Standard Model vacuum stability with a 125 GeV Higgs Stefano Di Vita Max Planck Institute for Physics, Munich October 9, 2014
Outline Standard Model vacuum stability 1 NNLO analysis: the gruesome details 2 NNLO analysis: the colorful plots 3
Outline Standard Model vacuum stability 1 NNLO analysis: the gruesome details 2 NNLO analysis: the colorful plots 3
SM symmetry-breaking sector Higgs potential L ψ j Φ + g ij V ( φ ) ∼ Λ 4 − µ 2 Φ † Φ + λ (Φ † Φ) 2 + Y ij ¯ L ψ jT ψ i Λ ψ i L ΦΦ T ◮ Cosmological constant problem (worst fine tuning problem ever!) ◮ Quadratic sensitivity to regularization cut-off (f.t. again. . . is it a true problem?) ◮ Quadratic sensitivity to heavy dof’s when matching onto UV theory ( do heavy dof’s exist?) ◮ Vacuum instability at large field values if λ < 0 ↔ M h ◮ Loss of perturbativity if λ > 4 π ↔ M h ◮ SM flavor problem + M ν : ◮ large unexplained hierarchy M t / M e ∼ 3 × 10 5 Y ij U ( 1 ) B ⊗ U ( 1 ) ( 3 ) ◮ U ( 3 ) 5 F − → L S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 1 / 24
SM symmetry-breaking sector Higgs potential L ψ j Φ + g ij V ( φ ) ∼ Λ 4 − µ 2 Φ † Φ + λ (Φ † Φ) 2 + Y ij ¯ L ψ jT ψ i Λ ψ i L ΦΦ T ◮ Cosmological constant problem (worst fine tuning problem ever!) ◮ Quadratic sensitivity to regularization cut-off (f.t. again. . . is it a true problem?) ◮ Quadratic sensitivity to heavy dof’s when matching onto UV theory ( do heavy dof’s exist?) ◮ Vacuum instability at large field values if λ < 0 ↔ M h ◮ Loss of perturbativity if λ > 4 π ↔ M h ◮ SM flavor problem + M ν : ◮ large unexplained hierarchy M t / M e ∼ 3 × 10 5 Y ij U ( 1 ) B ⊗ U ( 1 ) ( 3 ) ◮ U ( 3 ) 5 F − → L S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 1 / 24
The effective potential: single real scalar (1) V ( φ ) = m 2 L = 1 2 φ 2 + λ 2 ∂ µ φ∂ µ φ − V ( φ ) , 4 φ 4 ◮ Minimum of V ( φ ) gives φ c ≡ � φ � at the classical level ◮ we consider fluctuations around the minimum, φ → φ c + φ ◮ V ( φ ) gives the lowest order (classical) 1PI vertices and propagator Quantum corrections? [Coleman and E.Weinberg] ◮ V eff is the order-zero term in the derivative expansion of the effective action (gen. of full 1PI functions) ◮ For constant φ c , min of V eff ( φ ) gives φ c ≡ � φ � , the true quantum minimum (constant ↔ we don’t want to break Poincar´ e) S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 2 / 24
The effective potential: single real scalar (2) 1-loop computation [Coleman and E.Weinberg, Jackiw] and renormalization (e.g. MS or OS, µ is the ’t Hooft mass or the subtraction point): c + ( m 2 + 3 λφ 2 ln m 2 + 3 λφ 2 V eff ( φ c ) = m 2 c ) 2 c + λ c 2 φ 2 4 φ 4 64 π 2 µ 2 Consider e.g. m 2 = 0: 4 φ 4 ⇒ φ = 0 (min) ◮ V ( φ ) = λ � φ c = 0 max 4 φ 4 + 9 λ 2 φ 4 64 π 2 ln φ 2 ◮ V eff ( φ c ) = λ c µ 2 ⇒ c φ c : λ ln φ c µ ∼ − 8 9 π 2 min The min condition is for λ ln φ c µ ∼ O ( 1 ) , but higher orders contribute to V eff as λ ( λ ln φ c µ ) n . A weapon: dV eff d µ = 0 ⇒ resum logs with RGE S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 3 / 24
The SM effective potential eff ( φ ) ≃ m 2 ( µ ) φ ( µ ) 2 + λ ( µ ) λ ( µ ) φ ( µ ) 4 − V RGI φ ( µ ) 4 − − → 2 4 4 φ ≫ v ◮ The choice µ ∼ φ helps minimizing the large logs ◮ The shape of V RGI crucially depends on the running of λ eff gauge bosons loop ext. leg corrections scalar loop fermion loop � �� � � �� � � �� � � �� � d λ 1 8 g 4 + 3 9 ′ 4 + 3 + 24 N c λ 2 + λ ( 4 N c Y t − 9 g 2 − 3 g ′ 2 ) ′ 2 − 2 N c Y 4 4 g 2 g d ln µ = t + + . . . 8 g 16 π 2 � �� � ≡ B < 0 at EW scale If B = const , V eff unbounded from below at large φ , but B runs too!! S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 4 / 24
A few possibilities ◮ B ∼ 0 , M h large: Landau pole (or triviality problem: probably consistent continuum limit for φ 4 theory ⇔ λ R = 0) ◮ B < 0 at weak scale but does not run negative enough at large φ : V eff bounded from below (SM vacuum stable ) ◮ B < 0 at weak scale enough to stay negative at large φ : V eff unbounded from below (SM vacuum unstable , need NP) ◮ All SM parameters known ◮ B < 0 at weak scale but flips ◮ Assume no NP below M Pl sign at large φ : V eff develops another min (degenerate or ◮ 3-loop RGE lower) (SM vacuum metastable ) S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 5 / 24
A few possibilities ◮ B ∼ 0 , M h large: Landau pole 1.0 (or triviality problem: probably consistent g 3 y t continuum limit for φ 4 theory ⇔ λ R = 0) 0.8 ◮ B < 0 at weak scale but does g 2 0.6 SM couplings not run negative enough at g 1 large φ : V eff bounded from 0.4 below (SM vacuum stable ) 0.2 ◮ B < 0 at weak scale enough to Λ y b stay negative at large φ : V eff 0.0 10 10 10 12 10 14 10 16 10 18 10 20 10 2 10 4 10 6 10 8 unbounded from below (SM RGE scale Μ in GeV vacuum unstable , need NP) ◮ B < 0 at weak scale but flips ◮ All SM parameters known sign at large φ : V eff develops ◮ Assume no NP below M Pl another min (degenerate or ◮ 3-loop RGE lower) (SM vacuum metastable ) S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 5 / 24
Beware of the dog bowl! from A. Strumia Illustrative λ ( µ ) > 0 up to M Pl , i.e. stable very unstable → If your mexican hat turns out to be a dog bowl you have a problem... S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 6 / 24
Metastability V � Φ � Φ ◮ φ EW can be a false vacuum → quantum tunneling [Coleman; Callan, Coleman] ◮ compute bounce solution for Euclidean action ( ∼ WKB) τ 4 R 4 e − S B ( R ) for a bounce of size R , S B ( R ) = 8 π 2 ◮ tunneling p ∼ U 3 λ ( R − 1 ) ◮ dominated by bounce that maximizes the action, i.e. β λ ( R − 1 ) = 0 ◮ this scenario still ok if τ EW ≫ τ U � � 4 2600 e 140 e − | λ | / 0 . 01 ≪ 1 [Isidori, Ridolfi, Strumia 01] ◮ SM: p ∼ RM Pl ◮ higher dim. operators (e.g. Planck scale physics) could change the transition probability [Branchina, Messina 13] S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 7 / 24
Analysis strategy 1) compute V eff at n -loop level (not just λ ( µ ) φ ( µ ) 4 / 4) but one can’t trust it at large field values, even in λ stays perturbative 2) improve it with ( n + 1 ) -loop beta-functions now we can trust V RGI up to large scale since λ stays perturbative eff 3) but . . . how much are λ , y t at Λ EW ? we know m H , m t ! ( n + 1 ) -loop running up to M Pl , requires at least n -loop matching, can’t use just the tree √ √ level λ = G µ m 2 2 and y 2 t = 4 G µ m 2 H / t / 2 ◮ lower and upper bound on m h by requiring (meta)stability and perturbativity up to some scale Λ I [pre-Higgs times, either H or NP . . . ] ◮ instability scale Λ I as a function of m h or m t [gauge dependence . . . ] ◮ SM phase diag. in ( m h , m t ) plane: stable up to M Pl ? τ EW ≶ τ U ? S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 8 / 24
Higgs mass bounds at NLO in 2009 M t = 173 . 1 ± 1 . 3 GeV α s ( M Z ) = 0 . 1193 ± 0 . 0028 350 350 [GeV] [GeV] Perturbativity bound H H Stability bound M M 300 300 Finite-T metastability bound λ = 2 π Zero-T metastability bound λ = π Shown are 1 σ error bands, w/o theoretical errors 250 250 200 200 Tevatron exclusion at >95% CL 150 150 LEP exclusion at >95% CL 100 100 4 4 6 6 8 8 10 10 12 12 14 14 16 16 18 18 log log ( ( Λ Λ / / GeV) GeV) 10 10 one-loop V eff two-loop running one-loop matching [Ellis et al. 09] S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 9 / 24
SM phase diagram: LO vs NLO vs NNLO V eff < 0 before M Pl , τ EW < τ U instability metastability V eff < 0 before M Pl , τ EW > τ U stability V eff > 0 up to M Pl , i.e. stable Espinosa S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 10 / 24
Outline Standard Model vacuum stability 1 NNLO analysis: the gruesome details 2 NNLO analysis: the colorful plots 3
State of the art: SM vacuum stability at NNLO ◮ Complete two-loop effective potential [Ford, Jack, Jones 92,97; Martin 02] ◮ known since a long time but needed three-loop β ’s for RG improvement now three-loop known! [Martin 13] ◮ Complete three-loop beta-functions ◮ g i [Mihaila, Salomon, Steinhauser 12] ◮ Y t , b ,τ , λ, µ [Chetyrkin, Zoller 12,13; Bednyakov, Pikelner, Velizhanin 13] ◮ Two-loop matching conditions at the weak scale (large th. err, especially λ ) 1-loop 2-loop 3-loop g 1 , 2 full ? – O ( α 3 O ( αα s ) s ) y t full O ( αα s , α 2 ) λ full – O ( αα s ) [Bezrukov, Kalmykov, Kniehl, Shaposhnikov 12; Degrassi, Elias-Mir` o, Espinosa, Giudice, Isidori, Strumia, DV 12] O ( α 2 ) [Degrassi, Elias-Mir` o, Espinosa, Giudice, Isidori, Strumia, DV 12] S. Di Vita (MPI for Physics, Munich) SM vacuum stability with a 125 GeV H 11 / 24
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