IC/70/58

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

THE ROLE OP FORM FACTORS

IN HADRON RESONANCES

A.N. MITRA

INTERNATIONAL

ATOMIC ENERGYAGENCY

UNITED NATIONSEDUCATIONAL.

SCIENTIFICAND CULTURALORGANIZATION 1970 MIRAMARE-TRIESTE

IC/70/58

INTERNATIONAL ATOMIC ENERGY AGENCY

and

UNITED NATIONS EDUCATIONAL SCIENTIFIC AND CULTURAL ORGANIZATION

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THE ROLE OP FORM FACTORS

IN HADRON RESONANCES *

A.N. MITRA **

MIRAMARE - TRIESTE

July 1970

* To be submitted for publication.

* * Summer visitor to ICTP- Permanent addressi Dept. of Physics, University of Delhi, Delhi-7, India,

ABSTRACT

A phenomenological structure is proposed for the couplings of hadron

resonances to the 56 baryons and 36 me sons with a view to exploring the

possibilities of detailed dynamical applications of resonance data to several

other areas of particle physics where these have relevance. The group

structure used for hadron (H, HT ) classifications is that of SU(6) x O(3) ,

while the basic framework for a unified description of the HPHT andLi

HVHT interactions is provided by broken SU(3) x SU(3) , or partial sym-

metry, in conjunction with the language (not dynamics) of the quark model.

The coupling structures are expressed in terms of generalized Rarita-Schwinger fields and a Lorentz-invariant form factor (f ) for whose con-

Ij

struction a general set of criteria is proposed at the phenomenological level.

One explicit construction of the form x f satisfying the requirements of

crossing symmetry and consistent with Regge universality of the reduced

coupling constant (g ) together with a common form of parametrization forLi

BBT M and MM M couplings is obtained. The coupling struptures notLi LJ

only account satisfactorily for several "difficult" branching ratios in baryon

decays but also give i) the observed "transverse" angular distribution in

B -* UT decay and ii) a ratio (h' /h ) # 0. 57 of the helicity amplitudes for

A _» pur in rather good agreement with experiment. A PCAC relationbetween the A-.pT and pvir coupling constants which is derivable in a simple

way within this framework is found to be in very good agreement with the

value of the meson (L = 1} supermultiplet coupling constant g. determined

from the tensor meson decays.

The mathematics of an explicit application of the model, viz., u-channel

T -p scattering through the exchange of A-resonances in the spirit of the

Van Hove model, is worked out in some detail to bring out the practical applic-

ability of the model, especially in the intermediate energy region. Several

other applications, viz., ranging from photo- and electroproduction of re-

sonances to their effects on electromagnetic mass differences in hadrons,

are discussed with special reference to the role of the form factor.

- 1 -

I. INTRODUCTION

In spite of the discoveries of FESR [ 1] , duality [1, 2] and the

Veneziano model [3] , the interaction between theory and experiment in

the domain of resonance physics, apart from some notable exceptions [4] ,

seems to have been much weaker than, for example, in the realm of Regge

phenomenology [2, 5, 6] where various forms of parametrization for the

purpose of fitting high-energy data \ 7] have acquired a certain amount of

credibility, from the point of view of theory. The developments of the

idea of duality have taken on a very different course. Its theoretical base

has been expanding at a considerable rate through impressive sophistications

of the Veneziano model, successively through the stages of multi- Veneziano

structures [8] , duality diagrams [9] , operator formalism [10] , functional

methods [11], and so on. However, a corresponding degree of enthusiasm

for trying its validity at a more phenomenological level, through a more

detailed use of the resonance data,seems to be lacking. Thus the original

FESR spirit, viz. , the correlation of data in the (low-energy) resonance

region (s-channel) with those in the (high-energy) Regge region (t-channel)

in a quantitative fashion at the phenomenological level got largely buried

under the fast developments that followed. A similar fate also befell the

Van Hove model [12] whose strong similarity to the language of field theory

would make it a particularly attractive tool for the correlation of data in the

resonance region to those in the high-energy region, provided that the form

factors appearing in this model could be more closely related to the re-

sonance data. As things stand, this model never received more than formal

mathematical attention in the literature [13] , though its physical possibilities

would appear to be more substantial.

Development of resonance physics during the years has taken place

more or less independently of the main stream of high-energy physics. The

predictive powers of the dual models have been put to little quantitative use

in this regard except for the empirical fact of straight-line trajectories for

the resonances. At the experimental level [14] , the data have generally

been analysed in terms of phase space and SU(3) structures [15] , though in

more recent times experimentalists are showing greater awareness of the

role of centrifugal barrier factors [16] . From the point of view of theory,

- 2 -

moat investigations have been in the nature of i) classification according to

group theory [17] and ii) study of decay widths mainly for the purpose of

checking on the consistency of the assigned group structures [18] ,

Extensive calculations on these lines are available at the levels of the

SU{6) x O(3) quark model [19, 20] and more formal groups such as O(3, 1)

and 0(4, 2) with greater predictive powers [21, 22] . However, there is

little evidence so far in these attempts to show enough practical enthusiasm

for seeking more dynamical manifestations of the resonance data when

these are continued off the mass shell. The applications of the Veneziano

model which formally offers such scope,have so far been limited mostly

to purely mesonic processes, presumably because of its formal difficulties

with baryonic processes [24] .

We believe that the data from resonance physics, especially for baryons,

offer considerable scope for dynamical manifestation in the unphysical regions

through a fuller utilization of their coupling structures to the more familiar

hadrons. Unfortunately, at the present stage, the absence of a fully accept-

able theory necessitates the use of a certain degree of phenomenology.

However, we see little reason for objection to such a procedure in principle,

when we remember that many of the interesting predictions of the Regge

theory are based on a generous degree of parametrization [7] . The analogous

point of view at the domain of resonance physics would be to advocate greater

utilization of the "form factors" in the couplings of resonances to the £6

baryons (B and B*) and 36 mesons (P and V) , which must be given

suitable parametrizations within a well-defined framework and certain broad

guide-lines for their construction. (This has an obvious analogy in the

domain of nuclear physics, where potentials are fitted to the two-body phase

shifts, or prescriptions are given for continuing the T- matrix off

the energy shell, ostensibly for the purpose of application to three or many-

body problems.) The simplest language in this case is that of effective

Lagrangians with Lorentz-invariant structures. It is also possible to formu-

late broad guide-lines for • constructing the form factors at the pheno-

menological level without the immediate necessity of going into the sophistic-

ated framework of chiral lagrangian theories [25] with all their theoretical

implications. Now the idea of evaluating low-energy data starting from

- 3 -

parametrizations at the "Regge" end is not new. In more recent times

there have been interesting attempts to utilize the Khuri-Jones represent-

ation in the high-energy TT-N Regge amplitude to calculate the energy

dependence of the P. and S phase shifts, with considerable

success [26] . We are, in a sense, proposing to "reverse" this direction

of parametrization, viz., to start from the level of resonance physics by

mtroducingjconvenient structure for the form factors based on simple guide-

lines and then look for their off-shell manifestations in suitable areas in-

cluding the high-energy region.

For construction of the couplings we make the simplest assumpt-

ions on the group structures, viz. , that the baryons are given by the re-

presentations (56, even"**) and (70, odd") of the group SU(6) x O(3) [27] , together

with their radial excitations [28] , while the mesons are given by nonetC + +

structures [29] for each of J = (L±l) , L and L . The coupling

scheme, on the other hand, is conveniently expressed in the framework of

broken SU(3) iS SU(3) symmetry [30-33] . Unfortunately, the correlative

powers of such a symmetry are not strong enough for quantitative application

to resonance physics unless supplemented by additional assumptions. It

has also been suggested [34] that the violation of the Goldberger-Treiman

relations, especially for matrix elements of the currents between opposite

parity states, may be so large as to invalidate the practical usefulness of

the method. Additional assumptions may, for example, bear on the manner

of saturation of the algebra of SU(3) x SU(3) by suitable hadron states. One

such attempt has been made recently [35] with a certain amount of success,

but the essential point is that the correlative powers of any such theory are

limited to at least one parameter for a supermultiplet transition, in contrast

to a more detailed theory, possibly with non-compact group features.

We take a more pedagogical point of view in this respect. While keep-

ing the general framework of chiral SU(3) x SU(3) for the basis of coupling

structures we use the language of the quark model for their explicit construct-

ion. This construction is greatly facilitated by using a praticularly con-

venient formulation of (presumably) the same symmetry, outlined by

Schwinger 136],who gives it the name of "partial symmetry". The philosophy

- 4 -

of "partial symmetry", as also that of the quark model, is largely im-

material for this purpose. In the Schwinger scheme, which is ideally

suited to the non-relativistic quark language, one arranges the positive

parity components of the vector (V) and axial-vector (A) fields in the

form of a meson matrix (M) in the space of spin and SU(3), the coefficients

of the different terms in M being so adjusted as to reproduce the correspond-

ing terms in their respective free Lagrangians with correct normalizations1 2

when one evaluates — Tr M . The matrix elements of the meson matrix,o

evaluated between appropriate hadron (H) states, readily yield the couplings

of V and A mesons (and hence also of P-mesons via PCAC) to the cor-

responding baryon (B) and meson (M) states. Thus one obtains couplingsof the types _

B(V, A, P) BT and M(V, A, P) MT

where B and M represent the higher baryon and meson supermultiplets

of SU(6) x O{3) . It should be emphasised that this method need not be re-

garded as more than a simple pedagogical device for obtaining the correct

geometrical factors associated with the various coupling terms and has

little dynamical content beyond what is implied by the broken symmetry.

The real dynamics resides in the form factors for which (though these are

formally expressible in the quark model as the overlap integrals between the

spatial parts of the initial and final wave functions) we see no way to an ex-

plicit evaluation within the model without more detailed assumptions, such

as harmonic oscillator functions [37] , These quantities will be parametrized

directly in terms of certain general principles, without any reference to the

quark model.

For some time we have been trying to develop a coupling scheme of

the type outlined above [38-40] . It is built within the framework of multiple-

index tensor and Rarita-Schwinger fields, involving multiple derivatives.

Such structures,which were developed in the 'fifties [41],have been extensively

used in various connections, e. g., the electroproduction of resonances [42]

and the mathematical treatment of the Van Hove model [13] , The additional

feature that we wish to emphasise in our approach is the appearance of a

multiplying form factor which can be formally expressed in a Lorentz-

invariant manner and carries the main load of dynamics. Various criteria

-5-

can be used for its construction. For example, in the Veneziano modelit has a strong dependence on _J (or Jj) and hence on the mass of the re-

2sonance (via the linear relation MT = aL + b), but the predictions on the

Li

decay widths are not only qualitative 143] but seem to leave considerable

scope for parametrization. A more promising form (especially for applic-

ations off the mass shell involving integrations over virtual momenta) is

provided by the lessons of non-compact group theories [21] which have a

built-in mechanism for "toning down" the effect of strong momentum de-

pendence implicit in the multiple-derivative tensorial couplings. We have

found it convenient to adopt the latter point of view in conjunction with the

desirable requirement of "Regge universality" in the coupling structures,

manifested in the near equality of the "reduced" coupling constants multi-

plying the form factors. This last will not happen in general with any shape

of the form factor, but one which has this feature would naturally be preferred.

Indeed one such structure, exhibiting Regge universality for the A -reson-

ances,was found in the earlier treatment which,moreover, seemed to give

a fair amount of agreement with the data on decay widths of baryons [44]

and mesons [45] under a common form of parametrization for both.

This last feature may well be more general than its realization through

a particular empirical construction might suggest, so in this paper we wish

to examine somewhat more closely the problem of construction of the form

factor at the phenomenological level, by extending the guide-line to include

some additional requirements such as crossing symmetry in the momenta,

and reasonable simplicity of structure (consistent with fits to the data).

These points,which did not find any explicit emphasis in the earlier treat-

ments, are especially important for applications off the mass shell,which

represents the main theme of this paper. We shall therefore take the

opportunity in this paper to re-examine the earlier form of parametrization

in the light of extended guide-lines and see if some alternative structures

are also consistent with the data.

This paper has a two-fold objective: i) to discuss a more general set

of criteria for the construction of phenomenological form factors and ii) to

indicate the possibilities of application in several areas involving their off-

shell aspects. One such explicit construction for the form factors which

- 6 -

provides a reasonable description of the decay data will be given, and the

mathematics of an explicit application with this framework, viz. , the

Van Hove model for the u-channel ff-p scattering;will be described. The

scope for several other applications (including the nature of certain results

already obtained) will also be briefly indicated. However, no attempt will

be made in this paper towards either a detailed fit to the resonance decay

data or a detailed numerical analysis of the suggested applications, which

are best relegated to separate publications. Apart from this limitation,

the paper is designed to be a reasonably self-contained one with its emphasis

on the wide applicational aspect of the formalism as a practical means of

correlating the data on resonance physics with several other areas ranging

from intermediate to high energies, where resonances play a possible role.

In Sec. II we review the essential assumptions on our coupling scheme

developed earlier for BPB, and MPM. (without going into their mathe-

matical details), with a view to finding a more acceptable basis for their

rationalization, using some of the pertinent data from resonance physics.

Sec. Ill is devoted to the problem of constructing form factors in terms

of a list of several criteria which we consider desirable, partly from the

point of view of general theory and partly from the view point of possible

applications. One explicit construction is given and its predictions com-

pared with a small but selected list of some of the "more difficult" cases

among decay data. In particular, the new structure can account satisfactorily

for the experimental angular correlations in B -* WTT and A.-* P* decays,

as well as certain branching ratios in baryon decays which cannot be under-

stood in terms of SU(3) alone. The new form factor (fT)j which has aL

simple structure (/v x ) and is consistent with Regge universality for the

reduced coupling constant (g ), is employed in Sec. IV to make an exactLi

evaluation of TN scattering in the u-channel in the spirit of a Van Hove

model. Apart from the usual Regge features (including the "dip"-effects),

one now obtains an explicit structure for the residue function in terms of

the resonance parameters. The exact structure of the amplitude makes it

a convenient tool for application to the intermediate energy region. In

Sec. V we give an outline of the applications to electromagnetic interactions

through the VMD mechanism, and indicate the nature of some results al-

-7--

ready obtained (photoproduction of pions and the n-p mass difference).

Finally,Sec. VI is a summary of the possibilities and limitations of this

phenomenological approach.

II. A REASSESSMENT OF HADRON COUPLINGS

As the method of evaluation of these structures has been extensively

discussed elsewhere [20, 39] we shall not go into these details. The no-

tation is also the same as in earlier references, except that the vector

q. (or q J of Ref. 39 is relabelled as k. (or k J and an SU(3) general-

ization is made. Thus the M-matrix of Ref. 39 now reads:

8 r «, A- i

(2.1)

where the sets of indices (i, j , k , . . . ) , {ti,V,X,...) and (a, b, c , . . . ) stand

respectively for three-vector, four-vector and SU(3) labels^ and the euclidean

metric (A- B = A B = A* B + A , B . , A. = iAn) is used throughout. For

the P-meson terms in (2. 1) A are the usual Gell-Mann matrices• v ' a

(A = JTfTj). However, for the V-meson terms the notations A and A

will be concurrently used to stand for their w-like and ^-like counterparts

according to the ideal mixing angle, viz. ,

7L->\, = ° ' ° . \-»)u - V*

For transitions between Q states, the adjoint representations of the A-

matrices is necessary. The couplings of (2.1) to hadron states are given

by the matrix elements of the quark current

between appropriate QQQ or QQ states. The quantity f in (2. 3) is

inversely proportional to the PCAC constant f (f & f, * f ) for the cor-P "" is 'I

responding P-field <f> (x) defined by

- 8 -

•P ) <£pW, Sjr* - i r /v* > (2.4)

and is related to the NNT coupling constant G by

(2.5)

Let us restate some of our older prescriptions with a view to a better

re-appraisal in the light of the current experimental situation [14] which

seems to be much more confused than was the case even a year ago. In

this brief analysis we shall concentrate on some of the more pertinent

features of the data which have generally stood out, leaving the question of

detailed fits to a separate publication. The prescriptionsused in Refs. 20,

44 and 45 were the following:

A. A straightforward relativistic boosting from three-dimensional to

four-dimensional indices for both (L±l) wave couplings.

B. An additional prescription k> k -* k k to eliminate the extra threshold<- - n u

factor k-k appearing in the (L-l) wave terms.

C. A Van Royen-Weisskopf (V. W.) factor v#/m for transitions in-

volving the emission of a heavy meson (mass n) used as the radiation

quantum 146] .

D. A factor i/M/m or (2M) for BBT P and MM. P couplings respect-

ively, (M, m) being the masses of the parent and daughter hadron

respectively.

E. A form factor fJ, (JU/W ) — for (L±l) wave couplings.

There is little choice on A, which may be regarded almost as the

language for description of relativistic couplings. Assumption C is

probably also O. K., since it seems to have a raison d' etre in terms of QQ

relativistic wave equations [47] , Assumption D is much more tenuous,

having been introduced from considerations of energetics for emission of

a quantum in the exact PCAC limit. While it gives a better result for

A-+NT decay, its working is rather bad for the decays of strange baryons.

-9-

We are therefore inclined to drop this assumption for baryons, pretending

that better values for A -*NTT , etc., widths are more a matter of consider-

ing detailed physical effects such as the energy dependence of the width [48]

than one of modifying the basic coupling structures. We note in passing

that another possible (though not very attractive) mechanism for accounting

for the variation of the decimet widths from their SU(6) values is to con-

sider the effect of a small induced pseudoscalar term in the input axial

current between quark states. It is easy to show that while such a term

makes negligible contributions to the octet (B) couplings, it nevertheless

has a mechanism for increasing A-*Njr and decreasing ^ -» £ T T >

especially if its sign is opposite to that of the main axial term in the quark

current. The role of D- for ne son coupling is discussed later in this

section.

Assumption B:

This assumption is more substantial and can be examined partly on

its own and partly in the context of E. It was motivated by the necessity

to keep certain heavy meson modes of (L - 1) wave decays, suffering

from little phase space arid yet showing considerable enhancements (e. g.2T +3

Nu(1550) ->N^, AQ1 (1670) -*A(J> from collapsing like k ^ . In the

quark language, the extra term "k^k in k k can be formally interpreted

as the effect of the quark recoil. However, since the value of k k on the

mass shell is equal to {-** ) , it would give much too strong a heavy meson

enhancement {n :m ) for the experimental requirements which are largely

met by C. In the earlier work [38, 44, 45] it was therefore sought to "scale-2

away" this factor by an extra factor n appearing in the form factor for

the (L- 1) wave (ansatz E ). We now examine alternative possibilities in

the light of certain pertinent baryon data. For this purpose we discuss i)

the coupling ratio of Y(1405) to KN and T.TT and ii) the ratio of ^ (1670)

decays to Av\ and ZJT . The former works out as /m^/m [ILL/ M \ •

Clearly, the choice A* = ITL is out of the question since the ratio is too

large (/v20) for any "gentle structure" of the form factor. On the other2 2

hand, if instead of the assumption /u = m , we choose to continue the

quantity k k all the way along the P-meson trajectory from its value

-10-

2 P ~ 2 P +-m at J «0 to the value -m. at J «1 (viz., the correspondingP A 2

A-meson mass shell), the variation of the factor {-fx ) would be much

more gentle, so as to put much less load on the form fac tor^ We there-

fore revise our earlier prescription as follows:

Old: /U^= >^p I New: £ * ^ . (2.6)

With this prescription the ratio of the Y(1405) couplings becomes

(2. 7)

in contrast to our earlier prediction of Vm '/m = 1. 87 and the experimental

value [32] of *3. The model of Gell-Mann et al. [32] using the A -— — jj

coupling as Y A B gives (Y"N)/(Y- I ) » 2 , 3 , in surprisingly close

agreement with our empirical prescription,though based on a very different

mechanism for the coupling structures. We note in passing that equivalent

structures in the quark language can come about only through an ihduced

pseudoscalar term in the input quark current.

Interest in the Ani (1670) —»An, Z/r modes arises from the fact that its

SU(6) couplings to Ai and £ v states are identical in magnitude and phase,

irrespective of whether it belongs to J , , 8 or 1 , of 70 . This facta q a —

prevents any mixing effect from being of any help in increasing the ratio

of Ah to £JT modes from the phase space value of ~1/18 to the experi-

mental value of ** 0. 7 . Of this the V. W. factor supplies a factor of A-4. 04 4

while the new prescription gives an additional factor of m /m « 2 . 1 ,where m is the mass of the D(1285) meson chosen as the axial counter-

part of ?| . Thus, apart from^ossible effect of the form factor (and this

is small), our new prescription increases the ratio to ^0. 5 which is

reasonably close to experiment.

2 2

Because of the huge difference between m p and m , a relative

scale factor is now necessary between the (L±i) wave couplings of a given

supermultiplet. This can,however,be fixed only after taking account of the

effect of the form factor(E), to be discussed in the next section. When-11-

this is done, using a common form factor for (L±l) couplings, the scale

factor (S. F.), adjusted to the experimental Y{1405) -> £ff width as input,

= 42MeV (40 ± 10) , (2.8)

comes out as a surprisingly simple number, viz.,

S.F. * ' ' ^ i r / ^ ~, •"*/•»> . (2.9)

Its consequences on the angular correlations in B and A1 meson decays

turn out to be even more interesting, as we see below.

Meson decays

Before proceeding with a discussion of the form factors, let us see

the effect of the assumptions A —D , and their baryonic modifications, on

some meson couplings. One important difference in this case concerns

the role of assumption D , since some normalization factor is now formally

necessary for providing an acceptable translation from non-relativistic to

relativistic structures. As was found in Ref. 45, the factor (2M) did not

play a particularly helpful role for the V-meson decays to PP systems,

while its role in the decay of higher mesons (though it appeared helpful at

first sight) could not really be judged independently of the parametrization

of the form factors. Therefore,in the same spirit as employed for its

baryonic counterparts, we now propose to replace the factor 2M by

v 2M X 2m ; which would result if the problem of energy conservation in

the PCAC limit were not taken literally. However, for couplings to equally

massive particles (e. g. 7"r, KK) we are forced to replace rn_ by (M/2) in

the factor \/4Mm , a prescription which has really no deep principle behind

it, but is best regarded as a pedagogical device for preventing a violent dis-

agreement with the data on V -* PP decays. The decay widths for p -> irw*- —

K —> Kir, <j> -*KK and w -* 3*r are now respectively (in MeV)

(2.10)

where the numbers in parentheses denote the experimental figures. The

value of g = g __ now comes out as

- 1 2 -

(2.11)

in contrast to the previous value of (3. 6), We are unable to offer any formal

defence of this prescription but we believe that an analysis by Weisskopf

and collaborators [49] subsequent to the V. W. paper 146] is numerically

consistent with the above.

We now consider the effect of assumption B., together with the modi-2 2

fication m 4 m and the associated scale factor (m^/m ) obtained from

baryon data, on two crucial meson data, viz., the angular correlations in

B—¥ »i<i and A1 •-> Pw . The simple quark model predicts the BUJT and

A1 pfl1 couplings as

respectively. According to prescriptions A and B, we must first separate

the d- and s-wave components before writing relativistic structures.

Thus, we rewrite the Bujr coupling as T , , k . k . + - t k B ' « , where

- r , \ T> ,x . ^ 1 r T> . . (2.12)

with a similar reduction for A. P"" . The boosted structures for B<J?r and

A, pir, taking account of the scale factor (m /m ) ri the S-wave term,are1 * • n p

where

and

-13-

•4

From (2,13), the distribution in the angle X • cos w* k works out as pro-2 2

portional to (1 - 0. 76 cos X), compared with an almost pure sin X distri-

bution found experimentally [50] . This distribution is considerably better

than the one found in Ref. 45 with the older prescription. Similarly, from

(2. 14), the ratio of the helicity (1) to the helicity (0) amplitudes for A1 P*2 2

couplings works out as (taking m - 2m )

Z G-S7(2. 17)

where the quantity in parentheses represents the recent experimental number

translated into the helicity language [51] .

Ill, STRUCTURE OF THE FORM FACTOR

In this section we re-examine assumption E. on the parametrization

of the form factor (F, F ), keeping in view some pertinent data on baryon

decays for putting a particular choice to experimental test. In this respect

it is useful to remember two features of the latest Rosenfeld tables, viz. ,

i) less uncertainty in data on "strange" modes and ii) less confidence in

absolute rates than in the ratios of modes within an SU(3) multiplet. Since

the form factor (F F ) is empirical, it allows a considerable degree of free-

dom in its choice, and it is convenient to lay down certain broad guide-lines

for its construction. Some desirable features are :

a) It should be an explicitly Lorentz-invariant quantity.

b) It should have a dimension k in the momentum variable to offset

the effect of the multiple derivatives in the coupling structures for large

k , even off the mass shell.

c) It should be crossing symmetric in the momenta,

d) It should exhibit universality for the coupling strengths of successive

resonances which are believed to lie on a given Regge trajectory.

e) It should have a simple enough structure for easy applicability for

different processes which involve its behaviour off the mass shell for

one or more legs.

-14-

f) Its structure should show strong similarity for meson and baryon

couplings, so as to provide a unified framework for both.

g) It should lead to values of the "reduced" coupling constants for super-

multiplet transitions in harmony with the intuitive belief that super-

multiplets of the same representation exhibit much better overlaps

(e. g. , 56 with 56) than those with different representations (e. g.

56 with 70).

h) Lastly (and this is most important) it should exhibit the main experi-

mental features of various decay modes as a function of the physical

masses .

Unfortunately our ear l ier version of the P. P did not satisfy some of

these cr i ter ia , especially c). In particular,the undue emphasis on the

"quantum" P-meson for producing both the dimensional s tructure

ond the damping effect for large k_ , led to unusually large magnitudes for

heavy-baryon-cum-light-meson modes (especially A^ and ZTTT), Another

problem with this P F, was the relative enhancement of the couplings of

"unstretched s ta tes" compared with their fully "stretched" counterparts

(e .g . , N1(.(1688) -» NT versus A (1950) ->NT). Finally, the relative

magnitudes of the supermultiplet coupling constants g and g , v iz . ,2 2 .

g > 4g , did not appear to be in harmony with our cri terion (g) whichshould ra ther imply the opposite inequality.

We shall attempt to formulate a Class of F F. ' s designed to satisfy

the cr i te r ia a)-h) to a greater extent than was possible in the ear l ier t rea t -

ment. Though one explicit s tructure will be worked out in relation to some

selected data on baryons and mesons, the present treatment should be r e -

garded as largely illustrative from the point of view of fitting the data in

detail, a task which by itself is best relegated to a separate, more com-

prehensive, investigation.

Taking the four-momenta of the resonance (mass JM) daughter hadron

(mass nj) and quantum (mass j ^ as P , p and k ,respectively, we form

the invariants

-15-

quantities which are expressible, on the mass shell, in terms of the com-2 2 2

binations M ± m ± V . As we are primarily interested in application of

the F. F 's off the mass shell (for one or more legs), we must specify at

the outset the off-shell structures for the above quantities as well, For

this purpose, we choose the simplest ansatz: Eqs. (3.1) define the off-shell

extensions as they stand. This point of view was also taken in the earlier

version of the F F , the predictions of which in respect of several processes

viz., Tp -» pN 152] , 7p -*• ffN [53] and the (n-p) electromagnetic mass

difference [54], are in rather good accord with experiment. (We shall give

a brief discussion of the last two processes in a later section,) The chief

motivation for this ansatz is that this is the easiest way to ensure that the

contributions of successive (s-channel) resonances to a particular process

have a reasonable energy dependence. In the language of dispersion theory,

the effect of this ansatz is that contributions from successive s-channel re-

sonances in general imply more than mere "pole" contributions, though

formally they might look like the latter. This is because the residue function

in such a model is, so to say, a function of both energy and momentum trans-

fer, and not merely the latter. The same ansatz which, incidentally, keeps

our model closer to the spirit of conventional field theory than that of dispersion

theory, also prevents divergences from appearing in a calculation involving

integration over a virtual four-momentum, e. g., the electromagnetic self-

energy of a hadron [54] . There are, of course, problems of unf ore seen

singularities involved in an indiscriminate extension into the complex plane,

but these can be examined only in the context of a particular situation.

For comparison we exhibit the off-shell parametrization of our old

FF for (L±l) wave couplings explicitly in terms of the invariants (3. 1) as

. ( 3 - 2 )

While this structure no doubt has the desired features outlined in the pre-

ceding paragraph, the entire emphasis on the single invariant b out of a

list of four, makes it asymmetrical between the two outgoing particles.

Another disadvantage, from the applicational point of view, lies in its ex-

plicit dependence on the mass of the resonance, which in turn gives it a

-16-

much stronger L»-dependence than implied by the exponent, because of the2

(observed) linear relation between ML and L . In a way it violates

criterion e), which we really do not know how to formulate clearly except

through illustrations. Thus a structure which shows an explicit Li-depend-

ence only in the exponent, and none inside the argument, is clearly pre-

ferable to one which does not have this feature when it is remembered that

a summation over L is implied in any calculation where the resonance

appears as an intermediate state. The simplest way for the mass M to

appear is through a suitable combination of the invariants of (3. 1), which

have enough flexibility to show the correct dependence on the mass

shell, without giving up some obvious advantages off the mass shell. This

restriction does not of course apply to the physical masses of the decay

products.

The foregoing considerations leave us with the task of constructing an

F F of the form x , where x has the dimension of inverse momentum

and is built out of a suitable combination of the invariants of (3.1) and the

fixed masses (m, w) of the system. We should also prefer to have a

common parametrization for (L± 1) wave couplings, rather than separate

structures like (3. 2). Since crossing symmetry is an important require-

ment, preferably in all the three momenta, we are eventually led to consider-

ing the combinations y'abc and ,/ abed , which are of momentum (q) di-3/2 2

mensions q ' and q respectively. Since this factor has to come in the""• 1 / 2

denominator, the corresponding numerators must have dimensions q '

and q respectively. Several possible structures can be built out of the

masses m, 14 or a symmetrical combination, e. g., /m/u" . It is amusing

to note the following purely empirical relations involving the masses of the

corresponding V-meson which may serve as a further guide to the construct-

ion;

^ rf)Aw ' i j - "'CO ,(3.3)

- 1 7 -

A more attractive possibility for ensuring a gentler variation of the F F

with the masses is to make use of the masses of the axial counterparts of

the P-mesons as was suggested in Sec, II. More important, this proposal

offers much better prospects for a unified description of the F F 's of

BPBj and MPM. than, for example, structures of the type Jmfl which

show rather wide variations between meson and baryon couplings.

Baryon decays:

From the experimental point of view, fairly reliable guide-lines for

(h±l) wave modes are provided by the decay patterns of A (1520),

AQ5(1815), N15(1688), A37(1950) and A (2100) which (one hopes) are

comparatively free from mixing effects. Thus one observes a sharp increase

in the ratio of NK to T* modes of the successive A-resonances as the L-

value is increased. By a process of multiple feed-back between "guess"

structures and the above data, one acceptable form turns out to be

*-• r 1 Ljf U V "^A / G-v-C- j /o 4.\

where 2 is a scale factor to be adjusted from some absolute decay rates

which are fairly reliable. In (3. 4) we have suppressed some of the factors

which are already covered by the discussion in Sec. II, viz., the V. W. , SU(6)

and the coefficients of the Clebsch-Gordan expansion of the direct production

of spin and orbital functions. However, it has been written in a form which

permits a transition all the way to the L = 0 , showing the correct SU(6)

structures for the £6 couplings. The parametrization is now the same for

L±l couplings except for a scale factor for the latter, which we have also

shown in (3. 4). As to the dimensionless constant g , we expect it toLi _

satisfy our criterion d) which requires the following equalities:

g0 = g2 = g4 hi <3'5)

as also the criterion g) which requires (note that by definition g = 1)

-18-

To give a sample comparison with the data we first list certain pertinent

decay ratios within given L-values which are independent of the constants

g and o" , the quantities in parentheses denoting the experimental

numbers:

( l 5 ) [ i / ] = 1-1

A o l

To the extent that the comparison looks favourable we feel encouraged by the

general approach. In particular, these figures indicate that by our new

parametrization we seem to have succeeded in remedying two important

defects of the earlier parametrization mentioned in the beginning of this

section, viz., i) we now have more reasonable (smaller) magnitudes for

the £ JT and A?r modes and ii) the earlier suppression of the coupling

strengths of stretched states in relation to the unstretched ones no longer

exists. The physical reason for this is directly traceable to one less

power in the momentum structure of the form factor in the new parametrization

(3. 4) compared with the old form (3, 2), for (L + l) wave decays.

For the absolute rates, we determine o- from the experimental width

(12. 5 ± 2. 5) MeV of A(1815) -> I> which is free from heavy meson effects,

by using the equality (3. 5). This gives

CT- (1.8 ±04) . (3.8)

Next we fix the value of g by reference to the absolute width (6. 5 ± . 5 MeV)

of ^^1520) -+Zir decay, giving

g^ fc 0.62 ± 0 . 1 . (3,9)

- 1 9 -

It is pleasing to note that this value satisfies the inequality (3, 7). To check

on Eq. (3. 6), the value of A(2100) -*2> which comes out as 1. 0 MeV with

the parameters (3. 8) and (3. 9)j seems to be consistent with the experimental

value of 1 ± . 5 MeV.

Before concluding this brief discussion on baryon resonances we make

some comments on the Regge universality of the coupling structures for the

higher baryons, especially the 56 A-sequence. The strong L-dependence

of the decay widths is exhibited by the structure

(3. 10)

whose asymptotic representation for large L is

(3.11)

in qualitative agreement with the Veneziano features [43] and also with ex-

periment. Here we have used the empirical, but numerically accurate}

relation (all the way down to L = 0)

for the successive recurrence of the A-sequence. The data are quite com-

patible with the Regge universality implied by Eq. (3. 5) though a quantitative

comparison at this stage would not be in order. The same remarks apply

to (3. 6), where the data are even more inadequate.

Meson decays:

Since there are fewer (and probably less accurate) data for comparison

in the meson system, the extension of this analysis to this case is much

more limited. In this respect we are inclined to take seriously our criterion

- 2 0 -

f) and to this end check the same parameters as determined from the

baryonic decays for consistency {or lack of it) with the limited meson data.

Thus the form factor in this case has the same structure as (3. 4), except

for a separate constant g and the .normalization /4Mm or y2MLi

needed for the relativist'ic meson couplings for m f (t and m = v respect-

ively, as already postulated in Sec. II. Again, by definition, g = g = 1 ,

while g is expected to satisfy the inequality (3. 7). We determine f

from the piece of meson data which we consider more reliable than most

others, viz., K,-(1420) -* K*r which has an experimental decay width of

47 ± 4 MeV and find „gx ^ 0.47 ± 0 . 1 , (3.13)

The value of the decay width A1 -* P*" is found from (2.14) and (3. 13)

as 110 MeV (95 ± 35 MeV), while that of B ->UTT , obtained from (2.13) and

(3. 13) works out rather low at 38 MeV (102 ± 20 MeV), though we note that

the experimental value itself has come down appreciably, during the last

year. Similarly the f(1260) -*• nn and ^(1514) -> KK modes,on the assumption

of w-like and <£-like objects, come out as 92 MeV (150 ±25) and 89 MeV

(60 + 25) respectively. , (It may be noted that the last pair is extremely

sensitive even to small mixing effects.) Finally, the p (1660) -» fffr modeiN p

which can be calculated in this model assuming it to be an L = 2 , J =3recurrence of the p , worjcs out as ^40 MeV, while the quoted experimental

number is "dominant fraction of a total of 110 ± 30 MeV". The situation is

apparently too confused in this region to warrant further comments, but -

and this is important for our approach - the figures do not seem to be in-

consistent with the equality (3. 5) for the reduced meson coupling constants.

For the equality (3. 6) there is even less to go by. The general feature of

a strong decrease in the width with still higher p-recurrences (^ that is

present in the model in exact analogy with the AT -» N?r case (indeed the

p_ TT coupling is in some sense the exact mesic counterpart to the A NTL i •LI

coupling) is also not inconsistent with the qualitative experimental trends in

the "UXYZ" region,Before ending this section it is tempting to make a comment on the

2 _2relative magnitudes of the numbers g and g and their relevance (if any)

- 2 1 -

to the criterion f) for the form factors. Indeed, a simple consideration

based on a naive application of the quark model does seem to bear out their

relative magnitudes. Thus if we suppose that the spatial "quark wave

functions" of baryons (QQQ) and mesons (QQ) have essentially the same

structures, or at least play an unimportant role in the emission of a radiation

quantum, then a careful analysis of the overlap integral in terms of properly

normalized internal momentum variables, yields the simple ratio /2/3 :

/ i / 2 for the emission amplitudes involving QQQ and QQ states respect-

ively. It seems that these ratios are not only compatible with the values2 _2

of g and g given by (3.9) and (3.13) respectively, but even their absolute

values are very close to 2/3 and 1/2 respectively, giving some credibility to

the speculation that perhaps the spatial wave functions involved in the over-

lap integrals are influenced little by the difference between, for example,

56 and 70 symmetries, apart from purely geometrical factors which can

already be accounted for by the SU(6) x 0(3) structures. A fuller discussion

of this idea will be given elsewhere.

To summarize, we have been able to construct an explicit structure

for the form factor which has all the desirable features listed in a) to h).

In particular, the baryonic data are quite compatible with Regge universality

for the coupling constants, while the mesic data are at least not incompatible

with it. An equally interesting result is that the same parametrization for

the form factors (except for a relativistic normalization for the meson

couplings) gives quite a good description for hadronic resonances, on the

basis of a limited but pertinent set of data chosen for this analysis.

Presumably other form factors of a similar kind can be constructed, but the

one evaluated here appears to be a satisfactory candidate for applications

off the mass shell.

A PCAC relation:

As the simplest illustration of the application of our crossing-sym-

metric form factor off the mass shell, we note that it is ideally suited for

the derivation of a PCAC relation between the supermultiplet coupling

constants g\ and g by exploiting the coupling structure for A-P^ and

expressing it in terms of the pinr coupling in the limit of PCAC. A simple

- 2 2 -

method which is rather different in spirit from the conventional dispersion

approach [55] , has recently been described [56] and consists essentially

of the following steps. i) Regarding the full A.pff coupling as an effective

Lagrangian, identify the effective A current between ( x \ and / p>

states as the coefficient of the A field, and then divide by the universal

coupling g for the A-field to obtain the normalized A -current, ii) Relate

the latter to the pion field via PCAC and identify the result with <( * \ it \ p >2

in the limit when the (four-momentum) of A. tends to zero. This pre-

scription yields the relation (with L = 1)

\ c*where we have neglected (m^fm

2) in the form factor (3. 4) and used the

prescribed relativistic normalization /4m m for the A1P* coupling

(2.14). Finally, the Weinberg sum rule [57] g ^ 2g and the value

f z m //2 for the pion decay constant [49] yields,via (2,11), the PCACIT n

estimatev (0,75)2^ 0.55

which is in excellent agreement with the more empirical estimate (3.13)

from the point of view of supermultiplet (L = 1) transitions. This feature,

which was also present in the earlier form factor, appears to provide a

welcome check on the "numerical consistency" of the coupling structure and

gives reasonable confidence in the reliability of extending the form factor

considerably beyond the mass shell.

- 2 3 -

IV, APPLICATION TO HIGH-ENERGY SCATTERING

As a more substant ia l application of our coupling s t ruc tu re , we

shal l consider the process of TrU sca t t e r ing through the exchange of

baryon resonances in the s p i r i t of the Van Hove model* To describe the

essent ia l s t ructure of the theory in reasonable de ta i l we sha l l confine

our a t tent ion to a r e l a t i ve ly simple case, v i z , , backward Tr"p scattering^

which receives contributions only from A-exchange, unlike ir+p sca t te r ing ,

which i s dominated by U-exchange, Further , the t ra jec tory of the (56,

even+) ^ - s t a t e of J w L + 4 , being higher lying than any other , should be

the dominant contr ibutor to the process . This in terac t ion i s of the form

(suppressing the isotopio indices)

where

(4.2)

In making this translation k^—> q^, for the momentum factors we have kept

to the more conventional spirit [133 and this in any case makes no ^

difference for th-e predictions on the decay widths. For the process

ifp -• pTT~ , we take the initial and final (p***) momenta as (p^jk^

(p',k'), respectively, so that

?~ ^ /fO"3". (4.3)

Taking the propagator for AL in the standard form [133 the invariant

amplitude A-i is expressible as [58]

k = -i L 8*1

x

- 2 4 -

where

= U -(4.5)

(4.6)

^ S\^) (4.7)

and the "daughter" contributions have been neglected in view of the

inequality

(4.8)

The s»oond term in the curly brackets in (4*4) ~ the spin-flip part - makes

negligible contribution at high energy and will thus be dropped. For

MT , a oonvenient and numerically accurate representation is (3.12) which

works well al l the way from L « 0 onwards* For the summation over Lf

we use the following integral representations:

and

")- It") -

The summation over L which can now be carried out exactly, is expressible

in the following form:

-25-

\» w'4"v s!

" 6

( 4 * 1 5 )

These integrations,which lend themselves to rapid evaluation in the limit

of large j j t lead finally to

4 = ii 0-

A - A /s^C^) K^-t)3 £(-i« i S i + v ) {4#18)

k *" B ( i 4 - ^ i+au (4.19)

X "being the parameter oharacterizing the form factor. It is clear that

the above method of summation will work for any form factor of the type

X where x is independent of L. For completeness, the differential

cross-section in the backward direction at high energy is

where we have used our empirical relation between f and g :

>- ^ > . (4.21)

-26-

A few general remarks on "the structure of the u-channel amplitude

are probably in order* The Regge struo-bure whioh i s ,o f oourae, guaranteed

through the mechanism of the Van Hove model oan "be made more transparent

through the identif icat ion

CLtlX X <*AM~\ . (4.22)

The essential difference between conventional Regge phenomenologyand this model lies in the mode of parametrization of the Regge residuefunotion f(u). While,in the former, the choioe of this function isdirectly governed by the experimental features of the high-energydifferential cross-sections, the-present approach proposes to shift theemphasis to the experimental patterns exhibited in the decay channel,eventually leading to a "determination" of @(u) via the assumed structureof the form factors, (A formal advantage is that the present methodyields an amplitude which includes the "background integral" in the Reggelanguage and this may be of some value in the intermediate energy region,)However, once having "obtained" this function from the resonance data,one must depend entirely on i ts predictive role for the detailed fits tothe high energy data which should therefore provide a quantitative testfor the ohoioe of a particular structure, such as eq.,(3*4)» of the formfactor*

As for the general features of our amplitude (4.17-4.19)» one can,e,g., reoognize the out-structure (at u » 0) in the A-term which isdirectly traoeable to the mass terra of (M-iy.p) in the baryon propagator.To eliminate this i t is formally necessary to invoke the Gribov-Pomeranchuktheorem [59] on parity doublets, evidence for which does not seem to existin the baryon speotrum. On the other hand, i t has been found that the/u-out is in fact helpful in producing a better f i t to the backward pTrscattering data [60], Hbte also that the signature factor in front of(4,17) is just enough to prevent the B-amplitude from blowing up (orvanishing) at u»0, while the A-amplitude indeed vanishes at u-0 becauseof the effect of i t s multiplying beta-function. Further, as expected for7T*p backward scattering, the point a((u ) - - £ (whioh corresponds toa-u « - l ) exhibits no dip in the cross-section (4.20),

The method allows a straightforward extension to the exchange of

N-resonanoes whioh play the dominant role in tfp backward scattering. It

is clear that the "dip meohanism" in thiB case would be automatically

- 2 7 -

present in this model sinoe the signature faotor now turns out to be

4(1 + e" iauir) where au « *N(u) - £ as in (4*22). The numerical details

on this subject are currently "being investigated.

Behaviour in other channels

The struoture of the s-channel amplitude (whioh oan also "be

obtained by using a similar technique of summation over baryon resonances)

does not lead to any interesting structures in the high-energy region.

However, in view of the "exaot" structure of the amplitude obtained in this

manner, i t should be of considerable interest in the intermediate energy

region where the resonanoes are expected to play a more quantitative role.

In particular, i t may provide an alternative mechanism for the dip

struoture near the backward direction in yr p scattering at intermediate

energies and could therefore serve as a phenomenologioal test of duality.

Indeed a recent attempt on these lines has oome to our notice [61] and

we wish to remark that the present model provides a natural quantitative

base for such investigations.

Unfortunately, this model is inadequate for the description of

high-energy behaviour of amplitudes near the forward direction. This

requires the necessary Regge struoture in the t-ohannel, a feature which

cannot be obtained in this model without summing over t-channel (meson)

resonances. In principle, such a feature can be simulated by assuming,

e.g., universal ooupling of the suooessive Regge recurrences j>j(J=l,3,!>»*«•)

to TTtr and M systems, leading to ooupling structures of the respective

forms (suppressing the isospin indices)

< 4 > 2 3 )

L

(4.25)

where the quantities fL f include the form factors. The practical

difficulty l ies in the cut-structure of the la t ter in the t-channel. Themathematics goes through as before with the forms f ' ~ (x^ ^) , but i t

- 2 8 -

is now necessary to exert much more care in choosing appropriate cut—

structures for these functions in the light of stronger restrictions on

these structures dictated by general theory. Thus while the absence of

doublets in baryons allows a oertain degree of "tolerance" in the

construction of their form factors, the standards are, so to say, more

exaoting for their mesio counterparts for a theoretically acceptable

description of t-channel amplitudes* Prom this point of view the

representation (3*4) for the meson form factors must be regarded as more

tenuous than for baryons and the data on meson resonances (which are still

very qualitative) may well be compatible with stronger selection rules on

their form faotors than implied by our limited list (a-h) outlined in Sec.Ill

We have not yet succeeded in obtaining better structures of the general

form x compatible with general "cut" requirements in the t-ohannel, and

yet satisfying our "intuitive" criterion (f) which may well have to be

dropped or at least toned down to a more qualitative level. We close this

section with the remark that a oloser examination of purely mesonic

amplitudes may offer better insight into this question.

V . VECTOR AND ELECTROMAGNETIC INTERACTIONS

Another interesting area of application of our ooupling structure

off the mass shell is in the domain of vector meson couplings and especially

the electromagnetic interaction. The basic mechanism for the correlation

of V-meson to P-meson couplings, viz., SU(3) x SU(3) or partial symmetry,

is expressed by the structure of the M-matrix (2.1) which fixes the relative

strengths of the different terms. Relativistically invariant Bl V

interactions of four different varieties which were worked out in Ref,(3i)

as a straightforward generalization of the techniques for B^P couplings,

are reproduced below for easy reference.

V7

-29-

1 rL ^--^L. (5.3)

(5.4)

Corresponding structures for MM-V couplings could also be written down in

an analogous fashion but these are not of immediate physioal interest*

The correlative powers of the ooupling soheme are incorporated in the

assumption of a common coupling constant g-r and form factor f, for a

given supermultiplet transition B, —> B, through V or P mesons. A recent

application of this prinoiple to the evaluation of the differential cross-

section and density matrices for the prooess TTp -• pN shows a good fit

to the data (in magnitude and shape) with no free parameters [52] #

To introduce the e.m, interaction, an extra ingredient is necessary,

the simplest one being the vector meson dominance (VMD) [62] . In spito

of reoent claims about its failure on finer details, it is safe to assume

that VMD still represents the major mechanism for the ooupling of photons

to hadrons. The V-y coupling is described by the interaction

oL.,.. - HuA • 3.e 4V

which must be taken in conjunction with any one of the terms (5*l)~(5«4)

to obtain BB-rY couplings from BB_V, One important problem concerns the

question of gauge invarianoe. While it is not possible to go into the

full sophistications [63] of this point, perhaps the essential features of

gauge invariance can be incorporated without muoh difficulty even within

this phenomenologioal framework. Thus it is clear that the couplings (A)

and (D) are explicitly gauge invariant (Gfl) as they stand, (B) is gauge

invariant for equal-mass baryons, while for unequal masses the following

prescription is adequate:

-30-

For coupling (C), two points of view are possible* For an "external"photon lino (k*k» 0) it can "be regarded as gauge invariant in the (limited)

ttiat

senseAthe associated current is conserved* On the other hand, this is not

so for an "internal" photon line. A different point of view, which is more

in the spirit of the prescriptions for (L-l) wave BB,P couplings used in

Sec.II, is to take some "fixed" value of the k-k , say at the position of

the corresponding T-meson mass to which the photon is coupled via (5*5)*This reduoes type (C ) essentially to the form ^±y_ /• V "^suppressing the

(inessential) k^-factors and 4-vector indices. The other term under (C),

is also reducible to the same form on the mass ehell of thebaryons. Since this structure is no longer gauge invariant, a simple

prescription to make it so would beAanalogue of (5*6),

YS ] If (5.7)

This prescription does not of oourse work for equal masses. Fortunately

the negative evidence for parity doublets in "baryons is a help in this

regard*

Obvious candidates for the application of this e.m. interaction

are (i) Vp —» TtN via the s-ohannel, (ii) electroproduction of resonances

and (iii) e.m* masses of hadrons. Results of reoent calculations of the

first [533 and third [541 prooesses using the earlier version (3#2) of the

form factor are now available* While the details will be reported elsewhere,

it may be of interest to record some essential features of these results,

since the qualitative struoture of (3*2) is not very different from that of

eq,.(3.4)«

V-produotion of N*

This process has been considered by a number of authors [64] » K>51

in reoent times, with a view to understanding the following main experimental

features (6611

i ) The resonances 3). .,(1520) and F _(l688) contribute strongly to

total y-production oross^seotions but are absent in the forward and back-

ward directions.

i i ) Over a wide energy range 1 only the A-resonanoes are

prominent in the backward direotion*

i i i ) The TT+» cross-section is significantly larger than ir P near

the forward direotion.

-31-

The present oaloulation [53], which was also oriented towards an

understanding of the above features, seem to bear out the features (i ) and

(ii) fully and (iii) partially. Thus the fits to the experimental data

for 7r p and jrn are found to "be extremely good, except that the TT n

differential cross-section near the forward direotion falls short of

experiment by about (3O-4O)#, This last should probably be ascribed to

the role of pion exchange [64^ an effect which has not been included in the

calculation of Ref,53. However, for the other two features,(i) and (ii),

no special mechanism [65] seems to be called for, except for the coupling

structures implied by SU(2) z Stl(2) which automatically fixes the relative

strengths of the minimal and magnetio interactions of the photon with the

baryon*

(n~p) mass difference

The present model iB rather well suited to the problem of evaluation

of e,m, masses of hadrons,since the structure of the form factor (3»2) or

(3»4)| together with the propagators of p and u> (via VMD) are just

sufficient to make a prediction free from cut-off parameters • The

language is that of the old-fashioned Feynman diagram of the second order

t67 rather than dispersion theory [68],[69]» yet beoause of the role of

the form factor, which depends on both energy and momentum transfer, the

numerical effect of a formal "second-order.11 calculation extends far beyond

the naive interpretation in terms of strictly "pole" contributions of

different resonances. In particular, the role of "subtraction'may be

thought to have been effectively incorporated in the structure of the

form factor. Thus there is no formal problem of contradiction with the

analysis in terms of dispersion theory 169)t though it is difficult to

establish a one-to-one correspondence between the two methods.

The present model, which happens to give the right sign and

magnitude for the (n-p) mass difference,works essentially on the following

meohanism. For a given SU(6) 1 0(3) supermultiplet, the state J 0 L ± •§•

provide the "wrong" and "right" sign, respectively, for olfn-p). This

statement is independent of a detailed model for the form factor and

depends only on the algebraic structures of the respective couplings. The

form factor (3.2) on which the calculations are based [54] has the additional

property of giving a larger numerical contribution from (L-jjj-) than from

(L+j ) states, However, this effect is zero for L = 0 and small for L » 1,

It becomes pronounced in the range L a 2 to L o 4,which plays the crucial

-32-

role in bringing about the requisite change of signs, Beyond L o 4, the

overall magnitude declines rapidly toeoauee of the proportionality to a

factor B(L+1, L+l), Thus the moat significant contributors to the "right"

sign for (n-p) in this model are the P1^(l86O) as the L-|- counterpart of

F (1688) (and perhaps also the corresponding Regge recurrences of P.... (1450)),

To examine the "stability" of this result against alternative parametrizations

of the form factor, this region needs a more detailed investigation, "but

the essential mechanism seems to be veil founded* We note in this connection

that Cini and collaborators [70] have found the correct sign for S (n-p)

and several other sets through a different approach*

This model is also well suited to the calculation of electro-

production of ff*~resonanoes where the experimental data 171^ seem to indicate ap

very gentle fall of the differential cross-section with q , In aqualitative way, this feature comes about in the model through a super-

position of the following two effects 1

i) the momentum factorsc(kM) in the couplings (5«l)~(5»4)»

ii) the structure * / of the form factor, which was designed

mainly to tone down the effect of faotors (Qoff the mass shell.

Detailed calculations on this process are in progress.

VI. SUMMARY AKD CONCLUSIONS

We have tried to outline a phenomenological approach centred

round the physica of baryon resonances with a view to exploring its

dynamical manifestations in several areas requiring considerable extensions

off the mass shell. This point of view is opposite to the more conventional

approach which is centred round the Regge region for the parametrization of

data,and probably offers some differential advantage over Regge phenomenology

for the study of the intermediate energy region in a more quantitative form.

The language of classification of hadrons is that of SIT(6) x 0(3), while the

basic framework of interaction is provided by a pedagogical version of

chiral SU(3) x SU(3) going by the name of "partial symmetry". The quark model

is used as a convenient device for writing down the algebraic structure of

the interactions of BB, currents with P and V mesons, while no use is made

of its dynamical implications. The structure of the various interactions

-33-

is given by that of multiple-index Rarita-Sohwinger fields associated with

multiple derivatives of the P and V fields. The central role is played "by

a Lorentz-invariant multiplicative form factor fL for each supermultiplet

transition B. -> B, In the absence of any acceptable dynamical "basis for

its evaluation, a comprehensive set of general guidelines is used for

writing down its possible structures whose quality must be judged by their

capacityAprovide a sufficiently wide coverage of the main features of the

resonance data.

The problem of explioit construction of the form factor has been

dealt with in some detail. In particular, an explioit construction of

the form X is suggested to meet, among other things , the general requirements

of Regge universality, crossing symmetry, "gentle" behaviour for large

momenta off the mass shell and easy applicability to different phenomena.

This is achieved by chossing X to be explicitly independent of the

resonance mass (and hence of L) and of the dimensions of an inverse momentum

to "balance" the effect of the multiple derivatives in the Rarita-Schwinger

structure* This represents an improvement over an earlier form of

parametrization whioh, however, shares its main mathematical features. The

new form factor is found to provide a reasonably accurate description of

the main features of the hadronic data on the basis of a sample collection

of some of the more pertinent baryonic data. The same structure is found

to work broadly for the meson resonances as well, keeping in view the status

of the data on toe latter (which show a less dear pattern beyond "1,5 GeV),

Among the experimental successes of the new scheme are several ratios of

the baryonic coupling constants and decay widths, which are not easily

amenable to a simple treatment, and the reproduction of the observed angular

correlations in B -» WTT and A. -> pTT decays, A PCAC relation which is

derived between the A o7T and OTTTT couplings provides a further check on the

numerical consistency of the supermultiplet coupling constants for meson*.

Being phenomenological in character, the model cannot claim a

formal theoretical basis, but it is sought to compensate for this by its wide

applioational base; to which it is mainly oriented. From this point of view

the possibilities of several applications are discussed. Thus the model

provides a natural mechanism for the description of iTp scattering in the

u-channel, via the Van Hove mpdel. Some other areas of application are

indicated by s-channel processes such as PB -* PB, PB -* VB, yp -* BP in the

intermediate energy region, electroproduction of baryon resonances and e,m.

mass differences of baryons, all of which must make essential use of the off-

-34-

shell extensions of the form factors fL in various degrees* As most ofthese oaloulations are still under way, this aspect of the approach must "be

regarded as largely at the stage of a programme, and this has been indicatedat appropriate places in the text. However, to keep this aooount

reasonably self-contained, we have taken the opportunity to indicate the

nature of the results in respect of two processes, viz,, yp -?- nN and the

(n-p) mass difference (for whioh the numerical data are available with the

older version of the form factor), both of which seem to be fairly well

accounted for by the model*

One can also envisage applications to systems with baryon number

greater than unity. An interesting possibility is the explicit construction

it provides for the dKN* coupling as a potential candidate for understanding

certain phenomena associated with deuteron scattering, e.g., pp —» dn and

pd -» dp reactions [72], [73], Thus it has bean suggested [72] that the

magnitude of the backward peak in a pd —> dp process is too large to be

given by a simple nucleon exchange and that even a small percentage of

N* (1688) in the deuteron is adequate to explain the data* However,

the quantitative aspeots require a more careful evaluation of d M * coupling

for which the present model provides a very convenient framework using

standard techniques available in the literature [743,1751* A short derivationis given in the Appendix.

The formal similarity of parametrization of the baryon and meson

form factors would, off-hand,1 seem to suggest corresponding applications to

processes in which the coupling to the meson resonances plays the main role,

especially high-energy scattering near the forward direction (t-channel),

a feature which cannot be reproduced in our framework with baryon resonances

alone. However, the cut-structure for the mesic form factors must be

chosen much more carefully before such applications are possible in practice.

(This is because the theoretical disciplines on meson form factors are some-

what more exacting than those on their baryonic counterparts.),, Because of

this limitation we are inolined to regard our insistence on criterion (f),

viz., the desirability of formally similar structures for meson and baryon

form factors, with a certain degree of caution in .spite of its otherwise

intuitive appeal in the language of quarks.

As this is essentially an attempt to build a phenomenological

theory out of the experimental data on resonances, the structure automatically

shares the limitations and uncertainties of the latter. However, being

phenomenological in character, it has enough built-in flexibility to respond to

changes in the datafunlike a full-fledged theory which cannot stand ad hoc

-35-

modifications)* Its simple but comprehensive applioational base makes

it a convenient tool for the correlation of "resonance physios" with other

areas of particle physios* We hope that the quality of future data on

resonanoes will lend more meaning and purpose to such an approach*

ACKNOWLEDGMENTS

The author has benefitted from discussions with several colleagues

on this subject, especially Professor L, Van Hove,whose insistence on "betterfled to this work;

criteria for the construction of form factors^ and Drs* D, Plane andFerro-Luzzi who helped in ohecking several"guesses" on the form factors

in terms of their computer programme at CBRN, He is indebted to Drs.

F. Halzen and F, Buccella for helpful discussions on certain aspects of thia

paper in relation to their respective lines of approach and to (Miss)

R, Mehrotra and Dr, D* Choudhury for permission to quote from their results

prior to publication. This work was performed during the author's visit

to the International Centre for Theoretioal Physios, Trieste,in the summer

of 1970 (which was made possible by the Swedish International Development

Authority). He is grateful to Professors Abdus Salam and P. Budini as

well as the International Atomic Energy Agency and UNESCO for warm

hospitality at the Centre.

-36-

APPENDIX

We shall give here a short derivation of the dNN vertex in terms

of the model described in the text. For this purpose we also need the

relativistic dNN vertex which has been given by various authors 174, 76] .

Thus in the limit when the internal structure of the deuteron (four-momentum

d and polarization £J is neglected, this quantity is given by [74]

c5 ^ i [0+-h

C = -y^' CTgC = 7 5 (A. 2)

{ A ' 3 )

2

where a /m is the deuteron binding energy, (p , p ) are the four-

momenta of the two nucleons (mass m) and the on-shell value of 1c! (£: -a )

has been used. The quantity p is just the ratio of the asymptotic normal-

izations of the deuteron wave function [77] and

(A. 4)

A corresponding expression can be written down when the internal structure

of the deuteron is taken into account. Thus for the Yamaguchi [78] wave

function,which has recently received a certain degree of attention in the

literature1 on account of its interest in connection with the three-body

problem [79] , the corresponding form of the dNN vertex is:

m - iy- d r • 2— \

where C(k) and T(k) are the formal relativistic "notations" for the cor-

responding quantities C(p) and T(p) appearing in the Yamaguchi potential

- 3 7 -

g(p) [76,78]

g{p) = C(p) - -ft £ • 2) (£ * P) T(p) (A. 6)

T(p) =p" 2 T(p) (A. 7)

and the identification

p2 = - tt2 + | (m 2 -u ) , u = -"k2 (A. 8)

has been used. The new normalization constant N is given by

2*2 N"2 = f dq g2(q)(q2 + a2)'2 . (A. 9}

Now in any deuteron process (e. g., pd -»dp , pp -» * d) involving

nucldon exchange it is necessary to use a vertex of the form (A. 5) to re-

produce the essential features of a sharp decrease in the cross-section

away from the backward direction. However, for the evaluation of the

dNN* vertex, we shall use the simpler form (A. 1), in terms of the triangle

diagram of Fig. l(b). This is based on the assumption that while the u-

dependence of the N-exchange process comes mainly from the structure of

the deuteron wave function, the corresponding effect arising from N*~exchange

is. (hopefully) provided by the triangle diagram (Fig. l(b)) (since in any case

the latter is not expected to be the dominant effect).

For the evaluation of the dNN* vertex (whose kinematics i s shown

in Fig. I) the essential ingredients are the dNN vertex (A. 1), the NNir

vertex G ^iT_0 »5(Sees. II and III):

vertex G ^iT_0 *" and the N ""N vertex given by the general structureo

U(p2) %;" v *%*•. ^ { *>'« (A-10)

where f is a form factor of the type (3.4), a = 1 or iy_ , depending on

the quantum numbers of N , and

«„•*<"*.-V • (A11)

The effect of isospin is extremely simple; In terms of the triangle diagram,

the isospin indices lead to zero for dNA vertices and a factor of three for

- 3 8 -

,-fr. •

dNN vertices, so that only N-resonances need be considered. With this

understanding the isospin indices will be dropped. In the notation of

Yao 175] , the vertex structure corresponding to Fig. l(b) is given by

,

x fj °" q • • • q^ % . . . tf fp)

4For integration over d n. we closely follow the Yao approximation of "pole1

dominance in the two-nucleon propagators and transform to the variables2 2 2n , po , k and $ , where1 2

rest frame [73,75], so that:

2 2 2n , p , k and $ , where # is the azimuthal angle of k in the deuteron1 2 **~

2

= d(n^)

2' P 2 ** "]

u =

tn2

- P 2

!) d(k2)

2i K g^

= - ( d -

d0/8m[p

2-3m -u

P ) 2 .

(A. 13)

(A. 14)

(A. 15)

The form factor f does not have any singularities in the region2 2 2

(n m p_ » -m ) where the integral (A. 12) receives its main contribution.2Further, the k -integration is over a very restricted region of total range

2Ak « 4ia |p | . Collecting all these points we obtain finally:

where tf (p) = u(p)C , f is evaluated at the point (A. 14) and the quantityC JL

( ' " V " ) represents the azimuthal average of the indicated product in the

sense of integration described above. To evaluate the latter in a covariant

-39-

manner, the vector q can be resolved as

where q is essentially a two-vector determined by the conditions

leading to

sin20 q^ = [.(q. p) - (d- q)(d- p) d"2] p~2 p^ + [(q. d) - (d- p) (p. q) p ' 2 ] d"2

(A. 19)

sin2G «= 1 - (p- d)2 p " 2 d"2 . (A. 20)

The process of azimuthal averaging can now be carried out in a straight-

forward way in ierms of the tensor

Thus we have

"q2) V ' (A-22)

cyclic

(A. 23)

and so on, while the average of an odd number of q-factors vanishes. This

provides a convenient form for the application of the dNN* vertex.

-40-

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FIGURE CAPTIOUS

Fig* 1 (a) Backward 7T~p soattering diagram,

(la) Tlie dirar* vertex*

-44-

u

N(p)

N(P')

\

(a)

(b)

Fig. 1n inn

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