Scalar mass stability bound in a simple Yukawa-theory from renormalisation group equations

Transcription

1 Scalar mass stability bound in a simple Yuawa-theory from renormalisation group equations (arxiv: ) Antal Jaovác, István Kaposvári, András Patós Department of Atomic Physics Eötvös University, Budapest ACHT, 2015

2 Motivation Stability Higgs bounds with perturbative RGE in SM. dλ(q 2 ) d log Q 2 = 1 16π 2 ( 12λ 2 + 6λh 2 t 3h 4 t λ(3g2 2 + g 2 1) (2g4 2 + (g g 2 1) 2 ) ) SM couplings 1.0 yt g g 2 g Λ 0.0 y b RGE scale in GeV arxiv: [hep-ph] Higgs quartic coupling Λ Σ bands in Mt ± 0.6 GeV gray Α 3 M Z ± red M h ± 0.3 GeV blue RGE scale in GeV Mt GeV Αs M Z Αs M Z Mt GeV

3 Motivation dλ(q 2 ) d log Q = 1 ( 12λ 2 + 6λh π 2 t 3h 4 t λ(3g2 2 + g1) ) 16 (2g4 2 + (g2 2 + g1) 2 2 ) Non-perturbative studies in simplified models (see below): Lattice: Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, 2007 D.Y.-J. Chu, K. Jansen, B. Knippschild, C.-J. D. Lin and A. Nagy, 2015 FRG: H. Gies, C. Gneiting and R. Sondenheimer, 2014

4 Outline Functional Renormalization Group Equations Higgs-top model with discrete chiral symmetry Functional Renormalization Group Equations of the system The Local Potential Approximation (LPA) Consistent solutions

5 Functional Renormalization Group Equations The Wetterich equation Calculating the contributions to the effective action from momentum shell to momentum shell: t Γ = 1 ( ) 1 2 STr tr Γ (2) + R d The regulator term suppresses p < terms by giving them large mass. Γ contains the quantum fluctuations with momenta higher than. R

6 Functional Renormalization Group Equations The Optimized Regulator used in this paper for bosons for fermions R B (p) = ( 2 p 2) ( ) 2 Θ p 2 1 ( ) ( ) 2 R F (p) = /p 1 Θ p 2 p 2 1

7 Higgs-top model with discrete chiral symmetry Effective action of the Higgs-top toy model Classical action of a scalar-fermion Yuawa bound system: [ ] 1 S = d 4 x 2 ( µσ) 2 + U(ρ) + ψ / ψ + hσ ψψ, ρ = 1 2 σ2, I = σ ψψ ρ and I are invariant under the discrete chiral symmetry: σ(x) σ(x), ψ(x) γ 5 ψ(x), ψ(x) ψ(x)γ5 Scale-dependent effective-action for < Λ: [ ] Γ = d 4 Zσ, x 2 ( µσ) 2 + U (ρ) + Z ψ, ψ / ψ + h σ ψψ, ρ = Z σ, 2 σ2 Infrared observables (obtained by fine-tuning the initial conditions): v = Z 1/2 σ0 σ 0 = 246GeV, m ψ = h 0 σ 0 = 173GeV m 2 σ = U 0(ρ 0 ) + 2ρ 0 U 0 (ρ 0 )

8 Functional Renormalization Group Equations of the system The Wetterich equation t Γ = 1 ( ) 1 2 STr tr Γ (2) 1 + R = 2 ˆ t STr log(γ (2) + R ) The right hand side of the Wetterich equation: 1 STr log(γ(2) + R ) = 1 Tr log(γ(2) Tr log [1 ( Γ (2) σσ + R B Ψ T Ψ + RF ) + 1 Tr log(γ(2) σσ + R B ) 2 ) 1 ( ) ] 1 (2) Γ σψ Γ (2) Ψ T Ψ + RF (2) Γ Ψ T σ ψ-loop in σ-bgd σ-loop in σ-bgd mixed loop in ψ-bgd

9 The Local Potential Approximation (LPA) The Local Potential Approximation (LPA) Constant bacground values, no field renormalization: σ(x) v, ψ(x) ψ, ψ(x) ψ, Z σ, 1, Z ψ, 1 [U (ρ ) + h I ] = 1 2 ˆ m 2 σ = U (ρ ) + 2ρ U (ρ ), q [ 4 log(q 2 R + m 2 ψ) + log(q 2 R + m 2 σ)+ { } ] + log 1 h 2 1 2h I qr 2 + m2 σ qr 2 + m2 ψ. m 2 ψ = 2h 2 ρ At a given scale ρ and I are connected by the equation of motion of the scalar field: σ δγ = h I + 2ρ U δσ σ=v (ρ ) = 0.,ψ=ψ

10 Consistent solutions Consistent solutions The consistency of the two sides of the LPA equation can be ensured in different ways. Version A: Complete elimination of I on RHS h t = 0, t U (ρ ) = 1 { 2 ˆ t 5 log(qr 2 + m 2 ψ)+ q + log [ (q 2 R + m 2 σ)(q 2 R + m 2 ψ) + 4h 2 ρ U (ρ ) ] } After rewriting the equation using dimensonless variables and perform the ˆ t operation on RHS: t u r = (d 2)ρ r u r du r + ( + v d µ 2 ψ 1 + µ ) µ2 σ ψ (1 + µ 2 ψ )(1 + µ2 σ) + 4h 2 rρ r u r

11 Consistent solutions Consistent solutions ersion B: Linearization of RHS in I t (h I ) = 1 2 ˆ t t U (ρ ) = 1 2 ˆ t q q [ 2h 3 I (q 2 R + m2 ψ )(q2 R + m2 σ), 4 log(q 2 R + m 2 ψ) + log(q 2 R + m 2 σ)+ { + log 1 + h 2 1 qr 2 + m2 σ 1 2 ˆ t q } ] 4ρ U (ρ ) qr 2 + m2 ψ 4h 2 ρ U (ρ ) (q 2 R + m2 ψ )(q2 R + m2 σ). Gies et al. (2014): no ψ bacground, last two terms missing.

12 The potential used in the study for the symmetric (SYM) and symmetry broen (SB) regime respectively: U (SYM) (ρ ) = N p n=1 λ n ρ n, U (SB) n! (ρ ) = N p n=2 λ n (ρ κ ) n for λ 2 = 0.001, 1, 10, 50, 100 from bottom to top respectively: 1000 n! 800 m h Gev Λ Gev

13 The potential used in the study for the symmetric (SYM) and symmetry broen (SB) regime respectively: U (SYM) (ρ ) = N p n=1 λ n ρ n, U (SB) n! (ρ ) = N p n=2 Higgs mass stability and triviality bounds for N p = 2: λ n (ρ κ ) n n!

14 Including Z ψ, Z σ 1 leads to a percent level changes. The maximum allowed value of the cutoff N p = 2: GeV < Λ (2) max < GeV

15 Lower bound with quadratic and quartic truncations of U (ρ ) using Version A: The maximum allowed value of the cutoff N p = 2: GeV < Λ (2) max < GeV N p = 4: GeV < Λ (4) max < GeV

16 Summary The allowed range of the Higgs mass has been determined with FRG in presence of a pointlie composite fermion bacground. Close agreement of all approximation signals a robust determination of Λ max Outloo Investigation of a more general ansatz for Γ with the effects of heavy neutrinos of the seesaw mechanism Inclusion of the multiplet structure and the gauge interactions of the SM