Lecture 2
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)
First lets review what we
learned from lecture 1…
1.2 SUSY Algebra (N=1)
(Recap of Lecture 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:
Already saw significant
consequences of this SUSY
algebra:
OR
(Recap of Part 1)
Weyl representation:
Already saw significant
consequences of this SUSY
algebra:
decreases
spin
SUSY chiral supermultiplet with electron + selectron:
Simple case (not general solution) for illustration
Take an electron, with m= 0 (good approximation):
We have the states:
The spins of the new states given by the SUSY algebra
Extension of electron to SUSY theory,
2 superpartners with spin 0 to electron states
Electron spin 0 superpartners dubbed ‘selectrons’
Lecture 2
SUSY cross-sections
Supersymmetry is a symmetry of the S-matrix. So,
4E
So SUSY gives relations between processes involving the pariticles
and those with their superpartners. ) Very predictive.
Degrees of freedom
In SUSY:
number of fermionic degrees of freedom = number of bosonic degrees of freedom
Proof:
Witten
index
Degrees of freedom
In SUSY:
number of fermionic degrees of freedom = number of bosonic degrees of freedom
Proof:
Witten
index
Degrees of freedom
In SUSY:
number of fermionic degrees of freedom = number of bosonic degrees of freedom
Proof:
Witten
index
swap
Degrees of freedom
In SUSY:
number of fermionic degrees of freedom = number of bosonic degrees of freedom
Proof:
Witten
index
swap
Degrees of freedom
In SUSY:
number of fermionic degrees of freedom = number of bosonic degrees of freedom
Proof:
Witten
index
swap
Degrees of freedom
In SUSY:
number of fermionic degrees of freedom = number of bosonic degrees of freedom
Proof:
Witten
index
swap
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
Weyl representation:
Recall SUSY algebra lead to:
2 states from SM
fermion:
2 bosonic states
Electron spin 0 superpartners dubbed ‘selectrons’
Superpartners
Warning: Hand waving (details later)
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
Fermions
Vector bosons
Higgsino with
Sfermions with
Gauginos with
2. SUSY Lagrange density
How do we write down the most general SUSY invariant Lagrangian?
Brute force
– construct using two component Weyl spinors, by examining
the transformations of scalars, fermions and gauge boson
(See Steve Martin’s primer or Aitchison)*
superfields/ – work in a simpler formalism which treats the supersymmetry
superspace as an extension of spacetime and superpartners as
components of a superfield.
(Drees et al, Baer & Tata, our lectures)
z = (x ¹ ; µa ; µa_):
*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
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
z = (x ¹ ; µa ; µa_):
µ1 ; µ2; µ1 ; µ2
Grassmann
numbers
Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors
2 component
Weyl spinors
Dirac spinor
Left handed
Weyl spinor
Right handed
Weyl spinor
and
Form representaions
of lorentz group
Under Lorentz
transformation
Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors
2 component
Weyl spinors
Right handed spinor
Left handed spinor
and
Form representaions
of lorentz group
Under Lorentz
transformation
Dirac spinor
2 component
Weyl spinors
Bilinears
Lorentz scalar
Warning: take care with signs!
We define:
Note
Dirac spinor
Bilinears
2 component
Weyl spinors
Lorentz scalar
Warning: take care with signs!
Dirac spinor
Bilinears
2 component
Weyl spinors
Lorentz scalar
Warning: take care with signs!
Dirac spinor
Bilinears
2 component
Weyl spinors
Lorentz scalar
Warning: take care with signs!
Further Identities
Home Exercise:
prove identities!
Dirac spinor
2 component
Weyl spinors
Right handed spinor
Left handed spinor
Dirac spinor
2 component
Weyl spinors
Right handed spinor
Left handed spinor
Dirac spinor
2 component
Weyl spinors
Right handed spinor
Left handed spinor
Dirac spinor
2 component
Weyl spinors
Right handed spinor
Left handed spinor
For Majorana spinor:
Grassmann Numbers
Anti-commuting “c-numbers”
If
Similarly
Differentiation:
Integration:
{complex numbers }
{Grassmann numbers} then