MSSM Higgs self-couplings at two-loop

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1 MSSM Higgs self-couplings at two-loop Institut für Theoretische Teilchenphysik Karlsruhe Institute of Technology SUSY 2013, ICTP Trieste, Italy in collaboration with M. Spira and R. Gavin

2 Discovery of a new boson Observation of new boson φ by CMS and Atlas: mass: m φ = 125±0.2(stat) (sys) GeV Atlas spin-parity: J P φ = 0+ couplings to other particles: self-couplings: λ =??? Compatible with SM Higgs boson Also compatible with BSM Higgs Bosons? 1/2 or (g/2v) λ CMS Preliminary s = 7 TeV, L 5.1 fb s = 8 TeV, L 19.6 fb τ 68% CL 95% CL b W Z mass (GeV) t CMS

3 m φ = 125 GeV as an MSSM Higgs Boson Assume φ is the light CP-even scalar MSSM Higgs boson φ = h moderate to heavy stops m t GeV large stop mixing X t 1 2 (m t 1 + m t 2 ) decoupling limit: m A 250 GeV for tanβ 8 (or m A 350 GeV for tanβ 5) m h = 125 GeV h becomes SM-like in the decoupling limit BR(h WW, ZZ,γγ) Ellis, (1991) Haber, (1990) Hempfling,Hoang (1994) Heinemeyer, (1998) Zhang (1999) Slavich, (2001) Espinosa, Zhang (2000) Slavich, (2001) Heynemeyer, (2005) Martin (2007) Harlander, (2010) etc.

4 Motivation for Calculating Two-Loop Corrections to the Couplings Higgs self-couplings determine Higgs potential Higgs potential is responsible for EWSB need to measure Higgs self-interactions to understand EWSB SM: λ hhh difficult at LHC, e + e collider might be needed MSSM: five Higgs bosons promising process at LHC: g t, t H h λ hhh g h need high-precision predictions for trilinear couplings

5 Existing One-Loop Calculation λ O(α t ) hhh = 3m2 Z v + 9m4 t 2π 2 v 3 [ m t log 1 m t 2 m t 2 + X 2 t M 2 SUSY ( X 2 ) ] 1 t 12M SUSY 2 3 2, for tanβ 1, m A m Z, m t m t m h -max scenario, tan(beta) = 10 Barger, (1992) hhh Coupling in GeV tree-level m A in GeV 31% large corrections sizable uncertainties two-loop calculation needed.

6 Effective Potential Method Effective Potential V eff : Non-derivative part of the effective action correct in the limit of vanishing external momenta Generating functional of 1PI Greens functions with no external legs (vacuum diagrams) n-th derivative of V eff : sum of all 1PI diagrams with n external legs λ O(x) H i H j H k = 3 V O(x) H i H i H i effective coupling min,whereh i = (h, H, A, G)

7 Computing the Effective Potential First step: calculate δv α t, δv α tα s and δv α2 t 1-loop: O(α t ) t 1,2 t t 1,2 t 1,2 t t 2-loop: g g g t 1,2 O(α t α s ) t 1,2 t t 1,2 Zhang, (1999) Slavich, (2001)

8 Computing the Effective Potential First step: calculate δv α t, δv α tα s and δv α2 t q h ϕ 2-loop: ϕ q q O(α 2 t ) q q q q = (t, b) ϕ = (H, h, G, A, H ±, G ± ) h = ( h 0 1,2, h ± ) q q q ϕ q = ( t 1, t 2, b L ) Zhang, (1999) Slavich, (2001)

9 Renormalization The fully renormalized coupling can be calculated by λ O(α t+α t α s+α 2 t ) H i H j H k = 3 (V 0 +δv α t +δv α tα s +δv α2 t ) +δλ CT H i H j H k min The counterterm is obtained from derivatives where x i = δλ CT H i H j H k λ O(α t) H i H j H k x i δ αs+α t x i, H i H j H k. = i { } mt 2, m2 t, m 2 t, sin 1 2 2θ t, A t,µ, v are all parameters of the one-loop couplings that are renormalized at O(α s +α t ). Note that at O(α t ) the wave function of the external states is also renormalized.

10 Cancellation of Divergences For simplicity, start with DR-scheme: DR-counterterms δ DR x i are 1 ǫ -divergences O(ǫ)-terms in δv α t O(ǫ 0 )-terms in δv α t give finite contributions give 1 ǫ poles Non-trivial consistency check: All 1 ǫ 2 λ O(α t+α t α s+α 2 t ),DR H i H j H k and 1 ǫ is finite poles cancel.

11 Renormalization Scheme Can shift to any other scheme by adding finite counterterm. e.g. on-shell-scheme: λ O(α t+α t α s+α 2 t ),OS H i H j H k = λ O(α t+α t α s+α 2 t ),DR H i H j H k + λ CT,OS H i H j H k λ CT,OS H i H j H k = i λ O(α t ) H i H j H k x i αs+αt,os x i αs+α t,os x i : finite part of on-shell counterterm λ O(α t+α t α s+α 2 t ),OS H i H j H k is independent of the t Hooft scale Q.

12 Results: λ hhh m h -max scenario, tan(beta) = % 15% hhh Coupling in GeV tree-level m A in GeV

13 Results: λ hhh m h -max scenario, tan(beta) = 10 hhh Coupling in GeV tree-level m A in GeV 31% 8% 15%

14 Results: Other Self-couplings In the same way we obtained O(α t α s +α 2 t ) corrections for all other five trilinear neutral Higgs self-couplings: hhh, hhh, HHH, haa, HAA O(α t α s +α 2 t ) corrections for all nine quartic neutral Higgs self-couplings: hhhh, hhhh, hhhh, hhhh, HHHH, hhaa, hhaa, HHAA, AAAA Uncertainties are well under control.

15 Summary The effective potential method provides an efficient way to calculate two-loop corrections to Higgs self-interactions. The O(α t α s +α 2 t ) corrections to the hhh-coupling are small at the central scale M SUSY /2 and the theoretical uncertainty is reduced from 31% to 8%. stabilization Outlook: analytic formulae public code use these effective couplings to calculate a collider process (supplemented by process dependent corrections)

16 Backup: Mass tb = 2 tree-level tree-level m h in GeV 100 m h in GeV m A in GeV m A in GeV

17 Backup: Mass tb = 30 m h in GeV ) DR 1-loop O(a 60 t ) OS 2 ) OS tree-level m h in GeV ) DR 1-loop O(a 60 t ) OS 2 ) OS tree-level m A in GeV m A in GeV

18 Backup: Mass (m gluino, µ), tb = m h in GeV m h in GeV m gluino in GeV mu in GeV

19 Backup: Mass (tb, M SUSY ) m h in GeV ) DR ) OS M SUSY in GeV tan beta m h in GeV

20 Backup: Mass (X t ), tb = m h in GeV 130 m h in GeV X t in GeV X t in GeV

21 Backup: hhh Coupling (m gluino, µ), tb = 10 hhh Coupling in GeV l OS hhh Coupling in GeV l OS m gluino in GeV mu in GeV

22 Backup: hhh Coupling (X t ), tb = 10 hhh Coupling in GeV l OS hhh Coupling in GeV l OS X t in GeV X t in GeV

23 Backup: hhh Coupling (tb, M SUSY ) hhh Coupling in GeV loop O(a 100 s a t ) DR ) DR 1l OS 80 2 ) OS M SUSY in GeV tan beta hhh Coupling in GeV l OS

24 Backup: hhh Coupling tb = hhh Coupling in GeV ) DR ) OS tree-level m A in GeV hhh Coupling in GeV tree-level m A in GeV

25 Backup: hhh Coupling tb = 2 hhh Coupling in GeV tree-level hhh Coupling in GeV tree-level m A in GeV m A in GeV