Supersymmetry (SUSY) is a proposed symmetry between bosons and fermions that could help solve issues in the Standard Model such as the hierarchy problem. SUSY introduces new "quantum" dimensions beyond the usual 3 spatial and 1 time dimension. SUSY generators called Q transform fermions into bosons and vice versa. The SUSY algebra involves the generators Q satisfying anticommutation relations in addition to the usual commutation relations of generators like momentum P and angular momentum M. While experimental evidence for SUSY is still lacking, it is an attractive theoretical idea that may be discovered at energy scales below 1 TeV.

1. From First Principles June 2017 – R4.0
Maurice R. TREMBLAY
Standard Particles SUSY Particles
PART IX – SUPERSYMMETRY
Higgs Higgsino
Quarks SquarksLeptons SleptonsForce particles SUSY force
particles

2. Supersymmetry is a symmetry that unites particles of integer and half-integer spin in
common symmetry multiplets (e.g., a group of related subatomic particles) – it is a
symmetry connecting bosons and fermions. It is a possible symmetry of nature in four
space-time dimensions and it has the quality of uniqueness that physicists search for in
fundamental physical theories. There is an infinite number of Lie groups that can be
used to combine particles of the same spin in ordinary symmetry multiplets, but there
are only eight kinds of supersymmetry in four space-time dimensions, of which only one,
the simplest, could be directly relevant to observed particles. Unfortunately, there is still
no direct evidence for supersymmetry, as no pair of particles related to supersymmetry
transformations has yet been discovered. There is one significant piece of indirect
evidence for supersymmetry: the high-energy unification of the SU(3)C⊗SU(2)L⊗U(1)Y
gauge couplings works better with the extra particles called for by supersymmetry than
without them. Or maybe the problem might still reside with the point particle concept*.
Forward
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As with my other work, nothing of this is new or even developed first hand and frankly
it is a rearranged compilation of various quotes from various sources (c.f., References)
that aims to display an abridged but yet concise and straightforward mathematical
developmentof supersymmetry(and some higher-dimensional theories too) as I
understand it and wish it to be presented to the layman or to the inquisitive person. As a
matter of convention, I have included the setting h≡c≡1 in most of the equations and
ancillary theoretical discussions and I use the summation convention that implies the
summation over any repeated indices (typically subscript-superscript) in an equation.
* A point particle is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension.

3. Contents
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PART IX – SUPERSYMMETRY
Motivation
Introduction to Supersymmetry
The SUSY Algebra
Realizations of the SUSY Algebra
The Wess-Zumino Model
Lagrangian with Mass and Interaction
Terms
The Superpotential
Supersymmetric Gauge Theory
Spontaneous Breaking of
Supersymmetry
F-type SUSY Breaking
D-type SUSY Breaking
The Scale of SUSY Breaking
The SUSY Particle Spectrum
Supersymmetric Grand Unification
General Relativity
The Principle of Equivalence
General Coordinates
“Supersymmetry […] introduces, apart from the three obvious dimensions plus time, new ‘quantum’
dimensions that cannot be measured by numbers; they are ‘quantum’ (or ‘fermionic’) dimensions, like the
spin of the electron.” Edward Witten, The Quest for Supersymmetry talk at the Perimeter Institute.
Local Lorentz Frames
Local Lorentz Transformations
General Coordinate Transformations
Covariant Derivative
The Einstein Lagrangian
The Curvature Tensor
The Inclusion of Matter
The Newtonian Limit
Local Supersymmetry
A Pure SUGRA Lagrangian
Coupling SUGRA to Matter and Gauge
Fields
Higher-dimensional Theories
Compactification
The Kaluza Model of Electromagnetism
Non-Abelian Kaluza-Klein Theories
Kaluza-Klein Models and the Real World
N=1 SUGRA in Eleven Dimensions
References

4. The idea that there may be a symmetry called supersymmetry (SUSY) that interrelates
bosons and fermions is rather attractive. In particular, SUSY might help to solve the
hierarchy problem discussed in PART VIII – THE STANDARD MODEL: Hierarchy
Problem. We saw there that the origin of this problem is the difficulty of including
fundamental scalar Higgs fields in the theory. Scalars are the only fields that can have
nonzero vacuum expectation values (and so give spontaneous symmetry breaking)
without breaking the Lorentz invariance of a theory. On the other hand, the masses of
such scalars are subject to quadratically divergent renormalization corrections of the like
of ∆MH
2 ∝[g2/(2π)4]∫
Λ
(1/k2)d4k ~(g2/8π2)Λ2 and there is no natural way to sustain a light
Higgs of mass O(MW) together with a heavy Higgs of O(MX). There will be radiative
corrections to the light Higgs mass of O(MX) that automatically destroy the hierarchy.
Only by an unnatural fine tuning of parameters, order-by-order in perturbation theory,
could we keep a light Higgs of mass O(MW). A natural solution would be the existence of
a symmetry that requires that certain scalars must have zero mass. Chiral symmetry
ensures zero fermion masses by forbidding mass terms like mψRψL. This appears
unlikely to help with scalars because the scalar mass term, m2φφ*, always respects this
symmetry. However, in a supersymmetric model each scalar mass must be equal to that
of its fermion partner, which can be required to vanish by chiral symmetry. Moreover, the
scalar mass, MH, is then no longer quadratically divergent because the boson and
fermion loop corrections in ∆MH
2 above have opposite signs and cancel each other:
Motivation
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fermion
22
boson
222
H =ΛΛ∆ ggM
_

5. We begin by introducing the generators of SUSY transformations, Q, which turns
fermions F into bosons B and vice versa:
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Owing to the fermionic character of Q, any supersymmetric multiplet must contain
numbers of bosons and fermions of the same mass. Nature, however, is manifestly not
supersymmetric. The known bosons and fermions do not group themselves into such
mass-degenerate pairs. (N.B., the vacuum state is labelled by |0〉 regardless).
FBBF == QQ and
Although experimental support for supersymmetry is still lacking, many people believe
(e.g., the likes of Edward Witten!) that it is too beautiful to have been discarded by
nature, and that it is only a matter of time before evidence of SUSY, albeit in some
broken form, will be found. We need a finite Higgs mass of O(MW) to produce the obser-
ved electroweak symmetry breaking, so we want to break SUSY gently so that ∆MH
2
above becomes:
)( 2
F
2
B
22
H mmgM −∝∆
So if this last equation is to give ∆MH
2 < MH
2, supersymmetric partners of the ordinary
particles must be found with masses <1 TeV.~
~
In the subsequent chapters we will give an introduction of the main ingredients of
supersymmetric theories. For further technical details of the general formalism of
SUSY, we suggest the References and note that the definitive authoritative reference
seems to be Supersymmetry and Supergravity, 2nd Ed., Wess and Bagger (1992).
Introduction to Supersymmetry

6. Now, to illustrate how supersymmetry works, we begin with the following example,
which includes many of the most important features of SUSY theories. Consider a
simple harmonic oscillator with both bosonic and fermionic degrees of freedom.
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0],[],[1],[],[ ††††††
===≡≡− − bbbbbbbbbbbb and
while those of the fermion (i.e., f † and f ) satisfy the anticommutation relations:
0},{},{1},{],[ ††††††
===≡≡+ + ffffffffffff and
Since f †f †|0〉=0, two fermions cannot occupy the same state (i.e., Pauli’s exclusion
principle). In terms of these operators, the Hamiltonian takes the form:
],[ω½},{ω½ †
F
†
B ffbbH +=
where ωB and ωF are the classical frequencies of the boson and fermion oscillators,
respectively, and the allowed energies are:
)(ω)½(ω)½(ω FBFFBB nnnnE +=+++=
if ωB =ωF =ω. In this supersymmetry limit we have exact cancellation of the zero-point
energies! So, nF=0,1 are the only allowed eigenvalues of the fermion number
operator, f †f, and so all the energy levels are doubly degenerate, except for the
ground state (i.e., nB =nF=0). The system thus contains equal numbers of bosonic
and fermionic degrees of freedom!
The creation and annihilation operators of the boson (i.e., b†≡bcT and b, respectively,
with † the Hermitian conjugate symbol which means that one has to transpose T the matrix
then change the imaginary elements c from i to −i) satisfy the commutation relations:

7. This degeneracy indicates that there must exist some (super)symmetry of the
Hamiltonian. In fact, it is easy to check that the (annihilation and creation) charge
operators*:
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bfQfbQ †††
22 ω=ω= and
commute with the Hamiltonian H=P0 (e.g., Pµ =[P0,Pi]):
0],[],[ †
== HQHQ
with √(2ω) being just a simple normalization factor (Exercise). The operators Q and Q†
clearly have the effect of replacing a fermion by a boson, and vice versa, as in Q|F〉=|B〉
and Q|B〉=|F〉 above, and so they are supersymmetry generators. Furthermore, we find
that:
HQQ 2},{ †
=
and so the algebra of Q, Q†, and H closes if we include anticommunication as well. This
anticommutator is the essence of SUSY.
* We present a tabular representation of these operators to help in the association later on or to help in memorizing this fact.
bfQfbQ †††
22 ω=ω=
Replacing a: fermion by a boson boson by a fermion
Supersymmetric
charge generators
Operators:
This is
explained in
words by:
A fermion is
destroyed, f, and a
boson is created, b†.
A boson is destroyed,
b, and a fermion is
created, f †.
√(2ω) is a
normalization
factor.

8. In a 1967 paper, Sidney Coleman and Jeffrey Mandula (S.C.’s former Grad student)
adopted reasonable assumptions about the finiteness of the number of particle types
below any given mass, the existence of scattering at almost all energies, and the
analyticity of the S-matrix, and used them to show that the most general Lie algebra of
symmetry operators that commute with the S-matrix, that take single-particle states into
single-particle states, and that act on multiparticle states as the direct sum of their action
on single-particle states consists of the generators Pµ and Mµν of the Poincaré group,
plus ordinary internal symmetry generators Ta that act on one-particle states with
matrices that are diagonal in and independent of both momentum and spin.
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],,[],,[],,[],,[ 321030201321123123 KKKMMMJJJMMM =≡=≡ KJ and
So, Coleman and Mandula showed that, under very general assumptions, a Lie group
that contains both Poincaré group P and an internal symmetry G must be just a direct pro-
duct of P and G. The generators of the Poincaré group are the four-momentum Pµ=[H,P],
which produces space-time translations, and the antisymmetric tensor Mµν , which
generates space(J)-time(K) rotations:
The SUSY Algebra
where the angular momentum operator J≡Jk generates space rotations about the k-axis
and K≡Kk generates Lorentz boosts along the k-axis. So if the generators of the
Coleman-Mandula theorem requires that:
0],[],[ == aa TMTP µνµ
This no-go theorem shows the impossibility of combining space-time and internal
symmetries in any but a trivial way (or else commutators such as [.,.]=1 would exist).

9. Yet, formulating supersymmetry escapes this ‘no-go’ theorem because, in addition to
the generators Pµ, Mµν , Ta which satisfy commutation relations, it involves fermionic
generators Q that satisfy anticommutation relations. If we call the generators with these
properties even and odd, respectively, then the SUSY algebra has the general structure:
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odd]odd,even[even}odd,odd{even]even,even[ === and,
which is called a graded Lie algebra by mathematicians.
Without further ado we now present the simplest form of SUSY algebra. We introduce
four generators Qα (α =1,…,4),which form a four-componentMajorana spinor. Majorana
spinors are the simplest possible type of spinor. They are self-conjugate (i.e., ψ c =ψ ):
T
QCQQ c
==
and hence have only half as many degrees of freedom as a Dirac spinor. (N.B., we use
for Q the same convention as the adjoint [row] Dirac spinor, ψ =ψ †γ 0). Indeed, any Dirac
spinor ψ =[ψ1 ψ2 ψ 3 ψ 4]T may be written:
)(
2
1
21 ψψψ i+=
where:
are two independent Majorana spinors that satisfy ψ i =ψ i
c.
)(
2
)(
2
1
21
cc i
ψψψψψψ −−=+= and
__

10. Since Qα is a spinor, it must satisfy:
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βαβµνµνα QMQ )(
2
1
],[ Σ=
where the sum over β is implicit (i.e., summation convention for repeated indices). This
relation expresses the fact that the Qα transform as a spinor under the rotations
generated by Mµν (N.B., Σµν =½i(γ µγ ν −γ νγ µ )=½i[γ µ,γ ν ], when sandwiched between
spinors, transforms as an antisymmetric tensor). The Jacobi identity of commutators:
requires that Qα must be translationally invariant:
0]],,[[]],,[[]],,[[ =++ ανµµαννµα QPPPQPPPQ
0],[ =µα PQ
It is the remaining (anti)commutation relation (c.f., [Q,Q†]=2H of the Motivation chapter):
µαβ
µ
βα γ PQQ )(2},{ =
where the sum over µ is implicit, which we shall derive later on from first principles, and
closes the algebra, that has the most interesting consequences. Clearly this {Qα ,Qβ}
anticommutator has to yield an even generator, which might be either Pµ or Mµν . But a
term of the form Σµν Mµν (sum over µν) on the right-hand side would violate a
generalized Jacobi identity involving Qα, Qβ, and Pµ and the algebra would not close.
Indeed, if we go back to the ‘no-go’ theorem and allow for anticommutators as well as
commutators, we find that the only allowed supersymmetries are those based on the
graded Lie algebra defined by [Qα ,Mµν ], [Qα ,Pµ], and {Qα ,Qβ} above.
_
_

11. We choose Qα to be a Majorana spinor with four independent (real) parameters, but we
could have used a Weyl spinor with two complex components equally well. In fact, we
shall find it more convenient to work with a left-handed Weyl spinor χa with a=1,2, and
the chiral representation of the Dirac matrices in which:
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




−
=





=





−
=
I
I
I
I
0
0
0
0
0
0
50 γγ and,
σσσσ
σσσσ
γγγγ
The charge-conjugation matrix, C, defined by the final equation, satisfies the require-
ment:
Realizations of the SUSY Algebra






−
=





+





==
*0
*
0 2
0
Q
QQQ
σ
γ
i
CQQ c T
In this chiral representation we find:
c
RLLL C ψψψγψψ +=+= *
0
T






−
=
0
0
2
2
0
σ
σ
γ
i
i
C T
and:
Using the two-component Weyl spinor Qa, we can construct a Majorana spinor Qα as
in:
T
µµ γγ −=−
CC 1

12. We then look for possible SUSY representations that contain massless particles.
These should be the relevant multiplets, since the particles we observe are thought to
acquire their masses only as a result of spontaneous symmetry breaking. The procedure
we employ is to evaluate the anticommutator {Qα ,Qβ} of the The SUSY Algebra chapter
for a massless particle moving along the z-axis with Pµ =[E,0,0,E]. On substituting Q=Qc
= [Q −iσ2 Q*]T above into {Qα ,Qβ}, we find:
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E4},{0},{0},{ †
22
†
11
†
21 === QQQQQQ and,
with a=1,2, giving:
abba E )1(2},{ 3
†
σ−=QQ
We see that Q2
† and Q2 act as creation and annihilation operators, respectively, just like
f † and f in { f , f †}=1 and { f , f }={ f † , f †}= 0 of the Motivation chapter.
_
_

13. Now a massless particle of spin s can only have helicities λ=±s, so, starting from the
left-handed state |s,λ=−s〉, which is annihilated by Q2, only one new state can be formed
(i.e., Q2
†|s,−s〉). This describes a particle of spin s+½ and helicity −(s+½), and by virtue
of [Qα ,Pµ]=0 of the The SUSY Algebra chapter, it is also massless. Then, acting again
with Q1
† or with Q2
† gives states of zero norm by virtue of {Q1,Q2
†}=0, {Q1,Q1
†}=0, and
also {Q2 ,Q2
†}=4E above and note that Q2
†Q2
†=0 (which follows from the fermionic nature
of Q2). So the resulting massless irreducible representation consists of just two states.
Hence, the possible supersymmetric multiplets {|s,λ〉} of interest to us are:
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















½,½
2,1
0,0
½,½
gauginosfermion
bosongaugefermion
multipletgauge)(orVectormultipletChiral
To maintain CPT invariance we must add the antiparticle states that have opposite
helicity, thus giving a total of four states, |s+½, ±(s+½)〉, |s, ±s〉, in each multiplet.
All the particles in such multiplets must carry the same gauge quantum numbers. For
this reason, the known fermions (i.e., the quarks and leptons) must be partnered by
spin-0 particles (called sfermions), not spin-1 bosons. This is because the only spin-1
bosons allowed in a renormalizable theory are the gauge bosons and they have to
belong to the adjoint representation of the gauge group, but the quarks and leptons do
not. Instead, the gauge bosons are partnered by new spin-½ gauginos(spin-3/2 being
ruled out by the requirement of renormalizability).

14. There is of course no experimental evidence for the existence of such spin-0 sfermions
or spin-½ gauginos. The need to introduce new supersymmetric partners, rather than
interrelate the known bosons and fermions, is undoubtedly a major setback for SUSY.
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For completeness, we briefly consider also supermultiplets of particles with nonzero
mass M. In this case, in the particle’s rest frame, Pµ=[M,0,0,M], so the anticommutator
{Qα ,Qβ}=2(γ µ)αβ Pµ of the The SUSY Algebra chapter becomes:
abba Mδ2},{ †
=QQ
We see that Qa
†/√(2M) and Qa/√(2M) act as creation and annihilation operators,
respectively, for both a=1 and 2.
Starting from a spin state |s, s3〉, which is annihilated by the Qa, we can reach three
other states by the action of Q1
†, Q2
† and Q1
†Q2
† =−Q2
†Q1
†. For example, from the spin
states |½, ±½〉 we obtain:
and hence generate a SUSY multiplet consisting of one spin-0, one spin-1 and two
spin-½ particles, all of mass M.
|½, ½〉
|½, −½〉
|1, 1〉
|1, 0〉, |0, 0〉
|½, ½〉
|½, −½〉
|1, −1〉
χ1
†
χ2
†
χ1
†
χ2
†
χ2
†
χ1
†
χ2
†
χ1
†
_

15. In summary, supersymmetry is a symmetry that relates bosons to fermion (i.e.,
schematically a SUSY generator Q acts as Q(fermion)=boson and Q(boson)=fermion) and
so requires an equal number of fermionic and bosonic degrees of freedom.
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∫ ∂−Φ∂Φ−∂= )*(4
ψσψ µ
µ
µ
µ
ixdS
The simplest 4D system invariant under SUSY is a free theory with Weyl fermions ψα
and a complex scalar Φ, whose action is:
Now, if we consider the convention of a metric with signature [−,+,+,+] and the Weyl
spinor notation of Wess and Bagger, we have 2-component spinors with undotted and
dotted indices ψα ,ψ α , transforming in representation (½,0) and (0,½) of the Lorentz
group. A Dirac spinor contains two Weyl spinors, ΨD=(ψα ,χα ), and a Dirac mass term
reads ψ αχα +ψα χα. Some useful identities are ψχ =ψ αχα =−ψα χα =χαψα =χψ. One also
defines [σ µ
αα ]=[−I,σσσσ] and [σ µ
αα ]=[−I,−σσσσ], where I is the 2×2 identity matrix and σσσσ the
2×2 Pauli spin matrices.
⋅⋅⋅⋅
⋅⋅⋅⋅
⋅⋅⋅⋅
⋅⋅⋅⋅
⋅⋅⋅⋅ ⋅⋅⋅⋅
This system has a current which is conserved on-shell (i.e., upon use of the equations of
motion). This so-called supercurrent is:
0)*( =∂Φ∂= µ
αµα
µν
ν
µ
α ψσσ JJ with
which implies the conservation of the (super)charges:
∫∫ == 0303
αααα && JxdQJxdQ and
_
_
_
_

16. These charge generators have the unusual properties of being fermionic and
transforming as Weyl spinors under the Lorentz group – rather than scalars, as more
familiar symmetry generators. Also their algebra is generated by anticommutation
relations, rather than by commutators. Since both Q and Q are conserved, their
anticommutator should be a bosonic conserved quantity. The only candidate in the
space-time momentum Pµ, necessarily contracted with σ µ to have the right spinorial
structure. Indeed explicit computation leads to the (super)algebra:
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µ
µ
αααα σ PQQ && 2},{ =
with other (anti)commutators vanishing:
0],[],[},{},{ ==== µαµαβαβα PQPQQQQQ &&&
Remarkably, Qα and Qα are not generators of an internal symmetry, rather they
intertwine with the Poincaré algebra. You can compare this to the form obtained earlier:
⋅⋅⋅⋅
µαβ
µ
βα γ PQQ )(2},{ =
with:
These are similar but do not highlight the spinorial nature of the Wess and Bagger
notation (as it is widely used in the community). Since the content here is really to
show the features of supersymmetry I revert back to the original Majorana notation.






=





−
=
0
0
0
0
0
I
I
γand
σσσσ
σσσσ
γγγγ
_
_

17. We are now ready to consider the construction of supersymmetric field theories such as
the Wess-Zumino Model [Supergauge transformations in four dimensions, Nuclear
Physics B 70 (1): 39–50 (1974). Look up this title here: http://booksc.org] of the
massless spin-0–spin-½ multiplet, which in some way, is an alternative introduction to
SUSY. Indeed, probably the most intuitive way of introducing SUSY is to explore,
through this simple example, possible Fermi-Bose symmetries of the Lagrangian. It
could therefore equally well have been the starting point for our discussion of SUSY.
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ψγψ µ
µµ
µ
µ
µ
∂+∂∂+∂∂= iBBAA
2
1
))((
2
1
))((
2
1
L
The simplest multiplet in which to search for SUSY consists of a two-component Weyl
spinor (or equivalently a four-component Majorana spinor) together with two real scalar
fields. To be specific, we take a massless Majorana spinor field,ψ, massless scalar field,
A, and massless pseudoscalar field, B. The kinetic energy is:
The Wess-Zumino Model
with ψ =ψ †γ 0, as usual. The unfamiliar factor of ½ in the fermion term arises because ψ
is a Majorana spinor; a Dirac spinor ψ =[ψ1 ψ2 ψ3 ψ4]T is a linear combination of two
Majorana spinors (c.f., ψ =(1/√2)(ψ1+iψ2)).
The following bilinear identities are particularly useful when exploring SUSY. For any
two Majorana spinors ψ1, ψ2 we have:
1221 ψψηψψ Γ=Γ
where η=(1,2,−1,1,−1) for Γ={1,γ5 ,γµ ,γµγ5 ,Σµν}. These relations follow directly from
when we recall that Majorana spinors are self-conjugate, ψi
c=ψi.
_

18. To discover the Fermi-Bose symmetries of L, we make the following infinitesimal
transformations:
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ψδψψψδδ +=′→+=′→+=′→ and, BBBBAAAA
where:
where ε is a constant infinitesimal Majorana spinor that anticommutes with ψ and
commutes with A and B. So, overall, these transformations then look like:
εγγψδψγεδψεδ µ
µ
)( 55 BiAiiBA +∂−=== and,
_ _
These transformations are Lorentz-covariant but otherwise δ A and δ B are just fairly
obvious first guesses.
ψγε
ψε
5iBB
AA
+=′
+=′
and:
εγγψψ µ
µ
)( 5BiAi +∂−=′
The possibility of constructing two independent invariant quantities εψ and εγ5ψ and ψ
have mass dimensions 1 and 3/2, respectively, ε must have dimension −½. Hence, the
derivative in δψ is therefore required to match these dimensions.
We have also assumed that the transformations have to be linear in the fields.

19. Under the transformations δ A=εψ, δ B=iεγ5ψ, and δψ =−iγ µ∂µ(A+iγ5B)ε above the
change in L can be written in the form:
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where we have used the identities:






+∂/∂=












+∂−+∂∂=
+∂∂+∂+∂−∂∂+∂∂=
∂+∂+∂∂+∂∂=
∂+∂+∂∂+∂∂+∂∂+∂∂=
∂+∂+∂+∂∂+∂∂+∂∂+∂∂=
∂+∂∂+∂∂=
ψγγε
ψγγγγε
εγγγψψγγγεψγεψε
ψδγψψγψδδδ
ψδγψψγψδδδδδ
ψδγψψγδψψγψδδδδδ
ψγψδδδδ
µ
µ
ν
µνµ
µ
ν
ν
µ
µ
µν
µν
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µµ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µµ
µ
µ
µ
µ
µ
µ
µ
µ
µµ
µ
µ
µ
)]([
2
1
)(
2
1
)(
])([
2
1
)(
2
1
)(
2
1
)(
2
1
)()(
)]()[(
2
1
)]()([
2
1
)]()([
2
1
)]()()([
2
1
)]()([
2
1
)]()([
2
1
)(
2
1
)(
2
1
)(
2
1
5
55
555
BiA
BiABiA
BiABiABiA
iiBBAA
iBBBBAAAA
iBBBBAAAA
iBBAAL
εγψψγεεψψε 55
== and
of ψ1Γψ2 =ηψ2Γψ1 above where η=(1,2,−1,1,−1) for Γ={1,γ5,γµ,γµγ5,Σµν}. Since δ L
is a total derivative, it integrates to zero when we form the action. Hence, the action
is invariant under the combined global supersymmetric transformations δ A=εψ, δ B=
iεγ5ψ, and δψ =−iγ µ∂µ(A+iγ5B)ε that mix the fermion and boson fields. As usual,
global is used to indicate that ε is independent of space-time (also termed rigid).
__
__
_
_

20. We have remarked that the δψ transformation (i.e., −iγ µ∂µ(A+iγ5B)ε) contains a
derivative. It thus relates the Fermi-Bose symmetry to the Poincaré group. In particular,
the appearance of the time derivative (i.e., ∂µ≡∂/∂xµ =[∂/∂t,∇∇∇∇]) gives an absolute
significance to the total energy, which is normally absent in theories that do not involve
gravity. This could be relevant to the value of the cosmological constant Λ.
20
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When we extend the global supersymmetry of this chapter to local supersymmetry, the
presence of derivatives in the algebra will imply that derivatives at different points of
space-time are related. This has the implications for the metric of space-time and will
take us into the domain of general relativity. We shall find that when we come to
construct Lagrangians that are supersymmetric under local supersymmetry
transformations we shall be forced to introduce a spin-2 particle that can be identified
with the graviton. Thus, gravity naturally and automatically becomes unified with the
other forces of physics, which is why local supersymmetry is called supergravity
(SUGRA). This enhances the interest of super theories.
It is quite amazing that spin-2 particles (but none higher!) are naturally introduced by
these limited requirements of Fermion-Boson statistics, anticommutator-commutator
graded Lie algebras, and space-time locality. These very convincing arguments are used
to explain the rationale suggesting that nature was supersymmetric in its origin and that
what we see today is just the condensation of particle states (because the universe is so
cold!) and that spontaneous symmetry breaking occurred often in the past. Of course,
the mechanisms inherently used by particle interactions and the continuous and per-
sisting issue of finding a true universal vacuum occupies most of physics even today!

21. Returning our attention again to the global transformations δ A=εψ, δ B=iεγ5ψ, and δψ
=−iγ µ∂µ(A+iγ5B)ε above, we recall that the commutator of two successive
transformations of a symmetry group must itself be a symmetry transformation. In this
way we identified the algebra of the generators of the group. To obtain the
corresponding result for supersymmetry, we must therefore consider two successive
SUSY transformations like δ A, δ B, and δψ. For example, if for the scalar field A we
make a transformation δ A=εψ associated with parameter ε1, followed by another with
parameter ε2, then we obtain from δψ =−iγ µ∂µ(A+iγ5B)ε:
21
2017
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2511212 )()()( εγγεψεδδδ µ
µ
BiAA +∂−==
Hence, the commutator:
AP
AiAi
BiAiBiAiAA
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
εγε
εγεεγε
εγγεεγγεδδδδδδ
21
2121
152251122112
2
)(22
)()(],[)(
−=
∂−=∂−=
+∂++∂−==−
since the terms involving B cancel when we use identities for Majorana spinors (i.e., the
ψ1Γψ2 =ηψ2Γψ1 relation above where η=(1,2,−1,1,−1) for Γ={1,γ5,γµ,γµγ5,Σµν }) and, of
course, i∂µ=Pµ.
_ _
_
__

22. Now the generator of SUSY transformations Qα is a four-component Majorana spinor,
which we define by the requirement that (c.f., φ →exp(iα)φ ≈(1+iα)φ ≡φ +δφ and φ*→
exp(−iα)φ*≈(1−iα)φ*≡φ* +δφ*):
22
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AQA εδ =
To make this consistent with [δ2 ,δ1]A=−2ε1γ µε2 Pµ A above, we form the commutator:
AQQ
AQQQQ
AQQ
AQQA
},{
)(
],[
],[],[
21
2112
12
1212
βαβα
ββααααββ
εε
εεεε
εε
εεδδ
=
−=
=
=
using εψ =ψε and ε γ 5ψ =ψγ 5ε. Writing [δ2 ,δ1]A= −2ε1γ µε2 Pµ A in component form and
equating it with the last commutator, [δ2 ,δ1]A=−ε1α ε2β {Qα ,Qβ}A, reveals the basic
SUSY requirement:
µαβ
µ
βα γ PQQ )(2},{ =
which is indeed part of SUSY algebra {Qα ,Qβ} obtained earlier in the Motivation
chapter.
_
_ _
_
_ _
_
_
_

23. Without going into detail, the algebra closes when acting on the spinor field ψ :
23
22
aux
2
1
2
1
GFL +=
However, there is a problem with [δ2 ,δ1]ψ =−2iε1γ µε2 ∂µψ +iε1γ νε2 γν ∂ψ above
because it gives the required closure only when ψ satisfies the free Dirac equation, but
not for interacting fermions that are off the mass shell. The reason is that for off-mass-
shell particles the number of fermions and boson degrees of freedom no longer match
up. A and B still have two bosonic degrees of freedom, whereas the Majorana spinor ψ
has four. We can restore the symmetry by adding two extra bosonic fields, F and G
(called auxiliary fields because it appears without derivatives in the action hence no
dynamics associated to them), whose free Lagrangian takes the form:
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ψγεγεψεγε
εψγεγεψεεψγεγεψε
εγδγεγδγψδδ
ν
ν
µ
µ
µ
µ
µ
µ
∂/+∂−=
+∂/−+∂/−=
+∂−+∂−=
2121
251521152512
25115212
2
)()(
)()(],[
ii
iiiiii
BiAiBiAi
If we use the field equation ∂ψ =0 for a free massless fermion, the last term vanishes
identically and [δ2 ,δ1]ψ above has the same form [δ2 ,δ1]ψ =−2ε1γ µε2 Pµψ (same as
above for the field A) and hence we again obtain {Qα ,Qβ}ψ =−2(γ µ)αβ Pµ as above.
This gives the field equations F=G=0, so these new fields have no on-mass-shell
states.
_
_ _

24. From Laux =½ F 2 +½ G2 above they clearly must have mass dimension 2, so on
dimensional grounds their SUSY transformations can only take the forms:
24
ψγγεδψγεδ µ
µ
µ
µ
∂=∂−= 5
GiF and
and δψ =−iγ µ∂µ(A+iγ5B)ε above then becomes:
2017
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εγεγγψδ µ
µ
)()( 55 GF iBiAi +++∂−=
The mass dimensions prevent F and G from occurring in:
ψγεδψεδ 5iBA == and
Under these modified SUSY transformations (i.e., the unchanged δ A=εψ, δ B=iεγ5ψ,
and the corrected δψ =−iγ µ∂µ(A+iγ5B)ε +(F+iγ5G)ε above), we can show that the
unwanted term in [δ2 ,δ1]ψ = −2iε1γ µε2 ∂µψ +iε1γ νε2 γν ∂ψ above cancels and, moreover,
that:
FF µ
µ
εγεδδ ∂−= 2112 2],[ i
and similarly for G, as required by {Qα ,Qβ}ψ =−2(γ µ)αβ Pµ above.
In this way, we have obtained the spin-0–spin-½ realization of SUSY originally found by
Wess and Zumino in 1974 (c.f., http://booksc.org/book/16430412.)
_ _
__
_

25. We have found that the free Lagrangian:
25
that describes the multiplet (A,B,ψ, F, G), is invariant (up to a total derivative) under the
SUSY transformations:
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Lagrangian with Mass and Interaction Terms
22
o
2
1
2
1
2
1
2
1
2
1
GFL ++∂/+∂∂+∂∂= ψψµ
µ
µ
µ
iBBAA






−+= ψψ
2
1
BAmm GFL
and a cubic interaction term:
])(2[
2
5
22
nInteractio ψγψ BiABABA
g
−−+−= GFFL
Higher-order terms must be excluded because they are nonrenormalizable.
However, SUSY invariance is still preserved if the Lagrangian is extended to include a
quadratic mass term of the form:
ψγεδ
ψεδ
5iB
A
=
=
and
εγεγγψδ µ
µ
)()( 55 GF iBiAi +++∂−=

26. When we use the classical equation of motion:
26
These equations of motion are purely algebraic and so the dynamics is unchanged if we
use them to eliminate the auxiliary fields F and G from the Lagrangian. We then obtain:
2017
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0=
∂
∂
=
∂
∂
G
L
F
L
for the complete Lagrangian, L = Lo + Lm + LInteraction, we find:
0)(
2
22
=+++ BA
g
AmF
ψγψ
ψψψψµ
µ
µ
µ
)(
2
)(
4
1
)(
2
)(
2
1
2
1
22
1
2
1
5222222222
BiA
g
BAgBAAm
g
BAm
m
i
BBAA
−−+−+−+−
−∂/+∂∂+∂∂=L
Several features of this Lagrangian, which are characteristic of supersymmetric theories,
are worth noting. The masses of the scales and the fermion are all equal. There are
cubic and quartic couplings between the scalar fields, and also a Yukawa-type
interaction between the fermion ψ and the scalars A and B, yet in total there are only
two free parameters: m and g. This interaction between boson and fermion masses
and couplings is the essence of SUSY.
02 =++ BAgBmG
and:

27. The model can also be shown to have some remarkable renormalization properties in
that, despite the presence of the scalar fields, there is no renormalization of the mass
and coupling constant (although wave function renormalization is still necessary). The
divergences arising from boson loops are cancelled by those of fermion loops which
have the opposite sign. This is just the type of cancellation we need to stabilize the
gauge hierarchy.
27
2017
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These powerful nonrenormalization theorems make SUSY particularly compelling.
However, when we break SUSY, as we must given the absence of fermion-boson mass
degeneracy in nature, we have to be careful to preserve the relation between the
couplings of particles of different spin embodied in our L above.

28. To see how these results generalize with higher symmetries, it is convenient to work
entirely with left-handed fermion fields (c.f., PART VIII – THE STANDARD MODEL:
Possible Choices of the Grand Unified Group chapter). A Majorana spinor ψ can be
formed entirely from a left-handed Weyl spinor:
28
T
LL Cψψψ +=
where C is the charge-conjugation matrix and that the mass term is:
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The Superpotential
h.c.h.c. +=+= LLL
c
R Cmmm ψψψψψψ T
using ψL
c ≡ψL
c †γ0 =ψR*†γ0
T†C†γ0= −ψR
T C−1(=ψR
TC) and +h.c. means add the Hermitian
conjugate. For simplicity we have set −C−1 =C, which is valid in all the familiar
representations of the Dirac matrices.
_

29. We can rewrite the SUSY Lagrangian of the Lagrangian with Mass and Interaction
Terms chapter using just a single left-handed field ψL, and complex field φ and F for its
scalar partners:
29
2
* φφ gmF −−=
Then, using the equation of motion ∂ L/∂F=0, which gives:
2017
MRT
)(
2
1
)(
2
1
GF iFBiA −≡+≡ andφ
From L = Lo + Lm + LInteraction, we obtain:
h.c.)(
h.c.
2
1
**
2
+−+
+





−+
+∂/+∂∂=
LL
LL
LL
CFg
CFm
FFi
ψψφφ
ψψφ
ψψφφ µ
µ
T
T
L
we can eliminate the auxiliary field F* and so the Lagrangian becomes:






++−+−∂/+∂∂= h.c
2
1
*
2
2
LLLLLL CgCmgmi ψψφψψφφψψφφ µ
µ
TT
L
(c.f., Lagrangian of the Lagrangian with Mass and Interaction Terms chapter).

30. It is useful to re-expressthefirst Lagrangianof this chapter (i.e., L = Lo + Lm + LInteraction)
in terms of an analytic function W(φ), known as the superpotential:
30
where LKE denotes the sum of the kinetic energy terms of the φ and ψL fields. (N.B., W,
which is of dimension 3, depends on φ and not on φ*). Upon using ∂ L/∂F=∂ L/∂F*=0 to
eliminate the auxiliary fields, we find:
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







+
∂
∂
−
∂
∂
+
∂
∂
++= h.c.
2
1
*
*
** 2
2
KE LL C
WW
F
W
FFF ψψ
φφφ
T
LL








+
∂
∂
−
∂
∂
−= h.c.
2
1
2
22
KE LL C
WW
ψψ
φφ
T
LL
For a renormalizable theory W can be, at most, a cubic function of φ, since otherwise the
Lagrangian would contain couplings with dimension less than 0. Substituting:
32
3
1
2
1
φφ gmW +=
into our last Lagrangian L = LKE +|∂W/∂φ|2−½[(∂2W/∂φ2)ψL
T CψL +h.c.] above imme-
diately reproduces the L =∂µφ∂µφ*+iψL ∂ψL −|mφ +gφ2|2 −(½mψL
T CψL +gφψL
T CψL +h.c.)
Lagrangian derived earlier. The superpotential is the only free function in the SUSY
Lagrangian and determines both the potential of the scalar fields, and the masses and
couplings of the fermions and bosons.
_

31. In general there may be several chiral multiplets to consider. For example, if ψ i
belongs to a representation of an SU(N) symmetry group, we will have the
supermultiplets:
31
),( i
L
i
ψφ
where in the fundamental representation i=1,2,…,N. From the derived Lagrangian L =
LKE +|∂W/∂φ|2 −½[(∂2W/∂φ2)ψL
T CψL +h.c.] above we readily obtain a Lagrangian that is
under the additional symmetry and incorporates the new supermultiplets. It is:
2017
MRT








+
∂∂
∂
−
∂
∂
−∂/+∂= ∑∑∑∑ h.c.
2
1 22
2
Chiral
ji
j
L
i
Lji
i
i
i
i
L
i
L
i
i
C
WW
i ψψ
φφφ
ψψφµ
T
L
and the most general form of the superpotential W is:
kji
kji
ji
ji
i
i gmW φφφφφφλ
3
1
2
1
++=
where the coefficients m and g are completely symmetric under interchange of indices.
Since W must be invariant under SU(N) symmetry transformations the term linear in the
fields can only occur if a gauge-singlet field exists.

32. A combination of SUSY with gauge theory is clearly necessary if these ideas are to
make any contact with the real world. In addition to the chiral multiplet (φi,ψL
i) (i=1,2,…,
N) we must include the gauge supermultiplets:
32
),( aa
A χµ
with a=1,2,…,N2 −1 and where Aµ
a are the spin-1 gauge bosons of the gauge group G
(taken to be SU(N)) and χa are their Majorana fermion superpartners (the so-called
gauginos). These boson-fermion pairs, which in the absence of symmetry breaking are
assumed to be massless, belong to the adjoint representation of the gauge group. Our
task is to find a SUSY- and gauge-invariant Lagrangian containing all these chiral and
gauge supermultiplets.
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Supersymmetric Gauge Theory
The gauge multiplets are described by the Lagrangian (N.B., a,b,c=1,…, 8):
2
Gauge )(
2
1
)(
2
1
4
1 a
a
a
a
a
DDiFF +/+−= χχµν
µνL
where the gauge field-strength tensor is (c.f., PART VIII – THE STANDARD MODEL:
Quantum Chromodynamics (QCD) chapter – Ga
µν=∂µGa
ν −∂ν Ga
µ −gs fabcGb
µGc
ν ):
νµµννµµν
cbabcaaa AAfgAAF Gauge−∂−∂=
Dµ is the covariant derivative satisfying (c.f., op cit: Spontaneous Symmetry Breaking in
SU(5) chapter – (DµΦ)K= ∂µΦK +ig[(TI Aµ
I)KJ ΦJ] – N.B., I,J,K=1,…,24):
caabcaa AfgD χχχ µµµ
Gauge)( −∂=
and Da is an auxiliary field (similar to Fi of the chiral multiplet).

33. Actually, for this pure gauge Lagrangian the equation of motion, ∂ LGauge /∂Da =0,
implies Da =0; however, it will become nonzero when the chiral fields are coupled in. The
notation will be familiar: gGauge and fabc are the coupling and structure constants of the
gauge group, and in the equation for (Dµχ)a above, the matrices Tb representing the
generators in the adjoint representation have been replaced by (Tb)ac =i fabc. It is
straightforward to show that LGauge is invariant, and that the algebra closes, under SUSY
transformations:
33
εγχδ
χγγεδ
µν
µν
µµ
5
5
2
1 aa
aa
F
A
Σ−=
−=
where ε is a constant infinitesimal Majorana spinor. This transformation is analogous to
δ A=εψ, δ B=iεγ5ψ, and δψ =−iγ µ∂µ(A+iγ5B)ε (c.f., Wess-Zumino Model chapter) for
chiral multiplets.
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_ _
aa
DiD )( χεδ /−=
and

34. To include the chiral fields (φi,ψL
i), we add LChiral of the The Superpotential chapter but
substitute derivative Dµ for ∂µ in the kinetic energy terms:
34
aa
ATgiD µµµµ Gauge+∂=→∂
where Ta are the matrices representing the generators of the gauge group in the
representation to which (φi,ψi) belong. To ensure the supersymmetry of the combined
chiral + gauge Lagrangian, we must include two further terms, and write:
2017
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]h.c.)(2[)( *
Gauge
*
GaugeGaugeChiral ++−+= jLji
aa
i
a
jji
a
i PTgDTg ψχφφφLLL
where PL ≡½(1−γ 5), and also replace ∂µ in δ A=εψ, δ B=iεγ5ψ, and δψ =−iγ µ∂µ(A+iγ5B)ε
+(F+iγ5G)ε of the Wess-Zumino Model chapter by Dµ. Model building begins with this
SUSY Lagrangian. Using ∂ L /∂Da =0 to eliminate the auxiliary field gives:
jji
a
i
a
TgD φφ )(*
Gauge=
The terms in the Lagrangian that contribute to the potential for the scalar fields are
evident by inspection of L = LKE +|∂W/∂φ|2 −½[(∂2W/∂φ2)ψL
T CψL +h.c.] and LGauge =
−¼Fµν
aFµν
a+½iχa(Dχ)a +½(Da)2. They are:
∑ ∑∑ 







+
∂
∂
=+=
a ji
jji
a
i
i i
ai Tg
W
DFV
2
*
Gauge
2
22
)(
2
1
2
1
*),( φφ
φ
φφ
which are known as the F and D terms, respectively. This potential will play a central
role in the spontaneous breaking of SUSY and the gauge symmetry.
_ _
_

35. The particles observed in nature show no sign whatsoever of a degeneracy between
fermions and bosons. Even the photon and neutrino, which appear to be degenerate in
mass, cannot be SUSY partners. Hence, supersymmetry, if it is to be relevant to nature,
must be broken.
35
0],[ ≠HQα
The breaking could be either explicit or spontaneous:
2017
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Spontaneous Breaking of Supersymmetry
and so the violation would have to be small enough to preserve the good features of
SUSY and yet large enough to push the supersymmetric partners out of reach of current
experiments. However, we would inevitably lose the nice nonrenormalization theorems
and, even worse, any attempt to embrace gravity via local SUSY would be prohibited.
2. So instead we prefer to consider the spontaneous breaking of SUSY, not least
because this has proved so successful previously for breaking gauge symmetries.
Hence, we assume that the Lagrangian is supersymmetric but that the vacuum state is
not:
0],[ =HQα
1. Explicit breaking would be quite ad hoc. The SUSY generators would no longer
commute with the Hamiltonian:
and:
00 ≠αQ

36. A new feature arises here, however. The Higgs mechanism of spontaneous symmetry
breaking is not available in SUSY because, if we were to introduce a spin-0 field with
negative mass-squared, its fermionic superpartner would have an imaginary mass. Also,
using the anticommutator {Qα ,Qβ}=2(γ µ)αβ Pµ of the The SUSY Algebra chapter:
36
∑∑ +=
α
α
α
α
2
†2
00008 QQH
we can directly establish a general and important theorem. If we multiply this last
commutator by γ 0
αβ and sum over β and α, we obtain:
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µαβ
µ
βδδα γγ PQQ 2},{ 0†
=
HPQQ 88},{ 0
†
==∑α
αα
and hence:
It follows immediately that:
1. the vacuum energy must be greater that or equal to zero;
2. if the vacuum is supersymmetric (i.e., if Qα |0〉=Q†
α |0〉=0 for all α), the vacuum
energy is zero; and
3. conversely, if SUSY is spontaneously broken (i.e., if Qα |0〉≠0), then the vacuum
energy is positive.
_

37. These results have a disappointing consequence. Conclusion (1) gives an absolute
meaning to the zero of energy, a fact that is was hoped to use to explain why the
vacuum energy of the universe (represented by the cosmological constant Λ), is zero or
very close to zero. But now from (3) we see that broken SUSY implies a positive vacuum
energy. So small we find Λ=0 when we come to couple SUSY to gravity? Fortunately,
the situation can be redeemed because the coupling to gravity introduces non-positive-
definite terms in the scalar potential and a delicate cancellation can occur that may leave
Λ≅0; but the puzzle of why Λ is zero (i.e., 10−122!) to such high precision is still not
solved! [c.f., S. Weinberg, Rev. Mod. Phys., 61, 1 (1989)].
37
Leaving this aside we can see from (3) that SUSY breaking is rather special because it
requires the ground-state energy to be positive. In the classical approximation, the
energy of the ground state is given by the minimum of the scalar potential V(φ,φ*)=|Fi|2 +
½Dα
2 of the Supersymmetric Gauge Theory chapter:
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∑ ++
∂
∂
=
αβ
β
α
β δηφφ
φ
2
1
*
Gauge
2
])([
2
1
jjii
i
Tg
W
V
with:
kjikjijijiii gmW φφφφφφλ
2
1
2
1
++=
The sum overβ has been included to allow for the possibility of different gauge groups
with different couplings, and the constanttermη can only occur if β labels a U(1) factor.

38. It is evidently hard to break SUSY. The minimum V =0 will occur when φi =0 for all i
(and so SUSY will be unbroken) unless one of the following conditions applies:
38
ψδφψδψφδ ∂/+∂/ ~~~ FF and,
1. λi ≠0, that is, there exists a gauge-singlet field φi, so the superpotential W can
contain a linear term yet still be gauge invariant (F-type breaking);
2. η ≠0, so the gauge group contains an Abelian U(1) factor (D-type breaking). This is a
necessary but not a sufficient requirement. This mechanism cannot occur in GUTs
because they are based on simple gauge groups that do not have U(1) factors.
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There is an alternative way of seeing that the spontaneous symmetry breaking of
SUSY can only be accomplished by 〈F〉≠0 and/or 〈D〉≠0. If we look back at the multiplet
(φ,ψ,F), which takes the form:
and from δ Aµ
a =−ε γ µγ 5χa, δ χa =−½ΣµνFµν
aγ5ε +Daε, and δDa =−iε(Dχ)a of the
Supersymmetric Gauge Theory chapter for the gauge multiplet (Aµ ,χ,D), in which:
χδχδχγδ µν
µν
µµ ∂/+Σ ~~~ DDFA and,
and note that the vacuum expectation values of the spinor and tensor fields and ∂µφ
must be zero to preserve the Lorentz invariance of the vacuum, then it is only possible to
break the symmetry through nonzero vacuum expectation values of the auxiliary fields F
and D.
_ _

39. The spontaneous breaking of SUSY requires:
39
00 ≠αQ
and Qα |0〉 is necessarily a fermionic state, which we denote by |ψGauge〉. Since the Qα
commute with H, the state |ψGauge〉 must be degenerate with the vacuum. It must
therefore describe a massless fermion (with zero momentum). The situation is thus
exactly analogous to the spontaneous breaking of an ordinary global symmetry in which
massless Goldstone bosons are created out of the vacuum (c.f., PART VIII – THE
STANDARD MODEL: Spontaneous Symmetry Breaking (SSB) chapter). Here the
spontaneous breaking of global SUSY implies the existence of a massless fermion,
which is called the Goldstino.
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We next consider examples of these types of symmetry breaking, F-type and D-type
introduced in (1) and (2) above.

40. We now consider the O’Raifeartaigh (pronounced O’RAFFerty) Model which is a simple
example of SUSY breaking arising from the presence of a linear term in the
superpotential W:
40
2
BAgCBmAW ++−= λ
which contains three complex scalar fields A, B, and C. In this example the scalar
potential V =|∂W/∂φi|2 +½Σβα[gβ φi*(T α
β)ijφj +ηδβ1]2 (with W=λiφi +½φi φj +⅓gijkφi φj φk ) of
the Spontaneous Breaking of Supersymmetry chapter becomes:
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F-type SUSY Breaking
2222
222
22 BmBAgCmBAgCmBg
C
W
B
W
A
W
V ++++++−=
∂
∂
+
∂
∂
+
∂
∂
= λ
and see that V =0 is excluded because the last term is only zero if B =0 , but then the first
term is positive-definite. We conclude the V >0 and that SUSY is broken. Provided m2 >
2gλ the potential V has a minimum when B =C =0, independent of the value of A. For
simplicity, we set A =0 at the minimum.

41. As usual (c.f., ∂V/∂φi|φ =v =0 and ∂2V/∂φi∂φj|φ =v =Mij
2 >0 of the PART VIII – THE
STANDARD MODEL: Spontaneous Symmetry Breaking (SSB) chapter) the scalar
masses are determined by evaluating:
41
at the minimum. The only nonzero elements are:
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.
2
c&,
BA
V
VAB
∂∂
∂
≡
2
**** 2 mVVgVV CCBBBBBB ==−== andλ
We see that the scalar field A remains massless and that the field C has mass m. SUSY
breaking splits the mass of the complex B field because:
2
2
22
1
22
***
2
)2()2(**2 BgmBgmBVBBVBV BBBBBB λλ ++−=++
where B=(1/√2)(B1+iB2), and so the real scalar fields B1 and B2 have [mass]2 =m2 m2gλ,
respectively.

42. The fermion masses are obtained by evaluating ∂2W/∂A∂B, &c., at the minimum (c.f.,
LChiral of the The Superpotential chapter). From W=−λ A+mBC+ gAB2 above we find that
the only nonzero terms is:
42
and so the fermion mass matrix takes the form:
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m
CB
W
=
∂∂
∂2










=
00
00
000
F
m
mM
in the basis of the Majorana spinors ψA , ψB , ψC. The massless Goldstino state ψA is
evident, and the ‘off-diagonal’ structure signals that the two Majorana spinors ψB , ψC will
combine to give a single Dirac fermion of mass m. Despite the SUSY breaking, there is
still an equality between the sum and the [mass]2 of the bosons and that of the fermions.
Explicitly, for each degree of freedom we have the masses:
444 3444 2143421434214342143421
CBA
mmmmmmgm
CBA ψψψ
λ
,
2222222
.,,,,0,0,,,2,0,0 ±
FERMIONSBOSONS
Only B offers SUSY breaking since it is the only field that couples to the Goldstino;
its coupling gBψBψA appears when W=−λ A+mBC+ gAB2 above is inserted into Lchiral.
_

43. The value of the potential at the minimum can be written:
43
22
SMV ≡= λ
where the mass MS denotes the scale of SUSY breaking. The mass splittings within the
supermultiplet BOSONS/FERMIONS above are therefore:
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22
SMgm ≈∆
where g is the coupling to the Goldstino.
This simple model illustrates several more general results. The mass relation is a
particular example of the supertrace relation:
0)(Tr)12()1()(STr 222
=+−≡ ∑J
J
J
MJM
which holds whether SUSY is spontaneously broken or not. Here MJ is the mass matrix
for the field of spin J, and the sum is over all the physical particles of the theory.

44. The STr(M2) relation above holds in lowest-order perturbation theory. We say that it is
a tree-level result because it neglects corrections due to the diagrams containing loops.
This supertrace mass relation is important because it ensures that the scalars are not
subject to quadratically divergent renormalization. We may readily verify that the STr(M2)
relation above holds for an arbitrary multiplet structure. If there are several chiral
multiplets (φi,ψi), then it is convenient to arrange the scalar fields and their complex
conjugates as a column vector so that the boson mass terms have the matrix structure:
44












*
]*[ †
φ
φ
φφ
XY
YX
The block diagonal parts of the boson [mass]2 matrix, MB
2, have elements:
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ji
k
jkki
k kjkiji
ji MMMM
WWV
X )()()(
* *
FF
*
FF**
222
==










∂∂
∂








∂∂
∂
=
∂∂
∂
= ∑∑ φφφφφφ
where MF is the fermion mass matrix and so it follows that:
)(Tr2)(Tr 2
F
2
B MM =
at tree level.

45. We can also show that the fermion mass matrix has a zero eigenvalue and hence
identify the Goldstino. At the minimum of the potential:
45
Thus, the mass matrix MF annihilates the fermion state:
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∑∑∑ =








∂
∂








∂∂
∂
=










∂
∂
∂
∂
=
∂
∂
=
j
jji
j ijij iii
FM
WWWV *
F
*
2
2
)(0
φφφφφφ
∑=
j
jjF ψψ
*
Gauge
which is thus identified as the massless Goldstino. In our example,ψG=ψA since 〈FB〉=
〈FC〉=0.
However, the equality Tr(MB
2)=2Tr(MF
2) above, which is so desirable to ensure the
boson-fermion loop cancellations, is not supported experimentally. The difficulty is that in
these simple models the relation applies to each supermultiplet separately. Hence, for
the electron, for example, we require:
222
e2 BA mmm +=
which implies that one of the two scalar electrons (A,B) must have a mass less than or
equal to that of the electron. Such a particle would have been detected long ago if it
existed!

46. Various possible forms of V are shown in the Figure.
46
−+= φφmW
and so the scalar potential V=|Fi|2 +½Dα
2 of the Spontaneous Breaking of
Supersymmetry chapter becomes:
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We next consider the Fayet-Iliopoulos Model which is another simple example of SUSY
breaking but this time caused by the presence of a U(1) factor in the gauge group. It is a
SUSY version of QED with two chiral multiplets (φ+ ,ψ+) and (φ− ,ψ−), where subscripts
give the sign of the charge. The U(1) gauge-invariant superpotential is:
D-type SUSY Breaking
222222222
2222222
2
1
)()()(
2
1
])([
2
1
ηφηφηφφ
ηφφφφ
+−+++−=
+−++=
−+−+
−+−+
ememe
emmV
Possible forms of the scalar potential V when (Left) U(1) and SUSY are unbroken, (Middle) U(1)
unbroken and SUSY broken, (Right) both broken.
φ φ φ
V V V
0=η ηη em >≠ 2
0 & ηη em <≠ 2
0 &

47. Now, provided m2 >eη (where eη >0), the minimum occurs at:
47
0== −+ φφ
so U(1) gauge invariance is not spontaneously broken, but SUSY is broken since V≠0.
The boson masses are split, m±
2 =m2 ±eη, whereas the fermion masses are unaffected by
the breakdown of SUSY. Like the matrix MF =[::] in the F-type SUSY Breaking chapter
the off-diagonal form of the fermion mass matrix in the ψ+, ψ− Majorana basis implies
that these two states combine to give a Dirac fermion of mass m. The fermion-boson
mass splitting signals the breakdown of SUSY but the [mass]2 equality still holds, since:
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222
2mmm =+ −+
For m2 >eη, the U(1) symmetry is unbroken and the gauge multiplet (Aµ,χ) remains
massless. The fermion χ is the Goldstino arising from the spontaneous SUSY breaking.
The case m2 <eη is more interesting. The minimum of the potential now occur at:
00 == −+ φφ and
where e2v2 =(eη −m2). Now both the U(1) gauge symmetry and SUSY are
spontaneously broken (c.f., see previous Figure – Right). We find that the complex field
φ+ has [mass]2 =2m2, while one component of φ− is eaten by the usual Higgs mechanism
to give [mass]2 =2e2v2 to the vector gauge field Aµ, and the remaining component also
acquires [mass]2 =2e2v2. A linear combination of the ψ+ and χ Majorana fields forms the
massless Goldstino, whereas the two remaining combinations of ψ+, ψ−, and χ both
have [mass]2 =m2+2e2v2.

48. Even when SUSY is spontaneously broken:
48
remains true. Yet, to explain the absence of superpartners we must find a way to violate
this sum rule. We have noted that the above supertrace relation applies only to masses
evaluated from tree diagrams and may be broken if radiative corrections are included.
We explore this loophole next.
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The Scale of SUSY Breaking
0)(Tr)12()1()(STr 222
=+−= ∑J
J
J
MJM
It is hoped that SUSY will solve the hierarchy problem and naturally sustain the two
vastly different scales of symmetry breaking, MW and MX. With exact SUSY, the
quadratic divergences of the scalar (Higgs) masses are precisely canceled. They are not
renormalized. In fact, the nonrenormalization theorems are necessary for SUSY to exist
at all, for even if 〈V 〉=0 at tree level we would normally expect radiative corrections to
give 〈V 〉≠0 and destroy SUSY. Fortunately. it follows from the nonrenormalization
theorems that if SUSY is not broken at tree level then it will not be broken by
perturbative corrections either.

49. However, nature is not supersymmetric and SUSY must be spontaneously broken
somehow. A desirable scenario is for SU(2)⊗U(1)Y to be unbroken in the supersymmetric
limit and for SUSY breaking also to induce electroweak breaking. The aim is to break
SUSY spontaneously at some mass scale MS, in such a way that the boson-fermion
mass splittings within multiplets are of order:
49
2
W
2
Mm ≈∆
The splitting turns out to be:
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2
S
2
Mgm ≈∆
where g is the coupling of the Goldstino to the boson-fermion pair within the
supermultiplet. So, by varying the value of g, different MS can lead to the same MW.
We have discussed models with MS ≈MW and with g≈gGauge, that is, with tree-level
couplings between the Goldstino and the ordinary particles and their superpartners.
These have failed. Either STr(M2)=0 remains true or the models are plagued with other
problems. Instead, we consider MS >> MW and an F-type SUSY breaking that occurs in a
hidden sector that contains new gauge-singlets chiral supermultiplets of massive fields.
The SUSY breaking then trickles down to the ordinary low-energy sector via radiative
corrections, and may also induce electroweak breaking. The effective Goldstino coupling
to ordinary supermultiplets is very small, so that MW
2 ≈gMS
2 is satisfied.

50. If MS is sufficiently large, gravity can no longer be neglected. We then have the exciting
possibility that it is gravitational effects that are responsible for SUSY breaking. For
instance, suppose that the heavy hidden sector consists of particles of mass of order the
Planck mass, MP ≡√(hc/GN)=1.2×1019 GeV/c2, and that the Goldstino coupling g~O(1).
Since the heavy sector can only communicate with ordinary particles through
gravitational interactions, the light sector will have an effective Goldstino coupling g~
O(MW /MS), and hence the mass slitting ∆m2 ≈gMS
2 above within an ordinary
supermultiplet will be:
50
2W2
S
P
M
M
M
m ≈∆
Demanding that ∆m2 ≈MW
2 gives:
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211
W
2
)GeV10(≅≅ PS MMM
Local SUSY (i.e., supergravity), is then the appropriate framework. Supergravity has a
gauge supermultiplet that consists of a spin-2 graviton and a spin-3/2 gravitino, and, on
spontaneous breaking of symmetry, neither the Goldstino nor the gravitino remain
massless. There is a super-Higgs mechanism whereby:
and the Goldstino is absorbed to become the missing helicity ±½ components of the
massive gravitino.
)spin,0()spin,0()spin,0( 2
3
2
1
2
3
≠→=+= mmm

51. To estimate the mass acquired by the gravitino, recall that on spontaneous breaking of
symmetry of an ordinary local gauge symmetry (with coupling gG) the gauge boson
acquire a mass:
51
φGgM ≈
where 〈φ〉 is the vacuum expectation value of the scalar field that causes the breaking.
Similarly, it can be shown that in the super-Higgs mechanism the gravitino acquires a
mass:
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W
2
2
2/3 M
M
M
MGm
P
S
SN ≈=≈
where GN is Newton’s gravitational constant. Thus gravitational contributions proportional
to (m3/2)2 can no longer be neglected in the STr(M2) relation above because they are
comparable to the nongravitational terms.
In conclusion, it is now generally believed that MS >>MW and that SUSY is more likely
to occur as an effective low-energy limit of supergravity or some other similar model that
yields explicit soft SUSY breaking through the effective Lagrangian:
where LSoft consists of interactions of dimension <4 that do not lead to quadratic
divergences.
SoftSUSYGlobalEffective LLL +=

52. The Standard Model has 28 bosonic degrees of freedom (i.e., 12 massless gauge
bosons and 2 complex scalars) together with 90 fermionic degrees of freedom (i.e., 3
families each with 15 two-component Weyl fermions). To make this model
supersymmetric we must clearly introduce additional particles. In fact, since none of the
observed particles pair off, we have to double the number. In the Realizations of the
SUSY Algebra chapter, we saw that gauge bosons are partnered by spin-½ gauginos,
and these cannot be identified with any of the quarks and leptons. So the latter have to
be partnered by new spin-0 squarks and sleptons.
52
In the Standard Model the Higgs (φ) generates masses for the down-type quarks and
the charged leptons, while its charge conjugate (i.e., φc =iτ2φ*) gives masses to the up-
type quarks. Under charge conjugation the helicity of the spin-½ partner of the Higgs
(i.e., the Higgsino) is reversed, and so it proves impossible to use a single Higgs to give
masses to both up-type and down-type quarks. The second (complex) doublet is also
needed to cancel the anomalies that would arise if there were only one Higgsino. As in
the Standard Model, three of the Higgs fields are absorbed to make the W± and Z
bosons massive, and we are therefore left with two charged and three neutral massive
Higgs particles.
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The SUSY Particle Spectrum

53. The particle content of the supersymmetric Standard Model is shown in the Table.
53
There is no doubt that this Table is a setback for SUSY. To be economical, SUSY
ought to unite the known fermionic matter (i.e., quarks and leptons) with the vector
forces (i.e., γ, g, W, Z), but we have been compelled to keep them separate and to
introduce a new superpartner for each particle. A great deal of effort has gone into the
search for these superpartners but so far none has been found although an upgrade to
the Geneva Large Hadron Collider (LHC) might be in the running since its collision
energy will be nearly 4,000 GeV (circa 2017). To compare things, the machine that
discovered the Higgs the center of mass energy was about 14,000 GeV which means
that the center of mass energies of the partons – the quarks and gluons – within the
colliding protons was about 1,000 GeV. So following a two-year upgrade the LHC’s more
powerful electromagnets will be sufficient to accelerate two beams of protons to6,500
GeV increasing the potential collision energy from 8,000 GeV in 2012 to 13,000 GeV.
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Chiral Multiplets Gauge Multiplets
Spin ½ Spin 0 Spin 1 Spin ½
Quarks (qL, qR ) Squarks (qL, qR ) Photon (γ ) Photino (γ )
Leptons (lL, lR ) Sleptons ( lL, lR ) W, Z bosons Wino W, Zino Z
Higgsino (φ , φ ′) Higgs ( φ, φ′) Gluon ( g ) Gluino ( g )
~ ~
~ ~
~ ~
~~
~
~
The undetected superpartners are distinguished by the tilde ‘~’. The L, R subscripts on the spinless q and l refer to the
chirality of the fermionic partner.
Particle Multiplets in the Supersymmetric Standard Model.
~ ~

54. A major motivation for introducing SUSY was the hope that is would solve the hierarchy
problem of the Grand Unified Theories (GUTs), by allowing vastly different symmetry
breaking scales, MW and MX, without the incredibly fine tuning of the parameters needed
in the Higgs potential V(Φ,H)=αH†HTr(Φ2)+βH†Φ2H of the PART VIII – THE
STANDARD MODEL: Hierarchy Problem chapter. Can SUSY naturally sustain such a
hierarchy is fact?
54
To answer this question it is sufficient to study grand unified SU(5), in which the spin-½
matter fields fit neatly into ψ (5) and χ(10) representations, while the gauge bosons Aµ lie
in the 24 adjoint representation of the group (c.f., op cit: Grand Unified SU(5)). Grand
unified SU(5) is spontaneously broken to SU(3)⊗SU(2)⊗U(1) at the same MX ≈1014 GeV
by a superheavy Higgs Φ(24), and then electroweak breaking occurs at the scale MW ≈
102 GeV through the Higgs multiplet H(5) (c.f., op cit: Spontaneous Symmetry Breaking
in SU(5)). To make this model supersymmetric, we have to form chiral and gauge
multilets by introducing SUSY partners for all of the above particles (c.f., The SUSY
Particle Spectrum chapter). In addition, we must include a second Higgs multiplet H′(5)
to give mass to up-type and the down-type quarks.
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Supersymmetric Grand Unification
_
_

55. The most general SU(5)-invariant superpotential involving the scalar fields of the above
multiplets, up to and including cubic interactions as in W=λiφi +½φi φj +⅓gijkφi φj φk of the
Superpotential chapter, is of the form:
55
(where contraction of SU(5) indices is to be understood). The Yukawa counterpart of the
first terms gives masses to the quarks and leptons (c.f., PART VIII – THE STANDARD
MODEL: Fermion Masses Again – LY =GD(ψR
c)k(χL)klHl
†+¼GUεklmnp(χR
c)kl(χR)mnHp +h.c.).
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





Φ+Φ+′+Φ′′′+′+= 23
Tr
2
1
Tr
3
1
)(~~~~ MHMHHGHGW DU λλχψχχ
The first stage of the symmetry breaking is associated with the Φ(24), so for the
moment we ignore the other fields by pulling their vacuum expectation values equal to
zero, and seek a minimum of the scalar potential V under variation of the components
Φkl assembled in the form of a traceless 5×5 matrix Φ=(λI /√2)ΦI (I=1,…,24). With
SUSY, Φ is no longer Hermitian. However, the Hermitian and antihermitian parts of Φ
commute, and can therefore be diagonalized simultaneously by an SU(5) transformation.
Hence, without loss of generality, we can take the diagonal form:
),...,(diag 51 ee=Φ
but subject to the trace condition:
0Tr ==Φ ∑M
Me
with M=1,…,5.
_ _

56. The relevant part of the superpotential now becomes:
56
We seek a minimum of the scalar potential V(φ,φ*)=|Fi|2 +½Dα
2 of the Supersymmetric
Gauge Theory chapter:
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







+=





Φ+Φ= ∑∑ M
M
M
M eMeMW 2323
2
1
3
1
Tr
2
1
Tr
3
1
λλ
∑ ∂
∂
=
M Me
W
V
2
under variation of the eM. Clearly all eM =0 it has a minimum with V=0, so there is at least
one supersymmetric minimum.

57. To find whether there are other degenerate minima we need to know whether there are
other solutions of ∂W/∂eM =0, subject to the zero-trace constraint TrΦ=ΣMeM =0 above. If
this constraint is taken into account by using a Lagrange multiplier k, we need to solve:
57
called solutions (ii) and (iii), respectively, which manifestly satisfy TrΦ=ΣMeM =0 above.
Solution (i) does not break SU(5); solution (ii) breaks SU(5) down to SU(4)⊗U(1),
whereas solution (iii), with indices 1,2,3 associated with color and 4,5 with electroweak
SU(2), has just the property we require of spontaneously breaking SU(5) down to
SU(3)⊗SU(2)⊗U(1). All these solutions are degenerate and are supersymmetric, since
they all have V=0. At this stage there is no obvious reason why nature should choose
solution (iii), but if it does, then, provided we choose the breaking scale M~MX ~1014
GeV we have a chance of obtaining a realistic model.
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MeeMeee
M
e
M
eeee
32
3
4
3
54321
54321
−=====
−=====
and
and
00 2
=++⇒=








∂
∂
+
∂
∂
∑ λ
k
eMee
e
k
e
W
MM
N
N
MM
This is a quadratic equation with two roots, eM =r1,r2 say, that depend on the arbitrary
parameter k. If r1 =r2 then we can only have eM =eN for all pairs M,N and then the zero
trace implies that all eM =0; our previous solution, which we call solution (i). However, if r1
≠r2 then there are two further solutions, which we can write as:

58. We return to the superpotential W above and note that the nonzero vacuum
expectation value from (iii) above:
58
contributes to the masses of the H and H′ Higgs fields. The relevant part of the
superpotential is:
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)3,3,2,2,2(diag MMMMM −−=Φ
5,45,43,2,13,2,1 )3()2()( HHMMHHMMHMHW ′′+−′+′′+′=′+Φ′′= λλλ
In the last term is responsible for the standard-model electroweak breaking as scale MW,
so we require that:
MM 3≅′
to very high accuracy in order that the doublet components (4,5) have mass O(MW)
rather than O(MX). The color-triplet components of H, H′ have mass 5Mλ′, which we take
to be O(MX).
The fine tuning, whereby M′−3M=O(MW) while M and M′ are O(MX), is just the
hierarchy problem of PART VIII – THE STANDARD MODEL: Hierarchy Problem chapter
again, and it may ne wondered whether we have gained anything by introducing SUSY.
In fact we have, because of the lack of renormalization of the mass parameters in SUSY
theories. This means that the fine tuning is required just once, in the original Lagrangian,
and not in each order of perturbation theory separately. Moreover, if we suppose that it is
the breaking of SUSY that induces electroweak breaking, then we require the exact
equality M′=3M.

59. Does supersymmetrization ruin the attractive predictions of GUTs? Since we now have
more particles in the low-energy sector, the evolution of the coupling constant is
changed. The coefficients of the SU(3), SU(2), and U(1) β-functions become:
59
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HgHgg NNbNNbNb
10
3
2
2
1
269 123 +−=+−=−= and,
rather than b3 =11−(4/3)Ng, b2 =22/3−(4/3)Ng −(1/6)NH, and b1 = −(4/3)Ng −(1/10)NH of op
cit: General Consequences of Grand Unification, Ng being the number of generations of
fermions and NH the number of Higgs doublets in the electroweak sector. In the minimal
SU(5) SUSY model, with Ng =3 and NH =2, b3 becomes 3 rather than 7, so the evolution of
effective coupling constant α3 is slowed down. Consequently, the point MX, at which the
couplings are unified, is raised, and we find:
GeV102 16
X ×≈M
and αk (MX
2)~1/25. Fortunately, the successful prediction for mb/mτ =3 is hardly changed,
while the prediction for electroweak mixing angle, including higher-order corrections, is
increased slightly to:
003.0236.0)(sin 2
W
2
±=Mwθ
which is in good agreement with experiment.

60. In this chapter we shall describe the standard model of gravity (i.e., Einstein’s general
theory of relativity), which clearly has to be included in any complete description of the
forces of nature. It would perhaps be surprising if we were able to obtain a unified theory
of all the other forces but could not include gravity. Now, there are several further
indications that gravity ought to be incorporated into a unified theory:
60
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General Relativity
1. It is, as we shall see, a gauge theory, with a structure that is similar, though not
identical, to that of the theories in PART VIII – THE STANDARD MODEL: Gauge
Theories;
2. By itself, it does not yield a finite or renormalizable quantum field theory;
3. It arises naturally in some attempts to go beyond the Standard Model (e.g., through
local supersymmetry and superstrings);
4. Its basic mass scale, the so-called Planck mass, MP ≡(hc/GN)1/2 ≅1.2×1019 GeV/c2 is not
greatly different from the unification scale of the other forces which is generally found
to be around 1014-1016 GeV/c2;
5. General relativity is such a beautiful theory that it might suggest models for the other
forces of nature.
Although we shall try to give here a reasonably complete account of the fundamentals of
general relativity, starting from its basic principles, our treatment will necessary be rather
concise and to do so, we shall formulate the ideas mathematically through the tetrad
formalism. The advantage of this for our purpose are that it makes clear the manner in
which general relativity is a gauge theory, and that it provides the basis for discussing
the coupling of fermions to gravity, as we shall need to do when we study supergravity.

61. Gravity is unique among the forces of nature because it has the same effect on all
objects. This follows from the proportionality between the gravitational force on an object
and the mass of that object – a fact that is sometimes states as the equality of the
gravitational mass and the inertial mass. Precision tests of this equality were made by
Eötvös and, more recently, by Dicke, whose experiments show that a wide variety of
bodies experience the same acceleration in a given gravitational field regardless of their
mass or composition. One consequence of this equality is that the effect of any constant
gravitational field can be eliminated by working in a suitably accelerating coordinate
system (e.g., a freely falling elevator). This is called the weak equivalence principle.
61
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More generally, we can always choose coordinates such that locally the gravitational
field can be eliminated. This is the strong equivalence principle, which asserts that, in a
sufficiently small region of space, gravitational fields and accelerated frames of
reference have identical effects. From our particle physics perspective, where we are
concerned with the study of forces, it is probably better to think of the principle of
equivalence not as a way of eliminating gravity, but as allowing up to use any coordinate
system, not just inertial (i.e., nonaccelerating) system, to which we are restricted if
gravity is excluded.
The Principle of Equivalence

62. Einstein noted that an observer of mass m in a freely falling elevator (in a uniform
gravitational field g) would write down the same laws of nature as an observer in an
initial frame without gravity (i.e., minus mg hence the resultant weightlessness of the
observer).
62
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The coordinates of the accelerated observer (i.e., xi) are related to those of the inertial
observer by the familiar time-parabolic trajectory of kinematics:
)(2
2
jii
i
Vm
td
d
m xxg
x
−−= ∇∇∇∇
2
2
1
tii gxx +=
so that, plugging this into the equation of motion above, we get:
)(2
2
jii
i
V
td
d
m xx
x
−−= ∇∇∇∇
Einstein abstracted from this thought experiment a strong version of the equivalence
principle: The equations of motion have the same form in any frame, inertial or not.
In other words, it should be possible to write laws so that in two coordinate systems, xµ
and xµ(x), they take the same form.
_
_
Consider, for example, an elevator full of particles interacting through a potential
V(xi−xj) (the negative gradient of which, −∇∇∇∇i V, being the i-th force vector Fi). In the
inertial frame xi :

63. To express these ideas in mathematical form, we begin by choosing a set of coordinates
such that every point of space-time is labelled by {xµ}, with µ=0,1,2,3. In fact, none of
our results will be altered if we allow the number of space dimensions to be increased to
d−1>3, and this will be of importance in some of the later chapters.
63
µ
µ
nxdxd ˆ=
At each point of space-time we can define a set of four (or, generally d) vectors, nµ,
each of which is in the direction of one of our coordinate axes. This is illustrated, for two
dimensions, in the Figure. The nµ are unit vectors in the sense that their lengths
correspond to unit increments of the coordinates. Thus, the four-vector interval from the
point xµ to the adjacent point xµ +dxµ is given by:
General Coordinates
ˆ
A set of general coordinates on a plane.
The lines x0 =1, x0 = 2, x1 =1, x1 =2 are
shown. Also shown are the coordinate
unit vectors at a point [1,1].
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We now defined the metric-tensor gµν associated with these
coordinates by introducing a scalar product:
µννµµν gnng =⋅≡ ˆˆ
ˆ
where summation over repeated indices is implied.
Then the distance or interval ds, between xµ and xµ+dxµ, is
given by the scalar product:
νµ
µννµ xdxdgnnsd =⋅≡ ˆˆ2
x0 =1
x1 =1
x0 =2
x1 =2
n1ˆ
n0ˆ

64. At each point of space-time, xµ, we also introduce a local inertial coordinate system
– this is the free falling elevator system (i.e., the system in which there is no gravitational
force). We can define this system by a set of vectors ên (with Latin indices n=0,1,2,3,…),
called a tetrad (a term that is clearly appropriate when d=4), which satisfy:
64
nmnm η=⋅eˆeˆ
where ηmn is the usual flat-space metric tensor of Minkowski space:
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),1,1,1,1(diag K−−−+=nmη
Showing a choice of ê0, ê1 at the point of the
previous Figure. The components e0
0, e0
1 are
also shown.
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We can express any member of a tetrad in terms of the unit
vectors of the general coordinate system by putting (see Figure):
thereby introducing the vierbeins en
µ. (or, in higher-dimensional
space, vielbeins). On comparing nµ ⋅nν =gµν , êm ⋅ên =ηmn , and ên =
en
µ nµ , we find:
µ
µ
nenn ˆeˆ =
n1
ˆ
n0
ˆ ê0
ê1
e0
0
e0
1
ˆ ˆ
ˆ
nmnm eeg ηνµ
µν =

65. We expect that it will be possible to choose the êm to vary continuously from point to
point of space-time, so that the em
µ are differentiable functions of xµ. In any local region
this will be the case provided there are no discontinuous changes in the gravitational
field. However, depending on the topology of the manifold, such a choice may not be
possible globally (i.e., over the whole of the space-time manifold). As a trivial example of
such a topological restriction, we recall the fact that on the two-dimensional surface of a
sphere in three dimensions it is not even possible to define a unit-vector field
continuously over the surface (i.e., hairy ball cannot be combed smoothly). In such
cases we divide the manifold up into overlapping patches in each of which we define
continuously varying tetrads.
65
µ
νν
µ
δ=m
m ee
We now introduce the inverse vielbein, em
ν , by:
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where δ µ
ν is the Kronecker delta (i.e., equal to zero unless µ =ν, when it is 1). Then ên =
en
µnµ above gives:ˆ
n
n
en eˆˆ νν =
If we multiply this by em
ν and compare to ên =en
µnµ , we deduce that:ˆ
n
m
n
m ee δν
ν
=
Then, on multiplying gµν =em
µ en
ν ηmn above by em
µ en
ν and using this last equation, we find:
nm
nm
eeg ηνµµν =
Thus, the vielbein em
µ can be regarded as the square-root of the metric.
Local Lorentz Frames

66. It is useful now to introduce the flat metric ηmn, which is defined to be numerically
identical to ηmn =diag(+1,−1,−1,−1,…), but with superscripts indices. Then, clearly:
66
n
mmp
np
δηη =
Latin indices can now be raised and lowered by ηmn or ηmn , respectively. For example,
we can define:
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c.&,µµ
η n
nmm
ee =
Similarly, for the general space coordinate it is convenient to introduce the inverse
metric tensor, gµν, defined by:
µ
νλν
λµ
δ=gg
Greek indices are raised or lowered using gµν or gµν, respectively. For example, we
define:
µ
νµµ
µ
νµµ xgxege mm
== and
&c. It is usual to refer to the upper-index components (e.g., xµ) as being contravariant,
and lower-index components (e.g., xµ) as being covariant.

67. As a useful, and simple, exercise we can now show that gµν =em
µ en
ν ηmn above implies:
67
nm
nm eeg ηνµνµ
=
In all these expressions it is important to remember that there is a complete distinction to
be made between the flat (i.e., Minkowski space) indices, for which we are using the
Latin letters m, n, …, and the curved indices, which are denoted by Greek letters µ, ν, ….
The summation always involve two indices of the same alphabet, one upper and one
lower.
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So far in the chapter we have discussed a mathematical description of space-time.
Physics will enter through the hypothesis of the invariance of physical laws: in particular
that the laws of physics must be invariant under general coordinate transformations and
under local Lorentz transformations (i.e., under rotations or the tetrads). In other words,
the validity of the fundamental equations must not depend upon any particular choice of
the coordinates xµ or of the tetrads ên. It is the fact that the choice of tetrads can be
made independently at each point of space-time (i.e., that we can make local Lorentz
transformations that are functions of xµ) that provides the link between gravity and the
local gauge theories described PART VIII – THE STANDARD MODEL: Gauge Theories.

68. We consider, first, the effect of a local Lorentz transformation (LLT). This a rotation of the
tetrad:
68
n
n
mmm eˆeˆeˆ Λ=→
The condition that this simply rotates the tetrad is that êm ⋅ên =ηmn above remains true in
the new basis (i.e., that êm ⋅ên =ηmn) which implies that:
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Local Lorentz Transformations
nmqp
q
m
p
m ηη =ΛΛ
(N.B., If the η factor were replaced by Kronecker δ s this equation would tell us that Λm
n
is an orthogonal matrix. The η factors occur because Λm
n is a representation of O(1,3)
rather than O(4)). Using ηnpηmp =δ n
m we can write this last equation as:
k
npn
pk
δ=ΛΛ
A familiar example of such local Lorentz transformations is a boost by velocity v along
the 3-axis, where the corresponding rotation matrix is:












=Λ
γγβ
γβγ
00
0100
0010
00
n
m
with as usual:
2
1
1
β
γβ
−
≡≡ and
c
v

69. To find the corresponding transformation rule for the components of a vector we
consider, for example:
69
m
m
xx eˆ≡
In the transformed system, this becomes:
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n
n
m
m
m
m
xxx eˆeˆ Λ≡=
on using êm →êm =Λm
nên above. Comparing this last equation with x ≡xmêm above we find:
mn
m
n
xx Λ=
which, from Λm
pΛn
qηpq =ηmn, inverts to give:
nm
n
m
xx Λ=
All contravariant components of a vector transform in this way.
We now consider the transformation of the derivative of a scalar. We have:
m
k
k
mm
k
kmm
x
x
x
x
xxx ∂
∂
Λ=
∂
∂
∂
∂
=
∂
∂
→
∂
∂ φφφ
or, in a more concise notation:
)()( φφ k
k
mm ∂Λ=∂
Thus, a derivative with respect to xm transforms as a lower index (covariant) component.
_

70. From êm →êm =Λm
nên above we see that the vielbein emµ transforms covariantly:
70
µµ n
n
mm ee Λ=
from which it is easy to see that the em
µ transforms covariantly:
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km
k
m
ee µµ Λ=
as we expect from the position of the indices.
For many purposes it is adequate to consider only infinitesimal local Lorentz
transformations, that is, to put:
n
m
n
m
n
m λ+=Λ δ
where λm
n are small. Then, working only to first order, Λm
pΛn
qηpq =ηmn above yields:
nmmn
q
n
p
m
q
n
n
mqp λ−=λ=λ+λ or0)( δδη
We write the change in the vielbein under such an infinitesimal transformation as:
nm
n
m
ee µµδ λ=)(LLT

71. We turn now to the effect of a general coordinate transformation (GCT), which we write
as:
71
)(xxxx µµµµ
ξ+=→
where ξµ(x) are some continuous functions of x. The components of a vector dx
transform as:
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General Coordinate Transformations
ν
ν
µ
µ
ν
ν
ν
µ
µµ ξ
δ xd
xd
d
xd
xd
xd
xdxd 







+=







=→
This equation gives the transformation associated with any contravariant (upper) Greek
index. Thus, for example, under the transformation xµ →xµ =xµ +ξµ above we have:
)()()( xe
xd
d
xexe mmm
ν
ν
µ
µ
ν
µµ ξ
δ 







+=→
where we have kept only terms of first order in ξµ. So the change in em
µ is:
ν
ν
µ
µ ξ
δ mm e
x
e
∂
∂
=)(GCT
Using em
ν en
ν =δm
n above, which of course remains true in any system, we can then
readily deduce:
mm
e
x
e νµ
ν
µ
ξ
δ
∂
∂
−=)(GCT
_

72. The equations of physics will contain derivatives of tensor fields and it is therefore
necessary to define covariant derivative that have the correct transformation properties
under local Lorentz transformations and general coordinate transformations (c.f., Dµ ≡∂µ
+ieAµ(x), Dµ ≡∂µ + (ig/2)Σkτk Wk
µ, &c. of PART VIII – THE STANDARD MODEL).
Because we are concerned here both with two types of transformation, we will need two
connection fields. Thus, we define a covariant derivative of em
ν by:
72
mm
n
mmm
eeeeD νµρ
ρ
νµνµνµ ω+Γ−∂=
Here Γρ
µν is the connection associated with the general coordinate transformations. It is
referred to as a Christoffel symbol and, consistent with its lack of flat indices, we take it
to be a scalar under local Lorentz transformations (N.B., Γρ
µν is not a tensor under
general coordinate transformation):
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Covariant Derivative
0)( =Γρ
νµδLLT
Similarly ωµ
m
n , which is associated with local Lorentz transformations, is a vector
under a general coordinate transformation (i.e., under δGCT =−(∂ξν/∂xµ)em
ν above):
)()( m
n
m
n
x
νµ
ν
µ
ξ
δ ω
∂
∂
−=ωGCT
The fields ωµ
m
n are called the spin-connection (e.g., they are analogous to the vector
fields Wk
µ(x) introduced in Dµ ≡∂µ + (ig/2)Σkτk Wk
µ). Here the group of local transforma-
tions is the Lorentz group, whose elements are labeled by (m,n) (e.g., which play a
similar role to the index k in Wk
µ ).

73. As in op cit: Gauge Theories we now calculate how the ωµ
m
n transform under local
Lorentz transformations by requiring that Dµem
ν should have the correct transformation
property (i.e., in accordance with δLLT(em
µ)=λm
nen
µ and Dµ em
ν =∂µ em
ν −Γρ
µν em
ρ +ωµ
m
nen
ν
above):
73
)()()( pn
p
nnm
n
nm
n
m
eeeeDeD νµρ
ρ
νµνµνµνµδ ω+Γ−∂λ=λ=LLT
Alternatively, we may compute δLLT acting on each term of Dµ em
ν =∂µ em
ν −Γρ
µν em
ρ +
ωµ
m
nen
ν above separately. Using δLLT(em
µ)=λm
nen
µ and δLLT(Γρ
µν)=0 above we find:
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nm
n
pn
p
m
n
nm
n
nm
n
m
eeeeeD νµνµρ
ρ
µννµνµ δδ )()()( ω+λω+λΓ−λ∂= LLTLLT
Comparison of these two above equations yields:
m
p
p
n
p
n
m
p
m
n
m
n µµµµδ ωλ−ωλ+λ−∂=ω )(LLT
which, not surprisingly, has a similar form to the result quoted in δWk
µ =−(1/g)∂µαk(x)−
Σij αiWj
µ(x) of op cit: Yang-Mills Gauge Theories. We can make an analogous
calculation for Γρ
µν and we find:
σ
νµ
ρ
σ
ρ
σµ
σ
ν
ρ
νσ
σ
µ
ρ
νµ
ρ
νµ ξξξξδ Γ∂+Γ∂−Γ∂−∂−∂=Γ )()()()(GCT
The first term on the right-hand side of δLLT and δGCT above show that the connections
do not transform as tensors.

74. Before we attempt to write down the Lagrangian for the fields em
µ, ωµ
m
n and Γρ
µν we
have to consider that in Einstein’s theory of relativity these are not independent fields.
Instead, the two connections are postulated to be functions of the em
µ and the conditions
δLLT(ωµ
m
n) and δGCT(Γρ
µν) above then lead to unique expressions for these connections.
In particular, remembering gµν =em
µ en
ν ηmn, it is straightforward to show that:
74
)(
2
1
νµσσµνσνµ
σρρ
νµ gggg ∂−∂+∂=Γ
satisfies δGCT(Γρ
µν) above. This solution has the important symmetry property that:
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ρ
µν
ρ
νµ Γ=Γ
which implies:
φφ µννµ DDDD =
for any scalar field φ. When the general coordinate transformation connection satisfies
Γρ
µν=Γρ
νµ above, the space is said to have zero torsion.
We shall not write down the expression for ωµ
m
n analogous to Γρ
µν=½gρσ (∂µ gνσ +∂ν gµσ
− ∂σ gµν ) above, but shall instead regard ωµ
m
n as an independent field.

75. Recall that in PART VIII – THE STANDARD MODEL we introduced field-strength
tensors associated with the gauge vector potentials (e.g., Fµν=∂µ Aν − ∂ν Aµ and Wi
µν=
∂µ Wi
ν −∂ν Wi
µ −gΣjk εijk Wj
µWk
ν ). In a exactly analogous way we define it here:
75
npm
p
npm
p
mnmnmn
R µννµµννµνµ ωω+ωω+ω∂−ω∂=
This is a two-index tensor under both local Lorentz and general coordinate
transformations. (N.B., We do not need to introduce the Γρ
µν connections because they
would cancel, since Γρ
µν=Γρ
νµ whereas Rµν
mn is antisymmetric under µ ↔ ν).
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The Einstein Lagrangian
We now form a scalar quantity R that is first-order in the derivatives:
µν
νµ nm
mn
eeRR ≡
(R has dimensions [length]−2) and write for the Einstein Lagrangian:
2
2κ
Re
E −=L
(κ 2 has dimensions [length][energy]−1 or [mass]−1 when h=c=1) where e (N.B., compen-
sates for the Jacobian introduced by the transformation of x coordinates) is defined by:
)det(
)det(
1
)det( µνρρ g
e
ee
m
m
−==≡
from gµν =em
µ en
ν ηmn above, and is introduced so the LE is a scalar density. Since R is
invariant under general coordinate transformations we see that the (scalar) action is:
∫= EE xdS L4

76. The classical equations of motion for the fields ωµ
m
n and en
µ are obtained by
minimizing the action SE=∫d4x LE above with respect to variations in the field. On varying
with respect to ωµ
m
n, we find:
76
p
ppnmmmnnnmnm eeeeeeeeeee µρσσρ
σρ
µννµ
ν
µννµ
ν
µ )(
2
1
)(
2
1
)(
2
1
∂−∂−∂−∂−∂−∂=ω
This is a purely algebraic equation (because, in the Lagrangian, derivatives of ωµ
m
n only
appear in linear terms) and it allows us to eliminate the ωµ
m
n in favor of the em
µ.
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77. Minimization of SE =∫d4x LE above with respect to em
µ can immediately be seen to
require:
77
called the Einstein equation, where: 2017
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0)( =++ ν
µρ
ρµ
µ
µν
µ
νµ
δ
δ
m
np
np
n
mn
n
nm
e
e
eeReeReR
Then, using the general result that for any matrix M, δ [det(M)]/δMjk =det(M)(M−1)kj, we
have, from em
ν en
ν =δm
n and e=det(em
ρ)=[det(em
ρ)]−1 above:
m
m
ee
e
e
νν
δ
δ
−=
(N.B., The minus sign occurs because e is the inverse of the determinant of the em
ν ). It
follows from this last result that:
0
2
1
=− m
n
nm
eReR ν
µ
νµ
where we have used the antisymmetric properties of Rµν
mn, which follow from its
definition above. On multiplying this last equation by emσ , and using gµν =em
µ en
ν ηmn
above, the equation becomes:
0
2
1
=− νµνµ gRR
µ
ρ
σρνµ mn
nm
eeRR ≡

78. The equation Rµν −½Rgµν =0 is the Einstein equation for em
µ , and hence for the metric
gµν , in empty space. The equation is nonlinear and has many solutions, some of which
we shall meet later.
78
The procedure we have used in this chapter, in which ωµ
m
n and em
µ are regarded as
independent fields, is called the first-order or Palatini formalism. An alternate procedure
is to postulate that the spin connection is a function of the em
µ and seek a function that
satisfies δLLT (ωµ
m
n)=−∂µλm
n+λm
pωµ
p
n−λp
n ωµ
m
p above. The unique solution is that given
by ωµmn =½em
µ(∂µenν −∂νenµ)−½en
ν(∂µemν −∂ν emµ)−½em
ρen
σ(∂ρepσ −∂σ epρ )ep
µ above. If we
substitute this into the Lagrangian, we then have a Lagrangian that contains only the
field em
µ. This procedure is referred to as using the second-order formalism. At the
classical level, and in the absence of matter, these two procedures are identical.
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The quantized versions of the two formalisms are not necessary the same, however,
since the first-order formalism allows quantum fluctuations of ωµ
m
n about its classical
value. Also, with the inclusion of matter in the Lagrangian, the first order formalism gives
an ωµ
m
n that in general depends upon other fields, so the two formalisms then give
different results, even classically. However, these differences only show up in higher
orders of the gravitational coupling, and so affect only very short distances, of the order
of the Planck length lP =(hGN /c3)1/2 ≈1.6×10−35 m.
When the connections are chosen to satisfy Γρ
µν=½gρσ (∂µ gνσ +∂ν gµσ − ∂σ gµν ) and ωµmn
=½em
µ(∂µenν −∂νenµ)−½en
ν(∂µemν −∂ν emµ)−½em
ρen
σ(∂ρepσ −∂σ epρ )ep
µ above, then the
tetrad is covariantlyconstant (i.e.,Dµ em
ν =0).The converse is also true:tetrad postulate.

79. In this chapter we collect together a few properties of the R-symbols introduced earlier.
This will provide a link between our approach and more conventional treatments (c.f.,
PART II – MODERN PHYSICS).
79
σρνµσρνµ nm
nm
eeRR =
First, we remove the flat indices from Rµν
mn by defining:
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The Curvature Tensor
This quantity is called the Riemann-Christoffel curvature tensor.
Inserting ωµmn =½em
µ(∂µenν −∂ν enµ)−½en
ν(∂µemν −∂ν emµ)−½em
ρen
σ(∂ρepσ −∂σ epρ )ep
µ
above into the definition of Rµν
mn=∂µων
mn −∂νωµ
mn +ωµ
m
pων
pn +ων
m
pωµ
pn above, we are
able to express this tensor in terms of the emρ and hence in terms of the metric, and find:
τ
σνρµτ
τ
σµσµτρνσµρµσνσνρµσµρνµνρσ ΓΓ−ΓΓ+∂∂+∂∂−∂∂−∂∂= )(
2
1
ggggR
Some symmetry/antisymmetry relations ready follow:
ρσµνµνρσµνσρµνρσµρσνµνρσ RRRRRR =−=−= and,
A consequence of such relations is that, in four-dimensional space-time, only 20 of the
44 =256 components of Rµνρσ are independent. These 20 components completely define
the curvature properties of space-time. All the elements of the Riemann-Christoffel
tensor are zero if, and only if, the space is flat (i.e., if coordinates can be chosen so that
gµν =ηµν ). (N.B., If gµν =ηµν is not true in a given coordinate system we cannot assume
that the space is not flat – the only way we can be sure is to calculate the Rµνρσ and
show that at least one of the components differs from zero).

80. The two-index tensor Rµν ≡Rρν
mnen
ρemµ that appears in the Einstein equation Rµν −½Rgµν
=0 is called the Ricci tensor. To relate it to Rµνρσ we invert Rµνρσ=Rµν
mnemρenσ to obtain:
80
σρ
σρνµνµ
nmmn
eeRR =
Then Rµν ≡Rρν
mnen
ρemµ becomes:
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τσ
σµντµ
τσρ
σρντνµ gReeeeRR nm
nm
==
using em
µen
ν =δm
n above, or:
τ
τνµνµ RR =
The Ricci tensor is readily seen to be symmetric:
µννµ RR =
Contraction of the two indices of the Ricci tensor yields the scalar curvature R:
µ
µνµ
νµ
RRgR ==
The successive steps:
τν
ντσµντ
τσνµ
νµ
νµ
nm
mn
eeRRggRgR ===
show that R is the same as that introduced earlier as R≡Rµν
mnem
νen
µ above.
It is important to emphasize that only the Riemann-Christoffel tensor Rµνρσ with its
20 independent components contains all the information about space, since Rµν only
has 10 components and R just 1. Thus, when Rµνρσ is identically zero, both Rµν and R
are zero, but the converse does not necessarily hold. A space for which Rµν =0 is said
to be Ricci-flat, but a Ricci-flat space is not necessarily flat.

81. Finally, we give a useful relation for Rµν that follows from Rµν=Rρ
µνρ and Rµνρσ=
½(∂ν ∂ρ gµσ−∂µ ∂ρ gνσ−∂ν ∂σ gµρ+∂µ ∂σ gνρ)+Γτµσ Γτ
νρ−Γτµρ Γτ
νσ above:
81
σ
ρν
ρ
σµ
σ
σρ
ρ
νµ
ρ
νµρ
ρ
ρµννµ ΓΓ+ΓΓ−Γ∂−Γ∂=R
As a simple example to illustrate the use of this formalism we consider the two-
dimensional surface of a sphere in three-dimensional Euclidean space. If we denote by
yi (i=1,2,3) the Cartesian coordinates of this 3-space and use the metric {diag(+1,+1,
+1)}, then the surface of the sphere of radius a and center at the origin is given by:
22
ay
i
i =∑
Coordinates [r,θ] of the two-dimensional
surface of a sphere of radius a.
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To describe this two-dimensional space we use polar
coordinates [r,θ] as shown in the Figure. Then
where 0 ≤r≤1. A line element has length given by:
2
321 1sincos rayrayray −=== and, θθ
2222
2
2
22
1
)( θdrard
r
a
ydsd
i
i +
−
== ∑
It follows that the metric tensor is diagonal, with diagonal
elements:
22
2
2
1
1
1
ra
g
g
r
a
g
g rrrr ==
−
== θθθθand
y1
y2
y3
a ar
θ

82. From Γρ
µν=½gρσ (∂µ gνσ +∂ν gµσ − ∂σ gµν ) above we then calculate the elements of the
Christoffel symbol:
82
)1(0
0
1
0
1
2
rr
r
r
r
rr
r
r
r
rrrr
r
rr
−−=Γ=Γ=Γ
=Γ=Γ=Γ=Γ
−
=Γ
θθθθ
θ
θθ
θ
θ
θ
θ
θ
and
,,,
,
Hence, from Rµν =∂ν Γρ
µρ −∂ρ Γρ
µν −Γρ
µν Γσ
ρσ +Γρ
µσ Γσ
νρ above, the components of the Ricci
tensor become:
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2
2
1
1
rR
r
R rr −=
−
−= θθand
and the scalar curvature R=gµνRµν above is given by:
2
2
a
RgRgR rr
rr
−=+= θθ
θθ
which is a constant, as we expect for a sphere.
It is worth noting that, although clearly all points on this surface are equivalent, the
coordinate system does not reflect this equivalence. An unfortunate notational is the R is
called the curvature, but it is proportional to the inverse of the square of the radius of
curvature, a, via R=−2/a2.

83. In order to relate the purely geometrical concepts that we have discussed so far to
physics, we must introduce matter into the action (i.e., define a new action S=SE +SM
where SM describes the fermions, gauge fields, &c., with the requirement that SM also
respect the general coordinate and local Lorentz transformations).
83
∫= MM xdS L4
We can write SM in terms of the Lagrangian density:
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The Inclusion of Matter
and this can be expressed in terms of a scalar U as (c.f., LE =−eR/2κ 2):
UeM =L
where e=det(em
ρ)=[det(em
ρ)]−1. Then when we vary the action with respect to em
ν, we find:
ν
νν
δ
δ
δ
δ m
m
m
M eUe
e
U
exdS ∫ 







−= 4
where we have used δ e/δ em
ν =−eeν
m above. So, if we define Tµν by:








−−= Ue
e
U
eT m
m
m ννµµν
δ
δ
the Einstein equation Rµν −½Rgµν=0 is replaced by:
νµνµνµ κ TgRR 2
2
1
−=−

84. The factor κ 2, which provides the coupling between matter and the gravitational field,
was introduced by LE =−(e/2κ 2)R. Tµν is in fact the familiar energy-momentum tensor as
will become clearer when we consider specific examples. In the weak field limit, this last
equation reduces approximately to Newton’s theory of gravitation. Also, this last
equation shows that, in the presence of a nonzero energy-momentum tensor, Rµν ≠0 so
that the space cannot be Ricci-flat.
84
Λ= 2cosm
κ
e
L
First, we consider a few examples for LM. The simplest is to include a constant
background energy density:
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where Λ is the so-called cosmological constant. Then Tµν=−emµ(δ U/δ em
ν −em
ν U) above
yields:
Λ= 2
κ
νµ
νµ
g
T
and the Einstein equation becomes:
0
2
1
=Λ+− νµνµνµ ggRR
Note that gravity, because it couples to the energy content of the gravitational field, gives
a meaning to the absolute value of the potential energy, a meaning that is absent from
nongravity physics where it is only the forces (i.e., the derivatives of the potential
energies) that are significant.

85. As a second example of a matter Lagrangian, we consider a massless spin-½ field. In
flat space the Lagrangian is given by (c.f., PART VIII – THE STANDARD MODEL:
Fermions):
85
ψγψ µ
µ
∂= i)½(
L
In order to generalize this we must replace the derivative ∂µψ by the appropriate
covariant derivative Dµψ, which must transform as a spinor under a local Lorentz
transformation and as a vector under a general coordinate transformation:
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ψψδ µµ D
i
D mn
mn
Σλ−=
4
)(LLT
where the λmn are the infinitesimal parameters of Λm
n=δm
n +λm
n, and (c.f., δGCT(em
µ)=
−(∂ξ ν/∂xµ)em
ν ):
ψ
ξ
ψδ νµ
ν
µ D
x
D
∂
∂
−=)(GCT
Note that in δGCT(Dµψ) above, Σnm is defined in terms of the usual flat-space Dirac
matrices as in op cit: Fermions:
)(
2
1
mmmnmn i γγγγ −=Σ

86. These equations are satisfied by putting:
86
which explains why ωµ
mn is referred to as the spin connection. The new Lagrangian
density becomes:
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ψψψ µµµ mn
nmi
D Σ−∂= ω
4
ψγψ µ
µ
Dei=)½(
L
where e, defined in e=[−det(gµν)]1/2 above, is introduced because L is a scalar density,
and where the:
n
ne γγ µµ
=
are curved-space γ -matrices that, in general, are functions of xµ.

87. As our final example, we consider a single classical particle of mass m. The path of
such a particle can be written in parametric form:
87
)(τµµ
xx =
where the parameterτ labels the position on the trajectory. A possible choice forτ would
be the time coordinateτ =x0, but this is not essential. The scalar Lagrangian may be
taken as (c.f., op cit: The Lagrangian Formulation of the Field Equations):
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νµ
νµ xxgmLp &&−=
where the dot denotes a derivative with respect toτ. We can understand the origin of Lp
above if we consider the situation in which there are no gravitational fields so that we
can replace gµν by ηµν . Then, using x0 forτ, and working in the nonrelativistic limit (i.e.,
at small velocities), we have:
L&& ++−≅−−= 22
2
1
1 xx mmmLp
which, if we disregard the irrelevant constant rest mass m, is, as we expect from op cit:
The Lagrangian Formulation of the Field Equations, the classical kinetic energy.

88. The action:
88
∫−= νµ
νµτ xxgdmS p &&
is proportional to the length of the trajectory between its end points. This geometrical
interpretation ensures the readily verifiable reparametrization invariance of Sp (i.e.,
invariance under τ → f (τ)).
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In order to write the action as an integral over all space, we introduce a four-
dimensional delta function:
∫ ∫ −−= )]([44
τδτ νµ
νµ xxxxgdxdmS p &&
We can now calculate δ Sp/δ em
ν as before and we find for the single-particle energy-
momentum tensor:
∫ −= )]([4
τδτ
σρ
σρ
νµ
µν xx
xxg
xx
dmT p
&&
&&

89. By minimizing the action with respect to variations in xµ(τ), we determine the equation
for the path. This is most easily expressed when we chooseτ to be the proper time,
which is the path length defined by:
89
νµ
νµτ xdxdgd =
in the nonrelativistic limit, with zero gravitational field, τ becomes the ordinary time x0.
With this choice, we obtain:
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0
2
1
=
∂
∂
−
∂
∂
+ νµ
σ
νµνµ
ν
σµµ
σµ xx
x
g
xx
x
g
xg &&&&&&
On multiplying by gσρ this yields:
0
2
1
=





∂−∂+ νµ
νµσσµν
ρσρ
xxgggx &&&&
which, using Γρ
µν=½gρσ (∂µ gνσ +∂ν gµσ − ∂σ gµν ) above, can be written:
0=Γ+ νµρ
νµ
ρ
xxx &&&&
This equation generalizes Newton’s first law of motion (i.e., xρ =0) to describe, in an
arbitrary coordinate system, the motion of a particle in a gravitational field. The path
is a geodesic, which is the extremal length path between the end points. Length of
course is measured with respect to the metric gµν , and it is this that contains the
effect of gravity.
⋅⋅

90. In order to illustrate the relation between general relativity and Newton’s theory of
gravity, we shall consider the motion of a particle with mass m is a background
gravitational field (i.e., a field determined by some given external source). We ignore the
effect of the particle itself on the gravitational field since this contributes only a second-
order correction to the motion of the particle.
90
)1( <<+= νµνµνµνµ η hhg
The path of the particle is given by xµ(τ), and we shall again use τ =x0. Since we shall
consider nonrelativistic motion, we can neglect xi (i=1,2,3) in comparison to x0 ≡1. The
gravitational field is taken to be static, so that ∂0 gµν=0, and weak, which means that we
can choose an almost Cartesian coordinate system such that:
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The Newtonian Limit
⋅
Working to lowest order in hµν, the equation of motion xρ +gσρ(∂ν gµσ −½∂σ gµν)xµxν =0
above becomes:
⋅ ⋅
0
2
1 000
2
0
=
∂
∂








= x
x
h
d
xd
x
i
i
&&&& and
τ
In the usual 3-space notation, this last equation implies:
00
2
1
h∇∇∇∇−=x&&
Thus, the particle obeys Newton’s second law of motion with a potential given by:
00
2
1
hmV =

91. In order to find h00 we substitute gµν =ηµν +hµν into Γρ
µν=½gρσ (∂µ gνσ +∂ν gµσ − ∂σ gµν )
above and use Rµν=∂ν Γρ
µρ −∂ρ Γρ
µν −Γρ
µν Γσ
ρσ +Γρ
µσ Γσ
νρ above to evaluate Rµν to first
order in hµν. Then the Einstein equation Rµν −½Rgµν=−κ 2Tµν above becomes:
91






−−=∂∂−∂∂−∂∂+∂∂ λ
λνµνµσνρµµρσννµσρσρνµ
σρ
ηκη TThhhh
2
1
)(
2
1 2
Putting µ =ν =0, and using the fact that hµν is independent of x0, we find:
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





−=∇ λ
λκ TTh
2
1
2 00
2
00
2
For the source (i.e., T00) we consider a particle of mass M, fixed at the origin. Then
according to Tp
µν =m∫dτ [xµ xν /(gρσ xρ xσ )1/2]δ 4[x−x(τ)] above we have:⋅ ⋅ ⋅ ⋅
)(3
00 xδMT =
and the other components are zero, so ∇2h00 = 2κ 2(Tµν −½Tλ
λ) above yields:
xπ
−=
4
2
00
M
h
κ
or, from V=½mh00:
xπ
−=
8
2
mM
V
κ
Thus, we obtain Newton’s gravitational potential provided we identify κ 2/8π=GN, GN
being Newton’s gravitational constant.

92. We can rewrite Einstein’s equation Rµν−½Rgµν =−κ 2Tµν above with this constant as:
92
νµνµνµνµ TGggRR Nπ−=Λ+− 8
2
1
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in which we have also included the cosmological constant of Rµν −½Rgµν+Λgµν =0 above.
The magnitude of Newton’s constant is:
213
skgm −−−
×= 11
10)85(67259.6NG
but the error of 128 parts per million (ppm) is very large compared to that of the
electromagnetic coupling α ≡e2/4πεohc=[137.0359895(61)]−1 (dimensionless) which is
known to 0.045 ppm.

93. It is natural that we should try to turn the global supersymmetry of the first few chapters
of this work into a local supersymmetry, just as we have done with other symmetries by
allowing the spinorial parameter ε (e.g., δ A=εψ, δ B=iεγ5ψ, and δψ =−iγ µ∂µ(A+iγ5B)ε of
the The Wess-Zumino Model chapter) to become ε(x), a function of the coordinates. The
original Lagrangian is then no longer invariant under SUSY transformations, but the
invariance can be restored by adding terms involving extra fields, analogous to the
introduction of vector gauge fields in PART VIII –THESTANDARD MODEL:Local Gauge
Invariance in QED, Yang-Mills Gauge Theories and Quantum Electrodynamics (QCD).
93
εγγψδψγεδψεδ µ
µ
)]([ 55 BiAiiBA +∂−=== and,
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To see in more detail what is involved consider the SUSY Lagrangian:
Local Supersymmetry
ψψφφ µ
µ
∂/+∂∂= i
2
1
*
2
1
L
which describes a massless, free multiplet (φ,ψ,F), in which ψ is a Majorana spinor, φ
is a complex scalar field (i.e., φ =(1/√2)(A+iB) of the The Superpotential chapter), while
F, the auxiliary field, has been eliminated by its equation of motion (F=0). We showed
that, apart from an unimportant total derivative, this Lagrangian was invariant under the
global SUSY transformations, the unchanged δ A=εψ, δ B=iεγ5ψ, and the corrected δψ
=−iγ µ∂µ(A+iγ5B)ε +(F+iγ5G)ε of the The Wess-Zumino Model chapter:
In fact, we found that (c.f., The Wess-Zumino Model chapter):






+∂/∂= ψγγεδ µ
µ )]([
2
1
5BiAL
_ _
_ _

94. However, with a local SUSY transformation (i.e., in which ε depends on x) we obtain, in
addition to δ A, δ B and δψ above, a contribution:
94
2017
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and the dependence on ∂µε demonstrates that the Lagrangian L =∂µφ*∂µ φ +½iψ ∂ψ is
invariant under global SUSY transformations, but not under local SUSY transformations.
})]([{ 5 ψγγγψκ ν
µν
µ BiA+∂−=′L
})]([){( 5 ψγγγεδ ν
µν
µ BiA +∂∂=L
We can ensure local invariance too (c.f., op cit: Local Gauge Invariance in QED) by
adding to L a term:
where ψµ is a new Majorana field (N.B., we still have ψ =[ψ1 ψ2 ψ3 ψ4]T making ψµ =
[ψ1 ψ2 ψ3 ψ4]T
µ a 16-component field since µ =0,1,2,3) that transforms according to:
ε
κ
ψδ µµ ∂=
1
It is easy to see that the term proportional to ∂µε cancels when δ L ′ is added to δ L.
The massless field ψµ(x) plays the role of the vector gauge field Aµ, which we recall
transformed like δ A=−∂µα/e (c.f., Aµ →Aµ −(1/e)∂µα). There are however, two important
differences. First, since ψµ is a spinor with a vector index, it describes a spin-3/2 particle.
Second, unlike e, the coupling constant κ is not dimensionless; rather, it has dimensions
[mass]−1, as can be seen from L ′ above if we remember that ψµ has dimensions
[mass]3/2.
__
_

95. To complete the discussion, we must consider the change in L ′ due to the
transformations δ A=εψ, δ B=iεγ5ψ, and δψ =−iγ µ[∂µ(A+iγ5B)]ε above. We obtain:
95
2017
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where Tµν is the energy-momentum tensor:
ψγψφφηφφφφ νµλ
λ
µννµνµµν
∂+∂∂−∂∂+∂∂= iT **
εγψκ νµ
µν Ti=′L
This contribution to δ L ′ can only be canceled by adding to the Lagrangian another
term:
νµ
νµ Tgg −=L
in which we introduce a new (tensor) field gµν that requires to transform as:
εγψγψκδ µννµµν )(
2
1
+= ig
If we choose to identify gµν with the metric tensor gµν =em
µ en
ν ηmn of the General
Coordinates chapter, the expression Lg above is just the contribution of a scalar field to
the Lagrangian of general relativity (c.f., LM =eU of the The Inclusion of Matter chapter
with Tµν =emµem
ν U).
_ _

96. Remarkably, no other fields are required and it is now possible to construct a locally
supersymmetric Lagrangian using φ, ψ, ψµ , and gµν. The Noether method we are using
here (i.e., we add terms to compensate for changes in the Lagrangian), is not in fact the
best method of obtaining locally supersymmetric theories, but it does show that local
SUSY automatically requires the introduction of gravity into elementary particle theory. It
introduces a massless spin-2 particle (i.e., the graviton) together with its massless spin-
3/2 SUSY partner (i.e., the gravitino). This is an exciting development that suggests that
theory is now getting close to the goal which Einstein sought in the 1940s, the unification
of gravity with the other forces of nature. Just as the requirement of a local U(1) phase
invariance allowed us to derive Maxwell’s theory of electromagnetism in op cit: Local
Gauge Invariance in QED, so the requirement of local SUSY allows us to deduce
general relativity and gravitational interactions.
96
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The unification of gravity with SUSY could be beneficial to both theories. For, despite
considerable effort, the problem of finding a quantum theory of gravity by itself has
proved intractable, partly because the theory has a dimensional coupling constant and
so it is not renormalizable. It was hoped that, by combining gravity with SUSY, the
infinities would be canceled. Unfortunately, although the divergences are indeed
considered softened, this hope has not been fulfilled. However, the introduction of
gravity does help to solve some of the problems we encountered with SUSY; in
particular it can, by an analog of the Higgs mechanism, remove the unwanted massless
spin-½ Goldstino associated with the symmetry breaking.

97. We begin by discussing briefly the simplest possible SUGRA model that incorporates
just the gravity multiplet which consists of a spin-2 graviton (i.e., gµν) and a spin-3/2
Majorana gravitino (i.e., ψµ ).
97
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Recall that the Einstein Lagrangian is:
A Pure SUGRA Lagrangian
µν
νµ
κκ
nm
nm
eeR
eRe
22
)2(
22
−=−=L
where e=det(em
ρ)=[det(em
ρ)]−1 and Rµν
mn is given in terms of the spin connection ωµ
mn by
Rµν
mn=∂µων
mn −∂νωµ
mn +ωµ
m
pων
pn +ων
m
pωµ
pn, all of these results being from the The
Einstein Lagrangian chapter. Instead of gµν it is convenient to use the (vierbein) field em
µ,
which in the sense of gµν =em
µ en
ν ηmn may be regarded as the square-root of the metric. .
The kinetic-energy term for the massless spin-3/2 field ψµ is described by the (Rarita-
Schwinger) Lagrangian of PART VIII – THE STANDARD MODEL: Fermions:
σρνµ
σρνµ
ψγγψε ∂−= 5
)2/3(
2
1
L
When gravity is present, we must replace ∂ρ in this last Lagrangian by the covariant
derivative (c.f., The Inclusion of Matter chapter):
mn
nmi
D Σ−∂= µµµ ω
4
(N.B., Unlike Dµ em
ν =∂µ em
ν −Γρ
µν em
ρ +ωµ
m
nen
ν of the The Covariant Derivative chapter,
there is no need to include a term involving Γρ
µν since it would cancel due to εµνρσ ).

98. An obvious first guess for the SUGRA Lagrangian is to add L (2) and L (3/2), so that:
98
2017
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)2/3()2(
LLL +=
Remarkably, this works, since this Lagrangian is invariant under the local SUSY
transformations:
µµµµ ψγεκδε
κ
ψδ mm i
eD
2
1
−== and
which agrees with our earlier findings in δψµ=(1/κ)∂µ ε and δ gµν =(iκ/2)(ψµ γν +ψνψµ) of
the Local Supersymmetry chapter. It is important that the spin connection ωµ
mn (used in
Dµ) is the one which is obtained by minimizing the Lagrangian L above. The result is the
same as ωµmn =½em
µ(∂µenν −∂ν enµ)−½en
ν(∂µemν −∂ν emµ)−½em
ρen
σ(∂ρepσ −∂σ epρ )ep
µ of the
The Einstein Lagrangian chapter, except for the addition of terms involving ψµ that are of
higher order in κ. A consequence of this choice is that, although ωµ
mn varies under the
SUSY transformations, this variation does not affect L and so can be ignored in
verifying that the theory is supersymmetric.
_ _

99. The simple gravity multiplet (em
µ ,ψµ ) is adequate on-shell but it must be supplemented
by additional auxiliary fields in order that the off-shell algebra is closed. To determine the
number of auxiliary fields, we must count the number of fermionic and bosonic degrees
of freedom. Since the algebra contains gauge transformations, we are concerned only
with the gauge-invariant components. The field ψµ has 16 components, of which four can
be removed by local SUSY transformations, leaving 12 fermionic degrees of freedom.
However, for the field em
µ , which also has 16 components, four are removed by
translation and six by Lorentz gauge transformations, leaving only six bosonic degrees
of freedom. This mismatch of the fermionic and bosonic degrees of freedom means that
we need at least six auxiliary bosonic fields. It can be shown that the off-shell algebra
closes if we compensate these six missing boson fields by adding nonpropagating (np)
fields such as a scalar S, a pseudoscalar P, and an axial vector Aµ leading us to a pure
SUGRA Lagrangian of the form:
99
2017
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)(
32
1
2
222
52
np
)2/3()2(
mAPS
e
R
e
−+−∂−−=
++=
σρνµ
σρνµ
ψγγψε
κ
LLLL
where the index of Am is flat (i.e., Am =em
µ Aµ). Since only the squares of the auxiliary
fields occur, they can be eliminated, and so (in the absence of matter) L reduces to the
on-shell Lagrangian L = L (2)+ L (3/2).

100. To obtain a locally supersymmetric Yang-Mills theory we must couple the pure SUGRA
Lagrangian L =L (2)+ L (3/2)−(e/3)(S2 +P2 −Am
2) to a Lagrangian, such as:
100
2017
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where the gauge Lagrangian is:
Coupling SUGRA to Matter and Gauge Fields
]h.c.)(2[)( *
Gauge
*
GaugeGaugeChiral ++−+= jLji
aa
i
a
jji
a
i PTgDTg ψχφφφLLL








+
∂∂
∂
−
∂
∂
−∂/+∂= ∑∑∑∑ h.c.
2
1 22
2
Chiral
ji
j
L
i
Lji
i
i
i
i
L
i
L
i
i
C
WW
i ψψ
φφφ
ψψφµ
T
L
2
Gauge )(
2
1
)(
2
1
4
1 a
a
a
a
a
DDiFF +/+−= χχµν
µνL
both from the Supersymmetric Gauge Theory chapter and from the The Superpotential
chapter, the chiral matter Lagrangian:
that describes the gaugeand matter fields. This mess describesthe coupling of the gauge
supermultiplets (Aa
µ ,χa,Da) to the chiral matter multiplets (φi,ψ i,Fi). The first thing would
be to construct covariant derivatives, ∂ρ →Dµ (i.e., analogous to Dρ =∂ρ+(i/4)ωρ
mnΣmn for
ψµ in the A Pure SUGRA Lagrangian chapter) which enable the graviton to couple to the
spin-1, spin-½ and spin-0 fields. Then one adds to the messy Lagrangian the terms that
are required to ensure local SUSY. If you’ve survived this by now, you then eliminate all
the auxiliary fields (Fi,Da,S,P,Aµ ) in favor of the dynamical fields. Finally, one identifies
the potential of the scalar fields and examine the resultant breaking of the local SUSY.

101. Following through these steps is technically very difficult and the resulting Lagrangian
is quite complicated (c.f., E. Cremmer, B. Julia, J. Scherk, P. van Nieuwenhuisen, S.
Ferrara and L. Girardello, Phys. Lett. 79B (1978), P. 231):
101
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where details of the derivation can be found in the above reference.
χγγχ
ψγγχψγψε
χγγχψγγψψγψεχγγχ
χχχγψψψ
φφχχψγγψε
µ
µµµµ
µ
νµ
νσρν
µνρσ
µµ
µµ
µ
µ
ν
νµ
µ
µµσρνµ
σρνµ
5
2
555
2
2
*
5
)(
1
)(
4
ˆ
2
1
)(
8
1
)(ln
21)(
64
1
16
ˆˆ
ˆˆ
2
1ˆ
2
1
e
3
e22
1
2








∂−∂+∂−∂+
∂−+∂−∂−








+−





−+
















−++⋅
−
−Σ+








−−∂∂+/−−−=
jjj
ii
j
iij
i
jj
ii
jj
i
jj
ii
j
i
j
i
j
ia
acba
abc
iiij
i
ij
ii
j
i
i
j
i
G
j
i
j
i
Gi
ij
i
GG
G
GG
e
GeGG
G
Ge
e
e
GG
G
GG
G
G
G
e
G
GGe
GeD
e
DR
e
L

102. We now consider just those few terms which are essential for the later discussion:
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in which we have set the gravitational coupling κ equal to unity, and where:
ν
νµ
µ
µν
µνµµ ψψφφ Σ+−−−∂∂+−= − 212*
e)Re(
4
1
]3)([e
2
11 Gba
ab
ji
ji
G
i
ij
i FFfGGGGR
e
L
*
2
*
j
i
j
i
j
j
ii
G
G
G
G
G
G
φφφφ ∂∂
∂
≡
∂
∂
≡
∂
∂
≡ and,
(G−1)j
i is only used to denote the i,j element of the inverse of the matrix with elements Gj
i.
We now briefly explain how the various terms in (1/e) L above arise. In fact, the fields,
a real function G(φ i,φi*), called the Kähler potential, and an analytic function fab(φ i). We
see that the functions G and fab determine the general forms allowed for the kinetic
energy terms of the scalar fields φ i and of the gauge fields Aµ
a respectively. The scalar
kinetic-energy term demonstrates that Gi
j
plays the role of the metric in the space
spanned by the scalar fields. A metric Gi
j
of the form ∂2G/∂φ i ∂φj* is referred to as a
Kähler metric and it is for this reason that G is called the Kähler potential. In the absence
of gravity, Gi
j
→δ i
j
and fab →δ ab. The function G is invariant under transformations of the
gauge group, whereas fab transforms as a symmetric product of two adjoint
representations of the group. The remarkable thing is that the general form is restricted
to just two arbitrary functions (i.e., G and fab).

103. The Lagrangian (1/e) L above contains a scalar potential of the form:
103
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]3)([e*),( 1
−= − ji
ji
G
GGGV φφ
plus terms stemming from Da that we omit. The origin of the first term in the square
brackets is related to the elimination of the auxiliary field term |F|2 in global SUSY (c.f.,
V(φ,φ*)=|Fi|2 +½Dα
2 of the Supersymmetric Gauge Theory chapter), that is:
i
j
j
i
G
WW
F )(
* 1
*
2 −
∂
∂
∂
∂
→
φφ
where now we must include the Kähler metric. The second term in the square brackets
of V(φ,φ*) above comes from the elimination of the auxiliary scalar field terms, −|S+iP|2,
in the SUGRA part of the pure Lagrangian L =L (2)+ L (3/2)−(e/3)(S2 +P2 −Am
2). We shall
see that its negative sign has considerable importance. The exp(G) factor arises from the
(Weyl) rescaling of the emµ fields required to bring the first term in (1/e) L above into the
canonical Einstein form, −½R, of L (2) =−(e/2)R. This rescaling implies a redefinition of the
fermion fields and hence the factor exp(G/2) in the last term of (1/e) L above. Owing to
this term, when the local SUSY is spontaneously broken the gravitino acquires a mass:
2
2/3 eG
m =
G being evaluated at the minimum of the potential V(φ,φ*) above.

104. In general, the Kähler potential G has to satisfy certain conditions for the theory to be
well defined. For example, we require Gi
j
>0 so that the kinetic terms of the scalar fields
have the correct sign. A special choice is:
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2017
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2
3
*
2
)(
ln
1
*),(
M
W
M
G
i
i
i φ
φφφφ +=
in which we have reintroduced the (dimensional) gravitational coupling κ using:
π8
1 PM
M =≡
κ
and where W is the superpotential of The Superpotential chapter. This choice gives Gi
j
=
δ i
j
/M2 and hence the minimal kinetic terms as in global SUSY. Substitution of G(φ,φ*)
into V(φ,φ*) above yields:










−+
∂
∂
=
2
2
2
2
*1
3
e
*
2
W
MM
WW
V
i
i
M
i
i
φ
φ
φφ
In the large-M limit this reduces to the scalar potential of global SUSY, V=|∂W/∂φi|2,
which explains why we have introduced the last term of G(φ,φ*) above.

105. 105
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We live is a universe of, apparently, one time and three space dimensions. It could be,
however, that the world is really a space of higher dimensionality but that we are, for
some reason, limited in our ability to experience all its dimensions.
Suggestions that this might be the case have invoked in physics for a variety of
purposes:
Higher-dimensional Theories
1. To embed the curved space-time of general relativity in a flat space of higher
dimensions. Here the extra dimensions are introduced merely to aid visualization. so for
example we can regard our four dimensions to a curved surface in a higher-dimensional
space;
2. To unify electromagnetism (and, later, the other forces) with gravity by identifying
some of the extra components of the metric tensor with the gauge fields pf the four-
dimensional, physical space-time. This was the brilliant idea of Kaluza (published, after a
delay of two years, in 1921);
3. To help with the nonlocality problems of quantum theory;
4. To facilitate the construction of SUSY and SUGRA Lagrangians. For example, N=1
SUGRA in 10 or 11 dimensions has one 16-component Majorana generator that breaks
up into eight 2-component Majorana generators in four dimensions, thereby giving N=8
SUGRA. Thus, by starting with the N=1 SUGRA Lagrangian in the higher-dimensional
space, we have a powerful method of constructing Lagrangians for N>1 theories in
physical space;
5. To construct consistent string theories.

106. 106
The simplest method that has been proposed for making the extra dimensions
unobservable is to suppose that they are compactified, with a scale parameter (or
radius) that is smaller that we can resolve. For example, suppose there is one extra
space dimension, for which we use the coordinate y, which is compactified because the
points y and y+2πr are identified. Clearly, this is equivalent to saying that the y direction
is curled up into a circle of radius r, so that increasing y by 2πr corresponds to going
once around the circle and returning to the same point (see Figure). Of course, to use
the picture we must embed the y dimension in some imagined two-dimensional space
(e.g., like the surface of the page in the Figure). Such an embedding is not necessary for
physics, however, since all the properties follow just from making the identification.
(N.B., We are concerned here with global, topological properties of space in contrast to
the local properties which determine the metric).
Compactification
A compactified dimension. The points y
and y+2πr are the same point, for any
value of y.
2017
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In particular, if we consider, for example, a scalar field φ(x,y),
where x represents the usual space-time coordinates xµ, we can
require that:
)π2,(),( ryxyx += φφ
r
y=y+2πr
from which it follows that we can expand φ in the Fourier series:
∑
∞
−∞=
=
n
ryni
n xyx e)(),( φφ
It is then a consequence of quantum theory that, in a state with a
given n, the y components of the momentum must be O(|n|h/r).
Thus, for a sufficiently small r, only the n=0 state will appear in
the world of low-energy physics (i.e., E<<hc/r).

107. 107
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m35
3
106.1 −
×≈≡≈
c
G
r N
P
h
l
In his pioneering paper, Kaluza obtained the same effect by simply postulating that all
the fields were independent of y. He called this the cylinder condition. In 1926 Klein
realized that the result could be obtained from quantum theory by the above Fourier
series argument.
A common, and natural, proposal is to take the scale of compactification (e.g., r in the
simple example above) to be of the order of the Planck length:
so that the mass of the excited states (n≠0) would be of the order of the Planck mass,
MP ≈1019 GeV/c2. Such a scale would seem to ensure that the extra dimensions are
forever beyond the reach of direct observation, though in some cases the higher-mass
states might make a significant contribution to the low-energy effective Lagrangian.
If, on the other hand, the length scale of compactification is much larger, it could be
that as experiments achieve higher energies (and hence probe shorter distances) we
shall begin to observe the effects of extra dimensions directly.

108. 108
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0=
∂
∂
y
gMN
The generalization of local U(1) invariance to local non-Abelian invariance, made in 1954
by Yang and Mills and, independently, by Shaw, was an idea which, in retrospect, we
can see was too good not to be relevant to the physics. It took, however, about twenty
years for its importance to be properly appreciated. It may well be that the idea of Kaluza
in 1919, that electromagnetism can be regarded as a consequence of general relativity
in five-dimensional space-time, will one day be seen in a similar light. At present,
however, its relevance to physics, if any, remains unclear.
Kaluza considered a world of one time and four space dimensions in which the metric
gMN (M,N=0,…,4) of gMN ≡nM⋅nN =gNM, when viewed from physical space-time, contains
the following three parts:
The Kaluza Model of Electromagnetism
gµν (µ,ν =0,…,3) the standard metric of ordinary space-time;
gµ4 =g4µ a four-vector field;
g44 a scalar field.
In this chapter we use x ≡x0,…,x3 as usual, and y ≡x4 is the coordinate of the extra
dimension. In addition, Kaluza imposed the cylinder condition that:
ˆ ˆ

109. 109
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∑=
n
rynin
MNMN xgyxg e)(),( )(
The Einstein equation RMN −½RgMN =0 in five dimensions then reduces to the
corresponding equation in physical space-time, together with Maxwell’s equation if gµ4 is
identified with the vector potential, Aµ, of electromagnetism. Thus, Kaluza was able to
show that, remarkably, electromagnetism is already contained in the equations of
general relativity applied to a five-dimensional space-time. In fact, Kaluza’s original
derivation was made in the weak-field approximation, but Klein showed this restriction to
be unnecessary.
In order to see how Kaluza’s idea works, we expand gMN(x,y) analogously to φ(x,y)=
Σn=±∞φn(x)exp(iny/r) above:
and then parametrize g(0)
MN as:





 +
=
φφ
φφ
φ ν
µνµµν
A
AAAg
gMN 3/1
)0( 1
This form is clearly completely general, but the notation is of course chosen with the
benefit of hindsight!

110. 110
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∫−= )5()5(4
2
52
1
ReydxdS
κ
We now recall the Einstein action of general relativity, S(E) =∫d4x L (E) with the
Lagrangian density L (E)=−eR/2κ 2 and e=det(em
λ)=[det(em
λ)]−1=[−det(gµν)]1/2, in the form:
where κ5 is the Einstein gravitational constant in five-dimensional space (c.f., κ 2/8π=GN).
and e(5) and R(5) are to be calculated from g(0)
MN. In fact, from the matrix g(0)
MN =φ−1/3[::]
above and e=[−det(g(0)
MN)]1/2 we can readily show that:
3
)5(
φ
e
e =
where e is defined in terms of the physical 4×4 matrix gµν in e=[−det(gµν)]1/2. With
somewhat more effort we can calculate R(5) by calculating all Christoffel symbols ΓL
MN
using gMN then using them to calculate the Riemann curvature RMN and finally calculate
the curvature scalar R(5) =gMNRMN. Then the action integral S above simplifies to:
∫ 







∂∂++−= φφ
φ
φ
κ
µ
µµν
µν 22
5
4
6
1
4
1
2
)π2( FFR
e
xdrS
where R=gµνRµν is the usual 4-dimensional scalar curvature. The field strength tensor:
µννµµν AAF ∂−∂=
but the electromagnetic contribution to the action is scaled by a factor πrφ/κ5
2 as
compared to L = −¼FµνFµν −JµAµ.

111. 111
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rπ2
2
52 κ
κ =
The expression in S above is the Einstein action for gravity in 4-dimensional space,
together with electromagnetism and a massless scalar field, φ, which satisfies the Klein-
Gordon equation ∂µ∂µφ +m2φ ≡ 2φ +m2φ = 0. Since φ is the scale parameter of the fifth
dimension (c.f., g(0)
MN =φ−1/3[::]) it is sometimes called the dilaton field. The physical gra-
vitational constant can be obtained by comparing S above with S(E)=−[1/(2κ2)]∫d4xeR as:
In order to demonstrate how the Aµ field couples to matter, we introduce an additional
scalar field Φ. This adds to the action a kinetic energy term:
∫ Φ∂Φ∂=Φ )( )0()5(4
NM
MN
geydxdS
Here, g(0)MN is the inverse of g(0)
MN and, from g(0)
MN =φ−1/3[::], is given by:








+−
−
= −
ν
µµ
νµν
φ
φ
AAA
Ag
g MN
1
3/1)0(
Putting this into SΦ and expanding Φ as in φ(x,y)=Σn=±∞φn(x)exp(iny/r), we find:
∫ 







Φ−Φ





+∂Φ





+∂=Φ
2
2
2
4
nnn
r
n
A
r
ni
A
r
ni
gexdS
φ
ννµµ
µν
Thus, we obtain the usual locally invariant (minimal) coupling of the gauge field Aµ to
the scalar field Φn together with a mass term for the scalar fields.

112. 112
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φ
κ
φ
κ
φ
κ
φ
κ
r
n
r
n
rr
n
rr
n
Qn
1
2
22
2ππ
5
2
5
====
To find the charges of the scalar fields, we must first correctly normalize the photon
field Aµ by removing the factor πrφ/κ5
2 from the FµνFµν term in the action S above. Then,
on comparing the terms in round brackets in SΦ above to the usual replacement i∂µ →i∂µ
−eAµ, we find that the charge of the field Φn is (N.B., r√φ is due to metric g44 =φ scaling):
This apparent miracle, that electromagnetism is obtained from general relativity in five
dimensions can readily be understood by examining the symmetries of the theory. The
U(1) local gauge invariance of electromagnetism has its origin in the local coordinate
invariance of relativity, in particular in the invariance under rotations around the small
circle in y; the spatial symmetry has become an internal symmetry. Thus, if we make the
transformation:
)(xyy Λ+′→
and calculate the new metric g′MN by the standard rule:
PQN
Q
M
P
MN g
x
x
x
x
g
′∂
∂
′∂
∂
=′
we find that g(0)
MN =φ−1/3[::] above is unchanged apart from the gauge transformation:
)(xAA Λ∂+→ µµµ
of electromagnetism.

113. 113
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The n=0 states of gMN that we have considered so far have five degrees of freedom:
one for the scalar field φ, and two each for the (transverse) polarization states of the
massless spin-2 graviton and the massless photon. In the higher-order terms of the
expansion, all five degrees of freedom are used as the five helicity states of massive
spin-2 particles. The masses of the scalars in SΦ are quantized according to the
expression:
and they have quantized electric charge given by Qn=(nκ/r)√(2/φ) above.
φr
nMn
1
=

114. 114
2017
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We end this chapter with some conclusions and comments:
3. If we take r√φ to be the order of the Planck length lP≡√(hGN/c3), then Qn=(nκ/r)√(2/φ)
gives the correct order of magnitude for the fine-structure constant since, if we identify
Qn=e, we find, using κ2/8π=GN, that:
1. Electromagnetism can be derived from Einstein’s general relativity theory in five
dimensions. This remarkable fact encourages the belief that all the forces of nature
may have a geometrical origin and that all the internal symmetries might eventually be
understood as invariances under additional coordinate transformations.
2. Kaluza’s theory provides a natural explanation for charge quantization, the origin of
which is unclear in the Standard Model. Of course, the actual equation for the charges
(c.f., Qn=(nκ/r)√(2/φ)), when combined with that for the masses (c.f., Mn=|n|/(r√φ)), is
a disaster, since it implies that the gravitational force between two particles is equal to
the electrostatic force, whereas between typical elementary particles (e.g., quarks or
leptons of mass ∼1 GeV/c2) it is a factor O(10−40) weaker. The failure of the model to
produce light (i.e., n=0), charged particles was a considerable drawback to its accep-
tance. As we shall see in the next chapter, the unfortunate fact that the massless
particles are singlets of the gauge group does not hold in higher dimensions.
2
2
322
22
44
π4
2
π4 rcr
G
crc
e PN
φφφ
κ
α
lh
hh
===≡
is of order unity. A theory in which r√φ was determined would permit a precise
calculation of the electron’s charge.

115. 115
2017
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4. In Kaluza’s original work the extra dimension was not really taken very seriously.
Thus, he imposed the cylinder condition and put φ =constant (=1 without loss of
generality), without worrying about the origin of these properties. Hence, it is possible
to argue that his derivation of electromagnetism is just a mathematical curiosity,
amounting to little more than an alternative was of writing down Maxwell’s equations.
Today we would be more inclined to regard the extra dimension as real, and to expect
the Einstein equation Rµν −½Rgµν =0 to hold in the higher-dimensional space. Then we
could not impose the Kaluza condition φ =constant, since this is only compatible with
Einstein’s equation if Fµν =0.
5. One of the major problems of all theories that begin in higher dimensions is how to
choose the properties of the compact manifold. As we shall see, there are so many
constraints on the initial form of the fundamental Lagrangian that it may be (in some
sense) unique! However, the resultant physics is determined by the metrical and
topological properties of the compact manifold, too, and, even if we require that this
manifold must be a solution of the higher-dimensional equation of motion, there is still
so much freedom that all predictive power is lost.
In the next chapter we shall discuss in detail how the physical gauge group is
determined by the compact manifold.

116. 116
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





=
)(0
0
yg
gMN
αβ
µνη
Kaluza’s idea can be generalized to accommodate other gauge interactions, even non-
Abelian ones, by increasing the number of extra dimensions. To explain this we consider
a space of 4+D dimensions and begin by assuming that the metric of the vacuum has
the form:
where M,N=0 to 3+D, ηµν is the flat metric of special relativity (i.e., diag(+1,−1,−1,−1) in
Cartesian coordinates), and gαβ (y) is the metric of the compact, D-dimensional space
that is a generalization of the circle used in the previous chapter (which had D=1). The
D-dimensional space is denoted by B and the extra variables {y} that span this space
are all assumed to be spacelike.
Non-Abelian Kaluza-Klein Theories
As noted at the end of the last chapter, we should expect that the matrix gMN =[::]
above is a solution to the vacuum Einstein equation:
0=MNR
or maybe of the corresponding equation with a cosmological constant:
0
2
1
=Λ+− MNMNMN ggRR
Indeed, it should be, in some sense, the lowest-energy solution, although this
concept is hard to define if we wish to compare spaces with different topologies.

117. 117
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)()( yg
y
y
y
y
yg ′
∂
′∂
∂
′∂
= δγβ
δ
α
γ
αβ
But first, we need to define Killing vectors. The gauge symmetry that results when we
add other fields to gMN =[::] depends on the properties of B (i.e., the D-dimensional
space). For example, we might choose B to be a flat space with Cartesian coordinates yα
(α =1,2,…,D) in which we identify points yα and yα +2πrα. The resulting manifold B, a D-
torus, can be thought of as D small circles with radii rα, which is an obvious
generalization of the circle in the previous chapter. The symmetry group in this case is
[U(1)]D, corresponding to invariance under separate rotations around each of the circles.
More generally, the symmetry group that is obtained is the isometry group of the mani-
fold B (i.e., the group of transformations of coordinates that leave the metric unchan-
ged). To understand what this means, we recall that the metric gαβ associated with the
coordinate {yα } is related to that, g′αβ , associated with a transformed set {y′α} by:
Any transformation that satisfies this equation is called an isometry of the metric gαβ .
Using g′αβ =gαβ into gαβ (y)=(∂y′γ /∂yα)(∂y′δ /∂yβ) g′γδ (y′) above we obtain:
)()( yg
y
y
y
y
yg ′′
∂
′∂
∂
′∂
= δγβ
δ
α
γ
αβ
)()( ygyg αβαβ =′
In order that the metric be form-invariant we require that gαβ and g′αβ should be identical
functions:

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)(yKyy ααα
ε+=′
We now consider infinitesimal coordinate transformations, which we write as:
where ε is a small parameter and Kα(y) are the components of a vector field. By
substituting this in gαβ (y)=(∂y′γ /∂yα)(∂y′δ /∂yβ) gγδ (y′) above, and working to lowest order
in ε, we obtain:








∂
∂
+








∂
∂
+
∂
∂
+= χ
χ
δγ
δγβ
δ
γ
α
δ
βα
γ
δ
β
γ
ααβ εδεδεδδ K
y
g
yg
y
K
y
K
yg )()(
which is equal to gαβ (y) to first order in ε if:
0=
∂
∂
+
∂
∂
+
∂
∂
γ
αβγ
αγβ
γ
βγα
γ
y
g
Kg
y
K
g
y
K
This condition defines Kγ (y) to be the components of a Killing vector.
Killing vectors of the metric are the directions, at any given point in the manifold, in
which it is possible to move from that point which keeping the form of the metric
unchanged.

119. 119
0)( =∑a
aa yKc α
A set of Killing vectors Kα
a(y) is linearly independent if the equation:
for all y, can be satisfied only if all the constants ca are zero. It can be shown that there
are at most D(D+1)/2 Killing vectors for a manifold B of dimension D. Manifolds that
have this maximum number are said to be maximally symmetric. In general, the group of
transformations:
∑=
+=′→
Bn
a
a
a
yKyyy
1
)(αααα
ε
with nB independent Killing vectors Ka, defines the isometry group (of dimension nB) of
the metric.
The three independent Killing vectors of
the two-dimensional surface of a sphere
are chosen to be tangential to circles on
the sphere perpendicular to the three
Cartesian axes.
2017
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As an example of a nontrivial maximally symmetric manifold,
consider the two-dimensional surface of a sphere in three
dimensions. Here the isometry group is SO(3), reflecting the fact
that all points on the surface are equivalent. The three linearly
independent Killing vectors may be chosen as circles
perpendicular to three Cartesian axes (see Figure). (N.B., In this
example the space B is not flat, so that gMN =[::] cannot be a
solution of RMN =0, and instead a nonzero cosmological constant
is required as in RMN −½RgMN + ΛgMN =0 above).

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After this brief mathematical interlude we return to the Kaluza-Klein theory by writing
the metric gMN =[::] for the massless modes (i.e., the n=0 terms in expansions analogous
to φ(x,y)=Σn=±∞φn(x)exp(iny/r)) in the form:
This is an obvious generalization of g(0)
MN =φ−1/3[::] of the previous chapter with terms
such as Kα
a Aa
µ that are invariant under isometry transformations. It is, however, not a
solution of the Einstein field equations in (4+D)-dimensions except in the special case
where the Kα
a are independent of {y}. Thus, in using g(0)
MN =[::] above for general Kα
a
we are not treating the dynamics of the extra dimensions properly.







 +
=
αβν
β
αβ
µ
α
αβν
β
µ
α
αβµν
gAKg
AKgAKAKgg
g a
a
a
a
b
b
a
a
MN
)0(
If we make local coordinate transformations of the form:
∑=
+=′→
Bn
a
a
a
Kxyyy
1
)( αααα
ε
where now the ε a(x) are a set of infinitesimal parameters, then, because of the definition
of the Killing vectors, there will be no change in the gαβ part of the metric, and the change
in the other components is compensated by the transformation Aa
µ →Aa
µ +∂µε a(x) which
is just the gauge transformation Aµ→Aµ +∂µΛ(x) of the The Kaluza Model of Electro-
magnetism chapter. Thus, we see that the coordinate invariance of the (4+D)-dimen-
sional theory leads to a locally gauge-invariant theory, the gauge group being the
isometry group of the compact manifold.

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We now turn to the question of whether it is possible to construct a phenomenologically
satisfactory theory starting from Kaluza’s idea. Let us first see when freedom is
available. We can choose the number of dimensions D. When compactified, the theory
will automatically contain vector bosons (i.e., Aa
µ) and scalars (i.e., gMN =[::] like the φ of
g(0)
MN =φ−1/3[::]). In the spirit of the model, we do not expect to have to add any other
bosons to the original (4+D)-dimensions Lagrangian, though of course the fermions
have to be put in by hand.
The next step is to choose the compact manifold. This choice effectively determines
the observed low-energy physics and there are two important aspects, which we shall
discuss briefly, namely, the isometry group and the zero-mass fermion states.
Kaluza-Klein Models and the Real World
Since we do not want to have to add any extra vector bosons, the isometry group must
contain at least all the gauge bosons of the Standard Model. The simplest case,
therefore, is where the isometry group is precisely SU(3)⊗SU(2)⊗U(1). Witten (1981) has
shown that this requires the compact manifold to have at least seven dimensions. Thus,
in order that all the interactions of the Standard Model can be obtained geometrically
through the Kaluza method, we must live in a world of 11 or more dimensions.
The requirement that the 7-dimensional compact manifold has the Standard Model as
its isometry group does not determine it uniquely. Indeed, there is an infinite number of
possibilities, but none is a suitable candidate for the real world, as we shall find below.

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First, we generalize the discussion of fermion fields given in PART VIII – THE
STANDARD MODEL: Fermions to an arbitrary number of space dimensions. The Dirac
equation (iγ µ∂µ −m)ψ =0 becomes:
0)( =−∂Γ ψmi M
M
where M=0,1,2,…,D−1 for a space-time of D dimensions, and where the ΓM are unitary
matrices satisfying (c.f., {γ µ,γ ν}=2gµν ):
MNNM
η2},{ =ΓΓ
with η MN=diag(+1,−1,−1,…,−1).
In general, the Γ matrices have 2D/2 rows and columns, where D/2 is the largest integer
not greater than D/2. A simple method of constructing a particular representation is to
begin with D=2 and choose:
1
1
)2(2
0
)2( σσ i=Γ=Γ and
where the σi are the 2×2 Pauli matrices:






−
=




 −
=





=
10
01
0
0
01
10
321 σσσ and,
i
i
The D=2 analog of γ 5 introduced in γ 5 =iγ 0γ 1γ 2γ 3, is then:





−
=ΓΓ−=
10
011
)2(
0
)2(
5
)2(γ

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For D=3 we can now use:
( )2,1,0
0
0
)3(
)3(
)4( =








Γ
Γ
=Γ≡ µγ µ
µ
µµ
as our set of 2×2 Γ matrices. It is clear, however, that there is now no analog of γ 5
because the obvious choice:






−
−
=−=ΓΓΓ=
i
i
Ii
0
02
)3(
1
)3(
0
)3(
5
)3(γ
5
)2(
2
)3(
1
)2(
1
)3(
0
)2(
0
)3( γi=ΓΓ=ΓΓ=Γ and,
is a multiple of the unit matrix.
To construct the D=4 matrices we define:






−
=Γ≡
0
03
)4(
3
I
I
γ
and:
This gives a chiral representation in which:





−
=≡
I
I
i
0
032105
γγγγγ
This procedure can be generalized to produce a representation for arbitrary D.

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Whenever D is even, there is an analog of γ 5, so we can form Weyl spinors γ 5ψL,R=
mψL,R with 2D/2−1 components. On the other hand, Majorana spinors ψ c =ψ only exist for
the curious set of dimensions D=2,3,4,8,9 modulo 8 (i.e., D=2,3,4,8,9,10,11,12,16,17,18,
19,20,&c.). To understand the origin of these numbers, note that the Majorana condition:
ψψγψ =≡ *0
T
Cc
has the complex conjugate:
** †
0 ψψγ =C
If we eliminate ψ * from these equations, we obtain:
ψψγγ =*))(( 00
TT
CC
which is equivalent to the condition that the charge-conjugation operation performed
twice has no effect:
ψψ =cc
)(
However, the sign of a fermionic state is not an observable quantity, so in general we
only know that:
ψψ ±=cc
)(
that selects the allowed dimensionality for the existence of Majorana spinors. With
C−1γµ C=−γµ
T & CT=C†=C−1=−C the above is readily shown to be satisfied when D=4.
1* †
00 =γγ CC T
It is the requirement that we have the + sign in this last equation:

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The requirement that a spinor can be both Majorana and Weyl simultaneously is even
more restrictive, and it is not satisfied for D=4. On the other hand, for D=2, if was done
in op cit, beginning with the Weyl condition:
ψψγ =5
)2(
we find that:
cc
CCC ψψγψγψγψγ =Γ=Γ=Γ= *** *5
)2(
0
)2(
5
)2(
0
)2(
0
)2(
5
)2(
5
)2(
TTTT
which shows that one can impose the Weyl and Majorana conditions simultaneously. In
fact, this is possible for D=2 modulo 8 (i.e., D=2,10,18,16,&c.) and the case d=10 may
be especially significant.
We now turn to our D=11 model in which fermions are 32 component fields satisfying:
0=ΨΓ M
M
D
with ΓM ≡Γ(11)
M. It is convenient to multiply this equation by the four-dimensional chirality
operator defined by Γ(4) ≡iΓ0Γ1Γ2Γ3 and write it as:
0)
~~
( =ΨΓ+Γ α
α
µ
µ
DD
where the ΓM are defined by:
~
MM
ΓΓ≡Γ )4(~
where µ goes from 0 to 3 and α from 4 to D−1=10 in our case.

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The advantage of working with ΓM rather than ΓM is that all the Γα commute with all the
Γµ. Thus, we can introduce the product representation:
ααµµ
γγ ⊗=Γ⊗=Γ )4()7( ~~
II and
where γ µ are the standard 4×4 Dirac matrices if op cit, while the γ α are 8×8 matrices,
satisfying the 7-dimensional (all space dimensions) Dirac algebra, and the I(k) are unit
matrices of dimension k.
~ ~
~
We can expand the general solution of (Γµ Dµ +Γα Dα)Ψ=0 above in the form:
~ ~
∑=Ψ
i
ii yxyx )()(),( χψ
where the χi(y) are a complete set of solutions of:
iii mDi χχα
α
−=Γ
~
iii mDi ψψγ µ
µ
=
the mi being real constants. Putting Ψ(x, y) above into (Γµ Dµ +Γα Dα)Ψ=0 and using this
last equation gives:
that is, the usual 4-dimensional Dirac equation for fermions of mass mi.
~ ~
As before we expect the mi to be either zero or of O(MP), the only mass scale of the
theory. Thus, the fermions observed in low-energy physics must be zero-mass modes of
the Dirac operation iΓα Dα χi=−miχi above on the compact manifold.
~

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Now we meet a big problem. Real physics includes parity violation, so fermions of
opposite chirality must lie in different representations of the gauge group:
This, however, is impossible in the theory we are considering because there is no analog
of chirality in seven dimensions (i.e., in an odd-dimensional space the product of all the γ
matrices commutes – not anticommutes – with each of the γ matrices,so the analog of γ 5
exists only in an even number of dimensions, e.g., γ 5
(3)=−iI above). It follows that a 7-
dimensional mass is not forbidden by chirality, and since a 7-dimensional mass implies a
4-dimensional mass, too, from iΓα Dα χi=−miχi and iγ µDµψi=miψi above, it follows that a
4-dimensional mass is always possible (i.e., that the fermions are not chiral).
RR
L
R
L
du
d
u
e
e
ee
′





′




 −
− and,,,
ν
~
This chirality problem appears to rule out the 11-dimensional Kaluza model. But even if
we ignore this difficulty, it can be shown that none of Witten’s 7-dimensional manifolds
gives fermions in the required representations of SU(3)⊗SU(2)⊗U(1).

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In spite of the problems met in the last chapter, we shall briefly discuss a particular 11-
dimensional theory, namely N=1 SUGRA, because it boasts several features that make
it seem tantalizingly close to an acceptable theory-of-everything.
N=1 SUGRA in Eleven Dimensions
First, it is remarkable that D=11, which, we saw in the last chapter, is the minimum
dimension in which Kaluza’s mechanism can yield the Standard Model, is also the
maximum dimension for a consistent supergravity theory. To understand the reason for
this maximum, we recall that spinors in D>11 have 64 or more components. Thus, the
generator of N=1 SUSY contains at least 16 four-component SUSY generators in four
dimensions (i.e., it leads to N≥16 SUSY, which is not allowed if we restrict ourselves to
particles with spin 2 or less).
Second, it turns out that, provided we limit ourselves to second-order derivatives, the
Lagrangian of N=1 SUGRA in 11 dimensions is unique. In particular, there is only one
possible multiplet and it contains the following fields:
In this model all the symmetries, all the forces, and all the particles appear in this
multiplet and are therefore geometrical in origin. It is the complete fulfillment of
Kaluza’s idea, and of course it realizes Einstein’s hopes of unification in that there is
no room for an arbitrary, nongeometrical, right-hand side to the Einstein field equation
Rµν −½Rgµν =−κ 2Tµν.
Degrees of freedom .
Symmetric tensor gMN ½×10 ×9 −1=44
(Vector) spinor ψM 128
Antisymmetric tensor AMNP 84

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Third, as first demonstrated by Freund and Rubin (1980), there is a solution to the 11-
dimensional Einstein equation in which the 11 dimensions spontaneously separate into 4
+7, the 7-dimensional manifold being compact, which therefore provides a possible
reason why physical space-time has four dimensions. In order to explain how this comes
about, we first impose the condition that the vacuum expectation value of the gravitino
field is zero (which is certainly necessary if the compact manifold is maximally
symmetric). The field equations involve a generalization of Fµν defined by:
where the sum is over all antisymmetric permutations of M, P, Q, and R. Instead of RMN −
½RgMN =0 (c.f., the Non-Abelian Kaluza-Klein Theories chapter with Λ=0), the field
equations become:
∑=
AntiPerm
4 PQRMMPQR ADF






−=− PQRS
PQRSMN
PQR
NMPQRMNMN FFgFFgRR
8
1
3
1
2
1
where the right-hand side is the energy-momentum tensor TMN calculated from the N=1
SUGRA Lagrangian, and:
87654321
87654321
576
1
MMMMMMMM
PQRMMMMMMMMMPQR
M FFFD ε−=
where ε M1…M11 is the fully antisymmetric tensor with:
e(11) is the determinant defined in terms of the 11-dimension metric by e=[−det(gMN)]1/2.
1)11(10210
][ −
= eL
ε

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The Freund-Robin solution is given by:
where, as usual, µ,ν go from 0 to 3 and α,β from 4 to 10 and:






=
αβ
µν
g
g
gMN
0
0
µνρσµνρσ εCF =
where C is an arbitrary constant, and all other components of FMNPQ are zero. Here εµνρσ
is the fully antisymmetric tensor with:
13210 −
= eε
With this, DM FMPQR above is trivially satisfied since both sides are zero. Also we have:
22
24CggggCFF PQRS
PQRS −== ′′′′
′′′′
σρνµ
σσρρννµµ
µνρσ εε
and:
τσρν
ττσσρρ
µρστ εε ′′′
′′′
= gggCFF PQR
NMPQR
2
for (M,N)=(µ,ν), and zero otherwise. Hence:
2
6 CgFF PQR
PQR µννµ −=

131. 131
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So, RMN −½RgMN =⅓(FMPQR FN
PQR −⅛gMN FPQRS F PQRS) above becomes:
Therefore, on using the d=11 form of R=gMNRMN:
αβαβαβµνµνµν gCgRRgCgRR 22
2
1
2
1
=−−=− and






+=





+−= RCRCR
2
1
7
2
1
4 22
and so Rµν −½Rgµν =−C2gµν and Rαβ −½Rgαβ =C2gαβ above reduce to:
αβαβµνµν gCRgCR 22
3
2
3
4
=−= and
The 7-dimensional space with Rαβ satisfying Rαβ =⅔C2gαβ is necessarily a compact
space. The maximally symmetric solution Rµν =−(4C2/3)gµν gives an anti de Sitter space
(i.e., a de Sitter universe has no ordinary matter content but with a positive cosmological
constant Λ which sets the expansion rate, H, whereas an anti de Sitter space, AdSn,
means that it has a negative cosmological constant Λ=−4C2/3 instead of a positive one).
(N.B., Although the dimensionalities 4 and 7 are uniquely determined by the model – the
4 arising because FMPQR has four indices – it is an assumption that the time dimension
belongs in the 4- and not the 7-dimensional space; the model actually only predicts that
physical space has either 3 or 6 spatial dimensions).

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In order to make progress, we must now choose some particular compact manifold
satisfying Rαβ =⅔C2gαβ above, and it is here that we meet problems. All the manifolds,
discussed in the previous chapter, for which the gauge group contains the Standard
Model, have the effect of breaking supersymmetry completely. Thus, the model has no
means of preventing Higgs scalars becoming massive, with a mass of the order of the
compactification scale (≈MP).
If, on the other hand, we abandon the requirement that the Standard Model should be
contained in the isometry group, and choose instead the most symmetrical manifold
satisfying Rαβ =⅔C2gαβ , namely, the 7-sphere, S7, the resulting 4-dimensional theory
would have the full N=8 supersymmetry. Indeed, the zero-mass sector of this theory is
identical to the D=4 SUGRA. The gauge group is O(8), the isometry group S7. The 32-
component gravitino forms the required 8 four-component gravitinos ψµ , in D=4, and the
56 (=8×7) four-component spin-½ fields ψα .
So, although N=1 SUGRA in 11 dimensions once seemed as though it might be the
ultimate fulfillment of Kaluza’s idea, it does not in fact work.

133. 2017
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P.D.B. Collins, A.D. Martin, E.J. Squires, Particle Physics and Cosmology, Wiley, 1989.
University of Durham, England
This readable introduction to particle physics and cosmology discusses the interaction of these two fundamental branches of physics
and considers recent advances beyond the standard models. Eight chapters comprise a brief introduction to the gauge theories of
the strong and the electroweak interactions, the so-called grand unified theories, and general relativity. Ten more chapters address
recent concepts such as composite fermions and bosons, supersymmetry, quantum gravity, supergravity, and strings theories, and
relate them to modern cosmology and experimental astronomy.
M. Dine, Supersymmetry and String Theory – Beyond the Standard Model, Cambridge, 2007
University of California, Santa Cruz
The past decade has witnessed dramatic developments in the field of theoretical physics. This book is a comprehensive introduction
to these recent developments. It contains a review of the Standard Model, covering non-perturbative topics, and a discussion of
grand unified theories and magnetic monopoles. It introduces the basics of supersymmetry and its phenomenology, and includes
dynamics, dynamical supersymmetry breaking, and electric-magnetic duality. The book then covers general relativity and the big
bang theory, and the basic issues in inflationary cosmologies before discussing the spectra of known string theories and the features
of their interactions […]. This will be of great interest to graduates and researchers in the fields of particle theory, string theory,
astrophysics and cosmology.
S. Weinberg, The Quantum Theory of Fields, Volume III, Cambridge University Press, 2000.
Josey Regental Chair in Science at the University of Texas at Austin
Volume 3 (of 3) continues his masterly exposition of quantum field theory. This third volume of The Quantum Theory of Fields
presents a self-contained, up-to-date and comprehensive introduction to supersymmetry, a highly active area of theoretical physics
that is likely to be at the center of future progress in the physics of elementary particles and gravitation. The text introduces and
explains a broad range of topics, including supersymmetric algebras, supersymmetric field theories, extended supersymmetry,
supergraphs, nonperturbative results, theories of supersymmetry in higher dimensions, and supergravity. A thorough review is given
of the phenomenological implications of supersymmetry, including theories of both gauge and gravitationally-mediated
supersymmetry breaking. Also provided is an introduction to mathematical techniques, based on holomorphy and duality, that have
proved so fruitful in recent developments. This book contains much material not found in other books on supersymmetry, some of it
published here for the first time. [NDLR: This reference does not treat string theory in any detail, expept maybe historically.]
M. Kaku, Introduction to Superstrings, Springer-Verlag, 1988
City College of the CUNY
Superstrings – provocative, controversial, possibly untestable,but unarguably one of the most interesting and active areas of research
in current physics. Called by some, “the theory of everything,” superstrings may solve a problem which has eluded physicists for the
past 50 years – the final unification of the two great theories of the twentieth century, general relativity and quantum field theory. Now,
here is a course-tested, comprehensive introductory graduate text on superstrings which stresses the most current areas of
interest, not covered in other presentations, including: string field theory, multi loops, Teichmüller spaces, conformal field theory,
four-dimensional superstrings […]. Prerequisites are an acquaintance with quantum mechanics and relativity.
133
References / Study Guide

135. Supersymmetry is
an answer looking for a problem.
Anonymous