2008)
An effective quantum theory of gravitation in which gravity weakens at energies higher than
10
3
eV
is one way to accommodate the apparent smallness of the cosmological constant. Such a theory predicts
departures from the Newtonian inverse-square force law on distances below
0:05 mm. However, it is
shown that this modication also leads to changes in the long-range behavior of gravity and is inconsistent
with observed gravitational lenses.
DOI:
10.1103/PhysRevLett.100.031301
PACS numbers: 04.60. m, 04.50. h, 04.80.Cc
The discovery of the cosmic acceleration [
1
] has
prompted speculations of new physics. A leading hypothe-
sis is the existence of a cosmological constant, responsible
for the accelerated expansion. The milli-eV energy scale
implied by this phenomenon is difcult to understand in
terms of a fundamental theory [
2
]. The validity of
Einsteins general theory of relativity (GR) on cosmologi-
cal scales has thus come under suspicion. A novel solution
to this problem might be achieved if GR is a low-energy
effective theory in which gravity weakens at some energy
scale. In an effective theory of gravity there may exist a
threshold,
, beyond which gravitons cannot mediate mo-
mentum transfers. This behavior may be due to a fat
graviton, a minimal length scale associated with quantum
gravity, or possibly nonlinear effects which lter out high-
frequency interactions [
3 10
]. Such theories offer a novel
solution to the cosmological constant problem by regulat-
ing the contribution of vacuum uctuations to the cosmo-
logical constant. However, we show that this mechanism
may have already been explored and ruled out by gravita-
tional lensing on cosmological scales.
We estimate the energy scale of an effective theory of
gravitation by matching the predicted quantum vacuum
energy density with the energy density of a cosmological
constant,
, necessary to explain the accelerated cosmic
expansion. Following Zeldovich [
11
], the gravitating en-
ergy density of the particle physics vacuum as due to N
equivalent, massless scalar particles, is

N
2
Z d
3
k
2
@
3
kcf k :
(1)
We introduce the function f k
e
k=
to regulate the
momentum at the vertex where vacuum bubbles connect to
gravitons in order to limit the gravitating energy density.
We refer to
as a cutoff scale in the sense that the
standard gravitational interactions are severely weakened
above this scale. We match
crit
and obtain
0:0048
h
2
=N
1=4
eV=c as the desired cutoff scale.
Current measurements give
h
2
0:34
0:04 1
(see Ref. [
12
] and references therein) so that
0:0037 1
0:03 =N
1=4
eV=c. We now examine the con-
sequences of this cutoff.
We consider weak gravitational elds described by a
linearized, effective quantum theory of gravity [
13
]. The
interaction Lagrangian at lowest order is

L
I
1
2 h
T
;
(2)
where
32 G
p
, h
is the graviton eld, and T
is
the stress-energy tensor of the gravitating sources. Here,
we introduce an exponential cutoff at
on graviton
momenta.
Short-distance gravitational phenomena below the
length
0
@= 0:05 mm are affected by such a cutoff,
which we impose on the graviton four-momentum q
so
that q
2
q q <
2
. For real gravitons, q q
0 and
so the constraint is trivially satised. For virtual gravitons,
the cutoff may be imposed by suppressing the graviton
propagator in the ultraviolet [
14
]: 1=q
2
!
G q
2
=
2
=q
2
,
where
G is a function of the graviton momentum. For
example, our exponential cutoff follows if
G x
e
x
p
.
Such a modied propagator follows naturally from modi-
ed gravitational Lagrangians. This is clear upon inspec-
tion of the weak-eld, Coulomb gauge, gravitational
Lagrangian for a fading gravity model [
14
]:

L
g
2 h
1
2
h
G
1
=
2
h
;
(3)
where
is the DAlembertian operator. The sum of (
2
) and
(
3
) can be used to obtain the weak-eld equations of
motion.
An exponential cutoff to the momentum-space integral
for the virtual gravitons exchanged between two static
masses, m
1
and m
2
, changes the Newtonian potential to

V
8 Gm
1
m
2
Z d
3
q
2
3
@ 12q
2
e
i=
@ ~q ~x
1
~
x
2
f q
Gm
1
m
2
r
2 arctan r
0
:
(4)
Relativistic corrections to the potential are similarly modi-
PRL 100, 031301 (2008)
P H Y S I C A L
R E V I E W
L E T T E R S
week ending
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0031-9007= 08=100(3)=031301(4)
031301-1
© 2008 The American Physical Society ed [
15
,
16
]. The above expression asymptotes to the stan-
dard result for r 0
but reaches a nite minimum as
r=
0
! 0. Hence, static masses become free of gravitation
at short distances.
The possibility of new gravitational phenomena at sub-
millimeter distances has motivated laboratory tests of the
Newtonian force law [
17 22
]. These experiments look for
departures from the Newtonian force law, which are inter-
preted as bounds on a Yukawa-type modication of the
potential, V
Gm
1
m
2
r
1
e
r=
. The potential (
4
)
roughly corresponds to
1 and 0
. Recent mea-
surements show that the Newtonian force law holds down
to 56
m for j j
1 so that
> 0:0035 eV=c at the 95%
condence level [
22
]. These efforts are at the threshold of
the scale inferred from
.
Long-distance gravitational phenomena are also sensi-
tive to such modications and provide a tighter bound on
, the scale of new physics. The key is the limited range of
graviton momenta mediating the gravitational force ex-
erted by a massive body on a test particle. Considering
the deection of light as an elastic, quantum mechanical
scattering process, the photon energy is conserved, but its
momentum is redirected. A maximum graviton momentum
implies a maximum deection angle, and so j ~
k
i
~
k
f
j
2k
<
, where k is the photon momentum.
We perform a calculation of tree-level photon scattering
in linearized quantum gravity. We treat the lens as one
massive particle, as many constituent particles, or as the
source of an external gravitational eld. All approaches
yield the same result. The external eld offers the clearest
view. The cross section is

2
2
Z d
3
k
f
k
i
k
f
jhk
f
jMjk
i
ij
2
(5)
for a given photon polarization. The Maxwell tensor
T
F
F
1
4
F
F
is used in (
2
) to determine
the scattering vertex, and the matrix element is calculated
in the external-eld approximation, using h
for a weak
gravitational eld due to a point source of mass M.
Following Refs. [
23
,
24
] we obtain

hk
f
jMjk
i
i
8 GM
2 2
2
k
f
k
i
q
e
j ~
k
f
~
k
i
j=
j ~
k
f
~
k
i
j
2
e; k
e; k
1
2
p
^e
i
^e
f
3
^
k
i
^
k
i
^e
f
^
k
i
^e
i
^
k
f
;
(6)
where ^e is the photon polarization vector. Averaging over
incoming photon polarizations and summing over outgoing
polarizations, we obtain the differential cross-section in the
small angle limit

d
d
4GM
2
c
4
e
2 k =
:
(7)
In the absence of the cutoff, the cross section has the
familiar
4
dependence found in Coulomb scattering.
With the cutoff, we interpret the result to indicate that
high-energy photons nd a weaker gravitational lens than
low-energy photons. This stands in contrast with the ach-
romatic nature of lensing in general relativity.
It is not surprising that gravitational lensing can be
described by a tree-level diagram. As with Coulomb scat-
tering, a tree-level diagram is sufcient to reproduce the
classical result. We may also calculate the contribution of
higher-order Feynman diagrams in the eikonal limit,
wherein the total energy of the colliding particles vastly
exceeds the momentum transfer. This applies to astrophys-
ical gravitational lensing. In perturbative quantum gravity,
graviton loop diagrams are responsible for the nonrenor-
malizability of the theory and lead to a loss of predictive
power at high energies. In the eikonal limit, these diagrams
are negligible compared to the series of ladder and crossed-
ladder diagrams illustrated in Fig.
1
. As shown in
Refs. [
25
,
26
], the amplitude for gravitational scattering
of two massive scalar particles can then be summed to all
orders in perturbation theory. In the absence of a cutoff on
graviton momenta, this procedure yields the amplitude
multiplied by a divergent phase factor. Since the cross
section depends on j
Mj
2
, the Born approximation for the
cross section is exact. We generalize this result to the case
with the cutoff. We work in the rest frame of the massive
scatterer and include an exponential factor for the momen-
tum cutoff on each graviton propagator. The photon is
adequately treated as a massless scalar in the limit of small
deections. Then, following Ref. [
26
], the scattering am-
plitude due to an innite sum of ladder graphs in the
eikonal limit is

i
M
8 ME
q
2
e
q=
Z
1
0
dzzJ
0
z f k
IR
=
k
IR
=
2
zk
IR
=q
2
q
4i
1g:
(8)
As in QED, the infrared regulator k
IR
is necessary because
the asymptotic states assumed were plane waves, rather
than Coulombic wave functions. To proceed, we make a
series expansion in small k
IR
=
. Then, because
GME
1, the integral is found to be well approximated
by

i
M
i
M
Born;GR
e
q=
4k
2
IR
q
2
2i
1
2i
1
2i
e
iq=
;
(9)
where
M
Born;GR
32 GM
2
E
2
=q
2
for the gravitational
scattering of these two scalar particles. This nonperturba-
tive result consists of the exponentially suppressed Born
amplitude with an additional phase which does not affect
the scattering cross section. Thus our tree-level result is
exact in the eikonal limit.
As opposed to multiple graviton exchange in a single
scattering interaction, we may also consider multiple en-
counters along the particle trajectory. For photons im-
pinging on a target with an impact parameter b, the gravi-
PRL 100, 031301 (2008)
P H Y S I C A L
R E V I E W
L E T T E R S
week ending
25 JANUARY 2008
031301-2 tational interaction time is
b=c
. In comparison, the in-
terval during which the photon is i