Happy 60th Birthdays! Ishikawa-san & Kawamoto-san

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1 Glashow-Weinberg-Salam model on the lattice A construction 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 Happy 60th Birthdays! Ishikawa-san & Kawamoto-san

3 May 1986

4 We learned Mechanics and Quantum Mechanics from him! Ishikawa-san s nickname was first, Glue (from Glueball) later, Hall (from Quantum Hall Effect)

5 Izo: :q Ii Ta \. l),,11"- E!.,O

8 ( :b L Utz ) 1.,') tt tt) k7v13 = Xr'l

10 Univ. Festival 1985

11 Thank you very much, Ishikawa-sensei Univ. Festival 1985

12 Introduction to Lattice Gauge Theory

13 Species Doubling Problem VS Ishikawa-san

14 Problem of lattice fermions : Species doubling Dirac equation on the discrete space 3 1 H = α k + βm = H lat = i x k k=1 H lat eigenvalues E = ± 3 k=1 1 a 2 sin2 (p k a) + m 2, k ψ(x, t) = 3 1 α k 2i k=1 p 1, p 2, p 3 ( ) k k + β m ( ψ(x + ˆka, ) t) ψ(x, t) [ π a, π ] a W /a species doublers p k = π/a + q k, q k π/a a p ÅÅÅÅÅÅÅÅ p α k sin(p k a) ( α k )q k γ 5 = ( i)α 1 α 2 α 3 ( 1) n ( i)α 1 α 2 α 3 chirality flip!

15 Wilson-Dirac fermion S w = a 4 x ψ(x) ( γ µ 1 2 ( µ µ) + a 2 Explicit Breaking of chiral symmetry ) ( µ ) µ + m0 ψ(x) doubler mass : m 0 + µ a 2 ( 2 a sin k ) 2 µa m 0 + 2n 2 a n = numbers of π No-Go Theorem (Nielsen-Ninomiya) S = a 4 ψ(x) D ψ(x) = x π/a π/a d 4 k (2π) 4 ψ( k) D(k) ψ(k) D(k) is a periodic and analytic function of momentum k µ D(k) iγ µ k µ for k µ π/a D(k) is invertible for all except γ 5 D(k) + D(k)γ5 = 0 k µ k µ = 0 analyticity & locality: l k l D(k) = x e ikx (ix) l D(x) < = D(x) < Ce γ x

16 Matsuyama-san once told me, He was interested in the Species Doubling Problem of lattice fermions, when he was a graduate student. But Ishikawa-san did not like it and not allow him to study such a topic. He did not like the topics such as axiomatic approach to QFT, too. (I heard from Yabuki-san) So he studied not in his office (at the desk next to Ishikawa-san s) but in the library, hiding (^^; Maybe, Ishikawa-san s message was that one should be more physics-oriented, not mathematical formalism-oriented. Also, I think, he is very practical. But,...

17 Why Glashow-Weinberg-Salam model on the lattice?

18 the standard model = SU(3)xSU(2)xU(1) chiral gauge theory gauge symmetry & chiral symmetry

19 Chiral symmetry It was from Ishikawa-san that I first heard the name Nambu. It was when his collected papers was published from JPS in 1986(????).

20 Rep. of quarks and leptons (the first family) ( ν (x) e (x) ) Y = 1 2, ν + (x) Y =0 e + (x) Y = 1 ( u i (x) d i (x) ) Y = 1 6, u + i (x) Y =+ 2 3 d + i (x) Y = 1 3

21 Mass 1 TeV 1 GeV W "! b s Z t c W Weak Scale QCD Scale quark top charm up 174 GeV 1.25 GeV 1.5 MeV Q=+2/3 e 1 MeV e d u bottom 4.20 GeV strange 95 MeV down 3 MeV Q=-1/3 e lepton 1 kev 1 ev # Atomic Scale Charge " 1.78 GeV! 105 MeV e MeV Q=-e neutrino # "! e < 2 ev Q=0 Z 0 W!

22 LEP paradox (Naturalness) t H H W, Z, γ H H H H H top loop 3 8π λ 2 2 t Λ2 (2 TeV) 2 9 SU(2) gauge boson loops 64π g 2 Λ 2 (700 GeV) 2 2 Higgs loop 1 16π 2 λ 2 Λ 2 (500 GeV) 2. Fine-tuning required for a cutoff of 10 TeV For less than 10% fin tuning, 2 m h ~ (200 GeV) 2 tree top loops SM loop maximum mass of new particles top 2 TeV weak bosons 5 TeV Higgs 10 TeV gauge higgs M. Schmaltz, hep-ph/

Waht is behind the Higgs sector? New particles? New interactions? (dynamical mechanisms) New symmetries? Flavor problemsyukawa coupling, mixing & CP violations [Kobayashi-Maskawa (^^)], #generations,.

23 Waht is behind the Higgs sector? New particles? New interactions? (dynamical mechanisms) New symmetries? Flavor problemsyukawa coupling, mixing & CP violations [Kobayashi-Maskawa (^^)], #generations,.. GUT Cosmological problems Baryon number asymmetry, dark matter, dark energy Dynamics of chiral gauge theories may play important roles!! chiral fermions : fund. unit of matter realization of gauge & chiral symmetries Need some nonperturbative tools!

24 SU(3)xSU(2)xU(1), SO(10), etc.??? still hard like the question: What is the sound of one hand clapping? a koan from Zen Buddhism

25 How to construct Glashow-Weinberg-Salam model on the lattice?

26 the last part 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

27 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)

28 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

29 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

30 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

31 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 (99) direct proof of gauge anomaly cancellation in separate treatment of the Wilson line some non-perturbative applications? L 4 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., JHEP 0805:095 (2008), 0802:063 (2008)

32 possible applications of lattice EW theory a perturbative computation of the EW contributions to muon g-2 (one-loop check, beyond one-loop) a computation of the effect of quarks & leptons to the sphaleron rate at finite temp. (at one-loop, top quark?) a lattice construction of models of dynamical EW symmetry breaking SU(2) walking technicolor model can be put on the lattice (!) Appelquist et al., Dietrich, Sannino, Tuominen (05) [minimal] cf. recent activitiy to study QCD-like theories in/close to the conformal window T. Appelquist, G.T. Flemming and E.T. Neil (Yale Univ.) Phys.Rev.Lett. 100:171607,2008 (arxiv: )... a study of Electroweak phase transition(1st order possible?) some other non-perturbative applications?

33 related results N=(2,2) 2-dim. SuperQCD with exact chiral and Q symmetries Sugino-Kikukawa(2008) arxiv: reflection positivity in Lattice QCD with overlap Dirac operator Usui-Kikukawa(2009) in preparation... Strongly coupled fourth family and a first-order Electroweak phase transiton appear very soon Kohda-Yasuda-Kikukawa(2009) work in continuum (^^; lattice GWS model applicable!

34 Happy 60th Birthdays! Ishikawa-san & Kawamoto-san

q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices

q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices Grand Unification M.Dine, Supersymmetry And String Theory: Beyond the Standard Model 6 2009 2 24 by Standard Model Coupling constant θ-parameter 8 Charge quantization. hypercharge charge Gauge group. simple