Reparameterisation Invariance and RG equations:
Extension of the Local Potential Approximation
H. Osborn111ho@damtp.cam.ac.uk and D.E. Twigg222det28@cam.ac.uk
Department of Applied Mathematics and Theoretical Physics,
Wilberforce Road, Cambridge, CB3 0WA, England
Equations related to the Polchinski version of the exact renormalisation group equations for scalar fields which extend the local potential approximation to first order in a derivative expansion, and which maintain reparameterisation invariance, are postulated. Reparameterisation invariance ensures that the equations determine the anomalous dimension unambiguously and the equations are such that the result is exact to in an -expansion for any multi-critical fixed point. It is also straightforward to determine numerically. When the dimension numerical results for a wide range of critical exponents are obtained in theories with symmetry, for various and for a ranges of , are obtained within the local potential approximation. The associated , which follow from the derivative approximation described here, are found for various . The large limit of the equations is also analysed. A corresponding discussion is also given in a perturbative RG framework and scaling dimensions for derivative operators are calculated to first order in .
PACS:11.10.-z, 11.10.Gh, 64.60.Fr, 64.60Ak, 64.60.Kw, 68.35.Rh
Keywords:Exact Renormalisation Group, Derivative Expansion, Reparameterisation Invariance.
Exact functional renormalisation group flow equations [1, 2, 3, 4, 5, 6, 7, 8] allow, at least for scalar field theories, the possibility of a non perturbative analysis of fixed points and determination of critical exponents which control the RG flow near any fixed point. In all such equations there is a cut off function which is essentially arbitrary save for and vanishing sufficiently rapidly as . Any physical results, such as precise values for exponents, should be independent of the cut off although it may be feasible to optimise over different cut off functions . The exact RG flow equations are hard to handle except in some truncation or expanding in perturbation theory. The local potential approximation (LPA) neglects the spatial dependence of the fields and reduces the effective action from a highly non trivial functional to a simple function and the RG flow equations become a non linear differential equation for of the form where denotes the derivative with respect to , with the cut off scale (the equations are invariant under rescalings of ). The potential RG flow fixed points, as , are determined by requiring smooth solutions for all of . The critical exponents describing the RG flow in the neighbourhood of the fixed point, , may then be calculated by finding the eigenvalues , and eigenfunctions , for a corresponding linear differential operator depending on the fixed point solution for . For all the associated operators are relevant and it is necessary to tune to attain the fixed point under RG flow.
When applied to the Polchinski RG equation , for which has a very simple quadratic form, the LPA has the virtue that all dependence of the cut off can be removed by rescalings of and . Although rather crude the LPA is compatible with the global features of RG flow since, in cases that have been investigated, it realises the same fixed points as are present in the full quantum field theory for scalar fields that are found by other techniques (this is not manifestly true for more complicated theories with gauge fields and fermions ).
Despite describing the essential features of the landscape of critical points for scalar theories the LPA has, nevertheless, many limitations. In particular it is not possible to consistently determine , the anomalous dimension for . In theories with dimension , is generally small but clearly results for critical exponents must then have an error of at least , although results when the LPA is applied to different RG flow equations differ in general by rather more than this. Attempts to go beyond the LPA usually invoke an expansion in terms of derivatives of [11, 12, 13, 14, 15, 16, 17, 18]. To first order this introduces a function which is the coefficient of in an expansion of the effective action (for multi-component fields this becomes a symmetric tensor . and obey coupled equations which in principle allow to be determined by requiring non singular solutions for both and . However the dependence on the cut off becomes more severe in the derivative expansion. Applied to the Polchinski equation there are two constants which are essentially arbitrary . Apart from this arbitrariness the results also depend on the value chosen for in solving the coupled equations .
Exact RG equations, without approximations, are invariant under reparameterisations, including rescalings, of the fundamental fields . This property ensures that the full equations have a line of physically equivalent fixed points which may be parameterised by different values of , [15, 20, 5]. Physical results, such as , are independent of where on this line the fixed point solution is chosen. As a consequence of the line of equivalent fixed points the calculated exponents must include one which is exactly zero. The corresponding marginal operator is redundant, essentially one which vanishes on the equations of motion.
In the context of the Polchinski RG equation it was shown, for arbitrary dimensions , in  that for any local operator , such that gives an eigen-operator represented by an eigenfunction for the linearised equations with critical exponent , then it is possible to construct associated redundant operators with exponents
irrespective of any particular choice of a smooth cut off function. Furthermore the operator is a local operator determining an eigenfunction with and hence, applying (1.1) with , this directly shows that is a possible eigenvalue whose eigenfunction generates the marginal operator necessary for reparameterisation invariance.
Although reparameterisation invariance is a property of the full non linear RG equations it is generally lost in approximations such as the derivative expansion. There is no longer a fluctuation eigenfunction with exactly. Here we heuristically construct equations for , and also for multi-component generalisations, which maintain these desirable features. The equation for remains the same as in the LPA except for the introduction of . The associated equation for depends on the solution for and determines . Using an appropriate scalar product an integral expression may be found which may be used to find in a fashion which is manifestly independent of . The eigenvalues for the corresponding differential operator are in accord with (1.1) when and the zero mode eigenfunction can also be found explicitly.
For the purposes of comparison we also discuss results for derivative operators of the form using standard perturbation theory techniques and the -expansion to obtain results for the anomalous dimensions at the fixed point to first order in . For such derivative operators it is necessary to take account of mixing with scalar operators with the same dimension but there are also additional constraints on the associated -functions. Keeping only contributions just to first order in the coupled then an infinitesimal variation , for non-linear , in the lagrangian is equivalent to corresponding changes in . This leads to identities which show that the scaling dimensions satisfy relations of the same form as in (1.1) in general in a perturbative framework. We are also able to determine the scale dimensions to at each of the multi-critical fixed points for scalar theories when symmetry is imposed. Although such operators are irrelevant as far as RG flows they are of course of interest in determining the spectrum of operators and scale dimensions in the theory at its critical points.
In this paper we describe in the next section results for the simplest case of a single component field which corresponds to the Ising model and has been much discussed previously. For the equations are solved numerically and the associated eigenvalues determined for various values of . The appropriate value of necessary for a non singular solution of the -equation is also found. In section 3, we extend the discussion to multi-component fields, imposing symmetry so that simple equations, of similar form to those considered in section 2, are obtained. The eigenfunctions are then tensors. The irreducible representations are given by symmetric traceless tensors of rank and the corresponding eigenvalues depend also om . Numerical results are then given for various and . In section 4 we show how these equations may be solved in an -expansion recovering perturbative results at the various possible non-trivial multi-critical fixed points as the dimension is reduced. In section 5 we consider perturbatively the usual -functions in a loop expansion, extending results obtained in the single component case in . In section 6 these results are extended to derivative operators and mixing effects taken into account. In a conclusion we make some more general remarks concerning the status of the equations discussed in this paper. Although they have been motivated by requiring that they share general properties of the exact RG equations they serve to show how these may still be maintained in quite simple approximations. Various calculational details are relegated to four appendices. In appendix A we show how the equations can be solved for large and a formula for to obtained which is quite close to the exact large result. In appendix B we give some details of the perturbative results for -functions that are used in sections 5 and 6. In appendix C we give a general discussion using dimensional regularisation of the consequences of invariance of the regularised theory under variations . In the final appendix D we give some details of the nearest singularities that are found numerically when the solution of the local potential approximation for the Polchinski equation is extended to the complex plane.
2 Equations for a Single Component Field
It is simplest to consider first a single scalar field corresponding to the universality class for the Ising model. At a fixed point the equation for is
This is just the standard LPA for the Polchinski equation including the anomalous dimension . In general this is set to zero as there is no mechanism to determine this from (2.1). The two trivial solutions of (2.1) are , for the Gaussian fixed point, and , for the high temperature fixed point. Non trivial solutions even in , so that , which are non singular for all and
with the differential operator
It is easy to see, using (2.1), that
and hence we may construct two exact odd eigenfunctions
It is important for our later discussion to recognise that in (2.4) is hermitian with respect to a scalar product defined by
Extending the RG equations to we propose that, in conjunction with (2.1), the associated equation at the fixed point
Together with (2.1) this satisfies reparameterisation invariance so that is independent of the particular initial , unlike the case for other analogous derivative expansion equations. To ensure non singular solutions for all requires only a special choice for . Asymptotically, for large , solutions of (2.9) have the form
In general the value of the asymptotic constant depends on .
The proposed -equation (2.9) is similar to the derivative expansion result in . It differs in that the coefficient of the term is 2, rather than 4, and that on the right hand side there is a definite coefficient rather than an essentially arbitrary cut off dependent constant (in terms of the equations in  we are taking for the cut dependent constant the precise value ). In respect of these terms (2.9) is identical with an analogous equation obtained in  using an expansion in terms of scaling fields which is similar in spirit to the derivative expansion. In the scaling field approach the corresponding coefficient is determined precisely essentially by those divergencies in two point amplitudes which are universal, i.e. renormalisation scheme independent. If is the cut off dependent propagator, then we have for the following products
such that are constants, independent of the cut off function and depending only on the large behaviour of . According to the results obtained in 
The coefficients in RG equations such as (2.9) should not depend on the particular critical point, here labelled by , but may depend on the spatial dimension . Applying the -expansion to (2.9) with the particular coefficient ensures, as was shown in  and also subsequently here, that is correct to for all critical points Although results such as these for were obtained from Wilsonian RG equations as soon as they were first proposed, and were shown to be independent of the detailed cut off function , they are also identical, of course, with results from standard Feynman graph techniques which arise directly from the coefficients of the universal logarithmic divergencies for particular two point Feynman graphs. These logarithmic divergencies are equivalent to (2.11). In a sense compatibility with the -expansion may be regarded as an optimal choice for such constants as . However, in the scaling field derivation described in  there is no free constant to determine and agreement with the -expansion is not imposed but follows automatically.
In addition (2.9) differs from corresponding equations in  and  by the absence of a term. Removing such a contribution is essential to obtain subsequent results. In general in a derivative expansion there are also expected to be additional contributions on the right hand side of (2.1) involving but the exact form differs between  and  and also involves a cut off dependent constant which we are here essentially setting to zero.
It is easy to see that this decouples into pairs of equations for eigenvalues where is obtained from (2.3), with the corresponding determined in terms of by inverting , and also, with ,
For the eigenfunctions in (2.6) the corresponding functions are given by
representing the necessary marginal redundant operator present in the RG flow equations. Since is hermitian with respect to the scalar product in (2.7) we must have, for consistent solutions of (2.9),
When we may readily solve (2.1) numerically tuning so that the singularity in the solution arises for the largest possible value of compatible with numerical precision, (2.1) was written as two coupled first order equations and were integrated from using RK4. The limiting results when are shown in Figure 1.
Numerical results for the -equation (2.9) are also shown in Figure 2. These were obtained in a similar fashion as for in terms of corresponding first order differential equations. The solutions also develop singularities which are very sensitive to the value of , where the corresponding -solution of course has been used.
Having determined the eigenvalues are then determined numerically for small values of and by optimising the eigenvalue such that the eigenfunction blows up as slowly as possible withing the range where results for are reliable. The results are ordered such that with . For even,odd the associated eigenfunctions are even,odd in . From (2.6) and which provides a consistency check on our numerical results. For even the results are in Table 1 and for odd in Table 2. For the small values considered the dependence on is close to linear. For our results agree with the much more accurate determinations in [25, 26].
3 Multi-Component Fields
There are natural generalisations of the above equations to the case of a -component scalar field . Instead of (2.1) we have
for a symmetric anomalous dimension matrix and . The corresponding equation for critical exponents is just as in (2.3),
but with the differential operator
and we may now also allow tensorial eigenfunctions . Trivially so that is an exact eigenvalue. Just as in (2.6) we have exact vector eigenfunctions since
and we may choose a diagonal basis for . The corresponding scalar product to (2.7) is just
with additional tensorial contractions if required.
Extending (2.9) to we require
with used to denote symmetrisation of the indices. For fluctuations around the fixed point solutions we have (3.2) and also the associated coupled equation
As before it is easy to see that the possible eigenvalues are determined by (3.2) and given by
Hence we have an exact zero mode which may be obtained from (3.4)
In consequence for (3.6) to be solvable we must have
In order to obtain tractable equations we impose symmetry so that we need only deal with functions of and in this case we must have . Writing (3.1) becomes
At the origin we must have and asymptotically
To ensure non singular solutions as before it is necessary to fine tune . For critical exponents we consider, if , spherical harmonics which are expressible in terms of symmetric traceless tensors or equivalently
The eigenvalue equation (3.2) becomes
The relevant boundary conditions are that is analytic for and non singular for and as . Corresponding to (3.4)
so that . The scalar product, with respect to which is hermitian, becomes from (3.5)
When it is easy to see that this is identical with (2.18) since then .
To decompose (3.6) we write
we may reduce (3.6) to
The equation for thus decouples from that for so that (3.25) may be solved and the result used in (3.24) to then determine . Asymptotically approaches a constant, just as in (2.10), but vanishes. Since, as shown in (3.19), has a zero mode , given by (3.18), must be fixed to allow a non singular solution by,
we may also obtain
Combining (3.26) and (3.28) is equivalent to (3.21). If we restrict to scalar fluctuations, without any harmonics, then we may decompose in terms of in a similar fashion to (3.22) so that (3.8) reduces to
with and also
when . Manifestly the eigensolutions for in (3.29) are identical with the solutions of (3.16) for . The eigenvalues are related as expected from (1.1) so the solutions of (3.29) represent redundant operators. The eigenvalues determined by the -equation (3.30) give exponents corresponding to genuine physical scaling operators.
Numerically (3.13) and (3.16) can be solved straightforwardly for , as before after precisely tuning to ensure non singular solutions for all . The eigenvalues are denoted by where and we take . In terms of the single component results and . When some results are given in table 3, for they match as expected the results given earlier. For these agree with results in [27, 28]. In the large limit, taking , then .
For then and which is a useful check. Some other results are given in table 5. We also present some results for in table 6 and in table 7. An important observation is that whereas for . This reflects the instability of the symmetric fixed point against RG flow to one with just discrete cubic symmetry when . That the critical has been shown in very detailed multi-loop Feynman diagram calculations .