Journal ArticleIn the context of the Chew-Snider version of the Amati-Bertocchi-Fubini-Stanghellini-Tonin pion-exchange model, we investigate certain multiperipheral mechanisms that could account for those phenomena that have hitherto motivated the two-fireball model.  These mechanisms are (1) double diffractive  dissociation, (2) the presence of a sequence of neutral particles on the multiperipheral chain, and (3) statistical fluctuations in the log tanθ spectrum. At cosmic-ray energies they are of equal importance

DeTar, Carleton

Snider, Dale R.

English

The University of Utah: J. Willard Marriott Digital Library

VOLUME 25, NUMBER 6 PHYSICAL REVIEW LETTERS 10 AUGUST 1970 TWO-FIREBALL PHENOMENON AND THE MULTIPERIPHERAL MODEL* Carleton E. DeTar and Dale R. Snider Lawrence Radiation Laboratory, University of California, Berkeley, California 94720 (Received 8 June 1970) In the context of the Chew-Snider version of the Amati-Bertocchi-Fubini-Stanghellini-Tonin pion-exchange model, we investigate certain multiperipheral mechanisms that could account for those phenomena that have hitherto motivated the two-fireball model. These mechanisms are (1) double diffractive dissociation, (2) the presence of a sequence of neutral particles on the muitiperipheral chain, and (3) statistical fluctuations in the logtanil spectrum. At cosmic-ray energies they are of equal importance. The two-fireball model was proposed by vari-ous authorsl in the late 1950's to explain a re-markable property of cosmic-ray events observed in photographic emulsions, namely, that the spectrum of secondary particles for some events, when presented in log tang 1 a b, appeared to have large gaps, separating the secondary particles into "forward" and "backward" clusters. Accord-ing to the model, secondary particles were thought to arise from two well-defined centers, one moving in the projectile direction and one in the target direction in the center of mass. In interpreting these data in the context of the multiperipheral model, we shall stress what ap-pears to us to be the best-established empirical feature, namely, the frequent presence of sub-stantial gaps in the log tang 1 a b spectrum. We call this the "fireball effect." The decay distri-bution of the "fireballs," i.e., the forward and backward clusters, is empirically less well un-derstood, and it is here that the multiperipheral model is at variance with the two-fireball model. (In our model the fireball is not a well-defined entity. Its average mass increases without bound as the overall energy increases.) At first sight the fireball effect would seem to contradict predictions of the multiperipheral model. Consider a typical multiperipheral event described by the diagram in Fig. 1. All the two-particle subenergies2 Sij and momentum trans-fers t i are assumed to be small compared with the overall energy. The final particles all have small transverse momenta P J., and their longi-obi z )-L ... 1J ... -< A FIG. 1. Multiperipheral diagram for the process A+B-a+b+···+z. 410 B tudinal momenta in the laboratory frame P II are arranged in an approximately sequential order, more or less uniformly spaced in the variable lnp 11. 3 (Assuming that all particles have the same P J.., the uniform distribution in lnp II trans-forms into a uniform distribution in log tang.) However, when the model permits a broad dis-tribution in the subenergies so that a few can greatly exceed the average, individual events can deviate markedly from the above description. A large two-particle subenergy Sij will produce a "gap" in the P II distribution between Pi and P j, dividing the momenta into two groups, thereby producing a "fireball-type event." In the Amati-Bertocchi-Fubini-Stanghellini-Tonin (ABFST) multiperipheral model,4 the two-particle subener-gy distribution is proportional to the 1T1T elastic cross section which is dominated by resonances (chiefly the p), but which also has a small "high-energy tail" from Pomeranchukon exchange, ex-tending to quite high energies. Thus there is a small probability that an abnormally large suben-ergy (hence a gap) will occur. This is the multi-peripheral mechanism for double diffractive dis-sociation.5 A model employing this effect has been studied for other purposes by Chew and Sni-der6 ; we will use their model as a basis for mak-ing quantitative predictions. If one observes only charged secondaries, a sequence of neutral particles on the multiperiph-eral chain can also produce a "gap" in the mo-mentum distribution. This is the second source of fireball-type events in the multiperipheral model. Finally, for experiments in which only angles are measured and not momenta, there exists yet a third mechanism for producing gaps7 in log tang. Fluctuations in the transverse momentum P J. can produce gaps in log tang, even for a regular dis-tribution in lnPIi. We will calculate the probabili-ty for this effect from the experimental P J. distri-bution. Before calculating the probability for producing VOLUME 25, NUMBER 6 PHYSICAL REVIEW LETTERS 10 AUGUST 1970 "fireball-type events" from these three mecha-nisms, we must define our terms. We offer two definitions of a "two-fireball event." (1) The first relies upon a knowledge of only the direction of the produced secondary, which is usually the only reliable datum in emul-sion experiments. If the particle spectrum for an event in the variable log tan8 1 a b exhibits a gap larger than 1. 3, say, then we define it to be a two-fireball event. (2) For experiments (mainly in the future) that measure the momenta of the produced particle as well, we propose a second, more specific definition of a two-fireball event. If the laboratory momenta of an event can be di-vided into two groups, fast and slow, so that the squared invariant mass of all pairs of particles, one chosen from the fast group and one from the slow group, is greater than some minimum, say 3 GeV2 , then we call the event a two-fireball event with a "gap in momentum space.,,8 We will assume that whenever the subenergy of an adja-cent pair of particles on the chain exceeds such a large value, a gap in the momentum spectrum practically always results in accordance with our two-fireball criterion. Both definitions can be generalized in an obvious way for n-fireball events. To obtain a quantitative prediction for the prob-ability of fireball-type events due to the first mechanism, Pomeranchukon (P) exchange, we shall consider the model of the "schizophrenic" Pomeranchukon. 6 In this model the kernel is ap-proximated as a sum of two components, one with strength g R 4 representing the low-subenergy res-0nances in the 1111 elastic cross section, and the other with strength gp 4 representing the "high-6 0.6 E FIG. 2. Right scale, average multiplicity of Pome-ranchukon exchange per inelastic event, np; left scale, average (produced) pion multiplicity per inelastic event, n versus energy for 71'71' collisions, as predicted by the schizophrenic Pomeranchukon model (see Ref. 6). energy tail" of that cross section. 9 The strength of the high-subenergy part was determined in an-other paper5 where the 111T cross section, appear-ing in the kernel of the ABFST integral equation, was arbitrarily split into the two contributions above the g-meson peak at Sij = 3 GeV2 , hence the choice of 3 GeV2 in our definition of a two-fire-ball event. The probability of single P exchange per event, when it is small, is well approximated by the av-erage number of P exchanges per event. The lat-ter is readily calculated by differentiating the logarithm of the total cross section with respect to loggp 4 just as the average number of pion pairs is found by differentiating with respect to log(g ,/ +gp 4). We are excluding the special class of single diffractive-dissociation and elastic-scattering events from the "two-fireball events." These events are associated with P exchange at the very ends of the chain. This exclusion can be accomplished by not differentiating with respect to the gp2 factor that corresponds to P exchange at the ends. After some algebra, using the pa-rameters of Chew and Snider, we obtain the re-sult plotted in Fig. 2 for the average number np of fireballs per event in a 1T11 collision. Shown also, for comparison, is the average multiplicity n of pions, i.e., the average number of pions pro-duced per inelastic collision. It is possible to ex-tend this result to 1TP and pp collisions by means of a crude scaling law. The plot of np vs n in Fig. 3 is independent of this scale, however, and uni-versal to all reactions in this model. For n~ 7, the curve is fitted roughly by the expression np = O.10(n-6). (1 ) 0.7 r--,--,---,---,---,---.--. 0.6 0.5 0.4 np 0.3 0.2 0.1 n FIG. 3. Average multiplicity of Pomeranchukon ex-change per inelastic event versus average (produced) pion multiplicity per inelastic event. 411 VOLUME 25, NUMBER 6 PHYSICAL REVIEW LETTERS 10 AUGUST 1970 To estimate the frequency of neutral-particle gaps, we first find the average number of sequen-tial neutrals required to produce an apparent "gap" with a 3-Gey2 "subenergy." To do this, we shall use an approximate expression derived by Chew and Pignotti, 10 S/So"" II (sij/c). (2) all energie s From the rate at which the average multiplicity of secondaries grows with energy, it is possible to estimate the average value of the ratio S ij / c. Recent preliminary experimental results for sec-ondary multiplicities in pp collisions are fitted with the expression for a = 1.13. 11 Since from Eq. (2) In(s / so) "" n In(~ij/c), we estimate that In(sjc) "" 1/a = O. 9. (3) ( 4) (5 ) Comparison of the Chan-Loskiewicz-Allison mul-tiperipheral model12 with exp~rimental data indi-cates that a typical value of the average subener-gy is 0.5 Gey2.13 With this value for sij, we ob-tain C "" 0.2 Gey2. Corresponding to Eq. (2), there is an expres-sion relating the subenergy of two nonadjacent particles i and I to the intervening adjacent-parti-cle subenergies: (6) Thus when the number of intervening particles is two, the accumulated subenergy S j I is already 3 GeV2 on the average. Hence only two successive neutral particles are required to produce the ap-pearance of a large subenergy between adjacent particles. If it is further assumed that the prob-ability of a neutral particle is one-third per par-ticle and uncorrelated between adjacent particles, then we estimate the average multiplicity of neu-tral gaps to be (t)2 times the average multiplici-ty, Le., nIl =0.1(n-3). (7) (At least four produced particles are required to make two fireballs, each containing at least two particles, in addition to two neutral particles; hence the 3.) These two mechanisms, P exchange and neu-tral gaps, constitute our model for producing fireball events, according to our second defini-tion in terms of gaps in momentum space. Their 412 combined probability is np +n]r = 0.2n-0. 9. (8) When only the angles of the secondaries are measured, one must resort to our first definition in terms of gaps in logtan8. We show below that for particles with average transverse momenta, the two definitions are equivalent. We only need add to the above probability the probability that an average subenergy and abnormal transverse momenta produce a gap in logtan8. The average spacing in log tanB, L e., the aver-age value of K =log[(P /I /p J.j)(p J./p /I j)] is easily estimated from the expression for the average multiplicity. The total length of the logtan8 plot is logs + const. Therefore with P J.j "" Ii J.j, nit"" logs + const. Hence14 it"" log(p II /p II j) "" (loge )/a = 0.38. (9) To estimate the spacing when 8 jj > 3 Gey2, we can either use the rule of thumb, derived in the discussion of neutral-particle gaps, that three average gaps equal a P gap, or we can calculate the spacing directly, using the expression for S jj in terms of the momenta, when Pllj»Plli»PJ.i,PJ.j, Sij""(Plidp lIi)W/, where Wj = (pJ. j 2+ m7/)II2. Then for p.1.?m", K "" log(si/WjW j). (10) (11) If we use a typical experimental value P l = 0.15, then when Sjj > 3 Gey2, K?1. 3. For an average subenergy S ij "" O. 5, the gap ex-ceeds 1.3 only whenw jw j 50.025 [cf. Eq. (11)]. Assuming that the distribution in P .1.2 is approxi-mately uncorrelated with 8 ij and given by dN/dP.1.2 = exp(-pJ.2/0. 15), (12) the probability of such an occurrence is about 5 % per particle. However, this result is highly sen-sitive to the subenergy in this range. For Sij = 0.8 (a value less likely than 0.5 by a factor of about 2 13) the probability is 20 %. In order of magnitude the average number of gaps from this source is nIII "" 0.1 (n-1). (13) The net probability for gaps in logtan8 is then roughly 0.3n-1.0 for n?: 5. Although the multiperipheral model predicts that for individual events there may be gaps in the distributions, the combined distribution of many events will not have dips. The gaps occur with equal frequency anywhere along the chain. VOLUME 25, NUMBER 6 PHYSICAL REVIEW LETTERS 10 AUGUST 1970 We are encouraged by the observation reported by Dobrotin and Slavatinsky15 that in cosmic ray events at energies in the range 100-1000 GeV a marked asymmetry is observed in the particle distribution in the center of mass for some events. Moreover, the multiperipheral model tends to agree with their observations that some of the secondaries in the "decay" of the fireball have abnormally large energies in the fireball center of masso These would correspond to the left- and right-most particles in the fireball group on the multiperipheral chain. Furthermore, our results are not necessarily in disagreement with the detailed statistical anal-ysis of Gierula, Mifi!sowicz, and Zielinski and of Gierula and Wojner16 for bimodality in the logtane spectrum from emulsion experiments. A posi-tive D test for bimodality is not a positive test for two separate peaks. A trapezoidal distribu-tion gives a positive D test for bimodality. Because of the attractiveness of the two-fire-ball idea, many experimenters have hitherto fo-cused their attention on the structure of the fire-ball clusters and on the separation of the "cen-ters" of the clusters. From the standpoint of the multiperipheral model, however, a frequency distribution of gap sizes would be a useful analyt-ical tool. *Work supported in part by the U. S. Atomic Energy Commission. lp. Ciok, T. Coghen, J. Gierula, R. Holyiiski, A. Jur-ak, M. Miysowicz, T. Sanieski, and J. Pernegr, Nuovo Cimento 8, 166 (1958), and 10, 741 (1958); G. Coc-coni, PhYs. Rev. 111, 1699 (1958); K. Niu, Nuovo Ci-mento 10, 994 (1958). 2The two-particle subenergy is the invariant 8ij = (P i +PJ)2. 3For a discussion of this point, see C. E. DeTar, to be published. 4L. Bertocchi, S. Fubini, and M. Tonin, Nuovo Ci-mento 25, 626 (1962); D. Amati, A. Stanghellini, and S. Fubini, Nuovo Cimento 26, 896 (1962). 5G. F. Chew, T. Rogers,~d D. R. Snider, Univ. of Calif. Lawrence Radiation Laboratory Report No. UCRL-19457, 1970 (unpublished). 6G. F. Chew and D. R. Snider, Univ. of Calif. Law-rence Radiation Laboratory Report No. UCRL-19455, 1970 (unpublished). 1Czyzewski and Krzywicki have shown that an ideal-ized multiperipheral model which treats final particles independently and gives a rectangUlar distribution in logtanll can, on the average, through statistical fluctua-tions account for the experimental data. [See O. Czy-zewski and A. Krzywicki, Nuovo Cimento 30, 603 (1963).] We are concerned here with two-particle cor-relations. 8The numbers 1.3 in logtan(i and 3 GeV2 in subenergy were chosen so as to be roughly compatible when Ii J.. "='0.4 GeV/c. 9The multiperipheral model of the P. N. Lebedev In-stitute also allows for the production of fireballs. In their model, however, pion rather than P exchange divides the fireballs. Thus the gaps would presumably be quite small. [See V. N. Akimov, D. S. Chernavskil, 1. M. Dremin, and 1. 1. Roysen, Nucl. Phys. B14, 285 (1969).] -lOG. F. Chew and A. Pignotti, Phys. Rev. 176, 2112 (1968). -l1L. W. Jones et al., to be published. We thank D. Ly-on for supplying us with these data before publication. 12H._M. Chan, J. Loskiewicz, and W. W. M. Alli-son, Nuovo Cimento 57A, 93 (1968). 13Z . Ajduk, L. Michejda, and W. Wojcik, Acta Phys. Pol. A37, 285 (1970). 14In ~paring this with emulsion data, it must be kept in mind that multiplicities tend to be higher in collisions with nuclei. 15N. A. Dobrotin and S. A. Slavatinsky, in Proceedings a/the Tenth International Conference on Cosmic Rays, edited by J. R. Prescott et al. (Univ. of Calgary, Cal-gary, Alta., Canada, 1967), pt. A, p. 416. 16J. Gierula, M. Miysowicz, and P. Zieliiiski, Nuovo Cimento 18, 102 (1960); J. Gierula and E. Wojner, In-stitute for Nuclear Research, Warsaw, Report No. 1126/Vn/PH (to be published). 413 

Two-fireball phenomenon and the multiperipheral model

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Two-fireball phenomenon and the multiperipheral model

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