INTERACTION OF $\pi$-MESONS WITH NUCLEONS*

M. Gell-Mann and K. M. Watson

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Some physical considerations concerning $\pi$-mesons . . . . . . . . . . . 400
Scattering of $\pi$-mesons by nucleons . . . . . . . . . . . . . . . . . . . 407
Photoproduction of $\pi$-mesons on nucleons . . . . . . . . . . . . . . . . 415
Production of $\pi$-mesons in collisions of nucleons . . . . . . . . . . . 428
Meson theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

INTRODUCTION

During the seven years that have elapsed since the first observation of artificially produced $\pi$-mesons at Berkeley, we have witnessed a very rapid development of elementary-particle physics. There are now at least twelve accelerators on which $\pi$-mesons can be produced, and at several accelerators heavier particles are being generated, or are expected to be generated. It may be said that, as a result of work carried out on these accelerators, we have at least qualitatively clarified the picture of the elementary interaction of mesons with nucleons at low energies.

On the other hand, the understanding of the interaction of $\pi$-mesons with nucleons in a certain sense lags behind the factual data relating to these phenomena. It should apparently be expected that the application of field theories alone to the explanation of meson-nucleon processes will prove unsuccessful. Indeed, the existence of a large number of heavier unstable particles at present makes one doubt the success of field theories.

* Annual Review of Nuclear Science 4, 219—270 (1954). Translated by A. I. Lebedev and A. M. Baldin.

Despite difficulties of a fundamental nature, it proved possible to simplify the explanation of experimental facts by applying various phenomenological theories to meson reactions. In many cases this was no more than the use of certain general quantum-mechanical principles together with certain physical considerations borrowed from a more familiar field—nuclear physics. In some cases carefully developed models were proposed; some of them turned out to be very fruitful.

The principal task of this article is an attempt to collect the theoretical considerations that are used in studying the properties of the meson. We construct a model under three assumptions:

1) The radius of interaction of a meson with a nucleon is finite.
2) The phenomena under consideration satisfy the requirements of charge independence (i.e. the isotopic spin \(I\) is conserved).
3) An especially strong interaction (attraction) exists in the state of the meson–nucleon system \(I = 3/2\), \(j = 3/2\), \(l = 1\). Below it is shown that this model agrees well with the available experimental data at “low energies”; then the place of this model in more detailed and general theories, and the corresponding experimental data, are considered.

We shall begin with a phenomenological discussion combining the general principles of quantum mechanics with simple physical considerations (see the following section), and shall discuss the experimental data in the light of these principles. Then the results of more thorough approaches to the problem will be considered.

SOME PHYSICAL CONSIDERATIONS RELATING TO \(\pi\)-MESONS

General principles. First we shall consider the problem of collisions of two bodies, including the production (and absorption) and scattering of \(\pi\)-mesons on nucleons. The most studied of these processes fall into three classes:

Scattering \((S)\):

\[ \pi + N \to \pi + N \quad \text{(six)}. \]

(We use respectively the symbols \(\pi\) and \(N\) for the meson and the nucleon, without regard to their charge states.)

Photoproduction \((P_\gamma)\):

\[ \gamma + N \rightleftarrows \pi + N \quad \text{(four)}. \]

Production in collisions of a nucleon with a nucleon \((P_n)\):

\[ N + N \rightleftarrows \pi + N + N \quad \text{(seven)}. \]

INTERACTION OF \(\pi\)-MESONS WITH NUCLEONS

One may expect that at sufficiently high energies, in each of these reactions the production of additional mesons will be observed. In addition, other unstable particles may also be produced1. The numbers in parentheses to the right of the reaction equations indicate the number of possible meson and nucleon charge states.

We shall deal with these reactions mainly at low energies. By “low energies” we mean such energies at which the de Broglie wavelength of the meson \(\bar{\lambda}\) (in the center-of-inertia system) is not small in comparison with the radius of action of the forces. It is hardly to be expected, except in the case of the Coulomb interaction, that this radius will be much greater than the Compton wavelength

\[ \frac{\hbar}{\mu c}, \]

where \(\mu\) is the rest mass of the meson. Hence there follows the natural assumption that an orbital angular momentum greater than \(l\hbar\), where

\[ l=\frac{1}{\bar{\lambda}}\cdot \frac{\hbar}{\mu c}, \tag{1} \]

will not play an essential role in the reactions under consideration. For illustration, in Fig. 1 the value of \(l\), determined from formula (1), is plotted as a function of the meson energy in the laboratory system.

Fig. 1. Dependence of the greatest value of the orbital angular momentum (expressed in units of \(\hbar\)) in the scattering of a \(\pi\)-meson by a nucleon on the \(\pi\)-meson energy in the laboratory coordinate system.

for the scattering process (denoted above by \(S\)). It is evident that for many processes and for considerable energy regions it is sufficient to take into account one or two values of the quantity \(l\), if the assumption concerning the magnitude of the radius is valid.

of the action of the forces. From what follows it will be clear that this conclusion proves to be correct, and that only one state with unit orbital angular momentum (the \(P\)-state) plays an essential role.

Just as in nuclear physics, the assumption that the radius of the forces is finite leads to specific consequences. For example, if only one orbital-angular-momentum state plays an essential role in the emission (or absorption) of a particle, then the energy dependence of the cross section at low energies is determined uniquely *). The energy dependences of the processes of interest to us are presented in Table I. In this table \(q\) is the meson momentum in the center-of-mass system.

Table I

Dependence of the cross section on the momentum of the absorbed, emitted, or scattered meson at “low energies” \(^{1),\,2)}\)

Type of process	Absorption	Emission into a state of two particles	Emission into a state of three particles	Scattering
Dependence of the cross section on momentum	\(q^{2l-1}\)	\(q^{2l+1}\)	\(q^{2l+4}\)	\(q^{4l}\)
Dependence of the “matrix element” on momentum	\(q^l\)	\(q^l\)	\(q^l\)	\(q^{2l}\)

\(^{1)}\) \(q\) is the meson momentum, \(l\) is its angular momentum.
\(^{2)}\) If several emitted particles interact strongly with one another, the dependence on \(q\) may change, but usually in a simple way. (See the section “Production of \(\pi\)-mesons in nucleon–nucleon collisions.”)

When only one or two eigenvalues of the angular momentum are essential, one may expect a simple dependence of the cross section on the energy. (For more than one \(l\), one should expect a linear combination of the corresponding terms in Table I.) In Fig. 2

*) Let us show this. Let \(T(\mathbf r)\) be the scattering matrix describing the emission of a particle from the coordinate \(\mathbf r\). Suppose also that \(T(\mathbf r)\) vanishes for \(r>R\), and that only the state with angular momentum \(l\) plays an essential role. Then the probability amplitude that the emitted particle will be described by the plane wave \(\Phi_{\mathbf q}(\mathbf r)\), where \(\mathbf q\) is the momentum, is equal to

\[ A=\int \Phi_{\mathbf q}(\mathbf r)\,T(\mathbf r)\,d^3 r. \]

Since the \(l\)-th partial component \(\Phi_{\mathbf q}\) varies as \(\left(\dfrac{qr}{\hbar}\right)^l\) for \(r<\dfrac{\hbar}{q}\), we have \(A\sim q^l\) for \(\dfrac{\hbar}{q}>R\).

are compared with simple power laws as functions of the energy of the experimental total cross sections. With the exception of the process \(P_1(\gamma + p \to \pi^+ + n)\), the power law corresponding to a \(P\)-state,

Fig. 2.
a) Comparison of the cross section for scattering of \(\pi^+\)-mesons on protons with the law \(\sigma \sim q^4\), where \(q\) is the meson momentum in the center-of-mass system. The points are taken from the work: \(\Phi\)—Leonard S. and Stork D., Phys. Rev. 93, 568 (1954), and from the works indicated in Table III: \(\square\) \((d)\), \(\triangle\) \((e)\), \(\times\) \((g)\).

b) Measurement data of Durbin R., Loar H., and Steinberger J. [Phys. Rev. 84, 581 (1951)] for the reaction \(\pi^+ + d \to p + p\) are compared with the law \(\sigma \sim q\).

c) \(\sigma(\gamma + p \to \pi^0 + p)\), measured by Goldschmidt-Clermont Y., Osborne L., and Scott M. [Phys. Rev. 89, 329 (1953)], is compared with the law \(\sigma \sim q^3\).

d) The experimental cross section for the process \(\gamma + p \to \pi^+ + n\) is compared with the law \(\sigma \sim q\) (— — —) and with the law \(\sigma \sim q^3\) (—·—·—). Details are given in the section “Photoproduction of \(\pi\)-mesons on nucleons.”

turns out to be quite satisfactory. In the indicated case, however, a linear combination of \(S\)- and \(P\)-waves is required even for rather low \(\gamma\)-quantum energies.

A second consequence of our assumption about the essential role of a small number of values of the orbital angular momenta at low energies is that, for our discussion, the laws of conservation of angular momentum and parity will be important. This circumstance will be developed in greater detail when each reaction is considered separately.

There is, however, one especially important aspect of the laws of conservation of angular momentum and parity on which we shall dwell here. The \(\pi\)-meson is described by a pseudoscalar function \(^{2,3}\); this means (by definition) that the emission of a single meson by an individual nucleon must take place into a state with odd orbital angular momentum. The total angular momentum must, however, remain equal to \(j = 1/2\), since the nucleon had this value of angular momentum in the initial state. The only odd state of the meson–nucleon system with \(j = 1/2\) is \(l = 1\), i.e. a \(P\)-state. Thus, a simple process of emission (or absorption) must always occur in (or from) a \(P\)-state(s).

This, of course, is not a selection rule for physical processes, since simple emission and absorption cannot occur separately from other phenomena. On the other hand, if these simple emission and absorption processes are important stages in the physical picture of emission or absorption, then it is quite natural that the \(P\)-state should predominate (at low energies). This is explicitly confirmed by experiment (see Fig. 2 and the following sections) and, generally speaking, is not incompatible with many consequences of field theory.

We can now summarize the content of this section. Allowing for the possibility of exceptional cases, we assume that at low energies meson reactions proceed through a \(P\)-state. One should, of course, expect a small admixture of other states. The dependence of the cross section on energy at sufficiently low energies is unambiguously determined by the corresponding states of orbital angular momentum. Finally, selection rules and other consequences of the laws of conservation of angular momentum and parity must be taken into account. The aim of most phenomenological analyses of meson properties has been to determine the limits of applicability of this model.

Hypothesis of charge independence. The hypothesis of charge independence simplifies the study of meson phenomena. Apparently this hypothesis was first put forward in 1936 by Breit and Feenberg \(^{4}\), who believed that the nuclear forces \(n — n\), \(p — p\), and \(n — p\) are the same for states with equal angular momenta and parity (which, as is well known, is at least approximately true for low energies). This assumption was based on the closeness of the binding energies and the analogy of the scattering properties of neutrons and protons. In other words, if the state of a two-body system is classified by the quantum numbers \((j, S, \pi, Q)\), where \(Q\) is the charge, \(S\) —

spin, \(j\) is the total angular momentum, \(\pi\) is the parity, then the hypothesis of charge independence asserts that the interaction does not depend on \(Q\).

Kemmer \(^{5}\) showed how to construct a meson theory giving nuclear forces independent of charge. Then Gaitler \(^{6}\) pointed out that Kemmer’s theory must lead to selection rules and intensity relations in meson reactions. In view of the imperfection of meson theories, it is desirable to separate the hypothesis of charge invariance from meson theories. This can be done in a way that gives a simple interpretation of charge independence. In its most general form, charge independence means that neutrons and protons are physically completely equivalent if one disregards weak interactions (for example, electromagnetic interactions and \(\beta\)-interactions). This means that a wave function constructed as a linear combination of neutron and proton wave functions must be physically equivalent to the wave function of a neutron or a proton.

In mathematical form this means that the general unitary transformation which replaces the wave function of each neutron or proton of the system by a linear combination of neutron and proton wave functions must leave the physical properties of the system unchanged. Apart from a phase factor, this transformation is equivalent (isomorphic) to the spin representation of the group of rotations in three-dimensional space (this space was called “charge space”). Here one can draw a useful analogy. The invariance of a physical system with respect to rotations in ordinary space leads to the conservation of angular momentum, whose operators produce rotations. From invariance with respect to “charge rotations” follows the conservation of physical quantities described by the operators of this rotation. Their eigenvalues are called isotopic spin. From the analogy with angular momentum it is clear that these “rotation operators in charge space” formally coincide with angular-momentum operators, so that the same mathematical apparatus can be used without change.

In particular, one can introduce two components of the wave function for a nucleon, corresponding to the neutron and proton states. “Rotations in charge space” are described by three two-row matrices \(\tau_1\), \(\tau_2\), and \(\tau_3\), which are formally equivalent to the Pauli spin matrices. \(\tau_{1,2,3}\) are the components of the vector \(\boldsymbol{\tau}\) (with respect to charge rotations) in three-dimensional charge space. For a system of several nucleons the total isotopic spin \(I\) can be obtained in the same way as the total spin \(S\) in the case of ordinary spin. To choose a definite representation, we shall assume that the proton has isotopic spin directed “upward,” and the neutron “downward.” (This choice is, of course, arbitrary, and the reverse is often taken instead.)

The principle extends unambiguously to unstable particles, which may be emitted once or absorbed by nucleons. For example, consider the emission of a \(\pi^+\)-meson by a proton. Charge independence asserts that this emission process is not changed when, as a result of a “rotation,” the proton wave function is replaced by a linear combination of proton and neutron wave functions. It is obvious that such a “rotation” must replace the \(\pi^+\)-meson wave function by a linear combination of the \(\pi^+\)-, \(\pi^-\)-, and \(\pi^0\)-meson wave functions, since a neutron cannot emit a \(\pi^+\)-meson. This is a three-dimensional (irreducible \(^{7}\)) representation of the rotation group in three dimensions, the meson having an “isotopic spin” equal to unity, with isotopic-moment operator \(\mathbf{t}\). Thus, each meson has isotopic spin 1, each nucleon has isotopic spin \(1/2\). The states of a system of several nucleons and mesons can be decomposed into states of total isotopic spin \(I\), in complete analogy with the corresponding problem for ordinary angular momentum. The state \(I\) is \((2I+1)\)-fold degenerate, and the substates are dynamically equivalent.

If the third component of \(\mathbf{t}\) is diagonal, then the meson wave functions under rotations in charge space transform as spherical functions of first order:

\[ \begin{aligned} \Phi(\pi^+) &\to Y_1^1,\\ \Phi(\pi^0) &\to Y_1^0,\\ \Phi(\pi^-) &\to Y_1^{-1}. \end{aligned} \tag{2} \]

The charge operators of the meson and the nucleon, expressed in units of the proton charge, are respectively equal to

\[ \left. \begin{aligned} Q_\pi &= t_3,\\ Q_N &= \frac{1}{2}(1+\tau_3). \end{aligned} \right\} \tag{3} \]

The total spin \(I\) (“isotopic spin”) for several simple systems of mesons and nucleons is given in Table II. Each of these \(I\)-states is an integral of the motion for the corresponding systems. The \((2I+1)\) substates for each of them are equivalent, which means that the total number of different reactions is greatly reduced. For example, in the first part of this section (see “General principles”) it was indicated that there are six scattering reactions \((S)\). It follows from Table II that in this case there exist only two \(I\)-states; therefore only two, and not six, reactions need be investigated. From Table II it is seen that for the processes \((P_n)\), \(I = 0, 1, 1\), and, consequently, there are not seven but three independent processes.

Table II

Possible values of isotopic spin for some systems of mesons and nucleons

System	One nucleon	One $\pi$-meson	Two nucleons	$\pi$-meson, nucleon	$\pi$-meson, two nucleons	Two $\pi$-mesons	Two $\pi$-mesons, one nucleon
Value of isotopic spin $I$	$1/2$	$1$	$0,\ 1$	$1/2,\ 3/2$	$0,\ 1,\ 1,\ 2$	$0,\ 1,\ 2$	$1/2,\ 1/2,\ 3/2,\ 3/2,\ 5/2$

Finally, it should be noted that charge independence in its modern formulation is not exact because of the “weak interactions,” which violate it (electromagnetic interactions single out a direction in charge space, making it possible to distinguish protons and neutrons, etc.). This means that transitions will exist between different $I$-states; however, they probably play a secondary role in the phenomena that we shall consider (with the exception of special cases of electromagnetic interaction, such as, for example, photoproduction of mesons).

Hypothetical state of strong interaction of a meson–nucleon system. Let us denote the states of a system consisting of one meson and one nucleon, having isotopic spin $I$, angular momentum $j$, and orbital angular momentum $l$, as follows: $(I,\ j,\ l)$. It follows from meson theory that at certain energies the state $(3/2,\ 3/2,\ 1)$ should be a state of especially strong interaction. Brueckner$^{8}$ adopted this hypothesis, assuming that scattering in this state proceeds through a resonance at meson energies of about $200$ MeV (in the laboratory coordinate system). This hypothesis leads to certain consequences for meson phenomena, which will be considered below. The amount of evidence in favor of the resonance actually existing is steadily increasing.

SCATTERING OF $\pi$-MESONS BY NUCLEONS

The following elementary scattering processes have been studied:

\[ \pi^{+} + p \to \pi^{+} + p \qquad (S_{+}) \]

\[ \pi^{-} + p \to \pi^{-} + p \qquad (S_{-}) \]

\[ \pi^{-} + p \to \pi^{0} + n \qquad (S_{e}) \]

We shall denote the differential cross sections of these processes respectively by $\sigma^+$, $\sigma^-$, and $\sigma^e$, and for the total cross section we shall use the notations $\sigma_T^+$, $\sigma_T^-$, and $\sigma_T^e$. (In what follows we shall use the notations $\sigma$ and $\sigma_T$ respectively for differential and for total cross sections.)

In discussing Fig. 2 it was noted that the scattering of the $P$-wave in meson scattering plays a predominant role over a rather wide energy interval. This by no means implies that the other partial waves are unimportant; we shall return to this circumstance below.

Many papers have been devoted to the experimental study of meson scattering on protons. The data currently available on

Fig. 3. Total scattering cross section of $\pi^+$ and $\pi^-$ mesons on protons as a function of the meson energy in the laboratory coordinate system. In Table III references are given to the works whose data are used here. The points for $\pi^+$ mesons at energies 1.0 and 1.5 Bev were obtained from the formula $\sigma(\pi^-,d)-\sigma(\pi^-,p)$, see reference (k) in Table III. The solid curves correspond to the best approximation of the experimental data.

total cross sections are shown in Fig. 3. (Table III contains references to the works from which the points in Fig. 3 were taken.)

Of particular interest are the cross sections at an energy of about 200 Mev, recently measured at the Carnegie Institute of Technology (Table III, references $(f)$ and $(g)$), and at an energy of 300 Mev, measured at Brookhaven (Table III, references $(h)$, $(j)$, and $(k)$). The figure shows the curves $\sigma_T^+$ and $\sigma_T^{(-)} \equiv \sigma_T^-+\sigma_T^e$.

It should be noted that the cross sections shown in Fig. 3 include also inelastic scattering (i.e., scattering in which one or two additional mesons are produced). This becomes energetically possible at meson energies above 200 Mev in the laboratory ...

Table III

Works from which the experimental data for Fig. 3 were taken

Pion energy in the laboratory coordinate system, in MeV	References	Works (the measured cross sections are indicated in brackets)
40	(a)	J. P. Perry and C. E. Angell, Phys. Rev. 91, 1289 (1953) \([\sigma^{+};\) measured with a telescope of counters switched in coincidence with a telescope registering the incident beam]
40	(b)	S. Barnes, C. Angell, J. Perry, D. Miller, J. Ring and D. Nelson, Phys. Rev. 92, 1327 (1953) \([\sigma^{-};\) measured with a telescope of counters]
34	(c)	A. Roberts and J. Tinlot, Phys. Rev. 90, 951 (1953) \([\sigma^{e};\) at first measured only with a single \(\gamma\)-quantum counter, and then in coincidence]
58, 65	(d)	D. Bodansky, A. Sachs and J. Steinberger, Phys. Rev. 93, 1367 (1954) \([\sigma^{+},\ \sigma^{-},\ \sigma^{e};\) scintillation counters; liquid hydrogen served as the target]
78, 110, 120	(e)	E. Fermi, R. Martin and D. Nagle, Phys. Rev. 91, 155 (1953) \([\sigma^{+},\ \sigma^{-},\ \sigma^{e};\) scintillation counters; liquid hydrogen served as the target]
from 135 to 250	(f)	J. Ashkin, J. Blaser, F. Feiner, J. Gorman and M. Stern, Phys. Rev. 93, 1129 (1954) \([\sigma_{T}^{(-)};\) liquid hydrogen served as the target]
from 135 to 196	(g)	J. Ashkin, J. Blaser, F. Feiner, J. Gorman and M. Stern, Phys. Rev. 93, 1129 (1954) \([\sigma_{T}^{+};\) same as above]
28, 340, 450	(h)	S. J. Lindenbaum and L. C. Yuan, Phys. Rev. (in print), also Proceedings of 1954 Rochester Conference \([\sigma_{T}^{+};\) obtained by means of the subtraction method \(C—CH_{2}\)]
340, 450	(j)	S. J. Lindenbaum and L. C. Yuan, Phys. Rev. (in print) \([\sigma_{T}^{(-)};\) see above]
From 500 to 1000	(k)	R. Cool, L. Madansky and O. Piccioni, Phys. Rev. (to be published). \([\sigma_{T}^{+}\) and \(\sigma_{T}^{(-)},\) obtained by means of the subtraction method \(C—CH_{2}\) and \(C—CD_{2}\)]

This table in no way claims to be complete, but contains only the most recent works. A detailed bibliography can be found in the article by M. A. Ruderman, E. M. Henley, and J. Steinberger, in Ann. Rev. Nuclear Sci. 3 (1953).

coordinate system. In region 1 the cross section \(B_{\exists\theta}\) is probably due to a significant degree to inelastic scattering\(^9\).

It is of interest to resolve the cross section shown in Fig. 3 into the cross sections \(\sigma_{1/2}\) and \(\sigma_{3/2}\), corresponding respectively to pure substates \(I=1/2\) and \(3/2\) of isotopic spin. This is easily done by using the relations

\[ \sigma_T^{+}=\sigma_{3/2}, \tag{4} \]

\[ \sigma_T^{(-)}=\frac{1}{3}\left[\sigma_{3/2}+2\sigma_{1/2}\right] \tag{5} \]

or

\[ \sigma_{1/2}=\frac{1}{2}\left[3\sigma_T^{(-)}-\sigma_T^{+}\right]. \tag{6} \]

Let us note that these relations also hold for inelastic scattering.

Fig. 4. Experimental values of the cross sections for scattering of \(\pi\)-mesons on nucleons for pure isotopic-spin states: \(1/2\) and \(3/2\). The dashed curves represent the limiting cross sections for scattering in pure states \((j,l)\), calculated by formula (7).

Using the experimental values for \(\sigma_T^{+}\) and \(\sigma_T^{(-)}\) (Fig. 3), one can obtain \(\sigma_{3/2}\) and \(\sigma_{1/2}\), which are presented in Fig. 4. The solid curves are the best approximation to these cross sections. The dashed curves depict the upper limit of the scattering cross section in pure states with respect to \(j\) and \(l\). They are expressed as follows:

\[ \sigma(j,l)=2\pi\left(\frac{\hbar}{q}\right)^2\cdot [2j+1], \tag{7} \]

where \(q\) is the meson momentum in the center-of-inertia system. The presence of a sharp maximum near \(200\) MeV in \(\sigma_{3/2}\) and its absence in \(\sigma_{1/2}\) is striking and, apparently, excellently confirms both the hypothesis of charge independence and Brückner’s hypothesis\({}^{8}\) of a resonance in the state \((I=3/2,\ j=3/2,\ l=1)\). In particular, the height of the maximum is comparable with the value of \(\sigma(3/2,1)\), obtained from formula (7).

If the maximum in \(\sigma_{1/2}\) at an energy of about \(1\) BeV were due to scattering in only one angular-momentum state, this would mean that \(j \cong 5\) (see also Fig. 1). This seems unlikely, especially in view of the fact that the cross section here is mainly due to inelastic scattering. On the other hand, this maximum may indicate the predominance, at an energy of \(1\) BeV, of inelastic (i.e. with meson production) scattering in the state \(I=1/2\). If this is true, then charge independence gives the following relations for the differential cross sections:

\[ \sigma(p+\pi^- \to n+\pi^-+\pi^+) + \sigma(p+\pi^- \to n+\pi^+ +\pi^-) \]
\[ =2\sigma(p+\pi^- \to n+\pi^0+\pi^0)+\sigma(p+\pi^- \to p+\pi^-+\pi^0) \tag{8} \]

and

\[ \sigma(p+\pi^- \to p+\pi^0+\pi^-)=\sigma(p+\pi^- \to p+\pi^-+\pi^0). \]

For the total cross sections we have:

\[ 2\sigma_T(p+\pi^- \to n+\pi^-+\pi^+)= \]
\[ =\sigma_T(p+\pi^- \to n+\pi^0+\pi^0)+\sigma_T(p+\pi^- \to p+\pi^-+\pi^0). \tag{9} \]

These relations are valid only in the case where the cross sections for the production of \(\pi^+\)-mesons in collisions of \(\pi^+\)-mesons with protons are small, as was assumed above.

Weak scattering in the state with \(I=1/2\) at energies below \(400\) MeV was noted by Ashkin\({}^{10}\). The data obtained at Carnegie show this quite clearly if one plots on the same graph \(3\sigma_T^{(-)}\) and \(\sigma_T^{+}\), as is done in Fig. 5.

Brückner\({}^{8}\) proposed describing the state \((I=3/2,\ j=3/2,\ l=1)\) by a formula with one resonant level. Suppose that the total cross section \(\sigma_T^{+}=\sigma_{3/2}\), i.e. that it is entirely due only to this state. Then

\[ \sigma_T^{+}=\frac{2\pi\hbar^2}{q^2}\cdot \frac{\Gamma^2}{(E-E_0)^2+\dfrac{\Gamma^2}{4}}, \tag{10} \]

where

\[ \Gamma= \left[ \frac{2\cdot\left(\dfrac{qa}{\hbar}\right)^2} {1+\left(\dfrac{qa}{\hbar}\right)^2} \right]\cdot \gamma_\lambda^2 . \tag{11} \]

Here \(E_0\) is the “resonance energy” in the center-of-mass system, \(\gamma_\lambda^2\) is the reduced width, and \(a\) is the radius of the reaction channel. \(E\) is the energy of the meson and nucleon in the center-of-mass system. We have chosen

\[ \left. \begin{aligned} E_0 &= 159\ \text{MeV},\\ \gamma_\lambda^2 &= 58\ \text{MeV},\\ a &= 0.88\left(\frac{\hbar}{\mu c}\right). \end{aligned} \right\} \tag{12} \]

In Fig. 5 the cross section \(\sigma_T^+\), calculated from formula (10), is compared with the experimental cross sections. The agreement, as can be seen, is excellent.

Fig. 5. Comparison of the one-level resonance formula (10) for \(\sigma_T^+\) with experimental cross sections. The works used are indicated in Table III. The symbols \(\Phi\) denote the tripled cross section \(\sigma_T^{(-)}\), obtained by Ashkin et al.\(^{10}\)

The angular distribution in the scattering of \(\pi\)-mesons is also known (see the papers in Table III). Here the experimental data are considerably less complete than for total cross sections. If the scattering occurred only in the state \((I=3/2,\ j=3/2,\ l=1)\), then the angular distribution would have the form

\[ 1+3\cos^2\theta, \tag{13} \]

where \(\theta\) is the scattering angle in the center-of-mass system. Several experimental angular distributions for \(\pi^+\)-mesons are presented in Fig. 6. The asymmetry of the distribution with respect to the angle \(90^\circ\), evidently, is incompatible with formula (13). However, a relatively small admixture of \(S\)-scattering can lead to this asymmetry. The general

the expression for the scattering cross section of only \(S\)- and \(P\)-waves has the form

\[ \sigma = a + b \cos \theta + c \cos^2 \theta, \tag{14} \]

which apparently agrees with the observed angular distributions of the scattered mesons. Indeed, it proves possible to describe the known angular distribution by means of \(P\)-wave scattering only in the state \((I = {}^3/2,\ j = {}^3/2)\) and an admixture of \(S\)-wave scattering1.

Fig. 6. Angular distribution in the center-of-mass system for the process \(\pi^+ p \to \pi^+ + p\). The meson energies in the laboratory coordinate system are given in MeV. The experimental errors are not indicated in the figure. The curve for 45 MeV was obtained on the basis of the data of Orr J., Lord J., and Weaver A. [Phys. Rev. 93, 575, (1954)], for 65 MeV—on the basis of data from work (d) of Table III, for 120 and 135 MeV—on the basis of work (e) of Table III; the data for 260 MeV were obtained by Fowler W. B., Lea R., Sheppard W. D., Shutt R. P., Thorndike A. M., and Whittemore W. A. [Phys. Rev. 92, 832 (1953).]

A great deal of work has been done to determine the phase shifts of meson scattering on nucleons. For the energies at which it is necessary to take into account only \(S\)- and \(P\)-waves, according to the hypothesis of charge invariance, there are six different phase shifts. Following Anderson’s notation1, we denote the phase shifts of the \(S\)-wave respectively for states with \(I = {}^3/2\) and \({}^1/2\) by \(\alpha_3\) and \(\alpha_1\). Four phase shifts

Let us denote the \(P\)-wave phase shifts by \(\alpha_{33}, \alpha_{13}, \alpha_{31}\), and \(\alpha_{11}\), where the first subscript is equal to twice \(I\), and the second to twice \(j\). The determination of these six phase shifts from experimental data is highly ambiguous\(^{12}\), and we shall not go into details.

The simplest choice of phase shifts (and, from the point of view of modern theory, perhaps the most reasonable) is due to Bethe\(^{11}\), when \(\alpha_{13}, \alpha_{31}\), and \(\alpha_{11}\) are small, while \(\alpha_{33}\) passes through \(90^\circ\) at the meson energy \(E_\pi = 195\) Mev (in the laboratory coordinate system).

Table IV

Phase shifts in meson scattering on nucleons

\(E_\pi\), Mev	\(\alpha_{33}\)	\(\alpha_{3}\)	\(\alpha_{1}\)
120	\(30^\circ\)	\(-12^\circ\)	\(8^\circ\)
217	\(107^\circ\)	\(-20^\circ\)	\(-4^\circ\)
For \(E_\pi < 120\) Mev	\(\alpha_{33} \cong 16^\circ \left(\dfrac{q}{\mu c}\right)^3\)	\(\alpha_{33} \cong 16^\circ \left(\dfrac{q}{\mu c}\right)^3\)	\(\alpha_{33} \cong 16^\circ \left(\dfrac{q}{\mu c}\right)^3\)
Here are given the phase shifts of \(\pi\)-meson scattering on nucleons, obtained by Bethe\(^{11}\); \(\alpha_{33}\) and \(\alpha_{3}\) may be linearly extrapolated in the energy interval from 120 to 217 Mev. \(\alpha_{1}\) must be approximated in this energy interval by a parabola with derivative equal to zero at 120 Mev.	Here are given the phase shifts of \(\pi\)-meson scattering on nucleons, obtained by Bethe\(^{11}\); \(\alpha_{33}\) and \(\alpha_{3}\) may be linearly extrapolated in the energy interval from 120 to 217 Mev. \(\alpha_{1}\) must be approximated in this energy interval by a parabola with derivative equal to zero at 120 Mev.	Here are given the phase shifts of \(\pi\)-meson scattering on nucleons, obtained by Bethe\(^{11}\); \(\alpha_{33}\) and \(\alpha_{3}\) may be linearly extrapolated in the energy interval from 120 to 217 Mev. \(\alpha_{1}\) must be approximated in this energy interval by a parabola with derivative equal to zero at 120 Mev.	Here are given the phase shifts of \(\pi\)-meson scattering on nucleons, obtained by Bethe\(^{11}\); \(\alpha_{33}\) and \(\alpha_{3}\) may be linearly extrapolated in the energy interval from 120 to 217 Mev. \(\alpha_{1}\) must be approximated in this energy interval by a parabola with derivative equal to zero at 120 Mev.

The values of these phases are given in Table IV. The following arguments support such a choice:

a) Very good agreement of \(\sigma_{3/2}\) with the resonance formula (10) (see Fig. 5). If \(\alpha_{33}\) did not pass through \(90^\circ\), it would probably be necessary that at least two phases have a maximum at \(E_\pi \simeq 200\) Mev. For this argument, not only the dependence of the cross section on energy and angle is essential, but also its magnitude (see Fig. 4).

b) The coefficient \(b\) in formula (14) changes sign in the region \(E_\pi \simeq 180\) Mev (see Fig. 6). This is quite compatible with the fact that \(\alpha_{33}\) at this energy passes through \(90^\circ\).

c) Such behavior follows quite definitely from meson theory and indicates the reasons for the absence of a maximum in \(\sigma_{1/2}\) at an energy of 200 Mev (see Fig. 4).

d) The energy dependence and the change of sign of the interference term in the photoproduction of \(\pi^+\)-mesons (see the section “Photoproduction of \(\pi^+\)-mesons on nucleons”) indicate with great definiteness the existence of a resonance.

Further discussion of the phase analysis of scattering and its interpretation is contained in the section “Meson Theory.”

PHOTOPRODUCTION OF \(\pi\)-MESONS ON NUCLEONS

Four photoproduction processes are possible (as well as the inverse processes):

\[ \gamma + p \to \pi^+ + n \qquad (P_{\gamma +}) \]

\[ \gamma + n \to \pi^- + p \qquad (P_{\gamma -}) \]

\[ \gamma + p \to \pi^0 + p \qquad (P_{\gamma 0}) \]

\[ \gamma + n \to \pi^0 + n \qquad (P_{\gamma n0}) \]

The reactions \(P_{\gamma-}\) and \(P_{\gamma n0}\) have to be studied on bound neutrons (mainly on deuterium), so that the measurement of their cross sections is difficult and contains uncertainties.

Let us assume that for \(E_\gamma < 300 \div 400\) MeV (the energy of the \(\gamma\)-quanta in the laboratory coordinate system) mesons are emitted with appreciable probability only in \(S\) and \(P\) states relative to the nucleon. Arguments in favor of this point of view were given above in the section “Some physical considerations concerning \(\pi\)-mesons.” From Fig. 2 we concluded that the process \(P_{\gamma 0}\) proceeds mainly through emission in a \(P\)-state, whereas the process \(P_{\gamma +}\) proceeds at “low energies” through emission in \(S\)- and \(P\)-states. Then the general expression for the differential cross sections (in the center-of-inertia system) will be written in the form

\[ \sigma = A_0 + A_1 \cdot \cos\theta + A_2 \cdot \cos^2\theta . \tag{15} \]

We shall denote the differential cross sections of the four reactions mentioned, respectively, by

\[ \sigma(\gamma+),\ \sigma(\gamma-),\ \sigma(\gamma0)\ \text{and}\ \sigma(\gamma n0). \]

We shall write the total cross sections as \(\sigma_T(\gamma+)\), etc. The coefficient \(A_0\), in the general case, contains \(S\)- and \(P\)-waves and can be written in the form

\[ A_0 = A_0(S) + A_0(P), \tag{16} \]

where \(A_0(S)\) and \(A_0(P)\) refer respectively to the \(S\)- and \(P\)-states. The dependence of \(A\) on the energy at “low energies” (we shall take this to mean \(E_\gamma < 250\) MeV; the upper limit for “low energies” is, of course, not known a priori) can be obtained from Table I. For such energies we have:

\[ \left. \begin{aligned} A_0(S) &= \frac{\eta}{\nu}\, g_{0S},\\ A_0(P) &= \eta^3 \nu g_{0P},\\ A_1 &= -\eta^2 g_1,\\ A_2 &= -\eta^3 \nu g_2, \end{aligned} \right\} \tag{17} \]

where $\eta$ is the meson momentum and $\nu$ is the photon momentum in units of $\mu c$ in the center-of-inertia system. The dependence on $\nu$ is taken on the basis of meson theory. At energies sufficiently close to threshold, the dependence on $\nu$ is insignificant.

The cross section for the process $P_{\gamma}^{0}$ at an angle of $90^\circ$ in the laboratory coordinate system was measured by Silverman and Stearns$^{13}$ at Cornell. (The energy and emission angle of the recoil protons were measured for $E_\gamma$ in the interval from 200 MeV to 300 MeV.) An analogous investigation was carried out at the Massachusetts Institute of Technology by Osborne$^{14}$. This investigation included an analysis of the angular distribution; some of these data are shown in Fig. 2. At the California Institute of Technology*) the cross section of the reaction $P_{\gamma}^{0}$ was measured for several angles at photon energies $E_\gamma$ from 270 to 450 MeV. This work was done by Walker, Oakley, and Tollestrup$^{15}$; quite recently these same authors obtained further results. The method used was similar to that of the Cornell group (one of the $\gamma$ quanta from the decay of the $\pi$ meson was recorded in coincidence with the recoil proton).

The differential cross section for the process $P_{\gamma}^{+}$ was studied by Bernardini and Goldwasser$^{16}$ for $E_\gamma < 200$ MeV. Cross sections at low energies are evidently especially important for determining the quantities $g$ in formulas (17) and for determining multipole moments, which will be discussed below (in the part of this section entitled “Angular Distributions”).

At CIT the reaction $P_{\gamma}^{+}$ was studied for $E_\gamma$ from 200 to 400 MeV. Tollestrup, Keck, and Walker$^{17}$ used a telescope of scintillation counters to measure the ionization as a function of the residual range of $\pi^+$ mesons. Walker$^{18}$ measured the same cross sections, determining the energy and emission angle of the meson with a magnetic spectrometer. The CIT data$^{19}$ on the reaction $P_{\gamma}^{+}$ were analyzed by Bacher*) according to formula (15).

Total cross sections. We shall first consider the total cross sections for photoproduction of mesons. At sufficiently low energies the mesons must be emitted in an $S$ state; however, as has already been noted, the $S$-wave amplitude in the production of a $\pi^0$ meson is very small. This can be understood qualitatively on the basis of a simple model. Emission of pseudoscalar $\pi$ mesons in an $S$ state must occur through electric dipole absorption of the photon$^{20,21}$. If it is assumed that the amplitude of this process is proportional to the static dipole moment (in the center-of-inertia system)

) Instead of the full names Massachusetts Institute of Technology and California Institute of Technology, MIT and CIT are used below. (Translator’s note.)
*) We emphasize that the data on $\pi$ mesons and their analysis are preliminary in character.

of the corresponding meson–nucleon system in the final state, then

\[ \sigma(\gamma^-):\sigma(\gamma^+):\sigma(\gamma^0):\sigma(\gamma^{\prime 0}) = \left[1+\frac{\mu}{M}\right]^2:1:\frac{1}{2}\left(\frac{\mu}{M}\right)^2:0 \tag{18} \]

in the case when only the \(S\)-wave need be taken into account; here \(M\) is the nucleon mass.

This formula is consistent with the experimental fact of the small contribution of the \(S\)-wave to the production of a \(\pi^0\)-meson. From formula (18) we have:

\[ \frac{\sigma(\gamma^-)}{\sigma(\gamma^+)} = \left[1+\frac{\mu}{M}\right]^2 = 1.32. \tag{19} \]

This is also in agreement with Sands’ measurements\(^{22}\), where

\[ \frac{\sigma(\gamma^-)}{\sigma(\gamma^+)} \simeq 1.4 \tag{20} \]

for very small energies of the \(\pi\)-mesons.

In view of the successful application to scattering processes of formula (10) with one resonance level (Fig. 5), it is natural to try this approximation here as well, regarding photoproduction as a channel of the scattering reaction. If emission of \(\pi\)-mesons in the state \((l=3/2,\ j=3/2)\) predominates, then this approximation should be satisfactory for energies near the resonance, which is observed at \(E_\gamma \simeq 340\) Mev. Then\(^{13,23}\)

\[ \sigma_\gamma(\gamma^0) = 2\pi\left(\frac{\hbar}{k}\right)^2 \frac{\Gamma_\gamma \Gamma}{(E-E_0)^2+\frac{\Gamma^2}{4}}, \tag{21} \]

where \(k\) is the photon momentum in the center-of-inertia system, and \(\Gamma\), \(E\), and \(E_0\) denote the same quantities as in formula (10). For the width \(\Gamma_\gamma\) we have \(\left(\gamma=\dfrac{k}{\mu c}\right)\):

\[ \Gamma_\gamma = \frac{\gamma^2 f_\gamma}{1+\left(\dfrac{ak}{\hbar}\right)^2}, \tag{22} \]

where the channel radius \(a\) is given by relation (12). Here the coefficient \(f_\gamma\) plays the role of a reduced width and is the only arbitrary parameter in formula (21). We shall regard \(f_\gamma\) as a constant quantity

\[ f = 0.10\ \text{Mev}. \tag{23} \]

In Fig. 7, \(\sigma_T(\gamma^0)\), calculated by formula (21), is compared with experimental data. (The “experimental” values of \(\sigma_T\) were obtained from the observed differential cross sections under the assumption that the angular distribution has the form given in the section “Angular distributions.”) The values of \(\sigma_T(\gamma^0)\) thus determined from the experimental data agree well with formula (21) for \(0 < E_\gamma < 450\) MeV. The possible validity of formula (21) is of particular interest, since \(f_\gamma\) is the only free parameter.

We shall now consider the total cross section for the \(P_{\gamma^+}\) process, i.e. \(\sigma_T(\gamma^+)\). Here the behavior of the cross section at low energies indicates that

Fig. 7. Comparison of the cross section \(\sigma_T(\gamma + p \to \pi^0 + p)\), calculated by formula (21), with experiment. The first two points were calculated by formulas (17), and the remaining ones by formulas (19) and (20).

the \(S\)-wave predominates (see Fig. 2 and Ref. \(^{16}\)). Suppose that the only*) change in formula (21) consists in taking account of the \(S\)-wave, as predicted by pseudoscalar meson theory. Then

\[ \sigma_T(\gamma^+) = \sigma_0\left(\frac{q}{k}\right) \cdot \left[ \frac{1-\dfrac{k}{M \cdot c}} {\left(1+\dfrac{k}{M \cdot c}\right)^2} \right] + \frac{1}{2}\sigma_T(\gamma^0). \tag{24} \]

As has already been noted, the first term has the energy dependence predicted by meson theory. We choose \(\sigma_0\) to be a constant equal to

\[ \sigma_0 = 2.5 \cdot 10^{-28}\ \text{cm}^2. \tag{25} \]

*) Except that the contribution of the \(P\)-wave to \(\sigma_T(\gamma^+)\) is equal to one half of its contribution to \(\sigma_T(\gamma^0)\). The latter is a consequence of charge independence \(^{20}\).

In Fig. 8 formula (24) is compared with the experimental values of \(\sigma_T(\gamma^+)\). The agreement is not as good as it was for \(\sigma_T(\gamma^0)\). The calculated curve falls off insufficiently rapidly at high energies; however, this is most likely due to the unfortunate choice, in formula (24), of the term that takes account of the \(S\)-wave. This term alone exceeds the experimental cross section at energies above \(400\) MeV. In addition, the maximum of the calculated curve is apparently shifted somewhat toward higher energies. Formulas (21) and (24) simplify the picture, as will become clear in the discussion of the angular distribution. On the other hand, the agreement of these formulas with experiment, as is seen from Figs. 7 and 8, is not so bad. Consequently, apart from finer details, it may be considered that the state

\[ \left(I=\frac{3}{2},\quad j=\frac{3}{2}\right) \]

plays an essential role in photoproduction, and formulas (21) and (24) may serve as a reasonable approximation to the true cross sections.

Fig. 8. Comparison of the cross section \(\sigma_T(\gamma^+p\to \pi^+ + n)\), calculated by formula (24), with experiment. The first three points are taken from Ref. \(^{16}\); the remaining ones from Refs. \(^{17,18}\) and \(^{19}\). The dashed curve represents

\[ \sigma_T(\gamma^+) - \frac{1}{2}\sigma_T(\gamma^0). \]

Obviously, a priori there is no basis for choosing the radius of the photoproduction channel (22) to be the same as for the scattering channel (11). Indeed, in (17) we set this radius for \(\gamma\)-quanta equal to zero. Its finite value, used in (22), was taken by us in order to obtain a rapid decrease of the cross section after the resonance peak. It does not follow, of course, that \(f_\gamma\) should remain constant over a significant energy interval, so that at present it is difficult to say anything definite about the dependence of the cross section on \(k\) (except that it should not be strong at “low energies”).

It should also be noted that the term “resonance,” as applied to scattering and photoproduction, has not been precisely defined by us. It is desirable—

It would be useful to know not only that the phase \(\alpha_{33}\) passes through \(90^\circ\) at \(E_\pi \simeq 195\ \mathrm{MeV}\), but also its energy dependence in the region of such energies. Formulas (10) and (21) are based on an analogy with resonant nuclear reactions, but it is hardly possible to justify them for relativistic energies\(^{24}\). The most satisfactory treatment of processes with relativistic mesons was carried out by Sanc\(^{24}\). The expression he obtained for the “resonance” cross section is similar to formula (10).

Angular distributions. The angular distribution for the process \(P_\gamma^+\) was analyzed in the form (15) by Bacher\(^{19}\) for energies \(E_\gamma\) from 250 to 450 MeV. A similar analysis was carried out by Bernardini and Goldwasser\(^{25}\) for \(E_\gamma < 250\ \mathrm{MeV}\). Figure 9 presents

Fig. 9. Coefficients \(A_0, A_1\), and \(A_2\) for the angular distribution in the reaction \(\gamma + p \to \pi^+ + n\) (see formula (21)), calculated from experimental data. Three marked points are taken from Ref. \(^{16}\); points for energies above 250 MeV are taken from Ref. \(^{19}\).

the dependence of the coefficients of formula (15) on energy. The coefficients shown in Fig. 9 in fact differ somewhat from those obtained by the cited authors because, in the energy region \(E_\gamma < 250\ \mathrm{MeV}\), we used formulas (17) for the dependence of these coefficients on energy. Such a modification is compatible with the experimental data because of their inaccuracy. The values of the coefficients \(g\) in formulas (17) chosen by us are given in Table V.

Data on the angular distribution of the process \(P(\gamma^0)\) are still more limited. Table VI gives the values of the coefficients of for-

Table V

Coefficients \(g\) of formula (17) for the angular distribution of photomesons

Process	\(\gamma+p\to\pi^{+}+n\)	\(\gamma+p\to\pi^{0}+p\)
\(g_{OS}\)	10.5	—
\(g_{OP}\)	4.0	8.5
\(g_1\)	3.0	—
\(g_2\)	3.5	7.3

\(g\) are given in units of \(10^{-30}\ \text{cm}^2\). \(g_{OS}\) is given, probably, with an accuracy within 10%. The remaining values of \(g\) may have an error up to 25%, although their relative values are much more accurate.

The data on \(\pi^0\)-mesons are taken from Oakley and Walker \(^{24a}\) and from Osborn \(^{26}\).

mulas (15), obtained by Osborn \(^{26}\). The values \(g_{OP}\) and \(g_2\), obtained from formulas (17) and the data of the work carried out at the KTI, are given in Table V.

Table VI

Angular distribution for the reaction \(\gamma+p\to\pi^0+p\)*)

Energy of \(\gamma\)-quanta	from 220 to 280 MeV	from 280 to 330 MeV
\(A_0\)	\(9\pm1\)	\(18\pm1\)
\(A_1\)	\(-2.5\pm1\)	\(2.3\pm1\)
\(A_2\)	\(-7.5\pm2\)	\(-15\pm3\)

*) The coefficients \(A_0\), \(A_1\), and \(A_2\) are the same as in formula (31). They were obtained from preliminary data of the MTI.

Units — \(10^{-30}\dfrac{\text{cm}^2}{\text{steradian}}\).

Perhaps the most important consequence of Table V is that the contributions of the \(P\)-wave to \(\sigma(\gamma^0)\) and \(\sigma(\gamma^+)\) have the same angular dependence, and in magnitude are related approximately as \(2:1\).

This agrees with the simple “resonance” theory set forth in the first part of the section “Total Cross Sections.” We shall refine this theory.

A quantitative discussion of the angular distribution of photomesons is conveniently carried out by making use of the expansion of the amplitudes for absorption of \(\gamma\)-quanta in multipoles\(^{25,26}\).

Since the \(\pi\)-meson is a pseudoscalar particle, then, when it is emitted in an \(S\)-state, the transition must be an electric dipole one (we denote the corresponding matrix element by \(E_1\)). If, however, the meson is emitted in a \(P\)-state with \(j=\tfrac{1}{2}\), then such a transition is a magnetic dipole one (matrix element \(M_1\bigl(\tfrac{1}{2}\bigr)\)). A transition to a \(P\)-state with \(j=\tfrac{3}{2}\) may be either magnetic dipole (with matrix element \(M_1\bigl(\tfrac{3}{2}\bigr)\)) or electric quadrupole (with matrix element \(E_2\)). Since an arbitrary combination of these transitions is possible, the photoproduction amplitude \(T\) may be written briefly as follows\(^{20}\) (in the center-of-mass system):

\[ \begin{aligned} T={}& iE_1(\boldsymbol{\sigma}\hat{\mathbf e}) -M_1\!\left(\frac{1}{2}\right)\!\cdot \left\{([\hat{\mathbf k}\mathbf e]\mathbf q) -i(\boldsymbol{\sigma}[[\hat{\mathbf k}\mathbf e]\mathbf q])\right\}k^{-1}q^{-1} \\ &-M_1\!\left(\frac{3}{2}\right)\!\cdot \left\{2([\hat{\mathbf k}\mathbf e]\mathbf q) +i(\boldsymbol{\sigma}[[\hat{\mathbf k}\mathbf e]\mathbf q])\right\}k^{-1}q^{-1} \\ &+i\frac{1}{2}E_2 \left\{(\boldsymbol{\sigma}\mathbf k)(\hat{\mathbf e}\mathbf q) +(\boldsymbol{\sigma}\hat{\mathbf e})(\mathbf k\mathbf q)\right\} k^{-1}q^{-1}. \tag{26} \end{aligned} \]

Here \(\boldsymbol{\sigma}\) is the spin of the nucleon, \(\mathbf k\) the momentum of the photon, and \(\hat{\mathbf e}\) its polarization vector; \(\mathbf q\) is the momentum of the meson. As was already noted, there are four photoproduction processes, so that we have four amplitudes \(T\), which we shall denote by

\[ T^{+},\ T^{-},\ T^{0},\ T^{n0}, \]

and correspondingly there will be four sets of multipole moments: \(E_1^{+}\), \(E_1^{-}\), etc.

The differential cross section \(\sigma\) is obtained, as usual, by averaging \(|T|^2\) over all spin and polarization states:

\[ \begin{aligned} \sigma = W\Biggl\{& |E_1|^2+\left|M_1\!\left(\frac{1}{2}\right)\right|^2 +\left|M_1\!\left(\frac{3}{2}\right)\right|^2 \frac{1}{2}\,[5-3\cos^2\theta] \\ &+|E_2|^2\cdot\frac{1}{8}\,[1+\cos^2\theta] \\ &-2\operatorname{Re}\!\left[ E_1^{*}\left(M_1\!\left(\frac{3}{2}\right) -M_1\!\left(\frac{1}{2}\right)-\frac{1}{2}E_2\right) \right]\cos\theta \\ &-\frac{1}{2}\operatorname{Re}\!\left[ E_2^{*}\left(M_1\!\left(\frac{3}{2}\right) -M_1\!\left(\frac{1}{2}\right)\right) \right][3\cos^2\theta-1] \\ &-\operatorname{Re}\!\left[ M_1^{*}\!\left(\frac{3}{2}\right)\cdot M_1\!\left(\frac{1}{2}\right) \right]\cdot[3\cos^2\theta-1]\Biggr\}. \tag{27} \end{aligned} \]

Here \(\theta\) is the angle between \(\mathbf{k}\) and \(\mathbf{q}\) (all quantities are taken in the center-of-mass system). “\(Re(\ldots)\)” means “the real part of \((\ldots)\)”; \(W\) is a statistical weight, which is approximately equal to

\[ W=(2\pi)^4\frac{\eta\omega}{\left[1+\dfrac{k}{Mc}\right]^2}, \tag{28} \]

where \(\omega=[1+\gamma^2]^{1/2}\), and \(M\) is the nucleon mass.

For the four cross sections: \(\sigma(\gamma+)\), \(\sigma(\gamma-)\), etc., we have expressions analogous to (27). This means that there are 16 amplitudes. If the hypothesis of charge independence\({}^{7}\) is used, this number is reduced to 12 independent amplitudes. Although they are complex quantities, their complexity is trivial. Indeed, in the corresponding representation the complex phases of the multipole amplitudes can be expressed exactly in terms of the six phases \(\alpha\) (see “Scattering of \(\pi\)-mesons by nucleons”), which characterize the scattering of \(\pi\)-mesons\(*\). Thus we are left with twelve real parameters for describing the four reactions. These considerations and the explicit form of the amplitudes are given in the Appendix.

The coefficients \(A_0\), \(A_1\), and \(A_2\) of formula (15) can be expressed in terms of the multipole amplitudes with the aid of formula (27). We shall do this by Fermi’s method\({}^{27}\). Introduce the quantities

\[ \begin{aligned} X&\equiv \frac{3}{2}M_1\left(\frac{3}{2}\right)+\frac{1}{4}E_2,\\ Y&\equiv \frac{1}{2}\left[M_1\left(\frac{3}{2}\right)-\frac{1}{2}E_2\right]+M_1\left(\frac{1}{2}\right),\\ K&\equiv \left[M_1\left(\frac{3}{2}\right)-\frac{1}{2}E_2\right]-M_1\left(\frac{1}{2}\right). \end{aligned} \tag{29} \]

Then

\[ \begin{aligned} A_0&=W\left\{|E_1|^2+|X|^2+|Y|^2\right\},\\ A_1&=-W\left\{2Re\,[E_1^{*}K]\right\},\\ A_2&=W\left\{|K|^2-|X|^2-|Y|^2\right\}. \end{aligned} \tag{30} \]

From the form of the matrix elements for the individual multipoles (see the Appendix) it is clear that they are real at energies sufficiently close to threshold, when the phase shifts of the scattering are small (for example, for the process \(P_{\gamma+}\)) at \(E_\gamma<250\) MeV). This

\(*\) This fact was noted independently by K. Aizu, Fermi (unpublished work), and Watson\({}^{27a}\).

corresponds to the region of applicability (according to our assumption) of formulas (17). For these energies \(E_1\), \(X\), \(Y\), and \(K\) will be real, and, moreover, one should expect their dependence on energy to be simple. From formulas (30) one immediately obtains the Fermi relation \(^{27}\) between the coefficients \(g\) of formulas (17):

\[ g_1^2 = 4 g_{0S}\,[g_{0P}-g_2]. \tag{31} \]

Since each of the four coefficients \(g\) can be determined independently from experimental data, relation (31) can serve as a check on the model of meson photoproduction under consideration. Within the experimental errors for the process \(P_{\gamma+}\) (see Table V) this relation is satisfied.

The smallness of the quantity \(g_1^+\) in Table V, according to (31), shows that \(\sigma(\gamma^+)\) depends on the angle \(\theta\) approximately as \(\sin^2\theta\). Knowing \(g_{0S}^+\), one can directly determine the magnitude of the electric dipole matrix element near threshold. It is equal to

\[ \sqrt{\omega E_1^+}=\sqrt{\frac{\gamma_1}{\gamma}}\;3.3\cdot 10^{-15}\ \text{cm}. \tag{32} \]

Because of the smallness of \(g_1^+\), it is impossible to describe the cross section near threshold by only one term corresponding to the \(P\)-wave, contrary to the Brocker—Watson hypothesis \(^{20}\). Agreement with the experimental data can be obtained only if two of the three nonvanishing terms, \(M_1(^{3}/_{2})\), \(M_1(^{1}/_{2})\), and \(E_2\), are taken into account. There is no reason to suppose that all three terms are absent simultaneously.

Because of the smallness of \(E_1^0\) at threshold for the process \(P_{\gamma^0}\), relation (31) apparently will not be valid over a considerable energy interval. Here a more careful analysis is needed (see the footnote on p. 423).

Summarizing the discussion of the cross sections \(\sigma(\gamma^+)\) and \(\sigma(\gamma^0)\) at threshold energies, it should be noted that a noticeable contribution of the electric dipole transition occurs only in the cross section of the first process. The contributions of the \(P\)-waves to the cross sections of both processes depend on the angle approximately as \(\sin^2\theta\). The ratio \(2:1\), given by “resonance theory” for the contributions of the \(P\)-waves to the cross sections \(\sigma(\gamma^+)\) and \(\sigma(\gamma^0)\), is very well satisfied.

We shall now consider the experimental data for the resonance region. Since the resonance state \((I=^{3}/_{2},\ j=^{3}/_{2},\ l=1)\) contributes only to \(M_1(^{3}/_{2})\) and \(E_2\), one may expect precisely these terms to predominate. The expected predominance of the contribution of these terms in the resonance region, of course, does not mean that the terms corresponding to other multipoles are absent. However, if the resonance theory is valid, then in the first approximation the remaining terms should be

can be neglected. Then, on the basis of the formula for the multipole moments given in the Appendix, one can write, near the resonance energy:

\[ \left. \begin{aligned} \sqrt{\overline{W}}\,M_1\!\left(\frac{3}{2}\right) &= e^{i\alpha_{33}}\sin\alpha_{33}\, \begin{bmatrix} \dfrac{1}{2}\\[2mm] -\dfrac{3}{2} \end{bmatrix}_{\eta}^{\gamma}\,\mu_1,\\[3mm] \sqrt{\overline{W}}\,E_2 &= e^{i\alpha_{33}}\sin\alpha_{33}\, \begin{bmatrix} \dfrac{1}{2}\\[2mm] -\dfrac{3}{2} \end{bmatrix}_{\eta}^{\gamma}\,\varepsilon_2 . \end{aligned} \right\} \tag{33} \]

Here \(\mu_1\) and \(\varepsilon_2\) are constant real quantities. The complex factor \(e^{i\alpha_{33}}\) appears on the basis of the considerations given in the Appendix. The factor \(\sin\alpha_{33}\) represents a generalization of the energy dependence given by formula (21). The significance of this generalization is indicated in work \(^{23}\)*). For studying the process \(P_{\gamma}^{+}\), as is clear from Fig. 9, \(E_1^{+}\) is also essential. From the formula for \(E_1^{+}\) and for \(E_1^{0}\) given in the Appendix, and also from the fact that \(E_1^{0}\), apparently, is very small near threshold, it follows (in the notation of the Appendix)

\[ 2E_1^{(3)}=-\frac{1}{2}\left[E_1^{(1)}-2\delta E_1^{(1)}\right]. \tag{34} \]

We assume \(E_1^{0}=0\) at threshold, which seems a reasonable approximation. Using (34), one can write \(E_1^{+}\) in the form

\[ \sqrt{\overline{W}}\,E_1^{+} = \sqrt{\frac{\eta}{\gamma}}\, \frac{\varepsilon_1}{3} \left[e^{i\alpha_3}+2e^{i\alpha_1}\right]. \tag{35} \]

Here \(\varepsilon_1\) is determined by the relation

\[ \sqrt{2W}\,E_1^{(3)} = \sqrt{\frac{3}{\gamma}}\,\varepsilon_1 . \]

In what follows we shall assume that for \(E_\gamma<300\ \text{MeV}\) \(\varepsilon_1\) may be regarded as a constant (which, of course, may turn out to be incorrect).

*) Formulas (33) give a considerably more correct justification of the resonance theory than formula (21), but they are equivalent to this formula if \(\alpha_{33}\) has the form leading to formula (10). For example, in the more complicated theories of Sachs and Chew (see the section “Meson Theory”), photoproduction and scattering in the resonance region are connected by formula (33).

The formulas (30) can now be written, using (33) and (35), in the form

\[ \left. \begin{aligned} A_1^+ &= -\frac{2}{3}\frac{\sin\alpha_{33}}{\eta}\,\varepsilon_1 \left[\mu_1-\frac{1}{2}\varepsilon_2\right]\times \\ &\qquad \times \left[\cos(\alpha_{33}-\alpha_3)+2\cos(\alpha_{33}-\alpha_1)\right],\\[6pt] A_2^+ &= -\frac{\sin^2\alpha_{33}}{4}\left[\frac{\nu}{\eta^3}\right] \left\{ \left[3\mu_1+\frac{1}{2}\varepsilon_2\right]^2 -3\left[\mu_1-\frac{1}{2}\varepsilon_2\right]^2 \right\},\\[6pt] A_0^+(P) &= \frac{\sin^2\alpha_{33}}{4}\left[\frac{\nu}{\eta^3}\right] \left\{ \left[3\mu_1+\frac{1}{2}\varepsilon_2\right]^2 + \left[\mu_1-\frac{1}{2}\varepsilon_2\right]^2 \right\}. \end{aligned} \right\} \tag{36} \]

From (32) we obtain

\[ \varepsilon_1=3.3\cdot 10^{-15}\ \text{cm}. \tag{37} \]

From the data of Fig. 9 one can determine the quantities \(A\), and thus the formulas (36), for any given energy, are three equations for determining two parameters: \(\mu_1\) and \(\varepsilon_2\). From Fig. 8 it is seen that the dependence of the coefficients \(A\) on the energy agrees only in general outline with the dependence calculated on the basis of formulas (36)*).

It is interesting that \(A_1^+\) must change sign when
\[ [\cos(\alpha_{33}-\alpha_3)+2\cos(\alpha_{33}-\alpha_1)] \]
passes through zero. Taking the phases \(\alpha\) from Table IV, one can find that this bracket is equal to zero for \(E_\gamma=335\ \text{Mev}\). From Fig. 9 it was found that the experimental value of \(A_1\) goes to zero at \(E_\gamma\simeq 325\ \text{Mev}\). The agreement is evidently much better than is guaranteed by the accuracy of the experimental data. Since \(\varepsilon_1\) is known, in order to determine \(\left(\mu_1-\frac{1}{2}\varepsilon_2\right)\) one can use the slope in the dependence of \(A_1^+\) on the energy when this quantity passes through zero. We obtain:

\[ \mu_1-\frac{1}{2}\varepsilon_2=1.6\cdot 10^{-15}\ \text{cm}. \tag{38} \]

However, here one should exercise a certain caution because of the limited accuracy of the preliminary experiments to which we referred\({}^{19}\). Now one can determine \(\mu_1\) and \(\varepsilon_2\), using either

*) Chu and Bernardini pointed out that if one abandons the energy dependence \(\alpha_{33}\), which leads to formula (10), and uses instead the data of Table IV, then the results on scattering and photoproduction are described by (36) considerably better.

\(A_2^+\), \(A_0^+(P)\). This leads to the following values of \(\mu_1\) and \(\varepsilon_2\):

\[ \left. \begin{aligned} \mu_1 &= 2.5\cdot 10^{-15}\ \text{cm},\\ \varepsilon_2 &= 1.8\cdot 10^{-15}\ \text{cm}. \end{aligned} \right\} \tag{39} \]

Fortunately, each of the two independent paths gives values differing from the values (39) by no more than 10%. This is a convincing confirmation of our model (although perhaps not so spectacular as the vanishing of \(A_1^+\) at \(E_\gamma = 335\) MeV predicted by this theory).

A serious test of the resonance model is the prediction of the ratios

\[ \frac{A_2^0}{A_2^+}=\frac{A_0^0}{A_0^+(P)}=2. \]

These ratios are satisfied within the experimental errors (although the ratio obtained from experiment may be equal to 2.5). From Fig. 9 and Table VI it is seen that

\[ \frac{A_0^+(P)}{A_2^+}\simeq 1.35;\qquad \frac{A_0^0}{A_2^0}\simeq 1.3. \tag{40} \]

These ratios may, evidently, contain large experimental errors (\(\simeq 25\%\)). If there were no contribution from the electric quadrupole,\(^{20}\) then these ratios in both cases would have to be equal to \(\frac{5}{3}\simeq 1.67\). These ratios, as well as formulas (38) and (39), indicate that there is a contribution from the electric quadrupole transition.

If the amplitudes for the individual multipoles have already been determined in the “resonance region” for the process \(P_{\gamma+}\), then the theory unambiguously predicts the coefficients \(A_1^0\), \(A_0^0\), and \(A_2^0\) for the process \(P_{\gamma0}\). In our notation (see the Addendum and the footnote on p. 423)

\[ \left. \begin{aligned} A_0^0 &\simeq A_0^0(P)=\frac{1}{2}A_0^+(P),\\ A_2^0 &= \frac{1}{2}A_2^+,\\ A_1^0 &= -\frac{4}{3}\frac{\sin\alpha_{33}}{\eta}\,\varepsilon_1 \left[\mu_1-\frac{1}{2}\varepsilon_2\right]\times\\ &\quad \times\left[\cos(\alpha_{33}-\alpha_3)-\cos(\alpha_{33}-\alpha_1)\right]. \end{aligned} \right\} \tag{41} \]

When more detailed experimental data appear, these relations will make possible a serious test of the “resonance model.”

Before proceeding to the consideration of the more complicated phenomenon of meson production in nucleon collisions, let us summarize the discussion of the phenomena of scattering and photoproduction of mesons.

The three postulates from the section “Some physical considerations concerning \(\pi\)-mesons,” namely: (a) the finite radius of the forces; (b) the hypothesis of charge independence; (c) the hypothesis of the existence of a resonance, provide a reasonable basis for a successful description of the available data on meson phenomena at “low energies.” Such a treatment apparently reveals the inherent simplicity of the phenomenon and at the same time reduces the difficulties that will have to be overcome by a more justified and detailed theory.

PRODUCTION OF \(\pi\)-MESONS IN NUCLEON COLLISIONS \(^*)\)

In this section we shall consider the emission and absorption of \(\pi\)-mesons by a system of two nucleons. Experimentally, the emission process can be studied by recording \(\pi\)-mesons that appear when hydrogen is bombarded with neutrons or protons (or, equivalently, after subtracting the results obtained with polyethylene and with carbon). Absorption of \(\pi\)-mesons can be studied in the reactions

\[ \pi^{+}+D \to 2P \quad \text{and} \quad \pi^{-}+D \to 2N. \]

According to the hypothesis of charge independence, in the emission of a meson the isotopic spin \(I\) of two nucleons can undergo one of three transitions:

\[ I=1 \to I=0, \quad \text{the total cross section we shall denote by } \sigma_{10}; \]

\[ I=0 \to I=1, \quad \text{the total cross section we shall denote by } \sigma_{01}; \]

\[ I=1 \to I=1, \quad \text{the total cross section we shall denote by } \sigma_{11}. \]

The process \(I=0 \to I=0\) is forbidden, since the emitted meson carries away isotopic spin equal to unity.

Since for the deuteron \(I=0\), only the first of the processes listed above can lead to the formation of a deuteron. Therefore we shall write \(\sigma_{10}=\sigma_{10'}+\sigma_{10''}\), where \(\sigma_{10'}\) refers to the process with formation of a deuteron (a reaction with a “bound state”) and \(\sigma_{10''}\) to the formation of two free nucleons with \(I=0\) (a reaction with an “unbound state”).

The total cross sections for the various observed reactions that proceed with the formation of mesons can be expressed in terms of \(\sigma_{10'}\), \(\sigma_{10''}\), \(\sigma_{01}\), and \(\sigma_{11}\)

\(^*)\) This section overlaps with the article by A. Rosenfeld, Phys. Rev. 96, 139 (1954); the notation in this section coincides with Rosenfeld’s notation and differs somewhat from the notation of the preceding sections.

INTERACTION OF π-MESONS WITH NUCLEONS

in the following way:

\[ \begin{aligned} p + P &\to \pi^+ + D && \sigma=\sigma_{10'}\\ p + P &\to \pi^+ + N + P && \sigma=\sigma_{10''}+\sigma_{11}\\ p + P &\to \pi^0 + P + P && \sigma=\sigma_{11}\\ N + P &\to \pi^0 + D && \sigma=\frac{1}{2}\sigma_{10'}\\ N + P &\to \pi^0 + N + P && \sigma=\frac{1}{2}\sigma_{10''}+\frac{1}{2}\sigma_{01}\\ N + P &\to \pi^+ + N + N && \sigma=\frac{1}{2}\sigma_{11}+\frac{1}{2}\sigma_{01}\\ N + P &\to \pi^- + P + P && \sigma=\frac{1}{2}\sigma_{11}+\frac{1}{2}\sigma_{01} \end{aligned} \tag{42} \]

The factor \(\frac{1}{2}\) in the cross sections for processes in which a neutron and a proton take part arises because the \(N-P\) system has equal probabilities of possessing \(I=1\) and \(I=0\), whereas the \(P-P\) system always has \(I=1\).

The cross sections of the two observable absorption reactions can be expressed through \(\sigma_{10'}\), using the principle of detailed balance:

\[ \begin{aligned} \pi^+ + D &\to P + P\\ \pi^- + D &\to N + N \end{aligned} \qquad \sigma=\sigma_{10'}\frac{2}{3}\frac{p^2}{\mu^2 c^2}\frac{1}{\eta^2}. \tag{43} \]

Here \(\mu\) is the mass of the \(\pi\)-meson, \(c\) is the speed of light, \(p\) is the final momentum of the nucleon in the center-of-inertia system, and \(\eta\) is the momentum of the \(\pi\)-meson in the center-of-inertia system, expressed in units \(\mu c\).

All relations obtained on the basis of the hypothesis of charge independence require corrections for Coulomb forces, for the difference between the masses of the neutron and the proton, for the difference between the masses of charged and neutral \(\pi\)-mesons, and for other small charge-dependent effects (if one disregards possible substantial violations of the principle of charge independence).

In considering meson production we shall restrict ourselves to that range of energies of the incident particles in which \(\eta\) is always \(<1\) (energy less than \(450\) MeV). In this energy interval the energy of relative motion \(E\) of the two nucleons in the final state is always less than \(\mu c^2(\sqrt{2}-1)=57\) MeV and tends to decrease because the \(\pi\)-mesons carry away predominantly large momenta; we shall assume that, essentially, \(E<25\) MeV. It may be supposed that meson production occurs at a characteristic distance \(R\) from the center of collision and that \(R\) is of the order of magnitude of \(\frac{\hbar}{\mu c}\). Thus, in the energy interval under consideration, the product of \(R\) either by the final momentum of the neutron or by the final momentum

meson \(\ll \hbar\). Therefore one may assume that the meson will be emitted in an \(S\)- or \(P\)-state with respect to the two nucleons and that the two nucleons obtained as a result of the reaction will be in \(S\) or \(P\) states with respect to one another. For nucleons, the strong attraction in the \(S\)-state (in contrast to the relatively weak forces in the \(P\)-state at small energies) will lead to a strong increase in the role of the state with \(l=0\), and together with this to the predominance of small values of \(E\). Experiment indicates the predominance of the \(P\)-state of the meson (with the exception of energies close to threshold); this indicates that the mesons are produced owing to the interaction of the nucleon spin with the meson momentum, as in the pseudoscalar theory.

In experiments with absorption, the relative angular momentum of the initial system of two nucleons is always deuteron-like, i.e. with predominance of the \({}^{3}S_{1}\)-state with a small (\(\sim 4\%\)) admixture of the \({}^{3}D_{1}\) state. We shall again consider only energies for which \(\eta<1\), so that the main contribution to the process will be given by the \(S\)- and \(P\)-states of the mesons.

Let us consider the first of (42), the best-known reaction, which proceeds with the formation (or breakup) of the deuteron. The cross section for meson production with formation of the deuteron is denoted by \(\sigma_{10}\). In this process (if the meson is emitted in an \(S\)-state) the total angular momentum of the final state is simply equal to the angular momentum of the deuteron, \(J=1\). The parity in the final state is negative because of the pseudoscalar nature of the mesons. Thus in the initial state of the two nucleons one must have \(I=1\), \(J=1\), and negative parity. According to the Pauli principle, the wave function of two nucleons, symmetric in isotopic spin (\(I=1\)) and antisymmetric in spatial coordinates (negative parity), must be symmetric in spins (triplet state). Thus the only possible initial state is \({}^{3}P_{1}\).

If, however, the meson is emitted in a \(P\)-state, then in the final state the angular momentum may be \(J=0, 1\), or \(2\), and the parity positive. The initial state with \(I=1\) and positive parity must be a singlet, and then only the states \({}^{1}S_{0}\) and \({}^{1}D_{2}\) are possible.

Thus, at small energies there are three possibilities:

\[ (\mathrm{a}')\quad 2N({}^{3}P_{1})\to D({}^{3}S_{1})+\text{ meson in an }S\text{-state}, \]

\[ (\mathrm{b}')\quad 2N({}^{1}D_{2})\to D({}^{3}S_{1})+\text{ meson in a }P\text{-state}, \]

\[ (\mathrm{c}')\quad 2N({}^{1}S_{0})\to D({}^{3}S_{1})+\text{ meson in a }P\text{-state}. \]

Let us denote by \(\delta_{0}\) the ratio of the complex amplitude for process \((\mathrm{c}')\) to the complex amplitude of process \((\mathrm{b}')\); the subscript 0 notes the fact that for \((\mathrm{c}')\), \(J=0\). Similarly, by \(\delta_{1}\) we denote the ratio of the amplitudes of processes \((\mathrm{a}')\) and \((\mathrm{b}')\). These ratios of complex amplitudes are connected with the ordinary matrix elements of the \(S\)-matrix \(u_{0}, u_{1}\), and \(u_{2}\)

respectively for \(J=0\), \(J=1\), and \(J=2\) by the formulas:

\[ \begin{aligned} \delta_0&=-\frac{u_0}{\sqrt{5}\,u_2},\\ \delta_1&=-i\sqrt{\frac{3}{5}}\,\frac{u_1}{u_2}. \end{aligned} \tag{44} \]

Neither the differential nor the total cross section will contain interference terms of process \((a')\) with processes \((b')\) and \((c')\), since the initial state of process \((a')\) is a triplet state, whereas the processes \((b')\) and \((c')\) are singlet states. Consequently, we may consider separately the production of mesons in the \(S\)-state and in the \(P\)-state.

A reaction proceeding with emission of a meson in the \(S\)-state is characterized by an isotropic angular distribution, and its cross section (near threshold) must be proportional to the meson momentum. For small energies one may write:

\[ 4\pi\,\frac{d\sigma_{10'}}{d\Omega}\,(S\text{-wave})=a\eta, \tag{45} \]

where \(a\) depends neither on the angle nor on the energy. Using experimental data and carrying out a theoretical calculation, Brueckner, Serber, and Watson\(^3\) determined the value of the constant \(a\). We shall present their reasoning.

Panofsky, Aamodt, and Hadley\(^2\) measured the ratio of the yields of the two processes \(\pi^-+D\to 2N\) and \(\pi^-+D\to 2N+\gamma\), which occur as a result of absorption of \(\pi\)-mesons in deuterium. The \(\pi\)-mesons are apparently absorbed from the \(S\) orbit of the mesoatom of deuterium. The ratio obtained by Panofsky is approximately equal to the ratio of the corresponding cross sections for absorption of slow \(\pi\)-mesons. Using its value \(7/3\), we have for slow mesons:

\[ \sigma(\pi^-+D\to 2N)=\frac{7}{3}\,\sigma(\pi^-+D\to 2N+\gamma). \tag{46} \]

If one uses formula (43) and the fact that for slow mesons \(p^2=\mu c^2\cdot M\), where \(M\) is the nucleon mass, then

\[ \sigma_{10'}=\sigma(P+P\to \pi^++D)=\frac{3}{2}\eta^2\frac{\mu}{M}\,\sigma(\pi^-+D\to 2N). \tag{47} \]

Brueckner, Serber, and Watson, making an estimate, found that

\[ \sigma(\pi^-+D\to 2N+\gamma)=\frac{2}{3}\,\sigma(\gamma+N\to \pi^-+P). \tag{48} \]

From the principle of detailed balance one obtains the relation

\[ \sigma(\pi^-+P\to N+\gamma)=\frac{2}{\eta^2}\,\sigma(\gamma+N\to \pi^-+P). \tag{49} \]

The ratio of the photoproduction cross section of a \(\pi^-\)-meson to the photoproduction cross section of a \(\pi^+\)-meson on deuterium turns out\(^{22}\) to be equal to \(1.4\). This means

\[ \sigma(\gamma+N\to \pi^-+P)=1.4\sigma(\gamma+P\to \pi^++N). \tag{50} \]

Combining (46), (47), (48), (49), and (50), we obtain:

\[ \sigma_{10'}=\frac{14}{3}\,\frac{1.4\mu}{M}\, \sigma(\gamma+P\to \pi^++N)\simeq \sigma(\gamma+P\to \pi^++N). \tag{51} \]

Using Bernardini’s\(^{16}\) value \(\eta(0.14\ \text{mb})\) for the cross section \(\sigma(\gamma+P\to \pi^++N)\) in the threshold-energy region, and taking into account possible errors due to uncertainties in passing from (46) to (50), we obtain approximately:

\[ \sigma_{10'}(S\text{-wave})=\eta\cdot(0.14\pm0.05)\ \text{mb} \tag{52} \]

or

\[ \alpha=(0.14\pm0.05)\ \text{mb}. \tag{53} \]

In contrast to the \(S\)-wave, the contribution of the \(P\)-wave to \(\sigma_{10'}\) near threshold is characterized by an \(\eta^3\) dependence. The angular distribution depends on the complex number \(\delta_0\), which is equal to the ratio of the amplitude of process \((c')\) to the amplitude of process \((b')\). For the case of the \(P\)-wave it has the form \(X+\cos^2\theta\), where \(X\) is expressed by the formula

\[ X=\left[\left|\frac{2-\sqrt{2}\,\delta_0}{1+\sqrt{2}\delta_0}\right|^2-1\right]^{-1}. \tag{54} \]

Thus, near threshold the dependence of the cross section on energy and on angle in the case of the \(P\)-wave is expressed by the formula

\[ 4\pi\,\frac{d\sigma_{10'}}{d\Omega}(P\text{-wave}) =\beta\cdot\eta^3\,\frac{X+\cos^2\theta}{X+\frac{1}{3}}. \tag{55} \]

Combining the contributions from the \(S\)- and \(P\)-waves, we obtain for the total cross section

\[ \sigma_{10'}=\alpha\eta+\beta\eta^3, \tag{56} \]

and for the angular distribution

\[ A+\cos^2\theta, \]

where

\[ A=X+\frac{X+\frac{1}{3}}{\eta^2}\cdot\frac{\alpha}{\beta}. \tag{57} \]

Beginning at some energy above threshold, all parameters \(\alpha\), \(\beta\), and \(X\) will depend on \(\eta\). However, it is reasonable to assume that as long as the meson wavelength is greater than the critical distance \(R\) (i.e., \(\eta < 1\)), this dependence is weak. We shall try to interpret the experimental data by assuming that, in the energy interval under consideration, these parameters are constant.

It is now possible to determine \(\beta\) and \(X\) by comparing, at a definite energy, the total cross sections and angular distributions given by formulas (56) and (57) with the experimental ones. We shall use the data of Crawford and Stevenson\(^{28}\) (see Table VII), which were obtained for \(\eta\) equal to 0.58:

\[ \sigma_{\text{total}} = (0.269 \pm 0.026)\ \text{mb}, \]

\[ A = 0.29 \pm 0.08. \]

If, in accordance with (53), we take \(\alpha = 0.14\ \text{mb}\), then we obtain:

\[ \beta = 1.0\ \text{mb} \tag{58} \]

and

\[ X = 0.1. \tag{59} \]

We now compare our semiempirical formula, which includes the three parameters,

\[ 4\pi \frac{d\sigma}{d\Omega} = \left\{ 0.14\eta + 1.0\,\eta^{3} \frac{(0.1+\cos^{2}\theta)} {\left(0.1+\frac{1}{3}\right)} \right\} \ \text{mb}, \tag{60} \]

with the experimental data currently known for the reactions \(P+P \to \pi^{+}+D\) and \(\pi^{+}+D \to P+P\). This comparison is made in Table VII. We note that the absorption cross section of \(\pi\)-mesons has been recalculated into the corresponding cross section of the inverse process. In accordance with this, the energies have been converted into equivalent energies of bombarding protons.

The results of Cartwright et al. (Table VII, work (c)) and of Durbin et al. (Table VII, work (e)) at energies of about \(340\ \text{MeV}\) are the only data that seriously diverge from the semiempirical formula and, apparently, should be replaced by the results of Crawford and Stevenson.

There are two experimental results for the reaction \(N+P \to \pi^{0}+D\), which, in accordance with the charge-independence hypothesis, should be identical to the reaction \(P+P \to \pi^{+}+D\), except for a factor \(1/2\) in the absolute value of the cross section. Hildebrand\(^{29}\) gives the angular distribution (for this reaction) in the form \(0.21 \pm 0.06 + \cos^{2}\theta\) at \(\eta = 0.96\), while Shlyuter\(^{30}\) gives the total cross section,

Table VII

Experimental data on meson production in nucleon collisions

Reaction considered	Energy of incident protons in MeV	$\eta$	Measured value $\sigma_{10}$ ($\mu$b)	Calculated value $\sigma_{10}$ ($\mu$b)	Measured value $A$	Calculated value $A$	Literature
$P+P \to \pi^{+}D$	311	0.39	$0.100 \pm 0.013$	0.11	—	0.49	(a); (b)
$P+P \to \pi^{+}D$	315	0.42	$0.133 \pm 0.016$	0.13	—	0.44	(a); (b)
$P+P \to \pi^{+}D$	321	0.46	$0.168 \pm 0.018$	0.16	—	0.38	(a); (b)
$P+P \to \pi^{+}D$	324	0.48	$0.178 \pm 0.016$	0.17	$0.28 \pm 0.07$	0.36	(a); (b)
$P+P \to \pi^{+}D$	330	0.52	$0.228 \pm 0.017$	0.21	—	0.32	(a); (b)
$P+P \to \pi^{+}D$	332	0.54	$0.245 \pm 0.013$	0.23	$0.32 \pm 0.05$	0.31	(b)
$P+P \to \pi^{+}D$	336	0.56	$0.264 \pm 0.019$	0.25	—	0.29	(a); (b)
$P+P \to \pi^{+}D$	338	0.58	$0.269 \pm 0.026$	0.28	$0.29 \pm 0.08$	0.28	(b)
$P+P \to \pi^{+}D$	340	0.59	$0.18 \pm 0.06$	0.29	$0.11 \pm 0.06$	0.27	(c)
$\pi^{+}+D \to P+P$	341	0.59	$0.284 \pm 0.050$	0.29	—	0.27	(d)
$\pi^{+}+D \to P+P$	346	0.62	$0.22 \pm 0.02$	0.33	$0.19 \pm 0.09$	0.26	(e)
$\pi^{+}+D \to P+P$	382	0.82	$0.66 \pm 0.07$	0.67	$0.26 \pm 0.14$	0.19	(e)
$\pi^{+}+D \to P+P$	413	0.96	$0.97 \pm 0.10$	1.02	$0.18 \pm 0.15$	0.17	(e)
$P+P \to \pi^{+}+D$	437	1.05	$1.15 \pm 0.13$	1.30	$0.20 \pm 0.02$	0.15	(f)

(a): A. G. Schulz, U. S. Atomic Energy Commission Document UCRL—1756 (1952).
(b): F. S. Crawford and M. L. Stevenson, U. S. Atomic Energy Commission Document UCRL—2187—2188 (1953).
(c): W. F. Cartwright, C. Richman, M. N. Whitehead and H. A. Wilcox, Phys. Rev. 91, 677 (1953).
(d): D. L. Clark, A. Roberts and R. Wilson, Phys. Rev. 83, 649 (1951).
(e): R. Durbin, H. Loar and J. Steinberger, Phys. Rev. 84, 581 (1951).
(f): T. H. Fields, J. G. Fox, J. A. Kane, R. A. Stallwood and R. B. Sutton, Bull. Am. Phys. Soc. 29 (4) (1954).

equal to \(0.6 \pm 0.2\) millibarn for \(0.85 < \eta < 1.05\). Comparison with (60) shows that these data, within the experimental errors, do not contradict the hypothesis of charge independence.

Determination of the parameter \(X\) in the angular distribution of the \(P\)-wave does not fix the value of the complex number \(\delta_0\), equal to the ratio of the contributions \(J=0\) and \(J=2\) to meson production in the \(P\)-state. The value \(\delta_0\) is limited only by the fact that it lies in a certain circle in the complex plane. Indeed,

\[ \delta_0=-\frac{1}{\sqrt{2}}(1+3X)+\frac{3}{\sqrt{2}}X(X+1)e^{i\omega_0}. \tag{61} \]

For \(X=0.1\) we have:

\[ \delta_0=-0.92+0.70e^{i\omega_0}. \tag{62} \]

The magnitude of the phase angle cannot be determined from the data of the experiments considered. However, it can be found by measuring the polarization of deuterons in the reaction \(P+P\to \pi^+ + D\), or the asymmetry in the angular distribution of the reaction \(\pi^+ + D\to P+P\), when the target deuterons are polarized. In the case of the reaction \(P+P\to \pi^+ + D\), the deuterons produced by the \(P\)-wave part of the cross section are polarized perpendicular to the scattering plane with a degree of polarization \(P_p\), determined by Watson and Richman\({}^{31}\):

\[ P_p= \frac{2\sqrt{X(X+1)}\sin\theta\cos\theta}{X+\cos^2\theta} \cdot \frac{\sin\omega_0}{1+2X-2\sqrt{X(X+1)}\cos\omega_0}, \tag{63} \]

where \(-1 \leqq P_p \leqq 1\). Deuterons due to the \(S\)-wave part of the cross section do not possess such polarization.

We have not yet considered the relative phase of the amplitudes for meson production in the \(S\)-state and in the \(P\)-state. The complex parameter \(\delta_1\) gives the ratio of the \(S\)-wave amplitude (\(J=1\)) to the \(P\)-wave amplitude (\(J=2\)) of the mesons. The absolute value of \(\delta_1\) is determined by the relation

\[ \frac{|\delta_1|^2}{1+|\delta_0|^2}=\frac{\alpha}{\beta\eta^2}, \tag{64} \]

but the phase \(\delta_1\) has not entered into our discussion. It can be found by measuring the angular distribution of the reaction \(P+P\to \pi^+ + D\) with polarized protons, or by measuring the polarization of protons in the reaction \(\pi^+ + D\to P+P\). The first experiment is obviously simpler. If the beam of incident protons (flying in the direction of the \(z\) axis) is characterized by a degree of polarization \(P_1\) in the direction of the \(X\) axis, then the angular distribution of mesons and deuterons is given by the following formula\({}^{32}\):

\[ \frac{d\sigma_{10}}{d\Omega} = A+\cos^2\theta - P_1\cdot QA\cdot \sin\theta\sin\Phi, \tag{65} \]

where

\[ Q=-\frac{1}{\sqrt{2}}\frac{2b}{1+b^{2}}\sin(\psi-\tau_{1}), \tag{66} \]

where

\[ \delta_{1}=|\delta_{1}|\cdot e^{i\tau_{1}}, \tag{67} \]

\[ b=\frac{\left|\delta_{0}+\sqrt{\frac{1}{2}}\right|}{|\delta_{1}|}, \tag{68} \]

and

\[ \psi=\arg\left(\delta_{0}+\sqrt{\frac{1}{2}}\right). \tag{69} \]

We assume that the parameters do not depend substantially on the energy. Under this condition one should expect that, while \(|\delta_{1}|\) is inversely proportional to \(\eta\) [as is seen from (64)], the phase \(\tau_{1}\) is approximately constant for \(\eta \leq 1\). Further, the asymmetry parameter \(Q\) reaches its maximum value

\[ \frac{\sqrt{2}}{2}\cdot\sin(\psi-\tau_{1}), \]

when

\[ |\delta_{1}|=\left|\delta_{0}+\sqrt{\frac{1}{2}}\right|, \tag{70} \]

or, according to (64), when

\[ \eta=\eta_{c}=\sqrt{\frac{\alpha}{\beta}}\cdot \frac{\sqrt{1+|\delta_{0}|^{2}}}{\left|\delta_{0}+\sqrt{\frac{1}{2}}\right|}. \tag{71} \]

Using (64), (66), and (71), we obtain the dependence of the asymmetry on energy

\[ \frac{Q}{Q_{\max}}=\frac{2\eta\cdot\eta_{c}}{\eta^{2}+\eta_{c}^{2}}. \tag{72} \]

The energy at which the asymmetry is maximal corresponds to \(\eta_{c}\simeq 0.77\), independently of the magnitude of the phase \(\omega_{0}\). We take \(\frac{\alpha}{\beta}=0.14\), as in (60), and use (62) for \(\delta_{0}\). The maximum value of \(|Q|\) may vary between 0 and 0.71, depending on the phase \(\psi-\tau_{1}\). If this phase led to a significant asymmetry, then the study of reactions with polarized protons could provide information on the contribution of the \(S\)-wave and would aid in determining the phases*).

*) Preliminary results of Marshall J., Marshall L., and de Carvalho (private communication) indicate a very small asymmetry (\(Q=4\pm 6\) percent) at \(\eta=1\); the phase \(\psi-\tau_{1}\) must therefore be very small, and consequently this method cannot yield an estimate of the contribution of the \(S\)-wave.

Determination of the phases is very important, since they are closely connected with the phase shifts of proton scattering on a proton. We introduced:

\[ \delta_1=|\delta_1|e^{i\tau_1}, \tag{67} \]

analogously we write:

\[ \delta_0=|\delta_0|e^{i\tau_0}. \tag{73} \]

It can be shown that near the threshold of production of a \(\pi\)-meson \(\tau_1\) and \(\tau_0\) can be expressed through the \(P-P\) scattering phases \(\alpha({}^{1}S_0)\), \(\alpha({}^{1}D_2)\), and \(\alpha({}^{3}P_1)\) by means of the relations (where \(n\) and \(n'\) are integers):

\[ \tau_0=\alpha({}^{1}S_0)-\alpha({}^{1}D_2)+n\pi \tag{74} \]

and

\[ \tau_1=\alpha({}^{3}P_1)-\alpha({}^{1}D_2)+\left(n'+\frac{1}{2}\right)\pi . \tag{75} \]

Formulas (74) and (75) are obtained from considerations analogous to those used in the Appendix for determining the phases of the matrix elements of \(\pi\)-meson photoproduction through the phase shifts of meson scattering on nucleons. Formulas (74) and (75) lose their validity at energies for which there is appreciable elastic or inelastic scattering of mesons on deuterons. If the character of such scattering is clarified, then formulas (74) and (75) can be correspondingly corrected.

We now pass from the process of meson production with the formation of a deuteron to the corresponding reaction in which the two final nucleons turn out to be unbound; in the notation (42) we pass from \(\sigma_{10'}\) to \(\sigma_{10''}\). The final nucleons again have \(I=0\) and relative orbital angular momentum equal to zero; they are in the \({}^{3}S_1\) state, like the nucleons of the deuteron. As in the case of the deuteron, three processes are possible:

\[ (a'')\qquad 2N({}^{3}P_1)\to 2N({}^{3}S_1)+\text{meson in an }S\text{-state}, \]

\[ (b'')\qquad 2N({}^{1}D_2)\to 2N({}^{3}S_1)+\text{meson in a }P\text{-state}, \]

\[ (c'')\qquad 2N({}^{1}S_0)\to 2N({}^{3}S_1)+\text{meson in a }P\text{-state}. \]

It may be assumed, in accordance with the adopted point of view, that the basic matrix element for each of these processes at a given meson momentum depends only weakly on the energy of the final nucleons, and it may be regarded as approximately constant in the energy interval under consideration. In this sense one may consider that the matrix element for process \((a'')\) should be the same as for \((a')\) at the same meson momentum, and so on. Then from the semiempirical formula for \(\sigma_{10'}\) one can obtain the magnitude of \(\sigma_{10''}\) and the energy spectrum of the mesons formed in the “reaction with an unbound state.” The only factors that must be taken into account are the density of final states and the effect of the interaction of the nucleons in the final state.

Let the energy of the incident particles be such that the total kinetic energy of the particles in the center-of-mass system after the collision is equal to \(T_0\). Part of it, \(T=\mu c^2(\sqrt{1+\eta^2}-1)\), is carried away by the meson, and part \(E=T_0-T\) goes into the relative motion of the system of two nucleons (neglecting the recoil of this system). The differential cross section for the “reaction with an unbound state” is written\({}^{23}\) in the form:

\[ d\sigma_{10'}=(\alpha\eta+\beta\eta^3)\frac{\rho_E\,dE}{1}\cdot \frac{|\psi(R)|^2}{|\psi_D(R)|^2}, \tag{76} \]

where \(\alpha\eta+\beta\eta^3\) is the cross section for production of mesons having the same momentum \(\eta\) as is obtained in deuteron formation; \(\frac{\rho_E dE}{1}\) is the ratio of the number of final states in reactions “with an unbound state” and “with a bound state”; \(\frac{|\psi(R)|^2}{|\psi_D(R)|^2}\) is the ratio of the squares of the wave functions in the final state of two nucleons at the critical distance for meson production. The last factor expresses the effect of the increase in the meson-production cross section due to the attractive forces between the nucleons in the final state. If one adopts a simple model with zero range of the nuclear forces in the \({}^3S_1\)-state, with scattering length \(a\), expressed in terms of the deuteron binding energy \(B\) by the formula

\[ B=\frac{\hbar^2}{Ma^2}, \tag{77} \]

and if one takes \(R\simeq 0\), then one obtains

\[ \left|\frac{\psi(R)}{\psi_D(R)}\right|^2 = \frac{2\pi a\hbar^3}{M(B+E)V}, \tag{78} \]

where \(V\) is the normalization volume. The density of states is given by the formula

\[ \rho_E=\frac{V}{(2\pi\hbar)^3}\cdot 2\pi M^{\frac{3}{2}}\cdot E^{\frac{1}{2}}. \tag{79} \]

Then we obtain\({}^{23}\):

\[ \frac{d\sigma_{10'}}{dE} = (\alpha\eta+\beta\eta^3)\cdot \frac{1}{2\pi} \left(\frac{E}{B}\right)^{\frac{1}{2}} \frac{1}{E+B}, \tag{80} \]

whereas the cross section for deuteron formation at the same energies of the incident particles has the form

\[ \sigma_{10'}=\alpha\eta_D+\beta\cdot\eta_D^3, \tag{81} \]

where \(\eta_D\) is the momentum of the meson formed together with the deuteron, expressed in units of \(\mu c\).

Integration over the energy spectrum of mesons (80) and the use of (81) gives the value of the ratio

\[ \frac{\sigma_{10''}}{\sigma_{10}} = \frac{\sigma_{10''}}{\sigma_{10'}+\sigma_{10''}}, \]

which is presented in Table VIII as a function of the energy of the bombarding particles.

Table VIII

Value of the ratio \(\dfrac{\sigma_{10''}}{\sigma_{10}}\)

Measured quantity	Value of the ratio*)	Predicted ratio	Energy of incident particles in MeV	Literature
\(\dfrac{\sigma_{10''}}{\sigma_{10}}(P+P)\)	\(35\pm10\)	20	341	W. F. Cartwright (private communication to A. Rosenfeld)
\(\dfrac{\sigma_{10''}}{\sigma_{10}}(P+P)\)	\(45\pm10\)	20	341	V. Peterson, E. Iloff, D. Sherman, Phys. Rev. 84, 372, (1951)
\(\sigma_{10}(P+P)\)	\(40\pm30\)	21	345	S. Passman, M. M. Block and W. W. Havens, Phys. Rev. 88, 1239 (1952)
\(\sigma_{10}(P+P)\)	\(40\pm30\)	28	365	S. Passman, M. M. Block and W. W. Havens, Phys. Rev. 88, 1239 (1952)
\(\sigma_{10}(P+P)\)	\(55\pm30\)	29	381	S. Passman, M. M. Block and W. W. Havens, Phys. Rev. 88, 1239 (1952)
\(\dfrac{\sigma_{10''}}{\sigma_{10}}(N+P)\)	\(60\pm15\)	32	400	R. H. Hildebrand and A. H. Rosenfeld (private communication)
\(\sigma_{10}(P+P)\)	\(73\pm40\)	36	440	A. H. Rosenfeld (private communication)

*) The ratio is given in percent.

To find the experimental spectrum of mesons and the value of

\[ \frac{\sigma_{10''}}{\sigma_{10}}, \]

it is necessary to investigate the reaction \(P+P \to \pi^{+}+N+P\) or \(N+P \to \pi^{\circ}+N+P\) and to take into account that the cross section for the first reaction is equal to \(\sigma_{10}+\sigma_{11}\), and not simply \(\sigma_{10''}\), and for the second

the reaction is equal to \(\dfrac{\sigma_{10''}}{2}+\dfrac{\sigma_{01}}{2}\), and not simply \(\dfrac{\sigma_{10*}}{2}\). Fortunately, both \(\sigma_{11}\) and \(\sigma_{01}\) are small in our energy region (see below); therefore in Table VIII the experimental values of the ratios must be corrected by no more than \(10\%\). In those cases where \(\sigma_{10}\) was measured, and not \(\dfrac{\sigma_{10''}}{\sigma_{10}}\), the latter was calculated using the semiempirical formula (60) for \(\sigma_{10'}\). The agreement between the measured and calculated ratios is good, but the predicted values are usually smaller, if only the large experimental errors allow such a conclusion.

The results of measurements of the continuous meson spectrum given by formula (72) are not accurate enough for it to be meaningful to compare them with the theory. In any case, they do not plainly contradict the theory. At the same time, the existing experimental material on reactions giving a continuous spectrum does not exclude the possibility that the nucleons remain in a \(P\)-state in a considerable fraction of the total number of meson-production events (at least in the upper part of the energy interval under consideration). The most probable of these processes, apparently, are those in which the meson is emitted in a \(P\)-state:

\[ \begin{aligned} 2N({}^{3}P_{0}) &\to 2N({}^{1}P_{1})+\text{meson in a }P\text{-state},\\ 2N({}^{3}P_{1}) &\to 2N({}^{1}P_{1})+\text{meson in a }P\text{-state},\\ 2N({}^{3}P_{2}) &\to 2N({}^{1}P_{1})+\text{meson in a }P\text{-state},\\ 2N({}^{1}F_{2}) &\to 2N({}^{1}P_{1})+\text{meson in a }P\text{-state}. \end{aligned} \tag{82} \]

For such reactions the cross section per unit energy is proportional to \(\eta^{3}E^{\frac{1}{2}}\) and is not described by (80) if the effect of nuclear forces in the final state is neglected in it. (The nature of the nuclear forces in the \(P\)-state is not very well understood, but they are probably relatively weak.) Under the same assumptions, the complete excitation function for reactions (82) should be proportional to \(\eta_{\max}^{8}\), where \(\eta_{\max}\) is the maximum value of \(\eta\) for the given energy of the incident particles.

Up to now we have studied only the process \(I=1\to I=0\). Let us consider the following process \(I=1\to I=1\), which completely includes the reaction \(P+P\to P+P+\pi^{0}\). In the process that we have studied up to now, cases predominated in which the meson was emitted in a \(P\)-state and the nucleons remained in an \(S\)-state. In the process now under consideration, \(I=1\to I=1\), this is forbidden. If the nucleons remain in an \(S\)-state, then this can only be a singlet. Consequently, in order for the meson to be emitted in a \(P\)-state, the total angular momentum must be equal to 1 and the parity of the system must be positive. However, the initial states of positive parity are

\({}^1S_0,\ {}^1D_2,\ {}^1G_4\), etc., and not one of them has angular momentum equal to unity.

The only allowed process in which the final nucleons are in an \(S\)-state is

\[ \text{(d)} \qquad 2N({}^3P_0)\to 2N({}^1S_0)+\text{meson in an }S\text{-state}. \]

Process (d) is, of course, characterized by an isotropic angular distribution. Near threshold the cross section per unit energy must have the form

\[ \frac{d\sigma_{11}(\text{process (d)})}{dE} =\mathrm{const}\,\eta\,\frac{E^{1/2}}{E+B'} . \tag{83} \]

This corresponds to that part of formula (80) which describes the \(S\)-state, with the binding energy \(B\) of the deuteron replaced by the energy \(B'\) of the virtual \({}^1S_0\) level of two neutrons. Since \(B'\) is very small (\(\simeq 60\,K_{\mathrm{av}}\)), we shall neglect it. Then the total cross section near threshold will have the form

\[ \sigma_{11}(\text{process (d)})=\mathrm{const}\,\eta_{\max}^{2}. \tag{84} \]

As in the case of \(\sigma_{10''}\), one should not neglect the possibility that the nucleons may remain at \(P\)-distances. This applies especially to \(\sigma_{11}\), since the process giving the main contribution to \(\sigma_{10''}\) (the mesons are emitted in a \(P\)-state, the nucleons remain in an \(S\)-state) gives no contribution whatever to \(\sigma_{11}\). If the final state of the nucleons is a \(P\)-state and the mesons are emitted in a \(P\)-state, then the following possible processes, analogous to (82), occur:

\[ \begin{aligned} &2N({}^3P_0)\to 2N({}^3P_1)+\text{meson in a }P\text{-state},\\ &2N({}^3P_1)\to 2N({}^3P_0)+\text{meson in a }P\text{-state},\\ &2N({}^3P_1)\to 2N({}^3P_1)+\text{meson in a }P\text{-state},\\ &2N({}^3P_1)\to 2N({}^3P_2)+\text{meson in a }P\text{-state},\\ &2N({}^3P_2)\to 2N({}^3P_1)+\text{meson in a }P\text{-state},\\ &2N({}^3P_2)\to 2N({}^3P_2)+\text{meson in a }P\text{-state},\\ &2N({}^3F_2)\to 2N({}^3P_1)+\text{meson in a }P\text{-state},\\ &2N({}^3F_2)\to 2N({}^3P_2)+\text{meson in a }P\text{-state},\\ &2N({}^3F_3)\to 2N({}^3P_2)+\text{meson in a }P\text{-state}. \end{aligned} \tag{85} \]

For these processes, as also for the processes listed in (82), the cross section per unit energy interval is proportional to \(\eta^3 E^{1/2}\), and the total cross section is proportional to \(\eta_{\max}^{8}\). Whereas in the case of \(\sigma_{10''}\) the term \(\eta_{\max}^{8}\) in the total cross section was not detected experimentally (it is evidently suppressed by the predominant processes), in the case of \(\sigma_{11}\) there are convincing experimental grounds for the existence of this term. The reaction \(P+P\to \pi^+ + P+P\) was

investigated by Mather and Martinelli\(^{33}\) at an energy of 341 Mev (\(\eta_{\max}=0.66\)), and also by Marshall J. and Marshall L.\(^{34}\) at an energy of 440 Mev (\(\eta_{\max}=1.11\)). The values of the total cross sections were found to be, respectively, \((0.010\pm0.003)\) mb and \((0.45\pm0.15)\) mb. These results agree with the power law

\[ \sigma_{11}=(0.2\ \text{mb})\,\eta_{\max}^{8}. \tag{86} \]

In (86) we have completely ignored process (d), in which the nucleons remain in an \(S\)-state and the cross section varies as \(\eta_{\max}^{2}\). As we shall see below, it probably cannot be assumed that this process is absent. However, it will make a significant contribution to the total cross section only at low energies, where the cross section is very small and measurements are difficult.

Let us turn to the reaction \(I=0\to I=1\). In such a reaction, when the nucleons remain in an \(S\)-state, the only possible processes will be

\[ \begin{aligned} \text{(e)}\qquad &2N({}^{3}S_{1})\to 2N({}^{1}S_{0})+\text{meson in a }P\text{-state and}\\ \text{(f)}\qquad &2N({}^{3}D_{1})\to 2N({}^{1}S_{0})+\text{meson in a }P\text{-state,} \end{aligned} \]

For these processes one should expect that near threshold the cross section per unit energy interval has the form

\[ \frac{d\sigma_{01}\ \text{(processes (e) and (f))}}{dE} = \text{const}\,\eta^{3}\, \frac{E^{1/2}}{E+B'} . \tag{87} \]

As in (83), here one may omit \(B'\), and we obtain the following law for the variation of the cross section:

\[ \sigma_{01}\ \text{(processes (e) and (f))}=\gamma\eta_{\max}^{4}. \tag{88} \]

If the final nucleons are in a \(P\)-state and the mesons are emitted in a \(P\)-state, then the following possibilities exist:

\[ \left. \begin{aligned} &2N({}^{1}P_{1})\to 2N({}^{3}P_{0})+\text{meson in a }P\text{-state,}\\ &2N({}^{1}P_{1})\to 2N({}^{3}P_{1})+\text{meson in a }P\text{-state,}\\ &2N({}^{1}P_{1})\to 2N({}^{3}P_{2})+\text{meson in a }P\text{-state,}\\ &2N({}^{1}F_{3})\to 2N({}^{3}P_{2})+\text{meson in a }P\text{-state.} \end{aligned} \right\} \tag{89} \]

For these reactions, as usual, one should expect that the cross section will be proportional to \(\eta_{\max}^{8}\).

Unfortunately, \(\sigma_{01}\) has been measured only for one energy. From formula (42) it is seen that \(\sigma_{01}\) is always measured in combination with \(\sigma_{10}\),

or with \(\sigma_{11}\). Indeed:

\[ \sigma (N+P\to \pi^{+}+N+N)=\sigma (N+P\to \pi^{-}+P+P)= \]
\[ =\frac{1}{2}\sigma_{11}+\frac{1}{2}\sigma_{01} \tag{90} \]

and

\[ \sigma (N+P\to \pi^{0}+N+P)=\frac{1}{2}\sigma_{10'}+\frac{1}{2}\sigma_{01'}. \tag{91} \]

The total cross section of the process \(N+P\to \pi^{+}+N+N\) (or \(N+P\to \pi^{-}+P+P\)) at \(405\) MeV (\(\eta=0.915\)) was measured by Hodson \(^{35}\), who found the cross section \((0.22\pm 0.07)\) mb (see also \(^{36}\)). If we now estimate \(\sigma_{11}\) at this energy by interpolating a formula of the type (86), and then find \(\sigma_{01}\) from (90), then for \(\sigma_{01}\) at \(\eta=0.915\) we obtain, as a rough estimate, the value \(0.3\) mb. Despite the possibly large errors that should be attributed to this value of \(\sigma_{01}\), its smallness in comparison with the value of \(\sigma_{10'}\) at the same energy (\(\sigma_{10'}\gg 1\) mb) is evident.

Since \(\sigma_{01}\) at other energies is unknown to us, we cannot check whether processes (e) and (f) are the dominant processes in (89). Let us suppose, however, that the final nucleons appear predominantly in an \(S\)-state; in this case the constant in formula (88) is equal to

\[ \gamma \simeq 0.5\ \text{mb}. \tag{92} \]

It should be noted that relations (90) and (91) refer to total cross sections. In the case of reaction (91), the hypothesis of charge independence makes interference of states with two different isotopic spins impossible; therefore a similar formula is also valid for differential cross sections. Moreover, there is no forward–backward asymmetry in the angular distribution. (For a given state of isotopic spin the neutron and proton behave as indistinguishable particles.) However, in the reaction described by (90), interference between the two processes is possible and may lead to a forward–backward asymmetry.

The angular distribution of the reaction \(N+P\to \pi^{-}+P+P\) was studied by Hodson \(^{35}\) and by Rait and Schluter \(^{36}\). In their works a forward–backward asymmetry was found. If, in the reaction \(I=0\to I=1\), the final nucleons are mainly in an \(S\)-state, then, in order for such interference to arise, it is necessary that it occur at least partly also for the reaction \(I=1\to I=1\). Thus, process (d), apparently, takes place with appreciable probability. However, on the basis of the existing experimental facts it is difficult to draw any quantitative conclusions.

Now let us summarize the experimental data in the light of the interpretation considered. It turns out that the production of mesons with small energies has the following properties.

The cases in which the final nucleons are in an \(S\)-state:

\(I=1 \to I=0\): many mesons are produced in the \(P\)-state and few in the \(S\)-state. The angular distribution, due to the \(P\)-wave, has the form \(\simeq 0.1+\cos^2\vartheta\);

\(I=0 \to I=1\): production of a meson in the \(P\)-state is unlikely, and in the \(S\)-state is forbidden;

\(I=1 \to I=1\): production of a meson in the \(P\)-state is forbidden, and in the \(S\)-state is unlikely.

The cases in which the final nucleons are in a \(P\)-state. This state has to be invoked only in the case \(I=1 \to I=1\). Its role is not especially clear.

At present there are four experimentally determined quantities which must be predicted by a theory going beyond the simple considerations used above:

(A) The ratio of the contributions of the \(S\)- and \(P\)-waves in meson production in the reaction \(I=1 \to I=0\).

\[ \frac{\alpha}{\beta}\simeq \frac{1}{7}\quad [\text{see formulas (45), (53), (55), and (58)}]. \]

(B) The parameter of the angular distribution in the reaction \(I=1 \to I=0\).

\[ X\simeq 0.1\quad [\text{see formulas (55) and (59)}]. \]

(C) The ratio of the contribution of the \(P\)-wave in meson production in the reaction \(I=0 \to I=1\) to the contribution of the \(P\)-wave in the reaction \(I=1 \to I=0\).

\[ \frac{\sigma_{01}}{\sigma_{10''}}\lesssim 0.3\quad (\text{see the discussion preceding formula (92)}). \]

(D) The absolute magnitude of the meson-production cross section

\[ \beta\simeq 1\ \text{mb}\quad [\text{see formulas (55) and (58)}]. \]

If we try to predict these numbers on the basis of the pseudoscalar meson theory, we shall see that the fourth of them is directly connected with the interaction constant, i.e., with the details of the theory\(^{37}\). The other three are also connected with the details of the theory; however, they can be understood rather simply.

As for (A), it was noted in the introduction that pseudoscalar mesons must interact strongly with nucleons in the \(P\)-state, whereas the interaction in the \(S\)-state is, by its nature, a recoil correction. For meson production at th-

the threshold the principal matrix element for the emission of a pseudoscalar meson in the \(P\)-state is proportional to \((\boldsymbol{\sigma}\mathbf{k})\), where \(\boldsymbol{\sigma}\) is the spin of the recoil nucleon and \(\mathbf{k}=\mu\mathbf{v}\) is the meson momentum. If one introduces the correction for the motion of the nucleon, then \(\mathbf{v}\) must be replaced by \(\mathbf{v}-\langle\mathbf{v}\rangle_n\), where \(\langle\mathbf{v}\rangle_n\) is the mean value of the initial and final velocities of the nucleon. (The principle of invariance with respect to Galilean coordinate transformations requires that matrix elements depend on relative, and not on absolute, velocities.) Furthermore, the final velocity of the nucleon is close to zero, whereas the initial velocity \(\mathbf{v}_0\) (in the center-of-mass system) satisfies the relation

\[ Mv_0^2 \simeq \mu c^2, \tag{93} \]

since the kinetic energy of the colliding nucleons is converted into the rest energy of the meson. Thus, the recoil correction adds to the term \((\boldsymbol{\sigma}\mathbf{k})\) (representing meson production in the \(P\)-state) a term \(-\left(\boldsymbol{\sigma}\dfrac{\mu\mathbf{v}_0}{2}\right)\), representing meson production in the \(S\)-state, and the ratio of the intensities has the order of magnitude

\[ \frac{\mu^2 v_0^2}{4k^2} = \frac{\mu}{4M}\,\frac{1}{\eta^2}, \]

which should approximately correspond to \(\dfrac{\alpha}{\beta\eta^2}\). Hence it is seen that \(\dfrac{\alpha}{\beta}\) should be of order \(\dfrac{\mu}{M}\). This conclusion is confirmed by calculations in meson theory.

As for (B) and (C), the explanation of the smallness of \(\dfrac{\sigma_{01}}{\sigma_{10}}\) and of the closeness of \(X\) to \(1/3\) (it could, in principle, have any value between 0 and \(\infty\)!) was proposed by Eytken et al.\(^{37}\). They made use of the fact that between the meson and the nucleon there is a strong attraction in the state \((3/2,\,3/2)\), considered in the preceding sections. The basic idea was that this strong attraction between the \(\pi\)-meson and one of the nucleons in the state \((3/2,\,3/2)\) increases the matrix element of the transition to this state. (In a similar way we have seen that nucleon–nucleon forces increase the cross sections of reactions in which the final nucleons are in an \(S\)-state.)

Next, there are four processes which contribute to the cross sections for meson production in the \(P\)-state (with the nucleons remaining in an unbound \(S\)-state); we denoted them by \((b'')\), \((c'')\), \((e)\), and \((f)\). Of these, \((b'')\) and \((c'')\) contribute to \(\sigma_{10}\), whereas \((e)\) and \((f)\) contribute to \(\sigma_{01}\). The processes \((e)\) and \((f)\) cannot be enhanced by the effect of the resonant state \((3/2,\,3/2)\), since the total isotopic spin of the system in these cases is 0, whereas the system consisting of a nucleon and a \(\pi\)-meson in the state \(I=3/2\) and one more nucleon can have only \(I\) equal to 1 or 2. The cross section of process \(c''\) is not increased under the condition that, in meson formation, the nucleons approach one another to distances small in comparison with the meson wavelength. (We used this

(condition earlier, when only the lowest values of the orbital angular momenta of the emitted particles were taken into account.) In this case the state \(J=0\) cannot go over into the state of a system consisting of a \(\pi\)-meson and a nucleon with angular momentum \(J=3/2\), and one more nucleon in an \(S\)-state. The same considerations are also applicable to process \((c')\), in which a deuteron is formed. In the case of processes \((b'')\) and \((b')\), however, an increase of the cross section will occur, and therefore one may assume that \(\sigma_{01}\ll\sigma_{10'}\), and that the angular distribution of the process \(I=1\to I=0\) has approximately the same form as the angular distribution of the process \((b'')\) or \((b')\), i.e. \(1/3+\cos^2\theta\). We note that if this explanation of the angular distribution is accepted, then the phase angle \(\omega_0\) in (62) must be close to \(0^\circ\). Eytken\(^{37}\) formulated these considerations more rigorously, using meson theory.

MESON THEORY*)

Meson theory in the form proposed by Yukawa was constructed by analogy with quantum electrodynamics. Just as in electrodynamics the primary process is the virtual emission or absorption of one photon by an electron, so in meson theory such a process is the virtual emission or absorption of one \(\pi\)-meson by a nucleon. It is assumed that a “bare” nucleon (not interacting with the \(\pi\)-meson field) is described, like the electron, by the Dirac equation. The operator \(\Phi(x,t)\) of the pseudoscalar \(\pi\)-meson field is analogous to the vector (spin 1) field operator \(A_\mu(x,t)\), representing the quantized vector potential of the electromagnetic field. Just as \(A_\mu(x,t)\) is contracted with the vector Dirac operator \(\gamma_\mu\) for the electron, so \(\Phi(x,t)\) can be contracted with the pseudoscalar Dirac operator \(\gamma_5\) for the nucleon (\(PS\), or pseudoscalar, coupling), or the gradient \(\Phi(x,t)\), \(\dfrac{\partial \Phi}{\partial x_\mu}(x,t)\), can be contracted with the pseudovector Dirac operator \(\gamma_5\gamma_\mu\) for the nucleon (\(PV\), or pseudovector, coupling). No other simple couplings of the \(\pi\)-meson field with the nucleon, apart from these, are assumed. (It should be noted that, to describe the three charge states of the \(\pi\)-meson, a three-component field \(\Phi_i\) is required, whose three components form a vector in isotopic-spin space. In order that the requirement of charge independence be fulfilled, the “symmetric” theory\(^{5}\) is used, in which the vector \(\Phi_i\) is contracted with the isotopic-spin vector \(\tau_i\) of the nucleon.)

For each of the two types of coupling one can construct and study, by the method of perturbation theory, i.e. by expanding the observed quantities in powers of the coupling constant \(\dfrac{g^2}{4\pi\hbar c}\) (for \(PS\)) or \(\dfrac{f^2}{4\pi\hbar c}\)

*) In this section we shall usually take \(\hbar=c=1\).

(for \(PV\)), the complete relativistic theory of the interaction of a \(\pi\)-meson with a nucleon. As in quantum electrodynamics, the coefficients in such an expansion turn out to be infinite (with the exception of the lowest order). In electrodynamics these infinities disappear if the results are reformulated in terms of the observed mass and charge of the electron (the so-called renormalizations of charge and mass). As was shown, an analogous situation holds for the \(PS\) theory of \(\pi\)-mesons\(^{38}\)*). However, in the \(PV\) theory the infinities remain, and in order to obtain finite results one must modify the relativistic theory. A convenient modification is the introduction of a finite radius of the “bare” nucleon, of the order of magnitude equal to the Compton wavelength of the nucleon,

\[ \frac{\hbar}{M c}. \]

Then all integrals over the momenta of the virtual meson are cut off at momentum \(\hbar/a\), or at energy

\[ \omega_{\max}=\sqrt{\mu^2 c^2+\frac{\hbar^2}{a^2}} . \]

When the \(PV\) theory with a cutoff is applied, for calculations one usually uses the static approximation, i.e. the nucleons are considered fixed and all (or most) of the recoil effects of the nucleon are not taken into account. Such an approximation is not absolutely necessary, since in the \(PV\) theory one can in principle introduce a relativistically invariant cutoff and take into account the effects of the motion of the nucleons. However, usually when a cutoff is used, recoil is in one way or another taken into account poorly, and it is better not to consider it at all.

The static \(PV\) theory with a cutoff has been investigated in the approximations of weak and strong coupling by several authors (for references to earlier works see \(^{39}\), and to recent ones, \(^{40}\)). It was found that whereas both the very weak and very strong coupling approximations do not agree with the experimental data on \(\pi\)-meson phenomena, the case of neither very weak nor very strong coupling gives results in favor of which all experiments testify. The latter applies especially to the existence of strong attraction between \(\pi\)-mesons and nucleons in the state \((3/2,\,3/2)\). The case of not very weak coupling was studied in great detail by Chew\(^{40}\), and the results of the calculations were compared with experiment\(^{41}\). He found that with coupling constant

\[ \frac{f^2}{4\pi\hbar c}=0.058 \]

and cutoff energy \(\omega_{\max}=0.84M\), one can obtain rough quantitative agreement with the experimental data on the scattering of \(\pi\)-mesons, on photomeson effects at meson energies \(<250\) Mev, and on the anomalous magnetic moments of the neutron and proton, i.e. on all

*) In the case of the \(PS\) meson theory it is in fact necessary to introduce and renormalize one more parameter, describing the scattering of mesons by mesons (see \(^{47}\)).

phenomena involving one nucleon. Let us consider, in general terms, Chew’s theory.

The Hamiltonian in Chew’s theory has the form:

\[ H=H_\pi+\frac{f}{\mu}\sum_i \tau_i\left(\sigma\int \rho(x)\nabla\Phi_i(x)\,d^3x\right)+M. \tag{94} \]

Here \(H_\pi\) is the Hamiltonian of the meson field and \(\rho(x)\) is the function describing the nucleon as a source of finite dimensions; \(\rho\) is characterized by the cutoff energy \(\omega_c\) and by the condition

\[ \int \rho(x)\,d^3x=1. \tag{95} \]

The nucleon is located at the origin.

One should not think that this Hamiltonian is approximate with respect to the Hamiltonian of the relativistic \(PV\) theory. If we set ourselves the goal of constructing a symmetric theory of the interaction of pseudoscalar mesons with a fixed nucleon, and if we assume that the interaction must be linear in the meson field, so that in an elementary act only one meson is created or absorbed, then we arrive almost uniquely at expression (94).

If it is necessary to take into account the interaction with the photon field, then an additional term should be introduced into the Hamiltonian

\[ H_{add}=H_{ph}-\int (jA)\,d^3x+\frac{fe}{\mu}\int d^3x\,\rho(x)(\sigma A)(\Phi_1\tau_2-\Phi_2\tau_1). \tag{96} \]

Here the first term is the Hamiltonian of the photon field, the second is the interaction of the meson current with the electromagnetic field, and the third describes the direct process of photoproduction of mesons (it appears from the requirement of gauge invariance). We have neglected all Coulomb interactions, the interaction of the Dirac magnetic moment of the proton with the photon field, and terms due to the requirements of gauge invariance for the nucleon.

It is convenient to determine the coupling constant from the experimental data on the effect of photoproduction of \(\pi\)-mesons near threshold. In Chew’s theory this effect at low energies is entirely due to the third term in (96); moreover, it is very convenient to carry out charge renormalization in it, and the formula obtained in the lowest approximation of perturbation theory gives the exact solution:

\[ \sigma(\gamma^\pm)=8\pi\left(\frac{e^2}{4\pi\hbar c}\right)\left(\frac{f^2}{4\pi\hbar c}\right)\left(\frac{\hbar}{\mu c}\right)^2\frac{\eta}{\nu}\quad(\eta\ll1). \tag{97} \]

Here \(\eta\) and \(\nu\), as also in the section “Photoproduction of \(\pi\)-mesons on nucleons,” are respectively the momenta of the meson and the photon in units of \(\mu c\). Comparison with the experimental results of Ber-

Bernardini and Goldwasser\(^{16}\) gives for the coupling constant the value \(\frac{f^2}{4\pi \hbar c}=0.038\). Chew pointed out, however, that formula (97) can be corrected for certain kinematic effects of nucleon motion which are not included in the theory by themselves. As a result one obtains the formula

\[ \sigma(\gamma^\pm)=8\pi\left(\frac{e^2}{4\pi \hbar c}\right)\cdot \left(\frac{f^2}{4\pi \hbar c}\right)\cdot \left(\frac{\hbar}{\mu c}\right)^2 \frac{\eta}{\nu} \left(1 \mp \frac{\omega}{2Mc^2}\right)^2 \left(1+\frac{\mu}{M}\right)^{-1}\times \]

\[ \times \left(1+\frac{\omega}{Mc^2}\right)^{-1} \qquad (\eta \ll 1), \tag{98} \]

where \(\omega\) is the energy of the \(\pi\)-meson: \(\mu c^2\sqrt{1+\eta^2}\). If formula (98) is compared with the data of Bernardini and Goldwasser, the coupling constant is found to be \(0.058 \pm 0.015\). Formula (98) gives for \(\frac{\sigma(\gamma^-)}{\sigma(\gamma^+)}\) near threshold the value 1.3, which is in good agreement with the experimental value indicated in formula (20).

Using the value found for the coupling constant, Chew\(^{40,41}\) calculated the \(P\)-wave phase shifts for the scattering of \(\pi\)-mesons by nucleons. (It should be noted that Chew’s theory predicts the existence of scattering only in \(P\)-states. This is a rather serious difficulty, especially in view of the fact that it is impossible to interpret the observed \(S\)-scattering as a recoil effect; at present we in fact have no theoretical explanation for the existence of the \(S\)-wave in the scattering of a \(\pi\)-meson by a nucleon.)

In calculating the phase shifts of the \(P\)-wave it was found that perturbation theory is not adequate, especially for describing the state \((3/2,\,3/2)\). Although the coupling is not very weak and the second and fourth orders of perturbation theory in one way or another affect the results of calculations of the “effective potential” for the scattering of a \(\pi\)-meson by a nucleon, this “potential” itself is not so small in all states that it could be considered in the Born approximation. The word “potential” is put in quotation marks because the calculated operator is not a static potential \(V(r)\), but depends strongly on momentum and energy. Tamm\(^{42}\) and Dancoff\(^{43}\) were the first to propose calculating in problems of this type a nonstatic “potential” by the method of perturbation theory and then finding the scattering phases exactly, instead of obtaining the phase shifts in the form of an expansion in powers of the coupling constant. This method is now called the Tamm—Dancoff method, or the T—D method. It is inapplicable when the coupling strength is so large that even the “potential” cannot be correctly considered by perturbation theory.

Using a slightly modified T—D method and determining the phase shifts by numerical methods, Chew\(^{40}\), Gammel\(^{44}\), and Salzman with Snyder\(^{45}\) found, at \(\omega_{\max}\simeq 0.84M\), the following values of the phase shifts:

which are in satisfactory agreement with modern experimental data. The phase shifts obtained by Chew are presented in Table IX.

Table IX

Phase shifts of the \(P\)-wave in the scattering of \(\pi\)-mesons by nucleons, calculated by Chew

\(E_{\mathrm{lab}}\) (MeV)	\(\eta\)	\(a_{33}\)	\(a_{31}=a_{13}\)	\(a_{11}\)
38	0.67	\(3.7^\circ\)	—	—
57	0.83	\(8.0^\circ\)	\(-1.0^\circ\)	\(-2.3^\circ\)
78	0.98	\(14.7^\circ\)	—	—
99	1.12	\(24.9^\circ\)	\(-2.0^\circ\)	\(-4.3^\circ\)
122	1.25	\(39.4^\circ\)	—	—
144	1.38	\(56.9^\circ\)	\(-3.3^\circ\)	\(-6.4^\circ\)
167	1.50	\(73.2^\circ\)	—	—
190	1.62	\(85.8^\circ\)	\(-4.8^\circ\)	\(-8.4^\circ\)
215	1.73	\(94.6^\circ\)	—	—
240	1.85	\(99.5^\circ\)	\(-6.3^\circ\)	\(-10.3^\circ\)

It is seen from the table that scattering in the \((3/2,\,3/2)\) state is due to attraction and is “resonant”; in the other states it is very weak and is due to repulsion. Apparently, it would be unreasonable to compare the static theory with cutoff with experiment at energies much larger than those taken in the table, since at high energies recoil effects and the specific form of the form factor begin to play an essential role.

We shall now return to the effects of photoproduction of \(\pi\)-mesons and consider Chew’s theoretical results for energies at which formula (98) is inapplicable. Unfortunately, the corresponding calculations have not yet been fully brought to a satisfactory form; moreover, the static theory is inevitably unsatisfactory as regards the inclusion of the proton’s Dirac magnetic moment. Chew \(^{41*}\) reports, however, that preliminary calculations apparently agree rather well with the experimental data in the region of the “resonance.” The fact that the \((3/2,\,3/2)\) “resonance” is contained in the theory definitely confirms the applicability of the considerations used by us in the section “Photoproduction of \(\pi\)-mesons on nucleons.”

The contributions of meson currents to the magnetic moments of the proton and neutron (equal and opposite in sign) were calculated in the approximation of Chew’s perturbation theory \(^{41}\). As a result, for the very same parameters as were used above, in second order \(\pm 1.15\) nuclear Bohr magnetons was obtained, and \(\pm 1.48\), when

* A comparison of the theoretical results with experiment was published in Phys. Rev. 95, 1669 (1954). (Translator’s note.)

corrections of fourth order. The experimental values of the anomalous magnetic moments are, respectively, \(+1.79\) for the proton and \(-1.91\) for the neutron.

In addition to Chew’s program for investigating the one-nucleon problem, the static \(PV\) theory was applied to the problem of two nucleons, i.e. to the meson theory of nuclear forces. The static potentials of second and fourth order between a pair of nucleons were calculated by Taketani et al.,\(^{46}\) by Feynman and Lopes,\(^{47}\) by Brueckner and Watson,\(^{48}\) and by others. These potentials have a strong singularity at the origin of the coordinates and, in order to solve the Schrödinger equation, it is necessary to introduce a boundary condition at small distances. Usually one chooses a two-nucleon wave function that vanishes at distances of order

\[ \frac{1}{2}\frac{\hbar}{\mu c} \]

(the so-called “hard-core model,” first considered by Jastrow\(^{49}\)). It was found that, under this assumption and with a coupling constant not very different from Chew’s constant, all experimental parameters pertaining to the two-body problem at low energies are determined rather accurately. Moreover, somewhat later it was shown by Brueckner\(^{51}\) and Taketani\(^{50}\) that the qualitative features of nucleon–nucleon scattering at energies up to \(90\) MeV can be well explained with the aid of the very same potential. Of course, it is very important to show that higher-order effects do not destroy the agreement with experiment. The available data\(^{48,50}\) indicate that the potentials of sixth and higher orders, although they are very strong, are short-ranged and cannot have substantial significance outside the “core.” The same may also be true for effects connected with “new particles.”

We now pass from the static \(PV\) theory with a cut-off to the fully relativistic \(PS\) theory, which gives finite results without a cut-off. The results of the \(PS\) theory are not as transparent as the results of the theories that we have just discussed. It is quite clear that if the coupling constant is so small that ordinary perturbation theory can be applied, then the \(PS\) theory does not agree with experiment. Nor has it been possible to develop the strong-coupling approximation in such a way as to obtain even a remote resemblance to the experimental data. Various versions of the intermediate-coupling approximation have been proposed. At present, however, it is doubtful that any one of them reflects an accurate picture of the results of the \(PS\) theory. As for the agreement of these approximations with experiment, this agreement probably occurs only insofar as they resemble Chew’s static theory with a cut-off. It may be that Chew’s theory is in fact a good approximation to the \(PS\) theory with intermediate coupling; however, this supposition has by no means been proved.

The coupling constant, as in Chew’s theory, can be determined by comparing the cross sections for photoproduction of \(\pi\)-mesons near threshold with

experiment. Perturbation theory in lowest order gives for the \(PS\) theory formula (98), if in the latter one replaces \(f^2/4\pi\hbar c\) by

\[ \left(\frac{\mu}{2M}\right)^2 \frac{g^2}{4\pi\hbar c}, \]

where \(g^2/4\pi\hbar c\) is the coupling constant of the \(PS\) theory. In the \(PS\) theory this formula is not valid for all orders, in contrast to formula (97) in the Chyu theory. However, Kroll and Ruderman\({}^{52}\) showed that formula (98) is valid, with accuracy up to terms \(\mu/M\), for all orders; they also gave arguments in favor of the fact that such terms are indeed small\({}^{*}\). If in this way we determine \(g^2/4\pi\hbar c\), then we find a value approximately equal to 10. Hence it is clear why perturbation theory does not give satisfactory results for other processes!

In order to understand somewhat more deeply the structure of the \(PS\) theory, we shall apply to it the method of Foldy and Wouthuysen\({}^{53}\). These authors eliminated from the Hamiltonian, by means of successive canonical transformations, the odd Dirac matrices (anticommuting with \(\beta\)) in successive approximations in \(1/M\). The initial \(PS\) Hamiltonian, which for simplicity we write for one particle, has the form:

\[ H_{PS}=H_{\pi}+(\boldsymbol{\alpha}\mathbf{p})+\beta M+ig\beta\gamma_5\sum_i \tau_i\Phi_i(\mathbf{x}), \tag{99} \]

where \(\mathbf{x}\) and \(\mathbf{p}\) are, respectively, the coordinate and momentum of the nucleon. After the transformation, retaining terms of first order in \(1/M\), we obtain:

\[ H'_{PS}=H_{\pi}+\frac{g}{2M}(\boldsymbol{\sigma}\boldsymbol{\nabla})\sum_i \tau_i\Phi_i(\mathbf{x})+M+ \]

\[ +\frac{g^2}{2M}\sum_i \Phi_i^2(\mathbf{x})+\frac{p^2}{2M}. \tag{100} \]

If one compares (100) with formula (94) for the Chyu theory, then, putting \(f=g\mu/2M\), we see that the first three terms in (100) correspond exactly to the Chyu Hamiltonian; only the cutoff introduced there is absent. However, the term describing the kinetic energy of recoil, \(p^2/2M\), after renormalization gives a cutoff of the same type. The principal difference lies in the term

\[ \frac{g^2}{2M}\sum_i \Phi_i^2(x), \]

which corresponds

\({}^{*}\) Let us note that the validity of the result of perturbation theory as \(\mu/M\to 0\) depends on the choice of the prescription for charge renormalization.

to the interaction of mesons in the \(S\)-state with a nucleon through scattering and through the production and annihilation of meson pairs. At first glance it seems that formula (100) is a generalization of (94), since it gives the previous results for mesons in the \(P\)-state and, in addition, contains a description of meson scattering in the \(S\)-state, the absence of which was a weak point of Chew’s theory. However, if one calculates \(S\)-scattering using formula (100), agreement with experiment is not obtained. First of all, perturbation theory gives very strong \(S\)-scattering, which is wholly inconsistent with experiment. A refined consideration by Wentzel,^54 it is true, shows that this effect is very strongly suppressed by higher-order processes; the magnitude of the \(S\)-scattering is then obtained close to the observed one. However, the dependences of the \(S\)-scattering on isotopic spin and on energy will nevertheless contradict experiment. Thus, a theory based on the Hamiltonian (100) has practically no advantage over Chew’s theory.

However, (100) is only an approximation to the \(PS\) theory. Therefore one may hope that an exact treatment of the full \(PS\) Hamiltonian will give, at least, better agreement with the known properties of \(\pi\)-mesons in \(P\)-states and, moreover, will explain such effects as weak \(S\)-scattering depending on isotopic spin. In order to test the validity of this assumption, investigations were undertaken at Cornell by Bethe, Dyson, and others.^55 They use the Tamm–Dancoff method, the validity of which for the \(PS\) theory with coupling constant 10 has unfortunately not been proved. The calculations are being carried out for scattering and photoproduction cross sections of mesons, and the preliminary results seem encouraging in the sense of agreement with the experimental data on the \(P\)-wave.

If the exact \(PS\) theory proves correct, then one feature of the approximate Hamiltonian (100) should be preserved: the suppression of the \(S\)-wave. Some indications that this could occur in the relativistic \(PS\) theory were obtained as a result of very rough calculations by Brueckner, Gell-Mann, and Goldberger.^56

Bethe^55 expressed confidence that the \(PS\) theory will give a correct description of \(\pi\)-meson phenomena at sufficiently low energies. He attaches great importance to the fact that the \(PS\) theory is the only known relativistically invariant theory of pseudoscalar mesons that gives finite results after renormalization without a cut-off. Chew,^41 on the other hand, is inclined to think that attempts to improve the static theory with a cut-off should be more productive than the use of the \(PS\) theory, and should ultimately give a description of the “new unstable particles.” It is difficult, however, to believe that such a description is possible within the framework of any of the existing theories.

ADDENDUM

Calculation of the complex phases of photoproduction matrix elements

We shall briefly describe the method used in the works cited in the footnote on p. 423 for determining the phases in formulas (19) and (20). Consider transitions between eigenstates of the Hamiltonian \(H_0\), caused by the interaction \(V\).

The integral equation satisfied by the reaction matrix \(K\) has the form*):

\[ K = V + V \frac{1}{E - H_0} K, \tag{D-1} \]

where \(E\) is the energy of the system. If \(V\) and \(H_0\) are invariant with respect to the action of Wigner’s time-inversion operator \(\mathcal{K}\), then the same is true for \(K\), i.e.

\[ \mathcal{K} K \mathcal{K}^{-1} = K. \tag{D-2} \]

The integral equation establishing the relation between \(K\) and the transition amplitude \(T\) has the form:

\[ T = K + iKT, \tag{D-3} \]

where all quantities are specified on the energy surface (in contrast to the quantities in equation (D-1)). Denote the eigenfunctions of the operator \(H_0\) by

\[ \Phi_{rj}^{m}, \]

where \(r\) denotes the parameter of the reaction channel (i.e., it numbers the states into which the system can pass as a result of scattering) and \(j\) is the total angular momentum with projection on the \(z\)-axis equal to \(m\). (In the general case there will also be other degrees of freedom, which we ignore for brevity.) We can expand \(K\) in the functions \(\Phi\):

\[ K = \sum_{r,r'} \sum_{j,m} (r|K|r') \Phi_{rj}^{m}\Phi_{r'j}^{*m}. \tag{D-4} \]

Here \((r|K|r')\) is a function only of the energy and of \(r, r'\). If

) See, for example, the general theory of scattering — M. Gell-Mann, M. L. Goldberger, Phys. Rev. 91*, 398 (1953).

if one chooses the usual representation, then*):

\[ \mathcal{K}\Phi_{rj}^{m}=(i)^{2m}\Phi_{rj}^{-m} \tag{D-5} \]

and (since \(\mathcal{K}\) includes complex conjugation)

\[ \mathcal{K}K\mathcal{K}^{-1} = \sum_{r,r'}\sum_{j,m} (r|K|r')^{*}\, i^{2m}(-i)^{2m}\Phi_{rj}^{-m}\Phi_{r'j}^{*-m}. \tag{D-6} \]

Using formula (D-2), we obtain that

\[ (r|K|r')^{*}=(r|K|r')\equiv K_{0}. \tag{D-7} \]

Thus the matrix \((r|K|r')\) is real and symmetric (since it is Hermitian), which is a result depending on the choice of the functions \(\Phi\), which satisfy equation (D-5).

To illustrate the meaning of formula (D-7), let us consider meson-nucleon scattering for a pure \((j,l)\) state and photoproduction in the same state of the meson and nucleon. Then:

\[ K_{0}= \begin{pmatrix} 0 & \gamma\\ \gamma & \operatorname{tg}\hat{\delta} \end{pmatrix} \begin{array}{l} \text{(state: \(\gamma\)-quantum — nucleon),}\\ \text{(state: \(\pi\)-meson — nucleon),} \end{array} \tag{D-8} \]

where \(\hat{\delta}\) is the scattering phase shift and \(\gamma\) is a quantity characterizing the photon channel of the reaction. We regard \(\gamma\) as small (since it includes the electromagnetic interaction) and neglect the scattering of photons by the nucleon. (Therefore the element in the upper left corner of \(K_{0}\) is set equal to zero.)

Returning to equation (D-3), we expand \(T\) by analogy with (D-4), writing

\[ T_{0}\equiv (r|T|r'). \]

Formula (D-3), expressed in terms of \(T_{0}\) and \(K_{0}\), is written as

\[ T_{0}=K_{0}+iK_{0}T_{0}, \]

or

\[ T_{0}=(1-iK_{0})^{-1}K_{0}. \tag{D-9} \]

The product of matrices is easily calculated, and we find

\[ T_{0}= \begin{pmatrix} 0 & e^{i\hat{\delta}}\gamma\cos\hat{\delta}\\ e^{i\hat{\delta}}\gamma\cos\hat{\delta} & e^{i\hat{\delta}}\sin\hat{\delta} \end{pmatrix}. \tag{D-10} \]

*) Here the authors make an inaccuracy, assuming that (D-5) corresponds to the generally accepted representation. The usual requirement of the reality of the Clebsch–Gordan coefficients leads to the following definition of the time-inversion operation:
\[ K\Phi_{rj}^{m}=(-1)^{j-m}\Phi_{rj}^{-m}; \]
the use of (D-5) may lead to errors; see R. Huby, Proc. Phys. Soc. 67A, 1103 (1954). (Translator’s note.)

Here the off-diagonal matrix elements are the matrix elements of photoproduction corresponding to individual multipoles. Since \(\delta\) and \(\gamma\) are real, the matrix \(T_0\) has the form:

\(e^{i\delta}\), multiplied by a real quantity.

This result has a number of applications to meson photoproduction. They were examined in detail (see the footnote on p. 423). Here we shall give only the result of these works. The transition amplitudes for individual multipoles in formula (26) are expressed in terms of the amplitudes for transitions to pure \(I\) states. Then (here \(\alpha\) denotes the scattering phase shifts calculated for the energy of the \(\pi\)-meson and nucleon in the final state)

\[ \begin{aligned} E_1^{+} & = e^{i\alpha_3}\sqrt{2}\,E_1^{(3)} + e^{i\alpha_1}\frac{1}{\sqrt{2}}\left[E_1^{(1)}-2\delta E_1^{(1)}\right],\\ E_1^{0} & = e^{i\alpha_3}2E_1^{(3)} - e^{i\alpha_1}\frac{1}{2}\left[E_1^{(1)}-2\delta E_1^{(1)}\right],\\ M_1\left(\frac{1}{2}\right)^{+} & = e^{i\alpha_{31}}\sqrt{2}\,M_1\left(\frac{1}{2}\right)^{(3)} + e^{i\alpha_{11}}\frac{1}{\sqrt{2}} \left[ M_1\left(\frac{1}{2}\right)^{(1)} -2\delta M_1\left(\frac{1}{2}\right)^{(1)} \right],\\ M_1\left(\frac{1}{2}\right)^{0} & = e^{i\alpha_{31}}2M_1\left(\frac{1}{2}\right)^{(3)} - e^{i\alpha_{11}}\frac{1}{2} \left[ M_1\left(\frac{1}{2}\right)^{(1)} -2\delta M_1\left(\frac{1}{2}\right)^{(1)} \right],\\ M_1\left(\frac{3}{2}\right)^{+} & = e^{i\alpha_{33}}\sqrt{2}\,M_1\left(\frac{3}{2}\right)^{(3)} + e^{i\alpha_{13}}\frac{1}{\sqrt{2}} \left[ M_1\left(\frac{3}{2}\right)^{(1)} -2\delta M_1\left(\frac{3}{2}\right)^{(1)} \right],\\ M_1\left(\frac{3}{2}\right)^{0} & = e^{i\alpha_{33}}2M_1\left(\frac{3}{2}\right)^{(3)} - e^{i\alpha_{13}}\frac{1}{2} \left[ M_1\left(\frac{3}{2}\right)^{(1)} -2\delta M_1\left(\frac{3}{2}\right)^{(1)} \right],\\ E_2^{+} & = e^{i\alpha_{23}}\sqrt{2}\,E_2^{(3)} + e^{i\alpha_{13}}\frac{1}{\sqrt{2}} \left[E_2^{(1)}-2\delta E_2^{(1)}\right],\\ E_2^{0} & = e^{i\alpha_{23}}2E_2^{(3)} - e^{i\alpha_{13}}\frac{1}{2} \left[E_2^{(1)}-2\delta E_2^{(1)}\right]. \end{aligned} \tag{D-11} \]

The twelve quantities \(E_1^{(3)}\), \(M_1\left(\frac{1}{2}\right)^{(1)}\), \(\delta E_1^{(1)}\), etc., represent the transition amplitudes to pure \(I\)-states of the \(\pi\)-meson—nucleon system. The upper index is equal to twice the value of the isotopic spin \(I\). The main significance of these formulas is that the 12 quantities \(E^{(3)}\), etc., are real functions only of the energy of the \(\gamma\)-quanta.

Amplitudes corresponding to individual multipoles for \(P_{\gamma^-}\) are obtained from the amplitudes for \(P_{\gamma^+}\), and for \(P_{\gamma^{n0}}\) from the amplitudes for \(P_\gamma\), by a simple change of sign of \(\delta E_1^{(1)}\), \(\delta M_1\left(\dfrac{1}{2}\right)^{(1)}\), \(\delta M_1\left(\dfrac{3}{2}\right)^{(1)}\), and \(\delta E_2^{(1)}\) in formulas (D-11). Thus, the 16 complex amplitudes for the multipoles are expressed in terms of 12 real numbers. For further details see the paper cited in the footnote on p. 423. Only \(E_2^{(3)}\) and

\[ M_1\left(\frac{3}{2}\right)^{(3)} \]

contribute to the resonance state. It is expected that for \(E_\gamma \simeq 320\ \text{MeV}\) they will have the form \(^{23}\)

\[ M_1\left(\frac{3}{2}\right)^{(3)} \quad \text{and} \quad E_2^{(3)} = \frac{\sin \alpha_{33}}{\gamma^{1/2}} \times (\text{real constant}), \]

as is also indicated in formula (33).

REFERENCES

1. C. Dilworth, G. P. S. Occhialini and L. Scarsi, Ann. Rev. Nuclear Sci. 4, 271—314 (1954).
2. W. Panofsky, L. Aamodt and J. Hadley, Phys. Rev. 81, 656 (1951).
3. K. Brueckner, R. Serber and K. Watson, Phys. Rev. 81, 575 (1951).
4. G. Breit and E. Feenberg, Phys. Rev. 50, 850 (1936).
5. N. Kemmer, Proc. Cambridge Phil. Soc. 34, 354 (1938).
6. W. Heitler, Proc. Roy. Irish Acad. 51, 33 (1946).
7. K. Watson, Phys. Rev. 85, 852 (1952); K. Watson and K. Brueckner, Phys. Rev. 83, 1 (1951).
8. K. A. Brueckner, Phys. Rev. 86, 206 (1952).
9. J. Crussard, Proc. Conf. on High Energy Physics (Rochester, New York, 1954).
10. J. Askhin, Proc. Conf. on High Energy Physics (Rochester, New York, 1954).
11. F. de Hoffman, N. Metropolis, E. Alei and H. A. Bethe, Phys. Rev. (to be published).
12. H. Anderson, E. Fermi, R. Martin and D. Nagle, Phys. Rev. 91, 155 (1953); E. Fermi and N. Metropolis (unpublished work); R. L. Martin, Bull. Am. Phys. Soc. 29, Abstr. KA2 (1954); H. Bethe and M. Glicksman, Proc. Conf. High Energy Physics (Rochester, New York, 1954).
13. A. Silverman and M. Stearns, Phys. Rev. 88, 1225 (1952).
14. Y. Goldschmidt-Clermont, L. S. Osborne and M. B. Scott, Phys. Rev. 89, 329 (1953).
15. R. Walker, D. Oakley and A. Tollestrup, Phys. Rev. 89, 1301 (1953).
16. G. Bernardini and E. L. Goldwasser, Phys. Rev. 94, 729 (1954).
17. A. Tollestrup, J. Keck and R. Worlock, Phys. Rev. 92, 1090(A) (1953); and (private communication, January 1954).
18. R. Walker, J. Teasdale, V. Peterson and J. Vette, Phys. Rev. 92, 1090 (1953); and (private communication, January 1954).
19. R. F. Bacher, J. Keck, V. Peterson, J. Teasdale, A. Tollestrup, R. Walker and R. Worlock, Phys. Rev. 92, 1090 (1953).
20. K. A. Brueckner and K. M. Watson, Phys. Rev. 86, 923 (1952).

1. V. T. Feld, Phys. Rev. 89, 330 (1953).
2. M. Sands, J. Teasdale and R. Walker, Phys. Rev. 95, 592 (1954).
3. K. Watson, Phys. Rev. 88, 1163 (1952).
4. R. G. Sachs, Phys. Rev. (to be published) [see Phys. Rev. 95, 1065 (1954) (translator’s note)].
   24a. D. Oakley and R. D. Walker, Phys. Rev. (to be published) [see Phys. Rev. 97, 1283 (1955) (translator’s note)].
5. G. Bernardini and E. Goldwasser (private communication).
6. L. S. Osborne, Y. Goldschmidt-Clermont and M. Scott (unpublished work).
7. E. Fermi (private communication, April 1954).
   27a. K. Watson, Phys. Rev. 95, 228 (1954).
8. F. S. Crawford and M. L. Stevenson, U. S. Atomic Energy Commission Document UCRL—2187—88 (1953).
9. R. H. Hildebrand, Phys. Rev. 89, 1090 (1953).
10. R. A. Schluter, Bull. Am. Phys. Soc. 29 (4) (1954).
11. K. M. Watson and C. Richman, Phys. Rev. 83, 1257 (1951).
12. R. E. Marshak and A. M. Messiah, Nuovo Cimento 11, 337 (1954).
13. J. W. Mather and E. A. Martinelli, Phys. Rev. 92, 780 (1953).
14. J. Marshall and L. Marshall, Phys. Rev. 88, 632 (1952).
15. G. B. Yodh (private communication).
16. C. Wright and R. A. Schluter, Bull. Am. Phys. Soc. 29 (4) (1954); (private communication, May 1954).
17. A. L. Aitken, H. M. Mahmoud, E. M. Henley, M. A. Ruderman and K. M. Watson, Phys. Rev. 93, 1349 (1954).
18. J. C. Ward, Phys. Rev. 84, 897 (1951).
19. W. Pauli, Meson Theory of Nuclear Forces (Interscience Publishers, New York, 1946).
20. G. F. Chew, Phys. Rev. 89, 591 (1953); J. S. Blair and G. F. Chew, Phys. Rev. 90, 1065 (1953).
21. G. F. Chew, Phys. Rev. 95, 285 (1954).
22. J. Tamm, J. Phys. U. S. S. R. 9, 449 (1945).
23. S. Dancoff, Phys. Rev. 78, 382 (1950).
24. J. Gammel, Phys. Rev. 95, 209 (1954).
25. F. Salzman and J. Snyder, Phys. Rev. 95, 286 (1954).
26. M. Taketani et al., Progr. Theor. Phys. (Japan) 6, 581 (1951); M. Taketani et al., Progr. Theor. Phys. (Japan) 7, 45 (1952).
27. R. P. Feynman and J. Leite Lopes, Notas de Fisica, No. 2 (1954).
28. K. A. Brueckner and K. M. Watson, Phys. Rev. 92, 1023 (1953).
29. R. Jastrow, Phys. Rev. 81, 165 (1951).
30. S. Fujii, J. Iwadare, S. Otsuki, M. Taketani, S. Tani and W. Watari, Progr. Theor. Phys. (Japan) 11, 11 (1954).
31. K. A. Brueckner (private communication, May 1954).
32. N. Kroll and M. Ruderman, Phys. Rev. 93, 233 (1954).
33. L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950); F. J. Dyson, Phys. Rev. 73, 929 (1948).
34. G. Wentzel, Phys. Rev. 86, 802 (1952).
35. F. J. Dyson, S. S. Schweber and W. M. Visscher, Phys. Rev. 90, 372 (1953); M. Ross, Phys. Rev. 92, 454 (1954); H. A. Bethe and F. de Hoffman, Meson Physics, N. P. (1954).
36. K. A. Brueckner, M. Gell-Mann and M. Goldberger, Phys. Rev. 90, 476 (1953).

1. See Table III, reference (e). ↩↩↩