## Abstract

After a short elementary introduction to the exact renormalization group for the effective action, I discuss a particular truncation of the hierarchy of flow equations that allows for the determination of the full momentum of the *n*-point functions. Applications are then briefly presented, to critical *O*(*N*) models, to Bose–Einstein condensation and to finite-temperature field theory.

## 1. Introduction

The exact, or non-perturbative, renormalization group (RG) [1–5] stands out as a very promising formalism to address non-perturbative problems, i.e. problems whose solution does not appear to be expressible as an expansion in some small parameter. It leads to exact flow equations that cannot be solved in general, but which offer the possibility for new approximation schemes. It has been applied successfully to a variety of problems, in condensed matter, particle or nuclear physics (for reviews with various points of view on the subject, see e.g. [6–12]).

When only correlation functions at small momenta are needed, as is the case, for instance, in the calculation of critical exponents, a general approximation method to solve the infinite hierarchy of the flow equations has been developed [5,8,9]. This method is based on a derivative expansion (DE) of the effective action. However, in many situations, this is not enough: in order to calculate the quantities of physical interest, knowledge of the full momentum dependence of the correlation functions is mandatory.

This paper deals with this issue, and summarizes work that has been done following the original suggestion by Blaizot, Méndez-Galain and Wschebor (BMW) [13–15] to obtain the momentum dependence of *n*-point functions from the flow equations. The strategy put forward in Blaizot *et al*. [13] is based on the fact that the RG flow at scale *κ* involves the integration of fluctuations with momenta *q*≤*κ*, as ensured by the presence of a cutoff function *R*_{κ}(*q*). Since this cutoff function also guarantees that the vertex functions are smooth functions of the momenta, these can be expanded in powers of *q*^{2}/*κ*^{2}. The ‘leading order’ (LO) of the approximation scheme consists in a truncation at the level of the flow equation for the two-point function , setting *q*=0 in the vertices and that appear in this flow equation. Doing so, and working in a constant external field, one can then express and as derivatives of with respect to the background field, thereby closing the hierarchy of flow equations. Next-to-leading orders are defined by similarly truncating the hierarchy at the level of higher point functions.

The price to pay is that the flow equations become differential equations with respect to a uniform background field, with integral kernels that involve the solution itself. These nonlinear integro-differential equations are *a priori* difficult to solve. It is possible to do so, however, with a numerical effort comparable with that involved in solving the flow equations that result from the DE. We shall provide here examples of results obtained with this method in various contexts.

The outline of this paper is as follows. In §2, we briefly recall some basic features of the exact RG, and derive the flow equation for the effective action. We also digress on the use of the variational principle to obtain flow equations in the context of the Hamiltonian formalism often used in dealing with non-relativistic systems. Then, in §3, we discuss approximation schemes, such as the local potential approximation (LPA), and the BMW approximation scheme. Connections with the formalism of the two-particle irreducible effective action are also discussed. In §4, we present three applications within scalar field theory with *O*(*N*) symmetry in which the full momentum dependence of the two-point function plays an essential role. This paper ends with a short conclusion.

## 2. Exact renormalization group flow equations

In this section, we give an elementary introduction to the exact RG, focusing on the effective action. For most of the discussion (except for the digression in §2*b*), we shall restrict ourselves to the case of a scalar field theory with the classical action
2.1whose parameters *m* and *u* are defined at some ‘microscopic scale’ *Λ*, to be specified.

### (a) Flow equations for the *n*-point functions and their generating functional

The strategy of the version of the RG that we consider here is to build a family of theories indexed by a continuous parameter *κ*, with the dimension of a momentum, and such that fluctuations are smoothly taken into account as *κ* is lowered from the microscopic scale *Λ* down to 0. In practice, this is achieved by adding to the original Euclidean action *S* a (non-local) term, quadratic in the fields, of the form
2.2The regulator, or cutoff function, *R*_{κ}(*q*^{2}) is chosen so that: (i) *R*_{κ}(*q*≪*κ*)∼*κ*^{2}, which effectively suppresses the contribution of the modes by giving them a mass ∼*κ*; and (ii) it vanishes rapidly for , leaving the modes unaffected. The explicit form of the cutoff function *R*_{κ}(*q*) used in actual calculations will be specified later.

We consider then the ‘deformed’ (non-local) field theory with action *S*_{κ}≡*S*+Δ*S*_{κ}, and the functional integral that yields the generating functional of Green’s functions
2.3with . The expectation of the field in the presence of the source *j*(*x*) is given by
2.4Similarly,
2.5where we used here the fact that all the dependence on *κ* is contained in the regulator term Δ*S*_{κ}[*φ*]. Note that the expectation values in equations (2.4) and (2.5) are taken in the presence of the regulator, and depend therefore on *κ*. At this point, we define the effective action for the deformed theory, *Γ*_{κ}[*ϕ*], through the Legendre transform
2.6The functional *Γ*_{κ}[*ϕ*] is the generating functional of the one-line irreducible *n*-point functions of the deformed theory. Taking into account that, at fixed *ϕ*, *j* depends on *κ*, one easily obtains the flow of *Γ*_{κ}[*ϕ*] as
2.7It is in fact convenient to redefine *Γ*_{κ} by subtracting Δ*S*_{κ} from it. This subtraction has the advantage of making *Γ*_{κ} coincide with the classical action *S* at the microscopic scale *Λ* (rather than making it coincide with *S*_{Λ}=*S*+Δ*S*_{Λ}, as would be the case for the definition (2.6)). Furthermore, it modifies the flow equation in such a way that only the field fluctuations are involved. That is, once the subtraction is made, the flow is driven by the *connected* two-point function [1]:
2.8where
2.9is the full propagator in the presence of the background field *ϕ*. Using well-known properties of the Legendre transform, one can relate *G*(*q*,−*q*;*ϕ*) to , the second functional derivative of *Γ*_{κ}[*ϕ*] with respect to *ϕ*:
2.10

The initial conditions on the flow equation (2.8) are specified at the microscopic scale *κ*=*Λ* mentioned above. This scale *Λ* is the scale where fluctuations are damped by Δ*S*_{κ}, so that *Γ*_{κ=Λ}[*ϕ*]≈*S*[*ϕ*]. It should be emphasized, however, that this issue of the initial condition at the scale *Λ* is a subtle one. It is intimately related to that of renormalization and ultraviolet (UV) divergences, issues that will not be discussed here. The effective action of the original scalar field theory is obtained as the solution of equation (2.8) for , at which point *R*_{κ}(*q*^{2}) vanishes. Whether at that point the result obtained for *Γ*[*ϕ*] is independent of the choice of *R*_{κ}, that is, on the path joining the classical action to the full effective action, is another subtle question, whose answer depends on the approximation made. We shall comment on it further when we discuss specific approximations.

By taking successive functional derivatives of equation (2.8) with respect to *ϕ*, and then letting the field be constant, one gets the flow equation for the *n*-point functions,
2.11in a constant background field. Since the background is constant, these functions are invariant under translations of the coordinates, and it is convenient to factor out of the definition of their Fourier transform the *δ*-function that expresses the conservation of the total momentum. Thus, with an obvious abuse of notation, we define the *n*-point functions as
2.12We use here the convention of incoming momenta, and it is understood that in the sum of all momenta vanishes, so that is actually a function of *n*−1 momentum variables. We shall often use the simplified notation for the function .

When *ϕ* is constant, the functional *Γ*_{κ}[*ϕ*] itself reduces, to within a volume factor *Ω*, to the effective potential *V* _{κ}(*ϕ*):
2.13The flow equation for the effective potential *V* _{κ} follows from that of the effective action *Γ*_{κ}, equation (2.8), when restricted to constant *ϕ*. It reads
2.14where
2.15

By taking two derivatives of equation (2.8) with respect to *ϕ*, and then letting the field be constant, one obtains the equation for the two-point function:
2.16A diagrammatic illustration of this equation is given in figure 1. The flow equations for the *n*-point functions, such as equation (2.16), constitute an infinite tower of coupled equations. The coupling between equations occurs in two ways, upwards and downwards. Upwards: typically the equation for *Γ*^{(n)} involves *Γ*^{(n+1)} and *Γ*^{(n+2)}. Downwards: all the flow equations involve *Γ*^{(2)}, which is coupled successively to all the equations above it. There exist also consistency conditions that we illustrate here with *Γ*^{(2)} and the effective potential: *Γ*^{(2)}(*q*=0)=∂^{2}*V*/∂*ϕ*^{2}. Thus, *Γ*^{(2)}(*q*=0) can be calculated by taking the second derivative with respect to *ϕ* of the flow equation for the effective potential:
2.17Alternatively, it can be obtained from equation (2.16), where we set *p*=0. The two calculations agree owing to the facts that
2.18We shall consider later approximation schemes that treat at different levels of accuracy the equations for various *n*-point functions. This may lead to violations of such consistency conditions if one is not careful.

Relations such as equation (2.18) between *n*-point functions with some vanishing external momenta and derivatives of *n*-point functions of lower rank with respect to a constant background field play an important role in the approximation scheme presented below. In this context, we simply note here that the effective potential may be viewed as the generating functional of the *n*-point functions with vanishing external momenta. That is (for constant *ϕ*),
2.19In the same way, we may regard as the generating functional of *n*-point functions with one non-vanishing momentum,
2.20

### (b) Flow equation from the variational principle

The previous derivation of the flow equation did not refer to the interpretation of the various manipulations that were done in terms of ‘coarse graining’ or ‘elimination of degrees of freedom’, which are essential aspects of the RG. These are actually hidden in the specific functional form of the regulator *R*_{κ}(*q*), but nothing in the derivation of the flow equation actually depends on the choice of the regulator (aside from being quadratic in the fields, but even that can be generalized). Thus, one may view the flow equation as merely a tool to go continuously from a ‘simple’ action (the classical action) to the full effective action, by tuning an appropriate contribution to the action. This point of view opens new perspectives and suggests that the strategy of flow equations may have a larger flexibility than is presently exploited. As an illustration, we shall here indicate how the variational principle can be used to establish flow equations in terms of Hamiltonians and wave functions for non-relativistic systems.

The strategy we shall follow is identical to that of the previous section. The system is described by a Hamiltonian *H*, to which is added an ‘external field’ Δ*H*_{κ} that plays the role of the regulator. We assume that Δ*H*_{κ=0}=0, and that *H*_{Λ}≡*H*+Δ*H*_{Λ} is somehow ‘simple’, so that, for instance, one can determine its ground state. Then the flow equation will take us from the ground state of this simple Hamiltonian *H*_{Λ} to the ground state of the Hamiltonian *H*.

In this context, the initial condition of the flow is not restricted, as it is in most field theoretical applications, to be some ‘classical action’. Thus, it may be advantageous to depart from the standard approaches that would take *H*_{Λ} to be the Hamiltonian of independent particles, or quasi-particles, and include part of the effects of the interactions into *H*_{Λ}. In fact such a strategy was recently implemented by Machado & Dupuis [16], in their lattice RG, in which they separate the on-site physics from the long-range correlations, only the latter being treated by the RG. Much earlier, Parola & Reatto have developed a theory of liquids in which the short-range part of the interaction is treated (almost) exactly with hard-sphere systems, while the long-range correlations are treated by a flow equation [17–20] that actually resembles closely equation (2.8).

The flow equation for the effective action, equation (2.8), is general and could be used in the non-relativistic many-body context. However, to bring a new perspective to the discussion, we shall go through an extremely simple derivation of a flow equation that relies on the variational principle (J. P. Blaizot & T. Matsuura 2007, unpublished). For simplicity, we shall assume that Δ*H*_{κ} is a one-body operator, i.e. a quadratic form of creation and destruction operators, which depends on the continuous parameter *κ*:
2.21with and *a*_{q} the creation and destruction operators (of fermions or bosons). We wish to calculate the ground state energy of the Hamiltonian *H*_{κ}=*H*+Δ*H*_{κ}. We call |*Ψ*_{κ}〉 the corresponding eigenstate. In fact, we do not need the exact eigenstate; it is enough that |*Ψ*_{κ}〉 be determined from the variational principle,
2.22where it is understood that the regulator *R*_{κ} remains fixed in the variation. The flow equation for follows then immediately:
2.23where can be seen as an occupation factor. The equation above is the analogue of equation (2.5). It just describes the change in the ground state energy under the change of the parameters of the Hamiltonian. Of course, when some creation or destruction operators acquire expectation values (as in the case of Bose–Einstein condensation (BEC)), it may be convenient to work with the connected part of , as we did earlier. Note that the flow of the energy *E*_{κ}=〈*Ψ*_{κ}|*H*|*Ψ*_{κ}〉 reads
2.24This approach is useful if we can determine (approximately) the wave function, or at least the occupation factors, as a function of *κ*. It was applied in the study of the Bardeen–Cooper–Schrieffer to Bose–Einstein condensation (BCS–BEC) crossover (J. P. Blaizot & T. Matsuura 2007, unpublished): in this case the BCS wave function leads to a reasonably accurate estimate of the occupation factors and their dependence on *κ*, and this extremely simple scheme allowed us to reproduce qualitatively (and even semi-quantitatively) the results of more complete RG studies, such as those carried out in Birse *et al*. [21].

## 3. Approximation schemes

The flow equations for the *n*-point functions that have been presented in the previous section are exact. Their solution requires, in general, approximations. It is precisely one of the virtues of the formulation of field theory based on the exact RG to suggest approximations that are not easily derived in other, more conventional approaches. Particularly interesting are the approximation schemes for the effective action itself, that is, approximations that apply to the whole set of *n*-point functions at once. The approximation schemes to be discussed in this section have this property.

### (a) The local potential approximation

The LPA is the simplest of such approximations. It consists in assuming that for all values of *κ* the effective action takes the form
3.1Within this approximation, the two-point function, obtained by differentiating *Γ*_{κ}[*ϕ*] in equation (3.1) twice with respect to *ϕ*, and letting *ϕ* be constant, is of the form (*ρ*≡*ϕ*^{2}/2)
3.2The corresponding propagator is simply a massive propagator, with a *ϕ*-dependent mass *m*_{κ}(*ρ*). The flow equation for the effective potential *V* _{κ} becomes a closed equation that we write as follows:
3.3where *I*_{1}(*κ*,*ρ*) is given by
3.4Note that the form (3.3) of the equation for the effective potential is general: it would yield the exact effective potential if the propagator used to calculate *I*_{1} was the exact propagator, instead of the LPA propagator. Note also that, within the LPA, the equation for the two-point function at vanishing external momentum, given by equation (2.17), reduces to the closed equation
3.5since the derivatives of with respect to *ϕ* are identical to the derivatives of the effective potential, or, equivalently, to the derivatives of .

The LPA has been widely used, and, given its simplicity, the quality of the results obtained is quite good [9; 22–24]. Note that the LPA, by construction, yields *n*-point functions that have no intrinsic dependence on the external momenta: the generating functional of these *n*-point functions is given by equation (3.1) taken at *κ*=0, so that, for *n*>2, the LPA *n*-point functions are just derivatives of the effective potential (equation (2.19)). The LPA can be improved through an expansion in gradients of the fields, usually referred to as the derivative expansion (DE) [6,7,9,25,26]. Within the DE, the lowest *n*-point functions have polynomial dependence on the momenta (the degree of the polynomial corresponding to the order of the expansion in the derivatives). It follows, in particular, that the DE does not directly describe the anomalous behaviour of *n*-point functions at small momenta (anomalous dimensions can be recovered, for instance, from the scale dependence of the field normalization). In order to capture the full momentum dependence of the *n*-point function, a better truncation scheme is necessary: the BMW scheme [13], to be discussed next, achieves this goal.

### (b) The Blaizot, Méndez-Galain and Wschebor approximation

The BMW approximation scheme relies on two observations. First, the presence of the cutoff function *R*_{κ}(*q*) ensures that the *n*-point functions remain regular functions of the external momenta *p*_{i} as ; besides, it limits the internal momentum *q* in equations such as equation (2.16) to . In line with this observation, the approximation consists in neglecting the *q* dependence of the vertex functions on the right-hand side (r.h.s.) of the flow equations (e.g. set *q*=0 in *Γ*^{(3)} and *Γ*^{(4)} in equation (2.16)), while keeping the full dependence on the external momenta *p*_{i}. The second observation is that, for uniform fields, (a relation that we have already used, see equation (2.18)). This enables one to close the hierarchy of equations at some finite order. The order *m* of the scheme consists in keeping the full momentum dependence of all the *n*-point functions up to , and expressing and as derivatives of with respect to *ϕ*, after setting to zero the loop momenta that flows through and in the equation for . The accuracy of the scheme depends, of course, on the rank *m* at which one operates the truncation, but obviously the implementation becomes increasingly complicated as *m* grows. It is therefore gratifying that accurate results can be obtained with lowest-order truncations.

The approximations obtained by truncating the hierarchy at the lowest level, i.e. *m*=0, is identical to the LPA discussed in §3*a*. The next order of the approximation scheme, which we shall in fact refer to as the leading order of the BMW method (BMW-LO), consists in a truncation at the level *m*=2, i.e. at the level of the flow equation of the two-point function. The resulting equation is the closed equation
3.6where *J*_{3}(*p*,*κ*,*ρ*) and *I*_{2}(*κ*,*ρ*) are obtained from the general definitions in equation (3.4). This equation may be viewed as a generalization of equation (3.5) that takes into account the full momentum dependence of the two-point function. What the BMW approximation achieves is the possibility to factorize the vertices and take them out of the integrals. The momentum dependence that remains within the three- and four-point vertices is that of the two-point function itself (equation (2.20)). Interestingly, a very similar set of equations has been obtained much earlier in the context of the theory of liquids [17,18], but these were unknown to Blaizot *et al*. [13].

At this point, we recall the consistency condition discussed in connection with equation (2.17): the flow equation for can be obtained by taking the second derivative of the flow equation for the effective potential with respect to the background field. The resulting equation (2.17) does not coincide here with equation (3.6) in which we set *p*=0. This is because equation (3.6) results from the BMW truncation, which is not implemented in equation (2.17), and need not be. This results in a mismatch between the two ways of calculating *Γ*^{(2)}(*p*=0), whose origin can be traced back to the fact that the two-point function and the effective potential are not determined with the same accuracy: loosely speaking, the two-point function calculated from equation (3.6) is accurate to one-loop order, while the effective potential obtained from equation (3.3), with the BMW propagator, is accurate to two-loop order. In order to properly deal with this feature, we treat separately the zero momentum (*p*=0) and the non-zero momentum (*p*≠0) sectors, and write
3.7where Δ_{κ}(*p*=0,*ρ*)=0. Now, is obtained by solving the equation for the effective potential, while the equation for Δ_{κ}(*p*,*ρ*) can be easily deduced from that for , i.e. from equation (3.6), by subtracting the corresponding equation that holds for *p*=0. It reads
3.8where the symbol ^{′} denotes the derivative with respect to *ρ*, and .

This equation (3.8), together with that for the effective potential, equation (3.3), and that for the propagator, 3.9constitute a closed system of equations for , which can be solved with the initial condition .

### (c) Relation to the two-particle irreducible formalism

Because the exact RG formalism that we are using puts emphasis on the propagator, it is natural to ask for the connection with the two-particle irreducible (2PI) formalism [27–32]. Let us recall that the central quantity in this formalism is *Φ*[*G*], the sum of the 2PI ‘skeleton’ diagrams, a functional of the full propagator *G*. From *Φ*[*G*], one obtains the self-energy by functional differentiation (to within factors (2*π*)^{3}):
3.10This relation, together with Dyson’s equation
3.11defines the physical propagator and self-energy in a self-consistent way. We shall refer to equation (3.11), with *Σ*[*G*] given by equation (3.10), as the ‘gap equation’. A further differentiation of *Φ*[*G*] with respect to *G* yields the 2PI kernel,
3.12of a Bethe–Salpeter type equation,
3.13allowing calculation of the four-point function *Γ*^{(4)}(*q*,*p*)≡*Γ*^{(4)}(*q*,−*q*,*p*,−*p*): the quantity is the 2PI contribution to *Γ*^{(4)}(*q*,*p*) in one particular channel. If all skeletons are kept in *Φ*, these relations are exact. A *Φ*-derivable approximation [30,31] is obtained by selecting a class of skeletons in *Φ* and calculating *Σ* and *Γ*^{(4)} from the equations above.

The 2PI formalism provides a set of functional relations among the *n*-point functions that can be used to define a truncation of the flow equations [33]. Consider the equation for the two-point function for a vanishing background field, which reads:
3.14A possible truncation consists in using for on the r.h.s. of this equation, the extension of the relation (3.13) to the deformed theory, namely
3.15where the subscript *κ* on means that the functional derivative defining the kernel (equation (3.12)) is to be evaluated for , with . Since is a functional of the two-point function, the system of equations (3.14) and (3.15) is indeed closed.

One nice feature of this truncation scheme is that it is systematically improvable, by adding more skeletons to *Φ*: if all skeletons are included, the solution of the coupled system of equations (3.14) and (3.15) provides the exact two-point function as well as the exact four-point function for a particular configuration of the external momenta. A second attractive feature is that it preserves the property of the flow of being a total derivative with respect to the parameter *κ*. It is indeed not difficult to show, using equations (3.15) and (3.14), that
3.16where *Σ*_{κ}≡*Σ*[*G*_{κ}], with *Σ*[*G*] given by equation (3.10). This is a unique property of this truncation, which is not shared by most other popular truncations of the exact RG. (However, notable exceptions are the perturbative expansion, and the large-*N* approximation, e.g. [34]. A similar property of the flow equation was also obtained in the off-equilibrium context in Gasenzer & Pawlowski [35]. The BMW truncation does not respect this property.) Since it is a total derivative, the flow can be easily integrated to yield the gap equation whose solution is equivalent to a resummation of all the Feynman diagrams that are generated from the skeletons that are kept in the approximation considered. Because the solution of the gap equation corresponds to an exact resummation of selected Feynman diagrams, at the end of the flow where *κ*=0 and the regulator vanishes, the final result is rigorously independent of the choice of the regulator.

All these properties of the 2PI truncation may look at first disappointing from the point of view of the flow equations: indeed, all that the flow does in this particular truncation is to solve the 2PI equations! However, there is certainly interest in establishing direct connections between non-trivial non-perturbative approximations. In particular, because the 2PI truncations lead to flow equations that are exact derivatives, they could be used to test other approximations. Besides, from the point of view of the 2PI formalism, there is a practical advantage in reformulating the gap equation as a flow equation: this is because initial value problems are in general easier to solve than nonlinear gap equations. Furthermore, regarding the 2PI equations as flow equations sheds a new light on the renormalization of *Φ*-derivable approximations [36–38].

## 4. Applications

We turn now to a few specific applications of the non-perturbative RG using the BMW truncation scheme. Most of these applications concern scalar field theories with an *O*(*N*) symmetric action of the generic form
4.1The first application covered in this section may be seen as a ‘classic’ application within the context of the RG: it concerns the critical *O*(*N*) models, and the calculation of critical exponents and the scaling functions [39,40]. The second application refers to the shift of the transition temperature of the BEC in a dilute Bose gas: this again involves the *O*(*N*) field theory (with *N*=2), but the relevant quantity to be calculated is sensitive to the whole momentum range of the two-point function, and not only to the critical momentum region [34]. The last application concerns the thermodynamics of quantum fields. It shows how the exact RG allows one to circumvent the specific difficulties of perturbation theory in such systems. The physics motivation is the physics of the quark–gluon plasma, but some of the essential difficulties of perturbation theory in quantum chromodynamics (QCD) at finite temperature are well illustrated again by the scalar field [41,42].

Before turning to these applications, we need to comment on the dependence of the regulator in practical calculations. We shall use in our calculations a cutoff function of the generic form
4.2where *Z*_{κ} is a function of *κ* only. This factor *Z*_{κ} reflects the finite change in normalization of the field between the scale *Λ* and the scale *κ*. It can be defined as
4.3with *ρ*_{0} and *p*_{0} *a priori* arbitrary. In practice, one usually chooses *p*_{0}=0 and *ρ*_{0} to be the value of *ρ* at the minimum of the effective potential. The factor *Z*_{κ} enters the scaling to dimensionless variables used for the numerical solution in the critical region.

The results of the calculations presented below were obtained with an exponential regulator
4.4where *α* is a free parameter. As we shall see, physical quantities exhibit a small dependence on *α*. Since, in the absence of any approximation, they would be strictly independent of the cutoff function, a study of this spurious dependence provides an indication of the quality of the approximation [10,43].

### (a) Critical *O*(*N*) models

The equations of the BMW method have been solved first with additional approximations in Blaizot *et al*. [44] and Benitez *et al*. [45], but the results presented here were obtained by solving numerically the nonlinear integro-partial-differential equations (3.8) and (3.3) without any further approximation [39]. The numerical techniques used are described in Benitez *et al*. [39,40]. Here, we report only some selected results, and comment on important aspects of the approximation that are needed in order to gauge the quality of these results.

Table 1 contains results for the critical exponents *η* and *ν*, in dimension *d*=3 and for various values of *N*, together with some of the best estimates available in the literature. To appreciate the quality of these results, let us recall that these depend *a priori* on the parameters *α*, *p*_{0} and *ρ*_{0} (equations (4.2) and (4.3)). (They also depend on the functional form of the cutoff function, but no systematic exploration has been performed to study this dependence.) In all cases studied, we find the dependence on *p*_{0} and *ρ*_{0} to be much smaller than that on *α*, so that only the latter needs to be considered. As a function of *α*, physical quantities typically exhibit a single extremum *α** located near *α*=2. Moreover, we find this extremum generally pointing towards the best numerical estimates. Since, in the absence of any approximation, there should be no dependence on *α*, we regard these extremum values, being locally independent of *α*, as our best values, adopting in doing so a strategy often referred to as the principle of minimal sensitivity (PMS) [53]. Our numbers are then all given for the PMS values *α** of the regulator parameter, and the digits quoted are those which remain stable when *α* varies in the range . The quality of the values obtained for the critical exponents is obvious: for all *N* the results for *ν* agree with the best estimates to within less than 1 per cent; as for the values of *η*, it is typically at the same distance from the Monte Carlo (MC) and high-temperature series estimates as resummed perturbative data. For *N*=100, we find *η*=0.0023 and *ν*=0.990, which compare well with the values *η*=0.0027 and *ν*=0.989 obtained in the 1/*N* expansion [54]. Our numbers also compare favourably with those obtained at order ∂^{2} in the DE scheme [25,26].

The two-dimensional case, for which exact results exist, provides an even more stringent test of the BMW scheme. We have results at the moment only for the Ising model, i.e. for *N*=1, which exhibits a standard critical behaviour in *d*=2. The perturbative method that works well in *d*=3 fails here: for instance, the fixed-dimension expansion that provides the best results in *d*=3 yields, in *d*=2 and at five loops, *η*=0.145(14) [55], in contradiction with the exact value . We find instead *η*=0.254 and *ν*=1.00. Note, however, that no systematic study of the dependence of this result on the regulator parameter has been performed yet.

The BMW scheme yields the complete momentum dependence of the two-point function. All expected features of at criticality are observed: in the infrared (IR) regime *κ*≪*p*≪*u*, and this IR behaviour of can be used to extract the value of *η*; the value obtained directly from the momentum dependence of is in excellent agreement with that deduced from the *κ* dependence of the field normalization factor *Z*_{κ} (*Z*_{κ}∼*κ*^{−η}). The UV regime *κ*,*u*≪*p*≪*Λ* exists if *u* is sufficiently small; this regime can be studied perturbatively and one finds that, in LO, . The present approximation reproduces this logarithmic behaviour with, however, a prefactor 8 per cent larger than the two-loop result. Note that the complete two-loop behaviour can be recovered by a simple improvement of the BMW scheme [40].

The momentum dependence of the two-point function can be further tested by analysing the so-called scaling functions
4.5where *χ*^{−1}≡*Γ*^{(2)}(*q*=0) and *ξ* is the correlation length, which diverges close to criticality with the *ν* critical exponent. Here ± refers to the two phases, above and below the critical point, respectively. The functions *g*_{±} are universal. We consider here the case *N*=1. The scaling function *g*_{+}(*x*=*qξ*) has been calculated in the BMW approximation for different values of the correlation length. It is plotted in figure 2, where we can see that the scaling is perfectly reproduced. The agreement with the experimental data is also excellent, although this is somewhat deceptive since, in this range of momenta, the physics is dominated by mean-field effects, and the scaling function deviates only slightly from that predicted by the Ornstein–Zernicke approximation. A critical study is presented in Benitez *et al*. [40].

### (b) Temperature of Bose–Einstein condensation

A quantity particularly sensitive to the UV–IR crossover region is the shift, due to interactions, of the critical temperature of the dilute, weakly interacting, Bose gas [57]. The Hamiltonian describing such a system is typically of the form
4.6where *g*=4*πa*/*m*, with *a* the s-wave scattering length. (We are ignoring here a subtlety related to the UV divergences that are generated in calculations with a contact interaction and that require the introduction of a UV cutoff. This can be implemented in a standard fashion, but it plays no role in the present discussion.) The effective Hamiltonian (4.6) provides an accurate description of dilute systems, when the scattering length is small compared with the interparticle distance, i.e. *an*^{1/3}≪1. In the vicinity of the condensation, where *n*^{1/3}*λ*≃1, with the thermal wavelength, the diluteness condition reads *a*/*λ*≪1.

It has been shown that the shift Δ*T*_{c} of the BEC temperature is linear in *an*^{1/3} [57]:
4.7Here is the condensation temperature of the ideal gas, defined by the condition *nλ*^{3}=*ζ*(3/2)≈2.612, and *T*_{c} is the transition temperature of the interacting system at the same density. This result is non-trivial: although the shift is proportional to *a*, and hence is small if *a* is small, the result (4.7) cannot be obtained from perturbation theory. It is obtained through the following chain of arguments. In the limit of small coupling, the shift of the critical temperature may be obtained from the shift of the critical density, . The latter quantity is easier to calculate, and is dominated by the contribution of the zero Matsubara frequency part, *ψ*_{0}, of the bosonic field, whose dynamics is governed by a three-dimensional classical field theory with *O*(2) symmetry. To make the connection with the notation used in the rest of this section, we set , with *φ*_{1} and *φ*_{2} two real fields. The action for these real fields is given by equation (4.1), with the parameters *r* and *u* related to the parameters of the Hamiltonian (4.6) by
4.8The resulting density shift is then given by the change in the fluctuation of this classical field, with the coefficient *c* in equation (4.7) given by
4.9in the limit (and for *N*=2).

The calculation of is difficult for the reason already mentioned: although the coupling constant *u* can be arbitrarily small, perturbation theory cannot be used because of the IR divergences of the critical three-dimensional field theory. The best numerical estimates for Δ〈*φ*^{2}〉, and hence for *c*, are those which have been obtained using the lattice technique by two groups, with the results: *c*=1.32±0.02 [58,59] and *c*=1.29±0.05 [60,61]. The availability of these results has turned the calculation of *c* into a testing ground for other non-perturbative methods: expansion in 1/*N* [62,63], optimized perturbation theory [64–66], and resummed perturbative calculations to high loop orders [67].

To understand better the origin of the difficulty in the calculation of *c*, as well as the linearity in *a* of Δ*T*_{c}, let us write as the following integral:
4.10where *Σ*(*p*) is the self-energy at criticality, i.e. *Σ*(0)=0. In equation (4.10), the term within the square brackets is, to a very good approximation (when *p*/*u* is small enough), equal to *Σ*(*p*)/*p*, a function that is peaked in the region of intermediate momenta between the critical region and the high-momentum perturbative region. Thus, the difficulty in getting a precise evaluation of the integral (4.10) is that it requires an accurate determination of *Σ*(*p*) in a large region of momenta including the crossover region between two different physical regimes [62,68]. In that sense, the calculation of *c* can be viewed as a very stringent test of the approximation scheme, and, in fact, it was in order to obtain a reliable estimate of *c* that the BMW scheme was initially developed.

To see the origin of the linear relation between Δ*n*_{c} and *a*, we note first that the action (2.1) contains a single dimensionful parameter, *u*, *r* being adjusted for any given *u* to be at criticality. In fact, the effective three-dimensional theory is UV divergent, so there is *a priori* another parameter, a UV cutoff *Λ*∼1/*λ*, with *λ* the thermal wavelength. It follows then from dimensional analysis that *Σ*(*p*) can be written as *Σ*(*p*=*xu*)=*u*^{2}*σ*(*x*,*u*/*Λ*). Now, the diagrams involved in the calculation of *Σ* (at criticality) are UV convergent, so that the infinite cutoff limit can be taken. Note, however, that this requires that all momenta involved in the various integrations are small in comparison with *Λ* or, in other words, that the integrands are negligibly small for momenta *k*∼*λ*^{−1}. Only then can we ignore the effects of non-vanishing Matsubara frequencies, and finite cutoff effects. This implies that *uλ*∼*a*/*λ* is sufficiently small. In this region of validity of the classical field approximation, that is, for small enough *u*, *σ*(*x*,*u*/*Λ*) becomes a universal function *σ*(*x*), independent of *u*, and
4.11showing that the change in the critical density is indeed linear in *u*, and hence in *a*.

Table 2 contains our results for *c* together with some of the best estimates available in the literature. As was the case for the critical exponents in table 1, our numbers are all given for the PMS values *α** of the regulator parameter, and the digits quoted are those which remain stable when *α* varies in the range . For all *N* values where six-loop resummed calculations exist, our results for *c* are within the error bars. For *N*=100, we find *c*=2.36, which compares well with the exact large-*N* value *c*≃2.33 [62]. Our estimates for *c* are also comparable to those obtained from an approximation specifically designed for this quantity [14,15,70]. See also Ledowski *et al*. [71] for an earlier estimate of *c* using a different truncation of the RG equations, and also Floerchinger & Wetterich [72].

### (c) Thermodynamics of quantum fields

The last application that we shall consider concerns the thermodynamics of quantum fields at high temperature. This is motivated by the study of the quark–gluon plasma. In such a system, one could expect *a priori* perturbation theory to yield accurate results because the asymptotic freedom of quantum chromodynamics makes the effective coupling small at high temperature. However, strict perturbation theory does not work: it exhibits indeed very poor convergence properties, even in a range of values of the coupling constant where reasonable results are obtained at *T*=0. This difference in behaviour of perturbation theory at zero and finite temperature can be understood from the fact that, at finite temperature, the actual expansion parameter involves both the coupling constant and the magnitude of thermal fluctuations (for a recent review, see [73]; see also [74]). In that respect, the problem is not specific to QCD: similar poor convergence behaviour appears also in the simpler scalar field theory [75], and has also been observed in the case of large-*N* *φ*^{4} theory [76]. We focus here on the case of a scalar field with a *g*^{2}*φ*^{4} interaction (i.e. *g*^{2}≡*u*/24) [42].

Let us first recall how the effect of the interactions at a given scale depends on the magnitude of the relevant thermal fluctuations at that scale (and in some cases at a different scale as well). The thermal fluctuations of the field are given by the following integral:
4.12When we perform a perturbative calculation, we assume that the ‘kinetic energy’ ∼〈(∂*φ*)^{2}〉 is large when compared with the ‘potential energy’ ∼*g*^{2}〈*φ*^{4}〉. Obviously, this comparison depends not only on the strength of the coupling, but also on the typical wavelength, or momentum, of the fluctuations. To make things more precise, let us observe that the integral (4.12) is dominated by the largest values of *k* (in the absence of the statistical factor it would be quadratically divergent). One may then calculate the integral with an upper cutoff *κ* and refer to the corresponding value as to ‘the contribution of the fluctuations at scale *κ*’, and denote it by 〈*φ*^{2}〉_{κ}. In the same spirit, we may approximate the kinetic energy of modes at scale *κ* as 〈(∂*φ*)^{2}〉_{κ}≈*κ*^{2}〈*φ*^{2}〉_{κ}. Assuming furthermore that , one gets the expansion parameter (ratio of potential energy to kinetic energy ∼*κ*^{2}〈*φ*^{2}〉_{κ})
4.13

Let us examine the values of this parameter for several characteristic momenta. The fluctuations that dominate the energy density at weak coupling correspond to the plasma particles and have momenta *k*∼*T*. For these ‘hard’ fluctuations,
4.14Thus, at this scale, perturbation theory works as well as at zero temperature (with expansion parameter ∼*g*^{2}, or rather *α*=*g*^{2}/4*π*).

The next ‘natural’ scale, commonly referred to as the ‘soft scale’, corresponds to *κ*∼*gT*. We have then
4.15(In calculating 〈*φ*^{2}〉_{κ} for *κ*≪*T*, we have used the approximation *n*_{k}≈*T*/*k*, so that 〈*φ*^{2}〉_{κ≪T}∼*κT*.) Since *γ*_{gT}∼*g*, perturbation theory can still be used to describe the self-interactions of the soft modes. However, the perturbation theory is now an expansion in powers of *g* rather than *g*^{2}: it is therefore less rapidly convergent. The emergence of this new expansion parameter is the origin of odd powers of *g* in the perturbative expansion of the pressure (such as the plasmon term ∼*g*^{3}).

Another phenomenon occurs at the scale *gT*. While the expansion parameter *γ*_{gT}∼*g* that controls the self-interactions of the soft fluctuations is small, the coupling between the soft modes and the thermal fluctuations at scale *T* is not: indeed *g*^{2}〈*φ*^{2}〉_{T}∼(*gT*)^{2}, that is, the kinetic energy of the soft modes ∼(*gT*)^{2} is comparable with their interaction energy resulting from their coupling to the hard modes, ∼*g*^{2}〈*φ*^{2}〉_{T}. Thus the dynamics of soft modes is non-perturbatively renormalized by their coupling to hard modes. This particular coupling is encompassed by the so-called ‘hard thermal loops’ [77,78].

Finally, there is yet another scale, the ‘ultra-soft scale’ *κ*∼*g*^{2}*T*, at which perturbation theory completely breaks down. At this scale, we have indeed
4.16Thus the ultra-soft fluctuations remain strongly coupled for arbitrarily small values of the coupling constant. This situation does not occur for a scalar field since a mass is generated at scale *gT*, which renders the contribution of the fluctuations at the scale *g*^{2}*T* negligible. However, this situation is met in QCD for the long-wavelength, unscreened, magnetic fluctuations.

These considerations suggest that the main difficulty with thermal perturbation theory is not so much related to the magnitude of the coupling constant (for the relevant temperatures it is not that large), but it is rather due to the interplay of degrees of freedom with various wavelengths, possibly involving collective modes. In some sense, field theories at high temperature are multi-scale systems. At weak coupling the dynamically generated scales *T*, *gT* and *g*^{2}*T* are well separated. This allows, for instance, the organization of the calculation using effective field theory. However, the scale separation disappears when the coupling is not too small: then, the various degrees of freedom mix and the situation requires a different type of analysis.

The exact RG is ideally suited to cope with this type of situation. In particular, as we have seen, the BMW truncation scheme provides an excellent description of the momentum dependence of the two-point function, from the low momenta of the critical region, all the way up to the large momenta of the perturbative regime. One may then expect this method to capture accurately the contributions to the thermodynamical functions of thermal fluctuations from various momentum ranges, and hence to handle properly the mixing between degrees of freedom that takes place as the coupling grows. Since it involves also non-trivial momentum-dependent self-energies, the method also encompasses implicitly effects related to the damping of quasi-particles, or their coupling to complex multi-particle configurations.

Figure 3 displays the pressure as a function of the coupling constant at the scale *g*(2*πT*). The various diverging curves labelled indicate the results of perturbative calculations, up to order (for recent high-order calculations of the thermodynamics of the scalar field, see [79]). These curves clearly illustrate the poor behaviour of strict perturbation theory. The other curves correspond to various implementations of the exact RG, as well as to a 2PI calculation based on a simple two-loop skeleton [41]. The two curves labelled LPA correspond to two different choices of regulators: the Litim regulator [80–83], which is implemented only for three-momenta (with the untruncated sums over the Matsubara frequencies being performed analytically) [41], and an exponential regulator that affects both the momenta and the frequencies. The BMW approximation scheme is better justified when one uses a Euclidean symmetric four-dimensional regulator, and the calculations reported here have been done with the exponential regulator (4.2).

In contrast to the perturbative calculation, the calculations based on the RG show a remarkable stability, and a smooth extrapolation towards strong coupling. As it turns out, the results obtained are not too different from those of the LPA, nor from the simple two-loop 2PI approximation used in Blaizot *et al*. [41]. In physical terms, both the LPA and the two-loop 2PI approximation correspond to approximations where the degrees of freedom of the hot scalar plasma are massive quasi-particles. The new scheme goes beyond that simple picture. This stability of the results against improvements in the approximation suggests that the scheme that we are using to solve the RG equations may give already, at the level at which it is implemented here, an accurate representation of the exact pressure, and this over a wide range of coupling constants. It also indicates that for such a system the quasi-particle picture is presumably robust.

## 5. Conclusion

The few applications that are presented in the previous section illustrate the power of the RG, when coupled to an approximation scheme that allows for a determination of the full momentum dependence of the *n*-point functions. The equations that need to be solved are *a priori* complicated: these are flow equations which are at the same time partial differential equations (with partial derivatives with respect to the background field), with integral kernels that involve the solution itself. Still, they can be solved at a rather modest numerical cost, using elementary numerical techniques. The studies presented here involved only the LO of a systematic approximation scheme. In the absence of any small parameter controlling the magnitude of successive orders, a study of the next order would be necessary in order to quantify the accuracy that has been reached. However, the robustness of the LO results can already be gauged from their weak residual dependence on the choice of the regulator.

## Acknowledgements

Most of the work summarized here results from much enjoyable collaboration with F. Benitez, H. Chaté, B. Delamotte, A. Ipp, T. Matsuura, R. Méndez-Galain, J. Pawlowski, U. Reinosa and N. Wschebor, whose respective contributions to the various topics discussed here can be easily inferred from the references given. I would like to thank also the organizers for their invitation to the 2010 INT workshop on ‘New applications of the RG method’, and especially Y. Meurice and S.-W. Tsai for their persevering encouragement to complete this write-up.

## Footnotes

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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