VOLUME
48,
NUMBER
15
PHYSICAL
REVIEW
LETTERS
12
APRIL
1982
Three-Dimensional
Massive
Gauge
Theories
S. Deser
Department
of
Physics,
Brandeis
University,
Waltham,
Massachusetts
02254
R.
Jackiw
and S.
Templeton
Center
for
Theoretical
Physics, Laboratory
for
Nuclear
Science and
DePartment
of
Physics,
Massachusetts
Institute
of
Technology,
Cambridge,
Massachusetts 02IS9
(Received
22
January
1982)
Three-dimensional
Yang-Mills and
gravity
theories
augmented
by
gauge-invariant
mass
terms are
analyzed.
These
topologically
nontrivial
additions
profoundly
alter
the
particle
content of the models and lead
to
quantization
of
a
dimensionless
mass
—
coupling-constant
ratio.
The
vector
fieM
excitations
become
massive,
with
spin
1
(rather
than
massless
with
spin
0),
and
the mass
provides
an
infrared
cutoff.
The gravitation
acquires
mass,
mediates
finite-ra~~e
interactions,
and
has
spin
2
(rather
than
being
absent
altogether);
although
its
mass
term
is
of third derivative
order,
there
are
no
ghosts
or
acausalities.
PACS numbers:
11.15.-z, 04.60.
+n,
11.10.
6h
The
topology
of
odd-dimensional
spaces permits
the construction of
gauge
theories with
novel
and
at-
tractive
properties.
We
treat here the
non-Abelian
vector
and tensor models in
three
space-time
di-
mensions.
"'
In
both,
a
topological
structure,
the
Chem-Simons
secondary characteristic,
'
augments
the
usual action. These gauge-invariant
terms
dramatically
affect the nature
of
the
field
excitations,
changing
their
spin
and
endowing
them
with
a mass. For
the
(superrenormalizable) vector
model,
the
spin
changes
from 0
to
1
and
the mass
parameter provides
a
perturbative
infrared cutoff.
In the gravi-
tational
case,
the
dynamically
trivial Einstein action
becomes
that of a
finite-range spin-2
field;
de-
spite
the
third
derivative
order
of the
mass
term,
there
are neither
ghosts
nor
acausalities.
The
non-Abelian
mass term
also has nontrivial
homotopy
properties,
which
lead
to
quantization of a
dimen-
sionless
combination of mass and
coupling
constant.
The vector
field action is
I„=I„M+I„c,
=(g
'/2)
fd'x-trI 8I"
—
(kg
'/2)
fd'xe
8Itr[E„8A
-asA
A8A~]
in the
usual matrix notation for the
internal
indices. The
gravity
action
reads
Ia
=IE+Ioc,
s
=x
'fd'x g'
'R
—
(4y,
x')
'Jd'xe
(Ras„(uy'
+,
(u„„(us"
(uy,
').
(l.
a)
(lb)
Note the
sign
of the
Einstein
term,
which is
opposite
to
that
in four-dimensional
gravity.
The
coupling
constants
(g,
tc)
have
dimension
(mass)'~',
and
p
is
the
mass
parameter;
our
signature
is
(+
——
).
In
(lb),
&u„„=
—
&u
„~,
are
the
drei
bein
affinitieS
and
R
„8,
~
=
—
Saris„+
&ma
„&us',
—
(p
a)
iS
the
CurVature
ten-
sor. For
simplicity,
we
deal
with
the
source-free
case;
matter
coupling
may
be included in
the
stan-
dard
way,
and
its
effects will be noted.
The (P- and
T-nonconserving)
mass terms are the
Chern-
Simons
expressions.
They
are
closely
related
to
the
well-known
four-dimensional
(ct =0,
1,
2,
3)
Pontry-
agin
densities
X",
defined
by
s
=t
+s'"'z
=s
x
v
r
as
2
X(y y
P
-=t
*It.
R
"-=8
X
"
*R
8
-=~a
'R
g
r
ab
a8
e
c
ab
2
A,
aab
'
(2a)
(2b)
The integrals
of the
a
=3
components,
X',
of the
X
over
the
three-dimensional
subspace
excluding
x',
and with
x'
dependence
suppressed,
are
(up
to constants)
just
our
Ics.
The
source-free
equations
which follow
from
(1)
are
and
D Z
'+
p
+Z'=0,
+S'=-~e"
+p,
.
gn8+
+
zCns
0
ga8
—
R&8
~
ga8R
C&8
=g
~i2ea~oD
(R
8
Ad
8R)
(Sa)
(3b)
1982
The
American Physical Society
975
VOLUME
48&
NUMBER
15
PHYSICAL
REVIEW
LETTERS
12
ApRIL
1982
Despite
the
hei bein
dependence
of
I&cs,
its
variation
is
of
purely
metric
nature,
like
that
of the
Ein-
stein
action;
hence
the
metric form
of
(Sb).
We
have
made use
here
of
the
equivalence
between
Bie-
mann
and
Ricci tensors in three
dimensions,
which also
implies
that flat
space
is
the
only
solution
of
pure
Einstein
theory.
Each term in
(S)
separately
satsifies
a
covariant
conservation
identity.
In three
dimensions,
C""
is the conformal tensor
replacing
the normal
Weyl
tensor,
which
vanishes identical-
ly;
the
density
g'"C„"
is invariant
under
conformal
transformations
of
the metric
I.n addition
to
being
conserved,
C"'
is
symmetric
by
virtue of
the
usual
Bianchi
identity and
manifestly
traceless;
the
scalar curvature
R
therefore
vanishes in
the absence
of sources.
To demonstrate
the
massive nature
of
the
excitations,
the dual
of
the
vector
equation
(Sa),
0"8(p,
)+Fs-=(pg
+e"
~D~)
*E8
=0,
and
the
gravitational
ones,
may
be
iterated with
0(-
p)
to
yield
(g&D
+
p2)
g~&
g&8)'[
g~
gy
]
and,
using
the
R=O
property,
(0
6„+
p
)RBy
=
SRs~R~y-gsg~pR
"~.
(4a)
(4b)
(5a)
(5b)
In
linearized
approximation,
which
governs
the
kinematics
of our
theories, the
right-hand
sides
are
absent
and we
see
that,
whatever their
number
or
spin,
all
particles are
massive, with causal
rather
than
tachyonic
propagation.
The
latter fact
is not
controllable
since
the
action
depends
on
LL(, rather
than
g,
but
may
be
understood in
terms
of
the
a
priori
positivity
of the
energy,
and
Bel-Robinson
superenergy,
densities
of the
theories.
'
The nature of
the excitations
only
becomes
manifest
by,
more
detailed
analysis. The
linearized
ac-
tions
are
first
expressed
in
terms of
the
Abelian
gauge-invariant
components of
the
fields.
For
the
linearized
Yang-Mills
theory
[
equal
to the
topologically
massive
Maxwell
multiplet],
these
are the
transverse
components
of
the
spatial vector
potentials,
Ar'=e'~~,
p,
with
8~
-=8~
(-
&')
"',
and
the
longitudinal electric
fields. The
resulting constraint-free
action
is
of
Klein-Gordon
form
for
p;
it
describes
one
degree
of
freedom
per
color. For
linearized
gravity,
we use
the
following
three
gauge-
invariant
components of
a»
=
~
'(g»
—
rj»):
A
A
h A
q
=a"
~8,
8,
.
a',
y=v
a~+
28,
a
'
8,8&a„—
aq,
.
—
~=a'
(aja
—
s,
.
e,
a
').
In terms of
these,
the linearized
action
reads
I~~==
T
fd'x[(q
Clp+o'+xy)+p
'ao].
(6)
Note
that all
triple
time
derivatives
are
safely hidden in
the
linearized
Weyl-invariant
Lagrange
multi-
plier
and
constrained
fields
(A,
,
o)
and that
the
Einstein
(or topological)
term
alone
is
clearly
vacuous.
Elimination
of
A
immediately leads
to the
unconstrained
form
Igg=-
yfd
x
p(
+p
)p
(7)
which
is
identical to
the vector
action,
although the
respective
p
variables
have
different
Lorentz trans-
formation
properties.
Note
that
p
is
essentially the
conformal
part
of
the
metric, which
appears
with
ghost
sign
in
the
conventional
Einstein
action.
Accordingly,
the
overall
sign
of
(7)
is
the
physical one
because
the
sign
of
the Einstein
term
in
(1b)
is
opposite
to that
required in four
dimensions.
Calculation
of the
linearized
Poincard
generators
shows
further
that, despite
the
scalar
form
(7)
of
the
action, the
vector
theory
s
particle
has
spin &1,
while the
graviton
has
spin
&
2. The
spin
sign
is
correlated
to that
of
p,
in the
action.
'
[In
three
dimensions
spin
is a
rotational
pseudoscalar and
can
have
either
sign.
The value
of
the
spin
only
emerges
from
study
of the full
SO(2,
1)
generators,
and
it
is determined
by
the
(invisible)
temporal indices
of
the
field
variables.
]
As was
already
apparent
from
its
topological origins or from the
epsilon
tensor, a
single
spin
value
reflects
T
and
P
nonconservation.
[These
discrete
symmetries
may
be restored
by
considering a doublet
of
models and
opposite
mass
terms.
)
Our
excitation
spectra
differ
discontinuously
both
from
the
massless
ones
in
which the
vector
976
VOLUME
48)
NUMBER
15
PHYSICAL
REVIEW
LETTERS
12
APRIL
1982
particle
is
spinless
and
the
graviton
is
absent
altogether,
and
from those
of the traditional
gauge-non-
eovariant
massive
models,
which have two
degrees
of freedom
for either
case.
The
topological
terms
in the
Lagrangians change
by
the
total
divergences under
gauge
transforma-
tions;
in the
vector
case,
it is known that
l„cs
change
by
the
winding
number'
Mt
(U)
under
"large"
gauge
transf ormations U:
A~
-U
A~
U+U
BpU,
Ives
Iv-cs+8~
PZ
~(U)
w(U)
—
=
(24m')
'fd're"
&tr[
&„U(U
'&sU)(U '&~U)
Therefore
exp(iIvc,
)
is
gauge
invariant
only
if the
quantization
condition
holds.
Otherwise,
the
expectation
value of a
gauge-invariant
operator
0 would be
undefined,
as
can be
seen
from
the
functional
integral
repre-
sentation
(0)
=&
'
fnA
0
exp(iI
v),
with a
gauge-
invariant measure
S4
and
normalization
Z.
Un-
der
a
change
of variables
A-A",
where
A"
is
a
(large)
gauge
of transform
of
A,
(0)
would
change
by
the
phase
implied
by
(8),
which
would
only
be
unity
if
(9)
held.
'
The same
result
emerges
in
a
Euclidean formulation
since
I~cs
is
independent
of the
metric's
signature,
but
depends
only
on
the nature of the
maximal
compact
subgroup
of
the internal
gauge group.
The gravitational
ac-
tion's
change
under
large
local
Lorentz
trans-
formations
is
formally
identical,
but as
this
"in-
ternal"
group,
SO(2,
1),
has
the
trivial
compact
subgroup SO(2)
we
cannot
draw the same
conclu-
sion for
p
K
in
Minkowski
signature.
That is per-
haps
fortunate since that
quantity
may
have to be
infinitely
renormalized
in
the
quantum
theory.
Quantization
of
the
models is
straightforward.
For
the vector
theory,
already
superrenormaliza-
ble when
p
=0,
the
mass
term acts
as an
infrared
cutoff,
since the
propagator
behaves as
(p'
+pp)
'.
This
gain
of
one
power
of
momentum
should suffice to cut off the
low
frequencies
at
each
loop
order.
Explicit
loop
calculations
of both
Abelian and
non-Abelian
vector
models
coupled
to
fermions
have been
performed;
although every-
thing
is
finite,
some of
the finite results
are
actually
regularization
dependent.
'
In
gravity,
there
is a
dramatic
improvement
in
the ultraviolet behavior
of the eonformal
sector of
the
theory,
where the
propagator
now
decays
as
The model
would
be
manifestly
power-count-
ing
renormalizable
[with
&2'.
as
effective
coupling
constant] were
it
not for
the scale
factor,
i.
e.
,
the
part
pg
8
of
h„s
in
the linearized
approxima-
tion.
This
mode
retains
the conventional
P
prop-
agation
because
the
conformally
invariant
topolog-
U-1]
(8)
~
ical
part
does
not affect it.
'
There is a
close
parallel
with
four-dimensional
theory
when
the
eonformal term
-R»'
—
~R'
is
added
to the
Ein-
stein action.
'
However,
in four
dimensions,
ghosts
are also
introduced
in
the
process.
In
either
dimension, conformal invariance
is
remov-
able
by
further
addition
of a term
-
R',
but it
is
nonunitary.
It
appears
likely
that
supergravity
ean
be
con-
structed. When
we
adjoin
to
the
(P-
and
T-invari-
ant)
Barita-Schwinger
action
&i
fr
&$
„8
8$&
d'x
(which,
like
the
Einstein
action,
is trivial
by
it-
self),
the
supersymmetric
companion
of
X&'
de-
rived
from that
of
*RR
yields
an action
for a
propagating
massive
spin-
~
fermion.
Existence
of
supergravity
would also
provide
a
simple
proof
of
energy
positivity
of our full
nonlinear
gravity
model.
"
The
external-source
problem
has also
been
analyzed.
For
gravity,
the short-range
behavior
is exhibited
by
the
effective Yukawa
attraction,
H;,
,
—
—
fT„(-
&'+p')
'T„of
weak
static
T„
sources.
This
contrasts
with the
interactions
mediated
by
Einstein
gravity
(the
p
—
~
limit
of
our
theory). There,
the
linearized
propagator
leads to
a
coupling
-
f
dpp
'[T
„"(p)T„'(-p)
—
T
„"
(p)T,
"(-
p)].
Consequently,
there is
no
T"-T"
term at
all
there,
and the entire
interac-
tion is
purely
contact,
since the
p
'
pole
is
can-
celed
by
the momenta
implicit
in
T""
to
ensure
its
conservation.
Nevertheless
the Newtonian
component
of
the
metric, defined
as usual
from
the
linearized
part
of
G",
still behaves
asymptot-
ically
as
V
'T",
and the
total
energy
is
expressed
by
the
same
flux
integral
at spatial infinity
a,
s
in
ordinary gravity.
The vector
theory
also
implies
short-range
source interactions,
but differs
from
the
massless
case in
that the longitudinal electric
field
becomes
short
range
and
the total
charge
is
read
off
asymptotically
from the
magnetic
poten-
tial.
[In
both
cases,
it
is the
lowest
derivative
term
in the
constraint
equations
which carries
the
asymptotic
information
about
the
"charges.
"]
977
VOLUME
48,
NUMBER
15 P H
YSICAL
RE
VIE%'
LETTERS
12
APRiL
1982
A
number of questions
are
raised
by
these
re-
sults.
First,
it would
be
of
interest to
study
in
more
detail
the
infrared
properties
of
the
vector
model
to
verify
that the
mass
provides
a
cutoff
beyond
one
loop.
Second,
investigation
of the
ultraviolet
behavior
of
gravity
2nd
supergravity
should
reveal
the
extent
of the
improvement
due
to the
conformal
sector's
asymptotics.
This
could
also be done
by
determining
the form of the
(space-nonlocal)
self-interaction
V(C')
in the
full
nonlinear
action,
ls=
—
~
Jdx[@'(
+u')4'+V(@)],
ex-
pressed
in
terms of the
single
dynamical
variable
4'.
The
most
relevant
problem
is
clearly
the
rela-
tion
of
these
models to
four-dimensional
theory.
Although
the
normal terms (taken
in
Euclidean
signature)
can be
considered
as
high-temperature
limits,
"
or
dimensional
reductions of the
latter,
the
topological
terms
have
a different
origin,
in
the
0
vacuum
of
the
four-dimensional
physics.
"
However,
since
~
terms are
generally
present
in
four dimensions,
naturalness
suggests
that
the
corresponding
topological
terms be included
in
the
high-temperature
limit.
Similar
topologically
augmented
models
could
also be
introduced
in other
odd-dimensional
sys-
tems.
However,
as
is
clear
from their
origin,
the
topological
parts
will
be
of
higher
derivative
order and
powers
of the
fields,
unless
expressed
in
terms
of
higher
rank
gauge
fields. Such
vari-
ables and
terms
have arisen naturally
in
higher-
dimensional
supergravity
and
may perhaps
pro-
vide
a
way
of introducing
mass without
Higgs
scalar
s.
Details
of
this work will be
reported
elsewhere.
'
This work
is
supported
in
part
by
National
Science
Foundation
Grant
No.
Phy
78-09644
A02
and
by
U.
S.
Department
of
Energy
Contract No.
DE
AC02-76ER03069.
~Gauge-invariant
mass terms for vector
gauge
theor-
ies were
introduced
by
B.
Jackiw
and
S. Templeton,
Phys.
Rev.
D
23,
2291
(1981);
J.
Schonfeld,
Nucl.
Phys.
8185,
157 (1981). See
also
W.
Siegel,
Nucl.
Phys.
8156,
135
(1979).
More detailed
analysis,
summar-
ized in
this
Letter,
is
given
by
S.
Deser,
R.
Jackiw,
and
S. Templeton,
to be
published.
2Deser,
Jackiw,
and
Templeton,
Bef.
1.
3For
a description,
see S.
S.
Chem,
Comp/ex
Mani-
folds
without
Potential
Theory
{Springer,
Berlin, 1979),
2nd ed.
L.
P.
Eisenhart, Riemannian
Geometry
(Princeton
Univ.
Press,
Princeton, N.
J.
,
1949);
R.
Arnowitt,
S.
Deser,
and
C, W. Misner,
in
Gravitation,
edited
by
L.
Witten
(Wiley,
New
York, 1962);
J.
W.
York, Phys.
Rev.
Lett.
26,
1656
(1971).
5B.
Binegar,
to be
published.
B.
Jackiw
and C.
Rebbi,
Phys.
Rev.
Lett.
37,
172
(1976); R.
Jackiw,
Rev.
Mod.
Phys.
52,
661
(1980).
~An
alternative
argument
is
based
on
the
fact
that the
phase
exponential changes
by
f
(N)
—
=
exp'.
2~i@
(g
/4~)
'N]
as the integration (without
change
of variable)
ranges
over
gauge
copies,
with
relative
winding
number
N,
of
a given
configuration.
Without
quantization,
the sum
of these
contributions
is
the overall
factor
~&f
{K),
which
vanishes,
rather
than
—
~&1
which is
harm1essly
canceled
by
Z. However,
J. Schonfeld
(private
com-
munication)
suggests
that
the
"zero"
could
perhaps
similarly
be
canceled
by
Z;
whether this
is
acceptable
is
not clear
to
us.
We
thank
E.
Witten
for
pointing
out
an
error
in an
earlier
version
of
this
argument.
~K.
S. Stelle,
Phys.
Bev.
D
16,
953
(1977).
S.
Deser and
C.
Teitelboim,
Phys.
Rev.
Lett.
39,
249
(1977);
M.
Grisaru,
Phys.
Lett.
73B,
207
(1978);
E.Witten,
Commun. Math.
Phys.
80,
381
{1981).
S.
Weinberg,
in Understanding the Eundamental
In-
teractions,
edited
by
A. Zichichi
(Plenum,
New
York,
1978);
A.
Linde,
Bep.
Prog.
Phys. 42,
389
(1979);
D. J.
Gross,
R.
D.
Pisarski,
and L.
G.
Yaffe,
Bev.
Mod.
Phys.
53,
43
(1981).
For
gravity,
however,
the
sign
taken here
for the Einstein
term
is
opposite
to
that in
the
original
four-dimensional
theory.
'
For vector
theories,
see
Ref.
6;
for
gravity
theories,
see
S.
Deser,
M.
Duff,
and
C.
Isham,
Phys.
Lett.
93B,
419
(1980); C.
Isham,
International
Center
for Theoret-
ical
Physics
Report
No.
Sl/82-1
(to
be
published).
978