Lecture 5

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

Gauge invariant Kahler potential Lagrangian

But what about Kinetic terms for the gauge superfields?

Kinetic term of the vector superfield

The vector superfield is the generalisation of a gauge potential , but from this we still need to construct a field strength term which is invariant under supergauge transformations, to form the kinetic term for the gauge field.

Try

One can also show (home exercise) that,

Then one can then show that,

Finally note that from both the kinetic term for chiral and vector super fields the “auxilliary” fields Fi and Da have appeared without derivatives.

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

General SUSY invariant Lagrangian density

Superpotential:

Supersmymetric field strengths

Gauge invariant Kahler potential

OR

Recall:

Spontaneous SUSY breaking

O’Raighferataigh SUSY breaking: (F-term breaking example)

VEV breaks SUSY

VEV breaks SUSY

Masses

Fermion masses: Dirac mass term coupling

Goldstino: (required by Goldstone’s

theorem)

SUSY mass relations split by F-VEV!

Note Supertrace:

Hidden sector and SUSY breaking mediation

True generally for this type of breaking, If used directly to split SM particles from SUSY partners:

Solution: SUSY breaking transmitted indirectly:

Hidden sector and SUSY breaking mediation

Transmit breaking from Hidden sector where SUSY is spontaneously broken to visible sector.

Gravity Mediation (for example)

Soft SUSY breaking

Broken symmetries play a very important role in physics, e.g. Electroweak symmetry breaking and the Higgs Mechanism and can be well motivated.

None of the important motivations for SUSY, Hierarchy problem (HP), gauge coupling unifincation, dark matter, require it to be an exact symmetry of nature.

SUSY may be broken in such a way that the mass relations between superpartners are broken, but the relationship between couplings required to solve HP can be mainteined!

It is models of supersymmetry with Soft SUSY breaking which are motivate new physics at the TeV scale and are being searched for at the LHC right now.

Models of how supersymmetry is broken is also a very active area of theoretical research. More info from (where I stole some parts of the previous slides!) : http://iktp.tu-dresden.de/IKTP/Seminare/IS2011/SteveAbel.pdf

But here we evade the precise details of how SUSY is broken, and simply construct models without breaking prejudice, writiing down all possible ways SUSY can be softly broken. To construct such models we need:

Terms in the soft SUSY breaking Lagrangian [Shown to be soft to all orders, L. Girardello, M. Grisaru]

All dimension 3 or less, ) all coefficients have mass dimension!

) relationships between dimensionless couplings maintained!