Volume 81B, number 1 PI1YSICS LF, TTERS 29 January 1979

INSTANTONS IN QUARK PLASMA, MULTIBARYON HADRONS AND NEUTRON STARS

E.V. SHURYAK Institute o f Nuclear Physics, Norosibirsk, USSR

Received 8 November 1978

lnstantons cause the first order hadronic gas - quark plasma phase transition. As follows from the neutron star data, the multibaryon hadrons made of light quarks do not exist, hut they can be stabilized by the admixture of heavy quarks.

In our recent work [ 1 ] it was shown that in matter with a high enough baryon charge density n B (also at high temperature or strong field) the nonlinear fluc- tuations of the gauge field, tile instantons 12], are suppressed. The most important consequence of this phenomenon is the derivation of the bag-type struc- ture ofhadrons I3] directly from QCD.

The present work is devoted to a more detailed discussion of the equation of state of quark plasma in the intermediate density region n B ~ n 0 = 1 (fermi) -3. F'or snlaller n B tile nuclear matter calculations [4] are valid, while for larger ones the perturbative QCD calculations [5] may be used. The account of the in- stanton induced effects allows us to say more about the intermediate region where the hadronic gas- quark plasma phase transition takes place.

As we show below, this phase transition is of the first type with the finite density jump. This result is one more argument in favour of the bags with sharp boundaries. However, the discussion of the usual hadrons can only be made at the qualitative level [ I ], since they are not macroscopic and many corrections like surface energy must be included. Therefore we confine ourselves to the question whether there exist hadronic states with large baryon number, the multi- baryons. When this should be the case, the plasma and

vacuum pressure balance must be possible. In the naive bag model [3] this is possible for any number and combination of quarks, therefore many states appear some of them even in conflict with the known stabil- ity of normal nuclei. This defect of the bag model is automatically improved in our QCD approach, where the instanton indt, ced effects and the effective bag con-

stant are density dependent. So, depending on tile so far unknown suppression factor C (see [ 1 ] and below) the pressure balance may or may not be reached in the macroscopic limit. We have found that the existing data on neutron star masses seems to rule out the for- mer case and therefore the large bags with light quarks, if created, would blow out and decay into baryons. How- ever, as we show below, the small admixture of the heavy (say charmed) quarks essentially stabilizes tile system and theretk)re the "slightly charmed" multi- baryons seem to exist.

Let us begin with the qualitative discussion of the instanton induced effect starting from the nuclear matter side. The typical instanton radius in vacuum is known to be close to I fermi [6]. So, if the gap be- tween nucleons does not exceed this value there must be also some suppression of instantons inside the gap. As a result, the vacuum pressure [1] on the gap side of the nucleon is smaller and some effective attrac- tion appears. We can not say now what part of the

nuclear potential well Ueff ~ --40 MeV is due to this effect at nuclear density n B ~ 0.17 (fermi) -3, but we know that at n B ~ n O instantons are essentially suppressed and the effective potential well is much

deeper: Ueff ~ eo/n 0 = -450 MeV. llere e 0 is the vacuum energy density [7]:

e 0 = B = -0.0035 GeV 4 . (1)

So, the instantons make the equation of state to be softer for n B ~ n O .

The same conclusion can be reached starting from the quark plasma end. The energy density e and the pressure p can be represented as

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Volume 81 B, number l PIIYSICS LETTERS 29 January 1979

-5/3 e = e (P'T') + B - B ' C ' n l l

p = p(P.T.) _ B + ]B" C" n ~ 5/3 , (2)

where e (P'T'), p(P.T.) are those due to perturbative

effects. According to [51, one has for massless u, d, s quarks

(2°,°: ) p(P.T.) 3~r2/3 n 4/3 1 + + ... . . In o~ s +

= 4 - 3rr- n 2 . ""

I e(p.T.) ~ , ( 3 )

whe re a s = g2 (PF)/4~', PF = ( r r2nB) I/3. The second terms in (2) are subtracted vacuum

energy density and pressure (1), and the third ones re- present the cont r ibut ion of instantons in plasma ac- cording to estimates in [1 ] . Note that their signs are op- posite to the second ones, and that for n B - , oo they are very small, showing the ins tanton suppression. The parameter C is the ins tan ton suppression factor not yet known theoretically (see discussion in [1] ) and it is treated as a free parameter. Its value must be of the order of 1 (fermi) -5 .

The dependence P(nB) is shown at fig. 1 for various C values. One can see the min imum near n 0 being some trace of the bound states, the hadrons ~ t . For C~< 0.095 f - 5 the pressure curve crosses zero. If this is the case, the mul t ibaryons should exist. The calcula- tion of their energy shows that they are metastable since their decay into nucleons is energetically favour- able.

The formulae (2) are valid as far as the third terms are small compared to the second ones. otherwise the ins tanton interact ion nlust be taken into account . It remains unknown how to do this, but it is clear that some transit ion into the nucleon gas phase takes place here. This transit ion is probably of the second type (see [g] for the discussion of the high temperature case). The presumable behaviot, r of tire curves is shown by the dashed lines.

It is essential that the minima of the curves are reliable, at least for small C, and so the first order transi t ion is demonstrated. Therefore the unknown

We remind that for usual hadrons the pressure is balanced, which is possible because they arc made of different colour quarks and the kinetic energy is smaller than in (3).

10 ~

-~ o . i 102 / -~ :~ '~ .

-~- :--C. 3

n B

1 2 3

Fig. I. Pressure p (MeV (fermi) -31 versus barion charge den- sity n B ((fermi)-3). The solid lines correspond to (2), the num- ber near curves are the C value in (fermi) -s. The dashed area corresponds to nuclear matter calculations I4], the dashed curves represent their presumable behaviour. The dots at the lowest curve shows the limits of the unstable region. The curve labeled "n" corresponds to pion condensation, it is shown for comparison.

part of the curve lies in the unstable (and so unobserv- able) region t 2 .According to the well known Maxwell cottstruction, only the integral f p d V is needed in or- der to fix the transi t ion parameters. Therefore they (m particular, the density jump) are rather insensitive to the details of the curves.

The physical quant i ty very sensitive to the equat ion of state in the region in quest ion is the maximal mass of the cold stars. Solving numerical ly the T o l m a n - Oppenhe imer - Volkoff equat ion we have obtained the results shown at fig. 2. The curves correspond to dif- ferent C values and represent the connect ion between the star mass M and its central density n C. Note, that the region above max imum is unstable according to gravitational collapse. The lower part of fig. 2 summa- rizes the experimental restrictions [6] . The dashed line corresponds to the largest lower bound , so the curves at the plot which do not cross it contradict to

t2 Note that the usual confinement criterium, the infinite energy growth with quark separation, is unobservable in the same sense, since the quark pairs destroy the string already at finite separation.

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Volume 81B, number I PttYSICS LETfERS 29 January 1979

O. .0

' I :\i

I . . . . ~ , _ 1 2

M/M 1 I i

( 3 bO 900-40 (SMC X- I

Cen X-3 Her X - I

PSR 1913 + 16

. . . . . . . 4

Fig. 2. Dependence of the central density n c (fermi -3) on the star mass M (in units of Solar mass M,.:). The arrows in the lower part represent the experimental restrictions on the mas- ses of double pulsars [6]. The dashed line corresponds to the largest lower bound, the solid one and the arrow corresponds to the maximal mass according to all calculations. The dashed region is the most probable one for real value of pulsar masses 16].

data. This consideration makes it possible to conclude that the parameter C is larger than 0.15 (['ermi) -5, therefore no multibaryons made of light quarks exist.

Let us add, that according to these calculations the fraction of the star matter in the quark phase is rather small, but the energy yield and the density jump at the transition still may have some observable consequences.

The very interesting situation takes place if we add some number N n of heavy (say charmed) quarks into light quark plasma. The correlation energy is easily found to be Ecorr = ~g2 K where K-1 is the Debye length, so that the correction to pressure (2) is

6pip -x/2~75" ~- ^3/2 = -- % (Nn/NB) , (4)

where B is the total baryon charge. The numerical fac- tor is rather large, so the pressure balance may be reached even for rather small Nn/N B. It is therefore interesting to look for the charm 1, baryon number 2 (and more) resonances. In the limit of only heavy quarks such objects are similar to electron-hole liquid drops observed in semiconductors, and probably are even stable (only weak interaction decays), but it is

practically impossible to create such an object experi- mentally.

Let us also add, that soine phenomena discussed above can be seen in the heavy ion collisions at ener- gies ~> 1 GeV/nucleon. For example, the first order transition must show itself m the way discussed (in other framework)in [9].

Note added. After this work was completed we became aware of the work [10], the main idea of which: the instanton suppression inside hadrons as the origin of their bag-type structure, is the same as in our work [ 1 ]. Also the very important studies of the instanton interaction were made in [10] and the first order phase transition in external field was found, which resembles the transition discussed above. Also, the estimate of the vact, um pressure B is given in [10] which for their choice of the normalization (#= 73 MeV) gives B = 0.0027 GeV 4, which is very close to our value ( I ) and 6 times larger than the phenomenological value [3] (see discussion in [1]). Such an agreement between quite different and independent approaches is rather encouraging, especially because B is one of the most fundamental parameters of the strong interaction physics.

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