\errorcontextlines=10
\documentclass[11pt]{article}
\usepackage[dvips]{color}
\usepackage[dvips]{epsfig}
%\usepackage[english,russian]{babel}
%\RequirePackage[hyperindex,colorlinks,backref,dvips]{hyperref}
\RequirePackage{hyperref}
\begin{document}
%\documentstyle[art14,fullpage]{article}
%\topmargin=-2cm
%\textheight=25.8cm
%\parindent=20pt
%\thispagestyle{empty}
%\begin{document}
S.~Mandelstam, Phys. Rev. {\bf 112,} 1344 \hfill {\large \bf 1958}\\
\vspace{2cm}
\begin{center}
{\large \bf Determination of the Pion-Nucleon Scattering Amplitude
from Dispersion\\
Relations and Unitarity. General Theory }\\
\end{center}
\vspace{0.5cm}
\begin{center}
S. Mandelstam\\
{\it Department of Physics, Columbia University, New York, New York}\\
(Received June 27, 1958)\\
\end{center}
\vspace{1cm}
{\small A method is proposed for using relativistic relations, together with
unitarity, to determine the \mbox{pion-nucleon} scattering amplitude. The usual
dispersion relations by themselves are not sufficient and we have to assume a
representation which exhibits the analytic properties of the scattering
amplitude as a function of the energy and the momentum transfer. Unitarity
conditions for the two reactions $\pi + N \rightarrow \pi + N$ and $N + \bar{N}
\rightarrow 2 \pi$ will be required, and they will be approximated by neglecting
states with more than two particles. The method makes use of an interaction
procedure analogous to that used by Chew and Low for the corresponding problem
in the static theory. One has to introduce two coupling constants; the
\mbox{pion-pion} coupling constant can be found by fitting the sum of the
threshold scattering lengths with experiment. It is hoped that this method
avoids some of the formal difficulties of the Tamm-Dancoff and Bethe-Salpeter
methods and, in particular, the existence of ghost states. The assumptions
introduced are justified in perturbation theory.
As an incidental result, we find the precise limits of the region for which the
absorptive part of the scattering amplitude is an analytic function of the
momentum transfer, and hence the boundaries of the region in which the
partial-wave expansion is valid.}
\vspace{1cm}
\section*{
{\bf 1. Introduction}}
\vspace{0.5cm}
In recent years dispersion relations have been used to an increasing extent in
pion physics for phenomenological and semiphenomenological analyses of
experimental data,\footnote{Chew, Goldberger, Low, and
Namby, Phys. Rev. 106,
1345 (1957). This paper contains further references.}
%\cite{Chew1957}
and even for the
calculation of certain quantities in terms of the \mbox{pion-nucleon} scattering
amplitude.\footnote{Chew, Karplus, Gasiorowicz, and Zachariasen,
Phys. Rev., 110,
265 ( 1958).}
%\cite{Chew1958}
It is therefore tempting to ask the question whether or not the
dispersion relations can actually replace the more usual equations of field
theory and be used to calculate all observable quantities in terms of finite
number of coupling constants--a suggestion first made by Gell-Mann\footnote{M.
Gell--Mann, Proceeding of the Sixth Annual Rochester Conference High-Energy
Physics, 1956 (Interscience Publishers, Ins., New York, 1956), Sec. III, p. 30}
%\cite{Gell-Mann1956}
at first sight, this would appear to be unreasonable, since, although it is
necessary to use all the general principles of quantum field theory to derive
the dispersion relations, one does not make any assumption about the form of the
Hamiltonian other than that it be local and Lorentz-- invariant. However, in a
perturbation expansion these requirements are sufficient to specify the
Hamiltonian to within a small number of coupling constants if one demands that
the theory be renormalizable and therefore self- consistent. It is thus very
possible that, even without a perturbation expansion, these requirements are
sufficient to determine the theory. In fact, if the ``absorptive part'' of the
scattering amplitude, which appears under the integral sigh of the dispersion
relations, is expressed in terms of the scattering amplitude by means of the
unitarity condition, one obtains equations which are very similar to the
Chew-Low\footnote{G.F. Chew and F.E. Low, Phys. Rev. 101, 1570 (1956).}
%\cite{Chew1956}
equations in static theory. These equations have been used by Salzman and
Salzman\footnote{G.Salzman and F.Salzman, Phys. Rev. 108, 1619 (1957).}
%\cite{Salzman1957}
to obtain the \mbox{pion-nucleon} scattering phase shifts.
It is the object of this paper to find a relativistic analog of the
Chew-Low-Salzman method, which could be used to calculate the
\mbox{pion-nucleon} scattering amplitude in terms of two coupling constants
only. as in the static theory, the unitarity equation will involve the
transition amplitude for the production of an arbitrary number of mesons, and,
in this case, of nucleon pairs as well. In order to make the equations
manageable, it is necessary to neglect all but a finite number of processes; as
a first approximation, the ``one-meson'' approximation, we shall neglect all
processes expect elastic scattering.
The equations obtained from the dispersion relations and the \mbox{one-meson}
approximation differ from the static Chew-Low equations in two important
respects. Whereas, in the static theory, there was only \mbox{$P$-wave}
scattering, we now have an infinite number of angular momentum states, and the
crossing relation, if expressed in terms of angular momentum states, would not
converge. Further, in the relativistic theory, the dispersion relations involve
the scattering amplitude in the ``unphysical'' region, i.e., through angles
whose cosine is less than $-1$. For these reasons, the method of procedure will
be more involved than in the static theory. We shall require, not only the
analytic properties of the scattering amplitude as a function of energy for
fixed momentum transfer, which are expressed by the dispersion of both
variables. The required analytic properties have not yet been proved to be
consequences of microscopic causality. In order to carry out the proof, one
would almost certainly have to consider simultaneously several Green's functions
together with equation connecting them which follow from unitarity. It is
unlikely that such a program will be carried through in the immediate future.
However, if the solution obtained by the use of these analytic properties were
to be expanded in a perturbation series, we would obtain precisely those terms
of the usual perturbation series included in the \mbox{one-meson} approximation.
The assumed analytic properties are, therefore, probably correct, at any rate in
the \mbox{one-meson} approximation.
As we have to resort to perturbation theory in order to justify our assumptions,
we do not yet have a theory in which the general principles of quantum theory
are supplemented only by the assumption of microscopic causality. Nevertheless,
the approximation scheme used has several advantages over the approximation
previously applied to this problem, such as the Tamm-Dankoff or Bethe-Salpeter
approximation. It refers throughout only to renormalized masses and coupling
constants. The Tamm-Dankoff equations, by contract, are unrenormalizable in
higher approximation and the Bethe-Salpeter equations, while they are covariant
and therefore renormalizable in all approximations, present difficulties of
principle when one attempts to solve them. Further, we may hope that the
\mbox{one-meson} approximation is more accurate than the Tamm-Dankoff
approximations. The letter assumes that those components of the state vector
containing more than a certain number of bare mesons are negligibly small -- an
approximation that is known to be completely false for the experimental value of
the coupling constant. The \mbox{one-meson} approximation, on the other hand,
assumes that the cross section for the production of one or more {\it real }
mesons is small expect at high energies. while this approximation is certainly
not quantitatively correct, it is nevertheless probably a good deal more
accurate than the Tamm-Dankoff approximation. Finally, the \mbox{one-meson}
approximation, unlike the Tamm-Dankoff or Bether-Salpeter approximations,
possesses crossing symmetry. Now it is very probable that the ``ghost states''
which have been plaguing previous solutions of the field equations are due to
the neglect of crossing symmetry. As evidence of this, we may cite the case of
charged scalar theory without recoil, for which the \mbox{one meson}
approximation has been solved completely.\footnote{T.D.Lee and R.Serber (
unpublished).}
%\cite{Lee}
\footnote{Castillejo, Dalitz, and Dyson, Phys. Rev. 101, 453
(1956).}
%\cite{Castillejo1956}
The solution obtained with neglect of the crossing term possesses the
usual ghost state if the source radius is sufficiently small. The Lee
model,\footnote{K.W.Ford, Phys. Rev. 105, 320 (1957).}
%\cite{Ford1957}
Which has no crossing
symmetry, shows a similar behavior. If the crossing term in the charged scalar
model is included, however, there is no ghost state.
It has been pointed out by Castillejo, Dalitz, and Dyson$^7$ that the dispersion
relations, at any rate in the charged scalar model, do not possesses a unique
solution. This might have been expected, since it is possible to alter the
Hamiltonian without changing the dispersion relations. One simply has to
introduce into the theory a baryon whose mass is greater than the sum of the
masses of the meson and nucleon. Such a baryon would be unstable, and would
therefore not appear as a separate particle or contribute a term to the
dispersion relations. In perturbation theory, the simplest of the solutions
found by Castilejo, Dalitz, and Dyson, i.e., the solution without any zero in
the scattering amplitude, agrees with the solution obtained from a Hamiltonian
in which there are no unstable particles, and the more complicated solutions
correspond to the existence of unstable baryon. we shall assume that this is so
independently of perturbation theory, and shall concern ourselves with the
simplest solution. There is no physical reason why one of the other solutions
may not be the correct one, but it seems worthwhile to try to compare with
experiment the consequences of a theory without unstable particles. It should in
any case be emphasized that the ambiguity is not a specific feature of this
method of solution, but is inherent in the theory itself. The difference is
that, in other methods, it occurs in writing down the equations, whereas in this
method it occurs in solving them.
In Sec. 2 we shall discuss the analytic properties of the scattering amplitude,
and, in Sec. 3, we shall show now these properties can be used together with the
unitarity condition to solve the problem. We shall in this section ignore the
``subtraction terms'' in the dispersion relations. as in the
corresponding static
problem, we have to use an iteration procedure in which the crossing term is
taken from the result of the previous iteration. The details of this solution
will be entirely different from the static problem, the reason being that the
part of the amplitude corresponding to the lowest angular momentum states, which
is a polynomial in the momentum transfer, actually appears as a subtraction term
in the dispersion relation with respect to this variable and has thus not yet
been taken into account. In this and the next section we shall also be able to
specify details of the analytic representation that were left undermined in
Sec.2, in particular, we shall be able to give precise limits to the values of
the momentum transfer within which the \mbox{partial-wave expansion} converges.
In sec.4 we shall investigate the subtraction terms in the dispersion relations.
We shall find that, in order to determine them, we shall require the unitarity
condition for the lowest momentum states, not only in \mbox{pion-nucleon}
scattering, but also in the pair-annihilation reaction $N + \bar{N} \rightarrow
2 \pi$, which represented by the same Green's function. The coupling constant
for \mbox{meson-meson} scattering is thus introduced into the theory; as its
value is not known experimentally it will have to be determined by fitting one
of the results of the calculation, such as the sum of the \mbox{$S$-wave}
scattering lengths at threshold, with experiment. The calculations of these low
angular momentum states would be done in the same spirit as the Chew-Low
calculations, and the details will not be given in this paper. We thus have a
procedure in which the first few angular momentum states are calculated by
methods similar to those used in the static theory, while the remaining part of
the scattering amplitude, which will be called the ``residual part,'' is
calculated by a different procedure which does not make use of a
\mbox{partial-wave} expansion. Needless to say, the two parts of the calculation
become intermingled by the iteration procedure.
It is only in the calculation of the subtraction terms that $u$ $e$ has to be
made of the unitarity condition for the \mbox{pair-annihilation} reaction. For
the residual part, it is only necessary to use the unitarity condition for
\mbox{pion-nucleon} scattering. Had it been possible to use the unitarity
condition exactly instead of in the \mbox{one-meson} approximation, the result
would also satisfy the unitarity condition for the annihilation reaction in a
consistent theory. As it is, we find that the residual part consists of a number
of terms which correspond to various intermediate states in the annihilation
reaction. In Sec.5 it is pointed out that the calculation is greatly simplified
if we keep only those terms of the residual part corresponding to pair
annihilation through states with fewer than a certain number of particles. Such
an approximation has already been made in calculating the subtraction terms. The
unitarity condition for \mbox{pion-nucleon} scattering is no longer satisfied
except for the low angular momentum states. However, the terms neglected are of
the order of magnitude of, and probably less than, terms already neglected. The
two reactions of \mbox{pion-nucleon} scattering and pair annihilation are
now treated on an equivalent footing.
It will be found that the unitarity condition, in the \mbox{one-meson}
approximation, cannot be satisfied at all energies if crossing symmetry and the
analytic properties are to be maintained. The reason is that unitarity condition
for the scattering reaction is not completely independent of the unitarity
condition for the ``crossed'' reaction with the two pions interchanged, and they
contradict one another if an approximation is made. There is, of course, no
difficulty in the region where the \mbox{one-meson} approximation is exact. For
sufficiently small values of the coupling constant, we shall be able to obtain a
unique procedure. For values of the coupling constant actually encountered, one
part of the crossing term may have to be cut off at the threshold for pair
production in \mbox{pion-nucleon} scattering. It is unlikely that the result
will be sensitive to the form and the precise value of the cutoff.
\vspace{0.5cm}
\section*{
{\bf 2. Dispersion Relations and Analyticity
Properties of the Transition
Amplitude}}
\vspace{0.5cm}
The kinematical to be used in writing down the dispersion relations will be
similar to that of Chew et. al.$^1$ The momenta of the incoming and outgoing
pions will be denoted by $q_1$ and $q_2$, those of the incoming and outgoing
nucleus by $p_1$ and $p_2$. We can then define two invariant scalars
$$
\nu = - (p_1 + p_2)(q_1 + q_2)/4M, \eqno(2.1)
$$
$$
t = - \left( q_1 - q_2 \right)^2. \eqno(2.2)
$$
The latter is minus the square of the invariant momentum transfer. The
laboratory energy will be given by the equation
$$
\omega = \nu - \left( t/4M \right): \eqno(2.3a)
$$
It is more convenient to use, instead of the laboratory energy, the square
of
the \mbox{center-of-mass} energy (including both \mbox{rest-masses}), which
is
linearly related to it by the equation
$$
s = M^2 + \mu^2 + 2M \omega. \eqno(2.3b).
$$
The Green's function relevant to the process under consideration,
$$
\pi_1 + N_1 \rightarrow \pi_2 + N_2 \eqno({\rm I})
$$
also gives the processes
$$
\pi_2 + N_1 \rightarrow \pi_1 + N_2 \eqno({\rm II})
$$
and
$$
N_1 + \overline{N_2} \rightarrow \pi_1 + \pi_2. \eqno({\rm III})
$$
The matrix elements for the process II can be obtained from those for the
process I by crossing symmetry; the laboratory energy and the square of the
\mbox{center-of-mass} energy will now be
$$
\omega_c = - \nu - \left( t/4M \right) = - \omega - \left(
t/2M \right), \eqno(2.4a)
$$
$$
s_c = M^2 + \mu^2 + 2M \omega_c = 2M^2 + 2 \mu^2 - s - t. \eqno(2.4b)
$$
The square of the momentum transfer will be $-t$ as before. For the process
III,
the square of the \mbox{center-of-mass} energy will be $t$. The square of
the
momentum transfer between the nucleon $N_1$ and the pion $\pi_2$ will be
$s_c$
and that between the nucleon $N_1$ and the pion $\pi_1$ will be $s$.
The kinematics for the three reactions are represented diagrammatically in
Fig.1 in which $t$ has been plotted against $\nu$. $AB$
represents the line $s = (M
+ \mu)^2$, or $\omega = \mu$, and lines for which $s$ is constant will be
parallel to it. The region for which the process I is energetically possible
is
therefore that to the right of $AB$. However, only the shaded part of this
area
is the ``physical region''; in the unshaded part, though the energy of the
meson
is greater than its \mbox{rest-mass}, the cosine of the scattering angle is
not
between $-1$ and $+1$. The physical region is bounded above by the line $t =
0$, i.
e., the line of forward scattering, and below by the line of backward
scattering. Similarly $CD$ is the line $s_c = (M + \mu)^2$; the region for
which
the process II is energetically possible that to the left of $CD$, and the
shaded area represents the physical region for this reaction, Lines of of
constant energy for the reaction III are horizontal lines. The reaction will
be
energetically possible above the line $EF$, at which $t = 4M^2$, and again
the
shaded area represents the physical region.
We now examine the analytic properties of the scattering amplitude. To
simplify
the writing, we shall first neglect spin and isotopic spin; the transition
amplitude will then be a scalar function $A(\nu,t)$ of the two invariants
$\nu$
and $t$. Its analytic properties as a function of $\nu$ with $t$ constant,
are
exhibited by the usual dispersion relations
$$
A(\nu,~t) = \frac{g^2}{2M} \cdot \left( \frac{1}{\nu_B - \nu} + \frac{1}{\nu_B +
\nu}
\right) + \frac{1}{\pi}~ \int \limits^{\infty}_{\mu + (t/4M)}~
d \nu'~ \frac{A_1 (\nu',~t)}{\nu' - \nu} -
$$
$$
- \frac{1}{\pi}~ \int \limits^{- \mu - (t/4M)}_{- \infty}~ d \nu'~
\frac{A_2 (\nu', ~t,)}{\nu' - \nu}
\eqno(2.5)
$$
Where $\nu_B = - (\mu^2/2M) + (t/4M)$. In this and all subsequent
such equations, the energy denominators are taken to have a small imaginary
part. $A_1$ and $A_2$ are the ``absorptive parts'' associated with the
reactions
I and II, respectively, and are given by the equations
$$
(2 \pi)^4 A_1 (\nu,~t) \delta (p_1 + q_1 - p_2 - q_2) = (2 \pi)^6 \cdot
\left(\frac{4
p_{01} p_{02} q_{01} q_{02}}{M^2} \right)^{^1/_2} \times
$$
$$
\times \sum_n \langle N (p_1) \pi (q_1) \mid n \rangle \langle n \mid N
(p_2)
\pi (q_2) \rangle, \eqno(2.6)
$$
$$
(2 \pi)^4 A_2 (\nu,~t) \delta (p_1 + q_1 - p_2 - q_2) = (2 \pi)^6 \cdot \left(
\frac{4 p_{01} p_{02} q_{01}q_{02}}{M^2} \right)^{^1/_2} \times
$$
$$
\times \sum_n \langle N (p_1) \pi (- q_2) \mid n \rangle \langle n
\mid N (p_2) \pi (- q_1) \rangle. \eqno(2.7)
$$
The symbol $\langle N (p_1) \pi (q_1) \mid$ denotes a state with an ingoing
nucleon of momentum $p_1$ and an ingoing pion of momentum $q_1$. The sum
$\sum_
n$ is to be taken over all intermediate states. $A_1$ and $A_2$ are nonzero
to
the right of $AB$, and to the left of $CD$, respectively.
Equation (2.5) indicated that $A$ is an analytic function of $\nu$ in the
complex plane, with poles at $\pm \nu_B$, and cuts along the real axis from
$\mu
+ (t/4M)$ to $\infty$ and from $- \infty$ to $- \mu - (t/4M)$.
On Fig. 1, (2.5) will be represented by an integration along a horizontal
line
below the $\nu$ axis. The poles will occur where this line crosses the
dashed
lines; apart from them, the integrand will be zero between $AB$ and $CD$.
Except
for forward scattering, the region where the integrand in nonzero will be
lie
partly in the unphysical region, where the energy is above threshold but the
angle imaginary.
Equation (2.5) is only true as it stands if the functions $A$, $A_1$, and
$A_2$
tend to zero sufficiently rapidly as $\nu$ tends to infinity; otherwise it
will
be necessary to perform one or more subtractions in the usual way. Wherever
such a dispersion relation is written down,\\
%\vspace{1.5cm}
%\includegraphics{mandelstam1a.gif}
%\\Fig. 1\\
\begin{figure}[h]
\centerline{\epsfig{file=fig1r.eps}}
\caption{Kinematics of the reactions I, II, and III.}
\end{figure}
%\vspace{1.5cm}
The possibility of having to perform subtractions is implied.
We next wish to obtain analytic properties of $A$ as a function of $t$. In
order
to do this we shall write the scattering amplitude, not as the expectation
value
of the \mbox{time ordered} product of the two meson current operators
between
two \mbox{one-nucleon} states, as is done in the proof of the usual
dispersion
relations,\footnote{M.L.Goldberger, Phys. Rev. 99, 979
(1955).}
%\cite{Goldberger1955}
\footnote{R.H.Capps and G. Tekeda, Phys. Rev. 106, 1337 (1956).}
%\cite{Capps1956}
but as the expectation value of the product of a meson current operator and
a
nucleon current operator between a nucleon state and a meson state. Thus
$$
(2 \pi)^4 A \delta \cdot (p_1 + p_2 - q_1 - q_2) = (2 \pi)^3 \cdot
\left( \frac{2 p_{01}
q_{02}}{M} \right)^{^1/_2} i \int ~dx dx' \times
$$
$$
\times e^{ - i q_1 x + i p_2 x'} \langle N (p_1) \mid T \left\{ j (x)
\bar{a} (x') \right\} \mid \pi (q_2) \rangle, \eqno(2.8)
$$
Where $a(x')$ is a nucleon current operator. From this expression, we can
obtain
dispersion relations in which the momentum transfer between the incoming
nucleon
and the outgoing pion, rather than between the two nucleons, is kept
constant --
the proof is exactly the same as the usual heuristic proof is ordinary
dispersion relations.$^{9,10}$ As this momentum transfer is just $s_c$, we
obtain dispersion relations in which $s_c$ is kept constant; if $A$ is
written
as a function of $s_c$ and $t$, they take the from
$$
A(s_c,~t) = \frac{g^2}{s_c + t - M^2 - 2 \mu^2} - \frac{1}{\pi}
~\int \limits^{(M - \mu)^2 -s_c}_{- \infty} ~dt'~
\frac{A_1 (s_c, ~t')}{t' - t} +
$$
$$
+ \frac{1}{\pi}~ \int \limits^{\infty}_{4 \mu^2}~ dt'~
\frac{A_3(s_c,~t')}{t' - t}. \eqno(2.9)
$$
The absorptive parts in the integrand are as usual obtained by replacing the
\mbox{time-ordered} product in (2.8) by half the commutator. The first term,
in
which the operators are in the order $j(x)\bar{a}(x')$, is exactly $A_1$,
and
will therefore be nonzero to the right of $AB$ and have a $\delta$ function
along $IK$. The second term, however, in which the operators are in the
order
$\bar{a}(x')j(x)$, will now be related to the process III. It will be given
by
the equation
$$
(2\pi)^4 A_3(s_c,~t)\delta(p_1 + q_1 - p_2 - q_2) = (2 \pi)^6 \left(
\frac{4p_{01} p_{02} q_{01} q_{02}}{M^2} \right)^{^1/_2} \times
$$
$$
\times \sum_n \langle N(p_1) \bar{N} (- p_2) \mid n \rangle \langle n \mid
\pi
(- q_1) \pi (q_2) \rangle. \eqno(2.10)
$$
The state $n$ of lowest energy will now be the \mbox{two-meson} state. $A_3$
will therefore be nonzero above the line $t = 4 \mu^2$, represented by $GH$
in
Fig.1 (since $t$ is square of the \mbox{center-of-mass} energy of the
process
III). The dispersion relation (2.10) is represented by an integration along
a
line parallel to $CD$ and to the right of the line $s_c = 0$. It implies
that $
A$ is analytic function of $t$ for fixed $s_c$, with a pole at $t = M^2 +
\mu^2
- s_c$,
and cuts along the real axis from $- \infty$ to $(M - \mu)^2 - s_c$ and from
$4\mu^2$ to $\infty$.
As in the usual dispersion relation, part of the range of integration in
Eq.(2.9) will lie in the unphysical region. This region now includes,
besides
imaginary angles at permissible energies, the entire area between the lines
$t =
4\mu^2$ and $t = 4M^2$, where are contribution to $A_3$ from intermediate
states
with two or more pions. The rigorous proof of (2.9) is therefore much more
difficult than that of (2.5), and probably cannot be carried out without
introducing the unitarity equations.
By interchanging the two pions in the expression (2.8), we can obtain a
third
dispersion relation in which $s$ is kept constant:
$$
A(s,~t) = \frac{g^2}{s + t - M^2 - 2 \mu^2} - \frac{1}{\pi}~
\int \limits^{(M - \mu)^2 -s}_{- \infty}~dt'~ \frac{A_2(s,~t')}{t' - t} +
$$
$$
+ \frac{1}{\pi}~ \int \limits^{\infty}_{4 \mu^2}~
dt'~ \frac{A_3(s,~t')}{t' - t)} \eqno(2.11)
$$
On Fig.1, this would be represented by an integration along a line parallel
to
$AB$, and to the left of the line $s = 0$.
Let us now try to obtain the analytic properties of $A$ considered as a
function
of two complex variables. The simplest assumption we could make is that it
is
analytic in the entire space of the two variables except for cuts along
certain
hyperplanes. We can then determine the location of the cuts from the
requirement that $A$ must satisfy the dispersion relations (2.5), (2.9), and
(2.11); there will be a cut when $s$ is real and greater
than $(M + \mu)^2$, a cut
when $s_c$ is real and greater than $(M + \mu)$, and a cut when $t$ is real
and
greater than $4\mu^2$. The discontinuities across these cuts will be,
respectively, $2A_1$, $2A_2$, and $2A_3$. In addition, $A$ will have poles
when
$s = M^2$ and when $s_c = M^2$. By a double application of Cauchy's theorem,
it
can be shown that a function with cuts and poles in these positions can be
represented in the form
$$
A = \frac{g^2}{M^2 - s} + \frac{g^2}{M^2 - s_c} + \frac{1}{\pi^2}~
\int \limits^{\infty}_{(M + \mu)^2}~ ds'~
\frac{A_{13}(s',~t')}{(s' - s)(t' - t)} +
$$
$$
+ \frac{1}{\pi^2}~ \int \limits^{\infty}_{(M + \mu)^2}~ ds'_c ~
\int \limits^{\infty}_{4 \mu^2}~dt'~ \frac{A_{23}(s'_c,~t')}
{(s'_c - s_c)(t' - t)} +
$$
$$
+ \frac{1}{\pi^2} ~\int \limits^{\infty}_{(M + \mu)^2}~ ds'~
\int \limits^{\infty}_{(M + \mu)^2}
~ds'_c~ \frac{A_{12}(s',~s'_c)}{(s' - s)(s'_c - s_c)}. \eqno(2.12)
$$
This is a generalization of a representation first suggested by
Nambu.\footnote{Y.Nambu, Phys. Rev. 100, 394 (1955).}
%\cite{Nambu1955}
While we have for convenience
used the three variables $s$, $s_c$, and $t$, which are the energies of the
three processes, they are connected by the relation
$$
s + s_c + t = 2(M^2 + \mu^2),
\eqno(2.13)
$$
so that $A$ is really a function of two variables only. $A_{13}$, $A_{23}$
and $
A_{12}$, which will be referred to as the ``spectral functions'', are
nonzero in
the regions indicated at the top right, top left and bottom of Fig. 1. The
precise boundaries $C_{13}$, $C_{23}$, and $C_{12}$ of the regions will be
determined by unitarity in the following sections; from the reasoning given
up
till now, all that can be said is that the regions must lie within the
respective triangles as indicated, and that the boundary must approach the
sides
of the triangles asymptotically (or it could touch them at some finite
point).
The spectral functions are always zero in the physical region.
As in the case of ordinary dispersion relations, the representation (2.12)
will
not be true as it stands, but will require subtractions. The
subtractions
will modify one or both of the energy denominators in the usual way and, in
addition, they will require the addition of extra terms. These terms will
not
now be constants, but functions of one of the variables, e.g., if there is a
subtraction in the $s$ integration of the first term, the extra term will be
a
function of $t$. These functions must then have the necessary analytic
properties in their variables, so that they will have the form
$$
\frac{1}{\pi}~ \int \limits^{\infty}_{(M + \mu)^2} ~ds'~
\frac{f_1(s')}{s' - s} +
\frac{1}{\pi}~ \int \limits^{\infty}_{(M + \mu)^2}~ d s'_c ~
\frac{f_2(s'_c)}{s'_c - s_c} +
$$
$$
+ \frac{1}{\pi}~ \int \limits^{\infty}_{4 \mu^2} ~dt'~ \frac{f_3(t')}{t' -
t}.
\eqno(2.14)
$$
If more than one subtraction is involved, we may have similar terms
multiplied
by polynomials. Even if the spectral functions in (2.12) tend to zero as
one of
the variable is necessary, it is small not precluded that the corresponding
term
in (2.14) does not appear, as the function still has the required analytic
properties. For \mbox{pion-nucleon} scattering, however, there is no
undetermined \mbox{over-all} term, independent of both variables, to be
added,
as the requirement that the scattering amplitude for each angular momentum
wave
have the form $e^{i \delta} \sin \delta /k$, $\mbox{Im} \delta < 0$, forces
$A$
to tend to zero in the physical region when both $s$ and $t$ become
infinite.
The Nambu representations for the complete Green's functions are known to be
invalid, even in the lowest nontrivial order of perturbation theory. The
representation quoted here, however, restricts itself to the mass shells of
the
particles, and has not been shown to be invalid. In fact, in the case of
Compton
scattering, the \mbox{fourth-order} terms, which have been worked out by
Brown
and Feynman\footnote{L.M. Brown and R.P.Feynman, Phys. Rev. 85, 231 (1952).}
%\cite{Brown1952}
Are found to have this representation, and, as we have stated in the
introduction, all the perturbation terms included in the \mbox{one-meson}
approximation can be similarly represented.
The dispersion relations are an immediate consequence of the representation
(2.12). To obtain the usual dispersion relation (2.5), the third integral in
(2.12)
must be written as\footnote{When we make a change of variables, we imply of
course that the spectral functions still have the same value at the same
point,
and not that we must take the same function of the new variables.}
$$
- \frac{1}{\pi^2}~ \int \limits^{\infty}_{(M + \mu)^2} ~ ds'~
\int \limits^{t_2(s)}_{- \infty}~dt'\,
\frac{A_{12}(s',~t')}{(s' - s)(t' - t)}-
$$
$$ - \frac{1}{\pi^2} ~\int \limits^{\infty}_{(M + \mu)^2} ~ ds_c' ~
~\int \limits^{t_2(s_c)}_{- \infty}~dt'~
\frac{A_{12}(s_c',~t')}{(s'_c - s_c)(t' - t)}.
$$
It then follows that
$$
A = \frac{g^2}{M^2 - s} + \frac{g^2}{M^2 - s_c} + \frac{1}{\pi}
~\int \limits^{\infty}_{(M + \mu)^2} ~ds'~ \frac{A_1(s',~t)}{s' - s} +
$$
$$
+ \frac{1}{\pi} ~\int \limits^{\infty}_{(M + \mu)^2} ~ds'_c~
\frac{A_2(s'_c,~t)}
{s'_c - s_c}, \eqno(2.15)
$$
where
$$
A_1(s,~t) = \frac{1}{\pi} ~\int \limits^{\infty}_{t_1(s)}~dt'~
\frac{A_{13}(s,~t')}{t' - t} - \frac{1}{\pi}~
\int \limits^{t_2(s)}_{- \infty}~ dt'~ \frac{A_{12}(s,~t')}{t' - t},
\eqno(2.16)
$$
$$
A_2(s_c,~t) = \frac{1}{\pi}~ \int \limits^{\infty}_{t_1(s_c)}~ dt'~
\frac{A_{23}(s_c,~t')}{t' - t} - \frac{1}{\pi}~
\int \limits^{t_2(s_c)}_{- \infty}~dt' ~ \frac{A_{12}(s_c,~t')}{t' - t}.
\eqno(2.17)
$$
Equation (2.15) is, however, just the dispersion relation (2.5), since $s$,
$s_c$, and $\nu$ are connected by the relations (2.4) and $t$ is being kept
constant. we also see that the absorptive pairs $A_1$ and $A_2$ themselves
satisfy dispersion relations in $t$, with $s$ (or $s_c$) constant; the
imaginary
parts which appear in the integrand are now simply the spectral functions.
Equation (2.16) will be represented in Fig. 1, by an integration along a
line
parallel to $AB$ and to the right of it. The limits $t_1$ and $t_2$ are the
points at which this line crosses the curves $C_{13}$ and $C_{12}$. They
satisfy
the inequalities
$$
t_1 > 4 \mu^2, \eqno(2.18a),
$$
$$
t_2 < (M - \mu)^2 - s. \eqno(2.18b)
$$
$A_1$ will be nonzero for $s > (M + \mu)^2$, as it should, as long as the
curves
$C_{13}$ and $C_{12}$ approach the line $AB$ at some point and do not cross
it.
The dispersion relations (2.9) and (2.11) can be proved from (2.12) in
similar
way; the absorptive part $A_3$ will then satisfy a dispersion relation in
$\nu$
with $s$ constant:
$$
A_3 = \frac{1}{\pi} ~\int \limits_{\nu_3(t)}^{\infty}~ d \nu'~ \frac{A_{13}
(\nu',~t)}{\nu' - \nu} - \frac{1}{\pi}~ \int \limits^{- \nu_3(t)}_{-
\infty}~
d \nu'~ \frac{A_{23}(\nu',~t)}{\nu' - \nu}. \eqno(2.19)
$$
This dispersion relation will be represented by an integration along a
horizontal line above $GH$. $\nu_3$ and $-\nu_3$ will be the points at which
the line of integration crosses $C_{13}$ and $C_{23}$.
Finally, then, the scattering amplitude $A$ satisfies dispersion relations
in
which any of the quantities $t$, $s_c$, and $s$ are kept constant. Further,
it
follows from (2.12), by the reasoning just given, that the values of the
quantity which is being kept constant need no longer be resrticted in sign.
Thus, for example, we now know the analytic properties of $A$, as a function
of
momentum transfer, for fixed energy greater than (as well as less than)
$(M + \mu)^2$. They are given by analytic function of the square of the
momentum
transfer, with a pole at $t = M^2 + 2 \mu^2 - s$, and cuts along the real
axis
from $t = 4 \mu^2$ to $\infty$ and form $t = - \infty$ to $(M + \mu)^2 - s$.
For
$s > (M + \mu)^2$, these cuts and poles are entirely in the nonphysical
region.
It has already been shown rigorously by Lehmann\footnote{H.Lehmann (to be
published).}
%\cite{Lehmann1959}
that $A$ is analytic in $t$ in an area including the physical
region. The absorptive parts $A_1$, $A_2$ and $A_3$ will themselves satisfy
dispersion relations, provided that the correct variable be kept constant
($s,
s_c,$ and $t$ for $A_1$, $A_2$, and $A_3$, respectively). The weight
functions
for these dispersion relations are entirely in the nonphysical region, and
boundaries of the areas in which they are nonzero are yet to be determined.
In
particular, we see that the absorptive part $A_1$ has the same analytic
properties as a function of the momentum transfer [for $s$ constant and
grater
than $(M + \mu)^2$] as the scattering amplitude, except that there is now no
pole, and the cuts only extend from $t_1$ to $\infty$ and from $- \infty$ to
$t_
2$. According to the inequalities (2.14), these cuts do not reach as far
inward
as the cuts of $A$ considered as a function of the momentum transfer. This
agrees with another result of Lehmann$^{14}$ who showed that the region of
analyticity of $A_1$ as a function of $t$ was larger than the region of
analyticity of $A$ as a function of $t$.
The modifications introduced into the theory by spin and isotopic spin are
trivial. The transition amplitude will now be given by the expression
$$
- A + \frac{1}{2} \cdot i \gamma \cdot (q_1 + q_2) B, \eqno(2.20)
$$
and both $A$ and $B$ will have representations of the form (2.12). There
will,
further, be two amplitudes corresponding to isotopic spins of
$^1/_2$ and $^3/_2$. It is sometimes more convenient to use the combinations
$$
A^{(+)} = \frac{1}{3} \cdot \left( A^{(^1/_2)} + 2A^{(^3/_2)} \right),
\eqno(2.21a)
$$
$$
A^{(-)} = \frac{1}{3} \cdot \left( A^{(^1/_2)} - A^{(^3/_2)} \right)
\eqno(2.21b)
$$
and similar combinations $B^{(+)}$ and $B^{(-)}$. We then have the simple
crossing relations
$$
A^{(\pm)}(\nu,\,t) = \pm A^{(\pm)} \cdot (- \nu, \,t), \eqno(2.22a)
$$
$$
B^{(\pm)}(\nu,\,t) = \pm B^{(\pm)} \cdot (- \nu, \,t), \eqno(2.22b)
$$
or, in terms of the spectral functions,
$$
A^{(\pm)}_{1\,3} (s,~t) = \pm A^{(+)}_{23} (s_c,~t), \eqno(2.23a),
$$
$$
A^{(\pm)}_{1\,2} (s,~s_c) = \pm A^{(\pm)}_{12} (s_c,~s), \eqno(2.23b),
$$
$$
B^{(\pm)}_{1\,3} (s,~t) = \mp B^{(\pm)}_{23} (s_c,~t), \eqno(2.23c),
$$
$$
B^{(\pm)}_{1\,2} (s,~s_c) = \mp B^{(\pm)}_{12} (s_c,~s), \eqno(2.23d).
$$
The poles in (2.12) and in the dispersion relations will only occur in the
representation for $B^{(\pm)}$ (in pseudoscalar theory), and the second term
will have a minus or plus sign in the equations for $B^{(+)}$ and $B^{(-)}$,
respectively.
\vspace{0.5cm}
\section*{
{\bf 3. Combination of the Dispersion Relations
with the Unitarity Condition}}
\vspace{0.5cm}
The dispersion relations given in the previous section must now be combined
with
the unitarity equations in order to determine the scattering amplitude. We
shall
again begin by neglecting spin and isospin; the unitarity condition (2.7)
then
becomes, in the \mbox{one-meson} approximation,
$$
A_1 \cdot (s,~\cos \Theta_1) = \frac{1}{32 \pi^2} \cdot \frac{q}{W} \times
$$
$$
\times \int~ \sin \Theta_2~ d \Theta_2 ~d \phi_2 A^{\ast}\left( s,~
\cos \Theta_2 \right) A \cdot \left( s, ~
\cos(\overrightarrow{ \Theta_1},~\overrightarrow{\Theta_2}) \right),
$$
or
$$
A_1(s,~z_1) = \frac{1}{32 \pi^2} \cdot \frac{q}{W} \times
$$
$$
\times \int \limits^{+ 1}_{- 1}~ d z_2~ \int \limits^{2 \pi}_0~ d
\phi A^{\ast}
(s,~z_2)A(s,~z_1 z_2 +(1 - z^2_1)^{^1/_2}(1 - z'^2_2)^{^1/_2} \cos \phi),
\eqno(3.1)
$$
Where $z = \cos \Theta$ and $\overrightarrow{\Theta_i}(i = 1,~2)$ is unit
vector
in the $(\Theta_i,~ \phi_i)$ direction. W is the \mbox{center-of-mass}
energy
(equal to $\sqrt{s}$), and $q$ is the momentum in the \mbox{center-of-mass}
system, given by the equation
$$
q^2 = \left\{ s - (M + \mu)^2 \right\} \left\{ s - (M - \mu)^2 \right\}
/4 s. \eqno(3.2)
$$
$z$ is related to the momentum transfer by the simple relation
$$
z = 1 + (t/2 q^2). \eqno(3.3)
$$
The unitarity requirements only prove that Eq. (3.2) is true in the physical
region. $A_1$ must then
be obtained in the unphysical region. $A_1$ must then be obtained in the
unphysical region by analytic continuation. In order to do this, $A$ can be
expressed as an analytic function of $t$ or, equivalently, of $z$, by means
of
Eq. (2.11), in which the energy is kept fixed. Equation (3.3) shows that we
can
simply replace $t$ by $z$ in (2.12), so that we may write
$$
A^{\ast}(s,~z_2) = \frac{1}{\pi} \int~dz'_2~ \cdot \frac{A_2^{\ast} (s,~z'_2) +
A^{\ast}_3 (s,~z'_2)}{z'_2 - z_2}, \eqno(3.4a)
$$
$$
A \left\{s,~z_1 z_2 + (1 - z^2_1)^{^1/_2} (1 - z^2_2)^{^1/_2} \cos
\phi \right\} =
$$
$$
= \frac{1}{\pi}~\int ~d z'_3~ \frac{A_2(s,~z'_3) + A_3 (s,~ z'_3)}{z'_3 -
z_1 z_2
- (1 - z^2_1)^{^1/_2} (1 - z^2_2)^{^1/_2} \cos \phi}. \eqno(3.4b)
$$
For simplicity we have included the absorptive parts $A_2$ and $A_3$ under
the
same integral sign, but they will of course contribute in different regions
of
the variable of integration. $A_2(s,~z)$ will be nonzero only if $z<1-
\left\{s
- (M - \mu)^2 \right\}/2q^2$, apart from a $\delta$ function at $z = 1 - (s
- M^
2 - 2 \mu^2)/2q^2$, and $A_3(s,~z)$ will be nonzero only if $z > 1 + 2
\mu^2/q^
2$. The dispersion relations have been written down on the (incorrect)
assumption that there are no subtractions necessary; we shall in the
following
section how the theory must modified to take them into account.
On substituting (3.4) into (3.2) and performing the integrations over $z_2$
and
$\phi$, we are left with the equation
$$
A_1(s,~z_1) = \frac{1}{16 \pi^3} \cdot \frac{q}{W} ~\int~ dz'_2
~\int~dz'_3~ \frac{1}{\sqrt{k}} \cdot \ln~
\frac{z_1 - z'_2 z'_3 + \sqrt{k}}{z_1 - z'_2 z'_3 - \sqrt{k}} \times
$$
$$
\times \left\{ A^{\ast}_2(s,~z'_2) + A^{\ast}_3 (s,~z'_2) \right\} \left\{
A_2(
s,~z'_3) + A_3(s,~z'_3) \right\} \eqno(3.5)
$$
where
$$
k = z^2_1 + z^{\prime 2}_2 + z^{\prime 2}_3 - 1 - 2 z_1 z'_2 z'_3.
\eqno(3.6)
$$
We must take that branch of the logarithm which is real in the physical
region
$-1 < z_1 < 1$. Equation (3.5) then gives the value of $A_1$ in the entire
complex $z_1$ plane.
According to Eq. (2.16), $A(s,~z_1)$ must be an analytic function of $t$,
and
therefore of $z$, with discontinuities of magnitude $A_{13}$ and $A_{12}$ as
$z_
1$ crosses the positive and negative real axes. It is easily seen that the
expression for $A_1$ in (3.5) has this property, and, of identifying the
discontinuities along the real axis with $A_{13}$ and $A_{12}$, we arrive at
the
equations
$$
A_{13}(s,~z_1) = \frac{1}{8 \pi^2} \cdot
\frac{q}{W} ~ \int~ dz_2 ~\int~dz_3 K_1(z_1,
~z_2, ~z_3) \times
$$
$$
\times \left\{ A^{\ast}_3 (s,~z_2) A_3 (s,~z_3) + A^{\ast}_2 (s,~z_2) A_2
(s,~z_
3) \right\}, \eqno(3.7a)
$$
$$
A_{12}(s,~z_1) = \frac{1}{8 \pi^2} \cdot \frac{q}{W}~ \int~dz_2~
\int ~ dz_3 K_2(z_1,~z_2,~z_3)
\times
$$
$$
\times \left\{ A^{\ast}_2(s,~z_2) A_3 (s,~z_3) + A^{\ast}_3(s,~z_2)
A_2(s,~z_3)
\right\}. \eqno(3.7b)
$$
The primes on $z_2$ and $z_3$ have been suppressed. $K_1$ and $K_2$ are
defined
by the equations
$$
K_1(z_1,~z_2,~z_3) =
$$
$$
\left\{
\begin{array}{cll}
- 1/[k(z_1,~z_2,~z_3)]^{^1/_2},&
z_1 > z_2 z_3 + (z^2_2 - 1)^{^1/_2} \cdot (z^2_3 - 1)^{^1/_2},\\
0,& z_1 < z_2z_3 + (z^2_2 - 1)^{^1/_2} \cdot (z^2_3 - 1)^{^1/_2};
\end{array}
\right. \eqno(3.8a)
$$
$$
K_2(z_1,~z_2,~z_3) =
$$
$$
\left\{
\begin{array}{cll}
- 1/[k(z_1,~z_2,~z_3)]^{^1/_2},&
z_1 < z_2 z_3 + (z^2_2 - 1)^{^1/_2} \cdot (z^2_3 - 1)^{^1/_2},\\
0,& z_1 > z_2z_3 + (z^2_2 - 1)^{^1/_2} \cdot (z^2_3 - 1)^{^1/_2};
\end{array}
\right. \eqno(3.8b)
$$
The points $z_1 = z_2 z_3 \pm (z^2_2 - 1)^{^1/_2}$ are the points at which
$k$
changes sign.
Let us now transform back from $z$ to our original variables. As we shall
use
the dispersion relations (2.17) and (2.19), it is convenient to express
$A_2$
and $A_{12}$ as functions of $s$ and $s_c$ and $A_3$ and $A_{13}$ as
function of
$s$ and $t$. Equations (3.7) then become
$$
A_{13}(s,~t_1) =
$$
$$
= \frac{1}{32 \pi^2 q^3 W} \cdot \left[ \int~dt_2~\int~ dt_3 K_1
(s;~t_1,~t_2,~t_3)
A^{\ast}_3(s,~t_2) A_3(s,~t_3) + \right.
$$
$$
\left.
+ \int~ds_{c2} ~\int~ds_{c3} K_1 \cdot (s;~ t_1,~s_{c2},~ s_{c3}) A^{\ast}_2(s,~
s_{c2})A_2(s,~s_{c3})\right], \eqno(3.9a)
$$
$$
A_{12}(s,~s_{c1}) = \frac{1}{32 \pi^2 q^3 W} ~\int~dt_2~\int~ds_{c3} \cdot
K_2(s;~ s_{c1},~t_2,~s_{c3}) \times
$$
$$
\times \left[A^{\ast}_3(s,~t_2)A_2(s,~s_{c3}) + A^{\ast}_2(s,~s_{c3})
A_3(s,~t_
2) \right]. \eqno(3.9b)
$$
Note that $s$ is fixed in these equations, while $s_c$ and $t$ vary. $K$
must be
\mbox{re-expressed} as a function of the new variables by (3.3) and (2.13).
The use of Eq. (3.9), together with the dispersion relations, in order to
determine the spectral functions is greatly facilitated by the fact that $K$
is
zero unless the variables satisfy certain inequalities; for all $z$,
$$
\begin{array}{lllll}
K_1(s;~t_1,~t_2,~t_3)& = 0& \qquad \mbox{unless} \qquad& t^{^1/_2}_1 >
t^{^1/_2}_2 + t^{^1/_2}_3,&
~~~(3.10a)\\
K_1(s;~t_1,~s_{c2},~s_{c3})& = 0& \qquad \mbox{unless} \qquad
&t^{^1/_2}_1 > s^{^1/_2}_{c2} + s^{^1/_2}_{c3},&
~~~(3.10b)\\
K_1(s;~s_{c1},~t_2,~s_{c3})& = 0 &\qquad \mbox{unless} \qquad &
s^{^1/_2}_{c1} > t^{^1/_2}_2 + s^{^1/_2}_{c3},&
~~~(3.10â)\\
\end{array}
$$
(For any particular $s$, the restrictions on the variables could be
strengthened.) Equations (3.10) are true as long as $s_{c2}, s_{c3}, t_2$
and $t_3$ in the regions $s_c > M^2$, $t > 4\mu^2$, outside which $A_2$ and
$A_
3$ vanish. It follows from (3.9)
that, {\it for any given value of $t$ (or $s_c$)
$A_{13}(s, ~t)$} [or $A_{12}(s,~s_c$)]
{\it can be calculated in terms of
$A_3(s,~t')$ and $A_2(s,~s'_c)$, where the values of $t'$
and $s'_c$ involved are all less than
$t$ (or $s_c$).} On the other hand, by writing the dispersion relations
(2.17)
and (2.19) in the form
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
A_2(s,~s_c) = \frac{1}{\pi}~\int \limits^{\infty}_{s_2(s_c)}~
ds'~\frac{A_{12}(
s,~s_c)}{s' - s} + \frac{1}{\pi}~
\int \limits^{\infty}_{t_1(s_c)} dt'~ \frac{A_{23}(s_c,~t')}{t' - t},
\eqno(3.11a)
$$
$$
A_3(s,~t) = \frac{1}{\pi}~\int \limits^{\infty}_{s_3(t)}~ ds'~ \frac{A_{13}(
s',~t)}{s' - s} + \frac{1}{\pi}~ \int \limits^{\infty}_{s_3(t)} ds'_c~
\frac{A_{23}(s'_c,~t)}{s' - s_c}, \eqno(3.11b)
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
it is evident that $A_3(s,~t)$ and $A_2(s,s_c)$ can be found in terms of
$A_{
12}(s',~s_c)$ and $A_{13}(s', t)$, if for the moment we neglect the second
term
in these equations. We can therefore calculate $A_{13}$, $A_{12}$, $A_3$,
and $
A_2$ for all values of $s$ and successively longer values of $s_c$ and $t$.
The
lowest value of $s_c$ or $t$ for which either $A_2$ and $A_{12}$ in nonzero
is $
s_c = M^2$, at which there is a contribution of $g^2 \delta(s_c - M^2)$ to
$A_2$
from the \mbox{one-nucleon} state. From (3.9) and (3.10) it follows that
$A_{
13}$ and $A_{12}$ are zero if $t$ and $s_c$ are less than $4M^2$; for a
range of
values of $t$ above this, $A_{13}$ is nonzero and can be calculated by
inserting
the \mbox{$\delta$-function} contribution to $A_2$ into (3.9a). The rest of
$A_
2$ and $A_3$ will still not contribute owing to (3.10). Once we have the
procedure thus started, we can proceed to larger and larger values of $t$
and $
s_c$ by alternate application of (3.9) and (3.11).\footnote{It will be
noticed
that, though we have brought the pole in the crossing term from the
\mbox{one-
nucleon} intermediate state into our calculations, we have not yet
introduced
the pole in the direct term. This pole is actually
a subtraction term of Eq. (2.11) and will be treated in the following
section.}
Before discussing how to take the second terms of (3.11) into account, let
us
study in more detail the form of the functions $A_{13}$ and $A_{12}$
calculated
thus far. In order to do this require the precise values of $t$ and $s_c$,
at a
given value of $s$, for which the kernels $K$ vanish; we find that
%\vspace{1.5cm}
%\includegraphics{http://dbserv.ihep.su/images/mandelstam2a.gif}
%\\Fig. 2\\
%\vspace{1.5cm}
\begin{figure}[h]
\centerline{\epsfig{file=fig2r.eps}}
\caption{Properties of the spectral functions.}
\end{figure}
$$
K_1(s,~t_1,~t_2,~t_3) = 0
$$
$$
\mbox{unless} ~~~~~~ t^{^1/_2}_1 > t^{^1/_2}_2 \left( 1 + t_3/4q^2
\right)^{^1/_2} + t^{^1/_2}_3 \left( 1 + t_2/4q^2 \right)^{^1/_2},
\eqno(3.12a)
$$
$$
K_1(s,~t_1,~s_{c2},~s_{c3}) = 0
$$
$$
\mbox{unless} \quad t^{^1/_2}_1 >(s_{c2},~ - u)^{^1/_2} \left\{1 + (s_{c3} -
u)/4q^2 \right\}^{^1/_2} +
$$
$$
+ (s_{c3} - u)^{^1/_2} \left\{1 + (s_{c2} - u)/4q^2 \right\}^{^1/_2},
\eqno(3.
12b)
$$
$$
K_2(s;~ s_{c1},~t_2,~s_{c3}) = 0
$$
$$
\mbox{unless} \quad (s_1 - u)^{^1/_2} > t^{^1/_2}_2 \left( 1 + (s_{c3} -
u)/4q^2 \right\}^{^1/_2} +
$$
$$
+ (s_{c3} - u)^{^1/_2} \left( 1 + t_2/4q^2 \right\}^{^1/_2}, \eqno(3.12â)
$$
where
$$
u = (M^2 - \mu^2)^2/s. \eqno(3.13)
$$
As the smallest value of $s_c$ or $t$ which contributes to the integrand in
Eq.
(3.9a) is $s_c = M^2$, where $A_2$ has a \mbox{$\delta$-function}
singularity,
it follows from (3.12b) that the smallest value of $t$ for which
$A_{13}(s,~t)$
in nonzero (for any given value of $s$) is given by
$$
t^{^1/_2} = 2(M^2 - u)^{^1/_2} \left\{ 1 + (M^2 - u)/4q^2 \right\}^{^1/_2}.
\eqno(3.14)
$$
For very large $s$, this value of $t$ approaches $4\mu^2$, but, as $s$
decreases, $t$ becomes infinite. Equation (3.14) has been plotted as $C_1$
in
Fig. 2. $A_{13}$ will be nonzero above $C_1$, and near it, it will behave
like
$(t - t_0)^{-^1/_2}$, where $t_0$ is the value of $t$ given by (3.14). It
follows from (3.11b) that $A_3(s,~t)$ is nonzero if $t > 4 M^2$, and
behaves
like $(t - 4M^2)^{^1/_2}$ just above this limit. The value $t = 4M^2$ is
precisely the threshold for the process III, and we would have obtained the
same
results from our general reasoning in the previous section if we had
neglected
intermediate states containing two or more mesons but no nucleon pairs. This
indicated that our assumptions are probably correct, as we have not
considered
the process III explicitly in this section. when we treat the subtraction
terms
in the dispersion relations, we shall see that $A_{13}$ is also nonzero
between
$t = 4 \mu^2$ and $t = 4M^2$, and that the region in which $A_{13}$ is
nonzero
must be enlarged. The curve $C_1$ is therefore not yet the curve $C_{13}$ of
Fig. 1.
For a range of values of $t$ above the curve $C_1$, the entire contribution
to
the integrand in (3.9a) comes from the $\delta$ function in $A_2$. At a
certain
point, however, the other terms in $A_2$ and $A_3$ begin to contribute. If
for
the moment we neglect the second term in (3.9a), the new contribution begins
at
the value of $t$ obtained by putting $t_2 = t_3 = 4M^2$ in (3.12a), since
this (
at the present stage of the calculation) the lowest value of $t$ for which
$A_3$
is nonzero. The result has been plotted against $s$ in Fig. 2 to give the
curve
$C_2$. As this curve approaches the line $t = 16M^2$ asymptotically, there
will
be a corresponding new contribution to $A_3$ above this value, and, near it,
the
new contribution will behave like $(t - 16M^2)^{^1/_2}$ is just the
threshold
for the production of an additional nucleon pair in the process III, and
$A_3$
would be expected to show such a behaviour at this threshold.
We find similar discontinuities in the higher derivatives of $A_{13}$ at
series
of curve (there will now be more than one for each threshold) approaching
asymptotically the lines $t = 4n^2 M^2$, so that $A_3$ will have the
expected
behavior at the thresholds for producing $n$ nucleon pairs.
The functions
$A_{12}$ and $A_2$ will exhibit the same sort of characteristics. In Eq.
(3.9b),
the lowest values of $t_2$ and $s_{c3}$ which contribute to the integrand
are
$t_2 = 4M^2$, $s_{c3} = M^2$, so that the boundary of the region in which
$A_{
12}$ is nonzero is obtained by inserting these values into (3.12c). The
result
is represented by the curve $C_3$ in Fig. 2; it approaches the line $s_c =
9M^2$
as $s$ tends to infinity. As with $A_{13}$ , the region in which $A_{12}$ is
nonzero will be widened in the following section. From (3.19a), it follows
that
$A_2$ will (at present) be nonzero for $s_c > 9M^2$, which is the threshold
for
pair production in the reaction II. $A_{12}$ will also have discontinuities
in
the higher derivatives at series of curves such as $C_4$ which approach
asymptotically the lines $s_c = (2n + 1)^2 M^2$. Finally, it can be seen
that
the second term of (3.9a) will give rise to further curves at which the
higher
derivatives of $A_{13}$ are discontinuous, but these curves will all
approach
asymptotically the lines $t = 4n^2 M^2$.
We must now return to the second term in the Eq. (3.11), which we have so
far
neglect in the calculation. It can be taken into account by introducing the
requirement of crossing symmetry, which has not yet been used. As in the
static
theory, one now has to use an iteration procedure. The function $A_{23}$
which
only affect the crossing term in the dispersion relation (2.5), it first
neglect, and the calculation done as described. $A_{23}$ is then found from
the
calculated value of $A_{13}$ and the \mbox{crossing-symmetry} relations
(2.23),
and inserted into Eq. (3.11) for the next iteration. However, the scattering
amplitude calculated by this procedure would still not satisfy the equations
of
crossing symmetry since, while $A_{13}$ and $A_{23}$ are connected by
(2.23a), $
A_{12}$ does not satisfy (2.23b). We have seen that the dispersion relations
together with the equation of unitarity determine $A_{12}$ uniquely, and the
result is not a symmetric function of $s$ and $s_c$; even the region in
which it
is nonzero is not symmetric. It therefore appears that we cannot satisfy
simultaneously the requirements of analyticity, unitarity (in the
\mbox{one-meson} approximation), and crossing symmetry.
The reason why this is so is easily seen in perturbation theory. Among the
graphs included in the first iteration of the \mbox{one-meson} approximation
is
Fig. 3(a). The topologically similar graph Fig. 3(b) will also be included,
since Fig. 3(a) by itself would have square roots in the energy denominators
and
would not have the necessary analytic properties. If, therefore, crossing
symmetry is to be maintained Fig. 3(c) must also be included. It this graph,
however, there is an intermediate state of a nucleon and a pair, so that the
unitarity condition in the \mbox{one-meson} approximation is not satisfied.
This example also indicates how we should modify our iteration procedure. In
addition to inserting a term $A_{23}$, obtained by crossing symmetry from
the
previous iteration, into (3.11), we must insert a term $A'_{12}(s,~s_c)$
equal
to $A_{12}(s_c,~s)$ as calculated in the previous iteration. The
contribution
from this term is to be added to the contribution from $A_{12}(s,~s_c)$
calculated in the normal way. $A'_{12}$ will be nonzero above the curve
$C_5$ in
Fig. 2, and, in particular, it will be zero for all values of $s_c$ is $s$
is
less than $9M^2$. Complete crossing symmetry is now maintained, but the
addition
of $A_{12}$ violated the unitarity condition (in the \mbox{one-meson}
approximation) for values of $s$ greater than $9M^2$, and a perturbation
expansion would include graphs such as Fig. 3(c). As these graphs will
appear in
higher approximations, the fact that we are forced to include them here
should
not be considered a disadvantage of our method. In any case, the unitarity
condition is only violated where the \mbox{one-meson} approximation is far
from
correct.
The iteration procedure is found to give rise to further curves, like $C_2$
and
$C_4$ (Fig.2), at which the higher derivatives of the spectral function are
discontinuous. These new discontinuities correspond to the production of
mesons
together with nucleon pairs. We still do not have discontinuities at all
possible threshold.
The inclusion of the spin does not change any of the essential features of
the
theory, though the details are
%\vspace{1.5cm}
%\includegraphics{http://dbserv.ihep.su/images/mandelstam3a.gif}
%\\Fig. 3\\
\begin{figure}[h]
\centerline{\epsfig{file=fig3r.eps}}
\caption{Graphs which bring in intermediate states with pairs.}
\end{figure}
%\vspace{1.5cm}
rather more complicated. Following Chew et.al.,$^1$ we write the \mbox{pion-
nucleon} $T$ matrix in the form
$$
T = - \frac{2 \pi W}{E w} \cdot \left( a + \overrightarrow{\sigma}
\cdot \overrightarrow{q_2}
\overrightarrow{\sigma} \cdot \overrightarrow{q_1} b \right), \eqno(3.15)
$$
where $E$ is the \mbox{center-of-mass} energy of the nucleon and $w$ that of
the
pion. $a$ and $b$ are related to the quantities $A$ and $B$ in the
expression (
2.20) by the formulas
$$
a = \frac{E + M}{2W} \cdot \left( \frac{A + (W - M) B}{4 \pi} \right),
\eqno(3.16a)
$$
$$
b = \frac{E - M}{2W} \cdot \left( \frac{- A + (W + M) B}{4 \pi} \right).
\eqno(3.16b)
$$
The unitarity condition corresponding to (3.7) can now be worked out in
terms of
$a$ and $b$; equation obtained is
$$
a_{13(12)}(s,~z_1) = \sum_a \frac{q}{\pi} \int ~dz_2 ~\int~dz_3 K_{1(2)}
(z_1,~z_2,~z_3) \times
$$
$$
\times \left\{ a^{\ast}_a(s,~z_2) a_a(s,~z_3) + \frac{z_2 - z_3 z_1}{1 -
z^2_1} \cdot
b^{\ast}_a(s,~z_2) a_a(s,~z_3) + \right.
$$
$$
\left. + \frac{z_3 - z_2z_1}{1 - z^2_1} \cdot a^{\ast}_a(s,~z_2) b_a(s,~z_3)
\right\}
, \eqno(3.17a)
$$
$$
b_{13(12)}(s,~z_1) = \sum_a \frac{q}{\pi}~ \int~dz_2~\int~dz_3
K_{1(2)}(z_1,~z_
2,~z_3) \times
$$
$$
\times \left\{ \frac{z_3 - z_2z_1}{1 - z^2_1} \cdot
b^{\ast}_a(s,~z_2)a_a(s,~z_3)
+
\frac{z_2 - z_3z_1}{1 - z^2_1} \times \right.
$$
$$
\left. \times a^{\ast}_a(s,~z_2)b_a(s,~z_3) + b^{\ast}_a(s,~z_2)b_a(s,~z_3)
\right\},
\eqno(3.17b)
$$
where $\sum_a$ indicated that terms of the form $a^{\ast}_a a_a$ are to be
replaced by $a^{\ast}_2 a_2 + a^{\ast}_3 a_3$ in the calculation of $a_{13}$
and
$b_{13}$ and by $a^{\ast}_2 a_3 + a^{\ast}_3 a_2$ in the calculation of
$a_{12}$
and $b_{12}$, and $a_{13}$ and $b_{13}$ are related respectively to $A_2$
and
$B_2$, $A_3$ and $B_3$, $A_{12}$ and $B_{12}$, and $A_{13}$ and $B_{13}$ by
Eqs.
(3.16). The unitarity condition (3.17) can be rewritten in terms of $A$ and
$B$
it then becomes
$$
A_{13(12)}(s,~z_1) = \sum_a \frac{q}{4 \pi^2W}~\int~dz_2~\int~dz_3
K_{1(2)}(z_1,
~z_2,~z_3) \times
$$
$$
\times \left\{ \left( 1 - \frac{w}{2W} \cdot
\frac{1 - z_2 - z_3 + z_1}{1 + z_1}
\right) A^{\ast}_a(s,~z_2) A_a(s,~z_3) + \right.
$$
$$
+ \left( \frac{\omega}{2} \cdot
\frac{1 - z_2 + z_3 - z_1}{1 - z_1} + \frac{M_w}{2W} \cdot
\frac{1 - z_2 - z_3 + z_1}{ 1 + z_1} \right) A^{\ast}_a(s,~z_2) B_a(s,~z_3)
+
$$
$$
+ \left( \frac{\omega}{2} \cdot
\frac{1 + z_2 - z_3 - z_1}{1 - z_1} + \frac{M_w}{2W} \cdot
\frac{1 - z_2 - z_3 + z_1}{ 1 + z_1} \right) B^{\ast}_a(s,~z_2) B_a(s,~z_3)
+
$$
$$
\left. + \frac{W^2 - M^2}{2W} \cdot \frac{1 - z_2 - z_3 + z_1}{1 + z_1}
~B^{\ast}_a(s,~z_2) B_a(s,~z_3) \right\}, \eqno(3.18a)
$$
$$
B_{13(12)}(s,~z_1) = \sum_a ~\frac{q}{4 \pi^2W}~\int~dz_2~\int~dz_3
K_{1(2)}(z_1,
~z_2,~z_3) \times
$$
$$
\times \left\{ \frac{E}{2MW} \cdot \frac{1 - z_2 - z_3 + z_1}{1 + z_1}
A^{\ast}_a(s,~z_2) A_a(s,~z_3) + \right.
$$
$$
+ \left( \frac{1 - z_2 + z_3 - z_1}{1 - z_1} + \frac{E}{2W} \cdot
\frac{1 - z_2 - z_3 + z_1}{ 1 + z_1} \right) A^{\ast}_a(s,~z_2) B_a(s,~z_3)
+
$$
$$
+ \left( \frac{1 - z_2 + z_3 - z_1}{1 - z_1} + \frac{E}{2W} \cdot
\frac{1 - z_2 - z_3 + z_1}{ 1 + z_1} \right) B^{\ast}_a(s,~z_2) A_a(s,~z_3)
+
$$
$$
\left. \left( \omega - \frac{(w^2 - M^2)E}{2 MW} \cdot
\frac{1 - z_2 - z_3 + z_1}{ 1 + z_1}
\right) B^{\ast}_a(s,z_2)B_a(s,~z_3) \right\}. \eqno(3.18b)
$$
Equations (3.17) and (3.18) will hold separately for the amplitudes
corresponding to isotopic spin $^1/_2$ and $^3/_2$.
It remains to justify the claim that the result calculated by our procedure,
if
expanded in a perturbation series, would give a subset of the usual
perturbation
series. The proof is somewhat awkward because we were unable to satisfy the
unitarity condition in the \mbox{one-neson} approximation at all values of
the
energy.
Let us first ignore this. The $n$th term in the perturbation series
$A^{(n)}$ is
then determined uniquely in the physical region by the following two
requirements:
(i) For sufficiently small values of the momentum transfer
(less than $2 \mu
[\frac{2}{3}(2M + \mu)/(2M - \mu)]^{1/2}$), $A^{(n)}$ must satisfy
the dispersion relation (2.5), a result which has been proved
rigorously.$^{14}$
The absorptive part $A_1$ (and hence, by crossing symmetry, $A_2$) is known,
since it is determined by unitarity in terms of lower order perturbation
terms
in the physical region, and by analytic continuation (with $s$ constant)
outside
it.$^{14}$
(ii) For a fixed value of $s, A^{(n)}$ is an analytic function of the
momentum
transfer throughout the physical region.$^{14}$
As the function calculated by our method certainly fulfil these
requirements,
they must generate the correct perturbation series.
However, our result does not satisfy the unitarity condition in the
\mbox{one-
meson} approximation at all energies, and we must examine more closely how
$A_1$
is to be determined. Let us assume that our method gives the correct
perturbation series up to the ($n - 1$)th order. The reasoning developed in
this
section then shows that the \mbox{$n$th-order} contribution to $A_1$ will be
off
the form
$$
A^{(n)} = \frac{1}{\pi}~ \int~dt'~ \frac{A^{(n)}_{13}(s,~t')}{t' - t} -
\frac{1}{\pi} ~\int~dt'~\frac{A^{(n)}_{12}(s,~t')}{t' - t}, \eqno(3.19)
$$
where $A_{13}^{(n)}$ and $A_{12}^{(n)}$ are certainly zero below $C_1$ and
above
$C_3$, respectively, in Fig. 2. Inserting this expression into (2.5), we
find
that
$$
A_d^{(n)} = \frac{1}{\pi^2}~ \int~ds'~\int~dt'~
\frac{A^{(n)}_{13}(s',~t')}{(s' -
s) (t' - t)} -
$$
$$
- \frac{1}{\pi^2} ~\int~ds'~\int~dt'~ \frac{A^{(n)}_{12}(s',~t')}{(s' -
s)(t' -
t)}. \eqno(3.20)
$$
The suffix $d$ indicated that we are considering the direct and not the
crossing
term. The second term of (3.20) will not be an analytic function of $t$ in
the
physical region, but it will have a branch point at the largest value of $t$
for
which $A_{12}$ nonzero. We can make it analytic by adding to $A_2$ the
expression
$$
- \frac{1}{\pi}~\int~dt'~\frac{A^{(n)}_{12}(s_c,~t')}{t' - t},
\eqno(3.21)
$$
which we would expect from (2.17), if our representation is correct. By
inserting this into (2.5) and adding the result to the second term of
(3.20), we
obtain
$$
\frac{1}{\pi^2}~ \int~ds'~ \int~ ds'_c \frac{A^{(n)}_{12}(s',~s'_c)}{(s' -
s)(
s'_c - s)},
\eqno(3.22)
$$
which is analytic in the physical region. The contribution (3.21) to
$A_2^{(n)}$
is uniquely determined from the requirement that $A^{(n)}$ be an analytic
function of the momentum transfer in the physical region, and is nonzero
only
for $s_c > 9M^2$. It corresponds to adding a graph such as Fig. 3(b) to Fig.
3(
a); as $A_1$ for Fig. 3(c) is nonzero for $s > 9M^2$, $A_2$ for Fig. 3(b)
will
be nonzero for $s_c > 9 M^2$.
Finally, then, the \mbox{$n$th-order} perturbation term can be determined
from
the lower order perturbation terms without using any unproved properties of
the
scattering amplitude as follows:
(i) Calculate $A_1$ by unitarity, and extend it into the nonphysical region
for
momentum transfers less than $2\mu[ \frac{2}{3}(2M + \mu)/(2M _
\mu)]^{^1/_2}$
by analytic continuation.
(ii) Calculate a contribution $A_{2d}^{(n)}$ to $A_2^{(n)}$, for $s_c >
9M^2$,
from the requirement that if it, together with $A_1$ be inserted into (2.5),
the
resulting function $A_d^{(n)}$ must be an analytic function of the momentum
transfer in the physical region. By doing this we partially include
intermediate
states with nucleon pairs, which is necessary if we are to maintain the
required
analytic properties and crossing symmetry.
(iii) Now calculate $A_2^{(n)}$ and the extra contribution to $A_1^{(n)}$ by
crossing symmetry from $A_1^{(n)}$ and the extra contribution to
$A_2^{(n)}$.
(iv) Find $A^{(n)}$ from (2.5) for values of the momentum transfer less than
$2
\mu[\frac{2}{3}(2M + \mu)/(2M - \mu)]^{^1/_2}$, and calculate it in the rest
of
the physical region by analytic contribution in $t$.
This procedure defines a \mbox{one-meson} approximation in perturbation
theory.
>From what has been said, it is clear that our solution will precisely this
perturbation expansion, so that our assumptions are justified in
perturbation
theory.
\vspace{0.5cm}
\section*{
{\bf 4. Subtraction Terms in the
Dispersion Relations}}
\vspace{0.5cm}
We have thus far assumed that the dispersion relations are true without any
subtractions. As we have pointed out in the first section, by doing this we
neglect what is physically the most important part of the scattering
amplitude.
In this section we shall investigate how many subtractions are necessary for
each dispersion relation and shall outline how they can be calculated,
leaving
the details for a further paper.
Let us first consider Eqs. (2.11) and (2.16), which were used in obtaining
the
unitarity condition (3.9) [or (3.8) for nucleons with spin]. Even if these
dispersion relations are written with subtraction terms, it is found that
(3.9)
is unchanged, so that the subtraction terms are only needed in the final
evaluation of $A$ from $A_2$ and $A_3$ by means of (2.11), or of $A_1$ from
$A_{
12}$ and $A_{13}$ by means of (2.16). The number of subtractions will depend
on
the behavior of $A_{12}$, $A_{13}$, $A_2$ and $A_3$, as calculated by our
procedure, as $s_c$ and $t$ tend to infinity -- we shall have to perform at
least enough subtractions for (2.11) and (2.16) to converge.
It is difficult to make an estimate of the behavior of these functions at
infinity values of $s_c$ and $t$ from the equations determining them, and we
shall use indirect arguments which, though not rigorous, are very plausible.
We
shall find that, if the coupling constant is small enough, the functions
tend to
zero at infinity, so that one can write the dispersion relations without any
subtractions. For larger values of the coupling constant, more and more
subtractions will be needed. The reader who is prepared to accept this may
omit
the following two paragraphs.
We consider only the first iteration, since subsequent iterations proceed in
a
similar way and the results are unlikely to be qualitatively different. The
result can then be expanded in a perturbation series. If the solutions
obtained
for this problem by other methods, such as the \mbox{Tamm--Dankoff} or
\mbox{Bethe--Salpeter} methods, are expanded in a perturbation series.,
it is
found that the series for each angular momentum state converges as long as
the
coupling constant is within a certain radius of convergence, and that this
radius of a convergence tends to infinity with the angular
momentum.\footnote{
Note that the ``patential'' in the \mbox{Tamm--Dankoff} or \mbox{Bethe--
Salpeter} equation involved includes only the crossing term and not the
direct
term, which has still to be brought into the calculation.}
Our perturbation series would be different from the perturbation series
obtained
by these methods, partly because the intermediate states with paries which
we
include are not the same as those included by either of them, and partly
because, in calculating the subtraction terms (other than those at present
under
discussion), we shall not take into account terms corresponding to all
graphs
included by these approximations. Such differences would not be expected to
affect qualitatively the convergence properties of the angular momentum
states,
and we shall assume that the results quoted above are true for our
perturbation
series too.
The transition amplitude for the state of total angular momentum $j$ and
orbital
angular momentum $j \pm \frac{1}{2}$ can be shown to be
$$
f_{j\pm} = \int \limits^{+1}_{-1} ~dz~a(s,~z)P_{j \pm \frac{1}{2}}(z) +
\int \limits^{+1}_{-1}~dz~b(s,~z) P_{j\pm \frac{1}{2}}(z), \eqno(4.1)
$$
where $a$ and $b$ are functions defined in (3.15) and (3.16). Now it is
easily
seen that each term in the perturbation series for $a_2(s,~z),~ a_3(s,~z),
~b_2(
s,~z)$ and $b_3(s,~z)$ tends to zero like $1/z$ as $z$ tends infinity, so
that
the dispersion relation (2.11) for each term can be written down without any
subtractions. Hence
$$
f^{(n)}_{j\pm} = \int \limits^{+1}_{-1}~dz ~\int \limits~dz' \left\{
\frac{a_2^{(n)}(s,~z') + a_3^{(n)}(s,~z')}{z' - z} \cdot
P_{j \pm \frac{1}{2}}(z) + \right.
$$
$$
\left. + \frac{b_2^{(n)}(s,~z') + b_3^{(n)}(s,~z')}{z' - z} \cdot P_{j \pm
\frac{1}{
2}}(z) \right\},
\eqno(4.2)
$$
or
$$
f^{(n)}_{j \pm} = \int~dz'~\left\{ \left[a_2^{(n)}(s,~z') + a_3^{(n)}(s,~z')
\right] \phi_{j \pm \frac{1}{2}}(z') + \right.
$$
$$
\left. + \left[b^{(n)}_2(s,~z') + b^{(n)}_3(s,~z') \right] \phi_{j \pm
\frac{
1}{2}} (z') \right\},
\eqno(4.3)
$$
where
$$
\varphi_n(z') = \int \limits^{+1}_{-1}~ dz \frac{P_n(z)}{z' - z} \approx
1/z'^{n+1} \qquad \mbox{as} \qquad z' \rightarrow \infty.
\eqno(4.4)
$$
Let as suppose that the value of the coupling constant is such that the
perturbation series for states of angular momentum $j_1$ converges. If each
term
in the perturbation series for this angular momentum state is expressed by
(4.
3), and if we assume that we can interchange the order of summation and
integration, we arrive at the equation
$$
f_{j_1 \pm} = \int ~dz' \left\{ \sum_n \left[a^{(n)}_2(s,~z') +
a^{(n)}_3(s,~z')
\right] \phi_{j_1 \pm \frac{1}{2}}(z') +
\right.
$$
$$
\left. + \sum_n \left[b^{(n)}_2(s,~z') + b^{(n)}_3(s,~z') \right]
\phi_{j_1 \pm \frac{1}{2}} (z') \right\}.
\eqno(4.5)
$$
In order for the integrand to exist, we see from (4.4) that $a$ and $b$ must
be
smaller than $z$. The dispersion relations can therefore be written down
with
not more than $j - \frac{1}{2}$ subtractions. In particular if the coupling
constant is small enough the dispersion relations can be written down
without
any subtractions.\footnote{We should emphasize that it is only in the first
iteration that we relate the number of subtractions needed to the
convergence of
the angular momentum states. We say nothing at all about the convergence of
the
perturbation series in subsequent iterations, but assume simply that the
behavior of the spectral functions at infinite values of $z$ is not likely
to
be qualitatively different from their behavior in the first iteration.}
If the coupling constant is such that $n$ subtractions are required, the
unitarity condition for the states of angular momentum $\frac{1}{2}$ to
$n - \frac{1}{2}$ will have to be applied separately. The wave functions for
these states are polynomials of degree not greater than $n - 1$ in the
variable
$z$ (or $s_c$ and $t$), and are not determined from the absorptive parts in
the
dispersion relations (2.11) and (2.16).
The calculation must be done after each iteration, as the result will be
needed
for the next iteration. The details of the calculation will not be discussed
here, but they will in principle be similar to those of Chew and Low$^4$ and
Dalitz, Castillejo, and Dyson,$^5$, and will involve considering the
reciprocal
of the scattering amplitude. The analytic properties of the individual
angular
momentum states are not as simple as in the static theory, but they can be
determined from the assumed analytic properties of the transition amplitude,
and, as in the static theory, the singularities not on the positive real
axis
can be found from the previous iteration.
The precise number of subtractions required cannot be determined without
calculating the result, but it is almost certainly not less than two. It is
difficult to see how the observed resonant behavior of the $P_{\frac{3}{2}}$
state could be reproduced by means of the calculations described in the last
section, whereas it follows quite naturally from a \mbox{Chew-Low-type}
calculation. If the coupling constant were large enough to bind the (3,3)
resonance state, and for a certain range of values of the coupling constant
below this, we would definitely have to perform two subtractions. The
precise
range involved is difficult to determine, but it would be expected to
include
those values of the coupling constant for which the (3,3) state still has
the
appearance of an unstable isobar. Until we state otherwise, however, we
shall
suppose that the coupling constant is sufficiently small for the functions
$A(
s,~z)$ and $B(s,~z)$ to tend to zero at infinite $z$, as the situation with
regard to the other subtractions is much simpler in this case. Even then, we
would have to perform one subtraction for each of $A$ and, $B$, since the
calculations of the previous section did not include the pole of the
scattering
amplitude from the \mbox{one-nucleon} intermediate state: only the pole in
the
crossing term was included. The pole affects the states with
$j = \frac{1}{2}$ alone,
so that, if we apply the unitarity condition for these states separately by
the
\mbox{Chew-Low} method, we can include it correctly. We thereby change $A$
and $
B$ by a quantity independent of $z$.
When we calculate the scattering amplitudes for the states with
$j = \frac{1}{2}$ we
find a ghost state in the first iteration, just as in all other models. In
subsequent iterations, however, where the crossing terms contribute, it does
not
follow from the form of the equations that we shall necessary find a ghost
state, and, judging from the charged scalar model, we may hope that the
ghost
state does not in fact occur.
We now turn to consider the subtraction terms in the other dispersion
relations
used in the calculations, Eq. (3.11). By putting the
\mbox{$\delta$-function}
contribution to $A_2$ into (3.18), it can be seen that the lowest order term
in
$A_{13}(s,~t)$ tends to a constant as $s$ tends to infinity, whereas the
lowest
order term in $B_{13}(s,~t)$ behaves like $1/s$. For a certain range of
values
of $t$, only the lowest order term contributions to $A_{13}$ and $B_{13}$,
so
that there will certainly be one subtraction in Eq. (3.11b) for $A_3$, while
the
equation for $B_3$ could be written down without any subtractions. We find
similarly that both $A_{12}(s,~s_c)$ and $B_{12}(s,~s_c)$ tend to zero like
$1/s$
as $s$ tends to infinity. It would therefore appear that the dispersion
relations (3.11a) did not require any subtractions. However, we have seen
that $
A_1(s,~s_c)$ and $B_1(s,~s_c)$ behave like a constant for large $s_c$ with
$s$
constant, even for small values of the coupling constant, so that, by
crossing
symmetry, $A_2(s,~s_c)$ and $B_2(s,~s_c)$ will behave like a constant for
large
$s$. There will therefore be one subtraction term in Eqs. (3.11a) for
both
$A_2$ and $B_2$.
The determination of the subtraction terms in Eq. (3.11a) is not difficult,
since the contributions to $A_2$ and $B_2$ from the state with
$j = \frac{1}{2}$ (with
the energy $s_c$ of the reaction II kept constant) can be found by crossing
symmetry from the corresponding contributions to $A_1$ and $B_1$ in the
previous
iteration. However, for the subtraction terms in Eq. (3.11), we require the
unitarity condition for $A_3$ which involves the relation III. As there is
one
subtraction, only the $S$ waves will be involved. Again we have to limit the
intermediate states considered; in this first approximation we would
consider
the \mbox{two-meson} states (``\mbox{two-meson}'' approximation) and perhaps
the \mbox{nucleon-antinucleon} intermediate states (``\mbox{two-meson} plus
pair approximation'') as well. We shall then require the \mbox{meson-meson}
scattering amplitude (and the \mbox{nucleon-antinucleon} scattering
amplitude if
\mbox{nucleon-antinucleon} intermediate states are being considered). The
determination of these scattering amplitudes would be as extensive a
calculation
as the determination of the \mbox{pion-nucleon} scattering amplitude, but
neglect of the crossing term would probably not give to too great an error
in
our final result, in which case the \mbox{$S$-state} amplitudes could be
written
down immediately in the \mbox{two-meson} or \mbox{two-meson} plus pair
approximations. The \mbox{meson-meson} coupling constant is thereby
introduced
into the calculation, as has been mentioned in the introduction. Once the
\mbox{
meson-meson} and \mbox{nucleon-antinucleon} scattering amplitudes are
known,
the transition amplitude for the reaction III cab be calculated. Since the
integral equation is now linear, the details will be different from those of
the
\mbox{Chew-Low} calculations, but, as in their case, the solution could be
written down exactly if there were no other singularities of the transition
amplitude, and we can use as iteration procedure for the actual problem. The
iterations will again be interspersed between the iterations of the main
calculation. The \mbox{$S$-wave} portion of $A_3$, as calculated by this
procedure, will be nonzero for $t > 4 \mu^2$, so that the scattering
amplitude
now has the expected spectral properties. The boundaries of the regions in
which
the spectral functions are nonzero will thereby also be changed; this will
be
discussed in more detail at the end of the section.
We have seen that, as long as the coupling constant is sufficiently small,
we
require one subtraction for each of the dispersion relations except the
dispersion relation (3.11b) for $B_3$, for which we do not require any
subtractions. It is also easily seen that this behavior is
\mbox{consistent-the} functions as calculated in the last section. With
the calculations modified
by the subtraction terms, will not at any stage become too large at
infinity.
If, however, one were to make any additional subtractions, one would find
that,
on performing the calculations, one would need more subtractions as the work
progressed, and one could not obtain any final result. The number of
subtractions to be performed is therefore determined uniquely. There is one
exception to this statement: we could perform one subtraction in Eq. (3.11b)
for
$B_3$. Such a subtraction is, however, excluded by the requirement that the
theory remain consistent when the interaction with the electromagnetic field
is
introduced. If one were to make this subtraction, the scattering amplitude
would
behave like $f(t) \gamma (q_1 + q_2)$ for large values of $s$. If then
follows
from gauge invariance that the matrix element for the process
$$
\pi^{\pm} + n \rightarrow \pi^{\pm} + n + \gamma \qquad \mbox{or} \qquad
\pi^0
+ p \rightarrow \pi^0 + p + \gamma
$$
will contain a term which behave like $f(t)\gamma$ for large $s$,
where $t$ is now minus the square of the momentum transfer of the neutral
particle.\footnote{This can be shown by using a generalization of the Ward
identity due to H.S.Green, Proc. Phys. Soc. (London) 66, 873 (1953), and
T.D.Lee,
Phys.Rev. 95, 1329 (1954), and proved by Y.Takahashi, Nouvo cimento 6, 372 (
957).}
%\cite{Green1953}
%\cite{Lee1954}
%\cite{Takahashi1957}
The contribution to $B_1$ and $B_{13}$ from the $\pi - N - \gamma$
intermediate state therefore tends ti infinity at least as fast as $s$ for
infinite $s$, so that one would require two subtractions for the dispersion
relation in question and the theory would not be consistent.
Since the unitarity conditions for the two
$j = \frac{1}{2}$ states of the \mbox{pion-
nucleon} system, and for the $S$ state of the \mbox{pion-pion} system, have
to
be applied separately by the \mbox{Chew-Low} method, there will be \mbox{
Castillejo-Dalitz} ambiguities associated with these states. The ambiguities
will of course affect all states in subsequent iterations. They correspond
to
the existence of unstable baryons of spin
$\frac{1}{2}$ and either parity, or of heave
unstable mesons of spin zero. There are no ambiguities associated with
states of
higher angular momentum; this is in agreement with perturbation theory,
according to which it is impossible to renormalize systems containing
particles
of spin 1 or more. Had there been no interaction with the electromagnetic
field,
we could have introduced a further subtraction term which would have
necessitated a separate application of the unitarity condition for the $P$
state
of the \mbox{pion-pion} system. The resulting
\mbox{Castellejo--Dalitz--Dyson}
ambiguity would have been associated with a heavy unstable meson of spin 1.
This
corresponds to the \mbox{Bethe-Beard} mixture of vector and scalar mesons,
which
can be renormalized in perturbation theory as long as there is no
interaction
with the electromagnetic field.
Now let us consider the situation that occurs in practice, when the
coupling
constant is sufficiently large for the scattering amplitude and its
absorptive
parts to tend to infinity with $z$ (or $s_c$ and $t$) when $s$ remains
constant.
The function $A'_{12}$ which, according to our procedure, must be added to
$A_{
12}$ in iterations other than the first, will now tend to infinity with $s$,
so
that $A_2$, as calculated from (3.11a), would show a similar behavior. In
practice, when the unitarity condition for states with
$j = \frac{3}{2}$ as well as
with $j = \frac{1}{2}$ must be applied separately,
$A'_{12}(s,~s_c)$ and $A_{23}(s_c,
~t)$ will tend to infinity faster than $s$ or $t$, and the dispersion
relation (
3.11a) will require two subtractions. The subtraction terms can be
determined by
crossing symmetry as before. However, we have seen that, if $A_2$ tends to
infinity with $s$, we cannot consistently perform the calculation, so that
we
shall have to introduce some further modifications.
The reason for the difficulty is probably the inadequacy of the
\mbox{one-meson}
approximation. The breakdown occurs just at the value of the coupling
constant
for which the contribution to the scattering amplitude from $A'_{12}$ is
comparable to the remainder of the scattering amplitude when $s$ is large.
Since
that part of $A_1$ calculated from $A_{12}$ represents a partial effect of
states with one or more pairs, the contribution of these intermediate states
is
now important at high energies and it seems reasonable that, if one could
take
them into account properly, one could still perform the calculations for
values
of the coupling constant. In the \mbox{one-meson} approximation, one would
have
to make some sort of a cutoff to the contribution to $A_2$ from the crossing
term above $s = 9 M^2$. As this entails modifying the unitarity condition in
the
region where it is in any case inaccurate, it is consistent with our
approximations, and it may be hoped that the theory is not very sensitive to
the
precise location and form of the cutoff. If one were to go to further
approximations in which intermediate states with pairs were included, the
cutoff
would always be applied only at or above the threshold for processes which
were
neglected.
Once we are prepared to introduce cutoff into our approximations, we might
legitimately ask whether or not we should perform more than one subtraction
in
Eq. (3.11b). This could only be determined by examining the behavior of the
scattering amplitude and its absorptive parts at large values of $s$ when we
go
beyond the \mbox{one-meson} approximation. However, if $A$ and $B$ have the
behavior assumed thus far ($A$ remains constant and $B$ behaves like $1/s$),
the
cross section would tend to zero like $1/s$ at large $s$, whereas the
experimental results indicate that the cross section remains constant. It
therefore may be necessary to perform an additional subtraction and to
introduce
the unitarity condition of the reaction III in $P$ states.At first sight it
would seem as though there were \mbox{Castilejo-Dalitz-Dyson} ambiguities
associated with all states for which the unitarity condition has to be
applied
separately, not only with the
$j = \frac{1}{2}$ states. However, it is also possible
that only the solution without any of the extra terms in the higher angular
momentum waves would converge as we introduced more and more states into the
unitarity equations. This \\
%\vspace{1.5cm}
%\includegraphics{http://dbserv.ihep.su/images/mandelstam4a.gif}
%Fig. 4\\
\begin{figure}[h]
\centerline{\epsfig{file=fig4r.eps}}
\caption{Graphs involving the pion--pion interaction.}
\end{figure}
%\vspace{1.5cm}
solution would be an analytic continuation of the solution obtained for
small
values of the coupling constant, whereas the other solutions could not be
continued below a certain value of the coupling constant and would have no
perturbation expansion. While we can by no means exclude such a behavior, it
nevertheless gives us grounds to suppose that the ambiguity exists only for
\mbox{meson-nucleon} states with
$j = \frac{1}{2}$ and for
\mbox{$S$-wave} \mbox{meson-meson} states, even
when the coupling constant is large.
Before leaving this section, let us state the boundaries of the region in
which
the spectra functions $A_{13},$ $A_{23},$ and $A_{12}$ are nonzero, i.e.,
the
position of the curves $C_{13},$ $C_{23},$ and $C_{12}$ in Fig. 1. Since
$A_3$
is now nonzero for $t^2 > 4\mu^2$, $C_{13}$ in the \mbox{one-meson}
approximation is obtained by putting $t_2 = t_3 = 4\mu^2$ in (3.12a), so
that
$$
t^{^1/_2}_{1a} = 4 \mu \cdot \left( 1 + \mu^2/q^2 \right)^{^1/_2},
$$
or
$$
t_{1a} = \frac{16 \mu^2(s - M^2 + \mu^2)^2}
{[s - (M + \mu)^2][s -(M - \mu)^2]}. \eqno(4.6)
$$
For any given value of $s$, $A_{13}$ will be nonzero if $t > t_{1a}$. We
notice
that, as $t$ tends to infinity, $t_{1a}$ approach the value $16 \mu^2$. This
is
not expected result -- we have shown in Sec. 2 that it should approach the
value
$4 \mu^2$. The reason for the discrepancy is that, in our approximation, the
reaction III takes place purely through $S$ waves for $4 \mu^2 < t < 16
\mu^2$,
and $A_3$ will be a function only of $t$ in this region. Had it been
possible
for the reaction III to go through an intermediate state of one pion, $A_3$
would have had a $\delta$ function at $t = \mu^2$, and, on putting this
value
into (3.12a), we would have obtained the expected result. As it is, however,
we
shall have to go beyond the \mbox{one-meson} approximation to get the
correct
boundary of $A_{13}$.
The reaction $N + \bar{N} \rightarrow 3 \pi$ can go through a
\mbox{one-pion}
intermediate state by means of the process represented in Fig. 4(a). If,
therefore, we treat the outgoing pions in the reaction $N + \pi \rightarrow
N +
2 \pi$ as one particle with fixed energy and angular momentum, and represent
the
transition amplitude in the same way as we have represented the transition
amplitude for \mbox{pion-nucleon} scattering, the absorptive tart
corresponding
to $A_3$ will have a $\delta$ function at $t = \mu^2$. We can work out the
resulting contribution to $A_{13}$ (of \mbox{pion-nucleon} scattering
amplitude)
by unitarity in the same way as we worked out the contributions from the
\mbox{
one-meson} approximation. $z_2$ and $z_3$ in Eqs. (3.4)--(3.8) will now
refer to
the \mbox{center-of-mass} deflection of the nucleon in the production
reaction,
and will be connected with the momentum transfer by the relation
$$
z = \left\{q^2 + q^2_1 + t - [(M^2 + q^2)^{^1/_2} -
(M^2 + q_1^2)^{^1/_2}]^2 \right\}/2qq_1,
$$
where $q_1$, is the \mbox{center-of-mass} momentum of the outgoing nucleon.
The
value of $q_1$ will depend on the relative energy of the two pions; we shall
require the maximum value of $q_1$ (for a fixes $s$), which occurs when the
pions are at rest with respect to one another and is given by
$$
q^2_{1m} = \left\{[s - (M + 2 \mu)^2][s - (M - 2\mu)^2]\right\}/4s.
\eqno(4.7)
$$
We then find that the boundary of this contribution to $A_{13}$ has the
equation
$$
t_{1b} = \frac{4 \mu^2(s - M^2 - 2\mu^2)^2}{[s - (M + 2\mu)^2]
[s - (M - 2\mu)^2]}.
\eqno(4.8)
$$
The curve represented by (4.8) approaches asymptotically the lines $t = 4
\mu^2$
and $s = (M + 2 \mu)^2$. Thus, as would be expected, this contribution to
$A_{
13}$ only occurs above the threshold for pion production.
$A_{13}$ is therefore nonzero for $t > t_1$, where
$$
t_1 = t_{1a}, \qquad (M + \mu)^2 < s < (M + 2\mu)^2,
$$
$$
t_1 = \mbox{min}(t_{1a}, t_{1b}), \qquad (M + 2\mu^2) < s < \infty,
\eqno(4.9)
$$
and $t = t_1$ is the curve $C_{13}$ of Fig.1. We cannot be sure that
contribution from other intermediate states will not extend beyond this
curve,
but this is unlikely owing to the grater mass of these states.
The curve $C_{23}$ is obtained from $C_{13}$ simply by changing $s$ to
$s_c$.
$C_{12}$ can be calculated in a similar way; we find that
$$
s_{c2} = s_{c2a}, \qquad \qquad (M + \mu)^2 < s < (M + 2\mu)^2,
$$
$$
s_{c2} = \mbox{min}(s_{c2a},~s_{c2b}), \qquad \qquad (M + 2\mu)^2 < s <
\infty,
\eqno(4.10)
$$
where
$$
(s_{c2a} - u)^{^1/_2} = 2 \mu \left\{ \frac{s^2 - s (3M^2 + 2\mu^2) +
2(M^2 - \mu^2)}
{[s - (M + \mu)^2] [s - (M - u)^2]} \right\}^{^1/_2} +
$$
$$
+ \left\{ \frac{[M^2s - (M^2 - \mu^2)^2]
[s^2 - 2s(M^2 + 3 \mu^2) + (M^2 - \mu^2)^
2]}{s[s -(M + \mu)^2][s -(M - \mu)^2]}
\right\}^{^1/_2},
\eqno(4.11)
$$
$$
s_{c2b}(s) = s(s_{c2a}).
\eqno(4.12)
$$
The equation $s_c = s_{c2b}$ represents in fact the boundary of the region
in
which $A'_{12}$ is nonzero. We observe that, once the \mbox{pion-pion}
interaction has been included, this region approaches asymptotically the
line $s
= (M + 2 \mu)^2$ rather than the line $s = 9 M^2$. The reason is that
processes
represented by graphs such as Fig. 4(b) are now included in our
approximation,
so that the crossing term will include the contribution from Fig. 4(c), the
intermediate state of which involves a nucleon and two pions.
For a given real value of $s$, the absorptive part $A$ of the scattering
amplitude will be an analytic function of the momentum transfer as long as
$$
t_2 < t < t_1, \eqno(4.13a)
$$
where $t_1$ is given by (4.9), and $t_2$ by (4.10) and (2.13). The expansion
in
partial waves will converge if
$$
- t_1 - 4q^2 < t < t_1, \eqno(4.13b)
$$
as $-t_1 - 4 q^2$ is always greater than $t_2$.
We may note finally one interesting point concerning the spectral properties
of
the scattering amplitude. The unitarity condition should strictly, be used
in
the physical region only, and the results extended to the unphysical region
by
analytic continuation. This has actually been done for the reaction I, as
well
as for the reaction III with $t > 4 M^2$. For the reaction III in the region
$4
\mu^2 < t < 4 M^2$, we should apply the unitarity condition with the nucleon
masses taken, not on the mass shell, but at some smaller value where all the
momenta would be real. The result should then be continued analytically onto
the
mass shell. In our case this is found to make no difference, but if, in
addition
to the nucleon, we had a baryon whose mass $M_B$ satisfied the inequality
$$
M^2_B < M^2 - \mu^2, \eqno(4.14)
$$
it would be necessary to do the calculation in this way. On making the
continuation to the mass shell, it would be found that the absorptive part
$A_3$
extended below the limit $t^2 = 4 \mu^2$. It has been shown by several
workers\footnote{Kalplus, Sommerfield,
and Wichman, Phys. Rev. 111, 1187 (1958); Y.
Nambu, Nuovo cimento 9, 610 (1968); R. Oehme, Phys. Rev. 111, 1430 (1958).}
%\cite{Karplus1958}
%\cite{Nambu1958}
%\cite{Ochme1958}
that, if an inequality such as (4.14) is satisfied, the vertex function
would
show similar spectra properties. The simplest graph to exhibit them in our
case
would be Fig. 4(d), which will obviously have properties similar to those of
a
vertex graph. It is thus seen that these spectral abnormalities would not
limit
the applicability of our method, but, on the contrary, follow
from it.
\vspace{0.5cm}
\section*{
{\bf 5. Approximation Scheme for Obtaining
the Scattering Amplitude}}
\vspace{0.5cm}
In the methods developed in the previous sections, the unitarity condition
for
the reaction I is satisfied for all \mbox{angular-momentum} states in the
\mbox{
one-meson} approximation. The unitarity condition for the reaction III is
satisfied only for $S$ states in the \mbox{two-meson} or \mbox{two-meson}
plus
pair approximations. The unitarity condition for higher angular momentum
states
of the reaction III is not satisfied, but the scattering amplitude shows the
expected behavior at the threshold for competing real processes.
These properties suggest immediately a further approximation which would be
consistent with our other approximations. The major portion of the work, and
certainly the major part of the computing time, would be employed in
calculating
the spectral functions, as this involves finding double integrals which are
themselves functions of two variables. The calculations would therefore be
simplified if we neglected those contributions to the spectral functions
which
begin at the threshold for processes involving more than two particles. The
only
contributions to $A_{13}$ and $A_{23}$ left would be those beginning at $t =
4M^
2$, and they could be obtained by inserting the \mbox{$\delta$-function}
contribution to $B_2$ into (3.18). The spectral function $A_{12}$ would be
zero
in this approximation.
The unitarity condition for the higher angular momentum states of the
reaction I
is no longer satisfied. However, the terms neglected appear by their form to
arise from intermediate states of the reaction III with more than two
particles,
so that the approximation is in the spirit of the approximations already
made.
We have in fact made precisely this approximation in the unitarity condition
for
the $S$ waves of the reaction III. The unitarity condition for the low
angular
momentum states of the reaction I, and in particular for the states with
$j = \frac{1}{2}$ or $\frac{3}{2}$, is still satisfied, as
it has been introduced
separately. The present approximation treats the reactions I, II, and III on
the
same footing.
To summarize, then our method of procedure will be the following: The first
few
angular momentum states of $A_1$ and $A_3$ are found on the assumption that
each
angular momentum state is an analytic function of the square of the \mbox{
center-of-mass} energy except for the perturbation singularities and the
cuts on
the positive real axis. This calculation can be done exactly if the
discontinuity across the cut along the positive real axis is determined by
unitarity (complications arise, as the relations connecting $a$ and $b$ with
$A$
and $B$ involve square roots of kinematical factors, but the methods can be
modified accordingly). $A_{13}$ and $A_{12}$ are also found as just
described.
The analytic properties of the low angular momentum states are now
determined
from the analytic properties of the scattering amplitude given by (2.12).
The
singularities can be calculated in terms of $A_1,~A_2,~A_3.$ These
absorptive
parts can in turn be found from $A_{13}$ and $A_{23}$ by means of the
dispersion
relations (2.16), (2.17), (2.19), with subtraction terms which can be
obtained
from the \mbox{low-angular-momentum} states. In the next iteration, all the
singularities of the low angular momentum states except that along the
positive
real axis are found from the quantities calculated in the first iteration,
and
the singularity along the positive real axis
is redetermined from the unitarity
condition. The iteration procedure is repeated until it converges. As in the
calculations of Sec. 4, it is found necessary to cut off the absorptive
parts
$A_1,~A_2$ and $A_3$ at high energies, before calculating the singularities
of
the low angular momentum states in the next iteration. However, the cutoff
is
only applied above the threshold for processes neglected in the unitarity
condition, and in particular, above the threshold for pair production in the
reaction I.
This approximation could be regarded as the first of a series of
approximations
in which more and more of the contributions to the spectral functions are
included, until we ultimately reach a solution in which the unitarity
condition
in the \mbox{one-meson} approximation is satisfied for every angular
momentum
states. In the higher approximations the spectral functions are no longer
determined by perturbation theory, but, once contribution from the crossing
term
enters, they will have to be recalculated after each iteration. However, it
would be more worthwhile to go beyond the \mbox{one-meson} approximation at
the
same time as we took the higher contributions to the spectral functions into
account, In other words, we continue to put the reactions I, II, and III on
the
same footing, bringing in the higher intermediate states of all three
together.
If the approximation scheme converged, the exact unitarity condition of the
three reactions would finally be satisfied for all angular momentum states.
Needless to say, one would not in practice be able to go beyond the first
one or
two approximations.
The number of angular momentum states for which the unitarity condition is
applied separately will, as has been explained in the last section, depend
on
the behavior of $A$ and $B$ as $t$ (or $s_c$) tends to infinity with $s$
constant. However, in our first approximation, it should be sufficient to
treat
separately only states with $j = \frac{1}{2}$ and
$j = \frac{3}{2}$, as the other angular
momentum states will not be important below the threshold for pion
production.
If we went beyond the \mbox{one-meson} approximation we would probably have
to
treat some higher angular momentum states separately in any case, since, for
instance, two pions both in a (3,3) resonance states with a nucleon could
form a
$D_{\frac{5}{2}}$ states. For reaction III, one would have to treat
separately $
S$ states and possibly $P$ states as well.
If one neglected the \mbox{nucleon-antinucleon} intermediate state in the
reaction III and only took the \mbox{two-pion} intermediate states into
account,
all three spectral functions $A_{13},~A_{23}$, and $A_{12}$ would be zero,
since
they all begin above the threshold for processes which are being neglected.
The
entire scattering amplitude would then consist of ``subtraction terms'' for
one
or other of the dispersion relations. This may be the best first
approximation
from the point of view of the amount of work required and the accuracy of
the
result, as the \mbox{nucleon-antinucleon} intermediate state is a good deal
heavier than multipion states which are being neglected. Though the spectral
functions are not now brought in at all, it will of course be realized that
the
only justification for the approximation is that it is the first of a series
of
approximations which do involve the spectral functions. In this
approximation,
if the crossing term is neglected in the calculation of the \mbox{pion-pion}
scattering amplitude, only intermediate $S$ states occur in reaction III, so
that the unitarity condition for the $P$ states will not enter.
\vspace{0.5cm}
\section*{
{\bf Acknowledgements}}
\vspace{0.5cm}
The author would like to acknowledge helpful discussion with Professor N.M.
Kroll, Professor M.L. Goldberger, Professor R. Oehme, and Professor H.
Lehmann.
He also wishes to thank Columbia University for the award of a Boese
\mbox{Post-
Doctoral} Fellowship.
%SB. = ENCODED 3 MAY 1998 BY NIS;
\end{document}