Progress in Gauge-Higgs Unification on the Lattice

Transcription

1 Progress in Gauge-Higgs Unification on the Lattice Kyoko Yoneyama (Wuppertal University) in collaboration with Francesco Knechtli(Wuppertal University) ikos Irges(ational Technical University of Athens) Peter Dziennik(Wuppertal University) STROGnet2013, Graz, September 16th

2 Introduction Gauge Higgs Unification Model 5-dimensional gauge theory The Higgs fields is identified with some of the extra dimensional components of the gauge fields. [Gersdorff, Irges, Quiros 2002] [Antoniadis, Benakli and Quiros 2001] [Cheng, Matchev and Schmaltz 20002][Irges and Knechtli 2006, 2007] Higgs mass and potential are finite.(hierarchy problem is solved) Higgs potential can break gauge symmetry.(origin of Spontaneous Symmetry Breaking SSB) Lattice Gauge theory You can study non-perturbative region by Monte Carlo and Mean-Field. You can take gauge-invariant ultra-violet cut off (Λ = 1/a). [Hosotani 1983] Is there SSB without fermion in non-perturbative region? Torus boundary condition ; o, there is no SSB Orbifold boundary conditions ; Yes! there is SSB [Irges and Knechtli 2009] [Irges, Knechtli and Yoneyama 2012] I will talk about dimensional reduction and continuum limit 1/15

3 The Orbifold Boundary Conditions Reflection R : z =(x µ,x 5 ) z =(x µ, x 5 ) A M (z) α M A M ( z), α µ =1,α 5 = 1 Group conjugation C : A M (z) ga M (z)g 1, g 2 SU() x 5 =0 x 5 = πr x 5 = πr T = RC : A M (z) α M ga M ( z)g 1 Orbifold projection A M (z) =α M ga M ( z)g 1 = T A M (z) On the boundary (x 5 =0,πR), z = z A M (z) =α M ga M (z)g 1 only even components A a µ and Aâ5 are = 0 SU(2) case, we choose g = iσ 3. Then, only A 1 5, A 2 5 and A 3 µ are = 0 on the boundaries x 5 =0 generator gt a g 1 = T a gt âg 1 = T â x 5 = πr The symmetry break explicitly by boundary conditions ; SU(2) U(1) 2/15

4 Mean-Field Expansion [Drouffe and Zuber, 1983] The partition function of SU() gauge theory on lattice Z = DU e S G[U] SU() by Z = DV DHe S eff [V,H] gauge links U are replaced complex matrices V and Lagrange multipliers H S eff = S G [V ]+u(h) + (1/ )Re tr{vh} e u(h) (1/ )Re tr{uh} = DU e Saddle point solution (background) V (n, M) = S eff H(n, M) H(n,M), H(n, M) = S eff V (n, M) V (n,m) V v 0 1 H h 0 1 S eff [ V, H] = minimal Expansion in Gaussian fluctuations H = H + h V = V + v covariant gauge fixing on [Rühl, 1982] v 3/15

5 Five dimensional SU(2) lattice gauge theory for orbifold T L 3 5 lattice, gauge theory on orbifold boundary condition SU(2) Wilson plaquette action S W [U1,U2] = S W 1 [U1] + S W 2 [U1,U2] Where U 1 U(1), U 2 SU(2) S W 1 [U 1 ]= 1 4 S W 2 [U 2,U 1 ]= 1 2 β γ β γ n 5 =1 n 5 =0, ν,ρ tr{1 U(n, ν, ρ)} ν,ρ tr{1 U(n, ν, ρ)} + γ β 2 β = 2a 4 g 2 5 boundary n 5 =0, γ = a 4 a 5 tr{1 U(n, ν, 5)} ν (tree level) bulk gauge trans formation on a boundary link U(n, M) Ω (U(1)) (n)u(n, M)Ω (U(1)) (n + ˆM) ˆν on the bulk link U(n, M) Ω (SU(2)) (n)u(n, M)Ω (SU(2)) (n + ˆM) on a link whose one end is in the bulk and the other touches the boundary U(n, M) Ω (U(1)) (n)u(n, M)Ω (SU(2)) (n + ˆM) ˆµ n 5 =0 n 5 = 5 ˆ5 4/15

6 Five dimensional SU(2) lattice gauge theory for orbifold T L 3 5 lattice, gauge theory on orbifold boundary condition SU(2) Wilson plaquette action S W [U1,U2] = S W 1 [U1] + S W 2 [U1,U2] Where U 1 U(1), U 2 SU(2) S W 1 [U 1 ]= 1 4 S W 2 [U 2,U 1 ]= 1 2 β γ β γ n 5 =1 n 5 =0, ν,ρ tr{1 U(n, ν, ρ)} ν,ρ tr{1 U(n, ν, ρ)} + γ β 2 β = 2a 4 g 2 5 boundary n 5 =0, γ = a 4 a 5 tr{1 U(n, ν, 5)} ν (tree level) bulk gauge trans formation on a boundary link U(n, M) Ω (U(1)) (n)u(n, M)Ω (U(1)) (n + ˆM) on the bulk link U(n, M) Ω (SU(2)) (n)u(n, M)Ω (SU(2)) (n + ˆM) on a link whose one end is in the bulk and the other touches the boundary U(n, M) Ω (U(1)) (n)u(n, M)Ω (SU(2)) (n + ˆM) ˆν ˆµ n 5 =0 n 5 = 5 ˆ5 4/15

7 Five dimensional SU(2) lattice gauge theory for orbifold T L 3 5 lattice, gauge theory on orbifold boundary condition SU(2) Wilson plaquette action S W [U1,U2] = S W 1 [U1] + S W 2 [U1,U2] Where U 1 U(1), U 2 SU(2) S W 1 [U 1 ]= 1 4 S W 2 [U 2,U 1 ]= 1 2 β γ β γ n 5 =1 n 5 =0, ν,ρ tr{1 U(n, ν, ρ)} ν,ρ tr{1 U(n, ν, ρ)} + γ β 2 β = 2a 4 g 2 5 boundary n 5 =0, γ = a 4 a 5 tr{1 U(n, ν, 5)} ν (tree level) bulk gauge trans formation on a boundary link U(n, M) Ω (U(1)) (n)u(n, M)Ω (U(1)) (n + ˆM) on the bulk link U(n, M) Ω (SU(2)) (n)u(n, M)Ω (SU(2)) (n + ˆM) on a link whose one end is in the bulk and the other touches the boundary U(n, M) Ω (U(1)) (n)u(n, M)Ω (SU(2)) (n + ˆM) ˆν ˆµ n 5 =0 n 5 = 5 ˆ5 4/15

8 Five dimensional SU(2) Lattice Gauge Theory for Orbifold T L 3 5 lattice, gauge theory on orbifold boundary conditions SU(2) Wilson plaquette action S W [U 1,U 2 ]=S W 1 [U 1 ]+S W 2 [U 1,U 2 ] Where U 1 U(1), U 2 SU(2) S W 1 [U 1 ]= 1 4 S W 2 [U 2,U 1 ]= 1 2 β γ β γ n 5 =1 n 5 =0, ν,ρ tr{1 U(n, ν, ρ)} ν,ρ tr{1 U(n, ν, ρ)} + γ β 2 β = 2a 4 g 2 5 boundary n 5 =0, γ = a 4 a 5 tr{1 U(n, ν, 5)} ν (tree level) bulk gauge trans formation on a boundary link U(n, M) Ω (U(1)) (n)u(n, M)Ω (U(1)) (n + ˆM) on the bulk link U(n, M) Ω (SU(2)) (n)u(n, M)Ω (SU(2)) (n + ˆM) on a link whose one end is in the bulk and the other touches the boundary U(n, M) Ω (U(1)) (n)u(n, M)Ω (SU(2)) (n + ˆM) ˆµ ˆν n 5 =0 n 5 = 5 ˆ5 4/15

9 Five dimensional SU(2) Lattice Gauge Theory for Orbifold The action is expressed with complex matrices V and Lagrange multipliers H. S eff = 1 4 where β γ µ<ν boundary 1 β Re tr V p/ bound (n; µ, ν) bulk 2 γ n 5 =1 µ<ν γ β RetrV p/ bound (n; µ, 5) bulk 2 n 5 =0 µ + u 2 (ρ(n, µ)) + h α (n, µ)v α (n, µ) n 5 =1 µ α + u 2 (ρ(n, 5)) + h α (n, 5)v α (n, 5) n 5 =0 α + u 1 (ρ(n, µ)) + h α (n, µ)v α (n, µ) µ α n 5 =0, 5 u 1 (ρ(n, M)) = ln(i 0 (ρ)), Re tr V p bound (n; µ, ν) n 5 =0, 5 u 2 (ρ(n, M)) = 2 ρ I 1(ρ) ρ = (Re h 0 (n, M)) 2 + (Re h A (n, M)) 2 ˆν ˆµ n 5 =0 n 5 = 5 ˆ5 The parametrization of the field in the bulk 3 V (m, M) = v 0 (n, M)+i H(m, M) = h 0 (n, M) i on the boundaries A=1 3 A=1 V (n, M) = v 0 (n, M)+iv 3 (n, M)σ 3 v A (n, M)σ A H(n, M) = h 0 (n, M) ih 3 (n, M)σ 3 h A (n, M)σ A

10 The Mean-Field Background Then, we can get mean-field background from these minimization equations. S eff =0, H(n, M) H(n,M) S eff =0 V (n, M) V (n,m) The parametrization of the mean-field background (saddle point solution) for 4-dimensional links (n 5 =0, 1,..., 5 ) H(n, µ) = h 0 (n 5 ), V (n, µ) = v0 (n 5 ) ˆµ for extra-dimensional links H(n, 5) = h (n 5 =0, 1,..., 5 1) 0 (n 5 +1/2), V (n, 5) = v0 (n 5 +1/2) ˆν The mean-field back grounds depend on the position of 5th dimension. n 5 =0 v(1/2) v(3/2) v(5/2) v(0) v(1) v(2) ˆ5 6/15

11 The Phase Diagram The parameters are β, γ and confined deconfined β = 2a 4 g 2 5, γ = a 4 a 5 There are three phases. (tree level) 5 10 layered deconfined phase: v 0 (n 5 ) = 0 for all n confined phase: v 0 (n 5 ) = 0 for all n 5 layered phase: v 0 (n 5 ) = 0 for n 5 =0, 1,... 5 (along 4d hyperplanes) v 0 (n 5 ) = 0 for n 5 = 1 2, 3 2,..., (along the 5th dimension) γ 1 γ<1 ; compactification ; localization 7/15

12 The phase diagram = 0.7, 5 = 12, c = , k1 = , k2 = =12, Phase Diagram confined deconfined log(a 4 m H ) log(1 beta c./beta) 0.4 layered = 0.6, 5 = 12, c = , k1 = , k2 = a 4 m H = b(1 β c /β) ν log(a 4 m H ) critical exponential ν = k 1 1/2 second order phase transition log(1 beta c./beta) 8/15

13 Mean-Field Expansion Mean-field expansion of the expectation value of the observables O[V ]=O[V ]+ δo δv v V δ 2 O δv 2 V v O = 1 Z Dv = O[V ]+ 1 2 Z V = O[V ]+ 1 2 tr δ 2 O δv 2 Dh O[V ]+ 1 2 δ 2 O δv 2 V δ 2 O δv 2 v 2 e S eff [ V, H]+S (2) [v,h] V Dv Dh v 2 e S eff [ V, H]+S (2) [v,h] K 1 where the lattice propagator K = K (vh) K (hh) 1 K (vh) + K (vv) 1 δ 2 S eff δv 2 δ 2 S eff δv δh δ 2 S eff δh 2 V, H V, H V, H v 2 = v i K (vv) ij v j = v T K (vv) v vh = v i K (vh) ij h j = v T K (vh) h h 2 = h i K (hh) ij h j = h T K (hh) h 1st order of the expectation value of the observable O = O[V ]+ 1 2 tr δ 2 O δv 2 V K 1 9/15

14 Observables the expectation value of the observable g Higgs mass polyakov loops along 5th dimension g 1 O(t 0 + t)o(t 0 )= 1 L 6 T t 0 m m C(t) =< O(t 0 + t)o(t 0 ) > < O(t 0 + t) >< O(t 0 ) > tr P (0) (t 0, m ) tr P (0) (t 0 + t, m ) g g = iσ 3 Static potential wilson loop g 1 = C (0) (t)+c (1) (t)+... C (1) (t) = 1 δ (1,1) 2 tr (O(t 0 + t)o(t 0 )) δ 2 K 1 v C (1) (t) = λ m H = lim t ln c λ e E λt C(1) (t) C (1) (t 1) E 0 = m H t wilson loop : < O (t.r) W > t : e V (r)t < O (t.r) W > r V (r): static potential 10/15

15 Dimensional Reduction Conditions for the dimensional reduction to 4-dimensions 1) The fit to V (r) =const. + b e m z r r is possible with m Z = 0. This ensures that there is SSB, signaled by the presence of the massive U(1) gauge boson. Otherwise the gauge boson is massless and only a Coulomb fit is possible. 2) The quantities M H = a 4 m H and M Z = a 4 m Z < 1. The observables are not dominated by the cut off. 3) F 1 = m H R<1 and ρ MZ = m H /m Z > 1. The Higgs and the Z are lighter than 1/R and the Higgs is heavier than the Z. We will target the value ρ HZ = /15

16 The Line of Constant Physics We have 3 observables: M H (β,γ, 5 ), M Z (β,γ, 5 ) and M Z (β,γ, 5 ) =24, F1 = 0.61, = 0.485, on the boundary 4d Yukawa potential (a 4 y a 4 m Z /(a 4 m Z r+1)) V 4 = α e mr + C, α > 0 r F 4 = V4 = α e mr (m + 1 r r ) y = log(r 2 F 4 ) = log(α) mr + log(mr + 1) y = m + m mr r/a orbifold, L = 400, 5 =24, F1 = 0.61 ρ HZ =1.38 =m H /m Z on the boundary Z fit ( ) on the boundary Z fit ( ) = LCP; Change the Lattice spacing by keeping two dimensionless physical quantities fixed ρ HZ = m H m Z = 126/91.19 = 1.38 F 1 = m H R = means a 4 0 on the LCP ( 5 = πr a 5 = πr a 4 γ) 12/15

17 The Line of Constant Physics We have 3 observables: M H (β,γ, 5 ), M Z (β,γ, 5 ) and M Z (β,γ, 5 ) HZ =1.38, F 1 =0.61, on the boundary LCP; Change the Lattice spacing by keeping two dimensionless physical quantities fixed HZ =m H /m Z fit (a 4 m H ) HZ ρ HZ = m H m Z = 126/91.19 = 1.38 F 1 = m H R = means a 4 0 on the LCP ( 5 = πr a 5 = πr a 4 γ) a m 4 H Prediction of the Z mass 5 (i.e. a 4 = 0) on the LCP ρ HZ m Z = m H m Z = m H ρ HZ = = 126/ = 989 [GeV] Where χ 2 per degree of freedom is 0.025/3. 13/15

18 Summary Setup We studied 5-dimensional SU(2) pure gauge theory (Gauge Higgs Unification) on Euclidean lattice whose 5th dimension is orbifolded. Result from mean-field expansion We found there is SSB for orbifold boundary condition. This result is different from perturbative studies but support the Monte Carlo simulation [Irges and Knechtli 2007]. It means SSB occur even if there are no fermions and the Higgs mass can be large enough in the non-perturbative region. Also, it is possible to construct LCP s and predict the existence of a Z state with a mass around 1 TeV by taking the continuum limit for small γ. Monte Carlo Simulation Although the mean-field expansion works better in higher dimensions, there is still uncertainty of the convergence of the mean-field expansion. Therefore, we have to confirm the mean-field result by Monte Carlo simulation. ow, we are working on Monte Carlo Simulation and already have some result. 14/15

19 Monte Carlo Simulations 1 Higgs, =1, 64 x 32 3 x 5 Higgs mass, T = 64, L = 32, 5 = 4, γ = 1, β =1.68 higgs mass, T=64, L=32, 5=5, beta 4 =1.68, beta 5 = m H R ln λ(1) mh (t) λ (1) mh (t 1) ground state excited state ground, 1 loop PT t Z boson, =1, 64 x 32 3 x 5 Z-boson mass, T = 64, L = 32, 5 = 4, γ = 1, β = Z boson mass, T=64, L=32, 5=5, beta 4 =1.68, beta 5 =1.68 m Z R ground state excited state ln λ(1) mz (t) λ (1) mz (t 1) t 15/15