PROTON DECAY MATRIX ELEMENTS FROM LATTICE QCD. Yasumichi Aoki RIKEN BNL Research Center. 9/23/09 LBV09 at Madison

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1 PROTON DECAY MATRIX ELEMENTS FROM LATTICE QCD Yasumichi Aoki RIKEN BNL Research Center 9/23/09 LBV09 at Madison

2 Plan low energy matrix elements for N PS,l GUT QCD relation what is exactly needed to calculate in QCD? direct / indirect method Lattice QCD 3 flavor calculation with indirect method 3 flavor calculation with direct method (preliminary) further calculation plan summary

3 Lattice QCD: status u,d,s dynamical simulation (N f =3) [fully unquench] is being carried out unitary formulation for N, pi, K initial, final states and operators earlier calculation with quenched approx. (N f =0) or u,d dynamical (N f =2) had uncontrolled systematic errors

4 [p PS+l] low energy matrix element L GUT L SM + i C i (µ) O i (µ)+ O i (µ) = (qq) Γ (ql) Γ i labels chirality and flavor structure yields a decay: baryon meson + anti-lepton π 0,e + p GUT = i C i (µ) π 0,e + O i (µ) p SM π 0,e + (ud)(eu) p = v c e π 0 (ud)u p a convenient parametrization using 2 form factors π 0 (ud) Γ u L p = P L [W 0 i/q W 1 ]u p m p π 0,e + (ud) Γ (eu) L p = W 0 (v e,u p ) L + m e m p W 1 (v e,u p ) R partial width Γ(p π 0 + e + )= m p 32π 2 = W 0 (v e,u p ) L +O(m e /m p ); ( W 0 W 1 : ChPT, lattice) [ 1 ( mπ m p ) ] 2 2 C i W i 0(p π 0 ) i 2

5 QCD matrix element: indirect method π 0 (ud) Γ u L p = P L [W 0 i/q m p W 1 ]u p Γ = R, L approximation to lowest order in ChPT: Claudson, Wise, Hall, 1982 W 0 [ π 0 (ud) R u L p ] α (1 + D + F ) 2f W 0 [ π 0 (ud) L u L p ] β 2f (1 + D + F ) f : pion decay constant D + F = g A : nucleon axial charge 0 (ud) R u L p = αu p 0 (ud) L u L p = βu p Lattice: 2 tasks renormalization of operator & matching to the scheme & scale C i (μ) was calculated calculate low energy constants α, β with 3q operators

6 how to calculate alpha and beta (bare) 0 O RL p = αu p O RL = (u c P R d) P L u J p = (u c γ 5 d) u 0 O RL ( x, t) J p (0) 0 = x x x (color indices contracted with ε ijk to make singlet) :proton interpolation operator = i 0 O RL ( x, t) linear combination of products of 3 quark propagators quark propagators: inverse of domain-wall fermion Dirac operator engineering the interpolation operator necessary to have good S/N i i 1 2E i i J p (0) 0 e E it 1 2E i 0 O RL (0) i i J p (0) 0 (large t) e m pt 1 2m p 0 O RL (0) p p J p (0) 0 0 J p ( x, t) J p (0) 0 e m pt 1 2m p 0 J p (0) p p J p (0) 0

7 Direct method to calculate W0 (bare) π 0 O RL p = P L [W 0 i/q m p W 1 ]u p O RL = (u c P R d) P L u J p = (u c γ 5 d) u J 0 π = 1 2 (uγ 5 u dγ 5 d) :proton interpolation operator three point function with momentum injection to pion in proton s rest frame e i p ( y x) 0 J π 0( y, t ) O RL ( x, t) J p (0) 0 y (t t 0) e E π(t t) e m pt 1 2E π 1 2m p 0 J π 0 π 0 ( p) π 0 ( p) O RL (0) p p J p (0) 0 0 J p ( x, t) J p (0) 0 e m 1 pt 0 J p (0) p p J p (0) 0 2m p x e i p x 0 J π 0( x, t) J π 0(0) 0 e E 1 πt 0 J π 0(0) π 0 ( p) π 0 ( p) J π 0(0) 0 2E π x x :pion interpolation operator through some projection/subtraction, W 0 is obtained. computationally demanding: O(10) x [cost to calculate α, β]

8 Lattice computation Lattice gauge theory: gauge theory on discrete Euclidian space time (lattice spacing a) a regularization of gauge theory with manifest gauge invariance with finite volume and a, path integral can be performed using (super) computer continuum limit a 0 has to be performed, or discretization error must be estimated L(a) =L QCD + a i c (5) i O (5) i + a 2 j c (6) j O (6) j + all O (5) break chiral symmetry If the lattice action has chiral symmetry, no O(a) error! more continuum like Lattice action with chiral symmetry available Domain wall fermions (DWF) (Kaplan, Furman Shamir), overlap fermions (Neuberger) helps preserve continuum like structure of operator mixing

9 Lattice setup: input and output 3 flavor lattice QCD computation parameters: gauge coupling, m ud (degenerate), m s MC simulations are done at m (sim) s m (phys) s, m (sim) ud >m s /5 small m demanding: cost of some portion of simulation 1/m x : x>1 smaller m: larger finite volume effect fixed gauge counpling, tune m s so that it reproduces ratios of π, K, Ω mass. π, K: quark mass dependence best known: NNLO ChPT Ω (sss): no pion chiral logs at NLO: safe to apply linear chiral extrapolation all other quantities are predictions, ex: [RBC/UKQCD PRD78(08)114509] f π =124.1(6.9)MeV 130.7(0.1)(0.36)[exp], f K /f π =1.205(18) 1.223(12)[exp] one lattice spacing, estimate of O(a 2 ) systematic error was added. quark masses, B K...

10 nucleon related prediction: ga finite volume effect crutial [RBC/UKQCD T.Yamazaki, et al., PRL08, PRD09]

11 People involved for 3f DWF calculation RBC/UKQCD collaboration for gauge ensemble generation RBC: RIKEN, BNL, Columbia, Connecticut, Virginia UKQCD: Edinburgh, Southampton all togather ~40 people A small subgroup involved for proton decay RBC: YA, C. Dawson, T. Izubuchi, A. Lichtle, A. Soni some RBC people & QCDOC super UKQCD: P. Boyle, P. Cooney, L. Del Debbio, C. Maynard, R. Kenway, R. Tweedie

12 non-perturbative renormalization To avoid lattice perturbation theory, which has poor convergence property. RI/MOM scheme constructed: [YA et al., PRD75(07)014507] NLO matching to MSbar Exact chiral symmetry multiplicative renormalization DWF has small breaking In general, mixing occurs O RL = (u c P R d) P L s O LL = (u c P L d) P L s O A(LV ) = (u c γ µ γ 5 d) P L γ µ s Z: diagonal matrix: practically no mixing! Zq 3/2 Z 1 i,j diagonal off-diagonal +8% O(α s 2) error [RBC/UKQCD YA et al. PRD78(08)04505]

13 low energy constants [RBC/UKQCD YA et al. PRD78(08)04505] α, β calculated by taking the chiral limit of values at simulated u,d quark mass systematic error is estimated through difference: lightest point [included/not] gives largest contribution [18%] to total systematic error to final values s quark mass is kept fixed at near physical (still valid at the used order of approx.)

14 low energy constants: error budget low energy constants renormalized in MSbar NDR at μ=2 GeV with N f =3 α = ± (stat) ± (syst) GeV3, β = ± (stat) ± (syst) GeV3 Systematic errors non zero lattice spacing: likely negligible (is so in quenched approximation) finite volume: negligible Matching RI/MOM to MSbar O(α s2 ): 8% Chiral extrapolation: 18% [RBC/UKQCD YA et al. PRD78(08)04505]

15 direct method (preliminary) W 0 [ π 0 (ud) Γ u L p ] π 0 LL p Cγ 4 γ 5 p=(1,0,0)*2π/l p=(1,1,0)*2π/l physical kinematics 0 π 0 RL p Cγ 4 γ 5 p=(1,0,0)*2π/l p=(1,1,0)*2π/l physical kinematics a 2 W 0 a 2 W u,d s u,d s a m ud a m ud statistical error will be reduced by more sophisticated analysis fit using linear in u,d mass & p 2 to extrapolate/interpolate to physical kinematics systematic error will be studied once even lighter mass point is analyzed it seems hard to beat 20% total probably 30% is at best (indirect: 23% total error with unknown sys. error)

16 comparing with other results <π 0 (ud) R u L <π 0 (ud) R u L W 0 [GeV 2 ] phenomenology lattice QCD quench lattice QCD 2f lattice QCD 3f Wilson fermion JLQCD domain-wall fermion a=0.1 fm RBCRBC/UKQCD a->0 W 0 [GeV 2 ] Wilson fermion JLQCD a=0.1 fm a->0 phenomenology lattice QCD quench lattice QCD 2f lattice QCD 3f domain-wall fermion RBC RBC/UKQCD 0.1 preliminary direct method year preliminary direct method 2000 year st (preliminary) unquenched estimate of W 0 with direct method has been obtained! consistent with old quench result. smaller form factor for direct method than indirect caution!: indirect method underestimates proton lifetime

17 Why so differ? physical kinematics is far from the soft pion limit. Quench case [YA et al., PRD75(07)014507] Similar for N f =3

18 W0: how many are they? W 0 RR =W 0 LL W 0 RL =W 0 LR only connected part each process needs to be calculated separately with direct method We can calculate W 0 for all of these (in fact we have, in quenched approximation)

19 W0 summary in quenched approximation <π 0 (ud) <π 0 R u L (ud) <K 0 L u L (us) <K 0 R u L (us) <K + L u L (us) <K + R d L (us) <K + L d L (ud) <K + R s L (ud) <K + L s L (ds) <K + R u L (ds) L u L <η (ud) R u L <η (ud) L u L direct indirect W 0 [GeV 2 ] [YA et al., PRD75(07)014507] indirect method tends to overestimate W0, resulting in underestimate of lifetime We will have similar plot with fully unquenched results soon!

20 Summary Calculations of low energy matrix elements of proton decay was reviewed. Form factor W 0 of N PS is what we want. W o can be calculated either directly on the lattice or indirectly through approximation with ChPT and α, β calculated on the lattice. 3 flavor unquenched domain wall fermions are used for both methods. α, β obtained with ~20% error. preliminary direct estimate of W 0 for p π 0 +e + was obtained. direct estimate gives smaller form factor than indirect, which prolongs lifetime. soon we will get W 0 for all possible N PS transition with (qq)(ql) operator.