The electric dipole moment of the nucleon from lattice QCD with imaginary vacuum angle theta

Transcription

1 The electric dipole moment of the nucleon from lattice QCD with imaginary vacuum angle theta Yoshifumi Nakamura(NIC/DESY) for the theta collaboration S. Aoki(RBRC/Tsukuba), R. Horsley(Edinburgh), YN, D. Pleiter(NIC/DESY), P.E.L. Rakow(Liverpool), G. Schierholz(NIC/DESY), T. Izubuchi(Kanazawa/RBRC) and J. Zanotti(Edinburgh) July, 2008, Lattice 2008

2 Electric Dipole Moment(EDM) Permanent EDM is a signature of T(Time reversal symmetry, =CP) violation Various candidates of CP violations Electro Weak: CKM phase in quark mass matrix very small New Physics: SUSY, left-right, multi Higgs Vacuum angle θ In QCD gauge invariant CP odd terms are allowed S θ = iθ 1 32π 2 d 4 xf µν (x) F µν (x) = iθq

3 Since 50 years ago Neutron EDM Neutron EDM Upper Limit (e.cm) 1E-19 1E-20 1E-21 1E-22 1E-23 1E-24 1E-25 1E-26 Electromagnetic Weinberg multi-higgs Minimal SUSY Left-Right symmetric Participating institutions: ORNL-Harvard BNL-MIT ORNL-ILL... ILL-Sussex-RAL... LNPI St Petersburg 1E Year of Publication d n < e fm Harris (2007) Baker et al. (2007)

4 2 methods(so far) NEDM on lattice 89 Aoki-Gocksch N f =0 Wilson Electric Field 04 Berruto et al. N f =0 DWF F 3 05 Berruto et al. N f =2 DWF F 3 06 CP-PACS N f =0 DWF F 3 06 CP-PACS N f =0 DWF Electric Field 06 QCDSF N f =0 overlap F 3 06 Blum, Izubuchi, Doi N f =2 DWF Electric Field 08 CP-PACS N f =2 Clover Electric Field External Electric Field Measure a spin splitting of energy Electric form factor F 3 = in this talk Measure 3pt function with momenta p, and p 0 Dynamical simulations are important NEDM is very sensitive to sea quark mass, d n = 0 in the chiral limit Reweighting method have been used = noisy (needs large statistics) In usual QCD simulations, θ=0 In the real world θ is real But one can not do Full QCD HMC simulations with real θ

5 LQCD with imaginary θ

6 Motivation To calculate d N /θ In lattice QCD, θ is one of the input parameters d N /θ from lattice d n from ex. θ <? To check feasibility of lattice QCD simulations at imaginary θ

7 Simulations with imaginary θ There are 2 choices gluonic: θf F needed smearing/cooling fermionic: mθ ψγ 5 ψ by using anomalous chiral WT relation rotation by θ m me i θ N f γ 5

8 Action with θ We choose fermionic way S F + S θ = ψ {D + [cos(θ/n f ) + i sin(θ/n f ) γ 5 ]m} ψ m = cos(θ/n f ) m θ = tan(θ/n f ) N f S F = ψ { D + m + i ( θ/n f ) γ 5 m } ψ

9 Lattice action S F = ψ {D + m + i (θ R /N f ) Z m Z P γ 5 m} ψ Z m : the renormalization constant of the quark mass Z P : the renormalization constant the pseudoscalar density θ R : renormalized vacuum angle θ R = (Z m Z P ) 1 θ chiral fermion: nicer, definitely, extremely expensive Z m Z P = 1 and θ R = θ clover fermion: relatively cheap Z m Z P = 1 and θ R = θ in the continuum limit

10 We employ N f =2 flavors of dynamical clover fermions (chiral symmetry is violated) a(d + m) D lat = D Wilson + T clover a m 1 2κ 1 2κ c VWI quark D Wilson x,y = δ x,y κ µ { (1 γµ )U µ (x)δ x+µ,y + (1 + γ µ )U µ(x µ)δ x µ,y } T clover x,y = { i 2 c SWκσ µν F µν (x) + i(1 κ ) θ } R κ c 2 Z mz P γ 5 δ x,y In simulations with dynamical quarks, γ 5 Dγ 5 = D is required The vacuum angle θ is taken to be purely imaginary θ = i θ i θ I The Boltzmann weight positive definite

11 Lattice Size: β: 2.1(Iwasaki) κ: κ c : Lattice Spacing: a=0.1076(13) fm am π : (8) Simulation same action as CP-PACS θ I # of traj. P τ P int Q Q 2 Q 2 e H (13) 3.24(68) -0.06(31) 24.9(14) (25) (11) 3.30(73) -3.52(46) 24.1(15) (35) (17) 3.02(41) -7.35(36) 22.7(17) (37) (16) 4.1(19) (30) 21.7(15) (14) (16) 2.56(61) (37) 18.1(13) (28)

12 θ v =θ s θ v = θ v =θ s θ v =0 a m PS a m N θ I θ I

13 P(Q) is changed by θ I Topological charge distribution

14 Effective value of θ θ input = (1 κ κ θ c) θ I RZ θ mz θ P ex. Input parameter θ input = If κ θ c = and Z θ mz θ P = 1 θ I R = 0.4 We could define renormalized θ by Q(θ) θ I R θ I 0.75 One could also check the effect of θ in other hadronic observable m π (θ) = m π (0) cos(iθ/n f )? Brower et al. (2003), Aoki et al. (2007)

15 Nucleon form factors The electromagnetic current between nucleon states p, s J µ p, s = ū θ ( p, s ) J µ u θ ( p, s) J µ = γ µ F θ 1 (q 2 ) + σ µν q ν F θ 2 (q 2 ) + iθ 2m θ N [ (γq q µ γ µ q 2 ) γ 5 FA(q θ 2 F3 θ (q 2 ] ) ) + σ µν q ν γ 5 2m θ N q = p p, q 2 = (E E) 2 + ( p p) 2, γp = ieγ 4 + γ p Electric dipole moment d θ N = lim q 2 0 ef θ 3 (q 2 ) 2m θ N The Dirac spinors are modified by a phase in the θ vacuum u θ ( p, s) = e iα(θ)γ 5 u( p, s) ū θ ( p, s) = ū( p, s) e iα(θ)γ 5

16 Spinor relation is modified to ( ) u θ ( p, s )ū θ ( p, s) = e iα(θ)γ 5 iγp + m θ N s,s The lowest order in θ s,s 2E θ N e iα(θ)γ 5 u θ ( p, s )ū θ ( p, s) = iγp + m N(1 2α θ I γ 5 ) 2E N. We are primarily interested in the electric dipole moment in the limit θ 0, it is sufficient to consider the lowest order expansion only d n for θ I 0.4 in this work

17 Tr [G θ NN(t; 0)Γ 4 ] 1 2 Z N 2 e m Nt, Tr [G θ NN(t; 0)Γ 4 γ 5 ] α θ I 1 2 Z N 2 e m Nt, Γ 4 = (1 + γ 4 )/2 taking ratio R(t) = Tr [Gθ NN (t,0)γ 4γ 5 ] Tr [G θ NN (t;0)γ 4] α θ I θ I = 0.4

18 Nucleon form factors R µ (t, t; p, p) = Gθ NJ Γ µ N (t, t; p, p) Tr [G θ NN (t ; p )Γ 4 ] { Tr [G θ NN (t; p )Γ 4 ] Tr [G θ NN (t ; p )Γ 4 ] Tr [G θ NN (t t; p)γ 4 ] Tr [G θ NN (t; p)γ 4] Tr [G θ NN (t ; p)γ 4 ] Tr [G θ NN (t t; p )Γ 4 ] = E θ E θ (E θ + m θ N ) (Eθ + m θ N ) F(Γ, J µ), F(Γ, J µ ) = 1 [ 4 Tr Γ e iα(θ)γ 5 Eθ γ 4 i γ p + m θ ] N e iα(θ)γ E θ 5 [ J µ e iα(θ)γ 5 Eθ γ 4 i γ p + m θ ] N e iα(θ)γ E θ 5 J µ = γ µ F θ 1 (q 2 ) + σ µν q ν F θ 2 (q 2 ) + iθ 2m θ N [ (γq q µ γ µ q 2 ) γ 5 FA(q θ 2 F3 θ (q 2 ] ) ) + σ µν q ν γ 5 2m θ N } 1/2

19 momenta: quantized in units of 2π/L conventional (periodic) boundary condition(bc) p = 2π is not small noisier & large extrap. twisted boundary condision(tbc) Bedaque (2004) ψ(x k + L) = e i α k ψ(x k ), k = 1, 2, 3. the dispersion relation for the nucleon E = m 2 N + ( p + α)2 choice of twist angles α = 2π L α = 2π L α = 2π L α = 2π L (0, 0, 0) (0.36, 0, 0) (0.36, 0.36, 0) (0.36, 0.36, 0.36)

20 Preliminary results θ I = 0.2 θ I = 0.4 Fit using a dipole ansatz F θ 3 (q 2 ) = The renormalization constant Z V is needed F θ 3 (0) (1 + q 2 /M 2 ) 2

21 θ I = 0.2 θ I = 0.4 Z V cancels F θ 3 (0) = lim q 2 0 F θ 3 (q 2 ) F p 1 (q2 )

22 Anapole form factor θ I = 0.2 θ I = 0.4

23 fit d θ N = dθ N θ I θ I + c θ I 3

24 gives at θ I = 0 d θ N θ I = 0.080(10) stat+fit (?) sys [e fm] Proton d θ N θ I = 0.049(5) stat+fit (?) sys [e fm] Neutron θ < preliminary

25 θ I s = 0.4, θ I v = 0.0

26 Charge distribution and θ vacuum Q c Q 2 c θ I θ I Q n c i n n θ n lnz(θ)

27 S -3 K θ I θ I S = Q3 c Q 2 c K = Q4 c Q 2 c

28 Conclusions and future plans Have performed simulations of QCD with N f = 2 flavors of dynamical quarks at imaginary vacuum angle θ The use of partially twisted boundary conditions has allowed us to compute the relevant neutron form factor F 3 (q 2 ) with high precision over the entire range of momenta down to (aq) greatly facilitated the extrapolation to q 2 = 0 Successfully obtain signal disentangled from statistical noise Plan Sea quark mass dependence (χpt) EDM is zero in the chiral limit In this work m π is very heavy 1 GeV Volume dependence In this work V (1.7 fm) 3 small for baryon Dependence of the boundary conditions We used periodic BC for sea quark TBC for valence quark Gluonic operator instead of the pseudoscalar operator