# Implications

Fields, masses and coupling constants appearing in the Lagrangian are perturbatively divergent quantities which become acceptable by renormalization procedures.
As a consequence, the new quantities are function of the energy and the functional dependence is governed by the Renormalization Group Equations (RGE).
For the gauge coupling constants these equations have the form

$\mu\frac{d\alpha}{d\mu}=\alpha\beta(\alpha)$

where $\beta$ is the Callan-Symanzik function depending from the fields contained in the theory and $\alpha=\frac{g^2}{4\pi}$ indicates the generic constant.

Thus, the evolution of coupling constants will be different with respect to the case of the SM, as shown in the following figure,

where the SM coupling constants approach to about same values for energies $\mu\sim 10^{13}\div 10^{17}\ GeV$, even if they do not intersect to each other in the same point.
In the MSSM, in fact, the three constants seem to unify with great precision for $\mu \sim 2\times 10^{16}\ GeV$, under the hypothesis that SUSY effects could raise at the scale of $TeV$. This surprising result for the MSSM suggests an evidence of unification.

Together with the coupling constants, further implications regard the evolution of the mass parameters appearing in the MSSM Lagrangian. In the case of the masses of the gauginos, the RGE can be written as

$\mu\frac{d}{d\mu}\frac{M_i}{\alpha_i^2}=0$

and then they imply that the $\frac{M}{\alpha^2}$ ratio is independent by the energy scale $\mu$.
So, for masses approaching to the GUT scale where the three $\alpha$ coincide, even the $M_i$ masses will coincide:

$M_1=M_2=M_3\equiv M_{\frac{1}{2}}$,

with $M_{\frac{1}{2}}$ indicating the \textit{gauginos universal mass}. At electroweak scale, instead:

$M_1:M_2:M_3=\alpha_1^2:\alpha_2^2:\alpha_3^2 \sim 1:2:7$.

Analogue relations can also be obtained for the mass parameters of scalars. It could be interesting to note that $H_u$ assumes negative values for energies from the grand unification scale towards the electroweak scale, as shown in the following figure

It could be appropriate to point out that while in the SM the mass term of the Higgs Lagrangian is imposed “ad hoc”, in the supersymmetric scenario the spontaneous symmetry breaking of the electroweak symmetry occurs in a natural way; it exists in fact a field, among others, in which the mass term, due to the natural course of the RGE, presents the right sign to give rise to the spontaneous breaking of electroweak symmetry.

Another significative implication of the phenomenological MSSM is linked to the (probable) conservation of the R-parity. In fact, if such conservation exists:

• the mixing between SUSY particles and respective SM partners is not allowed;
• supersymmetric particles can be produced only in pairs formed by s–particles and the respective anti-s-particles, starting from the SM particles;
• supersymmetric particles can decay only into states containing other supersymmetric particles, provided that the number of superparticles is conserved at each stage;
• the Lightest Supersymmetric Particle (LSP) must be stable. Assuming that LSP is neutral and only weakly interactive, a good LSP candidate could be the $\tilde{\chi}_1^0$, as envisaged in many SUSY theories.
Thus, it could be a good WIMP candidate for the Dark Matter.

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