Department of Natural
Philosophy

University of Glasgow

Glasgow, Scotland

and

Department of Physics

Virginia Polytechnic
Institute and State University

Blacksburg, Virginia, USA

February 1974

Figures for the above paper

Over two decades ago Fermi and coworkers[1]
at the University of Chicago began a revolution in sub-nuclear physics by
measuring some differential cross sections for pion-nucleon (pN)
scattering and analyzing them in terms of partial waves. An unexplained strong
energy dependence was exhibited and the analysis showed that it was due to the
existence of a resonance at ~200 MeV pion laboratory kinetic energy (~1235 MeV total center-of-mass
energy) in the isospin I=3/2, total angular momentum J=3/2, and positive parity
[parity = -(-1)^{L} for pN, where L is the orbital
angular momentum, because the p has negative intrinsic
parity relative to N] partial wave. Since J=L±1/2 for pN,
we see that the positive parity quality of the partial wave is equivalent to
L=l, or a P-wave pN interaction, so we
designate the resonating partial wave as a P_{33} wave according to the
usual L_{2I,2J} symbol.

Although the resonance behavior of the P_{33}
wave is quite clear, even for crude data, the other partial waves were not so
easily determined. Various ambiguities were discovered[2],
and interest waned in pN partial-wave
analysis after it was shown that one of these ambiguities was due to lack of
data for the polarization of the recoil nucleon.

The 200 MeV P_{33} resonance is an extremely
sharp feature of the pN total cross section. Being
the easiest measurements to make, total pN cross sections appeared at
higher energies very soon after higher energy accelerators came on line, and
other less prominent but quite clear "resonance" bumps occurred at
~600 MeV and ~900 MeV pion laboratory kinetic energy (~1510 MeV and ~1685 MeV
total c.m. energy, respectively). Some crude analyses and evidence from pion
photo-production experiments indicated[3]
that the second (~600 MeV) bump was probably a D_{13} resonance, but
could not rule out a P_{13} resonance, and that the third (~900 MeV)
bump was probably an F_{l5} resonance.
Detailed analysis was impossible because of lack of any polarization
data and precise differential cross sections. The need for the data was
obvious, and many experimental groups were busily trying to obtain it.

This is the point at which the two authors of this
article came in. So, in the next two sections of this article we shall tell our
separate stories. Finally, at the end we shall tell how the history of the pN
analyses developed after the crucial stage described in our personal
narrations.

William M. (Bill) Layson and I were fellow teaching
assistants in the Junior Atomic Physics Laboratory at Massachusetts Institute
of Technology. I was impressed by his and another graduate student's comments
about thesis research work they were pursuing with Professor Bernard T. Feld.
Early in graduate school I thought I wanted to do research in quantum field
theory, but after about two years I came to the conclusion that I would be more
productive and happier with my feet on the ground rather than in the clouds.
Thus I gravitated toward particle physics phenomenology, Prof. Feld's
specialty. My first impressions of Prof. Feld as student advisor were greatly
enhanced when arrangements were made for Bill Layson to accompany Prof. Feld to
CERN in 1960-61 in order to finish his Ph.D. thesis work. When Prof. Feld
returned in the fall of 1961 I quickly approached him about being my advisor.

After tossing a few possible research problems
around, Prof. Feld and I soon decided to continue the work of Bill Layson[4].
Bill had obtained a set of p^{-}-p partial-wave
amplitudes by assuming the existence of the D_{13} and F_{15}
resonances at 600 and 900 MeV laboratory pion kinetic energy, respectively.
Prof. Feld felt that, with new data rapidly becoming available, we might be
able to separate out the isospin 1/2 and 3/2 partial waves by using both p^{-}-p
and p^{+}-p data. So I, with great enthusiasm, set out
on the arduous task of collecting all pion-nucleon scattering data. The
work was slowed somewhat by the necessity to study hard for my second, and
final, try at the MIT Physics Ph.D. General Examination. With that hurdle out
of the way by April of 1962, the future looked rosy when summer arrived. The
data available and promised for the near future looked abundant; all would be
well if I could just figure out how to manage the tremendous computing task of
fitting the data. And I had a "lucrative" job lined up for the
summer, to support my family of four, working with Dr. Michael J. Moravcsik at the
Lawrence Radiation Laboratory at Livermore (now called the Lawrence Livermore Laboratory).

Most of the summer with Dr. Moravcsik was spent
trying to determine the K-meson parity and yielded no results. But I gained
valuable computer experience and had many opportunities to discuss the
Livermore proton-proton scattering analysis and my proposed pion-nucleon
analysis with Dr. Moravcsik, Dr. H. Pierre Noyes,

Prof. Feld agreed to the condition set by Dr.
Moravcsik. So Robert Wright began developing the computer code and I began
getting the data into useable form and developing the equations and techniques
to be used in the analysis. Robert and I wrote each other approximately once
each week. I would send him new data, corrections to old data, and my ideas on
how we should proceed from that point in time; he would send me his latest
technique for doing a particular thing on the computer along with several
questions about the next step. At the same time, I was writing to Dr. Moravcsik
in Pakistan about twice a week, sending him my latest equations and thoughts
about the analysis. And he would answer every letter with his usual helpful
comments. And once or sometimes twice a week I would meet with Prof. Feld and
fill him in on my progress and ask him questions. Now, more than ten years
later, I realize much more than I did then how fortunate I was to have the
constant advice of two of the world's best particle physics phenomenologists.

Our approach was to put Breit-Wigner
resonances in certain partial waves and to parametrize the background for these
partial waves and all other partial waves by smooth functions of the energy.
(We used a power series in momentum for this parametrization.) We would then
determine which partial waves were resonating by trying resonances in various
partial waves. Of course, there was no question but that the P_{33}
state was the appropriate resonance state for the 200 MeV bump in the p^{±}-p total cross
sections. But there was some uncertainty about whether the D_{13}, P_{13}
or both states were resonating around 600 MeV. I decided to try all three
possibilities. (Our analysis showed that P_{13} did not resonate at 600
MeV.) All of the non-resonant states were started at the then known values of
the scattering lengths. Our goal was to analyze all four resonance regions: 0-350
MeV, 350-700 MeV, 700-1100 MeV and 1100-1500 MeV. There was
no problem in fitting the data in the first resonance region. But a foretaste
of the troubles ahead was provided by our first attempts to fit the data in the
second resonance region. We could not get good fits when we used all of the data.
Noticing that the total cross sections were badly fitted, we then did a fit to
the total cross sections alone. Then that solution was used as input for
fitting all of the data, and a good fit was obtained.

The
analysis took a huge amount of human and Livermore computer (IBM 7094) time.
MIT sponsored a two-week trip to California for me in March, 1963 to try
to speed things up. But, alas, the calculations still were not satisfactory by
graduation time. Fortunately, Livermore had already awarded me a postdoctoral
appointment in Moravcsik's particle physics group, so

Prof. Feld was anxiously awaiting these results at
the Siena Conference[5]
in Italy, so I fired off a long telegram giving the partial-waves'
behaviors, including strong evidence for a P_{ll} resonance near 600
MeV. (I later had a difficult time justifying that extravagant expenditure of
Livermore funds. This is probably a good place to admit that I regularly made
great demands for computer time at Livermore, with little recognition of the
fact that the taxpayers installed that gigantic computer complex there in order
to build bombs, not discover
resonances.) Later, in July, Livermore sent me back to MIT to successfully
defend my Ph.D. thesis.

The data were much better fitted by assuming a D_{13}
resonance at 600 MeV rather than a P_{13} or both D_{13} and P_{13}.
Unexpectedly, however, the P_{1l} state exhibited resonance-like
behavior near 600 MeV even though no resonance parametrization was used for it.
I tried many times to freeze the P_{11} state into some non-resonant
behavior while varying all other partial waves. But every time, upon release,
the P_{11} would change to look like a resonance. And a slightly better
fit would be achieved by assuming a resonance form for the P_{11}
state. Layson's work had indicated the possibility of a P_{11}
resonance at ~900 MeV and I had hoped to look for it when I got that high in
energy; but it was very surprising to observe its resonance behavior near 600
MeV because no one had ever hinted at it before and the P_{11}
scattering length is rather large and negative. I spent a much time trying to
eliminate the P_{11} resonance.

In December, 1963, several of we Livermore postdocs
went to the APS meeting at Cal Tech. Just before leaving, I received a
preliminary version of the long-awaited charge-exchange
differential cross section data measured in the second resonance region by Burton
Moyer's Laboratory at Berkeley. I hurriedly calculated my analysis' predictions
at their energies and plotted them on the data graphs while on the plane to
Pasadena. Almost without exception every data point included the computed curve
within its error bar! I was elated! I think that that was the precise time when
I __knew__ that my analysis was correct. After Dr. Moyer's talk about their
data at the APS meeting I introduced myself to him and showed him my curves
versus their data. I am not sure that he believed me. I later gave several
talks about my work at the Berkeley laboratory and spent many hours with the
Moyer's group experimentalists discussing my results and future pion-nucleon
experiments.

In January, 1964 I gave a post-deadline paper at the
APS Annual Meeting in New York City. After the talk, Frank Lin, a graduate
student working with Prof. Hull at Yale told me that they were also doing a
pion-nucleon scattering analysis[6].

In early 1964 a paper by Bareyre, et al.[7]
appeared which showed that the low-energy asymmetry of the 600 MeV bump
in the p^{-}-p total cross
sections is either due to a P_{ll} or an S_{ll} resonance. Mike
Moravcsik strongly urged that I quickly get a letter into Physical Review
Letters about the P_{ll} resonance.
He insisted that only my name should appear on this first paper[8]
announcing the P_{1l }resonance. He wrote a letter to Prof. Feld
expressing this opinion and Prof. Feld immediately sent back his agreement. The
later complete paper[9]
on the 0-700 MeV analysis contained the names of Robert Wright, Prof.
Feld and me. I have always felt that Mike Moravcsik's name should have been on
it also, but Mike did not think so. Wright and I later published a more
detailed analysis of the 0-350 MeV data[10]

During the various talks that I gave about the
analysis, snickers were usually rampant when I stated that we were fitting 1200
data with 100 variable parameters. The first unthinking comment was usually
"You can fit anything with 100 parameters." However, quite justified
questions were usually raised about the uniqueness of the fit. The constraints
of available computer time made extensive tests for uniqueness impossible, but
we were able to satisfy ourselves that the gross features of our partial-wave
amplitudes were unique. Later analysis by others confirmed this.

Our analysis would not have been possible without
the availability of the sophisticated nonlinear least-squares-fitting
program of Richard Arndt. Richard spent several years developing this program
at Livermore. I later in 1967 had the good fortune to join the faculty at
Virginia Polytechnic Institute (now Virginia Polytechnic Institute and State
University) along with Richard, who had earned a Ph.D. from Berkeley in 1965.

My attempts to extend the analysis above 700 MeV
were largely fruitless. In retrospect it appears that the trouble lay in not
having a S_{ll} resonance[11]
near 600 MeV and not including enough resonances[12]
near 900 MeV. If I had restudied Bill Layson's results more carefully at that
point I might have had better luck, because many of the resonances now known to
exist were strongly hinted at in his work.

Shortly after publishing our work I was delighted to
receive preprints from Bransden, Moorhouse, and O'Donnell at Rutherford
Laboratory in England which confirmed the basic features of our work. Also, the
Lin and Hull work confirmed our results.

The discovery of the P_{ll} resonance was
the spark that set off the baryon resonance explosion of the late 1960s. In
Tables 1 and Figure 1 are listed the pion-nucleon resonances of the
Dalitz-Horgan quark model fit to the known pion-nucleon resonance
masses. Several new resonances were predicted. In particular, notice that there
are nine resonances (six I=1/2 resonances and three I=3/2 resonances) near 900
MeV (~1700 MeV total c.m. energy). It remains to be seen whether scattering data
between 700 and 1100 MeV can uniquely determine this many resonances. There are
probably several more resonances belonging to higher SU6 supermultiplets besides those given in Tables 1 and
Figure 2 for the 1900 MeV total c.m. energy region.

Table l.
Pion-nucleon states (total c.m. energies) according to Dalitz and
Horgan for the lowest quark model SU6 supermultiplets.

__I=1/2 (N ^{*})__

__L (J) ^{P}__

3 (7/2)^{+} 1998

3 (5/2)^{+} 1707 1863 1962

2 (5/2)^{-} 1692

2 (3/2)^{-} 1535 1702

1 (3/2)^{+} 1704 1846 1891 1942

1 (1/2)^{+} 938 1443 1759 1937

0 (1/2)^{-} 1541 1708

__I=3/2
(____D)__

__L (J) ^{P}__

3 (7/2)^{+} 1927

3 (5/2)^{+} 1910 1927

2 (5/2)^{-} (l900)

2 (3/2)^{-} 1650

1 (3/2)^{+} 1233 1705 1910 1927

1 (1/2)^{+} 1879 1927

0 (1/2)^{-} 1650 (1900)

__Figure 1__

Pion-nucleon states
according to Dalitz and Horgan for the lowest quark model SU6 supermultiplets.

In 1963 it was already known that, in addition to
the P_{33} resonance of mass 1230 MeV (from here on all masses will be
given in terms of mc^{2}
in MeV) well established in the 1950's by Chicago pion-nucleon scattering
experiments and partial-wave analysis, there also existed resonances of
masses about 1520 MeV (most likely a D_{13} state),about 1690 MeV (an F_{15}
__or__ D_{15} state),and about 1920 MeV (states unknown then); the
evidence was from pion-nucleon scattering, both in total cross sections
and gross features of angular distributions, and from pion-photoproduction
experiments (photon + nucleon -> pion + nucleon). At that time, 1963, this
author was a staff member of the Rutherford Laboratory in England whose proton
accelerator was due to operate in 1964, one of the first planned experiments
being pion-nucleon scattering in the mass range 1500-1750 MeV.

There were many corridor discussions with the
experimenters, particularly Paul Murphy, in which the then recent and
forthcoming increase in world pionnucleon scattering data became evident.
Also, at a Rutherford Laboratory colloquium Claude Lovelace presented some
interesting theoretical work by himself and

Paul Auvil on information to be gained from those points where the pion-nucleon
scattering differential cross-section was very small. A stronger
theoretical strand was the contemporary interest in Regge-pole theory, and I
became interested in formulating the pion-nucleon scattering amplitude as
a superposition of 'Regge poles' in the direct channel (symbolically: pion +
nucleon -> 'Regge poles' -> pion + nucleon) since this would give an
ansatz on parametrization of these amplitudes as a continuous function of both
energy and scattering angle. Brian Bransden and I discussed analyzing the pion-nucleon
scattering data using such a parametrization which would give information on
the Regge poles, some of which could incorporate the above 'known' resonances.
Bransden, probably rather wisely, was not so enthusiastic, considering that
such an ansatz involved too many unproved assumptions. In July of 1963, at the
Scottish Universities Summer School, or perhaps earlier, I returned to Bransden
with a proposal that we formulate the scattering amplitude as a sum of partial
waves parametrizing the imaginary part of each wave as a continuous function of
energy and angle, the real part being expressed in terms of the imaginary part
through a partial-wave dispersion relation, thus incorporating some theoretical
constraints on the scattering amplitude. Almost immediately thereafter we wrote
the central part of a computer program to analyze pion-nucleon scattering
using this ansatz; we chose to use partial-wave dispersion relations in
the inverse, T^{-1}_{2I,2J}, of the partial-wave amplitudes, T_{2I,2J}, as suggested
by Bransden.

The time until the end of 1963 was occupied with
writing the complete program and with my trying to implement it on the
Rutherford Laboratory computer with help from the newly recruited third
collaborator Pat O'Donnell (who had just moved to Durham University from Glasgow
along with Bransden), but the effective computer job turn-around time for
us low-priority users was about three days and the debugging progress was
abysmally slow; great sufferers also were certain experimental physicists.
Fortunately, Bill Walkinshaw (a division Head at Rutherford) accepted a very
generous offer by the newly founded Deutsches Rechenzentrum at Darmstadt,
Germany to let those physicists compute at a nominal price on their new IBM
7090, for which there was little immediate German usage, and Walkinshaw let us
in on the tail of the experimental physicists. Pat O'Donnell and I arrived in
Darmstadt on Fassnacht in February of 1964 and immediately obtained about six
debugging runs per day and extensive long computer runs at night. He and I spent
the next few months commuting to Darmstadt for periods of two to three weeks,
sometimes alternately and sometimes together.

The first thing we did, as a test of our method, was
to analyze positive pion-proton (p^{+}p) elastic scattering from
threshold (1080 MeV) to about 1300 MeV center-of-mass energy; this
included the already well-known P_{33} resonance of 1230 MeV
mass. The positive pion proton system has the simplifying property of being in
a pure isospin state, I=3/2; also in this energy range, where the production of
a second pion (an inelastic effect) is kinematically possible, pion production
just happens not to occur, giving a further simplification by effectively
eliminating the need for inelastic parameters.

In these circumstances the choice of parametrization
as a function of energy for the various partial waves was rather easy, but it
was most heartening to find that our method and program achieved immediately a
creditable fit (c^{2} minimum[13])
to the data with values of the mass and width of the P_{33} resonance
within the range found by previous workers.

Our c^{2} was a complicated function
of the parameters, consequently no mathematical theory was available to
simplify the problem and, also, we had a large number of parameters, making
impossible a simple grid search by a computer. (If there are 30 parameters and
one evaluates c^{2} for 10 values of each
parameter with 1 second computer time for each evaluation then the time taken
would be 10^{30} seconds.) So the computer program which evaluates c^{2}
has to be linked, within the computer, to a computer program which makes a
rather sophisticated search in the parameter space to find minima of c^{2}.

In fact we were faced with an unusually large number
of parameters, about twenty to begin with, in our first simple minimization of c^{2}
for the P_{33} resonance region described above, and up to seventy
finally. It is surprising we were optimistic enough to commence a search for
minima of a complicated function in a fifty-dimensional, or more,
parameter space but we were helped as will be related, by the special (perhaps
against reasonable expectation) features of the pion-nucleon system.
Also, on a technical 1evel, we found a minimization program newly written by M.
J. D. Powell, a numerical analyst at a neighbouring atomic energy laboratory,
which satisfied all our requirements and has since been much used by a number
of elementary particle physicists.

Our fit to the p^{+}p data in the first
resonance region, besides confirming our method, helped to fix the partial-wave
amplitudes at the topmost energy 1300 MeV, which was to be the bottom-most
energy of our next step. There was already a careful partial-wave
analysis by Vik and Rugge[14]
at this single energy, 1300 MeV, (remember, our partial-wave analysis
covered a continuous range of energies) with three alternative solutions. We
already had some theoretical prejudice as to which of these solutions was the
best from the work of Jim Hamilton and Sandy Donnachie and of G. Kane and T. D.
Spearman[15]. Our first
analysis confirmed our prejudice and we used this particular Vik and Rugge
solution as a guide to our partial waves at 1300 MeV in the next stage which
was to fit both p^{+}p and p^{-}p
elastic scattering data from 1300 MeV to 1580 MeV, covering the region of the
second resonance D_{13}.

We decided to limit the D_{13}-wave
parameters of our model so that a resonance in that particular wave was forced,
but to allow the data-fitting program to determine the exact position and
width of the D_{13} resonance. At that time we had little or no idea that
there were any other resonances within this energy region and our main purpose
was to determine the exact D_{13} energy and width and then to step to
higher energies. Even with this assumption, however, it was not easy to obtain
any good fits to the data and many attempts were unsuccessful, with consequent
rethinking of the details of the parametrization and the ranges of the
parameters. Pat O'Donnell liked to leave the ranges of the parameters
relatively wide so that the computer minimization program was rather free to
make its own best choice. My reaction in case of difficulty was to restrict
certain key parameters to seemingly likely values on the hypothesis that we
knew better than the computer. This probably reflects some difference in our
psychology to my discredit. Over a period of four months from February to June
1964 we achieved one good fit to the data helped by the fact that the D_{13}
resonant wave is very prominent and that other waves are presumably largely
determined by their interference with it (though the exact details of this
determination lie hidden in the thought processes of the computer program) and
that we did not lightly allow other waves to become prominent. Our fit
displayed a large and resonant D_{13} wave, of course, but also large P_{ll}
and S_{ll} waves, whose interpretation was not immediately evident to
us.

Early during this period, February-June 1964,
a letter had been published by Bareyre, Valladas and others from Saclay
Laboratory, near Paris, which pointed out that, if one graphed the isospin
I=1/2 total cross section for pion-nucleon scattering as a function of energy
and subtracted from it the probable contribution of the D_{13}(1520)
resonance (which caused a bump in the total cross section at that energy), then
there remained another bump at ~1420 MeV which might be due to an S-wave
or a P-wave resonance. This was extremely interesting, though far from
evidence for another resonance. Then in April there appeared the Letter by
Roper, giving the results of his partial-wave analysis, with a resonance
in the P_{11} wave known almost immediately, and since, as the Roper
resonance. Bransden had been at the Siena Conference, so we knew from Feld's
report of the existence of the M.I.T.-Livermore analysis but did not know
its exact results. The publication of Roper's letter came at an advanced stage
of our own analysis and so did not affect our techniques or results, but it
inspired us to write a Rutherford Laboratory Report (July 1964) which was
published shortly thereafter, giving the results of our own analysis and
pointing out that, like Roper, we had a large P_{1l} wave but that a
resonance interpretation was not so evident from our results.

We gave our results in terms of d
(the real phase shift) and h (the inelasticity) of each
partial wave, as did Roper, and at this stage we had given little thought to
resonance theory and were looking for the phase shift d
to go through 90°, as an indication of resonance, as in the case of the P_{33}
and D_{13} resonances. At some time in July, I presented our results to
an evening seminar in Oxford with a vacation-time audience of about six
people and speculated about the possibility of P_{1l} and S_{1l}
resonances, whereupon Dick Dalitz introduced the idea (already known in other
contexts to some physicists) of plotting the amplitudes on an Argand diagram
(Figure 2).

__Figure 2__

Argand diagram for a
(complex) partial-wave amplitude T, a function of the energy E, of the
pion and nucleon in the elastic scattering process. If the scattering particles
have an ideal resonance in this partial wave then T describes a circle in the
Argand diagram, such as R, as E varies. In the diagram X represents a typical
point of T which may be alternatively specified by the phase shift angle, d,
and radial distance h/2, which are shown. The unitarity
(conservation of probability) conditions require that h__£__ 1, so that all points X must lie within or on the
unitarity circle, U.

In such a diagram for the amplitude to swiftly
traverse (as energy varies) a considerable portion of an anti-clockwise
circle is an indication of resonance; this technique of resonance spotting has
been of primary importance in pion-nucleon partial-wave analyses. More
work at Darmstadt led to a second fit to the data, with partial waves similar
to our first solution, and in November of 1964 we issued a Rutherford
Laboratory Report (later published in Physical Review) complete with Argand
diagrams (Figure 3) of the D_{13}, P_{ll} and S_{11}
waves and a discussion of possible resonances in the two waves (the D_{13}
resonance being beyond question); we were rather positive about an S_{1l}
resonance.

__Figure 3__

Argand diagrams of the D_{13},
P_{ll} and S_{ll} partial waves from the 1964 analysis of
Bransden, Moorhouse and O'Donnell. The D_{13} describes a rather good
circle as a function of energy, but the resonant circles in the other waves are
somewhat distorted by background (the P_{ll} curve is now know to
extend over the imaginary axis before curving back).

Shortly after the publication of our first Letter,
there appeared a Letter from some London University physicists, Paul Auvil,
Claude Lovelace, Sandy Donnachie, and Andrew Lea[16],
presenting a partial-wave analysis in the same energy region with similar
results. Unlike Roper's and our analyses the data were not fitted by continuous
functions of energy, but were fitted semiindependently, at each discrete
energy having data, with theoretical guidance before and after fitting from
partial-wave dispersion relations. Later, at a Royal Society of London
Discussion Meeting in February of 1965, Lovelace presented evidence for an S_{31}
resonance at 1650 MeV and a possible __higher__ P_{11} resonance,
given with his usual high-spirited and uninhibited attacks on other
research workers in the same field as is especially well-remembered by
one natural target who happened to be speaking at the same meeting.

One of my pleasures was to regularly attend the
Oxford Thursday evening particle physics seminars and one experimental talk
mentioned the strong eta-meson production in
the process p^{-}+proton-> eta+neutron
just above the threshold energy and consequently in the energy region of our S_{11}
phenomenon (Figure 3). As a result Archie Hendry and I used a two-channel
reaction formalism (the two channels being h+nucleon and p+nucleon,
both in an S-wave) to simultaneously fit the h-production
data (assumed from its angular distribution to be mainly s-wave) and also
the S_{ll} amplitude of our partial-wave analysis. Hendry and I
found about April of 1965 that this fitting process definitely indicated an s-wave
resonance at about 1530 MeV with formation from (and decay into) both the h-nucleon
and p-nucleon channels[17]

IV. __Rationalizing the Resonance Population
Explosion__

In the summer of 1965, Dick Dalitz at Oxford, who
was well aware of all these developments, was due to give a lecture series at a
Summer School at Les Houches in the French Alps and his mind turned to some
qualitative theoretical developments. In 1963 Gell-Mann, and also Zweig,
had introduced the idea of baryons, such as the nucleon, being formed of three
imaginary particles called quarks, and mesons being formed of one quark plus
one anti-quark. These quarks, which might indeed be purely mathematical
objects, served as bases for the newly discovered SU3 symmetry, and if endowed
with spin 1/2 would also serve as bases for the larger SU6 symmetry which
Radicatti and Pais pointed out produced a few remarkable agreements with hadron
properties. Morpurgo and Becchi had shown that regarding the quarks as real
objects undergoing nonrelativistic motion at the bottom of a deep potential
well gave agreement not only with some electromagnetic properties of the
nucleon but also with those of the P_{33} resonance, when similarly
regarded as a particle made out of three quarks. Greenberg had observed that
the 56-dimensional representation of SU6, which contains most of the
baryons that were well known at that time, such as the nucleon and the P_{33}
resonance, is symmetric in the quark coordinates and had hypothesized that
further baryons might also obey this symmetric rule.

In his Les Houches lectures[18]
(July,1965) Dalitz systematically developed the symmetric quark model,
including orbital motion of the quarks. The next higher energy state than the
orbital ground state L^{P}=0^{+}, of the quarks (P=parity) has orbital angular
momentum L^{P}=1^{-}
and, when combined with a 70dimensional SU6 representation, can make a
symmetric state. This L^{P}=1^{- }70dimensional representation, as shown in Figure
4, contained the old D_{13} resonance and the newly discovered S_{ll}
and S_{3l} resonances.

__Figure 4__

The {70} 1^{--} supermultiplet of the L-excitation
quark model. The {SU3}^{2S+1}L multiplets are shown split into the J^{P}
submultiplets. The nucleonic state corresponding to each submultiplet is
indicated in pion-nucleon scattering notation.

Also it contained a D_{15} resonance and a
possible further S_{ll} resonance (Figure 5) newly available to Dalitz
in a pre-preprint of a 0-1700 MeV discrete-energy partial-wave
analysis of Bareyre, Bricman, Stirling, and Villet[19].

__Figure 5__

The scattering amplitude
T(Sll) obtained by Bareyre et al. in their phase shift analysis of the pion-nucleon
scattering data is plotted on a Argand diagram as a function of pion laboratory
kinetic energy in MeV. After a strong cusp at the nn threshold, the amplitude
first follows a looped path (where the analysis of Hendry and Moorhousel7
indicates a resonance close to 600 MeV) and then rapidly traces out the upper
part of a second circular loop (which is interpreted to reflect the existence
of a resonant state at about 900 MeV).

Other resonances such as the Roper resonance could
be fitted into other multiplets. (Later on all the resonances of the 70-dimensional
L^{P}=1^{-}
multiplet would be discovered, with no further negative-parity resonances
below 1800 MeV.) At the Oxford International Conference[20]
in 1965 Peyrou gave the baryon resonance raporteur talk, concentrating largely
on pion-nucleon partial-wave analyses, particularly that of his
compatriots, Bareyre et al. Dalitz, also in a principal talk, presented his
work on the quark model interpretation.

With these presentations to an international
conference a certain revolution was completed and the succeeding years
consolidated the new regime[21].
The revolution was the large number of new pion-nucleon resonances
discovered, which brought about a new way of regarding resonances. Previously
explanations of, say pion-nucleon, resonances were sought in detailed
dynamical calculations involving the exchange of elementary particles between
the pion and the nucleon. Indeed, an explanation of the existence of the nucleon
itself was sought in this way, the nucleon being regarded as a P_{ll}
state of the pion-nucleon system. Such a calculation should certainly
have predicted the Roper, or P_{ll} resonance - but it did not[22].
This philosophy was not disproved or even totally abandoned - indeed it
could coexist with the quark model - but simply fell into relative disuse
over the succeeding years in the face of a naive quark model which has greater
predictive power though being still illogical and incomplete. The contribution
of the pion-nucleon partial-wave analyses to the revolution was the
discovery of the "hidden" low-angular-momentum
resonances, and eventually of, it appears, __all__ of the resonances up to a
certain energy. In fact the existence of the resonances of the 70-dimensional
L^{P}=1^{-}
multiplet, and no other 'low energy' negative-parity resonances is still
one of the most convincing pieces of evidence for the quark model -
along, incidentally, with the success of the quark-model predictions on
the photon + nucleon (i.e. electromagnetic) formation of these same resonances.
(Photoproduction is a whole other chapter in the resonance story.)

For the latest pi-nucleon analysis results see http://gwdac.phys.gwu.edu/analysis/pin_analysis.html.

[1] E.g., see H. L. Anderson, E. Fermi, R. L. Martin, and D. E. Nagle, Phys. Rev. 91, 155 (1953).

[2] E. Fermi, Phys. Rev. 91, 947 (1953). S. Minami, Progr. Theoret. Phys. (Kyoto) 11, 213 (1954).

[3] R. F. Peierls, Phys. Rev. __118__,
323 (1960).

[4] W. M. Layson, Nuovo Cim. 27, 724 (1963).

[5] B. T. Feld and L. D. Roper, __Proc.
of the Siena Intern. Conf. on Elem. Part.__ (Italian Phys. Soc., Bologna,
1963), p. 400.

[6] M. H. Hull, Jr. and F. C.
Lin, Phys. Rev. __139__, B630 (1965).

[7] P. Bareyre, C. Bricman, G Valladas, G. Villet, J. Bizard, and J. Sequinot, Phys. Letters 8, 137 (1964).

[8] L. D. Roper, Phys. Rev. Letters 12, 340 (1964).

[9] L. D. Roper, R. M. Wright,
and B. T. Feld, Phys. Rev. __138__, B190 (1965).

[10] L. D. Roper and R. M.
Wright, Phys. Rev. __138__, B921 (1965).

[11] B. H. Bransden, P. J.
ODonnell, and R. G. Moorhouse, Phys. Letters 11, 339 (1964); Phys. Rev. __139__,
B1566 (1965).

[12] R. Horgan, Nucl. Phys. __B71__,
514 (1974).

[13]
Fits to the data are
obtained by finding the minimum value of a certain wellknown quantity,
conventionally known as c^{2}, which measures the
agreement between the data values predicted by the physical model and the data
values found from experiment, with appropriate weighting for the experimental
errors:

_{
}

The
procedure is to vary the parameters of the physical model with the object of
obtaining as close as possible an agreement with the data which corresponds to
as small as possible a value of c^{2}. The attainment of such a
good agreement with experiment then signifies two things: (i) that, primae
faciae, the physical model used can represent reality and (ii) that the
parameters for which c^{2} is minimum are near their
real physical values.

[14] O. T. Vik and H. R. Rugge,
Phys. Rev. __129__, 2311 (1963).

[15] A. Donnachie, J. Hamilton,
and A. T. Lea, Phys. Rev. __135__, B515 (1964). G. L. Kane and T. D.
Spearman, Phys. Rev. Letters 11, 45 (1963).

[16] P. Auvil, A. Donnachie, A. T. Lea, and C. Lovelace, Phys. Letters 12, 76 (1964).

[17] A. W. Hendry and R. G. Moorhouse, Phys. Letters 18, 171 (1965).

[18] R. H. Dalitz, __Lectures in
Theoretical Physics__ (Gordon and Breach, New York, 1966).

[19] P. Bareyre, C. Bricman, A. V. Stirling, and G. Villet, Phys. Letters 18, 342 (1965).

[20]
C. Pevrou, __Proc.
1965 O~ford International Conference on Elementary Particles__, (Rutherford High Energy
Laboratory, (1966).

[21]
Of particular
importance was the raporteur's talk at the 1968 Heidelberg Conference in which
Claude Lovelace presented a large number of new resonances, including firm
evidence for all those classified in the quark model as belonging to the 70, L^{P}=1^{-} multiplet. These results
were mainly from the work of Lovelace, Donnachie, Kirsopp, and Lee at Cern and
Johnson, Grannis, Hansroul, Chamberlain, Shapiro, and Steiner at Berkeley.

[22] After its discovery, more complicated multichannel calculations on similar lines did "produce" the Roper resonance. See, e.g., E. N. Argyles and A. Rotsstein, Phys. Rev. 174, 1689 (1968).