# Introduction to the MSSM

Sudhir K. Vempati Centre for High Energy Physics, Indian Institute of Science,
Bangalore 560 012, India<EMAIL_ADDRESS>
###### Abstract

These lecture notes are based on a first course on the Minimal Supersymmetric
Standard Model. The level of the notes is introductory and pedagogical.
Standard Model, basic supersymmetry algebra and its representations are
considered as prerequisites. The topics covered include particle content,
structure of the lagrangian, supersymmetry breaking soft terms, electroweak
symmetry breaking and the sparticle mass spectrum. Popular supersymmetry
breaking models like minimal supergravity and gauge mediation are also
introduced.

##  I Prerequisites

These lectures111 Based on lectures presented at SERC school, held at IIT-
Bombay, Mumbai.  are devised as an introduction to the Minimal Supersymmetric
Standard Model. In the course of these lectures, we will introduce the basic
features of the Supersymmetric Standard Model, the particle content, the
structure of the lagrangian, feynman rules, supersymmetric breaking soft
terms, Electroweak symmetry breaking and the mass spectrum of the MSSM.
Supersymmetry is a vast subject and these lectures are definitely not a
comprehensive review and in fact, they are also not what one could term as an
introduction to supersymmetric algebra and supersymmetric gauge theories. The
prerequisites for this course are a good knowledge of the Standard Model and
also supersymmetry say, at the level of first eight chapters of Wess and
Baggerwessbagger : supersymmetry transformations, representations,
superfields, supersymmetric gauge theories and elements of supersymmetry
breaking. It is strongly recommended that readers keep a text book on basic
course of supersymmetry westetal ; sohnius ; onethousand with them all the
time for consulting, while going through this lecture notes.

The lectures are organised as follows : in the next section, we will give a
lightening introduction to the Standard Model and the structure of its
lagrangian. The reason for this being that we will like to introduce the MSSM
(Minimal Supersymmetric Standard Model ) in a similar organisational fashion,
which makes it easier to remember the MSSM lagrangian - as well as arranging
the differences and similarities , one expects in the supersymmetric theories
in a simpler way, if possible. The next section would introduce the basic form
of the MSSM lagrangian - the three functions of the chiral/vector superfields
- the superpotential, the Kahler potential and the field strength superfield
and the particle spectrum. The fourth section will be devoted to R-parity and
some sample feynman rules. Supersymmetry breaking and electroweak symmetry
breaking will be introduced in section 5 and the physical supersymmetric
particle mass spectrum will be done in section 6. Higgs sector will be
reviewed in section 7, while we close with some ‘standard’ models of
supersymmetry breaking in section 8.

Finally for the students not completely familiar with Standard Model, we point
out at some references with increasing order of difficulty in reading and
requirements of pre-requisites. These are : (a) Aitchison and Heyathey , Gauge
Field Theories, Vol I and Vol II (b)A good functional introduction to field
theory required for understanding Standard Model can be found in : M.
Srednicki, Quantum Field Theory srednicki (c) M. E. Peskin and D. Schroeder,
Quantum Field Theory peskin (d) E. Abers and B. W. Lee, Physics Reports on
Gauge Field Theories aberslee (e) T. Cheng and L. Li, Gauge theory of
elementary particle physicschengli (f) S. Weinberg, Quantum Theory of Fields,
Vol I -IIweinberg , (g) S. Pokorski, Gauge Field Theories pokorski and (h)
Donoghue, Golowich and Holstein, Dynamics of the Standard Model dynamics .

##  II Step 0 : A lightening recap of the Standard Model

The Standard Model (SM) is a spontaneously broken Yang-Mills quantum field
theory describing the strong and electroweak interactions. The theoretical
assumption on which the Standard Model rests on is the principle of local
gauge invariance with the gauge group given by

$G_{SM}\equiv SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y},$ (1)

where the subscript $c$ stands for color, $L$ stands for the ‘left-handed’
chiral group whereas $Y$ is the hypercharge. The particle spectrum and their
transformation properties under these gauge groups are given as,

$\displaystyle Q_{i}\equiv\left(\begin{array}[]{c}{u_{L}}_{i}\\\
{d_{L}}_{i}\end{array}\right)\sim\left(3,~{}2,~{}{1\over 6}\right)$
$\displaystyle U_{i}\equiv{u_{R}}_{i}\sim\left(\bar{3},~{}1,~{}{2\over
3}\right)$ $\displaystyle
D_{i}\equiv{d_{R}}_{i}\sim\left(\bar{3},~{}1,~{}-{1\over 3}\right)$
$\displaystyle L_{i}\equiv\left(\begin{array}[]{c}{\nu_{L}}_{i}\\\
{e_{L}}_{i}\end{array}\right)\sim\left(1,~{}2,~{}-{1\over 2}\right)$
$\displaystyle E_{i}\equiv{e_{R}}_{i}\sim\left(1,~{}1,~{}-1\right)$ (7)

In the above $i$ stands for the generation index, which runs over the there
generations $i=1,2,3$. $Q_{i}$ represents the left handed quark doublets
containing both the up and down quarks of each generation. Similarly, $L_{i}$
represents left handed lepton doublet, $U_{i},~{}D_{i},~{}E_{i}$ represent
right handed up-quark, down-quark and charged lepton singlets respectively.
The numbers in the parenthesis represent the transformation properties of the
particles under $G_{SM}$ in the order given in eq.(1). For example, the quark
doublet $Q$ transforms a triplet (3) under $SU(3)$ of strong interactions, a
doublet (2) under weak interactions gauge group and carry a hypercharge
$(Y/2)$ of 1/6 222Note that the hypercharges are fixed by the Gellman-
Nishijima relation $Y/2~{}=~{}Q-T_{3}$, where $Q$ stands for the charge of the
particle and $T_{3}$ is the eigenvalue of the third generation of the particle
under $SU(2)$.. In addition to the fermion spectra represented above, there is
also a fundamental scalar called Higgs whose transformation properties are
given as

$H\equiv\left(\begin{array}[]{c}H^{+}\\\
H^{0}\end{array}\right)\sim\left(1,~{}2,~{}1/2\right).$ (8)

However, the requirement of local gauge invariance will not be fulfilled
unless one includes the gauge boson fields also. Including them, the total
lagrangian with the above particle spectrum and gauge group can be represented
as,

${\cal L}_{SM}={\cal L}_{F}+{\cal L}_{YM}+{\cal L}_{yuk}+{\cal L}_{S}.$ (9)

The fermion part ${\cal L}_{F}$ gives the kinetic terms for the fermions as
well as their interactions with the gauge bosons. It is given as,

${\cal L}_{F}=i\bar{\Psi}\gamma^{\mu}{\cal{D}}_{\mu}\Psi,$ (10)

where $\Psi$ represents all the fermions in the model,

$\Psi=\left(Q_{i}~{}U_{i},~{}D_{i},~{}L_{i},~{}E_{i}\right)$ (11)

where ${\cal{D}}_{\mu}$ represents the covariant derivative of the field given
as,

${\cal{D}}_{\mu}=\partial\mu-ig_{s}G_{\mu}^{A}\lambda^{A}-i{g\over
2}W_{\mu}^{I}\tau^{I}-ig^{\prime}B_{\mu}Y$ (12)

Here $A=1,..,8$ with $G_{\mu}^{A}$ representing the $SU(3)_{c}$ gauge bosons,
$I=1,2,3$ with $W_{\mu}^{I}$ representing the $SU(2)_{L}$ gauge bosons. The
$U(1)_{Y}$ gauge field is represented by $B_{\mu}$. The kinetic terms for the
gauge fields and their self interactions are given by,

${\cal L}_{YM}=-{1\over 4}G^{\mu\nu A}G_{\mu\nu}^{A}-{1\over 4}W^{\mu\nu
I}W_{\mu\nu}^{I}-{1\over 4}B^{\mu\nu}B_{\mu\nu}$ (13)

with

$\displaystyle G_{\mu\nu}^{A}$ $\displaystyle=$
$\displaystyle\partial_{\mu}G_{\nu}^{A}-\partial_{\nu}G_{\mu}^{A}+g_{s}~{}f_{ABC}G_{\mu}^{B}G_{\nu}^{C}$
$\displaystyle F_{\mu\nu}^{I}$ $\displaystyle=$
$\displaystyle\partial_{\mu}W_{\nu}^{I}-\partial_{\nu}W_{\mu}^{I}+g~{}f_{IJK}W_{\mu}^{J}W_{\nu}^{K}$
$\displaystyle B_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu},$ (14)

where $f_{ABC(IJK)}$ represent the structure constants of the $SU(3)(SU(2))$
group.

In addition to the gauge bosons, the fermions also interact with the Higgs
boson, through the dimensionless Yukawa couplings given by

${\cal
L}_{yuk}=h^{u}_{ij}\bar{Q}_{i}U_{j}\tilde{H}+h^{d}_{ij}\bar{Q}_{i}D_{j}H+h^{e}_{ij}\bar{L}_{i}E_{j}H+H.c$
(15)

where $\tilde{H}=i\sigma^{2}H^{\star}$. These couplings are responsible for
the fermions to attain masses once the gauge symmetry is broken from
$G_{SM}~{}\rightarrow~{}SU(3)_{c}\times U(1)_{em}$. This itselves is achieved
by the scalar part of the lagrangian which undergoes spontaneous symmetry
breakdown. The scalar part of the lagrangian is given by,

${\cal L}_{S}=\left({\cal{D}}_{\mu}H\right)^{\dagger}{\cal{D}}_{\mu}H-V(H),$
(16)

where

$V(H)=\mu^{2}H^{\dagger}H+\lambda\left(H^{\dagger}H\right)^{2}$ (17)

For $\mu^{2}~{}<~{}0$, the Higgs field attains a vacuum expectation value
(vev) at the minimum of the potential. The resulting goldstone bosons are
‘eaten away’ by the gauge bosons making them massive through the so-called
Higgs mechanism. Only one degree of the Higgs field remains physical, the only
scalar particle of the SM - the Higgs boson. The fermions also attain their
masses through their Yukawa couplings, once the Higgs field attains a vev. The
only exception is the neutrinos which do not attain any mass due to the
absence of right handed neutrinos in the particle spectrum and thus the
corresponding Yukawa couplings. Finally, the Standard Model is renormalisable
and anomaly free. We would also insist that the Supersymmetric version of the
Standard Model keeps these features of the Standard Model intact.

####  II.0.1 Think it Over

Here are some important aspects of the Standard Model which have not found a
mention in the above. These are formulated in some sort of a problem mode,
which would require further study.

  * •

What is the experiment that showed that there are only three generations of
particles in the Standard Model ? Can one envisage a fourth generation ? If
so, what are the constraints this generation of particles expected to satisfy
?

  * •

The gauge bosons ‘mix’ at the tree level by an angle
$\tan~{}\theta_{W}=({g^{\prime}\over g})^{2}$. What happens at the 1-loop
level ? However are the relevant observables classified ? Some relevant
information can be found athollik .

  * •

What are the theoretical limits on the Higgs boson mass ? How sensitive is the
upper bound on the Higgs mass from precision measurements to the top quark
mass ? What is the lower limit on the Higgs mass from the LEP experiment ?
Some relevant information can be found atlephwg .

  * •

The LHC experiment has been rapidly constraining the allowed parameter space
of the Higgs boson. For latest information have a look at cerntwiki .

  * •

What is the CKM mixing ? How well are these angles measured ? What is the
present status after the results from various B-factories about the CP phase
in the SM ? What is the analogous mixing in the leptonic sector called ? In
comparison to the CKM matrix, how well are these angles measured ?

##  III Step 1 : Particle Spectrum of MSSM

What we aim to build over the course of next few lectures is a supersymmetric
version of the Standard Model, which means the lagrangian we construct should
not only be gauge invariant under the Standard Model gauge group $G_{SM}$ but
also now be supersymmetric invariant. Such a model is called Minimal
Supersymmetric Standard Model with the the word ’Minimal’ referring to minimal
choice of the particle spectrum required to make it work. Furthermore, we
would also like the MSSM to be renormalisable and anomaly free, just like the
Standard Model is.

Before we proceed to discuss about the particle spectrum, let us remind
ourselves that ordinary quantum fields are upgraded in supersymmetric333All
through this set of lectures, whenever we mention supersymmetry we mean N=1
SUSY ; only one set of SUSY generators. theories to so-called supermultiplets
or superfields444Superfields are functions (fields) written over a
‘superspace’ made of ordinary space ($x_{\mu}$) and two fermionic ‘directions’
($\theta$,$\bar{\theta}$); they are made up of quantum fields whose spins
differ by 1/2. To build interaction lagrangians one normally resorts to this
formalism, originally given by Salam and Strathdeesalamstrahdee , as
superfields simplify addition and multiplication of the representations. It
should be noted however that the component fields may always be recovered from
superfields by a power series expansion in grassman variable, $\theta$.
A chiral superfield has an expansion :
$\Phi=\phi+\sqrt{2}\theta\psi+\theta\theta F,$ (18) where $\phi$ is the scalar
component, $\psi$, the two component spin 1/2 fermion and $F$ the auxiliary
field. A vector superfield in (Wess-Zumino gauge) has an expansion :
$V=-\theta\sigma^{\mu}\bar{\theta}A_{\mu}+i\theta\theta\bar{\theta}\bar{\lambda}-i\bar{\theta}\bar{\theta}\theta\lambda+{1\over
2}\theta\theta\bar{\theta}\bar{\theta}D$ (19) . Given that supersymmetry
transforms a fermion into a boson and vice-versa, supermultiplets or
superfields are multiplets which collect fermion-boson pairs which transform
in to each other. We will deal with two kinds of superfields - vector
superfields and chiral superfields. A chiral superfield555Here we are
presenting the particle content in the off-shell formalism. contains a weyl
fermion, a scalar and and an auxiliary scalar field generally denoted by F. A
vector superfield contains a spin 1 boson, a spin 1/2 fermion and an auxiliary
scalar field called D.

The minimal supersymmetric extension of the Standard Model is built by
replacing every standard model matter field by a chiral superfield and every
vector field by a vector superfield. Thus the existing particle spectrum of
the Standard Model is doubled. The particle spectrum of the MSSM and their
transformation properties under $G_{SM}$ is given by,

$\displaystyle
Q_{i}\equiv\left(\begin{array}[]{cc}u_{L_{i}}&{\tilde{u}}_{L_{i}}\\\
d_{L_{i}}&\tilde{d}_{L_{i}}\end{array}\right)\sim\left(3,~{}2,~{}{1\over
6}\right)$ $\displaystyle
U_{i}^{c}\equiv\left(\begin{array}[]{cc}u_{i}^{c}&\tilde{u}^{c}_{i}\end{array}\right)\sim\left(\bar{3},~{}1,~{}-{2\over
3}\right)$ (23) $\displaystyle
D_{i}\equiv\left(\begin{array}[]{cc}d_{i}^{c}&\tilde{d}^{c}_{i}\end{array}\right)\sim\left(\bar{3},~{}1,~{}{1\over
3}\right)$ (25) $\displaystyle
L_{i}\equiv\left(\begin{array}[]{cc}\nu_{L_{i}}&\tilde{\nu}_{L_{i}}\\\
e_{L_{i}}&\tilde{e}_{L_{i}}\end{array}\right)\sim\left(1,~{}2,~{}-{1\over
2}\right)$ $\displaystyle
E_{i}\equiv\left(\begin{array}[]{cc}e_{i}^{c}&\tilde{e}^{c}_{i}\end{array}\right)\sim\left(1,~{}1,~{}1\right)$
(29)

The scalar partners of the quarks and the leptons are typically named as
‘s’quarks and ‘s’leptons. Together they are called sfermions. For example, the
scalar partner of the top quark is known as the ‘stop’. In the above, these
are represented by a ‘tilde’ on their SM counterparts. As in the earlier case,
the index $i$ stands for the generation index.

There are two distinct features in the spectrum of MSSM : (a) Note that we
have used the conjugates of the right handed particles, instead of the right
handed particles themselves. There is no additional conjugation on the
superfield itselves, the $c$ in the superscript just to remind ourselves that
this chiral superfield is made up of conjugates of SM quantum fields. In
eq.(23), $u^{c}~{}=~{}u_{R}^{\dagger}$ and
$\tilde{u}^{c}~{}=~{}\tilde{u}_{R}^{\star}$. This way of writing down the
particle spectrum is highly useful for reasons to be mentioned later in this
section. Secondly (b) At least two Higgs superfields are required to complete
the spectrum - one giving masses to the up-type quarks and the other giving
masses to the down type quarks and charged leptons. As mentioned earlier, this
is the minimal number of Higgs particles required for the model to be
consistent from a quantum field theory point of view666The Higgs field has a
fermionic partner, higgsino which contributes to the anomalies of the SM. At
least two such fields with opposite hyper-charges ($U(1)_{Y}$) should exist to
cancel the anomalies of the Standard Model.. These two Higgs superfields have
the following transformation properties under $G_{SM}$:

$\displaystyle H_{1}$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{cc}H^{0}_{1}&\tilde{H}_{1}^{0}\\\
H^{-}_{1}&\tilde{H}_{1}^{-}\end{array}\right)\sim\left(1,~{}2,~{}-{1\over
2}\right)$ (33) $\displaystyle H_{2}$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{cc}H^{+}_{2}&\tilde{H}_{2}^{+}\\\
H^{0}_{2}&\tilde{H}_{2}^{0}\end{array}\right)\sim\left(1,~{}2,~{}{1\over
2}\right)$ (36)

The Higgsinos are represented by a $\tilde{}$ on them. This completes the
matter spectrum of the MSSM. Then there are the gauge bosons and their super
particles. Remember that in supersymmetric theories, the gauge symmetry is
imposed by the transformations on matter superfields as :

$\Phi^{\prime}=e^{i\Lambda_{l}t_{l}}\Phi$ (37)

where $\Lambda_{l}$ is an arbitrary chiral superfield and $t_{l}$ represent
the generators of the gauge group which are $l$ in number and the index $l$ is
summed over777To be more specific, $t_{l}$ is just a number for the abelian
groups. For non-abelian groups, $t_{l}$ is a matrix and so is $\Lambda_{l}$,
with $\Lambda_{ij}=t^{l}_{ij}\Lambda_{l}$ Note that $V$ is also becomes a
matrix in this case.. The gauge invariance is restored in the kinetic part by
introducing a (real) vector superfield, $V$ such that the combination

$\Phi^{\dagger}e^{gV}\Phi$ (38)

remains gauge invariant. For this to happen, the vector superfield $V$
itselves transforms under the gauge symmetry as

$\delta V=i(\Lambda-\Lambda^{\dagger})$ (39)

The supersymmetric invariant kinetic part of the lagrangian is given by:

$\mathcal{L}_{kin}=\int
d\theta^{2}d\bar{\theta}^{2}\Phi^{\dagger}e^{gV}\Phi=\Phi^{\dagger}e^{gV}\Phi|_{\theta\theta\bar{\theta}\bar{\theta}}$
(40)

In the MSSM, corresponding to three gauge groups of the SM and for each of
their corresponding gauge bosons, we need to add a vector superfield which
transforms as the adjoint under the gauge group action. Each vector superfield
contains the gauge boson and its corresponding super partner called gaugino.
Thus in MSSM we have the following vector superfields and their corresponding
transformation properties under the gauge group, completing the particle
spectrum of the MSSM:

$\displaystyle V_{s}^{A}$ $\displaystyle:$
$\displaystyle\left(\begin{array}[]{cc}G^{\mu
A}&\tilde{G}^{A}\end{array}\right)~{}~{}\sim~{}~{}(8,1,0)$ (42) $\displaystyle
V_{W}^{I}$ $\displaystyle:$ $\displaystyle\left(\begin{array}[]{cc}W^{\mu
I}&\tilde{W}^{I}\end{array}\right)~{}~{}~{}\sim~{}~{}(1,3,0)$ (44)
$\displaystyle V_{Y}$ $\displaystyle:$
$\displaystyle\left(\begin{array}[]{cc}B^{\mu}&\tilde{B}\end{array}\right)~{}~{}\sim~{}~{}(1,1,0)$
(46)

The $G$’s ($G$ and $\tilde{G}$) represent the gluonic fields and their
superpartners called gluinos, the index $A$ runs from $1$ to $8$. The $W$’s
are the $SU(2)$ gauge bosons and their superpartners ‘Winos’, the index $I$
taking values from $1$ to $3$ and finally $B$s represents the $U(1)$ gauge
boson and its superpartner ‘Bino’. Together all the superpartners of the gauge
bosons are called ‘gauginos’. This completes the particle spectrum of the
MSSM.

##  IV Step 2: The superpotential and R-parity

The supersymmetric invariant lagrangian is constructed from functions of
superfields. In general there are three functions which are: (a) The Kähler
potential, $K$, which is a real function of the superfields (b) The
superpotential $W$, which is a holomorphic (analytic) function of the
superfields, and (c) the gauge kinetic function $f_{\alpha\beta}$ which
appears in supersymmetric gauge theories. This is the coefficient of the
product of field strength superfields,
$\mathcal{W}_{\alpha}\mathcal{W}^{\beta}$. The field strength superfield is
derived from the vector superfields contained in the model. $f_{\alpha\beta}$
determines the normalisation for the gauge kinetic terms. In MSSM,
$f_{\alpha\beta}=\delta_{\alpha\beta}$. The lagrangian of the MSSM is thus
given in terms of $G_{SM}$ gauge invariant functions $K$, $W$ and add the
field strength superfield $\mathcal{W}$, for each of the vector superfields in
the spectrum.

The gauge invariant Kähler potential has already been discussed in the
eqs.(40). For the MSSM case, the Kähler potential will contain all the three
vector superfields corresponding to the $G_{SM}$ given in the eq.(46). Thus we
have :

$\mathcal{L}_{kin}=\int
d\theta^{2}d\bar{\theta}^{2}\sum_{\scriptstyle{\text{SU(3)},\text{SU(2)},\text{U(1)}}}\Phi_{\beta}^{\dagger}~{}e^{gV}\Phi_{\beta}$
(47)

where the index $\beta$ runs over all the matter fields
$\Phi_{\beta}~{}=~{}\\{Q_{i},U^{c}_{i},D^{c}_{i},L_{i},e^{c}_{i},H_{1},H_{2}\\}$888The
indices $i,j,k$ always stand for the three generations through out this notes,
taking values between 1 and 3. in appropriate representations. Corresponding
to each of the gauge groups in $G_{SM}$, all the matter fields which transform
non-trivially under this gauge group999As given in the list of representations
in eqs. (23,33)  are individually taken and the grassman
($d\theta^{2}d{\bar{\theta}}^{2}$) integral is evaluated with the
corresponding vector superfields in the exponential101010Remember that the
function $e^{gV}$ truncates at ${1\over 2}g^{2}V^{2}$ in the Wess-Zumino
gauge. In fact, in this gauge, this function can be determined by noting:
$\exp
V_{WZ}=1-\theta\sigma^{\mu}\bar{\theta}A_{\mu}+i\theta\theta\bar{\theta}\bar{\lambda}-i\bar{\theta}\bar{\theta}\theta\lambda+{1\over
2}~{}\theta\theta\bar{\theta}\bar{\theta}~{}(D-{1\over 2}A^{\mu}A_{\mu}),$
(48) for an abelian Vector superfield. Here as usual $\lambda$ denotes the
gaugino field while $A_{\mu}$ represents the gauge field. D represents the
auxiliary field of the Vector multiplet. The extension to the non-abelian case
is straight forward.. After expanding and evaluating the integral, we get the
lagrangian which is supersymmetric invariant in terms of the ordinary quantum
fields - the SM particles and the superparticles. This part of the lagrangian
would give us the kinetic terms for the SM fermions, kinetic terms for the
sfermions and their interactions with the gauge bosons and in addition also
the interactions of the type: fermion-sfermion-gaugino which are structurally
like the Yukawa interactions ($ff\phi$), but carry gauge couplings. Similarly,
for the Higgs fields, this part of the lagrangian gives the kinetic terms for
the Higgs fields and their fermionic superpartners Higgsinos and the
interaction of the gauge bosons with the Higgs fields and Higgs-Higgsino-
gaugino vertices.

The second possible function of the superfields is the analytic or holomorphic
function111111This would mean that $W$ is purely a function of complex fields
($z_{1}z_{2}z_{3}$) or its conjugates
($z_{1}^{\star}z_{2}^{\star}z_{3}^{\star}$).  of the superfields called the
superpotential, $W$. This function essentially gives the interaction part of
the lagrangian which is independent of the gauge couplings, like the Yukawa
couplings. If renormalisability is demanded, the dimension of the
superpotential is restricted to be less than or equal to three,
$[W]~{}\leq~{}3$ i.e, only products of three or less number of chiral
superfields are allowed. Imposing this restriction of renormalisability the
most general $G_{SM}$ gauge invariant form of the $W$ for the matter spectrum
of MSSM (23,33) is given as

$W=W_{1}+W_{2},\\\ $ (49)

where

$\displaystyle W_{1}$ $\displaystyle=$ $\displaystyle
h^{u}_{ij}Q_{i}U_{j}^{c}H_{2}+h^{d}_{ij}Q_{i}D_{j}^{c}H_{1}+h_{ij}^{e}L_{i}E_{j}^{c}H_{1}+\mu
H_{1}H_{2}$ (50) $\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\epsilon_{i}L_{i}H_{2}+\lambda_{ijk}L_{i}L_{j}E_{k}^{c}+\lambda^{\prime}_{ijk}L_{i}Q_{j}D_{k}^{c}+\lambda^{\prime\prime}_{ijk}U_{i}^{c}D_{j}^{c}D_{k}^{c}.$
(51)

Here we have arranged the entire superpotential in to two parts, $W_{1}$ and
$W_{2}$ with a purpose. Though both these parts are gauge invariant, $W_{2}$
also violates the global lepton number and baryon quantum numbers. The
simultaneous presence of both these set of operators can lead to rapid proton
decay and thus can make the MSSM phenomenologically invalid. For these
reasons, one typically imposes an additional symmetry called R-parity in MSSM
which removes all the dangerous operators in $W_{2}$. We will deal with
R-parity in greater detail in the next section. For the present, let us just
set $W_{2}$ to be zero due to a symmetry called R-parity and just call $W_{1}$
as $W$. The lagrangian can be derived from the superpotential containing
(mostly) gauge invariant product of the three superfields by taking the
$\theta\theta$ component, which can be represented in the integral form as

$\mathcal{L}_{yuk}=\int d\theta^{2}~{}~{}W(\Phi)+\int
d\bar{\theta}^{2}~{}~{}\bar{W}(\bar{\Phi})$ (52)

This part gives121212The $\theta\theta$ components of the product of three
chiral superfields is given aswessbagger
$\Phi_{i}\Phi_{j}\Phi_{k}|_{\theta\theta}=-\psi_{i}\psi_{j}\phi_{k}-\psi_{j}\psi_{k}\phi_{i}-\psi_{k}\psi_{i}\phi_{j}+F_{i}\phi_{j}\phi_{k}+F_{j}\phi_{k}\phi_{i}+F_{k}\phi_{i}\phi_{j},$
(53) where as earlier, $\psi_{i}$ represents the fermionic, $\phi_{i}$ the
scalar and $F_{i}$ the auxiliary component of the chiral superfield
$\Phi_{i}$. Similarly for the product of two superfields on has :
$\Phi_{i}\Phi_{j}|_{\theta\theta}=-\psi_{i}\psi_{j}+F_{i}\phi_{j}+F_{j}\phi_{i}$
(54) the standard Yukawa couplings for the fermions with the Higgs, in
addition also give the fermion-sfermion-higgsino couplings and scalar terms.
For example, the coupling $h^{u}_{ij}~{}Q_{i}u_{j}^{c}H_{2}$ in the
superpotential has the following expansion in terms of the component fields :

$\displaystyle\mathcal{L}_{yuk}$ $\displaystyle\supset$ $\displaystyle
h^{u}_{ij}~{}Q_{i}u_{j}^{c}H_{2}~{}|_{\theta\theta}$ (55)
$\displaystyle\supset$ $\displaystyle
h^{u}_{ij}~{}(~{}u_{i}u_{j}^{c}H_{2}^{0}-d_{i}u_{j}^{c}H_{2}^{+}~{})~{}|_{\theta\theta}$
$\displaystyle\supset$ $\displaystyle
h^{u}_{ij}(\psi_{u_{i}}\psi_{u_{j}^{c}}\phi_{H_{2}^{0}}+\phi_{\tilde{u}_{i}}\psi_{u_{j}^{c}}\psi_{\tilde{H}_{2}^{0}}+\psi_{u_{i}}\phi_{\tilde{u}_{j}^{c}}\psi_{\tilde{H}_{2}^{0}}-\psi_{d_{i}}\psi_{u_{j}^{c}}\phi_{H_{2}^{+}}-\phi_{\tilde{d}_{i}}\psi_{u_{j}^{c}}\psi_{\tilde{H}_{2}^{+}}-\psi_{d_{i}}\phi_{\tilde{u}_{j}^{c}}\psi_{\tilde{H}_{2}^{+}})$
$\displaystyle\equiv$ $\displaystyle
h^{u}_{ij}~{}(~{}u_{i}u_{j}^{c}H_{2}^{0}+~{}\tilde{u}_{i}u_{j}^{c}\tilde{H}_{2}^{0}+u_{i}\tilde{u}_{j}^{c}\tilde{H}_{2}^{0}-d_{i}u_{j}^{c}H_{2}^{+}~{}-\tilde{d}_{i}u_{j}^{c}\tilde{H}_{2}^{+}~{}-d_{i}\tilde{u}_{j}^{c}\tilde{H}_{2}^{+}~{}),$
(56)

where in the last equation, we have used the same notation for the chiral
superfield as well as for its lowest component namely the scalar component.
Note that we have not written the F-terms which give rise to the scalar terms
in the potential. Similarly, there is the $\mu$ term which gives ‘Majorana’
type mass term for the Higgsino fields.

Finally, for every vector superfield (or a set of superfields) we have an
associated field strength superfield $\mathcal{W}^{\alpha}$, which gives the
kinetic terms for the gauginos and the field strength tensors for the gauge
fields. Given that it is a chiral superfield, the component expansion is given
by taking the $\theta\theta$ component of ‘square’ of that superfield131313In
the Wess-Zumino gauge, $\mathcal{W}_{\alpha}=-{1\over
4}\bar{\mathcal{D}}\bar{\mathcal{D}}\mathcal{D}_{\alpha}V_{WZ}$ wessbagger
($\mathcal{D}$ is the differential operator on superfields) and the lagrangian
has the form :  $\mathcal{L}\supset{1\over
4}\left(\mathcal{W}^{\alpha}\mathcal{W}_{\alpha}|_{\theta\theta}+\mathcal{W}^{\dot{\alpha}}\mathcal{W}_{\dot{\alpha}}|_{\bar{\theta}\bar{\theta}}\right)={1\over
2}D^{2}-{1\over
4}F_{\mu\nu}F^{\mu\nu}-i\lambda\sigma^{\mu}\partial_{\mu}\bar{\lambda}$ (57)
$D$ represents the auxiliary component of the vector superfields. The
extension to non-abelian vector superfields in straight forward.. In the MSSM,
we have to add the corresponding field strength $\mathcal{W}$ superfields for
electroweak vector superfields, $W$ and $B$ as well as for the gluonic $G$
vector superfields of eqs.(46).

So far we have kept the auxiliary fields ($D$ and $F$) of various chiral and
vector superfields in the component form of our lagrangian. However, given
that these fields are unphysical, they have to be removed from the lagrangian
to go “on-shell”. To eliminate the $D$ and $F$ fields, we have to use the
equations of motions of these fields which have simple solutions for the $F$
and $D$ as :

$F_{i}={\partial
W\over\partial\phi_{i}}~{}~{}~{}~{}\;;\;~{}~{}~{}~{}D_{A}=-g_{A}~{}~{}\phi^{\star}_{i}~{}T^{A}_{ij}~{}\phi_{j},$
(58)

where $\phi_{i}$ represents all the scalar fields present in MSSM. The index
$A$ runs over all the gauge groups in the model. For example, for $U(1)_{Y}$,
$T^{A}_{ij}=~{}(Y^{2}/2)\delta_{ij}$. The $F$ and $D$ terms together form the
scalar potential of the MSSM141414Later we will see that there are also
additional terms which contribute to the scalar potential which come from the
supersymmetry breaking sector. which is given as

$V=\sum_{i}~{}|F_{i}~{}|^{2}~{}+~{}{1\over 2}~{}D^{A}D_{A}$ (59)

Putting together, we see that the lagrangian of the MSSM with SUSY unbroken is
of the form :

${\cal L}_{MSSM}^{(0)}=\int\left(d\theta^{2}~{}W(\Phi)+H.c\right)+\int
d\theta^{2}~{}d{\bar{\theta}}^{2}~{}\Phi_{i}^{\dagger}~{}e^{gV}~{}\Phi_{i}+\int\left(d\theta^{2}~{}{\cal
W}^{\alpha}{\cal W}_{\alpha}+H.c\right).$ (60)

where all the functions appearing in (60) have been discussed in eqs.(47,50)
and (57).

####  IV.0.1 Think it over

  * •

The full supersymmetric lagrangian of the Standard Model can be constructed
from the prescription given in the above section. Identify the dominant one-
loop contributions for the Higgs particle. Note that SUSY is still unbroken.
What are the dominant 1-loop contributions for other scalar particles, say the
stop ? Compute the processes $\mu~{}\to~{}e+~{}\gamma$ and $K^{0}-\bar{K}^{0}$
mixing in this limit.

  * •

As we have seen, W is a holomorphic function and that there are two Higgs
doublets giving masses to up type and down type quarks separately. (a) Give
examples of operators which are gauge invariant but non-holomorphic ? (b) Show
that such operators involving the Higgs fields will lead to Yukawa like
couplings with the “wrong” Higgs. Study the implications of such couplings.

Historical Note
Supersymmetries were first introduced in the context of string theories by
Ramond. In quantum field theories, this symmetry is realised through fermionic
generators, thus escaping the no-go theorems of Coleman and Mandula weinbergv3
. The simplest Lagrangian realising this symmetry in four dimensions was built
by Wess and Zumino which contains a spin ${1\over 2}$ fermion and a scalar.

In particle physics, supersymmetry plays an important role in protecting the
Higgs mass. To understand how it protects the Higgs mass, let us consider the
hierarchy problem once again. The Higgs mass enters as a bare mass parameter
in the Standard Model lagrangian, eq.(16). Contributions from the self energy
diagrams of the Higgs are quadratically divergent pushing the Higgs mass up to
cut-off scale. In the absence of any new physics at the intermediate energies,
the cut-off scale is typically $M_{GUT}$ or $M_{planck}$. Cancellation of
these divergences with the bare mass parameter would require fine-tuning of
order one part in $10^{-36}$ rendering the theory ‘unnatural’natural . In a
complete GUT model like SU(5) this might reflect as a severe problem of
doublet-triplet splitting buras ; gildener . On the other hand, if one has
additional contributions, say, for example, for the diagram with the Higgs
self coupling, there is an additional contribution from a fermionic loop, with
the fermion carrying the same mass as the scalar, the contribution from this
additional diagram would now cancel the quadratically divergent part of the SM
diagram, with the total contribution now being only logarithmically divergent.
If this mechanism needs to work for all the diagrams, not just for the Higgs
self-coupling and for all orders in perturbation theory, it would require a
symmetry which would relate a fermion and a boson with same mass.
Supersymmetry is such a symmetry.

###  IV.1 R-parity

In the previous section, we have seen that there are terms in the
superpotential, eq.(51) which are invariant under the Standard Model gauge
group $G_{SM}$ but however violate baryon ($\mathrm{B}$) and individual lepton
numbers ($\mathrm{L}_{e,\mu,\tau}$). At the first sight, it is bit surprising
: the matter superfields carry the same quantum numbers under the $G_{SM}$
just like the ordinary matter fields do in the Standard Model and $\mathrm{B}$
and $\mathrm{L}_{e,\mu,\tau}$ violating terms are not present in the Standard
Model. The reason can be traced to the fact that in the MSSM, where matter
sector is represented in terms of superfields, there is no distinction between
the fermions and the bosons of the model. In the Standard Model, the Higgs
field is a boson and the leptons and quarks are fermions and they are
different representations of the Lorentz group. This distinction is lost in
the MSSM, the Higgs superfield, $H_{1}$ and the lepton superfields $L_{i}$
have the same quantum numbers under $G_{SM}$ and given that they are both
(chiral) superfields, there is no way of distinguishing them. For this reason,
the second part of the superpotential $W_{2}$ makes an appearance in
supersymmetric version of the Standard Model. In fact, the first three terms
of eq.(51) can be achieved by replacing $H_{1}~{}\to~{}L_{i}$ in the terms
containing $H_{1}$ of $W_{1}$.

The first three terms of the second part of the superpotential $W_{2}$
(eq.(51)), are lepton number violating whereas the last term is baryon number
violating. The simultaneous presence of both these interactions can lead to
proton decay, for example, through a squark exchange. An example of such an
process in given in Figure 1. Experimentally the proton is quite stable. In
fact its life time is pretty large
$\stackrel{{\scriptstyle\scriptstyle>}}{{\scriptstyle\sim}}~{}\mathcal{O}(10^{33})$
years skproton . Thus products of these couplings ($\lambda^{\prime\prime}$
and one of ($\lambda^{\prime}~{},\epsilon,~{}\lambda$) which can lead to
proton decay are severely constrained to be of the order of
$\mathcal{(}O)(10^{-20})$151515The magnitude of these constraints depends also
on the scale of supersymmetry breaking, which we will come to discuss only in
the next section. For a list of constraints on R-violating couplings, please
see G. Bhattacharyya gautamB .. Thus to make the MSSM phenomenologically
viable one should expect these couplings in $W_{2}$ to take such extremely
small values.

$u$$u$$u$$d$$\lambda^{{}^{\prime\prime}}$$\tilde{s}$$\lambda^{\prime}$$\bar{l}$$\bar{u}$
Figure 1: A sample diagram showing the decay of the proton in the presence of
R-parity violating couplings.

A more natural way of dealing with such small numbers for these couplings
would be to set them to be zero. This can be arrived at by imposing a discrete
symmetry on the lagrangian called R-parity. R-parity has been originally
introduced as a discrete R-symmetry 161616R-symmetries are symmetries under
which the $\theta$ parameter transform non-trivially. by Ferrar and Fayet
fayetfarrar and then later realised to be of the following form by Ferrar and
Weinberg weinfarrar acting on the component fields:

$R_{p}=(-1)^{3(B-L)+2s},$ (61)

where B and L represent the Baryon and Lepton number respectively and s
represents the spin of the particle. Under R-parity the transformation
properties of various superfields can be summarised as:

$\displaystyle\\{V_{s}^{A},V_{w}^{I},V_{y}\\}$ $\displaystyle\rightarrow$
$\displaystyle\\{V_{s}^{A},V_{w}^{I},V_{y}\\}$ $\displaystyle\theta$
$\displaystyle\rightarrow$ $\displaystyle-\theta^{\star}$
$\displaystyle\\{Q_{i},U^{c}_{i},D^{c}_{i},L_{i},E_{i}^{c}\\}$
$\displaystyle\rightarrow$
$\displaystyle-\\{Q_{i},U^{c}_{i},D_{i}^{c},L_{i},E_{i}^{c}\\}$
$\displaystyle\\{H_{1},H_{2}\\}$ $\displaystyle\rightarrow$
$\displaystyle\\{H_{1},H_{2}\\}$ (62)

Imposing these constraints on the superfields will now set all the couplings
in $W_{2}$ to zero.

Imposing R-parity has an advantage that it provides a natural candidate for
dark matter. This can be seen by observing that R-parity distinguishes a
particle from its superpartner. This ensures that every interaction vertex has
at least two supersymmetric partners when R-parity is conserved. The lightest
supersymmetric particle (LSP) cannot decay in to a pair of SM particles and
remains stable. R-parity can also be thought of as a remnant symmetry theories
with an additional $U(1)$ symmetry, which is natural in a large class of
supersymmetric Grand Unified theories. Finally, one curious fact about
R-parity : it should be noted that R-parity constraints baryon and lepton
number violating couplings of dimension four or rather only at the
renormalisable level. If one allows for non-renormalisable operators in the
MSSM, i.e that is terms of dimension more than three in the superpotential,
they can induce dim 6 operators which violate baryon and lepton numbers at the
lagrangian level and are still allowed by R-parity. Such operators are
typically suppressed by high mass scale $\sim M_{Pl}$ or $M_{GUT}$ and thus
are less dangerous. In the present set of lectures, we will always impose
R-parity in the MSSM so that the proton does not decay, though there are
alternatives to R-parity which can also make proton stable.

####  IV.1.1 Think it over

  * •

Is imposing R-parity the only way to get rid of the terms which lead to proton
decay ? (Hint: For proton decay to occur both $\mathrm{L}$ and $\mathrm{B}$
violating operators are required. R-parity removes both these sets of
operators which is unnecessary. We can think of discrete symmetries which can
remove only either $\mathrm{B}$ or $\mathrm{L}$ type of operators.) See for
examplerosshall .

##  V STEP 3: Supersymmetry breaking

So far, we have seen that the Supersymmetric Standard Model lagrangian can
also be organised in a similar way like the Standard Model lagrangian though
one uses functions of superfields now to get the lagrangian rather than the
ordinary fields. In the present section we will cover the last part (term) of
the total MSSM lagrangian

$\mathcal{L}_{\mbox{MSSM}}=\mathcal{L}_{\mbox{gauge/kinetic}}\left(K(\Phi,V)\right)+\mathcal{L}_{\mbox{yukawa}}\left(W(\Phi)\right)+\mathcal{L}_{\mbox{scalar}}\left(F^{2},D^{2}\right)+\mathcal{L}_{\mbox{SSB}}$
(63)

which we have left out so far and that concerns supersymmetry breaking (SSB).
Note that the first three terms are essentially from
$\mathcal{L}_{\mbox{MSSM}}^{(0)}$ of eq.(60). In Nature, we do not observe
supersymmetry. Supersymmetry breaking has to be incorporated in the MSSM to
make it realistic. In a general lagrangian, supersymmetry can be broken
spontaneously if the auxiliary fields F or D appearing in the definitions of
the chiral and vector superfields respectively attain a vacuum expectation
value (vev). If the $F$ fields get a vev, it is called $F$-breaking whereas if
the $D$ fields get a vev, it is called $D$-breaking.

Incorporation of spontaneous SUSY breaking in MSSM would mean that at least
one (or more) of the F-components corresponding to one ( or more) of the MSSM
chiral (matter) superfields would attain a vacuum expectation value. However,
this approach fails as this leads to phenomenologically unacceptable
prediction that at least one of the super-partner should be lighter (in mass)
than the ordinary particle. This is not valid phenomenologically as such a
light super partner (of SM particle) has been ruled out experimentally. One
has to think of a different approach for incorporating supersymmetry breaking
in to the MSSM Luty .

Figure 2: A schematic diagram showing SUSY breaking using Hidden sector models

One of the most popular and successful approaches has been to assume another
sector of the theory consisting of superfields which are not charged under the
Standard Model gauge group. Such a sector of the theory is called ‘Hidden
Sector’ as they cannot been ”seen” like the Standard Model particles and
remain hidden. Supersymmetry can be broken spontaneously in this sector. This
information is communicated to the visible sector or MSSM through a messenger
sector. The messenger sector can be made up of gravitational interactions or
ordinary gauge interactions. The communication of supersymmetry breaking leads
to supersymmetry breaking terms in MSSM. Thus, supersymmetry is not broken
spontaneously within the MSSM, but explicitly by adding supersymmetry breaking
terms in the lagrangian.

However, not all supersymmetric terms can be added. We need to add only those
terms which do not re-introduce quadratic divergences back into the
theory171717Interaction terms and other couplings which do not lead to
quadratically divergent (in cut-off $\Lambda$) terms in the theory once loop
corrections are taken in to consideration. It essentially means we only add
dimensional full couplings which are supersymmetry breaking.. It should be
noted that in most models of spontaneous supersymmetry breaking, only such
terms are generated. These terms which are called “soft” supersymmetry
breaking terms can be classified as follows:

  * •

a) Mass terms for the gauginos which are a part of the various vector
superfields of the MSSM.

  * •

b) Mass terms for the scalar particles,
$m^{2}_{\phi_{ij}}~{}\phi_{i}^{\star}\phi_{j}$ with $\phi_{i,j}$ representing
the scalar partners of chiral superfields of the MSSM.

  * •

c) Trilinear scalar interactions, $A_{ijk}\phi_{i}\phi_{j}\phi_{k}$
corresponding to the cubic terms in the superpotential.

  * •

d) Bilinear scalar interactions, $B_{ij}\phi_{i}\phi_{j}$ corresponding to the
bilinear terms in the superpotential.

Note that all the above terms are dimensionful. Adding these terms would make
the MSSM non-supersymmetric and thus realistic. The total MSSM lagrangian is
thus equal to

${\cal L}_{total}={\cal L}_{MSSM}^{(0)}+{\cal L}_{SSB}$ (64)

with ${\cal L}_{MSSM}^{(0)}$ given in eq.(60). Sometimes in literature we have
${\cal L}_{SSB}={\cal L}_{soft}$. Let us now see the complete list of all the
soft SUSY breaking terms one can incorporate in the MSSM:

  1. 1.

Gaugino Mass terms: Corresponding to the three vector superfields (for gauge
groups $U(1)$, $SU(2)$ and $SU(3)$) we have $\tilde{B},\tilde{W}$ and
$\tilde{G}$) we have three gaugino mass terms which are given as
$M_{1}\tilde{B}\tilde{B}$, $M_{2}\tilde{W}_{I}\tilde{W}_{I}$ and
$M_{3}\tilde{G}_{A}\tilde{G}_{A}$, where $I(A)$ runs over all the
$SU(2)(SU(3))$ group generators.

  2. 2.

Scalar Mass terms: For every scalar in each chiral (matter) superfield , we
can add a mass term of the form $m^{2}~{}\phi_{i}^{\star}\phi_{j}$. Note that
the generation indices $i,j$ need not be the same. Thus the mass terms can
violate flavour. Further, given that SUSY is broken prior to $SU(2)\times
U(1)$ breaking , all these mass terms for the scalar fields should be written
in terms of their ‘unbroken’ $SU(2)\times U(1)$ representations. Thus the
scalar mass terms are : $m_{Q_{ij}}^{2}\tilde{Q}_{i}^{\dagger}\tilde{Q}_{j}$ ,
$m_{u_{ij}}^{2}\tilde{u^{c}}_{i}^{\star}\tilde{u^{c}}_{j}$ ,
$m_{d_{ij}}^{2}\tilde{d^{c}}_{i}^{\star}\tilde{d^{c}}_{j}$ ,
$m_{L_{ij}}^{2}\tilde{L}_{i}^{\dagger}\tilde{L}_{j}$ ,
$m_{e_{ij}}^{2}\tilde{e^{c}}_{i}^{\star}\tilde{e^{c}}_{j}$ ,
$m_{H_{1}}^{2}H_{1}^{\dagger}H_{1}$ and $m_{H_{2}}^{2}H_{2}^{\dagger}H_{2}$.

  3. 3.

Trilinear Scalar Couplings: As mentioned again, there are only three types of
trilinear scalar couplings one can write which are $G_{SM}$ gauge invariant.
In fact, their form exactly follows from the Yukawa couplings. These are :
$A^{u}_{ij}\tilde{Q}_{i}\tilde{u}^{c}_{j}H_{2}$,
$A^{d}_{ij}\tilde{Q}_{i}\tilde{d}^{c}_{j}H_{1}$ and
$A^{e}_{ij}\tilde{L}_{i}\tilde{e}^{c}_{j}H_{1}$.

  4. 4.

Bilinear Scalar Couplings: Finally, there is only one bilinear scalar coupling
(other than the mass terms) which is gauge invariant. The form of this term
also follows from the superpotential. It is given as : $BH_{1}H_{2}$.

Adding all these terms completes the lagrangian for the MSSM. However, the
particles are still not in their ‘physical’ basis as $SU(2)\times U(1)$
breaking is not yet incorporated. Once incorporated the physical states of the
MSSM and their couplings could be derived.

##  VI STEP 4: $SU(2)\times U(1)$ breaking

As a starting point, it is important to realize that the MSSM is a two Higgs
doublet model i.e, SM with two Higgs doublets instead of one, with a different
set of couplings higgshunter . Just as in Standard Model, spontaneous breaking
of $SU(2)_{L}\times U(1)_{Y}~{}\to~{}U(1)_{EM}$ can be incorporated here too.
Doing this leads to constraints relating various parameters of the model. To
see this, consider the neutral Higgs part of the total scalar potential
including the soft terms. It is given as

$\displaystyle V_{scalar}$ $\displaystyle=$
$\displaystyle(m_{H_{1}}^{2}+\mu^{2})|H_{1}^{0}|^{2}+(m_{H_{2}}^{2}+\mu^{2})|H_{2}^{0}|^{2}-(B_{\mu}\mu
H_{1}^{0}H_{2}^{0}+H.c)$ (65) $\displaystyle+$ $\displaystyle{1\over
8}(g^{2}+g^{\prime 2})({H_{2}^{0}}^{2}-{H_{1}^{0}}^{2})^{2}+\ldots,$

where $H_{1}^{0},H_{2}^{0}$ stand for the neutral Higgs scalars and we have
parameterised the bilinear soft term $B\equiv B_{\mu}\mu$. Firstly, we should
require that the potential should be bounded from below. This gives the
condition (in field configurations where the D-term goes to zero, i.e, the
second line in eq.(65)):

$2B_{\mu}<2|\mu|^{2}~{}+~{}m_{H_{2}}^{2}~{}+m_{H_{1}}^{2}$ (66)

Secondly, the existence of a minima for the above potential would require at
least one of the Higgs mass squared to be negative giving the condition,
(determinant of the $2\times 2$ neutral Higgs mass squared matrix should be
negative)

$B_{\mu}^{2}>(|\mu|^{2}~{}+~{}m_{H_{2}}^{2}~{})~{}(|\mu|^{2}+m_{H_{1}}^{2})$
(67)

In addition to ensuring the existence of a minima, one would also require that
the minima should be able to reproduce the standard model relations i.e,
correct gauge boson masses. We insist that both the neutral Higgs attain
vacuum expectation values :

$<H^{0}_{1}>={v_{1}\over\sqrt{2}}\;\;\;\;;\;\;\;<H_{2}^{0}>={v_{2}\over\sqrt{2}}$
(68)

and furthermore we define

$~{}v_{1}^{2}+v_{2}^{2}~{}=~{}v^{2}~{}=~{}246^{2}~{}\mbox{GeV}^{2},$

where $v$ represents the vev of the Standard Model (SM) Higgs field. However,
these vevs should correspond to the minima of the MSSM potential. The minima
are derived by requiring $\partial V/\partial H_{1}^{0}~{}=~{}0$ and $\partial
V/H_{2}^{0}~{}=~{}0$ at the minimum, where the form of $V$ is given in
eq.(65). These derivative conditions lead to relations between the various
parameters of the model at the minimum of the potential. We have, using the
Higgs vev (68) and the formulae for181818In this lecture note, we will be
using $g_{2}~{}=g~{}=~{}g_{W}$ for the SU(2) coupling, whereas
$g^{\prime}=g_{1}$ for the $U(1)_{Y}$ coupling and $g_{s}=g_{3}$ for the SU(3)
strong coupling. $M_{Z}^{2}~{}=~{}{1\over 4}(g^{2}+g^{{}^{\prime}~{}2})v^{2}$,
the minimisation conditions can rewritten as

$\displaystyle{1\over 2}M_{Z}^{2}$ $\displaystyle=$
$\displaystyle{m_{H_{1}}^{2}-\tan^{2}\beta~{}m_{H_{2}}^{2}\over\tan^{2}\beta-1}-\mu^{2}$
$\displaystyle\mbox{Sin}2\beta$ $\displaystyle=$
$\displaystyle{2B_{\mu}~{}\mu~{}\over m_{H_{2}}^{2}+m_{H_{1}}^{2}+2\mu^{2}},$
(69)

where we have used the definition $\tan\beta=v_{2}/v_{1}$ as the ratio of the
vacuum expectation values of $H_{2}^{0}$ and $H_{1}^{0}$ respectively. Note
that the parameters $m_{H_{1}}^{2},~{}m_{H_{2}}^{2},B_{\mu}$ are all
supersymmetry breaking ‘soft’ terms. $\mu$ is the coupling which comes in the
superpotential giving the supersymmetry conserving masses to the Higgs
scalars. These are related to the Standard Model parameters $M_{Z}$ and a
ratio of vevs, parameterised by an angle tan$\beta$. Thus these conditions
relate SUSY breaking soft parameters with the SUSY conserving ones and the
Standard Model parameters. For any model of supersymmetry to make contact with
reality, the above two conditions (VI )need to be satisfied.

The above minimisation conditions are given for the ‘tree level’ potential
only. 1-loop corrections a ’la Coleman-Weinberg can significantly modify these
minima. We will discuss a part of them in later sections when we discuss the
Higgs spectrum. Finally we should mention that, in a more concrete approach,
one should consider the entire scalar potential including all the scalars in
the theory, not just confining ourselves to the neutral Higgs scalars. For
such a potential one should further demand that there are no deeper minima
which are color and charge breaking (which effectively means none of the
colored and charged scalar fields get vacuum expectation values). These
conditions lead to additional constraints on parameters of the MSSMcasasdimo .

####  VI.0.1 Think it over

  * •

In the MSSM, we have considered here contains two Higgs doublets. In addition
to $H_{1}$ and $H_{2}$, consider an additional Higgs field field $S$, which
transforms as a singlet under all the gauge groups of $G_{SM}$. Write down the
superpotential including the singlet field $S$ invariant under $G_{SM}$.
Derive the corresponding scalar potential including the soft SUSY breaking
terms. Minimise the neutral Higgs potential and derive the electro-weak
minimisation conditions. How many are there and what are they? (Hint: Assume
the $S$ field also develops a vev and that its vev is much larger than $v_{1}$
and $v_{2}$. )

##  VII Step 5: Mass spectrum

We have seen in the earlier section, supersymmetry breaking terms introduce
mass-splittings between ordinary particles and their super-partners. Given
that particles have zero masses in the limit of exact $G_{SM}$, only
superpartners are given soft mass terms. After the $SU(2)~{}\times U(1)$
breaking, ordinary particles as well as superparticles attain mass terms. For
the supersymmetric partners, these mass terms are either additional
contributions or mixing terms between the various super-particles. Thus, just
like in the case of ordinary SM fermions, where one has to diagonalise the
fermion mass matrices to write the lagrangian in the ‘on-shell’ format or the
physical basis, a similar diagonalisation has to be done for the super-
symmetric particles and their mass matrices.

####  VII.0.1 The Neutralino Sector

To begin with lets start with the gauge sector. The superpartners of the
neutral gauge bosons (neutral gauginos) and the fermionic partners of the
neutral higgs bosons (neutral higgsinos) mix to form Neutralinos. The
neutralino mass matrix in the basis

$\mathcal{L}~{}\supset~{}{1\over
2}~{}\Psi_{N}\mathcal{M}_{N}\Psi_{N}^{T}~{}+H.c$

where
$\Psi_{N}=\\{\tilde{B},~{}\tilde{W}^{0},\tilde{H}_{1}^{0},\tilde{H}_{2}^{0}\\}$
is given as :

$\mathcal{M}_{N}~{}=~{}\left(\matrix{M_{1}&0&-M_{Z}c\beta~{}s\theta_{W}&M_{Z}s\beta~{}s\theta_{W}\cr
0&M_{2}&M_{Z}c\beta~{}c\theta_{W}&M_{Z}s\beta~{}c\theta_{W}\cr-
M_{Z}c\beta~{}s\theta_{W}&M_{Z}c\beta~{}c\theta_{W}&0&-\mu\cr
M_{Z}s\beta~{}s\theta_{W}&-M_{Z}s\beta~{}c\theta_{W}&-\mu&0}\right),$ (70)

with $c\beta(s\beta)$ and $c\theta_{W}(s\theta_{W})$ standing for
$\cos\beta(\sin\beta)$ and $\cos\theta_{W}(\sin\theta_{W})$ respectively. As
mentioned earlier, $M_{1}$ and $M_{2}$ are the soft parameters, whereas $\mu$
is the superpotential parameter, thus SUSY conserving. The angle $\beta$ is
typically taken as a input parameter, $tan\beta={v_{2}/v_{1}}$ whereas
$\theta_{W}$ is the Weinberg angle given by the inverse tangent of the ratio
of the gauge couplings as in the SM. Note that the neutralino mass matrix
being a Majorana mass matrix is complex symmetric in nature. Hence it is
diagonalised by a unitary matrix $N$,

$\displaystyle N^{*}\cdot M_{\tilde{N}}\cdot
N^{\dagger}=\mbox{Diag.}(m_{\chi_{1}^{0}},m_{\chi_{2}^{0}},m_{\chi_{3}^{0}},m_{\chi_{4}^{0}})$
(71)

The states are rotated by $\chi_{i}^{0}=N^{\star}\Psi$ to go the physical
basis.

####  VII.0.2 The Chargino Sector

In a similar manner to the neutralino sector, all the fermionic partners of
the charged gauge bosons and of the charged Higgs bosons mix after electroweak
symmetry breaking. However, they combine in a such a way that a Wino-Higgsino
Weyl fermion pair forms a Dirac fermion called the chargino. This mass matrix
is given as

$\mathcal{L}\supset-\frac{1}{2}\left(\matrix{\tilde{W}^{-}&\tilde{H}_{1}^{-}}\right)\;\left(\matrix{M_{2}&\sqrt{2}M_{W}\sin\beta\cr\sqrt{2}M_{W}\cos\beta&\mu}\right)\left(\matrix{\tilde{W}^{+}\cr\tilde{H}_{2}^{+}}\right),$
(72)

Given the non-symmetric (non-hermitian) matrix nature of this matrix, it is
diagonalised by a bi-unitary transformation, $U^{*}\cdot M_{C}\cdot
V^{\dagger}=\mbox{Diag.}(m_{\chi_{1}^{+}},m_{\chi_{2}^{+}})$. The chargino
eigenstates are typically represented by $\chi^{\pm}$ with mass eigenvalues
$m_{\chi^{\pm}}$. The explicit forms for $U$ and $V$ can be found by the
eigenvectors of $M_{C}M_{C}^{\dagger}$ and $M_{C}^{\dagger}M_{C}$ respectively
haberkane .

####  VII.0.3 The Sfermion Sector

Next let us come to the sfermion sector. Remember that we have added different
scalar fields for the right and left handed fermions in the Standard Model.
After electroweak symmetry breaking, the sfermions corresponding to the left
fermion and the right fermion mix with each other. Furthermore while writing
down the mass matrix for the sfermions, we should remember that these terms
could break the flavour i.e, we can have mass terms which mix different
generation. Thus, in general the sfermion mass matrix is a $6\times 6$ mass
matrix given as :

$\xi^{\dagger}~{}M_{\tilde{f}}^{2}\xi~{}~{};~{}~{}~{}\xi=\\{{\tilde{f}_{L_{i}}},{\tilde{f}_{R_{i}}}\\}$

From the total scalar potential, the mass matrix for these sfermions can be
derived using standard definition given as

$m_{ij}^{2}=\left(\begin{array}[]{cc}{\partial^{2}V\over\partial\phi_{i}\partial\phi_{j}^{\star}}&{\partial^{2}V\over\partial\phi_{i}\partial\phi_{j}}\\\
{\partial^{2}V\over\partial\phi_{i}^{\star}\partial\phi_{j}^{\star}}&{\partial^{2}V\over\partial\phi_{i}^{\star}\partial\phi_{j}}\end{array}\right)$
(73)

Using this for sfermions, we have :

$M_{\tilde{f}}^{2}\;=\;\left(\matrix{m_{\tilde{f}_{\mbox{\scriptsize
LL}}}^{2}&m_{\tilde{f}_{\mbox{\scriptsize
LR}}}^{2}\cr{m_{\tilde{f}_{\mbox{\scriptsize
LR}}}^{2\,\dagger}}&m_{\tilde{f}_{\mbox{\scriptsize RR}}}^{2}}\right),$ (74)

where each of the above entries represents $3\times 3$ matrices in the
generation space. More specifically, they have the form (as usual, $i,j$ are
generation indices):

$\displaystyle m^{2}_{\tilde{f}_{L_{i}L_{j}}}$ $\displaystyle=$ $\displaystyle
M^{2}_{\tilde{f}_{L_{i}L_{j}}}+m^{2}_{f}\delta_{ij}+M_{Z}^{2}\cos
2\beta(T_{3}+\sin^{2}\theta_{W}Q_{\mbox{\scriptsize em}})\delta_{ij}$
$\displaystyle m_{{\tilde{f}}_{L_{i}R_{j}}}^{2}$ $\displaystyle=$
$\displaystyle\left(\big{(}Y^{A}_{f}\cdot^{v_{2}}_{v_{1}}-m_{f}\mu^{\tan\beta}_{\cot\beta}\big{)}\;\;\mbox{for}\;f=^{e,d}_{u}\right)\delta_{ij}$
$\displaystyle m_{\tilde{f}_{\mbox{\scriptsize RR}}}^{2}$ $\displaystyle=$
$\displaystyle M^{2}_{\tilde{f}_{R_{ij}}}+\left(m^{2}_{f}+M_{Z}^{2}\cos
2\beta\sin^{2}\theta_{W}Q_{\mbox{\scriptsize em}}\right)\delta_{ij}$ (75)

In the above, $M^{2}_{\tilde{f}_{L}}$ represents the soft mass term for the
corresponding fermion ($L$ for left, $R$ for right), $T_{3}$ is the eigenvalue
of the diagonal generator of $SU(2)$, $m_{f}$ is the mass of the fermion with
$Y$ and $Q_{em}$ representing the hypercharge and electromagnetic charge (in
units of the charge of the electron ) respectively. The sfermion mass matrices
are hermitian and are thus diagonalised by a unitary rotation,
$R_{\tilde{f}}R_{\tilde{f}}^{\dagger}=1$:

$R_{\tilde{f}}\cdot M_{\tilde{f}}\cdot
R_{\tilde{f}}^{\dagger}=\mbox{Diag.}(m_{\tilde{f}_{1}},m_{\tilde{f}_{2}},\dots,m_{\tilde{f}_{6}})$
(76)

####  VII.0.4 The Higgs sector

Now let us turn our attention to the Higgs fields. We will use again use the
standard formula of eq.(73), to derive the Higgs mass matrices. The eight
Higgs degrees of freedom form a $8\times 8$ Higgs mass matrix which breaks
down diagonally in to three $2\times 2$ mass matrices191919The discussion in
this section closely follows from the discussion presented in Ref.rohinibook .

The mass matrices are divided in to charged sector, CP odd neutral and CP even
neutral. This helps us in identifying the goldstone modes and the physical
spectrum in an simple manner. Before writing down the mass matrices, let us
first define the following parameters :

$m_{1}^{2}=m_{H_{1}}^{2}+\mu^{2},\;\;\;m_{2}^{2}=m_{H_{2}}^{2}+\mu^{2},\;\;\;m_{3}^{2}=B_{\mu}\mu.$

In terms of these parameters, the various mass matrices and the corresponding
physical states obtained after diagonalising the mass matrices are given
below:
Charged Higgs and Goldstone Modes:

$\left(\begin{array}[]{cc}H_{1}^{+}&H_{2}^{+}\end{array}\right)\left(\begin{array}[]{cc}m_{1}^{2}+{1\over
8}(g_{1}^{2}+g_{2}^{2})(v_{1}^{2}-v_{2}^{2})+{1\over
4}g_{2}^{2}v_{2}^{2}&m_{3}^{2}+{1\over 4}g_{2}^{2}v_{1}v_{2}\\\
m_{3}^{2}+{1\over 4}g_{2}^{2}v_{1}v_{2}&m_{2}^{2}-{1\over
8}(g_{1}^{2}+g_{2}^{2})(v_{1}^{2}-v_{2}^{2})+{1\over
4}g_{2}^{2}v_{2}^{2}\end{array}\right)\left(\begin{array}[]{c}H_{1}^{-}\\\
H_{2}^{-}\end{array}\right)$ (77)

Using the minimisation conditions (VI), this matrix becomes,

$\left(\begin{array}[]{cc}H_{1}^{+}&H_{2}^{+}\end{array}\right)({m_{3}^{2}\over
v_{1}v_{2}}+{1\over
4}g_{2}^{2})\left(\begin{array}[]{cc}v_{2}^{2}&v_{1}v_{2}\\\
v_{1}v_{2}&v_{1}^{2}\\\ \end{array}\right)\left(\begin{array}[]{c}H_{1}^{-}\\\
H_{2}^{-}\end{array}\right)$ (78)

which has determinant zero leading to the two eigenvalues as :

$\displaystyle m_{G^{\pm}}^{2}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle m_{H^{\pm}}^{2}$ $\displaystyle=$
$\displaystyle\left({m_{3}^{2}\over v_{1}v_{2}}+{1\over
4}g_{2}^{2}\right)(v_{1}^{2}+v_{2}^{2}),$ (79) $\displaystyle=$
$\displaystyle{2m_{3}^{2}\over sin2\beta}+M_{W}^{2}$ (80)

where $G^{\pm}$ represents the Goldstone mode. The physical states are
obtained just by rotating the original states in terms of the $H_{1},~{}H_{2}$
fields by an mixing angle. The mixing angle in the present case (in the
unitary gauge) is just tan$\beta$:

$\left(\begin{array}[]{c}H^{\pm}\\\
G^{\pm}\end{array}\right)=\left(\begin{array}[]{cc}sin\beta&cos\beta\\\
-cos\beta&sin\beta\end{array}\right)\left(\begin{array}[]{c}H^{\pm}\\\
G^{\pm}\end{array}\right)$ (81)

CP odd Higgs and Goldstone Modes:
Let us now turn our attention to the CP-odd Higgs sector. The mass matrices
can be written in a similar manner but this time for imaginary components of
the neutral Higgs.

$\left(\begin{array}[]{cc}ImH_{1}^{0}&ImH_{2}^{0}\end{array}\right)\left(\begin{array}[]{cc}m_{1}^{2}+{1\over
8}(g_{1}^{2}+g_{2}^{2})(v_{1}^{2}-v_{2}^{2})&m_{3}^{2}\\\
m_{3}^{2}&m_{2}^{2}-{1\over
8}(g_{1}^{2}+g_{2}^{2})(v_{1}^{2}-v_{2}^{2})\end{array}\right)\left(\begin{array}[]{c}ImH_{1}^{0}\\\
ImH_{2}^{0}\end{array}\right)$ (82)

As before, again using the minimisation conditions, this matrix becomes,

$\left(\begin{array}[]{cc}ImH_{1}^{0}&ImH_{2}^{0}\end{array}\right)m_{3}^{2}\left(\begin{array}[]{cc}v_{2}/v_{1}&1\\\
1&v_{1}/v_{2}\\\ \end{array}\right)\left(\begin{array}[]{c}ImH_{1}^{0}\\\
ImH_{2}^{0}\end{array}\right)$ (83)

which has determinant zero leading to the two eigenvalues as :

$\displaystyle m_{G^{0}}^{2}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle m_{A^{0}}^{2}$ $\displaystyle=$
$\displaystyle\left({m_{3}^{2}\over
v_{1}v_{2}}\right)(v_{1}^{2}+v_{2}^{2})~{}~{}=~{}~{}{2m_{3}^{2}\over
sin2\beta}$ (84)

Similar to the charged sector, the mixing angle between these two states in
the unitary gauge is again just tan$\beta$.

${1\over\sqrt{2}}\left(\begin{array}[]{c}A^{0}\\\
G^{0}\end{array}\right)=\left(\begin{array}[]{cc}sin\beta&cos\beta\\\
-cos\beta&sin\beta\end{array}\right)\left(\begin{array}[]{c}ImH^{0}_{1}\\\
ImH_{2}^{0}\end{array}\right)$ (85)

CP even Higgs:
Finally, let us come to the real part of the neutral Higgs sector. The mass
matrix in this case is given by the following.

$\left(\begin{array}[]{cc}ReH_{1}^{0}&ReH_{2}^{0}\end{array}\right)~{}{1\over
2}~{}\left(\begin{array}[]{cc}2m_{1}^{2}+{1\over
4}(g_{1}^{2}+g_{2}^{2})(3v_{1}^{2}-v_{2}^{2})&-2m_{3}^{2}-{1\over
4}v_{1}v_{2}(g_{1}^{2}+g_{2}^{2})\\\ -2m_{3}^{2}-{1\over
4}v_{1}v_{2}(g_{1}^{2}+g_{2}^{2})&2m_{2}^{2}+{1\over
4}(g_{1}^{2}+g_{2}^{2})(3v_{2}^{2}-v_{1}^{2})\end{array}\right)\left(\begin{array}[]{c}ReH_{1}^{0}\\\
ReH_{2}^{0}\end{array}\right)$ (86)

Note that in the present case, there is no Goldstone mode. As before, we will
use the minimisation conditions and further using the definition of
$m_{A}^{2}$ from eq.(84), we have :

$\left(\begin{array}[]{cc}ReH_{1}^{0}&ReH_{2}^{0}\end{array}\right)\left(\begin{array}[]{cc}m_{A}^{2}sin^{2}\beta+M_{z}^{2}cos\beta&-(m_{A}^{2}+m_{Z}^{2})sin\beta
cos\beta\\\ -(m_{A}^{2}+m_{Z}^{2})sin\beta
cos\beta&m_{A}^{2}cos^{2}\beta+M_{z}^{2}sin\beta\end{array}\right)\left(\begin{array}[]{c}ReH_{1}^{0}\\\
ReH_{2}^{0}\end{array}\right)$ (87)

The matrix has two eigenvalues which are given by the two signs of the
following equation:

$m^{2}_{H,h}={1\over
2}\left[m_{A}^{2}+m_{Z}^{2}\pm\\{(m_{A}^{2}+m_{Z}^{2})^{2}-4m_{Z}^{2}m_{A}^{2}cos^{2}2\beta\\}^{1/2}\right]$
(88)

The heavier eigenvalue $m_{H}^{2}$, is obtained by taken the positive sign,
whereas the lighter eigenvalue $m_{h}^{2}$ is obtained by taking the negative
sign respectively. The mixing angle between these two states can be read out
from the mass matrix of the above202020The mixing angle for a $2\times 2$
symmetric matrix, $C_{ij}$ is given by  $tan2\theta=2C_{12}/(C_{22}-C_{11}).$
as :

$tan~{}2\alpha={m_{A}^{2}+m_{Z}^{2}\over m_{A}^{2}-m_{Z}^{2}}~{}tan~{}2\beta$
(89)

Tree Level Catastrophe:
So far we have seen that out of the eight Higgs degrees of freedom, three of
them form the Goldstone modes after incorporating $SU(2)\times U(1)$ breaking
and there are five physical Higgs bosons fields in the MSSM spectrum. These
are the charged Higgs ($H^{\pm}$) a CP-odd Higgs ($A$) and two CP-even Higgs
bosons ($h,H$). From the mass spectrum analysis above, we have seen that the
mass eigenvalues of these Higgs bosons are related to each other. In fact,
putting together all the eigenvalue equations, we summarise the relations
between them as follows :

$\displaystyle m^{2}_{H^{\pm}}$ $\displaystyle=$ $\displaystyle
m_{A}^{2}+m_{W}^{2}>\max(M_{W}^{2},m_{A}^{2})$ $\displaystyle
m_{h}^{2}+m_{H}^{2}$ $\displaystyle=$ $\displaystyle m_{A}^{2}+m_{Z}^{2}$
$\displaystyle m_{H}$ $\displaystyle>$ $\displaystyle max(m_{A},m_{Z})$
$\displaystyle m_{h}$ $\displaystyle<$ $\displaystyle
min(m_{A},M_{Z})|cos2\beta|<min(m_{A},m_{Z})$ (90)

Let us concentrate on the last relation of the above eq.(VII.0.4). The
condition on the lightest CP even Higgs mass, $m_{h}$, tell us that it should
be equal to $m_{Z}$ in the limit tan$\beta$ is saturated to be maximum, such
that cos$2\beta~{}\rightarrow~{}1$ and $m_{A}~{}\rightarrow~{}\infty$. If
these limits are not saturated, it is evident that the light higgs mass is
less that $m_{Z}$. This is one of main predictions of MSSM which could make it
easily falsifiable from the current generation of experiments like LEP,
Tevatron and the upcoming LHC. Given that present day experiments have not
found a Higgs less that Z-boson mass, it is tempting to conclude that the MSSM
is not realised in Nature. However caution should be exercised before taking
such a route as our results are valid only at the tree level. In fact, in a
series of papers in the early nineties oneloophiggs , it has been shown that
large one-loop corrections to the Higgs mass can easily circumvent this limit.

The light Higgs Spectrum at 1-loop
As mentioned previously, radiative corrections can significantly modify the
mass relations which we have presented in the previous section. As is evident,
these corrections can be very important for the light Higgs boson mass. Along
with the 1-loop corrections previously, in the recent years dominant parts of
two-loop corrections have also been available slavich2loop with a more
complete version recently givenmartin2loop . In the following we will present
the one-loop corrections to the light Higgs mass and try to understand the
implications for the condition eq.(VII.0.4). Writing down the 1-loop
corrections to the CP-even part of the Higgs mass matrix as :

$M^{2}_{Re}=M^{2}_{Re}(0)+\delta M^{2}_{Re},$ (91)

where $M^{2}_{Re}(0)$ represents the tree level mass matrix given by eq.(87)
and $\delta M^{2}_{Re}$ represents its one-loop correction. The dominant one-
loop correction comes from the top quark and stop squark loops which can be
written in the following form:

$\delta M^{2}_{Re}=\left(\begin{array}[]{cc}\Delta_{11}&\Delta_{12}\\\
\Delta_{12}&\Delta_{22}\end{array}\right),$ (92)

where

$\displaystyle\Delta_{11}$ $\displaystyle=$
$\displaystyle{3G_{F}m_{t}^{4}\over
2\sqrt{2}\pi^{2}sin^{2}\beta}\left[{\mu(A_{t}+\mu cot\beta)\over
m_{\tilde{t}_{1}}^{2}-m_{\tilde{t}_{2}}^{2}}\right]^{2}\left(2-{m_{\tilde{t}_{1}}^{2}+m_{\tilde{t}_{2}}^{2}\over
m_{\tilde{t}_{1}}^{2}-m_{\tilde{t}_{2}}^{2}}ln{m_{\tilde{t}_{1}}^{2}\over
m_{\tilde{t}_{2}}^{2}}\right)$ $\displaystyle\Delta_{12}$ $\displaystyle=$
$\displaystyle{3G_{F}m_{t}^{4}\over
2\sqrt{2}\pi^{2}sin^{2}\beta}\left[{\mu(A_{t}+\mu cot\beta)\over
m_{\tilde{t}_{1}}^{2}-m_{\tilde{t}_{2}}^{2}}\right]ln{m_{\tilde{t}_{1}}^{2}\over
m_{\tilde{t}_{2}}^{2}}+{A_{t}\over\mu}\Delta_{11}$ $\displaystyle\Delta_{22}$
$\displaystyle=$
$\displaystyle{3G_{F}m_{t}^{4}\over\sqrt{2}\pi^{2}sin^{2}\beta}\left[ln{m_{\tilde{t}_{1}}^{2}m_{\tilde{t}_{2}}^{2}\over
m_{t}^{2}}+{A_{t}(A_{t}+\mu cot\beta)\over
m_{\tilde{t}_{1}}^{2}-m_{\tilde{t}_{2}}}ln{m_{\tilde{t}_{1}}^{2}\over
m_{\tilde{t}_{2}}^{2}}\right]+{A_{t}\over\mu}\Delta_{11}$ (93)

In the above $G_{F}$ represents Fermi Decay constant, $m_{t}$, the top mass,
$m_{\tilde{t}_{1}}^{2},~{}m_{\tilde{t}_{1}}^{2}$ are the eigenvalues of the
stop mass matrix and $A_{t}$ is the trilinear scalar coupling (corresponding
to the top Yukawa coupling) in the stop mass matrix. $\mu$ and the angle
$\beta$ have their usual meanings. Taking in to account these corrections, the
condition (VII.0.4) takes the form:

$m_{h}^{2}~{}<~{}m_{Z}^{2}\text{cos}^{2}2\beta+\Delta_{11}\text{cos}^{2}\beta+\Delta_{12}\text{sin}2\beta+\Delta_{22}\text{sin}^{2}\beta$
(94)

Given that $m_{t}$ is quite large, almost twice the $m_{Z}$ mass, for suitable
values of the stop masses, it is clear that the tree level upper limit on the
light Higgs mass is now evaded. However, a reasonable upper limit can still be
got by assuming reasonable values for the stop mass. For example assuming stop
masses to be around 1 TeV and maximal mixing the stop sector, one attains an
upper bound on the light Higgs mass as:

$m_{h}~{}\stackrel{{\scriptstyle\scriptstyle<}}{{\scriptstyle\sim}}~{}135~{}\text{GeV}.$
(95)

####  VII.0.5 Feynman Rules

In this section, we have written down all the mass matrices of the
superpartners, their eigenvalues and finally the eigenvectors which are
required to transform the superpartners in to their physical basis. The
feynman rules corresponding to the various vertices have to be written down in
this basis. Thus various soft supersymmetry breaking and supersymmetry
conserving parameters entering these mass matrices would now determine these
couplings as well as the masses, which in turn determine the strength of
various physical processes like crosssections and decay rates. A complete list
of the Feynman rules in the mass basis can be found in various references like
Physics Reports like Haber & Kane haberkane and D Chung et. alKaneking and
also in textbooks like Sparticles rohinibook and Baer & Tata baertata . A
complete set of Feynman rules is out of reach of this set of lectures. Here I
will just present two examples to illustrate the points I have been making
here.

$l_{i}$$\tilde{\chi}^{+}_{A}$$\tilde{v}_{X}$$\tilde{C}_{iAX}$ | $l_{i}$$\tilde{\chi}^{0}_{A}$$\tilde{l}_{X}$$\tilde{D}_{iAX}$
---|---
Figure 3: lepton-slepton-chargino and lepton-slepton-neutralino vertices.

Due to the mixing between the fermionic partners of the gauge bosons and the
fermionic partners of the Higgs bosons, the gauge and the yukawa vertices get
mixed in MSSM. We will present here the vertices of fermion-sfermion-chargino
and fermion-sfermion-neutralino where this is evident. These are presented in
Figure 3.
(i) Fermion-Sfermion-Chargino :
This is the first vertex on the left of the figure. The explicit structure of
this vertex is given by:

$\tilde{C}_{iAX}=C^{R}_{iAX}P_{R}+C^{L}_{iAX}P_{L}$ (96)

where $P_{L}(P_{R})$ are the project operators212121$P_{L}=(1-\gamma_{5})/2$
and $P_{R}=(1+\gamma_{5})/2$. and $C^{R}$ and $C^{L}$ are given by

$\displaystyle c^{R}_{iAX}$ $\displaystyle=$ $\displaystyle-
g_{2}(U)_{A1}R^{\nu}_{Xi}$ (97) $\displaystyle C^{L}_{iAX}$ $\displaystyle=$
$\displaystyle g_{2}{m_{l_{i}}\over\sqrt{2}m_{W}cos\beta}(V)_{A2}R^{\nu}_{Xi}$
(98)

In the above $U$ and $V$ are the diagonalising matrices of chargino mass
matrix $M_{C}$, $R^{\nu}$ is the diagonalising matrix of the sneutrino mass
matrix, $M_{\tilde{\nu}}^{2}$. And the indices $A$ and $X$ runs over the
dimensions of the respective matrices ($A=1,2$ for Charginos, $X=1,2,3$ for
sneutrinos), whereas $i$ as usual runs over the generations, $m_{l_{i}}$ is
the mass of the $i$ th lepton and rest of the parameters carry the standard
definitions.
(ii) Fermion-Sfermion-Neutralino :
In a similar manner, the fermion-sfermion-neutralino vertex is given by:

$\tilde{D}_{iAX}=D^{R}_{iAX}P_{R}+D^{L}_{iAX}P_{L}$ (99)

where $D^{L}$ and $D^{R}$ have the following forms:

$\displaystyle D^{R}_{iAX}$ $\displaystyle=$
$\displaystyle-{g_{2}\over\sqrt{2}}\left\\{\left[-N_{A2}-N_{A1}tan\theta_{W}\right]R^{l}_{Xi}+{m_{l_{i}}\over
m_{W}cos\beta}N_{A3}R^{l}_{X,i+3}\right\\}$ (100) $\displaystyle D^{L}_{iAX}$
$\displaystyle=$ $\displaystyle-{g_{2}\over\sqrt{2}}\left\\{{m_{l_{i}}\over
m_{W}cos\beta}N_{A3}R^{l}_{Xi}+2N_{A1}tan\theta_{W}R^{l}_{X,i+3}\right\\}$
(101)

In the above $N$ is diagonalising matrices of neutralino mass matrix $M_{N}$,
$R^{l}$ is the diagonalising matrix of the slepton mass matrix,
$M_{\tilde{l}}^{2}$. And the indices $A$ and $X$ runs over the dimensions of
the respective matrices ($A=1,..,4$ for neutralinos, $X=1,..,6$ for sleptons),
whereas $i$ as usual runs over the generations.

####  VII.0.6 Think it Over:

The LEP experiment at CERN searched for a light Higgs boson which has SM like
couplings through the process $e^{+}e^{-}~{}\to~{}ZH$ and has a put a limit on
the lightest Higgs boson mass as
$m_{h}~{}\stackrel{{\scriptstyle\scriptstyle>}}{{\scriptstyle\sim}}~{}114.2$GeV.
This limit applies to the light Higgs boson of the MSSM (except in some range
and in the presence of CP violation in the Higgs sector). Take the formula of
the 1-loop Higgs mass given by eq.(94) and simplify it by assuming the stop
masses are of the similar order $~{}M_{S}$ and the mixing between the stops is
maximal. Find out what is the least value of the $M_{S}$ which is consistent
with the Higgs mass. Now compute the 1-loop corrections to the minimisation
conditions and check what is the amount of fine-tuning required to obtain the
correct $M_{Z}$ mass. Show that a few percent fine tuning is already required
to satisfy the LEP limit on the light Higgs mass. The fine tuning rapidly
increases with increasing Higgs mass. This goes under the name Little
Hierarchy Problem.

##  VIII ‘Standard’ Models of Supersymmetry breaking

So far we have included supersymmetry breaking within the MSSM through a set
of explicit supersymmetry breaking soft terms however, at a more fundamental
we would like to understand the origins of these soft terms as coming from a
theory where supersymmetry is spontaneously broken. In a previous section, we
have mentioned that supersymmetry needs to be broken spontaneously in a hidden
sector and then communicated to the visible sector through a messenger sector.
In the below we will consider two main models for the messenger sector (a) the
gravitational interactions and (b) the gauge interactions. But before we
proceed to list problems with the general form soft supersymmetry breaking
terms as discussed in the previous section. This is essential to understand
what kind of constructions of supersymmetric breaking models are likely to be
realised in Nature and thus are consistent with phenomenology.

The way we have parameterised supersymmetry breaking in the MSSM, using a set
of gauge invariant soft terms, at the first sight, seems to be the most
natural thing to do in the absence of a complete theory of supersymmetry
breaking. However, this approach is itselves laden with problems as we realise
once we start confronting this model with phenomenology. The two main problems
can be listed as below:
(i). Large number of parameters
Compared to the SM, in MSSM, we have a set of more than 50 new particles;
writing down all possible gauge invariant and supersymmetry breaking soft
terms, limits the number of possible terms to about 105. All these terms are
completely arbitrary, there is no theoretical input on their magnitudes,
relative strengths, in short there is no theoretical guiding principle about
these terms. Given that these are large in number, they can significantly
effect the phenomenology. In fact, the MSSM in its softly broken form seems to
have lost predictive power except to say that there are some new particles
within a broad range in mass(energy) scale. The main culprit being the large
dimensional parameter space $\sim$ 105 dimensional space which determines the
couplings of the supersymmetric particles and their the masses. If there is a
model of supersymmetry breaking which can act as a guiding principle and
reduce the number of free parameters of the MSSM, it would only make MSSM more
predictive.

(ii). Large Flavour and CP violations. As mentioned previously, the soft mass
terms $m_{ij}^{2}$ and the trilinear scalar couplings $A_{ijk}$ can violate
flavour. This gives us new flavour violating structures beyond the standard
CKM structure of the quark sector which can also be incorporated in the MSSM.
Furthermore, all these couplings can also be complex and thus could serve as
new sources of CP violation in addition to the CKM phase present in the
Standard Model. Given that all these terms arbitrary and could be of any
magnitude close to weak scale, these terms can contribute dominantly compared
to the SM amplitudes to various flavour violating processes at the weak scale,
like flavour violating decays like $b\to s+\gamma$ or flavour oscillations
like $K^{0}\leftrightarrow\bar{K}^{0}$ etc and even flavour violating decays
which do not have any Standard Model counterparts like $\mu\to e+\gamma$ etc.
The CP violating phases can also contribute to electric dipole moments (EDM)s
which are precisely measured at experiments.

To analyse the phenomenological impact of these processes on these terms, an
useful and powerful tool is the so called Mass Insertion (MI) approximation.
In this approximation, we use flavour diagonal gaugino vertices and the
flavour changing is encoded in non-diagonal sfermion propagators. These
propagators are then expanded assuming that the flavour changing parts are
much smaller than the flavour diagonal ones. In this way we can isolate the
relevant elements of the sfermion mass matrix for a given flavour changing
process and it is not necessary to analyse the full $6\times 6$ sfermion mass
matrix. Using this method, the experimental limits lead to upper bounds on the
parameters (or combinations of)
$\delta_{ij}^{f}\equiv\Delta^{f}_{ij}/m_{\tilde{f}}^{2}$, known as mass
insertions; where $\Delta^{f}_{ij}$ is the flavour-violating off-diagonal
entry appearing in the $f=(u,d,l)$ sfermion mass matrices and
$m_{\tilde{f}}^{2}$ is the average sfermion mass. In addition, the mass-
insertions are further sub-divided into LL/LR/RL/RR types, labeled by the
chirality of the corresponding SM fermions. The limits on various $\delta$’s
coming from various flavour violating processes have been computed and
tabulate in the literature and can be found for instance in Ref.ourlectures .

These limits show that the flavour violating terms should be typically at
least a couple of orders of magnitude suppressed compared to the flavour
conserving soft terms222222The flavour problem could also be alleviated by
considering decoupling soft masses or alignment mechanisms.. While this is
true for the first two generations of soft terms, the recent results from
B-factories have started constraining flavour violating terms involving the
third generation too. In light of this stringent constraint, it is more
plausible to think that the fundamental supersymmetry breaking mechanism some
how suppresses these flavour violating entries. Similarly, this mechanism
should also reduce the number of parameters such that the MSSM could be easily
be confronted with phenomenology and make it more predictive. We will consider
two such models of supersymmetry breaking below which will use two different
kinds of messenger sectors.

###  VIII.1 Minimal Supergravity

In the minimal supergravity framework, gravitational interactions play the
role of messenger sector. Supersymmetry is broken spontaneously in the hidden
sector. This information is communicated to the MSSM sector through
gravitational sector leading to the soft terms. Since gravitational
interactions play an important role only at very high energies, $M_{p}\sim
O(10^{19})$ GeV, the breaking information is passed on to the visible sector
only at those scales. The strength of the soft terms is characterised roughly
by, $m_{\tilde{f}}^{2}~{}\approx~{}M_{S}^{2}/M_{planck}$, where $M_{S}$ is the
scale of supersymmetry breaking. These masses can be comparable to weak scale
for $M_{S}\sim 10^{10}$ GeV. This $M_{S}^{2}$ can correspond to the F-term vev
of the Hidden sector. The above mechanism of supersymmetry breaking is called
supergravity (SUGRA) mediated supersymmetry breaking.

A particular class of supergravity mediated supersymmetry breaking models are
those which go under the name of ”minimal” supergravity. This model has
special features that it reduces to total number of free parameters
determining the entire soft spectrum to five. Furthermore, it also removes the
dangerous flavour violating soft terms in the MSSM. The classic features of
this model are the following boundary conditions to the soft terms at the high
scale $\sim~{}M_{Planck}$ :

  * •

All the gaugino mass terms are equal at the high scale.

$M_{1}=M_{2}=M_{3}=M_{1/2}$

  * •

All the scalar mass terms at the high scale are equal.

$m_{\phi_{ij}}^{2}=m_{0}^{2}\delta_{ij}$

  * •

All the trilinear scalar interactions are equal at the high scale.

$A_{ijk}=Ah_{ijk}$

  * •

All bilinear scalar interactions are equal at the high scale.

$B_{ij}=B$

Using these boundary conditions, one evolves the soft terms to the weak scale
using renormalisation group equations. It is possible to construct
supergravity models which can give rise to such kind of strong universality in
soft terms close to Planck scale. This would require the Kahler potential of
the theory to be of the canonical form. As mentioned earlier, the advantage of
this model is that it drastically reduces the number of parameters of the
theory to about five, $m_{0},M$ (or equivalently $M_{2}$), ratio of the vevs
of the two Higgs, tan$\beta$, $A$, $B$. Thus, these models are also known as
‘Constrained’ MSSM in literature. The supersymmetric mass spectrum of these
models has been extensively studied in literature. The Lightest Supersymmetric
Particle (LSP) is mostly a neutralino in this case.

###  VIII.2 Gauge Mediated Supersymmetry breaking

In a more generic case, the Kahler potential need not have the required
canonical form. In particular, most low energy effective supergravities from
string theories do not posses such a Kahler potential. In such a case, large
FCNC’s and again large number of parameters are expected from supergravity
theories. An alternative mechanism has been proposed which tries to avoid
these problems in a natural way. The key idea is to use gauge interactions
instead of gravity to mediate the supersymmetry breaking from the hidden (also
called secluded sector sometimes) to the visible MSSM sector. In this case
supersymmetry breaking can be communicated at much lower energies $\sim 100$
TeV.

A typical model would contain a susy breaking sector called ‘messenger sector’
which contains a set of superfields transforming under a gauge group which
‘contains’ $G_{SM}$. Supersymmetry is broken spontaneously in this sector and
this breaking information is passed on to the ordinary sector through gauge
bosons and their fermionic partners in loops. The end-effect of this mechanism
also is to add the soft terms in to the lagrangian. But now these soft terms
are flavour diagonal as they are generated by gauge interactions. The soft
terms at the messenger scale also have simple expressions in terms of the susy
breaking parameters. In addition, in minimal models of gauge mediated
supersymmetry breaking, only one parameter can essentially determine the
entire soft spectrum.

In a similar manner as in the above, the low energy susy spectrum is
determined by the RG scaling of the soft parameters. But now the high scale is
around 100 TeV instead of $M_{GUT}$ as in the previous case. The mass spectrum
of these models has been studied in many papers. The lightest supersymmetric
particle in this case is mostly the gravitino in contrast to the mSUGRA case.

####  VIII.2.1 Think it Over

  * •

In both gravity mediated as well as gauge mediated supersymmetry breaking
models, we have seen that RG running effects have to included to study the
soft terms at the weak scale. Typically, the soft masses which appear at those
scales are positive at the high scale. But radiative corrections can
significantly modify the low scale values of these parameters; in particular,
making one of the Higgs mass to be negative at the weak scale leading to
spontaneous breaking of electroweak symmetry. This mechanism is called
radiative electroweak symmetry breaking. Consider two hypothetical situations
when (a) the top mass is twice its present value $m_{t}~{}=2~{}m_{t}$ (b) the
top mass is 1/10 th its present value $m_{t}~{}=~{}m_{t}/10$. In which case
there would be more efficient Electroweak symmetry breaking ?

  * •

The recent limits from LHC already put severe constraints on the lightest
squarks and gluino masses. They push their masses to be greater than 800 GeV -
1 TeV. In fact, this has severe constraints on mSUGRA model. For latest limits
have a look at cerntwiki .

##  IX Remarks

The present set of lectures are only a set of elementary introduction to the
MSSM. More detailed accounts can be found in various references which we have
listed at various places in the text. In preparing for these set of lectures,
I have greatly benefitted from various review articles and text books. I have
already listed some of them at various places in the text. Martin’s review
martin is perhaps the most comprehensive and popular references. It is also
constantly updated. Some other excellent reviews are peskin1 and bagger . A
concise introduction can also be found in csaki . For more formal aspects of
supersymmetry including a good introduction to supergravity please have a look
at vanproeyen and west . For Grand Unified theories and supersymmetry, please
have a look at mohapatra , yanagida , ramond and rosstextbook . For a
comprehensive introduction to supersymmetric dark matter, please see
kamionkowski . Finally, I would also recommend the original papers of anomaly
mediated supersymmetry breaking amsb . Happy Susying.

Acknowledgements We would like to thank Urjit Yajnik for detailed discussions
and consultations through out the teaching period. We would also like to thank
Ranjan Laha and Manimala Mitra for going through these lecture notes several
times and pointing out various typographical errors. The author is supported
by DST Ramanujan fellowship and DST project ”Complementarity between direct
and indirect searches of supersymmetry”.

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