Jan KalinowskiSupersymmetry, part 1 SUSY 1 Jan Kalinowski.
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Transcript of Jan KalinowskiSupersymmetry, part 1 SUSY 1 Jan Kalinowski.
Jan Kalinowski Supersymmetry, part 1
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
J. Wess, J. Bagger, Princeton Univ Press, 1992H. Haber, G. Kane, Phys.Rept.117 (1985) 75S.P Martin, arXiv:hep-ph/9709356H.K. Dreiner, H.E. Haber, S.P. Martin, arXiv:0812.1594M.E. Peskin, arXiv:0801.1928D. Bailin, A. Love, IoP Publishing, 1994M. Drees, R. Godbole, P. Roy, World Scientific 2004A. Signer, arXiv:0905.4630
and many others
Disclaimer: cannot guarantee that all signs are correct
Warning: be aware of many different conventions in the literature
Literature
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:
• Higgs mass unstable w.r.t. quantum corrections
• SM particles constitute a small part of the visible universe
WMAP
• neutrino oscillations
• mater-antimater asymmetry
• does not contain gravity
• can be valid only up to a certain scale
Hambye, Riesselmann
Jan Kalinowski Supersymmetry, part 1
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
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
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
Hironari Miyazawa (’68) first who considered mesons and baryons in the same multiplets
Gauge symmetries
Jan Kalinowski Supersymmetry, part 1
However, Coleman-Mandula theorem ‘67: direct product of Poincare and internal symmetry groups
Here all generators are of bosonic type (do not mix spins) and only commutators involved
we have to include generators of fermionic type that transform
|fermion> |boson> and allow for anticommutators
Particle states numerated by eigenvalues of commuting set of observables
Haag, Lopuszanski, Sohnius ’75: no direct symmetry transformation between states of integer spins
{a,b}=ab+ba
Jan Kalinowski Supersymmetry, part 1
Gol’fand, Likhtman ’71, Volkov, Akulov ’72, Wess Zumino ‘73
Graded Lie algebra, superalgebra or
Remarkably, standard QFT allows for supersymmetry without any additional assumptions
transforms like a fermion
Jan Kalinowski Supersymmetry, part 1
Jan Kalinowski Supersymmetry, part 1
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
Simplest case: N=1 supersymmetry
Jan Kalinowski Supersymmetry, part 1
Raising and lowering indices
using antisymmetric tensor
We will also need
Dirac matrices
Variables with fermionic nature with
Grassmann variables
Jan Kalinowski Supersymmetry, part 1
Technicalities:
Product of two spinoirs is defined as
For Dirac spinors Lorentz covariants
in particular
Jan Kalinowski 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
We will also use
Frequently used identities:
Jan Kalinowski Supersymmetry, part 1
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
Supersymmetry algebra
Jan Kalinowski 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
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
Supermultiplets
Jan Kalinowski Supersymmetry, part 1
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
Superspace and superfields
Jan Kalinowski Supersymmetry, part 1
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)
using Baker-Campbell-Hausdorff
i.e. under SUSY transformation
non-trivial transformation of the superspace
(dimensions!)
Jan Kalinowski Supersymmetry, part 1
In analogy to , we find a representation for generators
Convenient to introduce covariant derivatives
Check that satisfy SUSY algebra
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
• 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
note different dimensions of fields
Jan Kalinowski Supersymmetry, part 1
left-handed chiral superfield (LHxSF)
right-handed chiral superfield (RHxSF)
Invariant under SUSY transformation
Since
is LHxSF
Expanding in terms of components:
RHxSF:
contains one complex scalar (sfermion), one Weyl fermion and an auxiliary field F
(dimensions: )
Chiral superfields
Jan Kalinowski Supersymmetry, part 1
Transformation under infinitesimal SUSY transformation, component fields
boson fermionfermion bosonF total derivative
•The F term – a good candidate for a Lagrangian • Product of LHxSF’s is also a LHxSF
comparing with gives
Jan Kalinowski Supersymmetry, part 1
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
Vector superfields
Jan Kalinowski Supersymmetry, part 1
By a proper choice of gauge transformation we can go to the Wess-Zumino gauge
it is not invariant under susy, but after susy transformation we can again go to the Wess-Zumino gauge
Many unphysical fields have been „gauged away”
Jan Kalinowski Supersymmetry, part 1
Jan Kalinowski Supersymmetry, part 1
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
SUSY Lagrangians
Jan Kalinowski Supersymmetry, part 1
Consider one LHxSF
(using )
Introduce a superpotential
We also need a dynamical part
a D-term can be constructed out of
Kaehler potential
Example: Wess-Zumino model superfields
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 massAll couplings fixed by susy
Jan Kalinowski Supersymmetry, part 1
Generalising to more LHxSF
Yukawa-type interactions
couplings of equal strength
Alternatively, Lagrangian can be written as kinetic part and contribution from superpotential
D-terms only of the type
Terms of the type forbidden – superpotential has to be holomorphic
Jan Kalinowski Supersymmetry, part 1
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
Vector superfields
Jan Kalinowski Supersymmetry, part 1
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
Then the gauge interaction is invariant since
is also a LHxSF
(for Abelian)
Gauge theory: Abelian case
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 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
Jan Kalinowski 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
The VSF must be in adjoint representation of the gauge group
For matter xSF
Explicitly
Extending to non-Abelian case
Jan Kalinowski Supersymmetry, part 1
Feynman rules: relations among masses and couplings
Jan Kalinowski Supersymmetry, part 1
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 vev’s of heavy SF (spurions)
For global symmetry
Renormalised superpotential must be of
But must be regular
Only Kaehler potential gets renormalised
Non-renormalisation theorem
Jan Kalinowski Supersymmetry, part 1
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
Summary on constructing SUSY Lagrangians
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