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

Takumi Doi (Univ. of Kentucky / RBRC). Isospin breaking Study with Nf=2 domain-wall QCD + Quenched QED Simulation. In collaboration with. T.Blum (Univ. of Connecticut, RBRC) M.Hayakawa (Nagoya Univ.) T.Izubuchi (Kanazawa Univ., RBRC) N.Yamada (KEK). for RBC Collaboration.

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

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  1. Takumi Doi (Univ. of Kentucky / RBRC) Isospin breaking Study with Nf=2 domain-wall QCD + Quenched QED Simulation 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

  2. Introduction • Isospin breaking  important physics in QCD/QED. • Mass splitting: • p+ - p0 = +4.5936(5) MeV • K+ - K0 = - 3.972(27) MeV • p – n = - 1.2933317(5) MeV • S+ + S- - 2 S0 = 1.53(11) MeV • X- - X0 = 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  dominated by QED  QED +QCD (mu-md) Talk @ DWF 10yrs

  3. s B 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

  4. Qu=+2/3e, Qd=-1/3e QED configurations • Quenched non-compact QED • No photon self-coupling • free theory, coupling does not run • Generating QED configs: • Generate Am(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 (tint=0) even for arbitrary small coupling • Fourier inversion to x-space • Wilson line Um(em)=exp[iAm(em))] to connect next-neighbor-site A.Duncan, E.Eichten, H.Thacker, PRL76(1996)3894 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 (mp = 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

  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)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 p02g etc. 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 p0- h, p0–h’ (expected to be higher order) Talk @ DWF 10yrs

  8. Extract the mass difference • We focus on the mass difference directly. • P(e=0) = A(e=0) exp(-m(e=0)t) • P(e) = A(e) exp(-m(e) t) [For visibility] • R= P(e)/P(e=0) • R (1+ dA) – [ m(e)-m(e=0) ] t ,(dA=(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

  9. The QED effect on PS-meson (msea=0.03) (msea=0.04) 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(p0)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(p0)2-M(p+)2] sensitive to (mu-md), ms, insensitive to (mu+md) •  determine (mu-md) Talk @ DWF 10yrs

  11. Quark masses and splittings MILC w/o QED • 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 Kaon suffer from large systematic error Talk @ DWF 10yrs

  12. etc. Isospin breaking in baryons • Mass splitting between octets • p – n = - 1.2933317(5) MeV • S+ + S- - 2 S0 = 1.53(11) MeV • X- - X0 = 6.48(24) MeV • Two point correlation function with the operator • Forward and Backward propagation is averaged to increase statistics Talk @ DWF 10yrs

  13. Plot of P(proton)/P(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

  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: {Am(em)}  {+Am(em), -Am(em)} Very simple idea, but left unaware in the literature… Same Boltzmann Weight ! Talk @ DWF 10yrs

  15. Q= +e, -e trick Q=+e only Remarkable Improvement ! Talk @ DWF 10yrs Q= -e only

  16. Proton neutron mass difference from the QED effect proton-neutron at Q=physical Charge dependence msea=0.03 M(p)-M(n) M(p)-M(n) Physical a(em) The lattice result indicates M(p) > M(n) (QED) at each msea c.f. Cottingham formula: M(p)-M(n)(QED)= 0.76MeV (msea=mval) Need more statistics ? Finite V ? Talk @ DWF 10yrs

  17. Isospin breaking on S triplet • Insensitive to u/d quark mass difference • M(S+)+M(S-) – 2 M(S0) Only QED effect ! • M(S+)+M(S-) – 2M(S0) = O(e^2) + O(mu -- md) • When isospin symmetry breaks, mixing occurs between S0 , L8 and L1 c.f. L8(1116) < S0(1193) (<L1) • Diagonalize 3x3 correlation function matrix (variational method) (uus) (uds) (dds) +higher order terms mu md Talk @ DWF 10yrs

  18. Isospin breaking in S triplet Charge dependence [Variational method] msea=0.03 diagonalize (up to n-th excited state) a(em) S0 eigenvectors L chiral-extrapolation [Q=1.0] Talk @ DWF 10yrs t c.f. exp: 1.6MeV

  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 ! (for unquenched case) B.C.Tiburzi et al. NPA764(06)274 Talk @ DWF 10yrs

  20. Splittings with various (md-mu) Xi(-)-Xi(0) proton-neutron (msea=0.03) Talk @ DWF 10yrs

  21. Splittings with various (md-mu) Sig(+)-Sig(0) Sig(+)+Sig(-) – 2 Sig(0) (msea=0.03) 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 Talk @ DWF 10yrs

  23. 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

  24. 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 In the chiral limit, mres(u) = 0.001478(40) mres(d) = 0.001428(40) mres(QCD) = 0.001387(39) Talk @ DWF 10yrs

  25. The determination of LECs • Fit to the neutral mesons (NG-bosons) NLO EM-LECs NLO LECs (L5-2L8), (L4-2L6) Leading LEC Talk @ DWF 10yrs

  26. The determination of LECs • Pion mass splittings Using the lattice output for the pion mass splitting, LO EM-LEC NLO EM-LECs Talk @ DWF 10yrs

  27. Isospin breaking in X doublet chiral-extrapolation Charge dependence msea=0.03 a(em) Talk @ DWF 10yrs

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