- Email: [email protected]
Relaxing axions
13 May 1999
Physics Letters B 454 Ž1999. 92–97
Relaxing axions S.M. Barr
1
Bartol Research Institute, UniÕersity of Delaware, Newark, DE 19716, USA Received 1 March 1999 Editor: M. Cveticˇ
Abstract A mechanism for lifting the cosmological upper bound on the axion decay constant, f a , is proposed. It entails the near masslessness of the radial mode whose vacuum expectation value is f a. Energy in the coherent oscillations of the axion field in the early universe gets fed into the motion of the radial mode, from which it is then redshifted away. It is found that the initial value of f a can be at scales between 2 = 10 14 GeV and 10 16 GeV. This evolves with time to values close to the Planck scale. It is suggested that the nearly massless radial mode might play the role of quintessence. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 98.80.Cq; 11.30.Er; 14.80.Mz
1. Introduction The axion w1x idea is an extremely elegant approach to solving the Strong CP Problem. However, it is not without difficulties. The main difficulty is that axions have neither been observed directly in the laboratory, nor indirectly through astrophysical effects. This implies that the axion decay constant, f a , must be greater than about 10 10 GeV. Moreover, there is the cosmological bound, based on the socalled ‘‘axion energy problem’’ w2x, that f a is less than about 10 12 GeV. This means that the axion decay constant Žor, equivalently, the scale at which the Peccei–Quinn symmetry is broken. cannot be at any of the mass scales at which other physics is known or suspected to be based. In particular, it
1
E-mail: [email protected]
cannot be at the weak scale, the Planck scale, or the grand unification scale. A number of attempts w3x have been made to weaken or remove the cosmological bound on f a . In this paper we suggest an idea for doing this. The idea is that if the radial mode associated with the axion Žthat is, the scalar field whose expectation value is f a . has a nearly flat potential then it would have increased with time as soon as the coherent axion oscillations commenced. As a result of this, energy would have drained out of the axion oscillations into the nearly massless radial mode, and from thence been redshifted away by the cosmic expansion. We find that the initial value of f a can be as large as 10 16 GeV. In the course of cosmic evolution this would have ‘‘relaxed’’ to a value we presume to be near the Planck scale. Axions in this scenario would therefore be extremely hard if not impossible to detect. However, the radial mode may play the role of ‘‘quintessence’’ w4x.
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 3 6 8 - 8
S.M. Barr r Physics Letters B 454 (1999) 92–97
The plan of the paper is as follows. In Section 2, a rough sketch of the axion energy problem in conventional axion models is given. In Section 3, a simplified discussion of the relaxing axion scenario is given to show how energy is drained out of the axion field. In Section 4, a more detailed quantitative analysis is given, still assuming, however, that the radial direction is exactly flat. In Section 5, the radial potential is discussed, in particular how flat it must be and how such flatness might arise.
2. The usual axion energy problem In ordinary axion models, the Peccei–Quinn symmetry is spontaneously broken by a complex scalar field, which we shall denote F . The expectation value of F is called f a , while the phase of F is the axion mode. Thus one may write F s re i u , where ² r : s f a , and the axion field a is given by a s f a u . Since the Peccei–Quinn symmetry is anomalous, QCD instanton effects give the axion a mass. The instanton-generated potential has the form VI s ym40 cosŽ Nu ., where m 0 is of order the QCD scale, and N is an integer, which for present purposes can be taken to be 1. When the temperature in the early universe was well above the QCD scale the instanton-generated potential was not fully turned on, whereas after the temperature fell below the QCD scale it had essentially its full zero-temperature strength. Let us call the time when the instanton potential reached nearly full strength, t 0 . Since the temperature was then of order m 0 , t 0 ; MPrm20 . Assuming a prior epoch of inflation, the axion field at t 0 was approximately spatially constant, but barring an extremely unlikely coincidence, it had no reason to be sitting at the minimum of its potential Žwhich we take to be at u s 0.. Rather, it had some arbitrary value u Ž t 0 . s u 0 . Consequently, the axion field underwent coherent oscillations about its minimum. The energy density in these oscillations at time t 0 would have been approximately 12 m40 u 02 . This energy scaled as Ry3 , where R is the scale factor of the universe, and therefore the energy density in the coherent axion oscillations remained Žafter t 0 . in a constant ratio to the baryon energy density. Since at t 0 the baryon energy density was of order 10y1 0 m p m30 , the present
93
ratio of axion to baryon energy is roughly Ž 12 u 02 m 40 .r Ž10y1 0 m p m30 . ; 10 9u 02 . If the axions are not to overclose the universe, u 02 Q 10y7 . This bound on u 0 can be satisfied if f a is sufficiently small. The point is that the instanton-generated potential of the axion field did not turn on instantaneously. If the potential turned on slowly then the number of quanta in the coherent axion oscillations remained constant since it is an adiabatic invariant. This implies that as the mass of the axion increased the amplitude of the oscillations decreased, and therefore by t 0 , when the potential reached full strength, the amplitude u 0 could have been very small. A simple estimate gives that u 02 ; f arMP . Thus, to satisfy the bound on raxion one requires that f a Q 10 12 GeV.
3. The relaxing axion mechanism Let us suppose that the field F whose phase is the axion has a flat potential in the radial direction. Then the Lagrangian density can be written 2
L s 12 EmF q m40 cos u , 2
2
( 12 Ž Em r . q 12 r 2 Ž Em u . y 12 m40 u 2 .
Ž 1.
As before, F s re i u , and the axion field is a s ru . Ignoring spatial derivatives of the fields, the equations of motion of r and u are 0 s r¨ q 3 Hr˙ y ru˙ 2 , 0 s u¨q 3 H q 2
ž
r˙ r
/˙
u q m40 ry2u .
Ž 2.
The last term in the equation for the radial mode r is just the centrifugal force, and it is this that drives r to larger values as u oscillates. The energy densities in the radial and axion modes are given simply by rr s 12 r˙ 2 and ru s 12 r 2u˙ 2 q 12 m 40 u 2 . The energies in a comoving volume in these modes are given by Er s R 3rr and Eu s R 3ru . Using the equation of motion of u to eliminate u¨, it is easy to show that r˙ E˙u s R 3 y r 2u˙ 2 q 3 H Ž y 12 r 2u˙ 2 q 12 m40 u 2 . . r
Ž 3.
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S.M. Barr r Physics Letters B 454 (1999) 92–97
If the oscillator parameters evolve adiabatically, then averaged over many oscillations the kinetic and potential energy in the oscillator should be equal. That is, ² 12 r 2u˙ 2 : s ² 12 m 40 u 2 : s 12 ² ru :. Therefore, averaged over many oscillations, r˙ E˙u s y Eu r
Ž 4.
In a similar way, using the equation of motion of r to eliminate r, ¨ and averaging over many oscillations, one finds that r˙ E˙r s Eu y 3 HEr . r
Ž 5.
The interpretation is clear. The increase in r caused by the centrifugal force drains energy out of the axion oscillations and into the radial mode, while at the same time energy is redshifted away from the radial mode. To put it another way, as r increases the effective mass of the axions, m20rr, decreases, while the number of axions remains constant. Thus Eu ; 1rr, as implied also by Eq. Ž4.. Suppose that r ; t q , and Eu ; tyq . The exponent q can be determined by writing the equation of motion of r in terms of Eu as follows: 0 s r¨ q 3 Hr˙ y Ry3 ² Eu : ry1 . The first two terms go as t Ž qy2., while the last goes as tyŽ2 qq3r2., assuming that R ; t 1r2 . Therefore q s 1r6. Writing Eu s Eu 0 Ž trt 0 . y 1 r 6 , Eq. Ž 5 . is solved by E r s 18 Eu 0 Ž trt 0 .y1 r6 q cty3r2 . Thus, for large t, the radial energy is one-eighth of the energy in the coherent axion oscillations, and the energy in a comoving volume in either mode falls off as ty1 r6 .
4. A more detailed analysis So far we have not taken into account the temperature dependence of the instanton-generated potential of the axion field. According to the dilute-instantongas calculation of Gross, Pisarski, and Yaffe w5x, the instanton potential for u at high temperature goes as m4 ; T 4 expŽy8p 2rg 2 ŽT .. ; T 4 expŽ 13 Ž11 N y 2 Nf .lnT . ; T Žy7q2 N f r3.. We shall assume henceforth that m4 s m40 ŽT0rT . 2 k s m40 Ž trt 0 . k , for T 4 m 0 Ži.e. t < t 0 ., and that m 4 ( m40 for t 4 t 0 . The dilute-instanton-gas calculation suggests that k ( 5r2, but we shall keep k as a parameter.
If we change variables to t ' trt 0 , and denote ErEt by a dot, then we can write the equations of motion 3
0 s r¨ q
2t
0 s u¨q
r˙ y ru˙ 2 ,
3
ž
2t
q2
r˙ r
/˙
u q Ž m40 t 02rr 2 . t ku .
Ž 6.
The effective mass of the field u , then, is mu Žt . s Ž m 20 t 0rr Žt ..t k r2 . Therefore it is reasonable to make the ansatz that u Žt . has the form
u Ž t . s A Ž t . e i BŽt . ,
Ž 7.
where A and B are real and B˙ Ž t . s mu Ž t . s Ž m20 t 0rr Ž t . . t k r2 .
Ž 8.
Substituting into the equation of motion for u and taking the real and imaginary parts of the equation, ¨ ˙. one gets Žusing the fact that rrr ˙ s kr2t y BrB 0 s A¨q
¨ ˙ A, ˙ Ž 32 q k . rt y 2 BrB
0 s yB¨ q
˙ ˙ B. Ž 32 q k . rt q 2 ArA
Ž 9.
The first of these equations is exactly solved by ˙ ˙ 2 .t Ž3r2qk . s constant, while the second is Ž ArB ˙ 2 .tyŽ3r2qk . s constant. solved exactly by Ž BrA ˙ Eliminating B from these equations and solving gives A s u 0 Ž t Ž5r2qk . q C . .
y1 r3
B˙ s cXu 02 Ž t Ž5r2qk . q C .
,
y2 r3
t Ž3r2qk . ,
Ž 10 .
X
where c , C, and u 0 are integration constants. After many oscillations the integration constant C can be neglected, and one has, using Eqs. Ž7. and Ž10., a solution for u Žt . of the following form:
u Ž t . s u 0tyŽ5 r6qk r3. exp Ž ib 0t Ž5r6qk r3. q ibX0 . .
Ž 11 . X
Ž b0 s
c u 02 5r6 q kr3
.. Eqs. Ž8. and Ž10. directly give 2 the solution for the radial mode: r Žt . s m 0 t 0 Ž5r6q k r3 . b 0
Žagain, neglecting the integration constant C .. Note that for the case of k s 0, which corresponds to a fixed value of m4 , the radial variable increases as t 1r6 , in agreement with the result obtained in Section 3. =t Ž1r6qk r6.
S.M. Barr r Physics Letters B 454 (1999) 92–97
It must be checked that this solution for r Žt . and the solution for u Žt . given in Eq. Ž11. satisfy the equation of motion of r. Assuming that u˙ is dominated by the rapid oscillations of u rather than by the slow variation of its amplitude, one has that u˙ ( ib 0 Ž5r6 q kr3.u 0ty1 expŽ ib 0t Ž5r6qk r3. q ibX0 .. Averaged over many oscillations, therefore, ² u˙ 2 : s 12 b 02 Ž5r6 q kr3. 2u 02ty2 . Substituting this and the expression for r Žt . into the equation of motion for r Žsee Eq. Ž6.., one sees that that equation is satisfied if b 0 s ' 1q k Ž
.Ž4q k .
5q2 k
r Žt . s
uy1 0 . Thus we have that
6 m20 t 0
(Ž 1 q k . Ž 4 q k .
u 0t Ž1r6qk r6. .
Ž 12 .
The solutions given in Eqs. Ž11. and Ž12. apply to the period t - 1, in which the instanton potential was still turning on. The same expressions with k set to zero apply to the period t ) 1, after the instanton potential turned on. The true solution will smoothly interpolate between these in the period t ; 1. ŽNote that the k / 0 and k s 0 solutions for u actually agree at t s 1, while the k / 0 and k s 0 solutions for r differ at t s 1 by a factor of Ž 1 q k . Ž 4 q k . r2 f 2.4.. One is now able to estimate the energy in the axion oscillations. The crucial parameter is the value of r at the time when the axion oscillations started. We will call that time t i . The oscillations started ˙ when the effective mass of the u field, mu s B, became equal to the expansion rate of the universe, H s Ž2 t .y1 . That is, when Ž2 t i .y1 f Ž m20 t 0rr Ž t i .. =Ž t irt 0 . k r2 . Clearly, the smaller r Ž t i . was, the earlier the axion oscillations began. Turning this around, r Ž t i . f 2 m20 t 0 Ž t irt 0 .Ž1qk r2. ; MP Ž t irt 0 .Ž1qk r2.. At t i it is to be expected that u was of order unity, since it had not had time to be affected by the instanton potential. But according to Eq. Ž11., u Ž t i . ; u 0 Ž t irt 0 .y Ž5r6q k r3.. Thus, u 0 is of order Ž t irt 0 .Ž5r6qk r3., or, in terms of r Ž t i .,
(
95
turned on at t 0 . We will call this suppression factor S before . S before f u 02 ; Ž r Ž t i . rMP .
2 3
5q 2 k Ž
2q k
.
.
Ž 14 .
If this were the only suppression of the axion energy, solving the axion energy problem would require that u 0 Q 10y7 r2 . With k s 5r2 this gives r Ž t i . Q 2 = 10 14 GeV. That is, the initial value of ‘‘ f a’’ can have been quite near the grand unification scale. By t 0 this would have increased, according to Eq. Ž12., to a value r Ž t 0 . ; m20 t 0 u 0 ; u 0 MP ; 3 = 10 15 GeV. One possibility, which we will call Case I, is that the radial mode stopped evolving at that point, because its potential has a minimum there. Another possibility, which we will call Case II, is that r continued to increase to some final value near the Planck scale. That would mean that the axion energy would have been further suppressed by the evolution of r in the period t ) t 0 . Since r Ž t 0 . ; u 0 MP , and it is assumed that r Ž t f . ; MP , there is an increase of r by a factor of uy1 in this period. It is easily shown that the 0 energy of the axion field in a comoving volume varies inversely with r in this period Žas was seen already in Section 3., so that the further suppression of the axion energy, which we shall call Safter , is given by Safter f u 0 ,
Ž 15 .
or Stotal s S before Safter f u 03 .
Ž 16 .
In Case II, therefore, it is only necessary that u 0 Q 10y7 r3, meaning that r Ž t i . Q 10 16 GeV. Case I and Case II are, in a sense, the extreme cases. One can consider intermediate cases as well. But one sees that in general the relaxing axion scenario would have f a starting out in the range 10 14 to 10 16 GeV, near the grand unification scale, and evolving to higher values.
5. The flatness of the radial potential
5q 2 k
u 0 ; Ž r Ž t i . rMP .
Ž
6q3 k
.
.
Ž 13 .
As was seen in Section 2, the factor u 02 tells how much the energy of the coherent axion oscillations was suppressed by the time the axion potential fully
The mechanism described in the preceding sections depends crucially on the assumption that the potential in the radial direction is nearly flat. For the mechanism to work, the centrifugal term in the equation of motion for r had to have dominated over
96
S.M. Barr r Physics Letters B 454 (1999) 92–97
the force coming from the potential energy of r. That is, ru˙ 2 ) V X Ž r . . One can write this as raxion ) rV X Ž r . , where raxion is the energy in the coherent axion oscillations. At some point this condition was no longer satisfied and the radial field’s evolution was controlled by V Ž r .. Unless the radial field was at that point overdamped Žnot so in the cases of interest. it would have started to oscillate about the minimum of V Ž r .. For reasonable potentials Žwhere one assumes that the cosmological constant problem has somehow been solved. one would expect that V Ž r . ; rV X Ž r . , and therefore the energy in the coherent oscillations of the radial mode were also of that order. Consequently, when the coherent radial oscillations began, the energy in them was typically of the same order as the energy in the coherent axion oscillations. Thus the coherent radial oscillations do not in themselves pose a cosmological problem. However, for the mechanism to work at all it is necessary that the centrifugal term in the equation of motion of r dominated over the potential term for a sufficiently long time to solve the axion energy problem. This puts a constraint on the flatness of V Ž r .. In Case I, it is assumed that the centrifugal term drove r until t 0 , when the temperature was of order m 0 . At that time raxion had to have been less than about 10 2r B ; 10y8m30 m p ; 10y1 0 GeV 4 . Thus, at t 0 it must also have been that rV X Ž r . Q 10y10 GeV 4 . In Case II, the centrifugal term is assumed to have dominated until r got to be of order MP . Since r Ž t 0 . ; u 0 MP , and r grew as t 1r6 for t ) t 0 , this happened at a time t f ; uy6 0 t 0 . Assuming a radiation dominated universe, T Ž t f . ; u 03 T Ž t 0 . ; Stotal m 0 ; 10y7m 0 . Therefore, at t f it must have been that rV X Ž r . Q 10y8 T Ž t f . 3 m p ; 10y31 GeV 4 . How can the potential be that flat? Certainly it is trivial to arrange that the radial direction be flat in the supersymmetric limit. The real problem is to insulate the radial mode from supersymmetry breaking. This is easiest to do if supersymmetry is broken at low energies, as it is in models with gauge-mediated supersymmetry breaking w6x. In such a model, the ordinary quarks must be split from their supersymmetry partners by an amount that is of order 10 2 to 10 3 GeV, which scale we will call m 0 . Since the axion sector must couple directly or indirectly to the
quark sector for the Peccei–Quinn symmetry to have a QCD anomaly, supersymmetry breaking will be fed into the radial mode of the axion sector through quark loops. Typically, then, the radial mode of the axion sector will acquire a potential of the form e m40 lnŽ rrm 0 .. e is model-dependent and is smaller the more indirectly and the more weakly the axion sector couples to the quark sector. In Case I, one has that e m40 Q 10y10 GeV 4 , implying Žif m 0 ; 1 TeV. that e Q 10y2 2 . In Case II, one has the more severe constraint that e Q 10y4 3. To see how such small values of e might be achieved, consider first a conventional axion model where the radial mode has a tree-level potential. The relevant terms in the superpotential would have the following general structure: Waxion s gSQQq WS ,
Ž 17 .
where Q and Q are lefthanded quark and anti-quark superfields, and S is a superfield containing the axion. WS is some set of terms, generally involving other fields, which has the effect of fixing ² S : to have some value M. The phase of S is, however, assumed not to be fixed except by QCD instanton effects. One can write S s Ž M q S˜.e i u , where here we mean by S the bosonic component, and the axion field is Mu . The radial mode S˜ would typically have some mass of order M. Consider, now, a somewhat different model, with the corresponding terms in the superpotential being the following Waxion s gSQQq WS q g X Ž SA y M X B . Y ,
Ž 18 .
where S, A, B, and Y are all gauge singlets, and as before WS has the effect of making ² S : s M but leaving the phase of S undetermined. Suppose that both M and M X are very large compared to the scale of supersymmetry breaking. It is apparent that, since S has a Peccei–Quinn charge, so must either A or B. Therefore, if ² A: and ² B : are larger than M, they rather than ² S : control the value of f a . Assume that Y has no other couplings in the superpotential. Then one of the terms in the scalar 2 potential is g X Ž SA y M X B . . This term fixes only the ratio of A and B and leaves them otherwise undetermined. There is, therefore, a direction that is
S.M. Barr r Physics Letters B 454 (1999) 92–97
flat in the supersymmetric limit along which A and B can become arbitrarily large. This flat direction would play the role of r in this model. Writing S s Ž M q S˜.e i u , A s re i a , and B s Ž MMX r q B˜ .e iŽ aq uq d ., the aforementioned term can easily ˜ B, ˜ and be found to give a mass to the fields S, d˜ ' rd . In particular, one finds that < g X Ž SA y M X B .< 2 s g X 2 M 2d˜ 2 q g X 2 Ž² r : S˜y M X B˜ . 2 q terms of cubic and higher order. In addition, there is the mass term for S˜ coming from WS . The phase a is an exact goldstone mode, while u , since it corresponds to an anomalous UŽ1. by virtue of its coupling to the quarks, is the axion mode. The scalar field r has Žbefore supersymmetry breaking. a flat potential. It is easily seen that the Lagrangian terms for r and u have Žafter suitable rescaling of fields. essentially the same form as those shown in Eq. Ž1.. Thus this model is an implementation of the relaxing axion mechanism. The question is how large a potential r gets in this model. If supersymmetry breaking is mediated from some hidden sector by Standard Model gauge interactions, then only Q and Q will directly feel it. The splittings in the supermultiplet S will only arise through a quark loop, and the splittings in the multiplets A and B will only arise through diagrams involving both an S loop and a quark loop. One would expect that e ; Ž g 2 r16p 2 .Ž g X 2 r 16p 2 .. This can easily be as small as required. For example, if M and M X are near the unification scale, the coupling g could even be as small as 10y1 4 . It seems to be possible to shield the radial mode from supersymmetry breaking even more thoroughly. As an extreme case one could imagine a whole series of gauge-singlet sectors separating the quark sector from the axion sector: Squark l S1 l ...l Sn l Saxion , where the S’s denote various sectors, and the arrows represent a coupling in the superpotential between two sectors. In this case, supersymmetry breaking in the radial mode would be an n-loop effect. On the other hand, because of the couplings between sectors, there are fields in each sector that have non-trivial Peccei–Quinn charges. Thus the axion field, though removed by several steps from the quarks, will nonetheless get a potential from QCD instanton effects that goes as ym40 cosŽ Nu ., where N is an integer.
97
From these examples, it seems that there is no reason in principle why the radial mode could not be sufficiently flat to allow the relaxing axion mechanism to work. It is an interesting question whether this flat radial direction can be identified with other flat directions that have been discussed in recent years. Could it be, to mention two obvious examples, the dilaton of superstring theory or the ‘‘quintessence’’ field? In any event, given the increasing role that very flat potentials have played in particle physics and cosmology Žinflatons, quintessence, dilatons, moduli, and the axion itself., it is suggestive that a flat radial direction can allow the breaking of the Peccei–Quinn symmetry to take place at the unification scale or even higher.
Acknowledgements This work was supported in part by the Department of Energy under contract No. DE-FG02-91ER40626.
References w1x S. Weinberg, Phys. Rev. Lett. 40 Ž1978. 223; F. Wilczek, Phys. Rev. Lett. 40 Ž1978. 271; for a review see, The Strong CP Problem, by R.D. Peccei, in: C. Jarlskog ŽEd.., CP Violation, World Scientific, 1989, pp. 503–551. w2x J. Preskill, M. Wise, F. Wilczek, Phys. Lett. B 120 Ž1983. 127; L. Abbott, P. Sikivie, Phys. Lett. B 120 Ž1983. 133; M. Dine, W. Fischler, Phys. Lett. B Ž1983. 137. w3x P. Steinhardt, M. Turner, Phys. Lett. B 129 Ž1983. 51; G. Lazarides, R.K. Schaefer, D. Seckel, Q. Shafi, Nucl. Phys. B 346 Ž1990. 193; S.-Y. Pi, Phys. Rev. D 24 Ž1984. 1725; A. Linde, Phys. Lett. B 201 Ž1988. 437; G. Dvali, IFUP-TH 21r95, unpublished; S.M. Barr, K.S. Babu, D. Seckel, Phys. Lett. B 336 Ž1994. 213. w4x M. Bronstein, Physikalische Zeitschrift Sowjet Union 3 Ž1933. ¨ 73; M. Ozer, M.O. Taha, Nucl. Phys. B 287 Ž1987. 797; B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 Ž1988. 3406; J.A. Freiman, C.T. Hill, R. Watkins, Phys. Rev. D 46 Ž1992. 1226; R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80 Ž1998. 1582. w5x D.J. Gross, R.D. Pisarski, L.G. Yaffe, Rev. Mod. Phys. 53 Ž1981. 43. w6x For a review see A. Nelson, Nucl. Phys. B ŽProc. Suppl.. 62A-C Ž1998. 261.
Physics Letters B 454 Ž1999. 92–97
Relaxing axions S.M. Barr
1
Bartol Research Institute, UniÕersity of Delaware, Newark, DE 19716, USA Received 1 March 1999 Editor: M. Cveticˇ
Abstract A mechanism for lifting the cosmological upper bound on the axion decay constant, f a , is proposed. It entails the near masslessness of the radial mode whose vacuum expectation value is f a. Energy in the coherent oscillations of the axion field in the early universe gets fed into the motion of the radial mode, from which it is then redshifted away. It is found that the initial value of f a can be at scales between 2 = 10 14 GeV and 10 16 GeV. This evolves with time to values close to the Planck scale. It is suggested that the nearly massless radial mode might play the role of quintessence. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 98.80.Cq; 11.30.Er; 14.80.Mz
1. Introduction The axion w1x idea is an extremely elegant approach to solving the Strong CP Problem. However, it is not without difficulties. The main difficulty is that axions have neither been observed directly in the laboratory, nor indirectly through astrophysical effects. This implies that the axion decay constant, f a , must be greater than about 10 10 GeV. Moreover, there is the cosmological bound, based on the socalled ‘‘axion energy problem’’ w2x, that f a is less than about 10 12 GeV. This means that the axion decay constant Žor, equivalently, the scale at which the Peccei–Quinn symmetry is broken. cannot be at any of the mass scales at which other physics is known or suspected to be based. In particular, it
1
E-mail: [email protected]
cannot be at the weak scale, the Planck scale, or the grand unification scale. A number of attempts w3x have been made to weaken or remove the cosmological bound on f a . In this paper we suggest an idea for doing this. The idea is that if the radial mode associated with the axion Žthat is, the scalar field whose expectation value is f a . has a nearly flat potential then it would have increased with time as soon as the coherent axion oscillations commenced. As a result of this, energy would have drained out of the axion oscillations into the nearly massless radial mode, and from thence been redshifted away by the cosmic expansion. We find that the initial value of f a can be as large as 10 16 GeV. In the course of cosmic evolution this would have ‘‘relaxed’’ to a value we presume to be near the Planck scale. Axions in this scenario would therefore be extremely hard if not impossible to detect. However, the radial mode may play the role of ‘‘quintessence’’ w4x.
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 3 6 8 - 8
S.M. Barr r Physics Letters B 454 (1999) 92–97
The plan of the paper is as follows. In Section 2, a rough sketch of the axion energy problem in conventional axion models is given. In Section 3, a simplified discussion of the relaxing axion scenario is given to show how energy is drained out of the axion field. In Section 4, a more detailed quantitative analysis is given, still assuming, however, that the radial direction is exactly flat. In Section 5, the radial potential is discussed, in particular how flat it must be and how such flatness might arise.
2. The usual axion energy problem In ordinary axion models, the Peccei–Quinn symmetry is spontaneously broken by a complex scalar field, which we shall denote F . The expectation value of F is called f a , while the phase of F is the axion mode. Thus one may write F s re i u , where ² r : s f a , and the axion field a is given by a s f a u . Since the Peccei–Quinn symmetry is anomalous, QCD instanton effects give the axion a mass. The instanton-generated potential has the form VI s ym40 cosŽ Nu ., where m 0 is of order the QCD scale, and N is an integer, which for present purposes can be taken to be 1. When the temperature in the early universe was well above the QCD scale the instanton-generated potential was not fully turned on, whereas after the temperature fell below the QCD scale it had essentially its full zero-temperature strength. Let us call the time when the instanton potential reached nearly full strength, t 0 . Since the temperature was then of order m 0 , t 0 ; MPrm20 . Assuming a prior epoch of inflation, the axion field at t 0 was approximately spatially constant, but barring an extremely unlikely coincidence, it had no reason to be sitting at the minimum of its potential Žwhich we take to be at u s 0.. Rather, it had some arbitrary value u Ž t 0 . s u 0 . Consequently, the axion field underwent coherent oscillations about its minimum. The energy density in these oscillations at time t 0 would have been approximately 12 m40 u 02 . This energy scaled as Ry3 , where R is the scale factor of the universe, and therefore the energy density in the coherent axion oscillations remained Žafter t 0 . in a constant ratio to the baryon energy density. Since at t 0 the baryon energy density was of order 10y1 0 m p m30 , the present
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ratio of axion to baryon energy is roughly Ž 12 u 02 m 40 .r Ž10y1 0 m p m30 . ; 10 9u 02 . If the axions are not to overclose the universe, u 02 Q 10y7 . This bound on u 0 can be satisfied if f a is sufficiently small. The point is that the instanton-generated potential of the axion field did not turn on instantaneously. If the potential turned on slowly then the number of quanta in the coherent axion oscillations remained constant since it is an adiabatic invariant. This implies that as the mass of the axion increased the amplitude of the oscillations decreased, and therefore by t 0 , when the potential reached full strength, the amplitude u 0 could have been very small. A simple estimate gives that u 02 ; f arMP . Thus, to satisfy the bound on raxion one requires that f a Q 10 12 GeV.
3. The relaxing axion mechanism Let us suppose that the field F whose phase is the axion has a flat potential in the radial direction. Then the Lagrangian density can be written 2
L s 12 EmF q m40 cos u , 2
2
( 12 Ž Em r . q 12 r 2 Ž Em u . y 12 m40 u 2 .
Ž 1.
As before, F s re i u , and the axion field is a s ru . Ignoring spatial derivatives of the fields, the equations of motion of r and u are 0 s r¨ q 3 Hr˙ y ru˙ 2 , 0 s u¨q 3 H q 2
ž
r˙ r
/˙
u q m40 ry2u .
Ž 2.
The last term in the equation for the radial mode r is just the centrifugal force, and it is this that drives r to larger values as u oscillates. The energy densities in the radial and axion modes are given simply by rr s 12 r˙ 2 and ru s 12 r 2u˙ 2 q 12 m 40 u 2 . The energies in a comoving volume in these modes are given by Er s R 3rr and Eu s R 3ru . Using the equation of motion of u to eliminate u¨, it is easy to show that r˙ E˙u s R 3 y r 2u˙ 2 q 3 H Ž y 12 r 2u˙ 2 q 12 m40 u 2 . . r
Ž 3.
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S.M. Barr r Physics Letters B 454 (1999) 92–97
If the oscillator parameters evolve adiabatically, then averaged over many oscillations the kinetic and potential energy in the oscillator should be equal. That is, ² 12 r 2u˙ 2 : s ² 12 m 40 u 2 : s 12 ² ru :. Therefore, averaged over many oscillations, r˙ E˙u s y Eu r
Ž 4.
In a similar way, using the equation of motion of r to eliminate r, ¨ and averaging over many oscillations, one finds that r˙ E˙r s Eu y 3 HEr . r
Ž 5.
The interpretation is clear. The increase in r caused by the centrifugal force drains energy out of the axion oscillations and into the radial mode, while at the same time energy is redshifted away from the radial mode. To put it another way, as r increases the effective mass of the axions, m20rr, decreases, while the number of axions remains constant. Thus Eu ; 1rr, as implied also by Eq. Ž4.. Suppose that r ; t q , and Eu ; tyq . The exponent q can be determined by writing the equation of motion of r in terms of Eu as follows: 0 s r¨ q 3 Hr˙ y Ry3 ² Eu : ry1 . The first two terms go as t Ž qy2., while the last goes as tyŽ2 qq3r2., assuming that R ; t 1r2 . Therefore q s 1r6. Writing Eu s Eu 0 Ž trt 0 . y 1 r 6 , Eq. Ž 5 . is solved by E r s 18 Eu 0 Ž trt 0 .y1 r6 q cty3r2 . Thus, for large t, the radial energy is one-eighth of the energy in the coherent axion oscillations, and the energy in a comoving volume in either mode falls off as ty1 r6 .
4. A more detailed analysis So far we have not taken into account the temperature dependence of the instanton-generated potential of the axion field. According to the dilute-instantongas calculation of Gross, Pisarski, and Yaffe w5x, the instanton potential for u at high temperature goes as m4 ; T 4 expŽy8p 2rg 2 ŽT .. ; T 4 expŽ 13 Ž11 N y 2 Nf .lnT . ; T Žy7q2 N f r3.. We shall assume henceforth that m4 s m40 ŽT0rT . 2 k s m40 Ž trt 0 . k , for T 4 m 0 Ži.e. t < t 0 ., and that m 4 ( m40 for t 4 t 0 . The dilute-instanton-gas calculation suggests that k ( 5r2, but we shall keep k as a parameter.
If we change variables to t ' trt 0 , and denote ErEt by a dot, then we can write the equations of motion 3
0 s r¨ q
2t
0 s u¨q
r˙ y ru˙ 2 ,
3
ž
2t
q2
r˙ r
/˙
u q Ž m40 t 02rr 2 . t ku .
Ž 6.
The effective mass of the field u , then, is mu Žt . s Ž m 20 t 0rr Žt ..t k r2 . Therefore it is reasonable to make the ansatz that u Žt . has the form
u Ž t . s A Ž t . e i BŽt . ,
Ž 7.
where A and B are real and B˙ Ž t . s mu Ž t . s Ž m20 t 0rr Ž t . . t k r2 .
Ž 8.
Substituting into the equation of motion for u and taking the real and imaginary parts of the equation, ¨ ˙. one gets Žusing the fact that rrr ˙ s kr2t y BrB 0 s A¨q
¨ ˙ A, ˙ Ž 32 q k . rt y 2 BrB
0 s yB¨ q
˙ ˙ B. Ž 32 q k . rt q 2 ArA
Ž 9.
The first of these equations is exactly solved by ˙ ˙ 2 .t Ž3r2qk . s constant, while the second is Ž ArB ˙ 2 .tyŽ3r2qk . s constant. solved exactly by Ž BrA ˙ Eliminating B from these equations and solving gives A s u 0 Ž t Ž5r2qk . q C . .
y1 r3
B˙ s cXu 02 Ž t Ž5r2qk . q C .
,
y2 r3
t Ž3r2qk . ,
Ž 10 .
X
where c , C, and u 0 are integration constants. After many oscillations the integration constant C can be neglected, and one has, using Eqs. Ž7. and Ž10., a solution for u Žt . of the following form:
u Ž t . s u 0tyŽ5 r6qk r3. exp Ž ib 0t Ž5r6qk r3. q ibX0 . .
Ž 11 . X
Ž b0 s
c u 02 5r6 q kr3
.. Eqs. Ž8. and Ž10. directly give 2 the solution for the radial mode: r Žt . s m 0 t 0 Ž5r6q k r3 . b 0
Žagain, neglecting the integration constant C .. Note that for the case of k s 0, which corresponds to a fixed value of m4 , the radial variable increases as t 1r6 , in agreement with the result obtained in Section 3. =t Ž1r6qk r6.
S.M. Barr r Physics Letters B 454 (1999) 92–97
It must be checked that this solution for r Žt . and the solution for u Žt . given in Eq. Ž11. satisfy the equation of motion of r. Assuming that u˙ is dominated by the rapid oscillations of u rather than by the slow variation of its amplitude, one has that u˙ ( ib 0 Ž5r6 q kr3.u 0ty1 expŽ ib 0t Ž5r6qk r3. q ibX0 .. Averaged over many oscillations, therefore, ² u˙ 2 : s 12 b 02 Ž5r6 q kr3. 2u 02ty2 . Substituting this and the expression for r Žt . into the equation of motion for r Žsee Eq. Ž6.., one sees that that equation is satisfied if b 0 s ' 1q k Ž
.Ž4q k .
5q2 k
r Žt . s
uy1 0 . Thus we have that
6 m20 t 0
(Ž 1 q k . Ž 4 q k .
u 0t Ž1r6qk r6. .
Ž 12 .
The solutions given in Eqs. Ž11. and Ž12. apply to the period t - 1, in which the instanton potential was still turning on. The same expressions with k set to zero apply to the period t ) 1, after the instanton potential turned on. The true solution will smoothly interpolate between these in the period t ; 1. ŽNote that the k / 0 and k s 0 solutions for u actually agree at t s 1, while the k / 0 and k s 0 solutions for r differ at t s 1 by a factor of Ž 1 q k . Ž 4 q k . r2 f 2.4.. One is now able to estimate the energy in the axion oscillations. The crucial parameter is the value of r at the time when the axion oscillations started. We will call that time t i . The oscillations started ˙ when the effective mass of the u field, mu s B, became equal to the expansion rate of the universe, H s Ž2 t .y1 . That is, when Ž2 t i .y1 f Ž m20 t 0rr Ž t i .. =Ž t irt 0 . k r2 . Clearly, the smaller r Ž t i . was, the earlier the axion oscillations began. Turning this around, r Ž t i . f 2 m20 t 0 Ž t irt 0 .Ž1qk r2. ; MP Ž t irt 0 .Ž1qk r2.. At t i it is to be expected that u was of order unity, since it had not had time to be affected by the instanton potential. But according to Eq. Ž11., u Ž t i . ; u 0 Ž t irt 0 .y Ž5r6q k r3.. Thus, u 0 is of order Ž t irt 0 .Ž5r6qk r3., or, in terms of r Ž t i .,
(
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turned on at t 0 . We will call this suppression factor S before . S before f u 02 ; Ž r Ž t i . rMP .
2 3
5q 2 k Ž
2q k
.
.
Ž 14 .
If this were the only suppression of the axion energy, solving the axion energy problem would require that u 0 Q 10y7 r2 . With k s 5r2 this gives r Ž t i . Q 2 = 10 14 GeV. That is, the initial value of ‘‘ f a’’ can have been quite near the grand unification scale. By t 0 this would have increased, according to Eq. Ž12., to a value r Ž t 0 . ; m20 t 0 u 0 ; u 0 MP ; 3 = 10 15 GeV. One possibility, which we will call Case I, is that the radial mode stopped evolving at that point, because its potential has a minimum there. Another possibility, which we will call Case II, is that r continued to increase to some final value near the Planck scale. That would mean that the axion energy would have been further suppressed by the evolution of r in the period t ) t 0 . Since r Ž t 0 . ; u 0 MP , and it is assumed that r Ž t f . ; MP , there is an increase of r by a factor of uy1 in this period. It is easily shown that the 0 energy of the axion field in a comoving volume varies inversely with r in this period Žas was seen already in Section 3., so that the further suppression of the axion energy, which we shall call Safter , is given by Safter f u 0 ,
Ž 15 .
or Stotal s S before Safter f u 03 .
Ž 16 .
In Case II, therefore, it is only necessary that u 0 Q 10y7 r3, meaning that r Ž t i . Q 10 16 GeV. Case I and Case II are, in a sense, the extreme cases. One can consider intermediate cases as well. But one sees that in general the relaxing axion scenario would have f a starting out in the range 10 14 to 10 16 GeV, near the grand unification scale, and evolving to higher values.
5. The flatness of the radial potential
5q 2 k
u 0 ; Ž r Ž t i . rMP .
Ž
6q3 k
.
.
Ž 13 .
As was seen in Section 2, the factor u 02 tells how much the energy of the coherent axion oscillations was suppressed by the time the axion potential fully
The mechanism described in the preceding sections depends crucially on the assumption that the potential in the radial direction is nearly flat. For the mechanism to work, the centrifugal term in the equation of motion for r had to have dominated over
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the force coming from the potential energy of r. That is, ru˙ 2 ) V X Ž r . . One can write this as raxion ) rV X Ž r . , where raxion is the energy in the coherent axion oscillations. At some point this condition was no longer satisfied and the radial field’s evolution was controlled by V Ž r .. Unless the radial field was at that point overdamped Žnot so in the cases of interest. it would have started to oscillate about the minimum of V Ž r .. For reasonable potentials Žwhere one assumes that the cosmological constant problem has somehow been solved. one would expect that V Ž r . ; rV X Ž r . , and therefore the energy in the coherent oscillations of the radial mode were also of that order. Consequently, when the coherent radial oscillations began, the energy in them was typically of the same order as the energy in the coherent axion oscillations. Thus the coherent radial oscillations do not in themselves pose a cosmological problem. However, for the mechanism to work at all it is necessary that the centrifugal term in the equation of motion of r dominated over the potential term for a sufficiently long time to solve the axion energy problem. This puts a constraint on the flatness of V Ž r .. In Case I, it is assumed that the centrifugal term drove r until t 0 , when the temperature was of order m 0 . At that time raxion had to have been less than about 10 2r B ; 10y8m30 m p ; 10y1 0 GeV 4 . Thus, at t 0 it must also have been that rV X Ž r . Q 10y10 GeV 4 . In Case II, the centrifugal term is assumed to have dominated until r got to be of order MP . Since r Ž t 0 . ; u 0 MP , and r grew as t 1r6 for t ) t 0 , this happened at a time t f ; uy6 0 t 0 . Assuming a radiation dominated universe, T Ž t f . ; u 03 T Ž t 0 . ; Stotal m 0 ; 10y7m 0 . Therefore, at t f it must have been that rV X Ž r . Q 10y8 T Ž t f . 3 m p ; 10y31 GeV 4 . How can the potential be that flat? Certainly it is trivial to arrange that the radial direction be flat in the supersymmetric limit. The real problem is to insulate the radial mode from supersymmetry breaking. This is easiest to do if supersymmetry is broken at low energies, as it is in models with gauge-mediated supersymmetry breaking w6x. In such a model, the ordinary quarks must be split from their supersymmetry partners by an amount that is of order 10 2 to 10 3 GeV, which scale we will call m 0 . Since the axion sector must couple directly or indirectly to the
quark sector for the Peccei–Quinn symmetry to have a QCD anomaly, supersymmetry breaking will be fed into the radial mode of the axion sector through quark loops. Typically, then, the radial mode of the axion sector will acquire a potential of the form e m40 lnŽ rrm 0 .. e is model-dependent and is smaller the more indirectly and the more weakly the axion sector couples to the quark sector. In Case I, one has that e m40 Q 10y10 GeV 4 , implying Žif m 0 ; 1 TeV. that e Q 10y2 2 . In Case II, one has the more severe constraint that e Q 10y4 3. To see how such small values of e might be achieved, consider first a conventional axion model where the radial mode has a tree-level potential. The relevant terms in the superpotential would have the following general structure: Waxion s gSQQq WS ,
Ž 17 .
where Q and Q are lefthanded quark and anti-quark superfields, and S is a superfield containing the axion. WS is some set of terms, generally involving other fields, which has the effect of fixing ² S : to have some value M. The phase of S is, however, assumed not to be fixed except by QCD instanton effects. One can write S s Ž M q S˜.e i u , where here we mean by S the bosonic component, and the axion field is Mu . The radial mode S˜ would typically have some mass of order M. Consider, now, a somewhat different model, with the corresponding terms in the superpotential being the following Waxion s gSQQq WS q g X Ž SA y M X B . Y ,
Ž 18 .
where S, A, B, and Y are all gauge singlets, and as before WS has the effect of making ² S : s M but leaving the phase of S undetermined. Suppose that both M and M X are very large compared to the scale of supersymmetry breaking. It is apparent that, since S has a Peccei–Quinn charge, so must either A or B. Therefore, if ² A: and ² B : are larger than M, they rather than ² S : control the value of f a . Assume that Y has no other couplings in the superpotential. Then one of the terms in the scalar 2 potential is g X Ž SA y M X B . . This term fixes only the ratio of A and B and leaves them otherwise undetermined. There is, therefore, a direction that is
S.M. Barr r Physics Letters B 454 (1999) 92–97
flat in the supersymmetric limit along which A and B can become arbitrarily large. This flat direction would play the role of r in this model. Writing S s Ž M q S˜.e i u , A s re i a , and B s Ž MMX r q B˜ .e iŽ aq uq d ., the aforementioned term can easily ˜ B, ˜ and be found to give a mass to the fields S, d˜ ' rd . In particular, one finds that < g X Ž SA y M X B .< 2 s g X 2 M 2d˜ 2 q g X 2 Ž² r : S˜y M X B˜ . 2 q terms of cubic and higher order. In addition, there is the mass term for S˜ coming from WS . The phase a is an exact goldstone mode, while u , since it corresponds to an anomalous UŽ1. by virtue of its coupling to the quarks, is the axion mode. The scalar field r has Žbefore supersymmetry breaking. a flat potential. It is easily seen that the Lagrangian terms for r and u have Žafter suitable rescaling of fields. essentially the same form as those shown in Eq. Ž1.. Thus this model is an implementation of the relaxing axion mechanism. The question is how large a potential r gets in this model. If supersymmetry breaking is mediated from some hidden sector by Standard Model gauge interactions, then only Q and Q will directly feel it. The splittings in the supermultiplet S will only arise through a quark loop, and the splittings in the multiplets A and B will only arise through diagrams involving both an S loop and a quark loop. One would expect that e ; Ž g 2 r16p 2 .Ž g X 2 r 16p 2 .. This can easily be as small as required. For example, if M and M X are near the unification scale, the coupling g could even be as small as 10y1 4 . It seems to be possible to shield the radial mode from supersymmetry breaking even more thoroughly. As an extreme case one could imagine a whole series of gauge-singlet sectors separating the quark sector from the axion sector: Squark l S1 l ...l Sn l Saxion , where the S’s denote various sectors, and the arrows represent a coupling in the superpotential between two sectors. In this case, supersymmetry breaking in the radial mode would be an n-loop effect. On the other hand, because of the couplings between sectors, there are fields in each sector that have non-trivial Peccei–Quinn charges. Thus the axion field, though removed by several steps from the quarks, will nonetheless get a potential from QCD instanton effects that goes as ym40 cosŽ Nu ., where N is an integer.
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From these examples, it seems that there is no reason in principle why the radial mode could not be sufficiently flat to allow the relaxing axion mechanism to work. It is an interesting question whether this flat radial direction can be identified with other flat directions that have been discussed in recent years. Could it be, to mention two obvious examples, the dilaton of superstring theory or the ‘‘quintessence’’ field? In any event, given the increasing role that very flat potentials have played in particle physics and cosmology Žinflatons, quintessence, dilatons, moduli, and the axion itself., it is suggestive that a flat radial direction can allow the breaking of the Peccei–Quinn symmetry to take place at the unification scale or even higher.
Acknowledgements This work was supported in part by the Department of Energy under contract No. DE-FG02-91ER40626.
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