Mesons and Nuclear Forces

H. A. Bethe*)

The history of the solution of the problem of mesons and nuclear forces is, for contemporary scientific progress, a characteristic example both of the insight and of the naïveté of physicists.

The birth of the theory of nuclear forces dates to 1932 and is connected with the discovery of the neutron. This discovery made possible the development of consistent ideas about the structure of the atomic nucleus, according to which the nucleus consists of protons and neutrons, held together by very large forces, different from any other forces previously known in nature and exceeding them in magnitude. Only three years after the discovery of the neutron and the emergence of the theory of the nucleus, Yukawa proposed that the interaction between nuclear particles—the proton and the neutron—is due to the existence of other, at that time still undiscovered particles, now known as mesons. Yukawa predicted that such particles should exist, that their mass should be equal to 100–200 electron masses, that they should be charged and have an integral spin, probably equal to zero or one.

Three years later Yukawa’s prediction was confirmed. In cosmic radiation, two groups of physicists—Anderson and Neddermeyer at the California Institute of Technology, and Street and Stevenson at Harvard University—discovered new particles. The mass of these particles is close to 200 electron masses; they have a positive or negative charge, which agreed exactly with Yukawa’s expectations, and it seemed that they satisfied all his predictions rather well. In the following nine years experimenters discovered more and more new properties of these particles, while theorists calculated what their role in nuclear forces is. But both groups of physicists did not arrive at a co—

*) Text of the 22nd J. Henry Lecture, delivered on April 22, 1953.
Bethe H. A., J. Wash. Acad. Sci. 44, No. 4, 97–105, 1954.

in short: the theorists’ conclusions were completely at variance with the actual properties of these particles. According to the theorists’ predictions, mesons, after they had arisen, ought to interact very strongly with nuclei, i.e. they ought to be strongly absorbed, strongly scattered, and readily cause nuclear reactions. None of this was in fact characteristic of them. They did essentially nothing to nuclei. They moved through matter, slowed down like any other charged particles, and finally stopped and decayed in some way then still unknown. Nevertheless the theorists persisted in their belief in a connection between the Anderson particles of cosmic radiation and Yukawa’s predictions. However, the difference between the experimental results and the theoretical predictions was so great that it led to the appearance of different names for the particles. The experimentally observed particles came to be called mesotrons, whereas the particles predicted by the theory received the name mesons. Attempts to identify the two particles continued until, in 1947, the Italian physicists Conversi, Piccioni, and Pancini discovered that the mesons of cosmic radiation interact with nuclei even more weakly than had previously been suspected. Even when a meson was given the opportunity to remain close to a nucleus for a very long time (about a microsecond), it did not wish to take advantage of the favorable occasion and did not allow the nucleus to capture it. This showed conclusively that the two particles could not be identical. Several months later the solution to the problem was found by Occhialini, Powell, and Lattes, working in Bristol (England). They discovered the existence of another particle, which they named the \(\pi\)-meson (“primary” meson), which in a short time decays into a meson that had been discovered earlier and which they named the \(\mu\)-meson (“intermediate” meson).

The discovery of the \(\pi\)-meson confirmed Yukawa’s considerations. The \(\pi\)-meson interacts strongly with nuclei. After its formation it is readily scattered and readily absorbed by nuclei. Since 1947 we have regarded the \(\pi\)-meson as indeed the particle responsible for nuclear forces. Many properties of \(\pi\)-mesons were discovered by the Bristol group, which worked with cosmic-ray mesons and with photographic plates, but the greatest successes in the investigation of the properties of these particles were achieved after they had been produced artificially in accelerators, first at Berkeley and then in other laboratories. This was an example of the joint work of investigators from several countries: the powerful synchrocyclotron was located at Berkeley, but the Berkeley physicists were unable to detect the artificially produced \(\pi\)-meson until the Brazilian Lattes brought them from England the complex technique of photographic emulsions. Since then very much has been learned about \(\pi\)-mesons;

this will constitute the content of my lecture. However, before turning to the main topic, it is necessary to consider certain properties of $\pi$-mesons predicted by nuclear physics on the basis of purely theoretical considerations.

PREDICTIONS OF NUCLEAR PHYSICS

The first prediction of nuclear physics was that there must exist three types of mesons responsible for nuclear forces: positively charged, negatively charged, and neutral mesons. The positively and negatively charged varieties of mesons were easily discovered. Until 1949 the neutral particles were only a theoretical prediction, based on a very fundamental property of nuclear forces, namely their so-called charge independence. In 1935, in the department of terrestrial magnetism of the Carnegie Institution in Washington, it was discovered that the forces between two protons are exactly equal to the forces between a proton and a neutron, provided only that both pairs of particles are in one and the same state of motion. This fact was subsequently confirmed by many other experimental data and theoretical works of Breit and others, and is known as the theorem of charge independence. If only charged mesons existed, then only the following process predicted by Yukawa would be possible: a proton can emit a positive meson, thereby turning into a neutron; the positive meson can be absorbed by some nearby neutron, which consequently turns into a proton. Another proton located nearby cannot absorb the positive meson, since in doing so it would find itself with two positive charges and would constitute a particle whose existence is highly improbable. It is obvious that such a process can explain the interaction between a proton and a neutron, but not between two protons. When a proton and a neutron interact, they exchange charges. This type of interaction had been postulated in nuclear physics even before Yukawa and is known as exchange forces. And indeed, the exchange character of the forces was one of the initial facts on which Yukawa relied when developing his theory. It is easy, however, to see that by the exchange of charged mesons we cannot, in the first approximation, bring about the interaction between two protons or two neutrons: to carry this interaction it is necessary to have a neutral particle. The existence of the neutral meson was first postulated by Kemmer in England, who constructed a theory known as the “symmetric meson theory.” In this theory it is assumed that neutral and

charged mesons of both signs, and the magnitude of this coupling is assumed to be the same for each type of meson. Kemmer’s theory does indeed lead to the charge independence of nuclear forces.

After physicists had obtained artificial mesons, they did indeed discover, in addition to charged ones, also a neutral meson. Unfortunately, the neutral meson cannot be observed directly, since its lifetime is very short. It decays in a time of the order of $10^{-15}$ sec into two gamma quanta. The short lifetime can be explained from the standpoint of the theory, but I shall not go into consideration of this problem.

The second prediction of nuclear theory concerning mesons is that mesons cannot be scalar. A scalar particle has two distinguishing features: first, it has no spin, in which it differs from electrons and nucleons, whose spin is equal to one half; this means that they have angular momentum “about their own axis” and thereby resemble a spinning top. The term “scalar,” however, means more than the absence of spin. The second feature of a scalar particle is that the wave function of the particle remains unchanged if all space is reflected in a mirror plane. There is also the possibility that the wave function of a particle with zero spin changes sign under such a reflection. In this case the particle is called “pseudoscalar.” Scalar mesons must be excluded, since the nuclear forces caused by them are central forces, depending on the distance between two nucleons and on nothing else, whereas experiment shows that these forces also depend on the direction of the spins of the two nucleons. The third prediction of nuclear theory is that mesons must be pseudoscalar. This follows from the entire character of the dependence of nuclear forces on the direction of the nucleon spins, in particular from the sign of the quadrupole moment of the deuteron.

The fourth prediction came from a completely different area of physics—not from nuclear physics, but from investigations of the fundamental properties of fields, which had been carried out, not without success, especially after 1947. The problem consists in the mathematical treatment of the interaction of two fields, such as the electric field, the fields of mesons, nucleons, electrons, and so forth. Before 1947, in describing such interactions, infinities arose in the results; but since 1947 it became clear to us how to deal with these infinities. This became possible thanks to the so-called “renormalization theory,” in which the quantities used for eliminating infinities are reinterpreted as changes in the mass and charge of the particle. After such a reconsideration of the concepts, it proved possible to show that all physically observable quantities are in fact finite. This modern theory of fields proved—

...proved suitable for separating the grain from the chaff: in some theories finite results can be obtained by means of renormalization, whereas other theories do not lend themselves to this process. We can write down on paper the basic equations of these latter theories, but if we try, for example, to use them to calculate the probability of scattering of mesons by nucleons, we shall always inevitably arrive at an infinite result. Thus, the development of field theory has led us to establish a principle that makes it possible to choose between possible and impossible expressions for fields and their interactions. In particular, a field consisting of charged particles, such as mesons, which can be emitted and absorbed by other charged particles (nucleons), can have only zero spin. Thus, field theory confirms what had been postulated in nuclear physics, namely, the zero spin of mesons. It says still more about the connection between mesons and nucleons, but I shall return to this discussion later.

PROPERTIES OF FREE MESONS

Let us now consider experiments with mesons. There are many types of experiments that can be performed with particles. The experiments that we shall discuss first are the simplest. They can be interpreted without complicated calculations, since qualitative results are sufficient for obtaining the answer. First of all, the mass of the meson was measured very accurately; it is equal to 276 electron masses for the charged meson and 265 for the neutral one. The difference can be explained by the intrinsic electrical energy arising from the interaction of the electric charge of the charged meson with its electric field. Next, we know that mesons have a finite lifetime: charged mesons live about \(10^{-8}\) sec. This time is relatively long, which makes it possible to observe them. Indeed, a charged meson with such a lifetime will on average be able to travel 3 m before its decay; this distance is sufficient for its observation. On the other hand, a neutral meson, living about \(10^{-15}\) sec, travels only \(10^{-5}\) cm, which is not enough to observe its properties before decay. The third property of the meson, which is of greatest importance for our consideration, is the spin of the particle. It was determined experimentally for both charged and neutral mesons by various and very ingenious methods.

The determination of the spin of the charged meson is based on the statistical principle of detailed balance. From statistical mechanics and from quantum mechanics it is known that if we have a process that can proceed in a given direction, and a process that can proceed in the opposite direction, then the probabilities of these two processes are closely connected with one another.

For example, if all the types of particles participating in a process are at a very high temperature, so that there are many particles in each quantum state, then, in order to maintain equilibrium, the processes in both directions must occur equally often. What, then, is common between this principle of detailed balance and the definition of the spin of the meson?

If we have a particle with zero spin in an external field, then there is only one possibility for the behavior of this particle; it has nothing by which to be oriented with respect to the external field, it has only one quantum state. On the other hand, a particle with spin equal to unity may have three different orientations of its spin with respect to the external field. Now consider the process of formation and the process of absorption of a meson with spin equal to unity. If the meson is formed, then it may be formed with three different directions of spin. If it is absorbed, then we know that it disappears with one single direction of spin; thus, the ratio of the probabilities of absorption and formation must contain a coefficient equal to three, if the particle has spin one. This coefficient is called the “statistical weight.” On the other hand, if the particle has zero spin, the coefficient three is replaced by unity, so that the ratio of the probability of formation to the probability of absorption will be three times smaller. This principle, proposed by Marshak for the investigation of meson spin, was successfully used by two groups of experimenters, one from Rochester, the other from Columbia University. Both groups came to the conclusion that the meson indeed has zero spin, in agreement with the predictions of both nuclear physics and field theory.

It was somewhat more difficult to take the final step, namely, to find out whether the meson is scalar or pseudoscalar, and now I must explain in more detail what this means. You probably know that one of the characteristics of an atomic system is its parity. Parity tells us how the wave function of a system changes when the sign of all coordinates is changed, i.e., when \(x\) is changed to \(-x\), \(y\) to \(-y\), etc., which is known as inversion. In the case of zero angular momentum we can proceed still more simply, namely, draw a plane through the center of the atom and reflect all space in this plane, as though it were a mirror. Let us now ask how the wave function changes as a result of the inversion performed. We know that there are atomic states whose wave functions do not change under this operation; these states are called “even.” There also exist atomic states whose wave functions change sign under inversion. Such states are called “odd.” This property is called parity. For example, in optical spectra every permitted spectral line arises

as a result of a transition between even and odd states of the atom, or conversely. In an analogous way the parity of nuclei is defined. For example, the deuteron consists of a proton and a neutron and has an even wave function. At the same time the angular momentum of the deuteron is equal to unity, since the spins of the nucleons are parallel, so that we have an even state with spin one. Suppose now that the deuteron absorbs a positive meson; then it must turn into two protons. These two protons obey the Pauli principle, according to which the wave function of the complete system must be antisymmetric and from which, therefore, it follows that between the spin of the system and its parity there must be a definite relation. From the Pauli principle, in particular, it follows that it is impossible to obtain an even state of a system of two protons with spin one; only an odd state with spin one is possible. The derivation of this proposition from the general principles of quantum mechanics is somewhat difficult to present, and I shall not trouble you with it.

It is now clear that an experiment in which a meson is absorbed by a deuteron may serve as a method for the experimental determination of the parity of the meson. This process begins with the negative meson going over to the deepest Bohr orbit, whose angular momentum is zero, after which it is, in the end, captured by the deuteron. Since the meson is negatively charged, it transforms the deuteron into two neutrons, which are emitted in opposite directions. One may ask to what extent this capture process satisfies the conservation laws. As I have already said, a charged meson does not contribute angular momentum. Therefore the process begins with angular momentum equal to unity, and with an even state of the deuteron. At the end of the process we have two neutrons. But, just as two protons, two neutrons cannot be in an even state with angular momentum equal to unity. Thus this process is forbidden by the parity conservation rule and by the law of conservation of angular momentum. In reality, however, in experiment negative mesons are readily captured by the deuteron, and this leads to the formation of two neutrons. From this we conclude that the meson makes a certain contribution to parity, that it changes the parity of the system. This is precisely the property possessed by a pseudoscalar particle; such a particle changes the parity of a system of nucleons if it is absorbed or emitted by the system.

THE CONNECTION BETWEEN MESONS AND NUCLEONS

Thus, up to this point experiment gives an exact confirmation of the predictions of the theory of nuclear forces. The next problem, which is perhaps the most important question of meson theory, appears to be considerably more difficult. The question is whether

as mesons are coupled to nucleons. The coupling of mesons to nucleons, in general the coupling of any two fields, is expressed by the corresponding term in the Hamiltonian of the system, and I am afraid that at this point I cannot avoid special terms. The expression for the interaction must contain both interacting fields. Let the meson be described by a wave function, which I shall denote by \(\varphi\); let us denote the wave function of the nucleon by \(\psi\), and the so-called conjugate wave function by \(\bar{\psi}\). The coupling to a pseudoscalar meson must also contain the Dirac operator, denoted by \(\gamma_5\); the physical meaning of this operator is not easy to explain. The two most commonly used expressions for the coupling have the form:

\[ G\bar{\psi}\gamma_5\psi\varphi \qquad \text{— direct or pseudoscalar coupling,} \tag{1} \]

\[ g\bar{\psi}\gamma_5\gamma_\mu\psi\,\frac{\partial\varphi}{\partial x_\mu} \qquad \text{— pseudovector coupling.} \tag{2} \]

The first of these is known under the name of “direct” coupling; it contains directly the wave function of the meson. The second type of coupling, used almost exclusively in the literature before 1947, contains derivatives of the meson wave function with respect to the coordinates; it also contains one of the more or less usual Dirac operators \(\gamma_\mu\), where \(\mu\) takes values from 1 to 4. These two types of couplings are also known as pseudoscalar and pseudovector coupling.

Let us note that pseudoscalar coupling has one practical advantage, namely that it enables theorists to work with quantities more familiar to them. It has, however, a serious drawback, since it leads to a field theory with the aid of which, for any processes, you can calculate only the first-order approximation. Second approximations give infinite results for all the processes calculated. This is connected with the fact that the theory does not admit renormalization. A drawback of the pseudoscalar interaction is the use of the difficult-to-understand operator \(\gamma_5\), which has the strange property that it can transform a particle from a state with positive energy into a state with negative energy. In terms of the theory of “holes” this means that the formation of nucleon pairs is a very probable process. The use of this operator makes theoretical calculations somewhat more difficult, but, on the other hand, the great advantage of the pseudoscalar theory is that it admits renormalization in the sense of field theory and gives finite results for any processes in any approximation. The question arises which of these theories is correct, and theorists, of course, hope that the theory of pseudoscalar coupling is correct.

The coefficients \(G\) and \(g\) in (1) and (2) are simply constants determining the magnitude of the coupling. I have denoted these constants by larg-

\(G\) and small \(g\), respectively, since the first constant is larger than the second. It is clear that the determination of the coupling constants is a very important problem: they play the same role as the electric charge in the interaction between charged particles and the electromagnetic field. This latter interaction is governed by the so-called fine-structure constant \(\frac{e^2}{\hbar c}\), which is equal to \(1/137\), i.e., is very small. On the other hand, the corresponding dimensionless quantity of meson theory, \(\frac{G^2}{\hbar c}\), is close to 15. In contrast to \(\frac{e^2}{\hbar c}\), this is a large number, which is the main reason for the difficulties of meson theory. All the methods of quantum mechanics developed in the past were intended for weak coupling between fields and particles. This assumption of weak coupling is quite good for the electromagnetic field, and in this case we can predict phenomena with an accuracy up to \(10^{-9}\) and satisfy the experimental data by a simple expansion in powers of \(\frac{e^2}{\hbar c}\). However, it is not very convenient to carry out an expansion in powers of \(\frac{G^2}{\hbar c}\) if this quantity is equal to 15: each successive order of approximation will then give a larger result than the preceding one.

For the experimental investigation of the coupling between mesons and nucleons, one turns to the simplest phenomena that depend on this coupling. Among such phenomena is the scattering of mesons by nucleons, since in this phenomenon one meson and one nucleon take part. A somewhat more complicated phenomenon consists in the production of mesons in the interaction of electromagnetic radiation with nucleons—the so-called photoproduction of mesons. In this case it is necessary to take into account, in addition to the meson and the nucleon, the interaction with the electromagnetic field. This interaction is known and is sufficiently simple. The next, more complicated phenomenon is the one with which the theory began, namely, the interaction between nucleons that emit the meson field. This is obviously a more difficult problem, since here one must consider two nucleons and, at the very least, one meson. Finally, the most complicated of all the simplest phenomena is the production of mesons in collisions between two nucleons. In this case we have not only the meson that creates the force acting between the two nucleons, but, in addition, also the meson produced in the process itself.

SCATTERING OF MESONS BY NUCLEONS

The experimental study of the scattering of mesons by nucleons was carried out in several places. The most extensive investigations were undertaken at the University of Chicago. Columbia and Rochester Universities also made a significant contribution to this

problem. The results of these investigations proved, above all, extremely unpleasant for field theory, since it was shown that they agree with the assumption of a pseudovector coupling. In order to make such a comparison of theory with experiment, the scattering probability was calculated by the method of perturbation theory, i.e., using the first term of the expansion in powers of \(\dfrac{G^2}{\hbar c}\). In this approximation the pseudoscalar coupling gives a meson–nucleon scattering cross section which is almost independent of the energy, whereas the cross section obtained from the pseudoscalar coupling grows very rapidly with increasing energy. Experiment also gives a very rapid growth of the cross section with energy and thereby testifies in favor of the pseudovector coupling. However, in the approximation of perturbation theory (weak coupling), the pseudovector theory also leads to some other predictions.

Thus, for example, it predicts that a negative meson interacting with a proton should, generally speaking, simply be scattered. In principle the interaction between these two particles may also lead to the reaction

\[ \pi^- + p \to \pi^0 + n, \tag{3} \]

but both the pseudoscalar and the pseudovector theories in the weak-coupling approximation predict that the probability of such a reaction is very small. In reality this reaction has a very large probability: the Chicago experiments showed that it occurs twice as often as simple scattering of negative mesons. Further, both theories, in the weak-coupling approximation, predict that positive and negative mesons should be scattered by nucleons almost identically. This too is incorrect: even if we add the scattering with charge exchange (3) to the ordinary scattering of negative mesons, we obtain only about one third of the probability of scattering of positive mesons. From this one can draw only the conclusion that the approximation method is completely erroneous, that it is impossible to obtain the correct result by carrying out calculations only to the lowest power of \(\dfrac{G^2}{\hbar c}\). If \(\dfrac{G^2}{\hbar c}\) is equal to 15, this conclusion is not unexpected. The situation that arose is well described in a song composed by the University of Rochester experimentalist Arthur Roberts:

“There is a strong coupling and there is a weak coupling,
But we knew in advance that there is neither one nor the other.”

A more reasonable method was developed by Brueckner of Indiana University and his collaborators Case and Watson. Taking the strong interaction more seriously, they assumed that a meson and a nucleon can easily form a composite particle,

i.e., that there exists a virtual quantum state of the system consisting of a meson and a nucleon. By postulating such a state, they were able to use a number of calculations, carried out on the basis of strong-coupling theory in the period 1940–1945, in an attempt to explain the discrepancies between the observed properties of the $\pi$-meson and the theory. This theory predicted that stable “compound” states of the nucleon and meson should exist, and that the first of these states should have the following characteristics: the orbital angular momentum of the meson is equal to unity, and the total spin of the system is equal to $3/2$, so that this state may be denoted as a $P_{3/2}$-state. The system is characterized by one more quantity, $T$, which is called the isotopic spin; its value is also equal to $3/2$. I shall not go into an explanation of what this means.

Thus, Brueckner suggested that the phenomenon of meson scattering is determined by this excited state. It has been established experimentally that the energy of this state is 300 MeV greater than the energy of the ground state of the nucleon. The scattering cross section of both positive and negative mesons by protons should in this case have a resonant maximum near the stationary state and an energy dependence of the form shown in Fig. 1. It was to be expected that this maximum would be very broad, since there is a very large probability for the decay of the nucleon in the stationary state into a meson and a free nucleon, and a large decay probability is equivalent to a large width of the level.

Fig. 1.

The chief success of Brueckner’s theory was that it was able to predict the ratio of the cross sections for the scattering of positive and negative mesons. This ratio should be three to one, which is very close to the observed ratio. Brueckner and his collaborators were further able to show that charge-exchange scattering of negative mesons should be approximately twice as large as ordinary scattering of negative mesons, which again agrees with experiment. They were also able to show that the angular distribution of the scattered mesons is approximately described by the law $1 + 3\cos^2\vartheta$, and this again gives an acceptable approximation to the observed angular distribution, although it is far from a perfect approximation.

Scattering experiments are usually analyzed from the point of view of the phase shifts of definite partial waves describing the wave function of the particle. Such an analysis, carried out for scattering

mesons, showed that the most significant contribution is given by the state \(P_{3/2},\ T = {}^{3}/_{2}\). In addition to this state, there also exists a strong interaction in states with zero orbital angular momentum (in spectroscopic notation, in \(S\)-states), and in these states there is strong interaction both for the value of the isotopic spin \(^{3}/_{2}\) and for the spin \(^{1}/_{2}\). For \(^{3}/_{2}\) there is strong repulsion; for \(^{1}/_{2}\), a somewhat weaker attraction.

DEVELOPMENT OF THE THEORY

Although Brueckner’s theory proved very successful, it is purely phenomenological: the existence of an excited state is postulated, but nothing is said about its nature. It is desirable to return to the foundations of the theory in order to explain this state. The path for such a return was opened by the young French physicist Maurice Lévy, who about a year ago developed a meson theory of nuclear forces. Thus we return back to the very beginning of the history of the problem—to the theory of nuclear forces, which gave the first indication of a pseudoscalar interaction. By consistently using the pseudoscalar interaction, Lévy was able to explain the phenomena caused by nuclear forces. The most important discovery he made is a direct consequence of the pseudoscalar meson theory and consists in the fact that between two nucleons at small distances large repulsive forces act. This was an indication that had been absent in the preceding theories, in which two nucleons were always regarded as completely fixed in space. It followed from the theory that two such nucleons would always be strongly attracted, and this attraction is so great that they would fall onto one another without forming a stationary state with finite binding energy. Lévy’s discovery saved the situation, since he showed that at small distances a very strong repulsion arises, which prevents the two nucleons from falling onto one another.

After Lévy’s calculations, physicists began to take an interest in whether the pseudoscalar theory could shed any light at all on the experiments on the scattering of mesons by nucleons. As I have already said, attempts to explain these experiments by the theory of weak coupling proved completely unsuccessful. The first successes in the direct application of the pseudoscalar theory were obtained by Drell and Henley of Stanford University. They were able to show that between a meson and a nucleon there acts a potential of the same type as between two nucleons in Lévy’s theory, namely a potential leading to extraordinarily large repulsive forces at small distances. At somewhat larger distances there is an attrac-

attraction, mainly in the \(P_{3/2}\)-state. Strong repulsion does not depend on the angle and therefore acts primarily in states with zero angular momentum, i.e. in \(S\)-states. Now if you have a strong repulsive potential and compute the resulting cross section in the first Born approximation, you obtain an exceptionally important result. Since in our case the repulsive potential acts in the \(S\)-state, the scattering is isotropic. It is easy to see that it also should not depend on the energy. All these results agree exactly with the first-order approximation of which I spoke earlier. The merit of Drell and Henley was that they showed precisely why the first-order results are erroneous. Indeed, if you have a potential that gives strong repulsion at small distances and then a weaker attraction at larger distances, then the effect of the repulsion on the wave function will be that it will take the value zero at the point where the repulsion ceases. The most that can occur here is that a phase shift arises proportional to the radius of the region of repulsion, and this phase shift should not at all depend on the magnitude of the repulsive potential. Thus Drell and Henley showed not only that the theory of weak coupling is “a priori” incorrect, but also why it is incorrect and what should be taken instead of it.

Fig. 2.

The next major success was obtained at the University of Illinois by Chew, who did for the study of attraction what Drell and Henley did for repulsion: he showed how one can calculate, at least in principle, the influence of attractive forces without using perturbation theory. He was able to show that for the state \(P_{3/2}\), \(T = 3/2\), one should indeed expect a resonance, provided only that reasonable assumptions are made about the magnitude of the coupling constant. Chew used a pseudovector coupling for convenience of calculation, but his theory can be expressed simply also in the language of pseudoscalar coupling.

On the basis of all these works, last autumn at Cornell University we undertook the investigation of the problem from the very beginning, using a pseudoscalar interaction between the nucleon and the meson. We were able to give a qualitative explanation of a large part of the features of the scattering experiments. First of all, we obtain a strong repulsive interaction in the \(S\)-state, which gives only a moderately large \(S\)-wave in scattering, and this

precisely what follows from the experiments in Chicago and Columbia. Then the theory gives attraction in the state \(P_{3/2}, T = {}^{3}/_{2}\), and the phase shift in this state can be chosen so as to satisfy the experiment, with an appropriate choice of the coupling constant. Thus, in the theory there is one unknown, namely the coupling constant; if one adopts a coupling constant close to 15, then the observed phase shift, including its dependence on energy, can be explained well. Finally, we find from the theory that the phase shifts for all the other \(P\)-states are very small, and this again agrees with experiment. Only one point remains still unexplained: experiment indicates that in the \(S\)-state with isotopic spin \(1/2\) there is attraction. But we know that the theory is still imperfect, and although we know that it must be renormalized, we are only now beginning to understand how this can be done.

CONCLUSIONS

It seems to me that at present one can say that, although the pseudoscalar meson theory still cannot give a quantitative description of experiments on the scattering of mesons by nucleons, there is no reason not to believe this theory, since there is no qualitative discrepancy between the predictions of the theory and experiment.

The question of nuclear forces, as I have said, is considerably more complicated. Levy’s first attempt proved extremely valuable, since it showed that in principle the theory correctly describes the behavior of nuclear forces. Various theorists criticized the details of Levy’s work, and this is not unexpected. However, the theory can explain what forces hold particles in the nucleus, why these forces are large, and why nucleons do not fall onto one another. It predicts the interesting phenomenon of interaction among many bodies; this means that the interaction exists not only between two nucleons, but also among three or a larger number of nucleons, each of which is “surrounded” by its meson. Weisskopf and his collaborators showed that this many-body interaction is of great significance for understanding the saturation of nuclear forces, thanks to which heavy nuclei can exist. From the pseudoscalar theory follow the dependence of nuclear forces on spin and the existence of a quadrupole moment in the deuteron.

A few words should be said about the other mesons. I have considered only \(\pi\)-mesons, whose mass is close to 300 electron masses. There are a number of other mesons, with considerably larger masses. At the time when calculations were being made of interactions only of first order, some physicists assumed that perhaps these heavy mesons provide the stability of nuclei. It seems to me that this is incorrect, and that at the present time it is already clear that these heavy mesons have little influence on the structure

of the nucleus. This is again connected with the Levy potential, of which I spoke earlier and which gives a strong repulsion between nucleons at distances of about \(0.5\cdot 10^{-13}\ \text{cm}\). Heavy mesons may be responsible for forces acting at still smaller distances than these; therefore, whatever these forces may be, they will be suppressed by the large repulsive forces that exist in any case owing to the interaction of nucleons with \(\pi\)-mesons. It therefore seems to me that, in order to construct a satisfactory theory of nuclear forces, there is no need to know very much about heavy mesons. However, if two nucleons collide at very high energies, they may penetrate into the region of mutual repulsion, and in doing so—as we know from experiment—heavy mesons may be produced, which will appreciably affect the collision mechanism. But for nuclear forces at moderate energies, for example inside ordinary nuclei, it is apparently mainly \(\pi\)-mesons, which are coupled to nucleons by a pseudoscalar interaction, that are responsible. It seems to me that extracting from the theory the information contained in it is merely a matter of high mathematical skill.