IC/70/27

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

LOW-ENERGY ELECTROPRODUCTION

AND EQUAL-TIME COMMUTATORS

G. FURLAN

N. PAVER

and

C. VERZEGNASSI

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1970 MIRAMARE-TRIESTE

IC/70/27

INTERNATIONAL ATOMIC ENERGY AGENCY

and

UNITED NATIONS EDUCATIONAL SCIENTIFIC AND CULTURAL ORGANIZATION

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

LOW-ENERGY ELECTROPRODUCTION

AND EQUAL-TIME COMMUTATORS**

G. FURLAN

Istituto di Fisica Teorica dellijniversita di Triesteand International Centre for Theoretical Physics, Trieste, Italy,

and

N. PAVER and C. VERZEGNASSI

Istituto di Fisica Teorica dellTJniversita di Trieste.

ABSTRACT

We estimate low-energy photoproduction and electroproduction

of pions by using equaMime commutators between current densities

and divergences and by including low-lying intermediate states. Pre-

dictions are presented for charged photoproduction s- and p-wave

multipoles, in agreement with experiments and dispersion theory, and

for transversal and longitudinal electroproduction cross-sections near

threshold. The dependence of the results on the axial vector form

factor GA(t) is discussed.

MTRAMARE - TRIESTE

April 1970

* To be submitted for publication

* This work has been partially supported by the Istituto Nazionale di Fisica Nucleare,Sottosezione di Trieste, Italy.

1. The deep interconnection between equal-time

commutators and the behaviour of pion amplitudes in

the low-energy region has represented a fruitful

advance toward a simple description of this kind of

phenomena.

The first suggestion came from Nambu's fundamental

papers and it found further and more refined

versions in the framework of current algebra and

Actually the standard method relates an axial

charge commutator to the value of a massless pion

amplitude at the zero energy point. A generalization

of this approach to physical pions has been recently

proposed, where the equal-time commutator plays essen-

tially the role of a subtraction, constant in a dis-

persion relation performed along a line of variable

mass and energy.

The heart of the matter is that for low energies

such subtraction constant usually represents the

dominant contribution, as a consequence of the small-

ness of the pion mass.

This formulation has a twofold aspect: on one

hand one can use current algebra to (partially) predict

pion amplitudes? on the other^ one can try to infer,

from measured pion processes, some information about

equal-time commutator matrix elements. Recent calcula-

tions of pion-baryon scattering lengths are a good

(4)illustration of this point.

We want to report in this paper some improved

results of the application of those ideas to low-

energy electroproduction (in particular photoproduc-

-2-

tion) . In order to* recall what the real ingredients

of the approach are, we sketch briefly the derivation

of the relevant formulae.

Let us start by looking at the production of

physical pious at rest. The quickest way to get the

result is to introduce an operator

Q* =

L is the axial vector charge) with the property

(2)

.t.c. L ^ - L ; y ^ JConsider then the e.t.c. L VJ(- L j »</_, J and saturate

it between, one-nucleon states. Selection of the one-

pion disconnected contribution gives

til

where •*•/*- i s the amplitude for electroproduction of—*

a physical pion at rest, C^ — O . There is still

some freedom to choose in what frame of reference

of the external nucleons the pion is at rest,; it is

particularly convenient to work in the riucleon Breit

frame p. - •" r^ — f> and the main reason is that

both channels are treated symmetrically,.so that for

small I r 1 higher waves are equally depressed by

angular momentum and parity conservation.

-3-

On looking at the remaining part of the sum

rule it is clear that the final result will be a pre-

diction for 1 u_ in terms of matrix elements of

j U)

of nucleon form factors and higher states matrix

elements. These turn out to be of order 'VY^ as

a consequence of the partial conservation of the axial

<*- 'ri 7t and can be considered as a correc-

tion to the soft pion limit,

The explicit selection of the (two independent)

invariant amplitudes, together with a complete dis-

cussion and tentative estimate of the OC^n.) part of

the relation has been given in Ref.(5)> and we will

return to this later. A further \J) C'YVljt) term is

represented by the unknown (we mean: beyond the frame-

(*)work of current algebra) commutator

This term plays for the electroproduction pro

cess a role analogous to the 0^-term [ Q Q_ J

for pion-nucleon scattering. Its presence is an

(*) On the other hand, as a consequence of the electromagnetic current conservation

-4-

unavoidable feature of working, in the current algebra

framework, with physical pions of non^vanishing mass,

(indeed for massless pions Q , ~ O and only current

algebra commutators appear).

2. The generalization of the above formalism to

get predictions concerning amplitudes for (slowly)

moving pions is straightforward. Roughly speaking,

one has to work with local densities rather than with

integrated charges. Thus the relevant operator is

now

(7)

and

Saturation of the commutator matrix element in the

nucleon Breit frame

(9)

leads, after selection of the pion and nucleon poles,

to a determination of the electroproduction amplitude

in terms of the equal-time commutators, of the nucleon

-5-

form factors (weak and electromagnetic) plus a sum

over intermediate states. We reproduce here the final

result (the complete derivation can be found in

Ref.(6) ): f

{<(*)

Further symbols have been introducedthe virtual pion source

^ < T - (• - -Oand the "transverse" axial current

+ — :

with the common property

and we have exploited the simple relation

(*) The apex on the e.t.c. indicates that the overallfactor (-2,7C) O C"P +% — F - 1 nas a:t-readybeen selected.

-6-

Eq.(lO) can be rewritten in the compact form

-n *one nucieon) + CO FLu,

where

(13)

C ^ l l A ^ dcO' (14)

loo

(15)

«-Z

The advantage of this representation is to exhibit the

meaningful analogy with a dispersion relation subtracted

at the point CO — Cf, = O . In general, the factors

CJJ , a have the role of emphasising the dominance, in

the low energy region, of the nucleon Born term and of

the current algebra "potential" term

LA O 6^ ),Vu. C~^) J > corresponding to the A^

and pion exchange in the t-channel. In particular,as

Co , Q —> 0 the soft pion theorems are reproduced.

This separation between potential and resonance-

like scattering corresponds to the complementary des-

cription that current algebra and isobaric model give

-7-

••*' M •*>

of a pion process; the first is successful near

threshold or for waves where resonances are not present,

while the |p -waves are dominated, at least in the

resonance region, by the 33 contribution as a conse-

quence of the small denominator.

There is a feature of the representation (13)

which must be stressed. As Eq.(14) explicitly shows,

the dispersive integral is evaluated at fixed Q

(and *|p ) which implies that both the energy co'

and the "mass" Q1 = co' - , are varying.

Indeed.let us specify the kinematics. In the

Breit frame

In terms of scalar variables

V = e^'-T = to' ET

C 1 8 >

i. e., in the ( V , Q' , U, ) space the integration

is performed along the surface (l8). The electro-

production amplitude is}of course,evaluated at the

physical point

-8-

* t 4-fc -

In particular, as Q —* 0 , we regain the evaluation,

obtained by using charge commutators, of the electro-

production amplitude at the Breit threshold. (This

configuration corresponds to a 180** C M . pion of energy

On the other hand,by choosing a suitable value

of Q , the exact CM. threshold

can be directly reached, and it is easy to check that

this occurs for

7?* ,-"&(«&-•*') j 3.A = 22ii2!L^l (20)

In this configuration it is clear that, by

working out the standard kinematics and isospin de-

composition, we shall end with a determination of ther~ (±,O) T (±,O)

S -wave multipoles t. + , I , + in terms of

nucleon form factors and higher contributions.

-9-

3. We come now to a more detailed discussion of

the threshold calculation. In a previous work

the Breit frame threshold < ^ - 0 ("£>=£ 0 ) has

been studied and simple expressions given of the

electroproduction cross-sections. Comparison with

effective threshold required an extrapolation of those

results andjinoreover jthe evaluation of the fj (/irti )

contributions, mainly of the .S -wave continuum, was

not always easy.

The aim of the present calculation, directly

performed at threshold^by using the algebra of densities,

is to check and (hopefully) improve our previous esti-

mates.

According to Eq.(lO), we will assume the C.A.

commutator

[A* \T""M

plus the field algebra commutator

^ 0 • (22)

If one assumes c- number Schwinger terms, these

(*)are irrelevant for the present problem.

Therefore the " equal time " part will

(*) Actually by strictly adopting the originalfield- algebra scheme,the mixed Schwinger terms arezero. Possible operator Schwinger terms enter intothe determination of the imaginary part, so that theywould be required only in the above threshold cal-culation.

-10-

introduce into the representation (10) the axial vector

form factors G A (h) and ID (h) - - %^\ GA(jfc) + t G£

As far as the nucleon term is concerned, its

contribution will contain the electromagnetic form

factors -p"7, (V.S) and the axial ones (j, fQN

Finally, in the sura over intermediate states we

will explicitly take into account the following terms

only:

a) The S-wave continuum, evaluated with a simple

scattering length approximation.

it

b) The N contribution treated in the zero width

approximation but including all the form factors

for the transitions N—> J[ N . These have been

taken from Ref.(9).

c) The P , CO exchange in the 1ft -channel (analogous

to vector dominance graphs). This requires the

knowledge of the < £ IK* |~t»y J^transition matrix element.

Beside higher resonances and /- graphs,

we are neglecting the r\^ contribution coming from

the matrix elements involving the operator Cf • Ji ,

whose size should be;however, A ^ '^''/>TI and there-

fore negligible since our considerations will be

always limited to small Q ^ '^^rc

There is a general feature of the whole approach

which is particularly unpleasant and which we already men-

tioned. Since we are working out a variable mass dis-

-11-

persion relation^all the vertices ^1*fX l^^ are,in

general,energy dependent] since no information is

(*)available on their off-shell variation we will

take for them the physical values at W 1 ~ "

Analogously the vertices ^"HVu. i'n/> w i l 1 b e evaluated

at It 2r TW^-ft +Acj,-A . This is not unreasonable

since the integrals (14) are dominated by the low-

energy region, in particular Ayo , which also exhib-

its a good convergence factor. On the other hand,

Ou, j for which the approximation is not so good,

is multiplied by q } and this is so small as to make

the errors and the whole contribution unimportant

(for o ^

We now list our results and shall comment

on them later.

Photoproduction

We have, at the physical threshold^

CM. , r "1 *-

(23)

All calculations have been performed by choosing a

dipole fit for the nucleon form factors

(24)

in the timelike region!

-12-

and we allowed the axial shape to vary by considering

the values il^ = 6, 7, 8 /Wl7tp . (This affects only

the isospin antisymmetric multipoles which depend on

)• Furthermore, for the divergence form factor

(t) + t Gp(i), and the induced pseudoscalarw e have assumed simple pion dominance forms (*)

Correspondingly one finds for the cross-sections

the results of Table I:

do

10 «

doR = ——

do +

C M .

Th.

C M .

Th.

6mir

14.16 jjb/s

7mIT

14.51 fib/s

1.34

0.7

8mjr

14.73 fjb/s

exp

15.6±0.5>jb/s

1.265+0.075

Not well

established

Table I

(*) We do not include in ^ptx/ t h e Aj.-Contribution,which should be reasonable in the limited range of -fcwe are considering.

-13-

Electroproduction

It is convenient to recall the form of the

electroproduction differential cross-section at thres-

hold:

X^ \X/ d-£

where t_ e , t_ g are the electron energies in the

laboratory system and £ is the "polarization"

defined as

A very recent experimental result, corresponding

to the following set of values:

. do)gxvea

Our prediction is, for the same kinematical configura

tion, displayed in Table II.

-14-

MA

T i»n ^ •

——p-

lim j ^ dofff )

6mn

-313.45 10

0.57

2cm

(GeV/c)

7mir

4 84 10"31 C m 2

(GeV/c)

0.41

~0.6 io"31

8mIT

7.85 10"31

0.23

2era /(GeV/c)

2cm

(GeV/c)

2

Table II

In Table II we have also quoted the evaluation of

the threshold ratio;

(28)

Finally, we reproduce in Tables III and IV the

S -wave multipoles and the L / 6" ratio at thres-

hold, with a complete display of the various contri-

butions.

We think, at this point, that some comments

are appropriate. A glance at Table III shows first

of all the varying reliability of our predictions.

For t- o+ , 1— o+ *^e c u r r e n^ algebra plus nucleon- o+ , 1— o+

part (which is the one surviving in the limit CO

^ —* O ) represents the dominant contribution and

-15-

the rescattering plus vector dominance graphs amount

to a (not negligible) "correction" part *V/ 10 -r 30%.

Furthermore,no cancellations occur among the different

terms, which makes the final result fairly safe.

On the contrary^for the (+) multipoles,current ,

algebra and nucleon term are of the same order of

magnitude as the other contributions and strong can-

cellations occur. Since the (+) multipoles enter into

the prediction of the "ft* processes, it must be concluded

that for the neutral case our results can be considered,

at best, as a rough indication. This is not gratifying

but it seems to correspond to a situation largely shared

with other determinations of the "IT ° amplitudes, like

dispersion theory. On the other hand the status

of experiments does not appear to be in a better shape.

A general remark concerns the dependence of the

results on the "input" parameters we used,such as the

various form factors and coupling constants. In the

present calculation both the fit for the electromagnetic

form factors and the higher contribution parameters

have been considered fixed while 1 1 was allowed to

vary. As Table III shows, this dependence on H ^

becomes more crucial with increasing tC

One can ask what happens if we allow a similar

variation of the parameters exploited to evaluate the

rescattering and vector dominance corrections. If we

consider charged pion processes, then only the depends

ence on the P -meson part turns out to be rather un-

stable. As mentioned before, for that contribution the

estimate of the ^.P I X I " T / 5 / amplitude is required.

This has been done by taking the N, N , Tt exchange

-16-

graphs in the various channels and it is clear that

the final value for lu, depends on the form and

value of the various couplings and coupling constants

'JfTLJC > ^fMN* * ^ N N (we evaluated with the

vector dominance argument).

Again, the general feature is that the dependence

on these P-meson parameters becomes more crucial

for larger K and one can empirically estimate that

the "theoretical" error is,unfortunatelyj rather large,

reaching in some cases the 10$ of the total amplitude.

An unkind reader could ask at this point why we

are bothering with the evaluation, of painful higher

contributions when for photoproduction much simpler

current algebra calculations, based on the crude soft

pion limit, give similar (at least not worse) results.

The answer is that, in our opinion, the approach pre-

sented here gives the possibility of obtaining a unified

and theoretically well founded description of photo—

and electroproduction at threshold (and beyond thres-

hold, see next section) in the range 0 ^ \ \^J-^'Yriic-

The dependence on the relevant parameters is analyti-

cally simple,which means that the estimate of iu,

can be easily improved by sharpening the input informa-

tion, for instance about SJ^ (t) , and vice versa.

It is useful to compare the present current-

algebra-like formulation with other theoretical cal-

culations of electroproduction. Clearly the dispersion

relations approach, based on the solution of multipole

integral equations, remains the most complete theory.

However, the practical determination of the multipoles

and their dependence on the input parameters is not

-17-

easy and we believe that, in the low-energy region, the

present approach can be considered complementary to

the dispersion relations scheme. It is interesting to

notice in this context that the dispersive calculation

and the present one exhibit a dependence on different

quantities: for instance,in the Born approximation of

dispersion relations the pion form factor ^K is

usually included^ which does not appear here, where^ on

the other hand, CT^ t*fc) and G p ft) must' be supplied.

Finally, the use of strict soft pion theorems

(namely current algebra + nucleon pole) can be a quick

and successful solution for particularly simple confi-

gurations, like photoproduction, but their validity

remains to be ascertained and we defer a more detailed

comparison to the final remarks.

To conclude this section,let us look at the

comparison of the theory with available experimental

data. As far as photoproduction is concerned the

agreement is not terribly impressive. On the other

hand, according to the above discussion, a different

estimate of the vector dominance graphs can easily in-

crease the prediction for <%& (K*) . However, this

would lead to a simultaneous and strong increase in

(#) One can try anyway to exploit our determination ofthe electroproduction amplitude to learn somethingabout Tk (•«•*) . By evaluating ^)T^one finds z £

T \ ~ A + - ^ -—

where ^OTIJE is °ff shell on a pion line, at the point'* = 'Yn£ + Tl\% - k*".' I f we take p7c/c at *ne physical

value, the vector dominance prediction is, of course,reproduced.

-18-

the *• /fiT ' ratio for which experimental indi-

cations seem to require a value XS • 1 . Therefore we

have chosen a sort of "theoretical best fit" for the

evaluation of the P -contribution, where, without

drastically spoiling the photoproduction predictions,

the '•/(T ratio is maintained ^ 1 (see Table IV).' T

Then the agreement with the Amaldi experiment

for 7C is satisfactory and,furthermore, as we shall

see in the next section, a reasonable prediction of

the p - wave multipoles is allowed.

4. The general formula (lO) allows the determination

of the complete electroproduction amplitude also for

moving pions (i.e. beyond threshold) and we present in

this section some results for the range of values

Our considerations will be limited to the s-wave

and p - wave multipoles; the smallness of I Q |

(and I ""') will enable us to resort in many cases to

power expansions where only the first term in 1^ ' [

will be retained. For instance,in treating the p - wave

multipoles we shall find it convenient to assume the

validity of the following parametrization;

(*) The numerical evaluation given in Ref.(5), whereonly magnetic coupling has been assumed for the N N Pvertex, is somewhat larger than the one we find in thepresent paper, where the complete N N P vertex hasbeen employed,

(**) Remember that I ^ I indicates the pion three-momentumin the nucleon Breit frame rather than in the C M . system.

-19-

< 2 9 )

In this way our task is reduced to the evaluation

of the limits on the r.h.s. of (29)• This has been done

by inserting the same contributions and using the same

approximations as in the threshold case. In particular,

the value ' ' " ~f~ 'W-ir has been used.

Let us proceed now to a more detailed discussion

of the results.

Photoproduction

The C.M.S. photoproduction differential cross-

section can be parametrized, in the l,ow energy and for

not too small C.M.S. angles, retaining s - and p -waves

only. The resulting expression is

Jill d©" \CM- M<- r

\x/ J (30)

with

q (31)

x =

-20-

Although in principle all the required quantities

X , ^ , t can be evaluated from Eq.(7O), the explicit

calculation becomes rather complicated as far as y> is

concerned and we have preferred to limit ourselves to

the simple configuration *9 = 90°.

The numerical values of £-o+ , £ ,+ - M ,+ f

2>M-(+ + H J- a r e shown in Table V varying the laborat-

ory energy of the incoming photon, and the agreement

with the experimental data is fair, in the "TC case,

(no data exist for 71 *" ) up to C.^ = 260 MeV.

6

We have also compared our predictions (limited

to the (-) (0) multipoles) with those of dispersion

theory taken from Donnachie and again we believe the

correspondence of the results to be rather encouraging.(Table VI) .Electroproduction

If we retain 5 - and |?-wave multipoles only,

the differential cross-section for the electroproduction

process can be parametrized as follows:

(32)

-21-

where fL^ =• [Sf 4>) is the final M. - N C.M.S.

solid angle. The coefficients A, B , C have been

already given in Eq.(3l). Those remaining are

defined as:

T> = !_*• +• < * E = 2u_L o + /

H - L0+C^H-5a) + £o + (32)

I = UL (_4 it) 4- XV

In the explicit computation of U_ we are

faced by just the same difficulties as in the case

of X • Therefore we shall limit our predictions to

the quantities A , D , G and H , which can be easily

obtained .from Table VII at various I H , lC values,

using the same parametrization used in Eq.(29). Thus,

the purely transverse and longitudinal terms and the

transverse - longitudinal interference proportional to

Cos <4> can be determined at © = 90° • As far as

the transverse - longitudinal interference term propor-

tional to Cos 3, <& is concerned, it can be computed

in a quite general kinematical configuration.

-22-

5. We conclude our work by comparing the approach

and results of this paper with a recent current algebra(12)

discussion of electroproduction by Nambu and Yoshimura.

These authors have obtained an impressively simple fit

for the axial vector form factor G^(-t) , by matching

experimental data for electroproduction near threshold

(till l< -\- - 50 om^ ) with the original Nambu - Shrauner

formula. The result is

namely a dipole form with an axial meson mass around

1.34 BeV -*-' 9-8 Vnj , to be compared with our determina

tion M 4 <^>? <VY\K . (Notice that the higher M/^ is,

the most important Q? *{£•) results).

In the theory by Nambu and Shrauner, the thres-

hold multipoles fc. o+ , L. o+ are expressed in terms

of the axial vector vertex and of the nucleon pole

(evaluated in the Breit frame). Corrections for the

finite pion mass and gauge invariance require some

additional contact terms where,however,only nucleon

parameters appear. Anyway>in the estimate of Ref.(l2)

the finite pion mass effects turn out to be unimportant

(always smaller than

On the other hand,that calculation does not

include what we consider the main sources of correc-

tions, namely the s-wave rescattering and the

vector meson dominance graph. A glance at Table III

shows that these corrections (for charged TC's) are

always of the same sign as the fundamental C.A. + nuc-

leon term; it is therefore clear that if one keeps as

-23-

variable G A ( ^ ) and includes corrections, the fit to

experimental data can be achieved with a G> ("t-) smaller

than would come out from a calculation without higher

terms. This explains why in the present paper an al-

together satisfactory agreement has been achieved with

ri .^Tofli , to be compared with the higher value by

Nambu and Yoshimura.

As a check we have repeated some calculations

by using the fit (33) for G. C"t) and by taking only

equal-time commutators and nucleon into account. This

is shown in Table VIII j as expected, a good agreement

is obtained for photoproduction, while for electropro-

duction slightly higher values than ours are obtained.

It is interesting to remark the rather small value of

the i- f gr ratio and of course a clear-cut experimental

information on this quantity would be highly desirable.

Apart from the previous numerology, we think

that the following lesson can be learned from the above

comparison and considerations. Whilst the simple current

algebra formulation is useful for a first, reasonable

description of the process, the simultaneous use of

the experimental information to infer properties of

the theoretical input is not allowed , in general.

Even if we do not claim that our estimate of the cor-

rections is the best one can buy, we believe that the

present paper should represent a rather good illustra-

tion of the above statement.

ACKNOWLEDGMENTS

We thank B. Borgia, G. Stoppini, N. Dombey and M. von Gehlen for useful discussions.

One of us (G.F.) thanks Professor Abdus Salam, the International Atomic Energy Agency

and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

-24 -

REFERENCES

1.) Y.Nambu and D.Lurie: Phys.Rev. 12 5, 1429 (1962);

Y.Nambu and E.Shrauner: Phys.Rev. 128, 862 (1962).

2.) See S.L.Adler and R.Dashen: Current Algebras, W.A.

Benjamin (New York, 1968);

B.Renner: Current Algebras and their Applications,

Pergamon Press (1968).

3.) S.Fubini and G. Furl an: Ann. Phys. (NY) 48, 322

(1968).

4.) F. von Hippel and J.K.Kim: Phys.Rev.Letters 20,

1303 (1968) and Phys.Rev. to be published.

5.) G.Furlan, N.Paver and C.Verzegnassi: Nuovo Cimento,

62A, 519 (1969)-

6.) V. de Alfaro, S.Fubini, G.Furlan and C.Rossetti:

Nuovo Cimento 62A, 497 (1969).

See also G.Furlan: Lectures at the 1969 Schladming

Winter School.

7.) N.Paver and C.Verzegnassi: Nota interna N. AE 69/5

(1970) INFN- Sottosezione di Trieste. This paper

contains all the computational details and an

enlarged discussion of the various contributions.

-25-

8.) T.D.Lee, B.Zuraino and S.Weinberg: Phys.Rev.Letters

lji, 1029 (1967).

9.) A.J.Dufner and Y.S.Tsai: Phys.Rev. r68_, 1801 (1968)

10.) E.Amaldi et al.: Nuovo Cimento 6f5A, 377 (1970).

11.) F.A.Berends, A.Donnachie and D.L.Weaver: Nuclear

Physics B£ , 1 (1967) •

See also G. von Gehlen and M.G.Schmidt: Paper

presented at 4th International Symposium on Elec-

tron and Photon Interactions, Liverpool, September

1969 .

12.) M.S.Bhatia and P.Narayanaswamy: Phys.Rev. 172,

1742 (1968). Other calculations are quoted in

that paper.

P. De Baenst: An Improvement on the Kroll-Ruderman

Theorem using P.C.A.C. and Crossing Symmetry, pre-

print (1970).

Y.Nambu and M.Yoshimura: Phys.Rev.Letters 2^, 2 5

(1970).

-26-

0

- 3

- 6

- 9

-12

k*-

0

- 3

- 6

- 9

^ 1 2

\S

0

- 3

- 6

- 9

- 1 2

0,971

0,794

0,659

0,565

0,482

-0,075

-0,064

-0,056

-0,048

-0,042

r- c+;

-0,224

-0,218

-0,214

-0,199

-0,190

0,982

0,833

0,716

0,635

O,56l

CA.+N

-0,062

-0,053

-0,046

-0,040

-0,035

CA>N

+0,020

+0,069

+0,103

+0,128

+0,142

0,989

0,939

0,879

0,815

0,765

-0,010

-0,009

-0,008

-0,007

-0,006

+0,359

+0,310

+0,270

+0,242

+0,217

CA-+ N

0,821

0,665

0,544

0,466

0,391

GO

-0,003

-0,002

-0,002

-0,001

-0,001

?-0,597

-0,571

-0,545

-0,516

-0,488

0,831

0,701

0,597

0,531

0,465

0,634

0,285

0,199

0,162

0,142

-0,006

-0,026

-0,042

-0,053

-0,061

0,838

0,800

0,749

0,699

0,655

CA.+ tf

0,535

0,223

0,141

0,107

0,088

/ko

-0,004

-0,039

-0,043

-0,044

-0,038

I V G

5-wave

0,066

0,054

0,045

0,038

0,033

5 - \JCave.

0,043

0,020

0,014

0,011

0,010

CA.+ N

-0,069

-0,057

-0,048

-0,042

-0,034

0,067

0,057

0,049

0,043

0,038

s0,053

0,028

0,022

0,016

0,012

+0,086

+0,040

+0,028

+0,022

+0,020

0,067

0,064

0,060

0,055

0,052

} \ *

0,003

0,014

0,022

0,028

0,032

-0,020

-0,018

-0,017

-0,016

-0,014

f0,082

0,069

0,060

0,049

0,044

/ k o

-0,069

-0,057

-0,048

-0,042

-0,034

^NT

-0,001

-0,004

-0,006

-0,008

-0,010

[NT

0,002

0,006

0,010

^ 0,012

0,014

CA.+ rt

-0,061

-0,051

-0,043

-0,037

-0,031

-0,005

-0,004

-0,003

-0,003

-0,002

-

-0,003

-0,002

-0,002

-0,002

-0,001

Table III : S - wave multipoles at threshold ~ 1 ) •

tooo

-12TH*

0,29

0,37

0,49

0,72

0,26

0,31

0,38

0,52

0,19

0,20

0,22

0,27

0,48

0,72

0,99

1,35,

0,44

0,6l

0,80

1,02

0,35

0,42

0,50

0,57

Table IV : The ratio©V

at threshold.

• *

ICOCDt

160

180

2 00

2 2 0

2 4 0

2 6 0

160

180

2 0 0

2 2 0

2 40

2 6 0

L

rMeV

MeV

MeV

MeV

MeV

MeV

MeV

MeV

MeV

MeV

MeV

MeV

E£0,969

0,953

0,921

0,902

0,881

0,863c|s-(rc+K 3QO-N

4,78

7,75

9,62

11,50

13,69

16,37

0,115

0,218

0,319

0,416

0,512

0,612

e x p

e x p

e x p

exp ]

exp ]

exp 1

E

5

7

8

LI

L6

; : - <

-0,007

-0,013

-0,019

-0,02 5

-0,031

-0,037

t .5

± 1

-t 10

*-* 14

5-18

H18

-0,148

-0,280

-0,410

-0,534

-0,657

-0,786

-0 ,18.

-0,34-

-0,49.

-0,63.

-0,79.

-0,94-

10~3

10" 3

1O~3

1O~3

lO" 3

10"3

Table V : Photoproduction multipoles (unity l/y>| ) and differential cross-sections (,ub /ster) at & = 90° as a function of the incomingphoton laboratory energy. The experimental values for V JJl-are taken from Ref.(ll). £ o + has been taken practicallyconstant and equal to the threshold value C? -O.O75 .

CO

oI

L

<T

160 MeV

180 MeV

200 MeV

220 MeV

240 MeV

260 MeV

to+

1,008

0,930

0,868

0,817

0,775

0,737

jr C o )

t 0+

-0,060

-0,061

-0,062

-0,064

-0,065

-0,067

0,165

0,299

0,405

0,496

0,576

0,646

-0,007

-0,013

-0,017

-0,019

-0,022

-0,025

c-l a

-0,150

-0,306

-0,449

'-0,590

-0,730

-0,859

CO Co)

= 0

= 0

= 0

-0,004

-0,005

-0,007

Table VI : Photoproduction multipoles (unity l/iT)K) as afunction of the incoming photon, laboratoryenergy, as computed in Ref.(ll) .

It*

0

- 3- 6

— 9

-12

0

- 3

- 6

- 9

- 1 2

k*

0

- 3

- 6

- 9

- 1 2

E ^ - M,*

0,3450,211

0,168

0,140

0, 120

^ M ^ + M ; ; 1 ;

-0,443

-0,385-0,338

-0,294

-0,260

^,567

0,233

0,146

0,105

0,082

C.A +N

0,290

0,166

0,127

0,105

0,090

CA +M

-0,373

-0,319

-0,276

-0,241

-0,213

C.A +N

0,5330,221

0,139

0,101

0,079

_.

-

-

-

-

-0,001

-0,001

-0,001

-

-

s-ut>aVe

-

-

-

-

f

0,014

0,005

0,003

0,002

0,001

?-0,008

-0,006

-0,005

-0,004

-0,004

?0,034

0,012

0,007

0,004

0,003

0,041

0,040

0,038

0,033

0,029

-0,061

-0,052

-0,056

-0,049

-0,043

-

-

-

- .

-

-0,021

-0,018

-0,015

-0,013

-0,011

2.MS? + M;.O)

-0,001

-0,001

-0,002

-0,002

-0,003

+0,003

+0,002

+0,002

+0,001

+0,001

ft.

C A ¥fJ

-0,022

-0,019

-0,016

-0,014

-0,012

C.A. + H

-0,001

-0,001,

-0,002

-0,002

-0,003

C.A+U

+0,005

+0,004

+0,004

+0,003

+0,003

—

-

-

-

-

S-ufAve

-

-

-

-

-

-

-

-

-

00

+0,001

+0,001

+0,001

+0,001

+0,001

CO

-

-

-

-

-

CO

-0,002

-0,002

-0,002

-0,002

-0,002

Table VII : P - wave multipoles at threshold (»?„.= l), for different values of

I

CO

Photoproduction

1 M cio"(Jr+)

mi d ^c.

•to

Elect rop roducti on

In the kinematical

0

— o ory\ rc

- 9 ^

- 1 2 'VYt^,

1 ,

o,

o,

o,

o,

c

:'c«026

919

831

755

692

A. 5 *31L.

configuration

. M ,

t

0 E

-o,-o,- 0 ,

-o,

076

066

059

052

0 4 7

of Ref

o,

o,

0 ,

0 ,

0 ,

d*odec

. ( 1 0 ) :

W

533

2 2 1

139

1 0 1

079

1-)

J

\

6V

L o i c-vg- 0

- 0

- 0

- 0

- 0

,070

,061

,053

,Q47

,041

-

**- ..Table VIII : Results obtained from a pure

current algebra calculationwith the fit (33)•