Lecture 3
(Recap of Part 2)
Dirac spinor
2 component
Weyl spinors
Raising and lowering
of indices
(Recap of Part 2)
Dirac spinor
2 component
Weyl spinors
(Recap of Part 2)
Grassmann Numbers
Anti-commuting “c-numbers”
If
{complex numbers }
{Grassmann numbers} then
Superspace
Lorentz transformations act on Minkowski space-time:
In supersymmetric extensions of Minkowki space-time,
SUSY transformations act on a superspace:
8 coordinates, 4 space time, 4 fermionic
z = (x ¹ ; µa ; µa_):
µ1 ; µ2; µ1 ; µ2
Grassmann
numbers
Part 3
SUSY transformation
Recall Poincare transformation from lecture 1:
Similarly for SUSY:
But
Baker-Campbell-Hausdorff formula applies here!
Home exercise check:
SUSY transformation
Independent of x, so global
SUSY transformation
Excercise: for the enthusiastic
check these satisfy the SUSY
algebra given earlier
General Superfield
Scalar field
Scalar field
spinor
Vector
field
spinor
(where we have suppressed spinor indices)
spinor
Scalar field
SUSY transformation should give a function of the
same form, ) component fields transformations
Total derivative
General Superfield
Scalar field
Vector
field
Scalar field
spinor
spinor
(where we have suppressed spinor indices)
spinor
Scalar field
SUSY transformation should give a function of the
same form, ) component fields transformations
Invariant SUSY
contribution to action
Total derivative
2.3 General Superfield
Scalar field
Scalar field
spinor
Vector
field
spinor
(where we have suppressed spinor indices)
spinor
Scalar field
SUSY transformation should give a function of the
same form, ) component fields transformations
Total derivative
Notes:
1.) ±D is four divergence ) any such “D-term” in a Lagrangian
will yield an action invariant under supersymmetric transformations
2.) Linear combinations and products of superfields are also
superfields, e.g.
is a superfields if
are superfields.
3.) This is the general superfield, but it does not form an
Irreducible representation of SUSY.
4.) Irreducible representations of supersymmetry , chiral
superfields and vector superfields will now be discussed.
Chiral Superfields
- Irreducible multiplet,
- Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets
Try:
But
Chiral Superfields
- Irreducible multiplet,
- Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets
Try:
But
Chiral Superfields
- Irreducible multiplet,
- Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets
Try:
But
Home exercise
scalar
For example
spinor
scalar Note: 2 complex scalars ) 4 d.o.f.
1 complex spinor ) 4 d.o.f.
??
Auxilliary field (explained soon)
Different representations of the SUSY algebra
Working in the “ Chiral” representation, the SUSY transformation of a left chiral
superfield is given by,
Chiral
representation of
SUSY generators
Boson ! fermion
Fermion ! boson
Four-divergence, yields
invariant action under
SUSY
F-terms provide
contributions to
the Lagrangian
density