Fortschritte der Physik / Progress of Physics: Band 24, Heft 10 1976 [Reprint 2021 ed.] 9783112520543, 9783112520536

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HEFT IO • 1976 • BAND 24

A K A D E M I E - V E R L A G EVP 10,- M 31728

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B E R L I N

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Fortschritte der Physik 24, 5 2 9 - 5 5 4 (1976)

Hadron Production by a Thermodynamical Quark Bootstrap R . WOLF

Fachbereich Physik der Universität Kaiserslautern,

Kaiserslautem,

Federal Republic of Germany

(BRD)

Abstract The Gell-Mann-Zweig rule indicates how hadrons can be produced by quarks. This can be formulated in a bootstrap equation. A second equation which is necessary in order to eliminate one variable is found by a simple assumption about the behaviour of the quarks inside of the hadrons. Two models — a pure mesonic and a pure baryonic quark bootstrap — are presented in this paper.1)

The principal assumption of the bootstrap equation is that the mass spectrum of particles which are produced by a strong decay, is only a function of the mass of the decaying particle (quantum number conservation is at the beginning not taken into account). This density function g(m) was first calculated by HAGEDOBN [ J ] , who expressed the philosophy of the bootstrap equation by a self-consistent definition of a fireball that includes in particular every excited hadron. Every decay in further fireballs and every step in the decay chain differs from the preceding one by smaller and smaller masses of the again decaying fireballs. Of course such a simple concept is not able to give detailed predictions on single processes. So the results of the bootstrap equation can be expected to agree with experiments only, if the individual character of a special process is not taken into account. This situation is similar to classical statistical mechanics. There the individual collision processes are also not considered for the explanation of macroscopical relations. The bootstrap equation has a fundamentally statistical character. In fact FRAUTSCHI [2] first formulated is in the phase space. The solutions of the bootstrap equation for the mass spectrum always have the form 5

(1)

Here a is fixed by more detailed assumptions T0 is the ultimate temperature due to Hagedorn. In this article we set up a coupled system of bootstrap equations for quarks and hadrons. In this simple model a meson consists of two and a baryon of three similar quarks [3, 4]. x ) This material is a portion of the Ph. D. thesis submitted to the department of physics of the Universität Kaiserslautern.

40

Zeitschrift „Fortschritte der Physik", Heft 10

530

R . WOLF

Quark bootstrap with constant binding energy producing mesons This model is characterised by the following assumptions: There are free quarks; There are free quark fireballs; Quarks produce mesons subject to the Gell-Mann-Zweig rule. The bootstrap equation for a fireball can be found by giving the answer to the following questions: Into which particles (that is: resonances, fireballs and also particles in the groundstate) can a fireball decay under conservation of a given set of quantum numbers? What is the phase space for such a process? Then pout T

in( m 2 ) = r o u t (m 2 ) -f- low mass input [5]

Here x out (m 2 ) is the density of mass states of the decaying fireball, r i n (m a ) the density of mass states of the fireballs, which are produced by the decay of a fireball. Now there are in principle two different kinds of particles in this model: Quark fireballs and meson fireballs. Therefore it is necessary to define two different density functions. tjii(? a ) is the density function, describing the spectrum of the meson fireballs. rM(q2) (q2 — m 2 ) d*q is the number of states of the meson fireball in the four momentum interval (q, q-\- dq). (q2) describes the spectrum of the quark fireballs. The first equation which is used, is a simplification of Frautschi's bootstrap equation [2, 5]: T(TO) =

m«);

> ^•

(25)

In appendix M the essential properties of s{[j, A/2,TOQ)are derived. With this approximation eq. (20) is as follows: L(P)

~

exp ^ - j j

T Q (TO 2 ) i(/3, to) d m 2 .

The integral represents the four dimensional Laplace transform of ZQ(jS) = / rQ(m2) r0,

m>J.

(43)

[Z m (T)].[Zq(T)] On (xM (m2))] [in (x - '~ 2

[s(T)] s(Tn) Fig. 4. The partition function of the meaon fireballs and the partition function of the quark fireballs as a function of s(T). The branches which are connected by arrows, correspond to each other. The dotted lines represent the prohibited parts of the partition Junctions.

Fig. 5. Mass spectra-for mesons and quarks

The principal statement for this equation is, that there is a lower bound for quark fireball production at TO = A ¡2. (One has to keep in mind however that this formula was derived for the limit TO oo) (see Fig. 5). Finaly it is interesting to consider the branching ratio : lim

t q (to2)

(44)

It is easy to see by comparing the two densities (see appendix C (C.18) ... (C.21)) that li zl.

(59)

It begins, as it should, at M = A /3 = 388 M„ ~ TOq. For the branching ratio r this model yields: r = lim

[m-*oo T Q ( m 2 )

a

¿ ¿ m . = 10168.

(60)

This tremendeous number gives a convincing explanation (within this model) for the lack of quarks. Below 54 GeV there are no quarks at all. Above this limit the relative probability to find a quark fireball in a certain mass interval, is, compared to the probability of finding a baryon in the same mass interval, exceedingly small, and there is no hope of ever finding a quark. The essential condition which was used in applying the approximation formula, was T (see eqs. (23) and (C.4)). As this supposition is satisfied very well, this model has not the same disadvantage as the meson case considered before.

Conclusion The main result of the two models proposed, is, that a pure mesonic bootstrap is not realistic. The prediction, that quarks should be as numerous as mesons is obviously wrong. This is a consequence of the light quark masses. There are however many field theoretical calculations, which expect light quarks, e.g. m u ~ m d = 11 MeV, MS = 274 MeV [1J] or MU = MD = 300 MeV, MS + 400 MeV [12]. The shape of the mass spectrum agrees however with nature. A rather different result is presented by the baryon bootstrap. The mass spectrum has the expected shape and the immense value of r explains the complete suppression of quarks. If the assumption that there are free quark fireballs is correct, than a quark bootstrap model consisting of mesons and baryons is expected to give good results. This is more difficult than in a pure hadronic bootstrap, because the meson- and the baryon partition functions have to be extracted from a system of implicit equations and then the hadron partition function which is defined as: ZB(T)

has to be considered. An example :

:=

ZK(T)

ZB{T)

(61)

542

E . WOLF

If the baryons are produced according to eq. (53) and two, three, . . . ra-mesonstates (n = 1, . . . oo) are possible after a repeated application of eq. (2), a conceivable set of bootstrap equations is: t(ß, toq) = Z^ß) { 1 -

cosh (ZB(ß)) exp (Zu(ß)) -

ZB(ß) = V(/?).e4>>;

ZM(ß) = V

1)}

e**

(62) (63)

where Ax = 2mq — m p r o t 0 n a n d z J 2 = 2to q — m„. The eqs. (63) are consistent with the ideas developed in the first two models. There are other improvements possible by taking into account more complex diagrams, e.g. that of fig. 7. I t expresses the production of a single baryon and a pair of quark fireballs by a single quark fireball.

Fig. 7. A quark fireball decays in a baryon fireball and in a pair of (anti)quark fireballs.

Further improvements can be made by taking into account quantum number conservation. This yields matrix partition functions [o moreover it is (see eq. (M.16)): Urns ¡T,

= 0.

Thus the equation = 2M{T) (1 - M(T)}2 holds. Eq. (A. 10) gives for T

(A. 10)

0:

M(0) = 0;

Jf(0) = l .

(A.ll)

This is in contradiction to the assumption. On the physical branch there is a one to one correspondence between M and T. On the other hand there is a one to one correspondence between T and s, when A and niq are fixed. Consequently M is an unambiguous function of s. Therefore the equation (A. 10)

545

Hadron Production

can be written as follows: (A.12)

s(M) = +]/2M (1 - i l i ) .

It has the derivative :

ds dM

i - m pM

v

(A.13)

'

From this equation it follows lim s'(M) = + 0 0 2tf-«-0

and

lim s'(l — M) = — ]/2. M-J-O

s has a maximum for M = M0 = 1 / 3 and it is: (A. 14)

* ( J f 0 ) = : i 0 = ! • * ! = .544.

s(T) is infinitely differentiable and therefore s(M) is infinitely differentiable in the open interval (0,1). The first three derivatives of s(M) are necessary in the expansion of appendix C. They are (A.13), d2s _ _ + 1 (A. 15) dM* ~~ 2M ~\/2M and d3s _ 3~ßM{l +-M) dM3 {2Mf From the equation dT dT ds M = ~ds dM and from eq. (M.21) it follows that dT dM

= 0,

(A.16)

because M0 = M(T0), and thus (see (A.ll)) T0 4= 0. It is: ti2T dM2) = (l+c/l/6-d/l/3)mf 5 , MF2 -

(50)

{ F | — uA — CM8 —