(Received July 31, 1948)
The most attractive feature of Yukawa's theory of mesons is that its basis ideas are in good agreement with experiments, especially in three striking respects: the range of the two-nucleon potential and the mass of cosmic-ray charged mesons; the anomalous magnetic moment of a nucleon; and the nuclear β-decay and the β-decay of cosmic-ray cahrged mesons.
It was found however by the quantitative computation that the theoretical results involve grave difficulties: the magnetic moment of a nucleon and the two-nucleon potential are represented by divergent integrals; the observed lifetime of cosmic-ray mesons is 100 times longer than the theoretical one. Several ways of eliminating the latter defect were suggested but it seems that another way is necessary in order to agree with the recent sutdies of cosmic-ray phenomena. The search for remedies to the former defects has been tried by many authors but in vain.
There are rwo ways of the approach to the problems: the strong and weak coupling theories. According to the strong coupling theory with the extended source model, the magnetic moments of the proton and the neutron are approximately equal and of opposite sign, in contradiction to experience. The weak coupling theory with a λ-limiting device gives the proton a negative surplus magnetic moment and the neutron a positive moment both in the pseudoscalar and vector theories. These results are also in disagreement with the observed fact.
One of the divergent terms in the two-nucleon potential is a sort of direct interaction. It can be eliminated in a relativistically invariant way by adding suitable terms to the Lagrangian.
The divergent term of another kind is involved further in the spherically unsymmetric part of the potential given by the vector, the pseudovector, and the pseudoscalar theories. The above mentioned device can not remove such a divergent terms. This point should also be considered as one of grave difficulties of the meson theory though it has not yet been generally noticed. The scalar theory is only an exception. But this theory should be ruled out becouse it gives a repulsive potential to a deuteron system in addition to the other defect.
Such a divergent term has been usually omitted without reason. However there is an artificial device for liminating the devergent part where a strongly decreasing factor is multiplied to the integrand of the momentum integral and after the integration a limiting value of the integral is adopted such that the multiplier approaches to unity. By this device we can eliminate the divergent part in question, but we encounter the so-called r-3 difficulty as its result. Namely, a r-3 term in the spherically unsymmetric part of the potential admits the two-nucleon system no stable state.
In order to remove this difficulty Møller and Rosenfeld mixed the pseudoscalar and vector meson fields. They sliminated the spherically unsymmetric part altogether by assuming the same coupling constant and the same mass for two mesons. Unfortunately this mixture could not account for the electric quadrupole moment of deuteron in contradiction to their expectation.
Schwinger altered Møller-Rosenfeld's mixture so that the spherically unsymmetric part could be reserved and the r-3 term could be eliminated by assuming the mass of vector mesons larger than that of pseudoscalar ones. This assumption could be in accord with the sign of the quadrupole moment of deuteron, but it was not yet sufficient to account for its value. Moreover this mixture still involved the divergent term in the magnetic moment of nucleons.
Recently Tanikawa has proposed another way to eliminate the r-3 term from the two-nucleon potetntial without mixing fields. The above mentioned divergent part of the potential includes a function of the form -C/r-3 where C is a r-independent divergent integral. Tanikawa separates -r-3 from this divergent part and he adds it to the finite part. Then the divergent part takes the from -(C-1)/r-3. If this part is omitted according to Tanikawa (on account of some unknown reason) the finite part involves no r-3 term. However this theory never inprove the situation for the mangetic moment.
We have thus two unsolved foundmental difficulties which are directly related with experience: the divergencies of the magnetic moment of nucleons and the two-nucleon potential. The purpose of the present paper is to show that we can eliminate these difficulties altogether by assuming a mixture of the pseudoscalar and pseudovector fields.
We shall consider a mixture of two meson fields and assume the meson of spin 0 as one component in the Part I of the present paper. In order to account for the elelctric quadrupole moment of deuteron this component has to be pseudoscalar. We have therefore only two possibilities: pseudoscalar-vector or pseudoscalar-pseudovector mixtures. We have then to assume the mass of spin 1 mesons larger than pseudoscalar mesons in accordance with the sign of the quadrupole moment. As is mentioned above, the former mixture can not exclude the divergent part from the magnetic moment of a nucleon whereas the latter can give a finite value with a correct sign to the mangetic moment, as will be seen in the following. Schwinger's mixture should hence be ruled out. The pseudoscalar-pseudvector mixture eliminates further the divergent term and the r-3 term from the two-nucleon potential, as is in the case of Schwinger's.
In the case of pseudoscalar-pseudovector mixture the spherically symmetric part of the two-nucleon potential is repulsive for a small distance and is attractive for a large distance between two nucleons in the even state of the two-nucleon system. Consequently it has a shallow minimum. The present mixture differs from Schwinger's in this respect. This behaviour of the two-nucleon potential resembles to that of Bethe's neutral theory in the ground state of deuteron.
An essential difference between Bethe's and Schwinger's potentials is that the spherically tymmetric part is repulsive for hte ground state of deuteron in the former whereas it is attractive in the latter. This difference is the reason why Bethe's result is in good agreement with experience whereas Schwinger's is not fairly. Therefore the above mentioned feature of the potential of the present mixture provides us a possibilty of a fairly good agreement of the theory with experience.
The energy spectrum of the nuclear β-decay in the present theory is somewhat different from the result of the pseudoscalar theory. The theoretical lifetime of the β-decay of pseudoscalar mesons is nearly equal to that of cosmic-ray mesons.
The λ-limiting device changes the sign of the surplus magnetic moment of nucleons derived according to the perturbation theory. We shall see in the following that the surplus moment given by the pseudovector theory is of opposite sign to experience. Accordingly the pseudovector theory combined with the λ-limiting device gives a finite value with a correct sign to the surplus mangetic moment of nucleons. But the sign of the quadrupole moment of deuteron is incorrect in this case. Therefore this case should still be ruled out.
It is to be noted here that Pais and Sakata and Hara considered the mixed theory of photons and scalar mesons, that they obtained a finite value of the self-energy of charged Dirac's particles due to this mixed field, and that they could give an accout for the difference between masses of nucleons. According to Watanabe the scalar-vector and pseudoscalar-pseudovector mixtures are only two admissible forms from the standpoint of his five dimensional formalism.
It should be also stressed here that the present settlement is of provisory character, as will be easily seen in the following discussion: the divergency is eliminated only in the non-relativistic approximation and the general difficulty still remains without the fundamental solution.
URL : http://ptp.ipap.jp/link?PTP/3/422/
DOI : 10.1143/PTP.3.422