langrangian susy
TRANSCRIPT
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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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Weyl representation:
(Recap of Lecture 1)
Already saw significant
consequences of this SUSY
algebra:
OR
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Weyl representation:
(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.