Lecture 2

1. Lecture 2

2. Correction • Stockinger, - SUSY skript, • http://iktp.tu-dresden.de/Lehre/SS2010/SUSY/inhalt/SUSYSkript2010.pdf • Drees, Godbole, Roy - "Theory and Phenomenology of Sparticles" - World Scientific, 2004 • Baer, Tata - "Weak Scale Supersymmetry" - Cambridge University Press, 2006 • Aitchison - "Supersymmetry in Particle Physics. An Elementary Introduction" - Institute of Physics Publishing, Bristol and Philadelphia, 2007 • Martin -"A Supersymmetry Primer" hep-ph/9709356  • http://zippy.physics.niu.edu/primer.html Unfairly criticised: Now included full superfield chapter (as of 06/09/2011)

3. First lets review what we learned from lecture 1…

4. (Recap of Lecture 1) 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

5. (Recap of Lecture 1) Weyl representation: Immediate consequences of SUSY algebra: ) superpartners must have the same mass (unless SUSY is broken). Non-observation ) SUSY breaking (much) Later we will see how superpartner masses are split by (soft) SUSY breaking

6. (Recap of Lecture 1) Weyl representation: Already saw significant consequences of this SUSY algebra: OR

7. (Recap of Part 1) Weyl representation: Already saw significant consequences of this SUSY algebra: decreases spin

8. SUSY chiral supermultiplet with electron + selectron: Simple case (not general solution) for illustration Take an electron, with m= 0 (good approximation): We have the states: The spins of the new states given by the SUSY algebra Extension of electron to SUSY theory, 2 superpartners with spin 0 to electron states Electron spin 0 superpartners dubbed ‘selectrons’

9. Lecture 2

10. SUSY cross-sections Supersymmetry is a symmetry of the S-matrix. So, 4E So SUSY gives relations between processes involving the pariticles and those with their superpartners. ) Very predictive.

11. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index

12. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index

13. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index swap

14. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index swap

15. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index swap

16. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index swap Where we have used completeness of the set, , twice on the second term in lines 2 & 3 Note: proof assumes and may not be true in the ground state if SUSY is unbroken

17. Weyl representation: Recall SUSY algebra lead to: 2 states from SM fermion: 2 bosonic states Electron spin 0 superpartners dubbed ‘selectrons’

18. Superpartners Warning: Hand waving (details later) Analogously for a scalar boson, e.g. the Higgs, h, has a fermion partner state with either and a gauge boson with s = 1, -1, has a partner majorana fermion as superpartner Higgsino with Higgs, h, with Fermions Sfermions with Vector bosons Gauginos with

19. 2. SUSY Lagrange density How do we write down the most general SUSY invariant Lagrangian? Brute force – construct using two component Weyl spinors, by examining the transformations of scalars, fermions and gauge boson (See Steve Martin’s primer or Aitchison)* superfields/superspace – work in a simpler formalism which treats the supersymmetry as an extension of spacetime and superpartners as components of a superfield. (Drees et al, Baer & Tata, our lectures) *Martin now has a full chapter on superfields where he contructs the Lagrangian in a similar way to us, but maintains the brute force approach in earlier chapters

20. 2.2 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

21. Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors 2 component Weyl spinors Left handed Weyl spinor Dirac spinor Right handed Weyl spinor Form representaions of lorentz group Under Lorentz transformation and

22. Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors 2 component Weyl spinors Right handed spinor Left handed spinor Form representaions of lorentz group Under Lorentz transformation and

23. Dirac spinor 2 component Weyl spinors Bilinears Lorentz scalar Warning: take care with signs! We define: Note

24. Dirac spinor 2 component Weyl spinors Bilinears Lorentz scalar Warning: take care with signs!

25. Dirac spinor 2 component Weyl spinors Bilinears Lorentz scalar Warning: take care with signs!

26. Dirac spinor 2 component Weyl spinors Bilinears Lorentz scalar Warning: take care with signs! Further Identities Home Exercise: prove identities!

27. Dirac spinor 2 component Weyl spinors Right handed spinor Left handed spinor

28. Dirac spinor 2 component Weyl spinors Right handed spinor Left handed spinor

29. Dirac spinor 2 component Weyl spinors Right handed spinor Left handed spinor

30. Dirac spinor 2 component Weyl spinors Right handed spinor Left handed spinor For Majorana spinor:

31. Grassmann Numbers Anti-commuting “c-numbers” {complex numbers } If {Grassmann numbers} then Similarly Differentiation: Integration: