analysis of the a1

ANNALS OF PHYSICS 117, 56-88 (1979)

Analysis of the A,

R. AARON, R. S. LONGACRE, AND J. E. SACCO

Department of Physics, Northeastern University, Boston, Massachusetts 02115

Received January 13, 1978

We develop a unitary isobar model, and apply it to a study of partial wave analyses of the available three-pion and K*f? data.

1. INTRODUCTION

In the simple quark model, mesons are quark-antiquark systems. Considering the general success of this model, especially in light of recent discoveries of charmed particles, etc., it is hard to imagine that its predicted spectrum of the low-lying traditional meson states would be grossly violated. Nevertheless, while the S-states all have been found, the P-wave states are still largely undiscovered. The most striking vacancies occur in the two axial vector nonets (Fig. 1) where only two members are considered established- the B(1235) and the Q,( 1280), the latter being a linear combination of QA and QB . The first systematic search for the A, , the most famous

b/

l E0

1 ++

FIG. 1. Axial vector nonets.

0003-4916/79/010056-33$05.00/O Copyright Q 1979 by Academic Press, Inc. AU rights of reproduction in any form reserved.

56

ANALYSIS OF THE A, 57

of the “missing” mesons and the subject of this paper. was carried out by G Ascoli and collaborators [l] who did a phase-shift analysis of the reaction rr- -I- p - (nrn)p in the framework of the isobar model. A major result of their analysis was the presence of a broad I m= I, Jp = 11, pi S-wave enhancement at 1 I50 MeV, across which the associated isobar amplitude had a relatively small change in phase. Furthermore, the energy behavior of the magnitude of the amplitude was qualitatively described by the Deck model [2]. The latter result, and the lack of significant phase variation of the amplitude, has generally been interpreted as the absence of the A,

Two common features of the missing mesons described above are that they decay strongly into three mesons systems (nrrn, KKn, Knn, etc.), and are seen in channels that have large Deck backgrounds which dominate their own decays. The main point of this paper is that in a strongly interacting three-body system where a large background is interfering with a (usually) much smaller resonance contribution, our experience tells us neither what to expect, nor how to interpret the results. In this article we shall develop a formalism which can perform these tasks, and apply it to an analysis of the I- and O- three-pion (and K*R) data. We shall show, for example. that the very phase behavior interpreted by previous authors as demonstrating the absence of an 14, . in fact. demands the existence of the A,!

Almost all resonance-region multiparticle final-state data in elementary particle physics are analyzed using some version of the isobar model [3]. The essential ingredient of this model is the choice of a T-matrix element which factors into a part describing production of a particle and a correlated aggregate of particles (isobar) from an initial two-body state, and another part describing the propagation and subsequent decay of the isobar. After partial wave analysis, the T-matrix element for scattering in a particular state of Jp and Mv is shown in Fig. 2. In the standard isobar model, for fixed overall c.m. energy, the production part, represented by the circle in Fig. 2, is assumed, except for a barrier penetration factor. to depend only on the three-pion mass U’,, and the square of the momentum transfer t =-. (p - p’j2. The latter dependence is one of the more questionable aspects of the model and is partially associated with problems of unitarity [4]. In fact, it is well known that the standard isobar model is not unitary, and in order to make it so. one must include rescattering corrections. In Sections 2-5 we develop a unitary isobar model which incorporates both Deck background and resonance production. and in

FK;. 2. Representation of pn production (from a n-p initial state) in the isobar model for given total angular momentum J, parity P, spin projection M. and exchange parity 7 of the 377 system.

58 AARON, LONGACRE, AND SACCO

Section 6 apply it to a study of partial wave analyses of diffractively produced 33~ and K*K data. In particular, we consider the I = I, Jp = O- and l+ (43 channels. While much that appears in these sections is technically complex, the ideas involved and even the equations are often physically transparent. Therefore, we have tried to organize this article so that one not interested in the deails of the calculation may glean the basic content by reading only Section 1 (introduction), Section 7 (results), and Section 8 (conclusions). We now summarize the text of Sections 2-6, and hopefully provide the more general reader with a map to guide him through the inessential details.

In order to obtain a unitary and consistent formalism we must treat rescattering of the three pions properly, and therefore must develop a theory of the three-pion system. We proceed as follows: Initially, in Section 2, we consider rescattering induced by the known long-range scattering mechanisms in the 37~ system; i.e., one-pion exchange (OPE) between pm and/or EAT states (Fig. 3a in Section 2). In this case the rescattering can be expressed in terms of quasi-two-body amplitudes for pr and ET scattering which are usually derived from either dispersion theoretical techniques or Blankenbecler-Sugar methods [5]. In the latter approach, which is used exclusively in this article, the amplitudes are obtained by solving coupled Fredholm integral equations shown schematically in Fig. 3b of Section 2; they have the form

(1.1)

and for reasonable form factors at the nr vertices (i.e., form factors with smooth cutoffs ml-2 GeV) we find that in the A, channel, the Fredholm determinant of Eq. (1. I) satisfies

Det(1 - M) = 0.98, (1.2)

which is also obtained by Schult and Wyld 161. Not surprisingly, the latter authors find that the weak OPE effects just described cannot significantly distort the “Al phase.” Jn terms of our approach, Schult and Wyld have summed the diagrams in Figs. 6a and 6b of Section 6, where we describe diffractive production by single Pomeron exchange. Figure 6a represents the usual Deck mechanism (without rescattering).

We interpret the previous results as an indication that the long-range OPE mechanism is a weak force in the l+ channel and cannot be dynamically responsible for the existence of the A, . If indeed the A, does exist, it is the consequence of short-range forces. Thus we phenomenologically include a short-range mechanism in the I+ -- 37~ system by introducing coupling at primitive vertices of pn-, ET, and K*K(K*K) to a heavy particle which we call the bare A, (Fig. 4 of Section 3) and which hopefully simulates quark interactions, etc. New quasi-two-body amplitudes which include both long-range (OPE) and short-range interactions can now be obtained by solving (1.1) for the Xij and including all possible insertions of the bare A, in a manner first suggested by Bronzan [7]. This more extended model of the 37r system is discussed in detail in Section 3. Besides the usual Deck mechanism for 37~ production (Fig. 6a) and Deck plus rescattering through OPE (Fig. 6b), the introduction of the short-range


force results in the existence of Fig. 6c, and forces us to consider direct production of the A, via coupling of the Pomeron to the bare A, giving Fig. 6d. The full unitary production amplitude is now the sum of all the diagrams in Fig. 6. The vertex functions and A, propagator appearing in Figs. 6c and 6d are described in more detail by Fig. 5. If a phlviral A, exists, it will appear as a zero of the propagator function. We shall refer to any model which includes a direct productions mechanism (Fig. 6d) and a Deck contribution (Fig. 6a) as a two-component model (TCM); as far as we know, this type of model was first suggested by Bowler et al. [8]. We find that it is the two- component feature that is crucial in fitting the 3~- data. In Section 7.2 we discuss some general features of the TCM which might have important implications in other areas of elementary particle and nuclear physics.

The last problem in constructing the production amplitudes is to combine a properly parametrized Deck amplitude with the already calculated 3~ contributions. For reasons explained below we are posing a nontrivial challenge-from Fig. 6a, we may write schematically

P”,,, . or T,,,] &B[VA or 11, (1.3)

B = s,,v($ - tR)bl. (I .4)

tR = (k + k,)“, (1.5)

s,,,, = (p’ -F k,)“. (I .6)

In Eq. (1.3), T,,, and T,,, are the Deck amplitudes for p and E production with the factors describing the propagation and decay of the p or E divided out, and V is a (in general. off-shell) polarization vector defined in Section 7.2. In (I .4) i(swN) is the Pomeron propagator, and the momenta in (1 S) and (1.6) are defined in Fig. 6. If Eq. (1.3) described pure pion exchange, the Deck background could be combined neatly with our three-body amplitudes. Unfortunately, for s,,v large and t = ( p - p’)” near its minimum value, the very kinematical region under consideration, one may write [9]

s = (p + ky,

SA = (k, -; k, - k,)“, (1.X)

and thus the pion pole in (1.3) is largely cancelled! Rather than attempt to describe this complicated situation theoretically in our off-shell Blankenbecler-Sugar formalism, we choose to use a phenomenological Deck amplitude in our fitting procedure. This parametrization is discussed in detail in Section 4.

In Sec. 5 the definition of the S-matrix, phase space, and expressions for the cross sections are given. Combining the results of Sections 24 we here develop our version of the isobar model; the partial wave decomposition of the isobar amplitudes is also discussed. Finally, in Section 6 we discuss the fitting of our fully parametrized theoretical amplitude which sums the four diagrams of Fig. 6 to the available data,


which include O- and l+ p and E cross sections, l+ K*K cross sections, and relative phases of the isobar amplitudes averaged over phase space at cm. energies ranging from 950 to 1700 MeV. The results of our analysis are given in Section 7, and our conclusions in Section 8.

2. THREE-PION SYSTEM-OPE

A major input required for the construction of a unitary isobar model is a reasonable theory of the three-pion system. In this section we assume that the only interaction among the three pions occurs through one-pion exchange (OPE) between v~‘p and 7r~ states (Fig. 3a). We then choose as our dynamical framework the Blankenbecler- Sugar formalism which yields sets of coupled integral equations for amplitudes X,, , X,, , X,, , and X,, describing the quasi-two-body processes rr + p + n- + p, n- + p + P + E, etc., whose solutions are Lorentz invariant and satisfy two- and three-body unitarity and the cluster property. In operator formalism these equations have the structure (a schematic representation is shown in Fig. 3b),

x,,(w,) = bba( W/J + bO’-/,I GcW,d Xccz(W~)v a, b, c = p, E. (2.1)

al ‘0’ -Y------ \ \ -----\

P¶E 6)

p,c f-v'

!yI- ------

X = \ \ ---- ----\ P! E

+ I 5;;----&---

p:’ E” p:’ 6” P*E FIG. 3. (a) Long-range (OPE) mechanism. (b) Unitary sum of OPE diagrams in terms of coupled

integral equations.

In (2.1), WA is the overall c.m. energy of the three-pion system. For definite isotopic spin Z, momentum space matrix elements of various operators appearing in the above equation are now given:

cp,h’ , bI (wA) , pQ = XI cw) GYP, P’) J%N? V,(P, P’) L)P PI, Wp+,‘[ w2 - WA”] ’

(2.2)


where wz WY + loPiP’ --.. WY’ . wg _ (p2 -c ~“)W,

V(p, p’) = V’(p’, p) = p’ + “(P”. p”, p p’) p.

Lx(p”, p, p . p’) = ; - ‘E; FL!!& . 0 0

P, =- WY, f wp,y’ )

W” =~ p,? _ $,

In (2.2), h and h’ are the initial and final z-components of the p spins. The off-shell polarization vector V is defined in Ref. [lo]; as pointed out in the latter work, V2 is a Lorentz invariant and is equal to the square of the cm. momentum of the pions in the p rest frame. The form factors U, and v, are required for convergence in our integral equations and are taken to be functions of V2. Having defined all relevant quantities, we now display the remaining Born terms appearing in (2.1).

(2.3

(2.4)

The isospin recoupling coefficients xib appearing in (2.2)-(2.4) are given by

x;n = (-l)l+ek~ [(2ib + I)(2i, t- I)]‘/’ W(1111: ibi”), (2.5)

where i, and ib are the isospins of the correspondingly labeled isobars. The values of ,& are given in Table I. Finally, the propagators G, in (2.1) have the matrix elements

(P’ I GM,) I P> = 2w,(W3 S”(p - p’) D;‘(p, w,,.

D,(p, WA) = 0 - My2 - & j-” (fq @+,Yq) 0 w& - 4wq2] ;

(P’ ! G,(JJ’,) I P> = ‘%W~3 S”(p - p’) D,‘(p, Jv,),

(2.6)

with

(O)? D,(P, WA) = 0 - M, 1

-m ,, s o; 4 q”c,‘(q)

(2.7)

w&J - 4w,P]

(T = WA2 - 2w,w, + p”. (2.8)

As explained in Ref. [ll], b,, , b,, , etc., have the same residues at their poles as the corresponding Feynman-Born terms, but in addition have a cut structure which guarantees that all amplitudes satisfy two- and three-body unitarity. The propagators


TABLE I

Isotopic Spin Recoupling Coefficients for Born Terms

0 -1 0 0

1 + - 1/(3)'12 * 2 4 0 0

G, and G, are essentially the two-body scattering amplitudes and carry the phases of STVT scattering in the Z = 1, J = 1 (p) and Z = 0, J = 0 (E) channels, respectively. Note that all Born terms and propagators are given in terms of the form factors V, and u, ; these are chosen to be smooth functions of the form

Qm = ga15a2(q2 + Ba2>-1 (Q = PY 4. (2.9)

The values of g, , Iga , and MA” are determined by fitting on-shell nr scattering. The physics does not seem to depend on Pa , and we take all our /3’s to be ~OWJ,,~.

For application to data analysis we require the angular momentum projections of (2.1). Partial wave analysis yields the following set of matrix equations:

(2.10)

In the above equation J is the total angular momentum, 1 is the orbital angular momentum of the initial particle-isobar system, and I’ is the orbital angular momentum of the final particle-isobar system, where, for example,

(p’J1’ 1 X~,,(WA) 1 pJ1) = c (Z’lm’h’ ( JM)(llmh 1 JM) I dQg 1 dsZ, Ytjm@‘) m,A Wl’.A’

(2.11)

The partial wave decompositions of the various Born amplitudes are now given. The pp term has the basic structure


where asp, a;,, etc., are functions of p2, pt2, and p * p. Using (2.11), integrating over solid angles, and summing over Clebsch-Gordan coefficients, we obtain

<p’JI’ j b,,(W,) 1 pJ1) = [(21’ -f 1)(21 -1 l)]l’” ((1100 I JO)(Z’lOO I JO)

x (2.f + 11-l [p2(a,,k f p’2(av~p~)~ -I- p’p(ad,)d + ppy-1)3-‘-l c (000 j (10)(1’100 j AO>

‘I

x W(lAJ1; ll’)(a,,,),}. (2.13)

In (2.13), IV(lAJ1; U’) is a Recah coefficient [12]. In going from (2.12) to (2.13) we have expanded, for example,

aDp(p2, P”, P . P’> = C GhJt KW> Y&9 1.m

and thus

= gL (a,,), YhdA ~;“,W> (2.14)

(a,,)~ = 2~ f:’ dz PA4 a,, . (2.15)

Similar expressions may be obtained for the partial wave projections of b,, and b,, :

(p’JZ’ I b,,(W,) j pJ1) = (1100 / 1’0)[(21+ 1)/(21’ + l)]“”

x [P’(k7), + P(&,kl, (2.16)

where (p’ I b,, I pX) has the structural form

(P’ I b,,U+‘/d I PA> = 4,f’A (2.17)

with V in Eq. (2.17) and 01 in Eq. (2.16) defined in Eq. (2.2a). The matrix element for b,, may be written

(P’J~’ I b,,W,d I pJ1) = &(U (2.18) with

<q I U W,) I pi = BEE (2.19)

All partial wave projections in the above formulas use the normalizations defined by (2.14) and (2.15).

3. THREE-PION SYSTEM-INCLUSION OF SHORT-RANGE INTERACTIONS

We assume that short-range effects in the Z = 1, Jp = I+ three-boson system are mediated by an s-channel exchange,of a heavy particle called the bare A, between ET, prr, K*E, and i?*K states. We further choose the interaction to be of separable

595/117/I-5


form, so that in operator formalism the first-order (Born) terms, shown schematically in Fig. 4, may be written

bA,bdWA) = I b) (WA - dvl <a I > a, b = E, p, K*, iC* (3.1)

where MJ,O) is the mass of the bare A-particle. The momentum-helicity space matrix elements of (3.1) become

WA I bA.aa(WA) I p@ = <p/h’ I b) (W, - Mi”)-l <a I ph) (3.2)

I q 2

/ / / / ’ A,(o) ‘\,,

\ / \ / \ FIG. 4. Short-range mechanism.

with the vertex functions (a I ph), which describe decay of the isobar-u into its products, chosen to be of the form

(a I PA> = Y/J&) Y,.,($) (3.3)

with h the helicity of the bare A, I, the relative orbital angular momentum of the particle-isobar system, and ya an effective coupling constant. For p and K*, 1, = 0, and for E, 1, = 1. The form factors u, are taken to be the following smooth functions:

U,(P) = A%” + sP2)-1Y

e*(p) = S”K*/(P” + ff%*rl,

e*(P) = UK*(P), (3.4)

%(P) = PBE2/(P2 + lL2)-3’2.

The E form factor in (3.4) includes a P-wave barrier factor, and the fla)s are cutoff parameters. We find that the physics is independent of the value of the cutoff parameters pa for pa M 1-3 GeV, and in all our calculations we arbitrarily fix /?, at the particular value of 50m,2. The remaining parameters, Mj”) and the ya’s, become fitting parameters of our theory, and are determined by a x2 fit to the data.

A preliminary step in obtaining unitary three-boson amplitudes is the construction of auxiliary amplitudes raSaa which sum all diagrams except u-channel OPE. This amplitude has the perturbation expansion

t A.bo = bA.ba(WA) + bA.bc(WA) G,(W,) bA.ca(WA)

+ bA.bc(WA) G(WA) bA.cd(WA) ‘%WA) bA,dWA) + -“T (3.5)

ANALYSIS OF THE ki, 65

where for the case of E and p, G,( WA) is defined in Eq. (2.6). For K* (and K*)

(p’ I GdW.4) / p\ = 2w,(2+ S(p - p’) D$(p, WA)

DKe(p, W,,) = u - ,I@ - 1 z $X&*(q)

- s 6(2+ o T&-r) w*(K)[cr - X2] ’

The form factor oK* is taken to be of the form (2.9). Because b.d,8a has a simple separable form, (3.5) may be easily summed to obtain

rA,ba( WA) = I b) [Da”‘(W,)]-’ (a !, (‘7.7)

0% W.d = W, - M-F’ - ; $$ j” ;> u,?(q) G,(y, WA) (3.8) 0 *

with u,(q) coming from (3.4). Suppressing channel and isospin indices, the full T-matrix T which sums all dia-

grams may be expressed formally as the following operator expansion:

T = X + XGt, + t,GX + XGt,4GX -I- .... (3.9)

where X is defined in the previous section, G in Eq. (2.6), and T,, in Eq. (3.7). The rule in constructing the sum (3.9) is that no combination such as XGXor taCtA may appear in any term; terms of this type would result in double counting since, for example, X already contains the sum of all ladder diagrams. Equation (3.9) is an expansion of the Faddeev type, and we may sum the series using methods developed by Faddeev [ 131 for the nonrelativistic three-body problem; we first write

where

T = T”’ .+ T’2’ 1 (3.10)

and

T’l’ = x + J’GT’2’ (3.11)

Tf2’ = ta + t,GT’l’. (3.12)

Because tA is separable, we may obtain a closed-form expression for T by algebraic manipulations. Taking matrix elements in a-b space, and using (3.7), note that 7’t2) may be written

where

7’;;’ = ’ b) [D?‘( WA)]-’ (T ’ 0 * (3.13)

<T, I = <a I + (c I G,T,i’. (3.14)


Substituting (3.11) and (3.13) into (3.14) we obtain

(3.15)

Further, subsituting (3.15) into (3.13) gives Tc2) in closed form, and then substituting this expression for Tf2) into (3.11) gives P in closed form. Equation (3.10) now yields

Tim = xba + [x&c I c> + 1 b>l[(a 1 + (d 1 GXazl

DA WA) 9 (3.16)

where

DAWA) = D%C) - <c I GcXcdGd I 6. (3.17)

Suppressing spin and isospin indices, the momentum space matrix elements of (3.16) in the I = 1, I = 1, P = l+ channel become

(p’ 1 Ti,,(w,) 1 P> = <P’ 1 xb,(wA) 1 P> + rb(P’v WA) ra(&‘, WAYDA(J+‘A), (3.18)

with

ra(P, WA = Y&a(P) + 7 & joa g$ (P I &,W’A) I 4) GA WA) u&h (3.19)

DAWA) = ~!i”W,=i> - 5 $$ I m q2& o 2w, Gd Gc(qv WA)

x f ,9 e (4 I X~WA) x 4’) GA’, w,) dq’). (3.20)

The diagrammatic representations of (3.19) and (3.20) are given in Fig. 5. Three-body bound states or resonances will appear as zeros of DA( WA). Since X itself is a unitary

p,c,(K:i?)

FIG. 5. (a) Vertex function. (b) A, propagator.

ANALYSIS OF THE /i, 67

three-body amplitude, it may contain poles corresponding to bound states and resonances in the absence of the short-range interactions. When the latter interactions are turned on, the IY’/D term in (3.18) has poles which exactly cancel those in X, and the shifted bound state and/or resonance zeros occur in DA .

Finqlly, note that the energy denominator in (3.1) is linear in WA , while in the previous section such denominators were functions of s, = WA2. In Section 2 we maintained the dependence on WA 2 to be consistent and enable comparison with calculations that have been done in the past using off-shell relativistic three-body equations. The authors of the latter works realized that boson amplitudes should be analytic functions of WA2 and felt that it was worthwhile to maintain known reflection symmetry in WA at no extra cost. However, since crossing symmetry is, in general, violated by our relativistic three-body equations, the amplitudes at negative WA have little validity, and, furthermore, since it simplifies our labor in the fitting procedure to use denominators linear in W,, , we make the latter choice in developing our isobar model.

4. DECK AMPLITUDE

As mentioned in the Introduction, the Deck contribution to the quasi-three-body processes rr-p --+ en-p or prp behaves in a very complicated manner, and cannot be treated naturally by our off-shell relativistic three-body formalism. In this section we discuss the difficulties in more detail, and deveIop an approach which permits us to include a Deck amplitude together with the three-body amplitudes obtained in Sec- tions 2 and 3 in construction of the full t-matrix elements. For example the Deck matrix element describing r-p --+ ETP via Pomeron exchange has the form

g~dWMp) x (4 x 6 I TdW,d i k), (4.1 ,I

where g,,, is the NNP coupling constant, (is) is the Pomeron propagator, and T,,, is defined to be the amplitude describing rr-p ---, QTC via one-pion exchange. Using Feynman rules we obtain

s<k, I T,,,W,) I k) = Scvgwm(~2 - f&km (4.2)

with tR given by (1.5) and s,,~ by (1.6). It is at this point that two serious difficulties arise. The first has been mentioned earlier. For s and s,,,, large, and t = (p’ - JI)~ near its minimum value, exactly the conditions that hold for the diffractive data under consideration, Stodoisky [9] has shown that

S,N w s(p2 - tJ( WA” - p2). (4.3)

Thus, the factor (p2 - tR) coming from the Pomeron propagator cancels the pion pole in (4.2). This cancellation is not complete because we are not really at t = tmin . Nevertheless, (4.2) cannot be treated like a simple one-pion-exchange diagram. The second problem is that in order to adapt (4.2) for inclusion with our three-body


amplitudes, Feynman quantities must be replaced by the corresponding Blankenbecler- Sugar ones. We have already shown in Section 2 how this is done for the OPE Born term. On the other hand, we have no prescribed approach for obtaining the appropriate off-shell extrapolation of the Pomeron propagator. The latter problem is particularly sticky because of the pole cancellation mentioned earlier.

We finesse both of the above problems by treating (4.2) phenomenologically, and parametrizing the half-off-shell Born terms for rr + P -+ r + a (a = E, p) in the form

(qA j T,,,(W,J / p) = B(q, WA) x [l for a = E, or VT for a = p] (4.4a)

B(q, W.4) = [&(q, w,> + 3Blh W‘4bl/4~ (4.4b)

where z = $ * 4, and because we are treating the Pomeron as a real zero mass particle

P = w2 - cL2WW.4). (4.5)

The nature of the off-shell polarization vector V is discussed in Section 2. In terms of the quantities B, and B, in (4.4),

(4 I G,,WA) I P> = & for Jp = O-,

= 4 for Jp = l+,

<q I TdWzd I P> = P& + q’(~B, for Jp = O-,

= PBO + daBA for Jp = l+,

(4.6)

where ( )$ refers to the jth partial wave projection defined by (2.15), and 01 is defined in (2.2a). In our fitting procedure we further parametrize as functions of WA and q, i.e.,

Bdq, WA) = aJ%Nfo + x,S, + &SA~I/(SA - ~~1,

B,(q, WA) = g&(d[Yo + Y$A + Y,~A~II(~A - ~“1, (4.7)

where the P-wave barrier factor q has been explicitly included in FI, and the x’s and Y’s are fitting parameters. The subenergy dependence is contained in the functions F, and Fl ; the motivation and choice of these functions will be discussed in Section 7.3. Note from Eqs. (4.6) and (4.7) that B, and Bl are strongly correlated by the prr data.

5. CROSS SECTIONS AND ISOBAR AMPLITUDES

We now develop an isobar model to describe the specific process 7-p --f (mnr)p (or KKn-p) assuming that the reaction occurs via Pomeron exchange as shown in Fig. 6. In this case, the fully ofishell four-body production T-matrix element is written

(p’s’dv , b1, k2 , ksT2 I TV’) I PSTN, k7,)

= gNNn%‘(P’) d’>[~sl (‘&TI 3 ‘w2 3 &a 1 Tdw,) 1 P’ - P, kT,>, (5.1)


FIG. 6. (a) Deck diagram. (b) Deck plus rescattering through OPE. (c) Deck plus A, resonance rescattering. (d) Direct resonance production.

where the momenta labels refer to Fig. 6a, and Wis the overall c.m. energy; s and s’ are the z components of nucleon spins and the T’S are the third components of particle isospins. In terms of (5.1), the total cross section averaged over initial spins and summed over final spins is given by

with:the flux factor

F = 2pWIM (5.3)

and the expression for n-body phase space [14] is

where qi = 1 (2M) for bosons (fermions). It is the matrix element of TP appearing in (5.1) that will be related to the experi-

mental cross sections and that will be the subject of the remainder of this section. TP is a pseudo-T-matrix describing the three-body production processes rr + P -+ rr + ST + CT or K + R + x. Assuming that the final three mesons occur exclusively in the particle-isobar combinations NT, prr, K*R, and R*K, in the three-body c.m. system we define our isobar amplitudes as follows:

NT1 3 kzc 9 Jw3 I TP(WA) I P’ - P, kT,> = r,, + 7.0, + TK+ + TRq, (5.5)

70

where

AARON, LONGACRE, AND SACCO

In Eqs. (5.6)-(5.9), Vi, E V(p, ,pk) as defined in (2.2a). We are treating identical particles as in Ref. [lo], hence the factors of 1/(3)1/2 in Eqs. (5.6) and (5.7).

The Clebsch Gordan coefficients (CGC’s) in the above subsequent equations satisfy the conventions:

(1) The pion always appears in the first position.

(2) When coupling a particle and an isobar, the particle appears in the first position.

(3) When coupling orbital angular momentum 1 and channel spin j, I appears in the first position.

(4) In Eq. (5.8), f or example, the particles in the isospin CGC appear in the same order as they do in the polarization vector V.

With these conventions established, we now expand (5.6)-(5.9) in partial wave amplitudes which will contain the fitting parameters of the analysis:

I k) = c (kiJli I f,p(W,d I k, J = 0 Yrimi(&) Y,*,(&, (5.10) JM wb

I k> = C (kiJli I fpp(f+‘~) I k, J = I> JM

VT

with identical expressions for the expansions of fK* and fz* . The total cross section for the process rr + P -+ three bosons is defined as

W(TI , 72 , 73 ; TV> = s dpt3) l<k171, k2, k3T3 1 Tdw,) I p’ - p, kT,)j2. (5.12)


Using the expression for three-body phase space given by (5.4), and the partial wave expansions (5.10), (5.11), etc., for the case of production of three bosons from an initial pion-Pomeron state with a dejinite z-component of total angular momentum, (5.12) becomes

x Im (4 I &WA) I 4) P;l(q, WA)1 (4 1 .&WA) I k,, . (5.13)

In (5.13), the J-Z labels on the partial wave amplitudes in the previous equations have been replaced by Jp labels, and Im <b) is obtained, for example, by the replacement

w w,+,,[i;i;‘z _ wA2l -+ 2?7 a+(wA - WI7 - W,,)” - (P t P’)” - p2> (5.14)

in Eqs. (2.3) and (2.4). It is basically g’p in (5.13) that we will relate in subsequent sections to the results of partial wave analyses by Ascoli and other authors. We have checked that, except for overall sign, our overlap integrals agree with those of Ascoli [I]. Some of our overlaps, for example l+, prr-ET are smaller than Ascoli’s near threshold because, as mentioned in Section 7.1, we use an I = 0, J = 0 7~7~ phase which gives too small a scattering length (i.e., zero). On the other hand, in this case, Ascoli’s are also incorrect because he uses a simple Breit-Wigner parametrization to describe the E; this procedure is known to give too large an S-wave ~7r scattering length.

6. FINAL AMPLITUDES AND FITTING PROCEDURES

To compare (5.13) with data, we construct the partial wave amplitudes (q j fi’( W,,) j k) such that they include the four contributions shown in Fig. 6. We break up the amplitudes into sums of background (BG) terms corresponding to Figs. 6a and 6b, and resonance (res) terms corresponding to Figs. 6c and 6d, i.e.,

(4 I .f:‘W,, I k? = tfG(q, WA : Jpl + ti% W.J (6.1)

with k fixed on W, by Eq. (4.5), and

x P(P, WA) <P I $cWL) I k), (6.2)

t?(q, WA) = ra(q, WA) r&k, WA)/~AW’A), (6.3)


where

x K’(P, WA) (P I TDAWA) I k).

In the present work, the resonance or short-range term is assumed to contribute only in the Jp = I+ (A,) channel and we have thus suppressed spin labels in t(yes. All quantities in the above equations, except for yoD and up , have been defined previously. In (6.4) we have introduced a new parameter yp and a new form factor uIp which describe coupling of the pion and the Pomeron to the bare A-particle.

The parametrized amplitudes of (6.1) are now substituted into (5.131, and the values of the parameters are determined by fitting to the results of partial wave analyses using standard x2 techniques. These results include “cross sections” for pr, CT, and K*K production in the various Jp channels, and relative phases of the partial wave amplitudes given in (6.1). The “cross sections” are defined by (5.13); note that the latter expression contains terms involving overlap integrals of different partial wave amplitudes, and also terms involving the square of the absolute value of particular partial wave amplitudes. The partial wave cross section for production of a specific isobar u,“’ is defined by one of the latter integrals, e.g.,

s p IQ I f%Td I kj2 (8 Im Q?(q, WA)). (6.5) phasespace a

It is important to realize that in the standard isobar model used in previous partial wave analyses [I], the matrix elements (q [fi’( WA)lk) except for centrifugal barrier factors are assumed to be independent of q. So, there is a certain inconsistency in fitting our theory to results obtained using that model. However, recent investigations indicate that in most cases it is reasonable to interpret the q-independent amplitudes of the standard isobar model as equivalent to our amplitudes averaged over q. There is one case in the present analysis where we find the q-dependence is extreme; i.e., the O- pz-, and for this channel, all analyses agree that the partial wave amplitude is poorly determined by the standard isobar model. Thus, in some sense, our theory indicates the limits of validity of the standard model. Eventually we hope to fit our unitarized isobar model to the raw data and avoid questions such as those raised above.

We finally wish to discuss determination of the relative phases of the isobar amplitudes. From the previous discussion it is clear that the best we can hope to extract from our theory for comparison with standard results is an average phase. The definition gf an average phase is by no means unique. For example, the overlap integrals which appear in (5.13) could be used to define such a quantity. ‘For simplicity, we rather use a generalization of (6.5); in particular, we consider

s 9 <q I &WA) I k>* <q I &WA) I k> (8 Im D,(q, WAN. (6.6)

phasespace a


Lf (f$) can be shown to be relatively constant for the energy and subenergy regions in questions, we can take the phase of this amplitude as a reference, and the average relative phase of the other Jp amplitudes will be defined as the phase of the integral in (6.6). We find (6.6) attractive because it gives the phase at roughly the mass of the p (ImD, peaks strongly at the p mass) which is a major contributor to the three-pion final state. The main point we should like to stress is that if the phase has moderate variation with subenergy, the method used to obtain the average phase will not matter very much. On the other hand, if the phase varies rapidly with subenergy, no method of defining an average phase makes much sense, and one cannot use the predigested results of an analysis such as that of Ascoli. As mentioned previously, in this case, one must fit the raw data with a model which includes the possibility of subenergy dependence in the isobar amplitudes. Our results shall indicate that except for the case of the O- pn, the definition of an average phase is a reasonable procedure.

7. RESULTS

7. I. Three-Pion System-OPE

We have solved the set of integral equations given by (2.10) in the 1 = 0, Jp = 1 -- (w-meson) channel, the I = 1, Jp = O- (pion) channel, and the I = 1, Jp = If (A,) channel [15]. In the latter case we have neglected coupling to the D-wave amplitude which can be shown to be very small even at the highest energies under consideration. We obtain solutions for the amplitudes (p’J1’ I X&WA) I @I) with the left- and right-hand momenta both along a ray rotated an angle 0 into the lower half complex momentum plane. Because of the relative weakness of the OPE interaction mechanism, the solution along the ray extrapolates smoothly to real values of the momenta. We find that reasonable accuracy (10-15 %) is obtained by using angles 5” 5 0 5 lo”, and assuming that the values of the matrix elements along the ray are equal to their 6’ = 0 values. The more exact contour rotation and analytic continua- tion procedures of Shick and Hetherington [16], and of Aaron and Amado [ 171, are rather complicated for the situation at hand because for the case of one momentum real and one along the ray, production singularities pinch the contour of integration as 0 + 0, and we decided that the advantages of these approaches were not worth the additional effort.

In Fig. 7 we plot the (1= 0, and I = 1, J = 1) phase shifts which enter the integral equations for the X,, . These quantities are the phases of the propagator functions 0;’ and D;l defined in Eqs. (2.6) and (2.7). We have used an I = 0, J = 0 phase which fits the experimentally observed phase over a wide energy region (( 1 GeV), but actually has zero scattering length rather than the very small scattering lengths obtained from theoretical and experimental considerations. This choice is justified by the fact that small changes in the S-wave scattering length has almost no effect on our calculations. In Table II we display the Fredholm denominator functions of the integral equations at a few three-pion c.m. energies W, for the various isospin and spin-parity states. These results demonstrate the weakness of the OPE mechanism


f-

60°-

30" -

0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.1 1.2 0.7 0.8 0.9 f.0

TTTT c.m. energy IGeVl KT c. m. energy fGeVI

FIG. 7. WVT and KT phase shifts used in our calculations versus c.m. energy: (a) Z = 0, J = 0, nrr phase shift and Z = 1, J = 1 n?r phase shift. (b) Z = -& .7 = 1, KT phase shift.

TABLE II

Fredholm Denominator of Integral Equations for Particular Choices of Jp and Three Pion c.m. Energy WA

JP

O-

l- If

WA WV)

1050

780 1150

Fredholm denominator

0.8994 - io.1409

0.6877 - iO.0282 0.9787 - iO.0575

-recall that in the absence of interaction the denominator functions are unity. In Fig. 8 we plot the quasi-two-body amplitudes X,, , X,, , and X,, together with their Born approximations at W,,, = 1120 MeV as functions of momenta. One sees that the interaction only slightly modifies the Born terms. These results are consistent with those recently obtained by Schult and Wyld [6].

ANALYSIS OF THE A,

-\ /

i

8 % 8 s

--


7.2. Short-Range Interactions

Because of the relative weakness of the OPE mechanism in the three-pion system, any three-pion resonances must be consequences of short-range interactions. If we assume that the latter interactions are due to quark-antiquark forces, we have a picture more of less in line with contemporary views of elementary particle physics. In our theory, the short-range forces are described by coupling constants yp , yC, yO, yK+, corresponding form factors, and a bare mass parameter Ml’). The coupling constants and the bare mass are fitting parameters in any phenomenological analysis, and thus it is the experimental data which eventually determines the existence of any resonances. As in the previous discussion of Section 2, the physics seems independent of the cutoff parameters /3 appearing in the form factors for /3 M 1-3 GeV, and thus we arbitrarily set all fi2 = 50 n~.,~.

The discussion of Section 3 shows that the full quasi-two-body amplitudes which include all OPE and bare A-particle diagrams are convolutions of the OPE amplitudes obtained from the solution of integral equations given in Section 2, together with the various A-particle from factors and two-body scattering amplitudes. Fortunately, the inclusion of the short-range interaction does not require the solution of further integral equations. This fact is very important because it means that the fitting parameters do not enter the kernel of any integral equation; otherwise we would have the technically impossible task of solving a set of integral equations each time the x2 search routine stepped up a parameter. Nevertheless, fitting the model to the data is a formidable task; for example, calculation of the A, propagator (Eq. (3.20)) involves a two-dimensional integral over the fully off-shell amplitude X, which itself is obtained by solving a complicated integral equations (Eq. (2.10)). --

Note that in our theory we include K*K(K*K) states perturbatively in that we only couple these states to the bare A, and do not permit K or K exchange between isobars. This purely technical simplification shall be justified in our discussion of the Deck mechanism which follows shortly in this section.

We now wish to discuss some points of both immediate and general interest. If we think of the short-range forces as the consequences of interactions between quarks, it is the primitive soupling constants y,, , ye, yK*, etc., that will be related by some symmetry, e.g., SU(6). In the absence of the long-range (OPE) component of the 37~ force, the amplitudes TADn and TAG,, for the decay processes A, -+ pw and A, - E?T are given by Eq. (3.2) with one vertex removed,

(7.1)

In (7.1), X is the A-particle helicity, and for production via Pomeron exchange, the case under consideration, h = 0. The decay cross sections are given by

I %nax 42 dq uaw a = p, E. (7.2) o 2w, (wA _ MLv (8 Im G1(qy CN,


The superscript (0) in (7.2) refers to the fact that we have permitted no rescattering via OPE. In the presence of OPE, (7.1) and (7.2) are modified by the replacement

(7.3) (0)

(SAnx - uAa,,

with r, given by (3.19). Because the OPE mechanism is so weak, this modification will be small for the larger decay modes, but may be considerable in the case of the smaller ones. If we define an enhancement factor

&, z / (0) uAon uA,m > (7.3)

neglecting the K* decay mode, we see from (3.19) and (7.3) that 8, depends only on the ratio y,/yp . In Fig. 9 we plot 8, versus

3 01 GeV

-6 -4 -2 i 2 4 6

FIG. 9. Enhancement factor gp versus R.

(7.5)


where h = + 1 for r./rp > 0, and h = - 1 for yE/yp < 0. For j R [ 2 1, enhancement of the p cross section by rescattering is small and 8, w 1. But, as u$!J,, becomes much greater than u$‘J,, , for example, 1 R ] 2 5, the enhancement grows large. The reason is that even for very small direct coupling of the A1 to pr, we still can get pn final states by rescattering from an EAT state. In fact, C$ clearly approaches infinity as y,, + 0. In Fig. 9 we have given results for two 37r c.m. energies, 1.25 and 3.01 GeV. The moral to be gleaned from the above discussion is that the smaller decay modes of a particular particle may differ considerably from a symmetry consideration. On the other hand, when the kinetic energy available for the decay is large compared to the masses of the decay products, the enhancement effects seem to disappear, and symmetry predictions should presumably be correct. (It is not an accident that we chose 3.01 GeV, the mass of theJ/#particle, as our example!)InFig. 10 we give results for c?~. J?< and 8, behave differently because of the different spins of the decay products.

7.3. Data Analysis

FIG. 10. Enhancement factor 8’a versus R-l.

The parametrized solutions of our full model which sums the four diagrams of Fig. 6 have been fit to the 35~ results of Ascoli et al. [l], those of Antipov et al. [18], and to the K*K results of Otter et al. [19], using standard x2 fitting techniques. These results include O- and l+ “p” and “C’ cross sections as defined by Eq. (6.5), and relative phases of the isobar amplitudes averaged over phases space (Eq. (6.6)) at


c.m. energies ranging from 950 to 1700 MeV. The K*K states, whose importance was first emphasized by Basdevant and Berger [20], have been included only perturbatively -- in our model, in that K*K(K*K) intermediate states appear only in the A-particle bubbles; i.e., Deck terms involving K(x) exchange are excluded.The latter approximation is partially justified by the fact that the KErr differential cross section remains at 5 % of the 371 differential cross section for a wide range of initial beam momenta. Basdevant and Berger agrue that as s 4 co the cross section ratio of KErrlnrn should be + at t,in = 0, but presumably we are not in this domain because the data is much more peaked at threshold than expected from a K-exchange Deck mechanism with Stodolsky shielding.

We display our fits to the 40-GeV Serpukhov results [19] and the K*K results of Ref. [20] in Figs. I I and 12. We have presented elsewhere [21] similar fits to the results of Ascoli et al. The fits to the data presented in the figures correspond to different choices of subenergy dependence by appropriate choice of the functions F;, and FI in Eq. (4.7).

a

r

I .o 1.2 1.4 i .6

FIG. Ii. Soln 1 (-), Soln II (---): (a) I+ pi cross section in arbitrary units versus 3~ c.m. energy. Lower curves are corresponding Deck cross sections. (b) l+ ETT cross sections (with Deck). Cc) 1-l K*R cross section. (d) O- ET cross section. (e) O- p= cross section.


il _---

I-da-A 1 .o 1.2 1.4 1.6

s fMeV/

FIG. 12. l+ pn (upper curves) and l+ cm (lower curves) phases relative to O- l ; phases’of Deck plus rescattering are also shown. Soln I ( -) and Soln II (- - -).

For Soln I

For Soln II

(7.6)

(7.7)

The present fits are consistent with those published earlier [21], but with some interesting differences that will be discussed below. Our best fit to the cross sections and phases is Soln I (Figs. 11 and 12) where we obtain x2 = 68 for 48 degrees of freedom (DOF), so x2/DOF = 1.4. Solution I corresponds to x2/DOF = 1.8. It is satisfying that the better solution (Soln I) is the one most consistent with the isobar model in the sense that except for the O- p, the other amplitudes are reasonably constant as the subenergies vary over the Dalitz plot. This rapid behavior of the O- pn is shown in Fig. 13 where we plot the real versus imaginary parts of our amplitudes on an Argand diagram at a fixed cm. energy WA = 1220 MeV. In general, of course, all amplitudes have at least slight subenergy dependence, and those derived from standard isobar model fits must be interpreted as average amplitudes. It is striking that the one amplitude (O-p) that wejind to be small with rapidphase variation over the Dalitz plot, is poorly determined in previous partial wave analyses [l, 191. This behavior of the O- p amplitude is model independent and a consequence of Eq. (4.6). In this equation, B,, and Z3* have opposite sign because they represent a

ANALYSIS OF THE Al 81

O- p7r P-wave

FIG. 13. Real versus imaginary parts of the 0- prr amplitude defined by Eq. (6.1) at WA -- 1220 MeV. The off-shell momentum is varying from 0 to 4 in units of pion masses from one end of the arrow to the other; the origin is at the center of the circle.

u-channel exchange in adjacent partial waves, and the kinematics is such that if the 0- E and 1 + E are comparable in size (within a factor of 5 of one another) and independent of subenergy, the l+ p will also be reasonably independent of subenergy, but the O- p will necessarily be small and have rapid subenergy dependence. In our analysis of the Ascoli et al. results presented in Ref. [21], it was a solution with the subenergy behavior of the present Soln II which had the best x2 behavior. We feel that this is because our parametrization of the Deck amplitude works best at 40 GeV, an energy sufficiently high such that the Stodolsky shielding (described in Section 4) is almost complete. At the lower energies of Ascoli’s work, the shielding is only partial, and the Deck amplitudes can have complicated subenergy behavior. Also, at lower energies there is more N* contamination, etc., and thus, in general, we are more inclined to believe the details of our 40-GeV analysis when they differ from our previous one. We now present the results.

Examination of the A, propagator, DA( I+‘,) of Eq. (3.20) shows the presence of a well-behaved Breit-Wigner resonance. We have not prejudged the existence of this resonance by introduction of the bare A-particle. For example, a particular fit could easily place the physical A, at energies far above the data with a large width. and it would then be interpreted as a smooth background. The mass M,1 , width rtotnl and partial widths r,, , r,, , and rK+,e for Solns I and Soln II are displayed in Table III. It is interesting to note that the partial width to prr has changed from that in our previous analysis to agree qualitatively with the SU(3) predictions of Carnegie cl ul. [22]. Furthermore, even though the Breit-Wigner parameters give a resonance position of about 1500 MeV, because of dynamical effects, our direct pi production cross section peaks at about 1400 MeV and looks very much like the charge exchange results in Fig. 4 of Baltay et al. [23]. We compare our cross section with Baltay’s in


TABLE III

Results for Soln I and Soln II: A, mass, width, and partial widths to err, prr, and K*K (in MeV)

Soln I 1500 520 280 130 55 Soln II 1500 480 260 120 50

Fig. 14. We find the presence of a 1+ resonance with the above-mentioned parameters an essential ingredient in fitting the data, particularly the phase information. It is the very phase information interpreted by previous authors as showing that there is no A,, which we find, demends the A, . Finally, a parametrized Deck amplitude is obtained from the fit which may be used in other contexts. The pure Deck cross sections for Soln I and Soln II are shown in Fig. 11. Our liberally parametrized Deck amplitude can give a good fit to the cross sections, but even with rescattering it cannot fit the phases (see Fig. 12).

i+tpm,

75-----

FIG. 14. Our direct p= production cross section (via Pomeron exchange) and the charge-exchange cross section for n- + p + d++~+lr-& [14] versus c.m. energy.

1.4. Weak Points in Our Theory

(a) We do not include the S* in our S-wave nn phase shift.

(b) We have not included coupling to a D-wave amplitude in our l+ calculations. Although, all our investigations indicate that its effects are small, it might be useful to include it in a more careful manner.

(c) We have summarized the Z-N vertex by Pomeron emission. While this approximation seems reasonable, especially in the forward direction at 40 GeV, it might be that one must treat the pion nucleon interaction in a more precise manner.

(d) For reasons stated in Ref. 11 the use of the Blankenbecler-Sugar (BbS) procedure must always be questioned in any highly relativistic system. In this case


where unitarity is a crucial feature of the analysis, and complicated crossing singularities can (hopefully) be absorbed in the short-range interaction, the BbS formalism seems a highly desirable one.

(e) We have not yet put in the K-exchange Deck for K*R production.

8. SUMMARY AND CONCLUSIONS

It appears that the behavior of the Jp = l+ and O- partial wave amplitudes can be understood in terms of a production process involving interference between large nonresonance (Deck) and smaller resonance contributions. We shall refer to any model which includes coupling of the Pomeron to both long-range (background) and a short- range (resonance) mechanisms as a two-component model (TCM). As far as we know. the first such model of the three-pion system was proposed by Bowler et rrl. [8] and applied to analysis of the I+ prr channel. Our model, used in this paper, is a refined TCM which includes three-body unitarity for the first time, and thus takes resonance widths into account naturally and properly. The technical machinery in the two works referred to above is quite different; Bowler et al. use dispersion theory methods, while we use Blankenbecler-Sugar techniques. Recently, another approach, a dispersion theoretic model developed by Basdevant and Berger [20]. has been applied to the analysis of diffractively produced I-& pn and K*K data.

Shortly, we shall compare the above models and their results in some detail: first, however, we should like to give an historical perspective.

The situation in meson spectroscopy today is similar to that which existed in baryon spectroscopy in the early 1960’s when all the known resonances presented themselves as clear bumps in total cross sections. Within a short time, of course. many more resonances appeared from the study of interference effects through phase shift analyses [24]. The work of Ascoli et al. [I] using the isobar model was the first attempt at a multiparticle “phase-shift analysis.” Leaving aside questions concerning the validity of the isobar model, there still remained the difficult questions of interpreting the results because the theoretical problem of extracting subtk two-body information from three-body final states of strongly interacting particles was only beginning to be understood. For example, we showed in Section 7.3 that it was the very phase behavior interpreted by previous authors as demonstrating that there is no ~A1 . which demands the existence of the A, . The Reggeized Deck calculations of Ascoli et a/. [25] and (more recent) similar work in the Km-r system by Berger [26] represent a first generation attempt to explain the three-meson phase-shift analyses. While these calculations give important insight into the physics. any resonance information is

presumably being obtained through duality and cannot be of a detailed nature. In other words, at the low 3rr energies under consideration (l-2 GeV), the Dolew

Horn-Schmid duality principle [27] implies that Reggeized exchanges might yield an average description of direct channel resonances. To give a more complete picture one needs a dynamical theory such as ours, in which these resonances appear as complex poles in the scattering amplitudes. Particularly in such cases as the ..I, .


which is seen through its interference with a much larger Deck background, the full complexities of a three-body theory incorporating unitarity is necessary to extract resonance parameters. For example, we find that the neglect of rescattering, a relatively small effect, results in poorer fits to the data yielding a much wider A, .

Finally, let US return to the interference models disussed above. We feel that these models comprise a set of sophisticated “second generation” theories required to unravel the more subtle details of three meson systems that include comparable resonance and background contributions. It must be understood that in such cases, the results of the standard isobar model are not meaningful; i.e., the phase variations and peaks of the amplitudes cannot be interpreted in terms of resonance parameters. The first result of an interference model was that of Bowler et al. [8], who considering only the I+ p7r channel, combined a Reggeized Deck term with direct resonance production and concluded the existence of an A, with mass -1300 MeV and width ~240 MeV. Later, Morgan [28] argued strongly that including Reggeized Born terms in a unitarization procedure will in general result in severe double counting. Fitting the I+ pn- cross section using a non-Reggeized Deck background plus a resonance term he naturally arrived at an A, at approximately 1450 MeV with an ~200 MeV width. Because he did not have a real three-body theory, he could treat neither the ET channel nor rescattering properly, and was unable to explain the relative phases of the amplitudes. Considering possible difficulties with double counting in the work of Bowler et al., and the fact that both the previous authors and Morgan study only the I+ pr channel in the zero-width p limit, the results are in remarkable agreement with each other, and with those of this work. The main new feature appearing in all these calculations is a higher-energy A, than expected from simple theoretical argu- ments.

The dispersion theoretic model of Basdevant and Berger (BB) [20] has been applied recently to diffractively produced l+ px and K*K data. Only their solution B which has two channels and one A, pole is comparable with our results. Thus, we compare the two works by plotting in Fig. 15, (1) the full pn diffractive cross sections normalized at 1050 MeV, (2) the direct production and the Deck cross sections, and (3) the phases of the direct production terms. Even though the Deck cross sections are somewhat different, the direct production cross section and the phase of the prr amplitude are quite similar. BB’s Soln E which supports an A, with mass as low as 1200 MeV contains two poles. We interpret the second pole as corresponding to a low mass repulsive background in the I + channel. It serves to suppress the A, amplitude at low energies, and the phase would rise through 90”, if at all, at energies far above the pole position. Up to this point we have avoided such a repulsive mechanism in our formalism for technical simplicity and (perhaps misguided) aesthetic preference.

Finally, an interesting point has recently been made by Aitchison and Bowler (AB [29] and BB [2OJ). They show that for a one-channel production problem, for example, rf-p --f pnp, the TCM plus rescattering gives rise to an amplitude of the form

. Aez6:‘n ’ + TD,,eis cm S, (8.1)

ANALYSIS OF THE i‘f, 85

- Full Solution / I

- - Deck ---- Dire0 Term

FIG. 15. (a) Full prr diffractive cross sections of Basdevant and Berger (BB) and that of present work (not labeled) normalized at 1050 MeV. Corresponding Deck and direct terms are also given. We have compared specifically with solution B of BB. (b) Phases of direct terms.

where T,,, is defined earlier in Eq. (1.3) and 6 is the prr phase shift. For a spatially diffuse production process, as is the case in the problem being considered, AB show that the quantity A in Eq. (8.1) is very small compared to T,,, , and thus, except when 8 = 90” and the second term is identically zero, it will dominate the amplitude. (The question of whether 6 = 90” corresponds to a resonance position is one that requires further consideration in the presence of other channels and backgrounds.) However, the claimed eiscos 8 behavior can clearly be seen in our results-note in Fig. 1 la that near the position of the A, (~1450 MeV), the total cross section dips below the Deck and remains very small.

We are amazed at the general agreement among the different approaches in the research described above. There are, after all, important differences among the models. For example, Basdevant and Berger include a short-range background in the 1’ pv channel and neglect coupling to ~7r altogether, while we include ET coupling but do not have the short-range I+ p7-r background. Eventually, of course, both effects should be included. The crucial aspect of all the models is a proper accounting of interference


in a unitary framework. We and the authors of Ref. [8] also find the two-component feature of our models important for fitting the data. If particle widths can be neglected, all the above theories probably can give reliable results. On the other hand, if ET production, K*K production below threshold, and other such three-body effects cannot be neglected, then the model we have developed in this paper becomes the most trustworthy.

In conclusion, we feel that the analyses we have presented demonstrate with reasonable certainty the existence of the A, meson, and, further, explain the difficulties that have beset previous searches for this particle. On the other hand, it is forward charge-exchange experiments (proceeding through p-exchange) that are presently being performed at Argonne National Laboratory by a Carleton-McGill-Ohio State-Toronto collaboration that will establish the existence and properties of the A, beyond a reasonable doubt. This is because the amplitudes used to fit the data contain the same parameters as those used in the diffractive (Pomeron exchange) case, but the production mechanism is completely different. It is hard to imagine a simultaneous fit to the p-exchange and Pomeron exchange data occurring accidentally in view of the consistency requirements. For example, the p7~ coupling to the A, which appears in the direct production term is the same as that which shows up in the rescattering terms; different combinations of A, helicity states contribute to p-exchange than to Pomeron exchange, and so forth. We will present our predications of cross sections and phases for this analysis elsewhere.

Note added. We should like to discuss two new developments which occurred after this work went to press. First, we have used our formalism to analyze the 40-GeV Serpukhof results together with the coherent nuclear results of the CERN-TH-IC- Milan Collaboration [30]. As previously, we take the 0- 67~ partial wave as a reference. Our fits have x2 = 1.4 for 113 degrees of freedom, with A, parameters very similar to those obtained in the text for the 40-GeV data alone. We believe these results are particularly significant because the Deck background in the coherent nuclear situation is much different and better understood than for production from protons. We find this effect to be easily explained by the presence of the one-pion pole in the Deck mechanism. Also, in the heavy nucleus case, for the first time, there is measurable I+ D-wave p7-r production; this can be explained by the pion pole plus a small amount of resonance coupling to the D-wave prr system. The backgrounds in diffractive production off protons and coherent production off nuclei are so different that we are tempted to claim that the consistent results that we obtain cannot be accidental, and an A, exists with the parameters obtained from the fits. However, recent colliding beam data involving 7 decay suggests the presence of a low-mass A, (-1100 MeV) [31]. While the latter work is preliminary we feel that it is sufficiently impressive to warrant further search for a low-mass A, in the strong interaction data. It has been shown by several authors [6, 301 that a fit which contains a low-mass A, must necessarily have a 0- resonance at about the same mass. Previously such solutions were not taken serious- ly by most authors, but the colliding beam results force us to reevaluate the situation. We are proceeding to reanalyze the l+ and 0- 3~ and K*K data permitting a O-

ANALYSIS OF THE Al 87

resonance by including a short-range interaction (bare A-particle) in the O- channel. These results will be published elsewhere.

APPENDIX: WATSON FINAL STATE THEOREM FOR TWO-COMPONENT MODEL

Following Aitchison and Bowler [29], we assume a two-component isobar model describing the process r + P ---f r + p where p is considered to be a stable particle. We write schematically

T = Tre” + BP , (Al)

Tres = 1’$7-, , (A3

7c ::= $8 sin S/q. (A3)

In the above equations, BP is the (real) Deck amplitude, up is the isobar amplitude for A, production, and 71 production, and T, is the elastic p7~ scattering amplitude which is assumed to be pure S-wave, with 6 being the phase shift. The unitarity relations for T and 7,. are

2; Im T = T’(2iy) 7,. . (A4)

2i Im T, :::: 7,:(2iq) 7, . (AS)

where (2iq) is our defined normalized expression for the integral over two-body phase space. Noting that BP in (Al) is real, and subsituting (AI) and (A2) into (A4) and (A5), we may now write

Im(zpT,) = q($~f -- BP) ‘T,, . (Ah)

Im 7,. = L/ I 7,, ‘2. (A7)

Using the relation Im(ab) = a*Im(b) + bIm(a), and substituting (A7) in (A@ yields

Im up = qBP . (A81

We finally obtain the Aitchison-Bowler result by writing z:$ in terms of its real and imaginary parts, and rewriting (Al) using (A@, (A2) and (A3):

T = (Re I‘~ L iqBP) et6 sin S/q -- B,

= (Re c~) eid sin 6/q 4 BPei” cos 6.

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analysis of the a1

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