2)-heavy Kaon+EM ChPT Fit (preliminary)) 

[S.Uno, R. Zhou] [Hayakawa Uno, PTP 120(08) 413] 

• Treating Kaon as heavy particle (no chiral log from η). 

• Finite volume analysis is done. 

• Ultimately should give our main quote. 

• EM splitting NLO/LO is still large (∼ 50% at mq = 40 MeV) for Pion 

but small (∼ 10% at mq = 70 MeV) for Kaon. 

0.0018 

0.0017 

0.0016 

0.0015 

0.0014 

0.0013 

m Π 2 

SU2heavy kaon chipt, qi23,qj23, infinite 

Pion 

Kaon 

data 

mj 0.005 0.010 0.015 0.020 0.025 0.030 0.035 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

LO vs NLO, SU2heavy kaon chipt, pion, qi23,qj23 

m Π 2 NLO 

m Π 2 LO 

mj 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

LO vs NLO, SU2heavy kaon chipt, kaon, qi23,qj23 

m Π 2 NLO 

m Π 2 LO 

mj 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 


Quark mass determination 

• Using the LECs, B0, F0, Li, C0, Yi, from the fit, we could determine the quark masses 

mup, mdwn, mstr by the solving equations [PDG08] : 

MPS(mup, 2/3, mdwn, −1/3) = 139.57018(35)MeV 

MPS(mup, 2/3, mstr, −1/3) = 493.673(14)MeV 

MPS(mdwn, −1/3, mstr, −1/3) = 497.614(24)MeV 

• (mup − mdwn) is mainly determined by Kaon charge splittings, 

M 2 

K ± − M 2 

K0 = B0(mup − mdwn) + 2C 

(q1 − q3) 2 + NLO 

• π 0 mass is not used for now (disconnected quark loops). 

F 2 0 

• The term proportional to sea quark charge, −Y1 ¯ Q2χ13, is omitted. We will estimate 

the systematics by varying Y1. 


Quark mass results (Preliminary) [R. Zhou , S.Uno] 

• MS at 2 GeV.Non-perturbative technique for the mass renormalization constant is 

used. 

[RBC/UK<strong>QCD</strong>, PRD78(08)054510] 

• Quark mass have small finite size volume effects. SU(3)N F <strong>and</strong> SU(2)N F in infinite 

volume. 

• Uncertainties in <strong>QED</strong> LEC have small effect to quark mass. (π 0 is excluded) 

• Statistical error only. We use FPS(e = 0) in fitting LEC. 

c.f. SU(2) results mud = 3.72(16), ms =107.3(4.4) MeV [RBC/UK<strong>QCD</strong> PRD78(08) 

114509 ] due to this difference in two analyzes. 

lat mq range mu md ms mu/md ms/mud 

(2.7 fm) 3 SU(3)∞ 12-40 MeV 2.79(37) 4.84(52) 95.9(9.6) 0.57(1) 25.1(5) 

(2.7 fm) 3 SU(3)∞ 25-40 MeV 2.48(18) 4.77(30) 95(7) 0.52(3) 26.3(6) 

(2.7 fm) 3 

25-70 MeV 2.50(18) 4.81(30) 95(8) 0.52(3) 26.1(6) 

(1.8 fm) 3 

25-70 MeV 2.64(19) 4.81(32) 95(9) 0.55(4) 25.5(8) 

(2.7 fm) 3 SU(2)∞ 12-40 MeV 2.8(5) 4.7(1) 105(3) 

(2.7 fm) 3 SU(2)∞ 25-40 MeV 2.24(16) 4.62(24) 101(5) 


Quark masses results (preliminary) 

Particle Data Group 2008 

Up Quark mass [MeV] 

RBC09 Preliminary: 

2.80 (36) Stat. (21) FV,ChPT 

• Statistical + incomplete estimation of systematic errors only: 

Down Quark mass [MeV] 

RBC09 Preliminary: 

4.85 (57) Stat. (11) FV,ChPT 

• From PDG08, the world-averaging of each individual up <strong>and</strong> down quark masses were 

started. There are only 4 (2 lattice + 2 model) results so far. 

• PDG09 (PDGlive) world average (MS[NDR] 2GeV) 

mup = 2.70 ± 0.18MeVmdown = 5.00 ± 0.23MeV 


Quark masses (preliminary) 

red NF = 2 + 1 DWF blue NF =2 DWF (2.7 fm) 3 

(Only statistical errors are shown). 


Components of Kaon masses splittings 

• Reason why the iso doublet, (K + , K 0 ), has the mass splitting 

M K ± − M K 0 = −3.937(29) MeV, [PDG08] 

⊲ (m dwn − mup) : makes M K + − M K 0 negative. 

⊲ (qu − qd) : makes M K + − M K 0 positive. 

• Using the determined quark masses <strong>and</strong> SU(3) LEC, we could isolate (to O((mup − 

mdwn)α)) each of contributions, 

M 2 

PS (mup, 2/3, mstr, −1/3) − M 2 

PS (mdwn, −1/3, mstr, −1/3) 

M 2 

PS (mup, 0, mstr, 0) − M 2 

PS (mdwn, 0, mstr, 0) [∆M(mup − mdwn)] 

+M 2 

PS ( ¯mud, 2/3, ¯mud, −1/3) − M 2 

PS ( ¯mud, −1/3, mstr, −1/3) [∆M(qu − qd)] 

• ⊲ ∆M(mup − m dwn) = -5.7(1) MeV [145% in ∆M 2 (mup − m dwn)] 

⊲ ∆M(qu − qd) = 1.8(1) MeV [-45% in ∆M 2 (qu − qd)] 

Also SU(2) ChPT, ∆M(mup − mdwn)=-5.3(7) MeV <strong>and</strong> ∆M(qu − qd)=1.4(7) MeV. 

• Similar analysis for π is possible, but facing a difficulty of isolating sea strange quark 

terms. SU(2) analysis gives a reasonable value. 


Nucleon mass splitting in NF = 2, 2 + 1 (Preliminary) 

[R.Zhou, T.Doi] 

m p -m n [MeV] 

2 

1.5 

1 

0.5 

0 

(qu − qd) effect 

Cottingham formula 

Nf=2 (1.9 fm) 3 

Nf=2+1 (1.8 fm) 3 

Nf=2+1 (2.7 fm) 3 

m P - m N [MeV] 

0 

-20 

-40 

-60 

(mup − mdwn) effect 

-80 

0 20 40 60 80 

(m - m ) [MeV] 

u d 

0 0.1 0.2 0.3 0.4 0.5 

m 2 

ps [GeV2 -0.5 

] 

• Only EM effect, mu = md case, are shown. c.f. [Gasser Leutwyler, PR87(82)77] 

MN − Mp|EM = −0.76(30) MeV 

MN − Mp|quark mass = 2.05(30) MeV 

Wed Jun 10 02:05:49 <strong>2009</strong> 

(∼ 2 MeV at mup − mdwn) 


Systematic errors 

• Chiral extrapolation: mq ≤ 40 or 70 MeV. 

• <strong>QCD</strong>’s Zm : Λ<strong>QCD</strong> = 250 − 300 MeV, O(α) ∼ 1%. 

• π 0 : disconnected loops ( η ′ from DWF [K. Hashimoto TI PTP (08) ] ) 

• Quenched <strong>QED</strong> O(ααS): 

ChPT <strong>and</strong> a clever combinations of masses [Bijnens Danielsson, PRD75 (07) 014505 ] 

• One lattice spacing results, O(a 2 ). 

• Finite Size Effect from vector-saturation model: ∆π,EM = m 2 

π + − m 2 

π 0, to be 

∆π,EM(L) = 

3 α 

4π 

∆π,EM(∞) 

1 

a 2 

∆π,EM(L ≈ 1.9 fm) 

2 4 · π 2 

N 

X 

q∈ e Γ ′ 

= 1.10 . 

(amρ) 2 (amA) 2 

bq 2 (bq 2 + (amρ) 2 ) (bq 2 + (amA) 2 ) , 

Generally quark masses are stable against ∆π,EM ∼ 10 %, 

Finite volume for P-N case may be larger [1/3 closer by (1.8 fm) 3 −→ (2.7 fm) 3 .] 


• Systematic uncertainties due to 

Home works 

• SU(2)+Kaon+EM, O(m, α, αm, m 2 , α 2 ) 

• Omission of sea quark charges: Reweighting [T. Ishikawa] 

Y 

f=u,d,s 

n 

det[Df(e = 0)Df(e = 0) −1 o 

] 

• Omission of sea quark mass difference mu = md 

det 

n 

D(mud + ∆m)D(mud − ∆m)D(mud) 2o 

= 1 + O((ms − ml)α, α 2 ) 

= 1 + O(α∆m, ∆m 2 ) (1) 

• π 0 , (<strong>and</strong> η, η ′ ) need disconnected diagrams 

• Decay constant, Γ(π + → µ + νµ, µ + νµγ) + Vud(exp) 

f π + = 130.7 ± 0.1 ± 0.36MeV PDG 2004 

(the last error is due to the uncertainty in the part of O(α) radiative corrections.) 


Reweighting (calculation of det D ′ D −1 ) 

• For Ω = Dtarget(Dgeneration) −1 , reweighting factor is 

fl 

w = det Ω = 

fi 

e −ξ† (Ω−1)ξ 

ξ 

• There is, at least, two failure mode: 

• 1. Estimation for w is too hard. 

• 2. The fluctuation of w[U] is too large (insufficient overlap between target <strong>and</strong> original 

ensemble ) 

• The former could be fixed by stepping: 

Ω = 

w = 

nY 

i 

Ωi 

nY 

det Ωi = 

i 

nY 

i 

fi 

e −ξ† fl 

(Ωi−1)ξ ξ 

with Ωi close to unity. Eg. mass stepping Ωi = D(mi+1)D(mi) −1 

et.al.] or Ωi = [Ω] 1/n . [T. Ishikawa, Y.Aoki, TI LAT09] 

[A. Hasenfratz 


h 

H eff 

-1000 

-1050 

-1100 

-1150 

-1200 

-1100 

-1200 

-1300 

-1400 

-1500 

n=1 

0 100 200 

hit 

1 2 4 8 16 32 

n 

h eff 

h 

-340 

-350 

-360 

Reweighting (contd) 

n=4 

h eff 

0 100 200 

hit 

• Reweighting factor for Ls = 8 → 

h 

-86 

-87 

-88 

-89 

-90 

-91 

-92 

-93 

n=16 

h eff 

0 100 200 300 400 

hit 

h 

-43 

-44 

-45 

-46 

n=32 

h eff 

0 100 200 300 400 

hit 


Other works of isospin breaking effects on lattice 

• ETMC 2+1+1 (or 1+1+1+1) [K. Jansen’s talk] 

• [McNeile, Michael, Urbach (ETMC) PLB674(09) 286] ρ − ω mass splitting using twisted 

Wilson fermion. Discussed ρ − ω mixing from mup − md. Measure disconnected quark 

loop correlation. 

• [JL<strong>QCD</strong> PRL 101(08) 242001, PRD79(09)] Calculate ΠV −ΠA , derive the EM contribution 

to the pion’s charge splittings in quark massless limit <strong>and</strong> the S-parameter using overlap 

fermion. 

• [MILC Collaboration (S. Basak et al.) <strong>Lattice</strong>08 arXiv:0812.4486 ] EM spectrum using 

staggered ensemble to get the breaking of Dashen’s theorem 

∆M 2 

D 

= (M 2 

K ± − M 2 

K 0)em − (M 2 

π ± − M 2 

π 0)em 


Summary <strong>and</strong> Future perspective 

• Individual up, down, strange quark masses are determined using NF = 2 + 1 DWF <strong>QCD</strong> 

<strong>and</strong> non-compact quenched <strong>QED</strong>. 

• SU(2) + Kaon + EM Partially Quenched Chiral Perturbation are being finilized. 

• Break-ups of Kaon charge splitting <strong>and</strong> p-n splittings into electric charge/(mu − md) 

effects are examined. 

• Isospin breaking effects are interesting <strong>and</strong> inevitable for precise underst<strong>and</strong>ing of 

hadron physics, which could now be addressed by <strong>QCD</strong>+<strong>QED</strong> simulations from the first 

principle: quark masses, (mup 0 ?), mN − mP >0, ..., 

Future plans 

• Complete systematical errors estimation. 

• Analysis on the finer lattice, a ∼ 0.08 fm or larger volume. [T.Blum’s 1st talk] 

• EM splittings using the direct calculation of the <strong>QED</strong> diagrams [JL<strong>QCD</strong> OPE] . 

• Dynamical <strong>QED</strong> effects by reweighting [T.Ishikawa] 

• O(α) contribution to gµ − 2 (pure <strong>QED</strong>). O(α 3 ) contribution (light-by-light) to gµ − 2. 

Chiral magnetic effect in QGP. [T.Blum’s 2nd talk]

Lattice QED and QCD - 2009 Taipei Workshop on Lattice QCD

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