Fundamentalne problemy fizyki do rozstrzygniecia po LHC

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SUSY 1

• Jan Kalinowski

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• Three lectures
• Introduction to SUSY
• MSSM its structure, current status and LHC
  expectations
• Exploring SUSY at a Linear Collider

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Outline

• Whats good/wrong with the Standard Model?
• Symmetries
• SUSY algebra
• Constructing SUSY Lagrangian

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Literature

• J. Wess, J. Bagger, Princeton Univ Press, 1992
• H. Haber, G. Kane, Phys.Rept.117 (1985) 75
• S.P Martin, arXivhep-ph/9709356
• H.K. Dreiner, H.E. Haber, S.P. Martin,
  arXiv0812.1594
• M.E. Peskin, arXiv0801.1928
• D. Bailin, A. Love, IoP Publishing, 1994
• M. Drees, R. Godbole, P. Roy, World Scientific
  2004
• Signer, arXiv0905.4630
• and many others
  

Warning be aware of many different conventions
in the literature
Disclaimer cannot guarantee that all signs are
correct
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The Standard Model
Why do we believe it?
Why do we not believe it?
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Why do we believe the SM?

• 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 (1018
  m)

• the best theory we ever had

Why then we do not believe the Standard Model?
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The Standard Model
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

WMAP

• SM particles constitute a small part of
• the visible universe

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The hierarchy problem
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
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The hierarchy problem
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
We will consider supersymmetry
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Symmetries
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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
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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 1960ties 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
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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 fermiongt
bosongt and allow for
anticommutators
a,babba
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transforms like a fermion
Graded Lie algebra, superalgebra or
supersymmetry
Remarkably, standard QFT allows for supersymmetry
without any additional assumptions
Golfand, Likhtman 71, Volkov, Akulov 72, Wess
Zumino 73
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The SUSY algebra
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Simplest case N1 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
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Grassmann variables
Variables with fermionic nature
with
Raising and lowering indices
using antisymmetric tensor
We will also need
Dirac matrices
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Product of two spinoirs is defined as
in particular
Technicalities
For Dirac spinors
Lorentz covariants
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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
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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
non-vanishing vacuum energy
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SUSY multiplets massless representations
fermionic and bosonic states of equal mass
Since
Then
only
Equal number of bosonic and fermionic states in
supermultiplet
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Supermultiplets
Most relevant ones for constructing realistic
theory
Chiral spin 1/2 and
0 Weyl fermion
complex scalar
Vector spin 1 and
1/2 vector
(gauge) Weyl fermion (gaugino)
Gravity spin 2 and
3/2 graviton
gravitino
and CPT conjugate states
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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 Weyl auxiliary
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Derivatives with respect to Grassmann variable
one has to be very careful
since
Derivatives also anticommute with other Grassmann
variables
Integration defined as
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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
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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
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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 gt reducible
  representation
• 
  
• Too many components for fields with spin lt or 1

For the Minimal Supersymmertic extension of the
SM enough to consider chiral superfield
vector superfield
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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
RHxSF
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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 LHxSFs is also a LHxSF

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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
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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
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Constructing SUSY Lagrangians
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SUSY Lagrangians
Supersymmetric Lagrangians
F and D terms of LHxSF and VSF, respectively,
transform as total derivatives
Products of LHxSF are chiral superfields Product
s of VSF are vector superfields
Use F and D terms to construct an invariant action
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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
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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
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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
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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
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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)
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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 means evaluate at
But is a
spinor LHxSF since
In terms of component fields photino, photon
and an auxiliary D
Note that is gauge invariant, i.e.
does not change under
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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
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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
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Extending to non-Abelian case
The VSF must be in adjoint representation of the
gauge group
For matter xSF
Explicitly
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Feynman rules relations among masses and
couplings
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Non-renormalisation theorem
R-symmetry -- rotates superspace coordinate
Define R charge
Terms from Kaehler are invariant since
are real
For to be invariant
component fields of the SF have different R charge
Consider Wess-Zumino
Assume as vevs of heavy SF
(spurions)
For global
symmetry
Renormalised superpotential must be of
But must be regular
Only Kaehler potential gets renormalised
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Summary on constructing SUSY Lagrangians
Construct Lagrangians for N1 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 N2 more component fields in a
hypermultiplet
contains both ½ and ½ helicity fermions which
need to transform in
the same way under gauge symmetry
Ngt1 ?
non-chiral