The capture of a muon bound to an atomic nucleus is a singular phenomenon because it involves three forces: an electromagnetic force binds the muon to the nucleus and produces the muon's energy levels, including the hyperfine structure; the nuclear force produces the structure of the nucleus; and finally, the weak force is responsible for the capture. In this note, I describe the rather bizarre circumstances under which the first paper discussing the role of the hyperfine levels in this process was written in 1958.
I was recently straightening out a drawer in a cabinet when I came across a somewhat decayed looking preprint of a paper entitled, “The effect of the Hyperfine Splitting of a μ-Mesic Atom on Its Lifetime.” There is no date given but I happen to know, as I was one of its authors, that it was in the spring of 1958.1 My co-authors were T. D. Lee, C. N. Yang, and Henry Primakoff. This remarkable grouping, which I am going to explain, is an illustration of the Russian saying that living a life is not like crossing a field.
The events that led up to this preprint began a year or so earlier in Cambridge, Massachusetts. I was then the “house theorist” for the Harvard Cyclotron, a job that allowed me to work on anything I liked. My interest was in the more phenomenological aspects of what was then the theory of elementary particles. What I would call, broadly speaking, the field theory community in Cambridge was then quite small. At Harvard, besides myself, there were Ken Johnson, Roy Glauber, and of course Julian Schwinger. At M.I.T., there was Herman Feshbach and Viki Weisskopf, and Francis Low was a visitor at that time. We met for lunch more or less once a week on Wednesdays at Chez Dreyfus where Schwinger would present his latest ideas, usually on paper napkins, while the rest of us sat transfixed. An exception was Francis Low, who on one occasion turned to me and said in a rather loud voice, “He is wrong. The man is wrong.” He then explained to Schwinger.
Through these lunches I had gotten to know Viki Weisskopf fairly well, and after one such lunch he explained that he had a task for me. The young Austrian physicist Walter Thirring was visiting M.I.T. and he was trying to write a new English version of his book on quantum electrodynamics. Viki thought Walter might need some help with his English and felt that I was the right person for the job. Viki also told me that he thought the German edition was a masterpiece, comparable to Wolfgang Pauli's book on relativity, written when Pauli was about the same age as Walter was at that time.
Viki was very persuasive so I met Walter and agreed to help. It turned out that Walter's written English was as good as mine but we became friends and he was kind enough to give me an acknowledgement in his book. He also gave me a preprint to read that involved the hyperfine structure in μ-mesic atoms. I have since forgotten the conclusion of this preprint, but I was struck by the fact that compared to the hyperfine interactions in ordinary atoms those of μ-mesic atoms were a couple of orders of magnitude larger. Basically, these larger interactions are due to the muon in a hydrogen atom ground state being closer to the proton by a factor of . I then forgot all about it when I began my first year at the Institute for Advanced Study in the fall of 1957. However, I did bring Thirring's preprint along.
That year was an exciting one at the Institute. Lee and Yang won the Nobel Prize for their work on parity non-conservation. Lee had taken a leave of absence from Columbia and he and Yang continued their collaboration with a high volume of Chinese chatter emanating from one or other of their offices. I devoted much of my time trying to learn about the weak interactions, especially by reading their papers. In particular, I looked at what they had written about muon capture by protons. I also happened to look at Thirring's paper. I noticed that the Lee-Yang paper did not mention the hyperfine interaction and I wondered why. I had no idea what, if any, role it would play and even had no idea of what to look into. And there it would probably have remained except for one of those chance acts that life sometimes deals you.
It was a lovely spring Saturday morning and I was in my office reading both a paper by Lee and Yang and the one by Thirring. I was due to spend a long weekend in New York and was going to drive in early in the afternoon. I spotted Lee walking across the lawn to his office; I barely knew him it at all but thought it might be a chance to ask him about the role, if any, of the hyperfine splitting in these muon processes. I intercepted him before he reached his office and asked if he and Yang had looked into this. Lee said that there were no consequences. I asked him if he would mind explaining that to me and we went to his office. He went to the blackboard and began computing at a furious rate, saying things like “spin flipper.” I had no idea what he meant and I went to New York before I found out. When I came back on Monday afternoon I looked at my post box. I noticed what seemed to be a typed paper. It had three authors: Lee, Yang, and me! It had the same title as the preprint I saved. I was completely astounded. I had done nothing but ask a question that I had no idea how to answer and somehow this paper had appeared. The speed at which they worked was incomprehensible to me and their generosity was overwhelming. I was a more-or-less unknown junior physicist and they would have been perfectly within their rights simply to thank me for asking this question. What they did instead made an enormous difference to me. I am sure it played a role in my getting a second year at the Institute and eventually a National Science Foundation Fellowship, which allowed me to spend two years in Paris. To live a life…
Before I discuss the contents of the paper, I must explain how Primakoff got his name on it; this was also a matter of chance. That very week there was a meeting of the American Physical Society in New York and I went back to the city clutching our paper. I have no idea who I was planning to show it to but as luck would have it I showed it to the one person who understood it without even bothering to read it, Valentine Telegdi. Telegdi was a brilliant and sardonic professor at the University of Chicago whom I had met on his visits to Princeton. He was known as “Mister muon” because of the experiments he had performed. Telegdi told me that he knew all about this because of his contacts with Henry Primakoff, a deep-thinking theorist then at Washington University. Telegdi said that Primakoff had done the same work. Upon my return, I told my two collaborators; they were very surprised since nothing had been published. They called Primakoff and indeed he had considered the same problem and was preparing a manuscript that he would send. Primakoff's manuscript was gigantic and festooned with the sort of baroque symbols that he liked. I am not sure whether he had suggested an actual experiment, which is what Lee and Yang did, but he certainly had the same idea so he became a co-author. Later, we became good friends. From that point on, Telegdi began referring to our paper as BLYP (pronounced “blip”). I will now explain the contents of this paper.
The foundational idea is the role of the large muon mass compared to that of the electron, which enhances all the hyperfine effects. The hyperfine interaction is the coupling of the magnetic dipole of the nucleus with the magnetic field generated by the spin and orbital angular momentum of the electron, or, in this case, the muon. For the muon, the splitting of the hyperfine frequencies is much larger than the inverse lifetime of the μ-mesic atoms. This means that the hyperfine states de-cohere and the capture rates from the two states are in general different for reasons I will explain.2 The muon lifetime is ostensibly the same in the two states (there is a small wave function effect). What Lee and Yang had computed previously was the lifetime for which the hyperfine effect is irrelevant.
To compute the hyperfine effect Lee and Yang had to invoke a nuclear model. In this model, the muon is assumed to be in the K orbit, which is closest to the nucleus. The nucleus is supposed to have a non-vanishing spin . The two hyperfine states have angular momenta and ; for a single proton, this would be the triplet and singlet s states. For simplicity, our nucleus is assumed to consist of a single proton outside a core whose total spin is zero, so this proton carries angular momentum L and the two hyperfine states have angular momenta and . The capture interaction by the core itself is the same for the two hyperfine states. Again, for simplicity, the outside proton is taken to be free. If the capture interaction were spin-dependent, there would be a difference in the two rates. At the time our paper was written, the nature of the weak interaction was still being debated. So along with the vector and axial vector coupling of today, we also considered scalar and tensor coupling, which makes the results more complicated than they need be. That Lee and Yang could sort all this out in a little over a day is astonishing.
Apart from the scalar coupling, the other possibilities are all spin-dependent so they imply different capture rates for the two hyperfine states. This suggests an experiment—a very difficult experiment—the results of which were published by Roland Winston, a student of Telegdi's in 1963.3 Because of the different capture rates of the muons in the two hyperfine states, the number of muons as a function of time in each state is different. Hence, the decay electrons will not follow a simple exponential in time. Measuring the departure from this exponential was what was proposed and what as measured by Winston. As far as I can tell, other methods of measuring the hyperfine interaction have since been carried out and much more sophisticated nuclear models have been used.
Primakoff ended up publishing his gigantic paper as a review article.4 I eventually got to know him and even work with him a bit. He ended his career at the University of Pennsylvania. Primakoff died in 1983, Walter Thirring in 2014; the other three authors of the BLYP are still functioning. I might mention that Bohr happened to be visiting the Institute that spring. Oppenheimer arranged a little seminar where the various members were asked to speak about their work. I was asked to talk about what I had done with Lee and Yang; they had much more important things to tell Bohr. I was quite nervous and spoke very fast. Bohr listened impassively and then said he thought it was very interesting, which was Bohr language for saying that it was not. If he had been really interested, he would have engaged in a Socratic dialog. Living a life, as I have said, is not like crossing a field.
The author would like to thank Roland Winston for comments. This work was performed at the Aspen Center for Physics, which was supported by National Science Foundation Grant No. Phy-1066293.
APPENDIX: HOW THE HYPERFINE INTERACTION IN MUON CAPTURE WAS FIRST DESCRIBED
In this appendix, I want to fill in some of the technical details of the suggested BLYP experiment. We are only interested in negative muons with the decay mode . The lifetime for this decay, if the muon is free, is about 2.2 μs. We are interested in muons bound in μ-mesic atoms, which have different lifetimes, but we will neglect this relatively small effect.5 Let me call the free decay rate . In the absence of capture at any given time, the number of muons is given by
But the total number of muons plus electrons is conserved and equal to . Thus,
Next I want to consider the case in which the muon is captured by a proton at the rate via the reaction . In this reaction, the muon disappears and no electron results. As mentioned above, we considered the simplified model of the nucleus in which there was a single proton that carried the angular momentum around a core of protons that have no spin. There are two hyperfine states with total angular momentum and . Because of the spin-dependence of the capture mechanism, the capture rates from these two states, and , are different. There can also be a rapid transition from the higher-energy hyperfine state to the lower-energy state. One must also know how these states are initially populated.6 The precise values of the capture rates also depend on one's assumptions about the weak interactions. I will not go into the details here but refer the reader to our paper, noting once again that in light of our present knowledge of the vector and axial-vector nature of the weak interactions, the formulae we give in our paper are more complicated because we took all possibilities, including tensor coupling, into account.7
If we allow capture then Eq. (1) must be replaced by . Here, is a constant, is the capture rate, and is a parameter determined by the ratio , with the average capture rate including the protons in the core. For greater than 1, this ratio goes as , so to measure it one must use relatively light nuclei; it was first successfully measured in an isotope of fluorine. The rate depends on the hyperfine state. As time evolves, the muons will transition to the lower hyperfine state so is effectively . For Lee and Yang to have seen all this and to have written it up over a long weekend was for me nothing short of miraculous.
The quantum interference depends in and if the argument of the exponent is large the term oscillates to zero, which is the usual way decoherence is established.
There are a number of effects that must be considered. These include relativistic effects such as time dilation and the use of a bound state wave function. A relatively straightforward one is due to the effect on the muon mass due to its Coulomb binding. If is the Fermi constant and is the muon mass, then dimensional analysis shows that the decay rate is proportional to . The mass is reduced by a factor proportional to and hence the effect is small but contributes to the reduction of the decay rate.
One must distinguish between what I would refer to as “intrinsic” and “induced” couplings. The former are in the nature of the force itself while the latter are “induced” in the course of perturbative calculations. We considered only the former. An example of the latter is a negative muon emitting a virtual negative pion and a neutrino. The proton absorbs the pion, becoming a neutron; this appears as a pseudo-scalar in the capture rate. There is now a vast literature on these effects and their experimental consequences, but it is beyond the scope of this brief note to comment on it.