Model standardowy i stabilność próżni Marek Lewicki Instytut Fizyki teoretycznej, Wydzia l Fizyki, Uniwersytet Warszawski Sympozjum Doktoranckie Warszawa-Fizyka-Kraków, 4 Marca 2016, Kraków Na podstawie: Z. Lalak, ML and P. Olszewski, arxiv:1402.3826 Z. Lalak, ML and P. Olszewski, arxiv:1505.05505. Z. Lalak, ML and P. Olszewski, arxiv:1510.02797. ML, T. Rindler-Daller and J. D. Wells, arxiv:1601.01681.
Standard Model
Scalar Potential Classically V tree SM = m2 2 φ2 + λ 4 φ4 Quantum corrections V 1 loop SM + i = m2 2 φ2 + λ 4 φ4 [ ( ) ] n i M 2 64π 2 M4 i ln i C µ 2 i
V 1 loop SM = i [ ( ) ] n i M 2 64π 2 M4 i ln i C µ 2 i For large field values m 2 << φ 2 M i φ Expansion under control if µ = φ V SM (φ) λ eff (φ) φ 4 4
Tunneling Vacuum decay proceeds through nucleation of true vacuum bubbles within false vacuum. S. R. Coleman, Phys. Rev. D 15 (1977) 2929. C. G. Callan, Jr. and S. R. Coleman, Phys. Rev. D 16 (1977) 1762. Bubble:O(4) symmetric solution of euclidean EOM: φ + 3 s φ = V (φ) φ, s = t 2 + x 2. with φ(s = 0) = 0 at the true vacuum φ(s = ) = φ min at the false vacuum expected lifetime of the vacuum τ τ = Ae S E T U
Standard Model Anlytical solution for quartic potential (with λ = const < 0): V (φ) = λ 4 φ4 = S E = 8π2 3 λ K. M. Lee and E. J. Weinberg, Nucl. Phys. B 267 (1986) 181. Approximating SM with a constant λ: τ T U = 1 φ 4 (λ min )T 4 U e 8π2 3 λ min 10 596, we obtain a lower bound for φ that minimizes λ eff (φ).
Effective potential with nonrenormalisable interactions Nonrenormalisable couplings modify the potential around the Planck scale: V λ eff (φ) φ 4 + λ6 4 6! φ 6 + λ8 Mp 2 8! φ 8, Mp 4 Figure: Log 10 ( τ T u )
SM phase diagram SM (λ 6 = λ 8 = 0) 180 178 176 λ 6 (M p) = 1/2, λ 8 (M p) = 1 180 178 176 M t 174 M t 174 172 172 170 170 168 120 122 124 126 128 130 132 168 120 122 124 126 128 130 132 M h M t λ 6 (M p) = 1, λ 8 (M p) = 1/2 180 178 176 174 172 170 168 120 122 124 126 128 130 132 M h M h
Magnitude of the suppression scale Approximate lifetime: τ 1 = T U µ 4 (λ min )TU 4 e 8π2 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 12, 10 16 are, respectively, log 10 ( τ T U ) =, 1302, 581 while for the Standard Model log 10 ( τ T U ) = 540.
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 12, 10 16, are, respectively, log 10 ( τ T U ) = 45, 90, 110 while for the Standard Model log 10 ( τ T U ) = 540.
Conclusions We can constrain BSM theroies by requiring at least metastability of the electroweak vacuum. SM vacuum can be stabilized by new physics interactions only if they appear at low enough energy scale 10 10 10 11 GeV SM vacuum lifetime can be dramatically shortened by new physics at any scale
Baryogenesis Generating Baryon asymmetry requires Baryon number violation SU(2) sphalerons present in SM C and CP violation present in SM quark sector (needs enhancement... not a part of this talk though) Departure from thermal equilibrium I order phase transition BSM needed 150 000 100 000 SM with M h =125 GeV T=158.582>Tc T=157.793=Tc T=157.477<Tc T=157.004<Tc SM with H 6 /(750 GeV) 2 operator 3 10 6 2 10 6 T=129.01>Tc T=125.253=Tc T=124.626<Tc T=124.<Tc V [GeV] 4 50 000 0 V [GeV] 4 1 10 6 0-50 000-1 10 6-100 000 0 20 40 60 80 0 50 100 150 200 ϕ [GeV] ϕ [GeV]
Tunneling Decay probability dp of a volume d 3 x dp = dtd 3 x S 2 E 4π 2 det [ 2 + V (φ)] det[ 2 + V (φ 0 )] 1/2 Action of the bounce solution ( ) 1 S E = 2π 2 dss 3 2 φ 2 (s) + V (φ(s)). e S E. Simplifying: prefactor replaced with width of the barrier φ 4 (s = 0) volume of the universe approximated by T 3 U = (1010 yr) 3 Expected lifetime of the false vacuum (p(τ) = 1): τ = 1 T U φ 4 0 T U 4 e S E
Gauge Dependence tree-level potential with one-loop RGEs V λ(µ)zh 2 h 4 }{{} 4, Z 2 1 h = e Γ, Γ(µ) = λ eff (µ) RGE improvement induces gauge dependence γ = 1 16π 2 µ M t γ( µ) ln µ ( 3 20 ζg 2 1 + 3 4 ζg 2 2 + 9 4 g 2 2 + 9 20 g 2 1 3y 2 t 3y 2 b y 2 τ L. Di Luzio and L. Mihaila, JHEP 1406 (2014) 079 We need to normalize the field canonically L = 1 2 ( Z 1 2 h h) 2 + λ 1 eff 4 Z h 2 h 4 h=z 2 h ==== h L = 1 2 ( h) 2 + λ eff 4 h 4 λ eff = λ eff Zh =1 2800 ) 2600 2400 S 2200 2000 Z 1 Z=1 1800 1600-1.0-0.5 0.0 0.5 1.0 ξ
RG improvement Figure: Example solution with λ 6(M p) = 1 and λ 8(M p) = 0.1. Small correction to SM parameters β λ = λ6 m 2. 16π 2 One-loop beta functions of new couplings 16π 2 β λ6 = 10 ( m 2 9 7 λ8 + 18λ66λ 6λ6 M2 4 g 2 2 + 9 ) 20 g 1 2 3yt 2, 16π 2 β λ8 = 7 ( 9 5 28λ2 6 + 30λ 86λ 8λ 8 4 g 2 2 + 9 ) 20 g 1 2 3yt 2, M 2 p
Numerical vs Analytical with RG improvement Figure: Log 10 ( τ T u ) calculated numerically (left panel) and analytically (right panel).
Comparison Figure: Metastability boundary (τ = T u ) obtained using different methods.