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Lecture 3

Lecture 3. (Recap of Part 2). 2 component Weyl spinors. Dirac spinor. Raising and lowering of indices. (Recap of Part 2). 2 component Weyl spinors. Dirac spinor. Grassmann Numbers. (Recap of Part 2). Anti-commuting “c-numbers” {complex numbers }. If. {Grassmann numbers} then.

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Lecture 3

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  1. Lecture 3

  2. (Recap of Part 2) 2 component Weyl spinors Dirac spinor Raising and lowering of indices

  3. (Recap of Part 2) 2 component Weyl spinors Dirac spinor

  4. Grassmann Numbers (Recap of Part 2) Anti-commuting “c-numbers” {complex numbers } If {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

  5. Part 3

  6. SUSY transformation Recall Poincare transformation from lecture 1: Similarly for SUSY: But Baker-Campbell-Hausdorff formula applies here! Home exercise check:

  7. SUSY transformation Independent of x, so global SUSY transformation Excercise: for the enthusiastic check these satisfy the SUSY algebra given earlier

  8. General Superfield Vector field spinor Scalar field Scalar field spinor (where we have suppressed spinor indices) Scalar field spinor SUSY transformation should give a function of the same form, ) component fields transformations Total derivative

  9. General Superfield Vector field spinor Scalar field Scalar field spinor (where we have suppressed spinor indices) Scalar field spinor SUSY transformation should give a function of the same form, ) component fields transformations Invariant SUSY contribution to action Total derivative

  10. 2.3 General Superfield Vector field spinor Scalar field Scalar field spinor (where we have suppressed spinor indices) Scalar field spinor SUSY transformation should give a function of the same form, ) component fields transformations Total derivative

  11. 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.

  12. Chiral Superfields - Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets Try: But

  13. Chiral Superfields - Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets Try: But

  14. Chiral Superfields - Irreducible multiplet, - Describes lepton / slepton, quark / squark and Higgs / Higgsino multiplets Try: But

  15. Home exercise Note: 2 complex scalars ) 4 d.o.f. 1 complex spinor ) 4 d.o.f. scalar scalar spinor For example ?? Auxilliary field (explained soon) Different representations of the SUSY algebra

  16. Working in the “ Chiral” representation, the SUSY transformation of a left chiral superfield is given by, Chiral representation of SUSY generators F-terms provide contributions to the Lagrangian density Boson ! fermion Four-divergence, yields invariant action under SUSY Fermion ! boson

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