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)

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

Lecture 2

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!

Bilinears Lorentz scalar

Dirac spinor 2 component Weyl spinors

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

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 NumbersAnti-commuting “c-numbers” {complex numbers }

If {Grassmann numbers} then

Similarly

Differentiation:

Integration: