Unification ParadigmUnification Paradigm

• Unification of strong, weak and electromagnetic interactions within Grand Unified Theories is the new step in unification of all forces of Nature• Creation of a unified theory of everything based on string paradigm seems to be possible

3410 m

D=10

GUT

What is SUSY?What is SUSY?

• Supersymmetry is a boson-fermion symmetrythat is aimed to unify all forces in Nature including gravity within a singe framework

• Modern views on supersymmetry in particle physicsare based on string paradigm, though low energymanifestations of SUSY can be found (?) at moderncolliders and in non-accelerator experiments

| | | |Q boson fermion Q fermion boson

[ , ] 0, { , } 0 b b f f { , } 2 ( )ji ijQ Q P

{ , } 2 ( )ji ijQ Q P

First papers in 1971-1972

No evidence in particle physics yet

Motivation of SUSY in Motivation of SUSY in Particle PhysicsParticle Physics

Unification with Gravity

| | | |

2 3/2 1 1/2 0

Q boson fermion Q fermion boson

spin spin spin spin spin

| | | |

2 3/2 1 1/2 0

Q boson fermion Q fermion boson

spin spin spin spin spin

Unification of matter (fermions) with forces (bosons) naturally arisesfrom an attempt to unify gravity with the other interactions

{ , } 2 ( ) { , } 2( )

( ) local coordinate transf. (super)gravity

ji ijQ Q P P

x

{ , } 2 ( ) { , } 2( )

( ) local coordinate transf. (super)gravity

ji ijQ Q P P

x

Unification with Gravity Unification of gauge couplings Solution of the hierarchy problem Dark matter in the Universe Superstrings

Local supersymmetry = general relativity !

Motivation of SUSY in Motivation of SUSY in Particle PhysicsParticle Physics

Unification of gauge couplings

c L Y

3 2 1

SU (3) SU (2) U (1) (or + symm)

,

nGUT

GUT

Low Energy High Energy

G G

gluons W Z photon gauge bosons

quarks leptons fermions

g g g g

c L Y

3 2 1

SU (3) SU (2) U (1) (or + symm)

,

nGUT

GUT

Low Energy High Energy

G G

gluons W Z photon gauge bosons

quarks leptons fermions

g g g g

Running of the strong coupling

2

2( ) (distance)Qi i i

Motivation of SUSYMotivation of SUSY

1

2

( ) 128.978 0.027

sin 0.23146 0.00017

( ) 0.1184 0.0031

Z

MS

s Z

M

M

2 2 2 2 2, / 4 /16 , t=log(Q / )ii i i i i

db g

dt

1

2

3

0 4 / 3 1/10

: 22 / 3 4 / 3 1/ 6

11 4 / 3 0i Fam Higgs

b

SM b b N N

b

1

2

3

0 2 3/10

: 6 2 1/ 2

9 2 0i Fam Higgs

b

MSSM b b N N

b

RG Equations

Input

Output

3.4 0.9 0.4

15.8 0.3 0.1

-1GUT

10 GeV

10 GeV

26.3 1.9 1.0

SUSY

GUT

M

M

SUSY yields unification!

Unification of the Coupling Constantsin the SM and in the MSSM

Motivation of Motivation of SUSYSUSY

• Solution of the Hierarchy Problem

2

16

v 10 GeV

m V 10 GeV

Hm

-14 10 1Hm

m

Destruction of the hierarchy byradiative corrections

Cancellation of quadratic terms

2 2

bosons fermions

m m SUSY may also explain the originof the hierarchy due to radiativemechanism

Motivation of SUSYMotivation of SUSY• Dark Matter in the UniverseDark Matter in the Universe

SUSY provides a candidate for the Dark matter – a stable neutral particle

The flat rotation curves of spiral

galaxies provide the most direct

evidence for the existence of large

amount of the dark matter.

Spiral galaxies consist of a central

bulge and a very thin disc, and

surrounded by an approximately

spherical halo of dark matter

Cosmological Cosmological ConstraintsConstraints

New precise cosmological data

2 1

73%

23 4%

4%

vacuum

DarkMatter

Baryon

h

crit •Supernova Ia explosion• CMBR thermal fluctuations

(recent news from WMAP )

Dark Matter in the Universe:

Hot DM(not favoured by galaxy formation)

Cold DM(rotation curvesof Galaxies)

SUSYSUSY

SuperalgebraSuperalgebra

1 12 2

[ , ] 0,

( ) lg

Lorentz Algebra

SUSY

[ , ] ( ),

[ , ] ( ),

[ , ] [ , ] 0

,

[ , ] ( ) ,

Algebra

[ , ]

i i

i i i

P P P M i g P g P

M M i g M g M g M g M

Q P Q P

Q M Q Q M

Super A ebra

( ) ,

{ , } 2 ( ) ,

, , , 1, 2; , 1, 2,..., .

i

ji ij

Q

Q Q P

i j N

1 12 2

[ , ] 0,

( ) lg

Lorentz Algebra

SUSY

[ , ] ( ),

[ , ] ( ),

[ , ] [ , ] 0

,

[ , ] ( ) ,

Algebra

[ , ]

i i

i i i

P P P M i g P g P

M M i g M g M g M g M

Q P Q P

Q M Q Q M

Super A ebra

( ) ,

{ , } 2 ( ) ,

, , , 1, 2; , 1, 2,..., .

i

ji ij

Q

Q Q P

i j N

, ,

Superspace

x x , ,

Superspace

x x

Grassmannian parameters

, 1, 2

Q i

Q i

22 0, 0

SUSY Generators

The only possible graded Lie algebra that mixes integer and half-integer spins and changes statistics

22 0, 0Q Q

{ , } 2 ( )ji ijQ Q P

,

,

Supertranslation

x x i i

Quantum StatesQuantum StatesQuantum states: | ,E Vacuum = | , 0Q E

Energy helicity

State Expression # of states

vacuum 1

1-particle

2-particle

… … …

N-particle

| ,E

| , | , 1/ 2iQ E E

| , | , 1i jQ Q E E

1 2... | , | , / 2NQ Q Q E E N

1N N

( 1)2 2

N NN

1NN

Total # of states: 1 1

0

2 2 2N

N N N Nk

k

bosons fermions

[ , ] [ , ] 0i iQ P Q P

SUSY MultipletsSUSY MultipletsChiral multiplet

Vector multiplet

1, =0N helicity

# of states

-1/2 0 1/2

1 2 1

1, =1/2N helicity

# of states-1 -1/2 1/2 1 1 1 1 1

( , )

( , )A

scalar spinor

spinor vector

Members of a supermultiplet are called superpartners

N=4 SUSY YM helicity -1 –1/2 0 1/2 1

λ = -1 # of states 1 4 6 4 1

N=8 SUGRA helicity -2 –3/2 –1 –1/2 0 1/2 1 3/2 2

λ = -2 # of states 1 8 28 56 70 56 28 8 1

4N S spin4N 8N

For renormalizable theories (YM)

For (super)gravity

Simplest (N=1) SUSY Simplest (N=1) SUSY MultipletsMultiplets

( , ) ( , )A

Bosons and Fermions come in pairs

( , )g gSpin 0 Spin 1/2 Spin 1Spin 1/2 Spin 3/2 Spin 2

SUSY TransformationSUSY Transformation

14

( , ) ( ) 2 ( ) ( )

( ) ( ) ( )

2 ( ) / 2 ( ) ( )

y A y y F y

A x i A x A x

x i x F x

N=1 SUSY Chiral supermultiplet:

( )y x i

component fields

spin=0

spin=1/2

Auxiliary field

2 ,

2 2 ,

2

A

i A F

F i

2 ,

2 2 ,

2

A

i A F

F i

parameter of SUSY transformation(spinor)

(unphysical d.o.f. needed to close SYSY algebra )

SUSY multiplets Superfiled in Superspace

Expansion over grassmannian parametersuperfield

2 2 21 2 1 2, 0!

Gauge Gauge superfieldssuperfields

1 12 2

( , , ) ( ) ( ) ( ) ( ) ( )

( ) [ ( ) ( )] [ ( ) ( )]

[ ( ) ( )]

V x C x i x i x i M x i M x

v x i x i x i x i x

D x C x

Gauge transformationV V

*

*

2

2

( )

C C A A

i

M M iF

v v i A A

D D

*

*

2

2

( )

C C A A

i

M M iF

v v i A A

D D

Wess-Zumino gauge

0C M

physical fields

Field strength tensor

214

V VW D e D e

22 ( )iW i D F D

D i

Covariant derivatives

D i

Gauge superfield

0, d d

4 ( )Action d x L x 4 4 ( , , )d x d L x

How to write SUSY How to write SUSY LagrangiansLagrangians

Grassmannian integration in superspace

1st step

2nd step

( , , ) ( , )Field A Superfield V

3rd step

Take your favorite Lagrangian written in terms of fields

Replace

Replace

Superfield Superfield LagrangiansLagrangians

2 2 2 1 12 3 ( ) . .]i i i i ij i j ijk i j kL d d d m y h c

Grassmannian integration in superspace 0, d d

Gauge fields

2 2 21 1 14 2 4 L d W W d W W D F F i D

2 2 21 1 14 2 4 L d W W d W W D F F i D

Gauge transformation , , ( )ig ige e V V i

Gauge invariant interaction gVe

4 Action d x L 4 4 d x d L

Superpotential

Matter fields

Gauge Invariant SUSY Gauge Invariant SUSY LagrangianLagrangian

2 21 1 4 4

2 2 2 2

Tr(W ) Tr(W )

( ) ( ) ( )

SUSY YM

gV a bia ib i i

L d W d W

d d e d d

W W

1 1 4 2

†

† † †

2 2† 1 1

2 2† † †

( ) ( ) ( )

2 2

aa a a a aSUSY YM

a a a a a aii i i i i i

aa a a a aii i i i i i i

i ji i i ji i i j i j

L F F i D D D

A igv T A A igv T A i igv T

D gA T A i gA T i g T A F F

F FA A A A A A

W W W W

† †12, V=a a a a

i i i i ii

D gA T A F D D F FA

W