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
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
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
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
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
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
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
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
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
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
The Phase Diagram The parameters are β, γ and 5. 30 20 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 5 1.2 0 1 0.8 0.6 0.4 0.5 1 1.5 2 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,..., 5 1 2 (along the 5th dimension) γ 1 γ<1 ; compactification ; localization 7/15
The phase diagram = 0.7, 5 = 12, c = 1.4796862, k1 = 0.25084, k2 = 1.415 1 5 =12, Phase Diagram 1.5 1 0.8 0.6 confined deconfined log(a 4 m H ) 2 2.5 3 3.5 4 4.5 24 22 20 18 16 14 12 10 log(1 beta c./beta) 0.4 layered = 0.6, 5 = 12, c = 1.3702862, k1 = 0.47387, k2 = 1.9263 2.4 0.5 1 1.5 2 2.6 2.8 a 4 m H = b(1 β c /β) ν log(a 4 m H ) 3 3.2 3.4 critical exponential ν = k 1 1/2 second order phase transition 3.6 3.8 4 4.2 13 12.5 12 11.5 11 10.5 10 9.5 9 log(1 beta c./beta) 8/15
Mean-Field Expansion Mean-field expansion of the expectation value of the observables O[V ]=O[V ]+ δo δv v + 1 2 V δ 2 O δv 2 V v 2 +... 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
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
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 =1.38 11/15
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 ) 0.2 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)) 0.15 0.1 0.05 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 +1 0 0 25 50 75 100 125 150 r/a 4 1.6 1.4 orbifold, L = 400, 5 =24, F1 = 0.61 ρ HZ =1.38 =m H /m Z 1.2 1 0.8 0.6 0.4 0.2 on the boundary Z fit 12.4641 ( )+7.4173 on the boundary Z fit 1.0829 ( )+0.75583 =1.38 0 0.475 0.48 0.485 0.49 0.495 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 =0.61 5 means a 4 0 on the LCP ( 5 = πr a 5 = πr a 4 γ) 12/15
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 0.4 0.35 0.3 LCP; Change the Lattice spacing by keeping two dimensionless physical quantities fixed HZ =m H /m Z 0.25 0.2 0.15 0.1 0.05 fit 2.7124 (a 4 m H )+0.1272 HZ ρ HZ = m H m Z = 126/91.19 = 1.38 F 1 = m H R =0.61 5 means a 4 0 on the LCP ( 5 = πr a 5 = πr a 4 γ) 0 0 0.02 0.04 0.06 0.08 0.1 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 =0.1272 = 126/0.1272 = 989 [GeV] Where χ 2 per degree of freedom is 0.025/3. 13/15
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
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 =1.68 0.4 m H R 0.8 0.6 0.4 ln λ(1) mh (t) λ (1) mh (t 1) 0.3 0.2 0.1 0.2 ground state excited state ground, 1 loop PT 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 t 0 1.64 1.66 1.68 1.7 1.72 1.74 1 0.8 Z boson, =1, 64 x 32 3 x 5 Z-boson mass, T = 64, L = 32, 5 = 4, γ = 1, β =1.68 0.8 0.6 Z boson mass, T=64, L=32, 5=5, beta 4 =1.68, beta 5 =1.68 m Z R 0.6 0.4 0.2 ground state excited state 0 1.64 1.66 1.68 1.7 1.72 1.74 ln λ(1) mz (t) λ (1) mz (t 1) 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 t 15/15