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

 Based on 7 “doppel stunde” lectures I gave in Dresden, one hour lectures. However I can go faster or slower depending on prior knowledge, feedback is essential.  I will only cover N=1 global supersymmetry here, which is most relevant to phenomenology. For local or extended SUSY you will need to read further.  There are many conventions, but understanding shouldn’t depend on this. I will try to stick to one choice and be consistent, but I may slip on occasion, so beware.  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.  I already provided extensive discussion of the motivation in my SUSY talk (slides here: ) Here I just assume you are motivated to learn SUSY, but don’t know any of the details. Preliminary Remarks Cautions Comments

Literature Stockinger, - SUSY skript, 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/

The lectures will cover material necessary to give a reasonable understanding of SUSY and realistic supersymmetric models which might be detected at the LHC as well as prepare the audience to do SUSY phenomenology. Starting with the SUSY algebra for N=1 supersymmetry we will then introduce superfields (general, chiral and vector) which will then aid us in constructing a SUSY invariant Lagrangian. We will then discuss how supersymmetry can be broken softly to provide realistic models which still solve the Hierarchy problem. We will then use all of this to construct the Minimal Supersymmetric Standard Model (MSSM) and then discuss how electroweak symmetry breaking works there and how the gauge eigenstates are mixed to form mass eigenstates. Finally I will talk about a few topics which go beyond the standard scenarios. If I use “then” more than once please shoot me!

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

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 4 generators of translation: 6 Lorentz generators: One representaion of the Poincare group

Commutation relations For example orbital angular momentum is included: Generators for a scalar field Exercise for the enthusiastic: check explicit form of generators satisfy general commutation relations 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 operatorFulfills Poincare conditions for Fermions have spinor representation of Lorentz group, with transformation: Generators for a spinor

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: Coleman-Mandula “No-go theorem” [Stated here, without proof] This does not exclude a symmetry with fermionic generators! Haag, Lopuszanski and Sohnius extension: SUSY algebra! Supersymmetry is the only way to extend space-time symmetries! Space-timeinternal Extending with a new group which has generators that don’t commute with space time is impossible. [Coleman, Mandula Phys. Rev. 159, 1251 (1967).] [Gol’fand Y A and Likhtman E P 1971 JETP Lett ] [Haag R, Lopusanski J T and Sohnius M 1975 Nucl. Phys. B ]

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

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

Weyl representation: Immediate consequences of SUSY algebra: SUSY charges are spinors that carries ½ integer spin.

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

Weyl representation: Immediate consequences of SUSY algebra: OR SUSY breaking requires

1.Since Q is a spinor it carries ½ integer spin. 2.[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) Notes:

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

Extension of electron to SUSY theory, 2 superpartners with spin 0 to electron states We have the states: Electron spin 0 superpartners dubbed ‘selectrons’ We can also examine the spins of these states using the SUSY algebra