The Higgs sector in the MSSM with CP-phases at higher orders

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1 The Higgs sector in the MSSM with CP-phases at higher orders Paul-Scherrer-Institut in coll. with T. Hahn, S. Heinemeyer, W. Hollik and G. Weiglein

2 Outline Higgs sector in the MSSM with CP-phases Mass of the lightest Higgs boson Effective couplings

3 Higgs bosons At Born level: no CP-violation: one phase in the Higgs potential: V Higgs = + ɛ ij m3 2 eiϕ m 3 2 H1 i Hj elimination via Peccei-Quinn transformation phase difference of Higgs doublets: vanishes because of minimum condition Physical mass eigenstates (at Born level): 5 Higgs bosons: 3 neutral H 0, h 0, A 0 ; 2 charged H ± Masses of the Higgs bosons: not all independent: here: H ± -mass M H ± (and tan β) as free parameter tan β = v 2 v 1 : ratio of the Higgs vac. expect. values lightest Higgs boson: h 0

4 Mass of the lightest Higgs boson Upper theoretical Born mass bound: M h 0 M Z = 91 GeV with quantum corrections of higher orders: M h GeV dependent on the MSSM parameters: particularly on parameter phases Discovery of the Higgs boson: accurate measurement precise prediction mass of the Higgs boson strong bounds on the MSSM parameter, e.g. on tan β = v 2 v 1 i.e. inclusion of higher order corrections with full phase dependence Now, before the discovery: Exclusion of parts of the parameter space

5 Determination of the Higgs masses Two-point-function: iˆγ(p 2 ) = p 2 M(p 2 ) with the matrix: M 2 ˆΣ H M(p 2 Born 0 H 0 H 0(p2 ) ˆΣ H 0 h 0(p2 ) ˆΣ H 0 A 0(p2 ) )= ˆΣ H 0 h 0(p2 ) M 2 ˆΣ hborn 0 h 0 h 0(p2 ) ˆΣ h 0 A 0(p2 ) ˆΣ H 0 A 0(p2 ) ˆΣ h 0 A 0(p2 ) M 2 ˆΣ A 0 A 0 A 0(p2 ) Born Real parameters: ˆΣ H 0 A 0(p2 ) = ˆΣ h 0 A 0(p2 ) = 0 no mixing between CP-even and CP-odd states Calculate the zeros of the determinant of ˆΓ: det[p 2 M(p 2 )] = 0 Higgs masses M h1, M h2, M h3

6 Higgs masses at higher orders (incl. CP-phases) Status: Higgs masses at higher order without CP-phases good shape (up to leading 3-loop [S. Martin 07]) Including CP-phases: Eff. potential approach, up to two-loop leading-log contributions (sfermionic/fermionic contributions) [Pilaftsis, Wagner], [Demir], [Choi, Drees, Lee], [Carena, Ellis, Pilaftsis, Wagner] Gaugino contributions [Ibrahim, Nath] Effects of imaginary parts at one-loop [Ellis, Lee, Pilaftsis], [Choi, Kalinowski, Liao, Zerwas], [Bernabeu, Binosi, Papavassiliou] Here: full one-loop + dominant two-loop (Feynman diagrammatic) [M. Frank, et al.]

7 Renormalized Higgs self energies at one-loop + Parameters of the Higgs sector need to be defined at one-loop: define the H ± -, W - as well as the Z-mass as pole mass directly related to a physical observable δm (1) X = ReΣ XX (M 2 X ), X = {H±, W, Z} no shift of the minimum of the Higgs potential δt (1) φ = T φ DR-scheme for field and tan β renormalization δ tan (1) β = δ tan β DR, δz (1) H i = δz DR H i

8 Renormalized Higgs self energies at two-loop Calculation of the dominant two-loop contributions O(α t α s ) (α t = λ 2 t /(4π)): Extraction of the relevant terms (equiv. to eff. potential approach): use vanishing external momenta ˆΣ (2) (0) use vanishing electroweak gauge couplings g, g Parameters of the Higgs sector defined at two-loop: no shift of the minimum of the Higgs potential define the H ± -mass M H ± as the pole mass directly related to a physical observable Yukawa coupling Parameters of the top (bottom) sector defined at one-loop: top quark mass and top squark masses on-shell generalization of the mixing angle condition: ReˆΣ t12 (m 2 t 1 ) + ReˆΣ t12 (m 2 t 2 ) = 0

9 Phases in other sectors Sfermion sector: phase ϕaf of the trilinear coupling A f Higgsino sector: phase of µ (small), µ: Higgsino mass parameter constraints from measurements of electr. dipole moments Gaugino sector: soft breaking parameter phases of the gaugino mass parameters M1, M 2, M 3 one phase can be eliminated (R-Transformation), often ϕ M2 phase ϕ M3 is the phase of the gluino mass parameter enters into the Higgs sector at two-loop level

10 Results: ϕ At - versus ϕ Xt -dependence (large M H ±) size of the squark mixing: X t := A t µ cot β ents ϕ Xt π Mh1 [GeV] M H ± = 500 GeV O(α) A t = 1.7 TeV O(α + α t α s ) tan β = 10 PSfrag replacements 120 A t = 1.7 TeV 115 tan β = ϕ At π ϕ At π Mh1 [GeV] M H ± = 500 GeV O(α) X t = 1.6 TeV O(α + α t α s ) tan β = ϕ Xt π The qualitative behaviour of M h1 can change with the inclusion of quantum corrections of O(α t α s ). Quantum corrections tend to be smaller for constant absolute value of of the squark mixing, X t = const.

11 Results: ϕ Xt -dependence (small M H ±) M H ± = 150 GeV X t = 700 GeV ents Mh1 [GeV] PSfrag replacements 100 = 5 M h1 [GeV] = 5 ϕ X t = π M H ± = GeV X t = 700 GeV ϕ X t π + interpol. : tan β = 5 O(α) : tan β = 5 O(α + α t α s ) : tan β = 5 tan β = 15 tan β = 15 tan β = 15 Higgs mass M h1 does depend on the phase ϕ Xt, X t = 700 GeV. One-loop corrections are more sensitive to ϕ Xt for small M H ±.

12 Results: ϕ Xt -dependence (small M H ±) ents Mh1 [GeV] M H ± = 150 GeV X t = 700 GeV 100 PSfrag = 5 replacements = 5 M h1 [GeV] = 5 95 ϕ X t 15 0 π 15 M H ± = 150 GeV 15 X t = 700 GeV ϕ X t π Bands: Estimate of the size of the corrections of O(α b α s +α 2 t +α t α b +α 2 b ) [Slavich et al.], FeynHiggs Interpolation: Size of above corrections known for the MSSM with real parameters: Evaluate for ϕ Xt = 0 and ϕ Xt = π and interpolate O(α) : tan β = 5 O(α + α t α s ) : tan β = 5 O(α + α t α s ) + interpol. : tan β = 5 tan β = 15 tan β = 15 tan β = 15 Higgs mass M h1 does depend on the phase ϕ Xt, X t = 700 GeV. One-loop corrections are more sensitive to ϕ Xt for small M H ±.

13 Amplitudes with external Higgs bosons f Mixing between the Higgs bosons: φ i (DR/on-shell scheme) φ {i,j} = H 0, h 0, A 0 φ j f Finite wave function normalization factors needed: Ẑ i (Γ i + Ẑ ij Γ j + Ẑ ik Γ k ) Ẑ i ensures that residuum is set to 1 Ẑ ij describes transition i j Definition of mixing matrix (Ẑ ii = 1): Z ij = Ẑ i Ẑ ij Vertex with external Higgs boson: Z ii Γ i + Z ij Γ j + Z ik Γ k

14 Amplitudes with internal Higgs bosons Diagrams with internal Higgs bosons enter precision observables (W-mass,...): Calculation with Born states φ i = H 0, h 0, A 0 : no problem Calculation with φ i = h 1, h 2, h 3 Inclusion of higher order effects: One possibility: Use of effective couplings: Further approximations: Consider Z ij as mixing matrix: Problem: Z ij is a non-unitary matrix (no rotation matrix) effective potential approach: Z (ˆΣ(p 2 ) ) Z (ˆΣ(0) ) = R on-shell approximation: Z (ˆΣ(p 2 ) ) Z ( ReˆΣ(p 2 OS )) = U ˆΣ ii (p 2 os = M2 i Born ) ˆΣ ij `p2 os = (M 2 i Born + M 2 j Born )/2

15 Couplings One example: Coupling of two gauge bosons (V = W, Z) and one Higgs boson: g hi VV = [U i1 cos(β α) + U i2 sin(β α)] g HSM VV standard model coupling only CP-even components of the Higgs bosons couple to V all three Higgs bosons can have a CP-even component

16 Results: ϕ Xt -dependence of couplings ements gh1vv R, tan β = U, tan β = 5 M H ± = 150 GeV R, tan β = 15 U, tan β = 15 0 X t = 700 GeV ϕ X t π Here: g h1 VV is normalized to the standard model coupling. g h1 VV 2 does depend on the phase ϕ Xt, X t = 700 GeV. R p 2 =0 and U pos give similar results with only tiny differences.

17 Summary At Born level: no CP-violation in the Higgs sector Quantum corrections can induce CP-violation. Quantum corrections have to be taken into account: prediction of Higgs boson masses amplitudes with external Higgs bosons Z amplitudes with internal Higgs bosons R, U The dominant two-loop contributions O(α t α s ) with complete phase dependence are included into FeynHiggs (talk by T. Hahn).