Supersymmetry (SUSY)

1. Supersymmetry(SUSY) Lecture 2 (Theory Part 1) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

2. Literature • 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

3. Preliminary Remarks Cautions • Warning! As you may have noticed ;) these lectures are in power point. This is an experiment, feedback is essential. • These lectures are in English, but I speak with a Glaswegian accent. Ask me to speak slowly if you cannot understand. As always physics questions are very welcome. Comments • Lectures are based on hand written lecture notes from DominikStockinger with some changes drawn from other sources. • Supersymmetry is a deep and rich subject, I have been studying it for about 7 years, but I am still learning. We have only a few lectures and cannot teach it all. • We have not established that SUSY is realised in nature. SUSY is currently searched for at the Large Hadron Collider at CERN. • If such “low-energy” SUSY is discovered this will be tremendously exciting! However SUSY may be realised in nature in other ways (a symmetry broken at much higher energies) and can be significant for other reasons.

4. Basics of Supersymmetry 1 SUSY Algerbra 1.1 Poincare Algebra Rotations and Boosts from Special Relativity Lorentz Trasnformation scalar product invariant Translations Poincare Transformations Infinitesimal: ) 6 Independent entries in ) Lorentz group has 6 generators: 3 rotations, 3 boosts Poincare group has 10 generators: 4 translations + 6 Lorentz

5. 4 generators of translation: 6 Lorentz generators: Representation: Transforms the fields via 10 generators Infinitesimal: For example, a scalar: Generators for a scalar field Must obey the general commutation relations for Poincare generators: Commutation relations

6. Commutation relations Generators for a scalar field Exercise for the enthusiastic: check explicit form of generators satisfy general commutation relations For example orbital angular momentum is included: Recall for, we have A Lorentz scalar only has integer valued angular momentum but fermions also have 1/2 integer spin in addition to orbital angular momentum. Need Spin operator Fulfills Poincare conditions for Fermions have spinor representation of Lorentz group, with transformation: Generators for a spinor

7. Note: In these lectures we will use the Weyl representation of the clifford algebra. For example: Z-component of spin

8. Coleman-Mandula “No-go theorem” [Stated here, without proof] A Lie group containing the Poincare group and an internal group, e.g. the Standard Model gauge group, will be formed by the direct product Space-time internal Extending with a new group which has generators that don’t commute with space time is impossible. This does not exclude a symmetry with fermionic generators! Haag, Lopuszanski and Sohnius way out: Supersymmetry is the only way to extend space-time symmetries!

9. 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

10. Notes: Since Q is a spinor it carries ½ integer spin. [P^2, Q] = 0 )superpartners must have the same mass (unless SUSY is broken). 3. From anti-commutation relation Hamiltonian is +ve definite If SUSY is respected by the vacuum then If SUSY is broken then 4. Local SUSY !supergravitation, superstrings , Quantum theory of gravity. (beyond scope of current lectures)

11. 1.3 First Physical Consequences of SUSY SUSY Chiralmultiplet with electron + selectron: Try simple case (not general solution) for illustration Take an electron, with m= 0 (good approximation): 4 states: Just need 2 states: Electric charge = conserved quantity from internal U(1) symmetry that commutes with space-time symmetries, ) SUSY transformations can’t change charge.

12. We have the states: Since decreases spin Note switch of notation here So Extension of electron to SUSY theory, 2 superpartners with spin 0 to electron states Electron spin 0 superpartners dubbed ‘selectrons’

13. SUSY cross-sections Super symmetry 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.

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

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

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

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

18. 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

19. 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 majoranafermion as superpartner Higgsino with Higgs, h, with Fermions Sfermions with Vector bosons Gauginos with