Last modified 5 years ago Last modified on 04/06/12 16:33:02

The MSSM implementation

One of the most popular extensions of the Standard Model is TeV scale supersymmetry. Supersymmetry solves the problem of quadratically divergent corrections to the Higgs boson mass by the introduction of new bosonic particles having the same couplings as the Standard Model fermions, and new fermions having the same couplings as the Standard Model bosons, thus cancelling the loop contributions to the Higgs mass to all orders. The Minimal Supersymmetric Standard Model, MSSM, represents the minimal particle content for a supersymmetric extension of the Standard Model together with the maximum coupling space allowed by so-called soft supersymmetry breaking terms in the effective low-energy Lagrangean. These are constructed not to introduce new divergencies in any couplings, and therefore maintain the cancellations of quadratically divergent corrections to the Higgs mass. For an introduction to supersymmetry and the MSSM.

The implementation of the MSSM particles and vertices into MadGraph II was made in Cho:2006sx,Hagiwara:2005wg, following the conventions of Refs. Gunion:1984yn and Plehn:1998nh. Specifically, it is restricted to the minimal supersymmetric model conserving $R$-parity, without CP-violating phases and with diagonal CKM and MNS matrices. Higgs Yukawa couplings as well as mixing between right- and left-handed sfermions are implemented only for the third generation. However, no specific supersymmetry breaking scheme is assumed, so the spectrum and couplings of the supersymmetric particles can be produced with any spectrum generator regardless of the assumptions going into its calculations. The spectrum and couplings of the particles are read through SUSY Les Houches Accord files Skands:2003cj.

In order to consistently calculate decay widths and the dependent parameters, a model calculator for the MSSM is available. MSSMCalc takes a SUSY Les Houches Accord (SLHA) file Skands:2003cj from any Spectrum generator as input, and produces a MadEvent readable file, param_card.dat, with the missing Standard Model parameters, as well as decay widths for all supersymmetric particles (calculated at leading order by Sdecay Muhlleitner:2003vg, the Higgs particles and the top, $W^\pm$ and $Z$ particles. Care has been taken to ensure that the parameters used in the calculation of decay widths are as similar as possible to the parameters used in MadEvent, since the correct total decay widths are vital to get the correct tree-level cross-sections for processes involving decaying particles.

In the default run mode, MSSMCalc uses the Standard Model parameters given in the SUSY Les Houches accord ($\alpha_{em}$, $G_F$ and $M_Z$) to calculate the parameters $\sin\theta_W$ and $M_W$, which are stored in a MadEvent specific block MGSMPARAM in the resulting param_card.dat. The $b$ quark pole mass is calculated from the $\overline{MS}$ mass at 2-loop order. Another option is to extract the Standard Model parameters (and the vacuum expectation value ratio $\tan\beta$) from the chargino and neutralino mixing matrices, in order to ensure unitarity of ino-ino scattering at high energy. In this mode, also the Yukawa masses of the third generation fermions are extracted from the third generation sfermion mixing matrices. For a thorough essay discussion of this option, see section II C of Cho:2006sx.

The strong coupling $\alpha_s$ is calculated in MSSMCalc using 2-loop renormalisation group running in the $\overline{MS}$ scheme, at the scale specified in the GAUGE block statement. The value used for the strong coupling $g$ in the decay width calculations is stored for comparison in the block GAUGE, parameter 3. Note however, that the value of $\alpha_s$ used in MadEvent is given by the choice of parton distribution function and the scale chosen in the run.

If there are blocks missing in the SLHA file which are necessary for Mad\-Event, MSSMCalc will produce a param_card.dat file containing error messages.

The SUSY Les Houches blocks and parameters used by MadEvent are given in the table bellow. All blocks in the table should be provided by the user (and are indeed provided by most MSSM spectrum generators), except for the MGSMPARAM and the DECAY blocks which are produced by the parameter calculator MSSMCalc. Note that if parton density functions (PDFs) are used in the MadEvent run, the value for $\alpha_s$ at $M_Z$ and the order of its running is given by the PDF. Otherwise $\alpha_s(M_Z)$ is given by block SMINPUTS, parameter 3, and the order of running is taken to be 2-loop. The scale where $\alpha_s$ is evaluated is however always given by the scale parameter in the run_card.dat.

Block Comment
SMINPUTS Except for 5, the $b$ quark $\overline {MS}$ mass
MGSMPARAM Extra block with $\sin\theta_W$ and $M_W$, written by MSSMCalc
MASS Including 5, the $b$ quark pole mass
HMIX Only parameters 1 ($\mu$) and 2 ($\tan\beta$)
AU,AD,AE Only the third generation parameter 3 3
YU,YD,YE Only the third generation parameter 3 3
DECAY For all SUSY particles, Higgs bosons, top, $W^\pm$ and $Z$

-- Main.MichelHerquet - 09 Apr 2007