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Volume 157B, number 2,3 PHYSICS LETTERS 11 July 1985 QUARK NUGGETS OR BARYON NUGGETS? S.A. BONOMETTO a,b.c, P.A. MARCHETTI a,c and S. MATARRESE b a Dipartimento di Fisica "'G. Galilei", Via Marzolo, 8, 1 35100 Padua, Italy b SISSA, Strada Costiera 11, 1 34100 Miramare, Trieste, Italy c INFN, Sezione di Padooa, 1 35100 Padua, Italy Received 13 March 1985 We show that a process of concentration of baryon number (B) into restricted volumes, during the cosmological quark-hadron transition (Witten effect), should be present even if the transition temperature T~exceeds 100 MeV and, rather, is stronger for higher T~. If, however, quark nuggets do not form and B concentrations finally consist of large and localized baryon density peaks there still may be significant consequences for primeval nucleosynthesis and, in particular, present estimated limits on baryonic matter density, worked out from observed light element abundances, should be rediscussed. 1. Introduction. The suggestive idea of quark nug- gets, put forward by Witten [1], has led to a number of speculations [2] on the detectability of such pri- mordial objects. Among other proposed forms of dark matter (black holes, white dwarfs, jupiters, axions, massive neutrinos, supersymmetric inos, etc.) quark nuggest show the clear advantage that their average universal density is Pn = e-lpL (PL: luminous matter density), where, according to Witten computa- tions, 1 ~ e -1 <~ 102 naturally; in this case it is not necessary to invoke suitable fits between particle mas- ses and suppression factors or even more specific "ad hoe" coincidences to obtain such a ratio between dark and luminous matter. On the contrary, in dis- favour of quarks nuggets, there is the difficulty to fol- low the dynamics of their formation when l~/T>~ 1 (/,t is the chemical potential associated to baryon number B), and an admitted dependence ofe on the value of the quark-hadron transition temperature T c, which would greatly damp the mechanism of their formation if T¢ significantly exceeds 100 MeV. In this note we want to show that this latter prob- lem probably does not exist, being essentially due to the over-simplified treatment of the quark-gluon (Q) plasma and hadron (H) gas used by Witten to com- pute e: slightly more sophisticated treatments, taking into account precise physical elements, such as the presence of a proper volume of particles in the H- 216 phase, lead to a reinforcement of the Witten effect, consisting in the tendency shown by B (baryon num- ber) of flowing from the H-gas to the Q-plasma during a first-order phase transition, taking place reversibly in the presence of a large entropy per baryon. We also want to outline that the Witten effect might lead to relevant cosmological consequences even if the Q-H transition would lead to the forma- tion of. "baryon nuggets" (instead of quark nuggets) consisting of large B peaks around the points where quark bubbles fmally vanished. The size of these ef- fects would be mostly dependent on the distance d o between nuggets at the end of the Q-H transition. According to Witten d o ~ 1-10 cm. However, as we shall show, this inferior limit is to be lowered to 10-2-10 -1 cm (or even more). This is not without importance as d o ~ 1 cm shows to be a critical size; in fact, as we shall see, the ratio between the baryon diffusion length s and the average distance d of nugget centers at T~ 1 MeV, turns out to be (s/d)T=l MeV ~ ( 1 cm/d0) (1 mb/oN) 1/2 , where o N is the electromagnetic neutron cross section (whose value should slighly exceed 1 mb = 10 -27 cm2). As is known, at T~ 1 MeV, the proton-neutron ratio freezes down. Hence, afterwards, protons are 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Page 1: Quark nuggets or baryon nuggets?

Volume 157B, number 2,3 PHYSICS LETTERS 11 July 1985

Q U A R K N U G G E T S O R B A R Y O N N U G G E T S ?

S.A. B O N O M E T T O a,b.c, P.A. M A R C H E T T I a,c and S. M A T A R R E S E b

a Dipartimento di Fisica "'G. Galilei", Via Marzolo, 8, 1 35100 Padua, Italy b SISSA, Strada Costiera 11, 1 34100 Miramare, Trieste, Italy c INFN, Sezione di Padooa, 1 35100 Padua, Italy

Received 13 March 1985

We show that a process of concentration of baryon number (B) into restricted volumes, during the cosmological quark-hadron transition (Witten effect), should be present even if the transition temperature T~ exceeds 100 MeV and, rather, is stronger for higher T~. If, however, quark nuggets do not form and B concentrations finally consist of large and localized baryon density peaks there still may be significant consequences for primeval nucleosynthesis and, in particular, present estimated limits on baryonic matter density, worked out from observed light element abundances, should be rediscussed.

1. I n t r o d u c t i o n . The suggestive idea of quark nug- gets, put forward by Witten [1], has led to a number o f speculations [2] on the detectability of such pri- mordial objects. Among other proposed forms of dark matter (black holes, white dwarfs, jupiters, axions, massive neutrinos, supersymmetric inos, etc.) quark nuggest show the clear advantage that their average universal density is Pn = e - l p L (PL: luminous matter density), where, according to Witten computa- tions, 1 ~ e -1 <~ 102 naturally; in this case it is not necessary to invoke suitable fits between particle mas- ses and suppression factors or even more specific "ad hoe" coincidences to obtain such a ratio between dark and luminous matter. On the contrary, in dis- favour o f quarks nuggets, there is the difficulty to fol- low the dynamics o f their formation when l~/T>~ 1

(/,t is the chemical potential associated to baryon number B), and an admitted dependence o f e on the value o f the quark-hadron transition temperature T c, which would greatly damp the mechanism of their formation if T¢ significantly exceeds 100 MeV.

In this note we want to show that this latter prob- lem probably does not exist, being essentially due to the over-simplified treatment of the quark-gluon (Q) plasma and hadron (H) gas used by Witten to com- pute e: slightly more sophisticated treatments, taking into account precise physical elements, such as the presence of a proper volume of particles in the H-

216

phase, lead to a reinforcement o f the Witten effect, consisting in the tendency shown by B (baryon num- ber) o f flowing from the H-gas to the Q-plasma during a first-order phase transition, taking place reversibly in the presence o f a large entropy per baryon.

We also want to outline that the Witten effect might lead to relevant cosmological consequences even if the Q - H transition would lead to the forma- tion of. "baryon nuggets" (instead o f quark nuggets) consisting of large B peaks around the points where quark bubbles fmally vanished. The size o f these ef- fects would be mostly dependent on the distance d o between nuggets at the end of the Q - H transition. According to Witten d o ~ 1 - 1 0 cm. However, as we shall show, this inferior limit is to be lowered to

1 0 - 2 - 1 0 -1 cm (or even more). This is not without importance as d o ~ 1 cm shows to be a critical size; in fact, as we shall see, the ratio between the baryon diffusion length s and the average distance d of nugget centers at T ~ 1 MeV, turns out to be

( s / d ) T = l MeV ~ ( 1 cm/d0) (1 mb/oN) 1/2 ,

where o N is the electromagnetic neutron cross section (whose value should slighly exceed 1 mb = 10 -27 cm2).

As is known, at T ~ 1 MeV, the p ro ton-neu t ron ratio freezes down. Hence, afterwards, protons are

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Volume 157B, number 2,3 PHYSICS LETTERS 11 July 1985

more tightly trapped because of the large electron cross section and the requirement of charge neutrality. All that may lead to a definitely new outline of pri- mordial nucleosynthesis events, taking place when T ranged between 1 MeV and 0.1 MeV. In particular predicted abundances of light elements (D, 3He, 7 Be, etc.) can be affected, leading to a change of the pres- ent limits on the baryonic density parameter ~2 B, which are mostly related to the observed abundances of such light elements [3]. In view of that, the detail- ed dynamics of hadron bubble nucleation can offer a significant test of QCD forces through its cosmologic- al consequences.

2. An outline o f the Q - H transition. The cosmolo- gical Q - H transition has been treated in a number of recent papers. In most of them it is assumed that it is a first order one, in the presence of dynamical fer- mions lattice QCD computations still make no firm point about that * 1. However we shall make the same hypothesis here. Lattice QCD results can be used to follow the expansion of the universe since asymptotic freedom is lost; the expansion law during the transi- tion has been studied in ref. [5] (see also ref. [6] and the related comments in ref. [7]).

In refs. [8,5] it is shown that the expansion of the universe can take place at constant comoving entropy, even if the transition is a first order one, provided supercooling is negligible. The transition heat does then just account for the transfer of entropy from the quark-gluon into the lepton-radiation form, due to the sharp decrease of the number of spin degrees of freedom of strongly interacting matter. (For a tenta- tive treatment of the cosmological Q - H transition in the presence of supercooling see ref. [9]).

The transition then starts with the nucleation of H bubbles in a number of points, when T is just slightly below T e. When the bubbles begin to touch one an- other (after ~30% of the time taken by the transition) they may coalesce. Coalescence fixes the size of ha- dron bubbles if the average distance of nucleation points

~nuel.<d0min ~ 5 X lO-2al/3(m+r/Te)5/3 , (2.1)

where mTr is the rt meson mass, while o = aT 3 is the surface tension of the bubble (a is the main unknown in (2.1)).

,1 For a review of the situation see ref. [4].

In fact, surface tension forces F = oR acting on the mass M ~- 4pR 3, contained in a volume of size R, can lead to it taking a spherical form in a time t 2 [pR3/o] 1]2. The value of p (density of matter to be displaced) can be computed averaging between the densities in the Q-phase and in the H-phase (the dis- placement shall involve also radiation and leptons but for neutrinos). This leads to p ~ 12T 4 and then to t-~ Tc 1 7a-1/2(RTc) 3/2. This time ought to be a fraction y of the age t 1 of the universe at the beginning of the transition (the whole transition lasts ~ t l , (see e.g. ref. [10]). Requiring t < y t 1 -~ 4 X 10-2ym~ -1 X (mpl/m~r)(mTr/Tc) 2 and takingy ~ ~ yields R < 5 X 10 -2 al/3(m~/Te)5/3, which is the value o f d 0 min"

When more than 50% of the volume is filled with hadrons, they become the physically connected phase. Quark bubbles begin to shrink and the average dis- tance of their centers is therefore fixed either by do min (if)knuel ( d o rain) or by )~nuel itself (if knud > do min)" It will therefore turn out that d o >~ d o min" (In his paper Witten argued in this way and found do min ~ 1-10 cm, because of the neglect of a num- ber of numerical factors which all act in the same direction). The expansion of the universe then causes the gradual disappearance of quark bubbles in their original form. Around the points where they finally vanished B will tend to concentrate, forming either baryon or quark nuggets.

3. The Witten effect. The qualitative reason for the Witten effect is that, in the Q-phase,B is carried by quarks (whose mass mq ,~ To), while, in the H-phase, most hadrons are pions and B is carried by baryons (whose mass m b >> Tc); in this latter phase the baryon density is suppressed by a Boltzmann factor exp(--mb/Tc). However, according to Witten, if T e is not too low, such a factor can be at least partially balanced by the large number of baryon species whose mass does not exceed mp (proton mass) too much. The consequences of considering a number of baryon species is also shown in fig. 1 (see below).

However, the above argument is based on consider- ing that both phases are made of ideal gases of point- like particles. A major correction which will immedia- tely imply drastic effects at these temperatures con- fists in taking the proper volume of hadrons into ac- count. We shall also deal with the Q-phase using per- turbative results improved by means of renormaliza- tion group techniques.

217

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Volume 157B, number 2,3 PHYSICS LETTERS 11 July 1985

Taking the hadron volume into account implies that: (i) the room occupied by pions cannot be ex- ploited by baryons; (ii) increasing the number of ba- ryon species does not lead to a substantial increase o f the part of the total available volume taken by fer- mions, as higher-mass bosons shall also be considered. There may be some doubt whether considering a large numberg s o fhadron (spin) states can lead to a substantially greater total hadron density. In the model we shall use below, this increase cannot be very large as we require that there is a definite volume, to be at- tributed to each single hadron, inside which other ha- drons cannot penetrate. At temperatures next to To, such a model implies a numb©r density far less than proportional to gs, as it would be if hadrons were de- scribed as a perfect gas of pointlike particles. We do not claim that a proper volume is the only effect modi- fying the ideal gas description o f hadrons near to T c. Our claim is that this effect is likely to be dominant for a right evaluation of the Witten effect and that its over- coming would simply indicate a passage from the H- to the Q-phase.

According to Witten let us then define:

e = <B>hadxon/(B>quax k , (3.1)

where ( ) is an average made at T c in each one o f the two phases, at the same/a. A computation o f e requires a precise expression of the partition function for the two phases. For the hadron phase we shall take

C N +1 1

Z(T, la, V)=(2~r) -3 ~ ~-~-FI ~ f d 3 q i d m i N=0 N. i=1 Bi=-I

(3.2) X o(mi, Bi)exp [ - ( E i +Bi) /r ] ( V - 4NVo) N ,

(here E i = (q2 + m2)1/2; VO = (41r/3)m~-3 is the proper volume of hadrons which we shall take to be constant; C = V/4Vo; B i is the baryon number o f the ith particle whose mass is mi). This expression is a generalization to the case of B :/= 0 and of a specific spectrum of hadrons (represented by o(rn, B), see below) o f the finite volume model o f Karsh and Satz [11]. A similar generaliza- tion was also proposed by Kapusta [12] who, how- ever, considered a particle volume o:m i. Considering V 0 constant increases the importance o f high-mass hadrons and slightly softens the effect of the excluded volume approximation. This latter approximation can lead to two different phases (not to be confused with

218

Q- and H-phases, as both refer to the hadron state); in a "gas" phase it leads to the (V - 4NVo)N factor shown in (3.2); in a "solid" phase such a factor should be replaced by N! [(V/N) 1/3 - 182/3 V 1/3 ] 3N. It can be shown [13] that, at the relevant values of/a and T, the "gas" phase is thermodynamically favour- ed. The particle spectrum

K

p(m, B) = k~=lgk6 B, B(k )6(m - mk )

+ cO(m - mo)Ti~2m -7/2

X exp(m/T H - 77r2B2TH/30m), (3.3)

is composed of a discrete and a continuous compo- nent. We took all confirmed hadron species of the particle data list (1984) up to 1350 MeV (each one with its statistical weight gk), for the discrete compo- nent; the continuous component is assumed to start at m 0 = 1200 MeV, with T H (Hagedorn temperature) which will be assumed to coincide with T c. The con- tinuous Hagedorn component is taken from ref. [ 14]; the value o f c is not well determined experimentally (4 ~< c ~< 40). In (3.2) the sum over B i is from - 1 to +1 as higher IBI states are expected not to be in ther- modynamical equilibrium.

In the thermodynamical limit, at T e, the pressure o f the hadrons gas is found to be:

Ph(Te, #) = (3/16n)m 3 Teb ( X ) , (3.4)

where

b (X)exp [b (X)] = X

+1 _ 2 ~-I j ~ ( m [ T )

3rt 2 m 3 B=- I

X exp (-t.tB/Tc) (3.5)

(K 2 is the usual Bessel function). For the Q-phase we shall take the expression

Pq(T, u) --- [A~r 2 + ~-Tr2Nf +g2(T)(~ +~Nf ) ]T 4

+ Oa/T) 2 [~ Nf - g2(T)Nf/36~r2] + O((u/T)4), (3.6)

for the pressure, obtained for small la/T, withg2(T) = 4n [ 1 + (11/27r - Nf/37r) In (T/A)] -1 (A is the renor- malization point; here A will be chosen in order that Ph(T¢, O) =Pq(Tc, 0)).

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Volume 157B, number 2,3 PHYSICS LETTERS 11 July 1985

At T c the baryon-number densities for the two phases can be worked out according to

(B/V) = -~P(Tc, p)/ap ~-- pa2p(Tc, 0)/~p 2 , (3.7)

which can be inserted into (3.1); then owing to (3.4) and (3.6), w i thNf = 3, and taking into account that e h = eq, we find:

e = ((3/81r) (m,/Te)3b (x) [ 1 + b (x)] -1

× (Xf /X) /[~ + 4 b (x) (m~/Tc)a/(2n)3 ] )u=0, (3.8)

where

(Xf/X)u= 0 = ( fermi~ons gk(mk/Tc)2 K2(mk/Tc)

+ Or/2)l/2c(60]71r2)[1 -exp(-7rr2Tc/3Omo)] )

X ( ~ gk(mk/Tc)2K2(mk/Tc) \ Total

+ (rr/2)l/2c (Te/m 0

+(60/7~2)[1-exp(-7~2Tc/30mo] , (3.9)

In fig. 1 we report the value of e, computed nume- ricaily for different values of T c in the range 100-300 MeV. Different curves refer to'considering possible extreme values for c (4 and 40) and to including or

neglecting a specific contribution of the baryonic re- sonance at 1232 MeV, whose 32 spin states have a direct numerical effect. Typical values for e range about 10 -2 , which was the value obtained by Witten for T c --- 100 MeV, the numerical results also show that increasing T c leads to an amplified Witten effect (lower values of e) contrary to what one would expect on the basis of a model based on pointlike hadrons (for the sake of comparison a calculation of e based on ideal gases of pointlike hadrons and quarks is also plotted in fig. 1).

According to Witten, a fraction (1 + e) -1 of the baryon number will concentrate into small regions whose size is fixed by the range of validity of the re- quirement t~/T> 1. These regions will constitute the quark or baryon "nuggets". Therefore, according to the present computation, at the end of the Q - H tran- sition, ~ 99% of the baryon number should be concen- trated in these regions.

4. Conclusions. Baryon nuggets are clearly unstable

ld 1

100 150

I i I i i

i i 2oo 2so 3;o

T¢ (MeV)

Fig. 1. The dependence of e on T c is plotted for different values of the Hagedorn constant c (40 in curves a-e, 4 in curves b-d) and including (curves a-b) or excluding (curves e-d) the specific contribution of the baryon state A (1232 MeV). For the sake of comparison the values of e obtained considering both baryons and quarks as pointlike ideal gases (only baryon octet included) are also plotted (heavy line). The values obtained from our model, taking into account hadron proper volumes, are typically smaller than the ones obtained by Witten. For reasonable T c (130 < T c < 250) it turns out that 6.2 × 10 -3 < e < 7.6 X 10 -2. Also the trend of e dependence on T c is inverted. This is an effect of the increase of quark density with Tc, which is not com- pensated here by an equivalent increase of baryon density; kept nearly frozen by volume constraints (in a pointlike ba- ryon model this is over-compensated in the hadron phase by an increase of exponential Boltzmann factors.)

against neutron diffusion. Proton diffusion is clearly a slower process, for it implies electron diffusion as well. At T > 1 MeVwe can assume each baryon to be a proton (or a neutron) for 50% of the time. In a time t baryons can therefore diffuse over a distance s - (Dr) 1/2 where D = (90/41r 2) (grT3aN)-I is the baryon diffusion coefficient, taking account of the aver- age neutron electromagnetic cross section a N for col- lisions with electrons and photons (total statistical weight gr).

Let the distance of nuggets be d o at time t o when the Q - H transition ended; assuming a fully isentropic

219

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Volume 157B, number 2,3 PHYSICS LETTERS 11 July 1985

expansion afterwards, we obtain

s/d = 0.77(t0/2 X 10 -5 s)l/2(1 MeV/T) 3/2

X (1 mb/oN)l /2(1 cm/d0 ) . (4.1)

The neutron electromagnetic cross section is experi- mental ly determined by using data concerning nuclei. Results range between 2 and 30 mb, according to the nuclide considered; this wide range of variation seems not to be due to experimental reasons, but rather to detailed questions of nuclear physics. In a parton

17 2 model it would be easy to estimate o N = H oT(me/mq) (mq is the mass o f u and d quarks; o T is the Thomson cross section), yielding a value ~ 2 mb. Altogether (4.1) seems to indicate that baryon nuggets can be able to survive down to the beginning o f nucleosyn- thesis only i f their distance is fixed directly by nu- cleation (and not by coalescence) and is originally not much below 1 cm.

Apart from some ideas recently put forward by Van Hove [16], few at tempts were made to relate QCD parameters to the hadron bubble nucleation probabil i ty quantitatively. In our opinion this problem deserves more at tention from QCD research. Moreover it might be interesting to revisit primeval nucleosynthesis results, trying to verify which detail- ed consequences would ensue from assuming a highly non-homogeneous baryon-number distribution at T ~ 1 MeV.

Thanks are due to Professor L. Van Hove for a dis- cussion on possible observable remnants of the cos- mological Q - H transition, and other critical com- ments on the present work. Thanks are also due to

Professor M.J. Rees and to Dr. O. Pantano for discus- sions and comments. After completion o f the present work we received a paper by Applegate and Hogan [17], where the possible survival o f baryonic number concentration in baryon form, from the Q - H transi- t ion, down to nucleosynthesis, is also debated.

References

[1] E. Witten, Phys. Rev. D30 (1984) 272. [2] A. De Rfijula and S.L. Glashow, Nature 312 (1984) 734. [3] K. Olive, D.N. Sehramm, G. Steigman, M.S. Turner and

J. Jang, Astrophys. J. 246 (1981) 557. [4] E.g., H. Satz, 4th Intern. Conf. on Nucleus--nucleus

collisions (Helsinki, June 1984); Bielefeld preprint BI-TP 84/24 (1984).

[5 ] S.A. Bonometto and M. Sakeliariadou, Astrophys. J. 282 (1984) 370.

[6] J. Lodenquai and V. Dixit, Phys. Lett. 124B (1983) 317.

[7] S.A. Bonometto and S. Matarrese, Phys. Lett. 133B (1983) 77.

[8] S.A. Bonometto and O. Pantano, Astron. Astrophys. 130 (1984) 49.

[9] T. De Grand and K. Kajantie, Plays. Lett. 147B (1984) 273.

[10] S.A. Bonometto and L. Sokolowski, Phys. Lett. 107A (1985) 210.

[11] F. Karseh and H. Satz, Phys. Rev. D21 (1980) 1168. [12] JJ. Kapusta, Phys. Rev. D23 (1981) 2444. [13] S.A. Bonometto, Nuovo Cimento 74A (1983) 325. [14] J.I. Kapusta, Nucl. Phys. B196 (1982) 1. [15 ] OX. Kalaslmikov and V.V. Klimov, Phys. Lett. 88B

(1979) 328; O.K. Kalashnikov, Fortschr. Phys. 82 (1984) 529.

[16] L. Van Hove, Reports 1984 Varerma School, preprint CERN-TH 4055/84 (1984).

[17] J.H. Applegate and C.I. Hogan, preprint Caltech GRP- 032.

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