Meson Production by $\gamma$ Quanta

A. M. Baldin and V. V. Mikhailov

Recently, the question of meson production by $\gamma$ quanta has received considerable attention in the literature. The reason for this is that, given the present state of theory and experimental technique, meson production by photons is one of the simplest phenomena connected with mesons that is accessible to study. Existing experimental techniques make it possible to obtain more precise and more varied information about the nature of mesons and, consequently, about nuclear forces in meson production by photons than in meson production in nucleon–nucleon collisions[^1]. The theoretical treatment of meson production by photons also has a number of advantages over the case of meson production in nucleon–nucleon collisions. Indeed, in the case of meson production by photons, part of the process—the absorption of the photon—is described by well-studied electromagnetic interactions. These interactions may be regarded as weak and, consequently, the usual methods of perturbation theory may be used with confidence. It is not clear, however, how weak the interaction of nucleons with the meson field may be considered, although here too, in the pseudoscalar variant with pseudovector coupling, it is possible to obtain agreement with experiment for

$$ \frac{g^2}{\hbar c} \simeq \frac{1}{6}. $$

As will be shown below, the characteristic features of the process of meson production by photons are determined to a considerable extent by the interactions of mesons and nucleons with the electromagnetic field. Since different meson theories lead to different results for the angular distribution of mesons, for the dependence of cross sections on the energy of the $\gamma$ quanta, and for other essential features of the process, comparison of theoretical results with the experimental data available in the literature is of considerable interest.

I. EXPERIMENT

The experiments described below were carried out at the Berkeley synchrotron \(^{1-8}\). A beam of \(\gamma\)-quanta is produced as a result of bremsstrahlung from electrons accelerated to high energy (\(\sim 330\) MeV) \(^{1,2}\). This beam, having the form of a narrow ray with an angular aperture of a cone of less than one degree, is additionally collimated by two collimators and directed onto a target,

Fig. 1.

which serves as the source of mesons. (The second collimator is necessary in order to stop the electrons formed at the edges of the first.) (Fig. 1.)

a) Neutral mesons

The source of mesons in most experiments was a beryllium target \(^{1}\) of diameter \(5.1\) cm and length \(3.8\) cm. The method of registration was based on the fact (which is once again confirmed by the work discussed here) that the neutral meson decays into two \(\gamma\)-quanta. The detecting system consisted of two telescopes, each of which had three scintillation counters (see Fig. 1). Lead, usually \(0.63\) cm thick, was placed between crystals 1 and 2 of each telescope. An event is registered in the case where pulses are recorded simultaneously by the four outer crystals: 2, 3, \(2'\), and \(3'\) (counters 1 and \(1'\) are connected in anticoincidence with counters 2 and 3 and, respectively, \(2'\) and \(3'\). Thus, in order to register an event, it is necessary that crystals 1 and \(1'\) be penetrated simultaneously by neutral particles (photons), producing charged particles in the lead; the latter, passing through crystals 2 and 3 (respectively \(2'\) and \(3'\)), cause the recorded pulse. With an intensity of the \(\gamma\)-quantum beam of \(\sim 10^{11}\) MeV per minute, the counting rate was approximately 10 pulses per minute.

A. M. BALDIN AND V. V. MIKHAILOV

The nature of the particles incident on the lead was established in the following way[^1]. In the absence of lead between the crystals, the counting rate was almost zero. As the thickness of the lead was increased to \(0.63\ \text{cm}\), it rose rapidly, but with a further increase of the lead thickness to \(1.27\ \text{cm}\) the counting rate increased only slightly.

This dependence of the counting rate on the lead thickness is in agreement with what is expected from shower theory for photons with an energy of approximately \(100\ \text{MeV}\). If copper plates are placed instead of lead, then the latter, at a thickness of \(0.63\ \text{cm}\), give the same counting rate as \(0.32\ \text{cm}\) of lead, which is again in agreement with shower theory, since the number of shower units is the same for these thicknesses. The counting rate decreases by approximately a factor of 4 if plates of lead \(0.63\ \text{cm}\) thick are placed between the target and the crystals. This also agrees with what is expected if the particles incident on the lead are identified with \(\gamma\)-quanta. The energy of the electrons emerging from the lead was measured.

For this purpose, aluminum absorbers were placed between the last two crystals of one of the telescopes (for example, between \(2'\) and \(3'\)). The dependence of the counting rate on the electron energy required for them to pass through the absorber indicates that the mean energy of the electrons is \(\sim 50\ \text{MeV}\), i.e., is in agreement with the fact that the \(\gamma\)-quanta have an energy of about \(100\ \text{MeV}\) (the energy of \(100\ \text{MeV}\) \(\gamma\)-quanta corresponds to the decay of a meson having energy \(\sim 60\ \text{MeV}\)).

The specifically nuclear (and not bremsstrahlung Coulomb) origin of the photons was demonstrated in the following way. Instead of a beryllium target, a lead target was irradiated. In this case the counting rate increased only by a factor of 6, whereas the ordinary shower cross section increases by \(\sim 400\) times. When the maximum energy of the \(\gamma\)-quanta was decreased from \(330\ \text{MeV}\) to \(175\ \text{MeV}\), the counting rate (at angles \(\theta=\beta=90^\circ\)) decreased by at least a factor of 50. Such a dependence of the effective cross section on the energy of the incident \(\gamma\)-quanta is also observed in the production of charged mesons[^1].

Of interest is the dependence of the counting rate on the angles \(\theta\) and \(\beta\). Let us first consider the dependence on the angle \(\beta\) at fixed \(\theta\) (say, \(\theta=90^\circ\)). Coincidences at \(\beta=180^\circ\) are very rare; then, as \(\beta\) is decreased to \(\beta \simeq 90^\circ\), the counting rate increases, and with a further decrease of \(\beta\) it falls sharply. If one assumes that these coincidences are due to \(\gamma\)-quanta produced in the decay of a moving meson, then precisely such a distribution should be expected. Indeed, in the coordinate system in which the meson is at rest, the \(\gamma\)-quanta must have equal and oppositely directed momenta. From the laws of conservation of energy and momentum one can obtain the fol-

a relation:

\[ \sin \alpha=\frac{\mu c^{2}}{cP_{\mu}}\cdot \operatorname{ctg}\frac{\beta}{2}, \]

where \(\alpha\) is the angle, in the meson’s own coordinate system, between the direction of emission of the \(\gamma\)-quanta and the direction of the meson momentum, \(\mu\) is the mass of the meson, and \(P_{\mu}\) is the meson momentum. From this relation it is clear that the angle \(\beta\) can vary within the limits

\[ 2\arcsin\left(\frac{\mu c^{2}}{E_{\mu}}\right)\leq \beta \leq \pi, \]

where \(E_{\mu}\) is the total energy of the meson in the laboratory coordinate system (in reality, of course, mesons have different energies \(E_{\mu}\)). Further, if one assumes the angular distribution of the \(\gamma\)-quanta formed in meson decay to be isotropic in the coordinate system associated with the meson, then, using the relation between the angles \(\alpha\) and \(\beta\), one can immediately obtain the distribution with respect to the angle \(\beta\). It turns out that the \(\gamma\)-quanta formed in the decay of a meson with energy \(E_{\mu}\) will mainly have an angle \(\beta\) close to

\[ \beta_{\min}=2\arcsin\frac{\mu c^{2}}{E_{\mu}}. \]

Fig. 2.

Thus, for example, for mesons with energy \(E_{\mu}-\mu c^{2}=70\ \text{MeV}\), \(\beta_{\min}=84^\circ\), and the mean angle is \(92^\circ\). Therefore it will not be a large error to consider that each angle \(\beta\) corresponds to its own meson energy. Consequently, measurement of the distribution of \(\gamma\)-quanta with respect to the angle \(\beta\) will at the same time be a measurement of the energy distribution of neutral mesons. One such distribution (for \(\theta=90^\circ\)) is shown in Fig. 2 (the curve is constructed under the assumption that the \(\gamma\)-rays are the product of the decay of neutral mesons produced with the same energy distribution as the \(\pi^{+}\)-mesons born in hydrogen). Thus, if photons are regarded as the product of the decay of an intermediate particle, then from the angular distributions one can find an approximate value of the velocity of motion of this particle (in fact

\[ \beta_{\min}=2\arcsin\left(1-\frac{v^{2}}{c^{2}}\right)^{-\frac12}, \]

where \(v\) is the velocity of the intermediate particle). The angular distributions obtained

A. M. BALDIN AND V. V. MIKHAILOV

distributions show that \(\dfrac{v}{c} \simeq 0.8\). Hence it is concluded\(^1\) that the particle must have an “intermediate” mass (for particles with a large mass cannot acquire such a velocity at a \(\gamma\)-quantum energy of \(330\ \text{MeV}\)). In addition, the fact that at small angles \(\beta\) the counting rate is practically equal to zero indicates that the meson decays only into two photons\(^*\).

The distribution of neutral mesons with respect to the angle \(\theta\) is shown in Fig. 3 (it was obtained for the production of mesons on beryllium\(^1\)). The character

Fig. 3.

of the angular distribution of neutral mesons will be discussed in Section II.

The ratio of the production cross sections of neutral mesons on hydrogen \(\sigma_{\mathrm{H}\pi^0}\) and on carbon \(\sigma_{\mathrm{C}\pi^0}\) was found to be

\[ \frac{\sigma_{\mathrm{H}\pi^0}}{\sigma_{\mathrm{C}\pi^0}} = 0.12 \pm 0.03 \]

(at angles \(\theta \simeq \beta \simeq 90^\circ\)). Both the angular distribution and the value of the ratio \(\dfrac{\sigma_{\mathrm{H}\pi^0}}{\sigma_{\mathrm{C}\pi^0}}\) differ from the corresponding data for charged mesons (approximately isotropic angular distribution and the ratio \(\dfrac{\sigma_{\mathrm{H}\pi^+}}{\sigma_{\mathrm{C}\pi^+}} = 0.55\)). The small value of the ratio \(\dfrac{\sigma_{\mathrm{H}\pi^0}}{\sigma_{\mathrm{C}\pi^0}}\) indicates that neutral mesons are formed both on protons and on neutrons (whereas positive mesons are produced only on protons). In the case of production of a \(\pi^+\)-meson, the proton is transformed into a neutron and, consequently, in its vicinity

\(^*\) The latter result was also obtained in work \(^{10}\).

an excess neutron density is created. Therefore the restriction imposed by the Pauli principle greatly reduces the probability of the process. In the production of neutral mesons this restriction is not substantial, since the charge of the nucleons does not change.

The total cross sections for the production of neutral mesons are equal, in magnitude, to: $\sigma_{\mathrm{Be}} = 7.5 \cdot 10^{-28}\ \mathrm{cm}^2$ (on beryllium), $\sigma_{\mathrm{C}} = 10 \cdot 10^{-28}\ \mathrm{cm}^2$ (on carbon), and $\sigma_{\mathrm{H}} = 1.3 \cdot 10^{-28}\ \mathrm{cm}^2$ (on hydrogen). The absolute intensity of the $\gamma$-ray beam was determined with an accuracy of up to 10%, while the efficiency of the detecting system was known only to within a factor of 2; therefore, in determining the values of the total cross sections, a corresponding error may be allowed. Nevertheless, one may say that the cross sections for the production of positive and neutral mesons on hydrogen are approximately the same (for charged mesons $\sigma_{\mathrm{H}\pi^+} \simeq 3 \cdot 10^{-28}\ \mathrm{cm}^2$).

The experimental data presented above will be discussed below (see Section II).

b) Charged mesons

The production of charged mesons was also investigated at the Berkeley synchrotron2–8. The source of the mesons was a target, whose material differed in the different experiments. The production of mesons on hydrogen was studied by subtracting the cross sections obtained on paraffin and carbon; in the most recent experiments6 a liquid-hydrogen target was used. Mesons were registered by means of photographic plates or by means of counters. Details of the photographic-plate registration method may be found by the reader in the abstract9 of paper3. The method of registering positive mesons by means of counters4 is based on a technique well known in cosmic-ray studies and consists of the following.

A positive $\pi$-meson, stopping in matter, decays (having a lifetime of $(1.65 \pm 0.33)\,10^{-8}\ \mathrm{sec}$) with emission of a $\mu$-meson, and the latter in turn emits an electron (positron) with the well-known mean lifetime $2.1 \cdot 10^{-6}\ \mathrm{sec}$. The range of the $\mu$-meson is such that, if the decay $\pi \to \mu$ occurs in a scintillation crystal counter with linear dimensions of several centimeters, then a large fraction of the decay electrons will also appear in this same crystal. The decay $\mu \to e$ is used to detect mesons; delayed coincidences between $\mu$-mesons and their decay electrons are used for this purpose. Negative mesons are absorbed in matter and do not give decay electrons, and therefore cannot be registered by this method. The arrangement of the target, the telescopes of scintillation counters, and the absorbers is shown in Fig. 4. Mesons produced in the target pass through an aluminum absorber and

detected in a telescope made of three anthracene scintillation crystal counters. The telescope can be rotated around the target in a plane passing through the direction of incidence of the γ-quanta. A meson is recorded only in the case when it stops in crystal II, and there the decay electron also appears. In other words, it is necessary that first pulses be recorded in crystals I and II, but that there be no pulse in crystal III, and then, with a delay, a pulse appear in crystal II. There is a noticeable number (about 10–20%) of accidental delayed coincidences; however, they can be counted and subtracted from the total number. When the target was removed, the number of coincidences decreased by several hundred times. When the γ-quantum energy was reduced below the threshold, mesons were not recorded. After subtraction of the background, the dependence of the counting rate on the delay time makes it possible to obtain the lifetime of the μ-meson (\(2.14 \cdot 10^{-6}\) sec). At a γ-quantum beam intensity of about \(10^{10}\) Mev/sec and an aluminum absorber

Crystals detecting mesons
Lead
Collimators (Al)
Absorber, Al
Target
γ-quantum beam
Top view
Side view
\(0\quad 5\) cm

Fig. 4.

with a thickness of 2.5 cm, the counting rate was 15 pulses per minute. This makes it possible to carry out experiments more rapidly than with photographic plates. However, measurements of absolute cross sections may contain a large error because of insufficient knowledge of the detection efficiency.

The angular distributions of mesons obtained on carbon and hydrogen, in the laboratory coordinate system, are approximately isotropic (Fig. 5). A certain decrease in the value of the meson-production cross section on carbon at small angles \(\theta\) can evidently be explained by the stronger restriction imposed by the Pauli principle (since, as the angle \(\theta\) decreases, the recoil of the nucleons decreases). Angular distributions have been obtained\(^{5,6}\) for mesons produced by γ-quanta with an energy of 250 Mev. This can be done\(^{6}\) (despite the continuous character of the γ-quantum spectrum) owing to the fact that the energy of the incident γ-quantum is uniquely determined, with the aid of the laws of conservation of energy and momentum, by the energy and ang-

with the angle of emission of the meson, which can be measured simultaneously. The laws of conservation of energy and momentum are easily applicable for determining the energies of γ-quanta in the case of meson production on hydrogen and, of course, cannot effectively be used in the production of mesons in nuclei. The magnitude of the total cross section is approximately \(3 \cdot 10^{-28}\ \text{cm}^2\) per carbon nucleus. The ratio of the number of negative to the number of positive mesons observed at an angle of \(\sim 90^\circ\) to the γ-quantum beam and having energies in the interval \(30 \div 100\) MeV is \(1.7 \pm 0.2^{3,11}\).

Fig. 5.

In a later note\(^8\) it is reported that this same ratio, but averaged over the energies of the γ-quanta, turns out not to depend on the angle \(\theta\) and is equal to \(1.35\) (\(\pm 0.15\) for \(\theta = 90^\circ\) and \(\theta = 135^\circ\), and \(\pm 0.2\) for \(\theta = 45^\circ\)). The energy distribution of mesons produced at an angle of \(\sim 90^\circ\) to the beam is shown in Fig. 6 (the incident γ-quantum beam had a bremsstrahlung spectrum with maximum energy 330 MeV)\(^5\).

The dependence of the cross section for the production of \(\pi^+\)-mesons on hydrogen at an angle of \(90^\circ\) to the beam on the energy of the incident γ-quanta was also measured\(^6\) (Fig. 7). It is interesting to note that at γ-quantum energies of about \(260 \div 280\) MeV the cross section does not undergo rapid growth.

As we shall show below (see Section II), this indicates the absence of spin in the meson.

A study was carried out of the relative cross sections for the production of \(\pi^+\)-mesons by γ-quanta on various elements\(^7\).

The observation of mesons was carried out at an angle of \(90 \pm 8^\circ\) to the γ-quantum beam, whose maximum energy was 317 MeV. Mesons

Fig. 6.

Fig. 7.

were recorded with energies \(42 \pm 7\) MeV and \(75 \pm 6\) MeV. The values of the relative cross sections, calculated per proton in the nucleus \(\dfrac{\sigma_{\pi^+}}{Z}\), are given in Table I (in arbitrary units).

Table I

Element	Cross section (for meson energy 42 MeV)	Statistical error *) (in %)	Cross section (for meson energy 76 MeV)	Statistical error *) (in %)
H	6.6	17	8.07	11
Li	3.32	10	2.80	11
Be	2.82	11	2.13	10
B	3.02	11	2.28	15
C	2.60	6	1.93	5
Al	2.50	11	1.68	9
Cu	1.92	19	1.17	15
Sn	1.66	25	0.51	55
Pb	0.51	91	0.80	65

*) Additional non-statistical errors may be allowed, \(\sim 10\%\) for hydrogen, tin, and lead, and \(\sim 5\%\) for the remaining elements.

Analogous measurements were also carried out for neutral mesons with an energy of 75 MeV\(^7\). Relative cross sections were obtained for hydrogen, lithium, beryllium, carbon, aluminum, copper, and lead. It turned out that if one takes the cross sections per nucleon in the nucleus \(\dfrac{\sigma_{\pi^0}}{A}\) and requires equality of the cross sections \(\dfrac{\sigma_{\pi^+}}{Z}\) and \(\dfrac{\sigma_{\pi^0}}{A}\) for beryllium, then the cross sections for all the other elements, except hydrogen, will be approximately the same both for \(\pi^+\)-mesons and for \(\pi^0\)-mesons. The cross section for the production of \(\pi^0\)-mesons on hydrogen is approximately the same as the cross section per nucleon for lithium.

In conclusion, let us list the principal results on the production of neutral and charged mesons by \(\gamma\)-quanta with energies up to 330 MeV\(^{1—8}\).

As a result of the study of the production of neutral mesons, their existence has been proved and the following data have been obtained\(^{1,7}\).

1. Neutral mesons decay only into two γ-quanta.

2. The cross section for the production of neutral mesons is approximately equal in magnitude to the cross section for the production of charged mesons \((\sigma_{\mathrm{H}\pi^0} \approx 1.3 \cdot 10^{-28}\ \text{cm}^2)\).

3. The angular distribution of \(\pi^0\)-mesons obtained on beryllium has a maximum in the direction of the incident γ-quanta.

4. The ratio of the production cross sections of \(\pi^0\)-mesons on hydrogen and carbon (at an angle of \(90^\circ\) to the beam) is

\[ \frac{\sigma_{\mathrm{H}\pi^0}}{\sigma_{\mathrm{C}\pi^0}} = 0.12 \pm 0.03. \]

The study of the production of charged mesons led to the following results\(^{2-8}\):

1. The angular distribution of the mesons produced is approximately isotropic.

2. The graph of the dependence of the production cross section of \(\pi^+\)-mesons on hydrogen (at an angle of \(90^\circ\) to the beam) on the energy of the γ-quanta \((E_\gamma)\) has a maximum in the region \(E_\gamma \simeq 260 \div 280\ \text{MeV}\).

3. The cross section has a value of about \(3 \cdot 10^{-28}\) per carbon nucleus.

4. The ratio of the production cross sections of negative and positive mesons is

\[ \frac{\sigma^-}{\sigma^+} = 1.3 \pm 0.15. \]

1. The ratio of the production cross sections of \(\pi^+\)-mesons (measured at an angle of \(90^\circ\) to the γ-quantum beam) on hydrogen and carbon is equal to

\[ \frac{\sigma_{\mathrm{H}\pi^+}}{\sigma_{\mathrm{C}\pi^+}} = 0.55. \]

The experimental data cited (both for \(\pi^\pm\) and for \(\pi^0\)-mesons), as we shall see below, are well explained theoretically.

II. THEORY

Before mesons produced by photons were obtained under laboratory conditions, this question had been discussed (for example, in the work of A. Sakharov\(^{12}\)) as applied to cosmic rays. After the report on the first experiments carried out at the Berkeley synchrotron\(^{2}\), more detailed studies appeared of the angular distributions of the mesons produced, of the energy dependences of the cross sections, etc.\(^{11,13-16}\) We shall first present the principal results of these works, and then qualitative considerations, discussion, and comparison of the theoretical results with the experimental ones.

The cross sections for meson production by photons were calculated on the basis of specific meson theories. In doing so, perturbation theory was used, and it was assumed that the interactions of the meson and nucleon with light, as well as of the nucleon with the meson field, could be regarded as weak. With respect to the first two interactions such a

the assumption is legitimate. These interactions may be regarded as weak and one may neglect higher approximations (beyond the first non-vanishing one), as well as higher-order processes associated with this interaction (for example, meson production in photon scattering). With respect to the latter interaction, however, the legitimacy of such an assumption is doubtful. Taking higher approximations for this interaction into account shows that higher-order corrections make a large contribution to the cross section ^11. This is a well-known general shortcoming of existing methods of calculation. We note that calculations without the assumption of smallness of \(g^2/\hbar c\) (intermediate and strong coupling) lead ^25,26 to the same characteristic features of the process of production of scalar and pseudoscalar charged mesons by photons as do calculations by perturbation theory.

a) Neutral mesons

Since neutral mesons decay only into two \(\gamma\)-quanta, their spin must be equal to zero, unless spin values greater than unity are considered (particles with spin \(1/2\) and \(1\) cannot decay into \(2\ \gamma\)-quanta ^18). The authors of note ^17 report that they calculated the cross sections for the production of neutral mesons according to scalar, pseudoscalar, vector, and pseudovector theories in the lowest non-vanishing approximation of perturbation theory. In all variants it was found that the cross section for the production of neutral mesons is considerably smaller than the cross sections for the production of charged mesons (\(\sim (M/\mu)^2\) times, where \(\mu\) and \(M\) are the masses of the meson and the nucleon, respectively). It is hardly meaningful to calculate the cross sections for the production of neutral mesons according to vector and pseudovector theories, since they describe mesons with spin equal to unity. Let us also note that, on the neutron, in this approximation neutral mesons should not be produced at all. Thus, the results of calculations in the first non-vanishing approximation of perturbation theory were in clear contradiction with the experimental data. To get out of this difficulty, an attempt was made, as reported in the same note, to take into account the next approximation of perturbation theory. The authors found that the correction of the next approximation makes a large contribution to the cross section for the production of neutral mesons and does not change the order of magnitude of the cross section for the production of charged mesons. In order to obtain agreement between the magnitudes of the production cross sections for charged and neutral mesons, they had to set the coupling constant \(g^2/\hbar c \sim 10\). Although the question of the expansion parameter in calculations by perturbation theory is still not clear, such a large value of the coupling constant nevertheless casts doubt on the legitimacy of the calculations.

All the more so since the contribution given by the next approximation considerably exceeds the contribution of the first nonvanishing approximation.

The result that $\sigma_{\pi^0}$ in the first approximation, according to all theories, is $\left(\dfrac{M}{\mu}\right)^2$ times smaller than $\sigma_{\pi^+}$ (for the most interesting energy region) is due to the following. In calculating in the first approximation of perturbation theory, the magnitude of the cross section is determined mainly by the interactions of the electromagnetic field with the charges of mesons and nucleons (interactions with the particle current): in the production of a neutral meson, by the interaction of the electromagnetic field with the proton current, and in the production of a charged meson, by the meson current. The meson current, roughly speaking, is $\dfrac{M}{\mu}$ times greater than the proton current. Hence, in particular, it follows that (under such a consideration) neutral mesons should not be produced on a neutron (there are no currents). This will be discussed in more detail below.

Experiment$^{1}$ indicates rather definitely the approximate equality of the cross sections for the production of neutral and charged mesons, and the fact that neutral mesons are also produced on the neutron. This suggests that, in addition to the electromagnetic interactions with the particle current, an essential contribution is made by other electromagnetic interactions connected with the structure of the nucleons (since the neutral meson does not interact with light). In a consistent meson theory these interactions should be obtained automatically, as a result of the presence of a charged meson cloud around the nucleon. However, calculations according to existing meson theories do not give the correct magnitude of the interaction of the nucleon even with a static magnetic field, i.e. of the static anomalous magnetic moment of the nucleon$^{19}$. This may be explained either by the imperfection of the methods of calculation, or by the fact that mesons of several kinds are responsible for these interactions.

Proceeding from these considerations, it was assumed$^{16}$ that these electromagnetic interactions can be described by an additional term $\dfrac{1}{2}\mu_0 \psi^{+}\gamma_{\mu}\gamma_{\nu}\psi F_{\mu\nu}$, where $\gamma_{\mu}$ and $\gamma_{\nu}$ are Dirac matrices, $F_{\mu\nu}$ is the tensor of the electromagnetic field, $\psi$ is the nucleon wave function, and $\mu_0$ is the difference between the total and the normal magnetic moments of the nucleon. Generally speaking, the magnetic moment should depend in some way on the wavelength of the incident light. One may introduce some dependence of $\mu_0$ on the energy of the incident $\gamma$-quanta; this changes the calculations little. But one may also suppose that the anomalous magnetic moment is equal to the magnetic moment measured experimentally in a static field. The latter means that such a model of the nucleon is adopted in which the currents that give rise to the magnetic moment are concentrated in a region of considerable...

less than the wavelength of the electromagnetic radiation incident on the nucleon (for the region of γ-quantum energies of interest to us, \(\lambda \sim \frac{\hbar}{\mu c}\)).

The calculations \(^{16}\) were carried out according to the usual perturbation theory for the scalar and pseudoscalar variants of meson theories. (As noted above, it makes no sense to calculate according to other theories.) The cross section for the production of a pseudoscalar neutral meson on a proton, for the case of pseudovector coupling, was obtained in the form

\[ d\sigma_p^0 = \left(\frac{e^2}{\hbar c}\right) \left(\frac{g^2}{\hbar c}\right) \left(\frac{\hbar}{\mu c}\right)^2 \frac{\pi dE_\mu}{M c^2} \left\{ 2\frac{E_\mu^2}{E_\gamma^2} \left(1+\frac{\mu_p E_\gamma}{2Mc^2}\right)^2 - \right. \]

\[ \left. -\frac{ 4E_\mu (\mathbf P_\gamma \mathbf P_\mu)c^3 \left(E_\gamma-\frac{(\mathbf P_\gamma \mathbf P_N)}{2M}\right) \left(1+\frac{E_\mu}{2Mc^2}\right) \left(1+\frac{\mu_p E_\gamma}{2Mc^2}\right)\mu_p }{ E_\gamma^2 \left(E_\gamma-\frac{P_\gamma^2}{2M}\right) \left(E_\mu+\frac{P_\mu^2}{2M}\right) } + \frac{ \left(1+\frac{E_\mu}{2Mc^2}\right)^2 c^2 P_\mu^2 \mu_p^2 }{ \left(E_\gamma-\frac{P_\gamma^2}{2M}\right) \left(E_\mu+\frac{P_\mu^2}{2M}\right) } \times \right. \]

\[ \left. \times \left[ 1+\cos^2\theta+ \frac{(\mathbf P_\mu \mathbf P_\gamma)^2}{ 2M^3 \left(E_\gamma-\frac{P_\gamma^2}{2M}\right) \left(E_\mu+\frac{P_\mu^2}{2M}\right) } \right] \right\}, \tag{1} \]

where \(E_\mu\) is the meson energy, \(\theta\) is the angle between the meson momentum and the direction of incidence of the γ-quanta, \(\mu_p\) is the anomalous magnetic moment of the proton in nuclear magnetons; \(\mathbf P_\gamma\), \(\mathbf P_\mu\), \(\mathbf P_N\) are the momenta of the γ-quantum, the meson, and the recoil nucleon, respectively; \(E_\gamma\) is the energy of the incident γ-quanta. Here we have given the result of a calculation performed in the Pauli approximation (for the Pauli approximation see below). We also carried out the relativistic calculation; the result differs from the formula given by no more than 10% for γ-quantum energies up to \(2.4\mu c^2\) (\(\sim 340\) MeV). The cross section for the production of neutral mesons on a neutron is obtained from formula (1) if only the terms with \(\mu_p^2\) are retained there and \(\mu_p\) is replaced by \(\mu_N\), the anomalous magnetic moment of the neutron.

For agreement in the order of magnitude of the cross section with that experimentally observed at \(E_\gamma \approx 330\) MeV \(^{1}\), it is sufficient, instead of \(\mu_p\) and \(\mu_N\)

substitute the values of the static magnetic moments, and set \(\dfrac{g^2}{\hbar c}\) equal to \(1/6\). The angular distributions of neutral mesons produced on a neutron and a proton, for \(E_\gamma = 2.4\mu c^2\), are shown in Fig. 8, and the dependence of the total cross sections on the \(\gamma\)-quantum energy is shown in Fig. 9*).

Fig. 8.

Fig. 9.

*) Recent experiments\(^{27}\), in agreement with our calculations, have shown that the photoproduction cross section of \(\pi^0\)-mesons increases with the energy of the \(\gamma\)-quanta much more rapidly than the production cross section of charged mesons (cf. Fig. 9 and Fig. 15).

A completely different result was obtained in the calculation according to the scalar theory \(^{16}\). The introduction of an anomalous moment does not remedy the situation (the cross section remains \(10^3\) times smaller than the cross section observed experimentally, if one takes the same \(\mu_p\), \(\mu_N\), and \(\dfrac{g^2}{\hbar c}\) as in the pseudoscalar variant). This is easy to understand without carrying out detailed calculations. Let us consider the production of a neutral scalar meson on an infinitely heavy nucleon (neglecting recoil). Let the \(\gamma\)-quantum be absorbed owing to some interaction of the electromagnetic field with the nucleon, described in the Hamiltonian by the term \(H_e\). We shall not specify it—it may be an interaction with the magnetic moment or some other electromagnetic interaction. The production and absorption of the meson, however, occur through the interaction \(H_g = g\varphi\), where \(\varphi\) is the meson function. The matrix element in the lowest nonvanishing approximation of perturbation theory is written as:

\[ H_{k0} = \frac{H_{e ki} H_{g i0}}{E_0 - E_i} + \frac{H_{g kj} H_{e j0}}{E_0 - E_j} = \frac{H_{e ki} H_{g i0}}{-E_\mu} + \frac{H_{g kj} H_{e j0}}{E_\gamma} = 0, \tag{2} \]

since \(E_\gamma = E_\mu\), and \(H_g\) commutes with any \(H_e\). In the case of production of a pseudoscalar meson, however, the matrix \(\sigma\) enters into \(H_g\), and therefore \(H_g\) does not commute with \(H_e\), and the matrix element \(H_{k0}\) does not vanish. The consideration of meson production without taking account of the recoil of the nucleons is, of course, very crude. However, the expression obtained in such a consideration enters, in a somewhat modified form, also into the cross section calculated with recoil taken into account, and gives the principal contribution there.

The results obtained \(^{16}\) indicate that, apparently, only the pseudoscalar theory does not contradict the experimental data \(^{1}\). Below we shall dwell in greater detail on the comparison of theoretical results with experimental ones. For the present we shall proceed to consideration of the production of charged mesons.

b) Charged mesons

The most interesting characteristics of the process of production of charged mesons are the distribution of mesons with respect to the angle \(\theta\) and the dependence of the meson-production cross section on the energy of the incident \(\gamma\)-quanta. As will be shown below, these characteristics depend substantially on the properties of the mesons and their interactions with nucleons. These differences appear most distinctly when mesons are produced by photons of the highest \(\gamma\)-quantum energies attained under laboratory conditions \(^{2}\) \((E_\gamma \sim 2\mu c^2)\). Therefore, despite the preliminary character of the experiment, from the data given above one can already draw rather definite conclusions about the nature of \(\pi\)-mesons.

In the first works on the production of mesons by γ-quanta (with energies \(\sim 2\mu c^2\))\(^{13,14}\), of which \(^{13}\) should be noted as the most detailed investigation, the calculation was carried out without taking into account the recoil of the nucleons. This is a very rough treatment; however, already in this approximation the most characteristic features of the process for various meson theories are revealed. The most compact and easily interpreted expressions for the cross sections can be obtained only in a relativistic treatment. Calculations in the Pauli approximation for the nucleon give, in the region of γ-quantum energies of interest to us \((\sim 2\mu c^2)\), results differing from the corresponding results of relativistic calculations by less than \(10\%\). At present there are in the literature expressions for the cross sections of production of charged mesons by γ-quanta, calculated relativistically for all the principal meson theories and with all types of meson–nucleon interactions (with the exception of the pseudotensor interaction of the pseudovector theory). Therefore below we give the results of the relativistic calculations. (All cross sections refer to the laboratory coordinate system.)

Fig. 10.

Fig. 11.

In Figs. 10 and 11 are shown diagrams of the processes of production of \(\pi^+\)- and \(\pi^-\)-mesons for the case of meson–nucleon interactions not containing derivatives of the meson-field functions. The interactions causing the corresponding transitions are indicated near the arrows, namely: \(H_g\) is the interaction of the meson field with the nucleon, \(H_e^{I}\) is the interaction of the electromagnetic field with the meson, and \(H_e^{II}\) is the interaction of the electromagnetic field with the nucleon. In considering interactions containing derivatives of the meson field, to each of these diagrams one chain is added, representing a transition without an intermediate state.

Scalar theory. The part of the Lagrangian due to the interaction of the nucleon with the meson field has the form:

\[ g_1 \psi^{+}\psi(\varphi\tau_{pN}+\varphi^{*}\tau_{Np}) + g_2\frac{\hbar}{\mu c}\,\psi^{+}\gamma_{\nu}\psi \left( \frac{\partial\varphi}{\partial x_{\nu}}\tau_{pN} + \frac{\partial\varphi^{*}}{\partial x_{\nu}}\tau_{Np} \right), \]

where \(g_1\) and \(g_2\) are coupling constants, \(\psi\) and \(\varphi\) are the wave functions of nucleons and mesons, respectively, \(\tau_{Np}=\begin{pmatrix}0&1\\0&0\end{pmatrix}\) and \(\tau_{pN}=\begin{pmatrix}0&0\\1&0\end{pmatrix}\). In paper \(^{21}\) it was shown that, in the first approximation, the second type of coupling (the second term) vanishes. Therefore only the interaction of the first type is considered \(^{11,15}\).

The production cross section for positive mesons may be written in the form:

\[ d\sigma_{\mathrm{ck}}^{+} = \left(\frac{g_1^2}{\hbar c}\right) \left(\frac{e^2}{\hbar c}\right) \left(\frac{\hbar}{\mu c}\right)^2 \frac{\mu^2 c^4 c^3 P_\mu^2\, d\Omega} {4Mc^2 E_\gamma\left[cP_\mu\left(E_\gamma+Mc^2\right)-E_\gamma E_\mu\cos\theta\right]} \times \]

\[ \times \left\{ \frac{ 4c^2P_\mu^2\sin^2\theta \left(2M^3c^4-\frac{1}{2}\mu^2c^4\right)} {\left[2Mc^2(E_\gamma-E_\mu)+\mu^2c^4\right]^2} + \frac{2Mc^3E_\mu-\mu^2c^4}{2Mc^2E_\gamma} \right\}, \tag{3} \]

where \(d\Omega\) is the element of solid angle \((d\Omega=\sin\theta\, d\theta\, d\varphi)\). For the production cross section of negative mesons one obtains:

\[ d\sigma_{\mathrm{ck}}^{-} = \left[ \frac{2Mc^3E_\gamma}{2Mc^2E_\mu-\mu^2c^4} \right]^2 d\sigma_{\mathrm{ck}}^{+}. \tag{4} \]

The differential cross sections (3) and (4) were integrated in paper \(^{30}\)*). As a result, the following expressions are obtained for the total cross sections:

\[ \sigma_{\mathrm{ck}}^{+} = \left(\frac{e^2}{\hbar c}\right) \left(\frac{g_1^2}{\hbar c}\right) \left(\frac{\hbar}{\mu c}\right)^2 \frac{\pi\mu^2c^4}{2M^2c^4E_\gamma} \left\{ \left(1-\frac{4M^3c^4}{\mu^2c^4}\right) \times \right. \]

\[ \left. \times \left[ \frac{\mu^2c^4(2Mc^2E_\gamma+\mu^2c^4)}{2E_\gamma^2Mc^3} \ln \frac{2Mc^2E_\gamma+\mu^2c^4-2Mc^2R} {2Mc^2E_\gamma+\mu^2c^4+2Mc^2R} + \frac{2\mu^2c^4R}{E_\gamma^2} \right] + \frac{2Mc^2(E_\gamma+Mc^2)-\mu^3c^4}{(Mc^2+2E_\gamma)^2} -R \right\}, \tag{5} \]

\[ \sigma_{\mathrm{ck}}^{-} = \left(\frac{e^2}{\hbar c}\right) \left(\frac{g_1^2}{\hbar c}\right) \left(\frac{\hbar}{\mu c}\right)^2 \frac{\pi\mu^2c^4}{2M^3c^4E_\gamma} \left\{ \left(1-\frac{4M^3c^4}{\mu^2c^4}\right) \times \right. \]

\[ \left. \times \left[ \frac{\mu^2c^4(2Mc^2E_\gamma-\mu^2c^4)}{2E_\gamma^2Mc^2} \ln \frac{2Mc^2E_\gamma-\mu^2c^4-2Mc^2R} {2Mc^2E_\gamma-\mu^2c^4+2Mc^2R} + \frac{2\mu^2c^4R}{E_\gamma^2} \right] + Mc^2 \ln \frac{2Mc^2E_\gamma-\mu^2c^4+2M^2c^4+2Mc^2R} {2Mc^2E_\gamma-\mu^2c^4+2M^2c^4-2Mc^2R} \right\}, \tag{6} \]

*) Paper \(^{30}\) contains a small error. The charged-meson production cross sections obtained there should be reduced by a factor of two. Because of this error, the authors also obtained an incorrect value for the ratio of the cross sections \(\dfrac{\sigma^{0}}{\sigma_{\pi^{+}}}\).

where

\[ R=\sqrt{\left(E_\gamma+\mu c^2-\frac{\mu^2 c^4}{2Mc^2}\right)\left(E_\gamma-E_\gamma^{\text{threshold}}\right)}; \]

here the energy of the γ-quanta at threshold is

\[ E_\gamma^{\text{threshold}}=\mu c^2\left(1+\frac{\mu c^2}{2Mc^2}\right). \]

The angular distributions of mesons and the dependence of the total cross sections on the energy of the incident γ-quanta are shown in Figs. 12 and 13.

Fig. 12.

Fig. 13.

Pseudoscalar theory. In this case the interaction of the nucleon with the meson field is written in the form:

\[ f_1\bar{\psi}\tau_5\psi\left(\varphi\tau_{pN}+\varphi^*\tau_{Np}\right) +f_2\frac{\hbar}{\mu c}\,\bar{\psi}\tau_\nu\tau_5\psi \left(\frac{\partial\varphi}{\partial x_\nu}\tau_{pN} +\frac{\partial\varphi^*}{\partial x_\nu}\tau_{Np}\right), \]

where \(f_1\) and \(f_2\) are coupling constants; \(\tau_5=\tau_1\tau_2\tau_3\tau_4\), \(\tau_\nu\) is a Dirac matrix. The calculation of the cross sections was carried out with both types of coupling\(^{15,20}\).

However, according to the theorem on the equivalence of these couplings\(^ {21}\) (for the process under consideration), the results differ only by constant factors. The differential cross sections for meson production have the form:

\[ d\sigma_{\pi\text{-sk}}^{+} = \left(\frac{e^{2}}{\hbar c}\right) \left(\frac{f_{3}^{2}}{\hbar c}\right) \left(\frac{\hbar}{\mu c}\right)^{2} \frac{\mu^{3}c^{4}c^{3}P_{\mu}^{2}\,d\Omega} {4E_{\gamma}Mc^{2}\left[cP_{\mu}(E_{\gamma}+Mc^{2})-E_{\gamma}E_{\mu}\cos\theta\right]} \times \]

\[ \times \left\{ \frac{2Mc^{2}E_{\mu}-\mu^{2}c^{4}}{2Mc^{2}E_{\gamma}} - \frac{2\mu^{3}c^{4}c^{3}P_{\mu}^{2}\sin^{2}\theta} {\left[2Mc^{2}(E_{\gamma}-E_{\mu})+\mu^{2}c^{4}\right]^{2}} \right\}, \tag{7} \]

\[ \frac{(d\sigma_{\pi\text{-sk}}^{-})}{(d\sigma_{\pi\text{-sk}}^{+})} = \frac{(d\sigma_{\text{sk}}^{-})}{(d\sigma_{\text{sk}}^{+})} = \left[ \frac{2Mc^{2}E_{\gamma}} {2Mc^{2}E_{\mu}-\mu^{2}c^{4}} \right]^{2}, \tag{8} \]

where

\[ f_{3}=f_{1}+\frac{2M}{\mu}f_{2}. \]

The corresponding angular distributions are given in Fig. 14. The total cross sections are obtained in the form\(^ {20}\):

\[ \sigma_{\pi\text{-sk}}^{+} = \left(\frac{e^{2}}{\hbar c}\right) \left(\frac{f_{3}^{2}}{\hbar c}\right) \left(\frac{\hbar}{\mu c}\right)^{3} \frac{\pi\mu^{2}c^{4}}{2M^{2}c^{4}E_{\gamma}} \times \]

\[ \times \left[ \frac{\mu^{2}c^{4}(2Mc^{2}E_{\gamma}+\mu^{2}c^{4})}{2Mc^{2}E_{\gamma}^{2}} \ln \frac{2Mc^{2}E_{\gamma}+\mu^{2}c^{4}-2Mc^{2}R} {2Mc^{2}E_{\gamma}+\mu^{2}c^{4}+2Mc^{2}R} + \frac{2\mu^{2}c^{4}R}{E_{\gamma}^{2}} + \frac{2Mc^{2}(E_{\gamma}+Mc^{2})-\mu^{2}c^{4}} {(Mc^{2}+2E_{\gamma})^{2}}R \right]. \tag{9} \]

\[ \sigma_{\pi\text{-sk}}^{-} = \left(\frac{e^{2}}{\hbar c}\right) \left(\frac{f_{3}^{2}}{\hbar c}\right) \left(\frac{\hbar}{\mu c}\right)^{2} \frac{\pi\mu^{2}c^{4}}{2M^{2}c^{4}E_{\gamma}} \times \]

\[ \times \left[ \frac{\mu^{2}c^{4}(2Mc^{2}E_{\gamma}-\mu^{2}c^{4})}{2Mc^{2}E_{\gamma}^{2}} \ln \frac{2Mc^{2}E_{\gamma}-\mu^{2}c^{4}-2Mc^{2}R} {2Mc^{2}E_{\gamma}-\mu^{2}c^{4}+2Mc^{2}R} + \frac{2\mu^{2}c^{4}R}{E_{\gamma}^{2}} + Mc^{2}\ln \frac{2Mc^{2}E_{\gamma}-\mu^{2}c^{4}+2M^{2}c^{4}+2Mc^{2}R} {2Mc^{2}E_{\gamma}-\mu^{2}c^{4}+2M^{2}c^{4}-2Mc^{2}R} \right]. \tag{10} \]

The dependence of the total cross sections for the production of pseudoscalar mesons on the energy of the incident \(\gamma\)-quanta \((E_{\gamma})\) is shown in Fig. 15.

Vector theory. The interaction of mesons with nucleons is taken in the form:

\[ g_{1}\psi^{+}\gamma_{5}\psi(\varphi_{\mu}\tau_{pN}+\varphi_{\mu}^{*}\tau_{Np})+ \]

\[ +\,g_{2}\frac{\hbar}{\mu c}\frac{1}{2}\psi^{+}\gamma_{\mu}\gamma_{\nu}\psi \left[ \left(\frac{\partial\varphi_{\mu}}{\partial x_{\nu}}-\frac{\partial\varphi_{\nu}}{\partial x_{\mu}}\right)\tau_{pN} + \left(\frac{\partial\varphi_{\mu}^{*}}{\partial x_{\nu}}-\frac{\partial\varphi_{\nu}^{*}}{\partial x_{\mu}}\right)\tau_{Np} \right]. \]

Fig. 14.

Labels in the figure:

\[ \frac{d\sigma^{+}}{d\Omega}\times 10^{29} \]

\[ \frac{f_2^2}{\hbar c}=\frac{1}{6} \]

\[ E_\gamma=2.4\,\mu c^2 \quad \text{(without allowance for recoil)} \]

\[ E_\gamma=1.4\,\mu c^2 \]

\[ E_\gamma=2.4\,\mu c^2 \]

\[ \text{Curve I }(E_\gamma=2.4\,\mu c^2) \]

Horizontal axis:

\[ 0^\circ,\ 45^\circ,\ 90^\circ,\ 135^\circ,\ 180^\circ,\ \theta \]

Fig. 15.

Labels in the figure:

\[ \lg\left[\frac{\sigma_{n-\mathrm{sc}}}{\dfrac{\pi e^2 f_3^2}{2M^2c^4}}\right] \]

\[ \sigma^{+} \]

\[ \sigma^{-} \]

\[ \lg\frac{E_\gamma-E_\gamma^{\mathrm{threshold}}}{\mu c^2} \]

Horizontal-axis marks:

\[ -1.0,\ -0.5,\ 0,\ 0.5 \]

Vertical-axis marks:

\[ 0,\ -0.5 \]

FORMATION OF MESONS BY γ-QUANTA

“With the first type of coupling, calculations of the cross sections were carried out in work \(^{11}\), and with the second—in work \(^{15}\). The expressions for the cross sections obtained in this version of meson theory are very cumbersome and not very transparent, and therefore we shall not reproduce them here (the reader can find them in the cited works).

For the vector meson theory, a characteristic feature is the strong elongation of the angular distribution of mesons produced by photons in the direction of the momentum of the \(\gamma\)-quantum, and the rapid growth of the effective cross section with the energy of the \(\gamma\)-quanta. In Fig. 16 the angular distribution of mesons is shown for the case of vector \((g_1)\) coupling, and in Fig. 17—for the case of tensor \((g_2)\) coupling. Integration of the cross sections for the production of vector mesons over the angles \(\theta\) was not carried out,

Fig. 16.

In the figure: vertical axis \(\dfrac{d\sigma^+}{d\Omega}\,\dfrac{10^{31}}{(g^2/\hbar c)}\); curves \(E_\mu = 80\ \text{MeV}\), \(E_\mu = 40\ \text{MeV}\); horizontal axis \(\theta\).

Fig. 17.

In the figure: vertical axis \(\dfrac{d\sigma^+}{d\Omega}\times 10^{28}\); \(\dfrac{g^2}{\hbar c}=\dfrac{1}{6}\); curves \(E_\gamma = 2.4\,\mu c^2\), \(E_\gamma = 1.8\,\mu c^2\), \(E_\gamma = 1.4\,\mu c^2\); horizontal axis \(\theta\).

with the exception of \(^{13}\), where graphs are given for the dependence, on the energy of the \(\gamma\)-quanta, of the total cross section obtained without taking recoil of the nucleon into account. The rather crude growth of the total cross section with the energy \(E_\gamma\) may be estimated as \(\sim E_\gamma^2\). The ratio \(\dfrac{\sigma_{\mathrm{vect}}^{-}}{\sigma_{\mathrm{vect}}^{+}}\) is approximately the same as in the case of the scalar and pseudoscalar theory (Fig. 18).

Fig. 18.

Pseudovector theory. In the pseudovector theory the calculations were carried out in work \(^{11}\). The interaction of a nucleon with the meson field of the form

\[ f_1\psi^{+}\gamma_\nu\gamma_\delta\psi(\varphi_\gamma\tau_pN+\varphi_\gamma^{*}\tau_Np). \]

was considered. The angular distributions obtained are shown in Fig. 19, borrowed, as is Fig. 16, from work \(^{11}\), and representing the angular distributions of mesons of given energy \(E_\mu\), produced by a beam of \(\gamma\)-quanta having the spectrum \(\dfrac{dE_\gamma}{E_\gamma}\).

Fig. 19.

However, the character of these angular distributions and the character of the angular distributions of mesons produced by \(\gamma\)-quanta of a given energy \(E_\gamma\) are identical. For the pseudovector theory there is the same rapid growth of the cross section with the energy of the \(\gamma\)-quanta,

as also for the vector theory. The ratio \(\dfrac{\sigma_{\eta\text{-vect}}^-}{\sigma_{\eta\text{-vect}}^+}\) is very close to unity and its dependence on the angle \(\vartheta\) is quite different than in the case of the first three variants of meson theories (see Fig. 18).

c) Semiclassical treatment

For comparison of the results obtained from different theories, and for a clearer physical understanding of the process of meson production, it is useful to carry out the following semiclassical treatment.

In all meson theories the nucleon is customarily regarded as a particle possessing a certain meson charge. This means that the nucleon generates around itself a meson field, analogous to the way in which an electron generates an electromagnetic field. In the case of charged mesons, the meson field around the nucleon is associated with the so-called charged meson cloud, the existence of which gives a vivid interpretation of the presence of an anomalous magnetic moment in the nucleon. From the corpuscular point of view, this meson field is often regarded as the result of the fact that nucleons emit and absorb mesons (i.e., for example, the proton spends some fraction of the time in the state neutron \(+\) positive meson). Let a beam of \(\gamma\)-quanta fall on such a system. For the moment let us discard the interaction of the electromagnetic field with the anomalous magnetic moment of the nucleon and consider the case of production of a neutral meson. If in the laboratory coordinate system the proton is at rest, then it will interact with light only after it emits a meson and receives recoil. Denote the probability of emission of a meson (the relative residence time of the proton in the state of two particles) by \(W\). Then the total probability of production of a meson by a photon will be equal to the product of \(W\) by the probability of absorption of a \(\gamma\)-quantum by a moving proton. The latter may be taken proportional to the square of the energy of the electromagnetic interaction \((\mathbf{j}_p\mathbf{A})^2\) (here \(\mathbf{j}_p\) is the proton current, \(\mathbf{A}\) is the vector potential of the electromagnetic field). For the case of a relativistic particle this electromagnetic interaction is written in the form:

\[ \frac{e(\mathbf{v}\mathbf{A})}{1-\beta\cos\vartheta} = \frac{E_\gamma e(\mathbf{P}\mathbf{A})c^3}{E_\gamma(E+cP\cos\vartheta)} = -eE_\gamma\,\frac{(\mathbf{P}\mathbf{A})}{\left(\widehat{P}\widehat{P}_\gamma\right)}, \tag{11} \]

where \(\mathbf{v}\) is the velocity of the particle, \(\mathbf{P}\) is its momentum, \(\beta=\dfrac{v}{c}\). Here and below, for brevity of notation, we shall use four-vectors. \(\left(\widehat{A}\widehat{B}\right)\) denotes the scalar product of four-vectors:

\[ \left(\widehat{A}\widehat{B}\right)=A_4B_4+(\mathbf{A}\mathbf{B}). \]

Thus, the probability of production

of a neutral meson is proportional to:

\[ W_{\pi^0}\, e^2 E_\gamma^2 \frac{(\mathbf P_p \mathbf A)^2} {\left(\hat{\mathbf P}_p \hat{\mathbf P}_\gamma\right)^2}, \tag{12} \]

where \(\mathbf P_p\) is the proton momentum.

Let us consider, in the same way, the production of a positive meson. In the production of a positive meson, the probability of photon absorption will be determined by the square of its interaction with the meson current \((j_\mu A)^2\) (\(j_\mu\) is the meson current). In this case the electromagnetic field does not interact with the nucleon, since no charge transfer is associated with the latter. Therefore the probability of production of a \(\pi^+\)-meson will be proportional to the quantity:

\[ W_{\pi^+} e^2 E_\gamma^2 \frac{(\mathbf P_\mu \mathbf A)^2} {\left(\hat{\mathbf P}_\mu \hat{\mathbf P}_\gamma\right)^2}. \tag{13} \]

If one assumes that the nature of the interaction of the nucleon with charged and neutral meson fields is the same, then \(W_{\pi^0}\) may be set equal to \(W_{\pi^+}\). Then the ratio of the probabilities of production of neutral and positive mesons is written in the form:

\[ \frac{(\mathbf P_p \mathbf A)^2} {\left(\hat{\mathbf P}_p \hat{\mathbf P}_\gamma\right)^2} \cdot \frac{\left(\hat{\mathbf P}_\mu \hat{\mathbf P}_\gamma\right)^2} {(\mathbf P_\mu \mathbf A)^2} = \frac{\left(\hat{\mathbf P}_\mu \hat{\mathbf P}_\gamma\right)^2} {\left(\hat{\mathbf P}_p \hat{\mathbf P}_\gamma\right)^2}, \tag{14} \]

since, by virtue of the law of conservation of momentum \((\mathbf P_p=\mathbf P_\gamma-\mathbf P_\mu)\) and the transversality of the light wave \(((\mathbf P_\gamma \mathbf A)=0)\): \(-(\mathbf P_p \mathbf A)=(\mathbf P_\mu \mathbf A)\). It is interesting to note that precisely the same ratio of cross sections is obtained as the result of a consistent relativistic calculation according to the scalar and pseudoscalar meson theories. For \(\gamma\)-quantum energies near threshold, from (14) one may obtain that

\[ \frac{\sigma_{\pi^0}}{\sigma_{\pi^+}} \sim \left(\frac{\mu}{M}\right)^2 . \]

In the production of negative mesons the neutron is transformed into a proton and a \(\pi^-\)-meson; therefore the probability will depend both on the meson current and on the proton current:

\[ W_{\pi^-}\, e^2 E_\gamma^2 \left[ \frac{(\mathbf P_p \mathbf A)} {\left(\hat{\mathbf P}_p \hat{\mathbf P}_\gamma\right)} - \frac{(\mathbf P_\mu \mathbf A)} {\left(\hat{\mathbf P}_\mu \hat{\mathbf P}_\gamma\right)} \right]^2 . \tag{15} \]

Taking into account that \((\mathbf P_p \mathbf A)=-(\mathbf P_\mu \mathbf A)\) (and assuming \(W_{\pi^+}=W_{\pi^-}\)), we obtain for the ratio of the cross sections for the production of positive and negative mesons the following expression\({}^{11}\):

\[ \frac{(d\sigma^-)}{(d\sigma^+)} = \frac{ \left[ \left(\hat{\mathbf P}_\mu \hat{\mathbf P}_\gamma\right) + \left(\hat{\mathbf P}_p \hat{\mathbf P}_\gamma\right) \right]^2 } { \left(\hat{\mathbf P}_p \hat{\mathbf P}_\gamma\right)^2 } = \frac{M^2 c^4 E_\gamma^2} {\left(\hat{\mathbf P}_p \hat{\mathbf P}_\gamma\right)^2}. \tag{16} \]

Expression (16), after simple transformations, exactly coinci-

gives with expression (8), obtained in the relativistic calculation according to the scalar and pseudoscalar theories.

Let us now consider, using these intuitive representations, the results of calculations according to perturbation theory. The cross section for production of a neutral scalar meson (without taking into account the anomalous magnetic moment) can be obtained in the form:

\[ \left(\frac{e^{2}}{\hbar c}\right)\left(\frac{g^{2}}{\hbar c}\right)\left(\frac{\hbar}{\mu c}\right)^{2} \frac{\pi \mu^{2}c^{4}\,dE_{\mu}}{2Mc^{2}E_{\gamma}^{2}} \left\{ \left[ \frac{2Mc^{2}E_{\mu}-\mu^{2}c^{4}}{2Mc^{2}E_{\gamma}} + \frac{2Mc^{2}E_{\gamma}}{2Mc^{2}E_{\mu}-\mu^{2}c^{4}} -2 \right] - \frac{\mu^{2}c^{4}c^{2}P_{\mu}^{2}\sin^{2}\theta}{2\left(\hat{P}_{p}\hat{P}_{\gamma}\right)^{2}} + \left(\frac{2M}{\mu}\right)^{2} \frac{\mu^{2}c^{4}c^{2}P_{\mu}^{2}\sin^{2}\theta}{2\left(\hat{P}_{p}\hat{P}_{\gamma}\right)^{2}} \right\}. \tag{17} \]

For the practically interesting region of \(\gamma\)-quantum energies, the third term in the braces is considerably larger than all the others. It is not difficult to see that it is due to the interaction of the recoil nucleon current with the electromagnetic field, \((j_p A)^2\). The first term (in square brackets) is due to the interaction of light with the normal magnetic moment of the nucleon. To make this clearer, let us pass in formula (17) to \(\mu \to 0\); then we obtain:

\[ \frac{\pi e^{2}g^{2}}{2Mc^{2}E_{\gamma}^{2}}\,dE_{\mu} \left\{ \left[ \frac{E_{\mu}}{E_{\gamma}} + \frac{E_{\gamma}}{E_{\mu}} -2 \right] + 2\sin^{2}\theta \right\}. \tag{18} \]

Formula (18) almost coincides with the Klein–Nishina formula. The factor before the brace and the term in square brackets in these formulas coincide exactly (when \(g^{2}\) is replaced by \(e^{2}\)). If in the Klein–Nishina formula the second term represents the classical Thomson formula, then here this term is obtained from the semiclassical considerations given above.

The term in square brackets, common to both formulas, is due to the presence of spin in the particle and to its transitions into states with negative energy. This analogy of the formulas was also to be expected, proceeding from the analogy of the interactions of the meson field with the nucleon and of the electromagnetic field with the electron (of course, only in the sense of the analogy mentioned above). The second term in formula (17), which vanishes in the transition \(\mu \to 0\), differs from the latter only by the factor \(\left(\dfrac{2M}{\mu}\right)^2\). This term can be interpreted in the same way as is done in article 11 for charged mesons. The scalar meson cloud around the nucleon consists of two parts. One part extends over a region of order \(\left(\dfrac{\hbar}{\mu c}\right)\), while the other part of the cloud, closely bound to the nucleon, extends over a region of order \(\left(\dfrac{\hbar}{Mc}\right)\). Therefore the second part of the cloud gives in the cross-

...the formation of mesons gives a contribution smaller by \(\sim \left(\dfrac{\mu}{M}\right)^3\) times. The cross sections for the production of charged mesons differ from the cross sections for the production of neutral mesons only by a factor. Therefore, essentially the same considerations apply to them as to neutral mesons. The principal contribution to the cross section is again given by the third term, but now it takes a somewhat different form. In the case of production of positive mesons it will be due to \((j_\mu A)^2\), while for production of negative mesons it is due to \([- (j_\mu A) + (j_p A)]^2\). Thus, in the production of a charged scalar meson, the magnitude of the cross section is determined by the interaction of the electromagnetic field with the meson current (the proton current is small):

\[ (j_\mu A)^2 \to \frac{(P_\mu A)^2}{E_\mu^2(1-\beta_\mu \cos\theta)^2}. \]

From this follows the dipole character of the angular distribution of scalar mesons (since \((\mathbf{P}_\mu \mathbf{A})^2 \to \sin^2\theta\)) (see Fig. 12). The denominator \((1-\beta_\mu \cos\theta)^2\), caused by the retardation of the interaction of the electromagnetic field with a rapidly moving charge, leads to a distortion of the angular distribution. At high energies the angular distribution becomes elongated in the direction of the momentum of the \(\gamma\)-quantum. The character of the dependence of the total cross section on the energy of the \(\gamma\)-quanta is also clear. In the case of a scalar meson there is no such electromagnetic interaction of the system whose energy would rapidly increase with the energy of the incident photon.

The cross section for the production of a neutral pseudoscalar meson (without taking account of the anomalous magnetic moment) has the form:

\[ \left(\frac{e^2}{\hbar c}\right) \left(\frac{f_3^2}{\hbar c}\right) \left(\frac{\hbar}{\mu c}\right)^2 \frac{\pi \mu^2 c^4 dE_\mu}{2Mc^2 E_\gamma} \left\{ \left[ \frac{2Mc^2E_\mu-\mu^2c^4}{2Mc^2E_\gamma} + \frac{2Mc^2E_\gamma}{2Mc^2F_\mu-\mu^2c^4} -2 \right] - \frac{\mu^2c^4 c^2 P_\mu^2 \sin^2\theta} {2(\widehat{P}_p\widehat{P}_\gamma)^2} \right\}, \tag{19} \]

where

\[ f_3=f_1+\frac{2M}{\mu}f_2. \]

Comparing this expression with formula (17), one may conclude that in the case of the pseudoscalar theory the principal contribution to the cross section is given by the term due to the spin of the nucleon and its transition to a state with negative energy. This also indicates a strong coupling of the meson field with the nucleon and the fact that, in meson production, the interaction with the magnetic moment of the nucleon is essential. The cross section for the production of positive pseudoscalar mesons, just as in the case of the scalar theory, diff—

differs from the cross section for the production of neutral mesons (19) only by the factor

\[ \frac{\left(\widehat{P}_{p}\widehat{P}_{\gamma}\right)^{2}} {\left(\widehat{P}_{q}\widehat{P}_{\mu}\right)^{2}} . \]

Otherwise this cross section can be written in the form of formula (7). The main contribution to the cross section is made by the first term of formula (7), obtained from the expression in the square brackets of formula (19). It accounts for the approximate isotropy of the angular distribution. It is interesting to note that, when the calculation is performed without taking account of the recoil of the nucleon, the second term in formula (7) remains unchanged, while the first turns into unity. In this crude approximation it becomes especially clear that the second term is due to interaction with the meson current, while the first is due to the interaction \(eg(\boldsymbol{\sigma}\mathbf{A})\varphi\), which makes possible the process of meson production without an intermediate state. This interaction arises from the interaction of the nucleon with the meson field upon introduction of the electromagnetic field:

\[ g(\boldsymbol{\sigma}\nabla)\varphi \to g\left(\boldsymbol{\sigma},\, \nabla - \frac{ie}{\hbar c}\mathbf{A}\right)\varphi . \]

As is seen from formula (7), the expression in braces entering it does not undergo a rapid increase with the energy \(E_{\gamma}\), and the factor before the brace even decreases. Thus, for mesons with spin zero, a decrease of the cross section as a function of \(E_{\gamma}\) at high energies is characteristic.

Let us now consider the production of mesons possessing spin. For neutral mesons, as was already noted, such a consideration apparently has no meaning. Charged mesons having spin also have a magnetic moment, and therefore the interaction of the meson with the electromagnetic field must already be written differently. Namely: in addition to the interaction \((\mathbf{j}_{\mu}\mathbf{A})\), there will also be an interaction \((\boldsymbol{\mu}_{m}\mathbf{H})\), where \(\boldsymbol{\mu}_{m}\) is the magnetic moment of the meson and \(\mathbf{H}\) is the magnetic field of the light wave. Thus, if one considers it semiclassically, the probability of the process of production of a positive meson is written as

\[ W_{\pi+} \left[ \frac{ \frac{e}{c}(\mathbf{v}_{\mu}\mathbf{A})+(\mu_{m}\operatorname{rot}\mathbf{A}) }{ 1-\beta_{\mu}\cos\theta } \right]^{2}. \tag{20} \]

In the expression for the probability of production of a positive meson with zero spin there was present only the term \((\mathbf{v}\mathbf{A})\), and at \(\theta=0\) the probability vanished, since \(\mathbf{v}\parallel\mathbf{P}_{\gamma}\), while \(\mathbf{A}\perp\mathbf{P}_{\gamma}\). In the case under consideration, however, the numerator does not vanish, and the denominator (if the meson is a relativistic particle \((\beta_{\mu}\sim 1)\)) tends to zero at \(\theta=0\). Therefore, for charged mesons possessing spin, there should be observed an angular distribution strongly elongated in the direction \(\mathbf{P}_{\gamma}\) \((\theta=0)\). However, this remark is valid only in the case where the meson field does not

is strongly connected with the nucleon and the principal interaction determining the process is the photomeson interaction. The indicated character of the angular distribution is obtained from the vector theory. The magnitude of the cross section is determined by the term \(\dfrac{(\mu_m \operatorname{rot} A)^2}{(1-\beta_\mu \cos\theta)^2}\), or \(\dfrac{(\mu_m[AP_\gamma])^2}{(1-\beta_\mu \cos\theta)^2}\). Hence it follows that, if the meson possesses a moment, then the production cross section should increase rapidly with the energy of the \(\gamma\)-quanta in the region \(E_\gamma \sim 2\mu c^2\) (a further increase of the cross section may be connected with defects of the theory). As for the angular distribution of pseudovector mesons, here, apparently, just as for pseudoscalar mesons, owing to the presence of the matrix \(\gamma_5=\gamma_1\cdot\gamma_2\cdot\gamma_3\cdot\gamma_4\), there is a strong coupling of the meson field with the spin of the nucleon. The ratio \(\left(\dfrac{d\sigma^-}{d\sigma^+}\right)\) according to the vector theory is obtained close to that obtained according to the scalar and pseudoscalar theories. This is also easy to understand from semiclassical considerations. At small angles \(\theta\) the recoil momenta of the nucleon are small and, consequently, give \(\left(\dfrac{d\sigma^-}{d\sigma^+}\right)\sim 1\). As is seen from Fig. 18, the ratio

\[ \frac{[(j_\mu A)+(j_p A)]^2}{(j_\mu A)^2} \]

takes its greatest value at large angles \(\theta\). The interaction with the magnetic moment gives a large contribution only at small angles \(\theta\). Consequently, it brings the ratio \(\left(\dfrac{d\sigma^-}{d\sigma^+}\right)\) closer to unity in that region of angles \(\theta\) where this ratio is already close to 1, and changes the ratios little at large angles \(\theta\).

Thus, as is seen from this semiclassical analysis of the formulae, the character of the angular distributions, the energy dependences, and the relative magnitudes of the production cross sections of charged and neutral mesons depend essentially on electromagnetic interactions. From this consideration, in particular, it is clear why near threshold the production cross section of neutral mesons in all theories is obtained as \(\sim\left(\dfrac{M}{\mu}\right)^2\) times smaller than the production cross sections of charged mesons; and it is also seen that, if no other electromagnetic interactions are introduced (apart from interactions with the particle current), then \(d\sigma^0\) will be considerably smaller than \(d\sigma^+\). As we noted above, if one phenomenologically introduces an interaction with the anomalous magnetic moment, then the cross section \(d\sigma^0\) can be obtained of approximately the same magnitude as \(d\sigma^+\). However, in this case the question arises whether it is necessary to introduce an interaction with an additional magnetic moment of the nucleon when considering the production of charged mesons.

d) Introduction of the anomalous magnetic moment in calculating the process of charged-meson production

In introducing the moment in the case of production of \(\pi^0\)-mesons\(^{16}\), we proceeded from the fact that the first approximation of perturbation theory takes into account only the interaction with the charges of the particles, but does not take into account the interaction of the electromagnetic field with the anomalous magnetic moment of the nucleon or, more precisely, with the charged meson cloud around the nucleon. The process of production of a charged meson, however, already in the first approximation of perturbation theory substantially includes the interaction of the photon with the charged meson cloud. Therefore it is not clear what additional electromagnetic interactions must be introduced. In order to find out to what extent the introduction of these additional electromagnetic interactions may alter the results discussed above, we carried out calculations of the production cross sections of charged mesons, introducing the anomalous magnetic moment\(^{16}\). The calculations were performed in the Pauli approximation. The results are as follows.

The production cross section of a scalar positive meson in this case will be:

\[ \left( \frac{e^2}{\hbar c} \right) \left( \frac{g_1^2}{\hbar c} \right) \left( \frac{\hbar}{\mu c} \right)^3 \frac{M c^2 c^2 P_\mu^2 d\Omega} {E_\gamma \left[(E_\gamma+Mc^2)cP_\mu - E_\gamma E_\mu \cos\theta\right]} \times \]
\[ \times \left\{ \frac{2\mu^3 c^4 c^2 P_\mu^2 \sin^2\theta} {\left[2Mc^2(E_\gamma-E_\mu)+\mu^2 c^4\right]^2} + \right. \]
\[ \left. + \left(\frac{\mu}{2M}\right)^2 \frac{ \left[ \mu_N\left(E_\gamma-\frac{P_\gamma^2}{2M}\right) -\mu_p\left(E_\mu+\frac{P_\mu^2}{2M}\right) \right]^2 } { \left(1-\frac{E_\gamma}{2Mc^2}\right)^2 \left(E_\mu+\frac{P_\mu^2}{2M}\right)^3 } \right\}. \tag{21} \]

The corresponding graph of the angular distribution is shown in Fig. 20. From formula (21) it follows that taking account of the anomalous magnetic moment in the scalar theory does not change the characteristic features of the angular distribution and of the dependence of the cross section on the energy of the \(\gamma\)-quanta. It is true that the magnitude of the total cross section increases approximately twofold (for \(E_\gamma \sim 2\mu c^2\)). The ratio of the cross sections \(\left(\dfrac{d\sigma^-}{d\sigma^+}\right)\) for angles \(\theta\) greater than \(90^\circ\) becomes practically equal to unity, since the interaction with the anomalous magnetic moment gives in this range of angles a large and approximately equal contribution both to \(d\sigma^-\) and to \(d\sigma^+\). The latter should be

from the fact that, when the interaction with the magnetic moment is taken into account, two will be added to the basic chains (see above, Figs. 10 and 11),

Fig. 20.

both in the case of the production of a negative meson and in the case of the production of a positive meson (Figs. 21 and 22). In Figs. 21 and 22, the interactions causing the corresponding transition are shown next to the arrows: \(H_e^N\) is the interaction of the electromagnetic field with the magnetic moment of the neutron, \(H_e^p\) is the interaction of the electromagnetic field with the magnetic moment of the proton, \(H_g\) is the interaction of the nucleon with the meson field. From these chains it is clear that the addition due to the additional magnetic moment is approximately the same for negative as for positive mesons (indeed, everything reduces to the replacement \(\mu_N \to \mu_p\) and \(\mu_p \to \mu_N\) in the second term of formula (21)).

Fig. 21.

Fig. 22.

For angles \(\theta\) less than \(\sim 45^\circ\), \(\left(\dfrac{d\sigma^-}{d\sigma^+}\right)\) changes little upon the introduction of the anomalous magnetic moment. A large contribution to the cross section for scalar mesons from the additional magnetic moment can be

PRODUCTION OF MESONS BY $\gamma$-QUANTA

to be explained with the aid of the same considerations that we gave for explaining the small value of the cross section for the production of neutral scalar mesons (see formula (2)). For charged mesons all those considerations remain valid, only now (as is seen from the chains, Fig. 21) $H_{eki}$ will be the interaction with the magnetic moment of the proton, and $H_{ej_0}$ with the magnetic moment of the neutron. These moments have different signs. Therefore, if in the production of neutral mesons these chains were subtracted one from the other, now they will be added and will make a large contribution to the cross section.

Fig. 23.

The cross sections for the production of pseudoscalar charged mesons were also calculated (the pseudovector coupling was considered). However, because of the cumbersomeness of the formulas we shall not give them here and shall restrict ourselves only to the graph. The angular distribution of $\pi^+$ mesons, obtained with allowance for the anomalous magnetic moment, is shown in Fig. 23. There, for comparison, the angular distribution obtained from formula (7) is also given (dashed line). From this figure it is seen that the contribution from the additional magnetic moment to the cross section for the production of pseudoscalar charged mesons is small (less than 10%) and almost does not change either the magnitude of the total cross section or the other characteristic features of the process. Thus, in the production of charged mesons, for the pseudoscalar theory it is practically immaterial whether the anomalous magnetic moment is taken into account or not.

No calculation of the cross section for the production of charged vector mesons with allowance for the additional magnetic moment of the nucleon was carried out. However, for this version of the theory it is almost obvious that the magnetic moment gives a small contribution, since the order of magnitude of the cross section in this case is determined by the strong photomeson interaction.

III. DISCUSSION OF THE EXPERIMENTAL AND THEORETICAL RESULTS

Let us now compare the experimental and theoretical data presented above on the production of mesons by \(\gamma\)-quanta. The cross section for the production of positive mesons on hydrogen, measured experimentally\({}^{6}\), rapidly increases with the energy of the \(\gamma\)-quanta up to \(E_\gamma \simeq 2\mu c^2\), and then the growth of the cross section ceases (see Fig. 7). This, it seems to us, is the most convincing argument in favor of the absence of spin in the charged \(\pi\)-meson. For, as follows from calculations by perturbation theory and from semiclassical considerations, irrespective of the type of coupling of the meson with the nucleon, the production cross section of mesons having spin (and consequently also a magnetic moment) must increase rapidly with the energy of the \(\gamma\)-quanta (in the region \(E_\gamma \sim 2\mu c^2\)).

The dependence of the production cross section of charged mesons with spin zero on \(E_\gamma\), on the other hand, agrees well with what is observed experimentally (see Fig. 7 and Fig. 15).

Still more definite experimental evidence is available in favor of the spin of the neutral meson being equal to zero. It has been proved\({}^{1}\) that the neutral meson decays only into two photons. It follows from this that it must have spin either 0 or 2 and higher\({}^{18}\). Since the introduction of higher spins is an additional and, in general, unnecessary assumption, it is natural to stop at the value of the spin equal to zero.

On the basis of the considerations given above, one may say that the available experimental data do not agree with the vector and pseudovector meson theories.

For the angular distributions of charged mesons according to theories giving predominance to the photomeson interaction, with increasing energy of the \(\gamma\)-quanta there should be observed a characteristic tendency toward an increase in the probability of meson production at angles \(\theta\) close to \(\theta = 0^\circ\). This asymmetry in the angular distribution, as was noted above, is due to retardation in the interaction of the photon with the meson. Such a character of the angular distribution corresponds to the scalar and vector theories. In the case of strong interaction of the meson with the spin of the nucleon (the pseudoscalar variant of the theory), the angular distribution is approximately isotropic and does not change strongly with the energy of the \(\gamma\)-quanta. As preliminary experimental data show\({}^{5,6}\), the angular distribution of charged mesons is approximately isotropic.

Thus, only the pseudoscalar variant gives results, for the production of charged mesons by \(\gamma\)-quanta, that agree with experiment.

This same variant of the meson field is the only one suitable for describing the properties of the neutral meson. The remaining variants fall away, since they give either a value of the spin different from

from zero, or else (in the scalar theory) a value of the cross section \(d\sigma^{0}\) considerably smaller than \(d\sigma^{+}\). The pseudoscalar theory, when an anomalous magnetic moment is introduced, gives, in agreement with experiment, the same order of magnitude for these cross sections*).

As for the remaining experimental data, they concern the production of mesons in a nucleus. Therefore, in order to obtain theoretical results it is necessary to consider meson production on bound nucleons. Here there is a well-known difficulty due to the arbitrariness of model assumptions. Consequently, the remaining experimental results can be explained only qualitatively.

The experimentally observed excess of the cross section for the production of \(\pi^{-}\)-mesons over the cross section for the production of \(\pi^{+}\)-mesons agrees with the results obtained from the pseudoscalar theory\(^{22}\). In all meson theories positive and negative mesons enter symmetrically, as two components of one charged complex field. The only source of asymmetry in the production of charged mesons is the interaction of the electromagnetic field with the charge of the proton (and with the normal magnetic moment). It is true that there is one more source of asymmetry—the Coulomb interaction—but it has an appreciable effect only quite near the threshold for meson production. In the production of \(\pi^{+}\)-mesons on protons the Coulomb interaction is altogether absent (in the final state there is a \(\pi^{+}\)-meson and a neutron). In the production of \(\pi^{-}\)-mesons on neutrons (in the final state there is a \(\pi^{-}\)-meson and a proton) it gives a correction to the cross section of \(\sim 3\%\) for \(E_{\gamma}\sim 2\mu c^{2}\). As was noted in Ref. \(^{11}\), the ratio \(\left(\dfrac{d\sigma^{-}}{d\sigma^{+}}\right)\) would be equal to unity if electromagnetic interactions of the type of interactions with the anomalous magnetic moment of the nucleons played the principal role in the process of charged-meson production.

Despite the fact that the carbon nucleus contains six protons, the experimental value of the cross section for the production of \(\pi^{+}\)-mesons on carbon is only twice as large as on hydrogen. For neutral mesons, however, this ratio is

\[ \frac{\sigma^{0}_{\mathrm{H}}}{\sigma^{0}_{\mathrm{C}}}=0.12 \]

(measured at an angle \(\theta \simeq 90^\circ\)). This ratio can be obtained from the distributions of neutral-meson production on the proton and neutron given above (see Fig. 8), if it is assumed that the production cross section on carbon (at an angle of \(90^\circ\) to the beam) is equal to the sum of the production cross sections on six protons and six neutrons:

\[ \frac{\sigma^{0}_{\mathrm{H}}}{\sigma^{0}_{\mathrm{C}}}\simeq \frac{1}{6+\dfrac{1}{3}\cdot 6} \simeq 0.12 \]

(the factor \(1/3\) appeared because the cross section for the production of \(\pi^{0}\)-mesons at an angle \(\theta=90^\circ\) on the neutron

*) See also the footnote on p. 214.

approximately three times smaller than on the proton). In the case of the production of charged mesons on nuclei, however, the total cross section is no longer simply the sum of the production cross sections on the individual nucleons. This can be explained qualitatively by the fact that, in the production, for example, of a positive meson, a proton is transformed into a neutron, and a large local density of neutrons is thereby produced. In the presence of a large density of identical particles, the Pauli principle begins to play an essential role, restricting the possible transfer of momentum to the nucleon and thereby reducing the total cross section. In the production of neutral mesons, however, no transitions of neutrons into protons and protons into neutrons occur, i.e. the Pauli principle, even if it plays a role, plays only an insignificant one.

Thus, the difference between the ratios \(\frac{\sigma_H^0}{\sigma_C^0}\) and \(\frac{\sigma_H^+}{\sigma_C^+}\) should evidently be understood as a consequence chiefly of two factors: 1) neutral mesons are produced both on protons and on neutrons, whereas positive mesons are produced only on protons; 2) in the production of charged mesons the restrictions imposed by the Pauli principle are more substantial than in the case of the production of neutral mesons.

The angular distribution of neutral mesons obtained on beryllium may also be regarded as the result of a superposition of the angular distributions of mesons produced on five neutrons and four protons. In this case the angular distribution has, in agreement with experiment, a maximum in the direction of the momentum of the \(\gamma\)-quantum.

Above we have discussed experimental and theoretical data pertaining to the region of \(\gamma\)-quantum energies \(E_\gamma \sim 2\mu c^2\). This is apparently the most interesting energy region for the study of the production of \(\pi\)-mesons by \(\gamma\)-quanta. The production cross section of charged mesons has its maximum value in this region; the difference in the formation of mesons possessing and not possessing a magnetic moment, and consequently also spin, is also most clearly manifested. The characteristic features of the angular distributions obtained according to different meson theories at these \(\gamma\)-quantum energies are substantially different. One could also discuss meson production near the threshold, \(E_\gamma \sim \mu c^2,\ cP_\mu \ll \mu c^2\). In this case, as is evident from the formulas given above, the production cross section of scalar mesons would differ from the cross section for pseudoscalar mesons only by a constant factor. This is quite understandable, since at such energies the principal role is played by interactions not with the particle current but with the magnetic moment. Here one should expect a substantial difference in the formation of mesons having a magnetic moment and mesons with a moment equal to zero. However, in this region of \(\gamma\)-quantum energies the magnitude of the meson-production cross section is too small and their self-absorption in the source is large.

Summing up, one may say that, of all the existing experimental data\(^{1-8}\) on the production of both charged and neutral \(\pi\)-mesons by \(\gamma\)-quanta, only the pseudoscalar variant of the existing meson theories is consistent with them.

In conclusion we thank M. A. Markov for his great assistance and for discussion of the material presented.

ADDENDUM

Instead of an exact relativistic calculation in obtaining the cross sections for meson production by \(\gamma\)-quanta, one may in a number of cases restrict oneself to the Pauli approximation. For a qualitative estimate of the results one may also use an even cruder approximation, in which the nucleon mass is regarded as infinite (neglect of nucleon recoil). In all these approximations the mesons are treated as relativistic particles.

In calculations without taking nucleon recoil into account\(^{14}\), the correct orders of magnitude were obtained for the cross sections of charged-meson production according to the scalar and pseudoscalar theories, as well as the characteristic features of the angular distributions and of the dependence of the cross sections on the energy \(E_\gamma\). As we noted above, in this approximation it is fairly easy to understand, without carrying out calculations, that, for example, the cross section for production of scalar \(\pi^0\)-mesons on a neutron due to the interaction of the electromagnetic field with the anomalous magnetic moment of the nucleon must be small. However, the deviations from the results of the relativistic calculation then turn out to be quite substantial. For example, the cross sections for production of positive and negative mesons are obtained as equal, whereas the relativistically calculated \(d\sigma^-\) and \(d\sigma^+\) at some angles \(\theta\) may differ by almost a factor of three. A comparison of the angular distributions obtained in the different approximations is given in Fig. 14. As is seen from this figure, in a calculation without taking nucleon recoil into account the difference reaches \(\sim 300\%\). The results obtained in the Pauli approximation differ from the results of the relativistic calculation by no more than \(5\%\) (in Fig. 14 curve \(I\) was obtained in the Pauli approximation with account taken of the normal magnetic moment of the nucleon).

It is often remarked\(^{13,14}\) that expressions obtained without taking recoil into account are valid near the threshold of meson production. However, such a statement is inaccurate, since in calculations without taking nucleon recoil into account the interaction with the normal magnetic moment of the nucleon is ignored, which at energies \(E_\gamma - E_\gamma^{\mathrm{por}} \ll \mu c^2\) plays the principal role. This is easiest to see in the example of scalar mesons. Without taking nucleon recoil into account, in formula (3) there remains only the first term in the curly brackets\(^{14}\), which for \(c p_\mu \ll \mu c^2\) becomes much smaller than the second, due to the magnetic interaction. Thus, calculations without taking recoil into account near the threshold of meson production

photons (when \(c p_\mu \ll \mu c^3\)) are invalid. They give, only for the energies of \(\gamma\)-quanta \(E_\gamma - E_\gamma^{\text{threshold}} \sim \mu c^3\), results which agree more or less with calculations carried out relativistically.

In order to obtain the Hamiltonian in the Pauli approximation, it is sufficient to carry out the well-known transition from the Dirac equation to the Pauli equation. After this transition, the nucleon field interacting with the electromagnetic field will be described by the usual nonrelativistic Hamiltonian (including the interaction of the electromagnetic field with the normal magnetic moment). The interactions of the nucleon field with the meson field are obtained in the following form:

1) Scalar theory:

\[ H_g = c g_1(\varphi \tau_{pN}+\varphi^*\tau_{Np}), \]

\[ H_{eg}=0; \]

2) Pseudoscalar theory:

a) pseudoscalar coupling:

\[ H_g=f_1\frac{\hbar}{2Mc}\left[(\boldsymbol{\sigma}\nabla)\varphi \tau_{pN}+(\boldsymbol{\sigma}\nabla)\varphi^*\tau_{Np}\right], \]

\[ H_{eg}=-\frac{f_1 e}{2Mc^2}\, i(\boldsymbol{\sigma}\mathbf{A})\left[\varphi\tau_{pN}-\varphi^*\tau_{Np}\right]; \]

b) pseudovector coupling:

\[ H_g=-f_2\frac{\hbar}{\mu c}\left\{\left[(\boldsymbol{\sigma}\nabla)\varphi+ \frac{(\mathbf{s}\mathbf{p})\pi^*+\pi^*(\boldsymbol{\sigma}\mathbf{p})}{2Mc^2}\right]\tau_{pN}+\text{conj.}\right\}, \]

\[ H_{eg}=\frac{f_2 e}{\mu c^2}(\boldsymbol{\sigma}\mathbf{A})\left\{ i\left[\varphi+\frac{\hbar}{i}\frac{\pi^*}{2Mc^3}\right]\tau_{pN}+\text{conj.}\right\}. \]

3) Vector theory:

a) vector coupling:

\[ H_g=g_1\left\{\left[-\frac{\hbar^3}{\mu^2c}\operatorname{div}\boldsymbol{\pi}^* -\frac{(\boldsymbol{\sigma}\mathbf{p})(\boldsymbol{\sigma}\boldsymbol{\varphi})+(\boldsymbol{\sigma}\boldsymbol{\varphi})(\boldsymbol{\sigma}\mathbf{p})}{2Mc}\right]\tau_{pN}+\text{conj.}\right\}, \]

\[ H_{eg}=e g_1\left\{\left[\frac{(\boldsymbol{\sigma}\mathbf{A})(\boldsymbol{\sigma}\boldsymbol{\varphi})}{2Mc^2} -\frac{i\hbar}{\mu^2c^3}(\mathbf{A}\boldsymbol{\pi}^*)\right]\tau_{pN}+\text{conj.}\right\}; \]

b) tensor coupling:

\[ H_g=-g_2\frac{\hbar}{\mu c}\left\{\left[(\boldsymbol{\sigma}\operatorname{rot}\boldsymbol{\varphi}) +i\,\frac{(\boldsymbol{\sigma}\boldsymbol{\pi}^*)(\boldsymbol{\sigma}\mathbf{p})-(\boldsymbol{\sigma}\mathbf{p})(\boldsymbol{\sigma}\boldsymbol{\pi}^*)}{2M}\right]\tau_{pN}+\text{conj.}\right\}, \]

\[ H_{eg}=\frac{g_2 e}{\mu c^3}\left\{ i\left[(\boldsymbol{\sigma}[\mathbf{A}\boldsymbol{\varphi}]) -\frac{\hbar}{2M}(\boldsymbol{\sigma}\mathbf{A})(\boldsymbol{\sigma}\boldsymbol{\pi}^*)\right]\tau_{pN}+\text{conj.}\right\}. \]

Here \(\boldsymbol{\sigma}\) is the Pauli matrix, \(\boldsymbol{\varphi}\) is the meson field, \(\boldsymbol{\pi}\) is the field canonically conjugate to \(\boldsymbol{\varphi}\), \(\mathbf{A}\) is the vector potential of the electromagnetic field; for the remaining notation see above.

In deriving these formulas, the electromagnetic interaction with the anomalous magnetic moment of the nucleon was not taken into account. If this interaction is added to the relativistic Hamiltonian,

then, as a result of the transition to the Pauli equation, additional expressions are obtained in the formulas for \(H_{eg}\).

It is interesting to note that in the Pauli approximation the equivalence of the two types of coupling of a pseudoscalar field with a nucleon is especially clearly visible. At the same time, as was to be expected, the terms proportional to \(g^2\) that are discarded in the transition do not exhibit equivalence.

In carrying out relativistic calculations, recently the Feynman-Dyson method\(^{23,24}\) has been used almost exclusively; it considerably simplifies the calculations.

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