Summary of Lectures 1-8. By Anirbit Mukherjee, with a few corrections
by Sunil Mukhi.
1. Fix notations of dotted and undotted fermions and gauge fields and
field strengths and generators of SO(3,1), Weyl basis and creating
SU(2) invariants in 2-component notation. Define the principles of
building the standard model by stating which particle transforms under
which representation of which gauge group. Explain that giving masses
to fermions is hard because of parity violation as encoded in only the
right-handed particles lying as singlets of SU(2). Except that
neutrino allows a Majorana solution because it is uncharged. Define
the Higgs field and explain that the hypercharge of Higgs is not an
observable and can be absorbed in the coupling constant. Also explain
that Higgs getting a vev is not truly a spontaneous symmetry breaking
since the vev can be transformed by a choice of gauge. Therefore this
is instead a special realisation of gauge symmetry ("Higgs mode") and
there is no Goldstone boson. Fix the condition on the SU(2)xU(1)
transformations which leave the vev invariant. Do this symmetry
breaking to get the emergent massive gauge fields of W and Z and the
massless photon. Define the Weinberg angle.
2. Expresses the electric charge in terms of the coupling constants
and check that the hypercharge and I3 assignment ensures the electron
charge to be -1. Does classical "symmetry breaking" to give mass to
Higgs. Argues why in making 3 gauge bosons massive one lost 3 of the 4
possible Higgs field. Introduces the Yukawa coupling and mentions that
left handed spinors are undotted. Points out that if a quark is heavy,
then the Yukawa coupling increases and hence the Higgs interacts
strongly with heavier quarks. Mentions anomaly cancellation. 2 of the
8 triangle diagrams are non-trivial since their cancellation depends
on the hypercharge assignment. These are the ones with 2 SU(2) field
and 1 U(1) field and with 3 U(1) field. Anomaly cancellation forces
that if a single particle of a new generation is discovered then the
entire corresponding generation must exist. Introduces the CKM matrix
and chooses the convention by which the up, charm and top quarks are
always the mass eigenstates. Lists out the experimentally measured
standard model values of masses of the gauge fields and the Weinberg
angle and emphasizes that it was necessary that the mass of the
Z-boson came out larger than the mass of the W boson.
3. Explains that for physically relevant CP violation to happen one
has to argue that there is no basis of fermions in which all the
Yukawa couplings are real. Explains why the 3x3 CKM matrix will have 1
CP violating phase. Mentions that since there is a right-handed
neutrino there is a probability for CP violation from the lepton
sector. By the theta term there can be CP violation in the SU(3)
sector. Introduces supersymmetry transformations in the free
Wess-Zumino model. Using that derives the supersymmetry algebra and
motivates superspace. Shows another free supersymmetric Lagrangian
which couples a fermion to an U(1) field. Writes down the most general
function that can be written in superspace using no theta terms, or
single, double or triple or maximum 4 theta terms. The lecture ends
with writing the supersymmetry operators in terms of the superspace
variables.
4. Writes down the general form of a superfield. Uniqueness of the
4-theta term. Naive demand of analyticity on the superfield is not
supercovariant. So introduces the idea of chiral fields and show that
the superderivative operator and its adjoint and the supersymmetry
transformation operator and its adjoint all anticommute with each
other. Introduces the y-coordinates as the natural chiral
fields. Writes down the general chiral superfield expression and from
that derives that the mass dimension of theta is -1/2, of the spinor
is 3/2 and of the chiral superfield is 1. So the coefficient of the
2-theta term has to be an auxiliary field (i.e without a kinetic
term). Reviews Berezin integration in the superspace context. Thus
argues that the dimension of dtheta is 1/2. Introduces the D-term
(integrated with 2 dthetas and 2 dtheta bars) as the natural kinetic
term and the F-term (integrated with 2 dthetas) as the natural mass
term. Points out that the mass term goes as m (like fermions) instead
of m^2 like for bosons.
5. Explicitly calculates the D-terms and the F-terms. Points out that
typically auxiliary fields appear with a "wrong" sign in the
Lagrangian. Substituting solutions of the equations of motion for the
auxiliary field into the action is the same as integrating them out
since they have no derivative contribution. This produces mass for the
fermions. Argues that there are equal number of fermionic and bosonic
off-shell degrees of freedom. Points out that N=4 Super-Yang-Mills
field theory and N=8 Supergravity has no known auxiliary field
structure. Introduces the Kahler potential and the superpotential as
the natural generalization of D-terms and F-terms. "F/D terms of any
chiral/general superfield transform into a derivative under
SUSY". Mentions that supersymmetry can't get anomalous. Restricting to
real space, the Kahler potential term yields the non-linear sigma
model. The Kahler potential naturally satisfies the integrability
condition for the metric. Points out that in a supersymmetric theory
the Kahler potential is ambiguous upto the addition of a chiral and an
anti-chiral superfield. The superpotential on being restricted to the
real space gives rise to ordinary potential terms and Yukawa like
couplings.
6. Reviews the supersymmetry transformation of chiral
superfields. Points out that product of superfields is a superfield
and the chirality condition is not compatible with a reality
condition. Introduces the idea of real superfields as those which
are equal to their hermitian conjugate. Writes down the general form
of a vector superfield. Shows that if one adds the sum of a chiral
field and its hermitian conjugate to a vector superfield then its
4-vector component transforms like a gauge transformation. But one can
redefine such that the 3-theta field (lambda) and the 4-theta field
(D) remain gauge invariant. Introduces the Wess-Zumino gauge.
7. The full superspace Lagrangian for charged matter superfields
coupled to Abelian gauge superfield/vector superfield is written down
and expanded in terms of component fields. One identifies the F-term
and the D-term contributions separately. One notices the
s-fermion-D-term coupling which is proportional to the charge and the
fermion-gaugino-scalar coupling. It is pointed out that if the
superpotential has a 0 first derivative then it can be bounded below
by anything one likes. To write SQED one needs to use 2 different
chiral superfields, one for the left-handed-electron and one for the
left handed positron (these two are oppositely charged). As a general
principle one notes that a chiral superfield always contains a
left-handed-fermion. Here the corresponding chiral fields must
follow the SM structure of left-handed-electron being in a doublet and
the right-handed-electron being in a singlet.
Non-Abelian SUSY is formulated by using matrix valued vector
superfields of dimensions = dimension of the representation of the
gauge group. The gauge transformation need to be done by conjugation
action of e^{i gauge field} on e^{matrix valued vector
superfield}. Exponentials are not worrisome since in the WZ gauge the
power series stops at the third order and hence doesn't disrupt
renormalizability. It is shown that in the WZ gauge the variation of
the ordinary gauge field component goes exactly as in usual gauge
theory. A chiral field W = -1/4*(D-dagger)^2D V is constructed which
behaves as the strength of this vector superfield since it contains as
a component field the F_{\mu \nu} F^{\mu \nu} term. By taking trace of
this W the Lagrangian is constructed and it is seen that the
supersymmetry action naturally splits into a holomorphic and a
non-holomorphic part. One notes that since for SU(N) theory the
elements of its Lie algebra are traceless it can't have a
Fayet-Iliopoulos term. If the gauge group has factors then in general
there will be one Fayet-Iliopoulos term for each U(1)
factor. Explicitly the SQCD lagrangian is written down.
8. It is shown that complexifying the coupling constant of the
super-gauge-strength produces the \theta term of QCD. Refers to
Chapter 28 for motivations towards supersymmetry in LHC. 2 motivations
are pointed out both of which are coming from unification. If the
known theories are coming from some theory in the higher energy scale
then they would produce heavy particles which will unreasonably
influence low energy physics and these need to be cancelled. In this
way through loop running there can be a possible contribution to the
Higgs mass which is about 15 orders of magnitude higher than the Higgs
mass one expects. It will lead to the "fine-tuning" problem if by
counter-terms one tries to subtract off this large number and add in a
small number. This problem is cured if the theory in the high scale is
supersymmetric and boson-fermion superpartners being in the same
chiral multiplet will have the same mass and hence will pairwise
cancel the contribution to the loops. Georgi-Quinn-Weinberg mechanism
is allude to and it is pointed out that this large energy unification
of the coupling constant running is interesting but crucially depends
on fine-tuned parameters like the Higgs content of MSSM. Moreover in
large extra dimension scenarios it is simply a coincidence.
MSSM is motivated. It is pointed out that in the SM no two particles
can be superpartners of each other since all have different masses and
quantum numbers/interactions do not match. One writes down one chiral
multiplet for every left-handed-particle. Why chiral? Because fermions
in the SM are not in the adjoint representation but they would be
forced to be so if they come in the vector multiplet. Each of the
gauge fields is embedded into 1 vector multiplet. In MSSM one needs at
least 2 Higgs doublets. Motivation being that masses come from the
F-terms which couple only chiral superfields. But in the SM the
left-up-particles get mass by coupling to the Higgs field whereas the
right-down-particles get mass by coupling to the conjugate of
Higgs. This would force the absurd situation of having the Higgs field
and its conjugate to be both chiral. Hence one needs two Higgs
doublets. One of the Higgs doublet is given a vacuum expectation value
in the lower component and the other in the upper component and the
ratio is called "tan \beta". It is argued that giving vevs to the
scalar field does not break supersymmetry since to break supersymmetry
one must give a vev to the auxiliary field which is not the
case here. It is also argued from anomaly cancellation as to why 2
doublets of Higgs are required. It is pointed out that gauginos are the
superpartners of the gauge fields which are in the adjoint
representation and hence real and don't contribute to anomalies. That
Higgsino and gaugino can mix is an interesting MSSM effect. The MSSM
Lagrangian is written down. There is no quadratic self-coupling of the
Higgs but instead there is a cross coupling. There could have been
written down a few more three field couplings which are gauge
invariant and have hypercharge 1 but produce non-standard
interactions. These are ruled out by forcing a discrete symmetry under
parity under which the left-leptons, left-s-anti-leptons,
left-s-up-quarks, left-up-anti-quark and left-down-anti-quark are
positive parity and Higgs is given negative parity. Thus motivates
R-symmetry.