P H Y S I C A L R E V I E W V O L U M E 1 8 6 , N U M B E R 5 25 O C T O B E R 1 9 6 9

Scattering and Production of Excited Hadrons"^ RICHARD C. ARNOLD

High Energy Physics Division^ Argonne National Laboratory^ Argonne, Illinois 60439

AND

P. G. O. FREUND

Enrico Fermi Institute and Department of Physics^ University of Chicago, Chicago^ Illinois 60637 (Received 23 May 1969)

Models for heavy meson and baryon states contain consequences for their high-energy scattering on other particles. Features of production reactions are discussed which depend on these scattering properties, and specific proposals are made for experiments.

TH E most characteristic parameter for a high-energy hadronic collision is the slowly varying

value of the total cross section.^ Thus, pp cross sections are larger than wp cross sections roughly by a factor of f. The ratio is easily understood in an additive quark modeP or through universality considerations.^"^ Over the past few years, numerous excited hadrons have been discovered. The more general question is: What are the values of the high-energy total cross sections (THW of any two hadrons H and H''^

We shall first show that all available hadron classi­fication schemes and universality principles fall into two categories, as far as their predictions for (THH' are concerned. Schemes of the first category predict

CTMB^ and crBB^(^p (1)

for any (ground-state or excited) mesons (M) and baryons (B), In other words, the asymptotic total cross section in these schemes is independent of the degree of excitation of the incoming hadrons. Schemes of the second category, on the contrary, predict asymp­totic total cross sections increasing with the degree of excitation of the incoming hadrons. The higher the mass range (or multiplet) in which one finds a hadron, the larger a value of crtot will characterize its high-energy collisions with a given target in these schemes. Since anp is measurable through scattering in heavy nuclei from production-cross-section values and in angular-dis­tribution effects in production on lighter targets, we believe that meaningful new tests of hadron classifica­tion schemes are thus available.

We now present typical models for each category:

i . Additive quark model with orbital and radial excitations. In this model, all mesons (baryons) are

* Work performed under the auspices of the U. S. Atomic Energy Commission.

^Here and in the following we have in mind essentially the Pomeranchuk contribution to o-tot- At presently available energies it already accounts for the bulk (80%) of o-tot. I t is constant if Q;P(0)=:1 or very slowly decreasing if Q : P ( 0 ) < 1 . The remaining 20% of the cross section is accounted for by lower Regge poles, and cuts.

2 H. J. Lipkin and F. Scheck, Phys. Rev. Letters 16, 71 (1966). 3 P. G. O. Freund, Phys. Rev. Letters 16, 291 (1966). 4 P. G. O. Freund, Nuovo Cimento 46A, 563 (1966). 5 N. Cabibbo, L. Horwitz, and Y. Ne'eman, Phys. Letters 22,

3^6 (1966).

186

Q.Q. (QQQ) states characterized by their spin, imitary spin, orbital (L) and radial (n) quantum numbers. The hadron-hadron scattering amplitude is assumed^ to be in the form of a linear superposition of the individual qq, qq, and qq amplitudes (quark counting). Equations (1) will both hold, so that this is a scheme of the first category.

2. Pomeranchuk singularity that couples to the mass at L=0, All hadrons appear to lie on straight-line Regge trajectories. We can classify these hadrons in U{6)q)iU{6)qXO{i)L representations (dq,dq;L)j where dq, dqy and 2L+1 are the dimensionalities of the U(6)q, U(6)q, and 0(3)L representations. The Regge trajectory of such a multiplet can be conveniently ex­pressed as md^di{L)-md^d-i{0)-\-{a')~^L. Two models have been constructed^-^ in which the Pomeranchuk singularity couples to md^d-Jfi), If higher hadrons carry orbital but no qq excitations, then these universality schemes are of the first category.

J. Additive quark model with ^5i unitary-singlet pair excitations. Assume all higher hadrons to be constructed from the ground states by repeated addition of ^Si unitary-singlet qq pairs (coi's). The hadron spectrum in such a model would be identical to that of the orbital-radial excitation model but the total cross sections will not obey (1). Denote by a meson built of gg(coi)^, and by Bn a baryon built of qqq{oii)^. Then

and CTBmBn- ( l + f w ) ( 1 + f ^)o-pp . (2)

This scheme is of the second category. 4. Additive quark model with general qq pair excita­

tions, In such a model, in addition to the orbitally and radially excited hadrons discussed in model (1), there are higher mesons (baryons) that have the structure {qqj'^ijt'^^qj')' Higher unitary-spin multiplets are excited along with higher spins. Present hadron spectroscopic evidence tends to point against such a picture. I t is, nevertheless, clear that this is a model of the second category as far as the qq excited hadrons are concerned.

5. Pomeranchuk singularity that couples to the mass. A universality property of the Pomeranchon has been suggested^ in the form that its couplings ynn^ to 1631

1632 R . C . A R N O L D A N D P . G . 0 . F R E U N D 186

hadrons {H) be proportional to the central mass of the multiplets to which the hadrons belong. Thus, yTTTT^^ypp^, since the w and p, in spite of their large mass difference, belong in the same SU(6) multiplet; and ypp^/jTCT^-h since Wp^^^^^^Vw^r^ ''*' ^-!. In this scheme, excited hadrons belonging to multiplets with higher masses will have larger couplings to the Pomer-anchon and, therefore, give rise to larger cross sections. If the central masses squared of hadron multiplets are equally spaced (as would be the case if they lie on linear Regge trajectories), then this gives rise to

be described by a Regge-pole amplitude, parametrized as an exponential in t:

(TBB^ {fnE/ntBoYo-pp, (3)

where 7nM, w^, f^Mo, and MBO are, respectively, the central masses of the (5t / (6)XO(3)) multiplets to which M, B, TT, and p belong. This is also a scheme of the second category.

Note that in all these schemes the total cross sections for scattering of excited hadrons is equal to or greater than the corresponding ground-state scattering cross section, never less than it.

Before we discuss the experimental aspects of these alternatives, let us make an observation: Hadrons are found to lie on linearly rising Regge trajectories. Thus a particle with mass around 1 BeV may be dynamically related to a particle of mass around, say, 1000 GeV. What is it that confers on the latter such a large mass? Ever since Lorentz it has been believed that the mass of a particle (at least its order of magnitude) is deter­mined by the strongest interaction in which the par­ticle participates. According to this point of view these very heavy particles either exhibit ' 'superstrong" interactions that involve them only, or interact in a "normal" strong way with very many particles so that their mass is a cumulative effect of all these otherwise normal strong interactions. Whichever be the case in reality, it may not be surprising if the (superstrong or cumulative) effect of interaction would also manage to enhance some other observable, say ctot, besides the mass of these heavy particles.

Now consider production reactions for heavy hadrons. In models of the first category, the production and rescattering of heavy states will be similar to that of TT, p, etc., which form important noise contributions in any experiment. We thus expect similar momentum-transfer dependences for signal and noise in production experiments. In models of the second category, however, even if the primary production mechanisms are similar, the rescattering effects are much larger, and we expect considerable difference between the production char­acteristics of heavy particles compared to TT, p, etc.

In what follows, we will estimate the important qualitative properties of production at very high energies both on nucleons and nuclei of heavy hadrons. We assume the primary production mechanism can

G^oUt)=Xe'^^'l\ (4)

where X and RQ may be smoothly varying with s. To estimate rescattering effects, we presume that the high-energy elastic hadron-scattering processes have no important spin dependence; we suppress the helicity dependence of X throughout. A simplified multiple-scattering theory will be used, equivalent to applying absorptive corrections to the primary production mechanism (pole).^-^ First we must describe the elastic scattering amplitude of the incoming (produced) hadron from the other body in the initial (final) state. Suppose the primary mechanism for such scattering can also be considered as an amplitude,^ exponential in t,

Fpj(t)=iyjR/e^^^^^^ (j=l, incoming; 2, outgoing), (5)

with positive 7y. The elastic amplitudes are generated from this by multiple scattering.^ In particular, the heavy-hadron elastic scattering amplitude F2 is

where

F,{t)=i M6/o[H-0'' '][l-e^*' '^<' ' ' '], (6)

252(6') =n2exp( - iV2i?22) . (7)

In models of category 2, 72 should increase with mean multiplet mass.

For 72<1 , such as in Tp scattering where 7^^0.7, this amplitude is approximately an exponential in t for small t. For 7 2 ^ 1 , such as expected for heavy hadrons in models of category 2 (and/or when nuclei are present), this amplitude resembles that from black-disk diffraction with radius approximately R2.

The total cross section will be approximately pro­portional to R2^ when 7 2 ^ 1 , or to y2R2^ when 7 2 < 1 . In order that cross sections be large, it is necessary that R2 be large, compared to a ground-state scattering process. For example, in model 5 we expect 72i 2^ to be proportional to the mass of the heavy-hadron multiplet, since Pomeranchon coupling there is proportional to the mass.

6 R. C. Arnold, Phys. Rev. 153, 1523 (1967). 7 C. B. Chiu and J. Finkelstein, Nuovo Cimento 59A, 92 (1969);

R. C. Arnold, Argonne National Laboratory Report, 1968 (unpublished). The method is equivalent, for nonrelativistic nuclear problems, to Glauber's method with an infinitely composite system and uncorrected (product) wave functions; but since wave functions and constituent descriptions do not explicitly enter our formulas, they can describe hadron scattering in general at high energies and small angles.

^ Our somewhat unusual but very convenient normalization is such that o-tot==27rImF(0) and d(T/dt = l7r\F(t)\\

^ If Rj^ grows logarithmically with s, this is the model of S. C. Frautschi and B. Margolis, Nuovo Cimento 56A, 1155 (1968); also proposed in Ref. 7, Sec. IV. If the radii are constant, this is approximately the high-energy limit of the "hybrid model," Ref. 8; see also R. C. Arnold and M. L. Blackmon, Phys. Rev. 176, 2082 (1968).

186 S C A T T E R I N G A N D P R O D U C T I O N O F E X C I T E D H A D R O N S 1633

If the structure of heavy hadrons of category 2 is comparable to that of nuclei, then we expect y^aR^ and R^aA^^^, where A is the mass of the hadron. This yields, in our formula (6), a total cross section proportional to A'^^^ for ^ ^ 1 , but proportional to A for small A,

Given the amplitudes for scattering in the initial and final states, the effects of rescattering in production processes may be estimated by an "absorptive correc­tion'' formula^^:

where 8j is the (presumably imaginary) elastic-scatter­ing phase shift in the incoming (7 = 1) or outgoing (y = 2) channel, and

1 r'

2 7-00 dtJoLbi-ir^^G^oleit). (9 )

For 72<1 , we will approximate exp[i(52+5i)] in our cases by exp(2i5), where 5 is a phase shift appro­priate to the mean-square radius between initial and final radii. We thus take

exp[ i (52+5i) ]^ l - (ymy''e~''"''', (10a)

where R' = URi^+R2^)' ^ Alternatively, when 7 2 ^ 1 , we can consider 2 as in black-disk scattering with radius R2 and ignore the effects of the initial-state scattering; this yields

exp[i(52+5i)]^0, for b<R2

^ 1 , for b>R2. (10b)

These approximations enable us to obtain the relevant properties of the production amplitudes analytically.

The production amphtudes for 72 < 1 then become

where

with

G{t)=G^oUt)-Goit), (11)

G,(t)=\(yiy2y'KmRo')e^''''^', (12)

R^~'=Ro-'+R~';

while for 7 2 ^ 1 , we obtain

Git)=xRo-' [ bdb /o[K-0'^'>-^'^^^«^ (HO

J R2

This can be analytically evaluated at ^ = 0; we obtain

G ( 0 ) = x f xdxe--''' = \e-^'i^, (13)

10 See Sec. VII of Ref. 7, where this expression is obtained from a multichannel eikonal formulation; or F. Henyey, G. Kane, J. Pumplin, and M. Ross, Phys. Rev. 182, 1579 (1969), where a weak version of this appears with di, 52<^1.

with,s=i?2/-^o. For large 2,

(da/dt)o=i7r I G(0) 1 2 ^ j7rX^g-^22/i?o2 ^ (14)

which expresses the attenuation in rescattering through a mean-free-path exponential factor, as in nuclei.^^ The production of heavy hadrons on any target will therefore be strongly suppressed (at least for small momentum transfers) relative to production of lighter particles.

For ^ ^ 1 , the t dependence of (110 is approximately JoZR2(—ty^^2y characteristic of a source which is ring-shaped, as expected^^ from comparison with production processes in nuclei when the produced particle has a short mean free path in nuclear matter compared to the nuclear radius.

For the lighter of the heavy hadrons, e.g., Ai and ^2 , where we expect 72:^1, we can use (11). Their produc­tion cross section at i^=0 would be suppressed relative to T and p by a factor of

\:i~(y2yir'R'/(R'+Ro')J

[ l-7i i^iV(i^i^+i^o^)? (15)

where 71 refers to w scattering, and we have not taken into account possible differences in the primary produc­tion strength X. Note that 72 (and 71) should be less than unity in such an approximation. At the same time, for large R2, Rs is close to Ro, and the t dependences of pole and correction terms in (11) are very similar. The suppression continues until | | is large enough that the second term dominates. This requires

t{Rz'- - W ) > 1 . (16)

With - 2 large, the coefficient of t will be small. Thus, in models of category 2, production of such hadrons will be suppressed relative to the pole term (4) unless momentum transfers such that

\tRo'\ >Ro'/iR,'-'Ro')»R2'W (iorR2»Ro) (17)

are involved in the production. Such, a differential production cross section will

exhibit a minimum or break when Gc equals Gpoie. There would be a zero if the phases exactly match, but the cancellation will in general occur at slightly different t values for real and imaginary parts. Also, if more than one helicity amplitude is important in production, a break rather than a sharp minimum would be expected.^^

11 A. S. Goldhaber and C. J. Joachain, Phys. Rev. 171, 1566 (1968); J. Formanek and J. S. Trefil, NucL Phys. B4, 163 (1968), this paper contains references to the extensive literature con­cerning high-energy reactions in nuclei.

^ These features are common in elastic scattering and charge exchange when multiple-scattering corrections are included; cf. R. J. Glauber, in Proceedings of the Second International Conference on High-Energy Physics and Nuclear Structure^ Rehovoth^ Israel^ 1967y edited by G. Alexander (North-Holland Publishing Co., Amsterdam, 1969); V. Franco and R. J. Glauber, Phys. Rev. Letters 22, 370 (1969); and Ref. 11.

1634 R . C. A R N O L D A N D P . G. O. F R E U N D 186

We might use the estimate (17) for 72>1 (also for small t, provided R^ is not too large) since the qualita­tive behavior of the amplitude is similar. I t is more reasonable in that case to use R=R2, instead of the mean-square value, since the principal absorptive effects are associated with the final state and not with the initial state.

If the target (or one of the final-state bodies) is a nucleus, the pole residues will have t dependences con­taining the form factors of the nucleus,^^ This means the radii involved (Ro,RijR2) will be large, as they represent the nuclear extension. The results will be crude estimates of nuclear absorption effects, and may be considered as approximations to more involved calculations^^ which explicitly use nuclear wave functions and the Glauber formalism.

Although these estimates for G are appropriate only for very high energies, we can consider the rough order of magnitude of the effects discussed, as they may explain some features of presently available information about heavy-hadron production.

There is one class of experiments, missing mass in Tp—^Xp, which has clear evidence^^ for mesons {R,S,T,U,...) heavier than A2. Such data has been accumulated at — / values typically greater than 0.3 GeV^. If the square of the t/-meson scattering radius on a proton is about twice the square of the pion-proton radius (as would be suggested by models 3 or 4), we expect from (13) a signal suppression of a factor approximately e~^ at / = 0 compared to light-meson production.

Analyses of ^ 1 and/or A 2 production in heavy nuclei can, in principle, provide a measure of their total cross sections on nucleons. We do not regard the existing

determinations^^ as conclusive, and better experimental data would be welcome. We remark, however, that Ai, i^^(1300), and A^*(1400) production do not have clear interpretations at present because of ambiguity in identifying a well-defined resonance as compared to a continuum.

We suggest that a useful experiment would be co­herent production of Ai, A 2, and heavier mesons in deuterium, with an observation of the position and height of the second maximum in the differential pro­duction cross section. If their interactions with nucleons are of larger radius than for pions, as in category-2 models, the first secondary peak should be shifted [according to (11a)] to smaller values of t than those observed in pion elastic scattering from deuterium. If, on the other hand, they are found in similar places, this would be evidence for category-1 models. Such con­siderations do not depend essentially on the strength of the production mechanism, but only on the relative / dependences of the production and rescattering proc­esses. If RA2N^ = 2RT,N'^ and 7^2 = 1-0, we estimate a 30% decrease in the position of the break or minimum in d(x/dt.

I t is important to find the number of quarks and antiquarks in excited hadrons, since this would lead to a clearer understanding of their structure. The experi­ments we have discussed are the best way we know of to achieve this.^^

12 These results can be obtained formally using the methods of R. C. Arnold and S. Fenster, in Proceedings of the Topical Con­ference on High-Energy Hadron Collisions at CERN, 1968 (unpublished), Vol. II, p. 1. The "very weak-binding" case is applicable to nuclei.

M G, Chikovani et al., Phys. Letters 25B, 44 (1967).

15 A, S. Goldhaber, C. J. Joachain, H. J. Lubatti, and J. J. Veillet, Phys. Rev. Letters 22, 802 (1969). These authors state (Ref. 18) that a{AiN) ^2(r{7rN) within 4 standard deviations (s.d.), and (r(AiN) ©-(TriV") within two s.d. If these cases reflect probabilities of finding their results in a given experiment, given either a{Ai) = a-(ir) or cr(^i) = 2o-(7r) in nature, one would conclude that models of category 1 are favored by the experiment they analyzed. Their ' 'best" solution was quoted as a(AiN) ^0.5-0.2"^"^o-(xiV). We can offer no new interpretation of such a result if it should reflect the true situation; it would at least indicate that an additive quark model is not satisfactory.

1 For higher vector mesons, one could envisage measuring their total cross sections on nucleons by photoproducing them on nuclei. This method, however, is somewhat ambiguous, as one knows neither the photon-higher-vector-meson coupling nor the off-diagonal Pomeranchuk-lower-vector-meson higher-vector-meson couplings.