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Lecture 1
Preliminary Remarks
 Based on 7 “doppel stunde” lectures I gave in Dresden,
13-14 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.
• 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/9709356
If I use “then” more than once please shoot me!
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.
1 SUSY Algerbra
1.1 Poincare Algebra
Lorentz Trasnformation
Rotations and Boosts
from Special Relativity
scalar product invariant
) 6 Independent entries in
) Lorentz group : 3 rotations + 3 boosts
Poincare group : 4 translations + 6 Lorentz
Transforms the fields via 10 generators
4 generators of translation:
6 Lorentz generators:
Commutation relations
For example, a scalar:
One representaion of the Poincare group
Generators for a
scalar field
Must obey the general commutation relations for Poincare generators.
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:
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
Coleman-Mandula “No-go theorem”
[Coleman, Mandula Phys. Rev. 159, 1251 (1967).]
[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
with space time
which has generators
that don’t commute
is impossible.
This does not exclude a symmetry with fermionic generators!
[Gol’fand Y A and Likhtman E P 1971 JETP Lett. 13 323]
Haag, Lopuszanski and Sohnius extension: SUSY algebra!
[Haag R, Lopusanski J T and Sohnius M 1975 Nucl. Phys. B 88 257]
Supersymmetry is the only way to extend space-time symmetries!
Notational interlude
Note: In these lectures we will use the Weyl representation of the clifford algebra.
For example:
Z-component of spin
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
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:
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
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)
1.3 First Look at supermultiplets
SUSY chiral supermultiplet 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.
We have the states:
We can also examine the spins of these states using the SUSY algebra
Extension of electron to SUSY theory,
2 superpartners with spin 0 to electron states
Electron spin 0 superpartners dubbed ‘selectrons’