Stability of Electroweak Vacuum in the Standard Model and Beyond

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1 Stability of Electroweak Vacuum in the Standard Model and Beyond Takeo Moroi (Tokyo) Chigusa, TM, Shoji, PRL 119 ( 17) [ ] Chigusa, TM, Shoji, work in progress PACIFIC Workshop, Hokkaido, Japan,

2 1. Introduction

3 What we learn from the Higgs mass m h 125 GeV V = λ( H 2 v 2 ) 2 with λ(m h ) 0.13 λ becomes negative at a very high scale Pontential EW scale EW vacuum is not stable in the standard model (SM) λ is minimized at µ GeV H

4 Is the decay rate small enough so that t now 13.6 Gyr? (Probably) yes [Isidori, Ridolfi & Strumia; Degrassi et al.; Alekhin, Djouadi & Moch; Espinosa et al.; Plascencia & Tamarit; Lalak, Lewicki & Olszewski; Espinosa, Garny, Konstandin & Riotto; ] How precisely can we estimate the decay rate? Gauge-invariance of the result was unclear Effects of zero modes were not properly taken into account There has been progresses in the calculation of the decay rate of false vacuum [Endo, TM, Nojiri & Shoji; Chigusa, TM & Shoji; see also Andreassen, Frost & Schwartz]

5 Today, I discuss A calculation of the decay rate of EW vacuum Effects of extra matters Outline 1. Introduction 2. Bounce in the SM 3. Effects of Higgs Mode 4. Total Decay Rate 5. Case with Extra Matters 6. Summary

6 2. Bounce in the SM

7 The decay rate is related to 4D Euclidean partition function [Coleman; Callan & Coleman] Z = FV e HT FV exp(iγv T ) The path integral is dominated by the bounce Bounce: a saddle-point solution of classical EoM one-bounce t = bounce Z = t = Φ = exp [ ] γ 1 V T Im DΨ 1-bounce e S E DΨ 0-bounce e S E Ae B with B = S E (Bounce)

8 Main concern of this talk: calculation of the prefactor A A contains one-loop (and higher order) effects We expand the action around the bounce ϕ S E [ ϕ + Ψ] = S E [ ϕ] d 4 xψmψ + O(Ψ 3 ) S E [v + Ψ] = S E [v] d 4 xψ MΨ + O(Ψ 3 ) Prefactor A (for bosonic contribution) A 1 V T DetM Det M 1/2 i 1 ω i Sometimes M has zero eigenvalue A careful treatment is necessary with ω i = eigenvalue of M

9 Higgs potential in the SM: V = m 2 H H + λ(h H) 2 We consider very large Higgs amplitude for which λ < 0 We will neglect quadratic term because λ < 0 occurs at a scale much higher than the EW scale We use the following potential: V = λ (H H) 2 The bounce solution (Fubini-Lipatov instanton) H bounce = 1 2 e iσa θ a Explicit form of the bounce: ϕ(r) = 8 0 ϕ with r 2 ϕ + 3 r ϕ r + 3 λ ϕ 2 = 0 1 λ R R 2 r 2 with R = (free parameter)

10 Bounce action for the SM B = 8π2 3 λ Possible deformations of the bounce Dilatation: parameterized by R SU(2) transformation: parameterized by θ a Effects of zero modes in association with these transformations were not properly taken into account before Translation [Callan & Coleman] Expansion around the bounce: H = 1 2 e iσa θ a φ1 + iφ 2 ϕ + h iφ 3, Wµ a = wµ, a B µ = b µ

11 3. Effects of the Higgs Mode

12 We need to calculate the functional determinant of M (h) L 1 2 h ( 2 3 λ ϕ 2 ) h = 1 2 h M(h) h Expansion of h w.r.t. 4D spherical harmonics Y J,mA,m B h(x) = J,m A,m B,n J = 0, 1/2, 1, 3/2, ρ n,j : radial mode function α n,j,ma,m B ρ n,j (r) Y J,mA,m B (ˆr) α n,j,ma,m B : expansion coefficient (integration variable) Fluctuation operator for angular-momentum eigenstates: ( J + 3 λ ϕ 2) r r 4J(J + 1) r M (h) J r λ ϕ 2

13 Higgs-mode contribution to the prefactor A A (h) = DetM(h) Det M (h) 1/2 = J DetM(h) J Det M (h) J (2J+1) 2 /2 The ratio of the functional determinants can be evaluated with so-called Gelfand-Yaglom theorem Zero modes exist for M (h) Dilatational zero mode (for J = 0) ρ D (r) ϕ M (h) 0 ρ D (r) = 0 and ρ D (r ) = 0 R Translational zero modes (for J = 1/2) [Callan & Coleman]

14 Path integral over dilatational zero mode = integral over R H ϕ + h = ϕ + α D N ϕ D R + ϕ R R+αD + N D Dh (dilatation) DetM(h) 0 Det M (h) 0 1/2 dα D dr N D dr N D Det M (h) 0 Det M (h) 0 1/2 Det : zero eigenvalue is omitted from the Det Higgs-mode contribution: [Chigusa, TM & Shoji; Andreassen, Frost & Schwartz] A (h) d ln R 16π λ 1/2 J 1/2 DetM(h) J Det M (h) J (2J+1) 2 /2

15 Comment on gauge and NG contribution Gauge fixing function in old calculations F = µ B µ 2ξg 1 (ReH 0 )(ImH 0 ), Gauge and NG fields couple in the EoM General form of the bounce is more complicated We adopt the following gauge fixing function [Kusenko, Lee & Weinberg] F = µ B µ, F a = µ W a µ H bounce = 1 2 e iσa θ a Path integral over gauge zero modes = integral over θ a DetM(W,Z,NG) Det M (W,Z,NG) 1/2 V SU(2) 16π λ 3/2 0 ϕ 1/2 DetM(W,Z,NG) J (W,Z,NG) J 1/2 Det M J

16 4. Total Decay Rate

17 Decay rate: γ = d ln R [ I (h) I (W,Z,NG) I (t) e S C.T. e B] µ 1/R We derived complete and gauge-invariant expressions of I (X) I (h) : Higgs contribution I (W,Z,NG) : gauge and NG contribution I (t) : top contribution We take the renormalization scale as µ 1/R The effects of µ-dependent terms from higher loops, i.e., ln p (µr), are expected to be minimized This is important for the convergence of the integral

18 We use: m h = ± 0.24 GeV m t = ± 0.6 GeV α s (m Z ) = ± or 3-loop RGEs (with relevant threshold corrections) Decay rate of the EW vacuum (taking µ = 1/R) log 10 [ γ (Gyr 1 Gpc 3 ) ] For the present universe: Cosmic age: t Gyr Horizon scale: H Gpc

19 log 10 [γ (Gyr 1 Gpc 3 )] on m h vs. m t plane (with µ = 1/R) [Chigusa, TM & Shoji, in preparation] Instability: γ > H 4 now / Metastability: γ < H 4 now Absolute stability: λ > 0 - /

20 5. Case with Extra Matters

21 Let us consider vector-like fermions coupled to H L = L SM + y ψ Hψ L ψ R + y ψh ψl ψr + M ψ ψl ψ L + M ψ ψr ψ R + RGE for λ H H* λ(μ) H* H = SM ψ [ dλ dλ d ln µ = d ln µ ] SM 1 4π 2 ψ N (ψ) C y 4 ψ + With extra fermions, λ may become smaller (at high scale) Enhancement of the decay rate C.f., γ = Ae B with B = 8π2 3 λ

22 Case 1: Down-quark-like colored fermions ψ L (3, 2, 1/6) and ψ R ( 3, 1, 1/3) (max) ϕ C = M Pl PRELIMINARY Yukawa coupling larger than is dangerous

23 Case 2: Charged-lepton-like fermions ψ L (1, 2, 1/2) and ψ R (1, 1, 1) (max) ϕ C = M Pl PRELIMINARY

24 Case 3: Right-handed neutrino L = L SM + y ν Hl L ν c R M νν c Rν c R + (max) ϕ C = M Pl PRELIMINARY

25 6. Summary

26 I have discussed the decay rate of the EW vacuum Path integrals over the dilatational and gauge zero modes are properly performed Numerical result log 10 [ γ (Gyr 1 Gpc 3 ) ] The decay rate is extremely small: γ H 4 0 We will fall into another vacuum if we wait Gyr (assuming that the dark energy is cosmological constant) Extra fermions may change the above conclusion y > is dangerous

27 Back Up

28 Functional determinant for operators defined in 0 r r DetM ω n with n Mρ n = ω n ρ n with M = J + δw (r) ρ n (0) < ρ n (r ) = 0 We introduce a function f which obeys: Mf(r; ω) = ωf(r; ω) f(r = r ; ω) ω=ωn = 0 f (r; ω) Det(M ω) ω=ωn = 0 ω = ωn ω = ωn O r 8 r

29 Gelfand-Yaglom theorem [Gelfand & Yaglom; Coleman; Dashen, Hasslacher & Neveu; Kirsten & McKane; ] Det(M ω) Det( M ω) = f(r = r ; ω) f(r = r ; ω) with Mf(r; ω) = ωf(r; ω) M f(r; ω) = ω f(r; ω) f(r = 0) = f(r = 0) < Notice: LHS and RHS have the same analytic behavior LHS and RHS have same zeros and infinities LHS and RHS becomes equal to 1 when ω We take r at the end of calculation The results converge (in the case of our interest)