Langrangian Susy

7/31/2019 Langrangian Susy

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Lecture 3

(Theory Part 2)


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First lets review what we

learned from lecture 1


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(Recap of Part 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


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(Recap of Lecture 1)

Already saw significant

consequences of this SUSY

algebra:

OR


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(Recap of Part 1)

Already saw significant

consequences of this SUSY

algebra:

decreasesspin


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Superpartners

Analogously for a scalar boson, e.g. the Higgs, h, has a fermion partner state with eitherand a gauge boson with s = 1, -1, has a partner majorana fermion

as superpartner

Higgs, h, withHiggsino with

FermionsSfermions with

Vector bosons Gauginos with

Warning: Hand waving (details later)

(Recap of Lecture 1)


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Part 2


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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 Martins primer or Aitchison)

superfields/superspace

work in a simpler formalism which treats the supersymmetryas an extension of spacetime and superpartners as

components of a superfield.

(Drees et al, Baer & Tata, our lectures)

z = (x ; a; _a):


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2.2 Superspace

z = (x ; a; _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 1; 2; 1; 2

Grassmann

numbers


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Notational aside: 4component Dirac spinors to 2-component Weyl spinors

Dirac spinor 2 component

Weyl spinors

Under Lorentz

transformation

Form representaions of

lorentz group

We define:

and


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For Majorana spinor:

Bilinears Lorentz scalar


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Grassmann Numbers

Anti-commuting c-numbers {complex numbers }

If {Grassmann numbers} then

Note

Similarly


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Differentiation:

Integration:

SUSY transformation Independent of x, so global

SUSY transformation

Excercise: for the enthusiastic

check these satisfy the SUSY

algebra given earlier


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2.3 General Superfield

(where we have suppressed spinor indices)

Scalar field spinor Scalar fieldVector

field

spinor

spinor Scalar field

SUSY transformation should give a function of the

same form, ) component fields transformations

Total derivative


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Notes:

1.) D is four divergence )any such D-term will in a Lagrangian

will yield an action invariant under supersymmetric transformations

2.) Linear combinations and products of superfields are also

superfields, e.g.

is a superfields if are superfields.

3.) This is the general superfield, but it does not form anIrreducible representation of SUSY. For example the fermionic

degrees of freedom (16) bosonic degrees of freedom (12) if we

assume the vector field is real.

4.) Irreducible representations of supersymmetry , chiral

superfields and vector superfields will now be discussed.

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