SUSY 1
Jan Kalinowski
Jan Kalinowski
Supersymmetry, part 1
Three lectures:
1. Introduction to SUSY
2. MSSM: its structure, current status and LHC expectations
3. Exploring SUSY at a Linear Collider
Jan Kalinowski
Supersymmetry, part 1
Outline
What’s good/wrong with the Standard Model?
Symmetries
SUSY algebra
Constructing SUSY Lagrangian
Jan Kalinowski
Supersymmetry, part 1
Literature
J. Wess, J. Bagger, Princeton Univ Press, 1992
H. Haber, G. Kane, Phys.Rept.117 (1985) 75
S.P Martin, arXiv:hep-ph/9709356
H.K. Dreiner, H.E. Haber, S.P. Martin, arXiv:0812.1594
M.E. Peskin, arXiv:0801.1928
D. Bailin, A. Love, IoP Publishing, 1994
M. Drees, R. Godbole, P. Roy, World Scientific 2004
A. Signer, arXiv:0905.4630
and many others
Warning: be aware of many different conventions in the literature
Disclaimer: cannot guarantee that all signs are correct
Jan Kalinowski
Supersymmetry, part 1
Why do we believe it?
Why do we not believe it?
Jan Kalinowski
Supersymmetry, part 1


Renormalizable theory  predictive power
18 parameters (+ neutrinos):
• coupling constants
• quark and lepton masses
• quark mixing (+ neutrino)
• Z boson mass
• Higgs mass
 for more than 20 years we try to disprove it
 fits all experimental data very well
up to electroweak scale ~ 200 GeV (10–18 m)
 the best theory we ever had
Jan Kalinowski
Supersymmetry, part 1
inspite of all its successes cannot be the ultimate theory:
Hambye, Riesselmann
• does not contain gravity
• can be valid only up to a certain scale
• Higgs mass unstable w.r.t. quantum corrections
• neutrino oscillations
• mater-antimater asymmetry
• SM particles constitute a small part of
the visible universe
Jan Kalinowski
Supersymmetry, part 1
WMAP
Loop corrections to propagators
1. photon self-energy in QED
U(1) gauge invariance 
2. electron self-energy in QED
Chiral symmetry in the massless limit 
Mass hierarchy technically natural
Jan Kalinowski
Supersymmetry, part 1
3. scalar self-energy
Even if we tune
, two loop correction will be
quadratically divergent again
Presence of additional heavy states can affect cancellations of
quadratic divergencies  scalar mass sensitive to high scale
In the past significant effort in finding possible solutions of the hierarchy
problem
Jan Kalinowski
Supersymmetry, part 1
Jan Kalinowski
Supersymmetry, part 1
Noether theorem: continuous symmetry implies conserved quantity
In quantum mechanics symmetry under space rotations and translations
imply angular momentum and momentum conservation
Generators satisfy
Extending to Poincare we enlarge space to spacetime
Poincare algebra
Explicit form of generators depends on fields
Jan Kalinowski
Supersymmetry, part 1
Gauge symmetries
generators
fulfill certain algebra
Electroweak and strong interations described by gauge theories
invariance under internal symmetries imply existence of spin 1
Gravity described by general relativity: invariance under space-time
transformations -- graviton G, spin 2
In 1960’ties many attempts to combine spacetime and gauge symmetries,
e.g. SU(6) quark models that combined SU(3) of flavor with SU(2) of spin
Hironari Miyazawa (’68) first who considered mesons and baryons
in the same multiplets
Jan Kalinowski
Supersymmetry, part 1
However, Coleman-Mandula theorem ‘67:
direct product of Poincare and internal symmetry groups
Particle states numerated by eigenvalues of
commuting set of observables
Here all generators are of bosonic type (do not mix spins) and only
commutators involved
Haag, Lopuszanski, Sohnius ’75:
no direct symmetry transformation between states of integer spins
we have to include generators of fermionic type that transform
|fermion>
|boson> and allow for anticommutators
{a,b}=ab+ba
Jan Kalinowski
Supersymmetry, part 1
transforms like a fermion
Graded Lie algebra, superalgebra or
Remarkably, standard QFT allows for supersymmetry without
any additional assumptions
Gol’fand, Likhtman ’71, Volkov, Akulov ’72, Wess Zumino ‘73
Jan Kalinowski
Supersymmetry, part 1
Jan Kalinowski
Supersymmetry, part 1
Simplest case: N=1 supersymmetry
only one fermionic generator
and its conjugate
Reminder: two component Weyl spinors that transform under Lorentz
where
spinors
transform according to
spinors
transform according to
Dirac spinor requires two Weyl spinors
Jan Kalinowski
Supersymmetry, part 1
Grassmann variables
Variables with fermionic nature
with
Raising and lowering indices
using antisymmetric tensor
We will also need
Dirac matrices
Jan Kalinowski
Supersymmetry, part 1
Product of two spinoirs is defined as
in particular
Technicalities:
For Dirac spinors
Jan Kalinowski
Lorentz covariants
Supersymmetry, part 1
The Lagrangian for a free Dirac field in terms of Weyl
The Lagrangian for a free Majorana field in terms of Weyl
Frequently used identities:
We will also use
Jan Kalinowski
Supersymmetry, part 1
Supersymmetry algebra
or in terms of Majorana
Normalization, since
Spectrum bounded from below
If vacuum state is supersymmetric, i.e.
then
For spontaneous SUSY breaking
and
Jan Kalinowski
non-vanishing vacuum energy
Supersymmetry, part 1
SUSY multiplets – massless representations
fermionic and bosonic states of equal mass
Since
Then
only
Equal number of bosonic and fermionic states in supermultiplet
Jan Kalinowski
Supersymmetry, part 1
Supermultiplets
Most relevant ones for constructing realistic theory
Chiral:
Vector:
Gravity:
spin 1/2
Weyl fermion
and
spin 1
vector (gauge)
spin 2
graviton
0
complex scalar
and
and
1/2
Weyl fermion (gaugino)
3/2
gravitino
and CPT conjugate states
Jan Kalinowski
Supersymmetry, part 1
Superspace and superfields
Reminder: when going from Galileo to Lorentz we extended 3-dim
space to 4-dim spacetime
When extending to SUSY it is convenient to extend spacetime
to superspace with Grassmannian coordinates
and introduce a concept of superfields
Taylor expansion in superdimensions very easy, e.g.
scalar
Jan Kalinowski
Supersymmetry, part 1
Weyl
auxiliary
Derivatives with respect to Grassmann variable
one has to be very careful:
since
Derivatives also anticommute with other Grassmann variables
Integration defined as
Jan Kalinowski
Supersymmetry, part 1
With Grassmann variables SUSY algebra can be written as
like a Lie algebra with
anticommuting parameters
Reminder: for space-time shifts:
Extend to SUSY transformations (global)
(dimensions!)
using Baker-Campbell-Hausdorff
i.e. under SUSY transformation
non-trivial transformation of the superspace
Jan Kalinowski
Supersymmetry, part 1
In analogy to
, we find a representation for generators
Check that satisfy
SUSY algebra
Convenient to introduce covariant derivatives
transform the same way under SUSY
Properties:
Jan Kalinowski
Supersymmetry, part 1
Most general superfield in terms of components (in general complex)
Scalar fields
Vector field
Weyl spinors
note different dimensions of fields
• Not all fields mix under SUSY => reducible representation
• Too many components for fields with spin < or = 1
For the Minimal Supersymmertic extension of the SM enough to consider
chiral superfield
vector superfield
Jan Kalinowski
Supersymmetry, part 1
Chiral superfields
left-handed chiral superfield (LHxSF)
right-handed chiral superfield (RHxSF)
Invariant under SUSY transformation
Since
is LHxSF
Expanding in terms of components:
(dimensions:
)
contains one complex scalar (sfermion), one Weyl fermion and an auxiliary field F
Jan Kalinowski
Supersymmetry, part 1
RHxSF:
Transformation under infinitesimal SUSY transformation, component fields 
comparing with
gives
boson  fermion
fermion  boson
F  total derivative
•The F term – a good candidate for a Lagrangian
• Product of LHxSF’s is also a LHxSF
Jan Kalinowski
Supersymmetry, part 1
Vector superfields
General superfield
We need a real vector field (VSF)
 impose
and expand
(dimensions:
)
In gauge theory many components are unphysical
Important: under SUSY
a total derivative
Jan Kalinowski
Supersymmetry, part 1
By a proper choice of gauge transformation we can go to
the Wess-Zumino gauge
Many unphysical fields have been „gauged away”
it is not invariant under susy, but after susy transformation
we can again go to the Wess-Zumino gauge
Jan Kalinowski
Supersymmetry, part 1
Jan Kalinowski
Supersymmetry, part 1
SUSY Lagrangians
Supersymmetric Lagrangians
F and D terms of LHxSF and VSF, respectively, transform as total derivatives
Products of LHxSF are chiral superfields
Products of VSF are vector superfields
Use F and D terms to construct an invariant action
Jan Kalinowski
Supersymmetry, part 1
Example: Wess-Zumino model superfields
Consider one LHxSF
(using
Introduce a superpotential
We also need a dynamical part
a D-term can be constructed out of
Kaehler potential
Jan Kalinowski
Supersymmetry, part 1
)
Both scalar and spinor kinetic terms appear as needed.
However there is no kinetic term for the auxiliary field F.
F can be eliminaned from EOM
Terms containing the auxiliary fields read
Here superpotential as a
function of a scalar field
Finally
Scalar and fermion of equal mass
All couplings fixed by susy
Jan Kalinowski
Supersymmetry, part 1
Generalising to more LHxSF
Yukawa-type interactions
couplings of equal strength
D-terms only of the type
Terms of the type
forbidden –
superpotential has to be holomorphic
Alternatively, Lagrangian can be written as kinetic part and
contribution from superpotential
Jan Kalinowski
Supersymmetry, part 1
Vector superfields
General superfield
We need a real vector field (VSF)
 impose
and expand
(dimensions:
)
In gauge theory many components are unphysical
Important: under SUSY
a total derivative
Jan Kalinowski
Supersymmetry, part 1
Gauge theory: Abelian case
Remember that chiral superfield contains
with complex
Therefore define gauge transformation for the vector superfield
where
is a LHxSF with proper dimensionality
Now define gauge transformation for matter LHxSF
is also a LHxSF
Then the gauge interaction
is invariant since
(for Abelian)
Jan Kalinowski
Supersymmetry, part 1
General VSF contains a spin 1 component field
Products of VSF are also VSF but do not produce a kinetic term
Notice that the physical spinor can be singled out from VSF by
where
But
means evaluate at
is a spinor LHxSF since
In terms of component fields – photino, photon and an auxiliary D
Note that
Jan Kalinowski
is gauge invariant, i.e. does not change under
Supersymmetry, part 1
Drawing the lesson from the construction of chiral superfield theory
No kinetic term for D – auxilliary field like F
D field appears also in the interaction with LHxSF
For Abelian gauge symmetry one can also have a Fayet-Iliopoulos term
Now the auxiliary field D can be eliminated from EOM
Jan Kalinowski
Supersymmetry, part 1
But
, i.e. there are other terms
In the Wess-Zumino gauge expanding
Term with 1 contains kinetic terms for sfermion and fermion
The other two contain interactions of fermions and sfermions
with photon and photino
An Abelian gauge invariant and susy lagrangian then reads
Jan Kalinowski
Supersymmetry, part 1
Extending to non-Abelian case
The VSF must be in adjoint representation of the gauge group
For matter xSF
Explicitly
Jan Kalinowski
Supersymmetry, part 1
Feynman rules: relations among masses and couplings
Jan Kalinowski
Supersymmetry, part 1
Non-renormalisation theorem
R-symmetry -- rotates superspace coordinate
Define R charge
Terms from Kaehler are invariant since
For
are real
to be invariant
component fields of the SF have different R charge
Consider Wess-Zumino
Assume
as vev’s of heavy SF (spurions)
For
global
Renormalised superpotential must be of
But
must be regular
Only Kaehler potential gets renormalised
Jan Kalinowski
Supersymmetry, part 1
symmetry
Summary on constructing SUSY Lagrangians
Construct Lagrangians for N=1 from chiral and vector superfields
Multiplets containing fields of equal mass but differing in spin by ½
Fermion Yukawa and scalar quartic couplings from superpotential
Gauge symmetries determine couplings of gauge fields
 Many relations between couplings
Comment on N=2: more component fields in a hypermultiplet
contains both + ½ and – ½ helicity fermions which need to
transform in the same way under gauge symmetry
N>1  non-chiral
Jan Kalinowski
Supersymmetry, part 1