# The Minimal Supersymmetric extension of the Standard Model (MSSM)

The starting point for the construction of the Lagrangian of a such model is the SUSY Lagrangian, where the gauge invariance with respect to the group of the SM, $SU(3)_C\otimes SU(2)_L\otimes U(1)_Y$, has to be imposed.

In the minimal hypothesis, each particle included in the SM possesses a unique superpartner, i.e. formally a superfield has to replace a fermionic or bosonic field in the theory:

• the gauge bosons of the SM are promoted to gauge superfields composed by a vector field and a spinor;
• the fermion fields of the SM are promoted to chiral scalar superfields composed by a spinorial field ($s=\frac{1}{2}$) and complex scalar, with one superfield for each chirality of every SM fermion.

Furthermore, since the generator of the supersymmetric transformations commutes with the generators of the group $SU(3)_C\otimes SU(2)_L \otimes U(1)_Y$, the superparticles have the same quantum numbers of their SM partners. Fermionic superpartners are spin-0 particles and they are generally identified by prefixing to their name a s-, meaning scalar (s-quark, s-electron, etc …), while superpartners of bosons take the suffix -ino and then gauge bosons are called gauginos, Higgs bosons, higgsinos.

The usual Higgs doublet of the SM is promoted to a doublet of left-chiral superfields:

While in the SM an unique Higgs doublet is able to give the mass to each of the fermions, in the MSSM a doublet like $\hat{H}_u$ can give the mass just to the <em>u</em> quarks and not to the <em>d</em> ones. This is since the VEV of the scalar component of $\hat{h}_u^0$ gives mass to up-type quarks but, unlike in the SM, it cannot give a mass to the $T_3 = -\frac{1}{2}$ fermions.
The mass of fermions whose $T_3=-\frac{1}{2}$ is generated by the conjugate Higgs field and then the subsequent superfield would contain right-handed spinor terms, that are forbidden in the superpotential. For this reason it is necessary to introduce a new Higgs doublet

and its corresponding superpartner in the theory.

The SSB mechanism for electroweak symmetry causes the absorption of three degrees of freedom by the masses of intermediate vector bosons. As in MSSM there are two complex doublets, the degrees of freedom become $8-3=5$ and then there are 5 Higgs bosons,

$h,\ H,\ H^\pm,\ A$.

The $h$ and $H$ bosons are scalar and neutral, $A$ is neutral pseudo-scalar and $H^\pm$ are charged scalars; $h$ is associated with the Higgs boson of the SM. The SSB occurs when the Higgs fields are projected on the vacuum states:

where $H_d$ couples exclusively to down-type fermion pairs, while $H_u$ to up-type.
At the minimum for the Higgs potential, the VEVs associated with the two doublets are related to each other. The $tan\beta$ parameter is a free parameter of the model and it defined as the ratio between the VEVs of Higgs doublets:

$tan\beta\equiv\frac{v_u}{v_d}$

where the normalization is chosen so that

$v^2\equiv v_u^2+v_d^2=(246\ Gev)^2$.

By convention, the phases of the Higgs field are chosen such that $0\leq\beta\leq\frac{\pi}{2}$. Even the masses of the single Higgs bosons are related to each other and they are often depending on the mass of the $A$. The Higgs sector of the MSSM is then characterized by only two free parameters: $tan\beta$ and $m_A$. Finally, the complete Lagrangian for renormalizable supersymmetric gauge theory takes the form

where

• $\mathcal{D}$ indicates the gauge covariant derivative;
• $\mathcal{D}\kern-6.5pt\slash\$ expresses the gauge covariant derivative in the adjoint representation;
• $A$ indicates the gauge group index;
• $\psi_i$ and $\mathcal{S}_i$ are the fermionic and the scalar terms of the $i^{th}$ superfield;
• $F_{\mu\nu A}$ is kinetic tensor for the $A^{th}$ gauge field;
• $\lambda$ is a corresponding term for the gaugino;
• $t_A$ are the generators associated to the symmetry group.

The first four addenda in the MSSM Lagrangian represent the kinetic terms for each field in the MSSM; the fifth and the sixth terms rule the couplings between the fields. The $\mathcal{L}(\mathcal{W})$ function concerns the superpotential; its shape is not uniquely bound to the choice of superfields and it is therefore characteristic of the particular SUSY model chosen.

The superpotential chosen for the MSSM respects the conservation of the barionic and of the leptonic quantum numbers which are defined, as usual: $B=+\frac{1}{3}$ ($B=-\frac{1}{3}$) for quark (anti-quark) superfields and $L=+1$ ($L=-1$) for leptonic (antileptonic) superfield. Despite the case of the SM where the conservation of the baryonic and the leptonic numbers is ensured by the gauge invariance, in the MSSM this has to be taken into account in the choice of the potential.

As s-quarks and s-leptons have the same quantum number of their SM partners, it is possible to add to the MSSM potential some terms that violate the conservation of the baryonic and leptonic numbers even if these terms could be consistent with the $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y$ symmetry of the SM as well as of the SUSY.
However such violation is strongly constrained by experiment, so the presence of these terms is unacceptable: for instance, if conservation of baryon and lepton numbers are both violated, protons will decay at extremely rapid rates.

For this reason, in the spirit of minimality of new interactions, it is imposed that even in a supersymmetric scenario lepton and baryon numbers are conserved and the existence of a new parity is supposed, the R-parity.

As a consequence of the $BL$ invariance, the MSSM fulfits a multiplicative R-parity invariance, where for a particle of spin S the R quantum number is

$R=(-1)^{3(B-L)+2S}$.

This implies that all the ordinary SM particles have even R-parity, whereas the corresponding symmetric partners have odd R-parity.