Axions, Time Varying CP Violation, and Baryogenesis
Lawrence M. Krauss
arXiv:2201.01447v2 [hep-ph] 6 Jan 2022
The Origins Project Foundation, Phoenix AZ 85020∗
(Dated: January 10, 2022)
We derive two features of axion cosmology that may have cosmological implications, whether or not axions
are dark matter: For the full range of allowed axion masses, the evolution of a cosmic axion background allows
large CP violation until temperatures as low as ∼ 2 GeV, and once the axion field begins to oscillate, the cosmological axion field’s relaxation to its ground state can briefly provide a new departure from thermal equilibrium,
via time-varying CP violation. During both of these periods, the Strong CP violating parameter θ̄ can be as large
as O(1).
I.
INTRODUCTION
Sakharov first demonstrated [1] that the observed baryon
number of the universe may be dynamically generated as the
Universe evolves if three conditions are present: (1) baryon
number violating interactions, (2) departure from thermal
equilibrium, and (3) CP violation. Since his original proposal,
numerous scenarios have been proposed in which all three
conditions exist at early cosmic times. The standard model
of particle physics in fact incorporates all three possibilities
but in general cannot generate the observed baryon number
today because of two factors, small CP violation, and small
departures from thermal equilibrium. For this reason it has to
be supplemented by new physics in order to produce a phenomenologically acceptable value of nB today.
Axions [2] were proposed to explain the absence of observed CP violation in the Strong Interaction, by providing a
dynamical mechanism for θ̄—the effective Strong Interaction
CP violating parameter arising from a combination of a nonθ
µν
perturbative topological term 32π
and a possible
2 TrG µν G̃
′
imaginary phase θ that may appear in quark mass matrix—
to relax to zero by the present time. Specifically θ̄ becomes
proportional to the axion field, where the axion is a pseudoGoldstone boson associated with a new spontaneously broken
axial symmetry, the minimum of whose potential lies at a = 0
This same dynamics, however, implies that cosmological
axions inherently incorporate two of Sakharov’s three conditions in the early universe, and so may be relevant for baryogenesis or have other cosmological implications: possible large
CP violation until relatively low temperatures, and a departure from equilibrium involving time-varying CP violation as
θ̄ relaxes to zero.
II.
EARLY UNIVERSE AXION DYNAMICS
Axions arise from the breaking of a new global axial symmetry at high energy, which, through instanton effects associ-
∗ Electronic
address: lawrence@originsprojectfoundation.org
ated with the axial anomaly, turns the CP violating θ̄ into a dynamical quantity. The axion is the angular mode of a complex
scalar field whose non-zero radial expectation value breaks the
axial PQ symmetry at some scale ∼ fa . The angular potential
is initially flat and the value of the axion field is undetermined,
with the appropriately normalized dimensionless field taking
some value between -π and π. If some initially small region in
which the axion field is uniform inflates, a coherent uniform
background axion will result today. In non-inflationary models, quantum fluctuations in the axion field are unconstrained,
and while the value can therefore spatially vary, the RMS spatially averaged value of the field will be non-zero.
At late times, due to chiral symmetry breaking when the
temperature of the Universe is ∼ O(1 GeV), the PQ symmetry is explicitly broken, turning the axion into a PseudoGoldstone boson with mass calculable in chiral perturbation
theory to be [3]
ma ≈ 6 × 10−6 eV (
1012 GeV
)
fa
(1)
where fa is, by analogy to pion physics, the decay constant of
the axion.
¯ ≈ a(t) so that θ̄ dynamIn the Peccei-Quinn scenario, θ(t)
fa
ically relaxes to zero as the cosmological axion field a(t) relaxes to the minimum of its potential.
Note first that since initially a(t) is unconstrained, θ̄ can be
as large as order unity initially, allowing for large CP violating
effects. Moreover, as we now illustrate, these effects will in
general persist until temperatures well below the electroweak
phase transition.
A classical axion field with initial non-zero value a0 will
generically oscillate about 0 with characteristic frequency
∼ ma . Cosmologically, an axion field will evolve via the classical equation of motion
ä + 3H ȧ + m2a a = 0
(2)
This leads to the following canonical behavior of an axion field [4–6]. When H >> m, ȧ ≈ 0, while when m >> H,
a(t) ∼ a0 cos (mt). This can be understood heuristically as follows. When H >> m the age of the universe is much smaller
2
than the natural period of oscillation of the axion field, so it remains constant. When H << m the expansion of the universe
can be ignored, and the axion field oscillates at its natural frequency. To understand what happens when H ∼ m, we can
consider more generally the equations for energy density and
pressure associated with the axion field:
ρa =
1 2 1 2 2
ȧ + ma a
2
2
(3)
Pa =
1 2 1 2 2
ȧ − ma a
2
2
(4)
At early times, when ȧ = 0, the (constant) axion field behaves like a cosmological term, with P = −ρ. As a result,
its energy density doesn’t redshift, allowing it to become, in
principle, cosmologically significant at late times, so that axions can be dark matter candidates. However, once the axion
field begins to oscillate with period m, the pressure approaches
zero, meaning that the classical axion field has the equation of
state of a pressureless, non-relativistic gas. We know that the
energy density of such a gas redshifts inversely as the cube of
the scale factor R. Since the energy density of the axion field
is ∼ m2 a2 , this implies that a ∼ R−3/2 . Assuming that the expansion of the universe is radiation-dominated that this time,
then R ∼ t1/2 , implying that a ∼ t−3/4 .
Note also that because the axion field remains constant until H ∼ m, the energy density stored in the axion field also
remains roughly constant until H ∼ m. Thus the remnant axion energy density today turns out to be inversely proportional
to the axion mass, ignoring for the moment any temperature
dependence of the axion mass. For an initial value afa0 ∼ 1,
the cosmological energy density in axions would exceed the
inferred dark matter density today if fa > 4 × 1012 GeV [2].
At the same time, astrophysical constraints on cosmic axions,
particularly limits on white dwarf cooling [5, 7, 8], imply
fa > O(1010 ) GeV, seemingly squeezing the allowed range
of axion dark matter.
However, accepting the fact that Inflation generically leads
to a multiverse, it has been recognized that the actual value of
a0
fa in our universe could be smaller than O(1). Assuming, for
example, that the Peccei-Quinn scale and the Inflation scale
were both of O(1016 ) GeV, this would imply an axion mass of
O(10−10 ) eV, and in order not to overclose the universe today,
a0
−4
fa ≤ O (10 ) .
III.
LARGE CP VIOLATION FOR T >O(1) GEV
One might expect that the time when the axion field begins
to relax to its minimum, and thus CP violation would quickly
drop to zero, would vary inversely with axion mass. However,
in QCD, chiral symmetry breaking occurs at a temperature
close to the confinement scale ΛQCD ∼ O(1) GeV. Instanton
effects that give the axion a non-zero mass typically vanish at
temperatures well above this scale. Using a dilute-instantongas approximation one finds [4]
ma (T ) ∼ ma (0)T −4 [ln(T )]3 for T > Λ.
(5)
2
During the radiation dominated phase at T ∼ Λ, t ∼ MT pl ,
where M pl is the Planck mass. Thus, if the axion field begin to relax when ti ∼ ma (T ), then from eq. (5) we find that
for axions whose natural oscillation time is less than the time
when the universe undergoes its chiral phase transition, the
temperature at which axion oscillations begin scales as
T ∼ fa−1/6
(6)
rather than as fa−1/2 .
In standard cosmology, the age of the universe when chiral
symmetry breaks, when T ∼ 200 MeV, is approximately 5 ×
10−5 s. This corresponds to an oscillation period for an axion
with a mass ∼ 10−10 eV and fa ∼ 1016 GeV.
If we consider that astrophysics currently puts a lower
bound on fa of ∼ 1010 GeV, this means that for viable axion
models the axion field does not begin to relax from its initial
post-Inflationary value until at most a temperature T ∼ 2 GeV.
This means that θ̄, and hence Strong CP violation, remains relatively large, namely greater than ∼ 10−7 for fa < MPl and as
large as O(1) for fa < 1012 GeV, until well below even the
electroweak phase transition.
This upper bound on the relaxation temperature at which θ̄
begins to decrease means that even low energy baryogeneis
[9, 10] scenarios will benefit from relaxed constraints on new
sources of CP violation. The extent to which large Strong Interaction CP violation would affect different scenarios is of
course model-dependent. But this result suggests that if a
cosmological axion field exists, even if it does not comprise
the inferred dark matter dominating galaxies today, other new
sources of CP violation beyond the standard model may not
be required for successful baryogenesis.
IV. TIME VARYING CP VIOLATION WHEN H ∼ ma
An interesting new phenomenon occurs for axion models in
the allowed range, fa > 1010 GeV, corresponding to a relaxation temperature below 2 GeV. For these masses, the axion
field will begin to oscillate with a period comparable to the age
of the Universe at that time. For the range 1010 < fa < 1016
GeV the axion mass will be temperature dependent between 2
GeV and the chiral symmetry breaking scale as we have seen.
For lighter axions, which begin to relax below the chiral symmetry breaking scale, their mass will already be fixed by the
time oscillations start.
For simplicity, we consider here an axion field with fa ∼
1016 , which begins to relax just after the chiral symmetry
breaking transition is complete, although the effect we discuss will be qualitatively similar, if slightly more complex to
compute, for other masses. This scale is also of interest as it
corresponds to a possible scale of Grand Unification in various supersymmetric theories.
3
As the axion field begins to relax, via damped oscillations,
to its preferred ground state on a timescale comparable to the
age of the Universe (and the inverse Hubble constant), this
represents a non-equilibrium phenomenon, much like the latetime out-of-equilibrium decay of a long-lived particle. In this
case, however, the time-varying order parameter describing
the out-of-equilibrium transition is the θ̄ parameter of QCD.
If the field were merely oscillating and not redshifting, then
oscillations in θ̄ would average to zero. However, the damping
of oscillations implies that this will not be the case.
For processes that take place on a timescale commensurate
with H −1 we can visualize the oscillations in the axion field,
and hence in θ̄, as short non-equilibrium pulses of non-zero
CP violation. As shown in Section II, the amplitude of the
axion field’s oscillations, and thus θ̄, falls as t−3/4 once the
field begins to oscillate. We can estimate the average magnitude of the time-varying CP violation contribution to ongoing
processes during this period by adding up the the successive
oscillating non-zero CP pulses. The sum of a series of oscillating pulses falling off as t−3/4 is given by the Dirichlet η
function with argument 3/4:
η(3/4) =
+∞
X
−1n−1
n=1
t3/4
≈ 0.651
(7)
[1] Sakharov, A.D. Journal of Experimental and Theoretical
Physics Letters. 5, 24 (1967).
[2] Peccei, R. D. and Quinn, H. R., Phys. Rev. Lett. 38, 1440
(1977), S. Weinberg, Phys. Rev. Lett. 40, 223 (1978), F.
Wilczek, Phys. Rev. Lett. 40, 279 (1978).
[3] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978), F. Wilczek, Phys.
Rev. Lett. 40, 279 (1978).
[4] Abbott, L. and Sikivie, P., Phys. Lett. B120, 133 (1983),
Preskill, J., Wise, M.B., Wilczek, F., Phys. Lett B120, 127
(1983), Dine, M., Fischler, W.. Phys.Lett. B120 (1983) 137.
[5] Marsh, D.J.E., Phys.Rept. 643 (2016) 1-79
[6] Krauss, L. M., in High Energy Physics 1985, World Scientific
Thus, the effective magnitude of CP violation during this
non-equilibrium relaxation of the axion field is compared to
the initial value before the onset of oscillation.
V.
CONCLUSION
Independent of their mass, axions allow both significant
CP violation effects in the early universe and also allow for
a brief out-of-equilibrium relaxation period involving timevarying Strong CP violation. The former effect could significantly impact baryogenesis in the early universe, including
low-energy baryogenesis scenarios. Whether the latter period
of time-varying CP violation is merely an interesting curiosity,
or may have some residual impact on cosmological or astrophysical parameters may be worthy of further study.
(Singapore) (1985)
[7] Raffelt, G.G., Astrophysical Axion Bounds, Lect. Notes Phys.
741, 51–71 (2008)
[8] Tanabashi, M. et al. (Particle Data Group), Phys. Rev. D 98,
030001 (2018).
[9] The possiblity of large CP violation due to axions was exploited
in a low energy baryogenesis model by Servant, G., Phys. Rev.
Lett. 113, 171803 (2014).
[10] i.e. see Krauss, L.M., Trodden, M. Phys. Rev. Lett. 83, 1502
(1999)