1.1. Position of the problem

SUSY is a very attractive theory, yet it requires a lot of extra parameters to break it softly: enough not to see it, but not so much as to lose its advantages (like solving the hierarchy problem). One counts about 80 soft breaking terms, plus possibly 40 CP-violating phases in the MSSM (without mentionning MSSM extended to R-parity breaking).

That many free parameters isn't manageable when one consider a realistic physics simulation, from the Lagrangian to the detector efficiencies. One usually restricts the number of free parameters with GUT-like universality assumption: at some high-scale, where the gauge coupling unify, the soft-breaking term will also get common value. The most common set of universality assumptions is the minimal SUper-GRAvity (mSUGRA) model: one common value for gaugino masses, one for scalar masses, and one for trilinear coupling.

In order to extract the physical spectrum of this model, one has to ``run down'' the parameters from the Unification scale to the electroweak scale through the Renormalisation Group Equations (RGE). This is only a rough sketch: in fact, one must stop at all scales corresponding to each particle mass:

Pole masses

To lift the ambiguity on the scale at which one evaluate the running masses, one has to go to the definition of pole mass. The pole mass is a proper observable: renormalisation group invariant, gauge invariant, infrared safe.

Computing it involves two things:

• evaluating the running mass at the scale of the pole mass (leading logarithm)
• evaluating the self-energy finite correction.

Even forgetting the finite correction, this provide an unambiguous definition. This point require in principle a multiscale spectrum evaluator (with one scale for each mass to evaluate). In the particular case of mSUGRA, the masses most affected by the running are definitely the ones of squark and sgluino.

We can face two problems:

• extract the running mass from the (experimental) pole mass (for computing the Yukawa coupling of fermions), which is straightforward.
• compute the pole mass from the running one, which involve solving a non-linear equation.

In fact, one must provide one boundary condition for each equation of the RGE system of first order Ordinary Differential Equation. But all these boundary conditions are not defined at the same scale:

• the gauge coupling must unify at the GUT scale, and fit their experimental value at M_Z (in fact, this is overconstrained if one consider the 3 gauge couplings. As a consequence, one must choose between unification of strong coupling (trinification) or fitting its experimental value)
• the Yukawa couplings of the fermion are best defined at the pole mass of the related fermion
• the soft breaking terms unify at the GUT scale, with a value provided as input by the model builder
• the mu and b parameter are defined by the radiative ElectroWeak Symmetry Breaking (EWSB) condition at the EWSB scale as function of \tan\beta, M_Z, M^2_H_1 and M^2_H_2

Some of this conditions are trivial to implement (Soft Breaking Terms), but others are not even always defined (EWSB). We need to consider these conditions as a set of non-linear equations. The system of ODE is thus coupled not only through the beta functions (derivative of the parameter) but also by its boundary conditions.

When expressed at one-loop level, this conditions depends in turn on the spectrum to be evaluated: we end up with a feed-back problem. The usual solution to this problem is iterative.