03/15/2007 Talk @ DWF 10yrs 1
Isospin breaking Study with Nf=Isospin breaking Study with Nf=2 domain-wall QCD + Quenched 2 domain-wall QCD + Quenched
QED SimulationQED Simulation
Takumi Doi(Univ. of Kentucky / RBRC)
In collaboration withT.Blum (Univ. of Connecticut, RBRC) M.Hayakawa (Nagoya Univ.)T.Izubuchi (Kanazawa Univ., RBRC)N.Yamada (KEK)
for RBC Collaboration
03/15/2007 Talk @ DWF 10yrs 2
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
Introduction
dominated by QED QED +QCD
(mu-md)
03/15/2007 Talk @ DWF 10yrs 3
Introduction Precise theoretical calculation of muon g-2
Muon is expected to be sensitive to short-range New Physics ?
Large uncertainty from hadronic contribution
Bs
QCD+QED simulation !
03/15/2007 Talk @ DWF 10yrs 4
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 c
onnect next-neighbor-site
A.Duncan, E.Eichten, H.Thacker, PRL76(1996)3894
Qu=+2/3e, Qd=-1/3e
03/15/2007 Talk @ DWF 10yrs 5
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
03/15/2007 Talk @ DWF 10yrs 6
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)L
ds 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
02 etc.
03/15/2007 Talk @ DWF 10yrs 7
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)
03/15/2007 Talk @ DWF 10yrs 8
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
03/15/2007 Talk @ DWF 10yrs 9
The QED effect on PS-meson
(msea=0.04) (msea=0.03)
03/15/2007 Talk @ DWF 10yrs 10
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, ins
ensitive to (mu+md) determine (mu-md)
03/15/2007 Talk @ DWF 10yrs 11
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
03/15/2007 Talk @ DWF 10yrs 12
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
etc.
mtTtmmtTtm eCeCeCeCJtJ )(
0)(
0 )1()1()0()(
03/15/2007 Talk @ DWF 10yrs 13
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…
03/15/2007 Talk @ DWF 10yrs 14
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 !
03/15/2007 Talk @ DWF 10yrs 15
Q= +e, -e trick
Q= -e only
Q=+e only
Remarkable Improvement !
03/15/2007 Talk @ DWF 10yrs 16
Proton neutron mass difference from the QED effect
The lattice result indicatesM(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)
03/15/2007 Talk @ DWF 10yrs 17
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 and1
c.f. 8(1116) < 0(1193) (<1) Diagonalize 3x3 correlation function matr
ix (variational method)
mu md+higher order terms
(uus) (dds) (uds)
03/15/2007 Talk @ DWF 10yrs 18
Isospin breaking in triplet
c.f. exp: 1.6MeV
Charge dependence
[Variational method]
diagonalize
eig
en
vecto
rs
t
(up to n-th excited state)
0
[Q=1.0]
(em)
msea=0.03
chiral-extrapolation
03/15/2007 Talk @ DWF 10yrs 19
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)
03/15/2007 Talk @ DWF 10yrs 20
Splittings with various (md-mu)
(msea=0.03)
proton-neutron
Xi(-)-Xi(0)
03/15/2007 Talk @ DWF 10yrs 21
Splittings with various (md-mu)
(msea=0.03)
Sig(+)-Sig(0) Sig(+)+Sig(-) – 2 Sig(0)
03/15/2007 Talk @ DWF 10yrs 22
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
03/15/2007 Talk @ DWF 10yrs 23
Summary/Outlook We have investigated the isospin breaking effect on hadron spec
trum using Lattice QCD+QED simulation Determination of the LECs which appear in meson spectrum + e
xperimental 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
03/15/2007 Talk @ DWF 10yrs 24
Residual quark mass Because there exists explicit chiral symmetry break
ing (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,
03/15/2007 Talk @ DWF 10yrs 25
The determination of LECs Fit to the neutral mesons (NG-
bosons)
Leading LEC
NLO LECs (L5-2L8), (L4-2L
6)
NLO EM-LECs
03/15/2007 Talk @ DWF 10yrs 26
The determination of LECs Pion mass splittings
Using the lattice output for the pion mass splitting, NLO EM-LECsLO EM-LEC
03/15/2007 Talk @ DWF 10yrs 27
Isospin breaking in doubletCharge dependence
(em)
msea=0.03
chiral-extrapolation
