Decay Spectra of Particles and Resonances Produced in a Central Plateau

We consider the two-body decay of the spinless resonance or particle produced in a central plateau, with an arbitrary transverse-momentum spectrum. The spectrum of the decay products is calculated exactly as an integral over the spectrum of the centrally produced resonance or particle. Special forms applicable to large and small momentum transfer are presented along with an accurate inversion formula. We show how the large-transverse-momentum behavior of the resonance production is replicated in the decay products. The decay m y + y is considered in detail.

"H. Fritzsch, Nuovo Cimento 12A, 1016 We consider the two-body decay of the spinless resonance or particle produced in a central plateau, with an arbitrary transverse-momentum spectrum. The spectrum of the decay products is calculated exactly as an integral over the spectrum of the centrally produced resonance or particle. Special forms applicable to large and small momentum transfer are presented along with an accurate inversion formula. We show how the large-transverse-momentum behavior of the resonance production is replicated in the decay products. The decay m y + y is considered in detail.

I. INTRODUCTION
The recent verification of the existence of a central plateau in the CERN Intersecting Storage Rings (ISR) experiments' allows us to probe deeper into the detailed mechanism of pionization. In previous papers' ' we have investigated the properties of the pionization spectrum in q~r esulting from the internal-damping structure. As originally discussed by Amati, Stanghellini, and Fubini~( ASF), the pions in the central plateau arise from fireballs or resonances produced in a chain of peripheral pion exchanges. In this paper we calculate the inclusive spectrum of a particle resulting from decay of a spinless two-particle resonance which is peripherally produced in a central-plateau region. The generality of the calculation allows it to be applied also to the case of a m produced in the central plateau, which then decays into two photons. It can then be used to infer the m spectrum from the y spectrum.
Our work is an extension of the treatment of these problems as recently considered by others' " Our formulation (1) includes an exact treatment of the kinematics and integrations; (2) is applicable to any q~' spectrum of produced resonances or m" s; (3) applies to both large and small q~; (4) has the integrations performed analytically, not numerically; (5) gives a unified treatment of massive and massless final particles.
The formulation includes many of the earlier results as limiting or special cases. The calculation proceeds by considering a resonance of momentum q and mass q'-=m' being produced in a central plateau with a spectrum p(q ', q~') independent of longitudinal momentum.
This then decays into two particles of masses p. , and p. , so that q =q, +q, . Since only one particle q, is observed in the single-particle spectrum, we must integrate over the momentum of q~. It is convenient to work with q . =-(q, where the & denotes two-dimensional transverse vectors. The integration over q, is performed by converting to integrals over q, q"and the rapidity y, = sinh ' (q~~/q, '~'). The q, and y, integrations are performed exactly for infinite energy, and the integral over q, with the general function p(q) -= p(m2, q~') remains.
In Sec. II we formulate the problem and calculate the decay pionization spectrum for q,~m, '/4m'.
In Sec. III we study the large-q, approximation and show how the large q-behavior of p(q) replicates itself in the large-q, behavior. In Sec. IV the decay spectrum is calculated for q,~m ,4/4m'. The m 0-2y decay is presented in Sec. V.

II. RESONANCE DECAY SPECTRUM
We assume that a spinless resonance or particle 8 is produced in the central-plateau region and calculate the transverse-momentum distribution of its decay products. For notation we call the decay products g"and m2 of mass p. , and p. 2 and consider them as distinguishable.
In Sec. V, however, they are considered as massless photons.
The inclusive R production a+b-R(z, m2)+X (2 &) ( Fig. 1) contributes to the single-particle spectrum for w, in the central plateau": We assume that the inclusive integration and summation over X produces the spinless resonances g in a central-p1ateau region constant in the R's rapidity: (2.4) We can make this into a superposition of spectra for various masses m'=(q, +q, )' of the virtual resonance by introducing f dm a'(nP -(q, + q, )') = 1: (2.5) p(q) p(m', (q, =+q2)~2) .
(2.6) In the narrow-width limit I" «m~the integral in brackets becomes 1, and we evaluate the remaining integral at m'=m~'. For the remainder of this paper we will suppress this integral over the resonance virtual mass as well as tke dependence of p orms'.
We now convert (2.12) where the extra (2) is included for the doublevalued mapping q -+ 8». Rewrit'ng ritin this in terms of q"q, we have from (2.11) where the latter 8 function guarantees a physical angle between q, and q, .
We define the four roots in q2 of the denominator: (2.17) This gives "p (7) (m 4j4m', the right-hand side has the while for q, &m "m, opposite sign. Then for q, mp4, th b-&m 4 4m2 the substi u ion g =q t t co~sponds to c =d but not to R. MICHAEI BARNETT AND D. SILVERMAN d&q2& c. (2.22) b =c and is the lower limit of the q integration.
In this section we treat the case q, &m, »/4m' which for a p resonance is~q , (  (3.5) We are particularly interested in the connection between the asymptotic behavior of p(q) and that of p, (q, ). For the case q»m'» p, , ' we take the upper limit effectively infinite: E is a hypergeometric function which is 1 at y' =0.
It is related to elliptic integrals": The relation between the resonance spectrum p(q) and the decay spectrum p, (q, ) becomes even simpler for large q, . For q, &mo'/4m', we find from Fig. 2 two regions of q, g, space which give different g, limits. q, is fixed and for region I (Fig. 2) q &q&@+, b&c so d& q2 & c.
(4 1) But for region II A simple ASF model for resonance production with peripheral pion exchange gives p(q)~q -' due to the pion propagators alone. ' The resulting g, ' spectrum is inconsistent with data. In fact" internal form factors must also be included' to give p(q)~q -» and therefore p, (q, )~q, ' to fit the data.
In the first integral r'~1 , and in the second 1/r' &1.
We note that for q, «m, 4/4m' the curve r' =1 or b =c always occurs in the integration region, Fig.   2, and it is no longer possible to approximate the hypergeometric function by 1 throughout the entire region.
In this limit" p-0 limit. In   The decay m'-2y is of course characterized by a spinless decaying particle with no width, p, =0, q, =(q, )', m= m"o, and p, o(q) is the )yo production spectrum. The kinematics and results are obtained from the general case above by taking the For q, &m'/4, or iq', i&-, 'm"o we have from (4.4) We thank our colleagues at Irvine for helpful discussions.