The Higgs masses in the NMSSM at one- and two-loop level

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1 The Higgs masses in the NMSSM at one- and two-loop level Florian Staub (University of Würzburg) in collaboration with: W. Porod (Uni. Würzburg), B. Herrmann (DESY) Reference: JHEP10(2010)040, arxiv: International Workshop on Linear Colliders October 2010 Florian Staub (University of Würzburg) 1 / 12

2 Outline 1 The Next-to-minimal Supersymmetric Standard Model 2 Renormalization of the NMSSM Higgs sector 3 Numerical analysis 4 Summary Florian Staub (University of Würzburg) 2 / 12

3 The Next-to-minimal Supersymmetric Standard Model Introduction Motivation The NMSSM solves the µ-problem of the MSSM by adding a gauge singlet S to the particle content Florian Staub (University of Würzburg) 3 / 12

4 The Next-to-minimal Supersymmetric Standard Model Introduction Motivation The NMSSM solves the µ-problem of the MSSM by adding a gauge singlet S to the particle content Superpotential: W MSSM = ĤuˆqY u û + ĤdˆqY d ˆd + Ĥ dˆlye ê + µĥuĥd Florian Staub (University of Würzburg) 3 / 12

5 The Next-to-minimal Supersymmetric Standard Model Introduction Motivation The NMSSM solves the µ-problem of the MSSM by adding a gauge singlet S to the particle content Superpotential: W NMSSM = ĤuˆqY u û + ĤdˆqY d ˆd + Ĥ dˆlye ê + λĥuĥdŝ κŝŝŝ. Florian Staub (University of Würzburg) 3 / 12

6 The Next-to-minimal Supersymmetric Standard Model Introduction Motivation The NMSSM solves the µ-problem of the MSSM by adding a gauge singlet S to the particle content Superpotential: W NMSSM = ĤuˆqY u û + ĤdˆqY d ˆd + Ĥ dˆlye ê + λĥuĥdŝ κŝŝŝ. New soft-breaking terms: V SB,NMSSM = m 2 S S 2 + T λ H u H d S T κsss Florian Staub (University of Würzburg) 3 / 12

7 The Next-to-minimal Supersymmetric Standard Model Higgs sector of the NMSSM Scalar gauge singlet S receives a VEV v s after SUSY breaking S = 1 2 (φ s + iσ s + v s ) Florian Staub (University of Würzburg) 4 / 12

8 The Next-to-minimal Supersymmetric Standard Model Higgs sector of the NMSSM Scalar gauge singlet S receives a VEV v s after SUSY breaking S = 1 2 (φ s + iσ s + v s ) µ term The generated µ-term is of order SUSY-breaking scale µ eff = 1 2 λv s. Florian Staub (University of Würzburg) 4 / 12

9 The Next-to-minimal Supersymmetric Standard Model Higgs sector of the NMSSM Scalar gauge singlet S receives a VEV v s after SUSY breaking S = 1 2 (φ s + iσ s + v s ) µ term The generated µ-term is of order SUSY-breaking scale µ eff = 1 2 λv s. The Higgs sector of the NMSSM consists in total of Three CP-even scalar Higgs h i Two physical CP-odd pseudo scalar Higgs A 0 i Two physical charged Higgs H ± with H + = (H ) Florian Staub (University of Würzburg) 4 / 12

10 Renormalization of the NMSSM Higgs sector Renormalization of the Higgs sector Sketch of the procedure: Parameters are calculated by RGE running from GUT scale Higgs soft-breaking parameters fixed by tadpole equations Florian Staub (University of Würzburg) 5 / 12

11 Renormalization of the NMSSM Higgs sector Renormalization of the Higgs sector Sketch of the procedure: Parameters are calculated by RGE running from GUT scale Higgs soft-breaking parameters fixed by tadpole equations 1-loop tadpoles/self-energies calculated (DR, t Hooft-gauge) Florian Staub (University of Würzburg) 5 / 12

12 Renormalization of the NMSSM Higgs sector Renormalization of the Higgs sector Sketch of the procedure: Parameters are calculated by RGE running from GUT scale Higgs soft-breaking parameters fixed by tadpole equations 1-loop tadpoles/self-energies calculated (DR, t Hooft-gauge) 1-loop mass matrix given by m 2,h 1L (p2 ) = m 2,h T Π hh(p 2 ) 1-loop masses: real part of poles of propagator matrix [ ] Det p 2 i 1 m 2,h 1L (p2 ) = 0, iterative solution for external masses on-shell Florian Staub (University of Würzburg) 5 / 12

13 Renormalization of the NMSSM Higgs sector Two-loop contributions Dominant two-loop contributions known in literature G. Degrassi, P. Slavich, Nucl.Phys.B825: ,2010, arxiv: Includes the contributions of (s)top/(s)bottom Authors calculated also the one-loop corrections: Neglected Yukawa couplings of 1. and 2. generation: differences in the per-mille range to our results Complete agreement between both calculations in the limit they used Florian Staub (University of Würzburg) 6 / 12

14 Renormalization of the NMSSM Higgs sector Numerical analysis All analytical expressions for the NMSSM were calculated with the Mathematica Package SARAH [FS ( )] and exported to Fortran code Florian Staub (University of Würzburg) 7 / 12

15 Renormalization of the NMSSM Higgs sector Numerical analysis All analytical expressions for the NMSSM were calculated with the Mathematica Package SARAH [FS ( )] and exported to Fortran code Fortran routines were implemented in SPheno [Porod (hep-ph/ )]: Two-loop RGEs with complete flavor structure SUSY thresholds at EW scale are included All one-loop masses with external momenta on-shell calculated Calculations of two- and three body decays of SUSY particles Low energy constraints checked (e.g. µ eγ, M Bs,B d,... ) msugra like GUT scale conditions Florian Staub (University of Würzburg) 7 / 12

16 Renormalization of the NMSSM Higgs sector Numerical analysis All analytical expressions for the NMSSM were calculated with the Mathematica Package SARAH [FS ( )] and exported to Fortran code Fortran routines were implemented in SPheno [Porod (hep-ph/ )]: Two-loop RGEs with complete flavor structure SUSY thresholds at EW scale are included All one-loop masses with external momenta on-shell calculated Calculations of two- and three body decays of SUSY particles Low energy constraints checked (e.g. µ eγ, M Bs,B d,... ) msugra like GUT scale conditions Fortran routines for two-loop Higgs masses provided by Slavich & Degrassi were included Florian Staub (University of Würzburg) 7 / 12

17 Renormalization of the NMSSM Higgs sector Constraint NMSSM Constraint NMSSM (soft version) Free parameters Relations at the GUT Scale: m 0, M 1/2, A 0, tan β = v u v d, λ, κ, A λ, A κ, v s M 1 = M 2 = M 3 M 1/2, m 2 D = m 2 Ũ = m2 Q = m 2 Ẽ = m2 L m T u = A 0 Y u, T d = A 0 Y d, T e = A 0 Y e, T λ = A λ λ, and T κ = A κ κ. m 2 H d, m 2 H u, m 2 s fixed by tadpole equations at EW scale. Florian Staub (University of Würzburg) 8 / 12

18 Numerical analysis Numerical example Particle m T [GeV] m 1L [GeV] [%] m 2L [GeV] [%] h h h < < 0.1 A A Florian Staub (University of Würzburg) 9 / 12

19 Numerical analysis Comparison with literature A 0 = 1500 GeV, tan β = 10, λ SUSY = 0.1, A GUT κ = 33.45, µ eff > 0 mh m 0 [GeV] mh2, m 0 [GeV] SPheno 2-loop (plain), dominant 1-loop (dashed) NMSSM-Tools [Ellwanger,Hygonie (hep-ph/ )]: 2-loop (dotted), 1-loop (dotdashed) Florian Staub (University of Würzburg) 10 / 12

20 Numerical analysis Comparison with literature A 0 = 1500 GeV, tan β = 10, λ SUSY = 0.1, A GUT κ = 33.45, µ eff > 0 mh m 0 [GeV] mh2, m 0 [GeV] SPheno 2-loop (plain), dominant 1-loop (dashed) NMSSM-Tools [Ellwanger,Hygonie (hep-ph/ )]: 2-loop (dotted), 1-loop (dotdashed) Differences stemming from: 1 External momenta 2 Electroweak corrections Florian Staub (University of Würzburg) 10 / 12

21 Numerical analysis Comparison with literature A 0 = 1500 GeV, tan β = 10, λ SUSY = 0.1, A GUT κ = 33.45, µ eff > 0 m A m 0 [GeV] m A m 0 [GeV] SPheno 2-loop (plain), dominant 1-loop (dashed) NMSSM-Tools [Ellwanger,Hygonie (hep-ph/ )]: 2-loop (dotted), 1-loop (dotdashed) Differences stemming from: 1 External momenta 2 Electroweak corrections Florian Staub (University of Würzburg) 10 / 12

22 Numerical analysis Scale dependence m 0 = 180 GeV, m 1/2 = 500 GeV, A 0 = A GUT λ = 1500 GeV, A GUT κ κ GUT = 0.11, λ GUT = 0.1, vs = GeV. = 36 GeV, tan β = 10, mh Q[GeV] 1-loop, 2-loop Florian Staub (University of Würzburg) 11 / 12

23 Numerical analysis Scale dependence m 0 = 180 GeV, m 1/2 = 500 GeV, A 0 = A GUT λ κ GUT = 0.11, λ GUT = 0.1, vs = GeV. = 1500 GeV, A GUT κ = 36 GeV, tan β = 10, mh Q[GeV] Results: 1-loop, 2-loop 1 Scale dependence improves significantly at 2-loop level Florian Staub (University of Würzburg) 11 / 12

24 Numerical analysis Scale dependence m 0 = 180 GeV, m 1/2 = 500 GeV, A 0 = A GUT λ = 1500 GeV, A GUT κ κ GUT = 0.11, λ GUT = 0.1, vs = GeV. = 36 GeV, tan β = 10, mh mh Q[GeV] Q[GeV] Results: 1-loop, 2-loop 1 Scale dependence improves significantly at 2-loop level Florian Staub (University of Würzburg) 11 / 12

25 Numerical analysis Scale dependence m 0 = 180 GeV, m 1/2 = 500 GeV, A 0 = A GUT λ = 1500 GeV, A GUT κ κ GUT = 0.11, λ GUT = 0.1, vs = GeV. = 36 GeV, tan β = 10, mh mh Q[GeV] Q[GeV] Results: 1-loop, 2-loop 1 Scale dependence improves significantly at 2-loop level 2 For singlet states, 2-loop contributions of λ, κ needed Florian Staub (University of Würzburg) 11 / 12

26 Numerical analysis Scale dependence m 0 = 180 GeV, m 1/2 = 500 GeV, A 0 = A GUT λ = 1500 GeV, A GUT κ κ GUT = 0.11, λ GUT = 0.1, vs = GeV. = 36 GeV, tan β = 10, m A m A Q[GeV] Q[GeV] 1-loop, 2-loop Results: 1 Scale dependence improves significantly at 2-loop level 2 For singlet states, 2-loop contributions of λ, κ needed Florian Staub (University of Würzburg) 11 / 12

27 Summary Summary The NMSSM is an attractive extension of the MSSM The loop-corrections in the Higgs sector of the NMSSM are important as in case of the MSSM We performed a complete one-loop calculation of all masses without any approximation Implementation of a constraint GUT version in SPheno including dominant two-loop corrections Our results are consistent with literature Scale dependences improves significantly on 2-loop level for non-singlet states Florian Staub (University of Würzburg) 12 / 12