Baryon Annihilation and Exotic Exchange

3078 A . M c D O N A L D A N D L . R . R A M M O H A N 3 -

*Work supported by the U. S. Atomic Energy Commis- (1970) ; also see S. S. Shei, Phys. Rev. 188, 2274 (1969). sion under Contract No. AT(l1-1)-1428. %. CronstrBm and M. Noga, Nucl. Phys. z, 61 IS. Weinberg, Phys. Rev. 177, 2604 (1969); Phys. Rev. (1970); M. Noga and C. Cronstr8m, Phys. Rev. D I.,

Letters 22, 1023 (1969). 2414 (1970). 's. Fubini and G. Furlan, Lett. Nuovo Cimento 3, 168 4 ~ . Pasupathy, Phys. Rev. D 2, 357 (1970).

P I IYSICAL REVIEW D VOLUME 3 , N U M B E R 1 2 1 5 J U N E 1 9 7 1

Baryon Annihilation and Exotic Exchange*

Charles B. Chiu Culgorniu Institute of Technology, Pasade?za, Calgorniu 91109

and Institute f o r Theoveticul Physics, State University qf .Vezu York , Stony Bvook, A'ezc; York 11790

and

Rudolph C. Hwa Insti tutefov Theoretical Physics, State Uniijersity of Neu: Yovk, Stony Bvook, New Yovk 11790

(Received 1 October 1970; revised manuscript received 27 January 1971)

Within the framework of duality and quark diagrams, we suggest the importance of connect- ing the annihilation processes in BB collisions to the exchange of an ( E ) trajectory with qq@ quark content. It is argued that this E should he an exchange-degenerate pair with natural parity and I = 0. It is coupled strongly to the BB system at t = O . For the positive-t region, it is expected to be manifest as a physical vector meson with G = -1 at around 1 GeV. Available data do not rule out its existence.

I. INTRODUCTlON

In recent y e a r s the idea of duality has been proven to b e successful in the study of hadron phy- s ics . The relationship between the Regge poles in the s and t channels is made part icular ly t rans - parent by the u s e of quark diagrams. ' However, the application of the quark d iagrams to the baryon- antibaryon sys tem is hindered by the appearance of four-quark (qqqq) s ta tes , which cannot be inter - preted in the conventional duality picture based on the usual quark model. It is the purpose of this paper to give a specific interpretation f o r these four-quark objects i n t e r m s of the annihilation process in the s channel and to cor re la te them with exotic Regge t ra jec tor ies in the t channel.

L e t u s f i r s t review the s ta tus of o u r p resen t understanding about the exotics. All the well- known mesons have thus f a r been classified i n the 1 and the 8 representat ion of S U ( 3 ) . This regularity - suggests that these mesons a r e made out of a q7j pair . The underlying dynamical reason f o r having these mesons associated with only a single yi j pa i r instead of the exotic multi-q?j pa i r s is not c lear . In fact, the possibility f o r the existence of mesons made out of the qqqq exotic s ta tes h a s been specu- lated by various authors , for example, in connection with the quark graphs, and a l so in connection with the explanation of the A, ~ p l i t t i n g . ~ Experi-

mentally, much effort has also been devoted to the s e a r c h f o r the I - Y exotic part ic les , belonging to the 10, lo*, and 2 7 representat ions. No conclusive - - evidence has beenadvanced f o r their e ~ i s t e n c e . ~ On the other hand, i n the crossed-channel physical region, there s e e m s to b e s o m e indication in favor of the exchange of the I - Y exotic quantum number. The effect observed h e r e i s relatively weak and i t could alternatively b e attributed to the contribution of Regge cuts. It s e e m s that Regge t ra jec tor ies associated with the 10, lo*, and 27, if they exis t a t - - al l , a r e only weakly coupled to thehadronic system. Fur thermore , the lack of evidence f o r the c o r r e s - ponding physical par t ic les suggests that these t r a - jector ies may turn over in the positive-t region before reaching the lowest physical angular momentum, a ra ther common phenomenon i n potential scat ter ing.

The exotic t ra jec tor ies which we consider i n this paper differ f rom the ones discussed above. Their exis tence is motivated by the appreciable c r o s s sect ion of baryon-antibaryon annihilation processes , s o they mus t couple strongly t o the baryon-antibaryon sys tem a t l e a s t a t t = 0. They appear i n an exchange-degenerate pa i r , which we shal l label5 collectively by t. Some consequences of our p r o - posal a r e a s follows:

(1) The energy dependence of the baryon annihilation c r o s s sect ion should exhibit a Regge b e -

B A R Y O N A N N I H I L A T I O N AND E X O T I C E X C H A N G E 3079

havior with om, -s-lI2. (2) The annihilation c ros s sections for PP and

p~ should be comparable. (3) The conventional Regge-pole model for ba r -

yon-baryon and baryon-antibaryon scattering should be modified with the inclusion of the E contribution.

(4) There should be a new vector meson in the 1-GeV region, the existence of which i s also suggested in Ref. 6 in connection with the explanation of the width of the diffraction peak for NN scattering.

It. THE CONVENTIONAL REGGE-POLE MODEL

We review briefly the conventional Regge-pole description for the nucleon-nucleon and the nucleon- antinucleon total c ros s sections. The general features of the data a re illustrated schematically in Fig. 1. Here one assumes that

and

where for example the te rm up stands for the Pomeranchon contribution to the total c ross section,

with k being the c.m. momentum. The symbol A, stands for the t-channel helicity -nonflip amplitude which is parametrized by

with 7 , being the signature. Other t e rms in Eq. (1) a r e given by similar expressions. Since ap(0) = 1, the Pomeranchon contribution gives the asymptotic cross section. The other te rms with (r(O)< 1 contribute to the energy-dependent part of the cross section. Notice that the sum of the energy-dependent par t of the pp and @ cross sections i s much larger than the difference of the same cross sections. Also, the pp and the pn cross sections a r e relatively flat. These two features together imply that the f and w contributions (I, = 0) dominate over the p and the A, contributions (I, = I) , and that the conditions implied by exchange degeneracy a re approximately satisfied, i.e.,

For our discussion below we shall ignore the I, = 1 contribution and consider only the pp and pp c ross sections. For later convenience, we use R to designate f and w collectively. Thus from Eq. (4) we get

u(pfi) =up + a r - 0,- up,

and

a ( ~ p ) = o p + u R , w h e r e u , = o f + o , .

The quark diagrams' for the Pomeranchon exchange to both pp and $# scattering a re illustrated in Fig. 2(a). For 3p scattering with R exchange the diagram i s shown in Fig. 2(b). According to the Freund-Harari hypothesis, the Pomeranchon in the t channel i s dual to the background contribution in the s ~ h a n n e l . ~ The t-channel R trajec-

FIG. 1. A schematic illustration on the total cross sections for the nucleon-nucleon and the nucleon-antinucleon scattering. For detailed data, see Ref. 9.

FIG. 2 . Quark graphs for (a) the Pomeranchon exchange, (b) the R exchange, and (c) the 6 exchange.

3080 C . B . C H I U A N D R . C . HWA

tory by assumption i s made out of qq a s i l lustrated in Fig. 2(b). Notice that h e r e the s-channel intermediate s tate consis ts of the qq@ l ines . Unlike the situation for the meson-meson scattering, it i s well known that for the baryon-antibaryon case , the nonexotic mesons of one channel cannot "bootstrap" the nonexotic mesons in the c rossed channel. Also, in a conventional Regge-pole model, according to Eq. (5), the graph such as the one shown in Fig. 2(c) is d i smissed altogether.

111. THE ANNIHILATION CROSS SECTION AND EXOTIC EXCHANGE

In this section we presen t our argument that the diagram such a s Fig. 2(c) should not be discarded, in contrary to the usual Regge-pole model men- tioned above. T h i s diagram is expected to give an important contribution to the c r o s s section.

To s e e this , we s t a r t with the s-channel unitarity relation for baryon-antibaryon (BB) scat ter ing

The intermediate s ta tes in the above unitarity s u m a r e divided into two categories: those s ta tes (labeled a) without any BB p a i r s and those ( P ) with BB p a i r s . Thus A,, i s the annihilation amplitude while A,, is the elast ic o r production amplitude. Our task below is to motivate a relationship between the annihilation c r o s s section and the imaginary p a r t of Fig. 2(c).

We s t a r t with the annihilation amplitude for the

p rocess BB-MM, where M signifies a meson.8 A complete s e t of quark d iagrams for this amplitude i s i l lustrated i n Figs. 3(a)-3(c), which we denote, respectively, by Fa, F,, and F,. In the dual-resonance model the amplitude F for annihilation into two mesons is a sum of these th ree t e r m s :

It i s evident f rom Fig. 3(a) that F, represen ts the t e r m which has poles in the s and t channels, dual with respec t to each other. Similarly, F, is the s u t e r m , and F, i s the tu t e r m .

In computing the two-meson contribution to the f i r s t t e r m on the right-hand side of Eq. (6), we have

where the integration is over al l angles of the two- meson sys tem. F o r the purpose of computing the annihilation c r o s s section, the over-all t of A, , i s ze ro . In the energy region that we shall consider , each F in the integral has significant contribution only in the peripheral regions. Hence, the integral may be divided into two parts : One is over the forward region where the dominant contribution i s associated with the 2-channel baryon exchanges a r i s ing f rom the t e r m s Fa and F , ; the other is over the backward region where the u-channel b a r - yon exchanges due to F, and F, a r e important. Substituting Eq. (7) into Eq. (8) yields nine bilinear t e r m s in the integrand and, consequently, 18 pieces of integrations in general, counting forward and backward p a r t s separately. Per iphera l dominance implies that the t-u interference t e r m s a r e negligible; they a r e symbolically

FIG. 3. Quark diagrams for the BB - MM amplitude: (a) st dual diagram F a , (b) su dual diagrams F,, and (c) tu dual diagrams F , .

FIG. 4. Quark diagrams for (a) Fa F:, (b) Fa F $ - F, F: , and (c) F, F:.


where the labels F and B under the integral signs denote forward and backward regions, respectively. The t e rms

a r e small to the second degree, thus even more negligible. The only t e rms that we need to consider a r e then

IrnAii (MNI) = I , + I,. (11)

Using the quark diagrams in Fig. 3, we exhibit the four t e rms of I, in Fig. 4 in the same order. Iden- tical diagrams apply also to I, if the exchange channel of each half of each of the diagrams i s interpreted a s the u channel.

For annihilation into many mesons the quark diagrams corresponding to Fa, F,, and F, a re a s shown in Figs. 5(a), 5(b), and 5(c), respectively. In the dual model the production of the mesons may be described either by a multiperipheral a r ray , or a s decay products of resonances. It i s easy to see in the multiperipheral description that when these multimeson annihilation amplitudes a r e inserted

FIG. 5. Quark diagrams for the multiparticle annihilation amplitude: (a) st dual diagram G, , (b) su dual diagram G , , and (c) tu dual diagram G, .

into the unitarity equation, the dominant t e rms a r e those in which each pa i r of multi-Regge links a r e either both forwardly peaked or both backwardly peaked. Thus the two major pieces of unitarity integrals for each intermediate state of a fixed number of mesons a r e analogous to Eqs.(9) and (10). Alternatively, these expressions can also be obtained by considering diagrams with only one Regge link between two clusters of particles. Duality does not prefer one over the other mode of description. The four t e rms in (9) a r e consec- utively illustrated in Fig. 6.

The problem now i s to determine which quark diagrams of A i i among those in Fig. 2 have the appropriate imaginary par t s corresponding to the various t e rms in Figs. 4 and 6. We must admit at the very outset that we have no rigorous way of establishing this correspondence, since the application of unitarity to dual diagrams have never been worked out. Another way of expressing the difficulty is that we have no consistent rules to be applied to the quark diagrams when the intermediate state in the unitarity equation i s integrated and summed over. We suspect that this i s related to the difficulty of unitarizing the dual-resonance amplitude. Fortunately, what we need here is not a quantitative determination of the correspondence, but a qualitative motivation for the existence of the exotic exchange, Fig. 2(c). This we shall be able to do.

Let the BB state be divided into two types: those having one qq pair annihilation, li,), and those having two pa i rs annihilated, li,) . We need not consider those s ta tes that have no quark annihila-

FIG. 6. Quark diagrams for (a) G,G:, (b) G , G ~ + G,G:, and (c) G, G!.

C . B . C B I U A N D R . C . H W A - 3

tions since we a r e interested in the annihilation par t of the c r o s s section. Clearly, the diagram i n Fig. 2(b) corresponds to ( i , l ~ l i , ) while that i n Fig. 2(c) corresponds to (i,lAli,) . Presumably, taking the imaginary p a r t s of these amplitudes does not change the topology of the quark lines. Now, the right-hand side of the unitarity equation i s given by the sum of the d iagrams of the types shown i n Figs. 4 and 6. It i s obvious that the dia- g r a m s i n Figs. 4(a) and 6(a) should contribute t o ~ m ( i , l ~ l i d while those in Figs. 4(c) and 6(c) should contribute to ~m(i , /Ali , ) . The ambiguity l i es in the interpretat ion of Figs. 4(b) and 6(b). A c r o s s t e r m between / i,) and li,) cannot direct ly be identified with a meaningful diagram for BB scat ter ing. If one observes that the c r i s s c r o s s l ines in these d iagrams s t a r t and end at the s a m e BB s ta tes , and thereby in te rpre t s these d iagrams a s contributing to 1m(i,lAii2), then two fea tures ensue that a r e not a t t ract ive. F i r s t , a n li,) s ta te i s turned into a n ii,) s t a te by the integration over the phase space. Secondly, such a ru le i s not helpful in understanding the meson scat ter ing problem. Rejecting such an interpretation, we a r e forced to suppose that these c r o s s t e r m s contribute part ly to ~m(i,lA(i,) and part ly to ~ m ( i , l ~ l i , ) , the relat ive proportions being unknown.

Let us now at tempt a rough es t imate of the strength of the exotic exchange. In the in te res t of c lar i ty we confine our attention h e r e to the two- meson annihilation case , the extension to the multimeson c a s e being s traightforward. Moreover, to avoid repetition, we examine explicitly only the forward integral IF a s given by Eq. (9); the pro- cedure for studying the backward integral I, i s s imi la r . F o r fixed t and l a r g e s, the Regge behav- i o r s fo r Fa and F, a r e

We have included in Eq. (13) the s ignature factor . L e t the baryon be the nucleon, and the meson be the pion. Then the relevant t ra jec tor ies exchanged a r e the N, (even s ignature) and the A (odd signature) . The signature of the t rajectory affects only the over-al l sign of the c r o s s t e r m s F,FJ +F,F;. Evidently, Eqs. (12) and (13) imply

We s e e f rom Eqs . (14) and (16) that the contributions of F,F; and F , F: to 1m(i,l Ali,) and ~ m ( i , I ~ ( i , ) respectively, a r e equal and positive. The ambig- uous t e r m , Eq. (15), i s positive o r negative de-

pending upon the value of a . It is known empi r - icallyg that

Thus near 2=0 where the right-hand s ide of Eq. (15) i s g rea tes t in magnitude, i t s sign i s negative f o r both s ignature cases . Posi t ive value com- mences fo r t s -0.6 (GeV/r)' in the c a s e of N, exchange, but fo r t 5 -0.16 (GeVjc), in the c a s e of h exchange. In spi te of the s t rong damping of the residue factor P(t) a t l a rge -t , this oscillatory behavior does provide a suppression effect on the contribution of the c r o s s t e r m s to the integral over t i n Eq. (9), although not enough to make i t positive jn the net. We therefore a rgue that the contribution of the c r o s s t e r m s , when divided between ~ m ( i , ( Ali,) and 1m(i,i Ali,), does not upset the rough equality of the two established above. It is interest ing to note that in the meson-meson scat ter ing c a s e the c r o s s t e r m is proportional to c o s n a which i s very smal l near t = 0, and i s there- fo re insignificant in i t s contribution to the unitarity integral .

The multimeson contribution can be considered s imilar ly. We can then conclude f rom the above discussion that Im(i,lA li,) contributes to roughly half of the total annihilation c r o s s section. Actu- ally this is an underest imate. This is because there exis t additional contributions to 1rn(z,lA li,) ar i s ing f rom cer ta in t e r m s in the second sum of Eq. (6). Those t e r m s may be represented by dia- g r a m s s imi la r to the one in Fig. 6(a) except that one o r m o r e of the single closed loops a r e replaced by double concentric loops. These contributions a r e positive definite and therefore increase the above est imate.

The experimental annihilation c r o s s section tu rns out to be quite sizable. In fact, i t s magnitude is comparable to the difference between the $p and pp c r o s s sections. F o r example, we have, lo a t 3.28 ~ e v / c , oa,,(Pp) =33.8 i 3 . 2 mb and ~ ( p p ) - o@p) =32.2*2.3 mb, and at 7.0 GeV/c, ua,,(j5p)=23.6 1 3 . 4 mb and o(3fi)- o(pp) = 17.6* 1.2 mb. Over half of this annihilation c r o s s section comes f r o m the diagram in Fig. 2(c). A c r o s s section of such a magnitude mus t be accounted for by a significant t-channel exchange. We therefore propose that the qyqq object exchanged in the t-channel is to be represented by a Regge pole, labeled E , which i s dual to the s-channel annihilation processes . Thus, con t ra ry to the conventional Regge-pole model in which the en t i re difference between the $p and pp

, c r o s s sect ions i s explained by a pure qq exchange, h e r e we a s s e r t that a significant portion of this difference should be accounted f o r by the qqqq

3 - B A R Y O N A N N I H I L A T I O N AND E X O T I C E X C H A N G E 3083

exotic pole exchange. As will be shown below, this exotic pole i s expected to be high lying, in contrast to the usual notion of low-lying exotic cuts arising from the elastic iteration of ordinary poles. The departure originates in the completeness of our unitarity sum in which the number of annihilation channels is numerous even a t moderate energies, a circumstance where the usual Regge-cut pre- scription i s known to be inadequate.

IV. PROPERTIES OF E

1. J-parity. Since the E contributes to the total c r o s s section, i t has the J-parity, P(- 1)" = + 1.

2. Isospin. To investigate the isospin of the E ,

i t i s useful to look a t the quantity

where

For annihilation at r e s t , '' i t i s found that this rat io is 1.45 i0 .07 . It follows then that ~ ~ 0 . 2 . Con-

sequently, i f the notion that the annihilation c ros s section i s governed by E exchange is valid at XN threshold, then the isovector part of E must be small. In fact, i t should pers is t in being small even at higher energies for the following reason. The I, = 1 channel of NN consists of p, A, and the isovector t. The amplitude for this channel i s small because the c r o s s sections for pn- np and B - E n a r e both small. The p and A, contributions a r e by themselves small a s can be inferred inde- pendently from the K - p - p n c ros s section. This is because E does not couple strongly to KK a s will be discussed below under Sec. IV8. Thus there i s no compelling motivation for introducing an isovector E.'' Henceforth we shall regard E as an isoscalar object. Since the annihilation c ros s sections at high energies now receive contributions from the two isoscalar objects E and w , i t i s inter- esting to note that our analysis suggests

at high energies. 3. Exchange-degenerate pair . The absence of

the annihilation c ros s section in the pp channel together with the fact that f and w a r e exchange- degenerate implies that the E should be a pair of exchange-degenerate trajectories. We denote the odd signature one by E , and the even signature one by E , . Except for the quark content and a possible difference in SU(3) assignment, the E , trajectory has the same quantum number a s that of w and the

E~ that of j. The contribution of E , and E , to the 3p elastic amplitude at t = 0 i s given by

A, - -2pe-in a,(o) aE(o) 9 (22)

where a, (0)-$. (See Sec. IV 4 below.) Since annihilation c ros s section i s a positive-definite quantity, we must have Im A, >O. This in turn implies that a, does not choose nonsense at a, = 1; otherwise, we would have 0 < 0.13

4. Zero intercept a, (0). The low-energy pp differential c ros s section i s approximately satu- rated by the optical c r o s s section.I0 This implies that the $p elastic amplitude is roughly pure imaginary at t = 0. Now, we know that the contribution from the P i s purely imaginary, while that from the R i s also nearly so, since a,(O) =. g . Thus the E par t of the amplitude must also be imaginary. This agrees with the intuitive notion that the annihilation process i s purely absorptive. The phase factor of the E amplitude i s therefore

o r (23)

cu,(O)=$.

The difference between the and pp total c r o s s sections should be accounted for by a , and a , The observed energy dependence of this difference i s indeed consistent with the power law sa-', where a = L 1 4

2 .

5. Slope of a , (t). In our present scheme, the exchange of the E in the t channel is dual to the annihilation process in the s channel. From in- spection of Fig. 2(c) the lat ter can alternatively be described by the R trajectories and their daughters in the direct channel. If we assume the approxi- mate validity of local duality, then the slope of a , (t) in the negative t region near t = 0 should be similar to that of the ordinary trajectories.15 This argument on the slope is not applicable in the positive t region.

6. SU(3) assignment. The qqqq states can be in the 1,8,10,10*, and 27 representations. However,

- - - - since the E has the number I = 0 and Y = 0, i t cannot be in 10 o r lo*. From the SU(3) Clebsch- Gordan coefficients one finds that a 27 for E would lead to v=t, where r i s defined in ET (20). Thus Eq. (19) implies that q = -7. This is ruled out since the E contribution to u(3p) and o($n) must be positive definite. Hence E must belong to ei ther 1 o r 8. -

7 . Quark spin. For a qqqg system, there a r e three possible values for the total quark spins, j = 2, 1, and 0. Denote the relative angular momentum between qq and pg by L. We assume that the Regge recurrences correspond to increasing integer values of L. Then for L =0 , the total angu-

3084 C . B . C H I U A N D R . C . H W A 3 -

lar momenta J and the pari ty P of these s t a t e s a r e JP=2 ' , I+, and 0'; fo r L = l , they a r e J P = 3 - , 2-, I-, and 0-, e tc . Consider now the L = O s ta tes . The 2' s t a te cannot l i e on the t t ra jectory, s ince the associated 1- s ta te would have to be nonsense, a situation which violates Sec. I V 3 above. The 1' s ta te h a s the wrong J -par i ty ( see Sec. IV 1). The 0' s t a te can, and we suggest that i t does, l i e on the E t ra jectory. Now, according to Sec. IV4, we have a, (0) =i, s o the point a, = 0 o c c u r s in the negative t region. T h e r e can be no physical par t ic le there. The physical par t i c les on the exchange-degenerate t t ra jectory a r e then 1- a t L = 1, 2' a t L = 2, and 3- at L = 3 , etc. This t rajectory i s to be associated with the SU(3) octet o r singlet. Whether it i s in a pure 1 o r 8 o r a mixture of the two is to be deter- m i n e d b y the appearance o r absence of the other SU(3) par tners . We r e m a r k i n passing that if i t i s to be identified with the object needed in Ref. 6,

FIG. 7. The coupling of the E, meson to three pions via an i n t e r m e d i a t e f i loop.

double Reggeon exchange, the amplitude can be symbolically written a s

then i t must not be in a pure 8. Since t i s the + [A , +AE@(Ap+AR + A E ) ) . (25) leading qqqq t ra jectory, the symmetry implied by the quark model must be s o badly broken that the The t e r m s on the right-hand side involving the

t ra jec tor ies with the t r iplet quark-spin and other convolution (8 a r e cut contributions. Specifically,

SU(3) multiplets a r e much lower lying. We do not the t e r m Ap#3AE, for example, is i l lustrated i n

have a theory for such a breaking. Fig. 8 where the two shaded blobs contain the

8. Decay modes. We adhere to the r u l e s of "third" double-spectral functions. Within our p resen t scheme, in the s-channel Refs. 1 and 2 for the quark diagrams. In part icu -

physical region near t = 0, the duality picture can l a r , the t t ra jectory couples direct ly to the BB

be schematically represented a s follows: sys tem but not to the nonexotic meson sys tems . I t s coupling to these mesons can be achieved only through a second-order effect, e.g., through a BB loop. T h i s is i l lustrated in Fig. 7. If t h e r e is a vector meson on the E t ra jectory, we expect that i t decays weakly into n'n-no (or ap) and ny. The relat ive ra t io fo r these two decay modes depends on the m a s s of this 6, vector meson and can b e determined i n a manner s i m i l a r to that fo r the w.16

V. CUTS .4ND DUALITY

Within the pole approximation our proposal im- pl ies that

and (24)

Thus the difference between the pb and $p c r o s s sect ions i s given by oR + o E . Since the annihilation c r o s s sect ion we considered is approximately equal to this difference, i t appears f rom Eq. (24) that the R contribution should be a t mos t about half that obtained f r o m the usual Regge-pole model. In real i ty , our es t imate on the R and E contribution i s expected to be obscured somewhat by the cut contribution. F o r example, to the o rder of the

In Eq. (26) the t-channel t contribution together with the appropriate cu t s i s dual to the s-channel Regge t ra jec tor ies R and the i r daughters having the qQ quark content and p a r t of the production c r o s s section. The other energy-dependent piece in the t-channel amplitude given by the left-hand s ide of Eq. (27) is dual to t plus other contributions in the s channel. We do not expect that the E and i t s daughters could provide an adequate description for the qq$$ sys tem in the physical s region in spite of the fact that the t-channel t t ra jectory a t t = 0, being a high-lying J-plane singularity, does control the asymptotic behavior of the s-channel annihilation c r o s s section. In the physical region of the t-channel, near s = 0, the duality relat ion is identical to that given by Eqs. (26) and (27) except fo r the interchange of s and t .

VI. DETECTION OF THE e, MESON

As i n the c a s e of the 10, lo* and 27, where the - - t r a jec tor ies a r e conjectured to t u r n o v e r before reaching any physical angular momentum states , i t i s conceivable that the E being made out of qqqq could eventually a l so tu rn over. However, s ince


FIG. 8. A diagram for the P-E Regge cut.

a , (0) = 9 and the local duality picture favors a , ( t ) to have the usual slope in the negative t region near t = 0, we suggest that it should c ros s at least the J = 1 value. If one allows an uncertainty of 1 GeV2 in extrapolating the trajectory from a , = + to a , = 1, the trajectory a , ( t ) would c ross the J = 1 point somewhere between 0.7 and 1.2 GeV. This mass range i s compatible with what i s needed in Ref. 6. Provided that the NR vertex function does not have too strong a dependence on the nucleon four-momentum, we anticipate that such a vector meson should be detected. Because of the coupling scheme discussed in Sec. IV8, this meson can be effectively produced only in the baryon-antibaryon annihilation processesf7 and in backward scattering processes involving baryon exchange. We have looked a t some a+n-a' invariant-mass plots from 3 p annihilation. Also, in the reaction1' K-p- Aon+n-no, the n+.ir-no invariant-mass plot associated with the forward A has been compared with that of the backward A. Although there i s no obvious signal for E,, those data we have studied a r e relatively poor in statistics and they do not rule out the possibility for i t s existence. However, the data do seem to suggest that, if E , exists, i ts coupling to NX i s no stronger than that of ~ ' ( 9 5 8 ) .

The situation i s more uncertain in the energy region beyond 950 MeV. Future experiments involving a careful study of the 3n invariant-mass plots for the backward - a s opposed to the forward - produced baryons in the neighborhood of the 1-GeV region will be a crucial tes t deciding on the existence of such an object. Also, it might be of interest to study the neutral missing-mass spectrum again near the backward direction for reaction MB- B+ (missing mass)' in the event that cl has some unusual prominent decay modes.

V11. CONCLUSION

The duality picture for the baryon-antibaryon system i s investigated. We point out the importance of the special role which annihilation processes play in the BB system. We have seen that a significant par t of the annihilation processes a re dual to the exchange of an exotic E . We have studied the properties of this E and considered i t s effects on the phenomenology of the total c ros s sections. Available data on the relevant mass spectrum does not rule out i t s existence. Future experiments with better stat ist ics a re suggested for i t s detection.

ACKNOWLEDGMENTS

One of u s (C.B.C.) thanks Professor C.N. Yang for extending to him the hospitality of The Institute for Theoretical Physics at Stony Brook during his summer visit when this work was done. We thank R. Eisner, T . Ferbel, and K. W. Lai for helpful discussions.

*Work supported in part by U. S. Atomic Energy Com- mission under Contract Nos. AT(11-1)-68 and AT(30-1)- 3668B.

'H. Harari , Phys. Rev. Letters 22, 562 (1969); J. Ros- ne r , ihid. 22, 689 (1969).

2 ~ . G. 0. Freund, R. Waltz, and J. Rosner, Nucl. Phys. z, 237 (1969).

3 ~ . C. Arnold and J. Uretsky, Phys. Rev. Letters 23, 444 (1969); S. F. Tuan, ibid. 2, 1198 (1969).

4 ~ e e J. Rosner, "Review of Exotic Rlesons , " in P r o - ceedings of the Philadelphia Conference on Meson Spec- troscopy, 1970, edited by C. Baltay and A. Rosenfeld (Columbia Univ. P re s s , New York, 1970). he symbol E stands for "annihilation" ( ' E ~ ( Y $ & u ~ u L u )

o r "exotic" ( ' E ~ W T L K ~ U ) . We suggest that this designation should supersede the previous label for the "daughter" of p , which according to the Particle Data Tables should be refer red to a s the ~ ~ t ( 7 0 0 ) particle.

6 ~ . B. Chiu and R. C. Hwa, Phys. Rev. D (to be pub- lished).

7 ~ . G. 0. Freund, Phys. Rev. Letters 3, 235 (1968); H. Harari , ibid. 20, 1395 (1968).

do not consider he re the production of the exotic mesons ( E ) being proposed in this paper. As will be- come c lear in Sec. IV, their production cross section i s expected to be smal l compared to the total annihilation cross section.

9 ~ . Barger and D. Cline, Phys. Rev. Letters 16, 913 (1968). "For the j5p annihilation data, s ee T. Ferbel et a1 .,

Phys. Rev. 137, B1250 (1965); T . Ferbel et a1 ., Nuovo Cimento 38, 12 (1965). The value for the Fp annihilation quoted a t 7 GeV/c i s for annihilation into pions plus the njz + ~ + + r - - . . . states. The latter is of the order of 2 mb and i s comparable to the strange-particle annihilation cross section. The e r r o r quoted i s somewhat underesti- mated. We thank Dr. T. Ferbel for discussions on this point. For the total c ross section data, s ee B. Escoubes et d ., Phys. Letters 5, 132 (1963); W. Galbraith et a1 ., Phys. Rev. 9, B913 (1965); and D. V . Bugg e t al., ibid.

3086 C . B . C H I U A N D R . C . HWA 3 -

146, 980 (1966). - "A. Bettini et a t . , Nuovo Cimento m, 643 (1967). I21'he reasoning leading us to the conclusion that an iso-

vector E , i f existent at al l , i s low lying o r weakly coupled is based on phenomenological observations. This i s not surprising from the theoretical point of view. The inelastic process &-En i s in a pure I ,= 1 state. Annihilation contributions to its absorptive part does not add coherently, since it lacks the positivity condition which applies only to the imaginary part of the elastic amplitude. Indeed, Im[A (Fp - Kn)] ,,, a &[a,,($p) - u,,(@z)I, which i s smal l a s has already been pointed out.

I 3 ~ e r e we assume that, if a , were to choose nonsense, /3 would have a simple zero at a , = l . This i s s o in the simple Veneziano model, for example. I%. B. Chiu, S. Y. Chu, and L. L. Wang, Phys. Rcv.

161, 1563 (1967). T~. Schmid, Phys. Letters 20, 689 (1968); C. B. Cbiu

and A. Kotanski, Nucl. Phys. E, 615 (1968). I'M. Gell-Maim, D. Sharp, and W. G. Wagner, Phys.

Rev. Letters 8, 261 (1962). '%e note he re that resonance production cross section

in the tnultiparticle final states for $p annihilation a t r e s t has been explained in t e rms of the statistical model plus final-state interactions. [See L. Montanet, in Pro- ceedings of the Fifth International Conference on Ele- mentary Particles, Lund, Sweden, 1969, p. 218 (unpub- lished).] This suggests that the number of events involving t i formation in the final states having n pions (typically for n 2 5) i s directly correlated to the coupling of cl- 37r. Since r(tl- ST) << - 37r), we expect that the number of el events is suppressed as compared to that of w . The t i events should be much less contami- nated if one looks only at the peripheral processes in p 7 p - ~ O ~ ~ , & - t i p , K - P - I ~ ~ E ~ , etc.

I8we thank Dr. R. Eisner for providing us with the 3n invariant-mass plot from the reaction K -p - ~ n ' a - a ~ .

P I I Y S I C A L R E V I E W D V O L U M E 3 , N U M B E R 12 1 5 J U N E 1 9 7 1

Infinite Set of Quasipotential Equations from the Kadyshevsky Equation*

Robert J . Yaes Centev fov Pavticle Tlzeovy, Department of Physics, Lhzvevsity of Texas at Austin, Austin, Texas 78712

and Depautment of Physics, Illinois Institute of Technology, Chicago, Illinois 606167 (Received 22 February 1971)

We show that by modifying the propagator in the Kadyshevsky equation, we can obtain an infinite se t of quasipotential equations which satisfy both Lorentz covariance and elastic uni- tari ty and of which the Logunov-Tavkhelidze-Blankenbecler-Sugar-Alessandrini-Omnes equation and the Gross ecluation a r e special cases. We also show that the perturbation scheme of Chen and Raman, for using the quasipotential equation to obtain approximations to the Bethe- Salpeter equation, can be greatly simplified by the use of resolvent-identity-type arguments.

In potential theory, the off-shell T matrix satisfies the Lippmann-Schwinger equation, the integral form of the Schrddinger equation. Since the free-particle Green's function has the appropriate discontinuity, elastic unitarity i s guaranteed by the equation if the potential i s real and symmetric.'

In the relativistic case, we do not have a simple equation like the Schrddinger equation, so we must resort to the techniques of field theory. However, equations of the same form as the Lippmann- Schwinger equation can prove useful. The most common example i s the Bethe-Salpeter equation in the ladder approximation. The t e rms obtained from iterating this equation correspond to indivi- dual Feynman diagrams, the so-called ladder graphs. There i s thus a simple immediate connection with field theory. In addition, it can be shown that elastic unitarity i s exactly satisfied between the elastic threshold and the threshold for production.'

There a re , however, serious difficulties associated with the Bethe-Salpeter equation. Since it is a four-dimensional integral equation, it only r e - duces to a two-dimensional integral equation upon taking a partial-wave projection. In addition, there a re the difficulties associated with the indefinite- ness of the Lorentz metric, making the equation difficult t o deal with except in simple models such a s the Wick-Cutkosky model.

In order to circumvent these difficulties, a sim- pler equation has been proposed by Logunov and Tavkhelidze,"~ Blankenbecler and SugarJ4 and by Alessandrini and O m n e ~ . ~ This equation i s mani- festly covariant, and the Green's function i s chosen to have the discontinuity that will insure that the solution satisfies elastic unitarity for a real symmetric "potential." The equation, however, i s three-dimensional, and i t reduces to a form which i s very similar to the Lippmann-Schwinger equation in the center-of-mass system. The equation

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Baryon Annihilation and Exotic Exchange