Higher-order scalar interactions and SM vacuum stability

Transcription

1 Higher-order scalar interactions and SM vacuum stability Marek Lewicki Institute of Theoretical Physics, Faculty of Physics, University of Warsaw SUSY014, 4 July 014, Manchester based on: Z. Lalak, P. Olszewski and ML, JHEP05(014)119 The project International PhD Studies in Fundamental Problems of Quantum Gravity and Quantum Field Theory is realized within the MPD programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund

2 SM Effective potential Standard Model Effective potential V SM (µ) = m φ + λ 4 φ4 + i [ ( ) ] n i M 64π M4 i ln i µ C i For large field values m << φ and µ = φ the potential is very well approximated by { V SM (φ) φ 4 λ [ ( ) g ( 64π 6 ln 4 ( ) y ( ( ) 1 t y ln t 3 ) ( 3λ + ( ) g 5 ) ) ( ln ( 3λ V SM (φ) λ eff (φ) φ 4 4 ( g 1 + g ) ( ( g ln 1 + g ) 5 ) ) 3 ) ( ) λ ( ( ) λ + 3 ln 3 ) ] }

3 SM Metastability λ eff < 0 = Metastability D. Buttazzo, et al. [arxiv: ]. G. Degrassi, et al. [arxiv: ].

4 Tunneling Standard semiclassical formalism S. R. Coleman, Phys. Rev. D 15 (1977) 99. C. G. Callan, Jr. and S. R. Coleman, Phys. Rev. D 16 (1977) 176. with O(4) symmetric solution to euclidean equation of motion φ + 3 s φ = s = V (φ) φ, x + x 4. φ(s = 0) = 0 at the true vacuum φ(s = ) = φ min at the false vacuum

5 Tunneling Action of the bounce solution { S E = d ( ) φ(x) x x α + V (φ(x))} α=1 ( ) 1 = π dss 3 φ (s) + V (φ(s)), allows us to calculate decay probability dp of a volume d 3 x dp = dtd 3 x S E det [ + V 1/ (φ)] 4π det[ + V (φ 0 )] e S E. Simplifying normalisation factor replaced with width of the barrier φ 0 size of the universe is T U = yr we can calculate the lifetime of the false vacuum (p(τ) = 1) τ = 1 T U φ 4 0 T U 4 e S E.

6 Analytical solution K. M. Lee and E. J. Weinberg, Nucl. Phys. B 67 (1986) 181. Quartic potential : for λ < 0. V (φ) = λ 4 φ4 = S E = 8π 3 λ

7 Standard Model Approximating by a quartic potential: τ T U = φ 4 (λ min )TU 4 lifetime is minimal for φ that minimizes λ eff (φ). 1 e 8π 3 λ min

8 Effective potential with nonrenormalisable interactions We add new nonrenormalisable couplings (similar to V. Branchina and E. Messina, [arxiv: ].) V λ eff (φ) φ 4 + λ6 4 6! φ 6 + λ8 Mp 8! φ 8. Mp 4 That modify the potential around the Planck scale: Figure: effective potential with λ 6 = 1 and λ 8 = 1.

9 Standard Model with nonrenormalisable interactions Using simple quartic potential approximation: We minimize 4 V φ 4 = λsm eff (φ) + 4λ 6 6! φ Mp + 4 λ 8 8! φ 4 Mp 4.

10 Numerical calculations Equation we need to solve φ + 3 s φ = V (φ) φ, is an equation of motion of a particle in potential V (φ) with a time dependent friction 3 s φ. We used a simple Overshot Undershot algorithm

11 Numerical vs Analytical Figure: Decimal logatihm of lifetime of the universe in units of T U as a function of the nonrenormalisable λ 6 and λ 8 couplings, calculated numerically (left panel) and analytically (right panel).

12 RG improvement Figure: Example solution with λ 6(M p) = 1 and λ 8(M p) = 0.1. The correction to the running of the quatric Higgs coupling is of the form β λ = λ6 m. 16π M p One-loop beta functions of new couplings take the form 16π β λ6 = 10 ( m 9 7 λ8 + 18λ66λ 6λ6 M 4 g + 9 ) 0 g 1 3yt, 16π β λ8 = 7 ( 9 5 8λ λ 86λ 8λ 8 4 g + 9 ) 0 g 1 3yt,

13 Numerical vs Analytical again Figure: Decimal logatihm of lifetime of the universe in units of T U as a function of the nonrenormalisable λ 6 (M p ) and λ 8 (M p ) couplings, calculated numerically (left panel) and analytically (right panel).

14 Comparison Figure: Contours corresponding to metastability boundary (τ = T u ) obtained using four different methods.

15 SM phase diagram λ 6(M p) = 0, λ 8(M p) = 0 λ 6(M p) = 1/, λ 8(M p) = 1 λ 6(M p) = 1, λ 8(M p) = 1/

16 Conclusions Analytical approximation of vacuum lifetime is fairly accurate RG improvement stabilizes significant parts of the parameter space Standard Model vacuum lifetime can be significantly changed by high energy new physics

17 Analytical solution Analytical solutions for simple potentials K. M. Lee and E. J. Weinberg, Nucl. Phys. B 67 (1986) 181. Quartic potential: for λ < 0. V (φ) = λ 4 φ4 = S E = 8π 3 λ Quartic and linear potential : V η(φ) = { λ 4 φ4, φ η λ 4 η4 K (φ η), φ > η, = S 8π E = (1 (γ + 3 λ 1)4 ) γ = λ η3 K for λ < 0 and 1 < γ < 0

18 Standard Model with nonrenormalisable interactions Approximating by quartic and linear potential with: λ(η) = 4 V (η) φ 4 We still have to chose η: τ = 1 T U η 4 TU 4 e 8π 3 λ(η) (1 (γ+1)4), = λ SM λ6 η eff (η) + 4 6! M + 4 λ8 η 4 p 8! M. p 4 (1 (γ + 1) 4 ) = log 10 ( τ T U ) = (1 (γ + 1) 4 ) = log 10 ( τ T U ) = The difference comes from our arbitrary choice of η, the factor (1 (γ + 1) 4 ) is always negligible.

19 Standard Model with nonrenormalisable interactions Using simpler quartic potential approximation: We minimize 4 V φ 4 = λsm eff (φ) + 4λ 6 6! φ Mp + 4 λ 8 8! φ 4 Mp 4. log 10 ( τ T U ) = 189.6

20 Magnitude of the suppression scale Approximate lifetime: τ 1 = T U µ 4 (λ min )TU 4 e 8π 3 λ min. Positive λ 6 and λ 8 stabilizing the potential Figure: Scale dependence of λ eff 4 = V with λ φ 4 6 = λ 8 = 1 for different values of suppression scale M. The lifetimes corresponding to suppression scales M = 10 8, 10 1, are, respectively, log 10 ( τ T U ) =, 130, 581 while for the Standard Model log 10 ( τ T U ) = 540.

21 Magnitude of the suppression scale Positive λ 8 and negative λ 6 New Minimum Figure: Scale dependence of λ eff 4 = V φ with λ 4 6 = 1 and λ 8 = 1 for different values of suppression scale M. The lifetimes corresponding to suppression scales M = 10 8, 10 1, 10 16, are, respectively, log 10 ( τ T U ) = 45, 90, 110 while for the Standard Model log 10 ( τ T U ) = 540.