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Correction• 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
Unfairly criticised: Now included full superfield chapter (as of 06/09/2011)
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
(Recap of Lecture 1)
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
(Recap of Lecture 1)
Weyl representation:
(Recap of Lecture 1)
Already saw significant consequences of this SUSY
algebra:
OR
Weyl representation:
(Recap of Part 1)
Already saw significant consequences of this SUSY
algebra:
decreases spin
Extension of electron to SUSY theory, 2 superpartners with spin 0 to electron states
We have the states:
Electron spin 0 superpartners dubbed ‘selectrons’
The spins of the new states given by the SUSY algebra
SUSY chiral supermultiplet with electron + selectron:
Take an electron, with m= 0 (good approximation):
Simple case (not general solution) for illustration
Supersymmetry is a symmetry of the S-matrix. So,
So SUSY gives relations between processes involving the pariticles and those with their superpartners. ) Very predictive.
SUSY cross-sections
4E
Degrees of freedomIn SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom
Proof: Witten index
Degrees of freedomIn SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom
Proof: Witten index
Degrees of freedomIn SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom
swap
Proof: Witten index
Degrees of freedomIn SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom
swap
Proof: Witten index
Degrees of freedomIn SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom
swap
Proof: Witten index
Degrees of freedomIn SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom
swap
Proof:
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
Witten index
Weyl representation:
Recall SUSY algebra lead to:
2 states from SM fermion: 2 bosonic states
Electron spin 0 superpartners dubbed ‘selectrons’
Superpartners
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 majorana fermion as superpartner
Higgs, h, with Higgsino with
Fermions Sfermions with
Vector bosons Gauginos with
Warning: Hand waving (details later)
2. SUSY Lagrange density
How do we write down the most general SUSY invariant Lagrangian?
– construct using two component Weyl spinors, by examining the transformations of scalars, fermions and gauge boson
Brute force
(See Steve Martin’s primer or Aitchison)*
superfields/superspace
– work in a simpler formalism which treats the supersymmetry as an extension of spacetime and superpartners as components of a superfield.
(Drees et al, Baer & Tata, our lectures)
z = (x¹ ;µa;µ_a):
*Martin now has a full chapter on superfields where he contructs the Lagrangian in a similar way to us, but maintains the brute force approach in earlier chapters
2.2 Superspace
z = (x¹ ;µa;µ_a):
Lorentz transformations act on Minkowski space-time:
In supersymmetric extensions of Minkowki space-time, SUSY transformations act on a superspace:
8 coordinates, 4 space time, 4 fermionic µ1;µ2;µ1;µ2
Grassmann numbers
Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors
Dirac spinor
2 component Weyl spinors
Under Lorentz transformation
Form representaions of lorentz groupand
Left handed Weyl spinor
Right handed Weyl spinor
Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors
Under Lorentz transformation
Form representaions of lorentz groupand
2 component Weyl spinors
Right handed spinor
Left handed spinor
Dirac spinor 2 component Weyl spinors
We define:
Note
Bilinears Lorentz scalar
Warning: take care with signs!
Bilinears Lorentz scalar
Dirac spinor 2 component Weyl spinors
Warning: take care with signs!
Home Exercise: prove identities!
Further Identities
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