Does the existence of dark energy suggest that there is more to the graviton
than we think we know?Comment: 4 page

A. Zee

Banks

Montonen

Perlmutter

Riess

Vilenkin

Weinberg

Physics Letters B

English

arXiv

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Dark energy and the nature of the graviton

AbstractDoes the existence of dark energy suggest that there is more to the graviton than we think we know

Zee, A.

Elsevier - Publisher Connector 

Physics Letters B 594 (2004) 8–12
www.elsevier.com/locate/physletb
Dark energy and the nature of the graviton
A. Zee
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
Received 23 April 2004; accepted 30 April 2004
Available online 9 June 2004
Editor: H. Georgi
Abstract
Does the existence of dark energy suggest that there is more to the graviton than we think we know?
 2004 Published by Elsevier B.V. Open access under CC BY license.The word paradox has been emasculated by in-
discriminate usage in the physics literature. A real
paradox should involve a major and clear-cut discrep-
ancy between theoretical expectation and experimen-
tal measurement. The ultraviolet catastrophe, for ex-
ample, is a paradox, the resolution of which around the
dawn of the 20th century ushered in quantum physics.
Surely, the most egregious paradox of physics around
the dawn of the 21st century is the cosmological con-
stant paradox [1].
The root of the paradox lies in a fundamental clash
between Einstein’s view and the particle theorist’s
view of gravity. To particle physicists, the graviton
is just another particle, or a particular mode of the
vibrating string. Indeed, given that a massless spin-2
particle couples to the stress-energy tensor, one can
reconstruct Einstein’s theory. According to Einstein,
however, gravity has to do with the curvature of
spacetime, the arena in which all fields and particles
live in. The graviton is not just another particle.
E-mail address: zee@kitp.ucsb.edu (A. Zee).0370-2693 2004 Published by Elsevier B.V.
doi:10.1016/j.physletb.2004.04.087
Open access under CC BY lThe graviton is not just another particle—it knows
too much. The electromagnetic force knows about the
particles carrying charge, and the strong force knows
about the particles carrying color. But the gravitational
force knows about anything carrying energy and mo-
mentum, including an apparently innocuous constant
shift in the Lagrangian density L→L− Λ.
As is well known, the paradox can be easily de-
scribed. The natural value of Λ in particle physics
is expected by dimensional analysis to be µ4 =
µ/(µ−1)3 for some relevant mass scale µ where the
second form of writing µ4 reminds us that Λ is a
mass or energy density. Whether one associates µ
with grand unification, electroweak symmetry break-
ing, or the quark confinement transition and conse-
quently has a value of order 1019 GeV, 102 GeV, and
1 GeV, respectively, is immaterial. The natural value
Λ ∼ µ4 = µ/(µ−1)3 is outrageous even if we take the
smallest value for µ. We do not even have to put in
actual numbers to see that there is a humongous dis-
crepancy between theoretical expectation and observa-
tional reality. We know the universe is not permeated
with a mass density of order of 1 GeV on every cube
of size 1 (GeV)−1.
icense.
A. Zee / Physics Letters B 594 (2004) 8–12 9The cosmological constant paradox is basically an
enormous mismatch between the units natural to par-
ticle physics and natural to cosmology. Measured in
units of GeV4 the cosmological constant is so incred-
ibly tiny that particle physicists have traditionally as-
sumed that it must be mathematically zero, and have
looked in vain for a plausible mechanism to drive it to
zero. One of the disappointments of string theory is its
inability to resolve the cosmological constant paradox.
But Nature has a big surprise for us. While theorists
racked their brains trying to come up with a convinc-
ing argument that Λ = 0, observational cosmologists
[2,3] steadily refined their measurements and changed
their upper bound to an approximate equality1
(1)Λ ∼ (10−3 eV)4 (!!!).
The cosmological constant paradox deepens. The-
oretically, it is easier to explain why some quantity is
mathematically 0 than why it happens to be ∼ 10−124
in the units natural (if indeed they are) to the problem.
As is also well known, strictly speaking, we should
refer to the observation of the cosmological constant
as the observation of a hitherto unknown dark energy
since we do not know the equation of state associated
with the observed energy density (10−3 eV)4.
To make things worse, the energy density
(10−3 eV)4 happens to be the same order of magnitude
as the present matter density of the universe ρM . This
is sometimes refer as the cosmic coincidence problem.
The cosmological constant Λ is, within our traditional
understanding, a parameter in the Lagrangian. On the
other hand, since most of the mass density of the uni-
verse resides in the rest mass of baryons, as the uni-
verse expands ρM(t) decreases like (1/R(t))3, where
R(t) denotes the scale size of the universe. In the far
past, ρM was much larger than Λ, and in the far future,
it will be much smaller. It just so happens that in this
particular epoch of the universe, when we are around,
that ρM ∼ Λ. Or to be less anthropocentric, the epoch
1 The value 10−3 eV does not correspond to any known mass
scale in particle physics. The differences in mass squared of the
3 neutrinos have been measured experimentally to be |m23 − m22| ∼
3 × 10−3 eV2 and |m22 − m21| ∼ 7 × 10−5 eV2 and so it is
conceivable that the lightest neutrino has mass ∼ 10−3 eV and that
this may have something to do with the cosmological constant. In
a recent paper, using a particular ansatz [4] determined neutrino
masses to lie in the range 10−2 to 10−4 eV.when ρM ∼ Λ happens to be when galaxy formation
has been largely completed. In their desperation, some
theorists have even been driven to invoke anthropic se-
lection [5–7].
My impression is that theoretical physicists outside
the high energy community are not completely aware
of how desperate the situation is, but people are grasp-
ing at straws and a number of outlandish suggestions
have been aired. In this spirit I would like to offer a
thought I have entertained for some time but did not
“dare” to publish. When I wrote my recent field the-
ory textbook I sketched [8] what I had in mind in pass-
ing. It may be worthwhile to elaborate on what I wrote
there and to bring it to the attention of a broader audi-
ence.
In the development of physics, there have been nu-
merous instances of reasoning by historical precedent
or analogy. For example, when confronted with data
showing that the energy of the electron in an atom is
quantized physicists recalled that the vibrational fre-
quency of a violin string is also quantized. As we all
know, this turns out to be an apt analogy as both the en-
ergy of the electron and the vibrational frequency of a
string are given by the eigenvalues of linear partial dif-
ferential equations. I suggest that perhaps similarly we
can ask if historically there have been cases of a phys-
ical quantity initially thought to be 0 but then turned
out to be extremely small but not precisely 0. I sus-
pect that the proton decay rate may be an apt example
and that it may shed some light on the cosmological
constant paradox.
Let us go through the story of proton decay.
To make my point I will take some liberty with
history. Suppose that in 1953 some theorists were
to calculate the rate Γ for protons to decay in the
natural mode p → e+ + π0. The interaction of the
pion with the proton and the neutron was known to
be described by a term like gπn¯p in the Lagrangian
with g a dimensionless coupling of order 1. These
theorists would naturally construct a Lagrangian out
of the available fields, namely the proton field p, the
electron field e, and the pion field π , and thus write
down something like f πe¯p with some constant f .
Note that πe¯p has mass dimension 4 and hence
f is dimensionless just like g. Since πe¯p violates
isospin invariance, the theorists would expect f to be
suppressed relative to g by some measure of isospin
breaking, say the fine structure constant α. The natural
10 A. Zee / Physics Letters B 594 (2004) 8–12value for Γ would then come out to be many orders of
magnitude larger than the experimental upper bound
on Γ . The theorists would then set Γ = 0 and cast
about for an explanation. After an enormous struggle,
the theorists were unable to come up with a compelling
explanation and this failure became known as the
proton decay rate paradox.
Eventually, someone with great authority and pres-
tige in the community, namely Wigner, decreed the
law of baryon number conservation. Surely, even in the
unthinkably primitive days of 1953 this would have
been recognized as a pronouncement and not as an ex-
planation. (The pronouncement could be dressed up
formally by imposing a U(1) transformation under
which p → eiθp while e and π do not change and
requiring that the Lagrangian remains invariant.) But
there would have been no deep understanding of this
astonishing discrepancy between theoretical expecta-
tion and experimental upper bound.
Indeed, imagine an alternative history in which,
while other important particle physics experiments
were being performed in 1957, some intrepid exper-
imentalist, ignoring conventional theoretical wisdom,
actually went out and measured the proton decay rate
to be some tiny but non-zero value. The proton decay
rate paradox would have deepened, much as how the
cosmological constant paradox deepened with the dis-
covery of a tiny Λ.
Let us now review how the proton decay rate
paradox was resolved historically. The first remark is
that the eventual explanation did not emerge within
the orthodox theory fashionable in 1957, nor did
it come from an understanding of some kind of
mechanism causing protons to decay, but rather it
came totally from “left field”, from a study of baryon
spectroscopy, which led to the notion of quarks. The
correct degrees of freedom are not given by the proton
and pion fields p and π , but by the quark fields q .
The effective Lagrangian L is to be constructed out
of quark q and lepton l fields and must satisfy the
symmetries that we know. Three quarks disappear, so
we write down schematically qqq , but three spinors
do not a Lorentz scalar make. We have to include
a lepton field and write qqql. Since four fermion
fields are involved, these terms have mass dimension 6
and so in L they have to appear as (1/M2)qqql
with some mass M , corresponding to the mass scale
of the physics responsible for proton decay. Thus,the probability of proton decay is proportional to
(1/M2)2 = 1/M4. By dimensional reasoning, we
obtain the proton decay rate Γ ∼ (mp/M)4mp . The
absurdly small value of Γ is then naturally explained
by the fourth power of the small number mp/M for M
big enough. No mystery left!
Note that in principle all of this could be done as
soon as Gell-Mann introduced the notion of quarks in
1964, long before anybody even dreamed of a grand
unified theory with proton decay.
As long we are discussing revisionist, but possible,
history, we can imagine some brilliant theorist in
another civilization far away puzzling over the proton
decay rate paradox eventually realizing that the key to
explaining an absurdly small number is to promote the
dimension of the effective Lagrangian merely from 4
to 6. In hindsight, we can say that the extremely
long lifetime of the proton could have pointed to the
existence of quarks.
I would like to raise the question whether the
cosmological constant paradox might not be solved in
the same way. Perhaps the gravitational field gµν is
the analog of the proton and pion field p and π . The
high energy and more fundamental degrees of freedom
in the gravitational field may not be the metric gµν ,
but some mysterious analog of the quark field q . This
may emerge as a construct in string or M-theory, or
it could be something else completely. In the history
of the proton decay paradox as recounted by me,
there is an additional twist, namely that the degree of
freedom q is confined and not physical. Before the
advent of quantum chromodynamics, theorists could
only write p ∼ qqq , without any clear idea about what
the symbol ∼ might mean. We are in a similar position
here: the metric gµν might be a composite object, but
I certainly do not know what it is a composite of, and
the objects of which gµν is a composite may also be as
observable or as unobservable as the quarks.
The cosmological term Λ√g in the Lagrangian has
mass dimension 0 and we somehow have to promote 0
to a higher number. One difficulty with this view is of
course how we could possibly promote the dimension
of the cosmological term without at the same time
changing the mass dimension of the Einstein–Hilbert
term 1
G
√
gR. Our historical analogy may again be
helpful: the 1953 view that the pion nucleon coupling
term has dimension 4 turns out to be correct. While
the dimension 4 term πe¯p was replaced by the
A. Zee / Physics Letters B 594 (2004) 8–12 11dimension 6 term qqql the dimension 4 term πn¯p
was replaced by dimension 4 terms of the form q¯Aq
with A a gluon potential. The dimension of one of the
terms gets promoted while the dimension of the other
term remains the same. So it is entirely conceivable
to me that the cosmological constant term could end
up with a higher dimension while the Einstein–Hilbert
term either remains dimension 4 or is replaced by
dimension 4 terms. Thus, suppose the cosmological
constant term actually has dimension p > 0 so that it is
given in the Lagrangian by a term of the form 1
Mp−4O
with M some mass scale characteristic of the deeper
structure of the graviton, perhaps the same as the
Planck mass, perhaps not. The observed cosmological
constant would then be given by
Λ ∼ 1
Mp−4
〈O〉 =
(
m
M
)p
M4,
where the expectation value of the operator O in the
physical universe 〈O〉 = mp is set by physics at some
low energy scale m. With m small enough, and or p
big enough, we could easily get the suppression factor
we want.
As hinted above, I even suspect that the La-
grangian formalism, being a mathematical realization
of the quasi-theological (at least historically) varia-
tional principle, may well be wrong. The cause of all
our trouble is that the flat space Lagrangian L could
always be shifted by a constant L→ L − Λ without
changing its variation. But this also points to the mys-
tery of quantum mechanics2 because without quantum
mechanics we could have lived happily with equations
of motion without ever bothering with the Lagrangian.
Another possible way to nullify the physical con-
sequence of the shift L→ L− Λ is to postulate that
g = detgµν is not a dynamical variable. This was pro-
posed [9] 20 years ago as an explanation of why Λ is
mathematically zero. But with Λ now known to be tiny
but non-zero this avenue seems to me less promising.
2 When I discussed the speculative idea proposed in [9] with
R. Feynman, he asked me if I knew of a formalism in quantum
mechanics in which one could calculate the difference between
two energy levels in say an atom directly (which is after all
what experimentalists measure) without having to calculate the
two energy levels separately and then subtract one from the other.
Without gravity, only differences in energy matter.I also could not resist mentioning another wild
speculation [10]. Many years ago, inspired by the al-
most exact correspondence between Einstein’s post-
Newtonian equations of gravity and Maxwell’s equa-
tions of motion I proposed the gravitipole in analogy
with Dirac’s magnetic monopole. After Dirac there
was considerable debate on how a field theory of mag-
netic monopoles may be formulated. Eventually, ’t
Hooft and Polyakov showed that the magnetic mono-
pole exists as an extended solution in certain non-
abelian gauge theories. Most theorists now believe that
electromagnetism is merely a piece of a grand unified
theory and that magnetic monopoles exist. Might it not
turn out that Einstein’s theory is but a piece of a big-
ger theory and that gravitipoles exist? In grand unified
theory the electromagnetic field is a component of a
multiplet. Could it be that the gravitational field also
somehow carries an internal index and that the field we
observe is just a component of a multiplet? Throwing
caution to the wind, I also asked in [10] if the graviti-
pole and the graviton might not form a representation
under some dual group just as the magnetic monopole
and the photon form a triplet under the dual group of
Montonen and Olive [11].
Perhaps we do not know as much about the graviton
as we think we do.
Acknowledgement
This work was supported in part by the National
Science Foundation under grant number PHY 99-
07949.
References
[1] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1.
[2] A.G. Riess, et al., Astron. J. 116 (1998) 1009.
[3] S. Perlmutter, et al., Astrophys. J. 517 (1999) 565.
[4] X.G. He, A. Zee, hep-ph/0302201.
[5] T. Banks, Nucl. Phys. B 249 (1985) 332.
[6] S. Weinberg, Phys. Rev. Lett. 59 (1987) 2607;
S. Weinberg, Phys. Rev. D 61 (2000) 103505.
[7] A. Vilenkin, Phys. Rev. Lett. 74 (1995) 846.
[8] See p. 440 in: A. Zee, Quantum Field Theory in a Nutshell,
Princeton Univ. Press, Princeton, NJ, 2003.
[9] F. Wilczek, A. Zee, unpublished;
A. Zee, Remarks on the Cosmological Constant Paradox, in:
S.L. Mintz, A. Perlmutter (Eds.), High Energy Physics in
12 A. Zee / Physics Letters B 594 (2004) 8–12Honor of P.A.M. Dirac in his Eightieth Year, Proceedings of
the 20th Orbis Scientiae, Plenum, New York, 1983, p. 211. This
possibility was analyzed further in Section VI of [1].[10] A. Zee, Phys. Rev. Lett. 55 (1985) 2379.
[11] C. Montonen, D. Olive, Phys. Lett. B 72 (1977) 117.


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