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SUSY 1. Jan Kalinowski. Three lectures: Introduction to SUSY MSSM: its structure, current status and LHC expectations Exploring SUSY at a Linear Collider. Outline. What’s good/wrong with the Standard Model? Symmetries SUSY algebra Constructing SUSY Lagrangian. Literature.

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Jan kalinowski


Jan Kalinowski

Supersymmetry, part 1

  • Three lectures:

  • Introduction to SUSY

  • MSSM: its structure, current status and LHC expectations

  • Exploring SUSY at a Linear Collider

Supersymmetry, part 1


  • What’s good/wrong with the Standard Model?

  • Symmetries

  • SUSY algebra

  • Constructing SUSY Lagrangian

Supersymmetry, part 1


  • 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

  • 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

Supersymmetry, part 1


Standard Model

Why do we believe it?

Why do we not believe it?

Supersymmetry, part 1

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 (10–18 m)

  • the best theory we ever had

Why then we do not believe the Standard Model?

Supersymmetry, part 1

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


  • SM particles constitute a small part of

  • the visible universe

Supersymmetry, part 1

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

Supersymmetry, part 1

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

Supersymmetry, part 1


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

Supersymmetry, part 1

Gauge symmetries quantity

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

Supersymmetry, part 1

However, quantityColeman-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


Supersymmetry, part 1

transforms like a fermion quantity

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

Supersymmetry, part 1

The SUSY algebra quantity

Supersymmetry, part 1

Simplest case: N=1 supersymmetry quantity

only one fermionic generator and its conjugate

Reminder: two component Weyl spinors that transform under Lorentz


spinors transform according to

spinors transform according to

Dirac spinor requires two Weyl spinors

Supersymmetry, part 1

Grassmann variables quantity

Variables with fermionic nature with

Raising and lowering indices

using antisymmetric tensor

We will also need

Dirac matrices

Supersymmetry, part 1

Product of two spinoirs is defined as quantity

in particular


For Dirac spinors

Lorentz covariants

Supersymmetry, part 1

The Lagrangian for a free Dirac field in terms of Weyl quantity

The Lagrangian for a free Majorana field in terms of Weyl

Frequently used identities:

We will also use

Supersymmetry, part 1

Supersymmetry algebra quantity

or in terms of Majorana

Normalization, since

Spectrum bounded from below

If vacuum state is supersymmetric, i.e.


For spontaneous SUSY breaking


non-vanishing vacuum energy

Supersymmetry, part 1

SUSY multiplets – massless representations quantity

fermionic and bosonic states of equal mass




Equal number of bosonic and fermionic states in supermultiplet

Supersymmetry, part 1

Supermultiplets quantity

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

Supersymmetry, part 1

Superspace and superfields quantity

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

Supersymmetry, part 1

Derivatives with respect to Grassmann variable quantity

one has to be very careful:


Derivatives also anticommute with other Grassmann variables

Integration defined as

Supersymmetry, part 1

With Grassmann variables SUSY algebra can be written as quantity

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

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


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

Supersymmetry, part 1

Chiral superfields complex)

left-handed chiral superfield (LHxSF)

right-handed chiral superfield (RHxSF)

Invariant under SUSY transformation


is LHxSF

Expanding in terms of components:

(dimensions: )

contains one complex scalar (sfermion), one Weyl fermion and an auxiliary field F


Supersymmetry, part 1

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

Supersymmetry, part 1

Vector superfields component fields

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

Supersymmetry, part 1

By a proper choice of gauge transformation we can go to component fields

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

Supersymmetry, part 1

Constructing SUSY component fields


Supersymmetry, part 1

SUSY Lagrangians component fields

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

Supersymmetry, part 1

Example: Wess-Zumino model superfields component fields

Consider one LHxSF

(using )

Introduce a superpotential

We also need a dynamical part

a D-term can be constructed out of

Kaehler potential

Supersymmetry, part 1

Both scalar and spinor kinetic terms appear as needed. component fields

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


Scalar and fermion of equal mass

All couplings fixed by susy

Supersymmetry, part 1

Generalising to more LHxSF component fields

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

Supersymmetry, part 1

Vector superfields component fields

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

Supersymmetry, part 1

Gauge theory: Abelian case component fields

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)

Supersymmetry, part 1

General VSF contains a spin 1 component field component fields

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

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

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

Supersymmetry, part 1

Extending to non-Abelian case , i.e. there are other terms

The VSF must be in adjoint representation of the gauge group

For matter xSF


Supersymmetry, part 1

Feynman rules: relations among masses and couplings , i.e. there are other terms

Supersymmetry, part 1

Non-renormalisation theorem , i.e. there are other terms

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

Supersymmetry, part 1

Summary on constructing SUSY Lagrangians , i.e. there are other terms

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

Supersymmetry, part 1