Nobuhito Maru (Kobe)

Transcription

1 Calculable One-Loop Contributions to S and T Parameters in the Gauge-Higgs Unification Nobuhito Maru Kobe with C.S.Lim Kobe PRD [hep-ph/07007] 8/6/007 String theory & Univ.

2 Introduction Gauge-Higgs unification is is very interesting scenario because Higgs mass is is calculable and predictive regardless of the nonrenormalizable theory There are various explicit calculations of of Higgs mass: 5D QED on Hatanaka-Inami-Lim 5D Non-Abelian gauge theory on Gersdorff-Irges-Quiros 6D Non-Abelian gauge theory on Antoniadis-Benakli-Quiros 6D Scalar QED on Lim-Maru-Hasegawa 5D QED on Maru-Yamashita

3 The reason for finiteness is the following In the gauge-higgs unification, Higgs is identified with extra components of the higher dimensional gauge field M 5 A = A, A 5 ma Higgs The local mass term immediately is forbidden by the higher dimensional local gauge invariance No local counter term Finite mass is A A α x y +, m 4 A 5 g 6π R

4 The reason for finiteness is the following In the gauge-higgs unification, Higgs is identified with extra components of the higher dimensional gauge field M 5 A = A, A 5 ma Higgs The local mass term immediately is forbidden by the higher dimensional local gauge invariance No local counter term Finite mass is A A α x y +, m 4 A 5 g 6π R

5 Question: Is there any other finite physical observable? If there is, natural to guess in the gauge-higgs sector of the SM One of the candidates: S, T & U parameters ν ν S: H W H B, T : H D H H D H Naively, we can expect them to be finite similar to the Higgs mass These parameters are important physical quantities to test the SM and constrain the physics beyond the SM well-known that QCD-like technicolor is excluded

6 In In this work, we we investigate the structure of of divergence for -loop contributions to to S & T parameters in in the gauge-higgs unification Results: In In 5D case, S & T are both finite In In more than 5D case, both divergent Natural from the power counting argument However, the gauge-higgs unification predicts S 4 cosθw T becomes finite in in 6D cases because S & T are related by by the higher dim. gauge inv.

7 PLAN Introduction Operator Analysis Calculation of T-parameter Calculation of S-parameter Summary

8 Operator Analysis

9 S & T parameters are calculated as the coefficients of dimension six operators ν HW H B S, HDH HD H ν in in 4D sense for for T Higher dimensional gauge symmetry can forbid these local operators similar to Higgs mass??? NO

10 In gauge-higgs unification, S & T parameters are unified due to higher dim gauge inv 8 Tr D F D F m W + m WW L MN 4 4 L MN + + ν ν mpgνw B m pgν pp ν W B 6π 4π S = cosθ 8 m, T = 8m g g W 6π 4π ΔM S = Π, T = sin, m M g g M + θ Y W W tanθw sin θw W S 4 cosθw T becomes finite even in in more than 5D

11 Consider a minimal SU gauge-higgs model compactified on D with a triplet fermion M S Z Although this model is is NOT realistic, θ = 4 t b W θ W sin W exp : sin 0. m = 0, m = M SU x U U by <A5> assumed Enough to investigate the divergence structure for -loop contributions to S & T parameters

12 Lagrangian L MN M = Tr FMN F + iψγ DM Ψ MN M N N M D+ M N M M 5 M M M ψ, ψ, ψ T M Γ = [, ], 0,,,,5 F = F F ig A A M N = a a a : : Gell-Mann matrices D = ig A A = A λ λ Ψ= 5 γ, iγ A Boundary conditions: +, + + ψ R, ψ ψ ++, ++,,,, ++, ψl = ++, ++,,, AD+ =,, ++,, Ψ= ψl ++, + R,,, ++, ++, ++,, ψ, + +, + L R SU SU x U

13 Lagrangian L MN M = Tr FMN F + iψγ DM Ψ MN M N N M D+ M N M M 5 M M M ψ, ψ, ψ T M Γ = [, ], 0,,,,5 F = F F ig A A M N = a a a : : Gell-Mann matrices D = ig A A = A λ λ Ψ= 5 γ, iγ A Boundary conditions: Higgs is is identified with 0 mode of A5 A5 KK KK modes modes of of A5 A5are absorbed into into KK KK gauge gauge bosons +, + + ψ R, ψ ψ ++, ++,,,, ++, ψl = ++, ++,,, AD+ =,, ++,, Ψ= ψl ++, + R,,, ++, ++, ++,, ψ, + +, + L R

14 Lagrangian L MN M = Tr FMN F + iψγ DM Ψ MN M N N M D+ M N M M 5 M M M ψ, ψ, ψ T M Γ = [, ], 0,,,,5 F = F F ig A A M N = a a a : : Gell-Mann matrices D = ig A A = A λ λ Ψ= 5 γ, iγ A Boundary conditions: +, + + ψ R, ψ ψ ++, ++,,,, ++, ψl = ++, ++,,, AD+ =,, ++,, Ψ= ψl ++, + R,,, ++, ++, ++,, ψ, + +, + L R Chiral fermions are easily obtained

15 4D effective Lagrangian in terms of mass eigenbasis n iγ m 0 0 ψ n = ψ, ψ, ψ 0 iγ mn + m 0 ψ n= 0 0 iγ m n m ψ L 4D fermion B + + W + W W n ψ g W B W B + ψ, ψ, ψ W + γ ψ n W B W B ψ W + g + g + itlγ tl + b iγ m b + t γ LbW + bγ LtW + t γ Lt + bγ Lb W n ψ g 0 0 ψ t Lt b Lb b + γ + γ γ Rb B 6 ψ = 0 ψ 0 ψ ψ n g 5 L γ 5, R + γ 5, mn =, g =, m = gv = M W R π R

16 Calculation of T-parameter T

17 T-parameter is calculated from the mass squared difference between neutral and charged W-bosons Δ M δ M M δ ± W W ψ ψ ψ ψ δ = M W ψ ψ ψ ψ 0 Same as as the quantum correction to to the photon mass in in QED δ ± = + M W ψ ψ ψ ψ

18 Mode sum before Momentum integral T = T + T T div div sc = π 5 D D D D D Γ Γ D D D Γ Γ D 4π D t 4 D D D D π 4 D d ρ D sinh ρ D sinh ρ T = L dt sc 0 D α D π ρ cosh ρ ρ cosh ρ cosα D + ρ + 4t α ρ α L π R, ρ Lk, α Lm sinh ρ + 4t t L α α ρ + t t α t α cosh 4 cos α 4+ D D ρ sinh ρ + 4t t α + ρ + 4t t α cosh ρ + 4t t α cos tα

19 Finite value evaluation in 5D: Momentum integral before the mode sum 4 m π T D mr m m 5π m m 5 5 = n 0 n n= n Pole term vanishes because of 4 O m H D H H D H 0 dt t = 0 n O m Decoupling nature of KK modes T 0 for m 0 Custodial limit

20 Calculation of S-parameterS

21 W S-parameter is calculated from kinetic mixing of & ψ ψ ψ ψ ψ B iπ + = Y p g ν ψ ψ ψ ψ ψ S = S + S S div div sc = 5 D D + Γ Γ D 5 9π D D 4π Γ 5 D Γ D+ π R m D π D 4 D d ρ sinhρ sinhρ S = D L + sc D π ρ cosh ρ cosε 4ρ cosh ρ cosα 9 D 4 + dt t t + t t α ρ ρ D sinh ρ + 4t t α + + ρ 4t t α cosh ρ 4t t α cos t α

22 Finite value of S in 5D is calculated by expanding in terms of m/mn π 46 m π S 5D mr = π n= 5 mn 80 a a m H W H B ν ν O τ + 8 dtt t = 0 log divergence 0 O m n Decoupling nature of KK mode

23 In more than 5D T div S div = π = 4π D D D D D D 5 D D Γ Γ Γ Γ D 5 D D+ Γ Γ D 5 9π D D 4π D 5 Γ Γ D + π R m D π 5 6 D: S = T = T 8 R m 5 div div div Divergence is indeed canceled in S - T [S 4 cosθw T S - T SU model] D

24 Finite value is calculated by doing the momentum integral before the mode sum and expanding in terms of m/mn S T 6D n n m = + 5π m n= n n 6D m = + 5π m m m 4 0 n= n n m m S 6D T 6D = m R ζ 0π 4 n 0 n 0 st term in in S and T indicates the log divergence and is is canceled in in S - T

25 Comments : : In In our model, only one extra spatial dimension is is compactified as as Our arguments of of finiteness for 6D case is is meaningless?? because it it is isnot realistic However, our argument with respect to to UV divergence is is not affected by by the shape of of the compactified space because it it is is the IR IR property not UV one Finiteness of of S 4 cosθw cosθwt holds true even in in 6D theory compactified on on T^/Z, for example although the finite value itself might be be changed D M S Z

26 : For higher than 6 dimensions, the coefficients of the gauge invariant operators with mass dimensions > 6 diverge H H n ν S: H W HB n H H H H T : H D D H n Divergences from these operators No prediction in in more than 6D ν

27 Summary We have investigated the divergence structure of of one-loop contributions to to S & T parameters in in the gauge-higgs unification scenario S & T are finite in in 5D, but divergent in in more than 5D which are natural results from the power counting In 6D case, S 4 cosθw T cosθw becomes finite because S & T are related unified prediction!! Interesting to to study these parameters in in more realistic gauge-higgs unification models and obtain the predictions g- in in the gauge-higgs unification becomes finite in in any spacetime dimensions Adachi s Talk

28 Substituting KK mode expansions ++, 0 n A,5 x, y = A,5 x + A,5 cos y, π R n= R, n A,5 x, y = A,5 sin y, π R n= R ++, 0 n Ψ L, L,R x, y = ψ L, L,R x + ψ L, L,Rcos y, π R n= R, Ψ R, R,L xy, = ψ R, R,Lsin π R n= n R and integrating out 5 th coordinate y, i 5 making a chiral rotation 4 to remove iγ5, ψ y,, e πγ ψ,,

29 we obtain 4D effective Lagrangian n n n n i ψ, ψ, ψ γ ψ n= ψ L 4D fermion = ψ B + W + W 0 ψ g B + ψ, ψ, ψ W W + 0 γ ψ ψ n n n n 0 0 B mn 0 0 ψ ψ, ψ, ψ 0 mn m ψ 0 m m n ψ g + g + itlγ tl + b iγ m b + t γ LbW + bγ LtW + t γ Lt + bγ Lb W g t Lt b Lb b + γ + γ γ Rb B 6 n g L 5 R 5 mn g m R π R 5 γ, + γ, =, =, = = gv M W

30 Mixing occurs between SU doublet component & singlet component Each of mass eigenvalues has a periodicity with respect to m mn ± m+ = mn± ± m R Characteristic feature of gauge-higgs unification m + m n c.f. for UED

31 4: 4: Perturbativity Comparing n-loop graph to to n+-loop graph in in D+ dimensions, we we find the ratio Λ:cutoff scale D D n + -loop g g D π R D+ Λ Λ = = D+ D+ n-loop D+ D+ 4π Γ 4 π Γ For the perturbation to to make sense, the above ratio must be be less than one For example, D=4 case, g4 π R Λ 0 5 < R Λ< π Λ< O 4π Γ 5 R Cutoff scale Λ cannot be be largely separated from the compactification scale

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33 T Combining the neutral and the charged W boson contributions, we obtain KK mode contributions to T-parameter as D D n = dt n 0 D 0 D 4 Γ π M n W = mn + m + t mnm m + t m m mm n t mm n m + + m m + D mn + t mnm+ m This quantity vanishes in in the limit m 0, 0, which corresponds to the custodial symmetry limit in in our model

34 : : For higher than 6D, the coefficients of of the gauge inv. operators with mass dimensions > 6 diverge n ν S: H Wν HB H H n H H D H H T : H D H n Divergences from these operators No prediction in in more than 6D : : Brane localized terms spoil our results?? No problem! n ν n ν A5 A5 ν A5 S: H H H W HB W B T : n n H H H DH A HDH 5 A5DA5 A5D forbidden by by the shift symmetry A5 A5 A5 A5 + 5αx,y which is is still ν