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Fermionic decays of neutral MSSM Higgs bosons at the one-loop level

The results of a complete one-loop calculation for the fermionic decay width h^0, H^0, A^0 ->f bar(f) of the neutral MSSM Higgs bosons are presented and the dominant light Higgs decay channel h^0 ->b bar(b) is discussed in detail. The enhancement of the partial decay width h^0 ->b bar(b) compared to the standard Higgs decay is shown for pseudoscalar masses M_A<300 GeV. Simpler approximation formulae for the Higgs decays are given and their quality is discussed by introducing an effective neutral scalar mixing angle sin(alpha)_eff. Finally the Higgs branching ratios in b bar(b), c bar(c), tau^+ tau^- are calculated.


Introduction
Supersymmetry is at present the most predictive framework for physics beyond the standard model [1]. One theoretical motivation is the cancellation of quadratically divergent contributions to the mass of the scalar Higgs particle. This problem of naturalness is solved in supersymmetric theories. Supersymmetric models allow the unification of gauge couplings at the GUT scale O(10 15 GeV ) [2].
The minimal supersymmetric standard model (MSSM) is considered as the most general supersymmetric extension of the standard model (SM) at low energies [3]. The Higgs sector of the MSSM is that of a 2-Higgs-doublet model, where the coefficients of the Higgs potential are restricted by supersymmetry. As a consequence of the supersymmetric Higgs potential, a light Higgs boson exists with a tree level upper mass bound given by the Z 0 mass. Radiative corrections to the Higgs mass spectrum, however, predict an upper limit of the light Higgs mass O(130 GeV) [4,5]. Calculations were performed at the one-loop level using renormalization group technique [6], effective potential approximation [7] and one-loop calculations with top and stop contributions [8,9]. Two-loop effects to the upper limit of the lightest Higgs boson mass are discussed in [10].
Production and decay properties of the Higgs boson are charcteristic quantities for the experimental Higgs search at LEP 2, at a 500 GeV e + e − collider and LHC. Precise predictions of these quantities require the inclusion of radiative corrections. The one-loop Higgs production cross section and their respective branching ratios (decay width) at e + e − and pp colliders may allow to distinguish between a standard or MSSM Higgs sector. As a first step, complete on-shell renormalization schemes for the MSSM Higgs sector were presented [8,9,11,12].
In this article the complete one-loop partial decay width of the neutral MSSM Higgs bosons h 0 (H 0 , A 0 ) → ff is calculated within the on-shell scheme [12]. The dominant fermionic decay width of the light MSSM Higgs boson is discussed in detail and the branching ratios of the light scalar Higgs h 0 in bb, cc, τ + τ − are presented. The discussion points out the differences of the MSSM and the standard Higgs decay width and branching ratios.
Section 2 presents an overview of the Higgs production and decay mechanisms, including radiative corrections. The fermionic decay width with full one-loop corrections is calculated in section 3. Finally the numerical results for the decay width and branching ratios are discussed in section 4 for QED/QCD, gluino and weak MSSM virtual contributions. Vertex corrections and self energies are given in the appendix.

Decay channels of the neutral Higgs boson
Once a Higgs boson is found, it is of importance to investigate its characteristic decay properties. The observed cross section is composed by the production cross section for e + e − → Zh(H) or e + e − → Ah(H) and the branching ratios for the subsequent decays of the scalar bosons. The decay width (respectively the branching ratios) as well as the mass-width correlation are the quantities to differentiate between Higgs bosons of various origin. In the following we briefly review the decay modes of the neutral MSSM Higgs particles and discuss the most important fermionic decays in some more detail. Except from a small part of the parameter space, the fermionic decays are the only decay modes of the light Higgs allowed at tree level. The bosonic decays H 0 → ZZ, W W (which are dominant above the threshold in case of the standard Higgs) are kept small also for increasing M H 0 by the factor cos 2 (α − β).
In Tab. 1 we list the various decay channels of the neutral Higgs bosons indicating the level of the theoretical predictions by: full electroweak: complete 1-loop electroweak calculation performed and available QCD: QCD corrections performed and available improved Born: decay width is calculated including the complete 1-loop scalar 2point functions. The signature • denotes the corresponding decay mode as proceeding through 1-loop in lowest order.

Tree level structure
The Higgs sector of the MSSM consists of two scalar doublets H 1 , H 2 with opposite hypercharge Y 1 = −Y 2 = −1 and vacuum expectation values v 1 , v 2 [1,12]. The Higgs potential contains two independent free parameters, which can conveniently be chosen as tan β = v 2 /v 1 and M A , where M A is the mass of the A 0 boson.
Radiative corrections in the MSSM Higgs sector modify the tree-level decay rate Eq. (3.2) substantially, with the main effect from loops involving the top quark and its scalar partnert. The complete decay width comprises the following radiative corrections to (i) the physical neutral scalar MSSM Higgs (ii) the full one-loop decay amplitude H → ff (iii) ∆r in the MSSM.

One-loop structure H → ff
In this article the one-loop decay amplitudes H → ff are calculated within the complete one-loop renormalization scheme for the MSSM Higgs sector described in [12]. The one-loop decay widths for h 0 , H 0 , A 0 → ff with full electroweak MSSM corrections read:

4)
Z h 0 and Z H 0 are finite wave function renormalizations of the external light and heavy Higgs particles: .
The pseudoscalar Higgs self energy and A 0 G 0 mixing in the decay width Γ 1 (A 0 → ff ), Eq. (3.4), do not contribute, because the renormalization condition sets the residue of the pseudoscalar Higgs propagator equal to one. The Slavnov-Taylor identity yields: whereΣ A 0 Z 0 (M 2 A ) = 0 [12]. The renormalized vertex correction ∆T H in Eq. (3.4) is the sum of the one-loop vertex diagrams ∆T i given in appendix A and a counterterm CT: The counterterm CT reads: with the non-universal contribution from the fermion self energies (App. A): (3.8) and the universal part:

The one-loop mixing angle sin α
Introducing universal one-loop coupling coefficients via an effective universal one-loop mixing angle α 1−loop can be defined through the coefficients in Tab. 2 : A good approximation for this mixing angle α 1−loop , including only the leading oneloop term 12) follows from the diagonalization of the one-loop Higgs mass matrix [23] M Higgs = sin 2β 2 This approximate effective mixing angle α ef f is determined by where M h 0 ,ef f is the solution for the light Higgs mass [7] : The mixing angle α ef f corresponds to the effective potential approach with top and stop contributions.
Incorporating also the other correction terms of Eq. (3.4), a complete one-loop effective mixing angle α f can be defined with the full one-loop width Γ 1 , Eq. (3.4), normalized to the standard tree level width Γ SM,0 . Besides the universal mixing angle α 1 , defined in Eq. (3.11), ∆r M SSM is another universal contribution, whereas the residual, mainly vertex correction δ f V is fermion specific making the complete one-loop effective mixing angle flavour dependent. A momentum dependent mixing angle α has been defined in [24]. Figs. 1 a,b as a function of the renormalized light Higgs mass M h 0 and for fixed tan β values. Together with the complete result (dashed line), the approximations in Eqs. (3.11, 3.14) are also shown (dotted, full). In the range tan β ≤ 30 the full calculation is about 8% below the approximation (3.14).

Discussion
Radiative corrections to the partial decay width H → ff include QED/QCD, weak MSSM and virtual gluino contributions. The following subsections discuss these corrections and present the numerical sizes individually. QED/QCD corrections are identical with the standard Higgs decay H SM → ff contributions. Weak MSSM contributions give sizeable contributions to the partial decay width of the neutral light and heavy Higgs boson in the intermediate pseudoscalar Higgs mass range M A . Vertex corrections with virtual gluino exchange are discussed separatly for the decay channel h 0 (H 0 , A 0 ) → ff . Finally the branching ratios h 0 → bb, cc, τ + τ − are presented.

QED and QCD corrections
QED/QCD corrections to the partial decay width H → ff appear through vertex diagrams with virtual photons/gluons and final state photon/gluon radiation [14,15,16]: where δ QED for the scalar Higgs bosons reads: The pseudoscalar Higgs decay width A → ff receives a QED correction δ QED different from Eq. (4.2), [16] : where (4.4) In the region m 2 f ≪ M 2 H the QED correction for both scalar and pseudoscalar Higgs particles, sufficiently accurate, is given by: The renormalization group improved QCD corrected decay width reads [14]: where m q (M 2 H ) is the effective quark mass from the renormalization group equation [25] m q (q 2 ) = m q,0 β 0 α s (q 2 ) 2π and m q (m 2 q,0 ) = m q,0 is the on-shell mass. The coefficients in Eq. (4.7) are: (4.8) where ζ(3) ≈ 1.2020596... and N f = 6 for q 2 > 4m 2 t . The strong coupling constant α s with three-loop contributions is given by: (4.9) where L q = log(q 2 /Λ 2 QCD,N f ) and Λ QCD follows from the condition: The QED corrections are small for b-quarks (−δ QED < 0.4% for M H up to 1 TeV) and somewhat bigger for c-quarks (< 1.8%) and τ -leptons (< 4% ). QED contributions to top quark decays are ≤ 0.4% for M H > 500 GeV. Near the top production threshold, the Coulomb singularity appears and non-perturbative effects have to be taken into account (we do not consider these subtleties here). The fermionic decay width for a scalar Higgs vanishes for M H → 2m t [16], whereas for the pseudoscalar Higgs a finite contribution remains.
The QCD corrections to the hadronic decay width in terms of the on-shell masses are large: in the bb (cc) channel, δ QCD = −39% for a light Higgs < 2M W , increasing up to −60% (−75%) for 1 TeV Higgs boson. The dominant part can be absorbed in the running quark masses. For M H sufficiently above the tt threshold, the QCD corrections to the H → tt width are typically δ QCD ≤ 15%.

Weak MSSM corrections
The partial decay width h 0 → bb is the dominant decay channel for a light scalar MSSM Higgs and a standard Higgs with a mass below M H < 140 GeV. A precise prediction of the H → bb decay width and the fermionic bb, cc, τ + τ − branching ratios requires the inclusion of radiative corrections at the one-loop level. One-loop contributions to the bb decay channel are discussed in detail for the light (heavy) scalar and pseudoscalar Higgs decay width h 0 (H 0 , A 0 ) → bb. The cc, τ + τ − decay channels are presented within the context of the Higgs branching ratios in the following subsection.  [26]. In Figs. 2 a,b the upper limit of the light Higgs mass M h 0 increases ∼ m 4 t , as discussed in Eq. (3.15). For a light Higgs mass M h 0 < 80 GeV and tan β ≥ 5 the partial decay width h 0 → bb is almost insensitive on m t . Γ 1 (h 0 → bb) increases with ∼ m 4 t for a light Higgs mass M h 0 > 80 GeV. This m 4 t dependence of the partial decay width is a universal contribution of the external Higgs two-point functions, described by the one-loop mixing angle sin 2 α ef f , in Eq. (3.14). Figs. 2 c,d show the partial decay width as a function of the pseudoscalar Higgs mass M A . The decay width Γ 1 (h 0 → bb) reaches a maximum near 90 GeV < M A < 110 GeV and decreases for M A > 110 GeV. Figs. 2 a-d also show the standard Higgs partial decay width H SM → bb with one-loop weak corrections [15,27] For large pseudoscalar masses M A → ∞, however, the MSSM decay width approaches the standard model result closely. This behaviour is discussed in terms of Eq. (3.11), where sin 2 α → cos 2 β in the limit M A → ∞. In Fig. 2 the pseudoscalar Higgs mass range is chosen up to 300 GeV. The gap between the MSSM and the standard decay width is sizeable for M A = 300 GeV. Larger pseudoscalar masses tend to approach the standard model decay width as can be seen in Figs Eq. (3.15). Mixing effects from the left and right sfermions states are shown in Fig. 3 c for tan β = 2, 30. The off-diagonal mixing parameter A ′ t in Eq. (4.14) is A ′ t = 0, 100 GeV, 200 GeV, 300 GeV, 400 GeV. These mixings increase the light Higgs M h 0 and the partial decay width Γ 1 (h 0 → bb) simultaneously.
Effects from gaugino soft breaking parameters M, µ are displayed in Fig. 3 d. The dependence of the partial decay width Γ 1 (h 0 → bb) on M is shown for M = 100, 200 GeV to be very small. µ enters the sfermion mass matrix, Eq. (4.13), in the off-diagonal entries and in the Higgs-sfermion couplings. In Fig. 3 d no left-right mixing is present by fine-tuning the A parameter. Large tan β values (tan β ≥ 30), however, show sizeable effects for the partial decay width, since the parameter A increases with tan β, µ and contributes to the H-f L -f R couplings. For lower tan β values the partial decay width is almost insensitive on µ.

Vertex corrections of virtual gluinos
The supersymmetric partners of the SU (3) gluons, the gluinosg a , appear as virtual states in the H → qq vertex corrections (together with squarks) with the strong coupling constant α s . They contribute a shift δΓ Gl in the decay width for h 0 , H 0 , A 0 → qq. where is the one-loop gluino contribution to the decay width H → qq. The vertex correction ∆T H Gl and self energies Σ f S,Gl , Σ f V,Gl are given in Eq. (4.18, 4.22). Mixing effects from virtual squarks in the vertex corrections ∆T H Gl and quarks self energies Σ f Gl are described by a 2 × 2 squark mass matrix: with SUSY soft breaking parameters MQ, MŨ ,D , A q , and µ. The notation in the offdiagonal entries in Eq. (4.13): will be used. In the following discussion the soft breaking parameters are taken to be equal m sf = MQ = MŨ ,D . Up and down type squarks in (4.13) are distinguished by setting f=u,d and the {u, d} entries in the parenthesis. The parameter µ in the off-diagonal matrix elements in (4.13) is also present in the gaugino sector. The sfermion masses, obtained from diagonalizing (4.13) are: where the corresponding rotation matrices (4.16) are described by the sfermion mixing angle θq: .
( 4.17) The one-loop vertex correction for the scalar neutral Higgs decay h 0 (H 0 ) → qq is given by: (4.18) are the tree-level couplings, δ ij is the unit matrix and The pseudoscalar Higgs decay A 0 → qq yields the vertex correction: with T A 0 f f in Eq. (4.19) from Tab. 2 and ǫ 12 = −ǫ 21 = 1. T H i,j are the H-q i -q j couplings in the squark mass eigenstate fields, obtained by the transformation: where The first (second) column in the parentheses belongs to the up (down) squark coupling. In Eq. (4.12) the fermion self energy with virtual gluinos and squarks for the scalar (vector) components reads: The numerical analysis of the gluino contributions δΓ Gl is shown in

Fermionic branching ratios H → ff
Branching ratios of the fermionic Higgs decay channels are experimentally measurable, even if the partial decay width Γ(H → ff ) can not be measured directly. In the following the branching ratios of the light neutral scalar MSSM Higgs h 0 and the standard Higgs H SM in bb, cc, τ + τ − are presented. Here we restrict the discussion to the fermionic branching ratio R f , given by: where the light fermion contributions are negligible. Figs. 6 a,b show the light neutral MSSM branching ratios R f for bb (Fig. 6 a) and the cc, τ + τ − decay channels (Fig. 6  b), where the full one-loop contributions from section 4. The branching ratio R f in Eq. (4.23), where the approximation formulae Eq. (3.14) for the partial decay width is used instead:

Conclusions
The fermionic partial decay width Γ 1 (H → ff ) for the neutral MSSM Higgs bosons h 0 , H 0 , A 0 is calculated with full one-loop MSSM contributions for the decay channels bb, cc, τ + τ − . In the calculation, the renormalization scheme for the supersymmetric Higgs sector [12] was used. The tree-level decay width for down (up) type fermions is enhanced (suppressed) compared to the standard model Higgs decay width. One-loop corrections in the pseudoscalar Higgs mass range 80 GeV ≤ M A ≤ 110 GeV give large corrections ∼ m 4 t to the decay width Γ 1 (h 0 (H 0 ) → ff ). The diagonalization of the neutral scalar Higgs mass matrix, described by the mixing angle α 1−loop , receives the dominant contributions from top and stop loops O(m 4 t ). The mixing angle sin 2 α is calculated with full oneloop contributions (sin 2 α 1−loop ), in the effective potential approximation (sin 2 α ef f ) and as a flavour dependent effective mixing angle (sin 2 α f ). The mixing angles sin 2 α 1−loop , sin 2 α ef f and sin 2 α f are in agreement within 8%. For large pseudoscalar Higgs masses M A → ∞ the decay width Γ 1 (h 0 → ff ) approaches the standard model result. In this limit, non-universal model dependent one-loop contributions to the decay width Γ 1 can distinguish between a standard and MSSM Higgs boson and depend in detail on the chosen parameters. Virtual gluino vertex corrections give sizeable contributions to the H → bb decay width (branching ratios). The branching ratios for the light neutral scalar Higgs decay h 0 → bb, cc, τ + τ − are presented. The full calculation and the approximation formulae Eq. (4.24) are in agreement within 0.2 − 0.6%.

Acknowledgements.
I am grateful to W. Hollik for suggesting this topic and reading the manuscript and to S.
Pokorski for various discussions.

A Vertex corrections and self energies
The Feynman rules of the minimal supersymmetric standard model are given in [1]. All analytical formulae are calculated in the 't Hooft-Feynman gauge. The two-and three-point functions B 0 , B 1 , B 22 , C 0 , C 1 and C 2 are defined at the end of appendix A. f ′ denotes the isospin partner for the external fermion f in the same isodoublet.
• Scalar MSSM h 0 (H 0 ) → ff vertex corrections * : • Pseudoscalar MSSM A 0 → ff vertex correction: The chargino and neutralino mass matrix V ij , U ij , N ij are given in appendix B. The vertex correction diagrams are described by the following functions with masses and couplings in its arguments.
Fermion self energies are represented by the scalar functions Σ f S,V,A through the decomposition in the scalar, vector and axialvector part: Standard model and genuine SUSY self energy contributions are listed separately in the following: and the mass scale µ are the UV-parameters from dimensional regularization in Ddimensions.
with k 2 = (p − p ′ ) 2 and the denominators and for different masses m 1 , m 2 , m 3 : The analytic expression for the scalar vertex integral C 0 can be found in [28].

B Gaugino mass matrix
The chargino 2 × 2 mass matrix is given by with the SUSY soft breaking parameters µ and M in the diagonal matrix elements. The physical chargino mass statesχ ± i are the rotated wino and charged Higgsino states: V ij and U ij are unitary chargino mixing matrices obtained from the diagonalization of the mass matrix (B.1): The neutralino 4 × 4 mass matrix yields: B.4) where the diagonalization introduces the unitary matrix N ij by:           Figure 6: