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Supersymmetry (SUSY)
Lecture 1
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
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 operator Fulfills 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-time internal
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. 13 323]
[Haag R, Lopusanski J T and Sohnius M 1975 Nucl. Phys. B 88 257]
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