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