Particle ratios in energetic hadron collisions

9 - U N I V E R S A L S L O P E A N D T H E N O N L I N E A R R E G G E T R A J E C T O R I E S 1449

to r ies . This is due t o the multipole s t ruc ture of However, inclusion of cu t s i n the CHKZ r e p r e - the daughter t ra jec tor ies in the CHKZ represen ta - sentation does not change o u r resu l t s . This is tion. In this case, the condition Eq. (27) is not t r u e fo r the r a t h e r general c a s e where cuts a r e sat isf ied. constructed f r o m moving poles.

*Address after September 15: California Institute of Technology, Pasadena, California 91109.

'c. Schmid, Phys. Rev. Lett. g : ( 1 9 6 8 ) ; V. A . Alessan- drini, D. Amati, and E. J. Squires, Phys. Lett. z, 463 (1968).

'G. Veneziano, Nuovo Cimento c, 190 (1968). 3 ~ . A. Virasoro, Phys. Rev. 177, 2309 (1969). 4 ~ . Cohen-Tannoudji, F. Henyey, G. L. Kane, and

W. Zakrzewski, PhyS. Rev. Lett. 26, 112 (1971). 'L. Gonzales Mestres, Nuovo Cimento Lett. 2, 251

(19 71). 6 ~ . Mandelstam, Phys. Rev. 166, 1539 (1967). 'G. C. Joshi and A. Pagnamenta, Phys. Rev. 182, 1574

(1969). he constant k , , , is defined by the following equation:

where C l , , and d l , , are given by

P J ( Y ) = ~ C1, ,yr and

'G. C. Joshi and J. W. G. Wignall, Nuovo Cimento E, 483 (1973).

PHYSICAL REVIEW D VOLUME 9 . NUMBER 5 1 MARCH 1 9 7 4

Particle ratios in energetic hadron collisions*

J a m e s D. Bjorken Stanford Linear Accelerator Center, Stanford Universi ty , Stanford, California 94305

Glennys R. F a r r a r Culifornia Institute of Technology, Pasadena, Calqornia 91109

(Received 27 August 1973)

We construct a simple statistical model in order to estimate, for very energetic collisions, the ratios of particles with various quantum numbers produced with center-of-mass momenta less than a few GeV. Two conclusions are that (1) isobar decay can account for a large fraction of the SU(3) violation observed experimentally, and (2) at high transverse momentum the dominance of pions diminishes.

In this paper we construct an extremely s imple s tat is t ical picture of hadron production in the cen- t r a l region of rapidity. I t i l lus t ra tes how severa l fea tures of the produced-particle spec t rum in hadron collisions can be understood without resor t ing to sophisticated dynamical models. While we a r e not s u r e whether this is a model which can be used f o r se r ious quantitative study, we do believe it contains qualitative features of some generality.

Subsequent to carrying out this work, we realized that Anisovich and Shekhterl have developed a very s imi la r picture. We recommend that the interested

r e a d e r compare that work with this. Not s u r p r i s - ingly our vers ions differ in various ways: The ini- t ia l assumptions differ somewhat, and comparison of our resu l t s with the i r s provides an indication of the sensitivity of the bas ic idea to detai ls . Also, Anisovich and Shekhter considered not only the central rapidity region (for low-pL secondaries) but the t a rge t and projectile fragmentation regions as well. On the other hand, we have applied the idea to the production of par t i c les of high and low pL in the central rapidity region.

The model is very simple. Imagine that, as a re-

1450 J A M E S D . B J O R K E N A N D G L E N N Y S R . F A R R A R 9 -

suit of the high-energy collision, the interaction region is a chaotic r e s e r v o i r of partons: quarks and antiquarks. A quark leaving the zone of inter - action mus t pick up a t l eas t one neighbor parton f r o m the r e s e r v o i r in o r d e r to make a hadron. Assuming there a r e equal numbers of quarks and antiquarks of each variety in the rese rvo i r , the re is a 50% chance that the q g r a b s a q. We assume that a t th i s point the q q sys tem is satisfied and d e - p a r t s f o r infinity a s a meson M. In the other half of the c a s e s the qq sys tem mus t g rab again, and there is again a 50% chance of satisfaction, now a s a qqq baryon s ta te B. Proceeding thus, we have the following configurations:

Configuration M o r B Probability

of this rat io , probably tending to enhance i t , but perhaps this is a good f i r s t approximation. F o r instance, Anisovich and Shekhter used the rat io M / B = 6 because they considered the cor.figura- tion qq@ to be two mesons and q q a q to be a meson and baryon, etc.) Continuing our policy of making the s imples t possible assumption to s e e how f a r i t ge t s us , we assume that the M and B a r e produced i n the SU(6) 35 and 56 representat ions, according to s tat is t ical weights. Then computation of the part ic le ra t ios in the central rapidity plateau (integrating over p,) is reduced to bookkeeping. The ledger is presented in Table I. Combining the totals of Table I ( a ) (weighted by $) with the totals of Table I(c) (weighted by 6) gives the following numbers of par t ic les per emitted hadron:

Par t i c le n t no K+ KL 77 n

Number emitted 0.5 0.53 0.11 0.09 0.02 0.09 0.07

The substantial deviation f r o m exact SU(3) sym- Evidently the ra t io M / B equals 2. (Dynamics of a met ry comes about because much of the s tat is t ical m o r e sophisticated model could modify the value weight l i e s i n p , K*, A, and Y*, a l l of which pro-

TABLE I. Particle spectra integrated overp,: (a) The number of a* = r ' = r O , K+=K-, KL, and q per emitted 35 meson, allowing for resonance and Ks decay. (b) The number of a+ =T' =TO, p , n , A O , 2+= C-, E O = ~ - , and a-per emitted 2, averaged over baryons and antibaryons (this table includes the effects of resonance decays but not of weak decays, and therefore we call it a "bubble-chamber" ledger). (c) The number of a' =a+ , T O , p , and n per emitted 56, averaged over baryons and antibaryons, after all decays have taken place (we call it ailspec- trometer" ledger).

(a)

Parent Stat. wt. a + / M K - / M KL /M 7)/M

a , P , w 15 /35 0.65 0 0 0 K,K* 16 /35 0.42 0.25 0.25 0

$' 3 / 3 5 0.39 0.50 0.33 0 7) 1 / 3 5 0 0 0 1 .O

Weighted total 0.50 0.16 0.14 0.03

(b) Parent Stat.wt. a + / B p / B n / B AO/B Z + / B E O / B a-/B

N , A 20/56 0.27 0.25 0.25 0 0 0 0 A , z, Y* - - 20 /56 0.20 0 0 0.38 0.06 0 0

0 , 3 * 12 /56 0.22 0 0 0 0 0.25 0 n - 4 /56 0 o o 0 0 0 0.50 Weighted total 0.21 0.09 0.09 0.14 0.02 0.05 0.04

(c) Parent Stat. wt. T- / B r 0 / B P /B n /B

- -

N ,A 20/56 0.27 0.27 0.25 0.25 A , Z, Y* 20 /56 0.52 0.50 0.27 0.20

w c , E * 12 /56 0.81 1.06 0 .33 0.17 52 - 4 / 5 6 0.83 1.33 0.33 0.17

Weighted total 0.52 0.60 0.28 0.21

9 - P A R T I C L E R A T I O S I N E N E R G E T I C H A D R O N C O L L I S I O N S 1451

duce pions in their decay. We see that after the simply to isobar decay.3 resonance and weak decays have taken place the A similar exercise can be carried out for in- rat ios a r e K-/T-= 2% and F/K-= 890/0, not des- clusive distributions a t high p,. [In order to stay perately f a r from the experimental values of b the central rapidity plateau we must take -7% and - 1 0 6 , respectively.' Thus we draw our 2p,/&<<l, where n+/n--1, p/$- 1, etc.] This f i r s t conclusion: A large part of the observed time we will take account of the experimental fact1 SU(3) breaking in the central region may be due that the invariant c ross section falls rapidly with

TABLE 11. Particle spectra at highp,, but in the central rapidity region (2pL/&<<l), re- sulting from 35 and 56 production with an invariant cross section falling as p,-". Results a re given for n = 6,8,lO. We keep only contributions of 5% or more and therefore ignore three-body decays. (a) Relative numbers of r -=n+=no, K-=Kt , KL, and q at a given p , , per emitted 35 meson, allowing for resonance andKs decay. (b) "Bubble-chamber" ledger for baryon parents: relative numbers of n-=s+ = n o , p = n , Z+ = 72-, A', E0=Z-, and CL- at a given p , , per emitted 56, averaged over parent baryons and antibaryons . (c) "Spectrom- eter" ledger for baryon parents: relative numbers of n- = r + = T O , p and n at a g i v e n p ~ , per emitted 56, averaged over parent baryons and antibaryons.

(a) Parent Stat. wt. n n+/M K+/M KL/M 17/M

Weighted total 6 8

1 0

Parent Stat. wt. n r-,!I3

Weighted total 6 8

10

J A M E S D . B J O R K E N AND G L E N N Y S R . F A R R A R

TABLE I1 (continued)

(c )

Parent Stat. wt. n a-/B P /I3 n /B

N , A 20/56 6 0 0.14 0.14 8 0 0.12 0.12

10 0 0.10 0.10

Weighted total

increasing p,, apparently a s P;". The exponent may o r may not be the same for M and B, but we continue to assume that the 5 and 56 a r e produced according to statistical weights. The bookkeeping for decaying resonances i s a bit more tedious because of the steeply falling spectrum. If the probability of finding a child4 C of momentum xP,, given a parent of momentum P,, i s scale-invariant (a good approximation if P,* m, ), i.e.,

then a simple calculation shows that the rat io of children to parents a t a given p, i s

child ==a ""n-2&TP(.),

'independent of p,. In the two-body case gc (x) i s very simple: zero for x < x: and x > x: , and constant of value l/(x: - x t ) for x: < x < x: . If the parent mass i s Mand the children have masses m, and m,, then if m,= m , = m , x : = x ~ = ~ ~ / M ~ and x : = X U , = 1 - m 2 / M 2 . Another interesting case is mI2 <<rnZ2 , when x i =O,x:=l-m,2/M2and xh = r n Z 2 / ~ , , x:= 1. For three-body decays g, (x) i s generally quite small a t high x. Inasmuch a s n i s empirically large ( 2 8 ) , the high-p, region i s dominated by two-body decays in which the observed child i s emitted forward in the parent center-of-mass frame.

The high-p, ledger is given in Table I1 for n = 6, 8, and 10. We see that high-p, pions f romadecay , p decay, etc. a r e strongly suppressed. Conse- quently the fraction of kaons and baryons in the

charged-particle spectrum increases from low p, to high p,, and then stabilizes. For example, continuing to take M/B = 2 and assuming n = 8 for baryons and mesons, we find a t high p, that K - / n - -95%, q/nO " 57%, and $/K-(spectrometer) "6%, while if strange baryons a r e observed Too" 17% and ~1-/F-88%. It i s important to remember, when comparing these results with data, that they a r e only applicable in the central rapidity plateau. At CERN ISR energies a plateau i s not yet developed for the p ' s , so the asymptotic region has not quite been reached experimentally. Nonetheless the general trend, that a t high p, there a r e rela- tively more kaons and baryons, i s in line with the data.

Evidently by leaving the ratio M/B and/or the rat io of strange to nonstrange quarks in the res- ervoir arbi trary we could considerably improve the agreement with experiment, if that were our purpose. Instead, we choose to abstract thefollow- ing qualitative conclusions:

(1) In order to understand particle rat ios i t may be useful to take isobar production into considera- tion.

(2) Most of the SU (3) breaking in the central plateau can be accounted for by the known SU (3) violation in the decay of resonances.

(3) The spectrum at high p, i s r icher in kaons and baryons than that a t low p, , because a decay - ing isobar gives most of i t s momentum to the heavier child.

(4) There i s a parent-child relation a t high p,: If the parent distribution falls a s a power, P,'", and the decay distribution of children relative to

9 - P A R T I C L E R A T I O S IN E N E R G E T I C H A D R O N C O L L I S I O N S 1453

the parent obeys Feynman scaling a s in Eq. ( I ) , can be regarded a s produced from a single parent. then the distribution of children falls with the same While we have focused on hadron-hadron colli- power n. One implication of this parent-child r e - sions above, the same idea may have applicability lation i s that if all high-P, hadrons a r e progeny to hadron production in high-energy e'e- annihila- (via scale-invariant cascade processes) of the tion and in the plateau regions .in deep-inelastic same parent (e.g., quark parton o r gluon), then electroproduction and neutrino interactions. - they will all have the same power-law dependence

We thank David Horn for an interesting discussion. on PI. Hence measurement of the exponent n for various kinds of hadrons may tes t whether they

*Work supported by the U. S. Atomic Energy Commis- 1973 (unpublished). sion. 3 ~ a n y people have undoubtedly noticed this. We know

'v. Anisovich and V. Shekhter, Nucl. Phy s . z, 455 D. Horn and S. Nussinov have reached similar conclu- (1973). We a r e very grateful to the authors for bring- sions (private communication). ing their work to our attention. 4 ~ h e perceptive reader will guess that this word has

'A recent review with up-to-date references to the data been chosen with care. i s that of M. Jacob, CERN Report No. CERN-TH-1683,

P H Y S I C A L R E V I E W D V O L U M E 9 , N U M B E R 5 1 M A R C H 1 9 7 4

Hadron distributions in the reactions hadron + hadron -+ lepton pair + hadrons and two virtual photons+ hadrons, and tests of the parton model*

Min-Shih Chen Stanford Linear Accelerator Center, Stanford, California 94305

(Received 10 October 1973)

The parton-model description of final-state hadron distributions can be characterized in t e rms of a few fragmentation regions and plateaus. We discuss such distributions in the reactions hadron +hadron- lepton pair +hadrons and virtual photon +virtual photon- hadrons; we also compare different distributions a s tests of the parton model, and discuss kinematical conditions for the validity of the asymptotic descriptions of the distributions and the tests. From this discussion, we conclude that most of these asymptotic descriptions may be valid only at extremely high energies and a good nonasymptotic model is probably necded to describe these final-state hadron distributions at presently available energies.

The parton-model description of final-state hadron distributions has been discussed for various processes, namely deep-inelastic electroproduction,

lepton +hadron- lepton +hadrons , (1)

high-energy electron-positron a ~ i h i l a t i o n , ' - ~

electron +positron- hadrons , (2)

and deep hadron-hadron ~ c a t t e r i n g , ~

hadron +hadron- large-transverse-momentum

hadron +other hadrons (3)

in certain kinematical regions. Such distributions always involve some fragmentation regions and

plateaus,'-6 a s illustrated in Figs. l(a)-l(d), with the heights of the plateaus, a s well a s the shapes of the fragmentation regions, for different reactions related to one another. It i s interesting to see if such descriptions also can be naturally ex- tended to other reactions in t e rms of the existing fragmentation regions and plateaus. Furthermore, we indeed need to study some other reactions in order to provide more tes t s of the parton model. Except a t extremely high energies, there i s not enough phase space for the full s tructure of al l these regions, and therefore various regions may overlap. Thus it i s difficult to use the full s tructure of the hadron distribution a s a test of the model. On the other hand, the overlapping regions can st i l l be related to one another, and therefore

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Particle ratios in energetic hadron collisions