SFB/CPP-08-14, TTP08-08, WUB08-02

Higgs boson mass in supersymmetry to three loops

R.V. Harlander^{1}, P. Kant^{2}, L. Mihaila^{2}, M. Steinhauser^{2}

1 Fachbereich C, Theoretische Physik, Universität Wuppertal, 42097 Wuppertal, Germany 2 Institut für Theoretische Teilchenphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany (Dated: April 4, 2008)

Within the minimal supersymmetric extension of the Standard Model, the mass of the light CP-even Higgs boson is computed to three-loop accuracy, taking into account the next-to-next-to-leading order effects from supersymmetric Quantum Chromodynamics. We consider two different scenarios for the mass hierarchies of the supersymmetric spectrum. Our numerical results amount to corrections of about 500 MeV which is of the same order as the experimental accuracy expected at the CERN Large Hadron Collider (LHC).

PACS numbers: 11.30.Pb, 12.38.-t, 14.80.Cp

I. INTRODUCTION

Supersymmetry is currently the most-studied extension of the Standard Model (see, e.g., Ref. [? ]). It provides solutions to some profound theoretical problems of the Standard Model: the fine tuning of the Higgs mass, the (non-)unification of gauge couplings, a mechanism for spontaneous symmetry breaking, and a Cold Dark Matter candidate.

The minimal supersymmetric extension of the Standard Model (MSSM) is based on a two-Higgs-doublet model (2HDM) with five physical Higgs bosons: two CP-even h/H, one CP-odd A (also named the “pseudoscalar” Higgs), and two charged scalars H^{±}. Each particle of this 2HDM receives a SUSY partner of opposite spin-statistics, where left- and right-handed components of a Standard Model Dirac fermion are attributed with separate scalars f̃_{L/R} which mix to the physical mass eigenstates f̃_{1/2}.

Compared to the Standard Model, the MSSM Higgs sector is described by two additional parameters, usually chosen to be the pseudo-scalar mass M_{A} and the ratio of the vacuum expectation values of the two Higgs doublets, tan β = v_{2}/v_{1}. The masses of the other Higgs bosons are then fixed by SUSY constraints. In particular, the mass of the light CP-even Higgs boson, M_{h}, is bounded from above. At tree-level, it is M_{h} < M_{Z}. Radiative corrections to the Higgs pole masses raise this bound substantially to values that were inaccessible by LEP [? ? ? ]. The large numerical impact is due to a contribution ∼ α_{t}M_{t}^{2} ∼ M_{t}^{4} coming from top- and stop quark loops (M_{t} is the top quark mass and \sqrt{α_{t}} is proportional to the top Yukawa coupling).

The one-loop corrections to the Higgs pole masses are known without any approximations [? ? ? ? ]. They show that the bulk of the numerical effects can be obtained in the so-called effective-potential approach in the limit of vanishing external momentum. Motivated by this observation, all presumably relevant two-loop terms have since been evaluated in this approach (for reviews, see e.g. Refs. [? ? ]). More recently there has been quite some activity in the context of the MSSM with complex parameters which can lead to sizeable effects (see, e.g., Ref. [? ]). The two-loop results are implemented in the numerical programs FeynHiggs [? ] and CPsuperH [? ? ] using on-shell particle masses, and in SoftSusy [? ], SPheno [? ], and Suspect [? ] using \overline{DR} parameters. The influence of terms that go beyond the approximation of vanishing external momentum has been investigated in Ref. [? ].

Based mostly on the renormalization scale and scheme dependence, the theoretical uncertainty on the prediction of the light Higgs boson mass M_{h} has been estimated to 3-5 GeV [? ? ]. This is to be compared with the expected experimental uncertainty of a Higgs mass measurement at the LHC of the order of 100-200 MeV [? ]. At an International Linear Collider, this goes even down to roughly 50 MeV [? ]. These numbers clearly show the need for three-loop corrections to the SUSY Higgs bosons masses in order to fully exploit the physics potential of these colliders.

In fact, quite recently the leading and next-to-leading logarithmic terms in ln(M_{SUSY}/M_{t}) at three-loop level have been obtained, where M_{SUSY} is the typical scale of SUSY particle masses [? ]. In this letter, we want to present the first genuine three-loop calculation of the lightest Higgs boson mass, focusing on a few simplifying limiting cases for the sake of brevity. In particular, we consider effects of order α_{t}α_{s}^{2}, keep only the leading terms ∼ M_{t}^{4}, and neglect all mixing effects in the stop sector. More general results and their detailed phenomenological impacts shall be deferred to a later publication.

II. THE HIGGS BOSON MASS IN THE MSSM

At tree-level, the mass matrix of the neutral, CP-even Higgs bosons h, H has the following form:

M_{H,tree}^{2} = \frac{sin 2β}{2} ×  (1)

( \begin{matrix}M_{Z}^{2} cot β + M_{A}^{2} tan β & −M_{Z}^{2} − M_{A}^{2} \\ −M_{Z}^{2} − M_{A}^{2} & M_{Z}^{2} tan β + M_{A}^{2} cot β \end{matrix} ) .

The diagonalization of M_{H,tree}^{2} gives the tree-level result for M_{h} and M_{H}, and leads to the well-known bound M_{h} < M_{Z} which is approached in the limit tan β → ∞.

Quantum corrections to the Higgs boson masses are incorporated by evaluating the poles of the Higgs boson propagator at higher orders. As mentioned in the Introduction, the numerically dominant contributions can be obtained in the approximation of zero external momentum (see, e.g., Refs. [? ]) which we will adopt in the following. Furthermore, we will only consider corrections of order α_{t}α_{s}^{2}. Apart from the quark, squark, and gluino masses, there is another parameter with mass dimension, the trilinear coupling of the soft SUSY breaking terms, A_{t}. Before renormalization, we express it through the stop masses Mt̃_{1}, Mt̃_{2}, the stop mixing angle θ_{t}, and the bilinear Higgs parameter μ_{SUSY} as follows:

2M_{t}A_{t} = (M 2 ; t̃_{1} − M 2 ; t̃_{2}) sin 2θ_{t} + 2M_{t}μ_{SUSY} cot β . (2)

The mass matrix M_{H}^{2} is obtained from the quadratic terms in the Higgs boson potential constructed from the fields ϕ_{1} and ϕ_{2}. They are related to the physical Higgs mass eigenstates via a mixing angle α. Since ϕ_{1} does not couple directly to top quarks, it is convenient to perform the calculations of the Feynman diagrams in the (ϕ_{1}, ϕ_{2}) basis.

Including higher order corrections, one obtains the Higgs boson mass matrix

M_{H}^{2} = M_{H,tree}^{2} − ( \begin{matrix}Σ̂_{ϕ_{1}} & Σ̂_{ϕ_{1}ϕ_{2}} \\ Σ̂_{ϕ_{1}ϕ_{2}} & Σ̂_{ϕ_{2}} \end{matrix} ) ,  (3)

which again gives the physical Higgs boson masses upon diagonalization. The renormalized quantities Σ̂_{ϕ_{1}}, Σ̂_{ϕ_{2}} and Σ̂_{ϕ_{1}ϕ_{2}} are obtained from the self energies of the fields ϕ_{1}, ϕ_{2}, A, evaluated at zero external momentum, as well as from tadpole contributions of ϕ_{1} and ϕ_{2} (see, e.g., Ref. [? ]). Let us remark that if one sets Mt̃_{1} = Mt̃_{2} and A_{t} = 0, and evaluates only the leading contribution ∼ M_{t}^{4}, then only Σ̂_{ϕ_{2}} ≠ 0 and the matrix M_{H}^{2} − M_{H,tree}^{2} is diagonal. On the other hand, if we allow for non-zero A_{t}, also Σ̂_{ϕ_{1}} and Σ̂_{ϕ_{1}ϕ_{2}} contribute in general.

The calculation of Σ̂_{ϕ_{2}} is organized as follows: All Feynman diagrams are generated with QGRAF [? ]. In order to properly take into account the Majorana character of the gluino, the output is subsequently manipulated by a PERL script which applies the rules given in Ref. [? ]. The various diagram topologies are identified and transformed to FORM [? ] with the help of q2e and exp [? ? ]. The program exp is also used in order to apply the asymptotic expansion (see, e.g., Ref. [? ]) in the various mass hierarchies. The actual evaluation of the integrals is performed with the package MATAD [? ], resulting in an expansion in d − 4 for each diagram, where d is the space-time dimension. The total number of three-loop diagrams amounts to about 16,000.

At three-loop level we need to renormalize the top quark mass, the top squark mass, and the stop mixing angle at the two-loop order. In addition, the one-loop counterterm of the gluino mass is needed for the renormalization of the two-loop expression. We implement Dimensional Reduction (DRED) with the help of the socalled ϵ-scalars which appear for the first time at two loops. The renormalization of the ϵ-scalar mass is performed in the on-shell scheme, requiring that the renormalized mass is equal to zero. In the literature this is referred to as \overline{DR}′ scheme.

The one-loop on-shell counterterms are well-known (see, e.g., Refs. [? ? ? ? ]). As far as the two-loop counterterms for the squarks and quarks are concerned, one can find the results in Refs. [? ? ]. However, it is rather tedious to extract the results for the mass hierarchies we are interested in. Thus, we re-computed the corresponding corrections.

To our knowledge, the two-loop counterterm for the stop mixing angle is not yet available in the literature. It turns out that in our approximation, where Mt̃_{1} = Mt̃_{2} and A_{t} = 0, only the one-loop counterterm of the mixing angle enters the three-loop result.

As a cross check for our calculation, we re-calculated the exact two-loop result (in the limit of vanishing external momentum). The expressions are rather involved, so we settled for a numerical comparison which showed perfect agreement [? ]. Furthermore, the expansion of the exact expressions confirms the limiting cases discussed below. Both the two- and three-loop calculations are performed for a general QCD gauge parameter ξ_{S}. The independence of the final results on ξ_{S} serves as another welcome check on the correctness of our result.

We use anti-commuting γ_{5} which is allowed for fermion traces which involve an even number of γ_{5} matrices. It turns out that all traces involving an odd number of γ_{5} vanish because they contain less than four gamma matrices.

In the following we discuss three different cases for the mass hierarchy. In all cases we set the light quark masses to zero.

(i) Supersymmetric limit, i.e., M_{t} = Mt̃ and the gluino and other squarks are massless: Mg̃ = Mq̃ = 0. The quantum corrections to the Higgs boson mass vanish in this case, as required by supersymmetry. Still, the individual diagrams are different from zero and thus the calculation imposes a strong check on our setup.

(ii) Massless gluino, Mg̃ = 0. Expanding in the limit M_{t} ≪ Mt̃ = Mq̃ ≡ M_{SUSY}, we obtain for the leading