Resonance Decays Involving Vector Mesons

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

Resonance Decays Involving Vector Mesons*

W. P. Petersen and J. L. Rosnerf School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455

(Received 28 August 1972)

A picture of resonance decays using only very general features of the quark model i s applied to the case of final states. Recent experiments on 7d\i - Np provide a crucial tes t of the model, which predicts correctly the dominant spin of the Np system in several cases.

The search for symmetries of decay matrix ele- ments beyond SU(3) has been a part of resonance physics for nearly a decade. Only recently, however, has i t become c lear that certain relatively modest extensions of SU(3) describe a large body of data fairly

The picture to which we refer makes some min- imal use of the quark model.5 A hadron is a s - sumed to decay by the creation out of the vacuum of a qq pair with JPC = O w , i.e., a 3 ~ 0 unitary singlet pair. Each final hadron contains one member of this pair. Such a picture2f6 a r i s e s naturally when duality graphs7*' a r e endowed with spin.

The above model has SU(6), a s a limit cor re- sponding to L,=O for the initial hadrons (the z axis i s defined by the final part icles in the c.m. system). Physically, this corresponds to the assumption that quarks have no transverse momentum in- side a hadron. This assumption appears to con- tradict experimental data.

When a n admixture of L, # 0 in the initial hadron i s allowed, SU(6), i s broken in a natural way. The quark L of the initial hadron and that of the 3P0 pair (L = 1) couple to various values of final orbital angular momentum I. For final hadron pa i rs with no internal L, parity conservation requires I = L * 1. The amplitudes corresponding to these two values a r e related in SU(~),,' but freed from one another when both L, = 0 and L, # 0 decays a r e allowed. We shall cal l such a scheme "1,broken SU(6),." lo

The study of baryon resonance decays to itl- final s tates is a particularly interesting applica- tion of I-broken SU(6),. Three amplitudes contribute to these decays: two with one I (spin i and 2 ) and one with the other 1 (spin $). In SU(6), al l three of these a r e related to one another. In I- broken SU(6), the relations between spin-+ and spin-$ amplitudes for the I that has both of them remain valid. These a r e easy predictions to test. They do not rely on assumptions about centrifugal bar r ie rs , in contrast to most of the previous ca ses where I-broken SU(6), was

The recent availability of data on nN- nnN,ll al-

lowing a phase-shift analysis of nN - pN, makes our predictions of particular relevance a t present.

Should the following relations be found violated, the usefulness of the quark model for describing resonance decay^'*^"^ would be brought into s e r i - ous question.

Define a to be the natural parity of the initial baryon: a = r P , 7 = (-) .'-'I2 =signature, P =pari ty. Then the following partial waves contribute to itl- decays:

where S i s the spin of the 6'1- system. Ba r r i e r effects will usually suppress the higher part ial wave, so that only a = -states will involve detect- able S = $ and S = 5 admixtures. Exceptions a r e the ca ses J*=+-, for which only S = $ i s possible in 1 = 1- =0, and JP a$+, for which I- is unphysical and I+ should be observable, with both spins possible. To summarize, we expect useful spin-ratio predictions for 1' 1- decays of resonances with

We define decay amplitudes 3nk with h the vector- meson helicity (and the baryon helicity taken ++). The total itl- width i s

in our normalization. Various choices of multiplet assignment usually a r e possible for a resonance. For each choice, the amplitudes 311, may be calcu- lated by standard SU(6), methods.3s4 The rat ios 911, : 311, : m-, a r e well defined in each case.

Next one expands in partial-wave amplitudes [lls:

748 W . P . P E T E R S E N AND J . L . R O S N E R 7 -

TABLE I. Fractional contributions of different partial waves toNp decay predicted by SU(6),. Pa i r s of underlined numbers a r e expected to preserve their ratio in the presence of SU(6), breaking. The numbers in the second column refer to SU(6) representation, SU(3) x SU(2) representation, and quark orbital angular momentum, respectively.

Resonance Assignment

A(1950): ++ - 56, (10, 4), L=2 - 0.25 - 0.75 0 0

a The definitions of I + a r e given in Eqs. (1) and (2) .

The ratios of the three [Z], amplitudes a r e thus sistent with this result. The influence of the pre- specified in SU(6),. In I-broken SU(6),, the r a - dieted S = 5 admixture i s very different in the two tios [1],1/[1],2 for the 1 with two possible spin val- cases, however. For the former assignment, ues cont~nue to hold. SU(6), predicts X, = 0, while for the latter one ex-

The following ATp amplitudes in (3) a r e important pects X,/X, = - 3 / 0 . The neglect of the spin-$ experimentally" between 1700 and 2000 MeV: amplitude12 thus throws away considerable infor-

In each case a particular spin predominates though both a r e allowed.

Table I summarizes the predictions of SU(6), for fractional contributions of each partial wave to the total Np partial widths of the states in Eq. (6). It has some interesting features in comparison with the data:

a. Predominance of S =+ for A (1910). No choice between 56, (10, 4), L = 2 and 70, (10, 2), L = 0 is possible purely on the basis o E h e experimental dominance of spin $. Both assignments a r e con-

mation in this case. b , Favored assignment for N(1860). The experi-

mental dominance of spin $ in Eq. (7) i s consistent with only one assignment of this resonance to a pure state: It must belong to 56, (8,2), L = 2. This assignment, interestinglyenough, may help to explain the observed suppression of N(1860) -An, a s (by similar methods) i t predicts r,[hn]/ r[An] = A, rF[hn]/r[An] = & for this state. Even with the expected suppression of F waves, in fact, one might expect some F-wave An decay of this state. The fact that none i s seen" may indicate that small admixtures of other SU(6)@ O(3) repre- sentations a r e present in N(1860).

c . Predominance of s=$ for A(1950). The a s - signment shown i s the only reasonable one. The

TABLE 11. Predictions analogous to those of Table I but for a s yet unobserved states.

Resonance; Decay mode Assignment

a The definitions of I , a r e given in Eqs. (1) and (2).

R E S O N A N C E D E C A Y S I N V O L V I N G V E C T O R M E S O N S 749

TABLE 111. Comparison of relations for resonance photoproduction amplitudes on neutrons [ 'Xx(n)] in t e rms of those on protons [Fnx(p)]. Note the existence of one linear relation common to both approaches.

Resonance Refs. 12 and 14 Present w ~ r k (assuming vector

dominance)

result of Table I follows from two conditions which a r e expected to hold in general in the present model: 1 = 5 decays a r e absent, and 311, = O . As in the case of ~ ( 1 9 1 0 ) , information on the relative phase of the S = ', and the (smaller) S = 2 amplitude would be helpful here, to see if 311, really vanishes.

Further applications of the present method could be envisioned if data were good enough to see the amplitudes neglected in Ref. 11. We en- vision small contributions from the states listed in Table 11. Also presented a r e predictions for Nw decays. Since the w contains a unitary singlet piece, these predictions a r e based on the ansatz N&N@. Phase-shift analyses of the reaction n-p - wn, for example, with detection of the noy decay of the w, would thus provide a useful comple- ment to the analysis of Ref. 11. The prediction of dominance of S =$ for N(1860) - (No),=,, given our favored assignment, is one amusing result; recal l that S =* was found dominant for Np decay.

We close with some brief remarks relating our work to that of others,

A recent bootstrap c a l ~ u l a t i o n ' ~ predicts the dominance of S =s in al l the cases of Eq. '(6).

Hence, although our quark-model results may well follow from other "nonquark" arguments (and we would welcome such alternative descrip- tions) the correc t prediction of spin rat ios in such schemes is apparently not automatic.

The present method, when combined with the vector-dominance model (VDM), predicts relations among photoproduction amplitudes similar but not identical to those of Refs. 12 and 14. The results of the two approaches a r e compared in Table I11 for photoproduction of D,,(1520) and ~ ~ ~ ( 1 6 9 0 ) . One linear relation i s common to both approaches. In fact, we suspect that the relations of Refs. 12 and 14 may be correc t a t q2 = 0 while our relations should hold only a t q2 = m P 2 = nz:, where q2 i s the virtual photon mass. Presumably a smooth inter- polation holds between the two points. This con- jecture would provide a basis for predictions of resonance electroproduction, but requires accu- ra te neutron data for i t s verification.

We wish to thank Dr. Roger Cashmore for helpful discussions.

*Work supported in par t by the U. S. Atomic Energy Commission under Contragt No. AT(l1-1)-1764.

TAlfred P. Sloan ~ e s e a r c ~ ~ e l l o w , 1971-1973. 'E. W. Colglazier and J. L. Rosner, Nucl. Phys.

B27, 349 (1971). - 'L. Micu, Nucl. Phys. E, 521 (1969). 3 ~ . P. Petersen and J. L. Rosner, Phys. Rev. D 5,

820 (1972). *J. L. Rosner, Phys. Rev. D 5, 1781 (1972). 5 ~ . Gell-Mann, Phys. Letters 8, 214 (1964); G. Zweig,

CERN Reports No. TH-401 and No. TH-412, 1964 (unpublished).

6 ~ . L. Rosner, Phys. Rev. Letters 2, 689 (1969). 7 ~ . Imachi, T. Matsuoka, K. Ninomiya, and S. Sawada,

Progr. Theoret. Phys. @yoto) 40, 353 (1968); H. Harari, Phys. Rev. Letters 22, 562 (1969).

8 ~ . Carlitz and M. Kislinger, Phys. Rev. D 2, 336 (1970). his shortcoming has been stressed by S. Meshkov

(private communication). 'O1n a more phenomenological context, without the addi-

tional motivation based on the 3 ~ o picture, I-broken SU(6), has been used ear l ie r . See R. H. Capps, Phys. Rev. 158, 1433 (1967); 165, 1899 (1968); D. Faiman and D. E. Plane, Phys. Letters E, 358 (1972).

"R. J. Cashmore, D. W. G. S. Leith, D. J. Herndon, R. Longacre, I,. R. Miller, A. H. Rosenfeld, and G. Smadja, in Proceedings of the Sixteenth International

7 50 W . P . P E T E R S E N A N D J . L . R O S N E R

Conference on High Energy Physics 1972, Chicago, 1 3 ~ . Gustafson, Nucl. Phys. z, 205 (1972). Illinois (unpublished). I4F. E. Close and F. J. Gilman, Phys. Letters B,

"R. P. Feynman, M. Kislinger, and F. Ravndal, Phys. 541 (1972); F. E. Close, F. J. Gilman, and I. Karliner, Rev. D 2, 2706 (1971). Phys. Rev. D 5, 2533 (1972).

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

Linear Regge Trajectories with a Left-Hand Cut N. G. Antoniou*

CERN, Geneva, Switzerland

and

C. G. Georgalas and C. B. Kouris Nuclear Research Ce?zter "Democritus, " Aghia Paraskevi Attikis, Greece

(Received 14 June 1972)

The properties of boson Regge trajectories a ( s ) with a left-hand cut for s < 0 a r e studied, with the following assumptions: (a ) Rea(s) is linear in the physical regions of the s and t channels. @) The imaginary part is asymptotically smaller than the real part . (c) At s = 0, a ( s ) has a branch point due to the existence of a Regge cut a, (s) in the angular momentum plane, with a,(O) = ~ ( 0 ) . The branch point i s attributed to the collision of the physical pole a ( s ) with a second-sheet pole d (s ) , a t s = 0. (d) The function a ( s ) i s analytic in the s plane. Under these assumptions the imaginary part for each boson trajectory i s obtained in t e rms of a parameter A and a parameter-free universal function determined by solving an integral equation numerically. The parameter A i s determined by requiring that the first meson of each trajectory have the observed width. The model gives imaginary par ts in- creasing almost linearly with s for la rge / s j . The widths of the recurrences increase linearly with their mass. The results support exchange degeneracy p- f , K*-K,, w-A2. The imaginary part , for s c 0, i s compared with the phenomenological results of other authors.

I. INTRODUCTION

The possibility that Regge poles a ( s ) a r e complex for s <O h a s been investigated This means that Regge t ra jec tor ies develop a left- hand cut fo r s <O. The mechanism of production of th i s cut i s the collision of two Regge poles.'j2 In relat ivis t ic scat ter ing the mechanism of gener- ating complex poles i s strongly related to the existence of cu t s in the angular momentum plane. The colliding poles a r e expected, on physical grounds, to l ie on different Riemann sheets of the angular momentum plane f o r s >O, whereas f o r s <O they a r e complex conjugates of each other on the same sheet , which i s assumed i n th i s paper to be the physical ( f i rs t ) sheet . There is therefore a s t rong pole-cut relationship based on analyticity i n t h e angular momentum plane.2 Hence the study of complex poles i s connected to the investigation of the propert ies of the Regge cuts. In part icular , i t tu rns out that the effect of Regge cu ts c a n be simulated by a pair of complex poles i n a sa t i s - factory way.'

The new quantity enter ing into the Regge-pole description of the high-energy two-body react ions in the complex-pole model i s the imaginary par t of the t rajectory a , ( s ) , along the left-hand cut ( s <O). The form of the function a , ( s ) is unknown and only a crude phenomenological determination can be achieved. On the other hand, since the nature of the cu t s i n the angular momentum plane i s not yet known in detail , any information for a, (s) f r o m the pole-cut relationship i s necessar i ly model-dependent.

In the present work an integral equation for the discontinuity function h a s been established by the use of analyticity in the s plane together with some general propert ies of Regge poles and cuts . The solution of this equation is then determined nu- merical ly and the resu l t s a r e compared with the widths of the boson r e c u r r e n c e s and with the phenomenological left-hand absorptive part .

In Sec. I1 the assumptions a r e s tated and the equations of the model a r e derived and discussed. In Sec. 111 our resu l t s a r e compared with the experimental data and the phenomenological resu l t s

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Resonance Decays Involving Vector Mesons