Axion oscillations and the quark-hadron phase transition

Volume 214, number 4 PHYSICS LETTERS B 1 December 1988

A X I O N O S C I L L A T I O N S A N D T H E Q U A R K - H A D R O N P H A S E T R A N S I T I O N

N. D O W R I C K Department of Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK

and

N.A. M c D O U G A L L l Department of Physics, Tokai University, 1117 Kita-Kaname, Hiratsuka, Kanagawa 259-12, Japan

Received 8 June 1988; revised manuscript received 2 August 1988

We consider the possibility that the quark-hadron phase transition occurs when the axion field passes through the minimum of its potential during its oscillation cycle. If this were to occur, the axion field would gain no energy from the associated increase in mass thus permitting the cosmological bound on the axion decay constant to be raised. However, we find that the probability of this happening is small.

The most a t t rac t ive solut ion of the strong CP prob- lem is the existence o f a light pseudoscalar boson, the axion [ 1-3 ]. this couples to Fff, which permi ts the CP viola t ion arising f rom the Q C D vacuum angle 0 and the quark mass matr ix to be removed dynami- cally. The couplings o f the axion to mat te r (and, therefore, its mass) are suppressed by inverse powers of the axion decay constantfa. A severe const ra int on f~ is ob ta ined from the axion cont r ibu t ion to the cos- mological energy density, the s tandard analysis [ 4 - 6 ] producingfa ~< 4 X 1012 GeV. F r o m the study o f en- ergy loss f rom stars due to axion emission a lower bound o f 109 GeV is ob ta ined [7 ]. this means that the two most obvious scales, the electroweak scale 250 GeV and the G U T scale 10 ~ s GeV, are apparent ly ruled out (250 GeV is also unacceptable for o ther ex- per imenta l reasons) . Fur thermore , axions exist au- tomat ica l ly in the spect rum of superstr ing theories [ 8], and, in such theories, the natura l scale for the decay constant is a round 1016 GeV [9] . Thus, the upper bound on the axion decay constant is a mat te r o f some phenomenologica l impor tance . In this pa- per, we investigate in detai l a possible mechanism,

On leave of absence from Department of Theoretical Physics, Oxford University, Oxford OX 1 3NP, UK.

suggested by Unruh and Wald [ 10 ], by which the up- per bound on the axion decay constant could be raised.

The nonder iva t ive couplings o f the axion can be descr ibed by an effective lagrangian

LP=½(Ouq))2-f~m2F(f~), (1)

where ¢ is the axion field and F is a dimensionless function with per iod 2n and Taylor expansion F ( x ) = ½x2+ .... At very early t imes in the history o f the universe, when the Hubble pa ramete r H was greater than the axion mass ma, the axion is decoupled and behaves like a free massless scalar field. Therefore f luctuat ions in the value o f 0/fa are expected to be o f order one. Thus, when the universe cools to a tem- pera ture such that ma --~ H, the axion field will not be at the m i n i m u m of its potent ia l but will have a poten- t ial energy densi ty o f order f a 2 ma 2 when it begins to oscillate. The known axion couplings are so weak that energy diss ipat ion due to part icle product ion can be neglected.

As the universe expands, the equat ion satisfied by the zero m o m e n t u m mode of the axion field is

~ '+ 3 H ( t ) q~ + ma 2 ( t ) 0 = 0 , (2)

0370-2693 /88 /$ 03.50 © Elsevier Science Publishers B.V. ( Nor th -Hol land Physics Publishing Divis ion )

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Volume 214, n u m b e r 4 P H Y S I C S L E T T E R S B 1 D e c e m b e r 1988

which has the approximate solution, if H and/;?/a/ma are small compared to m,,

O=A(t) cos[m,(t)t] , (3)

where A (t) satisfies

d(m"A2) _ 3H(maA2). (4) dt

On integrating (4) we obtain (m.A2) - 3

(re,A2), -k,-~i] ' (5)

where R is the Robertson-Walker scale factor, that is, the zero-momentum mode of the axion field has en- ergy density p~= ½m~A 2, and may be regarded as a coherent state of axion particles at rest with number density n~ = ½ m~A 2, the number of zero-momentum axions per comoving volume in the expanding uni- verse being conserved.

The amplitude A begins to decay according to (5) at a temperature T, which satisfies

m,(T~) ~ 3 H ( t ) = (24ztGp)l/2= 39Ti2, (6) m p l

where mp~ = 1.2 × 1019 GeV is the Planck mass. Pres- kill et al. [ 4 ] derived an expression for the low tem- perature axion energy density

jCa mafa T3 Ai p~~2 ml.~ W ~ f ' (7) where it is expected that &/f .~ 1. The axion mass (assuming it arises from QCD instantons) may be calculated at high temperatures using the dilute in- stanton gas approximation [ 11 ], with the result

1/2 A 4 nT 3 ma(T)~ 15A~ (mumdms'~ ( ~ _ _ ~ ) [ l n ( - £ ) ]

\ A 3 ] (8)

where A ~ 200 MeV is the QCD scale parameter. Solving (6) and (8), it is found that Ti is a slowly decreasing function Offa, and therefore the axion en- ergy density at low temperatures increases as f, in- creases. Below the QCD chiral symmetry breaking scale, T< To the axion mass becomes first order in current quark masses [2,12 ]

1 (m.md) '/2 m . - f. (mu+ma)f~m~. (9)

Using ( 6 ) - ( 9 ), Preskill et al. [ 4 ] concluded that the energy density in the axion field exceeded the ob- served energy density of the universe unless f~ ~< 4 X 10 I: GeV.

In obtaining the above constraint on the axion de- cay constant, adiabaticity has been assumed during the chiral phase transition at T~ 200 MeV. However, as pointed out by Unruh and Wald [ 10 ], this need not necessarily be so. They suggested several mecha- nisms, including thermal and phase separation damping, which could lead to less energy going into the axion field. Their most interesting suggestion was that, although the axion mass increases during the phase transition, the axion energy does not increase since the mass turn-on occurs when the axion is at the minimum of its potential during the oscillation cycle, and it is this mechanism which we shall investigate here. I f the axion mass before (after) the phase tran- sition is m l (m2), then, if the phase transition is adi- abatic, the energy density of the axion field increases by the ratio of the axion masses m2/mL (since m~42R 3 is constant and the axion energy density pa = ½ m a 2A 2f 2). If tO estimate ml we put T=A ~ 200 MeV in ( 8 ), we find m tfa ~ 2 X 10- 4 GeV:, and, using ( 9 ) to estimate me, we find m2fa~ 6X 10 - 3 GeV2; there- fore, the increase in the axion mass is roughly by a factor of 30. Hence, if the phase transition occurs when the axion field is at the minimum of its poten- tial, and there is therefore no subsequent increase in the energy in the axion field, the upper bound on the decay constant could be raised, also by a factor of 30.

The reason the phase transition may occur when the axion is at the minimum of its potential is that as the axion field oscillates so does the critical temper- ature Tc, which reaches a maximum when the axion field is at the minimum of its potential [ 10]. Thus it is possible that as the universe cools the critical tem- perature is first reached when the axion field is at the minimum of its potential. However, Unruh and Wald did not quantitatively investigate this possibility.

In order to calculate the effect of the axion field on the transition temperature, let us consider the ther- modynamics of the axion field coupling to the vac- uum. Imagine the axion field is constant in time and uniform over space. Then the axion potential energy in volume Vis ½m~¢ 2 V. The work done on this vol- ume when the axion mass changes is

1 2 "~ d W = - ~ ¢ d ( m ~ V ) . (10)

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By considering the equilibrium conditions'for a mix- ture of the two phases, each with a different value of m 2 V but with 0 uniform throughout the mixture, we can derive the Clausius-Clapeyron equation for the system:

1 d ( 0 2 ) AS 2 d T - A ( m ~ V )

o r

(11)

dT _ TO Ama2 , (12) dO l

where I is the latent heat per unit volume of the tran- sition, satisfying

TAS [= - - ( 1 3 )

V

Eq. (12) determines the equilibrium temperature for a mixture of the two phases as a function of the axion field 0. The derivation proceeds in exactly the same way as for a p - V system. The uniform axion field 0 replaces the uniform pressure p, and the different values of m ~ take the place of the different specific volumes of the two phases in the usual case.

Is the assumption of a static uniform axion field justified? As regards the time dependence, the period of oscillation of the axion field is many orders of magnitude greater than a typical QCD time. Simi- larly the spatial inhomogeneities in the value of m a are likely to be on much smaller length scales than m~ -~ and 0 will remain spatially uniform. We have checked this last point by calculation and have found that scattering of axions into higher-momentum modes by the phase mixture is negligible.

If we consider the transition from the hadronic phase to the quark-gluon phase, then l > 0 and Am~ < 0, and hence as 0 2 increases Tc decreases (see fig. 1 ). Thus, Tc is a maximum when the axion field is at the minimum of its potential during the oscilla- tion cycle, and Tc will fluctuate with twice the fre- quency of the axion field. Provided the variation of the axion mass with temperature in the hadronic phase is not great, a good estimate of -Area 2 will be given by the zero temperature expression (9) for the axion mass i.e. -Arn~ __ m 2. Furthermore, for sim- plicity, we ignore the temperature dependence of the latent heat and estimate l~A4~ (200 MeV) 4. Thus, from ( 12 ) we have

Tc

To

Fig. 1. A graph showing the behaviour of the critical temperature T¢ as a function of the axion field 0. Tc decreases as 0 2 increases and so the transition is suppressed by the axion field.

f dT m~ f T = - A-a-3 0 d 0 , (14)

t t

TO 0

which leads to

1 m~02~ Tc=Zoexp 2 A 4 , ] ' (15)

where To is the transition temperature in the zero ax- ion field.

When the axion began to oscillate at temperature Ti, 0 2 " ~ f 2 and thereafter m.O 2 decreased as R -3 ( ~ T 3) as the universe cooled. Therefore 0 2 ( T ) ~

f 2 ( T / Ti) 3ma (Ti) / ma (T) and the exponent in ( 15 ) is considerably less than one. Hence we can write

m2fea m(Ti) T c = T 0 1 2 A 4 Tii - m - ~ c°s2ma(T)t

= To - AZc cos2ma (T) t . ( 16 )

I f either the amplitude or frequency of the fluctua- tion in Tc is sufficiently large, it is likely that, as the temperature of the universe falls due to expansion, Tc is first reached when the axion is at the minimum of its potential. This is shown schematically in fig. 2. Thus, in order for T¢ first to be reached when the ax- ion field is near the minimum of its potential, we require

d__~_ ATc ATcma(T) << - - - ( 1 7 )

T 7~

Since Toot -~/2, IdT /dtl = T /2t and, using expres- sion (6) for ma ( T~ ), the above condition becomes

3 9 , 2 m~-f~a T° T~t>> l . (18) 7~ A 4 Ti mm

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Volume 214, number 4 PHYSICSLETTERSB 1 December 1988

Temperature of universe T

~ (T - R-1 - - t-1/2)

Transition temperature

Tc

Time t

Fig. 2. A graph showing the behaviour of the critical temperature Tc as a function of time as the universe cools. If I dT/dt l << ATe/ r, Tc will first be reached when Tc ~ To and the axion field at the minimum of its potential.

Since the present upper bound on the axion decay constant isf~ ~ 1012 GeV, we choose to evaluate ( 18 ) for this value [we note that the only dependence on fa in (18) occurs in Ti and is therefore weak]. I f fa ~ 10 t 2 GeV, Ti = 800 MeV; and we choose To = 200 MeV. In the early universe, the relation between the energy density and time is

3 p = 32~Gt2 , (19)

and the radiation energy density at T~ 200 MeV gives t~ 8 × 10 - 6 S ( 1 × 1016 MeV-J ). Estimating the left- hand side of ( 18 ), we obtain 3 X 10- 3.

Thus, we see that the probability of the universe first cooling to Tc when the axion field is at the min- imum of its potential is small (the pressure the axion field exerts on the QCD vacuum is not significant). Of course, an abrupt transition could only take place if there had been significant supercooling; however, the above result has relevance to both an abrupt and slow phase transition. I f the bubble nucleation rate were low the phase transition would take place over many axion oscillation cycles, and if I d T / d t l due to the expansion of the universe were small, the temper- ature during the transition would remain close to To and portions of the hadronic phase would only be created when ~~ 0. However, the above calculation shows this would not happen. I f significant super- cooling occurs, then, although supercooling is a max- imum when ~ ~ 0, we consider there is little increased probability of the transition occurring when ¢ ~ 0 since ATc/T~~ 10 -6 , and the rate of change of the temperature of the universe due to expansion is a

more significant effect than the fluctuation in Tc with the axion field.

I f significant supercooling occurred, the chiral (or confinement) phase transition could still be relevant to solving the axion energy density problem: entropy generation due to reheating would permit the upper bound on the decay constant to be raised (this en- tropy generation could be as much as 106 without en- countering problems due to the dilution of baryon number density [ 13 ] ); and, recently, Voloshin [ 14 ] has pointed out that the rapid expansion of the bub- bles could lead to a very effective damping of the ax- ion oscillations provided the nucleation sites were sufficiently well separated. However, the dynamics of the chiral phase transition are not understood and the probability of supercooling is not easy to estimate. A better understanding of the chiral phase transition is clearly required.

N.D. was supported by the Science and Engineer- ing Research Council (UK) and Merton College, Oxford. N.A.McD. acknowledges support from the SERC and the award of a Research Fellowship from the Matsumae International Foundation (Japan), and thanks the staffand students of Tokai University for their hospitality while this work was completed.

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