A possible anthropic solution to the Strong CP problem
ABSTRACT We point out that the long-standing strong CP problem may be resolved by an anthropic argument. The key ideas are: (i) to allow explicit breaking(s) of the Peccei-Quinn symmetry which reduces the strong CP problem to the cosmological constant problem, and (ii) to conjecture that the probability distribution of the vacuum energy has a mild pressure towards higher values. The cosmological problems of the (s)axion with a large Peccei-Quinn scale are absent in our mechanism, since the axion acquires a large mass from the explicit breaking. Comment: 22 pages, 2 figures.v2: references added.v3:a version to appear in Progress of Theoretical Physics
arXiv:0804.2478v3 [hep-ph] 10 Apr 2009
A possible anthropic solution to the Strong CP problem
Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Chiba 277-8568, Japan
We point out that the long-standing strong CP problem may be resolved by an anthropic argu-
ment. The key ideas are: (i) to allow explicit breaking(s) of the Peccei-Quinn symmetry which
reduces the strong CP problem to the cosmological constant problem, and (ii) to conjecture that
the probability distribution of the vacuum energy has a mild pressure towards higher values.
The cosmological problems of the (s)axion with a large Peccei-Quinn scale are absent in our
mechanism, since the axion acquires a large mass from the explicit breaking.
One of the profound problems of the standard model (SM) is the strong CP problem.
In the quantum chromodynamics (QCD), there is no a priori reason to forbid the following
µν is the field strength of the SU(3)cgauge fields, and gsis the SU(3)cgauge
coupling. This operator contributes to the electric dipole moment of the neutron, and the
experimental measurements have severely limited the parameter θ as |θ| < 10−(9−10)≡
θ(exp). Such a tight constraint on θ is regarded as a fine-tuning; this is the strong CP
The Peccei-Quinn (PQ) mechanism provides a natural solution to the strong CP prob-
lem . In the mechanism, one introduces an axion [2, 3, 4], which is charged under the
PQ symmetry. Under the PQ transformation, the axion field a gets shifted as a → a+faǫ,
where fadenotes the axion decay constant (or the PQ scale), and ǫ is the transformation
parameter. In what follows we normalize the axion a by faso that a is dimensionless. The
axion is assumed to be coupled to the QCD anomaly,
After the QCD phase transition, the axion gets stabilized due to the QCD instanton effect,
satisfying a + θ = 0. Thus the strong CP problem is solved dynamically.
Since the PQ mechanism was proposed, a lot of efforts have been made to implement
the mechanism. The models proposed so far can be divided broadly into two categories.
One adopts a field theoretic approach using a U(1)PQsymmetry. In the DFSZ [5, 6] and
KSVZ (or hadronic) [7, 8] axion models, a global U(1)PQsymmetry is introduced, which
is spontaneously broken by a vacuum expectation value (VEV) of a scalar field. The
associated Nambu-Goldstone boson becomes an axion. Those models fall in this category.
The other identifies one of the axion-like fields in the string theory to be the QCD axion.
We focus on the latter category throughout this letter.
The string theory is currently the most promising candidate for a unified theory of all
forces including gravity . Moreover, it contains many axion-like fields associated with
the Green-Schwarz mechanism . Therefore, it is natural to seek for the QCD axion
in the string set-up. However, it turns out that there are severe cosmological problems
associated with the axion.
The PQ scale fais constrained as 109GeV ? fa? 1012GeV [12, 13, 14] from astro-
physical and cosmological considerations. The upper bound comes from the requirement
that the axion density should not exceed the observed amount of dark matter (DM), based
on an assumption that the initial displacement of the axion from the nearest minimum
is O(1). However, the PQ scale is expected to be as large as O(1016)GeV in the string
theory. If fais as large as 1016GeV, the axion abundance would exceed the observed DM
abundance by many orders of magnitudes. Although we may hope that the axion model
with smaller facan be constructed, currently it seems hard to make the value of famuch
smaller than 1016GeV . There are several solutions proposed so far; (i) to dilute the
axion abundance by the late-time entropy production ; (ii) to set the initial position
of the axion very close to the CP conserving minimum. However both are not completely
The first solution (i) is most easily realized by introducing the late-time decaying par-
ticle  or unstable topological defects , which produce enormous amount of the
entropy at the decay. However, since the pre-existing baryon asymmetry is also diluted,
we have to rely on a very efficient baryogenesis scenario such as the Affleck-Dine mech-
anism [18, 19, 20, 21, 22, 23]. We do not argue that it is impossible to have consistent
cosmology in this case, but the cosmology required by this solution is far from the simplest
one, making us feel that it is slightly contrived.
In the second solution (ii), we need to fine-tune the initial position of the axion. Since
the axion likely takes a randomly chosen value due to quantum fluctuations during infla-
tion, we need to indeed fine-tune the initial position by hand. One may hope that the
initial position of the axion might be selected in such a way that the axion abundance does
not exceed the DM abundance , based on the anthropic principle. When applied to the
cosmological constant, the anthropic principle was successful as shown in . However,
the recent analysis showed that the constraint on the DM abundance, therefore on the
initial position of the axion, is too loose based on the simple anthropic argument . On
the other hand, the authors of Ref.  performed much more detailed studies by taking
account of e.g. the comet impact rate in a universe with a larger amount of dark matter.
Their results showed that the anthopically favored value of the dark matter abundance is
very close to the observed one. While we agree that the comet impact rate can have an
important effect on the existence of life, it is not easy to estimate its effect precisely due
to our limited knowledge.
The bosonic supersymmetric (SUSY) partner of the axion, saxion, also leads to a severe
cosmological problem [28, 29, 30], which is similar to the notorious cosmological moduli
problem [31, 32, 33, 34]. One may be able to solve the problem in a similar fashion
described above, but the resultant cosmology again does not seem natural. Note also that
the anthropic argument on the (s)axion abundance cannot solve the problem, unless the
saxion is stable in cosmological time.
While a starting point is well grounded theoretically, i.e., the axion elegantly solves the
strong CP problem and the string theory seems to be the plausible candidate to implement
the PQ mechanism, we are nevertheless led to either apparently contrived cosmology or
the fine-tuning. Those tantalizing situation can be viewed as a hint that we might have
made a wrong assumption from the very beginning. That is to say, the dynamical solution
to the strong CP problem may not be the correct answer, if the axion is to be embedded
in the string theory.
In this letter, we give up the ordinary PQ mechanism, and instead, we consider what
happens if the PQ symmetry is explicitly broken other than the QCD instantons. The
beauty of the PQ mechanism has prevented most people to pursue this possibility seriously.
We find that the CP conserving minimum can be anthropically selected, if the probability
distribution of the vacuum energy excluding the contribution from the axion sector has
a pressure toward higher values. Whether the probability distribution possesses such a
property or not is tied to the cosmological measure problem, and we do not have a definite
answer at the moment. We will, however, give several possibilities that such a feature may
It is quite interesting to note that the axion can acquire a large mass due to the explicit
breaking, and it may be absent in the low-energy particle spectrum
a. This striking
feature has rich implications for cosmology. All the cosmological problems associated with
the (s)axion are solved, if the (s)axion mass is large enough. The axion may come to
dominate the energy density of the universe after inflation, and reheat the universe by its
decay. It is even possible to make the cosmological abundance of the axion negligible, if
the explicit breaking is large enough during inflation.
To summarize, with our conjecture on the probability distribution of the vacuum energy,
we arrive at the followings.
1. The strong CP problem is resolved by the anthropic reasoning.
2. The cosmological problems of the (s)axion with large facan be solved.
3. Interesting cosmological scenarios emerge: the axion may dominate and reheat the
universe; the axion may generate the cosmological density perturbations.
In the following sections, we will detail each point.
II.THE ANTHROPIC SOLUTION TO THE STRONG CP PROBLEM
Now let us explain how it works. The shift symmetry of the axion is violated by the
QCD instantons. After the QCD phase transition, the instantons generate the effective
potential of the axion,
VQCD(a) = Λ4
QCD(1 − cosa), (3)
where the axion field a is dimensionless, and we have chosen the CP conserving minimum
to be at a = 0 for simplicity. We drop numerical coefficients of order unity here and in what
We use the terminology, “axion”, although it is not the ordinary massless QCD axion in the PQ