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TRANSCRIPT
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)•