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LETTERS TO THE EDITOR
WHEN DOES A WEAK INTERACTION BECOME STRONG?
In my article “On Nonlocal and Nonlinear Field Theories,” published in the second issue of Uspekhi Fizicheskikh Nauk for 1957, the concept of a strong interaction was considered[^1]. In doing so, by a strong interaction was understood an interaction such that, during the collision time of the particles, their energy is concentrated mainly in their interaction energy, and not in their own kinetic energy.
On the basis of this criterion a number of examples were considered, in particular the electromagnetic interactions of electrons. However, the weak interaction of electrons involving $\mu$-mesons and neutrinos was not considered.
It turns out that this interaction can become strong in a definite higher sense.
Below a proof of this assertion is given, and this letter is thus an addition to the chapter of my article devoted to the physics of the strong interaction.
Let us consider the process of interaction of a neutrino and an electron with the conversion of the electron into a $\mu$-meson:
$$ \nu + e \to \mu + \nu'. \tag{1} $$
This is a peculiar “combinational” scattering of a neutrino by an electron. The energy density, to within order of magnitude, in this case is equal to
$$ W = g^{*}\psi_{e}^{+}\psi_{\mu}\psi_{\nu}\psi_{\nu'} \tag{2} $$
where $g^{*}$ is the Fermi constant, and $\psi_{e}, \psi_{\mu}, \psi_{\nu}$ are the wave functions of the electron, $\mu$-meson, and neutrino, respectively. The quantity $g^{*}$ may be written in the form
$$ g^{*}/\hbar c = \Lambda_{0}^{2}, \tag{3} $$
where $\Lambda_{0}$ is a certain length of order $\cong 10^{-16}\ \text{cm}$ (I. S. Shapiro[^2] drew attention to the possible value of this length in connection with nonconservation of parity). The density of kinetic energy, for example for electrons, will be
$$ \varepsilon_{e}=\bar{\psi}_{e}D\psi_{e}, \tag{4} $$
where $D=c\alpha p+\beta mc^{2}$ is the Dirac Hamiltonian. Hence, in order of magnitude,
$$ \bar{\psi}_{e}\psi_{e}\cong \frac{\varepsilon_{e}l}{\hbar c}, \tag{5} $$
where $l$ is the characteristic scale of the spatial region determining the magnitude of the gradients, so that $\frac{1}{c}\frac{\partial}{\partial t},\ \frac{\partial}{\partial x}\cong 1/l$. Consequently, the order of magnitude of $W$ will be
$$ W=\frac{g^{*}l^{2}}{\hbar^{3}c^{2}}\varepsilon_{e}^{1/2}\varepsilon_{\mu}^{1/2}\varepsilon_{\nu}. \tag{6} $$
Putting $\varepsilon_{e}=\alpha\varepsilon,\ \varepsilon_{\mu}=\beta\varepsilon,\ \varepsilon_{\nu}=\gamma\varepsilon$, where $\varepsilon$ is the total energy density, we find:
$$ \varepsilon \cong \varepsilon(\alpha+\beta+\gamma)+\frac{g^{*}l^{2}}{\hbar^{3}c^{2}}\varepsilon^{2}\alpha^{1/2}\beta^{1/2}\gamma. \tag{7} $$
In accordance with the definition, the interaction will be strong if, for
\(\alpha+\beta+\gamma \ll 1\) \((\alpha,\beta,\gamma>0)\),
\[ W=\frac{g^{*}l^{3}}{\hbar^{2}c^{2}}\left(\varepsilon^{2}a^{1/2}\beta^{1/2}\gamma\right)\simeq \varepsilon, \]
i.e.
\[ \varepsilon>\frac{\hbar^{2}c^{2}}{g^{*}l^{2}}=\frac{\hbar c}{\Lambda_{0}^{2}l^{2}}=\varepsilon_{\mathrm{cr}}. \tag{8} \]
Let us now consider a neutrino packet (in the center-of-mass system of the electron and the neutrino) with characteristic wavelength \(\bar{\lambda}\) and transverse dimensions \(a>\bar{\lambda}\), incident on an electron. The energy density \(\varepsilon\) in this case will be
\[ \varepsilon=\frac{\hbar\omega}{\bar{\lambda}a^{2}}=\frac{\hbar c}{\bar{\lambda}^{2}a^{2}}. \tag{9} \]
Further, \(l\simeq \bar{\lambda}\). Condition (8) now gives \(a^{2}<\Lambda_{0}^{2}\); since \(a>\bar{\lambda}\), the strong interaction of the electron and the neutrino sets in when
\[ \bar{\lambda}<\Lambda_{0}. \tag{10} \]
A direct calculation shows that the cross section for the process under consideration
\(\nu+e\to\mu+\nu'\) is, in order of magnitude,
\[ \sigma\simeq \Lambda_{0}^{2}\frac{\lambda_{0}^{2}}{\bar{\lambda}^{2}} \tag{11} \]
and, as is evident, becomes greater than \(\pi\bar{\lambda}^{2}\) for \(\bar{\lambda}<\Lambda_{0}\). In this connection one should expect that, at wavelengths of order \(\Lambda_{0}\), other effects may also occur which substantially change the electromagnetic interaction of electrons. It is precisely at small distances between electrons that an interaction arises which will lead to mutual scattering of electrons by means of the following process: first one of the electrons emits a pair, neutrino (or neutrino and antineutrino), and turns into a \(\mu\)-meson. The second electron absorbs these neutrinos and also turns into another \(\mu\)-meson. Then this meson emits neutrinos, which are absorbed by the first meson. As a result, two scattered electrons arise.
These same processes lead to a smearing of the electron charge, i.e. to the appearance of a “form factor” of the electron. This “form factor” will substantially change the Compton effect on an electron at high photon energies and the electromagnetic interaction of electrons. The origin of such smearing is easy to see from the fact that, in addition to the direct absorption and emission of real or virtual photons by an electron, absorption and emission of them by a \(\mu\)-meson is also possible, the \(\mu\)-meson arising in the temporary dissociation of an electron into a \(\mu\)-meson and a pair of neutrinos.
The situation is analogous to the emergence of a \(\pi\)-meson cloud around nucleons. This analogy, however, is incomplete, since in the case of the \(\pi\)-meson cloud its scales are determined by the Compton wavelength of the \(\pi\)-meson, whereas in the case of the electron the essential length is \(\Lambda_{0}\), and not the Compton wavelength of the \(\mu\)-meson.
The effects indicated here are again substantial at wavelengths of real or virtual photons close to \(\Lambda_{0}\).
In conclusion, one remark on the role of weak interactions of the type
\(p\to n+e^{+}+\nu\) in collisions of nucleons. As was noted in my paper, this interaction does not become strong at any nucleon energies.
It was assumed there that the energy of nucleons in the center-of-mass system is distributed in the volume of an ellipsoid
\[ V\simeq l_{0}^{3}\sqrt{\frac{Mc^{2}}{E}}, \]
where \(l_{0}\) is the Compton wavelength of \(\mu\)-mesons \(\left(\frac{\hbar}{\mu c}\right)\) or, possibly, of nucleons \(\left(\frac{\hbar}{Mc}\right)\); \(E\) is the nucleon energy in the laboratory coordinate system.
If, however, one assumes that the energy of a nucleon can be concentrated in an arbitrarily small region, then at nucleon wavelengths \(\bar{\lambda}<\Lambda_{0}\) (in the center-of-mass system) the weak interaction will become substantial.
This can be shown by arguments similar to those given above for the neutrino and the electron. The same is also seen directly from the theory of paired \(\beta\)-forces of Tamm—Ivanenko\(^3\).
The expression for the potential of these forces is
\[ V=\frac{1}{(2\pi)^{3}}\left(\frac{\Lambda_{0}}{R}\right)^{5}\frac{\hbar c}{\Lambda_{0}}, \tag{12} \]
where \(R\) is the distance between nucleons. For \(R<\Lambda_0\),
\[ V=\frac{\hbar c}{\Lambda_0}\gg Mc^2 . \]
In this case the nucleon is assumed to be pointlike. Thus, the estimate of the magnitude of weak interactions in nucleon collisions depends essentially on the reliability of the assumption that the self-energy of a resting nucleon is distributed in a volume no smaller than \(\left(\frac{\hbar}{Mc}\right)^3\). The theory of meson generation in collisions of energetic nucleons confirms this latter assumption\(^4\).
D. I. Blokhintsev
References Cited
- D. I. Blokhintsev, UFN 61, 137 (1957).
- I. S. Shapiro, UFN 61, 313 (1957).
- I. E. Tamm and D. D. Ivanenko, Nature 133, 981 (1934).
- S. Z. Belenkii, L. D. Landau, UFN 56, 309 (1955).