Gravitinos and hidden Supersymmetry at the LHC - Universität ...

2.2. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL
and the i, j are the corresponding family indices. All of the gauge indices are suppressed.
The µ term is the supersymmetric version of the Higgs boson mass and the only dimensionful
supersymmetric parameter, which we choose to be real.
Contrary to the case of the Standard Model, the Yukawa couplings presented so far are
not the most general couplings compatible with gauge invariance and renormalizability. Additional
potentially dangerous terms violating baryon and lepton number can be added. These
terms are usually forbidden by discrete symmetry, R-parity, which will be the topic of the
next chapter. Therefore, at this stage the MSSM is defined as the model with the minimal
field and coupling content.
The description of the MSSM is completed with the specification of the soft supersymmetry
breaking terms discussed in the previous section:
−L MSSM
soft
= 1 2 (M 3 g g + M 2 w w + M 1 b b + h.c.)
(
)
+ a u ˜qH u ˜ū + a d ˜qH d ˜¯d + a
e ˜lHd ˜ē + h.c.
+ ˜m 2 q ˜q †˜q + ˜m 2 l ˜l †˜l + ˜m
2
u ˜ū †˜ū + ˜m 2 d ˜¯d† ˜¯d + ˜m
2
e ˜ē †˜ē
+ m 2 u H † uH u + m 2 d H† d H d + (BH u H d + h.c.) . (2.25)
In eq. (2.25), M 3 , M 2 , and M 1 are the gluino, wino, and bino mass terms. We have suppressed
the adjoint representation gauge indices on the wino and gluino fields, and the gauge indices
on all of the chiral supermultiplet fields. The second line in eq. (2.25) contains the (scalar) 3
couplings. Each of a u , a d , a e is a complex 3 × 3 matrix in family space, with dimensions
of [mass]. They are in one-to-one correspondence with the Yukawa couplings of the superpotential.
The third line of eq. (2.25) consists of squark and slepton mass terms, the mass
matrices are in general 3 × 3 matrices in family space that can have complex entries, but they
must be hermitian so that the Lagrangian is real. In the last line of eq. (2.25) one has the
supersymmetry-breaking contributions to the Higgs potential; m 2 u and m 2 d
are squared-mass
terms of the m 2 type, while B is the only squared-mass term of the type b in e.q. (2.20)
that can occur in the MSSM 1 . The dagger on H u and H d indicates the contraction of the
doublets in contrast to terms like H u H d which should be read as ε ij H ui H dj , ε = iσ 2 being the
SU(2) metric. The soft breaking terms introduce many new parameters not present in the
Standard Model. It turns out that the MSSM Lagrangian has 105 physical masses, phases
and mixing angles, which have no counterpart in the Standard Model. This arbitrariness in
the Lagrangian is not inherent to supersymmetry but arises from the unknown mechanism of
supersymmetry breaking. Most of the new parameters imply flavor mixing or CP-violating
processes, which are restricted by experimental data. This is precisely the point attacked
by author of [95], as mentioned in the introduction to this chapter. Usually, it is assumed
that supersymmetry breaking is universal, meaning that the squark and slepton squared-mass
matrices are flavor blind. Additionally, one assumes that the scalar trilinear couplings are
proportional to Yukawa couplings and that the soft parameters do not introduce new complex
phases. These relations should result naturally from the specific model for the origin of SUSY
breaking. We will discuss such simplified models in the end of this chapter.
For the later discussion we will need the effects of the electroweak symmetry breaking in
the MSSM, which we introduce in the following section.
1 The parameter called B in this work is often denoted by b or Bµ.
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