Lecture 4

Before Christmas…

1.2 SUSY Algebra (N=1)

From the Haag, Lopuszanski and Sohnius extension of the Coleman-Mandula theorem we need to introduce fermionic operators as part of a “graded Lie algebra” or “superalgerba”

introduce spinor operators and

Weyl representation:

Note Q is Majorana

General Superfield

(where we have suppressed spinor indices)

Scalar field spinor Scalar fieldVector field

spinor

spinor Scalar field

Total derivative

Action

Chiral Superfields

- Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets

Scalar field

Fermion field

Auxilliary field

In the “symmetric” representation.

scalar scalarspinor

Auxilliary fields

Chiral Superfields

Chiral representation

- Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets

Switch between representations change of variables:

scalar scalarspinor

Auxilliary fields

Chiral Superfields

- Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets

Boson ! fermion Fermion ! bosonFour-divergence, yields invariant action under

SUSYF-terms provide contributions to the Lagrangian density

Action

The F-terms from products of chiral superfields will also contribute to the Lagrangian density:

Another SUSY invariant contribution to the action

Similarly from the triple product we can pick out the F-term:

Moreover: The F-term of any polynomial of chiral superfields contributes to the SUSY invariant action!

The Superpotential is a polynomial function of chiral superfields,

From which the F-term contributions to the SUSY Lagrangian density can be extracted.

Lagrangian density from Chiral superfields

Truncated at cubic term to keep only

renormalisable terms

We can now extract the F-terms for the SUSY lagrangian density,

The superpotential can also be a function of only the scalar components,

Allowing the Lagrangian density terms to be extracted via the recipe

As well as polonomials of chiral superfields we can also take the combination of ,

Invariant D-term

Note: 1) not a chiral superfield so the F-term (µµ term) does not transform as a total derivative, as can be checked by performing a SUSY transform or by comparison with a general superfield. 2) the D-term however does transform as a four divergence, and provides contributions to the SUSY Lagrangian density 3) the auxilliary fields Fi do not have a kinetic term (with derivatives) and hence are not really dynamical degrees of freedom, and will be fixed by their E-L eqn. 4) these terms are not present in the superpotential , but instead appear in the Kahler Potential, K(©1 … ©n, ©1

y … ©ny)

We can now extract the F-terms for the SUSY lagrangian density,

Are not dynamical degrees of freedom, eliminated by E-L eqns:

And obtain the kinetic parts from (Vector superfields)

We can now extract the F-terms for the SUSY lagrangian density,

Are not dynamical degrees of freedom, eliminated by E-L eqns:

And obtain the kinetic parts from (Vector superfields)

Sfermions (another glimpse)Recall in Lecture 1 we constructed states :

(via SUSY generators)

SUSY chiral supermultiplet with electron + selectron

And showed new states were spin zero:(using SUSY algebra)

Superpotential only a function of left chiral superfields!

Use:

(only renormalisable superpotential term allowed by charge conservation!)

SUSY mass relation!

Higgs + electron top chiral supermultiplets

Assume a superpotential

and

Higgs + electron top chiral supermultiplets

Assume a superpotential

and

Higgs + electron top chiral supermultiplets

Assume a superpotential

and

SM-like Yukawa coupling H-f-f

Higgs-squark-quark couplings with same Yukawa coupling!

Higgs + electron top chiral supermultiplets

Assume a superpotential

and

Quartic scalar couplings again from the same Yukawa coupling

SUSY gauge theory

But in SUSY Phase transform

superfield!

Abelian gauge transformation Abelian supergauge transformation

2.6 Vector Superfields

A Vector superfield obeys the constraint:

Note: still more degrees of freedom than needed for a vector boson. Some not physical!

Remember Vector bosons appears in gauge theories!

Supergauge invariance of superfields means many excess degrees of freedom!

Can fix gauge to Wess-Zumino gauge:

Note: 1) Wess-Zumino gauge has only gauge boson, gaugino and auxilliary D degrees of freedom.

Gauge boson Gaugino Auxilliary D

2) Wess-Zumino gauge does not fix the ordinary gauge freedom!

3) SUSY transforms will spoil Wess-Zumino gauge fixing constraints. Mainfest SUSY invariance lost in this gauge.

4) After each SUSY transform field dependent gauge transformation can restore us to Wess-Zumino gauge