A fresh look at the statistical bootstrap model

Volume 45B, number 5

PHYSICS LETTERS

20 August 1973

A FRESH LOOK AT THE STATISTICAL BOOTSTRAP MODEL M.I. GORENSTEIN, V.A. MIRANSKY, V.P. SHELEST and G.M. ZINOVJEV

Academy of Sciences of the Ukrainian SSR, Institute for Theoretical Physics, Kiev, USSR Received 6 June 1973 A new interpretation of the statistical bootstrap model is proposed, on the basis of which the statistical bootstrap equation is modified. A deep relationship between the statistical bootstrap model and the Pomeranchuk model for multiple hadron production is established and as a result the significance of the volume V in the statistical bootstrap model is elucidated. In this paper we focus on an interesting possibility to interpret the exact solution of the Hagedorn-Frautschi statistical bootstrap equation. This interpretation allows one to establish a deep relationship between this model and the old model of Pomeranchuk for multiple hadron production. Basing on this interpretation, we propose to reformulate the basic postulates of the statistical bootstrap model and modify to a certain extent the basic equation. All this enables one to make the statistical bootstrap model more perpicuous physically and more consistent logically. The following two postulates [1, 2] underlie the statistical bootstrap: 1) every hadron of mass m can be represented as a sphere of a radius of order 1/m o (m o is the n-meson mass), which contains lighter free hadrons; 2) the structure of every constituent hadron is also defined by postulate 1) - the so called bootstrap condition. When the strong bootstrap condition [2, 3] is satisfied, the mathematical content of these postulates is given by the following integral equation:

p(m) = d6(m - too) +

1n

Ivan-1

(1)

(m/)f P/ (m .~Po/)6 (~P/) ,,(

Here d determines the degeneracy of zr-mesons (for the sake of simplicity, we hence-forward set d = 1), V ~ ~ Zrmo ~, p(m) is the density of the hadron spectrum. The bootstrap equation (1) can be most easily investigated by going over to the so-called relativistic form

and thereby introducing an additional factor II7= 1 mi/Poi into the integrand in the nth term. Furthermore, the form of solving the equation in the limit m ~ oo remains unchanged. Using the method [3, 4], the equation obtained can be exactly solved:

p(m) = 6(m - rno) oo

+ ~ dnO(m-nmo)r(n)(m;mo . . . . . rno)

(2)

n=2 where

dn

~ n..~

2mn(v/aTr3) n-1 ~

X [8rr(ln4-1)]- 1/2n-3/Zexp [ - n l n ( l n 4 - 1 ) ] and

mo)=il-llfd41'iO(eoi)= n

r(n)(m;mo .....

X 6(p2-m2)6(m-~p~63(~ j= 1 9

P/)

(3)

\j=l

is the phase volume of n 7r-mesons connected with the hadron of mass m. It is possible to give the following interpretation to the relations (1) and (2). Let us divide both the sides of (1) and (2) into p(m). Then the nth term on the right-hand side of (1) determines the probability of decay of the hadron of mass m into n hadrons, and the relation (2) reflects the fact that all hadrons should eventually decay into stable 7r-mesons, and here the nth term determines the probability of discovering n 1r-mesons at the end of the decay:

475

Volume 45B, number 5

dn'c(n)(m;mo . . . .

PHYSICS LETTERS

(Vn I n-1 ~1

,m o) = \81r 3 ] d3p.

/

n

\

~--1fdmui .=

,

n

.

X6(r~-mo)f-~oti6~m-/~=lPoj)63(y~__lPi ) ~n =(dn/mn)l/(n-1)(n!)l/(n-1)

~

nV

(4) (5)

Since the average multiplicity of 7r-mesons in the statistical bootstrap model is proportional to the hadron mass m [5], in the dominant configurations Po eff will not depend on m, so that (4) can be interpreted as the density of the number of states of the microcanonicalensemble of n free jdentical particles with a volume Vneff ~ (mo/Po eff) gn~ n in the c.m.s. It is noteworthy that the proportionality of the volume tO the number of secondary n-mesons in fireball decay was postulated as far back as 1951 by Pomeranchuk [6, 7]. Redefining thus the volume in the Fermi model, he indicated that it should be defined as the volume of a system in which the thermodynamic equilibrium of free secondary particles is established [7]. When escaping, the secondary particles continue to interact until the distance between "all p,articles exceeds the interaction radius mo 1, and so Veff is of order n~ n m o 3. So, it seems that the bootstrap eq. (1) gives a specific realization to Pomeranchuk's idea. There is, however, a certain inconsistency here. From the point of view of the "decay" ideology the nth term in (1) determines the density of the number of states of n free hadrons into which the initial hadron decays. It is clear, however, that the volume, in which the thermodynamic equilibrium is established and in which the secondary hadrons can be treated as a microcanonical ensenthle of free particles, should also be proportional to their number n, Therefore, a consistent statistical equation which describes hadron decays should read as follows (we immediately use a relativistic form):

p(m)=6(m-mo) + n~=2~,8rr3]

(6)

and the postulates 1) and 2) in such an ideology must be replaced by 1') when escaping, n-hadrons, into which the ha&on can decay, form a microcanonical ensemble in the volume n V; 2') the density of states into which the hadrons of mass m can decay at m > m o , coincides with the spectral hadron density p(m), which determine~ the number of internal degrees of freedom, i.e. Pout(m) = p(m) - 8(m - rap). The physical meaning of both the postulates is quite perspicuous. The second of these postulates establishes the conservation of the number o f degrees of freedom of the hadron system in the process of decay, and is directly related to the conservation of entropy in dynamic processes. It replaces the bootstrap condition 2). The first one automatically removes the question (which inevitably arises regarding postulates 1) and 2)) of whether the representation of a hadron as a system of volume V made up of n > 1 free hadrons of the same volume is justified. Eq. (6) can be investigated by the well known methods [4, 8]. On performing a Laplace transformation, we obtain instead of (6) oo

Z=z o + ~ n=2

n

3 n

× i ~ ;dm2p(m~)~_::L~m_ff-~ P :~6 (~P~) n! i=l d t t"t 2¢'oi \ /=1 off \]=1 " 476

znn n- 1 n!

(7)

where

z=

8n 3

d4pp(m) exp(-pb)

where m 2 = p2,

b=(1/r, 0,0,0)

(8)

and z o coincides with (8), where p(m) is replaced by 8(m - mo). The basic difference of eq. (7) from that for the statistical sum in the usual Hagedorn-Frautschi formulation is the presence of a finite radius of convergence r = I/e in the series on the right-hand side of(7). Omitting detailed calculations, we present the basic conse-quences of eq. (7): The statistical sum Z has a root singularity in the complex plane of temperature T at Izl < 1/e. Therefore, the singularity has a thermodynamical significance of the limit temperature. For the asymptotic behaviour of p(m) we get the usual result

p(m) d3p.

20 August 1973

, const m -3 exp(m/ro)

(9)

m--~oo

The relative contribution of the nth term (i.e. the probability of decay into n-particles in the first act,

Volume 45B, number 5

PHYSICS LETTERS

P(n)) for eq. (6) will be different from that for eq. (1). For (1), P(n) decreases as 1/n! [2], whereas for (6) it decreases only as 1/cn. However, the monotone decrease of P(n) with increasing n takes place in this case, too. One can obtain an exact solution of eq. (6) such as (2), and the effective volume Veff will again be proportional to n. The relation of the volume V to mo and TO is such that the temperature TO ~ mo is attained at values of V several times less than Vo=~nmo3. This seems to reflect the fact that not only n-mesons, but also other heavier particles participate in strong interactions, and this naturally leads to a decrease in the effective size of hadrons. At first si),ht it seems surprising that when the coefficient of the nth term is multipled by n n- 1, the basic results of the statistical bootstrap model remain unaffected. One must bear in mind, however, that the terms with small n are dominant in eq. (1) and, moreover, the Boltzmann factor defines the multiplicity of final n-mesons as proportional to m, and their effective temperature as Telf -~. TO [5]. Since the n-mesons form a system which is thermodynamically equivalent to black radiation, the values n ~ m and Telf ~ TO uniquely define the volume Veff ~ nV, and, on the other hand, ~eff and n uniquely define Teff ~ To, so that the spectra of n-mesons in all theories with limit temperatures are similar. We emphasize that the main advantage of the "decay" ideology is the replacement of postulates 1) and 2) by the absolutely clear and physically justified postulates 1') and 2'). These postulates give the right to describe the systems of secondary hadrons after their large enough escape as a microcanonical ensemble of free particles, and establish in quite a natural way the conservation of the number of degrees of freedom of the hadron system in decay. The process of decay of the hadron of mass m is such a model can be figuratively described by dividing a box of volume V "- m, which is filled up by n ~ m free n-mesons, into all possible subboxes. It is quite clear that only the pecularities of decay dynamics can violate these post u l a t e s - m o r e specifically, the postulate 1') on a

20 August 1973

microcanonical ensemble formed by secondary hadrons. In particular, it might be that a "truncated equation" which includes only several first terms in (6) or eq. (1) itself would prove more correct, or, finally, the microcanaonical density p(m) would have to be replaced by the dynamic one pdyn(m) :/: p(m) [9]. It should be remembered, however, that eq. (6) is an initial, purely statistical equation, and the definition of these dynamic deviations from it will make it necessary to overstep the limits of purely statistical approach. We conclude by mentioning that the dynamics can, in principle, change the value of the volume for the nth term so that the singularity (not necessarily a rootsquare one) of an appropriate bootstrap equation will be defined by the divergence of the power series in z. First of all, this will lead to a change in the asymptotic form of the solution (9), and then the notion of a limit temperature will have an absolutely new physical significance. This possibility will be discussed at length in succeeding publications. The authors wish to thank Professor K. Nishijima and B.V. Struminsky for useful discussions.

References [1] R. Hagedorn, Nuovo Cimento Suppl. 3 (1965) 147. [2] S. Frautschi, Phys. Rev. D3 (1971) 2821; C.J. Hamer and S.C. Frautschi, Phys. Rev. D4 (1971) 2125. [3] J. Yellin, Nucl. Phys. 52B (1972) 583. [41 W. Nahm, Nucl. Phys. B45 (1972) 525. [5] S. Frautschi and C. Hamer, Nuovo Cimento 13A (1973) 645. [6] I.Ya. Pomeranchuk, Dokl. Acad. Nauk SSSR 78 (1951) 889. [7] E.L. Feinberg, Usp. Fiz. Nauk 104 (1971) 539; 1.N. Sissakyan, E.L. Eeinberg and D.S. Chernavsky, Tr. Fiz. Inst. Acad. Nauk SSSR 57 (1971) 170. [8] R. Hagedorn and I. Montvay, Preprint ref. T.H. 1610CERN. [9] M.I. Gorenstein, V.A. Miransky, V.P. Shelest, B.V. Struminsky and G.M. Zinovjev, Lett. al Nuovo Cim. 6 (1973) 325.

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