H
E

P
-P

H
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63

08

BA-94-21

June 14, 1994

Axion Dissipation Through the Mixing of
Goldstone Bosons

K.S. Babu, S.M. Barr and D. Seckel

Bartol Research Institute

University of Delaware, Newark, DE 19716

Abstract

By coupling axions strongly to a hidden sector, the energy density

in coherent axions may be converted to radiative degrees of freedom,

alleviating the \axion energy crisis". The strong coupling is achieved

by mixing the axion and some other Goldstone boson through their

kinetic energy terms, in a manner reminiscent of paraphoton models.

Even with the strong coupling it proves di�cult to relax the axion

energy density through particle absorption, due to the derivative na-

ture of Goldstone boson couplings and the e�ect of back reactions

on the evolution of the axion number density. However, the distri-

bution of other particle species in the hidden sector will be driven

from equilibrium by the axion �eld oscillations. Restoration of ther-

mal equilibrium results in energy being transferred from the axions to

massless particles, where it can redshift harmlessly without causing

any cosmological problems.



1 Introduction

Goldstone and pseudo{goldstone bosons play a large role in the speculations

of particle physicists and cosmologists. Some oft{discussed examples are

axions,1 majorons,2 familons,3 and the non{abelian goldstone �elds of texture

models.4 The salient feature of the dynamics of goldstone bosons is that they

couple derivatively. If the associated global continuous symmetryis broken at

a scale f, then the coupling always involves powers of f�1@�b, where b is the

goldstone boson. The consequence of this is that if f is very large compared

to the relevant masses and momenta, the goldstone boson decouples. This

fact is exploited in invisible axion models,5 where for fa >� 10
10 GeV the

axion's couplings are small enough for it to have evaded detection and avoided

con
ict with astrophysical observations.6 This fact also lies behind the well{

known \axion energy problem",7 which is that if fa >� 10
12 GeV the energy

in coherent oscillations of the axion �eld after the QCD phase transition in

the early universe is unable to dissipate due to the axion's weak coupling and

eventually overcloses the universe.

In this letter we show that di�erent goldstone �elds can mix with each

other through kinetic terms in the Lagrangian in a way reminiscentof photon{

paraphoton mixing,8 and that a goldstone boson can thus acquire e�ective

couplings at low energy to certain �elds which are much larger than the

typical p=f. After demonstrating this point we show how it makes possi-

ble scenarios in which the energy in coherent oscillations of the cosmological

axion �eld can be dissipated and the upper limit on fa removed.

1



2 Mixing of goldstone bosons

Consider a theory with a global symmetry U(1)a � U(1)b. Let 
 and � be

two scalar �elds with charges (1,0) and (0,1) under this symmetry. Suppose

that jh
ij � fa and jh�ij � fb, where fb is assumed to be much smaller than

fa, and call the resulting goldstone modes of 
 and �, `a' and `b' respectively.

Then it is clear that couplings of a will be suppressed by p=fa and those of

b by p=fb.

It can happen that by integrating out some �elds of mass M an e�ective

higher dimension operator will result of the U(1)a � U(1)b{invariant form

Lmix =
c

M2

y@�
�

y
@
�� + h:c: (1)

If 
 = (fa+~
)e
ia=fa and � = (fb+~�)e

ib=fb are substituted into this expression,

the result contains the term

La;b = �(@�a@�b) (2)

with

� � �2c
fafb

M2
: (3)

Through this mixing of a and b, the `axion' a can couple to particles which

have U(1)b charges with e�ective strength �p=fb, which can be much greater

than p=fa (see Fig. 1).

For example, a term such as in Eq. (1) could arise from integrating out a

�eld �, with charges (�1=2;�1=2), mass M, and coupling [�(
��2) + h:c:].

For simplicity let � have positive mass-squared and no VEV. Then the loop

shown in Fig. 2 will produce the following e�ective term (among others) for

2



momenta less than M:

�2

192�2M2
(
y@�
�

y
@
�� + h:c:) (4)

and therefore, from Eq. (3),

� = �
�2

96�2
fafb

M2
: (5)

Now, given that the term �(
��2) exists, it would require an arti�cial can-

cellation for M2 to be much less than �fafb, and hence for � to be much

larger than �=(96�2) � 10�3. There is also an absolute limit that � < 1,

since the kinetic terms of a and b have the form

1

2
( @�a @�b)

�
1 �

� 1

��
@�a

@�b

�

and a wrong{sign metric would otherwise result. There is no reason, however,

why � cannot be of order, though less than, unity.

If the higher dimension term in Eq. (1) were a \hard" term all the way up

to momenta of order fa, then the theory would become strongly coupled and

physics uncalculable unless M2 were � f2a and hence � were less than about

fb=fa. The e�ective coupling of a to particles with U(1)b charges through

mixing then would be of order �p=fb � p=fa. In other words, the coupling

of a would not be enhanced by this mixing. If, on the other hand, the term

in Eq. (1) arises through a loop such as that shown in Fig. 2 where the

intermediate particles have mass M2 � fafb, then it softens for momenta

above M and no problem arises in higher order.

3



3 Alleviating the axion energy crisis

We wish to make use of the mixing of goldstone bosons to let axions couple

strongly to some other particles and thus, somehow, allow the primordial

energy density in axions to be dissipated.

Let the �eld a be the axion that solves the strong CP problem. Then

U(1)a has a QCD anomaly and a gets a mass of order �
2

QCD=fa from instan-

ton e�ects. Several astrophysical arguments6 suggest that fa
>� 1010 GeV ,

otherwise the evolution of stars or their remnants would be drastically di�er-

ent than observed. Thus the coupling of the axion to ordinary matter, which

is proportional to 1=fa, is highly suppressed.

The axion energy crisis arises as a result of the small coupling and the

small mass of the axion. At temperatures above the QCD phase transition the

axion mass is small enough that _a � 0. As the universe cools, ma increases

until the axion �eld becomes dynamic, i.e. ma > H, where H is the expansion

rate. Barring a coincidence of alignment for the Peccei-Quinn angle, the

initial value of the axion �eld is a � fa. The axion number density is then

na � f2aH. After that point the number of axions in a comoving volume is

constant, while the axion mass increases from H to its zero temperature value

ma0. If the energy density of axions after the QCD phase transition exceeds a

critical value, �a;cr � 10�7�4QCD, then the the universe would become matter

dominated too early, i.e. at temperatures >� 10 eV . The energy density in

axions at T � �QCD is roughly �a(�QCD) � �4QCDfa=MPl, which results in

the constraint fa < 10
12 GeV . In non-in
ationary models there are bound

to be axionic strings - whose decay may9 or may not10 increase the axion

4



abundance 100-fold, lowering the cosmological bound to fa
<� 1010 GeV .

There have been several approaches to solving the axion energy crisis.

For example, entropy production11 may dilute the energy density in axions

relative to ordinary matter, or in
ation may �x the initial value of the Peccei-

Quinn angle to a suitably small value12. However, attempts to dissipate the

axion energy typically fail. Direct axion absorption presents several chal-

lenges. First, the goldstone boson nature of axions demands a factor of 1=fa

at each vertex. Second, since the axions to be absorbed are non-relativistic

the derivative couplings demand an additional factor of ma at each vertex.

Third, any process for single axion absorption must admit axion emission in

the time reversed channel. This leads to a reduction in the dissipation rate

by a factor of ma=T from a naive approach that would ignore the back reac-

tions. Flynn and Randall13 considered the non-derivative coupling of axions

to mesons14, but the �rst and third arguments above were su�cient to leave

axions undamped by several orders of magnitude.

In the present case, the mixing of axions with some other goldstone boson

allows the possibility of coupling axions strongly to some other non-standard

matter. However, without some explicit symmetry breaking of the U(1)b,

the axions must couple derivatively. The combination of powers of ma from

the couplings and from consideration of back reactions has caused all our

attempts at axion absorption to fail.

Nonetheless, there is a mechanism whereby axion oscillations can be

damped, allowing for fa of order the grand uni�ed scale MGUT , or even

the Planck scale MPl. Let the axion be mixed, as described above, with

strength � with a goldstone boson b arising from the breaking of U(1)b. The

5



axion coupling to `b{matter', as we will call it, will be vastly stronger than its

coupling to ordinary matter. Our basic idea is then that the oscillating axion

�eld causes the energies of b-particles to change in energy. If a thermal bath

of such b-particles were present in the early universe when the axion �eld

started to oscillate, then those oscillations would drive the b-distribution out

of thermal equilibrium. By adjusting the interaction rates of the b-particles

to a suitable value the axion energy can be dissipated into thermal energy of

b particles. The �nal step is to note that the b goldstone boson is massless,

so that all b energy can in principle redshift away as radiation. The axion

energy is thus put into a harmless form which never dominates the universe.

4 An explicit model

To illustrate how this may happen let us calculate in a simple model in which

the `b{matter' consists of the �eld � whose vacuum expectation value breaks

U(1)b and whose phase is b, and a scalar �eld � which has no VEV. Let

the U(1)b charges of � and � both be +1 and let them have a coupling

(1
4
g��

2

�2 + h:c:).

The terms of the Lagrangian that will be relevant for our analysis are

L =
1

2
(@�a)

2 �
1

2
m

2

aa
2 +

1

2
(@�b)

2 + �(@�a@
�
b) +

1

2
j@��j2 �

1

2
m2j�j2 �

1

4
�2
�
e�2ib=fb�2 + h:c:

�
: (6)

The last term arises from expanding � as (fb + ~�)e
ib=fb in the ��

2

�2 coupling

and de�ning �2 � gf2b . At this point we are free to choose the parameters

of L. For purposes of illustration it is convenient to choose fb � m � �QCD

6



and to choose � somewhat less than m. These values are not absolutely

necessary, but they clearly show the damping mechanism to work.

The �rst task is to resolve the mixing of a and b. To bring the kinetic terms

of a and b to canonical form we perform a non{orthogonal transformation

a
0 = a

p
1 � �2

b0 = b + �a : (7)

This gives

L =
1

2
(@�a

0)2 �
1

2
m

2

a(1 � �
2)�1a0

2

+
1

2
(@�b

0)2

+
1

2
j@��j2 �

1

2
m2j�j2 �

1

4
�2
 
e
2i=fb(�b0+ �a

0p
1��2

)

�2 + h:c:

!
: (8)

At this point it is convenient to drop the primes and work with the canon-

ically normalized �elds. For further convenience we represent the coherent

oscillation of the axion �eld by

a(t) = a0
p
1 � �2 sin(mat) : (9)

Then the e�ective Lagrangian for � in the presence of axion oscillations

becomes

Leff =
1

2
j@��j2 �

1

2
m2j�j2 �

1

4
�2
�
e2i�(t)�2 + h:c:

�
(10)

with �(t) � (�a0=fb)sin(mat).

It is important to note that even though a(t) is slowly varying (with

frequency ma), �(t) is rapidly varying: q � _�(t) = �a0mafb cos(mat). Since

a0ma = �
1=2
axion � �2QCD(fa=Mpl)1=2 (initially), q � ��QCD. For time scales

short compared to m�1a we may treat q as being constant and write L as

Leff =
1

2
j@��j2 �

1

2
m2j�j2 �

1

4
�2
h
e2iqt�2 + h:c

i
: (11)

7



The problem now is to �nd the evolution of the � �eld, a non-trivial task

because of the explicit time dependence. This is done in the Appendix, but

we give a simpli�ed version here. First, if �2 = 0 then for each momentum

~k there would be two degenerate eigenstates. One could choose to work

in a basis with linearly polarized or with circularly polarized states. With

degenerate states it is easy to show that the Hamiltonian is constant for

any linear combination. Next, consider the case �2 6= 0, but q = 0. Then

the two eigenstates have masses (m2 � �2)1=2, and correspond to the linearly

polarized states along the real and imaginary components of �; i.e. they

are A(x�) and B(x�), where �(x�) = A(x�) + iB(x�). Now, there is no

explicit time dependence so the Hamiltonian is still constant. This can be

seen explicitly by writing the Hamiltonian in terms of the linearly polarized

states and observing that there is no mixing of di�erent frequency states in

the Hamiltonian. Finally, for non{zero q, the Hamiltonian is time dependent

and the energy of the � �eld varies with time. For �2 small the evolution

is well approximated by the �2 = 0 eigenstates, but there is a qualitative

di�erence between the linearly polarized and circularly polarized solutions.

For the linearly polarized case, the potential in the A (or B) direction

has a curvature that varies sinusoidally in time: m2eff = m
2 � �2cos(2qt).

Since we will assume q � m, the variation in m2eff is adiabatic. In that

case one expects each quantum to have energy
q
m2 + ~k2 + �2cos(2qt) '

!0+
1

2

�2

!0
cos(2qt), where !0 �

q
m2 + ~k2. The number of quanta of oscillation

is an adiabatic invariant.

On the other hand, for the circularly polarized case, the state samples

the potential at all phases of �. Since m � q, this averaging takes place in

8



a time small compared to the evolution of the potential and there is little

time dependence in the energy. In the Appendix we show that, in fact, the

true eigenmodes are approximately circularly polarized. There is no time

dependence for pure states; however, in a thermal bath we do not expect pure

states unless there is some conserved quantum number. The corresponding

symmetry in our case is the b symmetry, which is broken, so that b charge

is not conserved. We therefore expect all polarizations to play a role, and

having identi�ed the linearly polarized states as having an energy dependence

driven by the axion �eld we look for dissipation of the axion energy through

those modes.

Time variation of the energy levels of the � �eld can lead to dissipation.

Suppose the rate at which a linearly polarized � particle of momentum ~k

scatters is given by 
(~k). Then the occupation number f~k, of that state will

obey the di�erential equation

_f~k = 
(
~k)

2
4f0

0
@E(~k;t)

T

1
A � f~k(t)

3
5 (12)

where E(~k;t) = !0+
1

2

�2

!0
cos(2qt) and f0 is the equilibriumoccupation number

(Bose{Einstein). Treating �E
T

= 1
2

�2

T!0
as small, this has solution

f~k(t) = f0

�
!0

T

�
+ f0

0

�
!0

T

�

p


2 + 4q2
�2

2T!0
cos(2qt + �) (13)

where f 0
0
� @f0=@x and tan� = �(2q)=
. The occupation numbers thus

`lag' by a phase � behind the equilibrium value as that oscillates in time. As


 ! 1, this lag goes to zero and f~k(t) ! f0
�
E(~k;t)

T

�
; the system remains

in equilibrium. As 
 ! 0, the interactions turn o� and there is no time

variation of f~k(t). Maximum dissipation occurs for 
 = 2q. The average

9



power dissipated over a period (2q)�1by the mode with momentum~k is given

by

P = �(2q)
Z

(2q)�1

0

dt _f~k(t)E(
~k;t)

= f 0
0

�
!0

T

�

(2q)2q

2 + (2q)2

�4

4!2
0
T
�

Z
(2q)�1

0

dt sin(2qt + �)cos(2qt) (14)

or

P = �f0
0

�
!0

T

�

(2q)2


2 + (2q)2
�4

8!2
0
T
: (15)

The total power dissipated per unit volume is then given by

P = �
1

3

Z
d3~k

(2�)3
f
0
0

�
!0

T

�"

(2q)2


2 + (2q)2

#
�4

8!2
0
T
: (16)

The factor 1=3 comes from averaging over all possible polarizations of the �

particles.

Equating the dissipation power with the change in axion energy density,

P = _�a, and using hq2i =
D
(��1=2a =fbcos(mat))

2

E
= 1

2
�2 1

f2
b

�a, the energy in

coherent axion oscillations evolves as

_�a '
1

4�2
C(m;T)

2�2�a
�
4


2f2b + 2�
2�a

; (17)

where C(m;T) =
R
dx x

2

c2+x2
f0
0
(
p
c2 + x2), with c � m=T and x � k=T . Let

us now make the approximation that whatever dissipation occurs takes place

in less than an expansion period - so that the functions 
 and C(m;T) may

be approximated as constant. Then, taking the integral
R
dt � 0:1Mpl=T2,

the �nal axion energy density is

�a;fe

2�
2
�a;f

f2
b

2 = �a;ie

2�
2
�a;i

f2
b

2

e

�10
�3

�
2
�
4
Mpl

f2
b

T2 (18)

10



This solution embodies a number of special cases, but we are mostly in-

terested in the cases where we can treat (2�2�a)=(f
2

b 
2) as small. Then,

the axion density decreases exponentially during one expansion time. Given

that the maximum initial value of �a is approximately �
4

QCD, as long as

(10�3�2�4Mpl)=(f
2

b 
T
2) > 20, the axion density will be reduced to accept-

able levels.

We take the temperature to be about �QCD. Dissipation is fastest if

fb is small. On the other hand, if fb < T , the U(1)b symmetry will be

unbroken and the axion will have no goldstone boson, b, to mix with, and this

dissipation mechanism will not operate. So the greatest dissipation should

exist for fb � T � �QCD, and

�
2

 
�

�QCD

!
4
 



�QCD

!�1
� 10�16 : (19)

On the other hand, the dissipation cuts o� for q � �2=m. So the energy in

the axion �eld cannot be reduced below

�f;min �
�4f2b
�2m2

(20)

by this mechanism. Demanding that this be less than � 10�8�4QCD in order

to comfortably solve the axion energy problem gives (�=�QCD)
4 � �210�8,

and from Eq. (19), �4(
=�QCD)
�1 � 10�8. These two constraints can easily

be satis�ed, for example by � � 10�3, 
 < 10�4�QCD, and �2 � 10�7�2QCD.

5 Discussion

We have established that it is possible to dissipate the primordial energy

density in axions through their interactions with other particles. A key fea-

11



ture of the process we envision is to mix the axion with some other goldstone

boson b, whose decay constant fb is much smaller than fa. The mixing may

be large, and this allow axions to couple strongly to `b matter'. By adjusting

the matter content of the b-sector we can arrange for the axion oscillations

to drive oscillations in the energy of b matter. This pushes the b matter out

of thermal equilibrium, and ultimately allows the axion energy density to

dissipate in the b-sector.

Although, we have constructed a model whereby the axions dissipate, the

addition of new particles may have other troublesome consequences, chief of

which is that b matter may itself come to dominate the universe. The easiest

way to avoid this is to adjust the masses and couplings of the b-sector so the

real part of the � �eld is the lightest b-particle other than the b goldstone

boson itself. Then as the universe cools, �rst all b matter will end up as �b's

which will then decay with a rate of order fb into b goldstone bosons.

We argued earlier that it is di�cult to dissipate axion energy through the

process of single axion absorption, yet, in the present scenario axion energy

is indeed absorbed. It is natural to ask if there is a \Feynman diagram"

explanation of our process. The answer is yes. Imagine a process whereby a

single axion is absorbed from the coherent state, as in Fig. (3a) and a second

process whereby two axions are absorbed, as in Fig. (3b). The ratio of the

rates for these two processes is

R = �2=�1 � na
�2m2a
f2b

; (21)

where na is the occupation number for the state. As stated earlier the e�ec-

tive single axion absorption rate is reduced by a factor of ma=T when back

12



reactions are included. However, this is also true of the two axion absorption

rate, and so the ratio in Eq. (21) is correct even in the presence of backre-

actions. Now, for the coherent state, na is approximated by na � f2a=H2, so

the ratio becomes R � �2M2pl=�2QCD, which is quite large. This leads to the

situtation that two axion absorption formally exceeds one axion absorption.

The three axion rate would be even larger... In this situation what one must

do is sum over diagrams with any number of axions attached to the particles

participating in the scattering process. This is equivalent to solving for the

propagator of these particles in the presence of the coherent oscillating axion

�eld, as we have done in this paper.

Along these lines, even in a normal axion model with no mixing each

additional axion brings a factor of Mpl=fa to the amplitude, so here also one

should not calculate individual absorption rates. Rather, one should calculate

the propagators for the other particles, including the time dependent axion

�eld, to arrive at the axion dissipation rate. We are presently considering the

question of what dissipation may result from the non-derivative couplings to

mesons, and plan to present results in a separate paper. Although derivative

couplings may seem important too, one must remember that such couplings

are less e�ective at damping the non-relativistic coherent axion oscillations

due to the factor of ma in the coupling.

Besides damping the coherent oscillations in the early universe, the kind

of mixing of axions with other goldstone bosons we have been discussing

could lead to other interesting e�ects. In particular, it is possible now to

contemplate that even invisible axions with fa � MGUT or MPl can have

sizable couplings to some kinds of matter, perhaps even to particles that

13



carry standard model gauge charges. This would mean that the \invisible

axion" may not be quite as invisible as was thought.

Also worth further investigation is whether similar mixing in familiar

majoron, familon or texture models could lead to interesting phenomena.

Appendix

The easiest way to �nd the eigenstates of the Lagrangian in Eq. (11) is to

make the �eld rede�nition  = eiqt�, after which the Lagrangian becomes

L0 =
1

2

�
j@� j2 � (m2 � q2) j j2 + iq( _  � �  _ � �

1

2
�2( 2 +  �2)

�
: (A1)

The equations of motion for L0 are

� �52 + (m2 � q2) � 2iq _ + �2 � = 0

� � �52 � + (m2 � q2) � + 2iq _ � + �2 = 0 : (A2)

The solutions to these equations are

 i = �i
�
ei(!+t�k�x) + �+e

�i(!+t�k�x)
�

or

= �i
�
e�i(!�t�k�x) + ��e

i(!
�
t�k�x)

�
; (A3)

where the index i denotes one of four solutions for a given wave vector ~k.

Due to the mixing of  and  �, to simultaneously solve both equations of

motion requires that �i be either pure real or pure imaginary, but otherwise

the normalization amplitude is arbitrary. The positive eigenfrequencies are

given by

!� �
�
!2
0
+ q2 �

h
4!2

0
q2 + �4

i
1=2
�
1=2

: (A4)

14



Negative frequency solutions for momentum ~k are equivalent to positive fre-

quency solutions with momentum �~k, and so are not distinct. For the case

when �i is real, the mixings are given by

�� =
��2

!2
0
� (!� � q)2

: (A5)

For the case where �i is imaginary, the sign of � is changed.

Counting !�, and the real and imaginary �i, there are eight eigenfunc-

tions combining the wavenumbers k and �k, which is the right number to

determine the �eld and its �rst time derivative at each point; so this is a

complete set of states for the  �eld.

For the � �eld, the states are given by �i = e
�iqt i. We can choose any

four of the eight eigenstates to characterize the momentumk (using the other

four for �k). It seems natural to chose a set where ! � q ! !0 as �2 ! 0,

since in that limit we must get back the free �eld theory for �. With this in

mind, the general solution that we will associate with momentum k is

� =
X
s=�

(�s+�s+ + i�s��s�) (A6)

where the �ss0 are real numbers and where

�ss0 � N
�1=2
ss0

h
(1 � �ss0) ei(!s�q)t�ik�x + �ss0e�i(!s+q)t+ik�x

i
: (A7)

Here

Nss0 � (1 � �ss0)2 + (�ss0)2

!s � s
�
!
2

0
+ q2 + s

h
4!2

0
q
2 + �4

i
1=2
�
1=2

�ss0 �
!2
0
+ s0�2 � (q � !s)2

4q!s
: (A8)

15



For �2 < mq, which is the limit taken in the text, Eqs. (A8) reduce to

!s ' sm + q (A9)

and

�ss0 '
ss0�2

4q(m + sq)
�

�4

16q2(m + sq)2
: (A10)

The energy density computed by just substituting Eq. (A6) into the Hamil-

tonian density is

� = �0 + Re
�
�2qe

2iqt
�
+ ::: (A11)

where the ellipses represent more rapidly varying terms (eg. terms varying

with frequency !+ � !� or 2!s) and

�0 =

2
4X
s;s0

(�ss0)
2

3
5!2

0
;

�2q = �
2 (�++ + i�+�)(��+ + i���) : (A12)

The approximately `linearly polarized' modes with arg(�) ' �=2 correspond

to (��+ + i���)=(�++ + i�+�) = e
i�, whereas the approximately `circularly

polarized' modes correspond to the cases (�++ + i�+�) = 0 and (��+ +

i���) = 0.

For linearly polarized modes

� /
�
!
2

0
+

1

2
�
2
cos(2qt)

�
(A13)

and the energy per quantum is

E = !0 +
1

2

�2

!0
cos(2qt) (A14)

as given in the text.

16



References

1. R.D. Peccei and H. Quinn, Phys. Rev. Lett. 38, 1440 (1977);

S. Weinberg, Phys. Rev. Lett. 40, 223 (1978);

F. Wilczek, Phys. Rev. Lett. 40, 271 (1978).

2. Y. Chikashige, R.N. Mohapatra and R. Peccei, Phys. Lett. 98B, 265

(1981);

G. Gelmini and M. Roncadelli, Phys. Lett. 99B, 411 (1981);

H. Georgi, S. Glashow, and S. Nussinov, Nucl. Phys. B193, 297 (1981).

3. F. Wilczek, Phys. Rev. Lett. 49, 1549 (1982);

D.B. Reiss, Phys. Lett. 115B, 217 (1982);

G. Gelmini, S. Nussinov and T. Yanagida, Nucl. Phys. B219, 31

(1983).

4. N. Turok, Phys. Rev. Lett. 63, 2625 (1989);

N. Turok and D. Spergel, Phys. Rev. Lett. 64, 2736 (1990).

5. J.E. Kim, Phys. Rev. Lett. 43, 103 (1979);

M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B166,

199 (1981);

M. Dine, W. Fischler and M. Srednicki, Phys. Lett. 104B, 199 (1981);

A.P. Zhitniskii, Sov. J. Nucl Phys. 31, 260 (1980).

6. For reviews, see, J.E. Kim, Phys. Rept. 149, 1 (1987);

M.S. Turner, Phys. Rept. 197, 67 (1990);

G.G. Ra�elt, Phys. Rept. 198, 1 (1990).

17



7. J. Preskill, M. Wise and F. Wilczek, Phys. Lett. 120B, 127 (1983);

L.F. Abbott and P. Sikivie, Phys. Lett. 120B, 133 (1983);

M. Dine and W. Fischler, Phys. Lett. 120B, 137 (1983).

8. B. Holdom, Phys. Lett. 166B, 196 (1986).

9. R.L. Davis, Phys. Rev. D32, 3172 (1985);

R.L. Davis and E.P.S. Shellard, Nucl. Phys. B324, 167 (1989).

10. D. Harari and P. Sikivie, Phys. Lett. 195B, 361 (1987).

11. P. Steinhardt and M.S. Turner, Phys. Lett. 129B, 51 (1983);

G. Lazarides, et al., Nucl. Phys. B346, 193 (1990).

12. S.-Y. Pi, Phys. Rev. D24, 1725 (1984);

A. Linde, Phys. Lett. 201B, 437 (1988).

13. J. Flynn and L. Randall, LBL preprint, LBL-25115 (1988).

14. H. Georgi, D. Kaplan, and L. Randall, Phys. Lett. 169B, 73 (1986).

18



Figure Captions

Figure 1. E�ective coupling to `b-matter' through goldstone boson mixing.

Figure 2. Mixing of the � and 
 scalars through a loop. The � particle

carries both a and b charges.

Figure 3. Similar processes involving the absorption of a) a single axion and

b) two axions. If the axions come from the coherent state describing the

classical �eld oscillation, then the rate for b) exceeds that for a).

19