Fortschritte der Physik / Progress of Physics: Band 25, Heft 11 1977 [Reprint 2021 ed.] 9783112519561, 9783112519554

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ISSN 0015 - 8208

FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 11 • 1977 • BAND 25

A K A D E M I E

- V E R L A G EVP 1 0 , - M 31728

•

B E R L I N

BEZUG SMÜGLICIIKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Ausliefermigsstelle KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1203 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, Postfach 160, oder an den Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4

Z e i t s c h r i f t „ F o r t s c h r i t t e der P h y s i k " H e r a u s g e h e n P r o f . D r . F r a n k K a s c h l u h n , P r o f . D r . A r t u r L ö s c h e , P r o f . D r . R u d o l f Ritsoli!, P r o f . D r . R o b e r t R o i u p e , i m A u f t r a g (irr P h y s i k a l i s c h e n Gesellschaft d e r D e u t s c h e n D e m o k r a t i s c h e n R e p u b l i k . Verlag: A k a d e m i e - V e r l a g , D D R - 1 0 N Berlin, Leipziger S t r a ß e 3 - 4 ; F e r n r u f : 2 2 3 6 2 2 1 u n d 2 2 3 6 2 2 9 ; T e l e x - N r . 114420; P o s t s c h e c k k o n t o : Herliii 3 3 0 2 1 ; B a n k : S t a a t s b a n k der D D R , Berlin, K o n t o - N r . 6836-26-20712. C h e f r e d a k t e u r : D r . Lutz. R o t h k i r c h . Auschril't der R e d a k t i o n : S e k t i o n P h y s i k der H u m b o l d t - U n i v e r s i t ä t zu Berlin, D D R • 104 Berlin, Hessische S t r a ß e 2. V e r ö f f e n t l i c h ! u n i e r der L i z e n / . n m n m e r 1324 des P r e s s e a m t e s heim V o r s i t z e n d e n de» Ministerrates der D e u t s c h e n D e m o k r a t i s c h e n Republik. G e s a m t h e r s t e l l u n g : VE1J D r u c k h a u s „ M a x i m G o r k i " , D D R - 74 A l t e n b u r g , C a r l - v o n - O s s i e t z k y - S t r a ß e 30/31. E r s c h e i n u n g s w e i s e : D i e Z e i t s c h r i f t „ F o r t s c h r i t t e d e r P h y s i k " e r s c h e i n t m o n a t l i c h . D i e 12 H e l t e eines J a l i r e s bilden eineu B a a d : B e z u g s p r e i s j e B u n d 180,— AI zuzüglich V c r s a n d s p e s e n (Preis fiir d i e D D R : 120,— AI). P r e i s j e H e l l 15,— AI (Preis i ü r die D D R . 1 0 , - AI). B e s t e l l n u m m e r dieses H e f t e s : 1027/25/11. (c) 1977 bv A k a d e m i e - V e r l a g B e r l i n . P r i n t e d in the G e r m a n D c m o c r a t i c R e p u b l i c . A N ( E D V ) 57 618

ISSN 0015 - 8208 Portschritte der Physik 25, 649—741 (1977)

Relativistic Hadron Couplings: A Unified Framework A . N . MITRA a n d SUDHIR SOOD

Department

of Physics and Astrophysics,

University

of Delhi, Delhi-110007,

India

Contents I. Introduction . 1.1. Form Factors in Sü(ñ)w x 0(3): Tensorial Framework 1.2. The RQ Hypothésis and Higher Unifying Principles 1.3. Problems of Relativization: Current vs. Constituent 8U(6)W 1.4. Scope of the Article II. Classification of Hadron Resonances and Kew Particles 11.1. Meson States 11.2. Baryon States (qqq) 11.3. Supermultiplet Patterns 11.4. Even Wave qq Forces III. P and V Meson Couplings of Hadrons 111.1. Non Relativistic Q+PQ Vertices 111.2. B+PB Couplings (for L = 0) 111.3. The Orbital Coupling Structures 111.4. The complete B+PBj Couplings 111.5. Meson Couplings (MPML, L > 0) : 111.6. Coupling with F-Meson as Radiation Quantum 111.7. Additional B+VBL and MVML Couplings 111.8. B+VBj Couplings 111.9. MVMj Couplings . IV. Relativistic Forms of Hadron Couplings IV. 1. Relativization of Baryon Couplings IV.2. BPBj Couplings IV.3. BVBj Couplings IV.4. Relativistic Forms for M(P, V) ML Couplings V. Unified Treatment of (P, V) Couplings: Partial Symmetry V.l. Formulation of Partial Symmetry V.2. Hadron Interactions for L — 0 V.3. Hadron Couplings for L > 0 V.4. Partial Symmetry with Jf-Matrix for Arbitrary Spin V.5. Comparison with SU(6)W and 3P0{qq) Models VI. Form Factors and Structure Function: Recent Developments VI. 1. Four Dimensional Approaches VI.2. Licht Pagnamenta Method for 3-dimensional Relativization VI.3. Modified L—P Argument VI.4. Applicability of L—P Formalism with Constituent Quarks 48

Zeitschrift „Fortschritte der Physik", Heft 11

'

650 651 652 653 654 657 657 660 660 661 662 663 .664 .666 668 671 673 675 676 678 679 679 680 681 683 686 686 688 690 691 693 695 695 696 697 698

650

A . N . MITRA a n d SUDHIR SOOD

VI.5. Structure Functions in Deep Inelastic Scattering VI.6. Comparison with High and Low q2 Data VII. QPC Couplings, E.M. Masses and Allied Phenomena VIL 1. VII.2. VII.3. VII.4. VII.5. VII.6.

. . .-

Predictive Powers of QPC Hadron Couplings in QPC Model Tests of Some QPC Coupling Predictions E.M. Masses of Hadrons E.M. Masses of Charmed Mesons Tj - > "in Decay . . :

VIII. Couplings of cc Mesons: Radiative Transitions VIII. 1. VIII.2. VIII.3. VIII.4. VIII.5. VIII.6.

Colour Gauge Theories and OZI Rule Violation O-Meson Dynamics Dual Unitarization Theories QPC Model and VMD for Radiative cc Decays El transitions : Comparison with Data Ml Transitions : y> —> rjcy etc. Decays

I X . Concluding Summary

699 702 704 704 706 709 710 712 712 712 713 714 716 717 719 720 721

Appendix A. qqq Wave Functions and SU(Q) Couplings of BPBL

722

Appendix B. Materials for Baryon Coupling Structures

726

Appendix C. qq Wave Functions and 8V{6) Couplings for M(P, V) ML

731

Appendix D. Normalised C.G. Expansions

735

References

737

I. Introduction The physics of resonances which gained universal recognition about a decade ago [7], now plays a central role [2] in the efforts to understand the basic structure of matter. At the experimental level, this fact is amply illustrated by the evergrowing stature of the particle Data Group Tables [3], to which the discovery of tp charmed-particles [4, 5] has added an extra dimension. The corresponding theoretical interest (though showing fluctuations because of preoccupation with ideas of more topical interest) has also, by and large, recognized the important role played by the resonances in the high energy domain. Initial theoretical interest in these particles started at the SU(3) level through a study of their decay characteristics which offered a direct probe into their couplings with decay products (two body systems like Nji, NK, An). This language for the chargehypercharge classification of these resonances has never looked back ever since its proposal by G E L L - M A N N [ 6 , 7] and N E ' E M A N [£]. At present the belief is generally held that a major part of $£7(3) symmetry violation comes from Phase space (which is accountable experimentally) and that the balance (hopefully a smaller numerical effect in comparison) should be ascribed to a dynamical theory. Studies of symmetries at levels higher than SU(3) are concerned with two aspects, firstly to study the experimental classification of an evergrowing number and variety of these resonances (the simple quark picture of SU(Q) X 0(3) [9] having gained an increasing degree of support over the years as the appropriate language of classification) and secondly to study the dynamical impact of these symmetries as manifested through their coupling structures. The latter cover a wide spectrum of approaches, ranging from the phenomelogical type which leave plenty of scope for parametrization, to elaborate relativistic groups (such as £7(12) [10], SL{6, C) [11] 0(4, 2) [12]) which leave little. Mainly concerned with the problem of imbedding SU(3) and SU(&) in the Lorentz group 0 ( 3 , 1), the latter type of theories encountered various formal difficulties in

Relativistic Hadron Couplings

651

view of the rigidity of their self consistency requirements (some examples are M C G L I N ' S theorem on the mass spectrum [ 7 3 ] and M I C H E L - O ' R A I F E A R T A I G H theorem [14] requiring more than four components for the energy momentum vector.) I t is not the purpose of this article to delve into the theory of, and the problems associated with, noncompact group theories for a critical description and analysis of which the interested reader is referred to the works of D Y S O N [15] and the (more recent) one of B A R U T and collaborators [12, 16). 1.1. Form Factors in SU(Q)W X 0(3): Tensorial Framework A more promising, but less ambitious, approach to relativistic formulations of hadron couplings opened up with the philosophy of current algebras pioneered by G E L L - M A N N [17]. In this approach, the algebras at various levels, such as SU(2) X SU(2), SU(3) X SU(?>) and Z7(6) X U(6), are useful for an understanding of the hadronic interaction parameters, independently of the question whether or not the dynamics of strong interactions is invariant under the corresponding groups. A less ambitious relativistic theory based merely on the current algebra philosophy could have the advantage of being free from many of the formal objections (cited above) that plagued the noncompact groups, and yet provide a convenient starting point for the relativistic extensions of the SU(3) and SU(Q) groups, albeit in a limited fashion. Of the various groups t h a t have been discussed in this connection [IS], we shall have particular occasion to deal with SU(&)w [19] which seems to have more successful contacts with experimental data on hadron resonances than some of the others. The important point to note is t h a t the less rigid nature of these groups offers greater scope for phenomenological pursuits than do the more formal relativistic group-theories. Phenomenological approaches, on the other hand, have the advantage of staying closer to experiment and hence of offering valuable pointers to the direction in which to look for experimentally desirable features in a more complete theory of the future. A very important development which offered scope for a more meaningful contact between theory and experiment in the realm of resonance physics was the emergence of F E S R [20] and duality [21] which for the first time provided a direct link between the role of hadron resonances and that of Regge phenomenology — hitherto two largely uncorrelated areas of research. Indeed it now became possible for the first time to suggest an alternative description of a high energy Regge exchange amplitude in terms of «-channel resonance contributions. However, the historical trend t h a t followed the discovery of F E S R and duality took a strongly theoretical (occasionally almost abstract) turn, without adequate emphasis on the comparison of these theoretical predictions with resonance data at the more phenomenological level. I n our view one of the important reasons for this unfortunate development from the point of view of wide-spread applications to experimental data has been a general - lack of enthusiasm for practical applications of hadron couplings in the overall duality spirit, using for definiteness a V A N H O V E [22, 23] or equivalent mechanism. Regge phenomenology did offer one such possibility. However, the quality of information on hadron coupling structures expected from a Regge approach via F E S R is likely to be poor in details because of the extensive parametrizations involved in practical calculations. On the other hand, a parametrization from the opposite direction, viz. in terms of the form factors governing the couplings of resonances [24] offers an alternative possibility of their measurement by considering the contributions of these resonances to various high energy reactions in a duality spirit. A survey of the contemporary literature, however, would lead one to think t h a t while the language of the "form factors" has found a general place in coupling schemes, the wide scope of its applications (far beyond the calculation of mere decay amplitudes) has not received adequate recognition in the literature. 48*

652

A . N . MITRA a n d SUDHER SOOD

One of the central objects of this article is to bring out the natural possibilities of the language of form factors as a powerful means of interaction between theory and experiment in the realm of resonance physics. The important ingredients of this language are the higher hadron supermultiplets and their coupling structures. In view of their predominantly strong decays, most of the physical interest is in their strong interaction. Interest in their electromagnetic interaction also stems fairly directly through the prospect of their photo- and electro-production, the radiative decays and the mass differences among the same isoplet members. The language of form factors finds a most natural expression in the so-called tensorial formulation which had been developed in the late sixties [23, 25, 26] but had remained largely at the abstract level of qualitative applications without much evidence of concrete numerical applications. The tensorial framework has not only the desired flexibility to accommodate variations in the input physical assumptions but also represents a versatile mathematical instrument for giving effect to wide ranging physical applications designed to bring out the so-called off-shell aspects of the form factor. 1.2. The R Q Hypothesis and Higher Unifying Principles The tensorial framework lends itself easily to a microscopic formulation of hadronic couplings starting from the quark level which also provides an independent motivation for the language of form factors. The quark level description involves single quark transition operators [27] whose matrix elements between specified qqqjqq states are given by the additivity principle, in close analogy to the matrix elements of single particle operators in nuclear physics. The operators themselves correspond to the emission and/or absorption of single quanta (pions, photons, V-mesons and so on) whose internal (quark) structure is neglected. This is the famous radiation quantum (RQ) hypothesis which has proved a most effective tool for the description of pionic and e.m. transitions of hadrons classified according to SU(&) X 0(3). Together with the quark-recoil effect [28], which is expressed by a correction ('recoil') term to the main ('direct') transition operator through the requirement of Galilean invariance, the R Q hypothesis also provides a realistic description of decays (mainly s-wave) which apparently violate SU(6)W X 0(3). The main disadvantage of the RQ hypothesis is its inherent asymmetry between'a (qq) meson and the RQ meson, and such asymmetry would especially show up in pure mesonic transitions through a lack of decision as to which of the two mesons emitted was the RQ. The subsequent discovery of duality diagrams [29] helped resolve this ambiguity formally, yet left much to be desired for a quantitative solution of the problem. Duality diagrams were developed further at the hands of Micu [30], CAKLITZK I S L M G E R [*° at 2.02 GeV and D*+ at 2.025 GeV respectively as has been argued forcefully by D E R U J U L A et al. [72]. It is quite likely, just as in the case of cc states, that the charmed meson states D, F, etc. admit of $ rich spectrum of orbital and radial excitations, as has been conjectured in recent literature [73].

A. N. M i t r a a n d

Sqdiilr

Sood

II.2. Baryon States (qqq) Because of their three-quark structures, baryons have received mostly non-relativistic treatments, until rather recently. There is a further problem of statistics in this case. We refer to certain extensive reviews of this subject for the details as well as the earlier references [74]. We shall ourselves choose the symmetrical version [75] partly because of its intrinsic appeal and partly because this convention will keep us closer to most of the important literature on baryon classification [50, 74\ A commonly used description of the baryon spectra is in terms of the harmonic oscillator potentials between the qq pairs [45], The order of energy levels of successive supermultiplets for harmonic oscillator potentials is represented by the notation iV[£i!7(6);i p (w)]

(2.4)

where n is the overall index for radial excitation (n = 0 for no radial excitation), and N is the principal quantum number which can be regarded collectively as a sum over two independent angular momenta (Z23, h) and two independent radial quantum numbers (w 2 3> »ii) in the sense of a harmonic oscillator description N=(k

+ 2ni) + (l23 + 2n23).

(2.5)

Essentially the samè supermultiplet structure holds for the relativistic harmonic oscillator model [50] whose true sophistication shows up mainly at the level of couplings. As is well known, there is a rapid rate of proliferation of the higher supermultiplets (N Sï 2), the general experimental status of which is still quite confused except that a reasonable degree of identification of certain individual members with higher /-values allows some speculative statements. Thus the existence of the A -sequence, with the spins of, its higher lying members identified through their Ji 2 -plot, is consistent with the belief in the existence of higher Regge recurrences of (56,0+) beyond (56,2+) perhaps upto L = 8. Similarly the existence of certain N and A states, presumably as Regge recurrences of the lower ones in (70, 1-) would encourage belief in the existence of (70,3"), (70,5~) etc. states, though states like (56,3") etc. need not be ruled out. II.3. Supermultiplet Patterns A simple picture that emerged from certain dynamical considerations consisted of a subset of supermultiplets given by [76] (56,. even+) ;

(70, odd")

(2.6)

in the notation of (2.4). In fact, upto quite recently, it was believed to be the likely framework for the baryon classification [2], Without going into the details, we merely list some of the general arguments that led to such a belief : (i) Its comparative economy of description compared to the more general classification according to Eq. (2.4), (ii) the occurrence of only the "natural parity" supermultiplets, (iii) its consistency with the pattern of Regge recurrences of the two basic supermultiplets (56,0+) and (70,1-). From a more dynamical point of view, an analysis of the qq forces [76] in terms of partial waves indicated that the s-wave, while automatically ruling out the unnatural parity (even- and odd+) L values, predicts precisely the series (2.6), as more depressed in mass than the complementary series (56, odd-);

(70, even+);

(2.7)

Relativistic Hadron Couplings

G61

thus the latter got somewhat relegated to the academic background. Still another argument for the existence of only the series (2.6) was advocated by BTTCCKLLA et al. [40] on the basis of their compatibility with the chiral SU(3) X SU(3) algebra. However, before one can accept such a picture with all its fundamental implications, it is necessary to look into other forms of guidance from a different set of principles such as the constraints of E X D [65], As is well known an E X D constraint implies the cancellation of the imaginary provided by various Regge exchanges in "exotic" channels. Thus if one considers M + B scattering in the backward direction (MB BM), one derives the E X D relations separately among the spin quartet (S = 3/2) and spin doublet (S = 1/2) states of (56, 21+) and (70, (21 -f- 1)-), as best exemplified by the famous A„ — Ay and £ d * — Ep* sequences, (and perhaps not so well by the series N„ — Nr or Ad — Np. The notation here is (a, y) for J = L + 1/2 (positive signature) and (5, /?) for / = L + 3/2 (negative signature) states, while the first and second letters for each case stand respectively for even and odd ¿-values. Experimentally the (mass)2 plots against J of the alternate parity states are known to coincide rather well, separately for the N, A and E* sequences. Even the coupling patterns of these states to Nn and KN systems show remarkable regularities [77]. However, if one considers, e.g., an 8„ -> (10)y pattern it is hardly found to respect EXD. To understand such anomalies, it was suggested by MANDULA et al. [65] that the quality of E X D predictions gets the poorer, the further removed is the trajectory from the threshold of the channel under study. In particular, they suggested that E X D constraints on baryons should not be extended beyond the MB -> BM channel to, scythe crossed channel MM —> BB, because of the high threshold of a BB system. A similar remark applies to BB —> BB as well. This point of view which has been termed "broken duality", rejects E X D constraints corresponding to high thresholds, and has generally received good support in the literature. This point of view favours the simple sequence (2.6). Another point of view, suggested subsequently by ROSNER [78], was to face the consequences of an "exact duality", by including the MM BB and BB BB processes BB within a wider E X D framework. A difficulty about this approach [52] is that a BB amplitude cannot be dominated by 8 and 1 meson exchanges at high energies, and yet be confined to 8 and 1. contributions in the direct channel, suggesting the possible existence of exotic mesons (qqqq). However if one is willing to pay the price of exotic mesons, it is possible to insist on an exact duality program [79] and obtain a different form of E X D pattern for supermultiplets [80] (56,0+); (70,1-); (56 and 70, 2+); (56 and 70, (21 + 1)) I ^ 1.

(2.8)

ROSNER [75] argued that a clear distinction between the simpler series (2.6) and the more comprehensive set (2.8) would be provided in the detection of a (70,2+) state which of course lies within the harmonic oscillator spectrum. Such a possibility. was indicated in the CERN [SI] and Saclay [82] phase shift studies, through the appearance of an i'^iV-resonance near 1983 MeV. Clearly there is no way of accommodating an Fi7 state in (2.6), while the simplest possibility in (2.8) is a (70,2+) state [S3]. II.4. Even Wave qq Forces We close this subsection with some remarks on at least another alternative approach of more recent origin. One of the natural predictions of the harmonic oscillator model is the existence of 20 states of Lp = 1+. Even after a decade of this prediction, there has been no evidence, direct or indirect, for any of the 20 states. A natural way to keep out the 20 states is through the s-wave qq forces. As already remarked above [76], this results in a mass spectrum with the (56, odd-) and (70, even+) lying much

662

A. £J. MITRA and SUDHIR SOOD

above the complementary cases of (56, even+) and (70, odd - ). However, extension of this idea to include all even partial waves, e.g., through a Serber-type h.c. potential, still keeps out the 20 states, while not ruling out the complementary series (2.8) which seem to be required by E X D arguments as well. While referring the interested to the recent literature for details [84, 8-5] we summarize here some of the more salient features of this theory. The spectrum of 56 states is exactly the same as predicted by the full h.o. model, but the 70 spectrum gets considerably modified. In particular there is a new ground state (70,0+) lying below the negative parity (70,l - ) states. This provides a natural understanding of the observed Roper-like states such as P n (1470) to be lower than the negative parity resonances. Next, there are now two sets of (70,1~) states termed lower (I) and upper (u), with excitation energies AM2 = 1/2« and j/3/2a respectively above the (70,0+) state, in contrast to the full h.o. prediction of AM2 ^ oc (1 GeV2) excitation for successively higher states. This prediction provides a natural understanding of the observed "bunching" of some (70,1") states, e.g. D13(1520) versus Z>15(1670), etc., on the M2 scale without having to resort to an elaborate mixing program at the purely phenomenological level [56], Similarly there is a lot more structure fov (70,2+) states than predicted by the full h.o. model, offering mass assignments which check against pionic and photo couplings of several "difficult" resonances in the conventional h.o. model. This is not to suggest that this theory predicts more states than the usual h.o. model, but merely to indicate that many of the 70 states of high N but low L in the full h.o. theory get strongly depressed in M2 so as to appear as relatively modest excitations in the new theory. These features are best exhibited, not in terms of the conventional h.o. variables