Pseudovector Coupling Research Articles

On the Exchange Magnetic Moment of the Two Nucleon System

The two body exchange magnetic moment operators are calculated in the pseudoscalar meson theory with non-relativistic pseudovector coupling. The method of canonical transformation is used and results are obtained to the order of eg4 in the power series expansion. The radial parts of all possible operators in the static approximation are given. The contributions of the exchange moment to the magnetic moment of the deuteron are calculated and it is found that in order to adjust the experimental moment µr=0.74 should be taken as the lower limit in the radial integral.

On the Recoil Correction to Adiabatic Nuclear Potential

Recoil correction to adiabatic nuclear potential in the symmetrical pseudoscalar meson theory with pseudovector coupling is calculated up to the first order in the nucleon velocities by means of the F-S.T formalism!) which is formulated by taking into account the normalization condition for the TammDancoff amplitude. Divergences which appear in the course of the calculations are subtracted by a sort of non-covariant renormalization procedure. Resultant correction consists of two parts, spin-orbit coupling part and velocity-independent part. The former is found to be of the correct sign and suitable magnitude as required by the theory of nuclear shell model.

Recoil Effect and Spin-Orbit Coupling

As is well known, the Tamm·Dancoff method has the difficulty of non-hermitic potential when applied to the derivation of the non-adiabatic cor· rectwn to the adiabatic potenttal. This difficulty is due to the fact that the Tamm-Dancofr wave function does not satisfy the normalization condition. Indeed the Tamm-Dancofr potential can hardly be called a potential in the ordinary sense of the word. It is not admissible to eliminate the energy dependence from the Tamm-Dancofr potential. In order to avoid this difficulty, Fukuda, Sawada and Taketanil) have given the general formalism for the construction of potential in the field theory, where the special attention has been paid to the normalization condition of the state vector. In this note, we shall derive according to their idea the first order velocity dependent correction to the adiabatic potential in the pseudoscalar meson theory with pseudovector coupling. Although it is possible in the F.S.T. theory to derive the potential in a way in which the perturbation approximation may be improved in some sense by treating the probability operator in !he fractional form, we have ;nade use of the usual perturbation expansion as a first step. Our calculation has been carried out only for the potential with velocity dependence specially important for the shell model. We can also obtain easily the static type corrections, which, however, will be discussed in a subsequent paper. The potential in the F.S.T. theory is given by

THE ANOMALOUS MAGNETIC MOMENT OF NUCLEONS

The nucleon magnetic moments and the neutron-electron interaction potential have been calculated using the theory of pseudo-scalar mesons with pseudo-vector coupling. The divergence in the neutron-electron interaction is removed after the vacuum polarization effect due to virtual mesons has been taken into account. The final result is the same as in the case of pseudo-scalar coupling.

Upper Bound of the Pseudoscalar Coupling Constant in Beta-decay

The [9-spectrum of RaE was analysed byPetschek and Marshak1) last year and it was conduded that P must be included among the other interactions. (We abbreviate the five interaction types in the Fermi theory of [9-decay as S, V, '!, A and P hereafter.) Though it was pointed out recently by one of the authors that this conclusion is not necessarily correct, P still seems necessary if the spin of RaE is zero.2) However, according to the relation of Ahrens, Feenberg and Primakoff3) the nuclear matrix element LBru in P is so small that we must take G p, the coupling constant of P, very large (-I G pi GTI 133) to make P match to T. Recently it was shown by Ruderman4) that, if nuclear force is mainly [9ru-type, J [91'u is markedly large, and consequently G p need not be so large as expected (I G pi-I GTI) to explain the [9-decay of RaE. However, if nuclear force arises chiefly from pseudovector coupling between 7l-mesons and nucleons, G p should still re~ain large. According to the latest 7l-meson theory in which the damping effect is taken into account,'-) it is very doubtful whether a large [9ru-type nuclear force exists, even if the coupling between 7l-mesons and nucleons is pseudoscalar one; though the conclusion is not to be taken too seriously, in view of many ambiguities in the present 7l-meson theory. We feel the largeness of G p not very agreeable aesthetically, and it seems probable thatGp is of the same order as GT • However, we should discuss this problem on the firmer foundations. The [9-spectrum of -RaE will be analysed by Takebe6) in the case of large G p taking into account another terms pointed out by Ahrens, Feenberg and Primakoff.3) In this paper, however, we shall find the upper bound of Gp from another viewpoint, i.e. from the fact that it very large Gp destroys the spectrum shapes of high energy allowed transitions. (He6 and B12) •

Damping Corrections in the Photo-Meson Process

An examination is made of the effect of damping on the photo-production of pseudoscalar charged and neutral π-mesons from protons, ignoring all self-energy effects. In the case of pseudoscalar coupling the damping corrections are large, but do not greatly remedy the disagreement of second order perturbation theory with experiment; while for pseudovector coupling damping effects are unimportant.

Symmetrical Pseudoscalar Meson Theory of Nuclear Forces

Nuclear forces yielded by the symmetrical pseudoscalar theory are discussed in terms of a perturbation expansion. It is shown that, up to terms in the square of the coupling constant, the pseudoscalar coupling is equivalent to a scalar pair coupling of the pseudoscalar field plus a pseudovector coupling of that field. The dominant contribution to the fourth-order nucleon potential is then obtained in a simple way by using this result.

Equivalence Theorems for Pseudoscalar Coupling

By a unitary transformation a rigorous equivalence theorem is established for the pseudoscalar coupling of pseudoscalar mesons (neutral and charged) to a second-quantized nucleon field. By the transformation the linear pseudoscalar coupling is eliminated in favor of a nonlinear pseudovector coupling term together with other terms. Among these is a term corresponding to a variation of the effective rest mass of the nucleons with position through its dependence on the meson potentials. The question of the connection of the nonlinear pseudovector coupling with heuristic proposals that such a coupling may account for the saturation of nuclear forces and the independence of single nucleon motions in nuclei is briefly discussed. The new representation of the Hamiltonian may have particular value in constructing a strong coupling theory of pseudoscalar coupled meson fields. Some theorems on a class of unitary transformations of which the present transformation is an example are stated and proved in an appendix.

Meson-Nucleon Scattering and Nucleon Isobars

The scattering (including charge exchange) of ${\ensuremath{\pi}}^{\ensuremath{-}}$ mesons in hydrogen rises from 18 millibarns at 60 Mev to a broad plateau of about 60 millibarns at 200 Mev, and is smaller than the ${\ensuremath{\pi}}^{+}$ scattering at 60 Mev in the ratio of 0.63\ifmmode\pm\else\textpm\fi{}0.09. The general features of the ${\ensuremath{\pi}}^{\ensuremath{-}}$ scattering, except for the high energy plateau, are given qualitatively by pseudoscalar theory with pseudovector coupling in the weak coupling limit; the ratio of ${\ensuremath{\pi}}^{\ensuremath{-}}$ to ${\ensuremath{\pi}}^{+}$ scattering predicted by this theory in the weak coupling limit is, however, 1.67, which is much higher than the experimental result. A phenomenological theory of the scattering is developed by using the methods of Wigner and Eisenbud and imposing the restrictions of charge symmetry. By using the qualitative assignment of the resonance levels parameters as given by weak and strong coupling theory, satisfactory agreement with experiment is obtained. It is concluded that the apparently anomalous features of the scattering can be interpreted to be an indication of a resonant meson-nucleon interaction corresponding to a nucleon isobar with spin $\frac{3}{2}$, isotopic spin $\frac{3}{2}$, and with an excitation of 277 Mev.

Remarks on the Adiabatic Nuclear Potential

We discuss on the validity and approximation method of the adiabatic nuclear potentials, especially of the fourth order nuclear potential derived from psendoscalar meson theory6l. In § 2, it is shown that effects of time component of pseudovector coupling on the pseudoscalar meson potential are not large when distance between two nucleons is larger than about half of the force range. In § 3, time variation of nucleon spin and or-spin are evaluated for phenomenological and meson potentials, and critical radii for their applicability are detived. In § 4, implications of the velocity.dependent forces in the interior region are discussed, and largeness of the purely quantum mechanical effects in the meson theory is pointed out. Hamiltonian for a two nucleon system in the intermediate coupling approximation is given. In § 5, features of the meson-nucleon interaction are compated with that of the photon-electron interaction. It is pointed out that the transition matrix of the nucleon-nucleon collision is not analytic with respect to nucleon energy at the threshold for meson production. An ambiguity occurs when one intend to calculate the sixth order nuclear potential_

On the Mesonic Correction to the -Decay

General types of the mesonic correction to the 8-decay are obtained from the invariance requirements. Explicit calculations are performed, using Feynman-Dyson method, up to the second order of meson-nucleon coupling constants for the syn'lmetrical pseudoscalar meson theory with both pseudoscalar and pseudovector couplings. It is shown that the mesonic correction is small and does not affect the selection rules and spectra in the Konopinski's forbidden theory.

Some Notes on Multiple-Boson Processes

Methods of calculation with nonlinear functions of quantized boson fields are developed during the discussion of two problems involving multiple boson processes. In the first of these a simple treatment is given of the multiple radiation of photons by classical current distributions, a special case of which, in effect, is the infrared catastrophe. In the second illustration, generalizations of the scalar and pseudoscalar meson theories are considered in which the interaction hamiltonian depends exponentially on the meson field. In the pseudoscalar case such hamiltonians are closely related to the familiar form of pseudovector coupling. Assuming the over-all coupling of the nucleon and meson fields to be weak, calculations are made of the nuclear forces, and of the multiple production of mesons in meson-nucleon and in nucleon-nucleon collisions. In the latter events statistical independence of meson emissions is found to prevail.

Production ofπ-Mesons in Nucleon-Nucleon Collisions

The production of $\ensuremath{\pi}$-mesons in nucleon-nucleon collisions as predicted by scalar meson theory, pseudoscalar meson theory with pseudoscalar and pseudovector coupling, and vector theory with vector coupling is compared with the experimental results of the workers at Berkeley. The calculations are made on the basis of third-order perturbation theory using the methods of Feynman and Dyson and also using a phenomenological treatment of the nucleon-nucleon interaction. Account is taken of the fact that the final nucleons are not in plane wave states. It is shown that pseudoscalar theory with pseudovector coupling gives qualitative agreement with experiment.

On the Equivalence of Pseudoscalar and Pseudovector Coupling Constants in Pseudoscalar Meson Theory

On the Equivalence of Pseudoscalar and Pseudovector Coupling Constants in Pseudoscalar Meson Theory

On the Equivalence of Pseudoscalar and Pseudovector Coupling Constants in Pseudoscalar Meson Theory

On the Equivalence of Pseudoscalar and Pseudovector Coupling Constants in Pseudoscalar Meson Theory

On the γ-Decay of Neutral Meson

Recently Tomonaga and Schwinger have independently developed a covariant formulation of quantumelectrodynamics (super-many-time theory), and have successfully applied it to the explanation of the Lamb shift in the hydrogen atom and the anomalous magnetic moment of the electron. They have shown that, although the present theory of fields, in general, gives infinte answers to such field reaction problems, since it cannot be formulated in a Lorentz- and gauge-covariant way without introduction of the singular delta function of Jordan and Pauli, it is nevertheless possible to avaid the divergences by amalgamating them into the mass and charge, and that the remaining finite term can well account for the experimental facts. But there remains the question, whether or not such finite term, as is separated from infinity, can be free from any ambiguity arising from the pathological character of delta function of Jordan and Pauli. One typical example of the appearance of such an ambiguity is the photon self energy. As first pointed out by Schwinger, the photon self energy should be zero from the gauge covariant point of view, while, completely against Schwinger's prediction, the recent calculation by Wentzel leads to a finite but non-gauge covariant result for the photon self energy. It is evident that this inconsistent result comes from the mathematical difficulty of obtaining a definite expression for the singularity of the light cone of Jordan-Pauli's delta function. A very similar situation is also encountered in the γ decay of neutral meson. By using the method of evaluation which has been applied by Schwinger to the calculation of the anomalous magnetic moment of the electron, we have obtained the convergent but non-gauge covariant result for the γ decay of neutral meson. Further our result is inconsistent with the recent discussions of Dyson, Sawada and Nanbu on the equivalence between the pseudoscalar and pseudovector couplings of the pseudoscalar meson field. Thus, in the present state of the field theory, we cannot give an unambiguous life-time for neutral meson. It is very desirable to find some general prescription which enables one to get the definite answer to field reaction problams.

Equivalence Theorems for Meson-Nucleon Couplings

The equivalence of pseudoscalar and pseudovector couplings of a pseudoscalar meson field with a nucleon field is examined. It is found that to the second order in the coupling constants the two couplings are equivalent except for two terms. One of these extra terms is predominantly an additional self-energy. The other occurs only for charged mesons and modifies the electromagnetic properties of a meson-nucleon system and the scattering of charged mesons. To this order (${g}^{2}$) it does not contribute to nuclear forces, Similarly, it is shown that the vector coupling of scalar mesons to nucleons can be replaced (to ${g}^{2}$) by a term identical with the last of the above-mentioned terms. These results show that the delta-function interactions which have been found in the past are actually not present.

On the Meson Theory of Nuclear Forces

It has generally been believed that the vector, pseuclovcctor, and pseudoscalar meson theories of nuclear forces involve a 1'-3 difficulty. Mixtures of two fields were considered first by M~ller and Rosenfeld(1) .and later by SchwingerC2l in order to eliminate such a term. Further it was noticed by the present authorCa) that they involve also a divergence difficulty. and another mixture was suggested by him. This mixture could also eliminate a divergent part from the magnetic moment of nucleons. However, if we examine the procedure of deriving the nuclear force we can find that the above mentioned difficulties are due to the inadequacy of the method of the derivation, and that they have no direct connection with the general defect of the contemporary field theory. For example, the pseudoscalar mesons interact with nucleons by the pseudovector and pseudoscalar couplings. As we shall see later on, the pseudovector coupling is equivalent, except for a factor 2M/ p., to the pseudoscalar one in a nonrelativistic approximation. If we make use of this relation we can get a non-relativistic nuclear force which is free from the above mentioned difficultie.li. It was first suggested by Nelson(1) that the pseuc1ovector coupling includes a term of a pscudoscalar-coupling type. Later, using a unitary transformation, the latter was derived from the former by Dyson.(o) One may imagine that this relation can not be applied to the cases of vector and pseudovector meson theories because the couplings are different in these cases. But the term in question comes from the matrix elemcnt of the same type in all cases of these three theories. And we can find that the above mentioned result still holds in the vector and pseudovcctor theories.

The Charge Density and Magnetic Moments of the Nucleons

Using pseudoscalar meson theory, the behavior of a single-nucleon system in an electric field is studied. From the electrostatic interaction, the interaction between neutrons and electrons is computed. From the spin-orbit interaction, the magnetic moments of neutron and proton are computed. The calculation is carried out relativistically, with nucleon and pseudoscalar meson fields both subjected to second quantization. The Hamiltonian of the nucleon-meson system is diagonalized to second order in the coupling parameters by two canonical transformations, and transitions induced by the electric field between single-nucleon states are investigated. For pseudoscalar coupling of 282 e.m. mesons, estimating the coupling constants for charge-symmetric theory from the observed singlet neutron-proton scattering length (using static nuclear forces), it is found that the volume integral of the neutron-electron interaction potential is about $\ensuremath{-}14 \mathrm{kev}\ifmmode\times\else\texttimes\fi{}(\frac{4\ensuremath{\pi}}{3}){(\frac{{e}^{2}}{\mathrm{m}{\mathrm{c}}^{2}})}^{3}$, and that ${\ensuremath{\mu}}_{P}$ and ${\ensuremath{\mu}}_{N}$ are about 1.7 and -5 nuclear magnetons respectively. Results are also given for pure charged theory. Pseudovector coupling is compared to pseudoscalar coupling, and is found to give the same magnetic moments, but a logarithmically divergent neutron-electron interaction. The influence of the contact interaction term is discussed.The results are compared with those of methods and with experimental values, and the neutron-electron interaction is computed approximately for several types of mesons with an improved non-relativistic method. Charge renormalization and the approximate distribution of the charge cloud around a neutron are discussed. The pseudoscalar meson contribution to the Lamb Shift of the $2S$ hydrogen level is estimated as +0.08 Mc.