hepthyymmnnn

The Sup ersymmetric Standard Mo del

a b c

Jan Louis Ilka Brunner and Stephan J Hub er

a

MartinLutherUniversitat Hal leWittenberg

FB Physik D Hal le Germany

emailjlouisphysikunihallede

b

Department of Physics and Astronomy

Rutgers University

Piscataway NJ USA

emailibrunnerphysicsrutgersedu

c

Universitat Heidelberg

Institut f ur Theoretische Physik

Philosophenweg

D Heidelberg Germany

emailshuberthphysuniheidelbergde

ABSTRACT

This set of lectures introduces at an elementary level the sup ersymmetric Stan

dard Mo del and discusses some of its phenomenological prop erties

Lectures given by J Louis at the summer school Grund lagen und neue Methoden

der theoretischen Physik Saalburg

November

We intend to regularly up date this review Therefore we would b e very grateful for your corrections comments and suggestions

Introduction

The Standard Mo del of Particle Physics is an extremely successful theory which

has b een tested exp erimentally to a high level of accuracy After the discov

ery of the top quark the Higgs b oson which is predicted to exist by the Standard

Mo del is the only missing ingredient that has not b een directly observed yet

However a number of theoretical prejudices suggest that the Standard Mo del

is not the nal answer of nature but rather an eective description valid up

to the weak scale of order O GeV The arbitrariness of the sp ectrum and

gauge group the large number of free parameters the smallness of the weak scale

compared to the Planck scale and the inability to turn on gravity suggest that at

higher energies shorter distances a more fundamental theory will b e necessary

to describ e nature Over the past years various extensions of the Standard

Mo del such as Technicolor Grand Unied Theories Sup ersymmetry

or String Theory have b een prop osed In recent years sup ersymmetric

extensions of the Standard Mo del b ecame very p opular also among exp erimen

talists not necessarily b ecause of their convincing solution of the ab ove problems

but rather b ecause most other contenders have b een more or less ruled out by

now Another reason for the p opularity of sup ersymmetric theories among the

orists is the fact that the low energy limit of sup erstring theory a promising

candidate for a unication of all interactions including gravity is by and large

sup ersymmetric

This set of lectures give an elementary introduction to the sup ersymmetric

Standard Mo del Section contains some of the necessary background on generic

sup ersymmetric eld theories while section develops sup ersymmetric extensions

of the Standard Mo del and discusses sp ontaneous breaking of sup ersymmetry In

section extensions of the Standard Mo del with softly broken sup ersymmetry are

presented and some of the phenomenological prop erties are discussed Section

contains a summary and our conventions which follow rather closely ref are

recorded in an app endix

These lectures are not meant to review the latest developments of the su

p ersymmetric Standard Mo del but rather attempts to give an elementary in

tro duction from a mo dern p oint of view Many excellent review articles on

sup ersymmetry and the sup ersymmetric Standard Mo del do exist and have b een

heavily used in these lectures In addition a collection of some of the

classic pap ers concerning the sub ject can b e found in ref

Introduction to Sup ersymmetry

Sup ersymmetry is a symmetry b etween b osons and fermions or more precisely it

is a symmetry b etween states of dierent spin For example a spin particle

particle under a sup ersymmetry transformation Thus is mapp ed to a spin

the generators Q Q of the sup ersymmetry transformation must transform in

representations of the Lorentz group These new fermionic generators the spin

form together with the fourmomentum P and the generators of the Lorentz

m

mn

transformations M a graded Lie algebra which features in addition to commu

tators also anticommutators in their dening relations The simplest N

sup ersymmetry algebra reads

m

fQ Q g P

m

fQ Q g f Q Q g

Q P Q P

m m

mn mn

Q Q M

mn

mn

Q Q M

m

where we used the notation and convention of ref are the Pauli matrices

mn y

and the are dened in the app endix

The particle states in a sup ersymmetric eld theory form representations su

p ermultiplets of the sup ersymmetry algebra We do not recall the entire

representation theory here see for example refs but only highlight a

few generic features

a There is an equal number of b osonic degrees of freedom n and fermionic

B

degrees of freedom n in a sup ermultiplet

F

n n

B F

I

y I

In general it is p ossible to have N sets of sup ersymmetry generators Q Q I N

_

in which case one refers to N extended sup ersymmetry Such extended sup eralgebras have

b een classied by Haag Lopuszanski and Sohnius generalizing earlier work of Coleman

and Mandula who showed that the p ossible b osonic symmetries of the Smatrix of a four

dimensional lo cal relativistic quantum eld theory consist of the generators of the Poincare

group and a nite number of generators of a compact Lie group which are Lorentz scalars In

ref this theorem was generalized to also include symmetry transformations generated by

fermionic op erators and all p ossible sup eralgebras were found In extensions of the Standard

Mo del N extended sup ersymmetries have played no role so far since they cannot accommo date

the chiral structure of the Standard Mo del

b The masses of all states in a sup ermultiplet are degenerate In particular

z

the masses of b osons and fermions are equal

m m

B F

c Q has mass dimension and thus the mass dimensions of the elds in a

sup ermultiplet dier by

The two irreducible multiplets which are imp ortant for constructing the su

p ersymmetric Standard Mo del are the chiral multiplet and the vector multiplet

which we discuss in turn now

The chiral sup ermultiplet

The chiral sup ermultiplet contains a complex scalar eld Ax of spin and

and mass dimension and an mass dimension a Weyl fermion x of spin

auxiliary complex scalar eld F x of spin and mass dimension

Ax x F x

has oshell four real b osonic degrees of freedom n and four real

B

fermionic degrees of freedom n in accord with The sup ersymme

F

try transformations act on the elds in the multiplet as follows

p

A

p p

m

A F i

m

p

m

F i

m

where we used the conventions of ref and the app endix The parameters of the

transformation are constant complex anticommuting Grassmann parameters

ob eying

The transformations can b e thought of as generated by the op erator

Q Q

with Q and Q ob eying This can b e explicitly checked by evaluating the

commutators on the elds A and F

z 2

This follows immediately from the fact that P is a Casimir op erator of the sup ersymmetry

2 2 mn

algebra P Q P M

m m

Exercise Show i by using and

m

Exercise Evaluate the commutator using for al l three elds A and

F and show that this is consistent with the results of the previous exercise

The eld F has the highest mass dimension of the members of the chiral

multiplet and therefore is called the highest comp onent As a consequence it

cannot transform into any other eld of the multiplet but only into their deriva

tives This is not only true for the chiral multiplet as can b e seen explicitly in

but holds for any sup ermultiplet This fact can b e used to construct La

grangian densities which transform into a total derivative under sup ersymmetry

transformations leaving the corresp onding actions invariant We do not review

here the metho d for systematically constructing sup ersymmetric actions which

is done most eciently using a sup erspace formalism Since these lectures fo cus

on the phenomenological prop erties of sup ersymmetry we refer the reader to the

literature for further details and only quote the results

For the chiral multiplet a sup ersymmetric and renormalizable Lagrangian is

given by

m m

A A F F LA F i

m m

mAF AF

Y A F A F A A

where m and Y are real parameters This action has the p eculiar prop erty that

no kinetic term for F app ears As a consequence the equations of motion for F

are purely algebraic

L L

F mA Y A F mA YA

F F

Thus F is a nondynamical auxiliary eld which can b e eliminated from the

action algebraically by using its equation of motion This yields

m m

LA F mA Y A i A A

m m

m

Y A A V A A

where V A A is the scalar p otential given by

V A A j mA YA j

m AA mY AA AA Y A A

F F j L

L

F

F

As can b e seen from and after elimination of F a standard renormaliz

able Lagrangian for a complex scalar A and a Weyl fermion emerges However

is not the most general renormalizable Lagrangian for such elds Instead it

satises the following prop erties

L only dep ends on two indep endent parameters the mass parameter m and

the dimensionless Yukawa coupling Y In particular the AA coupling

is not controlled by an indep endent parameter as it would b e in non

sup ersymmetric theories but determined by the Yukawa coupling Y

x

The masses for A and coincide in accord with

V is p ositive semidenite V

The vector sup ermultiplet

The vector sup ermultiplet V contains a gauge b oson v of spin and mass

m

dimension a Weyl fermion called the gaugino of spin and mass dimension

and a real scalar eld D of spin and mass dimension

V v x x D x

m

Similar to the chiral multiplet also the vector multiplet has n n

B F

The vector multiplet can b e used to gauge the action of the previous section

An imp ortant consequence of the theorems of refs is the fact that the

a

generators T of a compact gauge group G have to commute with the sup ersym

metry generators

a a

T Q T Q

Therefore all members of a chiral multiplet A F have to reside in the same

representation of the gauge group Similarly the members of the vector multiplet

have to transform in the adjoint representation of G and thus they all are Lie

algebra valued elds

a a a a a a

v v T T D D T

m

m

The sup ersymmetry transformations of the comp onents of the vector multiplet

are

a m m a a

i i v

m

a a mn a

i D F

mn

a m a a m

D D D

m m

x

As immediate consequence of this feature one notes that sup ersymmetry must b e explicitly

or sp ontaneously broken in nature

a a

The eld strength of the vector b osons F and the covariant derivative D

m

mn

are dened according to

a a a abc b c

F v v g f v v

m n

mn n m m n

a a abc b c

D g f v

m m

m

abc

where f are the structure constants of the Lie algebra and g is the gauge

coupling A gauge invariant renormalizable and sup ersymmetric Lagrangian for

the vector multiplet is given by

a a a mn a a m a

L F D D F i D

m

mn

As b efore the equation of motion for the auxiliary D eld is purely algebraic

a

D

A gauge invariant renormalizable Lagrangian containing a set of chiral mul

i i i

tiplets A F coupled to vector multiplets is found to b e

mn a a m a a a a a a i i i a

F i D F D D D LA F v

m

mn m

i i m i i i i m

A i D F F D A D

m m

p

j i j a a a i a

i A T g A T

ij ij

j i a a j i i j

A A T g D W W

ij ij

ij

i i

F W F W

i i

where the covariant derivatives are dened by

i i a a i

D A A ig v T A

m m

m ij

j a i i a

T D ig v

m m

ij m

W and W in are the derivatives of a holomorphic function W A called

i ij

the sup erp otential

i j i j k

m A A Y A A A W A

ij ij k

W

j j k

W m A Y A A

i ij ij k

i

A

W

k

m Y A W

ij ij k ij

i j

A A

By explicitly inserting into one observes that the m are mass param

ij

eters while the Y are Yukawa couplings Sup ersymmetry forces W to b e a

ij k

holomorphic function of the scalar elds A while renormalizability restricts W

to b e at most a cubic p olynomial of A Finally the parameters m and Y are

ij ij k

further constrained by gauge invariance

i a

As b efore F and D ob ey algebraic equations of motion which read

L

F W

i i

F

L

F W

i i

F

L

a i a j

D g A T A

ij

a

D

i a

They can b e used to eliminate the auxiliary elds F and D from the Lagrangian

and one obtains

i i a a a i a j

LA v F W D g A T A

i i

m ij

i i m i mn a a m a i m a

A i D F i D D A D F

m m m

mn

p

j i i j j i j a a a i a

V A A W W A T g A T i

ij ij

ij ij

where

j j i a i a

A A A T g A T W V A A W

i i

ij ij

i i a a

F F L D D j L

a

F D

As b efore the scalar p otential V A A is p ositive semidenite

Exercise Insert into and derive

A Sup ersymmetric Extension of the Standard

Mo del

The Standard Mo del

In this section we briey review some basic features of the Standard Mo del

The Standard Mo del is a quantum gauge eld theory with a chiral gauge group

G SU SU U The sp ectrum of particles includes three families

SM Y

of quarks and leptons the gauge b osons gluons W Z photon of G and

SM

one spin Higgs doublet In table the particle content and the corresp onding

SU SU U U

Y em

I

u

I L

quarks q

I

L

d

L

I

u

R

I

d

R

I

I L

leptons l

I L

e

L

I

e

R

h

Higgs h

h

gauge b osons G

W

B

Table The particle content of the Standard Mo del The index I lab els

I I I I I

All of them e and chiral leptons l d u the three families of chiral quarks q

R L R R L

are Weyl fermions and transform in the representation of the Lorentz group

they have an undotted spinor index The subscripts R L do not sp ecify the

representation of the Lorentz group but instead are used to indicate the dierent

transformation prop erties under the chiral gauge group SU U This

somewhat unconventional notation is used to make a smo oth transition to the

sup ersymmetric Standard Mo del later on The electromagnetic charge listed in

the last column is dened by Q T Q

em Y

SU

gauge quantum numbers are displayed The Lagrangian of the Standard Mo del

reads

X

b mn b m

F F D hD h L

m

a

mn

a

X

I I I I I I I I I I

iq D q iu D u id D d il D l ie D e

L L R R R R L L R R

I

X

I J I J I J

Y hq u Y hq d Y hl e hc V h h

u IJ d IJ l IJ

L R L R L R

IJ

m

where D D and the index a lab els the dierent factors in the gauge

m

group V h h is the scalar p otential for the Higgs doublet which is chosen to b e

V h h hh hh

In order to have a b ounded p otential has to hold For the elec

troweak gauge group SU U is sp ontaneously broken down to U

Y em

In this case the minimum of the p otential is not at hhi but at hhhi

Exercise Give explicitly al l covariant derivatives in

Exercise Check that the Lagrangian is gauge and Lorentz invariant

Sup ersymmetric Extensions

Let us now turn to the sup ersymmetric generalization of the Standard Mo del

The idea is to promote the Lagrangian to a sup ersymmetric Lagrangian

As we learned in the previous section sup ersymmetry requires the presence of

additional states which form sup ermultiplets with the known particles Since

all states of a sup ermultiplet carry the same gauge quantum numbers we need

at least a doubling of states For every eld of the SM one has to p ostulate

a sup erpartner with the exact same gauge quantum numbers and a spin such

that it can form an appropriate sup ermultiplet More sp ecically the quarks

and leptons are promoted to chiral multiplets by adding scalar spin squarks

I I I I I

q u d and sleptons l e to the sp ectrum The gauge b osons are promoted

L R R L R

to vector multiplets by adding the corresp onding spin gauginos G W B to

the sp ectrum Finally the Higgs b oson is also promoted to a chiral multiplet

Higgsino sup erpartner However the sup ersymmetric version of with a spin

the Standard Mo del cannot live with only one Higgs doublet and at least a

See also

second Higgs doublet has to b e added This can b e seen from the fact that

one cannot write down a sup ersymmetric version of the Yukawa interactions of

the Standard Mo del without introducing a second Higgs doublet The reason

is the denite chirality of the Higgsino Another way to see the necessity of a

second Higgs doublet is the fact that the Higgsino is a chiral fermion which carries

U hypercharge and hence it upsets the anomaly cancellation condition Thus

a second Higgsino with opp osite U charge is necessary and sup ersymmetry

k

then also requires a second spin Higgs doublet The precise sp ectrum of the

sup ersymmetric Standard Mo del is summarized in table

The Lagrangian for the sup ersymmetric Standard Mo del has to b e of the form

with an appropriate sup erp otential W It has to b e chosen such that the

Lagrangian of the nonsup ersymmetric Standard Mo del is contained This

is achieved by

X

I I J I J I

h h l l Y h d Y h q u Y h q W

u d l IJ d d IJ d u IJ u

R L R L R L

IJ

Once W it sp ecied also the scalar p otential is xed Of particular imp or

tance is the scalar p otential for the Higgs elds since it controls the electroweak

symmetry breaking Using and one derives the Higgs p otential for the

by setting all other scalars to zero h two neutral Higgs elds h

u d

jh j jh j V h h jj g g jh j jh j

d u d u u d

The coupling of the terms quartic in the Higgs elds is not an indep endent pa

rameter but instead determined by the gauge couplings g of U and g of

Y

SU Thus it seems that the number of parameters is reduced However now

i one more than in i hh there are two p ossible vacuum exp ectation values hh

d u

the Standard Mo del

Exercise Derive from and

In the last section we learned that the p otential of any sup ersymmetric theory

is p ositive semidenite and the Higgs p otential of eq is no exception as

can b e seen explicitly jj cannot b e chosen negative Thus the minimum of

V necessarily sits at hh i hh i which corresp onds to a vacuum with

u d

unbroken SU U Therefore the sup ersymmetric version of the Standard

k

Of course extensions with more Higgs doublets are also p ossible but two is the minimal

number

Note that the scalars can only b e set to zero in the p otential V but not in the sup erp otential

W since the computation of the p otential requires taking appropriate derivatives of W

sup ermultiplet F B SU SU U U

Y em

I

U

I I I L

quarks Q q q

I

L L L

D

L

I I I

U u u

R R R

I I I

D d d

R R R

I

N

L I I I

l leptons L l

I L L L

E

L

I I I

E e e

R R R

h h H

d d

Higgs H

d

h H

h

d d

h h H

u u

H

u

h H

h

u u

G G gauge G

b osons W W W

B B B

Table Particle content of the sup ersymmetric Standard Mo del The column

b elow F B denotes the fermionic b osonic content of the mo del

Mo del as it is dened so far the sp ectrum of table with interactions sp ecied

by the Lagrangian with the W of cannot accommo date a vacuum

with sp ontaneously broken electroweak symmetry A second phenomenological

problem is the presence of all the new sup ersymmetric states which have the

same mass as their sup erpartners but are not observed in nature As we said

b efore sup ersymmetry itself necessarily has to app ear in its broken phase and

as we will see electroweak symmetry breaking is closely tied to the breakdown of

sup ersymmetry

Before we close this section let us note that in addition to the couplings of

gauge and Lorentz invariance also allows terms in W which are of the form

h l l q d d d u l l e

u L L L R R R R L L R

These terms violate baryon or lepton number conservation and thus easily lead

to unacceptable physical consequences for example the proton could b ecome un

stable Such couplings can b e excluded by imp osing a discrete Rparity

Particles of the Standard Mo del includuing b oth Higgs doublets are assigned R

charge while all new sup ersymmetric particles are assigned Rcharge This

eliminates all terms of while the sup erp otential of is left invariant An

immediate consequence of this additional symmetry is the fact that the lightest

sup ersymmetric particle often denoted by the LSP is necessarily stable How

ever one should stress that Rparity is not a phenomenological necessity Viable

mo dels with broken Rparity can b e constructed and they also can have some

phenomenological app eal

Exercise Check the gauge and Lorentz invariance for each term in and

compute their Rcharge

Sp ontaneous breaking of sup ersymmetry

In the previous section we learned that in the simplest sup ersymmetric extension

of the Standard Mo del the electroweak symmetry is unbroken However so far we

constructed a manifestly sup ersymmetric extension but from the mass degeneracy

of each multiplet it is already clear that sup ersymmetry cannot b e an exact

symmetry in nature but has to b e either sp ontaneously or explicitly broken

Therefore we now turn to the question of sp ontaneous sup ersymmetry breaking

and return to the electroweak symmetry breaking afterwards

Let us rst recall the order parameter for sup ersymmetry breaking Multiply

m

ing the anticommutator fQ P of the sup ersymmetryalgebra Q g

m

n m n mn

with and using T r results in

n n

Q g P fQ

Thus the Hamiltonian H of a sup ersymmetric theory is expressed as the square

of the sup ercharges

H P Q Q Q Q Q Q Q Q

This implies that H is a p ositive semidenite op erator on the Hilb ert space

h jH j i

Sup ersymmetry is unbroken if the sup ercharges annihilate the vacuum Q ji

Q ji From we learn that also H annihilates a sup ersymmetric vacuum

H ji This in turn implies that the scalar p otential V of a sup ersymmet

ric eld theory which has a sup ersymmetric ground state has to vanish at its

minimum

hH i hV i V A Aj

min

i i a a

The general form of the scalar p otential V F F D D was given in

Since V is p ositive semidenite one immediately concludes from that in a

sup ersymmetric ground state

i i a a

hF i F j and hD i D j

min min

has to hold The converse is also true

i a

hF i or hD i V j Q j

min

i a

and sup ersymmetry is sp ontaneously broken Thus hF i and hD i are the or

der parameters of sup ersymmetry breaking in that nonvanishing F or D terms

signal sp ontaneous sup ersymmetry breaking

Sp ecic p otentials which do lead to nonvanishing D or F terms have b een

constructed For example the ORaifeartaigh mo del has three chiral

sup erelds A A A and the following sup erp otential

W A mA A g A A m g

By minimizing V it can b e shown that F j and therefore sup ersymmetry

min

is broken Furthermore the mass sp ectrum of the real b osons and the Weyl

fermions is found to b e

Bosons m m m g

Fermions m m

Thus the mass degeneracy b etween b osons and fermions is lifted but nevertheless

a mass sum rule still holds

X X

M M

b f

b osons fermions

Exercise Minimize V using and compute F j Verify the mass

i min

spectrum and the sum rule

Unfortunately the sum rule is not a coincidence but a sp ecial case of a

general sum rule which holds in any theory with sp ontaneously broken sup ersym

metry Let us therefore pro ceed and derive this sum rule In general the mass

matrix of the b osons has the following form

A

i j i j i j

V A Aj A A A A A A A AM

mass terms

ij

ij ij

A

where

ij

ij

M

ij ij

The entries in the mass matrix are determined by the appropriate derivatives of

the p otential evaluated at its minimum

j V j V j V

min min ij min

i j ij

ij ij ij

V

etc Using one derives where V

ij

i j

A A

a a

V W W D D

i i

a a

D V W W D

j ij i

j

a a a a

D D V W W D D

ij ik

j k

j

j k k

i j

where again the indices i j denote derivatives with resp ect to A A In

serted into one obtains for the trace of the mass matrix

a a a a

TrM Tr TrV j TrW W D D D D j

min ij ik min

ij

j

ij

k j k

For the fermion masses the relevant pieces of the Lagrangian are

p

i j i j i a j a a a i j

W W L i g A T T A

ij ij

ij ij

This can b e rewritten as

j

i a

hc L M

b

where

p

p

i a

a

g A T W i

D W i

ij

ij

ij j

p p

M

i b b

i i g A T D

ij

i

Thus we obtain

a a

j TrM M TrW W D D

min ik k j

j i

a

Already at this p oint we learn from and that for D j we have a

min

sum rule

X X

M M

b f

bosons f er mions

where in the sum real b osons are counted

a

For D j also the gauge symmetry is necessarily broken and some of

min

the gauge b osons b ecome massive From one obtains the mass matrix

of the gauge b osons

j a b k a b

M g A T T A D D

j l lk l

l

Combining and one arrives at the mass sum rule

X

a a J

g TrT D J Tr M Str M

J

J

where J is the spin of the particles The right hand side of vanishes for any

nonAb elian factor in the gauge group while for U factors it is prop ortional to

P

the sum of the U charges Q Whenever this sum is nonvanishing the

U

theory has a U traceanomaly In the sup ersymmetric Standard Mo del this

traceanomaly vanishes Finally by rep eating the steps of this section one can

show that holds over all eld space and not only at the minimum of V This

will play a rolein deriving the soft sup ersymmetry breaking terms

Exercise Verify and

P

in the supersymmetric Standard Model Exercise Compute Q

U

Y

Exercise Show that holds over al l eld space and not only at the minimum

of V

The sum rule is problematic for the sup ersymmetric Standard Mo del

Since non of the sup ersymmetric partners has b een observed yet they must b e

heavier than the particles of the Standard Mo del Close insp ection of shows

that this cannot b e arranged within a sp ontaneously broken sup ersymmetric

Standard Mo del

An additional problem is the presence of a massless Goldstone fermion Gold

stones theorem implies that any sp ontaneously broken global symmetry leads to

a massless state in the sp ectrum This also holds for sup ersymmetry where the

broken generator is a Weyl spinor and thus there is an additional massless Gold

stone fermion The presence of this state can b e seen explicitly from the condition

that at the minimum of the p otential one has

a a

V W W D D

j ij i

j

Let us consider for simplicity the case that sup ersymmetry is broken by a non

a yy

vanishing Fterm hF i W j while hD i From one learns

i i min

immediately that now W j has to have a zero eigenvalue Using this

ij min

implies that also the mass matrix of the fermions has to have a zero eigenvalue

which is the Goldstone fermion

To summarize the lesson of this section is that also sp ontaneously broken

sup ersymmetry runs into phenomenological diculties The only way out is an

explicit breaking of global sup ersymmetry

Extensions of the Standard Mo del with Softly

Broken Sup ersymmetry

The Hierarchy and Naturalness Problem

Before we continue in our endeavor to construct a phenomenologically viable

extension of the Standard Mo del let us briey review what is called the hierarchy

zz

and naturalness problem in the Standard Mo del

Consider the following nonsup ersymmetric Lagrangian of a complex scalar

A and a Weyl fermion

m m

L A A i m m AA

m m f

b

Y A A AA

From we learn that this Lagrangian is sup ersymmetric if m m and Y

f b

but let us not consider this choice of parameters at rst L has a chiral symmetry

for m given by

f

i i

A e A e

yy

The general case is discussed in ref

zz

The discussion of this section follows ref Y Y

mf

Figure The onelo op correction to the fermion mass

λ

Y Y

Figure The onelo op corrections to the b oson mass

This symmetry prohibits the generation of a fermion mass by quantum correc

tions For m the fermion mass do es receive radiative corrections but all

f

p ossible diagrams have to contain a mass insertion as can b e seen from the one

lo op diagram shown in Fig Since the propagator of the b oson upp er dashed

while the propagator of the fermion lower solid line line in the diagram is

k

is one obtains a mass correction which is prop ortional to m

f

k

m

f

m Y m ln

f f

where is the ultraviolet cuto Hence the mass of a chiral fermion do es not

receive large radiative corrections if the bare mass is small For that reason

t Ho oft calls fermion masses natural an extra symmetry app ears when the

mass is set to zero which in turn leads to a protection of the fermion mass by an

approximate chiral symmetry

This state of aairs is dierent for scalar elds The diagrams giving the

onelo op corrections to m are shown in Fig Both diagrams are quadratically

b

divergent but they have an opp osite sign b ecause in the second diagram fermions

are running in the lo op One nds

m Y

b

Thus in nonsup ersymmetric theories scalar elds receive large mass corrections

even if the bare mass is set to zero and small scalar masses are unnatural

They can only b e arranged by delicately netuning the bare mass

and the couplings Y This problem b ecomes apparent in extensions of the

Standard Mo del which apart from the weak scale M do have a second larger

Z

scale say M with M M In such theories the mass of the scalar

GUT GUT Z

b oson is naturally of the order of the largest mass parameter in the theory This

discussion applies to the Higgs b oson of the Standard Mo del and it is dicult to

understand the smallness of M and how it can b e kept stable against quantum

Z

corrections whenever the Standard Mo del is the low energy limit of a theory with

a large mass scale

A concrete example of this problem o ccurs in Grand Unied Theories GUTs

where the Standard Mo del is embedded into a single simple gauge group G

GUT

eg G SU The GUT gauge symmetry is broken by a Higgs mechanism

GUT

to the gauge group of the Standard Mo del and one has the following pattern of

symmetry breaking

M M

GUT Z

G SU SU U SU U

GUT em

where M GeV and thus M M

GUT GUT Z

There are basically two dierent suggestions for solving this problem The

rst class of mo dels assume that the Higgs b oson of the Standard Mo del is not

an elementary scalar but rather a condensate of strongly interacting techni

fermions These theories are called technicolor theories but in all such

theories it is dicult to arrange agreement with the electroweak precision mea

surements of this decade The second class of mo dels are sup ersymmetric

theories where the Higgs b oson is elementary but the quadratic divergence in

exactly cancels due to the sup ersymmetric relation Y

The cancellation of quadratic divergences is a general feature of sup ersym

metric quantum eld theories and a consequence of a more general nonrenorm

alization theorem The sup erp otential W of a sup ersymmetric quantum eld

theory is not renormalized in p erturbation theory and all quantum correc

tions solely arise from the gauge coupling and wavefunction renormalization

This nonrenormalization theorem only holds in p erturbation theory but nonp erturbative

corrections do app ear

The nonrenormalization theorem or in other words the taming of the quantum

corrections is one of the attractive features of sup ersymmetric quantum eld the

ories It leads among other things to the p ossibility of stab elizing the weak

scale M

Z

In that sense sup ersymmetry solves the naturalness problem in that it allows

for a small and stable weak scale without netuning However sup ersymmetry

do es not solve the hierarchy problem in that it do es not explain why the weak

scale is small in the rst place

Soft Breaking of Sup ersymmetry

As we have seen in section mo dels with sp ontaneously broken sup ersymme

try are phenomenologically not acceptable For example the mass formula

generally valid in such cases forbids that all sup ersymmetric particles acquire

masses large enough to make them invisible in present exp eriments One way to

overcome those diculties is to allow explicit sup ersymmetry breaking

In the last section we observed that the absence of quadratic divergences in

sup ersymmetric theories stabilizes the Higgs mass and thus the weak scale This

attractive feature of sup ersymmetric eld theories can b e maintained in theories

with explicitly broken sup ersymmetry if the sup ersymmetry breaking terms are

of a particular form Such terms which break sup ersymmetry explicitely and

generate no quadratic divergences are called soft breaking terms

One p ossibility to identify the soft breaking terms is to investigate the diver

gence structure of the eective p otential Consider a quantum eld theory of

a scalar eld in the presence of an external source J The generating functional

for the Greens functions is given by

Z Z

iE J

e D exp i d xLx J xx

The eective action is dened by the Legendre transformation

cl

Z

E J d xJ x x

cl cl

E J

where can b e expanded in p owers of momentum in p osition

cl cl

J x

space this expansion takes the form

Z

m

d xV Z

cl ef f cl m cl cl cl

The term without derivatives is called the eective p otential V It can b e

ef f cl

calculated in a p erturbation theory ofh

V V h V

ef f cl cl cl

where V is the tree level and V the onelo op contribution In a

cl cl

theory with scalars fermions and vector b osons the onelo op contribution takes

the form

Z Z

X

J

V d k Str lnk M J Tr d k lnk M

J

J

where M is the matrix of second derivatives of Lj at zero momentum for

k

J

y

scalars J fermions J and vector b osons J The UV

divergences of can b e displayed by expanding the integrand in p owers of

large k This leads to

Z Z

d k d k

V Str ln k StrM k

If a UVcuto is introduced the rst term in is O ln Its co ecient

Str n n vanishes in theories with a sup ersymmetric sp ectrum of par

B F

ticles cf The second term in is O and determines the presence

of quadratic divergences at onelo op level Therefore quadratic divergences are

absent if

StrM

More precisely one can also tolerate StrM const since this would corresp ond

to a shift of the zero p oint energy which without coupling to gravity is undeter

mined In theories with exact or sp ontaneously broken sup ersymmetry is

z

fullled whenever the traceanomaly vanishes as we learned in

The soft sup ersymmetry breaking terms are dened as those nonsup ersym

metric terms that can b e added to a sup ersymmetric Lagrangian without sp oiling

StrM const One nds the following p ossibilities

Holomorphic terms of the scalars prop ortional to A A and the corre

x

sp onding complex conjugates

Mass terms for the scalars prop ortional to AA

They only contribute a constant eld indep endent piece in StrM

y 2

M is not necessarily evaluated at the minimum of V Rather it is a function of the

ef f

J

2

scalar elds in the theory The mass matrix is obtained from M by inserting the vacuum

J

exp ectation values of the scalar elds

z

Indeed theories with a nonvanishing Dterm have b een shown to pro duce a quadratic

divergence at onelo op

x

Higher p owers of A are forbidden since they generate quadratic divergences at the lo op

level

Gaugino mass terms

A generic mass matrix of the fermions takes the form

p

b b

D D W W i

ij ij

i i

p

M

a a

i D D m

ab

j j

where according to

p

b

W i D

ij

i

p

a

D i

j

is the sup ersymmetric part of M Computing the sup ertrace of

reveals that StrM const requires W D while m can b e

arbitrary

Thus the most general Lagrangian with softly broken sup ersymmetry takes

the form

L L L

susy soft

where L is of the form and

susy

i j i j i j k

L m A A b A A a A A A hc

soft ij ij k

ij

a b

m hc

ab

and b are mass matrices for the scalars a are trilinear couplings often m

ij ij k

ij

called Aterms and m is a mass matrix for the gauginos

ab

The next step will b e to investigate if the more general Lagrangian can b e

used to construct viable phenomenology Before we do so let us mention that there

is an alternative way to motivate the relevance of softly broken sup ersymmetric

theories Ultimately one has to couple the sup ersymmetric Standard Mo del to

gravity This requires the promotion of global sup ersymmetry to a lo cal symme

try that is the parameter of the sup ersymmetry transformation x is no

longer constant but dep ends on the spacetime co ordinates x This de

mands the presence of an additional massless fermionic gauge eld the gravitino

with spin and an inhomogeneous transformation law

m

m m

The necessity of this transformation law can b e seen for example from the

sup ersymmetry transformation of A which now has an extra contribution

m

A Together with the metric g and

m m m m mn

auxiliary elds b M M the gravitino forms the sup ergravity multiplet

m m

g b M M

mn m m

The p otential for the scalar elds is mo died in the presence of sup ergravity

and found to b e

i h

a a AA

D D W jW j D W D V A A e

i

i

where

W

i

D W A W

i

i

M A

P l

The limit corresp onds to turning o gravity and in this limit one ob

W W

a a

tains indeed V D D in accord with Lo cal sup ersymmetry

i

i

A

A

is sp ontaneously broken if D W j for some i This can b e achieved by in

i min

tro ducing a hidden sector which only couples via nonrenormalizable interactions

to the observable sector of the sup ersymmetric Standard Mo del and which has a

sup erp otential W suitably chosen to ensure D W j In this

hid min

case the gravitino b ecomes massive through a sup ersymmetric Higgs eect

In the limit with the gravitino mass m kept xed the Lagrangian

for the elds in the observable sector lo oks precisely like eqs

Thus the sp ontaneous breakdown of sup ergravity in a hidden sector manifests

itself as explicit but soft breakdown of global sup ersymmetry in the low energy

limit of the observable sector

Finally a variant of this mechanism is to break sup ersymmetry dynamically

ie nonp erturbatively in an additional gauge sector with some asymptotically

free gauge theory In this case the sup ersymmetry breaking is com

municated to the observable sector by renormalizable interactions but as in the

previous case the breaking app ears in the observable sector as explicit but soft

The Sup ersymmetric Standard Mo del with Softly Bro

ken Sup ersymmetry

In the previous section we recalled the most general Lagrangian of a softly broken

sup ersymmetric gauge theory in eqs and For L we continue to take

susy

together with the sup erp otential sp ecied in For L only gauge

soft

invariance and Rparity is imp osed This leads to the following p ossible soft

terms

J I J I J I

bh h hc e l a h d a h q u L a h q

u d e IJ d d IJ d soft u IJ u

R L R L R L

X X

j i

A A m m hc

a a

ij

a

all scalars

Obviously a huge number of new parameters is introduced via L The pa

soft

rameters of L are the Yukawa couplings Y and the parameter in the Higgs

susy

p otential The Yukawa couplings are determined exp erimentally already in the

nonsup ersymmetric Standard Mo del In the softly broken sup ersymmetric Stan

dard Mo del the parameter space is enlarged by

a a a b m m

u IJ d IJ e IJ

a

ij

Not all of these parameters can b e arbitrary but quite a number of them are

exp erimentally constrained Some of these constraints we will see in the following

sections

Within this much larger parameter space it is p ossible to overcome several of

the problems encountered in the sup ersymmetric Standard Mo del For example

the sup ersymmetric particles can now easily b e heavy due to the arbitrariness of

the mass terms m and therefore out of reach of present exp eriments Further

ij

more the Higgs p otential is changed and vacua with sp ontaneous electroweak

symmetry breaking can b e arranged

However the soft breaking terms introduce their own set of diculties For

generic values of the parameters the contribution to avorchanging neutral

currents is unacceptably large additional and forbidden sources of CP

violation o ccur and nally the absence of vacua which break the U

em

andor SU is no longer automatic It is b eyond the scop e of these lectures

to review all of these asp ects in detail Let us therefore fo cus on a few selected

topics and refer the reader to the literature for further details and discussions

Electroweak Symmetry Breaking

In section we noticed that for unbroken or sp ontaneously broken sup ersym

metry the electroweak symmetry remains intact in the sup ersymmetric version

of the Standard Mo del Let us now review the situation in the presence of soft

breaking terms The Higgs sector of the sup ersymmetric Standard Mo del

consists of two SU doublets

h h

d u

h h

d u

h h

d u

which carry eight real degrees of freedom four of them neutral and four charged

Like in the Standard Mo del SU U will b e broken to U by non

L Y em

vanishing VEVs of the neutral Higgs b osons h and h For that purp ose consider

u d

their p otential which can b e derived from eqs

g g

V h h m jh j m jh j bh h h h jh j jh j

u d u u d d u d u d u d

where

m m jj

u u

m m jj

d d

Notice that the co ecient of the jh j term is exactly as in determined

ud

by the gauge couplings and not changed by soft breaking terms This term is

p ositive so that the p otential is b ounded from b elow for large values of h h as

u d

long as jh j jh j To secure this b ound also in the direction jh j jh j one has

u d u d

to imp ose the following constraint on the parameter space

m m jbj

u d

Exercise Verify formula using eqs

Exercise Verify the condition

The existence of a minimum of V with broken gauge symmetry requires that

the Hessian of V h h j

h h

u d

u

d

b m

u

b m

d

has at least one negative eigenvalue Together with this implies

b m m

d u

So the soft terms have to satisfy in order to induce electroweak sym

metry breaking but in addition also the masses of the Z and W b osons have to

come out correctly These masses are given by

M v v v g g g g

Z d u

g v v g v M

u d W

where

p p

hh i v v sin hh i v v cos

u d

u d

The electroweak symmetry breaking is parameterized by the two Higgs vacuum

exp ectation values v v which can b e chosen real or equivalently v and As in

u d

2

This is the generalization of the condition in the Standard Mo del to a Higgs sector

with two Higgs doublets

the Standard Mo del v has to b e chosen such that it repro duces the exp erimentally

measured Z and W masses on the other hand is an additional parameter

which arises as a consequence of the enlarged doublet Higgs sector

The Higgs exp ectation values are directly related to the parameters of the

Higgs p otential via the minimization conditions

V g g

v v v m v bv

u u d

u d u

h

u

V g g

m v bv v v v

d u d

d u d

h

d

This in turn can b e used to derive a more physical relationship among the pa

rameters Using and the minimization conditions can b e rewritten

as

sin m m b

u d

tan m m jj M

d u Z

tan

Finally the full Higgs p otential including all eight real degrees of freedom can

b e used to compute the mass matrix of all Higgs b osons After a somewhat

lengthy calculation one nds that this mass matrix has three eigenvalues zero

corresp onding to the three Goldstone mo des eaten by the W and the Z The

remaining ve degrees of freedom yield the physical Higgs b osons of the mo del

H charged Higgs b oson pair

A CPo dd neutral Higgs b oson

H h CPeven neutral Higgs b osons

Their treelevel masses are given by

m m m

A u d

M m m

W A H

q

h i

m m M m M m M cos

h A Z A Z A Z

q

h i

m M m M m M cos m

A Z A Z A Z H

The Higgs masses ob ey physically interesting mass relations From we learn

m M m M m M

W Z Z

H H h

Exercise Derive the mass relations from

Physically the most interesting is the last inequality since it predicts the exis

tence of a light Higgs b oson This prediction can b e directly traced to the fact

the quartic couplings in the Higgs p otential are xed by the measured gauge

couplings and are not free parameters as in the Standard Mo del However radia

tive corrections for this lightest Higgs b oson mass can b e large and after taking

is pushed up to ab out into account quantum corrections the upp er b ound on m

h

GeV However the prediction of one light neutral Higgs b oson remains

and is one of the characteristic features of the sup ersymmetric twodoublet Higgs

sector It even holds in the limit that all masses of the sup ersymmetric particles

are sent to innity In this limit one recovers the nonsup ersymmetric Standard

Mo del alb eit with a light Higgs

Finally a prop er minimization of the scalar p otential requires to take into

account all scalar elds and not truncate to the neutral Higgs b osons It is

p ossible and do es o ccur in certain regions of the sup ersymmetric parameter

space that there exist minima which not only break the electroweak symmetry

but also SU andor U An extended analysis of this asp ect can b e found

em

for example in ref

WeakScale Sup ersymmetry

Let us briey come back to the hierarchy and naturalness problem In section

we learned that sup ersymmetry sheds no light on the hierarchy problem ie

the question why M is so much smaller than M However b ecause of the

Z P l

absence of quadratic divergences it do es solve the naturalness problem that is it

provides a stable hierarchy in the presence of a light Higgs b oson To only add

soft sup ersymmetry breaking terms into the sup ersymmetric Standard Mo del was

precisely motivated by this feature However from eq we learn an imp ortant

additional constraint on the soft breaking parameters In order to introduce no

new netuning all soft terms in eq should b e of the same order of magnitude

ie of order O M or at most in the TeV range If this were not the case the

Z

soft parameters would have to b e delicately tuned in order to add up to M This

Z

in turn implies the breaking of sup ersymmetry should o ccur at the weak scale

k

and that most likely all sup ersymmetric particles have masses in that range

k

This argument is somewhat imprecise since not only the soft breaking terms m m but

u d

also have to b e O M However is not related to sup ersymmetry breaking in any obvious

Z

way but rather a parameter in the sup erp otential Thus one needs a mechanism which also

explains the approximate equality of with m m This is known as the problem

u d

This line of argument is used to motivate what is called weakscale sup er

symmetry and indeed the current LEP II exp eriments actively lo ok for sup er

symmetric particles with masses slightly ab ove the weak scale

Further Constraints on the Sup ersymmetric Parame

ter Space

The exp erimental searches for sup ersymmetric particles imp ose additional con

straints on the sup ersymmetric parameter space First and foremost the direct

lower b ounds on the masses of the sup ersymmetric particles exclude certain

regions of the parameter space The translations of exp erimental b ounds into

the sup ersymmetric parameter space is complicated by the fact that the states

which are listed in table are interaction eigenstates but not necessarily mass

eigenstates The only exception are the gluinos G and the mass b ounds directly

translate into b ounds on m On the other hand the three Winos W W

the B and the four Higgsinos h h h combine into a fourvector of neutral

ud d u

Weyl fermions consisting of N B W h h and two pairs of charged Weyl

u d

fermions C W h with the following set of mass matri C W h

u

d

ces

a T a T

L G C M C G m NM N hc

fmass C N

where

p

g v m i

u

p

M

C

i g v

d

and

i i

g v g v m

u d

i i B C

m g v g v

B C

u d

M B C

N

i i

A

g v g v

u u

i i

g v g v

d d

Thus the physical mass eigenstates of M and M are parameter dep endent

C N

linear combinations of the corresp onding interaction eigenstates and they are

termed charginos and neutralinos resp ectively

Exercise Derive the mass matrices and

The other sup ersymmetric particles are the spin partners of the quarks and

I I I I I

leptons the squarks q u d and the sleptons l e Their mass eigenstates are

L R R L R

derived from the following three and one mass matrices

y y y

UM U DM D EM E M

U D E

where

I

I I

I I I

D d d U u u E e e

R L L R R L

Explicit forms of these mass matrices in terms of the soft parameters can b e found

eg in ref Constraints on the soft parameters imp osed by the exp erimental

b ounds can b e found in

The Minimal Sup ersymmetric Standard

Mo del MSSM

The sup ersymmetric version of the Standard Mo del we discussed so far has a

huge parameter space and therefore very limited predictive p ower A much more

constrained version with less free parameters b ecame known as the Minimal

Sup ersymmetric Standard Mo del MSSM which is the topic of this section The

MSSM was motivated by the success of Grand Unied Theories combined with

a simple avor blind mechanism of sup ersymmetry breaking in a hidden sec

tor Over the last years this mo del went through a few alterations but

to day it is a well dened mo del with a very particular set of soft sup ersymme

try breaking terms which are avor blind and in some sense the minimal choice

of free parameters One imp oses that al l scalar masses are the

all gaugino masses are the same m m m m all m same m

ij

ij

aparameters are prop ortional to the Yukawa couplings with the same universal

prop ortionality constant a and nally that the bparameter is of a sp ecic form

Altogether one has

m m m m m m b b m

ij

ij

a a Y a a Y a a Y

u IJ u IJ d IJ d IJ l IJ l IJ

Thus the parameter space of the MSSM is spanned by the parameters

m m a b

which are sub ject to one further nontrivial constraint which ensures the

observed electroweak symmetry breaking

Of course the relations are meant to b e tree level relations and they do

enjoy quantum corrections which are governed by the appropriate renormalization

group equations The quantum corrections destroy the universality of the

soft parameters but the deviation from universality is small and so far in accord

with all measurements In particular the smallness of avor

changing neutral currents is naturally explained in the MSSM Thus

even without an underlying GUT theory this set of parameters seems to have

some phenomenological attraction

However one should also stress that sup ersymmetric GUTs are an extremely

viable p ossibility to day Among other things the LEP precision exp eriments

determined the gauge coupling constant g very precisely In light of these mea

surements one can ask to what extend the three gauge couplings g g g do

unify at some high energy scale M given the exp erimental input at M At

GUT Z

onelo op order the energy dep endence of the gauge couplings is given by

b M

a GUT

g M g M ln

Z GUT

a a

M

Z

where b are the co ecients of the onelo op b etafunction which dep end on

a

the massless sp ectrum of the theory Let us rst recall that the index T R

a

a b ab b

of a representation R is dened as Tr T T T R where T are the

R

a a

generators of the gauge group In terms of the indices b is given by

a

b T G T R n T R n

WF RS

a a a a

where G denotes the adjoint representation and n n counts the number

WF RS

of Weyl fermions real scalars in the representation R

For the nonsup ersymmetric Standard Mo del one nds

b b b

which do es not lead to a unication of coupling constants at any scale That is

one cannot nd an M where g M g M g M holds

GUT GUT GUT GUT

However in the sup ersymmetric Standard Mo del one nds

b b b

which do es lead to a unication of couplings at M GeV This can

GU T

b e taken as a strong hint for a sup ersymmetric GUT

Let us close this section with a discussion of electroweak symmetry breaking

in the MSSM At the tree level one now has m m and as a consequence the

d u

conditions for electroweak symmetry breaking and cannot b e satised

simultaneously

jbj m m m

d u

m m m jbj

u d

However quantum corrections alter this situation and naturally induce elec

troweak symmetry breaking Thus the MSSM naturally displays a sup er

symmetric version of the ColemanWeinberg mechanism where quantum cor

rections generate a nontrivial minimum in the Higgs p otential The electroweak

symmetry breaking scale is not put in hand but related to the scale of the su

p ersymmetry breakdown

Summary

Sup ersymmetry is a generalization of the spacetime symmetries of quantum eld

theories and it transforms fermions into b osons and vice versa In the sup ersym

metric Standard Mo del all particles of the Standard Mo del are accompanied by

sup erpartners with opp osite statistics Moreover it is necessary to enlarge the

Higgs sector and add a second Higgs doublet to the sp ectrum Sup ersymmetry

cannot b e an exact symmetry of nature and has to b e realized in its broken phase

if at all However sp ontaneously broken global sup ersymmetry is phenomeno

logically ruled out Sp ontaneously broken lo cal sup ersymmetry on the other

hand leads to mo dels which are still in agreement with all present observations

These mo dels do provide a solution to the naturalness problem as long as the

sup ersymmetric partners have masses not much bigger than T eV The sup er

symmetric Standard Mo del has one solid prediction a light neutral Higgs with

a mass smaller than GeV as well as additional charged but not necessarily

light Higgs b osons The gauge couplings of the sup ersymmetric Standard Mo del

do unify at a scale M GeV which may indicate a sup ersymmetric

GU T

GUT The MSSM motivated by GUTs contains only four new parameters

and is consistent with observations in large regions of this parameter space In

this mo del the electroweak symmetry breaking is driven by radiative corrections

realizing a sup ersymmetric version of the ColemanWeinberg mechanism Finally

the concept of sup ersymmetry also arises naturally in string theories which might

b e another hint towards its realization in nature

Exercise How can supersymmetry be veried or falsied Distinguish between

necessary and sucient conditions

App endixConventions and Notation

In these lectures the notation and conventions of ref are used The four

dimensional Lorentz metric is chosen as

diag

mn

Lorentz indices are lab eled by Latin indices m n which run from to Greek

indices are used to denote spinors A twocomponent Weyl spinor can transform

under the or the complex conjugate representation of the Lorentz

group and dotted or undotted indices are used to distinguish b etween these rep

resentations denotes a spinor transforming under the representation

representation of the Lorentz group The while transforms under the

spinor indices and can take the values and These indices can b e raised

and lowered using the skewsymmetric SU invariant tensor or

where

For dotted indices the analogous equations hold The pro duct

is a Lorentz scalar Spinors are anticommuting ob jects and one has the following

summation convention

The convention for the conjugate spinors are chosen such that it is consistent

with the conjugation of scalars

y y

m

The matrices are given by

i

i

The invariant tensor raises and lowers their indices

m m

and we have

The generators of the Lorentz group in the spinor representation are given by

n m m n nm n m m n nm

The Dirac matrices can b e written in terms of Weyl matrices

m

m

m

which fulll

m n mn

f g

A four comp onent Dirac spinor contains two Weyl spinors

D

Its conjugate is

y

D

D

The Dirac equation describing relativistic spin particles reads

n

i m

n D

It can b e decomp osed into two Weyl equations

n

i m

n

n

i m

n

Exercise Show the validity of the fol lowing identities

m m n n

Exercise Compute in terms of matrices the following matrices

P i P

R L

Exercise Express the fol lowing Lorentzinvariants in terms of Weylspinors

m m m n

D D D D D D D D D D

Acknowledgement We would like to thank the students and organizers of

the Saalburg Summers School for generating an inspiring scientic atmo

sphere Sp ecial thanks go to H G unther M Haack and M Klein for carefully

reading the manuscript and P Zerwas for guiding us through the CERN home

page JL also thanks the students of the University of Munich for their questions

and comments during a course on sup ersymmetry which was the seed of these

lecture notes

References

C Caso et al Particle Data Group Eur Phys J C

For a recent review see for example J Erler and P Langacker hep

ph and references therein

L Susskind Phys Rev D

S Weinberg Phys Rev D

S Dimop oulos and L Susskind Nucl Phys B

E Fahri and L Susskind Phys Rep C

IA DSouza and CS Kalman Preons World Scientic Singap ore

H Georgi and S Glashow Phys Rev Lett

G Ross Grand Unied Theories BenjaminCummings

J Wess and B Zumino Nucl Phys B

J Wess and J Bagger Supersymmetry and Supergravity Princeton Univer

sity Press

MB Green JS Schwarz and E Witten Superstring Theory Cambridge

University Press

D L ust and S Theisen Lectures on String Theory Springer Verlag

J Polchinski String Theory Cambridge University Press

HP Nilles Phys Rep

HE Hab er and GL Kane Phys Rep

MF Sohnius Phys Rep

HE Hab er in Recent Directions in Particle Theory From Superstrings and

Blach Holes to the Standard Model eds J Harvey and J Polchinski World

Scientic Singap ore

F Zwirner Trieste Pro ceedings High Energy Physics and Cosmology

vol

W Hollik R R uckl and J Wess eds Phenomenological Aspects of Super

symmetry Springer Lecture Notes

J Bagger in Boulder QCD and beyond hepph

M Dine hepph

JF Gunion hepph

M Peskin hepph

S Martin hepph

S Ferrara Supersymmetry World Scientic Singap ore

R Haag J Lopuszanski and M Sohnius Nucl Phys B

S Coleman and J Mandula Phys Rev

S Weinberg Phys Rev D

N Sakai and T Yanagida Nucl Phys B

P Fayet Nucl Phys B

GR Farrar and P Fayet Phys Lett B

For a recent review see for example H Dreiner hepph and ref

erences therein

P Fayet and J Iliop oulos Phys Lett B

L ORaifeartaigh Nucl Phys B

S Ferrara L Girardello and F Palumbo Phys Rev D

G t Ho oft in Recent Developments in Gauge Theories eds G t Ho oft et

al Penum New York

E Gildner Phys Rev D

MT Grisaru W Siegel and M Ro cek Nucl Phys B

I Aeck M Dine and N Seib erg Phys Rev Lett Nucl

Phys B

L Girardello and MT Grisaru Nucl Phys B

S Coleman and E Weinberg Phys Rev D

S Weinberg Phys Rev D

J Iliop oulos C Itzykson and A Martin Rev Mod Phys

W Fischler HP Nilles J Polchinski S Raby and L Susskind Phys Rev

Lett

For a review see P van Nieuwenhuizen Phys Rep

E Cremmer B Julia J Scherk S Ferrara L Girardello and P van

Nieuwenhuizen Nucl Phys B

E Cremmer S Ferrara L Girardello and A Van Proyen Nucl Phys B

E Witten and J Bagger Phys Lett B

R Barbieri S Ferrara and C Savoy Phys Lett B

L Hall J Lykken and S Weinberg Phys Rev D

SK Soni and H A Weldon Phys Lett B

GF Giudice and A Masiero Phys Lett B

M Dine A Nelson and Y Shirman Phys Rev D

S Dimop oulos G Dvali G Giudice and R Rattazzi Nucl Phys B

For a recent review on dynamical sup ersymmetry breaking see for example

E Poppitz and S Trivedi hepth and references therein

J Ellis and DV Nanop oulos Phys Lett B

R Barbieri and R Gatto Phys Lett B

J Hagelin S Kelley and T Kanaka Nucl Phys B

For a review see for example

A Masiero and L Silvestrini hepph M Misiak S Pokorski and

J Rosiek hepph and references therein

J Ellis S Ferrara and DV Nanop oulos Phys Lett B

W Buchmullerand D Wyler Phys Lett B

J Polchinski and MB Wise Phys Lett B

For a review see for example

Y Grossman Y Nir and R Rattazzi hepph

A Masiero and L Silvestrini hepph and references therein

See for example

JP Derendinger and C Savoy Nucl Phys B

G Gamberini G Ridol and F Zwirner Nucl Phys B

JA Casas A Lleyda and C Munoz Nucl Phys B

and references therein

J F Gunion H E Hab er G L Kane and S Dawson The Higgs Hunters

Guide Univ of California June

HE Hab er and R Hemping Phys Rev Lett

J Ellis G Ridol and F Zwirner Phys Lett B

M Carena JR Espinosa M Quiros and CEM Wagner Phys Lett

PH Chankowski and S Pokorski hepph to app ear in Perspectives

on Higgs Physics II ed GL Kane World Scientic Singap ore

HE Hab er hepph to app ear in Perspectives on Higgs Physics II

ed GL Kane World Scientic Singap ore

JR Espinosa in Surveys High Energ Phys hep

ph

E Witten Nucl Phys B

M Veltman Acta Phys Pol B

J Ellis K Enquist DV Nanop oulos and F Zwirner Mod Phys Lett A

R Barbieri and G Giudice Nucl Phys B

J Kim and HP Nilles Phys Lett B

The latest constraints can often b e found on the CERN homepage

httpwwwcernchCERNExperimentshtml at the four exp eriments

ALEPH DELPHI L OPAL

S Dimop oulos and H Georgi Nucl Phys B

L AlvarezGaume J Polchinski and M Wise Nucl Phys B

L Ibanez and C Lop ez Nucl Phys B

L Ibanez C Lop ez and C Munoz Nucl Phys B

SP Martin and MT Vaughn Phys Rev D

for a review see for example L Ibanez and GG Ross in Perspectives on

Higgs Physics

PH Chankowski and S Pokorski Phys Lett B hep

ph

S Bertolini F Borzumati and A Masiero Nucl Phys B

P Langacker and M Luo Phys Rev D

J Ellis S Kelley and DV Nanop oulos Phys Lett B

U Amaldi W de Bo er and H F urstenau Phys Lett B

P Langacker and N Polonsky Phys Rev D D

L Ibanezand G Ross Phys Lett B