MSSM Higgs self-couplings at two-loop mathias.brucherseifer@kit.edu Institut für Theoretische Teilchenphysik Karlsruhe Institute of Technology SUSY 2013, ICTP Trieste, Italy in collaboration with M. Spira and R. Gavin
Discovery of a new boson Observation of new boson φ by CMS and Atlas: mass: m φ = 125±0.2(stat) +0.5 0.6 (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 1-1 10-2 10-1 -1 CMS Preliminary s = 7 TeV, L 5.1 fb s = 8 TeV, L 19.6 fb τ 68% CL 95% CL b W Z 1 2 3 45 10 20 100 200 mass (GeV) t CMS
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 2 300 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.
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
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 250 200 150 100 50 0 tree-level 50 100 500 1000 m A in GeV 31% large corrections sizable uncertainties two-loop calculation needed.
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)
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)
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)
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.
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.
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.
Results: λ hhh m h -max scenario, tan(beta) = 10 250 200 31% 15% hhh Coupling in GeV 150 100 50 0 tree-level 50 100 500 1000 m A in GeV
Results: λ hhh m h -max scenario, tan(beta) = 10 hhh Coupling in GeV 250 200 150 100 50 0 tree-level 50 100 500 1000 m A in GeV 31% 8% 15%
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.
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)
Backup: Mass tb = 2 tree-level tree-level m h in GeV 100 m h in GeV 100 80 80 60 60 40 50 100 500 1000 40 50 100 500 1000 m A in GeV m A in GeV
Backup: Mass tb = 30 m h in GeV 100 80 2 ) DR 1-loop O(a 60 t ) OS 2 ) OS tree-level 40 50 100 500 1000 m h in GeV 100 80 2 ) DR 1-loop O(a 60 t ) OS 2 ) OS tree-level 40 50 100 500 1000 m A in GeV m A in GeV
Backup: Mass (m gluino, µ), tb = 20 150 145 170 m h in GeV 135 130 125 m h in GeV 150 130 115 300 1000 3000 m gluino in GeV 200 1000 3000 mu in GeV
Backup: Mass (tb, M SUSY ) m h in GeV 150 130 110 135 100 130 90 2 ) DR 125 80 2 ) OS 115 70 300 1000 3000 1 10 60 M SUSY in GeV tan beta m h in GeV 165 155 150 145
Backup: Mass (X t ), tb = 10 150 150 m h in GeV 130 m h in GeV 130 110 110 100 0 200 400 600 800 1000 0 0 100 0 500 1000 1500 2000 2500 X t in GeV X t in GeV
Backup: hhh Coupling (m gluino, µ), tb = 10 hhh Coupling in GeV 250 240 230 220 210 200 190 180 1l OS hhh Coupling in GeV 300 280 260 240 220 200 1l OS 170 300 1000 3000 m gluino in GeV 180 0 500 1000 1500 2000 2500 3000 mu in GeV
Backup: hhh Coupling (X t ), tb = 10 hhh Coupling in GeV 300 280 260 240 220 200 180 1l OS hhh Coupling in GeV 300 280 260 240 220 200 180 1l OS 200 400 600 800 1000 0 0 X t in GeV 500 1000 1500 2000 2500 3000 X t in GeV
Backup: hhh Coupling (tb, M SUSY ) hhh Coupling in GeV 260 240 220 200 180 220 200 2-loop O(a 100 s a t ) DR 180 2 ) DR 1l OS 80 2 ) OS 60 300 1000 3000 1 10 60 M SUSY in GeV tan beta hhh Coupling in GeV 320 300 280 260 240 1l OS
Backup: hhh Coupling tb = 10 40 40 20 20 hhh Coupling in GeV 0-20 -40-60 -80-100 - -60-80 2 ) DR -100 2 ) OS tree-level - 50 100 500 1000 m A in GeV hhh Coupling in GeV 0-20 -40 tree-level 50 100 500 1000 m A in GeV
Backup: hhh Coupling tb = 2 hhh Coupling in GeV 40 20 0-20 -40-60 -80 tree-level hhh Coupling in GeV 40 20 0-20 -40-60 -80 tree-level -100 - -100-50 100 500 1000 m A in GeV 50 100 500 1000 m A in GeV