Electroweak corrections to Neutralino and Chargino decays into a -boson in the (N)MSSM
We present the complete electroweak one-loop corrections to the partial widths for two-body decays of a chargino (neutralino) into a -boson and a neutralino (chargino). We perform the calculation for the minimal and the next-to-minimal supersymmetric standard model using an on-shell renormalization scheme. Particular attention is paid to the question of gauge invariance which is achieved using the so-called pinch technique. Furthermore we show that these corrections show a strong parameter dependence and usually are in the range of - percent if the neutralino involved is a higgsino or wino like state. However, in case of a bino-like or a singlino-like neutralino the corrections can go up to % and more. Moreover we present the public program CNNDecays performing these calculations.
A new energy frontier has been opened with the start of the LHC, namely the exploration of the tera-scale. An important part of the experimental program of the LHC is the search for supersymmetric particles Weiglein:2004hn; Ball:2007zza; Aad:2009wy. If they are indeed found, the determination of the underlying parameters will then be among the main tasks. Here the properties of neutralinos and charginos will play an important role as they occur in various steps in the cascade decays of squarks and gluinos which can be copiously produced at the LHC. These investigations will then most likely be complemented and continued at a prospective future linear collider (ILC) AguilarSaavedra:2001rg; Weiglein:2004hn and/or at a multi-TeV collider such as CLIC Accomando:2004sz.
The two-body decay of a chargino (neutralino) into a -boson and a neutralino (chargino) is one of the most important modes if kinematically allowed Bartl:1985fk; Baer:1988kx; Feng:1995zd. The determination of the corresponding branching ratio can give important information on the nature of the chargino and neutralino involved and therefore one will need at least an ILC for its precise value. This is one of the reasons to investigate the one-loop corrections to this decay mode. We will see below that the corrections can be as large as the LO decay width in certain regions of parameters space. A second important aspect is the question how to perform the complete electroweak corrections in a gauge invariant way. These decay modes offer an ideal play ground to tackle the technical questions involved.
We will use an on-shell renormalization scheme for masses and couplings. The on-shell renormalization of the mass matrices of charginos and neutralinos in the minimal supersymmetric standard model (MSSM) was first discussed in Eberl:2001eu; Fritzsche:2002bi and extensively addressed in Baro:2008bg. Moreover, electroweak corrections for these decays taking only into account third generation squarks and quarks within the MSSM have been calculated in Zhang:2001rd. Corrections of up to % have been found there. Note, that for these corrections gauge invariance is not an issue.
We will work in the general linear -gauge. An important aspect will be the question how to renormalize the mixing matrices of charginos and neutralinos, since it is already known from the Standard Model that the on-shell renormalization prescription according to Denner:1990yz; Denner:1991kt leads to a gauge dependent CKM-matrix Gambino:1998ec which in turn implies a gauge dependence of processes like at next-to-leading order level Yamada:2001px.
Since then, some papers proposed different solutions to the problem addressed above: whereas Espriu:2002xv argued that missing absorptive parts due to the unstable nature of the external particles have to be included in the calculation, Yamada:2001px proposed a method on how to construct a gauge invariant counterterm for the mixing matrices inspired by the pinch technique pinch, which defines gauge independent form factors for gauge bosons. Another perspective is presented in Pilaftsis:2002nc, where the gauge-variant on-shell renormalized mixing matrix is related to a gauge independent one in a generalized scheme of renormalization.
For the special case of the CKM matrix different methods were discussed in the literature: Gambino:1998ec proposed to set the momenta of the non-diagonal entries of the quark self-energies to zero, Denner:2004bm; Kniehl:2006bs; Kniehl:2006rc suggested new variants of renormalization in particular for the mixing matrices themselves partially based on physical processes which allows one to get gauge independent decay widths. For lepton or neutrino mass matrices also Diener:2001qt; Almasy:2009kn proposed useful renormalization conditions allowing gauge independent results.
In this paper we will apply the technique presented in Yamada:2001px. The technical features are the same for the MSSM and the next-to-minimal one (NMSSM). As for the processes under consideration the MSSM is a trivial sub-class of the NMSSM in the limit of heavy singlet states we will perform the calculation right from the start in the NMSSM. The paper is organized as follows: in Section 2 we fix our notation for both models and in Section 3 we summarize the tree-level results for the mass matrices of charginos and neutralinos as well as the tree-level decay widths for and . The virtual one-loop contributions and real corrections including the question of gauge invariance are discussed in Section LABEL:sec:oneloopdecaywidth. We present our numerical result in Section LABEL:sec:results and in Section LABEL:sec:conclusions our conclusions.
LABEL:sec:vertexcorr contains the generic formulas for the matrix element contributions of the vertex corrections. In LABEL:sec:photemission we give the formulas for the the real corrections which are non-factorizable as the higgsinos form a vector-like gauge state. The usage of the -gauge implies that all possible derivatives of the two-point one-loop functions show up in the calculation. Some of them are to our knowledge not available in analytical form in the literature and, thus, we provide them for completeness in LABEL:sec:oneandtwopointfunctions. In LABEL:sec:genericfortran we present the program CNNDecays for the numerical evaluation of these corrections.
2 The models
In this section we fix the notation for the models considered, the MSSM Chung:2003fi and the NMSSM Ellwanger:2009dp. Detailed formulas for mass-matrices and couplings can be found in Chung:2003fi; Staub:2010ty. Both model have in common the Yukawa part of the matter superfields coupled to the -doublet Higgs fields
where is the complete antisymmetric tensor with . In case of the MSSM the -term is added and the total superpotential reads as
whereas in the NMSSM the Higgs doublets are coupled to a gauge singlet superfield
If the scalar component of the gauge singlet gets a vacuum expectation value an effective -term is generated
In this way one gets naturally of the order of the electroweak scale Ellwanger:2009dp. To complete the models one has to add the soft-SUSY breaking terms, where the part common to both models reads as111We use the notation of Skands:2003cj; Allanach:2008qq.:
with as summation indices. The complete soft-SUSY breaking potential of the MSSM is then given by
and the NMSSM one by
3 Tree-level results
Here we collect the tree-level results which form the basis for the one-loop corrections considered in the subsequent sections.
3.1 Masses of Charginos and Neutralinos
Using Weyl spinors in the basis , the chargino mass matrix is given in both models by