Isospin breaking studies from Lattice QCD+QED. Taku Izubuchi, [RBC-UKQCD Collaboration]

Transcription

1 Isospin breaking studies from Lattice QCD+QED Taku Izubuchi, [RBC-UKQCD Collaboration] Tom Blum, Takumi Doi, Masashi Hayakawa, Tomomi Ishikawa, Chulwoo Jung, Eigo Shintani, Shunpei Uno, Norikazu Yamada, Ran Zhou RIKEN BNL Research Center Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

2 Introduction lattice QED+QCD and ChPT up, down and strange quark masses Isospin breaking in PS decay constants Isospin breaking in baryon masses QED reweighting Conclusion g-2 light-by-light & AMA new error reduction techniques Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

3 QCD+QED possible and necessary to extend the range of physical quantities [C.Sachrajda], = [Many Lattice talks in this conference] QED was the first Quantum Field Theory Hadronic light-by-light scattering contribution to the muon g 2 from l could be estimated by purely theoretical calculation. So far, it ha the hadronic picture [7, 8]. Thus the first principle calculation bas desirable. elastic scattering of two photons l 2 l 1 µ Figure 1: hadronic light-by-light scattering contributio Lattice QCD results are becomming very precise, [L.Lellouch s talk] e.g. err(f π ), err(f K ) 1%, err(f π /f K ) 0.5%. QED effects may not be negligible. The diagram in Fig. 1 evokes the following naive approach; relation function of four hadronic electromagnetic currents by independent four-momenta l 1, l 2 of off-shell photons, and integrat difficult to accomplish with use of supercomputers available inth Although QED part could be treated perturbatively (e.g. hadronic vacuum poralization in (g 2) µ ), not all of problems in QCD+QED system are conviniently solved by non-perturbative + perturbative treatments. A ground work towards (g 2) µ hadronic light-by-ligh diagram Here we propose a practical method to calculate the h-lbl (QCD + QED)simulation;wecompute quark QCD+quenched QED A quark Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9, 2012 amputate the external muon lines, and project the magnetic 3 form

4 Isospin symmetry In 1932, Werner Heisenberg introduced Isospin to explain the newly discovered particle, Neutron. Neutron s mass is nearly degenerated to Proton. Strong interactions of Neutron are almost equal to those of Proton. In the contemporary understanding, isospin symmetry is the SU(2) V SU(2) A flavor symmetry between up and down quarks. ( ) u d exp{i(θ a V + iθa A )τ a } ( ) u d Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

5 Isospin Breakings The effect of isospin breaking due to electromagnetic (EM) and the up, down quark mass difference has phenomenological impacts for accurate hadron spectrum, quark mass determination. Isospin breaking s are measured very accurately : u 2/3 e ¼ + (repulsive) d 1/3e m N m P = (4)MeV m π ± m π 0 = (5)MeV, q Q e ¼ 0 q -Q e m K ± m K 0 = 3.937(28)MeV, (attractive) The positive mass difference between Neutron (udd) and Proton (uud) stabilizes proton thus make our world as it is. One of the limiting factors for the precise understanding of nature from the current lattice QCD, especially so for u,d quark masses. [MILC 2004] [C.Bernard s talk] m u = 0 is considered to be a possible solution for Strong CP problem (but also see [M. Creutz] s arguments). Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

6 QCD+QED 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 GeV, lattice. [Duncan, Eichten, Thacker PRL76(96) 3894, PLB409(97) 387] Using N F = Dynamical DWF ensemble (RBC/UKQCD) would have benefits of chiral symmetry, such as better scaling and smaller quenching errors. Especially smaller systematic errors due to the the quark massless limits, m f m res (Q i ), has smaller Q i dependence than that of Wilson fermions, κ κ c (Q i ). Generate Feynman gauge fixed, quenched non-compact U(1) gauge action with β QED = 1. U EM µ = exp[ ia em µ (x)]. Quark propagator, S qi (x) with EM charge Q i = q i e source with Coulomb gauge fixed wall D [ (U EM µ ) Q i U SU(3) ] µ Sqi (x) = b src, (i = up,down) q up = 2/3, q down = 1/3 Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

7 photon field on lattice non-compact U(1) gauge is generated by using Fast Fourier Transformation (FFT). Feynman gauge with eliminating zero modes. Static lepton potential on lattice (β QED = 100, 4,000 confs) vs lattice Coulomb potential are shown. In our quenched QED simulation, QED coupling e is set by the static Coulomb potential in infinite volume limit to be, V (r) = e2 1 4π r = 1/137, e = Finite volume effects is checked by two volumes. dynamical QED (running coupling) will be intorduced by reweighting. Coulomb potential V(r)-V(1) ncu(1) simulation vs FFT prediction at beta=100 Finite size effect 0 wilson_vs_r.dat_shift V_t13.dat V_t14.dat V_t15.dat L=16 L=32 L=64 L= Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

8 Measurements lat m sea m val Trajectories N meas t src , 0.02, , , 0.02, , , 0.02, , {1,5}, 0.0{1,2,3} , 0.0{1,2,3} ,16,32, ,16,32,48 N F = DWF QCD ensemble generated by [RBC/UKQCD, PRD78:114509(08), in prep.] a 1 =1.784 (44) GeV, V = (16a = 1.76 fm) 3 and (24a = 2.65 fm) 3 m v = ( 9 MeV), ( 22 MeV), 0.01 ( 40 MeV), 0.02 ( 70 MeV),, 0.03 ( 100 MeV) m res = (46) ( 8.9 MeV) In total, 200 charge/mass combinations are measured. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

9 O(e) error reduction On the infinitely large statistical ensemble, term proportional to odd powers of e vanishes. But for finite statistics, O e = C 0 + C 1 e + C 2 e 2 + C 2n 1 could be finite and source of large statistical error as e 2n 1 vs e 2n. By averaging +e and e measurements on the same set of QCD+QED configuration, 1 2 [ O e + O e ] = C 0 + C 2 e 2 + O(e) is exactly canceled. More than a factor of 10 error reduction, corresponding to 100 measurements by only twice computational cost (vs naive reduction factor 2) sqrt(2) (q 1 -q 3 )/(e/3) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

10 EM splittings Axial WT identity with EM for massless quarks (N F = 3), L em = ea em µ (x) qq em γ µ q(x), Q em = diag(2/3, 1/3, 1/3) µ A a µ = iea em µ q [T a, Q em ] γ µ γ 5 q α ( 2π tr Q 2 em T a) F µν F em em µν, neutral currents, four A a µ (x), are conserved (ignoring O(α2 ) effects): π 0, K 0, K 0, η 8 are still a NG bosons. ChPT with EM at O(p 4, p 2 e 2 ) : M 2 π ± = 2mB 0 + 2e 2 C f 2 0 M 2 π 0 = 2mB 0 +O(m 2 log m, m 2 ) + I 0 e 2 m log m + K 0 e 2 m +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 (I ± I 0 )e 2 m log m + (K ± K 0 )e 2 m C, K ±, K 0 is a new low energy constant. I ±, I 0 is known in terms of them. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

11 ChPT+EM at NLO Double expansion of M 2 PS (m 1, q 1 ; m 3, q 3 ) in O(α), O(m q ). QCD LO: M 2 PS = χ 13 = B 0 (m 1 + m 3 ) QCD NLO: (1/F 2 0 ) (2L 6 L 4 )χ (2L 5 L 8 )χ 13 χ 1 + χ 13 QED LO: 2C (q 1 q 3 ) 2 F 2 0 (Dashen s term) QED NLO: ( Q 2 = q 2 sea i, no Q 1 in SU(3) NF ) I=1,3,π,η R I χ I log(χ I /Λ 2 χ ), Y 1 Q2 χ 13 + Y 2 (q 2 1 χ 1 + q 2 3 χ 3) + Y 3 q 2 13 χ 13 Y 4 q 1 q 3 χ 13 + Y 5 q 2 13 χ 1 +χ 13 log(χ 13 /Λ 2 χ )q B(χ γ, χ 13, χ 13 )q 2 13 χ 13 B 1 (χ γ, χ 13, χ 13 )q 2 13 χ 13 + QED LO adds mass to π ± at m q = 0, QED NLO changes slope,b 0, in m q. Partially quenched formula (m sea m val ) SU(3) NF [Bijnens Danielsson, PRD75 (07)] SU(2) NF +Kaon+FiniteV [Hayakawa Uno, PTP 120(08) 413] [RBC/UKQCD] (also [ C. Haefeli, M. A. Ivanov and M. Schmid, EPJ C53(08)549] ) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

12 The residual chiral symmetry breaking in QCD+QED Using DWF s PCAC relation, in terms of the mid-point correlator J 5q (L s /2), for the flavor off-diagonal current with same EM charge quarks, q i. Parametrize the EM charge dependence in terms of C 2 : m res (q i, q i ) = J a 5q ( x, t)πa (0) x, J a 5 ( x, t)πa (0) x Ls/ Ls-1 m res,i (q i, q i ) m res (0, 0) = e 2 C 2 q 2 i, m sea m res m res chiral limit (46) (15) N/A (16) (31) (15) (29) (16) (28) (15) q(l) U(L) mf Ω U(R) q(r) L s C 2 uū C 2 d d 16 3 lattice size (23) 2.532(22) (16) 0.301(16) 24 3 lattice size (7) 2.519(7) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

13 In the massless quark limit of QCD, m f = m res (0, 0), Neutral PS meson (should still be a NG boson upto α 2 ), has additive mass shift due to the additional chiral symmetry breaking from photon field, m res,i (q i, q i ) m res (0, 0). This effect is expressed in the DWF-ChPT as m 2 = M 2 PS 0 (e 0) M 2 PS 0 (e = 0) = BC 2 e 2 (q q2 3 ), where χ = 2Bm q is the LO PS mass squared. L s = 16 and 32 (partially quenched) consistent with DWF-PCAC Ls=16 and 32 result, fit range Ls=16 Ls=32 B0*C2*(...) dmres*(...) δm 2 (Lattice Unit) e m 2 ps Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

14 Q 2 scaling Staggered case [C.Bernard s talk] Similar counter part in Wilson s case Q 2 /a. Taste Splitting As charges increase, EM taste-violating effects start to become evident. 13 Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

15 [Bijnens Danielsson, PRD75 (07)] SU(3)+EM ChPT LEC By fitting charge splitting δm 2 = M 2 PS (m 1, q 1 ; m 2, q 2 ; m l ) M 2 PS (m 1, 0; m 2, 0; m l ) by SU(3) ChPT+EM formula at NLO, 3 QCD LECs (1 LO + 2 NLO), 5 QED LECs (1 LO + 4 NLO) are determined. Requiring m 1, m 3, m l 0.01 (0.02), 58 (124) partially quenched data for M PS (m 1, q 1 ; m 2, q 2 ; m l ) are used in the fit (to see NNLO effects). Finite volume effects are observed by repeating the fit on (1.8 fm) 3 and and (2.7 fm) unitary point 24 3 lat. fit range unitary point 16 3 lat. fit range: δm 2 (Lattice Unit) δm 2 (Lattice Unit) m 2 ps m 2 ps Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

16 [C. Bernard s talk] MILC-EM-ChPT fit Chiral Fit and Extrapolation Now subtract neutral NLO correction to the masses. Dashens s theorem : Perfect agreement of! splitting EM = (M 2 with K ± M 2 physical K 0 )/(M 2 value π ± M 2 π 0 ) is an accident: systematic EM = errors 0.65(7)(14)(...) are larger than (MILC the difference 2012) partial of purple sys. & error black lines (i.e., difference between Blum! 10 0 and (stat! ). error only),: EM = 0.75(5) for SU(3), Can now read off ratio of! EM = 0.63(5) for SU(2) and K splittings: =0.65(7) BMW (12) partial sys. error EM = 0.70(4)(8)(...) Difficulty in Covariant fit 20 Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

17 SU(2)+ Kaon+EM ChPT Fit M 2 K = M 2 4B(A 3 m 1 + A 4 (m 4 + m 5 ( )) ( ) ) +e 2 2 A (1,1) 5 + A (2,1) 5 q A(s,1,1) 5 q A(s,2) 5 q 1 q 3 e2 ( (4π) 2 F 2 (A (1,1) ) 5 + 3A (2,1) 5 )q A(s,2) 5 q 1 q 3 s=4,5 χ 1s log χ 1s µ 2 +e 2 m 1 ( x (K) 3 (q 1 + q 3 ) 2 + x (K) 4 (q 1 q 3 ) 2 + x (K) 5 (q 2 1 q2 3 ) ) +e 2 m 4 + m 5 2 +e 2 δ mres (q q2 3 ), ( x (K) 6 (q 1 + q 3 ) 2 + x (K) 7 (q 1 q 3 ) 2 + x (K) 8 (q 2 1 q2 3 ) ) EM splitting NLO/LO is still large ( 50% at m q = 40 MeV) for Pion but small ( 10% at m q = 70 MeV) for Kaon. But quark mass determination is stable under NLO correction. An accidental flat direction of χ 2 function in our data set (degenerate light quark) : increase light mass range (ml 0.02) or fix QED NLO LEC to zero to see the effects on quark mass (included in systematic error). Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

18 M M m f +m res m 1 +m res Left: Pion fit, ūd, ūu, dd from top. SU(2) fit is in solid curve and dashed curve is SU(3) fit. Right: Kaon fit for various charge combinations. Infinite volume fit formula are shown. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

19 Finite Volume effect on ChPT fits M M m f +m res m f +m res We use finite volume (FV) ChPT formula to fit data. Left: Pion unitary points. lower line: δm res, upper line: LO (Dashen s) term NLO contributions at simulation points are % LO. But only +2% contribution to m d m u from NLO. Left: Using FV fit on (2.7 fm) 3, dotted curve are predicted for (1.8 fm) 3, which overshoots the data by a factor of 2. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

20 Quark mass determinations Using the LECs, B 0, F 0, L i, C 0, Y i, from the fit, we could determine the quark masses m up, m dwn, m str by the solving equations [PDG] : M π ± = M PS (m up, 2/3, m dwn, 1/3) = (35)MeV M K ± = M PS (m up, 2/3, m str, 1/3) = (14)MeV M K 0 = M PS (m dwn, 1/3, m str, 1/3) = (24)MeV (m up m dwn ) is mainly determined by Kaon charge splittings, M 2 K ± M 2 K 0 = B 0 (m up m dwn ) + 2C (q F0 2 1 q 3 ) 2 + NLO π 0 mass is not used for now (disconnected quark loops). The term proportional to sea quark charge, Y 1 Q2 χ 13, is omitted. We will estimate the systematics by varying Y 1. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

21 Quark mass results MS at 2 GeV, using NPR, RI-SMOM γ µ scheme2 [C.Sturm et.al PRD (09) , Y.Aoki, PoS LAT , L. Almeida C.Sturm arxiv: , P.Boyle et. al. arxiv: , RBC/UKQCD in prep.] as a intermediate scheme. (10% 5% 2,3% error) m 1, m ( 40MeV), M ps 250 MeV SU(3) NF /SU(2) NF in infinite/finite volume. Uncertainties in QED LEC have small effect to quark mass. SU(3) SU(2) inf.v f.v inf.v. f.v. m u [MeV] 2.606(89) 2.318(91) 2.54(10) 2.37(10) m d [MeV] 4.50(16) 4.60(16) 4.53(15) 4.52(15) m s [MeV] 89.1(3.6) 89.1(3.6) 97.7(2.9) 97.7(2.9) m d m u [MeV] 1.900(99) 2.28(11) 1.993(67) 2.155(63) m ud [MeV] 3.55(12) 3.46(12) 3.54(12) 3.44(12) m u /m d 0.578(11) 0.503(12) (87) (93) m s /m ud 25.07(36) 25.73(36) 27.58(27) 28.34(29) Only statistical error shown above. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

22 Quark mass from QCD+QED simulation [PRD82 (2010) [47pages]] m u = 2.24 ± 0.10 ± 0.34 MeV m d = 4.65 ± 0.15 ± 0.32 MeV m s = 97.6 ± 2.9 ± 5.5 MeV m d m u = ± ± MeV m ud = 3.44 ± 0.12 ± 0.22 MeV m u /m d = ± ± m s /m ud = ± 0.29 ± 1.77, MS at 2 GeV using NPR/SMOM scheme. Particular to QCD+QED, finite volume error is large: 14% and 2% for m u and m d. This would be due to photon s non-confining feature (vs gluon). Volume, a 2, chiral extrapolation errors are being removed. Applications for Hadronic contribution to (g 2) µ in progress. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

23 PACS- CS [arxiv: [hep- lat]] (N. Ukita) Table summarizes our results for quark masses renormalized at μ=2gev. We neglect the QED correcaons to the renormalizaaon factor. Figure shows a raao of K 0 to K + propagators clarifying K 0 - K + mass difference, which is consistent with the experimental value. m π + = 137.7(8.0) [MeV] m K + = 492.3(4.7) [MeV] m K 0 = 497.4(3.7) [MeV] m K 0 m K + = 4.54(1.09) [MeV] ratio of K 0 to K + propagators m MS u = 2.57(26)(07) [MeV] m MS d = 3.68(29)(10) [MeV] m MS s = 83.60(58)(2.23) [MeV] m MS ud = 3.12(24)(08) [MeV] m u /m d = 0.698(51) m s /m ud = 26.8(2.0) experiment : 3.937(28)[MeV] t Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

24 PDG2012!"#$%&"'()*"+)$",-./±0-11(2"3343(56789:(;<(1-0=!"#$%&'%&""""""""""""""""""()"""""*$++"", EDB5(B:94C37I(4G8<-((&D9<(739(G4E(G9695J 573B8<(ED9(57I9(75(4>3(K;95EL(?78>95F )$*'+,-+,$$$$$$$$$$$$$$$%.$$$$&'(($$/!"##$$$$$$$$$$$$$$$$$$$%%$$$$$$$&'(( χ Z YTVS 10 T)&& 0-1 ')*#"P 10 T)&& 0-0 'QS#O$V"W 0X &%"Q 1-. '")O'+") 0/ &%"Q 0-0 U)S#O 0R &%"Q 0-0 S)PQO 0R T)&& 1-R O)+#PQO 0R &%"Q 0-R (((((((,-1 2[4G@B:9G69(T9?98(\(0-RR]= ^, ] R. / d-quark MASS (MeV) / New results from [BMW], smeared-wilson clover. New results from [PACS-CS]. On physics point, quenched QED + QED reweighting, as well as m u m d effects, N F = colover-wilson simulation. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

25 Error budget Statistic error is small, especially for ratios. Chiral fit error: m q 40 or 70 MeV (M ps 250 or 420 MeV). Finite Volume Effect by comparing (1.9 fm) 3 and (2.7 fm) 3. EM M 2 P S ( ) V.S.M EM M 2 P S (L 1.9 fm) V.S.M = FV ChPT overestimate the FV effect. Generally quark masses are stable against M P S O(10) %. (M π ±, M K ±, M K 0 inputs) stat. err (%) fit(%) fv(%) O(a 2 ) (%) QED qnch(%) renorm(%) m u m d m s m d m u m ud m u /m d m s /m ud QED Z m O(α) 1%. Error of m sea s 2 %. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

26 Origins of Isospin breaking in Kaon Reason why the iso doublet, (K +, K 0 ), has the mass splitting M K ± M K 0 = 3.937(29) MeV, [PDG] (m dwn m up ) : makes M K + M K 0 negative. (q u q d ) : makes M K + M K 0 positive. Using the determined quark masses and SU(3) LEC, we could isolate (to O((m up m dwn )α)) each of contributions, M 2 PS (m up, 2/3, m str, 1/3) M 2 PS (m dwn, 1/3, m str, 1/3) M 2 PS (m up, 0, m str, 0) M 2 PS (m dwn, 0, m str, 0) [ M(m up m dwn )] +M 2 PS ( m ud, 2/3, m ud, 1/3) M 2 PS ( m ud, 1/3, m str, 1/3) [ M(q u q d )] M(m up m dwn ) = (14) MeV [133(4)% in M 2 (m up m dwn )] M(q u q d ) = 1.327(37) MeV [-34(1)% in M 2 (q u q d )] Also SU(3) ChPT, M(m up m dwn )=-5.7(1) MeV and M(q u q d )=1.8(1) MeV. Similar analysis for π is possible, but facing a difficulty of isolating sea strange quark terms. m π ± m π 0 = 4.50(23) MeV (experiment: (5) MeV) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

27 Isospin violation in PS leptonic decays [discussion with A.Juttner, C.Sachrajda, G. Colangelo, L. CERN] f K /f π is getting very precise: f K /f π = 1.193(6) [0.5%] [WA by FlaviaNet Kaon WG 2010] CKM matrix elements ratio from charged π and K leptonic decay widths: Γ(K + l + ν(γ)) Γ(π + l + ν(γ)) = V us 2 V ud 2 f 2 K f 2 π m K(1 m 2 l /m2 K )2 m π (1 m 2 l /m2 π )2 (1+δ SU(2) + δ EM ) At which quark masses, f π and f K should be computed? f K : Should light quark mass be m u or m ud = (m u + m d )/2? m u /m ud f π : Is the π mass shift from EM effect totally removed by δ EM? m 0 π = 135 MeV vs m± π = 139 MeV? Which is the best way to correct isospin breakings in the V us /V ud extraction? Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

28 FK vs mx ms=0.03, mh= ml = 0.01 (~40 MeV) unitary (ml = mx) ml=0.005 (~ 22 MeV) ml= 0.7 mud, 1.0 mud, 1.3 mud unitary (ml=mx) ml=1.3 mud ml=mud ml = 0.7 mud K + = su (light sea quark mass: m l, light valence quark mass : m x ) f m l = m x = m ud : 149.6(7) MeV f m x = 0.7m ud, m l = m ud : δ SU(2) /2-0.15% vs the WA error, 0.5% f m l = m x = 0.7m ud : [-0.904%] ChPT analysis [Cirigliano, Neufeld 2011] says F K /F π would shift -0.22(6) % from (m u m d ), while it was found to be (4) % in Lattice study [RM123, 2012]. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

29 EM effects on PS decay (preliminary) f PS vs - 9 (q1 * q2) 2 32cube Iwasaki ml=mx=0.008 f PS (q1,q2) - f PS (0,0) [MeV] (2/3,2/3) Very preliminary Mpi = 420 MeV Pi+ (2/3, - 1/3) (1/3,1/3) (2/3,- 2/3) (1/3,- 1/3) K+ (2/3,- 1/3) Statistically well resolved (101 measurements) by the +e/ e averaging. c.f. [Bijnens Danielsson 2006] F π +,NLO /F 0 = F K +,NLO /F 0 = our preliminary results are smaller. Note heavy M π EM a<rac>ve - 9 (q1 * q2) EM replusive Decay constants with EM turned on, but m u = m d : δ EM /2 Wall-point 2pt A 4 (t)p (0) and P (t)p (0) Iwasaki DWFN F = (2.7 fm) 3, a GeV, m l = m x = Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

30 Baryon mass splitting in N F = 2, [A. Walker-Loud et. al] : new estimation for QED effects [R. Horsleyet. al (QCDSF-UKQCD)], octet baryon splittings due to (m u m d ) preliminary N-P splitting with Iwasaki-DSDR lattice N F = DWF (4.6 fm) 3 (q u q d ) effect m p -m n [MeV] Walker-Loud et al Cottingham formula Nf=2 (1.9 fm) 3 Nf=2+1 (1.8 fm) 3 Nf=2+1 (2.7 fm) 3 Nf=2+1 (4.6 fm) 3 DSDR m 2 ps [GeV2 ] (m up m dwn ) effect m p -m n (MeV) (m d -m u )/Zm (MeV) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

31 m u m d EM NPLQCD 2.26(72) BLUM 2.51(71) 0.54(24) RM (70) QCDSF-UKQCD 3.13(77) = M N M p = 2.14(42) MeV (experiment: (4) MeV) 2.68(35) 0.54(24) Also EM correction to Ω meson is found to be 1.26(6) MeV (statistical error only) (preliminary) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

32 QED reweighting [T. Ishikawa et al. arxiv: ]! Full QED (+QCD) from quenched QED (+QCD) [ Duncan et. al. PRD (2005) ] by computing the reweighting factor: w[u QCD,A]= det D[U QCD e iqea ] det D[U QCD ] on the dynamical QCD configuration O(e 2 )! Stochastic eval. via Root trick [T.Ishikawa et. al ] det Ω =(detω 1/n ) n = n e ξ i (Ω 1/n 1)ξ i ξi i=1!"#$%&'()*+&,-$(,#&.#$/0123$4536$578979:$ $;#<='>?(+$@#/&.ABCD$/AE4<"FFGHI$J553658K: $/"HELHE$/ $52895MN$ $ Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

33 Disconnected diagrams in HLbL! Missing disconnected diagrams! The second quark loop could be automatically evaluated as sea quark effect, if the sea quark electric charge effect is taken into account QED reweighting (or dynamics QCD+QED) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

34 x32, m l =0.01, m s =0.04, e=0 --> e=+e phys + e=-e phys, 96 hit u[m=0.01, q=+2e/3] d[m=0.01, q=-e/3] : m l =0.01, m s =0.04, 96 hit e=+e phys e=-e phys e=+e phys + e=-e phys unreweighted reweighted w/<w> U effective mass traj u[m=0.01, q=+2e/3] d[m=0.01, q=-e/3] : m l =0.01, m s =0.04, 96 hit t t t e=+e phys e=-e phys e=+e phys + e=-e phys th root 4 hits reweighted - unreweighted sea charges q u = 2/3, q d = q s = 1/3 for m u = m d Size of the sea charge LEC, Y 1, is roughly a ball park of other LEC, consistent with systematic error estimate t t t Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

35 Some results Full QED effect on PS meson correlator 1!10 6 C(t) =P (t)p (0) C(t)[e S = e phys ] C(t)[e S =0] C(t)[e S =0] w/o reweighting w reweighting ! (m v1,q v1 ) = (0.01, +2/3) (m v2,q v2 ) = (0.03, 1/3) ! t faint effect t Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

36 Separating the terms - A set of transformations T 1 :(m 1,q 1 ; m 3,q 3 ) (m 3,q 3 ; m 1,q 1 ), T 2 :(m 1,q 1 ; m 3,q 3 ) (m 1, q 1 ; m 3, q 3 ), T 3 :(m 1,q 1 ; m 3,q 3 ) (m 3, q 1 ; m 1, q 3 ). e.g. SU(2) formula (M SU(2) π ) 2 = 4e 2 s T 2 odd & T 3 even T 2 odd & T 3 odd +e s e v C F 4 0 Y 1 trq 2 s(2) + Y 1 8π 2 i=4,5 1(trQ s(2) ) 2 + Y1 q 6 trq s(2) χ 13 χ 1i ln χ 1i µ 2 χ 3i ln χ 3i µ 2 q i +4(χ 1 χ 3 ) JtrQ s(2) + J q 6 (q 1 q 3 ) +4e s e v KtrQs(2) + K q 6 (q1 + q 3 )χ 13, 14 T 2 even Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

37 Extraction of QED LEC s Separating the terms - The hierarchy problem is resolved and the difficulty of multi parameter fit is reduced using even/oddness of the transformations. actual data: total m 2 π (m 1,m 3 )=(0.01, 0.03) T 2 -even T 2 -odd, T 3 -even T 2 -odd, T 3 -odd = +! more than 10 2 (as expected) suppressed 15 Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

38 Extraction of QED LEC s ChPT fit e.g. SU(2) ChPT fit to e S e V (T 3 even) data e s e v (T 3 -even) term in M PS 2 (GeV) am 3 = am 3 = am 3 = (q 1,q 3 )=(+2/3, 1/3) a(m 1 + m res(qcd) ) - Infinite volume formulae are used, because quark mass parameter in this study is not so small that finite volume effects are significant. - Only minimal set of data with smaller valence quark masses is used in the each fit. 16 Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

39 QED LEC s Extraction of QED LEC s SU(3) ChPT SU(2) ChPT uncorr corr uncorr corr 10 7 C (qqed) 2.2(2.0) 18.3(1.8) 10 7 C 8.4(4.3) 8.3(4.7) 20(14) 15(21) 10 2 Y 1-5.0(3.6) -0.4(5.6) 10 2 Y 1-3.1(2.2) -0.2(3.4) -3.0(2.2) -0.2(3.4) 10 4 J -2.6(1.6) -3.3(2.8) 10 4 K -3.1(6.9) -3.7(7.8) SU(3) SU(2) Y 1 = Y 1 trq 2 s(3) Y 1 = Y 1 trq 2 s(2) + Y 1(trQ s(2) ) 2 + Y1 q 6 trq s(2) J = JtrQ s(2) + J q 6 K = JtrQ s(2) + K q 6 To fully obtain LEC s in SU(2) ChPT, at least 3 independent combinations of sea quark EM charges are required. 17 Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

40 QED LEC s Extraction of QED LEC s qqed (SU(3) ChPT) fqed (SU(3) ChPT) Y 2 Y 3 Y 4 Y 5 C C Y 1 Y 1 J K qqed (SU(2) ChPT) fqed (SU(2) ChPT) consistent : quenched QED full QED (reweighed) 18 Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

41 PACS- CS (N. Ukita) QED reweigh>ng [arxiv: [hep- lat]] U(1) gauge confs are generated on a 64 3 x128 la9ce and are averaged inside the 2 4 cell to reduce local fluctua>ons. - Reweigh>ng factor : square root trick D /D = ( D /D 2 ) 1/2, 426(=400+26) determinant breakup [Hasenfratz et. al, 2008], 12 noises for each breakup, block solver à factor of 3 4 speedup, 1) 400 breakup for U(1) charges + quark masses near the physical point, 2) 26 breakup for final tuning of hopping parameters to the phys. point. - Hadron measurement : 16 source points for each conf m + / m - experimant m K + / m - experimant m K 0 / m - experimant # noise # noise # noise Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

42 Conclusions Isospin breaking studies are interesting and inevitable as precision of lattice QCD is improved. quark masses Neutron-Pron splittings Other interesting quantities? D,B meson mass π 0 η η and ρ ω mixings K l 3 π 0 γγ K ππ and I = 1/2 rule Lattice QED +QED is also a ground work for (g 2) µ Hadronic light-by-light Statistical error reduction techniques are important for Lattice QED+QCD simulations: All Mode Averaging (AMA) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

43 g-2 light-by-light [A.Nyffeler talk]! Light-by-Light only needs the part of O( 3 )! Currently O( ), O( 2 ), and unwanted O( 3 ) are subtracted (T. Blum s talk) [ M. Hayakawa et.al PoS LAT ]! QED perturbative expansion works Order by Order Feynman diagram calculation on lattice : Aslash SeqSrc method Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

44 [T.Blum Lattice2012] a µ (HLbL) in2+1flavorlatticeqcd+qed Try larger lattice 24 3 ((2.7 fm) 3 ) Pion mass is smaller too, m π = 329 MeV Same muon mass two lowest values of Q 2 (0.11 and 0.18 GeV 2 ) Use All Mode Averaging (AMA) (Izubuchi, Shintani) 6 3 point sources/configuration (216) AMA approximation: sloppy CG, r stop = 10 4 Tom Blum (UConn and RIKEN BNL Research Center) The muon anomalous magnetic moment Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

45 All-mode-Averaging!"#$"%&'(")""!$*$+"%&'("(,''-".," "/01"2.34'05"&6(("7"89: A new class of variance reduction techniques using lattice symmetries Thomas Blum, 1, 2 Taku Izubuchi, 3, 2 Eigo Shintani, 2 and... 1 Physics Department, University of Connecticut, Storrs, CT , USA 2 RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA 3 Brookhaven National Laboratory, Upton, NY 11973, USA We present a general class of unbiased improved estimators for physical observables in lattice gauge theory computations which significantly reduces statistical errors at modest computational cost. The error reduction techniques, referred to as covariant approximation averaging, utilize approximations which are covariant under lattice symmetry transformations. We observed, the cost reduction of the new method compared to the traditional one for fixed statistical error are following : 35 times for Nucleon mass at M π 300 MeV (Doamin-Wall quark), 5-50 times for the hadronic vacuum polarization at M π 480 MeV (asqtad quark). PACS numbers: Ha,12.38.Gc,07.05.Tp As non-perturbative computations using lattice gauge theory are applied to a wider range of physically interesting observables, it is increasingly important to find numerical strategies that provide precise results. In Monte Carlo simulations our reach to important physics is still often limited by statistical uncertainties. Examples include hadronic contributions to the muon s anomalous magnetic moment [1], nucleon form factors and structure functions [2], including nucleon electric dipole moments [3 6], hadron matrix elements relevant to flavor physics gauge field configurations {U 1,,U Nconf } is generated randomly, according to the Boltzmann weight, e S[U], where S[U] is the lattice-regularized action. The expectation value of a primary, covariant observable, O, O = 1 N conf N conf i=1 O[U i ]+O 1 Nconf, (1) is estimated as the ensemble average, over a large number of configurations, N conf O( ). Here, we primarily consider observables made of fermion propagators Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

46 a µ (HLbL) in2+1flavorlatticeqcd+qed Try larger lattice 24 3 ((2.7 fm) 3 ) Pion mass is smaller too, m π = 329 MeV Same muon mass two lowest values of Q 2 (0.11 and 0.18 GeV 2 ) Use All Mode Averaging (AMA) (Izubuchi, Shintani) 6 3 point sources/configuration (216) AMA approximation: sloppy CG, r stop = 10 4 Tom Blum (UConn and RIKEN BNL Research Center) The muon anomalous magnetic moment Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

47 a µ (HLbL) in 2+1f lattice QCD+QED (PRELIMINARY) F 2 (Q 2 ) stable with additional measurements ( configs) Number of Measurements Q 2 = 0.11 (GeV 2 )t Q 2 = 0.18 (GeV 2 ) 24 3 lattice size Q 2 =0.11 and 0.18 GeV 2 m π 329 MeV m µ 190 MeV F 2 (Q 2 ) Tom Blum (UConn and RIKEN BNL Research Center) The muon anomalous magnetic moment Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

48 Related works Related recent works [BMW] [L.Lellouch s talk], light hadron masses on physics point [PACS-CS], (N. Ukita et al) PS mass including QED reweighting on physics point [UQCDSF-UKQCD], Octet baryon mass from (m u m d ) [Rome123], Hadron mass from (m u m d ) using perturbation [A. Walker-Loud], Isospin violation MILC [C.Bernard s talk] EM spectrum / ChPT fit [JLQCD] π 0 γγ [A.Nyffeler] g-2 light-by-light [T.Blum, T.Doi, M.Hayakawa, T.Izubuchi, S.Uno N.Yamada, and R.Zhou], Electromagnetic mass splittings of the low lying hadrons and quark masses from 2+1 flavor lattice QCD+QED, Phys. Rev.D82 (2010) arxiv: [hep-lat] (95 pages). [T.Izubuchi], Studies of the QCD and QED effects on Isospin breaking, PoS(KAON09) 034. [R.Zhou, T.Blum, T.Doi, M.Hayakawa, T.Izubuchi, and N.Yamada], Isospin symmetry breaking effects in the pion and nucleon masses PoS(LATTICE 2008) 131. [T. Blum, T. Doi, M. Hayakawa, TI, N. Yamada], Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

49 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) (38 pages) The isospin breaking effect on baryons with Nf=2 domain wall fermions PoS(LAT2006) 174 (7 pages) Electromagnetic properties of hadrons with two flavors of dynamical domain wall fermions PoS(LAT2005) 092 (6 pages) Hadronic light-by light scattering contribution to the muon g-2 from lattice QCD: Methodology PoS(LAT2005) 353(6 pages) Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

50 Why lattice QED? Since QED is weakly coupled, α = 1/137, the perturbation theory works well. One could extract the necessary quntities as QCD s matrix elements π(x)π(y) QCD+QED = π(x)π(y) QCD +α d 4 q π(x)v µ (q)v ν (q)π(y) QCD G photon µν (q)+ from which the QCD+QED physical observables would be obtained. Rather, we computed for full non-perturbative lattice QCD+QED system π(x)π(y) QCD+QED because of computational costs and higher order O(e 4 ) (see later A-Seq. method), its own interesting features, and as an exercises for (g 2) µ light-by-light calculation. Lattice QED has problems Finite volume effects from photon Landau ghost (but α(0) = 1/137 vs α(m Z ) 1/128) which will not be cured by switching the method to the QCD matrix element calculation. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

51 Other considerations and quantities A-Sequential source method. Compute each term of propagator in the e expansion. S(e) = S(0) + ies(0) AS(0) e 2 S(0) AS(0) AS(0) e 2 S(0)( A) 2 S(0) 1 1 e 2 : q 1 q 2 = 2 make the contraction to desired orders of wanted diagrams piece by piece. 2 G(t; q 1, q 2 ) = e 2 : q 1 q 2 q 2 1 q 2 2 e 4 : q 2 1q 2 2 q 2 1 q * No O(e 2n+1 ) noise to disturb O(e 2n ), can skip diagrams of lower orders than the target. * Value of q and e could be determined off-line. * # of solves are equal or less up to O(e 2 ), compared to the original methods, needs five solves (q = 0, ±2e/3, e/3). * Could use the e = 0 Eigen values/vectors. Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,

52 Various checks to make sure we understand systematics in light-by-light. The computation of quark propagators with EM will be shared among various quantities. O(α, α 2 ) : Vacuum polarizations Π µν = V µ V ν include the disconnected quark loops, which include. Quark condensate magnetic susceptibility qσ µν q F = eχ qq 0 F µν to constraint the short distance of π γ γ coupling Taku Izubuchi, Chiral Dyanmics 2012, Newport News, August 9,