Asymptotic solutions in asymptotic safety

Sergio Gonzalez-Martin, Tim R. Morris and Zoë H. Slade

Departamento de Física Teórica and Instituto de Física Teórica (IFT-UAM/CSIC),

Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain.

STAG Research Centre & Department of Physics and Astronomy,

University of Southampton, Highfield, Southampton, SO17 1BJ, U.K.,,

1 Introduction

The asymptotic safety approach to finding a quantum theory of gravity relies on finding a non-Gaussian ultraviolet fixed point of the gravitational renormalization group flow [1, 2]. This flow is couched in terms of a Wilsonian effective action [3], using the reformulation of the exact renormalization group in terms of an effective average action , the Legendre effective action with an infrared cutoff scale [4, 5, 6]. The existence of such a fixed point has been investigated and supported in a substantial number of approximations. For reviews and introductions see [7, 8, 9, 10, 11] and for some of the many recent approaches see refs. [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37].

In order to study this, it is necessary to work within some non-perturbative approximation scheme. The vast majority of these approximations are still formulated by keeping only a finite number of couplings in the effective action. A weakness of such an approach is that fixed points are effectively the solutions of polynomial equations in these couplings, which thus only allow for discrete solutions. But physical systems exist with lines or even higher dimensional surfaces of fixed points, parametrised by exactly marginal couplings (in supersymmetric theories these are common and termed moduli). Furthermore lines and planes of fixed points have also been found in approximations to asymptotic safety1 [39, 40, 23]. For isolated fixed points, careful treatment of polynomial approximations taken to high order, can allow extraction of convergent results, however one does not see in this way the singularities at finite field or asymptotic behaviour at diverging field, which are actually responsible for determining their high order behaviour, through the analytic structure they impose [41]. In fact such large field effects can invalidate deductions from polynomial truncations [41, 42, 43] and/or restrict or even exclude the existence of global solutions [44, 45, 46, 47]. A good example is provided by some of the most impressive evidence for asymptotic safety to date, polynomial expansions in scalar curvature including all powers up to [48, 49, 24], which are however derived from a differential equation for an fixed point Lagrangian [50] which was shown in ref. [39] to have no global solutions, as a consequence of fixed singularities at finite field.

Therefore if we are to believe that a putative isolated fixed point is not just an artifact of an insufficient approximation, we must go beyond polynomial truncations to approximations that keep an infinite number of couplings. Arguably the simplest such approximation is to keep a full function , making the ansatz:


and to date this is the only such approximation that has been investigated [51, 50, 52, 53, 12, 54, 55, 13, 14, 15, 16, 17, 18, 19], together with some closely related approximations in scalar-tensor [20, 21] and unimodular [22] gravity, and in three space-time dimensions [56]. It should be emphasised that this actually goes beyond keeping a countably infinite number of couplings, the Taylor expansion coefficients , because a priori the large field parts of contain degrees of freedom which are unrelated to all these .

The result of such an approximation is a fixed point equation which is either (depending on the implementation of the effective cutoff ) a third order or a second order, non-linear ODE (ordinary differential equation) for the dimensionless function , where


To be concrete we will give the discussion assuming the typical case where the equation is derived on a space of positive curvature (effectively the Euclidean four-sphere, the discussion is readily adapted to negative curvatures) in which case a fixed point corresponds to a smooth global solution over the domain . Now, to understand the solutions of these equations both physically and mathematically, it is crucial to develop the asymptotic solutions , as we explain below.

Although these ODEs are complicated, e.g. (2.1), the asymptotic solutions can fortunately be found analytically and in full generality [39, 23] by adopting techniques developed much earlier for scalar field theories [41, 44, 45]. These techniques apply to any functional truncation of the exact renormalization group fixed point equations, such that the result is an ODE or coupled set of ODEs (as e.g. in [23]), although to be concrete we will focus on solving for . Perhaps because these techniques were covered only briefly and without outlining the general treatment, they have yet to be entirely adopted, meaning that the functional solution spaces for many of the formulations [20, 21, 22, 56, 53, 12, 54, 55, 13, 14, 15, 16, 18, 19] remain unexplored or at best only partially explored. The main purpose of the present paper is to improve this situation, by describing in detail and with as much clarity as possible how the techniques allow to fully unfurl the asymptotic solutions. As an illustration we choose to apply these techniques to one example of a fixed point ODE [13] which fortuitously provides a zoo of asymptotic solutions of different types.

Perhaps another reason why the asymptotically large region may have been under-explored is that it has not been clear what meaning should be attached to this region when is larger than the physical size of the manifold, despite the fact that we know that the infrared cutoff is artificial and introduced by hand and the physical effective action,


is therefore only recovered when the cutoff is removed. This puzzle was brought into sharp relief in formulations that have a gap, i.e. a lowest eigenvalue which is positive, so that large then corresponds to being less than any eigenvalue [56, 13, 15, 18]. This issue was recently resolved in ref. [17] where it was shown to be intimately related to ensuring background independence (but in a way that can be resolved even for single-metric approximations, which is just as well since only the refs. [17, 16, 19] in the list above actually go beyond this approximation). Wilsonian renormalization group concepts do not apply to a single sphere. In particular, although in these formulations, can be low enough on a sphere of given curvature that there are no modes left to integrate out, the fixed point equation should be viewed as summarising the state of a continuous ensemble of spheres of different curvatures. From the point of view of the ensemble there is nothing special about the lowest mode on a particular sphere. The renormalization group should be smoothly applied to the whole ensemble, and it is for this reason that one must require that smooth solutions exist over the whole domain .

In the sec. 2, we introduce the ODE we will study, provide a compendium of our results, and discuss their meaning. In the secs. 3 and 4 we provide the details of how these are derived. We finish this introduction, by listing reasons why the asymptotic solution is so important.

1.1 Quantum fluctuations do not decouple

In the application to scalar field theory [41, 44, 45, 47], the leading asymptotic behaviour was always found by neglecting the right-hand side of the fixed point equation (more generally flow equation). This made physical sense since the right-hand side encodes the quantum fluctuations, and at large field one would expect that these are negligible in comparison. Therefore the asymptotic solution simply encodes the passage to mean field scaling, characteristic of the classical limit. We find that with functional approximations to quantum gravity, the situation is radically different. The leading asymptotic solution intimately depends on the right-hand side and never on the left-hand side alone. We will see this for the large solutions of the fixed point equation, (2.1), derived by Demmel et al [13]. This behaviour confirms what we already found in a different approximation in ref. [39], and also for a conformal truncation in ref. [23]. Again this actually makes physical sense because the analogue here of large field is large curvature which therefore shrinks the size of the space-time and thus forbids the decoupling of quantum fluctuations. In fact by Heisenberg’s uncertainty principle we must expect that the quantum fluctuations become ever wilder. We note that it is the conformal scalar contribution that is determining the leading behaviour [39, 13, 23] and appears to be related to the so-called conformal instability[58, 39, 23]. In any case, we see that for quantum gravity, the asymptotic solution encodes the deep non-perturbative quantum regime.

1.2 Physical part

The asymptotic solution contains the only physical part of the fixed point effective action. Recall that the effective infrared cutoff is added by hand and the physical Legendre effective action (1.3) is recovered only in the limit that this cutoff is removed. This is of course done while holding the physical quantities (rather than say scaled quantities) fixed. In normal field theory, e.g. scalar field theory, the analogous object is the universal scaling equation of state, which for a constant field precisely at the fixed point takes the simple form


where is the space-time dimension and is the full scaling dimension of the field (i.e. incorporating also the anomalous dimension). In the current case we keep fixed the the constant background scalar curvature . Thus by (1.1), the only physical part of the fixed point action in this approximation is:


The significance of this object is further discussed in sec. 2.4, in the light of the results we uncover.

1.3 Dimensionality of the fixed point solution space

For given values of the parameters, the fixed point ODEs are too complicated to solve analytically,2 and challenging to solve numerically. However the dimension, , of their solution space, namely whether the fixed points are discrete, form lines, or planar regions etc., can be found by inspecting the fixed singularities and the asymptotic solutions.

To see this we express the fixed point ODE in normal form by solving for the highest derivative:


where is the order of the ODE, and (right-hand side) contains only rational functions of and lower order differentials3 . The fixed singularities are found at points where this expression develops a pole for generic . For the solution to pass through the pole requires a boundary condition relating the , one for each pole. By a fixed singularity, we will mean one of these poles.

At the same time such non-linear ODEs suffer moveable singularities, points where diverges as a consequence of specific values for the . The number of these that operate in practice depends on the solution itself. However if the solution is to exist globally then it exists also for large , where we can determine it analytically in the form of its asymptotic solution , this being the central topic of this paper. The number of constraints implicit in is equal to , where is the number of free parameters in . This can be seen straightforwardly by noting that the maximum possible number of free parameters is ; if contains any less then this implies that there are relations between the at any large enough , which may be used as boundary conditions. Now the number of moveable singularities that operate for a solution with these asymptotics is also equal to , providing we have uncovered the full set of free parameters in the asymptotic solution, as has been explicitly verified by now in many cases [41, 44, 45, 42, 43, 46, 47, 39, 57, 23]. This follows because the moveable singularities can also occur at large where they influence the form of . Indeed linearising the ODE about , the perturbations can also be solved for analytically. The missing free parameters in correspond to perturbations that grow faster than , overwhelming it and invalidating the assumptions used to derive it in the first place. These perturbations can be understood to be the linearised expressions of these moveable singularities [41, 44, 45]. To summarise, if the number of fixed singularities operating in the solution domain is , then the dimension of the solution space is simply given by


where indicates a discrete solution set which may or may not be empty, and corresponds to being overconstrained, i.e. having no solutions. In sec. 2.5, we will illustrate this by working out the dimension of the fixed point solutions for eleven of the possible asymptotic behaviours. We discuss their significance in sec. 2.6.

At the same time the counting argument (1.7) aids in the numerical solution. For example it tells us where it is hopeless to look for global numerical solutions, namely where , and to improve the numerical accuracy if the numerical solution apparently enjoys more free parameters than allowed by [39].

1.4 Validation of the numerical solution

A priori one might think that the analytical solutions for can be dispensed with in favour of a thorough numerical investigation. The problem is that without knowledge of , there is no way to tell whether the numerical solution that is found is a global one, viz. exists and is smooth as , or will ultimately end at some large in a moveable singularity. In fact if the numerical solution is accessing a regime where the number of free asymptotic parameters , it will actually prove impossible to integrate numerically out to arbitrarily large . Instead the numerical integrator is guaranteed to fail at some critical value. The reason is that it requires infinite accuracy to avoid including one of the linearised perturbations that grow faster than which as we said, signal that the solution is about to end in a moveable singularity. On the other hand, if one can extend the solution far enough to provide a convincing fit to the analytical form of , then one confirms with the requisite numerical accuracy that the numerical solution has safely reached the asymptotic regime [39, 57, 23], after which its existence is established, and its form is known, over the whole domain. We will see an example of this in sec. 3.6 where we will see in fact that the numerical solution found in ref. [13] matches the power-law asymptotic expansion (2.11), but such that it would need to be integrated out twice as far in order to be sure of its asymptotic fate.

2 Overview

2.1 Fixed point equation

The fixed point equation that we will be studying is given by [13]:


where is defined in (1.2), and prime indicates differentiation with respect to . This gives the scaled dependence on the curvature of a Euclidean four-sphere. Therefore we are searching for smooth solutions defined on the domain . Each such solution is a fixed point of the renormalization group flow.

The coefficients and depend on and the endomorphism parameters4 and , and are given as


In ref. [13], the authors set and we will do the same.5

2.2 Fixed singularities

The ODE (2.1) is third order and thus admits a three-parameter set of solutions locally. As reviewed in 1.3, the fixed singularities will limit this parameter space. The positions of the fixed singularities are determined by casting the flow equation into normal form (1.6) (with ). The zeroes of the coefficient then give the points where the flow equation develops a pole. These poles are given by [13]:

Note that there is a double root and that the root , which is actually there for good physical reasons [52, 39], is always present, whereas the positions of the last 4 roots depend on the value takes. Different choices for will result in a different number of fixed singularities being present in the range , as shown in table 1.

Range of Singularities
Table 1: List of fixed singularities present for different choices for .

If no additional constraints emerge from the asymptotic behaviour of the solution then choosing leads by (1.7) to , which means that isolated fixed point solutions (or no solutions) can be expected. The authors of [13] choose for this reason. It is also noted in [13] that in addition this choice simplifies the numerical analysis. We will analyse the fixed point equation for general , both to uncover the extent to which the results depend on the particular choice and to demonstrate and explain the asymptotic methods in a large variety of examples. But since was chosen in ref. [13], we will pay special attention to this value.

If is chosen then , the ODE is overconstrained and global solutions do not exist. On the other hand non-positive give rise to continuous sets, again assuming that no extra constraints are coming from the solution at infinity. For example , would give rise to only 2 fixed singularities, resulting in a one-parameter set of solutions i.e. a line of fixed points, while gives us a plane of fixed points.

2.3 Asymptotic expansions

We now list all the asymptotic solutions that we found. They can have up to three parameters, which are always called , and .

(a) As covered in sec. 3.1, there exists a power-law solution where the leading power is . It takes the form


for all , where is given by (3.7).

(b) For , the subleading power is altered and the solution changes to


as explained in sec. 3.2.2. At this value of only this asymptotic solution is allowed.

(c) As covered in sec. 3.1, for a root of (3.8) such that , the asymptotic solution


with given by (3.10), exists for all , except as explained in sec. 3.2.2 for and , and except for the values


as explained at the end of sec. 3.1. When is complex, which happens for the parameter is in general also complex and the real part of (2.5) should be taken leading to type behaviour. The values are plotted in fig. 3.1.

(d) As explained at the end of sec. 3.1, for the values (2.6), is a root of (3.8), however the asymptotic solution is not given by (2.5) but


where is given by (3.12).

The techniques set out in secs. 3.5 and 4.4 need to be followed to find the missing parameters in the above solutions (a) – (d), before we can discover the dimension of their corresponding solution space. Of course we already know from table 1 that there are no solutions (i.e. ) for . In the remaining asymptotic solutions we also uncover the missing parameters.

(e) For generic the following solution:


where is a second free asymptotic parameter, forms the basis for the asymptotic solutions below. As explained in sec. 3.2 it fails to exist for , 1, and . The leading part is derived in sec. 3.1 and the subleading parts in sec. 3.3. The subleading coefficients are functions of and , where is given in (3.19) and the others are given in appendix A. As explained in sec. 3.4, exceptions develop at poles of these subleading coefficients where the corresponding term and subleading terms then develop an extra piece. As shown in sec. 3.5, the full asymptotic solution is then one of the following forms:


where is given by (3.23) and the ellipses stand for further subleading terms that will mix powers of the new piece and its free parameter, , with the powers of the terms in (2.8). The power is plotted in fig. 3.2. Since the authors of [13] use , the solution (2.11) is of particular interest. We show in sec. 3.6 that it provides a match to their numerical solution, as far as it was taken.

(f) Except for , and as in (2.6), as discussed in sec. 4.3, the asymptotic series


forms the basis for the asymptotic solutions below, where6


For , , the coefficient is derived in sec. 4.1 and is given in (4.2), while the other are derived in sec. 4.2 and are given in appendix B. As explained in sec. 4.3, for and the coefficients take different values as given in (B.2) respectively (B.3). The arguments in the logs in (2.14) can be replaced by as in (4.6) but as shown in sec. 4.2 this is not an extra parameter and can be absorbed into the free parameter in (2.13). The full asymptotic solution then takes the following forms:


where the positive square root is taken, and are defined in (4.10) and (4.11), in (4.27), and . The top three solutions are derived in sec. 4.4. However (2.15) required a separate analysis for and , in secs. 4.5.2 and 4.5.3 respectively. The next two are derived in sec. 4.5.1, (2.20)is derived in sec. 4.5.2, and (2.21) is derived in sec. 4.5.3.

2.4 Physical part

Using (1.5), we can now extract the corresponding physical parts. As noted in sec. 1.2, these give the universal equation of state precisely at the fixed point, analogous to (1.4) in a scalar field theory. We see that except for the cases (d) and (f) discussed below, the result vanishes:


The asymptotic solution (2.11) which matches the numerical solution in [13], thus also falls in this class. Similar results were obtained for cases in the conformal truncation model of [59, 23] where also divergent results were found. Perhaps these indicate that these do not give a sensible continuum limit, although a fuller understanding is needed for example by moving away from the fixed point by including relevant couplings.

For case (d), from (2.7) we get (for ):


This equation of state was also found in ref. [39] for the approximation given in [52], and for solutions found in ref. [15]. For any of the solutions for case (f), we get from (2.13) and (2.14) that


and is given by (4.2). Since we require , the second term is a positive logarithmic divergence. It is perhaps a signal of the asymptotic freedom of the coupling in this case where thus it should be treated as in ref. [18]. As we will see in the next section, global solutions with the asymptotics of case (f) exist only for and therefore is always positive.

2.5 Dimensionality of the fixed point solution spaces

Using (1.7) and table 1 we can read off the dimensionality of the corresponding fixed point solution spaces for cases (e) and (f).

(e) We see that since (2.8) has two free parameters, (2.9) only extends to global solutions for in the negative interval. When the fixed points, if any, form a discrete set, while lines of fixed points are found for . The other solutions, (2.10) – (2.12), all have three free parameters and thus provide no constraints on the dimension of the solution space, which thus follows the pattern discussed in sec. 2.2. In particular (2.10) extends to a global solution only for where it can have discrete fixed points or lines of fixed points depending on the sign of , (2.11) has discrete solutions, and (2.12) has planar regions of fixed points.

(f) Using (2.6), we see that since (2.14) has only one free parameter, solutions with asymptotic behaviour (2.15) and (2.19) do not exist since they are overconstrained by the fixed singularities, (2.16) has discrete solutions for and lines of fixed points for , (2.18) has discrete solutions, and (2.17), (2.20) and (2.21) all generate lines of fixed points.

2.6 Which fixed point?

From the point of view of the asymptotic safety programme, it would be phenomenologically preferable if the correct answer lay in only one of the discrete sets: (2.9) for , (2.10) for , (2.11) (the choice made in ref. [13]), (2.16) for , or (2.18). However we need a convincing argument for choosing one solution over the others.

Inspection of the form of the cutoff functions used in the derivation of (2.1), see eqn. (3.13) in ref. [13], shows that corresponds to cases where some scalar modes never get integrated out, no matter how small we take . One therefore could argue that for the Wilsonian renormalization group is undermined. Continuous sets of solutions [39] were also found with another approach to the approximation [52], and there also there is a scalar mode that never gets integrated out. Clearly it would be very useful to know if this correlation is found for other formulations [20, 21, 22, 56, 53, 12, 54, 55, 13, 14, 15, 16, 18, 19] in the literature.

For these continuous solutions it could also be, like in ref. [39], that the approximation is breaking down there, such that the whole eigenspace becomes redundant [40]. To check this would require developing the full numerical solutions.

It could also be that these are artefacts caused by violations of background independence [57]. We saw there that at in the Local Potential Approximation, providing the modified split Ward identity that reunites the fluctuation scalar field with its background counterpart , is satisfied, the spurious behaviour is cured. The implementation of background scale independence in refs. [17, 16, 19] is arguably the equivalent step for the approximation, since it reunites the constant background curvature with the multiplicative constant conformal factor piece of the fluctuations. We saw there that the resulting formulations can be close to the single-metric approximation used to derive (2.1), in the sense that minimum changes are needed e.g. setting space-time dimension to six, or choosing a pure cutoff, to convert the fixed point equation into a background scale independent version. It could therefore be promising to investigate formulations with these changes.

On the other hand, continuous solutions were found in the conformal truncation model of ref. [23], where a clear cause was found in the conformal factor instability [58]. The cutoff implementation did not introduce fixed singularities, background independence was incorporated [59], all modes were integrated out, and an analogous breakdown to the approximation [40] was either not there or not possible. This suggests that the issues go deeper.

In the next two sections, we provide the details of how the asymptotic solutions were discovered and developed.

3 Asymptotic expansion of power law solutions

We now give the detailed study of the asymptotic behaviour of the solution in the IR limit . (This is equivalent to the large background curvature limit for fixed , by (1.2). For fixed , corresponds to what we more commonly refer to as the IR limit, .) Finding such an asymptotic solution initially requires a degree of guesswork. A profitable place to start is to assume that the asymptotic series starts with a power, i.e. 


where is typically an arbitrary coefficient, and subsequent terms need not be powers but are successively smaller than the leading term for large .

Now, requiring that (3.1) satisfies the fixed point equation (2.1), means that at large , the leading piece in this equation must itself satisfy the equation. In this way we typically determine and sometimes also . The leading piece of the fixed point equation will be satisfied either because the left-hand side and the right-hand side provide such a piece and for appropriate values of and these are then equal, or because only one side of the equation has such a leading piece but this can be forced to vanish by appropriate values of and . In this sense we require the leading terms to ‘balance’ in the fixed point equation. As we will see, requiring then the sub-leading terms also to balance will determine the form of the corrections needed in (3.1).

3.1 Leading behaviour

We begin by finding the leading behaviour of i.e. solving for the power in (3.1). We build the asymptotic series leaving unspecified for the reasons given at the end of sec. 2.2. We plug the solution ansatz (3.1) into the fixed point equation (2.1), expand about and keep only the leading terms in the large limit. Since the coefficients (2.2) are expanded along with everything else and only their leading parts are kept, it is useful to introduce the following definitions:


where we have rewritten and using . For functions and , means that . We note that for certain values of , the leading behaviour of the coefficients will be different from those given in (3.1) since the leading coefficients will vanish. We discuss this in section 3.2, and comment there and below on the case of .

Inserting ansatz (3.1) into the fixed point equation (2.1), we find that the leading piece on the left-hand side as is simply


and the leading piece on the right-hand side is given by


where we substituted in the first fraction. The important observation here is that the left-hand side goes like whereas the right-hand side goes like , and thus which side dominates will be determined by whether is less than, greater than or equal to 2. Below we investigate these possible scenarios to determine the power . We recognise that the scaling behaviour of the right-hand side could differ from if cancellations were to occur in either the denominators or the numerators. This is discussed in section 3.2.

Scenario (1): For , the left-hand side (3.3) dominates and thus we require that be chosen to set the left-hand side to zero. However we see immediately that this is only true if and thus we reach a contradiction.

Scenario (2): For , the leading right-hand side part, (3.4), must vanish on its own. For this is because the right-hand side dominates, while for it must be so because we have just seen that in that case the left-side vanishes on its own. Thus for this scenario, we require to be such that the coefficient in (3.4) vanishes. Solving this we find for generic , four solutions for , which we now describe.

One solution is


We will pursue this asymptotic solution in sec. 3.3, where, as we will see, it allows us to build a legitimate asymptotic series. We work this out in detail to demonstrate the general method. As we will see in section 3.5 there are values of for which this choice of scaling is not allowed. However we can say that for generic of one possibility for the leading term in an asymptotic expansion (3.1) is therefore


This leading behaviour agrees with the quantum scaling found in [13], but is in conflict with the classical and balanced scaling that the authors ultimately use to approximate the large behaviour of their solution. In fact the full solution we find, namely (2.11), does not agree with that suggested in ref. [13], but can match quite acceptably the numerical solution they found, as we show in sec. 3.6.

Another solution is . This is already clear from (2.1) and follows from the fact that the numerators depend only on differentials of . Setting , we have arranged that the leading term vanishes, so now we turn to the subleading term. We see that the left-hand side of (2.1) will provide an piece. For to be the beginning of an asymptotic series we will need to balance this piece with a term on the right-hand side. Whatever subleading term we add, it will generically no longer be annihilated by the numerators in (2.1). Meanwhile the denominators will go like a constant for large (from the undifferentiated parts). Thus, using (3.1), we see by inspection that the subleading piece goes like . By expanding (2.1) in an asymptotic expansion and matching coefficients we thus find eqn. (2.3) with


We could continue to investigate this asymptotic solution, developing further subleading terms and finding out how many parameters it ultimately contains, but this solution should be a very poor fit to the numerical solution found in ref. [13] which numerically shows behaviour identified in ref. [13] as for large .

Figure 3.1: Plot of the two solutions given by (3.8).

The last two solutions are functions of and are given by the roots of the quadratic


The solutions (3.8) are plotted in fig. 3.1. These roots take complex values when and so the allowed solutions which we want are those for which Re. The only region where there is not a solution Re is from the point where Re crosses the line in this range, namely at , through to the point where the ‘blue’ root diverges, namely .

In order to know whether the Re solutions of (3.8) really lead to valid asymptotic series we need to take the expansion to the next order and check that the next order is genuinely sub-leading. We pursue this in a similar way to the case above. We know that the left-hand side of (2.1) and this term will need balancing by terms on the right-hand side. On the right-hand side the denominators also , whereas the numerators, which would have gone like have had the corresponding coefficient cancelled by choosing (3.8). Therefore the subleading term we need to add is a piece such that


where we have trialled just the first term in the first numerator on the right-hand side of (2.1) and compared it to the left-hand side. Solving this gives . Since , we see that this is genuinely subleading only if Re. By inspection, such a power law solution then works out for the full numerators on the right-hand side of (2.1). Substituting these first two terms of the fledgling asymptotic series into (2.1) and expanding for large we find the coefficient and thus confirm our last two power-law asymptotic solutions (2.5) where


where we have set


When (3.8) has real roots, those with are taken. When complex, the real part of (2.5) should be taken leading to type behaviour.

For , both the left and right-hand side scale as and therefore could be expected to balance. However, as we have already seen, if the left-hand side is identically zero and so again we require the right-hand side (3.4) to vanish, but now with fixed to . As we will see in sec. 4 this impasse gives a clue however to a non-power law asymptotic solution. Pursuing for now the power law case (3.1) but with fixed , we find that this equation is satisfied either for where is defined in (2.6), or apparently for , since then all the and vanish. The exceptional case of is discussed in section 3.2. The solutions (2.6) are just values of such that one of the roots of (3.8) is indeed . Substituting in (2.6) gives a quartic in , but the other two roots, , , are cancelled at by the denominator in (3.4). The remaining quadratic is in fact the one that appears in the denominator of coefficients (4.10) and (4.11) that we will come across later.

To demonstrate the validity of the solution (2.6), we need to show that the expansion can be taken to the next order. We know the expansion for general given in (2.5) breaks down for . In fact in this case, since the left-hand side of (2.1) already vanishes, the subleading term comes from the next term on the right-hand side in a large expansion. In this way we see that the asymptotic solution is (2.7) where


and is either root in (2.6).

We see that overall there are in general four types of power-law asymptotic solutions given by the power being one of the roots (3.8) providing is such that Re 2, or two independent cases: and .

3.2 Exceptions

In general, as mentioned previously, we recognise that certain choices for alter the scaling behaviour of the right-hand side of the fixed point equation such that it differs from in the limit .

Exceptions from the denominators

For instance, the leading behaviour could increase to if the terms in one or both of the denominators were to cancel amongst themselves. These cancellations occur in the first and second fractions respectively when




We will concentrate on the asymptotic solution with . For this value, (3.13) and (3.14) are satisfied when and respectively. For these values of , a leading power of is not allowed and instead to find the leading behaviour in these cases we must treat and separately from the start. As we will see in these cases we just recover the other three power-law solutions.

: the leading piece in the limit on the left-hand side is still of course given by (3.3). Also, since does not correspond to one of the values at which the leading parts of the coefficients vanish, see (3.1), the leading piece on the right-hand side will still be given by (3.4), but now with set to 1:


Note that even though has been identified as a value at which the terms in the first denominator of (3.4) vanish, this is only when and so here we still see the right-hand side scaling as . The scaling of the right and left-hand sides is the same as that in section 3.1 and so by the same reasoning we see that must be less than 2 and therefore (3.15) must vanish in order to satisfy the fixed point equation for large . The right-hand side (3.15) vanishes when and also trivially for . Indeed these are the remaining power-law solutions, in particular the former pair are the values for given by (3.8) with as expected, while the latter is the solution (2.3).

: the right-hand side of the fixed point equation again scales like in the large limit, as this is also not one of the exceptional values appearing in (3.1). We see that again must be less than 2 and that the right-hand side must vanish. Once again, the values of just correspond to (3.8) when . And we still also have the solution (2.3).

Exceptions from the numerators

Exceptions to the leading behaviour also arise from particular choices for reducing the powers of appearing in the coefficients and which could result in an overall decrease in the leading power on the right-hand side of the fixed point equation. The values of for which the leading power of in the coefficients vanishes can readily be read from (3.1). Notably however, both numerators in (3.4) are satisfied independently for . This means that for the values for which only one of the fractions becomes sub-dominant the -independent solution remains valid.

: although we are concentrating on the solution, for completeness we note that does present an exception for the general power solutions (3.8). From (3.8), we would expect to find asymptotic series with leading powers . However when , we see from (3.1) that , and all vanish. Then from (3.4) we see that in this case the only solutions left for are the and cases established in sec. 3.1.

: in this case the leading powers of in all the coefficients vanish, cf. (3.1). (We see the implications of having in section 3.3 where it represents a pole of all but one of the coefficients in the asymptotic expansion given in appendix A.) As a result, the right-hand side of the fixed point equation (2.1) in general no longer scales as but instead goes like . This apparently implies two solutions: either the asymptotic series is , since satisfies the left-hand side on its own, or the asymptotic series takes the form , with the term then balancing both sides of the equation. However substituting into the right-hand side of (2.1) (with ) we find the leading term is:


which thus presents an exception for both of these cases! The reason for this is that happen to be precisely the two powers that reduce the leading power in second denominator in this case, as can be seen from (3.14). Thus actually when or , the second term on the right-hand side of (2.1) contributes . Since it does so now with no free parameters ( and having been fixed), neither suggested asymptotic solution will work: for because the correction is larger than the supposed leading term, while for there is nothing to balance it since the left-hand side vanishes identically. Finally, we consider general . For , the left-hand side dominates and we require that it vanishes for the fixed point equation to be satisfied, but this only gives us the already excluded solution. For Re the right hand side dominates and (3.16) must vanish on its own. The cubic in the numerator has no roots in this region and thus we are left with only an solution, namely (2.4). Note that this differs from (2.3), in particular the first subleading power is now .

To summarise for the asymptotic series in particular, we have seen that this fails to exist at , 1, and . However as we will see, in general these exceptions do not obstruct the construction of the subleading terms or the subsequent determination of the missing parameters.

3.3 Sub-leading behaviour

In this section we present the method for determining subsequent terms in the asymptotic series (3.1). As we stated, we will concentrate on the solution. We have seen in the previous section that for generic the fixed point equation scales like for large . Choosing causes the contribution to disappear, therefore satisfying the equation in the large limit. This is precisely because a leading power of is what is required to make the coefficient multiplying the term be identically zero.

Once the terms have vanished, the new leading large behaviour of the fixed point equation is , coming from the undifferentiated on the left-hand side of (2.1). The next term in the solution should be such that it now cancels the pieces contributing to the new leading behaviour. It is with this in mind that we proceed to build the sub-leading terms of the solution.

We denote the next term in the solution by a function such that


where grows more slowly than the leading term. This implies that we can if necessary find its leading corrections algorithmically, by taking large enough to allow linearising the fixed point equation in . This will give us a linear differential equation for , where we keep only the leading parts in a large expansion of its coefficients. This equation is set equal to the new leading piece, namely the piece discussed above, and is straightforward to solve since we only want the leading part of the particular integral. In fact inspection of the fixed point equation shows that this contribution can only come from the right-hand side and that it requires . Indeed with this choice, the leading contribution from the numerators on the right-hand side are then terms which scale like . The leading terms from the denominators scale like (providing no exceptional cases arise). Together these give an overall contribution of to the right-hand side as required. Including the next term in the solution we now have


To find the coefficient , we substitute the solution as given above into the fixed point equation and take the large limit. Collecting all terms on one side of the equation, the leading terms go like as expected, multiplied by a coefficient containing . We require this coefficient to vanish in order to satisfy the fixed point equation and so must take the following form


The next terms in the series are found by repeating this procedure. After five iterations the solution becomes (2.8) where is given in (3.19) and the more lengthy expressions for the other coefficients are given in appendix A. Note that a second constant , independent of , is found as a result, through particular integrals containing logs. At this point our solution therefore contains two independent parameters in total.

3.4 Exceptions

The solution (2.8) will break down at values of corresponding to poles of the coefficients . As can be seen straight-away, is one such value. This has already been flagged-up as problematic in section 3.2 where the trouble was traced back to the fact that when , the leading part of each of the coefficients in the fixed point equation (3.1) vanishes. This means that (3.4) no longer represents the true asymptotic scaling behaviour of the fixed point equation and should not be used to derive the leading behaviour of the solution.

There are other poles in the coefficients besides at , as can be readily seen from the full form of the coefficients given in appendix A. We find that as we build the asymptotic series, new coefficients contain new poles, not featured in earlier terms. In this way we will build a countable, but apparently infinite, set of exceptional values of . It is not clear how these exceptional values are distributed (for example whether they lie within some bounded region or not), but since the real numbers are uncountable, we are always guaranteed values of for which there are no poles present in the series.

If we do happen to choose a value for that gives rise to a pole then this signals that the term in the solution containing the pole does not have the correct scaling behaviour in order to satisfy the fixed point equation in this instance. Take the first coefficient as an example. At a pole of (all except ), adding a piece on to the solution that goes like does not result in an contribution on the right-hand side of the fixed point equation as required, because the coefficient automatically vanishes in this case.

Instead we must look for a different sub-leading term. A less simple choice but one that works nonetheless is . The reason for this is that if all the derivatives hit the factor and not the factor then again the piece on the right-hand side must vanish identically, since it is as though the is just a constant multiplier for these pieces. We therefore know that in the asymptotic expansion at this order, the only terms that survive have the term differentiated. But this then maps which is the power-law dependence we desired for the differentiated term. We will apply the same strategy in sec. 4.

Taking this as the sub-leading term, such that the solution now goes , gives rise to the desired term on the right-hand side, but now without the same pole (i.e. a different ) and we can continue to build the series solution from there. This suggests that for a solution with leading behaviour , each new set of poles associated with a new coefficient gives rise to further appearances of in that sub-leading term and therefore a plethora of different possible solutions dependent on these exceptional values for .

3.5 Finding the missing parameters

The asymptotic solution (2.8) contains only two parameters, and yet we are solving a third order ordinary differential equation. Local to a generic value of we know there is a three parameter set of solutions. In this section we linearise about the leading solution (2.8) to uncover the missing terms [41, 44, 45, 39, 23]. We do so by writing such that the solution becomes


where and is some arbitrary function of . Since the constant introduced in (3.1) can take any value, we are permitted to change it by any constant amount. Thus a constant should be a solution in the asymptotic limit . We use this reasoning to help find the missing parameters. In sec. 4 we will follow a related but different strategy.

We insert (3.20), complete with all modified sub-leading terms, into the fixed point equation (2.1) and expand about . We know already that at the fixed point equation is satisfied for large , since at this order (3.20) is equivalent to the original solution (2.8). At in the large limit we obtain a third order ODE for :




Initially we would expect to have another term on the left-hand side of (3.21) that looks like times to some power. However the coefficient vanishes up to the order of approximation of the solution we are working to. In fact this had to be so, since otherwise would not satisfy the equation. This means that what would have been be a third order ODE is instead a second order equation in . This idea is analogous to the Wronskian method for differential equations. The solution contains two independent parameters, and , and so we already know two independent solutions of fixed point equation. These solutions can be used to build a Wronskian that satisfies a first order differential equation and which can then be used to find the unknown solution. We will not need this full machinery however.

The differential equation (3.21) is invariant under changes of scale and thus has power law solutions. Setting we find three solutions for : the trivial solution as required for consistency with the possibility of , and


The complete solution for is then given by a linear combination of these powers:


where we have introduced infinitesimal parameters ,