DWF 10yrs1 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

DWF 10yrs2 Isospin breaking  important physics in QCD/QED. Mass splitting:     = (5) MeV K + - K 0 = (27) MeV p – n = (5) MeV  + +   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 )

DWF 10yrs3 Introduction Precise theoretical calculation of muon g-2 Muon is expected to be sensitive to short-range  New Physics ? Large uncertainty from hadronic contribution B s QCD+QED simulation !

DWF 10yrs4 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 A 0 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 Qu=+2/3e, Qd=-1/3e

DWF 10yrs5 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=16 3 X32, L s =12  L 3 = (2fm) 3 domain-wall height M 5 =1.8 sea quark mass=0.02, 0.03, 0.04 m q = 1/2 m s – m s (m  = MeV), m s = About 200 configs with 25 trajectories separation Manifest flavor structure We will use Nf=2+1 confs as well in near future

DWF 10yrs6 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) = T 3 +T 8 /sqrt(3) Axial WT identity SU(2) R ds X SU(2) L ds X U(1) em X U(1) V  SU(2) V ds X U(1) em X U(1) V 3 NG-bosons for massless quark  0  2  etc.

DWF 10yrs7 Meson masses QED parametrization + NLO QCD  NG-boson  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)

DWF 10yrs8 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

DWF 10yrs9 The QED effect on PS-meson (msea=0.04) (msea=0.03)

DWF 10yrs10 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(K 0 ) 2  sensitive to ms, (mu+md), insensitive to (mu-md)  determine ms [M(K 0 ) 2 -M(K + ) 2 ] - [M(  0 ) 2 -M(  + )2]  sensitive to (mu-md), ms, insensitive to (mu+md)  determine (mu-md)

DWF 10yrs11 Quark masses and splittings Masses By employing RBC nonperturbative 1/Z m =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

DWF 10yrs12 Isospin breaking in baryons Mass splitting between octets p – n = (5) MeV  + +   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.

DWF 10yrs13 Plot of  (proton)/  (neutron) The negative slope corresponds to m(p) > m(n) from the QED effect (m u =m d ) 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 …

DWF 10yrs14 The idea for the S/N improvement Q= +e, -e trick Physical observables are expected to (Perturbatively, only O(e 2n ) 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 !

DWF 10yrs15 Q= +e, -e trick Q= -e only Q=+e only Remarkable Improvement !

DWF 10yrs16 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)  (em) Physical msea=0.03 (msea=mval)

DWF 10yrs17 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)

DWF 10yrs18 Isospin breaking in  triplet c.f. exp: 1.6MeV Charge dependence [Variational method] diagonalize eigenvectors t (up to n-th excited state) 00  [Q=1.0]  (em) msea=0.03 chiral-extrapolation

DWF 10yrs19 The QCD part ([md-mu] effect) ChPT for baryons (HQchiPT) LO  linear in quark mass NLO  m q ^(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)

DWF 10yrs20 Splittings with various (md-mu) (msea=0.03) proton-neutron Xi(-)-Xi(0)

DWF 10yrs21 Splittings with various (md-mu) (msea=0.03) Sig(+)-Sig(0)Sig(+)+Sig(-) – 2 Sig(0)

DWF 10yrs22 The isospin breaking from QCD p – n  (18)(51) MeV Xi(-) - Xi(0)  +3.86(11)(77) MeV Sig(+) – Sig(0)  (12)(66) MeV Sig(-) – Sig(0)  +3.04(11)(61) MeV Sig(+) – Sig(-)  (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/ p – n = (57)(42)(10) MeV

DWF 10yrs23 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

DWF 10yrs24 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) = (40) mres(d) = (40) mres(QCD) = (39) In the chiral limit,

DWF 10yrs25 The determination of LECs Fit to the neutral mesons (NG-bosons) Leading LEC NLO LECs (L 5 -2L 8 ), (L 4 -2L 6 ) NLO EM-LECs

DWF 10yrs26 The determination of LECs Pion mass splittings Using the lattice output for the pion mass splitting, NLO EM-LECs LO EM-LEC

DWF 10yrs27 Isospin breaking in  doublet Charge dependence  (em) msea=0.03 chiral-extrapolation