Baryon evaporation from cosmic quark-gluon plasma

Volume 225, number 1,2 PHYSICS LETTERS B 13 July 1989

BARYON EVAPORATION FROM COSMIC QUARK-GLUON PLASMA

K. SUMIYOSHI, K. KUSAKA, T. KAMIO and T. KAJINO Department of Physics, Tokyo Metropolitan University, Setagaya, Tokyo 158, Japan

Received 18 April 1989

Baryon evaporation from cosmic quark-gluon plasma is studied in a chromoelectric flux tube model which is an effective phenomenological model of QCD. The probability for baryons to penetrate through phase boundaries of the plasma is found to be very small compared with that of mesons. As a remnant of universal QCD phase transition, the baryon number density distri- bution is expected to be largely inhomogeneous.

The present universe is presumed to have under- gone several successive phase transitions associated with breaking symmetries at a very early stage of the hot big bang. A phase transition in quantum chro- modynamics (QCD) is of particular interest because it determines the late time evolution of baryons by nucleosynthesis. The QCD phase transition is expected to be first order from the lattice QCD calculations [1], which makes the two phases, quark- gluons and hadrons to coexist at a critical temperature around T= 100-300 MeV. As the phase transition proceeds the excess quarks are progressively concentrated inside the shrinking quark-gluon bubbles, leaving baryon number density fluctuations at the end of the phase transition [ 2 ]. The realized initial condition for nucleosynthesis is far from homo- geneity [ 3 ] against that assumed in the standard big bang model. Several recent works [4-6] on baryon inhomogeneous cosmologies have shown that the QCD phase transition drastically changes the nucleosynthesis, suggesting a baryonic dark matter for miss- ing mass and even a cosmological rapid-neutron process which could have created heavy elements in the big bang.

These attractive consequences of baryon inhomogeneous cosmologies, however, depend on assumed baryon fluctuation shapes which have not been studied very well. One of the most critical dynamics that determines the fluctuation shapes is baryon evaporation from cosmic quark-gluon plasma through the phase boundaries. Although meson evaporation has

been theoretically studied in a lot of works based on QCD, the baryon evaporation is left untouched mainly because searches of quark-gluon plasma formation in heavy-ion collisions have concentrated on measuring meson multiplicities, in particular of strange mesons. A theoretical description of the dynamics of baryon evaporation also is not so easy as that of mesons. If one goes back to the early universe, baryons take the key to universal evolution. The first purpose of this Letter therefore is to theoretically predict the baryonization rate from quark-gluon plasma in a chromoelectric flux tube model [7], which is a phenomenological model of QCD and en- joys a success for mesons. The second purpose is to discuss cosmological QCD phase transition based on the calculated results.

Let us start our discussion by considering the had- ronization mechanism on the surface of the quark- gluon plasma. The chromoelectric flux tube model assumes that as a quark or antiquark passes through the surface, a tube is built up behind it at the shortest distance independently of its transverse motion and that the strings of color flux, having a constant energy a per unit length, are confined to the narrow tube connecting opposite color charges. Meson production is associated with a q-q pair creation inside the tube [7,8]. A fission probability due to q-O pair creation per unit four volume is given by

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P (q -q ) = 4~-z @21n[1--P(PT)] 0

az ~__ l ~ e x p ( ~ ; 2n) - - 4~-3 --1 n-~ - - '

(1)

where a is set equal to 0.177 GeV 2 which is deduced from Regge trajectories of hadrons, and mq is the quark mass. A constant value of a leads to a linear potential between quarks. We consider here three flavors of quarks u, d and s. P(PT) is the probability for tunneling of a virtual q-el pair to a real q-el pair as a function of the transverse momentum Pv

p(pv) =exp zr(p-r + mq) _ , (2) O"

which is calculated in the WKB method by taking account of the exact energy conservation. As claimed by Glendenning and Matsui [9 ], the energy conservation law, generally neglected in QED for e +-e - pair creation in the strong external electric field [ 10], is very important in QCD because a created q-el pair must generate a color field of the same strength as that of the existing field with opposite sign in order to screen the color field completely and materialize a fission object to be a physical meson.

The energy conservation becomes even crucial for baryon production. Although Casher, Neuberger and Nussinov [7] proposed a dynamical process of se- quential q-Ct pair creation with different colors for baryon production, this process has a vanishing fission probability. In the chromoelectric flux tube model, there is only one unique way to create baryons satisfying exactly the energy conservation law; it is by way of diquark-antidiquark pair creation and fission in the tube. Since the energy conservation law is quite essential in the chromoelectric flux tube model, we adopt this mechanism for baryon production in the present study. Andersson, Gustafson and Sjostrand [ 11 ] have studied in the Lurid model the jet phenomenology observed in e+-e - and p -p col- lision experiments. They found that the observed baryon multiplicities are well explained by assuming the probabilities of diquark-antidiquark pair creation and fission as

P ( q q - ~ ) :p(q-cl) = 1:0.065, (3a)

p(u-f i ) : p ( d - a ) : p ( s - g ) = 1 : 1:0.37, (3b)

and

P(ql q l -q l q, ) :P(ql s-q1S ) :p (SS-Sg)

= 1:0.058:0.0007, (3C)

where p ( q q - ~ ) is defined in a similar way as eq. ( 1 ) by taking account of the statistics properly, and ql represents symbolically the light quarks u and d. We adopt the same values to constrain the probabilities of diquark-antidiquark pair creation in the color flux tube, leaving single quark masses mu = md and m, as free parameters in order to satisfy eq. (3b). When the light quarks are approximated by a current mass mu = rod----- 0 GeV, eq. ( 3b ) is fulfilled by adopt- ing m~=0.2 GeV. However, there remains a contro- versial problem on the quark masses. I f one takes an effective theory of constituent quarks which de- scribes quarkonium, for example, quarks are massive of the order one-third for the baryon mass, MJ3 ~ 0.3 GeV. We therefore adopt massive quarks mu =md = 0.3 GeV and ms=0.38 GeV satisfying eq. (3b). It is also suggested theoretically in several studies [ 12,13 ] of chiral transition that the constituent mass may change dynamically from 0 GeV up to 0.3 GeV de- pending on temperature and density. In this article, we assume that the quark masses are independent of temperature.

Having calculated probabilities of real q-el and qq- qq pair creation and fission in the flux tube model, we can calculate the probability that the string associated with leading quark or antiquark passing through the surface of quark-gluon plasma will finally fission with a physical hadron moving outward and a shorter string coming back to the plasma. The flux of evaporating hadrons is defined by [ 8 ]

. f ( Jn= E J q , H = E 7 d3koexp - - q q L q

x f dk"~ f dE HdzP(k°'kz°;k~'EH) dk~dEn , (4)

taking the thermal average of the initial quark momentum/to in an environment of equilibrium at the temperature T in the plasma. In this equation, nq is the number density of leading quarks, Zq is

Zq= f d3koexp(- ~ ) , (5)

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2 2 Eo = ~ and k~o are the energy and longitudi- nal momentum of the leading quark or antiquark when it passes through the phase boundaries, and E n and k~ n are those of the final hadron with mass MH. The integrand ofeq. (4) is the evaporation probability of hadrons as a function o f E n and k~ n for a given ko and k~0, defined by

d2P(ko, kz0; k z , E H dkz n dE H

1 k ~ , E )O(kp)O(k~o-kP) - k2 R(ko, kzo; n n

M 0(EMAx --ETH )0 (EMAx - - E H ) O(E H -ETH ) ,

(6)

R(ko, kzo, kz , E n) =exp - (kzo - k p )

1 ( k:+Ez l + (kpE~-k~oEzo)+m21nk~o+E~oj j j ,

(7)

2 2 H 2 2 where E~o = x/kzo + m q, Ez = ~ , and the 0's are the step functions arising from the kinemati- cal conditions including thresholds for hadron production:

ETH = x / ( k n ) Z + k 2 " 2 -kzo +MH , (8a)

Eo EMAX -- (Ezo - - m q ) . (8b)

Ezo

In these equations, kc is the characteristic momentum that the leading quark loses until fission of the tube,

k~= N/ 30.3 (9) 2~acp '

where p is the fission probability p (q-el) and p (qq- qq) defined by eqs. ( 1 ) and (3), and ce~ is the QCD coupling constant which is set equal to ac = 2 as in previous studies [ 7,8 ]. This value % = 2 corresponds to a flux tube of a diameter of 1 fm with the fixed string tension ~= 0.177 GeV 2. The flux of evaporating hadrons (4) depends on the flavor of the leading quark or antiquark through the quark mass mq, the dynamics of q-C1 and q q - ~ ) pair creations and flavor combinations through p, and the hadron mass MH.

Fig. 1 displays the ratios of the calculated flux of penetrating pions and nucleons to the flux of leading quarks JH/JQ, where JQ is defined by

. f ( JQ ----- 2 Z d3k° exp -- (10) q=u,d,s Eoo

In order to calculate JH/JQ for various hadrons, it is required to know the ratios of each quark number density nu:nd:ns= 1 : 1 : exp( - -Am/T) with Am= ms-- mu, which is the consequence of the equilibrium condition for weak decays. For pions and nucleons we consider two flavors u and d here. The pions to nucleons ratio in this quantity is smaller than expected in jet phenomenology [ 11 ] at higher temperatures T> 1 GeV. This is because only a single fission is taken into account for hadron evaporation without introducing a cascade process: Excited hadrons may eventually fission to many hadrons. This tendency is more remarkable for mesons than for baryons, as it should be. This defect, however, does not prevent our analysis of baryon evaporation from cosmic quark- gluon plasma because a coexisting temperature T= 100-300 MeV of the two phases is too low to al- low easily successive baryonic fission from a single tube. The thermodynamical weight for successive baryonic fission is exp ( - 2MB/T) ~ 10 - 5 at T= 200 MeV which is to be compared with that for a single fission e x p ( - M t s / T ) ~ 10 -2 at the same temperature. The advantage of such a situation is that an evaporating baryon conserves exactly the baryon

0.5

0 4

0.3 [-., <

0.2 01/ 0.0

1 2

T (GeV)

Fig. 1. The calculated ratios of flux of pions and nucleons penetrating through the phase boundaries of quark-gluon plasma to flux of leading quarks, using mu =md = 0 GeV and ms = 0.2 GeV.

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number no matter how it decays to baryon plus several mesons. We do not care for mesons in the present study. It is to be noted here that the chromoelectric flux tube model works very well even at temperatures as low as that in cosmic quark-gluon plasma since only the high energy tail of quark and antiquark dis- tribution functions is responsible for formation of the tubes containing enough energy > 1 GeV for baryon production.

The ratios JB/JQ of the flux of penetrating baryons to that of leading quarks are displayed in fig. 2, at temperatures T= 100-300 MeV of astrophysical interest, distinguished by strangeness quantum number. Four solid curves denoted by s= 0, - 1, - 2 and - 3 are those for p + n + A , A ° + E + + Y ° + Z -, .E°+E , and f2-, respectively. The mass differences among baryons were taken into account exactly. Non- strange baryons have the largest evaporation rate among them, but it is at most a few percent of that for mesons at these temperatures.

Evolution of the baryon number density nB in the quark-gluon plasma phase is approximately given by [4]

/P ?v~ ( l l a )

10 -~

s=O ~

j ~ I0° ~ J I0~ ~ J ~ 10"

10 '

10' i

1 O0 200 300

T (McV)

Fig. 2. The calculated ratios of flux of baryons penetrating through the phase boundaries of quark-gluon plasma to flux of leading quarks, using m u = m d = 0 GeV and ms=0 .2 GeV. Four different curves are the sum of the penetration rates for baryons having strangeness s = 0 , - 1 , - 2 and - 3 ; p + n + A ( s = 0 ) , A ° + E + + Z ° + Z ( s = - l ) , E ° + . E - ( s = - 2 ) , a n d f ~ - ( s = - 3 ) .

~ - 3 na+ 31r~l nB, ( l l b ) r

where J.q ~ B is the baryon number transfer rate through the phase boundaries which was taken to be a free parameter in the previous calculations [ 4 ] but is now calculated from Jn/Jo, Vis the horizon volume of the expanding universe, andfv is the volume fraction of the quark-gluon plasma phase. We here estimate an order of each term in eq. ( 1 la): Ik/V is the volume red-shift factor andj?v/fv is the rate of the shrinking volume fraction. Vandfv are known functions of time as solution of Einstein equation. Assuming that all of the shrinking quark-gluon bubbles have in common a mean radius r and a mean velocity i of the phase boundaries, we finally obtain ( 11 b). The calculated small baryon evaporation r a t e JB/JQ< 10 - 3 infers that the second term of eq. ( 11 b) dominates over the time evolution ofnB and the solution is an increasing function of time, namely the concentration of baryon number density becomes larger. Although the number of evaporating baryons is small, they are mostly nucleons which consist of u and d quarks because strange baryons have much smaller penetration rates with decreasing strangeness as shown in fig. 2. How- ever, this does not mean that the concentration of strange baryons is stronger than the others. During the phase transition a fraction of the strange quark number density in the quark-gluon plasma phase remains constant

ns exp ( - Am / T) nu+nd+ns 2+exp(-Am/T)

due to equilibrium of the weak process. At the final stage of the phase transition, these strange quarks may play an important role in the baryon number density fluctuations. If the peak baryon density exceeds nuclear matter density, these strange quarks may form strange quark matter [ 14 ]. If not, they also contrib- ute to form a fossil of high baryon density zones because strange baryons decay to nucleons in ~ 10- ' ° s. The shape of the fossil of baryon number density, of course, depends on the size of quark-gluon bubbles just after an epoch when the hadron phase dominates over the quark-gluon plasma phase and the velocity of phase boundaries [ 15 ]. An analysis ad- dressing these problems awaits theoretical studies of

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the dynamics of phase transi t ion and will be reported elsewhere [ 16 ].

We finally comment on the quark mass dependence of calculated results. Baryon evaporation rates become systematically smaller than those displayed in fig. 2 by almost an order when one uses consti tuent

masses m u = m d = 0 . 3 GeV and ms=0.38 GeV in- stead of current masses m u = m d = 0 . 0 GeV and ms=0.2 GeV. This is a consequence of smaller fission probabilities p (q-(:l) and p (qq -qq ) which have a strong quark mass dependence as shown in eq. ( 1 ). With the use of smaller baryon evaporation rates in eq. ( 11 ), we will obtain larger concentrat ion of baryon number density. Therefore, all of the numbers in figs. 1 and 2 which were calculated by using mu = md = 0 GeV and ms = 0.2 GeV give a lower l imit of the baryon number concentrat ion at the end of cosmic QCD phase transition.

To summarize, we have calculated baryon penetration probabilities through the phase boundaries of cosmic quark-gluon plasma in a chromoelectric flux tube model. The probabilities are very small compared with those of mesons at temperatures of astrophysical interest T = 100-300 MeV. This tendency is more remarkable for strange baryons. As a result, baryon number density fluctuations, as remnants of a first order QCD phase transition, are expected to be large.

We acknowledge useful discussions with T. Ohta, T. Kobayashi, H. Toki, K. Kubo, H. Minakata, M. Hosoda, S. Hirenzaki and G.J. Mathews. This work has been performed in US-Japan Joint Research

Project under auspices of Japan Society for the Pro- mot ion of Science.

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Baryon evaporation from cosmic quark-gluon plasma