IL NIgOVO CIMENTO VoL. 12 A, N. 1 1 Novembre 1972

A ~ Annihilation at Rest in the Statistical

C . J . I-IAMEI%

California Institute o/ Technology - Pasadena, Cal.

(ricevuto il 23 Matzo 1972)

Bootstrap Model (*).

S u m m a r y . - - Fermi's statistical model of high-energy hadron collisions is modified to take full account of resonance production, using the statistical bootstrap model of Hagedorn and Frautschi. Comparisons are made with experiments on A~A ~ annihilations at rest, the most favorable testing ground available. After adjustment of the single para- meter R (the <~ interaction radius ~>), it is found that the multiplicity distributions and branching ratios are generally well fitted, except that events in which KK pairs are emitted occur less frequently than pre- dicted, indicating that these channels are dynamically suppressed. The expected size of statistical fluctuations is discussed, as a basis for judging the results. The best fit is obtained with R = (1.6 =~ 0.25)fro, a little bigger than expected, but much more reasonable than the very large radii which were required in previous models. The maximum tempera- ture T o = (110 :E 20) MeV corresponding to the fitted value of R is lower than the figure usually quoted in fits to high-energy transverse-nmmentum distributions, but still matches reasonably well with these distributions when the full chain of secondary decays is properly included in tim analysis. Some general features of resonance decay which follow from the model are discussed: in particular, the linear dependence of the multiplieity of decay products on the mass of the resonance.

1 . - I n t r o d u c t i o n .

I n the o r ig ina l F e r m i s t a t i s t i c a l mode l (1) of h i gh - e ne r gy h a d r o n collisions,

i t was p roposed thsJt t he e n e r g y re leased would d i s t r i b u t e i t se l f according to

(*) Work supported in part by the U.S. Atomic Energy Commission. Prepared under contract AT911-1)-68 for the San Francisco Operations Office, U.S. Atomic Energy Commission. (x) E. F~RMI: Progr. Theor. Phys., 5, 570 (1950).

162

J ~ P ~ ANNIHILATION AT I:~EST IN THE STATISTICAL BOOTSTt~&P MODEL 163

st,~tistical laws ~mong the var ious degrees of f reedom awdlable wi th in a hudronic

(~ in terac t ion vo lume )) V, of order 4z/3(h/m~c)a= ,(2o in size. Hence the tel-

a t ive p robab i l i t y of producing each final s ta te can be predic ted. Nueleon-}mtinueleon annihi l~t ion at res t was recognized as a p r ime tes t ing

ground for these ideas ("), since var ious secondary questions which ~re impor t an t

in other processes do not ar ise: sl~ch as whether to app ly a Lorentz contract ion

factor to the in te rac t ion volume corresponding to the re la t ive veloci ty of the

incoming par t ic les , or how to account for per iphera l in teract ions and s tates of high angular m o m e n t u m , or whe ther to use a phenomenologicul inelas t ic i ty

f~ctor. The energy released is quite high, ~,nd so is the mul t ip l ic i ty of reaction

products , so t ha t s ta t is t ical considerat ions should be applicable. Fur the rmore ,

the exper imen ta l d~t~ are ab l m dan t ('.~).

The object of the s ta t i s t iea l model is to reproduce the mul t ip l ic i ty dis-

t r ibut ions, m o m e n t u m spectra and angular correlations of the annihi lat ion

products (mainly pions). There is only one p a r a m e t e r avai lable within the sim-

plest vers ion of the model , n 'mle ly the volume V. Even failures of the model

m,~y be useful: t hey can isolate i m p o r t a n t dynamica l effects which have not

been t aken into considerat ion, in analogy to the way tha t resonances appear as bumps super imposed on the (( s ta t is t ical ~) phase space curves when mass spectra are p lo t t ed for a single react ion channel.

To s t a r t with, only s ta tes consist ing of nonin terac t ing pions (and kaons) were t aken to comprise the (~ degrees of f reedom ~ ment ioned above. But then

the in te rac t ion volume required to ma tch the exper iment~l pion mult ipl ic i ty

turns out to be of order 6~9o, which is too b~rge to be plausible physically. I t

w~s soon recognized ('~) t ha t this was due to strong a t t rac t ions be tween the

pions, which allow more of t h e m to be squeezed into a given volume than the

nonin te rac t ing model predicts . ]~ELV,~-Ku (4) adap ted "m old a rgument of B E ~ and UI~LE]~ECK (5) tO show th'~t wi th in the (( nar row reson,~nce ~) approxi- ma t ion the two-body in teract ions could be t~ken care of b y counting resonant states , us well as s table part icles such ~s pions, ~mong the possible const i tuents wi th in the in te rac t ion volume.

CERm~US (~) was the first to app ly this idea to the annihi lat ion process. Fie

considered the effect of including p-mesons among the const i tuents , and man-

(2) For a review of early work see M. KRETZSCnMAR: Ann. Rev. Nucl. Sci., 11, 1 (1961). (b For experimental reviews see I{. ARMENTERO8 and B. FRENCH: High-Energy Physics, Vol. 4, edited by E. B]mnoP (New York, 1969), p. 237; L. MONTANET: Proceedings o] the Lund Internatiot~al Con]erence (Lund, 1969), p. 189. (a) S. Z. BEL]~NKY: Nucl. Phys., 2, 259 (1956). (~) E. BETH nnd G. E. UttLENBECK: Physica, 4, 915 (1937). For recent treatments, see L. LANDAU and E. LIFSHITZ: Statistical Physics, 2rid ed., Sect. 77 (Oxford, 1969); ]~. DAStIEN, S. ~[A ~lld H. J . BERNSTEIN: Phys. Rev., 187, 345 (1969). (6) F . CERULUS: NUO~;O Ci'mento, 14, 827 (1959).

164 c . J . HA~ER

aged to fit the exper imenta l mult ipl ic i ty using V = Qo, at the price of giving an unphysical ly low mass to the p (which at tha t t ime was no more than a gleam in the theorists ' eye). Using the correct mass for the p, the volume required was much larger. Since then, i t has been proposed from t ime to t ime (7) tha t all the pseudoscalar and vector mesons should be included, and B ~ A - S~:ENK0V, MA~TSEV and ZINOVIEV (s) have actual ly done this, subject to the simplifying approximat ion of complete S U3 symmet ry for the part icle masses and the i r couplings. They found reasonable agreement with the exper imenta l multiplicit ies using a fixed volume V----/2o, and fur thermore were able to make predict ions for several specific final states including resonances; bu t again i t mus t be emphasized tha t t hey found i t necessary to use unphysical

resonance masses. Wi th the adven t of the stat is t ical boots t rap model of HAGEDORN (~) and

FRAUTSC]~ (~0), i t has now become possible to carry Belenky 's idea to its logical conclusion. The stat is t ical boots t rap provides a model for the spectrum of resonances above those which are present ly known (n), so one can now include all resonances of muss lower than the energy released by the annihilat ion process. This paper implements a suggestion to tha t effect by F~AVTSC~:I (~). In Sect. 2 we will give a more detai led descript ion of the model. In Sect. 3 some general features of resonance decay are discussed; and in Sect. 4 the results for the annihi lat ion process are set out and compared with experiment . Section 5 contains a discussion of s tat is t ical fluctuations. Our results and conclusions are su~nmarized in Sect. 6.

2. - Descr ipt ion o f model .

The original Fe rmi s tat is t ical model took no account of interactions between the pions. The idea of BETK and U~ENBECK (~), aS adapted b y BELENKY (t), is t ha t the two-body interact ions between them can be approximate ly taken care of by counting all resonances as independent possible consti tuents. Our

pic ture of the decay process then becomes a (( democrat ic ~ one, in which i t is

assumed tha t the JV2V sys tem decays into any given combinat ion of hadrons

at a ra te propor t ional to the phase space available to those hadrons within the 2V~ ~( in teract ion volume )) V, and no dist inct ion is made between ~( stable ~>

particles such as the 7: and K mesons, and the unstable resonant states.

(7) J. J. SAKVRA~: Ann. Phys., 11, 1 (1960); B. MAGLIC: unpublished. (s) V. S. ]~ARASHENKOV, V. M. 1V[ALTSEV and G. M. ZINOVIEV: Acta Phys. Polonica, 33, 315 (1968). (9) R. ItAGEDORN: Suppl. •uovo Cimento, 3, 147 (1965). (lo) S. C. F•AVTSCHI: Phys. Rev. D, 3, 2821 (1971). (11) C. J. HAM]~R and S. C. F~Av~scI~: Phys. Rev. D, 4, 2125 (1971). (12) S. C. FRAVTSCHI: NUOVO Cimento, 12A, 133 (1972).

v ~ p ~ 5 A N N I H I L A T I O N A T R E S T I N T H E S T A T I S T I C A L B O O T S T R A P M O D E L 1 6 ~

In order to apply this idea, i t is necessary to know the spect rum of resonant states up to the (Ar3~) mass. The stat is t ical boots t rap model of ttAGv, DOI~ (9) and FttAUTSC~ (lo) provides a model for calculating this spectrum, in an ap- proximate fashion. The assumption is t ha t the level dens i ty of hadrons Qo,~(m) is given by the phase space available for a rb i t r a ry numbers n = 2 , 3, ... co of noninteract ing const i tuents ( themselves with level densi ty ~.(m~)) in a hadron-sized box of volume Y':

(1) 1 |__| I-I Id3p~ldm,o~(m~) ~ p~ (5 m - - E,

where the const i tuents are mere ly hadrons themselves:

(2) ~n(m) = Ooo~(m) § o (low-mass inpu t ) .

This type of s ta t is t ical approach is not expec ted to be v e r y accurate a t low masses, and in par t icu lar i t is necessary to use a low-mass input spectrum, which cannot i tself be boots t rapped within this f ramework, in order to s tar t generat ing states at higher mass.

I f one pictures AP~ annihi lat ion at res t as taking place vi~ resonant inter- mediate states, as in the Bohr << compound nucleus )> model of nuclear interac- tions, then our s tat is t ical model of the decay process, and the Hagedorn- Frautschi model for generat ing the hadron spectrum, are seen to be but two sides of the same coin. The dens i ty of resonant states a t a given mass is just equal to the dens i ty of << scat ter ing states ,) into which the resonances m a y decay. I n par t icular , the A ~ <<interaction volume )> V and the << hadronic volume ~) P" are expec ted to be one and the same. The annihi lat ion process will proceed via the mechanism shown in Fig. ] , wi th the final-state pious

Fig. 1.

/ / / / / /

/ / / / / /

/ / / + + + + + + + + , , t + i t + - / . . . . . . . . . . / /

\ \ \ \ \ \ \

\ N

/ ~ \ \ \

(or kaons) being produced at various stages of a long decay chain (see Sect. 3). Since the dens i ty of resonance states rises in exponent ia l fashion with mass

(like ceb'~/m ~ asymptot ica l ly (11.13)), the number of possible in termedia te res- onances is large.

(13) W. !NTAHM: ~Yucl. Phys. 45 B, 525 (1972).

166 c . J . n A M ~

A method of generating a realistic meson spectrum was discussed by HAMER and FRAU~SCKI (11). The pseudoscalar and vector meson SU3 multiplets, with each mass taken to the nearest multiple of m=, were fed into the bootstrap equations as low-mass input, and a spectrum of resonances at higher masses was generated, which bore a reasonable resemblance to that presently revealed by experiment (14) at those masses where our knowledge is anywhere near

complete. Exotic states were excluded by the use of appropriate SU3 isoscalar factors in the bootstrap equations; and S U3 symmetric couplings were assumed. A least squares fit at high masses gave a value (132 MeV) -1 for the parameter b (the inverse of Hagedorn~s ~( nlaximuln temperature )) To (9)), when a box radius /~ = 1.3 fm was used. The value of b is expected to change proportion- ally to ~ (11). ~ow ]=[AGEDORN (o) has shown that the transverse-momentum distributions of particles produced in high-energy hadron collisions can be fitted using an ((effective temperature ~> (Sect. 3) roughly equal to 160 MeV. For consistency, then, we ought to be able to fit the A~A ~ annihilation data with a radius R corresponding to the same temperature.

The statistical branching ratios for decay of one of these (~ bootstrapped ~ resonances, in the first generation, can be read off almost immediately. The density of resonances generated at a given mass is equal to the total phase space available, within V, to all the possible constituent states. So the branching ratio into any given combination of constituents is just the proportion of phase-space contributed by that combination to the total. These first-geners~tion reaction products will then decay in their turn, and so on down the chain; but the bran- ching ratios into the various ultimate final states (with stable constituents only) can be simply deduced from those of the first-generation decay products, as- suming these are known. In practice, the decay modes of the input mesons were taken from the Particle Data Group tables (14), and then the decay modes of the resonances generated by the bootstrap were calculated for successively higher masses by the phase-space prescription above.

The branching ratios for 2~JV annihilation are assumed to be the same

as those for the decay of meson resonances of the same mass and quantum

numbers. Some assumptions were made about the relative probability of an-

nihilation in various S U3 channels, with the proportions being

(3)

~ n = 1 x [8, I = 11 ,

~p =~x[a,Z=l]+~x x[8,

The reasoning behind these figures is:

I = 0 ] + 5 x [ 1 , I 0 ] ) .

(14) PARTICLE DATA GROUP: Phys. Lett., 33 B, 1 (1970).

, ) ~ ANNIHILATION AT REST IN TIIE STATISTICAL BOOTSTRAP MODEL 167

i) DESAI (15) has shown tha t the Couh)mb field will produce t ransi t ions be tween different isospin (and ~U3) s ta tes of the J~',N , sys tem at a much more rapid ra te t han the occurrence of annihi lat ion, so tha t i t is possible to assume tha t annihi la t ion only takes l)lace via ((nonexotic ~ SU~ channels which are dominated by resonances.

ii) The evidence f rom ~(1 :~nnihilation processes (3) is t ha t the ra t io of

the ( I = 0) to ( I 1) ra tes is abou t 5:3, as used above. Lacking a n y fu r the r

exper imenta l informat ion , we h:~ve also a ssunled t ha t the ra t io of [8, I : 0] to

[1, I 0] r~tes for Pt) is jus t equ,1 to the r:~tio of the squares of the

8Ua isosealar faetors eonneet ing the ]~i) s ta te to the 8U3 s ta tes in question.

Final ly, G-par i ty selection rules h:~ve been app rox ima te ly t aken into account. For an ~(',W sta te ,

(4) G = ( - - 1) L~s+'

and i t is known tha t annihi la t ions a t res t occur p r edominan t ly f rom S-wave

s tates (~6.s), so t ha t in a given isospin ehannel the G-par i ty is de te rmined b y

the spin. Since the electrom:~gnetic spin-flip t rans i t ions in the JTJ~ sys tem

occur a t a slower ra te than annihi la t ion (14)5 We assume tha t the ra t io of singlet

to t r ip le t annihi la t ions t~tkes the s ta t is t ical wdue of 1:3 which is appropr ia te

to the ini t ia l formati()n of the ,N)A '' bound s ta te before annihilat ion. Then

the branching ra, tios into final s ta tes containing even numbers of pions (G + 1)

or odd numbers (G -- 1), and with a given isospin, are mul t ip l ied by factors �89 or ~- accordingly.

The J~',N" b ranch ing rat ios can now be calculated. I t is wor th not ing t ha t the resul ts are op(,n to several sources of error:

i) The assunipt ions as to internM q u a n t u m numbers which are outl ined

above are somewhat etude. F o r t u n a t e l y the resul ts are not ve ry sensi t ive

to the t r e a t m e n t of SU:, q u a n t u m number s ; bu t the G-par i ty selection rule obviously will have i m p o r t a n t effects.

ii) No account has been t aken of a n g u l a r - m o m e n t u m conserw~tion. This

would res t r ic t the phase space avai lable in any given channel, bu t i t also restr icts

the to ta l phrase space s imi lar ly (1), and the two effects t end to cancel each other out in the b ranch ing rat ios.

(15) B. R. DESAI: Phys. Bey., 119, 1385, 1390 (1960). (16) l{. ARMENTEROS et al.: Proceedings o] the Inlernational Con]erence on High-Energy Physics, edited by J. PRENTKI (Geneva, 1962), p. 351. (17) This has been known for a long time: see, e.g., ref. (2) or S. C. FRAUTSCHI: Progr. Theor. Phys., 22, 15 (1959).

168 c.g. HAMiER

iii) I t turns out t ha t there is a dynamical mechanism at work to suppress channels involving K K product ion (17), which has not been taken into account.

iv) Pure ly calculational errors and approximations m ay occur, such as were discussed by HA~E~ and FgAvTsc~I (11) in connection with the spectrum calculation.

Fur the rmore , one expects ((statistical fluctuations ~ to be apparen t in the data, as in the nuclear-physics case. Some consideration of this effect is neces- sary in order to judge our results , as emphasized b y FRAv~sc~ (1~): i t will be discussed more ful ly in Sect. 5.

3. - General features o f resonance decay.

Before turning to the detai led analysis of JV ~ annihi lat ion at rest , i t is of

in teres t to discuss some general features of resonance decay, as they are pre- dicted by the s tat is t ical boots t rap model (18). I t was shown by F~AvTscm (10)

tha t in the first generat ion:

i) The probabi l i ty of decay into n const i tuents is given by

(5) P(n) = (in 2)" ~/(n--1) !

so tha t two-body decays occur 69% of the t ime, three-body decays 24%, and only in 7 ~o of all decays are more than three secondary part icles emit ted.

ii) Of these const i tuents , one tends to be heavy, and the remainder light. The kinet ic energy released is small.

For instance, a large propor t ion of the f irst-generation decays result in a

s ta te consisting of one pion plus a heavy resonance. The contr ibut ion of these

states to the to ta l phase space at mass m (in units /g = c = m~ = 1) is pro- por t ional to

(6)

m - - 1 co

1 0

+ p~ + m~ - - m) ,,~"-~

co

e exp [bin] P "~ (n J p~dp m . - o . vo

0

expV-- b +

(~s) These predictions are not expected to be accurate for resonances of low mass or high angular momentum.

~ AINN1HILATION AT REST IN THE STATISTICAL BOOTSTRAP MOD:EL 1 ~

The probabi l i ty of finding one of these pions wi th m o m en tu m p is therefore given by

(7) P(p) ~ p-~ exp[ - - b~/1 -- p2].

Hence the average kinet ic energy per pion can be calculated in terms of modified

Bessel funct ions:

3K3(b) d- Kl(b) (s) iT=> = 1 . 4K2(b)

In the nonrela t iv is t ic l imit , this reduces to

(9)

which is just the �89 per degree of f reedom one would expect in classical thermodynamics .

In this model, then, the decay of a heavy resonance will typica l ly proceed via a long decay chain, as shown in Fig. 1. In nons t range decays (1"), the pions emi t t ed at each stage have a fixed averaged kinet ic energy of order _3 T 2 o, and the recoil of the heavy secondary resonances can be neglected unt i l the

ve r y end of the decay chain. Therefore the average number of pions emi t ted should be s imply propor t ional to the mass of the ini t ial resonance: this was first r emarked by HAGEDORN (co).

This behavior can be checked numerically, using the model described in Sect. 2. The resul ts are shown in Fig. 2, the radius R having been adjusted to

/k 4 / "

' 1'o ' 2b MaSS m/rn~

Fig. 2. - Average number of pions in the final state after nonstrange decay of reso- nances with strangeness zero. Experimental points are from the Particle Data Group tables (la). Box radius R = 1.6 fm. o experimental points (input); �9 points computed via the statistical bootstrap.

(19) That is, decays in which no K-mesons occur in the final state. (20) R. HAGEDOI~N: unpublished lectures CERN 71-12 (1971).

1 7 0 c . J . frAMeR

fit A ~ annihilation at rest (Sect. 4). I t can be seen that a linear relationship between (n=} and the mass of the resonance is established immediately, with

(10) <n=> = 0.6 + 0.30 (re~m=).

I t would be interesting to see whether the multiplicity (n=} does in fact rise in this fashion for p~ annihilation in ]light over the first severe,1 hundred MeV/c above threshold, a region where the main assumptions of the present model should still hold reasonably ~.ell. Unfortunately no experimental data seem to be presently available in this range. At higher momenta the model is expected to break down, as the emergence of forward and backward peaks in the distri- bution of reaction products signals the increasing importance of coherence among partial waves, which is not present in a purely statistical model (~2).

The slope of the line shown in Fig. 2 is about ]1% less than one would have estimated from eq. (8). There are several factors which might contribute to this effect:

i) Most important is the emission of other light particles beside the (such as ~, p, o~, etc.), which may occur at any stage along the deca,y chain. These particles will carry away appleeiable kinetic energies (again, of order ~ To) be]ore disintegrating into pious, and thus will tend to increase the average kinetic energy <T=}.

ii) The decay modes of the low-mass input st,~tes have been put in by hand, and may deviate from the thermodynamic or statistical predictions. But a glance at Fig. 2 shows that the deviation is not large.

iii) The kinetic energy of the heavy secondary resonances relative to the original center of ~nass will tend to increase from one generation to the next; but this is only important towards the end of the chain, and will have a neg- ligible effect on <T~> when the initial mass is large.

iv) The internal temperature of a finite-mass resonance actually tends to be higher than To, as was recently shown by R. CA~LI~Z (Phys. l~ev. D, 5, 3231 (1972)). Again, this effect dies away as the initial mass gets large.

This discrepancy in slope will have important practical consequences. I t means that the (~ effective temperature ~) obtained by fitting a Planck-type distribution

(11) P(p~) ~: p~ exp [-- ~/m 2 + p~/Toff]

to the experimental high-energy momentum spectra will be about 16 % larger than To. In particular, Hagedorn's results for high-energy transverse-mo- mentum distributions (~) show ~ 1 6 0 M e V : this would correspond to To --~ 138 MeV, and a box radius ~ very close to 1.3 fro.

, J ~ r ~ A N N I H I L A T I O N AT REST IN THE STATISTICAL BOOTSTRAP MODEL 171

HAGEDORN himself has noted (~) tha t decays in second ~nd later generations

would tend to raise To, above To, i.e. to broaden the m o m e n t u m spectrum

of emit ted particles, t~ut up till now he has used a mass spectrum ~(m)

cm~exp Ibm] with a - - 25 (rather than - - 3), which implies a logarithmic in-

crease in the mult ipl ic i ty of first-generation decay products as a funct ion of the

mass of the decaying resonance. This makes it very difficult to perform phase-

space calculations, and to compute effects beyond the first generation. He was

thus unable to make ~ny qmmti ta t ive est imate of the difference between T f~ and

To. In the present ~ strong boots t rap ,> version, on the other hand, one ean neg-

lect states with more than three consti tuents (~o.u) emerging from each ver-

tex, and the phase-sp~('e calculations are much more simple.

Finally, let us consider the form of the mult ipl ici ty distr ibution for the

emit ted pions. Now the kinetic-energy distribution of the pions emit ted f,,t

any single ver tex can be deduced from eq. (11); but when one averages over

all the pions emit ted in a long decay chain, then by the central-limit theorem

the dis tr ibut ion in a-cerage kinetic energy will be a G~mssian. Therefore, the

multiplici ty distr ibution will also be Gaussian {presuming it to be fairly n,~rrow).

The numerical results for ~p annihilat ion at rest i l lustrate this behavior (21)

(Fig. 3). The stand~rd derivation of this mult ipl ici ty distribution should va ry

//

/

10

0

/

3 I I

D~

\

\\t 6

Fig. 3. - Calculated multiplicity distribution of pions in nonstrange final states of ]~p annihilations. Also shown is a Gaussian fit (dashed line). Box radius R- -1 .6 fm. - - - - - - calculated multiplicity distribution; - - - - - - f i t ( l /~ /~a )exp [(--n~--~)2/2a 2] with ~=--4.61 and (r 2 0.95.

(21) Since the number of pious emitted is still low, however, deviations from the Gaussian shape occur. For instance, the events with n~-- 6 are not numerous enough. Because of the sources of error mentioned in Sect. 2, we certainly need not expect these details to be faithfully reproduced in the experimental results.

172 c . J . HAMEn

like the square root of (n~:}, i.e. like the square root of the mass, and once again this is borne out by the numerical calculations. The standard deviation quickly settles down to a form

(12) ~.~ - 0.26 V /n /m~ .

We note that the multiplicity distribution comes out much the same whatever the SU3 quantum numbers of the initial state (provided of course that it has strangeness zero).

The multiplicities of other particles (e.g., p-mesons, KK pairs, JV~ pairs) emitted in the decay of a heavy resonance should follow similar laws. HAGEDORN has already shown (~) that the probability of emission of a given type of particle, or particle pair, goes down exponentially with its mass.

4. - Compar i son wi th exper iment for annihi lat ions at rest .

4"1. Pion multiplicities. - The most important point of comparison between theory and experiment is the average mmaber of pious emitted in nonstrange annihilations, which we have denoted (n=}. The lone parameter V in the statistical model must be adjusted to fit the (( experimental ~) value of (n=}, which is 4.6 ~= 0.1 for ~p (although the result is slightly model dependent). In Fig. 4 the dependence of (n=} on V is shown for various versions of the statistical model. For the old Fermi model, the volume required to match experiment is about 6S2o, too large to be physically meaningful. For an (~ inter- mediate ~) model including the effects of vector mesons as well as pseudoscalar constituents, we estimate (93) that once again too large a volume is required. This conclusion differs from that of BA~ASHE~KOV et al. (8), presumably because they assumed SUa symmetry for the masses, while we use the actual experi- mental values.

For the statistical bootstrap model, which takes into account higher-mass resonances and thus favors higher multiplicities, it is found that a volume of ( 1 . 5 i 0 . 4 ) Do, corresponding to a radius (1.14~0.10) ~ , suffices to match (n=}.x,~. This is still a little higher than expected, but we feel that it does fall within the bounds of physical plausibility. The corresponding value of To is (112 =[= 10)~eV, compatible with the value of 138 MeV which we deduced in Sect. 3 from Hagedorn's results.

(22) R. HAGEDORN: Suppl. Nuovo Cimento, 6, 311 (1968). (as) The calculation took only crude account of contributions from channels with 4 or more constituents.

, ~ A N N I H I L A T I O N AT REST IN THE STATISTICAL BOOTSTRAP M O D E L 173

A modification of the model is possible, using a <( covaria.nt >> phase-space expression ins tead of eq. (1), namely

(13) ] /V\ '~ ~ ~ F ~ m~ (5 a

n=2 \ / i=1 J J

votume v/# o 10 0 101

00t5%~ ~Q//// / �9 .... 5 ~" /~ ; ,o0"L~- .~e~ ~//e" /

.i- /- .11 // expetiment

,:o 2:0 radius R/~[~

Fig. 4. - Average pion multiplicity as a function of volume V, for various versions of statistical model.

The use of this type of expression was first advocated by S~IVASTAVX and SO-D~t{S~ (24), and i t has since become popular for its cMculational con- venience, bu t there is no theoret ica l reason to prefer i t (25). The extra factors

(m~/E~) favor low kinet ic energies and thus high final-state multiplicities, and so one finds tha t the volume needed in this version is only (0.4 -4- 0.3) [2o, cor-

responding to a radius (0.75 4- 0.18) ~ (Fig. 4). This would be ve ry satisfac-

tory, bu t for the fact tha t the corresponding spect rum of ou tput resonances is much too sparse to be compatible with exper iment . For this and other reasons

to be discussed below, the original (( noncovar iant ~) version is strongly favored. Unless otherwise s ta ted, we shall refer solely to this version in what follows.

(24) p. p. SRIVASTAVA and G. SUDARSHAN: Phys. Rev., l l0 , 765 (1958). (2~) R. HAGEDOR•: Relativistic Kinematics (New York, 1963), p. 89.

1 7 4 c . J . HAMER

4"2. Charge distributions. - The b r a n c h i n g ra t ios i n to specific charge s tu tes

were deduced f rom the m u l t i p l i c i t y d i s t r i b u t i o n (Fig. 3) u s ing the s t~ t i s t i cu l

mode l for charges (28) of laAIs (27). The resu l t s ure comp,~red wi th e x p e r i m e n t

in Tab les I a n d I I .

The p r ed i c t i ons for the averuge n u m b e r of charged p ions , u n d the churged-

p r o n g m u l t i p l i c i t y d i s t r i b u t i o n , ure accu ra t e to w i t h i n u conple of p e r c e n t for

TABLE I. - Relative branching ratios /or Op annihilation into pions.

Channel Branching ratio o/ (Jo) (*) Experiment (rcf. ( 2 s ) ) l 'rediction (R = 1.6 fro)

0 prong 3.4 4- 0.5 1.4

2 prong 44.7 • 1.2 46.3

~+r:- 0.34 • 6.03 0.15

~+~-~o 8.2 4- 0.9 10.9

v:+w-mr: ~ (m:> 1) 36.2 4- 1.3 35.2

4 prong 48.0 4- 1.1 49.1

2r:+2r:- 6.1 4- 0.3 12.7

2::+2r:-rc ~ 19.6 4- 0.9 28.0

2r:+2T:-mT: ~ ( m ~ 1) 22.3 4- 1.2 8.4

6 prong 4.0 ~: 0.2 3.3

3::+3::- 2.0 :~ 0.2 2.0

3w+3r:-rc~ 1.7 4- 0.3 1.1

3::+3r:-m~ ~ ( m ~ 1) 0.3 4- 0.1 0.2

<n~) 3.05 • 0.04 3.08

(*) As a percentage of all events in which no K-mesons are emitted.

(26) An alternative procedure would have been to include SU 2 Clcbseh-Gordan coef- ficients in the bootstrap equations (eq. (42) of ref. (11)), and keep track of charges there. This is a tedious business; and checks indicated that thc results w,mld haw ~, been quite similar anyway, so that the present procedure should be adcquatc within the accuracy of the calculations. (~7) A. PAIs: Ann. o] Phys., 9, 548 (1960); see also F. C~mmvs: b'uppl. Nuovo Cimento, 15, 402 (1960). (2s) C. BALTAY, t ). FRANZINI, G. Lt~TJENS, J. C. SEVERIENS, D. TYCKO and D. ZANELLO : Phys. Re~'., 145, 1103 (1966).

~ ANNIHILATION AT REST IN THE STATISTICAL ]~OOTSTR, AP MODEL

TABLE II . - Relative branchi~g ratios /or pn annihilation into pions.

175

Channel Branch ing ralio (~

E x p e r i m e n t t ' r cd ic t ion ( /~= 1.6 fro)

1 prong 16.4 ~ 0.5 17.7 ~_ 0.7 17.2

r~-~ ~ <_ 0.7 0.75 -~ 0.15 0.45

3 prong 59.7 ~-1.2 59.1 • 66.8

2r:-r: + 1.57 • 0.21 2.3 ~ 0.3 2.7

2rc-r:Sr: 9 21.8 -- 2.2 13.7 ~ 2.0 44.5

5 p rong 23.4 ~ 0.7 22.8 ~ 0.9 15.5

3r:-2r: + 5.15 -~ 0.47 4.2 • 0.2 5.4

3~-2r~+r: ~ 15.1 ~ 1.0 6.9 • 9.0

7 prong 0.39 • 0.07 0.35 ~ 0.03 0.4

< ~ > 3.15 ~ 0.03 2.98

(*) As ~ percentage of all events in which no K-mesons are emitted.

~p annihilations. For ~n annihilations, the predicted value for (n~} is too low, and the prong distribution reflects this fact. But such a result is to be expected, since the ~n experimental values were actually obtained in deuterium, where approximately 16% of events involve 3-body interactions (a~) which will tend to increase the average pion multiplicity (for instance, ~ annihilations in emulsions gave (n~}~_5.3, compared with (n~} = 4.6 for annihilations in hydrogen). So we attach most importa.nce to the ~p results, a.nd conclude that the prong distributions are reproduced satisfactorily by the model.

When it comes to the branching ratios into specific charge states, however, a few important discrepancies occur. For instance, the model predicts that ~n -+ 2=-~+~ ~ over 40 % of the time, and that this should be the most important single final state: the reason being that n~- -4 and 5 are much the most common multiplicities (Fig. 3), and n~- -4 is favored over % = 5 by a

(29) A. BE'rTINI, M. CRESTI, S. LI)IENTANI, L. PERUZZO, 1{. SANTANGI~LO, S. SARTORI,

L. BERTANZA, A. BI(~I, R. ()APR.aRA, R. CASALI and P. LARICC1A: .u Cimento, 47 A, 642 (1967).

(ao) T. E. KALOGEROPOULO8: Symposium on J%"~ Interactions (Argonne, tll . , 1968), p. 17; ~nd privat.e commml ica t ion of resul ts due ~o P. ttAGERTY a.nd L. GRAY.

176 c .J . I~AMER

factor 3 to 1 due to the G-parity selection rule discussed in the previous Section. These arguments seem hard to evade, and it is therefore disturbing to find that the experimental branching ratio is only 20 % or less. Now in identifying an event as belonging to this channel, various (( cut-off ~ criteria have been applied in the experiment (~9), and one might suspect that a significant fraction of events

were thus thrown away: but the quoted error seems to exclude this possibility. The discrepancy thus remains a puzzle. Similarly in ~p annihilation the model predicts that the 2r:+2~- and 2::+2n-~: ~ channel should be (50--100)% more common than they are found to be experimentally (*). For other channels, the model is in reasonable agreement with experiment.

We note at this stage that very similar distributions are obtained in any version of the statistical model, including Fermi's original one, once the average multiplicity is adjusted to the same value. So the successes and failures above are not peculiar to the statistical bootstrap.

4"3. S t r a n g e - p a r t i c l e p r o d u c t i o n . - Statistical considerations generally predict that K-mesons will be emitted in too large a fraction of the annihilation events (17). This is true once again in the present model, where KK pairs are

predicted to occur in 24% of ~p annihilations and 23~o of ~n annihilations, whereas the experimental figure is around 5 to 7 ~o (3) in both cases.

I t seems, therefore, that there must be some dynamical mechanism which suppresses such events. At least par t of the answer may lie in Zweig's rule (3~), which forbids processes involving disconnected quark diagrams. In the present case, this would forbid the production of o-mesons, for instance; whereas the statistical bootstrap predicts that the o-meson will be produced, and then decay into a KK pair, in 10 ~o of all events (comprising nearly half of all the annihilations with strange particles in the final state). Experimental evidence

for this suppression has been given by the CER17-CdF collaboration (3,): they find that

R(~p --> o?n+n-) ---- 143 • 28

R(pp --> ?~:+7:-)

(3:) R. BIZZARRI, G. CIAPETTI, U. DORE, E. C. FOWLER, P. GUIDONI, I. LAAKSO, F. MAR~ ZANO, G. C. MOI~ETI, D. ZA:NELLO, L. GRAY, P. HAGERTY, T. KALOGEROPOULOS and S. ZE~ONE: Lett. 2~uovo Cimento, 2, 431 (1969); G. C~AP]~TTI, R. BIZZARRI, U. DORE, G. C. GIALANELLA, G. C. ~/[ONETI, P. GUIDONI, P. ANNINOS, L. GRAY, P. HAGGERTY, E. HAI~TH, T. KALOGERorouLos and S. ZENONE: Suppl. Nuovo Cimento, 3, 1208 (1965). (*) Recent experimental evidence (C. GHESQUII~R]~: Aix-en.Provence Con]erence on Elementary Particles (September 1970)) indicates that the discrepancies discussed above are symptomatic of the fact that the multiplicity of neutral pions is about 30% higher than predicted by the statistical model. This effect is conceivably due to Bose statistics, which have been neglected in our treatment. (.2) G. ZWEIG: CERN Reports TH 401 and TH 412 (1964), unpublished. (33) Z. BIZZARRI, M. FOSTER, PH. GAVILLET, G. LABROSSE, L. MONTANET, R. SAL- MERON, P. VILLEMOES, C. GHESQUI~:RE and E. LILLESTOL: Nucl. Phys., 14 B, 169 (1969).

d~Od~ O ANNIt I I I , ATION AT REST IN 'FIIE STATISTICAL BOOTSTRAP MODEL

TABLE I I I . - Relative branehittg ratios /or strange pI1 annihilations.

177

Channel Branching rat io (~0) (~)

Expcr imct t t (b) l ' r ed ic t ion (R = 1.6 fro)

K K 4.4 ~ 0.~ 2

KOK - 4.4 • 0.6 2

K K ~ 25.6 ~ 2. l 29

KOK-T: ~ 1t).6 • 1.2 6

Koko~ - 15.0 ~ 1.6 23

K K 2 = (53) 63

KOK-n+7: - 10.1 • 1.0 7

K~ ~ (5) 3

K~176 ~ (30) 66

K~ 7.3 • 0.8 7

KK3T: (17) 5

KOK T:+~-~ ~ 9.9 ~: 1.I ~ 0

K ~ ~ (2) ~ 0

K~ ~ (2) 2

K~176 2.2 • 0.8 3

K~ ~ 2r:-7: ~ 0.5 :~ 0.3 ~ 0

KK4r : 1

<n=> (! .83) 1.79

(a) As a percen tage of ~ll s t range ~nnihilations. (b) Dat~ ex t r ac t ed f rom ref. U'). The figures in brackets a re no t d i rect measu remen t s , b u t h a v e been crudely es t imated using stat ist ical ~ssumptions such as R ( K K x + ~ - ) = 2 • R(K~K~ -) = ~ R ( K ~ etc.

(34) A. Bn'rT1NI, M. CRES'rI, S. |,IMENTANI. L. P~n u zzo . R. SANTANGELO, S. SARTOR],

L. BERTANZA, A. BIGI, R. (~ARRARA. |{. CASALI, P. LARICC[ A and C. DETRI : ~UOVO Cimeuto, 62 A, 1038 (1969). Wc have not p resen ted similar resull~s for Op annihi la t ion, because the da t a the re are plent i ful but not comple te : m e a s u r e m e n t s of channels involving K + K pairs, in par t icular , are laekin~z.

12 - II Nuovo Cimento A.

178 c. ~. HAMER

whereas a s ta t i s t ica l model would pred ic t these two ra tes to have the same order of magni tude .

In spi te of this large ini t ia l d iscrepancy be tween theory and exper iment as

to the overal l ra tes , i t is of in teres t to make more detai led comparisons for the

relative ra tes in specific KKnw channels. In Table I I I the exper imenta l re-

sults for ~n annihi la t ion (34) are compared wi th rough values obta ined f rom

the model b y the procedure ment ioned in footnote (~6) (n = 2 only). I t can be

seen t h a t the model predic ts a d i spropor t iona te ly large n u m b e r of events in-

volving neutral K K pairs , in line wi th the conclusion t h a t q > ~ K K events

and others l ike t h e m are suppressed. :But the average n u m b e r of pions emi t t ed wi th the K K pairs , and the i r mul t ip l ic i ty distribution~ are correctly reproduced.

Once again these resul ts would occur in other versions of the model besides the

s ta t i s t i ca l boo t s t r ap .

4'4. Nonstrange 2-body annihilation channels. - Over the pas t several years

a g rea t deal of in format ion has been accumula ted on resonance product ion

in 2TJT annihi la t ion a t rest . Comparisons wi th these results can be used to dis-

t inguish be tween various versions of the s ta t is t ical model , which up till now have all g iven r a the r s imilar resul ts once the vohLme V was adjus ted to fit <n=>.

The original Fe rmi version, for ins tance , took no account of resonance product ion a t all. The (~intermediate ~ vers ion (7.s) can account for the product ion of

pseudoscalar and vector mesons, bu t tends to over-es t imate the branching rat ios for channels (especially 2-body channels) involving these part ic les; i t has noth- ing to say abou t channels involving other resonances. The s tat is t ical boot- s t rap p ic ture , on the o ther hand, is an a t t e m p t to t ake all resonances into account

in an average , s ta t i s t ica l way: the branching ra t io into any par t icular channel

can be found b y the prescr ip t ion of Sect. 2 (~s).

I n Table I V the model resul ts are compared wi th exper imen t for var ious

2-body channels. Now in mak ing this comparison we are bese t b y var ious dif-

ficulties:

i) I n an exper iment , the resonance contr ibut ion always has to be separa ted

f rom a background, which is difficult to do unambiguously . Thus~ the two

groups quoted in Table I V which have inves t iga ted ~p annihi la t ion channels

somet imes disagree wi th each o ther b y factors of 3 or more.

(3s) At this point, it may again be asked how we can justify neglecting the effects of angular-momentum conservation. The answer is that angular-menmntum barrier effects are not important in the particular channels discusscd, bccause the kinetic energy is large. Rough estimates indicate that the inclusion of conservation of angular momentum

o, good deal less than the would alter our branching ratios by amounts of order 30/~, a size of the expected ~ statistical fluctuations ~>. I t would also tend to reduce the required interaction radius somewhat (F. CERULUS: Nuovo Cimento, 22, 958(1961)). Our overall conclusions should not be affected.

~ O J ~ ANNIHILATION AT REST IN THE STATISTICAL BOOTSTRAP MODEL

TABLE IV. B;anching ratios /or nonstrange 2-body annihilations.

179

Channel Branching ratio o~ (/o) (*)

Experiment Theory

rcf. (2s,36) CERN-CdF <~ Noncovariant )) <, Covariant ,) (ref. (3)) R = 1.6 fm R = 1.1 fm

PP

n+n - 0.33 J_ 0.03 0.15 0.04

pOn~ 1.5 i 0.2 2.0 ~= 0.3 0.4 0.9

p• T 2.8 ~_ 0.4 4.1 ~: 0.3 1.7 3.3

(~~176 0.8 1.6

B-Er: :F 0.79 ~= 0.26 0.8 =k 0.l 0.9 3.5

fOn~ 0.25 • 0.05 0.9 3.6

A~n :F 2.1 • 0.3 1.1 4.4

~~176 0.23 i 0.18 0.7 J= 0.2 0.2 1.5

popo 0.4 ~ 0.3 0.13 ~ 0.t3 0.4 3.8

(o~ ~ 0.7 :~ 0.3 2.4 ~ 0.2 2.1 23.0

Totals 6.7 4 0.7 5.9 36.0

t2.5 • 0.6 7.9 44.0

]~n rcf. (29) ref. (3o)

nOr: - ~< 0.7 0.75 :t: 0.15 0.45

p~ 0.63 0.05 J: 0.05 0.4

(o~ - 0.41 :L 0.08 0.33 =z 0.04 2.4

q~n- 0.07 ~z 0.004 1.2

f~ 0.94 1.l ~: 0.4 0.7 0 A2n- < 3.3 < 0.5 1.8

(*) As a percentage of tall events in which ilo K-mesons are emi t ted .

ii) A s t a t i s t i c a l mode l c a n n o t p r e t e n d to p r e d i c t t he coup l ing of a n in-

dividual 2-body c h a n n e l to t he J V ~ s y s t e m w i t h a n y c e r t a i n t y (see Sect. 5).

U p o n t a k i n g t he sum over m a n y channe l s , however , (( r a n d o m )) v a r i a t i o n s in

the coupl ings should ave rage out , a n d s t a t i s t i cu l factors should become dom-

i n a n t : o therwise u s t a t i s t i c a l t r e a t m e n t is wor th less .

Because of these facts , i t is no t su rp r i s i ng to f ind t h a t the t h e o r y fai ls to

p red ic t the i n d i v i d u a l 2 -body b r a n c h i n g ra t ios accura te ly . The d iscrepancies

(36) C. BALTAY, ,[. C. SEVERIENS, N . Y E H a n d D . ZANELLO: Phys. Rev. Lett., 18, 93

(1967).

180 c . J . ~ . M ~

show no systematic pattern, though, and when one looks at the totals over all ~p channels the model has more success. The theoretical totals are actually too low by 10 to 40 %, but this is not a very significant amount statistically, since the rates in individual channels are off by factors of up to 4. For the sake of completeness, nevertheless, it is worth noting that a 20 % increase in the theoretical totals could be achieved by lowering the radius/~ by about 0.1 fro, which is within the allowed limits set in Subsect. 4"1.

A similar comparison with the (( covariant ~) version of the statistical boot- strap exposes some severe shortcomings (Table IV). This model clearly gives undue importance to channels involving high-mass particles, and predicts totals over all the measured two-body channels which are far too high. (This complements the previous statement (Subsect. 3"1) that the resonance spectrum was too sparsely populated at high masses in this version: the small m~nbers in the high-mass spectrum are compensated by their undue relative weight in the branching ratios, leading to the same final pion multiplicities as in the (~ noncovariant )~ version.)

5 . - S t a t i s t i c a l f l u c t u a t i o n s .

In any statistical treatment, one must expect the data to show statistical fluctuations or deviations from the theoretically predicted values. These fluctuations are an intrinsic part of the theory, and some understanding of them is necessary in order to judge our results (12).

This problem may be treated using the methods of the statistical theory of compound nuclear reactions (37). The rate at which an ~ pair annihilates into a given final state / can be written

ne

(xa) -> I ) Xr o ,ol%o,1 , c J c = l

where ~ denotes the sum over all channels c contributing to the final state I, o

and ~ denotes the sum over all resonances j~ contributing to the channel e So

at that energy (i.e. within an energy interval of order of the width of a resonance, which we shall take to be about m~). The $'s are coupling constants determined by the chance values of many approximately independent dynamical variables, and are taken to vary randomly.

But since the intermediate resonances j are taken to form a complete set of states in this problem, then corresponding to any partition of the J ~ an- nihilation products into states / (for example, i) first-generation decay products,

(aT) See, e.g., M. A. PRESTO~: Physics o/the Nucleus, Chap. 17 (Reading, Mass., 1962).

~ a ~ A N N I H I L A T I O N AT REST IN T t t E STATISTICAL BOOTSTRAP M O D E L 1 8 1

or ii) ultimate final states consisting of (~ stable ~) particles) we can choose a set of basis states {~i} such that ]~-->/ only, and

h e !

i t ! " c ~ct=l

For heavy resonances, where the number of decay channels is large, the total widths should become uni/orm (aT). Then the expected relative fluctuations in R ( A ~ - ~ / ) are proportional to (~ ncj) -t, while one expects R(X~'-->/)oc

c

oc ~ nc~ itself. Therefore, the expected relative fluctuations decrease as one c

on the square root of R ( J ~ - ~ / ) . This argument ignores interference terms between resonances, variations

in SUe couplings, and so on, as is usually done in the Bohr model. Now the spectrum generated in the numerical analysis, when one takes

account of the various quantum number conservation laws (including J and G), contains of order 60 different resonances which should couple to the A~A ~ system at rest (i.e. within AE = m~ of that energy). The predicted branching ratios A ~ - ~ (2-body final state) are all of order 0.01; so each 2-body final state couples to 0.6 resonances, on the average. Therefore, the expected fiuctuations in these branching ratios should be of a similar order of magnitude to the fluc- tuations in the single-channel partial widths F~v._~x~, as one goes from resonance to resonance in the Particle Data Group tables (14). This checks out quite well: in both cases the fluctuations are of order 100 ~ , corresponding to factors of order 2.

From this starting point, the expected fluctuations in the branching ratios for the various ultimate reaction products can now be estimated, l~or instance, the predicted branching ratio into KKn~ final states is about 0.25, so the ex- pected percentage fluctuation away from this figure is only ~ of the 2-body fluctuations, i.e. around 20%. In fact, the experimental branching ratio is only about 0.07, too small by a factor of 4--this is clearly outside the expected error limits, and indicates the presence of some systematic or dynamical effect of the type already discussed in Sect. 4.

The predicted branching ratios for ~p -> 2-prong and 4-prong (nonstrange) final states approach 0.5 each: so the expected error in these branching ratios would be about 15 ~ , on the basis of the above ideas. In fact, these branching ratios are predicted to within an accuracy of less than 4 ~o. The reason for the greater accuracy is presumably the fact that we have adjusted the volume V to fit <n=>. If we had been given the (( correct ~) value of V a priori, the above arguments suggest that the predicted value of <n.) could have been in error by about 4 %. Turning this reasoning around, it follows that the experimental multiplicity <n.)_~4.6 m~y correspond to quite a large range of values of R, namely /~ ~--(1.14 • if one allows for statistical fluctuations.

182 c . J . nAMES

6. - S u m m a r y and d i s c u s s i o n .

A statistical bootstrap model for ~JV annihilations at rest has been com- pared with experiment. After adjusting the single parameter V to a suitable value, the following results were obtained:

i) The overall pion multiplicity, and the charged-prong frequencies, are well fitted.

ii) The branching ratios into specific multipion charge states are fitted satisfactorily, except for two or three channels which are theoretically expected to be heavily populated, but where the experimental figures turn out too low. The reason for this cannot be ascertained, but we feel that these few discrep- ancies are not enough to destroy the overall agreement.

iii) The model predicts too many annihilations into final states containing KK pairs. The suppression of these channels seems largely to be due to Zweig's famous rule (32), which forbids processes involving disconnected quark graphs: e.g., the production of o-mesons, in the present case. The pion multiplicities associated with strange events are fitted quite well.

iv) The branching ratios into specific 2-body resonance channels are fitted, on the average, about as well as could be expected.

We conclude that the model is generally successful as a phenomenologicM description of JVJV annihilation at rest, except that some account needs to be taken of the dynamical effect which suppresses strange-particle production.

In all the aspects i) through iii), the original Fermi model and its subsequent modifications gave very similar results to those of the statistical bootstrap, except that implausibly large values of the (( interaction radius )~ were required to fit experiment. I t is only when one looks at resonance production (point iv)) that one is able to distinguish between the various models on phenomenological grounds. Here the advantage of the statistical bootstrap is clearly evident, in that it endeavors to include the effects of all the resonances in the spectrum, and treats them all on an equal footing. From this point of view, previous models have been definitely incomplete.

A brief discussion of statistical fluctuations was given. These determine the expected error associated with the statistical predictions, and form an intrinsic part of the theory. Rough estimates of their magnitude were used as a basis for judgement in the experimental comparisons above.

Because of these statistical fluctuations, our fit to the JV~ annihilation results is not expected to be an accurate way of determining the volume V.

~ ANNIHILATION AT REST IN THE STATISTICAL BOOTSTRAP MODEL l ~

The box radius required to fit the exper imenta l pion mult ipl ici ty w~s 1.6 fm, but there is a l ikely error of =L 0.25 fm associated wi th the result. Never- theless, this radius is much more plausible physically th~,n the v a lu e /~ = 2.6 fm required in the old Fermi model.

The value of V is re la ted to the <~ maximum tempera tu re ~> To = b -~ in this model (n), and the present volume corresponds to a t empera tu re of (112-]- :k 20)HeV. For consistency of the model, we require tha t this should agree with Hagedorn 's de terminat ion f rom t ransverse-momentum distributions in

high-energy h,~Aron collisions. Now HAGEDORN uses a t empera tu re of 160 MeV; but this is an <~effective ,> tempera ture , corresponding to a single-pion mo- m e n t u m dis t r ibut ion

(11)

The results of Sect. 3 suggest t ha t the t rue value of To will be about 16 % less than T , , namely To = 138 MeV (the difference arising f rom the secondary decays of emi t t ed ~-mesons, etc.). This is quite compatible with the value we have obta ined above, and corresponds to ,~ box radius ve ry close to 1.3 fm.

Finally, some general fe,%tures of resonance decay have been discussed. The mode] predicts theft when one sums over the whole decay chain:

i) The average number of pions emi t ted will rise l inearly with the mass of the reson'mee.

ii) The mult ipl ic i ty dis tr ibut ion will have a Gaussian form. For non- strange resonances, a fit gives

(lo) (n~} = 0.6 ~ 0.30 (m/m~)

-md

(12} (~,,~ = 0.26 (m/m~) t .

The multiplicit ies of o ther decay products should follow similar laws. Unfor-

tuna te ly these results probably will not apply to resonances of low mass or high angular momentum, i .e. those lying on the leading Regge trajectories .

They might be tes ted, however, in experiments on iTS ' annihi lat ion in flight.

The author would like to t hank Prof. S. FI~AIJTSOHI for his guidance and

encouragement th roughout this work, and for m a n y valuable suggestions. Helpful advice and crit icism was also given by R. HEI~A~N.

1 ~ C . J . HAMER

�9 R I A S S U N T 0 (*)

Si modifica il modcllo s tat is t ico di F e r m i delle eollisioni adroniche di a l ta energia per tenere conto appieno della produzione di r isonanze, facendo uso del modello di boots t rap stat is t ico di Hagedorn e Frautschi . Si fanno confront i con gli esper iment i sulle anni- chilazioni A ~ in quiete, il pi~ favorevole banco di p rova disponibile. Dopo aver regolato il solo pa ramet ro R (il ~ raggio di in terazione ~)), si t rova che le distr ibuzioni delle mol te- pl ici t~ ed i rappor t i di suddivisione sono in generale ben approssimati , salvo che gli event i in cui sono emesse coppie K K seeadono meno di frequen~e di qusn~o p r c d e ~ o , indicando che questi cana]i sono d inamicsmen te soppressi. Si discute la dimensione p rev i s t s delle f lu t tusz ioni s ts t is t iche, come base per giudicsre i r isul ts t i . L s migliore approssimazione si o t t iene con R : (1.6 ~ 0.25)fro, un po' maggiore del previs to , ma molto pi~ ragionevole dei grandiss imi rsggi richiesti dai modell i precedenti . L s tem- p e r s t u r s mass ims T o = (110 • 20 )MeV corr ispondente sl vs lorc sppross imato di R inferiore al vs lore c i t s to di solito nelle sppross imszioni alle distr ibuzioni del l ' impulso t r ssversa le di a l ta energia, m s si accords ancors rag ionevolmente bene con queste distri- buzioni quando si inc luds oppor tunsmen te nell 'anslisi l ' in te rs c s t ens di decadiment i secondari . Si discutono alcune cara t ter i s t iche generali del decadimento di r i sonsnza che seguono dal model lo; in psr t ico lare ls d ipendenza lineare della molteplici t~ dei p rodo t t i di decadimento della masss della r isonanza.

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