1 Effective Theories Enter Nuclear Physics

Like many others, I was immediately stricken by Weinberg’s seminal works on the the use of effective field theory ideas to nuclear physics [1, 2] when they first appeared. Could it really be that all the famously messy nuclear phenomenology could be reduced down to a simple organizing principle? Could the connection of nuclear physics to QCD finally be established? I also had a personal interest on this question, besides the obvious intellectual one. I had just finished my PhD working under a particle physicist, supported by a high energy grant so I consider myself a particle physicist. But I was a postdoc at the nuclear theory group at MIT. I felt very much an outsider in the nuclear community, ignorant of the central questions and methods of what was supposed to be my own field. I was familiar—and fascinated—by effective theories and how the “correct” way of understanding field theory was to think of them as effective field theories [3]. Weinberg’s papers explained how nuclear physics worked in a language that was simple and familiar to me. To my young eyes, this seemed like a unique opportunity: Weinberg was inviting us to reinvent nuclear physics from the ground up instead of painstakingly learning the lore accumulated over decades. It fit perfectly to my theoretical prejudices and life contingency. This was an attitude shared by many of my colleagues—if they are honest enough to admit it!—who were also making the switch from particle to nuclear physics at the time. As ridiculous as it sounds to me now, it did serve the purpose of injecting us with motivation and confidence. After all, Weinberg (and K. Wilson) could be wrong on details but not on the big picture and all we had to do was to go through the road they had paved for us.

The archetypical effective field theory is Chiral Effective Theory (\(\chi P T \)), the effective theory describing pions (and kaons) at momenta well below the QCD scale (\(\approx 1\) GeV) [4,5,6]. The predictions of \(\chi P T \) can be stated as fairly rigorous “low-energy theorems” and, at the time, I was very impressed that this kind of rigor was possible in low energy QCD. I would be even more impressed if I knew of some lattice results of the 2010’s confirming that even for realistic non-zero pion masses, the \(\chi P T \) expansion works very well. Pion \(\chi P T \) was the archetypical example of a whole philosophy of how to think about renormalizability in field theory (\(\chi P T \) is not renormalizable in the most traditional sense) that slowly permeated the particle physics community. “Slowly” is the operative word here. I remember that around that time I attended a talk by S. Weinberg where he presented the “field theory is effective field theory” perspective, essentially as described in [7]. Leaving the seminar I overheard two comments that stuck to my memory. The first, from a very distinguished senior particle theorist ironically wondering if Weinberg had finally read Wilson’s work. Another, equally distinguished senior theorist, was apparently shocked by what he heard and completely unconvinced. Was Weiberg’s message old news of too novel? And that was in 1994! A similar process of slow assimilation, sometimes moved one retirement at a time, was beginning to occur in the nuclear community.

The inclusion of one single nucleon makes the problem much more complicated and \(\chi P T \) less rigorous. Should the \(\Delta \)’s be included explicitly or not? How? Still, a fairly sophisticated formalism was created, incorporating some of the lessons of heavy quark physics, and some of the phenomenology came outright. That was encouraging. If the level of rigor was not quite the same as in the meson sector, it looked more solid than the wild modeling that existed before. At that point in time the natural next step would be to include two, or more, nucleons, that is, to create a \(\chi PT\) for nuclear physics. But the generalization is not so obvious and took the insight of Weinberg to initiate the process. The complication is that any theory describing nuclear physics must be non-perturbative since bound states do not arise from free particles at any order in perturbation theory. This is in contrast to \(\chi P T \) in the zero or one-nucleon sectors, where the low energy/chiral expansion involves, at any fixed order, Feynmann diagrams with a finite number of loops. In order to argue, as Weinberg did, that cutoff dependences from loops are matched by cutoff dependences of the counterterms, order-by-order in the low energy expansion, one would have to understand the ultraviolet behavior of the theory non-perturbatively. This is a task that is still being worked out. This difficulty was bypassed by Weinberg by working out the low energy expansion for the nuclear potential instead of the scattering amplitudes. The potential is given, at any finite order, by a finite number of diagrams and allows for a more straightforward generalization of \(\chi PT\).

Of course, the moment the effective theory nuclear potential was actually used for the (numerical) computation of bound states and phase shifts, the problem Weingerg cleverly avoided resurges. Since the nuclear potentials are singular at short distances, the Schrödinger equation is not well defined and requires regularization. The dependence on the regularization has to be absorbed by short-distance interactions that are themselves regularization dependent. This regularization dependence is in addition to the one appearing in the construction of the nuclear potential and so it is not a priori clear that it can be absorbed in the same contact interactions used in the construction of the potential. The nuclear potential has another feature interweaved with this. Nucleon-nucleon s-wave scattering lengths are fairly large, much larger than one would expect from the size of nucleons, the range of nuclear forces given by the pion mass, or typical QCD scales (\(\approx 1\) fm). This suggests a fine-tuning in the short distance interactions, an anathema to effective field theories that are based on assumptions of naturality. A tremendous amount of research appeared at the time trying to figure out the renormalization structure of the nuclear effective theory and its interplay with the s-wave fine-tuning, complicated by the fact that the most elegant regularization method used in \(\chi PT\), dimensional regularization, does not apply to non-perturbative calculations. There were so many suggestions and calculations that at some point it became hard to identify real progress. Comparison with data did not help. Many models/theories can fit s-wave phase shifts and their dependence on quark masses, in principle predicted by the chiral effective theory, was completely unknown at the time. The discussion on these topics quickly descended to comparing fits to “data” (phase shift analysis) and comparing \(\chi ^2\) values. Unfortunately, many ill-motivated models fit the data well with few parameters and it was unclear if effective field theories had any advantage over traditional methods.

2 Pinball Wizards

Given those circumstances, it seemed that studying three-nucleon systems was sensible. One could fix low energy constants in the two-nucleon sector and use those for make parameter-free predictions in the three-nucleon sector. This was made possible in practice because of the fine-tuning mentioned above. The fact tha s-wave scattering length a were unnaturally large created a hierarchy of momentum scales: \(1/a \ll m_\pi \ll \Lambda _{QCD}\) (\(m_\pi \) is the pion mass). The regime \(p\sim m_\pi \ll \Lambda _{QCD}\) is the regime where \(\chi PT\) is valid. But in the lower momentum regime \(p\sim 1/a \ll m_\pi \) a simpler, friendlier effective theory arises where only nucleons, but not the pions, appear as explicit degrees of freedom. This effective theory is subtle and non-perturbative in the sense that, even at leading order in the low energy expansion, an infinite number of diagrams have to be resumed. Yet, this “pionless effective theory” (\(/\!\!\!\pi EFT\)) was worked out in detail and by 1997 it was well-known [8, 15], at least in Seattle where, by that time, I had moved to. I remember David Kaplan suggesting I looked into three-nucleon bound states in \(/\!\!\!\pi EFT\) and showing me a derivation on the board of the relevant equations. I did not understand it or believed it. But I re-derive them in a more usual field theory language, as a resummation of an infinite series of diagrams—the “pinball diagrams” (see Fig. 1)—by the use of an integral equation. Working with Bira van Kolck we solved it numerically to postdict the neutron-deuteron scattering length in the \(J=3/2\) channel with surprising precision [9]. The precision was not surprising if one takes into account that our equation was accurate to next-to-next-to-leading order and generalized the old result of Skorniakov–Ter–Martirosian [10] by including finite effective-range effects. That was very exciting for me and other EFT’ers, but failed to impress some of the older nuclear theorists. They knew that any reasonable model of nuclear forces would also arrive at the same result. In fact, this had to be the case if our EFT was indeed universal among theories with the same two-nucleon phase shifts! Soon afterward we were joined by H. W. Hammer who taught us that the numerical stable way of solving the integral equation for finite initial energy was to find the K-matrix (worked extended later to energies above the breakup threshold using the elegant Hetterington–Schick [11] contour deformationFootnote 1). That worked well too but the data was not good enough for a precise comparison.

Fig. 1
figure 1

Pictorial representation of the integral equation satisfied by the half-off-shell scattering amplitude T summing up all leading order diagrams, including the three-body force

Full size image

Things started getting really interesting when the \(J=1/2\) channel, where the triton/\({}^3\)He lives, was approached the same way. The initial results made no sense: an ultraviolet cutoff had to be used for numerical reasons and the answer depended wildly on its value. And that dependence did not diminish no matter how large the cutoff was. It was clear the cutoff was merely a numerical kludge but necessary to make sense of the integral equation. The scattering amplitude oscillated (to be more precise, the combination \(k\cot \delta (k)\) of incoming energy \(k^2/M\) and phase shifts \(\delta (k)\), oscillated) between positive and negative infinity. That completely stumped us. We chatted about this result with several few-body experts; no one had a clue about the origin of these oscillations. The cutoff dependence expected from the usual theoretical arguments was very different. If one looked diagram-by-diagram, one would expect the same cutoff dependence as in the \(J=3/2\) channel, that is, a small dependence suppressed by two powers of \(\Lambda \) (\(\sim 1/\Lambda ^2\)). After all, we are summing the same diagrams in both channels and the kernel of the integral equation that sums them is the same, up to normalization. If, instead, we resum the two-body interactions and use them to estimate the ultraviolet behavior of the three-nucleon amplitudes the cutoff dependence would be even more suppressed. What the numerics suggested, however, was a leading order \(\Lambda \)-dependence. The only thing we understood was that there was something really interesting and novel going on here and that was worth taking the risk of pursuing it. That was not a trivial decision. The years were passing fast for Bira and I (Hans was a little younger), we still had temporary jobs, and even 6 months without publishing a paper could have sad consequences in the job market. But we bet that we should put all our effort in figuring this out. At least in my personal case, I thank the nurturing support I had at the INT (Institute for Nuclear Theory at the University of Washington) for encouraging this kind of attitude. I try to do the same with students/postdocs working with me now.

The situation began to clear up when we discovered an old paper by Minlos and Faddeev analyzing the same equation for three-body scattering with short-range potentials [13]. They made the observation that in the regime \(q\gg 1/a\) (q being a loop momentum) the equation had scale invariance (since the only scale, 1/a, could be neglected) and power-law solutions of the form \(q^s\) could be found. The spectrum of allowed values of s was given by the solutions of a trigonometric equation and included one imaginary value \(s=\pm i s_0\), with \(s_0\approx 1.0064\ldots \). That led us to conclude a number of facts corroborated by the numerics. First, the half-off-shell amplitude as a function of the (off-shell/loop) momentum q was oscillatory in the logarithm of q, since \( q^{\pm is_0} \sim e^{i s_0 \log q}\) and the period matched Minlos and Faddeev’s prediction. Secondly, the half-off-shell amplitude had an asymptotic behavior that could not be guessed by looking at Feynmann diagrams individually. It was, in fact, much harder than what the perturbative analysis suggested. No surprise then that the cutoff dependence was stronger than expected. Finally, the off-shell scattering amplitude was a superposition of two functions, \(q^{is_0}\) and \(q^{-is_0}\) and so it depends on two constants. Matching to the infrared (\(q\sim 1/a\)) region fixes one of them. The other constant is determined by the matching to the ultraviolet region \(q\sim \Lambda \) and, therefore, depends on the value and shape of the cutoff. So the origin of the cutoff dependence became less mysterious. The consequence for the \(/\!\!\!\pi EFT\) was that this cutoff dependence renormalized the three-body force that had, therefore, to be a leading-order effect, contrary to everybody’s expectations and the Weinberg power counting valid for \(\pi EFT\). All this implied that it was unnatural to assume a vanishing three-body force even at the leading order of the low energy expansion. A quick and dirty argument gave us the dependence of the three-nucleon force on the cutoff. It was not only enhanced by two powers of the cutoff \(\Lambda \) but it also oscillated [14]:

$$\begin{aligned} H(\Lambda ) = -\frac{1}{\Lambda ^2}\frac{\sin (\Lambda /\bar{\Lambda }) -\text {arccot} s_0}{\sin (\Lambda /\bar{\Lambda }) +\text {arccot} s_0}. \end{aligned}$$
(1)

The scale \(\bar{\Lambda }\) is the analogue of \(\Lambda _{QCD}\) where quantum effects also break scale invariance.Footnote 2 What excited me at the time is that we could fit the value of \(\bar{\Lambda }\), say, to the triton binding energy \(B_3\) (not to \({}^3He\) because we didn’t include Coulomb corrections) and predict not only the \(J=1/2\) neutron-deuteron scattering length \(a_{1/2}\) but even the phase shifts to energies even well above the breakup threshold. Also, it clarified the observation that while all kinds of potential models predicted different values of \(B_3\) and \(a_{1/2}\), the predictions were correlated and clustered around a line—the Phillips line [17]—in the \(B_3 \times a_{1/2}\) plane. In fact, the effective theory can also be seen as an effective theory for each one of these potential models. Since they were tuned to reproduce two-nucleon scattering and nothing else, the presence of an unconstrained three-nucleon force parametrized by one and only one number (\(\bar{\Lambda }\)) explains the reason for a one-parameter family of results, the Phillips line. Another reason to be excited about these results was that, at the time, it looked like the proposal by Kaplan et al. [15] (“KSW”) of a different “pionfull” nuclear effective theory, whose leading order coincided with the pionless effective theory, was blossoming and looked that it would last. Thus, our observation that three-body forces were a leading-order effect could have a much larger validity, extending to the whole of nuclear physics, in direct contradiction to Weinberg’s explanation of the hierarchy among multi-nucleon forces (and the experience of the model builders). It turns out that it was quickly found out that the KSW proposal, that treated pions as perturbative, was not successful [16] and our result has a more limited application. Still, that paper is cited today at the same rate as it was 20 years ago. I believe its longevity is due to reasons I could not possibly have guessed at the time and has little to do with merely changing the power counting in a theory effective valid in a small corner of nuclear physics.

3 Unforeseen influence

One reason for the longevity of this work on the three-body problem had was that limit cycles in renormalization group flows were thought to be impossible. I personally never gave much importance to this. The c-theorem [18] (or the more recent a-theorem [19]) do not forbid this behavior [22] and the physics seemed straightforward: as the momentum cutoff is increased the phase shifts stay fixed but more bound states appear periodically at the bottom of the spectrum. Still, it attracted the attention of K. Wilson who, in his early work on the renormalization group, had considered and then discarded the possibility of limit cycles. After our paper was published he and S. Glazek discussed this possibility both in the three-particle case and also in other models [23]. The other reason for not being surprised about the oscillatory behavior of the three-nucleon force was that they started being seen everywhere, including in the quantum mechanical treatment of singular potentials (steeper than \(\sim 1/r^2\) at the origin). In fact, our observation of an unexpected renormalization of the three-body force led us [24] and many other authors [25] to revisit the power counting of the (pionful) nuclear effective theory since pion exchange leads to very singular potentials between nucleons. The lessons found there were similar to what we found in the three-body sector: short distances interactions dependence on the cutoff is different than one would naively (perturbatively) estimate and can be a fractional or even complex power of the cutoff (in which case it exhibits cyclic behavior). This seems like a path for completing Weinberg’s work by understanding how to use chiral potentials non-perturbatively and renormalizing amplitudes—not potentials—consistently. Many pieces of this puzzle are known but, to my knowledge, the big picture is still lacking. That did not stop the chiral potential, obtained as recommended by Weinberg, to become the nucleon-nucleon potential used in nuclear physics. The price to pay for this boldness is that some modeling (that is, unjustified assumptions) sips into these purportedly model-free calculations, but the community seems to accept this bargain.

I have told the development of these ideas from my personal perspective but the real history of the problem predated us by a lot of time. Already in the early days of quantum mechanics, Thomas pointed out that the hamiltonian of three bosons (or fermions with different quantum numbers) and short distance interactions was not bounded from below; there are states with arbitrarily large binding energy where the three particles collapse into one point [27]. The paper by Minlos and Fadeev was already mentioned and it was the first that made use of the scale invariance in the regime between the infrared (1/a) and ultraviolet (the range of the forces R) scales. Danilov and Lebedev already in 1961 had understood that the 3-body integral equation did not determine a unique solution but matching the result to another piece of data in the three-nucleon sector, the binding energy of the triton, for instance, was enough to fix a unique solution [26]. Starting in the 1970s, apparently unaware of Danilov and Lebedev results, Efimov pointed out the infrared counterpart of this phenomena: the accumulation of three-particle bound states near the threshold as the two-particle scattering length a diverges (usually called the “Efimov effect”) [28,29,30]. The three-particle spectrum has a logarithmic form with successive states appearing with a binding energy ratio of \(E_{n+1}/E_n =e^{2\pi /s_0}\) between the scales \(1/Ma^2 \ll E \ll 1/MR^2\). Efimov did not make the connection with the renormalization framework but fully understood the physics of the problem and the origin of the new, three-body scale \(\bar{\Lambda }\). The Efimov effect remained a curiosity, a “pathology” [31] of the zero-range model with no connections to other physical systems. Only after the Efimov effect was incorporated into the larger world of ideas it was fully appreciated. From 1970 to 1997, Efimov’s original article was cited about 60 times; in the next 27 years, it was cited another 550 times. This is due, in part, to the advent of cold atom physics and the realization that they form a perfect physical realization of the “pionless” EFT, with unnaturally large scattering lengths and all [32, 33].

Over the last 23 years, a number of developments occurred in the field. After some discussion, it seems settled that there is another, independent three-body force only at next-to-next-to-leading order in the low energy expansion [34,35,36], although a next-to-leading order a dependent component exists [37, 38]. The \(/\!\!\! \pi EFT\) has found applications well beyond the orginal context of few nucleons physics. Besides cold atoms [39, 40], D meson “molecules” [41] and \({}^4\)He atoms are also described by \(/\!\!\! \pi EFT\). Renormalization group limit cycles also appeared elsewhere and now it seems they are generic in on-perturbative RG flows [20, 21].

As I mentioned above, K. Wilson, one of my scientific heroes, did take an interest in our work on the three-body problem and I had the opportunity to talk to him about it, in what was truly a highlight in my career. But I don’t know if Weinberg ever knew about our limit cycles. Of course, fully aware of the large role he played in the Physics community and as a public intellectual and not particularly interested in nuclear physics beyond that brief period in the 90’s, he may never have had the time or the opportunity. I have a feeling he would have found it, at least, amusing.