Personal recollections on chiral symmetry breaking

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The author’s work on the mass of pseudoscalar mesons is briefly reviewed. The emergence of the study of CP violation in the renormalizable gauge theory from consideration of chiral symmetry in the quark model is discussed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index B02, B52, B60


Introduction
Yoichiro Nambu spent most of his research life in the United States. His frequent visits to Japan, however, had a great impact on the Japanese scientific community. Through these opportunities, I had the privilege of being acquainted with him. Among many memories, one unforgettable thing is that I shared the 2008 Physics Nobel Prize with him, together with Toshihide Maskawa. Needless to say, this is a great honor for me. I was stunned when I heard this news in the telephone call from Stockholm and I could not believe it for a while. The citation of the Nobel Prize for Nambu is "for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics," while ours is "for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature." Although the two citations have the key words "broken symmetry" in common, the general view is that there is no direct relation between them. Nevertheless, there is a narrow path connecting the two works in my mind.
I entered the graduate school of Nagoya University in 1967. At that time, chiral dynamics was beginning to attract attention. In this approach the soft pion amplitude based on the current algebra and partial conservation of the axial current (PCAC) method can be reproduced by a Lagrangian method very easily. It was demonstrated that what had been done by current algebra and PCAC were mostly the result of spontaneously broken chiral symmetry. The implication of the PCAC relation as spontaneously broken symmetry was first noticed by Nambu around 1960 [1], but it took quite some time for people to understand this.
When I entered the graduate course, Maskawa was studying chiral dynamics together with a few graduate students. I joined this group and began my research. We wrote a couple of papers related to chiral dynamics and then we studied the mass of pseudoscalar mesons from the viewpoint of the quark model.
At that time, the theoretical particle physics group of Nagoya University was led by Shoichi Sakata. In 1956, Sakata proposed the so-called Sakata model [2], in which all hadrons are supposed to be composite states made of the fundamental triplet: the proton, the neutron, and the lambda. It turned out, however, that the Sakata model is not correct as a model to explain the real world. The Sakata model was replaced by the quark model proposed by Gell-Mann in 1964.
When I started our research, many people doubted the reality of the quarks, because of their curious property of a fractional charge and the fact that quarks had not been found. However, the viewpoint of the Sakata group was somewhat different. They maintained the composite model approach even after they abandoned the original Sakata model. So when we considered chiral symmetry in the quark model, our viewpoint on the reality of the quark was positive, following that of the Sakata group.

Mass of pseudoscalar mesons
In the quark model, the quark mass terms are the natural source of the SU (3) symmetry breaking. The transformation property of the quark mass term under the flavor SU (3) transformation is 1 + 8, and the octet breaking can explain the basic structure of the hadron masses.
In order to discuss the mass of pseudoscalar mesons, we need to consider the chiral transformation, too. Under the chiral SU (3) × SU (3) transformation, the quark mass term behaves as (3, 3 * ) + (3 * , 3), and all the axial currents have non-vanishing divergence. This explains the masses of the pions and the kaons naturally. The smallness of the pion mass is related to the smallness of the u and d quark masses.
The problem is the masses of iso-singlet mesons. To discuss their mass, we consider the SU (3) singlet axial current or U (1) A axial current together with the eighth component of the octet axial current. If the chiral symmetry is broken only by the quark mass terms, then their divergences are given by Today, we know that there is an anomaly term in the divergence of A μ 0 , but what we are considering here are the developments before the appearance of QCD. Natural results obtained from the above relations are the so-called nonet mass relations, which imply that one of the iso-singlet mesons has the same mass as the pion. However, the lightest iso-singlet meson in the real world is η, whose mass is 548 MeV/c 2 .
To solve this problem, we considered that the Hamiltonian would have an extra term violating the U (1) A axial symmetry but invariant under the chiral SU (3) × SU (3) transformation [3]. In this case, the above formula for the divergence of A μ 0 has an extra term in addition to the contribution of the quark mass term. From this modified relation, we obtained the following mass relation: where X is another iso-singlet meson. For the physical values of m π , m K , and m η , this relation gives m X = 1.6 GeV/c 2 . Although the result is not so good quantitatively, we thought that we were on the right track. Then the next problem was how to construct such an extra term of the Hamiltonian from the quark field. Our answer to this problem is the product of six quark fields given by So far I have described what we did and published in 1970 [3]. Follow-up arguments can be seen in Ref. [4]. After this, a possible relation between the U (1) problem and the QCD anomaly was noticed, and 't Hooft discovered the instanton in 1976 [5]. The instanton causes a chirality flip, and the effective interactions around the instanton are essentially the same as the above form of the product of six quark fields.
Our paper aimed to express chiral symmetry breaking in terms of quark fields and did not discuss the particular form of the strong interaction. However, our basic viewpoint was that quarks obey field theory and strong interactions should be described by some kind of field theory, which maintains the chiral symmetry.

Up to the six-quark model
In the meantime, the renormalizability of non-Abelian gauge theory was proved. In particular, the Weinberg-Salam-Glashow model attracted attention as a possible theory describing the weak and electromagnetic interactions in a unified manner. To apply this model to the quark sector, it was necessary to consider a GIM-type extension. This was, however, favorable for us because a cosmic ray group led by Niu found new events in emulsion exposed to the cosmic rays some time before and we were investigating them with the four-quark model. So, we thought that there are basically no problems in describing weak interactions of quarks and leptons in terms of the Weinberg-Salam-Glashow model.
Then, a natural question was whether all the interactions can be described by gauge theories. The first problem was the strong interaction. We considered a color gauge theory as a possibility, but an obvious problem was the mass of gauge bosons. They are naively massless particles, while there are no such particle in the real world. We could not solve this problem. We could not think of asymptotic freedom nor infrared slavery.
We also considered another possibility: that the strong interaction is caused by massive Abelian vector meson exchange. In this case, a problem is the symmetric property of the wave function of baryons. At that time, Kamefuti and Ohnuki were investigating para-Fermi theory and I learned from them that order three para-Fermi theory looks like a U (3) symmetric fermion system [6]. So I was vaguely thinking that this could be a possible solution.
Anyhow, we did not reach any conclusion for the strong interaction, and turned our attention to another interaction. When I entered the graduate course in 1967, CP violation was already an established experimental fact. However, the mechanism of CP violation had not yet been clarified. So, our question was whether CP violation can be described within the renormalizable gauge theories [7].
I obtained my PhD from Nagoya University in 1972 and moved to Kyoto University. Maskawa also moved from Nagoya to Kyoto a couple of years before. We started this work soon after I moved to Kyoto.
It was not so difficult to show that the minimal system consisting of four quarks cannot accommodate CP violation, providing that the strong interaction has good properties concerning CP symmetry. This implied that there must still be unknown particles.
It is easy to construct a CP-violating interaction by introducing additional particles. Among many possibilities we considered there was a six-quark model. Although it was not a unique solution, it was an interesting possibility, because it has only one CP-violating parameter and the interaction is a simple extension of a familiar one. There was no hesitation in predicting the existence of so many quarks, but we did not expect the subsequent rapid development of experiments. 3

Remarks
Through the study of spontaneously broken chiral symmetry, we became familiar with the field theoretical approach for the fundamental interactions. We were trying to find a strong interaction compatible with chiral symmetry, and we also studied field theoretical descriptions of spontaneously broken systems.
These studies were quite helpful in understanding the implication of 't Hooft's work on the renormalizability of non-Abelian gauge theories. I was not so much interested in the calculation technique, but rather in applying the new scheme to build up an entire picture of interactions. This was probably the influence of the Sakata group. As mentioned above, the Sakata group accepted the quark model as a realistic one. For the model to be realistic, it was necessary that we could draw an entire picture based on the model. It seems that this kind of thought prompted us to focus on CP violation in the new scheme.
On the other hand, Sakata was somewhat critical of field theory, and in particular of renormalization theory. He thought that renormalization is a tentative solution for the difficulty of divergence and he was expecting a more radical change of the fundamental theory. This belief could be correct even today, but our work was greatly dependent on renormalizability. He passed away in 1970, without seeing the development triggered by the discovery of renormalizability of non-Abelian gauge theories.
To conclude, our work on CP violation is benefited by many things. Timing is one of the most important elements. We came across a revolutionary period of particle physics in our early research career. It is also important that we learned at Nagoya University. Last but not least, I realize that we followed the road paved by Nambu.