Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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for the gauge field and six associated scalars 

(all in

the adjoint representations of the gauge group G)is

N¼4

¼

xtr

 F



þ2



I<J

2½

;







32

xtr F









þfermionic terms ½11

Consider for simplicity the case G =SU(2). The

scalar potential has flat directions along which the

six 

commute. By an SO(6) R-symmetry rotation,

we can set all but one of them to zero, and let

<tr(



)> = v

in the vacuum. In this ‘‘Coulomb

phase’’ of the theory, a U(1) gauge multiplet stays

massless, while the charged states become massive

by the Higgs effect. The theory admits furthermore

smooth magnetic-monopole and dyon solutions, and

there is an elegant formula for their mass:

M ¼ vjn

þ n

j; where  ¼



2

4i

½12

and n

) denotes the quantized electric (mag-

netic) charge. This is a BPS formula that receives

no quantum corrections. It exhibits the SL(2, Z)

covariance of the theory,

 !

a þ b

c þ d

and

ðn

; n

Þ!ðn

; n



1

½13

Here a, b, c, d are integers subject to the condition

ad  bc = 1. Of special importance is the transfor-

mation  !1=, which exchanges electric and

magnetic charges and (at least for  = 0) the strong-

with the weak-coupling regimes. For more details

see the review by Harvey (1996).

The extension of these ideas to string theory can be

illustrated with the strong/weak- coupling duality

between the type I theory, and the Spin(32)=Z

heterotic string. Both have the same massless spec-

trum and low-energy action, whose form is dictated

entirely by supersymmetry. The only difference lies in

the relations between the string and supergravity

parameters. Eliminating the latter, one finds



het

2

and 

het

ﬃﬃﬃ





½14

It is thus tempting to conjecture that the strongly

coupled type I theory has a dual description as a

weakly coupled heterotic string. These are, indeed,

the only known ultraviolet completions of the

theory [7]. Furthermore, for 

 1, the D1 brane

of the type I theory becomes light, and could be

plausibly identified with the heterotic string. This

conjecture has been tested successfully by comparing

various supersymmetry-protected quantities (such as

the tensions of BPS excitations and special higher-

derivative terms in the effective action), which can be

calculated exactly either semiclassically, or at a given

order in the perturbative expansion. Testing the duality

for nonprotected quantities is a hard and important

problem, which looks currently out of reach.

The other three string theories have also well-

motivated dual descriptions at strong coupling .

The type IIB theory is believed to have an SL(2, Z)

symmetry, similar to that of the N = 4 super Yang–

Mills. (Note that  is a dynamical parameter, that

changes with the vacuum expectation value of the

dilaton <>. Thus, dualities are discrete gauge

symmetries of string theory.) The type IIA theory

has a more surprising strong-coupling limit: it grows

one extra dimension (of radius R

= 1=

ﬃﬃﬃﬃﬃ



), and

can be approximated at low energy by the maximal

11-dimensional supergravity of Cremmer, Julia, and

Scherk. The latter is a very economical theory – its

massless bosonic fields are only the graviton and a

3-form potential A

. The bosonic part of the action

reads

11D

2

ﬃﬃﬃﬃﬃﬃﬃﬃ

G

ðR 



12

^ F

½15

The electric and magnetic charges of the 3-form are a

(fundamental?) membrane and a solitonic 5-brane.

Standard Kaluza–Klein reduction on a circle maps S

11D

to the IIA supergravity action [6],whereG



, ,andC

descend from the 11-dimensional graviton, and B

and

from the 3-form A

.Furthermore,allBPSexcita-

tions of the type IIA string theory have a counterpart in

11 dimensions, as summarized in Table 1. Finally, if

one compactifies the eleventh dimension on an interval

(rather than a circle), one finds the conjectured strong-

coupling limit of the E

 E

heterotic string.

The web of duality relations can be extended by

compactifying further to D  9 dimensions. Readers

interested in more details should consult Polchinski

(1998) or one of the many existing reviews of the

subject(Townsend(1996),seealso‘‘FurtherRead-

ing’’section).Inninedimensions,inparticular,the

two type II theories, as well as the two heterotic

superstrings, are pairwise T-dual. T-duality is a

perturbative symmetry (thus firmly established, not

138 Superstring Theories

only conjectured) which exchanges momentum and

winding modes. Putting together all the links one

arrives at the fully connected web of Figure 4. This

makes the point that all five consistent superstrings,

and also 11-dimensional supergravity, are limits of a

unique underlying structure called M theory. (For

lack of a better definition, ‘‘M’’ is sometimes also

used to denote the D = 11 supergravity plus

supermembranes, as in Figure 4.) A background-

independent definition of M theory has remained

elusive. Attempts to define it as a matrix model of

D0 branes, or by quantizing a fundamental mem-

brane, proved interesting but incomplete. A diffi-

culty stems from the fact that in a generic

background, or in D = 11 Minkowski spacetime,

there is only a dimensionful parameter fixing the

scale at which the theory becomes strongly coupled.

Other Developments and Outlook

We have not discussed in this brief review some

important developments covered in other contribu-

tions to the encyclopedia. For the reader’s conve-

nience, and for completeness, we enumerate (some

of) them giving the appropriate cross-references:

Compactification. To make contact with the

standard model of particle physics, one has to

compactify string theory on a six-dimensional

manifold. There is an embarassment of riches,

but no completely realistic vacuum and, more

significantly, no guiding dynamical principle to

help us decide (see Compactification of Superstring

Theory). The controlled (and phenomenologically

required) breaking of spacetime supersymmetry is

also a problem.

Conformal field theory and quantum geometry.

The algebraic tools of 2D conformal field theory,

both bulk and boundary (see Two-Dimensional

Conformal Field Theory and Vertex Operator

Algebras), play an important role in string theory.

They allow, in certain cases, a resummation of 

effects, thereby probing the regime where classical

geometric notions do not apply.

Microscopic models of black holes. Charged extre-

mal black holes can be modeled in string theory by BPS

configurations of D-branes. This has led to the first

microscopic derivation of the Bekenstein–Hawking

entropy formula, a result expected from any consistent

theory of quantum gravity. As with the tests of duality,

the extension of these results to neutral black holes is a

difficult open problem – see Branes and Black Hole

Statistical Mechanics.

AdS/CFT and holography. A new type of (holo-

graphic) duality is the one that relates supersym-

metric gauge theories in four dimensions to string

theory in asymptotically anti-de Sitter spacetimes.

The sharpest and best-tested version of this duality

relates N = 4 super Yang–Mills to string theory in

AdS

 S

. Solving the -model in this latter back-

ground is one of the keys to further progress in the

subject (see AdS/CFT Correspondence).

String phenomenology. Finding an experimental

confirmation of string theory is clearly one of the most

pressing outstanding questions. There exist several

interesting possibilities for this – cosmic strings, large

extra dimensions, modifications of gravity, primordial

cosmology (see String Theory: Phenomenology for a

Table 1 BPS excitations of type IIA string theory, and their counterparts in M theory compactified on a circle of radius R

Tension Type IIA M on S

Tension

(

ﬃﬃﬃ



=

)(2

ﬃﬃﬃﬃﬃ



)

D0 brane K–K excitation 1=R

= (2

)

1

String Wrapped membrane 2R

2

=



1=3

(

ﬃﬃﬃ



=

)(2

ﬃﬃﬃﬃﬃ



) D2 brane Membrane T

= 2

=



1=3

(

ﬃﬃﬃ



=

)(2

ﬃﬃﬃﬃﬃ



)

1

D4 brane Wrapped 5-brane R

2

=



2=3

(=

)(2

) NS-5-brane 5-brane 1=2ðÞ2

=



2=3

(

ﬃﬃﬃ



=

)(2

ﬃﬃﬃﬃﬃ



)

3

D6 brane K–K monopole 2

=

From Bachas CP (1997) Lectures on D-branes. In: Olive DI and West PC (eds.) Duality and Supersymmetric Theories, Proceedings,

Easter School, Newton Institute, Euroconference, Cambridge, UK, April 7–18. With permission of Cambridge University Press.

II A

II B

HE HO

Strong/weak Strong/weak

Figure 4 Web of dualities in nine dimensions. From Bachas CP

(1997) Lectures on D-branes. In: Olive DI and West PC (eds.)

Duality and Supersymmetric Theories, Proceedings, Easter School,

Newton Institute, Euroconference, Cambridge, UK, April 7–18. With

permission of Cambridge University Press.

Superstring Theories 139

review). Here we point out the one supporting piece

of experimental evidence: the unification of the

gauge couplings of the (supersymmetric, minimal)

standard model at a scale close to, but below the

Planck scale, as illustrated in Figure 5.Thisisa

generic ‘‘prediction’’ of string theory, especially in its

heterotic version.

See also: AdS/CFT Correspondence; Boundary

Conformal Field Theory; Brane Construction of Gauge

Theories; Brane Worlds; Branes and Black Hole

Statistical Mechanics; Compactification of Superstring

Theory; Derived Categories; Electroweak Theory;

Gauge Theories from Strings; Noncommutative

Geometry from Strings; Supermanifolds; String Field

Theory; String Theory: Phenomenology; Supergravity;

Two-Dimensional Conformal Field Theory and Vertex

Operator Algebras; Wheeler–DeWitt Theory.

Further Reading

Aharony O, Gubser SS, Maldacena JM, Ooguri H, and Oz Y

(2000) Large N field theories, string theory and gravity.

Physics Report 323: 183 (arXiv:hep-th/9905111).

Angelantonj C and Sagnotti A (2002) Open strings. Physics

Report 371: 1.

Angelantonj C and Sagnotti A (2003) Open strings – erratum.

Physics Report 376: 339 (arXiv:hep-th/0204089).

Bachas CP (1997) Lectures on D-branes. In: Olive DI and West

PC (eds.) Duality and Supersymmetric Theories, Proceedings,

Easter School, Newton Institute, Euroconference, Cambridge,

UK, April 7–18, arXiv:hep-th/9806199.

Berkovits N (2002) ICTP lectures on covariant quantization of the

superstring, arXiv:hep-th/0209059.

D’Hoker E and Phong DH (2002) Lectures on two-loop super-

strings, arXiv:hep-th/0211111.

Green MB, Schwarz JH, and Witten E (1987) Superstring Theory.

Vol. 1: Introduction; Vol. 2: Loop Amplitudes, Anomalies and

Phenomenology. Cambridge: Cambridge University Press.

Harvey JA (1996) Magnetic monopoles, duality, and super-

symmetry. arXiv:hep-th/9603086.

Kiritsis E (2004) D-branes in standard model building, gravity

and cosmology. Fortschritte der Physik 52: 200 (arXiv:hep-th/

0310001).

Lust D (2004) Intersecting brane worlds: A path to the standard

model? Classical and Quantum Gravity 21: S1399 (arXiv:hep-

th/0401156).

Obers NA and Pioline B (1999) U-duality and M-theory. Physics

Report 318: 113 (arXiv:hep-th/9809039).

Polchinski J (1998) String Theory. Vol. 1: An Introduction to the

Bosonic String; Vol. 2: Superstring Theory and Beyond.

Cambridge: Cambridge University Press.

Salam A and Sezgin E (1989) Supergravities in Diverse Dimen-

sions. Amsterdam: North-Holland/World Scientific.

Sen A (1997) An introduction to non-perturbative string theory,

arXiv:hep-th/9802051.

The following URL of the HEP-SPIRES database: http://www.slac.-

stanford.edu collects many popular string theory reviews.

Townsend PK (1996) Four lectures on M-theory, arXiv:hep-th/

9612121.

Zwiebach B (2004) A First Course in String Theory. Cambridge:

Cambridge University Press.

Supersymmetric Particle Models

S Pokorski, Warsaw University, Warsaw, Poland

Introduction

Supersymmetric quantum field theories (see Super-

gravity) are characterized by the existence of one

(N = 1 supersymmetry) or several (N > 1 extended

supersymmetry) conserved Noether-like charges

A = 1, ..., N, which establish symmetry links

between particle states of different spin. Super-

symmetry ensures equal numbers of bosonic and

fermionic particle states. If it is exact, bosons and

fermions related by supersymmetry transformations

have equal masses. Moreover, supersymmetry

imposes stringent relations between interactions

which involve particles of different spin. This gives

rise to a special ultraviolet behavior of supersym-

metric theories. Their ultraviolet divergences are

much softer than in nonsupersymmetric theories. In

particular, N = 4 supersymmetric quantum field

theories are finite and for any N they are free from

quadratic divergences plaguing ordinary theories

with elementary scalars. N > 4 supersymmetric

theories necessarily involve particles of spin higher

than 1 and are not renormalizable. Supersymmetry

promoted to a local symmetry includes gravity.

Only N = 1 supersymmetric theories allow for

chiral fermions which are the fundamental objects in

elementary particle interactions (see Standard Model

of Particle Physics). This is because parity and

log E (GeV)

317

Figure 5 The unification of couplings.

140 Supersymmetric Particle Models

charge conjugation symmetries are violated in weak

interactions. Therefore, N > 1 theories may not be

of immediate phenomenological relevance. How-

ever, they may be useful for constructing super-

symmetric theories in more than four dimensions

(more than three spatial dimensions). Chiral (effec-

tive) theory in four dimensions can be then obtained

after compactification of extra dimensions. For

instance, N = 2 theory in five dimensions (x



, y)

compactified on a circle with reflection symmetry

y !y (orbifold compactification) gives chiral

N = 1 theory in four dimensions.

Absence of quadratic divergences in supersym-

metric theories is the main argument supporting the

belief that fundamental interactions of elementary

particles at energies not higher that O(1 TeV) should

be described by an (approximately) N = 1 super-

symmetric extension of the standard model (SM).

Indeed, supersymmetric models elegantly solve the

so-called hierarchy problem of the SM. At present,

supersymmetry remains a theoretical hypothesis.

No experimental evidence for it has been found yet

(for experimental lower bounds on the masses of

supersymmetric particles see Eidelman et al. (2004)).

Supersymmetric models will be tested experimentally

at the Large Linear Collider at CERN (Geneva), after

the completion of its construction in 2007. Super-

gravity theories may be physically relevant as an

intermediate step in constructing phenomenologically

viable models from superstring theories.

The essence of the hierarchy problem of the

standard model (SM) – the successful SU(3)



SU(2)

U( 1)

gauge theory of interactions of

quarks and leptons at energies up to about 100 GeV –

is the following. By itself, the SM does not explain the

value of the Fermi scale v of the electroweak

SU(2)

U( 1)

symmetry breaking (v G

1=2

where

is the Fermi constant determined by the life time

of the muon). Indeed, in the SM, the electroweak

symmetry breaking is realized by an elementary Higgs

field H (an SU(2) doublet) with a potential

V ¼m

H þ



ðH

HÞ

½1

where m and  are free parameters of the SM. When

< 0 is chosen, the minimum of the potential

occurs when

Hi¼





½2

that is, the Higgs doublet acquires SU(2) U(1)

breaking vacuum expectation value v which is just

the Fermi scale. The masses of the intermediate

vector bosons W



and Z

are proportional to v and

depend also on the gauge couplings. Within the SM

understood as a theory with the momentum cut-off



, quantum corrections to the mass parameter m

in eqn [1] are quadratically divergent :

m

64

þ g

þ   8y





þ ½3

Here, g

, g

, and y

are the gauge couplings of the

groups U(1)

, SU(2)

, and the top-quark Yukawa

coupling, respectively. This means that if, above the

energy scale 

, the SM is replaced by some more

fundamental theory, in which there are particles of

masses M & 

, the quantum corrections to m

are

quadratically dependent on the new mass scale M.

For M v, this is very unnatural even if the original

parameter m

remains a free parameter of this

underlying theory and particul arly difficult to accept

if in the underlying theory m

is fixed by some more

fundamental considerations. If the SM was the

correct theory up to, for example, the mass scale

suggested by the see-saw mechanism for the neu-

trino masses, 

10

GeV

jm

j10

GeV

10

Clearly, this excludes the possibility of understand-

ing the magnitude of the Fermi scale v in any

sensible way. Thus, for naturalness of the Higgs

mechanism in the SM there should exist a new mass

scale M & v, say only one order of magnitude higher

than v and the theory describing the physics above

that scale should be free of quadratic divergences.

(Approximate) supersymmetry is at present the most

elegant a nd theoretically most complete solution to

the hierarchy problem of the SM.

Supersymmetric Extensions of the SM

In supersymmetry, the gauge fields A



are promoted

to vector superfields

= (A



, 

, D

), one for each

gauge symmetry group generator, where 

’s are

Weyl fermions (called gauginos) and D

’s are

nondynamical auxiliary fields. A renormalizable

supersymmetric gauge theory is completely defined

(see, e.g., Sohnius (1985) and Wess and Bagger

(1992)) by specifying the gauge group, the set of

chiral supermultiplets



= (

, F

) representing

matter fields, and the superpotential – a holo-

morphic polynomial function of at most third

order in the chiral superfields which determines

Yukawa couplings of the fermions

and scalars 

Auxiliary fields D

and F

can be eliminated via their

(algebraic) equations of motion.

The so-called minimal supersymmetric SM

(MSSM) encodes the main features of any super-

symmetric extension of the SM. Its gauge group is

Supersymmetric Particle Models 141

SU(3) SU(2) U(1) – the same as in the SM – and

the chiral superfie lds are associated to each of the

SM quark and lepton fields. Thus, quarks and

leptons get scalar spin zero superpartners, the

squarks and sleptons, carrying the same quantum

numbers as their corresponding fermions and the

vector superfields provide spin 1/2 superpartners for

the gauge fields – the gluinos, the winos, and the

bino. The SM Higgs doublet with weak hypercharge

Y = 1=2 becomes a scalar component of a chiral

superfield

which contains in addition one

doublet of Weyl fermions – the Higgsinos. The

chiral anomaly cancelation condition requires that

there be a lso a second Higgs chiral superfield

with Y = 1 =2. Such a superfield is also required for

giving masses to all flavors of quarks; because of the

holomorphicity of the superpotential the same Higgs

doublet cannot couple simultaneously to all quarks.

With the MSSM superfield content, the most

general renormalizable superpotential consistent

with the gauge symmetry has the form

W ¼ Y

þY

þ

þ

L þ

þ

½4

(flavor indices are suppressed) where the superfield

contains the SU(2) quark doublet Q and its scalar

superpartner

Q and similarly for the lepton doublet

L, quark singlets

D, and lepton singlet

E super-

fields. The three first terms in [4] give the SM-like

Yukawa couplings of quarks and leptons to the Higgs

fields together with Yukawa coupling s of the corre-

sponding superpar tners. The fourth term has no SM

analogy; it gives supersymmetric masses to the Higgs

scalar and Higgsinos. The interactions in the second

line do not conserve baryon and lepton numbers,

respectively B and L, and should be forbidden (or

strongly suppressed) by some additional symmetry of

the theory as they would lead to rapid proton decay. A

discrete symmetry, called R-parity R= (1)

2Sþ3(BL)

where S is the spin of the field, is an interesting

possibility. R-parity acts differently on the different

components of the superfields: it is even for all SM

particles and odd for their superpartners. Its conserva-

tion implies that superpartners must appear in pairs in

any interaction vertex. Thus, with R-parity imposed,

the lightest supersymmetric particle is stable and it is an

excellent candidate for the dark matter in the universe.

Supersymmetry cannot be an exact symmetry of

nature because there do not exist elementary fermions

and bosons degenerate in mass. The superpotential

[4] does not break supersymmetry spontaneously but

even if it did the elementary fermions and bos ons

would on average have equal masses (they would

satisfy some mass sum rule) which is also

contradicted by the experimental data. Therefore, in

the MSSM, supersymmetry has to be broken expli-

citly but in such a way that the soft ultraviolet

behavior remains intact. Remarkably, the super-

symmetry breaking terms which can be added to the

MSSM Lagrangian without reintrodu cing quadratic

divergences make heavy just those fields which are

opposite statistics superpartners of the SM gauge

bosons and fermions. These so-called soft terms are:

soft

¼



m

ðH

þc:c:Þ

þA

½5

and yield gaugino (gluino

G, wino

W, and bino

and scalar mass terms as well as explicit trilinear

couplings between scalars (scalar mass terms and

A-terms are 3 3 matrices in the flavor space). As a

result, supersymmetry is broken in the mass spectra

but not in the dimensionless couplings.

The origin of the soft supersymmetry breaking

remains an open issue. Terms [5] are most probably

remnants of the spontaneous supersymmetry break-

ing in the so-called ‘‘hidden’’ sector – a hypothetical

set of fields that do not interact directly with the

MSSM fields. For example, in the popular scenario,

they interact with the MSSM fields only gravitation-

ally and spontaneous supersymmetry breaking in the

hidden sector is communicated to the MSSM sector

by gravitational interactions giving rise to terms [5].

Several other mechanisms of supersymmetry break-

ing transmission have also been proposed (gauge

mediation, anomaly mediation, etc.).

The mass parameters and A-terms in [5] are free

parameters of the low-energy supersymmetric theory

and, combined with the interactions like Q

originating from supersymmetric kinetic terms, may

be a new, troublesome, source of flavor changing

neutral currents and of CP violation.

Higgs Sector of the MSSM

The MSSM Higgs potential reads

V ¼m

þ m

ðH

þ c:c:Þ

þ g

jH



½6

Its quartic part is uniquely determined by the

structure of the supersymmetric gauge theory. The

parameters m

, m

,andm

are determined by

142 Supersymmetric Particle Models

the soft supersymmetry breaking Higgs boson

masses [5] and the  pa rameter in [4]. The potential

[6] is bounded from below for m

þ m

> 2m

, and

for m

 m

< 0 it has the electroweak symmetry

breaking minimum at v

= hH

i 6¼ 0, v

= hH

i 6¼ 0.

The ratio v

 tan  is then phen omenologically

a very important parameter.

Quantum corrections to the mass parameters in

[6] are controlled by the mass scale M

soft

of the

supersymmetry breaking terms [5]; at the one-loop

level instead of [3], one finds

m

1;2



16

þ g

 12y

b;t



soft



NEW

soft

½7

where y

and y

are the bottom- and top-quark

Yukawa couplings, respectively and 

NEW

is the scale

at which the soft supersymmetry breaking terms are

generated by the putative supersymmetry breaking

transmission mechanism. In gravity mediation scenar-

ios, 

NEW

M

. In gauge mediation scenarios, 

NEW

is low but it is a new scale, introduced by hand.

In the softly broken supersymmetric models, the

hierarchy problem is solved for M

soft

. O(10)v.

Moreover, eqn [7] shows that via quantum correc-

tions the large top-quark Yukawa coupling y

drives

the mass parameter m

to a negative value, inducing

the electroweak symmetry breaking. This means that

in supersymmetric models the electroweak scale is

calculable in terms of the known coupling constants

and the (unknown) scales M

soft

and cutoff scale



NEW

to the MSSM. If M

soft

. O(10)v, the correct

electroweak scale is obtained for 

NEW

M

GUT

This nicely fits with unification of the gauge

couplings.

In supersymmetric models, the quartic couplings

in the Higgs potential are restricted. This typically

leads to a strong uppe r bound on the mass of the

lightest Higgs particl e. In the minimal model with

the potential [6], at the tree level

Higgs

< M

91 GeV ½8

This bound is substantially modified by quantum

corrections. They depend quadratically on the top-

quark mass and logarit hmically on the stop mass

scale M

M

soft

Higgs

<v

½9

where  is given by

 ¼

þ g



cos

2 þ 

with  ¼

8

½10

For M

< 1 TeV, M

Higgs

< 130 GeV.

The minimal-model bound on the Higgs mass can

be relaxed in models with extended Higgs sector.

For instance, if an additional gauge group singlet

chiral superfield couples to the Higgs doublets, the

Higgs self-coupling  in [9] receives additional

contributions. Explicit calculations show that in

such and other models, with M

soft

. 1 TeV, the

bound on the Higgs mass cannot be raised above

150 GeV if one wants to preserve perturbative

gauge coupling unification.

Supersymmetric Grand Unified Theories

There are two striking aspects of the matter

spectrum in the SM. One is the chiral anomalies

cancelation (Weinberg 1996–2000, Pokorski 2000),

which is necessary for a unitary (and renormaliz-

able) theory, and occurs thanks to certain conspiracy

between quarks and leptons suggesting a deeper link

between them. The second one is that the spectrum

fits into simple representations of the SU(5) and

SO(10) groups (Ross 1985). Indeed, each generati on

of the SM matter fills 5



þ 10 þ 1 (if the right-handed

neutrino is included into the spectrum) representations

of SU(5) and for SO(10), 16 = 5



þ 10 þ 1.The

assignment of fermions to the SU(5) or SO(10)

representations fixes the normalization of the U(1)

generator. Both facts suggest unification of strong and

electroweak elementary forces in a grand unified

theory with some bigger gauge symmetry group. Such

unification implies that all the SM gauge forces

become of equal strength at some unification scale.

Their strength is measured by the running gauge

couplings 

= g

=4, i = 1, 2, 3, of the three group

factors SU(3)

SU(2)

U(1 )

. The energy scale

dependence of 

is governed by the renormalization

group equations. In the first nontrivial approximation,

they read:



ðQÞ



ðM



ðiÞ

2



½11

Here, 1=

) = (58.98  0.04, 29.57  0.03,

8.40  0.14) are the experimental values of the

gauge couplings at the Fermi scale and b

(i)

are the

coefficients which depend on the matter content of

the theory. They are

; 

; 11 þ



in the SM and

þ 2N

; 5 þ 2N

; 9 þ 2N



in the MSSM, where N

is the number of fermion

generations. In the SM, the running gauge couplings

Supersymmetric Particle Models 143

approach each other at high scale of order 10

GeV

but never unify.

In the MSSM, with sparticle spectrum character-

ized by M

soft

1 TeV and for the initial Fermi scale

values given above, the three gauge couplings unify

with high precision at the scale M

GUT

10

GeV.

Therefore, the MSSM can be embedded into super-

symmetric grand unified theories with no hierarchy

problem for the Fermi scale (it is stable with respect

to radiative corrections generated by particles with

masses M

GUT

) and no conflict with the measured

values of the gauge couplings.

In the SM, the baryon number is (perturbatively)

conserved since there are no renormalizable couplings

violating this symmetry. Experimental search for

proton decay, for example, p !e



, p !K

,is

one of the most fundamental tests for particle physics.

The present limit on the proton life time is 

yr. In grand unified theories, baryon number

conservation is violated by interactions mediated by

the heavy gauge bosons corresponding to the enlarged

gauge symmetry (e.g., SU(5)), spontaneously broken at

GUT

to the SM gauge symmetry. Such interactions

manifest themselves at low energy as additional,

nonrenormalizable interactions added to the SM

Lagrangian. Proton decay is then induced by the set

of dimension-6 operators of the form

ð6Þ

qqql ½12

where q, l denote quarks and leptons, respectively.

For c

(6)



GUT

1=25, the experimental limit on



requires M

(6)

& 10

GeV, consistently with

GUT

= 10

GeV in supersymmetric GUTs. How-

ever, in supersymmetric GUTs, there is still another,

genuinely supers ymmetric, source of contributions

to the pro ton decay amplitudes. These are the

dimension-5 operators

ð5Þ

l ½13

where

l denote squarks and sleptons, respectively.

Such operators originate from the exchange of the

color triplet scalars present in the Higgs boson GUT

multiplets, with M

(5)

M

GUT

10

GeV, and

(5)

& 10

7

is given by the Yukawa couplings.

Inserted into diagrams with gaugino exchanges they

give rise to dimension-6 operators of the form [12].

One then gets c

(6)

= 

GUT

(5)

, M

(6)

= M

(5)

SUSY

Given various uncertainties, for example, in the

unknown squark, gaugino, and heavy Higgs boson

mass spectrum, such contributions in supersym-

metric GUT models predict the proton life time to

be consistent with but close to the present experi-

mental limits.

Summary

Supersymmetry is distinct in several very important

points from all other proposed solutions to the

hierarchy problem. First of all, it provides a general

theoretical framework which allows one to address

many physical questions. Supersymmetric models,

like the MSSM or its simple extensions, satisfy a

very important criterion of ‘‘perturbative calculabil-

ity.’’ In particular, they are easily consistent with

the precision electroweak data. The SM is their

low-energy approximation in the sense of the

Appelquist–Carazzone decoupling, so most of the

successful structure of the SM is built into super-

symmetric models. The quadratically divergent quan-

tum corrections to the Higgs mass parameter (the

origin of the hierarchy problem in the SM) are absent

in any order of perturbation theory. Therefore, the

cutoff to a supersymmetric theory can be as high as

the Planck scale, and ‘ ‘small’ ’ scale of the electroweak

breaking is still natural. Supersymmetry is not only

consistent with grand unification of elementary forces

but, in fact, makes it very successful. And, finally,

supersymmetry is needed for string theory.

However, there are also some problems to be solved:

the hierarchy problem of the electroweak scale is solved

but the origin of the soft supersymmetry breaking scale

soft

remains an open question: spontaneous super-

symmetry breaking and its transmission to the visible

sector is a difficult problem and a fully satisfactory

mechanism which would yield M

soft

hierarchically

smaller than the Planck (string) scale has not yet been

found. On the phenomenological side, there are new

potential sources of flavor-changing neutral current

transitions and of CP violation, and baryon and lepton

numbers are not automatically conserved by the

renormalizable couplings. But even those problems

can at least be discussed in a concrete quantitative way.

See also: Brane Construction of Gauge Theories;

Perturbation Theory and its Techniques; Seiberg–Witten

Theory; Standard Model of Particle Physics;

Supergravity; Supermanifolds.

Further Reading

Eidelman S et al. (The Particle Data Group) (2004) Review of

particle physics. Physics Letters B 592: 1.

Kane GL (ed.) (1998a) Perspectives in Supersymmetry. Singapore:

World Scientific.

Kane GL (ed.) (1998b) Perspectives on Higgs Physics II.

Singapore: World Scientific.

Nilles H-P (1984) Physics Reports C 110: 1.

144 Supersymmetric Particle Models

Pokorski S (2000) Gauge Field Theories, Cambridge Monographs

on Mathematical Physics, 2nd edn. Cambridge: Cambridge

University Press.

Ross GG (1985) Grand Unified Theories. Redwood City, CA:

Addison-Wesley.

Sohnius M-F (1985) Physics Reports C 128: 39.

Weinberg S (1996–2000) The Quantum Theory of Fields,

vols. I–III. Cambridge: Cambridge University Press.

Wess J and Bagger J (1992) Supersymmetry and Supergravity,

Princeton Series in Physics, 2nd edn. Princeton: Princeton

University Press.

Supersymmetric Quantum Mechanics

J-W van Holten, NIKHEF, Amsterdam,

The Netherlands

Introduction

Supersymmetric quantum mechanics is a specific

extension of quantum mechanics with fermionic

degrees of freedom. In quantum field theory and

many-body theory, a fermionic degree of freedom is

one which is subject to Pauli’s principle: any

nondegenerate quantum state associated with a

fermionic degree of freedom can be occupied at

most once at any time. Similarly, in quantum

mechanics, one associates a fermionic degree of

freedom with an observable, the eigenvalue spec-

trum of which is restricted to the discrete set (0, 1).

The simplest example of a purely fermionic

quantum system is the fermionic oscillator. It is

represented by conjugate operators (f , f

) such that

¼ 0; f

¼ 0; ff

þ f

f ¼ 1 ½1

with a Hamiltonian H given by the bilinear

expression

¼ "

þ h!f

f ½2

The state space of this system is spanned by two

independent state vectors j0i and j1i, such that

f j0i¼0; f

j0i¼j1i

f j1i¼j0i; f

j1i¼0

½3

By constru ction, the states jn

i are eigenstates of

fermion number,

¼ f

f ; N

¼ N

½4

with eigenvalue n

= (0, 1); this implements the Pauli

principle. The states have energy eigenvalues

¼ "

þ n

h!; n

¼ð0; 1Þ½5

differing in energy by E =h!. Physically, the system

can be identified with a single fixed magnetic dipole in

an external magnetic field, the only polarization states

of the dipole being spin up or spin down.

In the Schro¨ dinger representation of quantum

mechanics (wave mechanics), fermionic degrees of

freedom are represented by anticommuting Grassmann

variables. These have no immediate classical analog,

but can be used to construct quasiclassical obser-

vables like spin.

A supersymmetric quantum system is a system

possessing both fermionic and bosonic degrees of

freedom, characterized by a degeneracy between

states with even and odd fermion number. In the

Schro¨ dinger representation, this is manifest in a

symmetry transforming bosonic (Grassmann-even)

into fermionic (Grassmann-odd) variables. The

generators of the supersymmetry transformations

square to the Hamiltonian of the system.

The Supersymmetric Oscillator

An elementary example of a supersymmetric quan-

tum system is the supersymmetric oscillator. It is a

physical system combin ing a standard bosonic

quantum oscillator with a fermionic oscillator of

the same frequency. The ordinary harmonic oscilla-

tor is described by the pair of lowering and raising

operators (b, b

), with commutator

 b

b ¼ 1 ½6

and the Hamiltonian

¼ "

þ h!b

b ½7

In this case, the eigenvalue spectrum of the occupa-

tion number

¼ b

b ½8

consists of all non-negative integers n

= 0, 1, 2, ...,

with corresponding energy eigenvalues. To construct

the supersymmetric oscillator, the harmonic oscilla-

tor is combined with a fermionic oscillator [2] of the

same frequency:

¼ "

þ h! b

b þ f



½9

Supersymmetric Quantum Mechanics 145

where "

= "

þ "

. The ground state of this system is

the state annihilated by both b and f:

bj0; 0i¼f j0; 0i¼0 ½10

The full set of energy eigenstates of the system is

constructed by taking

; n

i¼

ﬃﬃﬃﬃﬃﬃﬃ

j0; 0i

¼ð0; 1; 2; ...Þ; n

¼ð0; 1Þ

½11

with the energy eigenvalue spectrum

Eðn

; n

Þ¼"

þ nh!; n ¼ n

þ n

½12

Clearly, there is a degeneracy in energy between the

states jn

þ 1, 0i and jn

,1i, which have the same

total occupation number n, but differ in the bosonic

and fermionic occupation number by one unit. This

is illustrated in Figure 1. Such pairs of states which

are degenerate in energy can be transformed into

each other by the operators

Q ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2h!

f ; Q

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2h!

b ½13

The explicit transformations are

þ 1; 0i¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðn

þ 1Þh!

Qjn

; 1i

; 1i¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðn

þ 1Þh!

þ 1; 0i

½14

The operations [14] are called supersymmetry

transformations, and the operators Q and Q

are

called supercharges.

As the zero point of energy is arbitrary in systems

without gravitational inte ractions, it is customary to

take "

= 0, that is, "

=  "

; with the normal-

ization [13], the Hamiltonian H is then the symme-

trized absolute square of the supercharges:

þ Q

Q ¼ 2H ½15

whilst

¼ Q

¼ 0 ½16

The above relations suffice to guarantee that the

supercharges (Q, Q

) are conserved:

½Q; H¼ Q

; H



¼ 0 ½17

a result re-expressing the degener acy between states

with the same n but different n

and n

. The real

form of the supercharges is

Q þ Q



; Q

Q  Q



½18

In this representation

H ¼ Q

þ Q

½19

An important observation is that the ground state is the

only state annihilated by both supersymmetry operators:

Qj0; 0i¼0; Q

j0; 0i¼0 ½20

Indeed, it is the only state with zero energy

eigenvalue, and only such a state can be an

invariant supersinglet; all other states have positive

energy and t hey necessarily occur in supersymmetry

pairs.

Anticommuting Variables

Fermionic degrees of freedom can be described in a

pseudoclassical formulation by anticommuting vari-

ables  taking values in an infinite-dimensional

Grassmann algebra:



þ 

 ¼ 0 ½21

With an anticommuting variable , we can associate

a derivative operator @=@, which is an element of

another Grassmann algebra such that

@

;



@

 þ 

@

¼ 1;

@

¼ 0 ½22

This extends the original Grassmann algebra to a

Clifford algebra. Integration with respect to an

anticommuting variable is defined in the same

way:

d   ¼ 1;

d  1 ¼ 0 ½23

that is, integration i s t he same as differentiation for

anticommuting variables. With these definitions,

we can represent the fermionic raising and lowering

operators in terms of anticommuting variables as

! ; f !

@

½24

430

(0, 0)

(0, 1) (1, 0)

(1, 1)

(2, 0)

(2, 1)

(3, 0)

(3, 1) (4, 0)

States (n

, n

)

hω

Figure 1 Spectrum of states of the supersymmetric oscillator.

146 Supersymmetric Quantum Mechanics

and the states by

j0i!1; j1i! ½ 25

Then an arbitrary state takes the form of a linear

superposition

ji¼

j0iþ

j1i!ðÞ¼

 ½26

and the standard positive-semidefinite inner product

on the state space is represented on the wave

functions by the double integral

hji¼

d d



 e









ÞðÞ¼



þ 



½27

By construction, f

=  and f = @=@ are conjugates

with respect to this inner product:

d d



 e









ÞðÞ¼



 e





@

@







ÞðÞ½28

The real (self-conjugate) forms of the fermion

operators are, therefore, defined by



¼  þ

@



;

¼ i  

@



½29

which satisfy the Pauli–Dirac anticommutation

relations



þ 



¼ 2

½30

By taking the product, we obtain



¼i



¼ 1  2

@

¼ 1  2N

, N

1  

ðÞ ½31

Thus, we may think of the wave functions as two-

component spinors, the components being labeled

either by the eigenvalues of the spin operator 

,or

equivalently by the fermion number N

, which is a

projection operator on the states with negative spin.

The action of the Hamiltonian on a wave function

() is represented by the integral

½HðÞ¼

d



 e



ð

Þ

Hð;



Þð

Þ½32

where H(,



) is the ordered symbol of the Hamilt onian:

Hð;



Þ¼"

þ h!



 ½33

This expression is to be considered as the classical

Hamiltonian of the system. In particular, the

exponent of the action

S ¼

dt i h



  Hð;



Þ



¼ h

dt i



 þ !







þ "

ðt

 t

Þ½34

provides the integrand for the path-inte gral repre-

sentation of the evolution operator in the quantum

theory. The proof is not given here; the reader is

referred to the literature. In passing, note that as the

anticommuting variables (,



) are taken to be

dimentionless, one actually should identify the

momentum conjugate to  with  =  ih



;inthe

quantum theory, this is replaced by the operator

ih@=@.

Classical Supersymmetry

The classical action for the supersymmetric oscilla-

tor with bosonic amplitude x and fermionic ampli-

tude  is

S ¼



þ i



 þ !







½35

As inferred from the quantum theory, it is a

combination of a linear harmonic oscillator and a

fermionic oscillator of the same frequency. A factor

ﬃﬃﬃ

h

is also absorbed in  and



; equivalently, we can

use natural units in which h = 1. In the following,

we use this convention.

The action [35] is invariant under infinitesimal

symmetry transformations

x ¼i



 þ 





 ¼ð

x þ i!xÞ; 



 ¼ð

x  i!xÞ





½36

with (



, ) Grassmann-odd parameters. The Noether

theorem then implies that there are conserved

fermionic charges

Q ¼ðp  i!xÞ;



Q ¼ðp þ i!xÞ



 ½37



with the momentum defined by p =

x. The other

conserved quantity is the energy, represented by the

Hamiltonian

H ¼

þ !



þ !



 ½38

The canonical phase-space formulation is obtained

by defining brackets of two functions (A, B) on the

phase space (x, p ; ,



)by

fA; Bg¼



þ ið1Þ

@



@



½39

where (1)

is the Grassmann parity of A. In terms

of these brackets, the time evolution a nd super-

symmetry transformations take the form

A ¼ H; A

;A ¼ i



Q þ 



Q; A



½40

Supersymmetric Quantum Mechanics 147