I give an outline of recent applications of the renormalization group to effective theories of nuclear forces, focusing on the use of a Wilsonian approach to analyse systems of two or three non-relativistic particles.
The last 20 years have seen a major shift in how we describe the forces between nucleons. This was triggered by the suggestion of Weinberg [1,2] that the ideas of effective field theory (EFT) could be applied to these strongly interacting systems. This approach offers the possibility of a systematic, model-independent treatment that, ultimately, could form a bridge to the underlying theory of the strong interaction, quantum chromodynamics (QCD).
EFTs are built out of fields corresponding to the appropriate low-energy degrees of freedom and contain all possible terms consistent with the symmetries of the underlying dynamics. To have predictive power, such a theory must be expandable in powers of ratios of low-energy scales to those of the underlying physics. For nuclear physics, the low-energy scales, denoted generically by Q, include the momenta of the nucleons and the mass of the pion. The underlying scales of QCD, denoted by Λ0, include 4πFπ (the scale associated with the hidden chiral symmetry) as well as the masses of the nucleons and ρ, ω and other mesons.
Since Weinberg made his original proposal, debate has raged within the community over the appropriate power counting to use in organizing this expansion in powers of low-energy scales. Extensive reviews can be found in the articles by Beane et al. , Bedaque & van Kolck , Epelbaum  and Epelbaum et al. , and more recent summaries from two different points of view have been given by Epelbaum & Gegelia  and Birse . The main bones of contention have been: how do we renormalize non-perturbative systems consistently, and which pieces of the interaction should we iterate to all orders?
To get unambiguous answers to these questions, we need a rigorous tool to analyse the scale dependences of physical systems, and this is provided by the renormalization group (RG). I concentrate here on the version developed in Manchester [9–13], which uses a Wilsonian approach to construct functional RG equations. These can be solved exactly, at least in simple cases. However, many of the results discussed here were first obtained from more heuristic RG equations with momentum–space cutoffs [14–24], by using dimensional regularization with subtraction of power-law divergences [25–30] or with a Bogoliubov–Parasiuk–Hepp–Zimmermann subtractive renormalization scheme [31,32].
In addition, there is a powerful functional RG based on the Legendre-transformed effective action [48,49]. This has recently been applied to few-body (as well as many-body) systems of strongly interacting, non-relativistic particles [50–58]. As generally implemented, this is less rigorous than the versions of the RG mentioned above because it relies on truncations of the effective action that are not based on any systematic power counting. Nonetheless, it is proving very useful in situations where exact solutions of other RG equations cannot be found, such as four-body systems and dense matter.
Lastly, I should mention the approach known as Vlow-k [59,60] and related applications of the similarity RG to nuclear forces [61–64]. Both of these lead to effective interactions that evolve according to RG equations. However, the structures of the resulting equations are more complicated than those of the approaches mentioned above and, at least so far, it has not been possible to analyse their scaling behaviours.
Whatever regulator or subtraction scheme we choose, we should first identify fixed-point solutions of the RG equations. If we expand a general solution around one of these points, we can use the linearized RG equations to classify the perturbations as relevant, marginal or irrelevant. The eigenvalues of the linear equations give the anomalous dimensions of the corresponding operators, and hence can be used to construct a power-counting scheme. This counting is what makes it possible to define a systematic expansion of the corresponding EFT.
In systems whose particles interact very strongly at low energies, the terms in the potential can have large anomalous dimensions. Their scaling behaviour, and hence the power counting for the corresponding EFT, is then quite different from naive dimensional analysis. This is particularly true of systems close to the ‘unitary limit’, that is, with very large two-body scattering lengths. Nuclear forces are a prime example of this, since the nucleon–nucleon (NN) scattering lengths are of the order of 5–20 fm, much larger than the range of the interactions.
In such cases, the leading two-body interaction can become a relevant term, and other two-body operators are strongly promoted compared with naive expectations. The two-body sectors of the resulting effective theories are really just versions of the much older effective-range expansions [65–68]. The benefit of the modern field-theoretic framework is that it can provide consistent effective current operators as well as three and more body forces.
The first application of RG methods to three-body forces in theories with two-body contact interactions was to channels of three fermions with mixed-symmetry spatial wave functions. These give rise to a repulsive ‘particle-exchange’ force between an interacting pair and the third particle [15,69]. This work showed that three-body forces were not required to describe the low-energy physics in these systems, implying that there are no large, negative anomalous dimensions. Subsequent, more detailed RG analyses in momentum space  and in coordinate space  have determined the exact anomalous dimensions for these repulsive three-body channels.
In systems of three bosons or three fermions with a fully symmetric spatial wave function, the particle-exchange force is attractive. This can have dramatic consequences, most notably the Efimov effect [70,71]: a tower of bound states with energies in a geometric sequence. The RG flow in these systems displays a limit-cycle behaviour [11,17–19,72,73], which corresponds to anomalous breaking of scale invariance to a discrete remnant of the symmetry. The leading three-body force in these systems is a marginal term, corresponding to the starting point on the limit cycle. This means that one piece of three-body physics is sufficient to determine their low-energy behaviour, as has long been known from the Phillips line [74,75] that shows a correlation between the triton-binding energy and neutron–deuteron scattering length for forces fitted to the two-body scattering data.
In contrast, four-body systems show no evidence for relevant or marginal forces, even in cases where the three-body subsystems display Efimov behaviour [21,24,76]. This means that their low-energy observables are determined purely by two- and three-body physics, providing an explanation for the ‘Tjon line’ correlation between 4He- and triton-binding energies . No exact results are available for the scaling behaviour in four-body systems, but functional RG methods are now being used to estimate anomalous dimensions for both bosonic and fermionic systems [52,58].
All of the RG techniques mentioned above are applicable to any system of strongly interacting, non-relativistic particles, where they can be used to determine the pertinent power counting and hence to set up a consistent EFT description. Other areas where these ideas are now being applied within nuclear and hadron physics are weakly bound ‘halo’ nuclei [78–80] and the interactions of mesons containing heavy quarks [81–84]. Looking further afield, they are being used increasingly in studies of ultracold atomic systems (e.g. ).
2. Scales and scaling
Underpinning any viable EFT is an expansion in powers of the low-energy scales for a system. The RG can help to elucidate this scale dependence but, first, we need to identify all the relevant scales, Q. The most obvious ones for any low-energy theory are particles’ momenta, both on-shell and off-shell.
Typical momenta in nuclear systems are often of the order of 100 or 200 MeV, which is comparable to the mass of the pions. These mesons are not only the approximate Goldstone bosons of QCD, they also give rise to the longest range forces between nucleons. If we wish to describe physics on this scale, we need to include pions in our EFT, and to count their mass as one of our low-energy scales.
So far, this list of scales is just the same as in chiral perturbation theory (ChPT), Weinberg’s original EFT for low-energy meson physics . However, in contrast to the case of pions, where the hidden chiral symmetry ensures that its interactions are weak at low energies, nucleons interact strongly, forming bound states (nuclei).
To see why this can lead to problems with extending ChPT to two or more nucleons, we should look at the non-relativistic loop diagram for NN scattering in figure 1. For contact interactions, this has the form 2.1 As noted by Weinberg [1,2], this is enhanced to order Q, instead of Q2 as in the relativistic case. Nonetheless, the leading terms of the potential are of order Q0 (one-pion exchange (OPE) and the simplest contact interaction), and so each iteration is suppressed by a power of Q/Λ0. The theory is therefore still perturbative, provided Q<Λ0.
In fact, the analytical part of the integral (2.1) is linearly divergent and so we need to either cut it off or subtract it at some scale q=Λ. Iterating the potential then leads to contributions with powers of Λ/Λ0. These will again be perturbative provided we keep our cutoff within the domain of our EFT, Λ<Λ0, and so they cannot generate bound states.
We therefore need to identify further low-energy scales. Of particular interest are any that promote some of the interactions to order Q−1 (making them marginal terms in RG language) since these can, and indeed must, be treated non-perturbatively. The first examples to be identified in NN scattering were provided by the S-wave scattering lengths [9,15,16,25,26].
In addition, there are scales associated with long-range interactions. A simple example is provided by the Coulomb potential between two charged particles, such as two protons. Here, after we scale the nucleon mass MN out of the Hamiltonian, the strength of interaction can be expressed in terms of the inverse Bohr radius, 2.2
The long-range pion-exchange forces can be expanded using the methods of ChPT. The leading piece is OPE, whose strength can be expressed in terms of the momentum scale 2.3 This is built out of high-energy scales in ChPT, 4πFπ and MN. Counting it as a high-energy scale would lead us to a perturbative treatment of OPE, as developed by Kaplan et al. [25,26]. Numerically, however, the value of λπNN is only about twice mπ, suggesting that it may be better viewed as a low-energy scale. If we do so, OPE is promoted to order Q−1, implying that it should be iterated.
Following a Wilsonian approach, the next step in the RG is to cut off our EFT at some arbitrary scale Λ, lying above the low-energy scales Q, but below the scale Λ0 of the underlying physics, as in figure 2. (This assumes good separation of these scales, as required for an EFT with a convergent expansion.)
Then, we can follow the evolution of our theory as we ‘integrate out’ more and more of the physics by lowering Λ. As we vary our arbitrary cutoff, we demand that physics (for example, the scattering matrix) be independent of Λ. This means that the couplings in our EFT must run with Λ to compensate for the physics we are integrating out. Ultimately, for Λ≪Λ0, we lose all memory of the underlying physics and the only scale left is Λ.
Finally, we rescale the theory by expressing all dimensioned quantities in units of Λ. At one of the end points of the RG flow, all couplings are then just numbers independent of Λ. We have arrived at a fixed point of the RG: a theory that describes a scale-free system. Two are shown in figure 3. The one on the left is stable: any nearby theory will flow towards it as the cutoff is lowered. In contrast, the one on the right has an unstable direction: the flow can take theories away from the fixed point unless they lie on the ‘critical surface’.
Close to a fixed point, we can find perturbations that scale with definite powers of Λ. They can be classified into three types.
— Λ−ν: relevant (super-renormalizable in the language of particle physics), for example, mass terms in quantum field theories like quantum electrodynamics;
— Λ0: marginal (renormalizable), for example, the couplings familiar in gauge theories like the Standard Model (typically these show a dependence on the cutoff); and
— Λ+ν: irrelevant (non-renormalizable), for example, the interactions in mesonic ChPT.
We now can expand our EFT around one of the fixed points using these perturbations. Their RG scaling maps directly onto the order in the usual power counting , an interaction running as Λν corresponding to a term in the EFT of order Qd where d=ν−1.
3. Short-range forces
To illustrate how RG methods can be applied to the scattering of non-relativistic particles, let us look at a system of two particles at energies where the range of the forces is not resolved (for example, two nucleons with an energy below about 10 MeV). This can be described by an effective Lagrangian with two-body contact interactions or, equivalently, a Hamiltonian with δ-function potentials. In momentum space, the S-wave potential can be written 3.1 where k and k′ denote the initial and final relative momenta, and the energy dependence is expressed in terms of the on-shell momentum .
Scattering can be described by the reactance matrix, defined similarly to the scattering matrix but with standing-wave boundary conditions. This has the advantage that it is real below the particle-production threshold. For S-wave scattering, it satisfies the Lippmann–Schwinger equation 3.2 where denotes the principal value. This integral equation sums up chains of the bubble diagrams in figure 1 to all orders.
With contact interactions, the integral over the momentum q of the virtual states is divergent and so we need to regulate it. Here, I follow the method developed by Birse et al.  and simply cut the integral off at q=Λ. Demanding that the off-shell K matrix be independent of Λ, 3.3 ensures that scattering observables will be independent of the arbitrary cutoff. If we write the integral equation for K in the schematic form 3.4 differentiating, it gives 3.5 where implies differentiation with respect to the cutoff on the integral. Since this involves the off-shell K matrix, we can use the integral equation for K to convert it into the form 3.6 This describes the evolution of the potential as the cutoff is lowered and states are integrated out of the effective theory. Written out explicitly, it is 3.7 Note that the use of the fully off-shell K matrix was essential to obtaining an equation involving only the potential. The corresponding equation for the evolution of Vlow-k [59,60] is based on the half-off-shell T matrix and so still involves the scattering matrix.
Our equation for the cutoff dependence of the effective potential is still not quite an RG equation: the final step is to express all dimensioned quantities in units of Λ. Rescaled momentum variables (denoted with hats) are defined by , etc., and a rescaled potential by 3.8 (The factor M in this corresponds to dividing an overall factor of 1/M out of the Schrödinger equation.) This satisfies the RG equation 3.9 The sum of logarithmic derivatives is similar to the structure of analogous RG equations used in other areas of physics; it counts the powers of low-energy scales present in the potential. The boundary conditions on its solutions are that they should be analytical functions of , and (since they should arise from an effective Lagrangian constructed out of ∂/∂t and ∇2).
Having constructed our RG equation, the first thing we need to do is to look for its fixed points—solutions that are independent of Λ. Let us start with the obvious one, the trivial fixed point, 3.10 Since this gives no scattering, it obviously describes a scale-free system.
To describe more interesting physics, we need to expand around the fixed point, looking for perturbations that scale with definite powers of Λ. These are eigenfunctions of the linearized RG equation. They have the form 3.11 and they satisfy the eigenvalue equation 3.12 Its solutions are 3.13 with k,l,m≥0 since only non-negative, even powers satisfy the boundary condition. The corresponding eigenvalues are 3.14 These are all positive and so the fixed point is stable. The eigenvalues ν simply count the powers of low-energy scales and can be written ν=d+1, where d is the ‘engineering dimension’ of an operator in the potential.
In addition, there are various non-trivial fixed points, all of which are unstable. The most interesting one is purely energy dependent. To study it, let us focus on potentials of the form V (p,Λ). The RG equation for these can be rewritten in the form of a linear equation for , 3.15
To find its fixed point, we set the left-hand side of this equation to zero. The resulting ODE can then be integrated easily. The solution that satisfies our boundary condition is 3.16 The precise form of this is regulator dependent, but the presence of a negative constant of order unity is universal.
Since this potential has no momentum dependence, the integral equation for K simplifies to an algebraic equation. In rescaled, dimensionless form, it can be written 3.17 The integral here is, up to a sign, the same as the one in itself and hence we get 3.18 The corresponding T matrix has a pole at p=0 and so the fixed point describes a system with a bound state at exactly zero energy. This is often called the ‘unitary limit’ of two-body scattering and it forms another example of a scale-free system.
More general systems can be described by perturbing around the fixed point. In particular, energy-dependent perturbations can be found by substituting 3.19 into the RG equation. The functions satisfy the eigenvalue equation 3.20 The solutions to this are powers of the energy, 3.21 with eigenvalues 3.22 The RG eigenvalues for these perturbations have been shifted by −2 compared with the simple ‘engineering’ power counting. There is one negative eigenvalue, and so the fixed point is unstable.
A slice through the RG flow for equation (3.9) is shown in figure 4. The two fixed points can be seen, as well as the critical line through the non-trivial one. As in the generic flow in figure 3, potentials close to this line initially flow towards the fixed point as we lower the cutoff, but are then diverted away from it. A potential to the right of the line is not quite strong enough to produce a bound state. As Λ passes through the scale associated with the virtual state, the flow turns to approach the trivial fixed point from the weakly attractive side. In contrast, a potential to the left of the critical line generates a finite-energy bound state. This state drops out of our low-energy effective theory when the cutoff reaches the corresponding momentum scale. As this happens, the RG flow takes the potential to infinity and it then reappears from the right, ultimately approaching the trivial fixed point from the weakly repulsive side.
Physical observables are given by the on-shell K matrix. Returning to physical units, this is 3.23 where the Cn are the coefficients of the RG eigenfunctions in . Comparing this with 3.24 we see that this expansion is just the effective-range expansion [65,66]. Note that the terms in the expansion of our effective theory correspond directly to scattering observables.
The discussion above deals only with energy-dependent perturbations around the non-trivial fixed point. There are also RG eigenfunctions that depend on the off-shell momenta. However, in contrast to the expansion around the trivial point, these do not appear at the same orders as the corresponding on-shell terms . The momentum-dependent terms have larger scaling dimensions and so appear at higher orders in the EFT.
It is worth noting that the promotion of terms in the expansion around the unitary limit can be understood from the form of the wave functions at short distances. Two particles in the unitary limit are described by irregular solutions of the Schrödinger equation. At small radii (in S waves), these behave as ψ(r)∝r−1. Any cutoff smears a contact interaction over range R∼Λ−1. If we require observables to be independent of Λ, we therefore need the extra factor of Λ−2 in the interaction to cancel the cutoff dependence from |ψ(R)|2∝Λ2 in its matrix elements. This provides the ‘anomalous dimension’ of −2 for the on-shell contact interactions.
4. Long-range forces
Exactly the same techniques can be applied to renormalize short-range interactions in systems with known long-range forces. Details can be found in the papers of Barford & Birse [10,11] and Birse [12,13]. The parameters of the resulting EFT again have a direct connection to scattering observables, either via a distorted-wave Born expansion (for weakly interacting systems) or via a distorted-wave effective-range expansion (for strong short-range interactions) [65,67,68].
The simplest example of these is the 1/r2 centrifugal barrier in partial waves with non-zero angular momentum L . This leads to wave functions that behave as rL for small r and so, near the trivial fixed point, short-range interactions scale with an additional power of Λ2L compared with S waves. This matches exactly with the number of derivatives needed to produce a contact interaction with a non-zero contribution in these waves.
The RG flow in these higher partial waves can also end up at fixed points with multiple unstable directions. However, these turn out not to be physically realizable, even with enough fine tuning: either the wave functions are not normalizable  or causality is not respected [88,89].
In three-body systems with short-range pairwise interactions, long-range forces are generated by exchange of one particle between an interacting pair and the third particle. Close to the unitary limit, these can lead to scaling behaviour of three-body interactions that is quite different from what naive dimensional analysis would suggest. The reasons for this can be seen most clearly in position space if we work in hyperspherical coordinates. At short distances, the ‘particle-exchange’ potential has a 1/R2 form, where R is the hyper-radius (the radial coordinate that is zero when all three particles coincide).
In three-body channels where this long-range potential is repulsive, it acts just like a centrifugal term but with a non-integer ‘angular momentum’. This leads to wave functions that vanish as powers of the hyper-radius as it tends to zero and hence to irrational anomalous dimensions for the three-body contact interactions . The values for these have also been obtained from a momentum–space treatment  and by Werner & Castin [90,91] from the eigenvalues of three-body systems trapped in a harmonic oscillator potential.
Systems of three bosons or three distinct fermions with a completely symmetric spatial wave function behave very differently. Close to the unitary limit, these display the towers of geometrically spaced bound states known as the Efimov effect [70,71]. This is a consequence of the attractive 1/R2 potential in these systems, which leads to wave functions with the form ψ(R)∝R−2±is0, with s0≃1.006 as R→0. As a result, the leading three-body force is promoted to a marginal term, of order Q−1. The oscillatory behaviour associated with the imaginary part of the exponent is the origin of the Efimov effect. It causes the RG flow in these systems to tend to a limit cycle instead of a fixed point [11,17–19,72].
In the presence of the Coulomb potential, the scaling behaviour of the short-range interactions is the same as that described in §3 for systems without long-range forces [10,30]. This is because the 1/r singularity of the long-range potential is not strong enough to change the power-law behaviour of the wave functions at short distances. For strongly interacting systems, such as proton–proton scattering, the resulting EFT embodies the Coulomb distorted-wave effective-range expansion [28,29].
Perhaps the most long-range important potential for nuclear physics is OPE. Although formally of order Q0 within the framework of ChPT, the large value of the pion–nucleon coupling means that it plays a central role in nuclear forces. The unnaturally small value of the scale λπNN associated with OPE suggests that it should be added to our list of low-energy scales. This implies that OPE should be treated non-perturbatively.
The central piece of OPE is the only one that contributes to scattering in spin-singlet waves. Like the Coulomb potential, this has a 1/r singularity, and so it does not alter the power-law forms of the wave functions at small r. The scattering in singlet waves with L≥1 is weak, and so the corresponding effective potential can be expanded using naive dimensional analysis. In contrast, the 1S0 channel has a low-energy virtual state, and so the expansion of its short-range potential is like the one around the unitary fixed point.
The tensor piece of OPE is important in spin-triplet waves. It has a much stronger, 1/r3, singularity at the origin. The resulting short-distance wave functions have the form ψ(r)∝r−1/4, multiplied by either a sine or an exponential of (λπNNr)−1/2. As a result, short-range interactions are strongly promoted in these waves and so a new power counting is needed, as observed by Nogga et al. . An RG analysis shows that the leading contact interaction is of order Q−1/2 in waves with L=1 or 2 . In the 3S1 wave, there is a further enhancement of the short-range interactions, analogous to that in the 1S0 wave, associated with the deuteron bound state.
Turning now to three-nucleon forces, two-pion exchange interactions are purely long-range and so are not renormalized by the effects discussed here. In contrast, other interactions, involving two- or three-nucleon contact operators, are affected by the short-distance behaviour of the wave functions. For example, OPE terms of the type represented by figure 5 contain two-nucleon-one-pion contact operators, and these are substantially enhanced if either the initial or final pair is in an S wave. The most important of these operators is one that couples the 1S0 and 3S1 NN channels. This is promoted to order Q5/4 by the non-perturbative treatment of the two-body forces in these channels. In addition, there can be strong promotion of operators that couple S and P waves, and of those that couple various combinations of P and D waves. The effect of tensor OPE on three-body contact interactions is currently unknown. The leading term is expected to be promoted, albeit less dramatically than in the case of pure short-range forces. Determining the anomalous dimension for this force will entail solving the three-body problem with 1/r3 two-body potentials. I have recently summarized the results obtained from a range of RG studies of two- and three-nucleon forces .
Finally, I should mention four-body systems. So far at least, these have not yielded to a detailed RG analysis of the sort outlined here. Numerical treatments of these systems have found no signs of promotion of four-body forces to relevant or marginal terms [21,24,76], but the anomalous dimensions of these forces remain to be determined. In this context, the RG for the Legendre-transformed effective action  may provide some help. Applications of this rely on truncations of the effective action to a small number of local terms and, while less rigorous than truncations based on a specific power counting, these have proved remarkably successful for a wide variety of systems . Recently, the first applications have been made to few-body systems that make it possible to estimate the scaling dimensions of three- and four-body forces in both bosonic and fermionic systems close to the unitary limit [52,58].
I am grateful to my collaborators, especially T. Barford, J. McGovern, D. Phillips and K. Richardson, for their contributions to the ideas summarized here. I must also give special thanks to Y. Meurice and S.-W. Tsai, not only for the organization of the 2010 INT Workshop on ‘New applications of the renormalization group method’ but also for their repeated encouragements to keep working on this review. I am indebted to the Institute for Nuclear Theory for its hospitality and support during several visits over the last 2 years. This work was supported by the UK STFC under grant ST/F012047/1.
One contribution of 11 to a Theme Issue ‘New applications of the renormalization group in nuclear, particle and condensed matter physics’.
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