A construction of the Glashow-Weinberg-Salam model on the lattice with exact gauge invariance

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1 A construction of the Glashow-Weinberg-Salam model on the lattice with exact gauge invariance Y. Kikukawa Institute of Physics, University of Tokyo based on : D. Kadoh and Y.K., JHEP 0805:095 (2008), 0802:063 (2008) D.~Kadoh, Y.~Nakayama and Y.K., JHEP 0412, 006 (2004) Y. Nakayama and Y.K., Nucl. Phys. B597, 519 (2001)

2 the Glashow-Weinberg-Salam model (^^; (SU(2)x U(1) sector of the standard model without SU(3) color int.) a chiral gauge theory with SU(2)L x U(1)Y gauge symmetry breaking via Higgs mechanism baryon number violation due to chiral anomaly but... etc. Weakly coupled theory, Still, non-perturbative dynamics may be relevant no gauge-invariant regularization is known (cf. dimensional reg.) non-perturbative definition is missing

3 previous attempts to put on the lattice... Eichten-Preskill approach (symmetry/symmetry breaking) Wilson-Yukawa model (Smit, Swift, Aoki) Rome (gauge-fixing) approach (Testa et al, Golterman-Shamir) domain-wall + Eichten-Preskill hybrid (Creutz) Mirror GW fermion approach (Poppitz) etc. in this talk... a gauge-invariant construction of GWS model on the lattice use of overlap Dirac operator (the Ginsparg-Wilson relation) cf. U(1) chiral gauge theory with exact gauge invariance Luscher (99) the first invariant / non-perturbative regularization of the model all SU(2) togological sectors with vanishing U(1) magnetic fluxes

4 plan of this talk 1. chiral lattice gauge theories based on overlap D / the G-W rel. 2. gauge anomalies in the lattice SU(2)L x U(1)Y chiral gauge theory 3. topology of the space of SU(2)xU(1) lattice gauge fields 4. our approach & results explicit construction of the smooth measure term proof of the global integrability conditions [reconstruction theorem] 5. discussion an extention to the standard model (the inclusion of SU(3) ) possible applications

5 overlap Dirac op. / the GW rel. D = 1 2a Neuberger(1997,98) ( ) H w 1+γ 5 H 2 w chiral operator Luscher ; Hasenfratz, Niedermayer(1998) ˆγ 5 γ 5 (1 2aD) = H w H 2 w γ 5 D + Dγ 5 =2aDγ 5 D Path Integral Quantization ψ (x) = i c i v i (x) chiral fermion ˆγ 5 ψ ± (x) =± ψ ± (x) ψ ± (x)γ 5 = ψ ± (x) Path Integral Measure depends on gauge fields! det ψ (x) = Q ( ) ṽ v i (x)c i Z = D[ψ ]D[ ψ ]e P a4 ψ i (x) =v j (x) Q 1 x Dψ (x) ji i c i = Q ij c j = dc i d c j e P ij c jm ji c i i j complex phase! {v i (x) ˆγ 5 v i (x) = v i (x) (i =1,,N )} { v i (x) v i (x)γ 5 = + v i (x) (i =1,, N )} = det M ji M ji = a 4 x overlap formula v j Dv i (x) Narayanan-Neuberger(1993)

6 variation of effective action & gauge anomaly Γ eff = ln det( v k Dv j ) δ η U(x, µ) =iη µ (x)u(x, µ) δ η Γ eff = Tr {(δ η D) ˆP } D 1 P + + (v i,δ η v i ) i measure term = itrωγ 5 (1 D) i i (v i,δ ω v i ) η µ (x) = i µ ω(x) the gauge-field dependence must be fixed... Luscher(99) locality? gauge invariance? integrability? [ admissibility cond. cf. Hernandez, Jansen, Luscher(98) ] [ gauge anomaly cancellations ] [ topology of the space of gauge fields non-trivial due to Admissibility cond. ] * different situation from Dirac fermions in Vector-like theories like QCD

7 applying this formulation to quarks and leptons... our results on the lattice GWS model : 1. explicit construction of the smooth measure term, which fulfills requirements of locality, gauge invariance & local integrability L η = i i (v i,δ η v i )= x η µ (x)j µ (x) η µ (x) =η (2) µ (x) η (1) µ (x) 2. proof of the reconstruction theorem (global integrability conditions) key issues... SU(2)xU(1) gauge anomaly topology of space of SU(2)xU(1) gauge fields

8 gauge anomaly in the SU(2)xU(1) chiral gauge theory η µ (x) =η (2) µ (x) η (1) µ (x) δ η Γ eff = Tr {(δ η D) ˆP } D 1 P + + i = itrωγ 5 (1 D) i i η µ (x) = i µ ω(x) (v i,δ ω v i ) (v i,δ η v i ) Y 3 R L Y U(1) Y U(1) Y U(1) Y 3 =0 Y U(1)! a! b Y =0, doublet(l) SU(2) SU(2) Y =0 singlet(r) SU(2) 3 gauge anomaly pseudo reality of SU(2) measure term vanishes identically for a pair of doublets (a,b) v (a) j (x) =v j (x) v (b) j (x) = ( γ 5 C 1 ) iσ 2 [vj (x)] SU(2) 2 x U(1), U(1) 3 gauge anomaly cohomological analysis in Γ4 x Γ 4 Y α q(x) U (1) {U (1) } Yα α = α Y α q(x) U (2) + α Y 3 α 1 32π 2 ɛ µνλρf µν (x)f λρ (x +ˆµ +ˆν)+ µk µ (x) = µk µ (x) cf. Suzuki et al. (01) Kadoh-Nakayama-YK(04)

9 topology of the space of lattice SU(2)xU(1) gauge fields finite volume case Γ 4 = {x =(x 0,,x 3 ) Z 4 0 x µ <L} = L 4 m [U(1)] admissibility condi. 1 U (2) ɛ 1 {U (1) }6Y ɛ ɛ< 1 30 topological charges m µν = 1 2πi s,t ln U (1) µν (x + sˆµ + tˆν) Q Q = x Γ 4 tr{γ 5 (1 D)(x, x)} U (2) U(1) ~ T^n U(1) gauge fields U µ (x) =e iat µ (x) g(x)g(x +ˆµ) 1 U [w] (x, µ)v [m] (x, µ) T n [U(1)] M[SU(2)] F µν (x) = µ A T ν (x) ν A T µ (x)+ 2πm µν L 2 SU(2) topological structure of SU(2) space is not known yet!

10 our approach pure SU(2) theory measure defined globally! a pair of doublets (a,b) v (a) j (x) =v j (x) v (b) j (x) = ( γ 5 C 1 ) iσ 2 [vj (x)] U(1) degrees of freedom cf. Nuberger(98) Bar-Campos (00) m [U(1)] Q [SU(2 U µ (x) =e iat µ (x) g(x)g(x +ˆµ) 1 U [w] (x, µ)v [m] (x, µ) measure term smooth on T n [U(1)] M[SU(2)] U(1) ~ T^n proof of the global integrability condition SU(2) non-contractible loops

11 measure term in the SU(2)xU(1) chiral gauge theory η µ (x) =η (2) µ (x) η (1) µ (x) Kadoh-YK (08) local counter term! Wilson line contr. explicit constr. with two simplifications direct proof of gauge anomaly cancellation in finite volume cf. Luscher(98) U(1) ~ T^n separate treatment of the Wilson lines SU(2)

12 1. jµ(x),j a µ (x) are defined for all admissible SU(2)xU(1) gauge fields in the given topological sectors and depends smoothly on the link variables 2. jµ(x),j a µ (x) are gauge-covariant / invariant, respectively and both transforms as axial vector currents under lattice symmetries 3. The linear functional L η = { η a µ (x)jµ(x)+η a µ (x)j µ (x) } is a x solution of the integrability condition, { } δ η L ζ δ ζ L η + L [η,ζ] = itr ˆP [δ η ˆP,δ ζ ˆP ] 4. The anomalous conservation laws hold, { µj µ } a (x) = tr{t a γ 5 (1 D)(x, x)} { } + itr ˆP+ [δ η ˆP+,δ ζ ˆP+ ] µj µ (x) = tr{y γ 5 (1 D L )(x, x)} tr{y + γ 5 (1 D L )(x, x)} where and Y = diag(1, 1, 1, 3) Y + = diag(4, 2,, 0, 6)

13 Reconstruction theorem In the topological sectors with vanishing U(1) magnetic flux, if there exist local current jµ(x) a (a =1, 2, 3),j µ (x) which satisfy the following four properties, it is then possible to reconstruct the fermion measure ( the basis {v j (x)} ) which depends smoothly on the gauge fields and fulfills the fundamental requirements such as locality, gauge-invariance, integrability and lattice symmetries:

14 the Glashow-Weinberg-Salam model on the lattice finite volume case covers all SU(2) topological sectors with vanishing U(1) magnetic fluxes global integrability is proved rigorously some non-perturbative applications? ex. a computation of the effect of quarks & leptons to the sphaleron rate at finite temp. (at one-loop) infinite volume case a local counter term constructed non-perturbatively the first gauge-invariant regularization of the EW theory (cf. dimensional reg. ) may be used in perturbation theory ex. computations of higher order EW contr. to muon g-2

15 possible applications of the lattice EW theory a computation of the effect of quarks & leptons to the sphaleron rate at finite temp. (at one-loop) introduction of Higgs field & Yukawa-couplings S EW = S G + S F + { µ φ µ φ + V (φ)} { x } y t Q φt+ (x)+y b Q φb + (x)+c.c. x sphaleron on the lattice fermion fluctuation det. cf. Moore (96) κ F (v, λ, y t, ) det M/ det M 0 q,l ω n ( ) ( vk Dv M t = j ) y t ( v k φuj ) y t (ū φ k v j ) (ū k Du j ) U (2) µ (x),u (1) µ (x),φ(x)(x L 3 ) (b) (e) lo~~y i 0 ~ (a) x (b) x (C) saddle point cooling Perez- van Baal (96) sum over matsubara freq. one-loop renormarizations dependence on the Higgs, Yukawa coupling comparison to other methods cf. Bodeker et. (00)

16 the Glashow-Weinberg-Salam model on the lattice in finite volume covers all SU(2) topological sectors with vanishing U(1) magnetic fluxes global integrability is proved rigorously even number of SU(2) doublets, U(1) Wilson line parts explicit with two simplifications cf. U(1), Luscher (98) direct proof of gauge anomaly cancellation in separate treatment of the Wilson line some non-perturbative applications? L 4 based on : Y.~Nakayama and Y.K., Nucl. Phys. B597, 519 (2001) D.~Kadoh, Y.~Nakayama and Y.K., JHEP 0412, 006 (2004) D.~Kadoh and Y.K., in preparation

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