Lecture 3 Recap of Part 2 Dirac spinor

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

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 Grassmann numbers (Recap of Part 2)

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 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 (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 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