Unitary-Symmetrical Theory of Multiple Particle Production

Fortschritte der Physik 16, 719-736 (196s)

Unitary -Symmetrical Theory of Multiple Particle Production

V. S. BARASHENKOV and G. M. ZINOVJEV

Joint Institute for Nuclear Research,

Laboratory of Theoretical Physics, Dubna

1. Introduction

The experimental information available by the present time gives a sufficiently complete and consistent picture of the inelastic interactions of elementary particles a t high energies. This picture may be thought to remain unaffected up to energies of the order of a few hundreds of GeV [Z, 21. One of the most characteristic features of high energy inelastic interactions is the multiple production of new particles. On the average, three new particles are created in one inelastic TCN or N N interaction already a t T N 10 GeV1). With increasing energy T, the number of particles in the final state grows as T1j4 and for T N 1000 GeV it is 15-20. I n some reaction channels the number of particle produced may be several times larger. From this point of view it is safe to say that any further theory of inelastic interactions must be with necessity of statistical nature. Since the discover of the multiple particle production at high energies theoretists have suggested many various statistical models of elementary particle inelastic interactions. All the theories of multiple particle production, exact some original ones based on very simple and primative suggestions about the properties of strong interaction lagrangians, have, to a large extent, phenomenological charac- ter and contain a definite number of arbitrary parameters chosen from the comparison with experiment. It is essential to bear in mind that each model has its specific application depend- ing on which features of the phenomenon considered are chosen as the main ones. Therefore care should be taken when comparing models and especially when choosing the “best” model among the available ones. For example, in the late fifties i t was found out that the Fermi statistical model of multiple particle production does not account for a number of rather important details of inelastic interactions which are easily explained in the model of peripheral interactions. So, some physicists have put a slight upon the Fermi model while, in reality,

1) Here and in what follows T is the kinetic energy of an incident particle in the lab. system.

720 V. S. BARASHEXKOV and G. M. ZINOVJEV

this model and the model of peripheral interactions have different domains of applicability and essentially supplement one another. Owing to an ever-increasing number of experimental data on inelastic particle interactions it is advisable t’o have a simple even very rough model which would be capable of giving rather reliable estimates and solving many problems arising when preparing experiments and treating the results. The Fermi model is very convenient from this point of view since it allows to calculate simply very important’ characteristics of inelastic interactions such as distribution over the mult,i- plicity of produced particles, average momentum spectra of these particles aiid their average angular distributions in the lab. system. The Fermi model is urii- versa1 enough and describes the interactions of particles of various kinds: N N , x N and K N interactions, annihilat’ion processes and so on. At the same time from the theoretical point of view this model is rat,her general because the statistical approximat’ion underlying it can be formulated in terms of the S-matrix scattcr- ing theory. However, for a long time there were many uncertainties in this model. It was shown by many authors [4-71 that taking into account only one resonance N* (1236 MeV) and one new parameter (the “effective coupling constant” of K- mesons A N 0.1) we can bring in agreement the average characteristics of inelastic x N and N N interactions with the appropriate experimental data in the whole region of accelerator energies T > 1 GeV. In addition, to N* (1236 MeV) many other resonances, in particular, a number of long-lived pion resonances with smaller masses are presently available. It is quite necessary to take these resonance into account in order t o get agreement between the N f l annihilation calculated by the statistical model and the appropriate experimental data. However, in the case of x N and N N interactions the account of the resonances leads to an overestimated multiplicity. The detailed calculations have shown that it is impossible to eliminate this con- tradiction within the framework of the usual statistical model. Great calculation difficulties arise due to the fact that it is necessary to take into account a large number of possible resonance channels for a high-energy inelast’ic process. These difficulties become more serious if in such a statistical model one takes into account the possibility of bhe strange particle production. The above troubles and contradictions may be essentially reduced by formulating a statistical approach which takes into account the unitary symmetry of strong interactions. In subsequent sections the unitary-symmetrical statistical theory of multiple particle production is considered in detail ; approximations used in this model are formulated ; the calculation procedure is described and the results of calculation of different characteristics of inelastic interactions are sketched.

2. Unitary-symmetrical statistical theory of multiple-particle production

As is known [ S - l o ] , the complete set of operators necessary for an unambiguous determination of the states of a given irreducible representation of the group SU(3) consists of

G3, F2, I, I,, Y

the eigenvalues of the Casimir operators G3 and P 2 being “external” quantum numbers of the states indicating what is the dimensionality of the representation

Unitm-y-Symmetrical Theory of Multiple Particle Production 72 1

which the states belong to. For the same purpose one often applies the numbers p and q (the number of upper and lower indices for tensors respectively). The eigenvalues of the operators of the total isotopic spin I its third projection I , and hypercharge Y are “internal” quantum numbers distinguishing the states from one another inside the given representation. Therefore particles belonging to some supermultiplets of the group S U ( 3 ) may be specified by the ket vectors of the form

[ p , q ; I , I , Y ) = j N ; I ? I , , Y } (1)

since there is a correspondence between bhe numbers p , q and the N-represent>at,ion dimensionality. States corresponding to the direct product of the representations (pl, ql) 0 ( p z , q2) transform, generally speaking, according to a certain reducible representation of the group XU(3). These states can be reduced to a superposition of states transforming according to irreducible representations of the definite dimensionality. It is clear that states corresponding to the direct product of two representations may be given by the eigenvalues of ten linearly independent commuting operators

where the lower indices 1 and 2 distinguish between the operators of the complete set of two different representations. However, if the operator of the total unitary spin is introduced the choice of operators may be as follows

However this set is not complete enough since, as compared, to the set ( 2 ) it is short of one operator. This situation corresponds to that the representation charac- terised by identical numbers p and q may enter the direct product of two representations more than one time. Therefore, in addition, one includes into (3) the tenth operator I’ distinguishing from one another the representations of the same dimensionality. It should be noted that the operator r is not contained inside the S U ( 3 ) representations group. All this enables us to write the state vector of two part,icles in the form

where y is the eigenvalue of the operator I’. According to the SU(3) symmetry all the particles belonging to one and the same supermultiplet have identical mass, baryon number, spin J and space parity. Therefore we may formulate a statistical approximation to the problem of multiple particle production from the point of view of the S-matrix scatterinq theory. One of the main assumptions in analysirig the statistical model by means of the S-matrix scattering theory is the assumption on the statistical independence of secondary particles. As is known, the idea of this is that the expression for the square of the S matrix has no interference terms. Moreover, each matrix element, of the S niat’rix leading to the formation of n paltides in the final state is a product of a few factors among which there arc such which take into account various laws of conservation. In the scatt’ering problem this reduces to the invariancr condition for amplitudes with respect to different transformation groups : the

722 V. S. BARASHENKOV and G. M. ZINOVJEV

Lorentz group, group of all possible gauge transformations and so on. For example, a consistent consideration of the isotopic symmetry of elementary particles and the above assumption about the statistical independence leads to that the probability of inelastic interaction with n-particle production is the sum of the prob - abilities of a number of channels which contribution is determined not only by the phase volume but also by the “isotopic weight” Pn (I). Taking into account the unitary symmetry of particles and keeping the main assumption of the Fermi theory about the statistical equilibrium in the Lorentz contracted domain of strong interaction one can obtain the following expression for the probability of transition from the initial state corresponding to the representation ( p ) q) of the group S U ( 3 ) to the final state containing n particle

(5 )

where E, is the total energy in the c.m.s. of colliding particles. The space factor V , (E , ) is of the same form as in the former model [3, 4) 61

[11-131 Sn Gn Wn(Eo; P, 4 ) = “n(Eo) - un(p> Q; 1210, n8> 121) n n ( E o )

here m, is the mass of a x meson, ,ul and pz are the masses of colliding particles, The factor ‘%,(E,) is the level density in the final state and is given by the former expwssion

but in this case “particles” are the supermultiplets of the S U (3) group, Mi are the masses of these secondary “particles” and 1, are their momenta. As far as in calculating the energy weight 9Xn(E,) the “particle” masses Mi are the average masses of supermultiplets, the model suggested is best applicable in that region xvhere total energy spent for the new particle production exceeds essentially the experimental mass differences in supermultiplets. It is obvious that practically this condition is quite well fulfilled even a t energies higher than a few GeV. The spin factor 8, has the same form as before, since all the particles entering the supermultiplet have the idmtical spin. Therefore

n

The factor of identity G, keeping its form

G, = n,! n,!. . . . (9)

has changed its content. Supermultiplets possessing identical spins, space parity and baryon number are now assumed to be identical. In calculations of refs. [Il, 131 made on the basis of this model one has taken into account the octet of baryons with positive parity, the decuplet of baryon resonances with positive parity, the octet of pseudoscalar mesons with negative parity and the octet of vector mesons with negative parity. The consideration of only these supermultiplets is due to the

Unitary-Symmetrical Theory of Multiple Particle Production 723

f a d t,liat they are well determined, that is they well obey t’he Gell-Mann mass relations. The large number of ot’her baryons and meson resonances available can be also taken into account in the niodel considered if t,hey are wcll classified according to the representations X U (3) [ I d ] . The basic expression for the probability of TL part>icle production in the find state ( 7 ) differs formally from the corresponding expression of the usual statistical model 14, 61 only by the replacement of the “isotopic weight” by the “unit’ary weight” where n,,, n,8 and n, 1s the number of decuplet, octet and singlet “partic- Iw” respectively. It is clear that U,(p, y;, T L ~ , , , Y L , ~ ) is independent on the number of singlet “particles”. The term “unitary weight” means the number of various possibilities t,o obtain the desired irreducible representation ( p , q) by taking the direct product of n multiplets n = nl, + rig + n,

(3 .0) @ .... @ (3,O) @ (1,l) @ . . . . @ (1,l) 0 (0,O) 0 (0,O) @ .... @ (0,O). (10) b d - “A-

n10 n8 fil

I t is obvious t>hat the determination of the “unitary weight” U,(pl q) from the Clcbsch-Gordan series (10) is only due to the fact that till now all bhe available particles and resonances were classified only according to tlhe representations (3 ,O). ( 1 , l ) and (0,O). It is natural t’liat for the 27-plet the factor U,(p, q) will be dekrmined from the product containing also n27 factors ctc. In the statistical model of multiple particle production taking into account o d y isotopic symmetry the factor of the “isotopic weight’ for each channel of reactions separatcly was independent of I, and was dependent on I only. The dependence of the unitary weight only on the external quautum numbers p and q for each rc.action channel takes place in the new model as well. Besides, the unitary weight depends also on what supermultiplcts enter the final state and what is their number. It is not necessary to indicate the dependence of tlie unitary weight on the number of singlets since it is clear that the addition of any number of t,liem does not change the unitary weight,. Since for all pract,ically interesting cases t,he initial state is represented by two octets then the reaction can proceed only through representations which enter the direct product of two octets. Knowing the total isotopic spin and the hypercharge of the init,iaI state it is necessary to find out in u-hat channels the reaction proceeds tlirii to calculate the reactions probability for each channel (represent’ation) separa- tely a i d to find the total reaction probability by summing up the probabilities of all possible channels ( p , q) then

wliere the coefficients L&,(I, I,, Y) determine the relative ueight of a given state (I, 13, Y ) in the channel corresponding to the representation ( p , q ) . Table 1 gives the coefficients k&ql ( I , I,, Y) for the most interesting cases of particle interaction. Up to now the problem of multiple production vas considered in the framework of the statistical model taking into account the cxact 8 U (3) symmetry. However, the existence of mass differences of particlei entering one and tlie same super-

31 Zeitsrhnft , E’ortichritti, ilcr Ph>sik“, Heft 11/12


multiplets point out that in fact the unitary symmetry is broken. The real dyna- mics of the process is far more complicated than that which follows from purvly statistical considerations and the group properties of strong interactions. In particular, it is kno\\,ri from the earlier statistical calculations [6 , 71 that the

Table 1

hilations The values of the coefficient I<?, ,,,)(I, I,, Y ) for xiV, N N interactions and antinucleon anni-

Represen- ( p , q) tation

~ _ _ _ _ _ _ _ ~ _ _ _ _ _ _ _

nF E2 (p , , , ) ( I = 1, I, = - 1, P = 0) - 3/10 116 1jO 116 1,s

k&,, (I = 1, I , = 0, Y = 0) - 3/10 ljli 116 l / O 1;5

kt1,,(,) ( I = 0, 1, = 0, I’ = 0 ) 114 1/10 ijs - - 3,”O PP

effective coupling constants of n and K mesons are very different. In other ords, attempts should be made to take into account in (7 ) the splitting of the effective coupling constants for various kinds of produced particles. In the phenomenological statistical theory these constants approximate the unknoun part of the matrix elements and may be dependent on thc kind of particles. However, the only important fact for us is that this part depends neakly on energy and may be replaced by a constant. It is known that in statistical model calculations for channels involving K-nieaons an additional “cutoff” factor /I N 0.1 was introduced by means of which one has succeeded in obtaining agreement betwcen the calculated probabilities for strange particle production and the experimental ones. In the statistical model taking into account the exact S U (3) symmetry it turns out also that the calculated u o s s section for strange particle production exceeds the experimental one. We consider the splitting of the effective coupliiig coilstant for I<-mesons aloi-e since, due to the conservation of strangeness, the hyperon production is accompanied by thc K-nieson production of K-mesons are a h ays produced hy pairs KE. To this end, as in the previous calculations, we introduce into (16) a cutoff factor i f i k where nk-the number of K-mesons produced in the reaction and the constant 1, is taken from the cornparison with experiment. The introduction of the factor 7.”. for the reaction involving strange particles may be considered as a certain violation of the unitary symmetry which reduccs to the consideration of the differenrc in the interaction cf x and K-mesons. It is possible that the constant i used in the model can give some information about the constant of a moderately strong

Unitary-Symmetrical Theory of Multiplc Particle Production 725

interaction violating the S U (3) symmetry since the value 2 ‘v 0.04 obtained from the mass formulas [I51 is very close to thc value of 3, used in our model for the case of n N interactions A N 0.06 and for the N 8 annihilation 1 = 0.034. Another effect due to the S U (3) summetry violation that is the y - w mixing effect [9, 161 has been considered in studying inelastic interactions in the framework of the model presented here. This effect would influence the total crosb section for strange particle production which is a very important characteristic of inelastic processes. This is seen from the fact that a y meson decays mainly with the K E pair emission and w meson with the pion emission. It is obvious that the account of a particle in the statistical model either as a singlet or the octet I = Y = 0 component is not the same. However, the account of the y - o mixing with an angle of 0 = 39’40’ in the calculations for x p interactions has weakly affected the inelastic interaction characteristics of interest to us.

3. “Unitary weight”, methods of calculation

As was mentioned above out main expression (7) for the probability of n particle production in the final state differs from the corresponding expressioi; in the former substitution of the “unitary weight” U,(p, q ) for the isotopic weight. Therefore we clear up in more detail the meaning of this factor and give different methods for finding it. Taking as an example the two-particle elastic interaction we consider what restrie- tions are imposed on the reaction amplitudes by the S U (3) invariance. It is not difficult to extend then this consideration in the general form to the case of inelastic interactions with n particle production. We suppose that in the reaction

a + b --f c + d

the particles a, 6, c and d belong to the irreducible representations of the S CT (3) group. Then the S U (3) invariance leads to that the reaction amplitudes must be unitary invariants or in other words the collision operators S (scattering matrix) must commute with the unitary spin. Expanding the initial and final two-par- tide states in the states of irrcducible representations we obtain

(cdlSlab) =c z(n, 7% y , na, nb, Ina) nba, nb, 777+,) i.: 7n,n,y m’,n’,y’

x {n‘) m’, y ‘ , n,, n d I n,, mC, nd, ??id) (n‘, m‘, y ‘ , n,, n, I S In, m, y , n,, nb) (12)

where n denotes the set of external quantum numbersp, q and m- the set of internal quantum numbers I , I,, Y . It is known that the invariance S with respect to the subgroup X U (3) of the form S U ( 2 ) , @ UY ( I ) (which appears to be the real symmetry group of strong interactioiis) requires that only those elements of (12) which have m = rn’ should not vanish. Show that the XU(3) invariance leads also to the equality n = n‘. From the S U ( 3 ) invariance it follows that the Casimir operators G3, P2 should also commute with the S-matrix i.e.

(13) [P2, Sl] = [a3. 91 = 0 .

52*

726 V. S. BARAZHENKOV and G. M. ZINOVJEV

Taking the matrix element of these commutators between the same states as in (12) we get the two relations

[G3(p', 2') - G 3 ( p , 2)1 (TL', m', Y', %, n d I I n, 7% y ; n,, 1%) = 0 .

It is seen that the matrix elements of the X matrix are non-zero only in the case if F z (p ' , q ' ) = P 2 ( p , q ) , Q3 ( p ' , 4') = G3 ( p , q ) this is possible only if p' = p ,

Thus, the requirement of X U (3) invariance for the reaction amplitudes leads to that only transitions between initial and final states which correspond to identical irreducible representations ( p , q) are possible i. e. (12) is rewritten in the form

q' = 4.

x (72, n ~ , y' , n,, nd, I n,, n i t , nd, m d ) (n, in, y ' , n,, nd I S 1 n, na, y ; n,, 7 4 . (15)

If u c assume the particles a, b, c and d to be the octet components then all thc previous rcsults lead t o that only eight amplitudes are possible

(l.i)z - (l.l)l (3.0) ---r (3.0) (0.3) * (0.3) (2.2) - (2.2)

as far as it is known [lo] that the Clebsch-Gordan series for the direct product of two octets is of the form

(1.1) @ (1.1) = (0.0) + ( l . l ) l + (l.l)2 + (3.0) + (0.3) + (2.2). (17)

I t is clear that only seven of these eight amplitudes are independent for the elastic reaction considered, since due tlo invariance under time inversion thc amplitudes (l.ll) --f (l.l)2 and (l.l)z -+ (l.l)l are equal to each ot'her. The number of amplitudes might, be also restricted by a possible existence of the so called R invariance of strong interactions [ l 7 ] . The R reflection is usually defined as the inversion operation of weight diagrams. This operation which docs not enter the S U (3) group can br used to distinguish brtwcen the representations (l.l)l and (1.1)2 since for self-conjugat8e representations (2 .2) , (l . l) l , (l.l)z and (0.0) the states in the centre of the weight diagramme are the eigenfunctions with an eigenvalue equal to unity for the (3.2), ( l . l . ) l and (0.0) rcpresentations and to (- 1) for the (1 representation. Therefore. the e,xistencr of the R-invariance would forbid transitions between octets with different indices. However a t present tlierc is no evidence for the existence of the R-invariancc of strong interactions

Lf now we attempt to generalize (15) t'o the case of three particles in the final state then all the amplitudes (16) will have different multiplicity. If in this case the third particle is also a component on the octet then the multiplicity of the amplitudes will be the one with which the corresponding representation enters the Clebsch-Gordan series for the product of three octet's. If otherwise this t'hircl

[ lSJ


particle belongs to the decuplet then the multiplicity will be equal to that of the corresponding representations in the Clebsch-Gordan series for (1.1) 0 (1 . I ) (3 @ (3.0). When the number of particles in the final state is increased the finding of the multiplicity of the corresponding channel reduces simply to the calculation of the multiplicity of the corresponding representation ( p , q ) in the Clebsch- Gordan series for the product the same number of representations as the number of particles considered in the final state. The “unitary weight” is determined as the multiplicity with which the appropriate representation ( p , q ) enters the Clebseh- Gordan series for the direct product of the corresponding representations. Since a t present only the representations (0.0\, (1.1) and (3.0) of the group X U(3) are used for classification possible reaction channels are restricted only by the representations in which the (1.1) @ (1.1) and (1.1) @ (3.0) products arc expen- ded. Processes involving particles from the decuplet are of no practical interest, therefore the set of channels (16) is only used. Froni (15) it follows that if thc initial state is fixed, i.e. if the initial particle? are given by the vectors 11.1; I, I,, Y) , 11.1, If, 1’, Y } then not all the channels (16) are to be considered as possible ones, but only those which contain the state with

I;‘ = I , + I;; Y” = 1’ + y’

and for the isotopic spin I“ of this state the triangle inequality is fulfilled

II - I’j <I” < I + I’ So, for example, in the case of nip interaction the state I = 312, I, = 313, I’ = I enters only the (3.0) and ( 2 . 2 ) representations of the Clebsch-Gordan series (1.1) @ (1.1). Now it is possible to calculate simply the unitary weight U , ( p , q ; nlo, ns) for the case, say, of three particles in the final state belonging to the octets. The appropriate Clebsch-Gordan series is

(1.1) @ (1.1) @ (1.1) = 2(0.0) + g(1.1) + 4(3.0) + 4(0,3) + 6(8.8) + 2(4.1)

+ 2(1.4) + (3.3)

and the multiplicities with which the (3.0) and (2.8) representations enter this series are the desired unitary weights, i.e.

To calculate the factor unitary weight sense of which we have clearedup it is necessary to able to find the Clebsch-Gordan series (10). This problem can be solved using the wellknown technique of weight diagrams of the representations and the Spciser scheme [19]. Note that this method can be applied to all the irreducible R(:(3) group representations and not only for the representations of the group “eight fold way” used. However, as was already noted, the Clebsch-Gordan series (10) alone is of piacticsl interest. Therefore it is very important to find a simple way of reduction only for the ( p , q ) @ (1.1) and ( p , q) @ (3.0) products. The use of the weight diagram technique and the Speiscr scheme allows to obtain a rather simple rule for the reduction of the direct product ( p , q ) @ (1.1).


This rule strictly proved in [20] shows that the reduetion of the direct product ( p , q ) 0 (1.1) contaiiis the following irreducible represeritations

( p + 3. q - 1) once unless q = 0

( p + 1, q - 3 ) once unless

(11 + 1 , q $- 1) once

( p - 1, q - 1) once unlew p = 0 or q = 0

q = 0 or q = 1

( p - 1, q + 2)

( p - 3, q + 1 )

once unless p = 0

once unless p = 0 or p = 1

( p . q) twice unless p = 0 and q = 0

onre if p = 0 and q + 0 or if p $. 0 and q = 0 not a t all if y = q = 0.

This simple rule allowed us to calcrilate the uiiitary weights for the product of ten octets. The results are given in Table 2. It is seen that tlie values of U,(O.O) starting with n = 3 arc equal to the values of U n ( l . l ) for tlie foregoing valuc of n. This regularity follows from the fact that the (0.0) representation can be obtai-

Table 2 The values of the unitary weight Un(p , a) for the octet product,ions

~~

'2*4P, n) (0.0) (1.1) (3.0) , (0.3) (2.2)

0 Y I 2 1 1 1 3 1 8 4 4 6 4 8 32 20 20 33 5 32 145 100 I00 180 6 146 703 536 525 999 7 702 3 598 2 856 3 856 5670 8 3 698 19180 15834 15 834 32 284 9 19180 105910 90390 90 390 173766

10 105910 685 546 511 179 511 179 1088 220

>

ned from the direct product ( p , q) @ (1 .1) only if ( p , q ) is also an octet. The equality of the unitary weights for thc (3.0) and (0.3) representations may be easily under- stood if we take into consideration the symmetry in (18) with which different representations enter the reduction ( p , q ) 0 (1,i). A similar rule may be established [21] for the reduction ( p , q ) 0 (3,O) as well. It will contain the following irreduciblc representations

( p + 3, 4 ) once ( p + 2, q - I) once unless q f 0

( p + 1, q - 2) ( P , P - 3)

once unless once unless

q+0 or q+ 1 q $= 0 or q + 1 or q $= 2

( p + 1, q + 1) once unless p + O (19) (P, q) once unless p + 0 and q+ 0

Unitary-Symmetrical Theory of Afultiple Particle Production

once unless once unless p + 0,l

once unless p + 0,l or q + 0 once unless p + 0, 1, 3 .

729

(p -- 1, y - 1) ( p - 1, q + 2 ) ( p - 2, q + 1) ( p - 3, q 4- 3)

p + 0 or q + 0,i

This rule allom to obtain the unitary weight for many-particle reactions with dccuplets. Table 3 gives the unitary weights of the products involving one or two decuplets. The characteristic feature of Table 3 is that for reactions with one

Table 3 Tlic valiies of the unitary weight CJi,(p, q) for the productions of the octet and decuplet

Q I

3 4 5 6 7 8 9

10

0 1 4

20 100 525

2 856 15834 90390

1 4

20 100 525

3 856 15834 90390

527013

1 4

17 86

451 2 406

14084 84 062

486462

Table 3

0 2

12 70

400 2310

12762 78 507

465 583

1 5

27 150 855

4881 39 020

168830 1035447

nI(TJ3 q) n,, = 2 (0.0) (1.1) (3.0) (0.3) (2.2)

2 0 0 0 1 1 3 0 3 3 2 4 4 2 12 12 10 23 5 12 70 66 56 126 6 70 400 374 330 341 7 400 2310 2 166 1886 4 432 8 2310 12762 12032 11562 26445 9 13 762 78507 721 133 67950 153426

10 78 507 4G5 583 448 846 422 700 973433

decuplet the unitary weight for channel (1 , l ) coincides with that for channels (3.0) and (0.3) in Table2 for all values of n. This follows from the fact that the representations (3.0) and (0.3) in the reduction ( p , q ) @ (1.1) occurs for these p and q for which (1.1) occurs in the ( p , g) x (3.0) reduction. It is quite obvious that the values of the unitary weights given in Tables 2 and 3 may be used in studying all practically interesting processes of inelastic interactions of particles in the framework of the statistical model. The cho ce of particles in the initial state leads simply to that taking into account the conservation of the baryon number we must choose the necessary values of the “unitary weights” from the corresponding table (naturally, within the limits of those n, nlo for which these tables are made).

730 V. S. BARASHEKKOV and G . M. ZINOVJEV

4. Distribution of secondaries over the charge and hypercharge in unitary symmetrical statistical model

As is known, when multiplying the two representations of dimensionalitier N , and N , in the N , N , dimension space we may choose as the basis vectors either the products of the vectors of states (I) , or state vector (4) of different irreducible representations entering the direct product of the considered S U (3) representations. These two sets of the orthoiiormalized basis vectors are connected by the unitary transformation. The condition of completeness of (4) has the form

d T' In., m; n', n", y ) (n, m ; n', n", y I = 1. (no) l2,?17,7

By multiplying both sides of this equality by t'he vectors of the stat,e (10) wc get

j n', nz.') In", m,") = 2 In, m ; n,', m", y\ {n, m ; n', n", y In', wz', n", nt"). (21)

This is the relation which just defines the transition from one set of basis vectors to another and the transition coefficients

(22)

are tjhe Clebsch-Gordan coefficients of the X U (3) group. If arbitrary phases inside isot'opic multiplets and between them are fixed according to the conditions of ref. [ lo] then the Clebsch-Gordan coefficients are real matrices and the expression inverse to (21) will contain the same Clebsch-Gordan coefficients as (21). Taking into consideratior all the above-said and transforming the initial exprw- sion for the matrix element we may obtain the following expression for the multiple particle production probabiliby

?l !?lZ,?

(n, nt, n', n" , y I n', m', n", m,")

W , ,- I (n,, m,, . . . , n,, m,, I Jl j n, m ; n', n", y , ai) 1, =

= IC (n,, m,, ..., n,,m, ln ,m. ,n ' , nrr , y , B ) :< (23) B

:-: (n, m, n', n", y , N ~ , B I Jf I n,, nz, n,', n", y , ei) 12

where (Y a.re all the quantum numbers which are necessary for describing particles in the usual space. According to the statistical hypothesis we get

where the summand is naturally independent on the configuration in p space and is the square of the Clebscli-Qordan coefficient. Then it follows from (34) that the probability of production in any channel of particles with definite strangeness, isotopic spin and charge e.g.

X + X+ no n n+n+KoA * n+x+K+S-


is obtained by multiplying the statistical weight of a given channel %&(E,) by the squares of the corresponding Clebsch-Gordan coefficients of the X U (3) group. It is known that these coefficients are expanded into the Clebsch-Gordan coefficients of the S U (2) group and the isoscalar factors i. e. in our notations we have

(n, ni, y, n,, I n,, m,, n2, ni2> = C ( I , I’, I”, I,, I;, I;) U ( n , m, n’, m’, n”, ni”, y )

(’5)

where C ( 1 , I t , I“, 13, I;, I;) are the Clebsch-Gordan coefficients of the isospin group S U (2) and U(n, m, n’, m’, nrr, m”, y ) are the isoscalar factors independent of the third components of isospin. Therefore if mre are not interested in the charge distributions then we should use only isoscalar parts of the Clebsch-Gordan coefficients. It is obvious that a successful use of the unitary symmetrical statistical theory of multiple particle production in analysing the available experimental data on inelastic interactions a t high energies is closely connected with the calculation of the Clebsch-Gordan coefficients for the product of a large number of representations and even not of coefficients but only of their isoscalar parts. The solution of this problem, i e. derivation of simple closed formulas for the case of direct product of a large number of arbitrary representations is very complicated. In some cases, e.g. multiplication by an octet or decuplet, simple algebraic tables of isoscalar vectors are constructed [20, 811 by means of which one suceeds in obtaining the squares of isoscalar factors (see Table 4). In calculating the isoscalar factors according to the Table [ZO, 211 we use their two well-known properties [lo]

U ( p , y ; 1 , l ; p ’ , q ’ ; I, Y; I”, Y“,I’, Y’) = x ( - l ) I + P - I ’ > ,

>: U ( l , 1 ; p , y ; p‘ , 4 ‘ ; I”, Y”, I, Y , 1’, Y‘) (xi) where x = & I following the choice of the phases, x being independent on the internal quantum numbers of the representation and is determined only by the numbers (P, n), (P’, Y’) and (1, 1)

N ( p , q ) (21’ + 1) 1/2 U ( n ; 1,1; n‘; I, Z’, I”, Y, Y’, Y”) ( 2 7 )

N (p’ , 4’) ( 2 1 + 1) 1 where I”, Y“ are the isotopic spin and the hypercharge respectively from represtn- tation (1.1), N ( p , y) and N ( p ’ ; y’) are the dimensionalities of the ( p , q) and (p’ , q’) representations. For the calculated probabilities of some channels Rkq ( I , Y ) of inelastic rraction with n particle production in the final state ( I , Y ) the equality naturally hold which is similar to that in the model taking into account only the isotopic symmetry provided that the probabilities are not normalized to unity

732 V. S. BARASHENEOV and G. M. ZINOVJEV

Table 4

For the ease (8) @ (8 ) @ (8) in the final state I = l ; Y = O

~ ~~

(1 12. 1 ) (112. -1) (1.0) 0.3646 0.4456 0.4456 0.5313

(1.0) (1.0) (1.0) 0.0871 0.1203 0.1203 0.1853 (1.0) (1.0) (0.0) 0.1264 0.1167 0.1167 0.0817 ( 1 .O) (0.0) (0.0) 0.1647 0.1162 0.1152 0.0727

(1p.l) (lp. -1) (0.0) 0.2572 0.2022 0.3023 0.1890

(1/2. 1) (1.0) (1.0) 0.4643 0.5417 (l/"l, (112.1) (l/2. -1) 0.1961 0.2083 ( 1 D.1) (1.0) (0.0) 0.3393 0.2500

(11, Yl) ( 1 2 , Y2) (139 1 1 3 ) (271 {lo*} (8,) (821

(li2.1) (1.0) (1.0) 0.2540 0.3021 0.43135 0.3803 (112. 1 ) (lj2.1) (I/& -1) 0.2446 0.2916 0.3292 0.2052 (113.1) (1.0) (0.0) 0.3653 0.3125 0.1875 0.300 (1 12.1 ) (0.0) (0.0) 0.1461 0.0938 0.0468 0.1156

For the case (10) @ {8] @ (8) in the final state I = 3 / 2 I ' = 1

(3/d.l) ( 112 .I) (112. -1) 0.2877 0.3155 ( 1 .O) (1.0) (1/2.1) 0.1750 0.1537 ( 1 .O) (0.0) (112.1) 0.0856 0.0565 (312.1) (1.0) (1 .O) 0.3392 0.3204 (312.1) (1.0) (0.0) 0.1437 0.0744 {3/2.1) (0.0) (0.0) 0.0331 0.040'2 (1/2, -1) (112.1) (V2.1) 0.0357 0.0393

I = l /2; Y = 1

( I , , I'll v2, y2) v3, (271 {lo*} (8,) (82)

(3j2. 1) (112.1) (112. -1) 0.1358 0.2912 0.2518 0.1333 (3/6.1) ( 1 .O) ( 1 .O) 0.0764 0.1476 0.3186 0.2000 (312.1) (1.0) (0.0) 0.1636 0.1965 0.1112 0.2000 (1.0) (1.0) (1P.I) 0.3682 0.1658 0.1848 0.2833 ( 1 .O) (0.0) (112.1) 0.0960 0.1165 0.0744 0.0500 (lp. -1) (112.1) (112.1) 0.1600 0.0825 0.0592 0.1333

Unitary.Symmetrica1 Theory of Multiple Particle Production 733

For the case {8} @ { S } 0 (8} @ (8) in the final state I = 312; Y = 1

(zi* 1'1) ( 1 2 3 y z ) ( 1 3 , Ya) (149 y4) {27}

(112.1) (1/2.1) (1/2.-1) (1.0) 0.3769 0.3905 (li2.1) (1.0) (1.0) (1.0) 0.2447 0.2915 ( 1 13.1) (1.0) (1.0) (0.0) 0.2129 0.1748

(lj2.t) (1.0) (0.0) (0.0) 0.0755 0.0643 (112.11 (1/2.1) (1/'2.-1) (0.0) 0.0900 0.0789

I = 1p; 1' = 1

(11, 1'1) ( 1 2 2 1'2) ( 1 3 3 Y,) ( 1 4 9 Y4) (571 (10"l (8,) (821

(112.1) (1p.1) (1/2.-1) (1.0) 0.3536 0.3010 0.3750 0.2858 (1/2.1) (li5.1) (1/2.-1) (0.0) 0.2004 0.1833 0.1452 0.1784 ( 112.1 ) (1.0) (1.0) (1.0) 0.1371 0.1614 0.2340 0.1860 ( lj2.l) (1.0) (1.0) (0.0) 0.2403 0.2030 0.1765 0.3336 (l/"l) (1.0) (0.0) (0.0) 0.0934 0.0791 0.0557 0.0872 (li2.1) (0.0) (0.0) (0.0) 0.0273 0.0202 0.0136 0.0290

For the case {lo} x (8j >: (S} x 18) in the final state I = 3 / 2 ; I'L 1

( 1 1 , 1'1) ( 1 2 . Y2) ( 1 3 , Y3) (14, Y4) (27) (10)

(32 .1 ) (3:'.3. 1 ) (3,'2.1) (3 id. 1 ) (3/2.1) ( 3 / a , 1 ) (1.0) (1.0) (t.0) (1.01 ( l i d . -1) (1'3. -1) (0. -2)

(lj2.1) (1/2.1) (1.0) (1.0) (1.0) (0.0) ( 1 M . 1 ) (Il2.l) (li2.l) (t12.1) (1/2.1) ( 1 p .1) (112.1)

(1.0) 0.3144 (0.0) 0.0951 (1.0) 0.0994

(0.0) 0.0205

(112-1) 0.0604 (0.0) 0.0644

(1.0) 0.0639

(0.0) 0.1007

(0.0) 0.0071

(0.0) 0.0150 (1.0) 0.1321

(0.0) 0.0176 (112.1) 0.0094

I = 1/2; 1'= I

0.3510 0.0800 0.1320 0.0771 0.0215 0.0040 0.0642 0.0561 0.0135 0.1214 0.0541 0.0157 0.0094

( 1 1 3 I'J v.2. Y2) ( 1 3 9 Y 3 ) (147 Y4) (27) (10"l (82)

(3/3.1) (lj2.1) (112. -1) (1.0) 0.1890 0.2431 0.3146 0.2031 (312.1) (lp.1) (112, -1) (0.0) 0.0713 0.0775 0.0646 0.0783 (3P2.1) (1.0) (1.0) (1.0) 0.0500 0.0716 0.1327 0.0722 (316.1) (1.0) (1.0) (0.0) 0.0740 0.0875 0.0617 0.1058 (312.1) (1.0) (0.0) (0.0) 0.0294 0.0280 0.0250 0.0333 ( 1 .O) (lp2.1) (lp2.1) (112. -1) 0.1100 0.0774 0.0719 0.0678 (1.0) (112.1) (1.0) (0.0) 0.1147 0.1041 0.1008 0.1205 ( 1 .0) (1/2.1) (0.0) (0.0) 0.0267 0.0200 0.0134 0.0249 (1.0) (112.1) (1.0) (1.0) 0.1G06 0.1391 0.1123 0.1157 (1/2. -1) (112.1) (1/2.1) (1.0) 0.1001 0.0841 0.0630 0.0881 (l/%--l) (1/2.1) (1/2.1) (0.0) 0.0535 0.0451 0.0313 0.0583 (0. -2) (1/2.1) (1/2.1) (1/2.1) 0.0207 0.0225 0.0187 0.0326


where the sum is taken over all the channels with different particles allowed in the given representation ( p , 4 ) by the fixed values of the I and Y initial state and U,@, y) is the unitary weight of the given reaction in the representation (P, Y). Using Table 4 i t is necessary to remember that when going over from the states of the X U (3) representations to particles we should naturally take into account all possible permutations of the particles between the states I, I,, Y of a given channel in order to find probabilities of separate channels in the case of particles of different kinds. The permutation should concern only particles belonging to identical representations.

5. Inelastic xN interactions at high energies and NN annihilation at rest in the uni-

tary symmetrical statistical theory of multiple particle production

Here we sketch the main results of calculation of different characteristic of inelastic nN interactions a t T N 10 GeV and of nucleon-antinucleon annihilation a t rest in the framework of the new model [ 1 I - I 3 ] . The calculated average multiplicities of Z produced particles are close to the exyeri- mental ones both for xfp and for x-p interactions and change a little with varying constant 1. Practically, the average multiplicity ?i would be calculated a t A = 1 . In calculating ?i and the average multiplicity of charged particles 2 channels were separated which contain q and w mesons, since they disintegrate mainly into a larger number of particle than other resonances belonging t o the same aupermultiplets. For the decay rates of resonances their experimental values have been used [22]. It is worth noting that agreement between theory and experiment is much better for %' than for the average multiplicity % the calculated values of which are by about 10-20 percent larger than the experimental ones. However the latter are not experimental values in the true sense of this word since they are obtained from the measured values of 7i* using some assumptions on the relationsliip of charged and neutral particles [ I ] . Agreement between experimental and theoretical values of ?i* is naturally more convincing. The probability of creation of stars with different number of prongs calculated for the case of inelastic n-p interaction is also in good agreement with experiment. In order to bring in agreement the theoretical values for the strange particle production in inelastic nip interactions and the experimental ones the parameter 1 was chosen to be 0.06. The study of the behaviour of some channels of inelastic interactions shows that the calculated results for many-particle channels are in good agreement with experiment, disagreement occurs only for channcls with small number of particles and in the region of low energies. Nucleon-antinucleon annihilation a t rest or a t low kinetic energy is a very suitable process for checking the degree of validity of the main assumptions of the statistical model. This followsfirst of allfrom that the annihilatioiiprocessrelcasesanenergy sufficient for production of a large number of mesons, and on the other hand, the inelasticity of the process is 100 percent, i.e. only bosons arc produced. The most surprising results of the experimcntal study of nucleon-antinucleon annihilation a t rest were always the high multiplicity of secondary pions and comparatively small probability for the K-meson production.


One succeeds in fitting these most important characteristies calculated in the framework of the unitary symmetrical statistical model of multiple particle production with experiment a t 1 = 0.034. An agreement is also obtained between

Table 5 Distribution over the number of prongs

0 prong x+x- x+x-xo X+X-XO 3x+2n- 2x+3n-xo 2x+2x-x0 6 prong

Quark model 12.9 0 7,4 22,3 25,6 28,5 3.1 0.1 ( 0!0 1 Experiment 3.2 & 0.33 * 7.8 f 34.8 f 5,s & 18.7 & 21,3 & 3,8 (%I f 0.5 f 0.03 & 0.9 & 1.2 & 0.3 f 0.9 & 1.1 f 0.2

Unitary symmetrical statistical 3.9 1.25 6.7 43.4 5.0 16.6 18.3 1 model (yo)

experimental and calculated data on the many-particle channel probabilities both for pp and n g annihilation. For channels with a small number of secondaries as in the case of xN-interactions agreement wibh caxperiment is not observed. We give for comparison the results of pp annihilation obtained in the mentioned model and the results obtained in the quark model [23, 241 which was widely discussed in the literature.

6. Conclusion

The comparison of the results of calculations of different characteristics of in- clast'ic interactions performed in the framework of the unitary-symmet'rical statistical model of the multiple particle produetioh with experiment, alloivs to conclude that the above model simplifies essentially the statistical calculations and is a way of taking into account consistently strange particles in stabistical models. Thus this model is an effectjive method of calculation of the reactions with a large number of particles. Let us st>ress, however, that we imply here only average values since using the statistical approach we may not claim to an.ything larger. The application of the unitary symmetrical statist'ical model which is so useful in analysing inelastic interactions in the energy range up to 10 GeV appears t>o meet a t T N 1000 GeV the same difficulties as the first version of the Fermi model, i.e. the necessary of taking into account a large number of channels and of newly produced supermultiplets. Therefore in a similar manner it would seem to be possible to take into account higher symmetries of elementary particles. This would allow again to simplify essentially the calculations and would give the possibility to take consistently into account the conservation of some quaiiturn numbers. In this sense the S U ( 6 ) symmetry of elenieiitary particles is very attractive. However, a simple generalization of the statistical model with the account of the S U (6) symmetry would meet serious difficulties and would need t'he use of a number of quit,e nnjustified assumptions


References

[l] V. S. BARASHENKOV, V. M. MALTSEV, I. PATERA, V. D. TONEEV, Fortschr. Phys. 14,

121 V. 6. BARASHENKOV, V. M. MALTSIW, Fortschr. Phys. 15, 435 (1967). [3] E. FERMI, Progr. Theor. Phys. 16, 570 (1950). [4 S. Z. BELENKY, V. M. MAKSIMENKO, A. I. NIKISHOV, I. L. ROSENTAL, Uzpekhi k'iz. Xi,xuk

[5] R. HAGEDORN, E. C. G. SUDARSHAN, Reports in the Proe. of the Intern. Conference on

[GI &I. KRETZCHBZAR, Ann. Rev. Nuel. Science 11, 1 (1961). [7] V. S. BARASHENKOV, Fortschr. Phys. 9, 29 (1961). [A'] V. G. KADYSHEVSKY, R. &I. A~URADYAN, preprint JINR P-2124, Dubna 1965. [9] NGUYEN VAN HIEU, Lectures on the unitary symmetry of the elementary particles, Atom-

357 (1966).

62, I (1957).

Theor. Aspects of Very High Energy Phenomena, CERN, 1961.

izdat, Moscow 1967. [lo] J. DE SWART, Rev. Mod. Phys. 35, 916 (1964). [I11 V. 8. BARASHENXOV, G. M. ZIKOVJEV, V. M. MALTSEV, preprint JINR P2-2956, Dubna

[I21 V. 8. BARASHENHOV, G I . &l. ZINOVJEV, V. M. MALTSEV, Yadern. Phys. 6, 412 (1967). [I31 V. S. BARASHENXOV, G. M. ZINOVJEV, V. 81. MALTSEV, Aeta Phys. Polon. 39, 315 ( 1 9 W [I41 L. JENKOVSKY, V. I. KUKHTIN, NCUYEN THI HONG, Yadern Phys. 5, 891 (1967). [I51 R. DALITZ, Lecture delivered a t Les Hoiiehes During the 1965 Session of the Summer

School of Theor. Phys. "High Energy Physics", Gordon and Breach Science Pub11dic.r~. [ I S ] J. J. SAKURAI, Phys. Rev. 132,434 (1963). [I71 P. FRUEND, H. RUEGG, D. SPEISER, N. MORALES, Xuovo Cimento 25, 307 (1962). [I81 V. I. Ogievetsky, I. V. Polubarinov, Lectures a t the Winter School, v. 3, Dubna 1964. [I91 D. SPEISER, Helv. Phys. Acta 38, 73 (1965). [ZO] J. G. KURIYAN, D. LURIE, A. J. NACFARLANE, J. Math. Phys. 6, 722 (1965). [el] L. M. MUXUNDA, N. PANDIT, J. Math. Phys. 6, 1547 (1965). [22] A. H. ROSENFELD, A. BARBARO-GALTIERI, W. PODOLSKY et al., Rev. Mod. Phsz. 39,

[23] Z. R. RUBINSTEIN, H. STERN, Phys. Letters 31,447 (1966). [ 2 4 J. HARTE, R. H. SOCOLOV, J. VANDERMEULEN, Nuovo Cimento 49, 555 (1967).

1966.

l(1957).

Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Ur. Artur Laschc, Prof. Dr. Rudolf Ritschl und Prof. Ur. Robrrt Rompe. Manuskripte siud zu richten an die Schriftleitung: Dr. Lutz Rothkirch, Sektinn Physik dcr Ilinii- boldt-Universitat zu Berlin, 104 Berlin. IKessischc Str. 2. Vcrlag: Aliademie-Verlag GinhH, 108 Iserlin, Leipziger Str. 3/4, lpernruf 220441, Telex-Kr. 0112020, Postschecklionto: Berlin 35021. M e Zeitschrift ..k.ort- schritte der Physik" erscheint nionatlich; Rezugspreis dieses Heftes 16,- (Sonderprcis fiir die DDR 12.- 31). Bestellnummer diesesaeftcs: 1005/16/11-12.- Satz und Druck: Vl2B Ururkhaus ,,Maxim Gorki", 74 Altenbiirg. Bez. Leipzig, Carl-von-Ossietzky-Str. 30131. - VerBffentlicht unter der Lieenmummer 1321 des P r c w n i i i t c b

beim Vorsi tzcnden des Ministerrates der Deutschen Dcmokratischen Ilepublik.

Documents

Unitary-Symmetrical Theory of Multiple Particle Production