Some remarks on anthropic approaches to the strong CP problem
Accepted: May
Some remarks on anthropic approaches to the strong
Michael Dine 0 1 2 5
Laurel Stephenson Haskins 0 1 2 3
Lorenzo Ubaldi 0 1 2 4
Di Xu 0 1 2 5
0 Via Bonomea 265 , 34136 Trieste , Italy
1 Hebrew University , Jerusalem 91904 , Israel
2 University of California at Santa Cruz , Santa Cruz CA 95064 , U.S.A
3 Racah Institute of Physics
4 SISSA International School for Advanced Studies
5 Santa Cruz Institute for Particle Physics and Department of Physics
The peculiar value of is a challenge to the notion of an anthropic landscape. We brie y review the possibility that a suitable axion might arise from an anthropic requirement of dark matter. We then consider an alternative suggestion of Kaloper and Terning that might be correlated with the cosmological constant. We note that in a landscape one expects that is determined by the expectation value of one or more axions. We discuss how a discretuum of values of
QCD Phenomenology

nated by QCD, and
nd the requirements to be quite stringent. Given such a discretuum,
we
nd no circumstances where small
might be selected by anthropic requirements on
the cosmological constant.
1 Introduction
2
3
4
5
3.1
3.2
3.3
3.4
4.1
4.2
Fine
Coarse
Conclusions
Anthropic axions
Models which achieve a discretuum
Prelude: the irrational axion
Models with a single axion
Models with multiple axions
A stringy variant
Canceling the cosmological constant in di erent parameter ranges
A A clockwork construction
1
Introduction
At present, however frustrating it may be, the most compelling solution we have to the
cosmological constant (c.c.) problem is provided by the anthropic landscape [1{4]. The
picture scored an enormous success with the discovery of the dark energy [5, 6], with a value
only somewhat smaller than expected from the simplest version of Weinberg's argument.
More re ned versions of the argument may come closer. This success has raised the specter
that anthropic considerations may play an important role in determining all of the laws of
physics. But there are good reasons for skepticism. One might expect, in such a picture,
that the parameters of the Standard Model should either be anthropically determined or
should be random numbers. A principled objection to these ideas, then, emerges from
the fact that some parameters of the Standard Model appear to be neither random nor
anthropically constrained [7]. Possibly the most dramatic of these is the
parameter [7{9].
If a landscape picture is ultimately to make sense, one needs to
nd correlations between
the
parameter and other quantities that are anthropically constrained. For example, it
is conceivable that dark matter is an anthropic requirement [10], and it might be that the
most e cient way to obtain dark matter in a landscape is through an axion. This is a tall
order. In particular, the requirement of a dark matter axion does not necessarily imply a
PecceiQuinn symmetry of su cient quality to explain
< 10 10 [11].
The notion of a landscape can hardly be considered welldeveloped; we don't have
gravity theories in which we can reliably demonstrate the existence of even a small number
{ 1 {
Lagrangian parameters, P ( ; m2H ; gi; yff ; ), where
yff are the Yukawa couplings, m2H is the Higgs mass, and
is the c.c., gi are the gauge couplings,
is the QCD
parameter.
It seems plausible, as Weinberg assumed, that the cosmological constant is a uniformly
distributed random variable near its observed value. The question of naturalness of the
Higgs mass is the question of whether the same is true for the Higgs mass. If the distribution
is uniform, then one would seem to require an anthropic explanation of the Higgs mass, or
alternatively rely on an extraordinary piece of luck. On the other hand, supersymmetry
or dynamics could enhance the probability that one nds the mass near zero, providing a
realization of naturalness in a landscape framework.
The problem of
in a landscape is that, absent a light axion, it is not clear why a
distribution of
should peak at
= 0. As we will explain below, in a model, say, like
that of KKLT [4], one would expect that
is a discrete, uniformly distributed random
variable. As we have noted, there is no obvious anthropic preference for extremely small
. For critics of the landscape program, this is perhaps the most principled argument that
the landscape idea may not be correct. For it to survive, one almost certainly has to nd
correlations between
and other quantities which are anthropically constrained.
It is possible that small
is selected by some other consideration, dark matter being
one possible candidate. Recently, Kaloper and Terning (KT) [12] have put forward another
proposal to account for small , which would correlate the value of
with the problem of the
cosmological constant. In this note, we will attempt to esh out this proposal, determining
what is required at a microscopic level to realize their picture, and what might be the
parameters of such a model and their possible distributions. Then we ask in what range of
parameters one would in fact account for a small value of .
KT assume that the cosmological constant has two contributions, one from a structure
similar to that of Bousso and Polchinski (BP) [2], which we will denote by
BP, and
another, independent, one from QCD.
=
BP +
some fundamental scale raised to the fourth power (say Mp4) divided by the number of
states. KT assume that
is a continuous variable, and argue that, for a range of
, a
small, negative value of
BP will be compensated by a small , bringing the c.c. into the
anthropically allowed range.
But this picture raises several puzzles. In our generic landscape picture, above, we
would expect that
is some combination of axion expectation values in the underlying
theory. If
can be considered to be independent of the values of the uxes, and if QCD is
the dominant source of the
potential, then we would seem to have a conventional axion.
{ 2 {
On the other hand, we might expect the axion expectation values to be determined, in
a scenario like that of BP, by values of uxes or other quantities which label the di erent
states. In this picture, the
angles would take on random, discrete values, not obviously
correlated with other quantities. The KKLT [4] model provides a sharp implementation
of this picture. In that model, the axion is not light. There,
is xed by the expectation
values of Kahler moduli, which are themselves xed by the (random) expectation value of
the superpotential of complex structure moduli. This superpotential, hW i, itself is complex.
In order that
vanish, one might require that the expectation value of the Kahler modulus
be real. This would be the case among the tiny subset of uxes (a fraction of order (1=2)N ,
where N is the number of ux types) which conserve CP [13]. But this might not be enough
to play any role in determining the c.c., one needs, in this case, very
small steps in ,
. We call a the upper limit of the anthropic window, which we assume
close to the observed value, i.e. a
10 47 GeV4. Then a minimal requirement on
is
that it leads to steps in V ,
V , small enough to bracket a:
yielding
j V j
1
2
several models of this type, in the spirit of outlining some of the ingredients required to
achieve a correlation between
and the c.c. Modest numbers of states can be accounted
for within conventional eld theory principles. The extension of these models to account
for small enough
requires features which are not particularly plausible. Conceivably
there is some more plausible structure which could give rise to these features.
Allowing for such a structure, we then ask: does the requirement that the observed
cosmological constant lie within the anthropic window, 0 <
<
a, favor any particular
range of ? We will survey a twoparameter space:
, the typical spacing of cosmological
constants in the
ux vacua; and
. We
nd that throughout this parameter space, the
anthropic constraint on the dark energy favors large .
{ 3 {
The rest of this paper is organized as follows. In section 2, we review the problem of
obtaining suitable axions assuming an anthropic requirement of dark matter [11]. We consider
models where a discrete ZN symmetry accounts for an accidental PecceiQuinn symmetry,
and ask whether minimal requirements for dark matter yield an N large enough to account
for
< 10 10. We also mention the (possibly di erent) expectations for string theory.
In section 3, we consider models which yield a discretuum for . We rst note that the
irrational axion[18] has many of the desired features. It is not clear that such a structure
ever arises in some more fundamental theory (string theory), and we will see that, in any
case, it would tend to predict large . We then construct two models which implement
at least some aspects of the KT program. One involves a single axion coupled to a very
large additional gauge group; the other involves multiple axions and requires an intricate
discrete symmetry. These models are useful in that they do allow us to address a subset
of the questions we have raised:
1. In the theory
is discrete, and the potential behaves as V =
m2 f 2 cos .
2. The system is described in terms of two parameters,
, the typical spacing of the
BP contribution to the c.c.,1 and
, the spacings in .
We will describe another model in the appendix. In section 4, we will ask, in terms of these
parameters, where are the bulk of the states which satisfy the anthropic condition. We
will see that throughout the parameter space, cancellation of the c.c. (to within anthropic
constraints) is most e ective at
1.
2
Anthropic axions
It is conceivable that dark matter, with something close to its observed density, is an
anthropic requirement [10]. In that case, the question becomes: does the requirement of
dark matter lead to a PecceiQuinn symmetry of high enough quality to account for the
smallness of
[11]? One might imagine that in a landscape setting, an axion might be
a favorable dark matter candidate. Models based on string constructions, for example,
often have many axions, and there may be a signi cant fraction of the space of vacua in
which one or more of these is very light. We might model this by a eld, , subject to a
ZN symmetry,
2 i
! e N ;
! e
i :
{ 4 {
leading to an approximate U(
1
) symmetry,
We will assume suppression of higher dimension operators by the scale Mp. Suppose that
the leading PQ symmetryviolating operator is:
V = Mp4 N
N + c:c: :
1We will use \BP" to refer more generally to features of the theory, other than , which allow for many
possible values of the c.c.
(2.1)
(2.2)
(2.3)
minimal requirement is that the lifetime of the axion should be longer than the age of the
universe.2 Again with fa = 1011 GeV, the requirement is ma < 10 3 GeV, or N > 9. This
is a weaker requirement on N than the demands of . More stringent requirements can
arise from a detailed cosmological picture. For example, things may be di erent if there
is some approximate supersymmetry and the universe is dominated by a saxion for some
period. Depending on the details, the requirement of dark matter can sometimes account
for a small enough . It is also possible, of course, that one has a bit of luck  that N is
somewhat larger than it needs to be.
String theory suggests di erent possibilities [11]. The PQ symmetryviolating operator
in eq. (2.3) may be suppressed by a small parameter, A, e.g A = e 2
suppression of the mass by A2, for example, this might also be su cient to account for the
quality of the PecceiQuinn symmetry.
One concludes from this that it is plausible that anthropic considerations could favor
an axion suitable to solve the strong CP problem, but that it is by no means certain; many
cosmological and microphysical details would need to be understood to settle the question.
3
Models which achieve a discretuum
The basic structure of the potential of Kaloper and Terning is rather puzzling. In their
picture,
is continuous, and its potential, for any choice of the uxes, has a minimum at
= 0. The QCD contribution to the potential dominates over other microscopic
contributions. It is not clear how , as a continuous variable subject to a superselection rule,
scans. (KT refer to an earlier paper of Linde's [20] which does not really provide a precise
picture). But if somehow
is selected from this distribution, it must be compatible with
the anthropic bound for the c.c. Then, at least for some range of parameters, they argue,
small cosmological constant implies small .
In this section, instead, we consider models which create a discretuum of values of ,
and ask about the distribution of ground state energies with . High energy dynamics give
rise to a large number of degenerate vacua; the degeneracy is lifted by QCD.
2Conceivably, the lifetime could be slightly shorter, being constrained only by the requirement of
structure formation. Alternatively, e ects of the radiation due to decaying axions could yield a stronger
constraint. Observationally, the constraint is signi cantly stronger [19]. This range translates into about six
orders of magnitude in m2a, which is typically a change of 1 or 2 in the constraint on N , for a reasonable
range of fa.
{ 5 {
We start by revisiting the idea of the irrational axion[18], noting that this is, in fact, a
possible setting for these ideas. However, it is not clear whether the irrational axion is
realized in any underlying theory, so we then consider two other possible models. These models
are more concrete. On the other hand, while these theories generate a discretuum with a
nonzero , to actually play a role in the cosmological constant problem, the discretuum
must be extremely ne, and the theories then exhibit some rather implausible features. For
example, the model with the simplest eld content requires an SU(N ) gauge group with
N > 1022, a number vastly larger than any appearing in proposed landscape models.
3.1
Prelude: the irrational axion
The irrational axion is a hypothetical setting with many vacua with di erent
were realized in an underlying theory, it might provide a setting for the ideas of [12]. In the
irrational axion proposal, the
potential receives contributions with di erent periodicities,
which are not rational multiples of one another. This could arise from two groups, for
example, with couplings to the same axion,
HJEP05(218)7
2
X
i=1
a
where q1 is not a rational multiple of q2.
group with scale M 4
Then, taking the group 2 to be the Standard Model SU(3), and the group 1 another
For simplicity take q1 = x, with x irrational, and q2 = 1. Then the system has an in nity
of nearly degenerate vacua with
In this case, is zero.
2xn + 0, hence a true discretuum of values of QCD.
One has, then, a picture where for each value of the uxes, there is a distribution of
states of di erent , with minimal energy at the point at which the strong interactions
preserve CP. The crucial element here is the absence of corrections to the potential of
eq. (3.2), other than those from QCD, which lift the degeneracy. Lacking a model, it is
di cult to address the question of what sorts of corrections might arise to eq. (3.2).
One would have a similar picture if, say, q2=q1 were not irrational, but a ratio of two
extremely large primes. In any case, as discussed in [18] and subsequently by others, it is
not clear if such an axion actually emerges in string theory. We will see shortly, however,
that in this model, selection for the cosmological constant favors large, rather than small, .
Ref. [21] does not directly address our questions here, but explores the interesting
possibility that a theory with many axions might realize some features of the irrational
axion. In this situation, the number of states may be exponentially large. One obtains
bands of cosmological constant.
However, small c.c. and small
are not immediately
correlated. We will discuss a variant with many axions in section 3.3.
{ 6 {
(3.1)
(3.2)
We consider in this section a more concrete model for the small
's required to tune the
cosmological constant. The model (if it is to yield extremely small
) is not particularly
plausible, but illustrates the main ingredients required to implement the KT solution. We
will see that for a limited range of parameters, one can account simultaneously for both
the observed c.c. and the limits on . Conceivably there exists a more compelling structure
with the features we describe below. We will o er another model in an appendix.
The model has two sectors, actual QCD and an SU(N ) gauge theory with a single
adjoint fermion,
(N will be extremely large), and a ZP symmetry acting on . We
include a complex scalar, , coupled to
and to a (heavy) quark:
(3.3)
(3.4)
(3.5)
(3.6)
Under the ZP symmetry,
! e 2Pi . For general P , the discrete symmetry is anomalous
with respect to both groups. We will comment on this in a moment. We assume that
develops a vev, breaking the approximate PecceiQuinn symmetry of the model,
Integrating out the heavy elds yields F F~ couplings to both groups:
This yields a potential
N
= f ei :
Both terms in 3.6, for general P , violate the ZP symmetry. The phenomenon of discrete
symmetries with di erent apparent anomalies does occur in string theory. In such cases,
these anomalies are cancelled by a GreenSchwarz mechanism involving multiple axions.
In principle, these axions need not be light if there role is to cancel a discrete anomaly, but
actual realizations in string constructions involve light scalars.
The huge value of N is puzzling from a string theory perspective. Even focusing on
the Cartan subalgebra of an SU(N ) group with an exponentially large value of N , we are
not aware of string theory constructions with such vast numbers of vector elds. So these
models do not appear extremely plausible. Such vast numbers of elds would likely have
other signi cant physics implications, which we won't explore here.
Note here we do not have to assume an alignment of the CP conserving points in the
two theories. Rather we need to assume that any additional contributions to the potential
from other sources (such as other gauge groups) behaving as, say,
M 4 < 10 10m2 f 2. This is similar to the requirement in
theories with a light axion of a PecceiQuinn symmetry of su cient quality to solve the
strong CP problem. It imposes a minimum on P .
Neglecting e ects of QCD, the minima of
lie at points:
= 2Nk . So steps in , are
of size
=
2
N
contribution to the vacuum energy
with k an integer, which reduces to eq. (1.2) for small
. To scan the c.c. nely enough,
the requirement
< 10 22 here corresponds to N > 1022.
In the appendix, we provide an alternative model, based on the clockwork axion
idea [23, 24], which does not require such huge gauge groups.
3.3
Models with multiple axions
Taking our clue from the irrational axion idea, we can proceed in another direction. If one
is willing to pay the price of a large number of gauge groups (say 10), with large fermion
representations (more generally with large anomaly coe cient), one can avoid the gigantic
single gauge group. There are still stringent requirements regarding discrete symmetries.
The idea is to have, say, M gauge groups (plus one more, QCD) and M approximate PQ
symmetries (so M axions). The M approximate discrete symmetries arise as a consequence
of M ZP types symmetries. There are M axions. Take the fermions to be in the adjoint
representation of the groups (or possibly larger representations  the point is to have big
anomalies, so large cos(N ) type terms). More generally, one has something like
This has a large number of degenerate solutions,
{ 8 {
which can readily be huge.
An example that is simple to analyze contains M groups, SU(Ni), i = 1; : : : ; M ; M
discrete (and approximate continuous) symmetries; and M scalars. Under the symmetries,
2 ik
i ! e ni i
;
i = fiei i ;
i =
ai
fi
:
Each scalar couples to adjoint fermions, i i i. The resulting potential is
(3.8)
(3.9)
HJEP05(218)7
(3.10)
(3.11)
(3.12)
(3.13)
The (degenerate) vacua have
i =
The discrete symmetries are anomaly free if ni = p Ni, with integer p (with di erent
conditions if the fermions are in representations other than the adjoint representation).
Whether this is a condition we need to impose will be discussed later.
It is important that the vacua be degenerate to a high degree of approximation, with
splittings smaller than 10 10m2 f 2. As it stands, the symmetries allow couplings
L =
Mpni 4
ni + c:c::
Depending on fi, one obtains di erent conditions on ni, but inevitably ni must be rather
large  typically 12 or more.
Now suppose the QCD axion arises from a eld
QCD with ZN charges (q1; : : : ; qm),
with couplings to a heavy quark
QQ. Also suppose
QCD
i
. Then the QCD
contribution is a small perturbation, of the desired type, lifting the degeneracy among the
vacua. Note
where
QCD = ei QCD ;
QCD =
X qi i:
This construction is also quite complicated. An elaborate set of elds and couplings will
be required to generate a vev for
QCD, and to avoid additional approximate global
symmetries. But at least there is no group SU(Avogadro's number).
3.4
A stringy variant
Rather than postulate a large set of gauge groups, we might consider a string theory with
a large number of axions, and suppose that some nonperturbative e ect (e.g. instantons
in the string theory) gives rise to a potential for the axions. Such a possibility has been
considered as an alternative to
uxes to obtain a large discretuum of states in [21]. Here
we are essentially considering both
uxes and multiple axions as sources of the c.c. (this
will be quite explicit in the next few sections). This potential might be a sum of terms:
N
X cos
a=1
M
X
i=1
!
a
iri
Mp4e sa ;
where sa might be the (assumed large) expectation value of some modulus, and ria are
some integers. We have rather arbitrarily chosen Mp as the fundamental scale. The sum is
over those terms which are large compared to QCD. If M = N , there are a large (discrete)
number of degenerate minima. If M < N , there are not a large number of degenerate
solutions (this is essentially the idea in [21] for generating a discretuum of cosmological
constants). If M > N , there are one or more light axions.
{ 9 {
(3.14)
(3.15)
HJEP05(218)7
(3.16)
(3.17)
(3.18)
In this section, we assume that the underlying theory has a discretuum of states of di erent
, and we ask whether anthropic selection for the c.c. leads to small . We start from
eq. (1.1), and we treat both
and
BP as discrete parameters. We write
= k
, with q taking discrete values, not necessarily integer,
ranging from order 1 to large values, in order to stress that the spacing of states in the
BoussoPolchinski landscape is not uniform. The cosmological constant,
=
q
cos(k
)m2 f 2
q
+
2m2 f 2 ;
must satisfy the anthropic bound
states [labeled by (k; q)] that satisfy the bound of eq. (4.2). The question we wish to ask
is: are these states characterized by large or small ?
Given that the cosmological constant is a sum, in this picture, of two independent
parameters, one might be skeptical as to whether or how small
might be favored. Indeed,
as we will explain in more detail below, if scanning is arbitrarily precise in both
and
, then small
will not be favored. It is perhaps slightly less obvious what happens if
scanning is coarse in both parameters. The rest of this section enumerates and explores
the various possibilities.
At a simpleminded level, there are a few constraints on the parameters that appear
in eq. (4.1). There is a maximum possible k, kmax, given by kmax
= 2
(there are
also degeneracies in k, which will not be particularly important in what follows). Also,
< m2 f 2, or we won't, in general, have any possibility of canceling o the
We organize the discussion according to whether we are scanning nely or coarsely in
we require
c.c. as we scan in k.
the
BP direction:
4.1
For
rst
Fine
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
Fine scanning
Coarse scanning
<
a ;
a <
< m2 f 2 :
<
a, we write y =
q
and treat y as a continuous parameter. Considering
10 22, we can approximate
as continuous as well. The c.c. is
Because in this case we scan nely in
, it makes sense to de ne:
ymax =
a
m2 f 2 2;
ymin =
anthropically allowed value for the c.c., as determined by eq. (4.5). The red shading is the portion
of this region for which
< 10 10 [not drawn to scale]. This shows that values of
of order one
are favored by the c.c. anthropic selection. Right: zoomed in to very small , drawn to scale.
We expect a uniform distribution in y and , so the fraction of states satisfying the anthropic
condition is:
Z
Z
d dy
(ymax
y) (y
ymin) =
d (ymax
ymin);
(4.7)
which is independent of . So large
> 10 22 and we treat it as a discrete variable.
Also in this case there is a larger number of states with large , and the energy density of
any such state can be canceled to the desired accuracy by y [see gure 1].
4.2
Coarse
convenient to de ne
In this subsection, we consider the case
a <
< m2 f 2. Here, because we do not scan
nely in
, we can't treat q as a continuous variable as in our previous analysis. It is
a
As the theta potential goes as k2, there can be a value, k0, above which also the scanning
in the theta direction becomes coarse (larger than
a). k0 is obtained from:
V (k0) = m2 f 2k0
2 =
a ;
)
k0 =
1
2a
We distinguish 2 cases. In the rst
we scan nely for all values of k; k
kmax. In the second
1 < k0 < kmax, so the scanning becomes coarse for k > k0.
0 < a <
1
2 m2 f 2
a ;
a
1
2 m2 f 2 < a <
1
2
;
(4.8)
(4.9)
(4.10)
(4.11)
the anthropic constraint is
N (q)
k+
k
1
2paqb
< 1 ;
Treating q and
as approximately continuous, q
satisfying the anthropic condition is larger at large
2, and we see that the number of states
(q) by pq / .
Now take the case of eq. (4.11). Let's start with
xing large q, and correspondingly
large , where, for general q there is not a choice of k satisfying the anthropic condition.
In other words, for
r qb
a
> k0
)
P (q)
k+
k
1
2paqb
< 1
1
2paqb
< 1 ;
the window k+
k in this case is smaller than 1. Still, there is a small chance, for any
given q, to nd an integer k in such a window. The probability is
k
=
r qb
a
;
k+ =
r qb
a
+
1
a
r qb
a
+
1
2paqb
:
Consider, rst, the case of eq. (4.10). Then the number of states, for xed q, satisfying
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
Eq. (4.2) in terms of a and b becomes
qb
a
< k2 <
qb
a
+
1
a
Now we want to ask: are there more (k; q) states satisfying the above inequalities at small
or large k?
with
We can write eq. (4.12) as which decreases with increasing q ( ). The point, however, is that at large q the number of possible states increases. We can estimate the number of large q states which satisfy the anthropic condition as
Nlarge =
Z ab km2ax
a k2
P (q)dq = p2ab
1
s
Note that when a saturates the upper bound of (4.10), Nlarge = 0, and remains zero for
smaller a.
Next, consider small q:
The expansion in (4.14) still holds, because qb > 1. Now, however, we have
r qb
a
< k0
)
1
2paqb
> 1 :
k+
k
1
2paqb
> 1 ;
implying there is at least one integer k which satis es (4.13), for any q which satis es (4.19).
Thus, we can estimate the number of states which satisfy the anthropic condition in this
case as
Nsmall
Z ab k02
1
dq =
1
4ab
In the rst case of eq. (4.10) we have Nsmall > Nlarge = 0, and the favored states have
< k0
. However, k0
> 1 in this case, so large theta is favored. In the second case
of eq. (4.11), Nsmall and Nlarge can be comparable as long as a is close to the lower bound
1
2 m2af2 . This, again, favors
which also favors large theta.
1. For larger values of a we quickly obtain Nlarge
Nsmall,
When a > 12 the chance of canceling the c.c. is always smaller than one, and decreases
going to larger values of q. However, as in the analysis above, the bulk of the states which
satisfy the anthropic condition are at large q, so large
Arguably if one cannot nd a suitable solution to the strong CP problem in a landscape
framework, the landscape idea may be unsupportable. As a result, it is important to study
any proposal to understand how the value of
might be correlated with other physical
quantities which might be anthropically determined. One possibility is that axions are
selected by an anthropic requirement for dark matter; the main issue is whether the
PecceiQuinn symmetry is of su cient quality to account for the small value of . We have reviewed
the challenges to such a possibility, and concluded that such an anthropic explanation of
is plausible, but that whether it is realized depends on questions about the microphysical
theory and cosmology.
Kaloper and Terning propose to correlate
with the value of the cosmological constant.
We have attempted to esh out this proposal. Rather than a continuous range of , which,
as we have explained, is likely to correspond to a conventional axion, we have argued that
one should consider the possibility that, absent QCD, there is a massive axion with a vast
number of nearly degenerate ground states. QCD then lifts this degeneracy. We have put
forward models in which
takes on a discretuum of discrete values, with a potential on
this discretuum of the desired type. Having reduced the system to a discrete system, we
were able to assess the probability of nding larger or smaller , in the sense of asking: are
most of the states with anthropically favored c.c. at large or small ? The system has two
parameters; on all of the space,
1 is favored. A skeptical reader might have expected
such a result from the start. But it is perhaps not completely obvious, at rst glance, if
scanning in
and
are not arbitrarily
ne. Here we have seen that, throughout the
range of parameters,
of order one is more likely.
It is conceivable that some other consideration might favor small
within a
discretuum. The models we have proposed to obtain such a discretuum are not particularly
attractive; indeed they are hardly plausible. The irrational axion [18] has some of the
desired features of such a system, but it is not clear that such axions actually arise in any
theory of quantum gravity, and, in any case, this would predict
1. Another class of
models involves a huge gauge group and a large discrete symmetry; others a very large
number of elds.
As for the anthropic axion, in addition to the question of axion quality, mentioned
above, it should be stressed that existing anthropic arguments for the dark matter density
are interesting but arguably not compelling. While these are serious concerns, given the
challenges of tying small
to the cosmological constant, an anthropic axion would seem a
more plausible solution of the strong CP problem in this framework. The reader, of course,
is free to view all of this as reason for skepticism about the landscape program altogether.
Acknowledgments
This work was supported in part by the U.S. Department of Energy grant number
DEFG0204ER41286. L.U. acknowledges support from the PRIN project \Search for the
Fundamental Laws and Constituents" (2015P5SBHT 002). L.S.H. is supported by the
Israel Science Foundation under grant no. 1112/17. We thank Patrick Draper for extensive
conversations. M.D. also thanks Savas Dimopoulos and Peter Graham for a helpful
conversation. We especially thank Cli ord Cheung and Prashant Saraswat for correcting a
misconception which seemed to allow an anthropic solution for a narrow range of
and .
A
A clockwork construction
The construction relies on the potential [24]
j 's are complex scalar elds. The terms in the rst sum respect a global U(
1
)N+1
symmetry, while the second sum explicitly breaks it to a U(
1
). The elds
j have charges
Q = 1; 13 ; 19 ; : : : ; 31N under the unbroken U(
1
). We take
2 > 0, so all the U(
1
)'s are
spontaneously broken at a scale f =
. All the radial modes then have a mass
of order f . Neglecting the second sum in the potential, we have N + 1 massless
NambuGoldstone bosons (NGBs). Taking into account the second sum, with the explicit breaking
parameter
1, we nd that N of these NGBs get a mass of order p
remains massless. The latter corresponds to the linear combination
Here N is a normalization factor.
Now, we can write the Yukawa coupling
N+1 N+1 N+1, where
N+1 is a fermion
that lives at the site N + 1 and is in the fundamental representation of SU(3)c, the QCD
gauge group. This is a KSVZ axion model, and the QCD anomaly leads to the coupling
f , while one, ,
= N 1 13 19 : : : 31N .
HJEP05(218)7
(A.2)
(A.3)
which in turn gives us the cosine potential
VQCD = m2 f 2 1
= m2 f 2 (1
cos ) :
Here we have de ned
F , and we have F = 3N f from the clockwork construction.
s
8 F
GG~ ;
cos
F
Next, we introduce fermions
1 in the fundamental of a new gauge group SU(N1),
with con nement scale
, and we write the Yukawa coupling
1 1 1. Again there is an
anomaly with respect to SU(N1), from which we get the potential
f
V1 =
4
1
cos
=
Note that, as this potential arises from a coupling at the rst clockwork site where the
axion
is mostly localized, the periodicity is given by f .
The sum VQCD + V1 gives the desired structure of eq. (3.6).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
HJEP05(218)7
[INSPIRE].
[INSPIRE].
058 [hepth/0309170] [INSPIRE].
[2] R. Bousso and J. Polchinski, Quantization of four form
uxes and dynamical neutralization
of the cosmological constant, JHEP 06 (2000) 006 [hepth/0004134] [INSPIRE].
[3] L. Susskind, The anthropic landscape of string theory, hepth/0302219 [INSPIRE].
[4] S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys.
Rev. D 68 (2003) 046005 [hepth/0301240] [INSPIRE].
[5] Supernova Search Team collaboration, A.G. Riess et al., Observational evidence from
supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998)
1009 [astroph/9805201] [INSPIRE].
[6] Supernova Cosmology Project collaboration, S. Perlmutter et al., Measurements of
and
from 42 high redshift supernovae, Astrophys. J. 517 (1999) 565 [astroph/9812133]
[7] T. Banks, M. Dine and E. Gorbatov, Is there a string theory landscape?, JHEP 08 (2004)
D 69 (2004) 129901] [hepth/0310203] [INSPIRE].
Rev. D 81 (2010) 025011 [arXiv:0811.1599] [INSPIRE].
[9] L. Ubaldi, E ects of theta on the deuteron binding energy and the triplealpha process, Phys.
[10] M. Tegmark, A. Aguirre, M. Rees and F. Wilczek, Dimensionless constants, cosmology and
other dark matters, Phys. Rev. D 73 (2006) 023505 [astroph/0511774] [INSPIRE].
[11] L.M. Carpenter, M. Dine and G. Festuccia, Dynamics of the Peccei Quinn scale, Phys. Rev.
D 80 (2009) 125017 [arXiv:0906.1273] [INSPIRE].
[12] N. Kaloper and J. Terning, Landscaping the strong CP problem, arXiv:1710.01740
[13] M. Dine and Z. Sun, R symmetries in the landscape, JHEP 01 (2006) 129 [hepth/0506246]
(2016) 023522 [arXiv:1510.06388] [INSPIRE].
[arXiv:1212.4371] [INSPIRE].
HJEP05(218)7
observations, JHEP 11 (2013) 193 [arXiv:1309.4091] [INSPIRE].
[1] S. Weinberg , Anthropic bound on the cosmological constant , Phys. Rev. Lett . 59 ( 1987 ) 2607 [8] J.F. Donoghue , Dynamics of Mtheory vacua , Phys. Rev. D 69 ( 2004 ) 106012 [Erratum ibid . [16] A. Arvanitaki et al., String axiverse , Phys. Rev. D 81 ( 2010 ) 123530 [arXiv: 0905 .4720] [17] M. Dine and L. StephensonHaskins , Hybrid in ation with Planck scale elds , JHEP 09 Lett . 53 ( 1984 ) 329 [INSPIRE]. [14] A.E. Nelson , Naturally weak CPviolation , Phys. Lett. 136B ( 1984 ) 387 [INSPIRE]. [15] S.M. Barr , Solving the strong CP problem without the PecceiQuinn symmetry , Phys. Rev.
[18] T. Banks , M. Dine and N. Seiberg , Irrational axions as a solution of the strong CP problem in an eternal universe , Phys. Lett. B 273 ( 1991 ) 105 [ hep th/9109040] [INSPIRE].