Abstract
This review is devoted to a discussion of both experimental studies of the scattering of $\pi$-mesons by nucleons and attempts at a theoretical interpretation of the regularities observed in the experiments. An attempt is made to find an extrapolation formula for the dependence of the total cross section on energy. A comparison is made with experimental data of the existing meson theory of the interaction of $\pi$-mesons and nucleons in the weak- and strong-coupling approximations, as well as taking damping into account. In doing so, only one variant of the theory is considered, namely the symmetric pseudoscalar meson theory with pseudovector (gradient) coupling.
Full Text
SCATTERING OF $\pi$ MESONS BY NUCLEONS
V. P. Silin and V. Ya. Fainberg
INTRODUCTION
Recently, experimental work on the scattering of $\pi$ mesons by nucleons has attracted considerable interest. This is explained by the fact that the problem of the interaction of elementary particles, and in particular the interaction of heavy nuclear particles—nucleons (nuclear forces)—is a central problem of modern physics. We still know very little about the nature of the forces acting between nucleons. Since these forces are, at least in part, due to $\pi$ mesons, elucidating the character of the interaction of the latter with nucleons is of great importance for constructing a consistent theory of nuclear interactions.
In this question, a special place belongs to experiments on the scattering of $\pi$ mesons by nucleons, since in them, unlike, for example, the photoproduction of $\pi$ mesons on nucleons[^1], at sufficiently high energies ($\gtrsim 20$ MeV) the interaction of the particles with the electromagnetic field is negligibly small, and purely nuclear scattering plays the principal role.
The present review is devoted to a discussion both of experimental work on the scattering of $\pi$ mesons by nucleons and of attempts at a theoretical interpretation of the regularities discovered in the experiments.
In § 1 the results of experimental work on the scattering of $\pi$ mesons by hydrogen and deuterium are collected.
The main results of the experiments are as follows: first, the ratio of the scattering cross sections of positive and negative (including charge exchange) $\pi$ mesons on hydrogen noticeably exceeds unity at 60 MeV, while at 150 MeV this ratio reaches approximately 3.
The scattering of positive and negative $\pi$ mesons on deuterium proves to be the same.
Second, the $S$ wave makes a noticeable contribution to the scattering (in connection with which terms $\sim \cos \theta$ appear in the angular distribution). Third, the scattering cross sections increase rapidly with energy; at energies $\sim 200$ MeV the cross section for negative $\pi$ mesons reaches saturation (for
$\pi^+$-mesons the cross section has been measured approximately up to 150 MeV). The ratio obtained in the experiments between the cross sections for $\pi^+$- and $\pi^-$-mesons in hydrogen and deuterium apparently confirms the hypothesis of the charge independence of nuclear forces. In its most general form, independently of the specific form of the interaction (assuming charge independence), the indicated relations between cross sections can be obtained in the theory of isotopic spin. This question is treated in § 2.
Despite the absence of a consistent theory of the interaction of $\pi$-mesons and nucleons (see, for example, $^3$), important conclusions about the character of the forces acting between these particles can be drawn on the basis of a phenomenological analysis of the experimental scattering data. In § 3 such an analysis is carried out, starting from very general propositions of the phase theory of scattering. This analysis makes it possible to find the relation between the contributions to scattering from the various states participating in the scattering ($S$- and $P$-waves). The most substantial result here is that the contribution to the scattering from the $S$-wave turns out to be comparable with the contribution from the $P$-waves.
At the end of this section an attempt is made to find an extrapolation formula for the dependence of the total cross section on the energy.
Finally, § 4 is devoted to a comparison with the experimental data of the existing meson theory of the interaction of $\pi$-mesons and nucleons in the approximations of weak and strong coupling, and also with allowance for damping. In doing this, only one version of the theory is considered, namely the symmetric pseudoscalar meson theory with pseudovector (gradient) coupling. This restriction is due to the fact that from a number of experiments one may conclude$^{1,2,2a}$ that $\pi$-mesons are pseudoscalar particles. Other experiments, in particular the rapid increase of the $\pi$-meson scattering cross section with energy, can in part be understood from the point of view of gradient coupling.
A comparison of the results of investigations of $\pi$-meson scattering published to date, based on meson-field theory, with the experimental data leads to discouraging conclusions. It is especially important to emphasize that, in the approximations considered, meson theory, which assumes gradient coupling of mesons with nucleons, is not able to explain the appearance of a large $S$-wave in scattering.
At the end of § 4 some possible ways of explaining the experimental results are discussed.
§ 1. EXPERIMENTAL RESULTS
The number of experimental works on $\pi$-meson scattering available at the present time is small. Nevertheless, a number of interesting conclusions can already be drawn from the available data.
We shall not consider the results of experimental works in which the scattering of $\pi$-mesons by complex nuclei was studied,
since the influence of the structure of the nucleus considerably complicates this phenomenon, we shall confine ourselves only to considering the scattering of \(\pi\)-mesons by protons and by deuterons. At the same time, we shall consider the scattering of \(\pi\)-mesons by deuterons only insofar as, in a number of cases, the interaction of the nucleons in the deuteron can be neglected.
The richest in results are papers \(^{4-8}\). In these papers the energy dependence of the total cross sections for the scattering of negative and positive \(\pi\)-mesons by protons \(^{5}\) and by deuterium \(^{6}\) was obtained. In addition, the ratio of the cross sections for elastic scattering of negative \(\pi\)-mesons by protons and for their scattering with charge exchange \(^{7}\) was obtained, and the angular dependence of the scattering of negative and positive \(\pi\)-mesons by protons \(^{8}\) was also studied.
This same range of questions was also investigated at lower energies by other authors, both by similar \(^{9,10,11}\) and by other methods \(^{12,13}\).
In papers \(^{4-8}\) \(\pi\)-mesons were produced in collisions with copper or beryllium targets of fast protons accelerated to an energy of \(450\) MeV. After being produced, the \(\pi\)-mesons were deflected by the magnetic field of the phasotron and entered channels leading the mesons out of the phasotron (Fig. 1). In this case the beam of positive \(\pi\)-mesons proved to be less intense than the beam of negative \(\pi\)-mesons. The latter is connected with the fact that the negative \(\pi\)-mesons taken out of the phasotron are emitted in the direction of the proton beam, whereas the positive ones are emitted in the opposite direction, and the intensity of \(\pi\)-meson emission backward is smaller than the intensity of emission forward. The beam of \(\pi\)-mesons extracted from the phasotron was monochromatized and purified, giving a well-collimated beam of \(\pi\)-mesons with an energy determined with an accuracy of \(\pm 3\%\). In addition to \(\pi\)-mesons, the beam contained \(\mu\)-mesons and electrons with the same momentum. In this connection, appropriate corrections were introduced, taking into account the Coulomb scattering of \(\mu\)-mesons and electrons. The number of \(\mu\)-mesons varied between five and ten percent, while the number of electrons for beams with energy greater than \(100\) MeV was negligibly small. At lower energies the number of electrons increased rapidly. Therefore, for measurements at low energies a beam of \(\pi\)-mesons with an energy of \(122\) MeV was used, in whose path a beryllium absorber was placed, lowering the beam energy.
Fig. 1. Schematic of the experimental setup.
For measuring total scattering cross sections the following method was used. The \(\pi\)-meson beam was registered by two scintillation-
with crystalline counters of area \(\sim 6.5\ \text{cm}^2\), located at a distance of one meter from each other. Coincidences of these two counters indicated the number of particles entering the scattering chamber. The chamber could
Table I
Total scattering cross section of positive \(\pi\)-mesons on hydrogen
| Energy [Mev] | Cross section \((10^{-27}\ \text{cm}^2)\) |
|---|---|
| \(56 \pm 8\) | \(20 \pm 10\) |
| \(82 \pm 7\) | \(50 \pm 13\) |
| \(118 \pm 6\) | \(91 \pm 6\) |
| \(136 \pm 6\) | \(152 \pm 14\) |
Table II
Total scattering cross section of negative \(\pi\)-mesons on hydrogen
| Energy [Mev] | Cross section \((10^{-27}\ \text{cm}^2)\) |
|---|---|
| \(89 \pm 8\) | \(21 \pm 8\) |
| \(112 \pm 6\) | \(31 \pm 9\) |
| \(135 \pm 6\) | \(52 \pm 6\) |
| \(176 \pm 6\) | \(66 \pm 6\) |
| \(217 \pm 6\) | \(60 \pm 6\) |
be filled with liquid hydrogen and emptied. Particles not removed from the beam by the scatterer were recorded by the coincidence of two scintillation counters, one of which had a diameter of \(7.6\ \text{cm}\), and the other \(10\ \text{cm}\). In the experiment, double coincidences of the first two counters and simultaneously quadruple coincidences of all four counters were registered. The attenuation of the beam was determined from a comparison of the ratio of quadruple and double coincidences in the case of a chamber filled with the scattering substance and in the case of an empty chamber. The total effective scattering cross sections for positive and negative \(\pi\)-mesons on hydrogen obtained from this are presented in Tables I and II and in Fig. 2.
Fig. 2. Total effective scattering cross section of positive \((+)\) and negative \((\square)\) \(\pi\)-mesons on hydrogen as a function of the energy of the incident mesons.
Scattering of positive \(\pi\)-mesons was measured for energies from \(60\ \text{Mev}\) to \(150\ \text{Mev}\), and of negative \(\pi\)-mesons—from \(80\ \text{Mev}\) to \(230\ \text{Mev}\). In the energy region \(>100\ \text{Mev}\) the scattering cross section of positive \(\pi\)-mesons proves to be approximately three times larger than the scattering cross section of negative \(\pi\)-mesons. The total scattering cross section of negative \(\pi\)-mesons on hydrogen increases rapidly and at
SCATTERING OF $\pi$-MESONS ON NUCLEONS
$150\ \mathrm{MeV}$ reaches the “geometrical” value $\pi(\hbar/\mu c)^2$, after which it remains approximately constant.
Figure 2 also gives the results of works $^{9-11}$ and $^{13}$. In works $^{9-11}$ the effective cross section for scattering of $\pi$-mesons on hydrogen was determined from the difference in the attenuation of a $\pi$-meson beam in carbon and polyethylene, and counters were used for particle registration. Thus, in $^{9}$, for the total scattering cross section of negative $\pi$-mesons with energy $85\ \mathrm{MeV}$, the value
\[
\sigma(p^-)=(13.3\pm 1.1)\cdot 10^{-27}\ \mathrm{cm}^2
\]
was obtained. In $^{10}$, for the mean meson energy $58\ \mathrm{MeV}$, it was found that
\[
\sigma(p^+)=(27.8\pm 2.5)\cdot 10^{-27}\ \mathrm{cm}^2,\quad
\sigma(p^-)=(17.6\pm 2.2)\cdot 10^{-27}\ \mathrm{cm}^2.
\]
At a mean beam energy of $37\ \mathrm{MeV}$, in $^{11}$ the total scattering cross section of positive $\pi$-mesons on hydrogen was studied. The measured cross section proved to be equal to $(12.4\pm 3)\cdot 10^{-27}\ \mathrm{cm}^2$. The counter geometry excluded the possibility of registering particles scattered at small angles. Therefore, to obtain the total cross section it was necessary to make an assumption about the character of the angular distribution. The assumption of isotropy of the cross section in the center-of-inertia system leads to $\sigma=(16.6\pm 4)\cdot 10^{-27}\ \mathrm{cm}^2$; the distribution law $\cos^2\vartheta$ gives $20.8\cdot 10^{-27}\ \mathrm{cm}^2$, while $\sin^2\vartheta$ gives $\sigma=14.9\cdot 10^{-27}\ \mathrm{cm}^2$.
In works $^{12}$ and $^{13}$ the scattering was observed in a Wilson chamber. In $^{13}$, at an energy of $53\ \mathrm{MeV}$, for the scattering of positive $\pi$-mesons by protons, a cross section of magnitude $(20\pm 4)\cdot 10^{-27}\ \mathrm{cm}^2$ was obtained. At an energy of $60\ \mathrm{MeV}$, in $^{12}$ the scattering of negative $\pi$-mesons on hydrogen was observed. The cross section obtained there is equal to $\sim 3\cdot 10^{-27}\ \mathrm{cm}^2$. It should be noted that this cross section corresponds to a process different from that observed in the method using beam attenuation.
Indeed, the total scattering cross section of negative $\pi$-mesons is composed of the cross sections of three processes
\[
\pi^-+p\to \pi^-+p,
\tag{1.1}
\]
\[
\pi^-+p\to \pi^0+n\to 2\gamma+n,
\tag{1.2}
\]
\[
\pi^-+p\to n+\gamma.
\tag{1.3}
\]
In the method using beam attenuation, the cross section of all three processes is measured. On the contrary, in $^{12}$ only the cross section of process (1.1) is measured (elastic scattering of negative $\pi$-mesons).
In studying the scattering of $\pi$-mesons on deuterium $^{6}$, the difference in the attenuation of a $\pi$-meson beam in $\mathrm{H_2O}$ and $\mathrm{D_2O}$ targets was measured. In both cases the scattering chamber had the same shape and dimensions and contained approximately the same number of atoms per $1\ \mathrm{cm}^2$. This led to the fact that the energy losses, Coulomb scattering, and also nuclear effects due to the presence of oxygen were approximately the same in both cases. Consequently, half the difference of the observed effective scattering cross sections on $\mathrm{D_2O}$ and $\mathrm{H_2O}$ gives the difference
cross sections for the scattering of π-mesons on deuterium and hydrogen. This difference \(\sigma_D-\sigma_H\), with corrections taken into account, is given in Table III. In the first column of this table is indicated the solid angle at which the last counter is seen from the scattering target. This angle characterizes the quality of the “geometry” of the experiment. It is important that the values of the scattering cross section for different geometries (different solid angles) agree within the accuracy of the experimental errors.
Table III
Effective scattering cross section of positive and negative π-mesons on deuterium
| Solid angle (steradians) | Energy (\(M_{\mathrm{eV}}\)) | \(\sigma_D-\sigma_H\) \((10^{-27}\ \mathrm{cm}^2)\) | \(\sigma_H\) \((10^{-27}\ \mathrm{cm}^2)\) | \(\sigma_D\) \((10^{-27}\ \mathrm{cm}^2)\) |
|---|---|---|---|---|
| \(\pi^-\) | \(\pi^-\) | \(\pi^-\) | ||
| 0.63 | \(79\pm10\) | \(34\pm10\) | \(48\pm10\) | \(54\pm13\) |
| 0.63 | \(109\pm15\) | \(72\pm5\) | \(80\pm10\) | \(103\pm10\) |
| 0.088 | \(115\pm9\) | \(88\pm7\) | \(95\pm15\) | \(124\pm11\) |
| 0.63 | \(115\pm9\) | \(84\pm18\) | — | — |
| 0.43 | \(127\pm15\) | \(84\pm8\) | \(125\pm15\) | \(129\pm11\) |
| 0.63 | \(133\pm9\) | \(76\pm15\) | \(135\pm15\) | \(128\pm16\) |
| 0.088 | \(164\pm9\) | \(139\pm13\) | — | \(198\pm12\) |
| 0.63 | \(164\pm9\) | \(128\pm14\) | — | — |
| 0.088 | \(179\pm9\) | \(172\pm10\) | — | \(234\pm12\) |
| 0.63 | \(179\pm9\) | \(163\pm12\) | — | — |
| 0.43 | \(209\pm15\) | \(131\pm25\) | — | \(192\pm26\) |
| \(\pi^+\) | \(\pi^+\) | \(\pi^+\) | ||
| 0.43 | \(72\pm17\) | \(24\pm6\) | \(15\pm8\) | \(60\pm9\) |
| 0.63 | \(79\pm10\) | \(31\pm13\) | \(20\pm8\) | \(79\pm15\) |
| 0.43 | \(109\pm15\) | \(29\pm12\) | \(31\pm9\) | \(109\pm16\) |
| 0.43 | \(127\pm15\) | \(26\pm15\) | \(45\pm10\) | \(151\pm21\) |
In the second column are given the energies of the π-mesons. The effective scattering cross section of π-mesons on deuterium is obtained by adding \(\sigma_H\) to \(\sigma_D-\sigma_H\). Within the accuracy of the experimental errors it proves to be independent of the sign of the charge of the π-mesons. The same result was obtained in \(^{10}\) for the scattering on deuterium of mesons with mean energy \(58\ M_{\mathrm{eV}}\). Namely, there it was obtained:
\[ \sigma(d^+)=(38.3\pm3.1)\cdot10^{-27}\ \mathrm{cm}^2;\quad \sigma(d^-)=(32.8\pm3.1)\cdot10^{-27}\ \mathrm{cm}^2. \]
In Fig. 3 the cross section for scattering of π-mesons on deuterium is presented, and, for comparison, the sum \(\sigma_{\mathrm{H}}(\pi^+) + \sigma_{\mathrm{H}}(\pi^-)\) is also given. We shall return below to a comparison of these data.
Let us return again to the scattering of negative π-mesons. Estimates based on the principle of detailed balance indicate that the cross section of process (1.3), which is the inverse process to the photoproduction of a π-meson by a γ-quantum on a neutron\(^1\), is of the order of several units of \(10^{-27}\ \text{cm}^2\), and, consequently, is much smaller than the total scattering cross section of negative π-mesons, which at an energy of \(120\ \text{MeV}\) is approximately \(40 \cdot 10^{-27}\ \text{cm}^2\). Further, it is important to separate the contributions to this quantity from processes (1.1) and (1.2).
Fig. 3. Total effective cross section for scattering of π-mesons on deuterium. For comparison, points corresponding to \(\sigma_{\mathrm{H}}(\pi^+) + \sigma_{\mathrm{H}}(\pi^-)\) are plotted.
The experimental method\(^7\) for scattering with charge exchange [process (1.2)] differed only slightly from that used in the study of total scattering, chiefly in that the third and fourth counters were no longer placed in the path of the beam, but were arranged at an angle of \(90^\circ\) to the π-meson beam. To record γ-quanta, a lead plate of thickness \(\sim 0.6\ \text{cm}\) was placed in front of the third counter. The measurement was reduced to determining the ratio of the numbers of scattered and incident particles in the presence of hydrogen in the scattering chamber and without it, and also with lead and without it. In this way it is possible to separate events due to scattered negative π-mesons and events due to γ-quanta. For the solid angle under which the second two counters are seen from the scatterer, the ratio of quadruple coincidences to double coincidences due to π-mesons scattered with energy \(118\ \text{MeV}\) is equal to \((0.34 \pm 0.12)\cdot 10^{-4}\), while that due to photons is \((1.41 \pm 0.32)\cdot 10^{-4}\). To obtain the total cross sections of processes (1.1) and (1.2), it is necessary to make an assumption about the character of the angular distribution. If the angular distribution in the center-of-inertia system is taken to be isotropic and it is taken into account that γ-quanta are formed in pairs in the decay of the neutral π-meson, then the cross sections of processes (1.1) and (1.2) turn out to be, respectively, \((10 \pm 4)\cdot 10^{-27}\ \text{cm}^2\) and \((20 \pm 5)\cdot 10^{-27}\ \text{cm}^2\). Thus, scattering with charge exchange has a cross section approximately twice as large as the cross section of elastic scattering of negative π-mesons. This result is also confirmed in the experiment on the angular distribution (Table IV, see p. 332).
Measurements of the differential effective cross sections in\(^8\) were carried out on the basis of the same method. The second pair of counters was loca-
Table IV
| Energy \([MeV]\) | Process | \(a\ \left(10^{-27}\ \dfrac{cm^2}{sr}\right)\) | \(b\ \left(10^{-27}\ \dfrac{cm^2}{sr}\right)\) | \(c\ \left(10^{-27}\ \dfrac{cm^2}{sr}\right)\) | \(\displaystyle \int \dfrac{d\sigma}{d\Omega}\,d\Omega\ \left(10^{-27}\ cm^2\right)\) |
|---|---|---|---|---|---|
| 110 | \(\pi^+ \to \pi^+\) | \(3.5 \pm 0.6\) | \(-4.6 \pm 0.8\) | \(7.2 \pm 1.8\) | \(74.5 \pm 5.4\) |
| 135 | \(\pi^+ \to \pi^+\) | \(3.8 \pm 2.2\) | \(-6.8 \pm 2.7\) | \(17.5 \pm 6.6\) | \(121 \pm 19\) |
| 135 | \(\pi^- \to \pi^-\) | \(1.2 \pm 0.2\) | \(-0.1 \pm 0.3\) | \(0.3 \pm 0.7\) | \(16.2 \pm 2.3\) |
| 135 | \(\pi^- \to \pi^0\) | \(1.1 \pm 0.6\) | \(-2.5 \pm 0.5\) | \(6.3 \pm 1.9\) | \(40.6 \pm 2.3\) |
was placed at that angle to the \(\pi\)-meson beam at which the scattering products were studied. Charge-exchange scattering was distinguished from elastic scattering of negative \(\pi\)-mesons by placing a lead plate in front of the second pair of counters. The angular distribution of elastic scattering of positive \(\pi\)-mesons at \(110\ MeV\) and \(135\ MeV\) was measured, as well as elastic scattering and charge-exchange scattering of negative \(\pi\)-mesons at an energy of \(135\ MeV\). The observations were made at angles to the meson beam (in the laboratory coordinate system) of \(45^\circ\), \(90^\circ\), and \(135^\circ\).
Fig. 4. Angular distribution of positive \(\pi\)-mesons scattered by protons at an energy of \(53\ MeV\).
In the center-of-mass system the results can be expressed by the formula:
\[ \frac{d\sigma}{d\Omega}=a+b\cos\vartheta+c\cos^2\vartheta . \tag{1.4} \]
The coefficients \(a\), \(b\), and \(c\) are given in Table IV together with the statistical errors. The total cross sections obtained by integrating (1.4) over the angles are in good agreement with the values given above (see Tables I and II).
The angular distribution in the scattering of positive \(\pi\)-mesons with an energy of \(53\ MeV\) by protons was observed in a Wilson chamber\({}^{13}\). The differential cross section obtained in this case (Fig. 4) was determined with a considerable error. Nevertheless, it apparently indicates that the angular distribution deviates noticeably from isotropic.
§ 2. THEORY OF ISOTOPIC SPIN
In interactions with charged \(\pi\)-mesons, the proton and neutron can transform into one another. (For example, a proton, absorbing a negative \(\pi\)-meson, is transformed into a neutron.) In considering such transformations it is convenient to treat the proton and the neutron as two states of one and the same particle, which is called the nucleon. Both states of the nucleon are characterized by the value of the charge variable (charge), which is equal to 1 in the proton state of the nucleon and to 0 in the neutron state.
The wave function of the nucleon, in addition to coordinates and spin, must depend on the charge variable, which takes two values. By analogy with ordinary spin, this dependence can be introduced by representing the wave function of the nucleon in the form
\[ \psi = \begin{pmatrix} \psi_1\\ \psi_2 \end{pmatrix}. \tag{2.1} \]
The condition that the wave function be normalized to unity imposes the requirement
\[ |\psi_1|^2+|\psi_2|^2=1. \]
If we now assume that in the proton state of the nucleon the upper component \(\psi\) is nonzero, and in the neutron state the lower one, then the wave functions of these states will have the following form:
\[ \psi_p = \begin{pmatrix} 1\\ 0 \end{pmatrix}, \quad \psi_N = \begin{pmatrix} 0\\ 1 \end{pmatrix}. \tag{2.2} \]
By analogy with spin theory, the operators acting on the charge variable (or, what is the same thing, on the two-component functions (2.1) and (2.2)) are called operators of isotopic spin*). One may introduce three matrices \(\frac12\tau_x\), \(\frac12\tau_y\), and \(\frac12\tau_z\), forming the isotopic-spin vector \(\tau\),
\[ \frac12\tau_x=\frac12 \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}; \quad \frac12\tau_y=\frac12 \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}; \]
\[ \frac12\tau_z=\frac12 \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}, \tag{2.3} \]
which in appearance completely coincide with the matrices \(\frac12\sigma_x\), \(\frac12\sigma_y\), and \(\frac12\sigma_z\) of ordinary spin, but have a physical meaning entirely different from them. Whereas in spin theory the eigenvalues of the operator \(\frac12\sigma_z\) determine the projection of the particle spin on the \(z\)-axis, the operator \(\frac12\tau_z\) in the theory of isotopic spin directly
*) Isotopic spin was first introduced into the theory in the consideration of \(\beta\)-decay\(^{14}\).
has no physical meaning. However, with its aid one can construct the operator of the nucleon charge \(q\). Indeed, it is not difficult to verify that the operator
\[ q=\frac{1+\tau_z}{2}= \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix}, \tag{2.4} \]
when applied to the function \(\psi_p\), gives unity, and to the function \(\psi_N\), zero. Consequently, the eigenvalues of this operator are \(\lambda=1\) and \(\lambda=0\), depending on whether the nucleon is in the proton or neutron states (the nucleon charge is equal to \(e\lambda\)).
Instead of the two other operators \(\tau_x\) and \(\tau_y\), in the theory one more often uses the operators of creation and annihilation of charge \(\tau\) and \(\tau^*\) \(^{17,18}\), which are expressed in the form of a linear combination of \(\tau_x\) and \(\tau_y\),
\[ \tau=\frac{1}{2}(\tau_x+i\tau_y)= \begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}, \qquad \tau^*=\frac{1}{2}(\tau_x-i\tau_y)= \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}. \tag{2.5} \]
The operators \(\tau\) and \(\tau^*\) act on \(\psi_p\) and \(\psi_N\) in the following way:
\[ \tau\psi_p= \begin{pmatrix} 0&0\\ 1&0 \end{pmatrix} \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} 0\\ 1 \end{pmatrix} =\psi_N,\quad \tau\psi_N=0, \qquad \tau^*\psi_p=0;\quad \tau^*\psi_N= \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix} \begin{pmatrix} 0\\ 1 \end{pmatrix} = \begin{pmatrix} 1\\ 0 \end{pmatrix} =\psi_p. \tag{2.6} \]
The operator \(\tau\) transforms a proton into a neutron, while the operator \(\tau^*\) transforms a neutron into a proton.
Let us note that, if one regards the \(\psi\)-function of a nucleon as a spinor in a certain three-dimensional isotopic space, then \(i\tau_x\), \(i\tau_y\), \(i\tau_z\) will represent the operators of an infinitesimal rotation of the spinor \(\psi\). In the language of group theory \(^{19}\) one says that the matrices \(\tau_x,\tau_y,\tau_z\) and the identity matrix form a representation of the two-dimensional group of rotations in three-dimensional isotopic space.
In the case when nucleons interact with both charged and neutral mesons, it proves fruitful, in addition to the isotopic spin of the nucleon, to introduce also the operators of isotopic spin of the meson \(^{20,21}\). Indeed, the three kinds of \(\pi\)-mesons: \(\pi^+\), \(\pi^-\), and \(\pi^0\) can be regarded as three different states of one and the same particle. The wave function of such a particle, in contrast to the wave function of the nucleon \(^{21}\), must possess three components
\[ \varphi= \begin{pmatrix} \varphi_1\\ \varphi_2\\ \varphi_3 \end{pmatrix}, \tag{2.7} \]
corresponding to the three possible charge states of the \(\pi\)-meson. Moreover, taking normalization into account, for the positive, negative,
and neutral \(\pi\)-mesons \(\varphi\) has the form
\[ \varphi_+= \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix},\quad \varphi_-= \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix},\quad \varphi_0= \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix}. \tag{2.8} \]
Analogously to the isotopic spin of the nucleon \(\tau\), one may introduce the vector of the isotopic spin of the meson \(\mathbf T\). Since the wave function (2.7) has three components, the projections of the operator on different axes will be matrices with three rows and columns. These matrices have the following form \(^{16*}\):
\[ T_x= \begin{pmatrix} 0 & 1/\sqrt{2} & 0\\ 1/\sqrt{2} & 0 & 1/\sqrt{2}\\ 0 & 1/\sqrt{2} & 0 \end{pmatrix},\quad T_y= \begin{pmatrix} 0 & i/\sqrt{2} & 0\\ -i/\sqrt{2} & 0 & i/\sqrt{2}\\ 0 & -i/\sqrt{2} & 0 \end{pmatrix}, \]
\[ T_z= \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -1 \end{pmatrix}. \tag{2.9} \]
In form they exactly coincide with the matrices \(-M_x, M_y, M_z\) of the orbital angular momentum in a state with \(l=1\) (the orbital angular momentum is equal to unity) \(^{16}\). Therefore the operator of the isotopic spin of the meson is sometimes called the operator of the orbital isotopic angular momentum.
The eigenvalues of the operator \(T_z\) in the states \(\psi_+\), \(\psi_-\), and \(\psi_0\) are, respectively, \(+1\), \(-1\), and \(0\).
We thus see that for the meson, in contrast to the case of the nucleon, the charge operator is simply equal to \(T_z\)—the projection of the meson isotopic spin on the \(Z\) axis.
The meson wave function (2.7) may be regarded as a vector \(\varphi\) in isotopic space. The components of this vector \(\varphi_x\), \(\varphi_y\), and \(\varphi_z\) can be expressed in terms of the wave functions \(\varphi_+\), \(\varphi_-\), and \(\varphi_0\) of positive, negative, and neutral mesons. Namely:
\[ \varphi_x=\frac{1}{\sqrt{2}}(\varphi_+ + \varphi_-),\quad \varphi_y=\frac{i}{\sqrt{2}}(\varphi_- - \varphi_+),\quad \varphi_z=\varphi_0. \tag{2.10} \]
Under different transformations in isotopic space these quantities transform into one another.
The projections of the vector of the isotopic spin of the meson \(\mathbf T\), when multiplied by \(i\), are the operators of an infinitesimal rotation of the vector \(\varphi\) in isotopic space and, together with the unit matrix
*) The operator \(\mathbf T\) can also be expressed through the operators of absorption and emission of mesons \(a^*\) and \(a\): \(\mathbf T=i(a^*a)\), where the projections \(a\) (respectively \(a^*\)) refer to the different charge states of the \(\pi\)-meson \(^{22}\).
define a three-dimensional representation of the three-dimensional rotation group in this space.
Let us now turn to the consideration of the isotopic properties of the simplest system, formed from a $\pi$-meson*) and a nucleon.
Such a system, as well as each particle entering into it, can be characterized by a total isotopic moment $\mathbf{I}$, whose operator will be composed of the operator of the isotopic spin of the nucleon $\frac{1}{2}\boldsymbol{\tau}$ and the operator of the isotopic spin of the meson $\mathbf{T}$
\[ \mathbf{I}=\frac{1}{2}\boldsymbol{\tau}+\mathbf{T}. \tag{2.11} \]
The operator of the total charge of the meson—nucleon system will be equal to the sum of the charge operators of each particle
\[ \varphi=q+T_z=\frac{1}{2}+I_z. \tag{2.12} \]
In order to obtain the eigenfunctions of a system consisting of a $\pi$-meson and a nucleon, we must construct all possible products of the functions $\psi$ and $\varphi$. In all there are six distinct eigenfunctions:
\[ \left. \begin{array}{lll} (p^+) = \varphi_+\psi_p, & (p^-) = \varphi_-\psi_p, & (p^0)=\varphi_0\psi_p,\\ (n^+) = \varphi_+\psi_N, & (n^-) = \varphi_-\psi_N, & (n^0)=\varphi_0\psi_N . \end{array} \right\} \tag{2.13} \]
Here by $(p^+)$ is denoted the function of a system consisting of a positive meson and a proton, etc. The functions written out, generally speaking, are not eigenfunctions of the square of the operator of the total isotopic moment of the system $(I)^2$.
However, by taking certain linear combinations of the functions (2.13), one can construct eigenfunctions of $(I)^2$ which simultaneously belong to a given value of the projection $I_z$ of the total moment on the $z$ axis.
There will likewise be six such functions in all, two belonging to the state in which $(I)^2=\frac{1}{2}\left(\frac{1}{2}+1\right)$ (for simplicity we shall below say that $I=\frac{1}{2}$), and $I_z=\pm\frac{1}{2}$, and four to the state in which $I=\frac{3}{2}$ and $I_z$ takes the values $\pm\frac{1}{2},\ \pm\frac{3}{2}$; these functions have the following form$^{21,23}$**):
\[ \left. \begin{array}{l} \Phi^{1/2}_{1/2}=-\sqrt{\frac{1}{3}}\,\varphi_0\psi_p+\sqrt{\frac{2}{3}}\,\varphi_+\psi_N,\\[4pt] \Phi^{1/2}_{-1/2}=-\sqrt{\frac{2}{3}}\,\varphi_-\psi_p+\sqrt{\frac{1}{3}}\,\varphi_0\psi_N. \end{array} \right\} \tag{2.14a} \]
*) By a $\pi$-meson we here mean, as above, a particle which may be in three charge states.
**) That $(I)^2\Phi^{3/2}_{I_z}=\frac{3}{2}\left(\frac{3}{2}+1\right)\Phi^{3/2}_{I_z}$ (respectively for $\Phi^{1/2}_{I_z}$) can be verified by direct calculation.
for states in which \(I=1/2\), and
\[ \begin{aligned} \Phi^{3/2}_{3/2}&=\varphi_{+}\psi_p; \qquad &\Phi^{3/2}_{1/2}&=\sqrt{2/3}\,\varphi_0\psi_p+\sqrt{1/3}\,\varphi_{+}\psi_N,\\ \Phi^{3/2}_{-3/2}&=\varphi_{-}\psi_N; \qquad &\Phi^{3/2}_{-1/2}&=\sqrt{1/3}\,\varphi_{-}\psi_p+\sqrt{2/3}\,\varphi_0\psi_N \end{aligned} \tag{2.14b} \]
for states with \(I=3/2\). The upper index in these formulas indicates the value of \(I\), and the lower one the value of \(I_z\). All the functions \(\Phi^{I}_{I_z}\) are mutually orthogonal and normalized to unity. From formulas (2.14), in particular, it follows that in a system consisting of a positive \(\pi\)-meson and a proton, or of a negative \(\pi\)-meson and a neutron, both the total isotopic moment \(I\) and its projection \(I_z\) have definite values.
The functions (2.13), in turn, may be represented, with the aid of formulas (2.14a) and (2.14b), in the form of a superposition of states with definite \(I\) and \(I_z\):
\[ \begin{aligned} (p^+)&=\Phi^{3/2}_{3/2}; \qquad &(p^-)&=\sqrt{1/3}\,\Phi^{3/2}_{-1/2}-\sqrt{2/3}\,\Phi^{1/2}_{-1/2};\\ (p^0)&=\sqrt{2/3}\,\Phi^{3/2}_{+1/2}-\sqrt{1/3}\,\Phi^{1/2}_{+1/2}; \qquad &(n^+)&=\sqrt{1/3}\,\Phi^{3/2}_{1/2}+\sqrt{2/3}\,\Phi^{1/2}_{1/2};\\ (n^-)&=\Phi^{3/2}_{-3/2}; \qquad &(n^0)&=\sqrt{2/3}\,\Phi^{3/2}_{-1/2}+\sqrt{1/3}\,\Phi^{1/2}_{-1/2}. \end{aligned} \tag{2.15} \]
In the general case, for an arbitrary interaction of a \(\pi\)-meson with a nucleon, the relations obtained above, (2.14) and (2.15), are of little use. On the contrary, in the case of the so-called symmetric interaction \(^{24}\), leading, in particular, to charge independence of nuclear forces*), from formulas (2.14) and (2.15) one can obtain a number of interesting results. Specific to the symmetric theory of interaction is the assumption that the proton and neutron have identical interaction constants with charged mesons, while the constants of their interaction with neutral mesons have opposite signs. It follows from this that the interaction Hamiltonian in the symmetric theory is proportional to the scalar product \(\boldsymbol{\tau}\boldsymbol{\varphi}\) of the isotopic-spin operator of the nucleon \(\boldsymbol{\tau}\) and the vector \(\boldsymbol{\varphi}\), and is invariant under rotations in isotopic space. It is known, however, that invariance of the Hamiltonian with respect to some transformation means that, in the interaction process, a certain physical quantity is conserved. In the present case the invariance of the Hamiltonian with respect to rotations in isotopic
*) By charge independence here is meant equality of the forces between nucleons in different charge states \(^{25}\). The charge independence of nuclear forces is apparently confirmed by experiment.
space is connected with the conservation of the total isotopic moment of the system formed from the \(\pi\)-meson and the nucleon. Let us note that conservation of the total isotopic moment means something more than simply conservation of the total charge of the system. The latter is conserved under any interactions, whereas for the conservation of \(I\) invariance of the Hamiltonian with respect to rotations in isotopic space is required\(*\).
There are grounds for assuming that conservation of the total isotopic moment in nuclear interactions in fact occurs in nature\(**\). It is therefore natural to take as the basis of the further discussion not a concrete form of the interaction Hamiltonian characteristic of a symmetric theory, but precisely the assumption of conservation of the total isotopic moment \(I\) in any processes of nuclear interaction of \(\pi\)-mesons with nucleons\(**\*\).
Let us apply this proposition to the scattering of \(\pi\)-mesons by nucleons. We shall denote the matrix elements of the transition from the initial state to the final one as follows: \((p^+,p^+)\) for scattering of a positive \(\pi\)-meson by a proton; \((p^-,p^-)\) for scattering of a negative \(\pi\)-meson by a proton; \((n^0,p^-)\) for scattering of a negative \(\pi\)-meson with charge exchange, and analogously for scattering by a neutron. Using the conservation law of the total isotopic moment (which will be manifested in the fact that the matrix elements of transitions between states with different \(I\) and \(I_z\) must vanish), with the aid of the wave functions (2.14) we find:
\[ \left. \begin{aligned} (p^+,p^+) &= (\Phi^{3/2}\!\mid\!\Phi^{3/2}),\\ (p^-,p^-) &= \frac{1}{3}\{(\Phi^{3/2}\!\mid\!\Phi^{3/2})+2(\Phi^{1/2}\!\mid\!\Phi^{1/2})\},\\ (n^0,p^-) &= \frac{\sqrt{2}}{3}\{(\Phi^{3/2}\!\mid\!\Phi^{3/2})-(\Phi^{1/2}\!\mid\!\Phi^{1/2})\}. \end{aligned} \right\} \tag{2.16} \]
and analogous formulae for scattering by a neutron. The projection of the isotopic spin is not written out, since the interaction does not depend on \(I_z\) for a given \(I\).
\(*\) In the first nonvanishing approximation of perturbation theory, the symmetric theory leads to a potential for the interaction of two nucleons which contains only invariant combinations of the type \((\tau_1\tau_2)\), where \(\tau_i\) is the operator of the isotopic spin of the \(i\)-th nucleon \((i=1,2)\).
\(**\) Let us note that in the case of interaction with an electromagnetic field the law of conservation of the total isotopic moment is violated. For example, two protons in an \(S\)-state interact differently than two neutrons in the same state, owing to the presence of Coulomb forces between the protons.
\(**\*\) The only requirement imposed in this case on the interaction Hamiltonians is invariance with respect to rotations in isotopic space. Such an interaction we shall in the general case call charge-independent.
For cross sections that are proportional to the squares of the matrix elements, one obtains\(^{21,26}\)
\[ \left. \begin{aligned} d\sigma(p^+,p^+) &= d\sigma^{3/2},\\ d\sigma(p^-,p^-) &= \frac{1}{9}\left[d\sigma^{3/2}+4d\sigma^{1/2} +4\left(d\sigma^{3/2}d\sigma^{1/2}\right)^{1/2}\cos\varphi\right],\\ d\sigma(n^0,p^-) &= \frac{2}{9}\left[d\sigma^{3/2}+d\sigma^{1/2} -2\left(d\sigma^{3/2}d\sigma^{1/2}\right)^{1/2}\cos\varphi\right], \end{aligned} \right\} \tag{2.17} \]
where:
\[ d\sigma^{3/2}=\left|(\Phi^{3/2}\mid\Phi^{3/2})\right|^2,\qquad d\sigma^{1/2}=\left|(\Phi^{1/2}\mid\Phi^{1/2})\right|^2, \]
\[ 2\left(d\sigma^{3/2}d\sigma^{1/2}\right)^{1/2}\cos\varphi = \]
\[ =(\Phi^{3/2}\mid\Phi^{3/2})^*(\Phi^{1/2}\mid\Phi^{1/2}) (\Phi^{1/2}\mid\Phi^{1/2})^*(\Phi^{3/2}\mid\Phi^{3/2}), \]
\(\cos\varphi\) is an undetermined phase factor depending on the energy and angles.
Let us note that the terms proportional to \(\cos\varphi\) appear as a result of interference in the scattering of \(\Phi\)-waves corresponding to the values of the total isotopic spin \(I=3/2\) and \(1/2\). The total cross section for negative \(\pi\)-mesons, however, does not depend on \(\varphi\).
From formulas (2.17) it is evident that, generally speaking, \(d\sigma(p^+,p^+)\ne d\sigma(p^-,p^-)\) for an arbitrary interaction invariant with respect to rotations in isotopic space. The ratio of the cross sections for \(\pi^-\)- and \(\pi^+\)-mesons on the proton according to (2.17) is equal to
\[ \frac{d\sigma(p^-,p^-)+d\sigma(n^0,p^-)} {d\sigma(p^+,p^+)} =\frac{1}{3}+\frac{2}{3}\left(d\sigma^{1/2}/d\sigma^{3/2}\right) \tag{2.18} \]
and may vary over wide limits, from \(1/3\) to an arbitrarily large value, depending on the magnitude of the interaction in the state with \(I=1/2\).
The experimental data (see § 1) indicate that this ratio is close to \(1/3\). It follows from this that the principal contribution to the scattering of \(\pi\)-mesons by nucleons is given by states with total isotopic spin \(I=3/2\), whereas in the state with \(I=1/2\) the scattering is evidently appreciably weaker.
If one assumes that \(d\sigma^{1/2}=0\) (the interaction in states with \(I=1/2\) is completely absent), then from (2.17) it follows that
\[ d\sigma(p^+,p^+):d\sigma(p^-,p^-):d\sigma(n^0,p^-)=9:1:2. \tag{2.19} \]
Relation (2.19) is, in all probability, confirmed experimentally, which testifies in favor of a charge-independent (and, in particular, symmetric) interaction.
For the cross sections of scattering of \(\pi\)-mesons on the neutron, one can obtain formulas entirely analogous to formulas (2.17), and it turns out that
it is obtained that
\[ \left. \begin{aligned} d\sigma(n^-,n^-)&=d\sigma(p^+,p^+),\\ d\sigma(n^+,n^+)&=d\sigma(p^-,p^-),\\ d\sigma(p^0,n^+)&=d\sigma(n^0,p^-). \end{aligned} \right\} \tag{2.20} \]
Consequently, between these cross sections the relations (2.18) and (2.19) hold with \(\pi^+\) replaced in them by \(\pi^-\).
The formulas (2.20) can be checked in experiments on the scattering of \(\pi\)-mesons by deuterons. If it is assumed that the particles of the deuteron scatter mesons independently, then the total scattering cross section on the deuteron will be equal to the sum of the cross sections on the proton and neutron.*)
\[ \left. \begin{aligned} d\sigma(d^+,d^+)&=d\sigma(p^+,p^+)+d\sigma(n^+,n^+),\\ d\sigma(pp^0,d^+)&=d\sigma(p^0,n^+),\\ d\sigma(d^-,d^-)&=d\sigma(p^-,p^-)+d\sigma(n^-,n^-),\\ d\sigma(nn^0,d^-)&=d\sigma(n^0,p^-). \end{aligned} \right\} \tag{2.21} \]
By virtue of (2.20), it follows from (2.21) that
\[ d\sigma(d^+,d^+)+d\sigma(pp^0,d^+)=d\sigma(d^-,d^-)+d\sigma(nn^0,p^-). \tag{2.22} \]
Thus, under the assumption of independent scattering of \(\pi\)-mesons by the nucleons of the deuteron, charge-independent and, in particular, symmetrical theory lead to equality of the total scattering cross sections of positive and negative \(\pi\)-mesons by the deuteron. Such equality is apparently confirmed by experiment (§ 1).
With the aid of formulas (2.17), (2.20), and (2.22), the total scattering cross section of charged \(\pi\)-mesons by the deuteron can be represented in the form
\[ d\sigma(d^+)=d\sigma(d^-)=\frac{4}{3}d\sigma^{3/2}+\frac{2}{3}d\sigma^{1/2}. \tag{2.23} \]
Under the assumption that the interaction in states with \(I=\frac{1}{2}\) is weak, the second term in (2.23) may be neglected. Then one obtains
\[ d\sigma(d^+):d\sigma(p^+,p^+):[d\sigma(p^-,p^-)+d\sigma(n^0,p^-)]=4:3:1. \tag{2.24} \]
We note that the accuracy of the experimental data available at present is still insufficient to draw an unambiguous conclusion about the role in scattering of states with total isotopic moment \(I=\frac{1}{2}\).
Everything done above can be generalized to the case of systems consisting of an arbitrary number of mesons and nucleons.
*) Here and below, reactions of the type \((2p,d^+)\) and \((2p,d^-)\), which proceed with a relatively small cross section at energies \(\sim \mu c^2\), are not taken into account.
§ 3. GENERAL THEORY OF SCATTERING OF π-MESONS BY NUCLEONS AND INTERPRETATION OF EXPERIMENTAL DATA
In order to make it possible to interpret the results of experiments on the scattering of π-mesons by nucleons, it is necessary to have at one’s disposal certain basic formulas of scattering theory. First of all we shall obtain formulas for the effective cross sections for the scattering of π-mesons by nucleons, expressed in terms of phases. The dependence of the phases on energy can be obtained from comparatively simple qualitative assumptions. Below, expressions will also be obtained for the effective cross sections for the scattering of π-mesons as functions of the energy of the colliding particles.
The problem of the scattering of π-mesons by nucleons reduces to a one-body problem by means of the usual transition to the coordinate system in which the center of inertia of the colliding particles is at rest. We shall denote the momentum and the total energy of the meson in this coordinate system by \(k\) and \(\mathcal E\), and the scattering angle by \(\vartheta\). For quantities in the laboratory coordinate system, in which the nucleon is at rest, we use the following notation: \(x\) is the momentum of the incident meson, \(E=\sqrt{x^2+\mu^2}-\mu\) is the kinetic energy of the meson, \(\mu\) is the mass of the meson (throughout we adopt the system of units \(\hbar=c=1\)), \(\vartheta_1\) is the scattering angle of the π-meson, and \(\vartheta_2\) is the angle between the direction of motion of the recoil nucleon and the momentum of the incident π-meson.
\(\mathcal E\) and \(k\) are related in the following way to the energy and angles of deflection of the particles in the laboratory coordinate system*):
\[ k=E^{1/2}(E+2\mu)^{1/2} \left[1+2\frac{E}{M}+2\frac{\mu}{M}+\left(\frac{\mu}{M}\right)^2\right]^{-1/2}, \tag{3.1} \]
\[ \cos\vartheta_1= \frac{\mathcal E_1(E+\mu+M)-(E+\mu)M-\mu^2} {E^{1/2}(E+2\mu)^{1/2}k}, \tag{3.2} \]
\[ \cos\vartheta_2= \frac{(E+\mu+M)(\mathcal E_2-M)} {E^{1/2}(E+2\mu)^{1/2}k}, \tag{3.3} \]
where
\[ \mathcal E_1 = E+\mu- \frac{ME(E+2\mu)} {\mu^2+M^2+2ME+2M\mu} (1-\cos\vartheta), \tag{3.4} \]
\[ \mathcal E_2=M+E+\mu-\mathcal E_1. \tag{3.5} \]
Below we shall everywhere use the coordinate system in which the center of inertia is at rest, and formulas (3.1)—(3.5) should be used for conversion in passing from the laboratory coordinate system.
The wave function of the system consisting of a π-meson and a nucleon is a multicomponent function. In this case, as we
*) In connection with these formulas see \(^{27}\).
As we have already seen in the preceding paragraph, it can be represented in the form of a superposition of six functions corresponding to the six possible charge states. Further, each state with definite values of the isotopic quantum numbers (the total isotopic angular momentum and its projection on the \(z\)-axis), in addition, must be characterized by the direction of the nucleon spin and by the sign of its energy. Therefore each of the charge functions, by itself, must be a Dirac bispinor. For simplicity, we shall at first restrict ourselves to considering one charge state. Then the wave function describing scattering in such a charge state has four components \(\psi_\mu^{I,I_z}\), \(\psi_2^{I,I_z}\), \(\psi_3^{I,I_z}\), \(\psi_4^{I,I_z}\), whose asymptotic behavior at large distances is determined by the formula\(^{28}\)
\[ \psi_\mu^{I,I_z}\simeq a_\mu^{I,I_z}e^{ikz}+ \frac{f_\mu^{I,I_z}(\vartheta,\varphi)e^{ikr}}{r} \qquad (\mu=1,2,3,4). \tag{3.6} \]
If the nucleons are considered nonrelativistically, then one may restrict oneself to two-component functions corresponding to Pauli spinors. However, even in a relativistic treatment of nucleons in the scattering problem one may restrict oneself to only two-component functions. This is due to the fact that the four quantities \(a_\mu\), and, in turn, the four quantities \(f_\mu\), are not mutually independent, only two components being independent, corresponding to the two possible orientations of the nucleon spin\(^*\). Taking this into account, we obtain the following expression for the differential scattering cross section
\[ d\sigma^{I,I_z}= \frac{\left|f_1^{I,I_z}\right|^2+\left|f_2^{I,I_z}\right|^2} {\left|a_1\right|^2+\left|a_2\right|^2}\,d\Omega . \tag{3.7} \]
The wave function of the system consisting of a \(\pi\)-meson and a nucleon can be represented in the form of an expansion in a series in eigenfunctions of the total angular momentum. For two-component spinor functions such an expansion has the following form\(^{29}\):
\[ \psi^{I,I_z} = \sum_{m;\,l=0}^{\infty} a_{l+\,m}^{I,I_z} \begin{pmatrix} \sqrt{\,l+m+\dfrac{1}{2}\,}\;Y_{l,m-1/2}(\vartheta,\varphi) \\[4pt] -\sqrt{\,l-m+\dfrac{1}{2}\,}\;Y_{l,m+1/2}(\vartheta,\varphi) \end{pmatrix} R_{l+}^{I,I_z}(r) + \]
\[ + \sum_{m;\,l=1}^{\infty} a_{l-\,m}^{I,I_z} \begin{pmatrix} \sqrt{\,l-m+\dfrac{1}{2}\,}\;Y_{l,m-1/2}(\vartheta,\varphi) \\[4pt] \sqrt{\,l+m+\dfrac{1}{2}\,}\;Y_{l,m+1/2}(\vartheta,\varphi) \end{pmatrix} R_{l-}^{I,I_z}(r). \tag{3.8} \]
\[ \text{---} \]
\(^*\) In the coordinate system in which the meson is at rest, the function of the incident particle can be represented as a solution of the Dirac equation for a plane wave. Then \(|a_1/a_3|=|a_2/a_4|\). An analogous relation can also be written for the quantities \(\psi_\mu\)\(^{23,28}\).
Here \(a_{l\pm,m}^{I,I_z}\) are coefficients that are functions of the energy and of the eigenvalues of the angular momentum \((j)\) and its projection on the \(z\)-axis \((m)\), as well as of the isotopic quantum numbers \(I\) and \(I_z\); \(l\) is the orbital angular momentum associated with the total angular momentum \(j\) by the relation \(l=j\mp \frac12\), in accordance with the two possible mutual orientations of the nucleon spin and of the relative orbital angular momentum of the nucleon and the \(\pi\)-meson. The radial functions \(R_{l\pm}^{I,I_z}(r)\) must be determined from the corresponding wave equation. The problem of establishing such an equation for the system of a nucleon and a \(\pi\)-meson, in a rigorous formulation, cannot be regarded as solved. If, however, the nucleons are treated nonrelativistically, then, proceeding from the existing quantum theory of wave fields, one can arrive at an equation of the form\(^*\)
\[ \left[E^2-\mu^2+\left(1+\frac{E}{M}\right)\Delta+V\right]\psi=0, \tag{3.9} \]
where \(V\) is a complicated integral operator, acting both on the coordinates and on the spin variables of the wave function and characterizing the interaction of the \(\pi\)-meson and the nucleon.
From (3.9), for \(R_{l\pm}^{I,I_z}(r)\) one obtains the equation
\[ \frac{d^2 R_{l\pm}^{I,I_z}}{dr^2} +\frac{2}{r}\frac{dR_{l\pm}^{I,I_z}}{dr} -\frac{l(l+1)}{r^2}R_{l\pm}^{I,I_z} +k^2 R_{l\pm}^{I,I_z} = V_{l\pm}^{I,I_z}R_{l\pm}^{I,I_z}, \tag{3.10} \]
where
\[ k^2=(E^2-\mu^2)\left(1+\frac{E}{M}\right)^{-1}. \tag{3.11} \]
The forces acting between the \(\pi\)-meson and the nucleon have a certain finite range of action, which in order of magnitude is evidently equal to \(\hbar/\mu c\) (for a \(\pi\)-meson with mass 277 electron masses, \(\hbar/\mu c \simeq 1.4\cdot 10^{-13}\,\mathrm{cm}\)). We neglect the electrical Coulomb interaction, since in the experimentally investigated region of \(\pi\)-meson energies the Coulomb forces are significant only for scattering through small angles. The inclusion of the electrical interaction may be carried out in a manner completely analogous to that used in considering the scattering of protons by protons\({}^{25}\). The finite range of the forces leads to the following asymptotic form of the radial functions:
\[ R_{l\pm}^{I,I_z}\simeq \frac{\sin\left(kr-\frac{l\pi}{2}+\delta_{l\pm}^{I,I_z}\right)}{kr}. \tag{3.12} \]
\[
{}^*
\]
Equation (3.9), for \(V=0\) and \(M\to\infty\), goes over into the equation of a free particle with spin zero.
The phases \(\delta^{I,I_z}_{l\pm}\) are functions of the energy, generally speaking different for states with different \(I, I_z\) and \(j=l\pm \tfrac12\).
Further, the asymptotic form of the wave function of a free nucleon and \(\pi\)-meson can be represented in the following form:
\[ \begin{pmatrix} a^{I,I_z}_{1}\\ a^{I,I_z}_{2} \end{pmatrix} e^{ikr}\simeq \]
\[ \simeq \begin{pmatrix} a^{I,I_z}_{1}\\ a^{I,I_z}_{2} \end{pmatrix} \sum_{l=0}^{\infty} i^l(2l+1)P_l(\cos\vartheta)\, \frac{\sin\left(kr-\frac{\pi l}{2}\right)}{kr}. \tag{3.13} \]
According to formula (3.6), the difference between the wave function describing the scattering of a \(\pi\)-meson by a nucleon and (3.13) at large distances must represent an outgoing wave. From this condition it follows*) that the coefficients \(a^{I,I_z}_{l\pm,m}\) in the expansion (3.8) of the function describing scattering are different from zero only for \(m=\pm \tfrac12\) and have the following form:
\[ \left. \begin{aligned} a^{I,I_z}_{l-,\,1/2} &=\sqrt{2\pi}\,a^{I,I_z}_{1}\sqrt{l}\, \exp\left(i\delta^{I,I_z}_{l-}+i\frac{\pi l}{2}\right),\\ a^{I,I_z}_{l-,\,-1/2} &=\sqrt{2\pi}\,a^{I,I_z}_{2}\sqrt{l}\, \exp\left(i\delta^{I,I_z}_{l-}+i\frac{\pi l}{2}\right),\\ a^{I,I_z}_{l+,\,1/2} &=\sqrt{2\pi}\,a^{I,I_z}_{1}\sqrt{l+1}\, \exp\left(i\delta^{I,I_z}_{l+}+i\frac{\pi l}{2}\right),\\ a^{I,I_z}_{l+,\,-1/2} &=-\sqrt{2\pi}\,a^{I,I_z}_{2}\sqrt{l+1}\, \exp\left(i\delta^{I,I_z}_{l+}+i\frac{\pi l}{2}\right). \end{aligned} \right\} \tag{3.14} \]
The scattering amplitude turns out to be equal to (see also \(^{30}\))
\[ f^{I,I_z}(\vartheta,\varphi) = \frac{\sqrt{2\pi}}{2ik} \sum_{l=0}^{\infty} \left\{ \begin{pmatrix} a^{I,I_z}_{1}\\ a^{I,I_z}_{2} \end{pmatrix} \left[ (l+1)\exp\left(i2\delta^{I,I_z}_{l+}\right) + l\exp\left(i2\delta^{I,I_z}_{l-}\right) -2l-1 \right]Y_{l,0}(\vartheta,\varphi) + \sqrt{l(l+1)} \left[ \exp\left(i2\delta^{I,I_z}_{l-}\right) - \exp\left(i2\delta^{I,I_z}_{l+}\right) \right] \begin{pmatrix} a^{I,I_z}_{2}Y_{l,-1}(\vartheta,\varphi)\\ a^{I,I_z}_{1}Y_{l,1}(\vartheta,\varphi) \end{pmatrix} \right\}. \tag{3.15} \]
*) The considerations used in this derivation are quite similar to those applied in obtaining the scattering amplitude, expressed in terms of phases, in the case of the motion of a nonrelativistic particle in a central field \(^{15,16}\).
We shall be interested below in the scattering of \(\pi\)-mesons by unpolarized nucleons; therefore, in calculating its cross section one may average over the spin states of the nucleon. The cross section obtained [see (3.7)] then has the following form:
\[ \begin{aligned} d\sigma^{I,I_z} &= \frac{1}{k^2}\sum_{l,l'} \Biggl\{ \left[(l'+1)\exp\!\left(-i\delta_{l'+}^{I,I_z}\right)\sin\delta_{l'+}^{I,I_z} + l'\exp\!\left(-i\delta_{l'-}^{I,I_z}\right)\sin\delta_{l'-}^{I,I_z}\right] \\ &\qquad\qquad \times \left[(l+1)\exp\!\left(i\delta_{l+}^{I,I_z}\right)\sin\delta_{l+}^{I,I_z} + l\exp\!\left(i\delta_{l-}^{I,I_z}\right)\sin\delta_{l-}^{I,I_z}\right] P_{l',0}(\cos\vartheta)P_{l,0}(\cos\vartheta) \\ &\qquad +\sqrt{l'l(l'+1)(l+1)} \left[\exp\!\left(-i\delta_{l'-}^{I,I_z}\right)\sin\delta_{l'-}^{I,I_z} -\exp\!\left(-i\delta_{l'+}^{I,I_z}\right)\sin\delta_{l'+}^{I,I_z}\right] \\ &\qquad\qquad \times \left[\exp\!\left(i\delta_{l-}^{I,I_z}\right)\sin\delta_{l-}^{I,I_z} -\exp\!\left(i\delta_{l+}^{I,I_z}\right)\sin\delta_{l+}^{I,I_z}\right] P_{l',1}(\cos\vartheta)P_{l,1}(\cos\vartheta) \Biggr\}\,d\Omega . \end{aligned} \tag{3.16} \]
Integrating (3.16) over the angles, we obtain the following expression for the total scattering cross section:
\[ \sigma^{I,I_z} = \frac{4\pi}{k^2} \sum_{l=0}^{\infty} \left\{(l+1)\sin^2\delta_{l+}^{I,I_z} + l\sin^2\delta_{l-}^{I,I_z}\right\}. \tag{3.17} \]
For \(\delta_{l+}=\delta_{l-}\), formula (3.17) goes over into the well-known expression for the scattering cross section of a spinless particle\(^{15,16}\)
\[ \sigma = \frac{4\pi}{k^2} \sum_{l=0}^{\infty} (2l+1)\sin^2\delta_l . \]
Formulas (3.16)—(3.17), obtained under the assumption that there is only one charge state, are applicable for describing the scattering of positive \(\pi\)-mesons by protons, since this case corresponds to an eigenstate of isotopic spin \((I=I_z=3/2)\). For the scattering of negative \(\pi\)-mesons by protons, when the state of the system is a superposition of states with total isotopic spins \(3/2\) and \(1/2\), the formulas prove to be somewhat more complicated. The wave function of an arbitrary charge state of the system consisting of a \(\pi\)-meson and a nucleon can be represented as a superposition of functions of states with definite values of the isotopic spin
\[ \psi=\sum_{I,I_z} C_{I,I_z}\Phi_{I_z}^{I}. \tag{3.18} \]
The scattering amplitude also has a similar form
\[ f(\vartheta,\varphi)=\sum_{I,I_z} C_{I,I_z} f^{I,I_z}(\vartheta,\varphi), \tag{3.19} \]
where \(f^{I,I_z}(\vartheta,\varphi)\) is determined by formula (3.15), and in the latter the quantities depending on \(I\) and \(I_z\) are the phases \(\delta\). A special feature of the charge-independent interaction is that the phases with the same \(I\) and different \(I_z\) turn out to be equal (just as the phases depend only on the magnitude of the total angular momentum \(j=l\pm \frac12\) and do not depend on the values of its projection \(m\)). Below we shall assume that the interaction is charge-independent.
Let us consider the case of scattering of negative \(\pi\)-mesons by protons. The asymptotic behavior of the wave function is determined by the formula
\[ \frac{1}{\sqrt{3}} \left[ \Phi_{-\frac12}^{\frac32}+\sqrt{2}\Phi_{-\frac12}^{\frac12} \right] \begin{pmatrix} \alpha_1\\ \alpha_2 \end{pmatrix} e^{ikz} + \]
\[ +\frac{e^{ikr}}{r}\frac{1}{\sqrt{3}} \left[ \Phi_{-\frac12}^{\frac32} f^{\frac32,-\frac12}(\vartheta,\varphi) +\sqrt{2}\Phi_{-\frac12}^{\frac12} f^{\frac12,-\frac12}(\vartheta,\varphi) \right]. \tag{3.20} \]
The second term in (3.20) represents a diverging wave. It can be represented in the form of a superposition of waves corresponding to the charge states \((\pi^-,p)\) and \((\pi^0,n)\). Taking into account the form of the charge functions of these states [see (2.15)], we obtain the following expressions for the corresponding scattering amplitudes*):
\[ f(\pi^-,p)=\frac{1}{3\sqrt{3}} \left[ \Phi_{-\frac12}^{\frac32}+\sqrt{2}\Phi_{-\frac12}^{\frac12} \right] \left[ f^{\frac32,-\frac12}(\vartheta,\varphi) + \right. \]
\[ \left. +2f^{\frac12,-\frac12}(\vartheta,\varphi) \right], \tag{3.21} \]
\[ f(\pi^0,n)=\frac{\sqrt{2}}{3\sqrt{3}} \left[ \sqrt{2}\Phi_{-\frac12}^{\frac32}-\Phi_{-\frac12}^{\frac12} \right] \left[ f^{\frac32,-\frac12}(\vartheta,\varphi)- \right. \]
\[ \left. -f^{\frac12,-\frac12}(\vartheta,\varphi) \right]. \tag{3.22} \]
Of course,
\[ f(\pi^-,p)+f(\pi^0,n)=\frac{1}{\sqrt{3}} \left[ \Phi_{-\frac12}^{\frac32} f^{\frac32,-\frac12} + \right. \]
\[ \left. +\sqrt{2}\Phi_{-\frac12}^{\frac12} f^{\frac12,-\frac12} \right]. \tag{3.23} \]
From (3.22), in particular, it follows that if in the isotopic states \(I=\frac32\) and \(I=\frac12\) the operators of the interaction of the \(\pi\)-meson and the nucleon turn out to be identical, then in the scattering of negative \(\pi\)-mesons by protons no charge exchange will occur. The squares
*) Neglecting the process \(\pi^-+p\to n+\gamma\).
absolute values (3.21) and (3.22), give the differential cross sections of elastic scattering of negative \(\pi\)-mesons and of scattering with charge exchange. The corresponding formulas are rather cumbersome, but, generally speaking, can easily be obtained on the basis of (3.15), (3.21), and (3.22). The expressions for the total scattering cross sections are somewhat simpler:
\[ \sigma(p^-,p^-)= \]
\[ =\frac{4\pi}{3k^2}\sum_{l=0}^{\infty} \left\{(l+1)\left[\sin^2\delta_{l+}^{3/2}+2\sin^2\delta_{l+}^{1/2} -\frac{2}{3}\sin^2\left(\delta_{l+}^{3/2}-\delta_{l+}^{1/2}\right)\right]+\right. \]
\[ \left.+\,l\left[\sin^2\delta_{l-}^{3/2}+2\sin^2\delta_{l-}^{1/2} -\frac{2}{3}\sin^2\left(\delta_{l-}^{3/2}-\delta_{l-}^{1/2}\right)\right]\right\}. \tag{3.24} \]
\[ \sigma(p^-,n^0)= \]
\[ =\frac{8\pi}{9k^2}\sum_{l=0}^{\infty} \left\{(l+1)\sin^2\left(\delta_{l+}^{3/2}-\delta_{l+}^{1/2}\right) +l\sin^2\left(\delta_{l-}^{3/2}-\delta_{l-}^{1/2}\right)\right\}. \tag{3.25} \]
The total scattering cross section of negative \(\pi\)-mesons on protons is the sum of the cross sections (3.24) and (3.25) and is equal to:
\[ \sigma(p^-)=\sigma(p^-,p^-)+\sigma(p^-,n^0)= \]
\[ =\frac{4\pi}{3k^2}\sum_{l=0}^{\infty} \left\{(l+1)\left(\sin^2\delta_{l+}^{3/2}+2\sin^2\delta_{l+}^{1/2}\right)+ l\left(\sin^2\delta_{l-}^{3/2}+2\sin^2\delta_{l-}^{1/2}\right)\right\}. \tag{3.26} \]
Comparing (3.26) and (3.17), we obtain:
\[ \sigma(p^-)=\frac{1}{3}\sigma^{3/2}+\frac{2}{3}\sigma^{1/2}, \tag{3.27} \]
which is a manifestation of the general theorem based on the assumption of charge independence of the interaction [see (2.18)].
In the region of low energies, when the wavelength of the meson considerably exceeds the radius of action of the forces between the meson and the nucleon, the non-electromagnetic interaction leads to appreciable scattering only in the \(S_{1/2}\)-state, i.e. only in head-on collisions, since the impact parameters of particles colliding with nonzero angular momentum considerably exceed the radius of action of the forces. With increasing energy, scattering in \(P\)-states also becomes appreciable.
The experimental data can be satisfactorily interpreted by taking into account only states with \(l\leq 1\). Therefore, in what follows
we shall confine ourselves to taking into account the \(S_{1/2}\)-, \(P_{1/2}\)-, and \(P_{3/2}\)-states. The expressions for the differential scattering cross sections then take the form
\[ \frac{k^{2}\,d\sigma}{d\Omega}=A+B\cos\vartheta+C\cos^{2}\vartheta, \tag{3.28} \]
where the coefficients \(A\), \(B\), \(C\) are functions of the phases. In the case of scattering of positive \(\pi\)-mesons by protons they have the following form:
\[ A(p^{+},p^{+})=\sin^{2}\delta_{0}^{3/2} +\sin^{2}\bigl(\delta_{1+}^{3/2}-\delta_{1-}^{3/2}\bigr), \tag{3.29a} \]
\[ \begin{aligned} B(p^{+},p^{+})={}&3\sin^{2}\delta_{0}^{3/2} +\sin^{2}\delta_{1-}^{3/2} +2\sin^{2}\delta_{1+}^{3/2}\\ &-\sin^{2}\bigl(\delta_{1-}^{3/2}-\delta_{0}^{3/2}\bigr) -2\sin^{2}\bigl(\delta_{1+}^{3/2}-\delta_{0}^{3/2}\bigr), \end{aligned} \tag{3.29b} \]
\[ \frac{1}{3}\,C(p^{+},p^{+}) =\sin^{2}\delta_{1-}^{3/2} +2\sin^{2}\delta_{1+}^{3/2} -\sin^{2}\bigl(\delta_{1+}^{3/2}-\delta_{1-}^{3/2}\bigr). \tag{3.29c} \]
If, in addition, \(\delta_{0}^{3/2}\) is taken equal to zero, then \(B\) also vanishes. Therefore the most striking manifestation of \(S\)-scattering is the presence in the angular distribution of a term proportional to \(\cos\vartheta\), leading to an asymmetry of forward and backward scattering. This also applies to the processes of scattering of negative \(\pi\)-mesons. For elastic scattering of \(\pi^{-}\)-mesons \(A\), \(B\), and \(C\) are determined as follows:
\[ \begin{aligned} A(p^{-},p^{-})={}& \frac{1}{3}\sin^{2}\delta_{0}^{3/2} +\frac{2}{3}\sin^{2}\delta_{0}^{1/2} -\frac{2}{9}\sin^{2}\bigl(\delta_{0}^{3/2}-\delta_{0}^{1/2}\bigr)\\ &+\frac{1}{9}\sin^{2}\bigl(\delta_{1+}^{3/2}-\delta_{1-}^{3/2}\bigr) -\frac{2}{9}\sin^{2}\bigl(\delta_{1-}^{3/2}-\delta_{1-}^{1/2}\bigr)\\ &+\frac{2}{9}\sin^{2}\bigl(\delta_{1-}^{3/2}-\delta_{1+}^{1/2}\bigr) +\frac{2}{9}\sin^{2}\bigl(\delta_{1+}^{3/2}-\delta_{1-}^{1/2}\bigr)\\ &-\frac{2}{9}\sin^{2}\bigl(\delta_{1+}^{3/2}-\delta_{1+}^{1/2}\bigr) +\frac{4}{9}\sin^{2}\bigl(\delta_{1-}^{1/2}-\delta_{1+}^{1/2}\bigr), \end{aligned} \tag{3.30a} \]
\[ \begin{aligned} B(p^{-},p^{-})={}& \sin^{2}\delta_{0}^{3/2} +\frac{1}{3}\sin^{2}\delta_{1-}^{3/2} +\frac{2}{3}\sin^{2}\delta_{1-}^{1/2} +\frac{2}{3}\sin^{2}\delta_{1+}^{3/2}\\ &+\frac{4}{3}\sin^{2}\delta_{1+}^{1/2} +2\sin^{2}\delta_{0}^{1/2} -\frac{1}{9}\sin^{2}\bigl(\delta_{0}^{3/2}-\delta_{1-}^{3/2}\bigr)\\ &-\frac{2}{9}\sin^{2}\bigl(\delta_{0}^{3/2}-\delta_{1-}^{1/2}\bigr) -\frac{2}{9}\sin^{2}\bigl(\delta_{0}^{3/2}-\delta_{1+}^{3/2}\bigr)\\ &-\frac{4}{9}\sin^{2}\bigl(\delta_{0}^{3/2}-\delta_{1+}^{1/2}\bigr) -\frac{2}{9}\sin^{2}\bigl(\delta_{0}^{1/2}-\delta_{1-}^{3/2}\bigr)\\ &-\frac{4}{9}\sin^{2}\bigl(\delta_{0}^{1/2}-\delta_{1-}^{1/2}\bigr) -\frac{4}{9}\sin^{2}\bigl(\delta_{0}^{1/2}-\delta_{1+}^{3/2}\bigr)\\ &-\frac{8}{9}\sin^{2}\bigl(\delta_{0}^{1/2}-\delta_{1+}^{1/2}\bigr), \end{aligned} \tag{3.30b} \]
\[ \begin{aligned} C(p^{-},p^{-})={}&\sin^{2}\delta^{3/2}_{1-}+2\sin^{2}\delta^{1/2}_{1-}+2\sin^{2}\delta^{3/2}_{1+}+4\sin^{2}\delta^{1/2}_{1+}-\\ &-\frac{1}{3}\sin^{2}\bigl(\delta^{3/2}_{1+}-\delta^{3/2}_{1-}\bigr) -\frac{2}{3}\sin^{2}\bigl(\delta^{3/2}_{1-}-\delta^{1/2}_{1+}\bigr)-\\ &-\frac{2}{3}\sin^{2}\bigl(\delta^{3/2}_{1+}-\delta^{1/2}_{1-}\bigr) -\frac{2}{3}\sin^{2}\bigl(\delta^{1/2}_{1+}-\delta^{1/2}_{1-}\bigr)-\\ &-\frac{2}{3}\sin^{2}\bigl(\delta^{3/2}_{1+}-\delta^{1/2}_{1+}\bigr). \end{aligned} \tag{3.30в} \]
In the case of scattering with charge exchange, the corresponding formulas turn out to be the following:
\[ \begin{aligned} \frac{9}{2}A(n^{0},p^{-})={}&\sin^{2}\bigl(\delta^{3/2}_{0}-\delta^{1/2}_{0}\bigr) +\sin^{2}\bigl(\delta^{3/2}_{1-}-\delta^{1/2}_{1-}\bigr)+\\ &+\sin^{2}\bigl(\delta^{3/2}_{1-}-\delta^{3/2}_{1+}\bigr) -\sin^{2}\bigl(\delta^{3/2}_{1-}-\delta^{1/2}_{1+}\bigr) -\sin^{2}\bigl(\delta^{1/2}_{1-}-\delta^{3/2}_{1+}\bigr)+\\ &+\sin^{2}\bigl(\delta^{1/2}_{1-}-\delta^{1/2}_{1+}\bigr) +\sin^{2}\bigl(\delta^{3/2}_{1+}-\delta^{1/2}_{1+}\bigr), \end{aligned} \tag{3.31а} \]
\[ \begin{aligned} \frac{9}{2}B(n^{0},p^{-})={}&\sin^{2}\bigl(\delta^{3/2}_{0}-\delta^{1/2}_{1-}\bigr) -\sin^{2}\bigl(\delta^{3/2}_{0}-\delta^{3/2}_{1-}\bigr)-\\ &-2\sin^{2}\bigl(\delta^{3/2}_{0}-\delta^{3/2}_{1+}\bigr) +2\sin^{2}\bigl(\delta^{3/2}_{0}-\delta^{1/2}_{1+}\bigr) +\sin^{2}\bigl(\delta^{1/2}_{0}-\delta^{3/2}_{1-}\bigr)-\\ &-\sin^{2}\bigl(\delta^{1/2}_{0}-\delta^{1/2}_{1-}\bigr) +2\sin^{2}\bigl(\delta^{1/2}_{0}-\delta^{3/2}_{1+}\bigr) -2\sin^{2}\bigl(\delta^{1/2}_{0}-\delta^{1/2}_{1+}\bigr), \end{aligned} \tag{3.31б} \]
\[ \begin{aligned} -\frac{3}{2}C(n^{0},p^{-})={}& \sin^{2}\bigl(\delta^{3/2}_{1+}-\delta^{3/2}_{1-}\bigr) -\sin^{2}\bigl(\delta^{3/2}_{1+}-\delta^{1/2}_{1-}\bigr)-\\ &-\sin^{2}\bigl(\delta^{3/2}_{1+}-\delta^{1/2}_{1+}\bigr) -\sin^{2}\bigl(\delta^{1/2}_{1+}-\delta^{3/2}_{1-}\bigr) +\sin^{2}\bigl(\delta^{1/2}_{1+}-\delta^{1/2}_{1-}\bigr). \end{aligned} \tag{3.31в} \]
If the interaction of the \(\pi\)-meson and the nucleon is not charge-independent, then the phases of states with isotopic spin \(3/2\), entering into the cross section for scattering of \(\pi^{+}\)-mesons and into the cross section for \(\pi^{-}\)-mesons, may turn out to be different. This corresponds to the fact that the former refer to states with \(I_z=3/2\), and the latter to states with \(I_z=1/2\). As was already said above, we take the phases to be independent of the projection of the isotopic angular momentum. Below we shall see that this assumption does not contradict the experimental data, and consequently confirms the hypothesis of charge independence of the interaction of \(\pi\)-mesons and nucleons.
The values of the phases can be obtained without particular difficulty on the basis of the measured angular distribution of the scattered \(\pi\)-mesons. From comparison of formulas (3.28) and (1.4) it follows that the coefficients \(A, B, C\) in (3.28) are related to \(a, b, c\) in Table IV by the relations \(A=k^{2}a,\ B=k^{2}b,\ C=k^{2}c\). Solving equations (3.29), with \(A, B\), and \(C\) thus determined, for the phases, we obtain, in the case of scattering of \(\pi^{+}\)-mesons with energy \(110\ \mathrm{MeV}\) by protons, two solutions. The first solution (denoted below by I) leads to the following phases\(^8\):
\[ \delta^{3/2}_{0}=\pm 15^{\circ},\qquad \delta^{3/2}_{1-}=0^{\circ},\qquad \delta^{3/2}_{1+}=\pm 24^{\circ}. \]
The second solution (II) gives the same value of the phase of \(S\)-scattering, but different phases of the \(P\)-waves:
\[ \delta^{3/2}_{0}=\pm 15^\circ,\qquad \delta^{3/2}_{1-}=\mp 34^\circ,\qquad \delta^{3/2}_{1+}=\mp 9^\circ . \]
For \(\pi\)-mesons with energy \(135\) MeV, the angular distribution of three processes has been measured: scattering of \(\pi^+\)-mesons on protons, elastic scattering of negative \(\pi\)-mesons on protons, and their scattering with charge exchange. The corresponding nine coefficients \(a\), \(b\), and \(c\) are given in Table IV. For the interpretation of these nine experimental quantities, a theory which is restricted to taking into account scattering only in states with \(l \leq 1\) and assumes charge independence of the interactions has at its disposal six parameters—six phases \(\bigl(\delta^{3/2}_{0}, \delta^{3/2}_{1-}, \delta^{3/2}_{1+}, \delta^{1/2}_{0}, \delta^{1/2}_{1-}, \delta^{1/2}_{1+}\bigr)\). These six phases are related to the experimental data by means of the nine equations (3.29), (3.30), and (3.31).
Despite the fact that the system of equations thus obtained is overdetermined, all nine equations turn out to be compatible. This result should be regarded as good confirmation of the hypothesis of charge independence. The values of the phases for the energy \(135\) MeV are given in Table V.
Table V
| \(E\) (MeV) | \(S_{1/2}\) I | \(S_{1/2}\) II | \(P_{1/2}\) I | \(P_{1/2}\) II | \(P_{3/2}\) I | \(P_{3/2}\) II | |
|---|---|---|---|---|---|---|---|
| \(I=3/2\) | 135 | \(\pm 25^\circ\) | \(26^\circ\) | \(\mp 10^\circ\) | \(-46^\circ\) | \(\mp 35^\circ\) | \(-20^\circ\) |
| \(I=3/2\) | 110 | \(\pm 15^\circ\) | \(\pm 15^\circ\) | \(0^\circ\) | \(\mp 34^\circ\) | \(\mp 24^\circ\) | \(\mp 9^\circ\) |
| \(I=1/2\) | 135 | \(\pm 1^\circ\) | \(-1^\circ\) | \(\pm 19^\circ\) | \(-3^\circ\) | \(\pm 1^\circ\) | \(-15^\circ\) |
The calculation of phases constitutes the first step toward a theoretical interpretation of experiments on \(\pi\)-meson scattering. In addition to confirming the hypothesis of charge independence, on the basis of the phase values found one can obtain information about the magnitude of the interaction in the various states. It is true that an ambiguity arises with respect to scattering in \(P\)-states. As for \(S\)-scattering, the corresponding phases are calculated uniquely. In the state with isotopic spin \(3/2\), the \(S\)-phase proves to be comparable with the phases of the \(P\)-waves (see Table V). Such a large value of the \(S\)-phase finds its interpretation in the theoretical studies, existing at the present time, of the interaction of mesons and nucleons. For the \(P\)-phase, owing to the ambiguity mentioned above, the situation is somewhat more complicated, since it is not clear which of
obtained in calculating the phase shifts should be taken as the true ones. However, even in this case one may say that \(P\)-scattering also does not agree with the predictions of meson field theory. We shall discuss this in detail in the next paragraph, where it will be shown that as yet there is no unified theoretical scheme that explains the experiments on \(\pi\)-meson scattering in any reasonably complete way. In this respect the situation is somewhat similar to that which has arisen in the study of the interaction of nucleons.
For a further understanding of the data on \(\pi\)-meson scattering it is necessary to clarify the possible dependence of the phases on the energy. Since our information about the nature of the interaction between a \(\pi\)-meson and a nucleon is very limited, it is desirable to proceed along a path that uses the smallest number of concrete assumptions about the law of the forces acting between the \(\pi\)-meson and the nucleon.
In doing so one may start from the concepts used in explaining nucleon scattering \(^{25,31}\).
The point is that equation (3.10) for the radial functions of the system consisting of a \(\pi\)-meson and a nucleon is similar to the corresponding equation occurring in nonrelativistic quantum mechanics for particles without spin. The difference lies in the dependence of \(V_l^{\pm}\) on \(k\). If, however, in the region of small energies \(V_l^{\pm}\) remains finite as \(k \to 0\), then for sufficiently small \(k\) this dependence may be neglected. There are grounds to suppose that for \(S\)- and \(P\)-states at \(k=0\), \(V\) differs from zero. Then, in the region of small energies,
\[ \tg \delta_l \sim k^{2l+1} \tag{3.32} \]
(\(l\) is the orbital angular momentum). Thus, for the \(S\)- and \(P\)-waves considered by us, in the region of small energies, and consequently of large wavelengths (much larger than the radius of action of the forces), the relations hold
\[ k \ctg \delta_0 = a_0^{-1}, \quad k^3 \ctg \delta_{1+} = a_{1+}^{-3}, \quad k^3 \ctg \delta_{1-} = a_{1-}^{-3}. \tag{3.33} \]
These relations are valid both for \(I=3/2\) and for \(I=1/2\). Here \(a\) are the effective scattering lengths, which in general depend on the isotopic number*).
In the energy region where the effective cross section for \(\pi\)-meson scattering has been measured, the wavelength becomes comparable with the radius of the forces acting between the \(\pi\)-meson and the nucleon \(\left(\lambda = k^{-1} = \dfrac{\hbar}{\mu c}\ \text{at}\ E = 0.41\,\mu c^2\right)\). Therefore formulas (3.33) in the indicated region are no longer exact. The right-hand sides of relations (3.33) in this case cannot be regarded as constants, but must be considered as
* According to \(^{8a}\), \(a_0(^{3}/_{2}) = (3.8 \pm 0.7)\,10^{-14}\ \text{cm}\) and \(a_0(^{1}/_{2}) = (0 \pm 1)\,10^{-14}\ \text{cm}\).
some functions of \(k\). Approximately, this dependence can be taken into account by expanding the right-hand sides in powers of \(k^2\) and retaining the first two terms of the expansion. Then, instead of (3.33), we obtain:
\[ k\operatorname{ctg}\delta_{0}^{3/2}=a_{0}^{-1}\left({}^{3}/_{2}\right)+k^{2}r_{0}\left({}^{3}/_{2}\right), \tag{3.34а} \]
\[ k^{3}\operatorname{ctg}\delta_{1-}^{3/2}=a_{1-}^{-3}\left({}^{3}/_{2}\right)+ \frac{k^{2}}{r_{1-}\left({}^{3}/_{2}\right)}, \tag{3.34б} \]
\[ k^{3}\operatorname{ctg}\delta_{1+}^{3/2}=a_{1+}^{-3}\left({}^{3}/_{2}\right)+ \frac{k^{2}}{r_{1+}\left({}^{3}/_{2}\right)}, \tag{3.34в} \]
\[ k\operatorname{ctg}\delta_{0}^{1/2}=a_{0}^{-1}\left({}^{1}/_{2}\right)+k^{2}r_{0}\left({}^{1}/_{2}\right), \tag{3.34г} \]
\[ k^{3}\operatorname{ctg}\delta_{1-}^{1/2}=a_{1-}^{-3}\left({}^{1}/_{2}\right)+ \frac{k^{2}}{r_{1-}\left({}^{1}/_{2}\right)}, \tag{3.34д} \]
\[ k^{3}\operatorname{ctg}\delta_{1+}^{1/2}=a_{1+}^{-3}\left({}^{1}/_{2}\right)+ \frac{k^{2}}{r_{1+}\left({}^{1}/_{2}\right)}. \tag{3.34е} \]
Knowing the angular distribution, and consequently also the phases for scattering processes at two energies, one can calculate the effective scattering lengths \(a\) and the effective radii of action of the forces \(r\). For \(\pi\)-meson scattering processes the angular distribution has been measured at two energies only for the case of scattering of positive \(\pi\)-mesons by protons. Therefore, from experiment one can determine only the constants corresponding to scattering in states with isotopic spin \(3/2\). It should also be noted that the measurements were carried out at relatively close energies and with considerable errors. Therefore, extrapolation of the energy dependence of the phases by means of formulas (3.34) can hardly prove accurate*). Nevertheless, as an example of the application of formulas (3.34), we shall consider what conclusions they lead to for the cross section \((\pi^{+},\pi^{+})\). From formulas (3.17) and (3.24) it follows that
\[ \sigma(\pi^{+},\pi^{+})= \frac{4\pi}{k^{2}+\left[a_{0}^{-1}\left({}^{3}/_{2}\right)+k^{2}r_{0}\left({}^{3}/_{2}\right)\right]^{2}} + \frac{4\pi k^{4}}{ k^{6}+\left[a_{1-}^{-3}\left({}^{3}/_{2}\right)+ \frac{k^{2}}{r_{1-}\left({}^{3}/_{2}\right)}\right]^{2}} + \frac{8\pi k^{4}}{ k^{6}+\left[a_{1+}^{-3}\left({}^{3}/_{2}\right)+ \frac{k^{2}}{r_{1+}\left({}^{3}/_{2}\right)}\right]^{2}} . \tag{3.35} \]
*) One could have restricted oneself to considering scattering only in a single state, for example in the \(P_{3/2}\) state for \(I=3/2\). Such an approach to the study allows one to determine the effective length and effective radius on the basis of the energy dependence. However, this does not take into account the contribution to scattering from other states, which, as follows from the phase values given above, is not small.
The ambiguity indicated above in the determination of the scattering phases also leads to ambiguity in the determination of the constants entering into (3.35). Thus, using two solutions of equations (3.29), we obtain two different energy dependences of the cross section for the scattering of positive \(\pi\)-mesons by protons. For the phase values we shall take those given above, although they too are determined with a large error (\(\pm 5^\circ\)).
The cross sections obtained in this way are shown in Fig. 5; the experimental points are plotted there as well for comparison. It should be pointed out that the energy behavior of the cross sections depends strongly on changes in the phase values from which the parameters entering formula (3.35) are calculated. Thus, for example, changing the \(S\)-phase from 15 to \(20^\circ\) shifts the maximum of the cross section \(\sigma(p^+)\) by 60 MeV. It is therefore clear that the curves shown in Fig. 5 are mainly of a qualitative character.
Fig. 5. Dependence of the total cross section on energy according to the extrapolation formulas (3.35) for positive \(\pi\)-mesons, for the first (I) and second (II) solutions of the phase equations (see Table V, § 3).
For the scattering of positive \(\pi\)-mesons both solutions give predictions that do not contradict experiment. The situation is somewhat different for the scattering of negative \(\pi\)-mesons by protons. The contribution from states with isotopic spin \(3/2\) to the total scattering cross section of \(\pi^-\)-mesons is one third of expression (3.35) [see (3.26)]. Since from Table V, and also from the ratio of the scattering cross sections of \(\pi^+\)- and \(\pi^-\)-mesons, it follows that in states with \(I=1/2\) the interaction is weak, then in a first rough approach these states may be neglected altogether.
Figure 6 gives a comparison of the experimental values of the total cross section \((\pi^-,p)\) with the contribution to this cross section from states with \(I=3/2\). As is seen from the figure, the first solution (I) gives good agreement with experiment. Conversely, the second solution (II) leads to a strong discrepancy with the experimental data and, for example, at an energy of 176 MeV gives a cross-section value almost twice as large as the observed one.
In connection with this, one may think that in reality solution I of equation (3.29) is realized, i.e., that the main contribution to the scattering is made by the \(P_{3/2}\) state with \(I=\ ^3/_2\). However, it should be said that the poor behavior of the cross section with energy, predicted in the case of solution II, may also be due to the inaccuracy of formulas (3.34). A refinement of these results can probably be obtained in the near future, when a larger number of experimental data will be at our disposal.
Fig. 6. The same as Fig. 5 for negative \(\pi\)-mesons.
In conclusion of this paragraph let us dwell on an interesting model proposed in \(^{49}\) for the interpretation of \(S\)-scattering in the state with isotopic spin \(3/2\). In this paper the following experimental values of the phase \(\delta_0(3/2)\) are given:
\[ \begin{array}{ll} \text{at } 135\ \text{MeV} & \pm 21^\circ,\\ \text{at } 113\ \text{MeV} & \pm 13^\circ,\\ \text{at } \phantom{1}78\ \text{MeV} & \pm 6^\circ. \end{array} \]
Comparison of such a rapid growth of the \(S\)-phase with that obtained from experiments on the capture of \(\pi\)-mesons in hydrogen, the value
\[ a_0(3/2)=k^{-1}\operatorname{tg}\delta_0(3/2)\big|_{k=0} \]
leads to the idea that in the energy region of mesons \(E_0 \simeq 30—50\ \text{MeV}\), \(\delta_0(3/2)\) changes sign. Such an energy dependence of the \(S\)-scattering phase of \(\pi\)-mesons by nucleons is similar to the energy dependence of the \(S\)-phase of nucleon–nucleon scattering \(^{50}\). This similarity permits one to suppose that at large distances the forces acting between the \(\pi\)-meson and the nucleon are attractive forces, while at small distances repulsive forces occur (cf. \(^{50}\)). In \(^{49}\), calculations are given of the phases of \(S\)-scattering of \(\pi\)-mesons by nucleons under various assumptions about the law of the forces acting between the \(\pi\)-meson and the nucleon. If, moreover, an effective potential with repulsion at small distances also occurs for \(P\)-states with \(I=\ ^3/_2\), it may be thought that the corresponding \(P\)-scattering phases will turn to zero at energies greater than \(E_0\). Then, for \(E<E_0\), the angular distribution will be characterized by a maximum of scattering forward.
In the energy region exceeding \(E_0\), but still less than that energy at which the \(P\)-phases vanish, the maximum of the angular distribution of meson scattering will turn out to be reversed. This picture can be tested directly in experiment.
§ 4. THEORY OF WEAK AND STRONG COUPLING
Up to now we have considered the scattering of \(\pi\)-mesons by nucleons exclusively from a phenomenological point of view, without specifying any particular form of the interaction of these particles with one another. This is not accidental, since at present there is no completed theory of the interaction of nucleons with \(\pi\)-mesons. The creation of such a theory encounters serious difficulties, and, unlike in electrodynamics, these difficulties apparently have a deeper nature. It is known that in electrodynamics the interaction is weak. This is manifested, in particular, in the fact that in a number of cases, for scattering processes, all physical results can be represented in the form of a series in powers of the fine-structure constant \(\alpha\), where, significantly,
\[ \alpha = 1/137 \ll 1 \]
(perturbation theory; \(\alpha=\dfrac{e^2}{\hbar c}=e^2\) in our units). The weakness of the interaction is due to the specific form of the interaction Hamiltonian
\[ H=-\hat{j}\hat{A}, \tag{1} \]
which is the scalar product of the current 4-vector \(\hat{j}\) and the 4-vector potential of the electromagnetic field \(\hat{A}\). Expressions for the cross sections of various processes, calculated in the first nonvanishing approximation of perturbation theory (Compton effect, bremsstrahlung radiation, etc.), are in brilliant agreement with experiment in the presently attained energy range.
The difficulties inherent in modern electrodynamics are connected mainly with the problem of the infinite self-energy of the electron. In overcoming these difficulties, definite progress has been achieved in recent years. With the help of a covariant method of writing the equations of electrodynamics, it has been possible to carry out an unambiguous elimination of the infinite expressions arising in the theory in the higher approximations of perturbation theory, and to interpret such subtraction as a renormalization (redefinition) of the known physical constants—the mass and charge of the electron. The greatest success of the theory in this direction should be considered the calculation of the hyperfine splitting in the hydrogen atom and of the anomalous magnetic moment of the electron \(^{33,34}\).
The applicability of perturbation theory is essentially connected with the form of the interaction Hamiltonian (1). If in electrodynamics in
if the interaction Hamiltonian included terms of the form \(F_{ik}\gamma_i\gamma_k^{*}\), where
\[ F_{ik}=\frac{\partial A_k}{\partial x_i}-\frac{\partial A_i}{\partial x_k} \]
is the electromagnetic-field tensor, then perturbation theory in this case would be inapplicable, since, for example, for the cross section of light scattering by an electron one would obtain expressions increasing without bound with the energy, independently of the magnitude of the interaction constant\(^{35,36}\). The specific difficulties of mesodynamics are connected with the fact that the interaction Hamiltonian contains expressions involving derivatives of the wave functions of the meson field\(^{35}\).
For example, the interaction Hamiltonian in the symmetric pseudoscalar theory with pseudovector (gradient) coupling has the following form:
\[ g\tau_\alpha\left\{\boldsymbol{\sigma}\cdot\nabla\varphi_\alpha+\rho_1\frac{\partial\varphi_\alpha}{\partial t}\right\}, \tag{11} \]
where \(g\) is the interaction constant, and \(\boldsymbol{\sigma}\) and \(\rho_1\) are known four-row matrices. The cross sections for meson scattering by a nucleon, calculated with such a Hamiltonian in the first nonvanishing approximation of perturbation theory, will increase without bound as the energy increases, which is physically inadmissible. In contrast to electrodynamics, here it is likewise impossible to carry out unambiguously the removal of the infinities that appear in higher approximations of perturbation theory.
All this indicates that in mesodynamics, in the case of gradient coupling, the interaction cannot be regarded as weak\(^{**}\), and that, for the correct solution of the equations, one must seek other methods, different from the usual perturbation theory (the theory of “weak coupling” is inapplicable). Such attempts have been made in two directions: first, in the damping theory, in which, without violating the relativistic invariance of the theory, attempts are made to take into account effects due to the reaction of the radiated field on the motion of the nucleon, but all inertial effects associated with the reaction of the nucleon’s own field are arbitrarily discarded\(^{***}\); second, in the theory of extended sources (including the case of so-called “strong coupling”), in which finite dimensions are assigned to the nucleons; in such a theory, in addition to the reaction of the radiated field, it is possible to take into account effects due to the inertia of the spin; however, the assumption of finite dimensions of the nucleon violates the relativistic invariance of the theory.
*) This would occur if the electron possessed its own anomalous magnetic moment.
**) The strength of the interaction in the case of derivative coupling is determined not so much by the magnitude of the constant \(g\) as by the fact that the interaction grows with increasing energy.
***) If the meson interaction with the nucleon depends on the spins, then additional terms associated with the inertia of the nucleon spin appear in the self-energy.
SCATTERING OF π-MESONS BY NUCLEONS
Below we shall dwell on the results obtained by the methods indicated above (including weak coupling) as applied to the processes of scattering of π-mesons by a nucleon. In doing so we shall restrict ourselves to considering only the pseudoscalar symmetric theory with gradient coupling, since, as was noted in the introduction, there are grounds for believing that the π-meson field is pseudoscalar (p.-s.), and their coupling with nucleons is pseudovector (p.-v.).*) In the theory of weak coupling, the scattering of p.-s. mesons with p.-v. coupling has been investigated by various authors.37–40 Everywhere in these calculations one had to restrict oneself to the first nonvanishing approximation of perturbation theory, since attempts to take into account the next approximations in the theory with gradient coupling, as has already been noted, lead to nonrenormalizable divergent expressions.
In the first nonvanishing approximation of perturbation theory the scattering cross sections turn out to grow without bound with increasing energy of the incident mesons. Therefore the results obtained here can be valid only in the region of small energies \((\ll \mu)\). For such energies the symmetric theory gives the following expressions for the differential cross section of ordinary and exchange scattering of π-mesons by nucleons38–39 **) (in the center-of-inertia system)
\[ \frac{d\sigma(p^{+},p^{+})}{d\Omega} = \left(\frac{g}{\mu}\right)^{4}\frac{k^{4}}{\omega^{2}}, \tag{4.1a} \]
\[ \frac{d\sigma(p^{-},p^{-})}{d\Omega} = \left(\frac{g}{\mu}\right)^{4}\frac{k^{4}}{\omega^{2}}, \tag{4.1б} \]
\[ \frac{d\sigma(n^{0},p^{-})}{d\Omega} = 2\left(\frac{g}{\mu}\right)^{4}\frac{k^{4}}{\omega^{2}}\cos^{2}\vartheta . \tag{4.1в} \]
(Here and below the notation of the preceding sections is used; in ordinary units \(g^{2}=g'^{\,2}/\hbar c\).) In these expressions all terms \(\sim \omega/M\) and of higher order of smallness have been omitted. Analogous formulas hold for
\[ \frac{d\sigma(n^{\pm},n^{\pm})}{d\Omega} \quad \text{and} \quad \frac{d\sigma(p^{0},n^{+})}{d\Omega}. \]
Let us note that at sufficiently low energies the terms \(\sim(\omega/M)^{3}\) can no longer be discarded, since they remain finite as \(k\to 0\), whereas the terms \(\sim k^{4}\) vanish. Nevertheless, taking them into account is superfluous, since in the region of small energies where they give a noticeable contribution in comparison with the terms \(\sim k^{4}\), the Coulomb interaction of π-mesons with nucleons begins to play the main role.
*) It is not excluded that the intrinsic parity of nucleons is different. However, in this case as well the interaction will have the form (II). Speaking of pseudoscalar mesons with pseudovector coupling, we shall mean precisely this form of interaction.
**) In these works the scattering cross sections of π-mesons by nucleons were also calculated with account of the pseudoscalar coupling and the Coulomb interaction of π-mesons with nucleons.
From formulas (4.1a, b, c) it is seen that at nonrelativistic energies the differential cross sections \(d\sigma(p^+,p^+)\) and \(d\sigma(p^-,p^-)\) do not depend on the angles, i.e., the scattering is isotropic. The isotropy is obtained as a result of the superposition of \(P_{1/2}\)- and \(P_{3/2}\)-waves. The fact that the isotropy of the cross section is not connected with the \(S\)-wave can be verified, for example, from the fact that the cross sections (4.1a and b) are proportional to \(k^4\) (in the preceding paragraph it was shown that just such a dependence is characteristic precisely of \(P_{1/2}\)- and \(P_{3/2}\)-waves).
The charge-exchange cross section (4.1c) is proportional to \(\cos^2\vartheta\). Comparison with experiment (see §§ 1 and 3) shows that the angular distribution in the weak-coupling theory is not in agreement with experiment.
The most substantial discrepancy is connected with the absence, in the predictions of the theory, of a noticeable contribution to the cross section from the \(S\)-wave. According to the weak-coupling theory this contribution is \(\sim(\omega/M)^2\); it is relativistic in nature and leads to too small a length of \(S\)-scattering\(^{8a}\).
The total cross section for all three processes is easily obtained by integrating (4.1a, b, c) over the angles:
\[ \sigma(p^+,p^+)=4\pi\left(\frac{g}{\mu}\right)^4\frac{k^4}{\omega^2}, \tag{4.2a} \]
\[ \sigma(p^-,p^-)=4\pi\left(\frac{g}{\mu}\right)^4\frac{k^4}{\omega^2}, \tag{4.2b} \]
\[ \sigma(n^0,p^-)=\frac{4\pi}{3}\left(\frac{g}{\mu}\right)^4\frac{k^4}{\omega^2}. \tag{4.2c} \]
It follows from this that
\[ \frac{\sigma(p^+,p^+)} {\sigma(n^0,p^-)+\sigma(p^-,p^-)} = \frac{3}{5}, \tag{4.3} \]
which is in striking contradiction with experiment.
Experimentally (see Fig. 2), already at energies \(\sim 60\) MeV this ratio becomes noticeably greater than unity, while in the region of 150 MeV it reaches approximately 3.
The equality of the cross sections \(\sigma(p^+,p^+)\) and \(\sigma(p^-,p^-)\) according to the weak-coupling theory indicates that in this theory \(d^{1/2}\) is small in comparison with \(d^{3/2}\) (see formula (2.18), § 2).
The energy dependence of the total cross section according to the weak-coupling theory apparently does not contradict the experimental data at low energies. However, at low energies the intensity of \(\pi\)-mesons is so small and the measurement errors so large that it is difficult to draw unambiguous conclusions on this point. In addition, it should be noted that analogous behavior of the cross section with energy at low energies will occur in any theory with a gradient coupling of the meson field to nucleons, since in this case one should expect a strong interaction in the \(P\)-state.
Thus, the results of perturbation theory in the charge-symmetric theory with pseudovector coupling, with respect to the scattering of \(\pi\)-mesons
on nucleons do not agree with the majority of experimental data and are in obvious contradiction to experiment at energies \(\gg \mu\).
Attempts to eliminate the unbounded growth of the cross section with energy in the weak-coupling theory (see formulas (4.1) and (4.2)) are based on the assumption that the interaction in fact cannot be regarded as weak, and that one must take into account the damping of the initial state due to the back reaction of the emitted meson field on the motion of the nucleon (damping theory\(^{41-44}\)). The inclusion of terms due to the action on the motion of the nucleon of its own meson field (spin inertia), which, generally speaking, are substantial\(^{35}\), cannot be carried out consistently within the relativistic scheme, since all higher approximations of perturbation theory that include inertial effects lead to infinite expressions. Therefore, in damping theory the effects connected with the inertia of the nucleon spin are not taken into account.
In the static approximation (the nucleons are at rest) the damping theory leads to the following expression for the total cross section for scattering of \(\pi^{+}\)-mesons by a proton in the symmetric variant*):
\[ \sigma(p^{+},p^{+})= \]
\[ =4\pi\left(\frac{g}{\mu}\right)^{4}\frac{k^{4}}{\omega^{2}} \left[ \frac{ 1+\dfrac{4}{3}\left(\dfrac{g}{\mu}\right)^{4}\dfrac{k^{6}}{\omega^{2}} }{ \left(1+4\left(\dfrac{g}{\mu}\right)^{4}\dfrac{k^{6}}{\omega^{2}}\right) \left(1+\left(\dfrac{g}{\mu}\right)^{4}\dfrac{k^{6}}{\omega^{2}}\right) } \right]. \tag{4.4} \]
A characteristic difference between this expression and formula (4.2) is the presence in it of an additional term in the square brackets, due to the inclusion of damping (\(g^{2}\) in the denominator).
For large values of \(k\) \(\left(k\gg \dfrac{\mu}{g}\right)\) the cross section (4.3) decreases proportionally to \(k^{-2}\); in the region of small \(k\) it goes over into formula (4.2a).
For \(\sigma(p^{-},p^{-})\) and \(\sigma(n^{0},p^{-})\) in the symmetric theory, rather similar expressions are obtained, which we shall not write out here. We note only that they, just like \(\sigma(p^{+},p^{+})\), at small \(k\) go over into the corresponding formulas of the weak-coupling theory (4.2b and c). The course of the dependence of the cross section on energy in the damping theory qualitatively resembles the picture obtained experimentally. The cross section reaches a maximum at energies of the order of several \(\mu\) and then falls as the energy increases.
Nevertheless, inclusion of the terms responsible for damping does not bring the theory into complete agreement with experiment. Indeed, the ratio (4.3) in the symmetric theory with damping differs hardly at all from the corresponding ratio in the weak-coupling theory in the energy region \(<\mu\). For \(g^{2}\simeq 0.3\) and for \(k\sim\mu\) it approaches unity.
*) These results were communicated to us by V. Mikhailov. In work\(^{45}\) these calculations were not carried through to the end in the energy region of interest to us.
With a further increase in energy this ratio becomes somewhat greater than unity, but the formulae themselves then turn out to be no longer applicable (the recoil of the nucleons is not taken into account and the role of higher approximations is quite unclear).
The angular distribution calculated with damping taken into account likewise does not agree with experiment, since, as in the theory with weak coupling, the contribution from \(S\)-states to the scattering is of a purely relativistic character.
Let us note that, since the interaction in the damping theory is not assumed to be weak, an essential contribution to the physical results may be made by inertia terms, which are arbitrarily discarded in this theory.
Therefore it is precisely in that energy region \((>\mu)\), where damping effects begin to play a noticeable role, that the correct expressions for the cross sections of scattering of \(\pi\)-mesons by nucleons may have little in common with formula (4.4) and with the analogous formulae for \(\sigma(p^{-},p^{-})\) and \(\sigma(n^{0},p^{-})\).
We shall now dwell briefly on the results to which a theory leads that is based on the assumption that heavy particles have finite dimensions \(^{17}\) (a new physical constant \(a\) is introduced—the radius of the nucleon). In such a theory it proves possible to take into account effects connected both with the reaction of the radiated meson field and with the inertia of the nucleon spin. In this theory the nucleons are assumed to be at rest. Mesons may be treated relativistically (however, their energy must remain \(\ll M\)).
If the spin operators \(\boldsymbol{\sigma}\) and the isotopic spin \(\boldsymbol{\tau}\) of the nucleon are regarded as classical vectors, then the equations of motion in the theory of extended sources can be solved by a classical method.
The expressions for the cross sections for scattering of \(\pi\)-mesons by nucleons in the symmetric pseudoscalar theory with pseudovector coupling were calculated in works \(^{20,46*}\). In the general case of an arbitrary nucleon radius the cross sections have a rather cumbersome form. Since, however, the characteristic features of the expression for the meson scattering cross section in the theory of extended sources are common to all variants of the pseudoscalar theory with gradient coupling, we shall restrict ourselves to analyzing the simplest case. Let us consider the scattering of a neutral pseudoscalar meson with gradient coupling. The total scattering cross section for arbitrary \(a\) in this case will be
\[ \sigma = 4\pi \cdot \left(^{8}/_{9}\right)\left(\frac{g}{\mu}\right)^{4} \frac{k^{4}}{\omega^{2}} \frac{1+\xi^{2}+\eta^{2}}{(1-\xi^{2}-\eta^{2})^{2}+4\xi^{2}} |G(k)|^{2}; \tag{4.4a} \]
\[ \overline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]
*) In works \(^{46}\) the scattering of \(\pi\)-mesons in the symmetric theory was calculated by a classical method. The results obtained there agree qualitatively with the results of the quantum-mechanical treatment carried out in \(^{20,47}\).
here
\[ \left. \begin{aligned} \xi &= \frac{2}{3}\frac{g^2}{\omega}\left(\frac{\omega^2}{a}-\mu^3\right),\\ \eta &= \frac{2}{3}g^2\frac{k^2}{\omega}, \end{aligned} \right\} \tag{4.46} \]
and \(G(k)\)—the nucleon form factor—is a function that depends on the specific assumptions about the form of the nucleon. The constant \(a\) is related to \(G(k)\) in the following way:
\[ \frac{1}{a}=\frac{2}{\pi}\int G(k)\,dk. \]
In formula (4.4a) the terms \(\eta^2\) owe their origin to taking account of the reaction of the emitted meson field on the motion of the nucleon; the terms \(\xi^2\) are connected with the inertia of the spin in the scattering of mesons by nucleons. For a point source \((a\to 0)\) the terms \(\xi^2\) become infinite.
In the limiting case where \(\frac{1}{a}\gg \mu\) (the radius of the nucleon is much smaller than the Compton wavelength of the meson; the case of strong coupling), and the energy is subject to the restriction \(\omega\ll \frac{1}{a}\), \(G(k)\) in (4.4a) may be replaced by unity and neglected in comparison with \(\xi\). Then one obtains
\[ \sigma=4\pi\frac{8}{9}\left(\frac{g}{\mu}\right)^4\frac{k^4}{\omega^2} \left\{ \frac{1+\left(\dfrac{2}{3}g^2\dfrac{\omega}{a\mu^2}\right)^2} {\left[1-\left(\dfrac{2}{3}g^2\dfrac{\omega}{a\mu^2}\right)^2\right]^3} \right\}. \tag{4.5a} \]
If, in addition, the coupling constant is so large that \(\frac{g^2\omega}{\mu^2 a}\gg 1\), then unity may be neglected in comparison with \(\left(\frac{2}{3}\frac{g^2\omega}{\mu^2 a}\right)\). The total cross section in this case turns out to be independent of \(g^2\) and equal to
\[ \sigma=8\pi\frac{k^4}{\omega^4}a^2. \tag{4.5b} \]
Since \((k^4/\omega^4)\) in (4.5b) is always less than unity, and \(a\), by assumption, is \(\ll \frac{1}{\mu}\), we come to the conclusion that in the theory of strong coupling the cross section for scattering of \(\pi\)-mesons by nucleons is much smaller than the geometrical one. For intermediate values of the quantity \(g^2\omega/a\mu^2\), formula (4.5a) leads to a typically resonant dependence of the scattering cross section on the energy of the incident mesons, with the width of the resonance curve determined by the ratio of the quantities \(\eta\) and \(\xi\), and when \(\eta\sim \xi\) becoming close to that observed in experiments on the scattering of \(\pi\)-mesons by nucleons. The ratio of the scattering cross sections of positive and negative \(\pi\)-mesons in the theory of extended sources, apparently,
can be brought into agreement with experiment both for the quantum\(^{47}\) and the classical\(^{46}\) solution of the problem.
However, the angular distribution obtained in this way does not contain terms \(\sim \cos \theta\), which arise owing to the interference of the \(S\)- and \(P\)-waves in scattering.
In principle, by taking recoil into account, the \(S\)-wave can be obtained in the theory of extended sources. However, it will have a relativistic origin and will give a negligibly small contribution to the cross section in the region of energies \(\leqslant \mu\), which is not in agreement with the experimental data (see §§ 1 and 3).
In another limiting case
\[ \frac{1}{a}\ll \mu \]
(including the case \(\frac{1}{a}=0\), when all inertial effects vanish), in formula (4.4a) one may neglect \(\xi\) in comparison with \(\eta\). Now the principal role is played by the terms responsible for damping. In this case the expressions obtained in the theory of extended sources, for \(\eta^2\gg 1\), coincide, up to a numerical factor, with the corresponding formulas of the theory of damping.
Summarizing briefly the results presented in this paragraph, one may say that the existing investigations based on meson-field theory are not able to give a complete explanation of the experiments on the scattering of \(\pi\)-mesons by nucleons. In the theoretical attempts at such an explanation undertaken recently\(^{26, 46, 47}\), notions of isobars are used. It should be emphasized that the isobars introduced in this way differ from the isobars usually considered in theories of strong coupling\(^{20, 35}\) by their instability, connected with the fact that their energy is greater than \(Mc^2+\mu c^2\). The width of the energy levels of such isobaric states proves to be large.
Let us note that, from experiments on the scattering of \(\pi\)-mesons by nucleons, it is not yet possible to draw a definitive conclusion about the existence of a maximum (see Fig. 2) in the energy dependence of the total cross section on energy. However, the presence of such a maximum should be expected, since at energies \(\gg 10^{10}\) MeV the cross section for the interaction of \(\pi\)-mesons with nucleons is close to the geometrical one, whereas at \(E \sim 150\) MeV \(\sigma(p^+,\ p^+)\) is approximately three times greater than the geometrical cross section. If the width of such a maximum proves to be sufficiently narrow\(^*\), then one will be able to speak of an isobaric state. Such isobars, apparently, can be obtained in the so-called theory of intermediate coupling\(^{48}\).
One may think that with the aid of the theory of intermediate coupling it will be possible to achieve agreement between theory and experiment. To be sure, for a pseudovector interaction, at any rate in the modern formulation of the theory of intermediate coupling, it is hardly possible to obtain
\(^*\) That is, the width of the isobaric level proves to be much smaller than its height above the ground state.
a considerable \(S\)-wave. Therefore, the introduction of a pseudoscalar coupling, leading to appreciable \(S\)-scattering, may prove useful\(^{38,39}\).
Of considerable interest is the solution of the problem of the scattering of \(\pi\)-mesons by nucleons by means of the method of I. E. Tamm\(^{51}\), proposed by him at one time for the relativistic study of the problem of the interaction of two particles. At present various attempts are being made to use this method for the theoretical study of the scattering of \(\pi\)-mesons by nucleons\(^{52,53}\). However, a complete investigation of the problem in this direction is still far from completion. Another possible path for the theoretical interpretation of the experiment may be a phenomenological description of the selected states with the aid of a theory of particles with several spin and charge states.
In conclusion, however, it should be emphasized that, in order to construct a complete theory of the scattering of \(\pi\)-mesons by nucleons, a considerable refinement of the available experimental data is necessary, as well as the obtaining of experimental results in the region of high \(\pi\)-meson energies.
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