DAMTP-2007-77

Epsilon Expansion for Multicritical Fixed Points

and Exact Renormalisation Group Equations

J<EMAIL_ADDRESS>and H<EMAIL_ADDRESS>

Department of Applied Mathematics and Theoretical Physics,

Wilberforce Road, Cambridge, CB3 0WA, England

The Polchinski version of the exact renormalisation group equations is applied
to multicritical fixed points, which are present for dimensions between two
and four, for scalar theories using both the local potential approximation and
its extension, the derivative expansion. The results are compared with the
epsilon expansion by showing that the non linear differential equations may be
linearised at each multicritical point and the epsilon expansion treated as a
perturbative expansion. The results for critical exponents are compared with
corresponding epsilon expansion results from standard perturbation theory. The
results provide a test for the validity of the local potential approximation
and also the derivative expansion. An alternative truncation of the exact RG
equation leads to equations which are similar to those found in the derivative
expansion but which gives correct results for critical exponents to order
$\varepsilon$ and also for the field anomalous dimension to order
$\varepsilon^{2}$. An exact marginal operator for the full RG equations is
also constructed.

PACS:11.10.-z, 11.10.Gh, 64.60.Fr, 64.60Ak, 64.60.Kw, 68.35.Rh
Keywords:Epsilon Expansion, Exact Renormalisation Group, Multicritical points.

##  1 Introduction

A fundamental development in quantum field theory was understanding the role
of the renormalisation scale induced by the presence of a cut off, or any
other regularisation ensuring finiteness, and the associated flow of the
couplings of the theory under changes of scale. The RG flow equations
therefore reflect the essential arbitrariness of the renormalisation scale.
Nevertheless the global nature of the renormalisation flows in the space of
couplings and the various fixed points that are present are crucial properties
of any particular quantum field theory of physical interest, although in
general their analysis is beyond the scope of conventional perturbation
theory. Since the time of Wilson [1, 2, 3, 4] various exact RG equations have
been formulated which in principle transcend perturbation theory and allow the
determination of fixed points and also the critical exponents that determine
the flow of the couplings in the neighbourhood of fixed points, for recent
reviews see [5, 6, 7, 8] and for a critical discussion [9].

For theories involving just scalar fields, when a cut off function is
introduced in the quadratic part of the action, these have been extensively
explored. At a rigorous level they may be used to provide an alternative proof
of the renormalisability of such theories [4, 10]. On the other hand outside
the perturbative domain it is necessary to resort to approximations when the
functional differential equations for the RG flow of the effective action,
which are in principle exact, are reduced to non linear coupled differential
equations which may then be analysed numerically. The simplest approximation
is when the effective action, in general a nonlocal functional of local
fields, is restricted to a function just of the scalar field without any
derivatives, the local potential approximation (LPA) [11]. Beyond the LPA it
is possible to consider a derivative expansion to second and potentially
higher orders in the number of derivatives. However these approximations are
essentially uncontrolled. The resulting equations depend in detail on the form
of the cut off function and it is unclear whether there is any systematic
procedure for improving, in principle, order by order the accuracy of results
for critical exponents which should be independent of the particular form of
the cut off.

Despite such difficulties the numerical results are often impressive and are
in good agreement with other methods of determining critical exponents for
appropriate statistical field theories in three dimensions. The LPA is
applicable to various different versions of the exact renormalisation group.
In general the resulting equations are inequivalent but the LPA for the
Polchinski equation [4] with scalar fields, where the cut off dependence can
be removed by simple rescalings and so is absent from calculated critical
exponents, the results are identical to the LPA ERG equations for the one
particle irreducible generating function with a particular smooth cut off
function [12, 13]. Expanding the action as an integral over local functions of
the fields with increasing numbers of derivatives then at other than zeroth
order there is an intrinsic dependence on the cut off in the resulting
truncated equations which cannot be removed by redefinitions. For the
Polchinski equation this involves at each order just a finite set of
parameters which are essentially arbitrary.

Nevertheless the basic LPA, yielding a simple nonlinear differential flow
equation for a potential $V(\phi)$, encapsulates the essential fixed point
structure of such scalar theories. As the dimension $d$ is reduced a new fixed
point is generated whenever the operator $\phi^{2n}$, for $n=2,3,\dots$,
becomes marginal. In the neighbourhood of each fixed point the flow equations
determine various critical exponents which may be compared with results from
other calculational methods. A not yet fully realised goal is whether it is
possible to improve the LPA, while restricting to just a tractable finite set
of coupled partial differential equations but with a systematic prescription
for the determination of any parameters present, so as to ensure that results
for critical exponents should be quantitatively improved, closer to the
results of the particular quantum field theory, for all fixed points.

As a possible procedure for understanding how far the LPA and its extensions
are valid we consider here the connection with the $\varepsilon$-expansion. As
originally shown by Wilson and Fisher [14] this provides a method whereby
standard quantum field theory calculations of $\beta$-functions and related
anomalous dimensions as a loop expansion in $d=4-\varepsilon$ dimensions may
be applied to determine critical exponents for $d=3$ as an asymptotic power
expansion in $\varepsilon$. For an extensive discussion in the context of
standard quantum field theory see [15]. The $\varepsilon$-expansion can also
be obtained directly from exact RG equations, as was the case historically,
since for $\varepsilon\to 0$ the equations become linear and the non linear
terms may be treated perturbatively. An interesting question is then the
extent to which the $\varepsilon$-expansion results are compatible with those
from the LPA. Although this has been considered previously we here attempt a
systematic discussion in relation to the Polchinski RG equation. Initially
this is applied for just the LPA itself but we also consider derivative
expansion extensions to see whether any improvements in the domain of joint
validity is feasible. A similar discussion for $2<d<4$ is undertaken for the
hierarchical RG in [16].

An alternative approximation for the exact RG flow equations is to consider
expanding the effective action in terms of translation invariant functions of
the basic fields which are eigenfunctions of the linearised RG flow functional
differential operator which are referred to as scaling fields [17]. The non
linear part of the RG flow equation may then be expanded in this basis. This
gives a set of coupled equations which in the simplest approximation is
equivalent to the LPA and at the next order is very similar to the derivative
approximation. However in this approach the dependence on the cut off function
is more controlled and in the $\varepsilon$-expansion it is possible to get
the correct result for the critical exponent $\eta$ at order ${\rm
O}(\varepsilon^{2})$, unlike in the usual derivative expansion.

In this paper in Section 2 we first consider standard perturbative
calculations, with the aid of the background field method, for determining
critical exponents in the $\varepsilon$-expansion at all multicritical points
for a single scalar field. This is applied both for scalar operators with no
and also two derivatives. Some higher order results, which involve multi-loop
calculations, are obtained in Appendix A. Although the methods used are very
different from exact RG calculations they provide results which are useful
comparison for later approximations. In Section 3 we consider the LPA. It is
shown how at ${\rm O}(\varepsilon)$ the solution for each multicritical point
is a single Hermite polynomial, whose coefficient is determined by the
nonlinear terms, and at ${\rm O}(\varepsilon^{2})$ it is just a finite sum. In
Section 4 the results are worked out in more detail for the first three
critical points and graphical comparisons are made between the approximate
analytic solution and numerical solutions for various $d$. In 5 the
corresponding critical exponents, within the LPA, are found at ${\rm
O}(\varepsilon)$ and also ${\rm O}(\varepsilon^{2})$, where they disagree with
the perturbative results.

The LPA is well known to be of restricted validity, it requires that the
critical exponent $\eta$, which is essentially the anomalous dimension of the
elementary scalar field, is zero. The derivative expansion attempts to
overcome these limitations and we consider this in the context of the
$\varepsilon$-expansion in Section 6. The coupled equations now depend on two
cut off function dependent constants $A,B$ but they now allow $\eta$ to be
determined. The solutions in terms of Hermite polynomials may also be extended
to this case with some modifications. Following this in 7 we use these results
to determine critical exponents at ${\rm O}(\varepsilon)$ for two classes of
scalar operators. For one class the results are the same as in the LPA case
and agree with perturbation theory, for the other set of operators which
involve derivatives the calculated exponents depend on $A,B$.

The scaling field approach based on the exact RG flow equation is considered
in Section 8. A similar truncation to the derivative expansion is possible
leading to equations which also may be solved simply in the
$\varepsilon$-expansion. In this case the dependence on the cut off function
resides in various integrals. In special cases these are independent of the
precise cut off function and they then determine universal results for
critical exponents to ${\rm O}(\varepsilon)$ and also $\eta$ to ${\rm
O}(\varepsilon^{2})$. The relevant integrals are discussed in Appendix C where
the cut off function independent values are shown to be related to logarithmic
divergences in two vertex Feynman integrals. In Section 9 the resulting
equations are recast as coupled differential equations which are very similar,
although different in detail, to those arising in the derivative expansion.
Some more general remarks are contained in a conclusion. In Appendix D we
obtain some exact results for perturbations of the full RG flow equations and
show how to construct an exact marginal operator. The existence of such an
operator, leading to a line of equivalent fixed points, ensures that the RG
equations determine $\eta$.

##  2 Perturbation Calculations

We here discuss for the purposes of comparison a conventional quantum field
theory calculation of critical exponents at multicritical fixed points in the
$\varepsilon$-expansion. We initially consider just the basic Lagrangian

${\cal L}={\textstyle\frac{1}{2}}(\partial\phi)^{2}+V(\phi)\,.$ (2.1)

To determine the counterterms to ensure a finite theory it is sufficient as
usual to consider connected one particle irreducible graphs. We adopt as the
basic propagator

$G_{0}(x)=\frac{1}{4\pi}\,\frac{\Gamma(\nu)}{\pi^{\nu}}\,\frac{1}{(x^{2})^{\nu}}\,,\qquad\nu={\textstyle\frac{1}{2}}d-1\,,$
(2.2)

satisfying $-\partial^{2}G_{0}(x)=\delta^{d}(x)$, and also use a background
field approach, following similar methods used for four dimensional theories
in [18], where

$\phi=\varphi+f\,,$ (2.3)

with $f$ the quantum field. Only vacuum graphs are then necessary and since
with dimensional regularisation $G_{0}(x)|_{x=0}=0$ no graphs with lines
involving a single vertex need be included.

At lowest order for the one particle irreducible functional $W$ we have

$W_{1}=\sum_{r\geq 2}\frac{1}{2r!}\int{\rm d}^{d}x_{1}\,{\rm
d}^{d}x_{2}\,V^{(r)}(\varphi_{1})\,G_{0}(x_{12})^{r}\,V^{(r)}(\varphi_{2})\,,\qquad
x_{ij}=x_{i}-x_{j}\,,\ \varphi_{i}=\varphi(x_{i})\,.$ (2.4)

To evaluate this we note that

$\int{\rm d}^{d}x\,e^{ik\cdot
x}\,G_{0}(x)^{r}=\frac{1}{(4\pi)^{r}}\,\frac{\Gamma(\nu)^{r}}{\Gamma(r\nu)}\
\Gamma\big{(}1-(r-1)\nu\big{)}\,\bigg{(}\frac{k^{2}}{4\pi}\bigg{)}^{(r-1)\nu-1}\,.$
(2.5)

This has a pole whenever $(r-1)\nu=1,2,\dots$ so that writing

$d=d_{n}-\varepsilon\,,\qquad d_{n}=\frac{2n}{n-1}\,,$ (2.6)

it is easy to see from (2.5) that for $\varepsilon\to 0$

$\displaystyle G_{0}(x)^{n}\sim{}$
$\displaystyle\frac{2}{\varepsilon}\,\frac{1}{(4\pi)^{n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,\delta^{d}(x)\,,$
(2.7a) $\displaystyle G_{0}(x)^{2n-1}\sim{}$
$\displaystyle\frac{1}{\varepsilon}\,\frac{1}{(4\pi)^{2n}}\,\frac{n-1}{n}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\,\partial^{2}\delta^{d}(x)\,.$
(2.7b)

From (2.4), assuming $r\leq 2n$, the necessary counterterms are then

$\displaystyle{\cal L}_{\rm{c.t.}1}={}$
$\displaystyle\frac{1}{\varepsilon}\,\frac{1}{(4\pi)^{n}}\,\frac{1}{n!}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,V^{(n)}(\phi)^{2}$
$\displaystyle{}-\frac{1}{\varepsilon}\,\frac{1}{(4\pi)^{2n}}\,\frac{n-1}{(2n)!}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\,V^{(2n)}(\phi)^{2}(\partial\phi)^{2}\,,$
(2.8)

where the two terms arise from (2.7a) and (2.7b) at $n-1$ and $2n-2$ loops
respectively. The theory is therefore renormalisable for $V(\phi)$ a
polynomial of degree $2n$. For $\mu$ a regularisation scale and
$\mu^{-\varepsilon}({\cal L}+{\cal L}_{\rm{c.t.}})={\cal
L}_{0}={\textstyle\frac{1}{2}}(\partial\phi_{0})^{2}+V_{0}(\phi_{0})$ the
usual perturbative $\beta$-functions and anomalous dimensions may be defined
by

$\mu\frac{{\rm d}}{{\rm d}\mu}{\cal L}\Big{|}_{{\cal
L}_{0}}=-{\hat{\gamma}}_{\phi}\,\phi\frac{\partial}{\partial\phi}{\cal
L}+{\hat{\beta}}^{V}\\!(\phi)\,,$ (2.9)

where

${\hat{\gamma}}_{\phi}=-{\textstyle\frac{1}{2}}\varepsilon+\gamma_{\phi}\,,\qquad{\hat{\beta}}^{V}\\!(\phi)=-\varepsilon\big{(}{\textstyle\frac{1}{2}}\phi
V^{\prime}(\phi)-V(\phi)\big{)}+{\beta}^{V}\\!(\phi)\,.$ (2.10)

At lowest order (2.8) then gives

$\beta_{1}^{V}\\!(\phi)=\frac{1}{(4\pi)^{n}}\,\frac{n-1}{n!}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,V^{(n)}(\phi)^{2}\,.$
(2.11)

Assuming

$V(\phi)=\frac{1}{(2n)!}\,g\,\phi^{2n}+\sum_{k}\frac{1}{k!}\,g_{k}\,\phi^{k}\,,$
(2.12)

then $\gamma_{\phi}=\gamma_{\phi}(g)$ and

$\displaystyle{\hat{\beta}}^{V}(\phi)={}$
$\displaystyle\frac{1}{(2n)!}\,{\hat{\beta}}^{g}(g)\,\phi^{2n}+\sum_{k}\frac{1}{k!}\,{\hat{\gamma}}_{k}(g)\,g_{k}\,\phi^{k}+{\rm
O}(g_{k}g_{k^{\prime}})\,$ $\displaystyle{\hat{\beta}}^{g}(g)={}$
$\displaystyle-\varepsilon\,(n-1)g+\beta^{g}(g)\,,\qquad{\hat{\gamma}}_{k}(g)=-\varepsilon\,{\textstyle\frac{1}{2}}(k-2)+\gamma_{k}(g)\,.$
(2.13)

In general ${\hat{\gamma}}_{1}(g)=-{\hat{\gamma}}_{\phi}(g)$. Applying (2.10)
and (2.11) to ${\rm O}(g)$ we then have

$\gamma_{k,1}(g)=\frac{2g}{(4\pi)^{n}}\,\frac{n-1}{n!^{2}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,\frac{k!}{(k-n)!}\,.$
(2.14)

As usual in the $\varepsilon$-expansion, there is a fixed point such that
${\hat{\beta}}^{g}(g_{*})=0$ with $g_{*}$ expressible perturbatively as a
power series in $\varepsilon$. At lowest order it is easy to see from (2.11)
and (2.13) that

$\frac{g_{*}}{(4\pi)^{n}}\,\frac{(2n)!}{n!^{3}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}=\varepsilon\,,$
(2.15)

and the result (2.14) then gives

${\hat{\gamma}}_{k}(g_{*})=-{\textstyle\frac{1}{2}}(k-2)\varepsilon+2(n-1)\,\frac{n!}{(2n)!}\,\frac{k!}{(k-n)!}\,\varepsilon+{\rm
O}(\varepsilon^{2})\,.$ (2.16)

We may also use (2.8) to determine the anomalous dimension $\gamma_{\phi}(g)$
of the field $\phi$ which is non zero at $2(n-1)$ loops,

$\gamma_{\phi,1}(g)=\frac{g^{2}}{(4\pi)^{2n}}\,\frac{2(n-1)^{2}}{(2n)!}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\,.$
(2.17)

At the fixed point the critical exponent $\eta=2\gamma_{\phi}(g_{*})$ so that
from (2.17)

$\eta=4(n-1)^{2}\,\frac{n!^{6}}{(2n)!^{3}}\,\varepsilon^{2}+{\rm
O}(\varepsilon^{3})\,.$ (2.18)

The $\varepsilon$-expansion at multicritical points using standard quantum
field theory was considered in [19] who obtained (2.15) and (2.18). In three
dimensions, corresponding to $n=3$, results equivalent to (2.14) were obtained
in [20].

The local operators in the basic quantum field include also those with
derivatives as well as just $\phi^{k}$. To extend the above discussion we
consider in addition to (2.1)

${\cal L}^{Z}={\textstyle\frac{1}{2}}Z(\phi)(\partial\phi)^{2}\,,\qquad
Z(\phi)=\sum_{k\geq 2n}\frac{1}{(k-2n)!}\,h_{k}\,\phi^{k-2n}\,,$ (2.19)

where we keep only graphs which involve one $Z$-vertex. At the same order as
(2.4) we have

$\displaystyle W^{Z}_{1}=\sum_{r\geq 2}\frac{1}{2r!}$ $\displaystyle\int{\rm
d}^{d}x_{1}\,{\rm
d}^{d}x_{2}\,\Big{\\{}Z^{(r)}(\varphi_{1})(\partial\varphi_{1})^{2}\,G_{0}(x_{12})^{r}$
$\displaystyle{}+2r\,Z^{(r-1)}(\varphi_{1})\,\partial\varphi_{1}\cdot\partial
G_{0}(x_{12})G_{0}(x_{12})^{r-1}$
$\displaystyle{}+r(r-1)\,Z^{(r-2)}(\varphi_{1})\,\big{(}\partial
G_{0}(x_{12})\big{)}^{2}G_{0}(x_{12})^{r-2}\Big{\\}}V^{(r)}(\varphi_{2})\,.$
(2.20)

Using (2.7a) with

$\displaystyle\partial G_{0}(x)\,G_{0}(x)^{n-1}\sim{}$
$\displaystyle\frac{2}{\varepsilon}\,\frac{1}{n}\,\frac{1}{(4\pi)^{n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,\partial\,\delta^{d}(x)\,,$
$\displaystyle\big{(}\partial G_{0}(x)\big{)}^{2}\,G_{0}(x)^{n-2}\sim{}$
$\displaystyle\frac{2}{\varepsilon}\,\frac{1}{n(n-1)}\,\frac{1}{(4\pi)^{n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,\partial^{2}\delta^{d}(x)\,,$
(2.21)

shows that in addition to (2.8) the counterterms are

$\displaystyle{\cal L}^{Z}_{\rm{c.t.}1}={}$
$\displaystyle\frac{1}{\varepsilon}\,\frac{1}{(4\pi)^{n}}\,\frac{1}{n!}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,\Big{(}Z^{(n)}(\phi)V^{(n)}(\phi)+Z^{(n-1)}(\phi)V^{(n+1)}(\phi)\Big{)}(\partial\phi)^{2}\,.$
(2.22)

With $\mu^{-\varepsilon}({\cal L}^{Z}+{\cal L}^{Z}_{\rm{c.t.}})={\cal
L}_{0}^{Z}$ the corresponding $\beta$-function is given by

$\mu\frac{{\rm d}}{{\rm d}\mu}{\cal L}^{Z}\Big{|}_{{\cal L}_{0}+{\cal
L}_{0}^{Z}}=-{\hat{\gamma}}_{\phi}\,\phi\frac{\partial}{\partial\phi}{\cal
L}^{Z}+{\textstyle\frac{1}{2}}\,{\hat{\beta}}^{Z}(\phi)(\partial\phi)^{2}\,,\qquad{\hat{\beta}}^{Z}(\phi)=-\varepsilon{\textstyle\frac{1}{2}}\phi
Z^{\prime}(\phi)+{\beta}^{Z}(\phi)\,,$ (2.23)

which can be expanded as

$\displaystyle{\hat{\beta}}^{Z}(\phi)={}$ $\displaystyle\sum_{k\geq
2n}\frac{1}{(k-2n)!}\,\big{(}{\hat{\gamma}}^{hh}_{k}(g)\,h_{k}+{\gamma}^{hg}_{k}(g)\,g_{k}\big{)}\,\phi^{k-2n}\,,$
$\displaystyle{\hat{\gamma}}^{hh}_{k}(g)=-\varepsilon\,{\textstyle\frac{1}{2}}(k-2n)+\gamma^{hh}_{k}(g)\,.$
(2.24)

We must also extend (2.13) to include mixing effects if $k\geq 2n$ to the form

${\hat{\beta}}^{V}(\phi)=\frac{1}{(2n)!}\,{\hat{\beta}}^{g}(g)\,\phi^{2n}+\sum_{k}\frac{1}{k!}\,\big{(}{\hat{\gamma}}^{gg}_{k}(g)\,g_{k}+{\gamma}^{gh}_{k}(g)\,h_{k}\big{)}\,\phi^{k}\,,\quad{\hat{\gamma}}^{gg}_{k}(g)={\hat{\gamma}}_{k}(g)\,.$
(2.25)

The terms in ${\cal L}^{Z}$ involving $g_{2n},h_{2n}$ may be absorbed in
${\cal L}$ by a redefinition of $\phi,g$ giving for this special case

$\displaystyle{\hat{\gamma}}^{gg}_{2n}(g)={}$
$\displaystyle{\hat{\beta}}^{g}{}^{\prime}(g)-2ng\,{\hat{\gamma}}_{\phi}{}^{\prime}(g)\,,\quad\gamma^{gh}_{2n}(g)=-ng\,{\hat{\gamma}}^{gg}_{2n}(g)+n{\hat{\beta}}^{g}(g)\,,$
$\displaystyle\gamma^{hg}_{2n}(g)={}$
$\displaystyle-2{\hat{\gamma}}_{\phi}{}^{\prime}(g)\,,\hskip
52.63777pt{\hat{\gamma}}^{hh}_{2n}(g)=2ng\,{\hat{\gamma}}_{\phi}{}^{\prime}(g)\,.$
(2.26)

The anomalous dimensions of operators at the fixed point are then given by the
eigenvalues $\omega_{k,1},\,\omega_{k,2}$ of the matrix

$\left(\begin{array}[]{cc}{\hat{\gamma}}^{gg}_{k}(g_{*})&{\gamma}^{gh}_{k}(g_{*})\\\
\vskip 3.0pt\cr{\gamma}^{hg}_{k}(g_{*})&{\hat{\gamma}}^{hh}_{k}(g_{*})\\\
\end{array}\right)\,,$ (2.27)

for $k=2n,\dots,4n-3$. It is easy to see from (2.26) that
$\omega_{2n,1}={\hat{\beta}}^{g}{}^{\prime}(g_{*}),\ \omega_{2n,2}=0$. More
generally, using the equations of motion,
$\omega_{k,2}=\omega_{k-2n+1,1}-{\textstyle\frac{1}{2}}\varepsilon+{\textstyle\frac{1}{2}}\eta$.

Since $\beta^{Z}_{1}=(n-1)\varepsilon{\cal L}^{Z}_{\rm{c.t.}1}$ then from
(2.22) we have at lowest order

${\gamma}^{hh}_{k,1}(g)=\frac{2g}{(4\pi)^{n}}\,\frac{n-1}{n!^{2}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\frac{(k-2n+1)!}{(k-3n+1)!}\,,$
(2.28)

and ${\gamma}^{gg}_{k,1}(g)={\gamma}_{k,1}(g)$ as in (2.14). For the off
diagonal parts of the anomalous dimension matrix we may note that (2.8)
implies a $2(n-1)$-loop contribution to $\beta^{Z}$,

$\Delta\beta^{Z}_{1}(\phi)=-\frac{1}{(4\pi)^{2n}}\,\frac{4(n-1)^{2}}{(2n)!}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\,V^{(2n)}(\phi)^{2}\,,$
(2.29)

and also we have an additional term at $(n-1)$-loops to that in (2.11),

$\Delta\beta_{1}^{V}(\phi)=-\frac{1}{(4\pi)^{n}}\,\frac{1}{(n-2)!}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,V^{(n)}(\phi)^{2}Z(\phi)\,.$
(2.30)

Thus the lowest order non zero off diagonal terms are

$\displaystyle{\gamma}^{gh}_{k,1}(g)={}$
$\displaystyle-\frac{g^{2}}{(4\pi)^{n}}\,\frac{1}{(n-2)!n!^{2}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,\frac{k!}{(k-2n)!}\,,$
$\displaystyle{\gamma}^{hg}_{k,1}(g)={}$
$\displaystyle-\frac{g}{(4\pi)^{2n}}\,\frac{8(n-1)^{2}}{(2n)!}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\,.$
(2.31)

Thus $\gamma^{gh}_{k,1}(g_{*})={\rm O}(\varepsilon^{2})$ and
${\gamma}^{hg}_{k,1}(g_{*})={\rm O}(\varepsilon)$ so that the lowest order
eigenvalues of (2.27) are given by (2.16) for $\omega_{k,1}$ and

$\omega_{k,2}={\hat{\gamma}}^{hh}_{k}(g_{*})+{\rm
O}(\varepsilon^{2})=-{\textstyle\frac{1}{2}}(k-2n)\varepsilon+2(n-1)\,\frac{n!}{(2n)!}\,\frac{(k-2n+1)!}{(k-3n+1)!}\,\varepsilon+{\rm
O}(\varepsilon^{2})\,.$ (2.32)

Further perturbative results for $\beta^{V}$ and anomalous dimensions are
obtained in Appendix A.

##  3 Local Potential Approximation

In the LPA, the Polchinski RG equation in $d$ dimensions may be reduced by an
appropriate rescaling to the following renormalisation flow for a potential
$V(\phi,t)$,

$\dot{V}(\phi,t)=V^{\prime\prime}(\phi,t)-V^{\prime}(\phi,t)^{2}+dV(\phi,t)-{\textstyle\frac{1}{2}}(d-2)\,\phi
V^{\prime}(\phi,t)\,,$ (3.1)

where $t=-\log\Lambda$ and $\Lambda$ is a cut off scale. At a fixed point,
$V(\phi,t)\to V_{*}(\phi)$ which solves

$V_{*}^{\prime\prime}(\phi)-V_{*}^{\prime}(\phi)^{2}+dV_{*}(\phi)-{\textstyle\frac{1}{2}}(d-2)\,\phi
V_{*}^{\prime}(\phi)=0\,.$ (3.2)

This equation has been extensively analysed, both numerically and analytically
[21, 22]. There are two trivial solutions $V_{*}=0$, the Gaussian fixed point,
and $V_{*}(\phi)={\textstyle\frac{1}{2}}\phi^{2}-\frac{1}{d}$, the so called
high temperature fixed point. For non trivial solutions even in $\phi$, with
$V_{*}{\\!}^{\prime}(0)=0$, and bounded below, it is necessary to fine tune
$V_{*}(0)$ to ensure that there are no singularities for all $\phi$. Such
solutions appear whenever $d$ is reduced below $2n/(n-1)$ for $n=2,3,\dots$.

For our purposes it is convenient to consider a further rescaling by defining

$x={\textstyle\frac{1}{2}}(d-2)^{\frac{1}{2}}\phi\,,\qquad
v(x,t)=V(\phi,t)\,,\ \ v_{*}(x)=V_{*}(\phi)\,.$ (3.3)

Then (3.2) becomes

$v_{*}^{\prime\prime}-2xv_{*}^{\prime}+\frac{4d}{d-2}\,v_{*}=v_{*}^{\prime
2}\,,$ (3.4)

where $v_{*}^{\prime}=\frac{{\rm d}v_{*}}{{\rm d}x}$.

We first consider the linearised form of (3.4),

$\frac{{\rm d}^{2}v}{{\rm d}x^{2}}-2x\frac{{\rm d}v}{{\rm
d}x}+\frac{4d}{d-2}\,v=0\,,$ (3.5)

which becomes a valid approximation when $v(x)$ is small. Global solutions for
all $x$ which are bounded by a power for large $x$ are only possible for
Hermite polynomials $H_{n}(x)$, $n=0,1,2,\dots$ which satisfy

$(D+n)H_{n}=0\,,\qquad D={\textstyle\frac{1}{2}}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}-x\frac{{\rm d}}{{\rm d}x}\,.$ (3.6)

Hence (3.5) has solutions with the appropriate behaviour

$v(x)=cH_{n}(x)$ (3.7)

only when $d$ is restricted to

$n=\frac{2d}{d-2}.$ (3.8)

With the further requirement that $v(x)$ be bounded below, we must restrict to
$n$ to be even and $c>0$. Relabelling $n\rightarrow 2n$ we then have solutions
of the linearised equation

$v(x)=cH_{2n}(x)\quad\hbox{for}\quad d=d_{n}\,,$ (3.9)

with $d_{n}$ defined in (2.6). This may then used as a starting point for the
analysis of the full non linear fixed point equation.

For the non-linear equation (3.4) we consider the case when $d$ is close to
the value in (3.9), where the linearised solution holds, and may be written as
in (2.6). We therefore seek solutions as $\varepsilon\to 0$ of the form

$v_{*}(x)=v_{n}(x)+{\rm O}(\varepsilon^{2})\,,\qquad v_{n}(x)=c_{n}\varepsilon
H_{2n}(x)\,.$ (3.10)

The right hand side of (3.4) is clearly ${\rm O}(\varepsilon^{2})$. Writing

$\frac{4d}{d-2}=4n+2(n-1)^{2}\varepsilon+(n-1)^{3}\varepsilon^{2}+{\rm
O}(\varepsilon^{3}).$ (3.11)

the leading ${\rm O}(\varepsilon)$ terms on the left hand side of (3.4) are
then absent. To ${\rm O}(\varepsilon^{2})$ we may determine $c_{n}$ in (3.10)
by noting that the differential operator $D+2n$, which is hermitian with
respect to the measure ${\rm d}x\,e^{-x^{2}}$, generates only functions
orthogonal to $H_{2n}$ so we must require at this order

$2(n-1)^{2}\varepsilon\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{2n}(x)v_{n}(x)=\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{2n}(x)v_{n}{\\!}^{\prime}(x)^{2}\,.$ (3.12)

Both sides may be evaluated using the integrals for Hermite polynomials

$N_{k}\equiv\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}{H_{k}(x)}^{2}=2^{k}\pi^{\frac{1}{2}}\,k!\,,$ (3.13)

and also for three $H_{n}$’s,

$\displaystyle G_{klm}\equiv{}$
$\displaystyle\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}{H_{k}(x)}{H_{l}(x)}{H_{m}}(x)=2^{s}\pi^{\frac{1}{2}}\frac{k!\,l!\,m!}{(s-k)!(s-l)!(s-m)!}\,,$
$\displaystyle{}s={\textstyle\frac{1}{2}}(k+l+m)\,,\qquad k,l,m\leq s\,,$
(3.14)

with $s$ required to be an integer. On the right hand side of (3.12)

$\displaystyle\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{2n}(x)v_{n}{\\!}^{\prime}(x)^{2}$
$\displaystyle=c_{n}{\\!}^{2}\varepsilon^{2}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{2n}(x){H_{2n}{\\!}^{\prime}(x)}^{2}$
$\displaystyle=-{\textstyle\frac{1}{2}}c_{n}{\\!}^{2}\varepsilon^{2}\int_{-\infty}^{\infty}{\rm
d}x\,{H_{2n}(x)}^{2}\frac{{\rm d}}{{\rm
d}x}\big{(}e^{-x^{2}}H_{2n}{\\!\\!}^{\prime}(x)\big{)}$
$\displaystyle=2nc_{n}{\\!}^{2}\varepsilon^{2}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,{H_{2n}(x)}^{3}=2nc_{n}{\\!}^{2}\varepsilon^{2}G_{2n\,2n\,2n}\,,$ (3.15)

using standard identities. Hence (3.12) determines a value for $c_{n}$

$2(n-1)^{2}c_{n}N_{2n}=2nc_{n}{\\!}^{2}\,G_{2n\,2n\,2n}\quad\Rightarrow\quad
c_{n}=\frac{(n-1)^{2}}{2^{n}n}\,\frac{n!^{3}}{(2n)!^{2}}\,.$ (3.16)

Since $c_{n}>0$ then for relevant solutions in (3.10) we must have
$\varepsilon>0$.

We may now extend the solution (3.10) to ${\rm O}(\varepsilon^{2})$ by
assuming the form

$v_{*}(x)=c_{n}\varepsilon
H_{2n}(x)+c_{n}\varepsilon^{2}\sum_{m}a_{m}H_{m}(x)+{\rm
O}(\varepsilon^{3})\,.$ (3.17)

Inserting in (3.4) with (3.11) and keeping only terms which are ${\rm
O}(\varepsilon^{2})$ gives

$(n-1)^{2}H_{2n}(x)+\sum_{m}(4n-2m)a_{m}H_{m}(x)=c_{n}H_{2n}{\\!}^{\prime}(x)^{2}\,,$
(3.18)

and hence using the orthogonality properties of Hermite polynomials we may
determine $a_{m}$ for $m\neq 2n$,

$N_{m}(4n-2m)a_{m}=c_{n}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{m}(x)H_{2n}{\\!}^{\prime}(x)^{2}\,.$ (3.19)

The integral on the right hand side may be calculated by using
$H_{n}{\\!}^{\prime}=2nH_{n-1}$ or with the following judicious integrations
by parts

$\displaystyle\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{m}{H_{2n}{\\!}^{\prime}}^{2}$
$\displaystyle=-\frac{1}{2}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{m}{\\!}^{\prime}\frac{{\rm d}}{{\rm
d}x}\big{(}H_{2n}{\\!}^{2}\big{)}-\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,H_{m}H_{2n}\frac{{\rm d}}{{\rm
d}x}\big{(}e^{-x^{2}}H_{2n}{\\!}^{\prime}\big{)}$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,\frac{{\rm d}}{{\rm
d}x}\big{(}e^{-x^{2}}H_{m}{\\!}^{\prime}\big{)}H_{2n}{\\!}^{2}+4n\int_{-\infty}^{\infty}{\rm
d}x\,e^{-x^{2}}H_{m}H_{2n}{\\!}^{2}$
$\displaystyle=(4n-m)\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{m}H_{2n}{\\!}^{2}$ $\displaystyle=(4n-m)\,G_{m\,2n\,2n}\,,$
(3.20)

which is zero unless $m$ is even. Hence taking $m=2p$,

$\displaystyle a_{2p}$
$\displaystyle=\frac{c_{n}}{N_{2p}}\,\frac{2n-p}{2(n-p)}\,G_{2p\,2n\,2n}$
$\displaystyle=2^{n-p-1}\frac{(n-1)^{2}}{n}\,\frac{2n-p}{n-p}\,\frac{n!^{3}}{p!^{2}(2n-p)!}\,,\quad
p=0,1,\dots 2n-1,\,p\neq n\,.$ (3.21)

Finally we compute $a_{2n}$ by using (3.17) in (3.4) and imposing
orthogonality with $H_{2n}$ to ${\rm O}(\varepsilon^{3})$, just as $c_{n}$ was
determined in (3.12).

$\displaystyle(n-1)^{2}a_{2n}N_{2n}\ $ $\displaystyle+\
{\textstyle\frac{1}{2}}(n-1)^{3}N_{2n}$
$\displaystyle=c_{n}\sum_{p}a_{2p}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{2n}H_{2n}{\\!}^{\prime}H_{2p}{\\!}^{\prime}$
$\displaystyle=2c_{n}\sum_{p}pa_{2p}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{2n}{\\!}^{2}H_{2p}$ $\displaystyle=2c_{n}\sum_{p\neq
n}pa_{2p}G_{2n\,2n\,2p}+2nc_{n}a_{2n}G_{2n\,2n\,2n}.$ (3.22)

Using (3.16), (3.22) becomes

$\displaystyle(n-1)^{2}a_{2n}N_{2n}={\textstyle\frac{1}{2}}(n-1)^{3}N_{2n}-2c_{n}\sum_{p\neq
n}pa_{2p}G_{2n\,2n\,2p},$ (3.23)

and so

$a_{2n}=\frac{1}{2}(n-1)-\frac{2c_{n}}{(n-1)^{2}}\sum_{p\neq
n}p\,a_{2p}\,2^{p}\frac{(2n)!(2p)!}{p!^{2}(2n-p)!},$ (3.24)

where we may use (3.16) and (3.21).

Since $H_{n}(x)\sim 2^{n}x^{n}$ for large $x$ then for $\varepsilon x^{2}={\rm
O}(1)$ the next to leading terms in (3.17) are comparable with the leading
$H_{2n}(x)$ term so the $\varepsilon$ expansion for $v_{*}(x)$ breaks down.
From (3.21) $a_{2p}<0$ for $p>n$ so that the result given by (3.17), which is
a polynomial of degree $4n-2$, is negative for sufficiently large $x$. For the
exact solution to (3.4) the nonlinear terms play a crucial role for large $x$
and we have $v_{*}(x)\sim\frac{2}{d-2}x^{2}$.

##  4 Applications in Particular Cases

The results obtained for $v_{*}$ obtained above are here considered in more
detail for $n=2,3,4$ and compared with results of numerical calculations. In
particular $V_{*}(\phi)$ was calculated numerically in the LPA in [23] for
these cases and various values of the dimension $d$. The basic approximation
from (3.17) is

$v_{*}(x)=c_{n}\varepsilon
H_{2n}(x)+c_{n}\varepsilon^{2}\sum_{p=0}^{2n-1}{a}_{2p}H_{2p}(x),$ (4.1)

where $a_{2p}$ is given by (3.21) and (3.24).

The first case of interest is $n=2$, corresponding to the Wilson-Fisher fixed
point. From (3.21) we have

$k_{2}=\frac{1}{4!^{2}}\,,\qquad{a}_{2p}=\frac{4-p}{2-p}\,\frac{2^{3-p}}{p!^{2}(4-p)!}\,,\quad
p=0,1,3\,,$ (4.2)

and hence

${a}_{0}=\frac{2}{3}\,,\quad{a}_{2}=2\,,\quad{a}_{6}=-\frac{1}{36}\,.$ (4.3)

From these, and (3.24), we have that

${a}_{4}=\frac{3}{2}\,.$ (4.4)

Using the following results for Hermite polynomials,

$\displaystyle H_{2n}(0)=(-1)^{n}\frac{(2n)!}{n!}\,,\qquad
H^{\prime\prime}_{n}(0)=-2nH_{n}(0)\,,$ (4.5)

we may then obtain from (4.1) in this case

$v_{*}(0)=\frac{1}{48}\varepsilon\left(1+\frac{3}{2}\varepsilon\right)\,,\qquad
v^{\prime\prime}_{*}(0)=-\frac{1}{6}\varepsilon\left(1+\frac{7}{4}\varepsilon\right)\,.$
(4.6)

As a consequence of (3.3) for the original $V_{*}(\phi)$ we have
$V_{*}(0)=v_{*}(0)$ and

$\displaystyle V_{*}^{\prime\prime}(0)$
$\displaystyle={\textstyle\frac{1}{4}}(d-2)v_{*}^{\prime\prime}(0)$
$\displaystyle=-\frac{\varepsilon}{12}\left(1+\frac{5\varepsilon}{4}\right)+{\rm
O}(\varepsilon^{3})\,.$ (4.7)

This results satisfy the consistency check

$\frac{V_{*}^{\prime\prime}(0)}{V_{*}(0)}=-4\left(1-\frac{1}{4}\varepsilon\right)=-d\,,$
(4.8)

as follows directly from (3.2). As remarked earlier for solutions of the fixed
point equation without singularities it is necessary to fine tune
$V_{*}^{\prime\prime}(0)$, or equivalently $V_{*}(0)$. In Table 1 we compare
the results from (4.7) to ${\rm O}(\varepsilon^{2})$ with those from numerical
calculation, as contained in [23], for various $d$. The detailed form of the
approximate solution in comparison with numerical results is shown in Figure 1
for various $d$. For small $\varepsilon$ the agreement is good.

$d$ | Numerical |  $\varepsilon$-expansion at ${\rm O}(\varepsilon^{2})$
---|---|---
4 | $0$ | $0\qquad$
3.9 | $-0.009$ | $-0.009\qquad$
3.8 | $-0.021$ | $-0.021\qquad$
3.7 | $-0.035$ | $-0.034\qquad$
3.6 | $-0.051$ | $-0.050\qquad$
3.5 | $-0.070$ | $-0.068\qquad$
3.4 | $-0.092$ | $-0.088\qquad$
3.3 | $-0.119$ | $-0.109\qquad$
3.2 | $-0.149$ | $-0.133\qquad$
3.1 | $-0.186$ | $-0.159\qquad$
3.0 | $-0.229$ | $-0.188\qquad$
Table 1: Comparison of ERG numerical and analytical results for
$V_{*}^{\prime\prime}(0)$ for the case $n=2$.

Figure 1: Graphs for $v_{*}(x)$ for $n=2$ from $\varepsilon$-expansion at
${\rm O}(\varepsilon^{2})$ and numerical solution with
$v_{*}(0)=-V_{*}^{\prime\prime}(0)/d$ from Table 1 with $d=3.9$, $3.8$, $3.6$,
$3.4$, $3.2$, $3.0$.

Similarly we consider the multi-critical fixed points obtained for $n=3$ and
$n=4$, which correspond to $\varepsilon$-expansions for dimensions $d=3$ and
$d=\frac{8}{3}$ respectively. When $n=3$ the results are

$k_{3}=\frac{1}{5!^{2}}\,,\qquad{a}_{2p}=\frac{6-p}{3-p}\,\frac{9.2^{7-p}}{p!^{2}(6-p)!}\,,\quad
p=0,1,2,4,5\,.$ (4.9)

Hence

${a}_{0}=\frac{16}{5}\,,\quad{a}_{2}=12\,,\quad{a}_{4}=12\,,\quad{a}_{8}=-\frac{1}{8}\,,\quad{a}_{10}=-\frac{1}{800}\,.$
(4.10)

From these, we can determine from (3.24)

${a}_{6}=\frac{173}{20}\,.$ (4.11)

At the origin we then obtain

$v_{*}(0)=-\frac{1}{120}\varepsilon\left(1+\frac{1087}{120}\varepsilon^{2}\right)\,,\qquad
v^{\prime\prime}_{*}(0)=\frac{1}{10}\varepsilon\left(1+\frac{389}{40}\varepsilon\right)\,,$
(4.12)

and hence

$V_{*}^{\prime\prime}(0)=\frac{1}{40}\varepsilon\left(1+\frac{349}{40}\varepsilon\right)\,.$
(4.13)

It is easy to verify that $V_{*}^{\prime\prime}(0)/V_{*}(0)=-d$ to this order
again. In Table 2 the result (4.13) is compared with the numerical results of
[23] in this case and in Figure 2 a graphical comparison of
$\varepsilon$-expansion and numerical solutions is made.

$d$ | Numerical |  $\varepsilon$-expansion at ${\rm O}(\varepsilon^{2})$
---|---|---
3 | 0 | $0\qquad$
2.9 | 0.005 | $0.005\qquad$
2.8 | 0.015 | $0.014\qquad$
2.7 | 0.032 | $0.027\qquad$
2.6 | 0.062 | $0.045\qquad$
2.5 | 0.108 | $0.067\qquad$
Table 2: Comparison of ERG numerical and analytical results for
$V_{*}^{\prime\prime}(0)$ for the $n=3$ fixed point.

Figure 2: $n=3$ $v_{*}(x)$ from $\varepsilon$-expansion at ${\rm
O}(\varepsilon^{2})$ and numerical solution for $d=2.9$, $2.8$ and $2.6$.

Finally, we consider the $n=4$ case. Following the same programme as above,
the coefficients in the expansion are

$\displaystyle{a}_{0}=$
$\displaystyle\frac{432}{35}\,,\quad{a}_{2}=\frac{288}{5}\,,\quad{a}_{4}=\frac{324}{5}\,,\quad{a}_{6}=36\,,$
$\displaystyle{a}_{10}=$
$\displaystyle-\frac{27}{100}\,,\quad{a}_{12}=-\frac{3}{800}\,,\quad{a}_{14}=-\frac{1}{39200}\,,$
(4.14)

and

$a_{8}=\frac{23904}{875}\,.$ (4.15)

The fixed point solution at order $\varepsilon^{2}$ then determines

$V_{*}^{\prime\prime}(0)=-\frac{3}{560}\varepsilon\left(1+\frac{99441}{3500}\varepsilon\right)\,.$
(4.16)

Table 3 compares this result with [23] and with some corresponding graphs
exhibited in Figure 3.

$d$ | Numerical |  $\varepsilon$-expansion at ${\rm O}(\varepsilon)$
---|---|---
$\frac{8}{3}$ | 0 | $0\qquad$
2.6 | $-0.001$ | $-0.001\qquad$
2.5 | $-0.008$ | $-0.005\qquad$
2.4 | $-0.028$ | $-0.012\qquad$
2.3 | $-0.082$ | $-0.022\qquad$
2.2 | $-0.226$ | $-0.035\qquad$
Table 3: Comparison of ERG numerical and analytical results for
$V_{*}^{\prime\prime}(0)$ for the $n=4$ fixed point.

Figure 3: $n=4$ $v_{*}(x)$ from $\varepsilon$-expansion at ${\rm
O}(\varepsilon^{2})$ and numerical solution for $d=2.6$, $2.5$ and $2.4$.

##  5 Critical Exponents

Having computed $v_{*}(x)$ to order $\varepsilon^{2}$ for the fixed points
below each critical dimension $d=2n/(n-1)$, we now consider the RG flow near
these fixed points and compute certain critical exponents. In the local
potential approximation, the ERG flow is given by (3.1) and with the change of
variables as in (3.3) we now have the following RG flow equation for $v(x,t)$

$\frac{4}{d-2}\big{(}\dot{v}(x,t)-dv(x,t)\big{)}=v^{\prime\prime}(x,t)-2xv^{\prime}(x,t)-v^{\prime}(x,t)^{2}\,.$
(5.1)

In the neighbourhood of a fixed point

$v(x,t)=v_{*}(x)+e^{\lambda t}f(x)\,,$ (5.2)

where $f(x)$ therefore satisfies the following linear eigenvalue equation

$\frac{4}{d-2}(\lambda-d)=f^{\prime\prime}-2xf^{\prime}-2v^{\prime}_{*}f^{\prime}\,.$
(5.3)

This may be rewritten in the form

$-Df(x)+v_{*}{\\!}^{\prime}(x)f^{\prime}(x)=\hat{\lambda}f(x)\,,\qquad\hat{\lambda}=-\frac{2}{d-2}(\lambda-d)\,,$
(5.4)

with $D$ the differential operator in (3.6).

In the case of the Gaussian fixed point $v_{*}=0$, the eigenvalues are
$\hat{\lambda}_{k}=k$ and the associated eigenfunctions $f_{k}$ are just
Hermite polynomials, so that

$f_{k}(x)=H_{k}(x)\,,\qquad\lambda_{k}=d-{\textstyle\frac{1}{2}}k(d-2)\,,\qquad
k=0,1,2,\dots\,.$ (5.5)

These correspond to the operators $\phi^{k}$ where $\phi$ has dimensions
${\textstyle\frac{1}{2}}(d-2)$. For the high temperature fixed point then in
(5.3) $v_{*}^{\prime}(x)=\frac{4}{d-2}x$ and the eigenfunctions are again
Hermite polynomials with rescaled variable and
$\lambda_{k}=d-{\textstyle\frac{1}{2}}k(d+2)$.

For non trivial critical points we require $v_{*}$ to be a non singular
solution of (3.4) For dimension $d$ as in (2.6) we may then consider a
perturbation expansion in $\varepsilon$, so that

$\displaystyle f_{k}(x)$ $\displaystyle=H_{k}(x)+{\rm O}(\varepsilon)\,,$
$\displaystyle\hat{\lambda}_{k}$
$\displaystyle=k+\varepsilon\hat{\lambda}_{k}{\\!}^{(1)}+\varepsilon^{2}\hat{\lambda}_{k}{\\!}^{(2)}+{\rm
O}(\varepsilon^{3})\,.$ (5.6)

To ${\rm O}(\varepsilon)$ it is sufficient to take in (5.4)

$v_{*}(x)=c_{n}\varepsilon H_{2n}(x)\,.$ (5.7)

To extract $\hat{\lambda}_{k}{\\!}^{(1)}$ we use standard first-order
perturbation theory using the basis of eigenfunctions of $D$,

$\displaystyle N_{k}\hat{\lambda}_{k}{\\!}^{(1)}$
$\displaystyle=c_{n}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{k}(x)H_{2n}{\\!}^{\prime}(x)H_{k}{\\!}^{\prime}(x)$
$\displaystyle=-{\textstyle\frac{1}{2}}c_{n}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,H_{k}(x)^{2}\frac{{\rm d}}{{\rm
d}x}\left(e^{-x^{2}}H_{2n}{\\!}^{\prime}(x)\right)$
$\displaystyle=2nc_{n}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{k}(x)^{2}H_{2n}(x)$ $\displaystyle=2nc_{n}\,G_{kk\,2n}\,.$
(5.8)

This gives

$\hat{\lambda}_{k}{\\!}^{(1)}=2(n-1)^{2}\frac{n!}{(2n)!}\frac{k!}{(k-n)!}\,,$
(5.9)

and in terms of the exponents $\lambda_{k}$, we have

$\displaystyle\lambda_{k}$
$\displaystyle=d-{\textstyle\frac{1}{2}}(d-2)\hat{\lambda}_{k}$
$\displaystyle=\frac{2n-k}{n-1}+\varepsilon\left(\frac{1}{2}k-1-2(n-1)\frac{n!}{(2n)!}\frac{k!}{(k-n)!}\right)+{\rm
O}(\varepsilon^{2})\,.$ (5.10)

The results (5.9) or equivalently (5.10) were found in the beginning of the RG
analysis of critical points in [24], using an approximation to the Wegner-
Houghton RG equation, see also [17], and (up to misprints) in [25], using the
LPA for the Wegner-Houghton equation. They are identical with the perturbative
result (2.18).

For higher order calculations it is convenient to modify the eigenvalue
equation in (5.4) by considering the transformed differential operator

$\Delta=e^{-v_{*}(x)}\Big{(}-D+v_{*}{\\!}^{\prime}(x)\frac{{\rm d}}{{\rm
d}x}\Big{)}e^{v_{*}(x)}=-D+\frac{2d}{d-2}\,v_{*}(x)\,,\\\ $ (5.11)

using that $v_{*}(x)$ satisfies the fixed point equation (3.4). It is obvious
that the eigenvalue equation

$\Delta{\hat{f}}={\hat{\lambda}}{\hat{f}}\,,$ (5.12)

is equivalent to (5.4) and furthermore the operator $\Delta$ is hermitian111We
may also note $e^{-\frac{1}{2}x^{2}}\Delta\,e^{\frac{1}{2}x^{2}}$ is a
Scrödinger operator with potential
$U(x)={\textstyle\frac{1}{2}}x^{2}-{\textstyle\frac{1}{2}}+\frac{2d}{d-2}\,v_{*}(x)$.
Asymptotically
$U(x)\sim{\textstyle\frac{1}{2}}\frac{(d+2)^{2}}{(d-2)^{2}}\,x^{2}$. with
respect to the measure ${\rm d}x\,e^{-x^{2}}$.

To ${\rm O}(\varepsilon)$ it is easy to see that, with $v_{*}$ given by (5.7)
and using the expansion (3.11), that the eigenvalues are the same as (5.9). To
${\rm O}(\varepsilon^{2})$ and using (4.1) second-order perturbation theory
gives

$\displaystyle\hat{\lambda}_{k}{\\!}^{(2)}$
$\displaystyle=(n-1)^{2}\frac{c_{n}}{N_{k}}G_{2n\,kk}+\frac{2nc_{n}}{N_{k}}\sum_{p=0}^{2n-1}a_{2p}G_{2p\,kk}-(2nc_{n})^{2}\sum_{m\neq
k}\frac{1}{N_{m}N_{k}}\frac{G_{2n\,mk}^{\,2}}{m-k}\,,$ (5.13)

where the first two terms arise from the ${\rm O}(\varepsilon^{2})$ terms in
the operator itself, and the final term is the usual second order perturbation
expression for a perturbative potential $2nc_{n}\varepsilon H_{2n}$.
Substituting the expressions for $a_{2n}$ and $a_{2p}(p\neq n)$, (3.24) and
(3.21) respectively, the expression above becomes

$\displaystyle\hat{\lambda}_{k}{\\!}^{(2)}$
$\displaystyle=\left((n-1)^{2}+n(n-1)\right)\frac{c_{n}}{N_{k}}G_{2n\,kk}-\frac{4n}{(n-1)^{2}}\frac{c_{n}}{N_{2n}}\sum_{p\neq
n}\frac{p(2n-p)}{2(n-p)}\frac{G_{2n\,2n\,2p}^{\,2}}{N_{2p}}$ $\displaystyle\ \
+2nc_{n}^{2}\sum_{p\neq
n}\frac{2n-p}{2(n-p)}\frac{G_{2p\,kk}G_{2n\,2n\,2p}}{N_{2p}N_{k}}-(2nc_{n})^{2}\sum_{m\neq
k}\frac{1}{N_{m}N_{k}}\frac{G_{2n\,mk}^{\,2}}{m-k}\,.$ (5.14)

Finally, substituting for $N_{m}$ from (3.13), $G_{klm}$ from (3.14) and
$c_{n}$ from (3.16), we have

$\displaystyle\hat{\lambda}_{k}{\\!}^{(2)}$
$\displaystyle=(n-1)^{3}(2n-1)\frac{1}{n}\frac{n!k!}{(2n)!(k-n)!}$
$\displaystyle\ \
-2(n-1)^{4}\frac{1}{n^{2}}\frac{n!^{7}}{(2n)!^{2}}\frac{k!}{(k-n)!}\sum_{p\neq
n}\frac{p(2n-p)}{n-p}\frac{(2p)!}{p!^{4}(2n-p)!^{2}}$ $\displaystyle\ \
+(n-1)^{4}\frac{1}{n}\frac{n!^{6}}{(2n)!^{2}}k!\sum_{p\neq
n}\frac{2n-p}{n-p}\frac{(2p)!}{p!^{4}(k-p)!(2n-p)!}$ $\displaystyle\ \
-2(n-1)^{4}\frac{n!^{6}}{(2n)!^{2}}k!\sum_{s\neq
n}\frac{1}{n-s}\frac{(2n+k-2s)!}{(k-s)!^{2}s!^{2}(2n-s)!^{2}}\,,$ (5.15)

where $2s=2n+k-m$. While this expression is somewhat complicated, for any
particular choices of $n$ it can be simplified to a polynomial in $k$ of order
$2n-1$. Here we give the results for the first two fixed points

$\displaystyle\hat{\lambda}_{k}{\\!}^{(2)}\big{|}_{n=2}=$
$\displaystyle-\frac{1}{12}k(k-1)(k-4)\,,$ (5.16a)
$\displaystyle\hat{\lambda}_{k}{\\!}^{(2)}\big{|}_{n=3}=$
$\displaystyle-\frac{1}{600}k(k-1)(13k^{3}-17k^{2}-424k+800)\,.$ (5.16b)

A similar, but not identical result to (5.16a) was obtained in [25], again
using the LPA for the Wegner-Houghton RG equation rather than the Polchinski
equation.

Having computed the eigenvalues $\hat{\lambda}$ to ${\rm O}(\varepsilon^{2})$,
we may now extend (5.10) to calculate the corresponding critical exponents to
${\rm O}(\varepsilon^{2})$. The exponents $\lambda_{k}$ are given in terms of
the eigenvalues as

$\displaystyle\lambda_{k}$
$\displaystyle=\frac{2n-k}{n-1}+\varepsilon\left(\frac{k}{2}-1-2(n-1)\frac{n!}{(2n)!}\frac{k!}{(k-n)!}\right)$
$\displaystyle\ \
+\varepsilon^{2}\left((n-1)^{2}\frac{n!}{(2n)!}\frac{k!}{(k-n)!}-\frac{1}{n-1}\hat{\lambda}_{k}{\\!}^{(2)}\right)+{\rm
O}(\varepsilon^{3})\,.$ (5.17)

This gives

$\displaystyle\lambda_{k}\big{|}_{n=2}={}$ $\displaystyle
4-k+\varepsilon\big{(}{\textstyle\frac{1}{2}}k-1-\textstyle{\frac{1}{6}}k(k-1)\big{)}+\varepsilon^{2}\,\textstyle{\frac{1}{12}}k(k-1)(k-3)+{\rm
O}(\varepsilon^{3})\,,$ $\displaystyle\lambda_{k}\big{|}_{n=3}={}$
$\displaystyle{\textstyle\frac{1}{2}}(6-k)+\varepsilon\big{(}{\textstyle\frac{1}{2}}k-1-\textstyle{\frac{1}{30}}k(k-1)(k-2)\big{)}$
$\displaystyle{}+\varepsilon^{2}\,\textstyle{\frac{1}{1200}}\,k(k-1)(k-5)(13k^{2}+48k-144)+{\rm
O}(\varepsilon^{3})\,.$ (5.18)

###  5.1 Exact Exponents

The results (5.9) and (5.16a), (5.16b) show that ${\hat{\lambda}}_{k}=0$ for
$k=0,1$ to ${\rm O}(\varepsilon^{2})$, at least for $n=2,3$. For general $n$
from (5.15)

$\displaystyle\hat{\lambda}_{1}{\\!}^{(2)}$
$\displaystyle=(n-1)^{4}\frac{1}{n}\frac{n!^{6}}{(2n)!^{2}}\sum_{p=0}^{1}\frac{2n-p}{n-p}\frac{(2p)!}{p!^{4}(1-p)!(2n-p)!}$
$\displaystyle\ \ \
-2(n-1)^{4}\frac{n!^{6}}{(2n)!^{2}}\sum_{s=0}^{1}\frac{1}{n-s}\frac{(2n+1-2s)!}{(1-s)!^{2}s!^{2}(2n-s)!^{2}}$
$\displaystyle=2(n-1)^{4}\frac{n!^{6}}{(2n)!^{2}}\frac{1}{n(n-1)(2n)!}\left((n-1)(-2n)+2n(2n-1)-2n^{2}\right)$
$\displaystyle=0\,.$ (5.19)

These results follow in general since it is possible to find exact
eigenfunctions for $k=0,1$ in (5.4). Firstly we have the trivial case,

$f_{0}=1\,,\qquad\hat{\lambda}_{0}=0\,.$ (5.20)

For $k=1$ we also have

$f_{1}(x)=v^{\prime}_{*}(x)-\frac{4x}{d-2}\,,\qquad\hat{\lambda}_{1}=1\,.$
(5.21)

To verify this we first obtain from (3.4)

$Dv_{*}{\\!}^{\prime}+\frac{d+2}{d-2}\,v_{*}{\\!}^{\prime}=v_{*}{\\!}^{\prime}v_{*}{\\!}^{\prime\prime}\,,$
(5.22)

since then

$Df_{1}(x)-v_{*}{\\!}^{\prime}(x)f_{1}(x)=-v_{*}{\\!}^{\prime}(x)+\frac{4x}{d-2}=-f_{1}(x)\,.$
(5.23)

The vanishing of $\hat{\lambda}_{1}$ reflects that there is no anomalous
dimension for the field $\phi$ in the LPA.

Additionally from (5.22) we have another exact eigenfunction and eigenvalue,

$f_{2n-1}(x)=v^{\prime}_{*}(x)\,,\qquad\hat{\lambda}_{2n-1}=\frac{d+2}{d-2}\,.$
(5.24)

Expanding the exact eigenvalue $\hat{\lambda}_{2n-1}$ in powers of
$\varepsilon$ gives

$\hat{\lambda}_{2n-1}=2n-1+(n-1)^{2}\varepsilon+{\textstyle\frac{1}{2}}(n-1)^{3}\varepsilon^{2}+\dots\,,$
(5.25)

which is consistent with (5.9) since

$\hat{\lambda}^{(1)}_{2n-1}=(n-1)^{2}\,.$ (5.26)

It is also in accord with the second-order result, (5.15). Relabelling
$s=2n-p$,

$\displaystyle\hat{\lambda}_{2n-1}^{(2)}$
$\displaystyle=(n-1)^{3}(2n-1)\frac{1}{n}\frac{n!(2n-1)!}{(2n)!(n-1)!}$
$\displaystyle\ \ \
-2(n-1)^{4}\frac{1}{n^{2}}\frac{n!^{7}}{(2n)!^{2}}\frac{(2n-1)!}{(n-1)!}\sum_{p\neq
n}\frac{p(2n-p)}{n-p}\frac{(2p)!}{p!^{4}(2n-p)!^{2}}$ $\displaystyle\ \ \
+(n-1)^{4}\frac{1}{n}\frac{n!^{6}}{(2n)!^{2}}(2n-1)!\sum_{p\neq
n}\frac{2n-p}{n-p}\frac{(2p)!}{p!^{4}(2n-1-p)!(2n-p)!}$ $\displaystyle\ \ \
+2(n-1)^{4}\frac{n!^{6}}{(2n)!^{2}}(2n-1)!\sum_{p\neq
n}\frac{1}{n-p}\frac{(2p-1)!}{(p-1)!^{2}p!^{2}(2n-p)!^{2}}$
$\displaystyle=\frac{1}{2}(n-1)^{3}+\frac{(n-1)^{4}}{2n^{2}}\frac{n!^{6}}{(2n)!}\sum_{p=0}^{2n}(4n-3p)\frac{(2p)!}{p!^{4}(2n-p)!^{2}}\,.$
(5.27)

Provided the final sum is identically zero for all $n$ this agrees with the
exact result. We demonstrate that the sum vanishes in Appendix B. In this case
the relevant operator $\phi^{2n-1}\propto\partial^{2}\phi$ by the equations of
motion and has dimension ${\textstyle\frac{1}{2}}(d+2)$.

##  6 Beyond the Local Potential Approximation

Although the LPA captures the essential features of the fixed point structure
in the space of couplings for scalar theories it neglects the momentum
dependence of vertices and is therefore limited in terms of calculating
critical exponents quantitatively, the anomalous dimension $\eta$ of the
scalar field is undetermined and set to zero. Although at the Wilson-Fisher
fixed point $\eta$ is small and in the $\varepsilon$-expansion $\eta={\rm
O}(\varepsilon^{2})$ it is of course necessary to take $\eta$ into account in
more systematic treatments. To this end a natural extension of the LPA is to
assume a solution of the exact RG flow equations which is expressible in terms
of a local functional of the fields and their derivatives, the derivative
expansion [26], for a recent discussion see [27]. In the derivative expansion
there is a necessary dependence on the form of the cut off function but in
application to the Polchinski RG equation, with terms quadratic in
derivatives, there are two constants, see [28, 29].

At the second order in the derivative expansion, the Polchinski equation may
be reduced [28] to a following pair of coupled ODEs, extending (3.1), for a
potential $V(\phi,t)$ and also the coefficient $Z(\phi,t)$ of
${\textstyle\frac{1}{2}}(\partial\phi)^{2}$ in the derivative expansion,

$\displaystyle\dot{V}$
$\displaystyle=V^{\prime\prime}-{\textstyle\frac{1}{2}}(d-2+\eta)\phi
V^{\prime}+dV-V^{\prime 2}+AZ\,,$ (6.1a) $\displaystyle\dot{Z}$
$\displaystyle=Z^{\prime\prime}-{\textstyle\frac{1}{2}}(d-2+\eta)\phi
Z^{\prime}-2V^{\prime}Z^{\prime}-4V^{\prime\prime}Z-\eta
Z-\eta+2BV^{\prime\prime 2}\,,$ (6.1b)

where $A$ and $B$ are the two cut off dependent constants which cannot be
eliminated by any rescaling (essentially the same equations were obtained from
the Wilson RG equations in [30]).

As earlier it is convenient in our discussion to introduce a rescaled variable

$x={\textstyle\frac{1}{2}}(d-2+\eta)^{\frac{1}{2}}\phi\,,\qquad
v(x,t)=V(\phi,t)\,,\quad z(x,t)=Z(\phi,t),$ (6.2)

so that the coupled equations (6.1a), (6.1b) become

$\displaystyle\frac{2}{d-2+\eta}\,\dot{v}$
$\displaystyle=(D+K)v-{\textstyle\frac{1}{2}}\,v^{\prime 2}+\tilde{A}z\,,$
(6.3a) $\displaystyle\frac{2}{d-2+\eta}\,\dot{z}$
$\displaystyle=(D-L)z-v^{\prime}z^{\prime}-2v^{\prime\prime}z-L+{\textstyle\frac{1}{2}}\tilde{B}\,v^{\prime\prime
2}\,,$ (6.3b)

where

$K=\frac{2d}{d-2+\eta}\,,\quad
L=\frac{2\eta}{d-2+\eta}\,,\qquad\tilde{A}=\frac{2}{d-2+\eta}A\,,\quad\tilde{B}={\textstyle\frac{1}{2}}(d-2+\eta)B\,.$
(6.4)

At a fixed point, $v(x,t)\to v_{*}(x),\,z(x,t)\to z_{*}(x)$ which satisfy the
equations

$\displaystyle(D+K)v_{*}$
$\displaystyle={\textstyle\frac{1}{2}}\,v_{*}{\\!}^{\prime
2}-\tilde{A}z_{*}\,,\quad$ (6.5a)
$\displaystyle(D-L)z_{*}-v_{*}{\\!}^{\prime}z_{*}{\\!}^{\prime}-2v_{*}{\\!}^{\prime\prime}z_{*}$
$\displaystyle=L-{\textstyle\frac{1}{2}}\tilde{B}\,v_{*}{\\!}^{\prime\prime
2}\,.$ (6.5b)

Assuming (2.6) then in an $\varepsilon$-expansion, assuming $A,B$ are $O(1)$,
a consistent solution is obtained by requiring $v_{*}(x)={\rm O}(\varepsilon)$
and $\eta,z_{*}(x)={\rm O}(\varepsilon^{2})$. To lowest order then
$v_{*}\to\varepsilon c_{n}H_{2n}$ as in (3.10). With this result then (6.5b)
becomes

$Dz_{*}=(n-1)\eta-\varepsilon^{2}{\textstyle\frac{1}{2}}\tilde{B}c_{n}{\\!}^{2}H_{2n}{\\!\\!}^{\prime\prime
2}+{\rm O}(\varepsilon^{3})\,.$ (6.6)

This determines $\eta=\varepsilon^{2}\,\eta^{(2)}+{\rm O}(\varepsilon^{3})$
since the left hand side of (6.6) is orthogonal to 1 so that

$(n-1)\eta^{(2)}N_{0}={\textstyle\frac{1}{2}}\tilde{B}c_{n}{\\!}^{2}\int^{\infty}_{-\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{2n}{\\!\\!}^{\prime\prime 2}\,,$ (6.7)

which implies

$\eta^{(2)}=2^{2n+2}\frac{n(2n-1)}{n-1}(2n)!\,\tilde{B}c_{n}{\\!}^{2}\,.$
(6.8)

We consider this further later but first we may then use (6.6) to determine
$z_{*}(x)$ to lowest order by using the expansion

$z_{*}=\varepsilon^{2}c_{n}{\\!}^{2}\sum_{p}b_{2p}H_{2p}+{\rm
O}(\varepsilon^{3})\,.$ (6.9)

Since $DH_{2p}=-4pH_{2p}$ and with
$H_{2n}{\\!\\!}^{\prime\prime}=8n(2n-1)H_{2n-2}$ we may obtain

$\displaystyle b_{2p}={}$ $\displaystyle
16\tilde{B}\,n^{2}(2n-1)^{2}\frac{G_{2p\,2n-2\,2n-2}}{pN_{2p}}$
$\displaystyle={}$ $\displaystyle
2^{2n-p}\tilde{B}\frac{(2n)!^{2}}{p(2n-2-p)!p!^{2}}\,,\quad p=1,\dots,2n-2\,.$
(6.10)

We note that $b_{0}$ is not constrained by (6.6). Nevertheless $b_{0}$ may be
determined by imposing $z_{*}(0)=0$ [28]. This ensures that (6.5a), (6.5b)
have well defined solutions for all $x$ only for a specific choice of $\eta$
and $v_{*}(0)$.

With these results (6.5a) becomes

$(D+2n)v_{*}+(n-1)^{2}\varepsilon
v_{*}=\varepsilon^{2}c_{n}{\\!}^{2}\bigg{(}H_{2n}{\\!\\!}^{\prime
2}-{\tilde{A}}\sum_{p=0}^{2n-2}b_{2p}H_{2p}\bigg{)}+{\rm
O}(\varepsilon^{3})\,,$ (6.11)

where $v_{*}$ is expressible as in (4.1) and $b_{2p}$ is given by (6.10). Just
as in (3.12) and (3.16) this determines normalisation coefficient $c_{n}$,

$(n-1)^{2}c_{n}N_{2n}=c_{n}{\\!}^{2}\big{(}nG_{2n\,2n\,2n}-{\tilde{A}}N_{2n}b_{2n}\big{)}\quad\Rightarrow\quad
c_{n}\Big{(}1-\frac{n-1}{n}\,{\tilde{A}}{\tilde{B}}\Big{)}=\frac{(n-1)^{2}}{2^{n}n}\,\frac{n!^{3}}{(2n)!^{2}}\,.$
(6.12)

Furthermore in (4.1) the expansion coefficients are also determined by (6.11)

$\displaystyle a_{2p}$ $\displaystyle=c_{n}\frac{1}{2(n-p)}\
\Big{(}(2n-p)\,\frac{G_{2p\,2n\,2n}}{N_{2p}}-{\tilde{A}}\,b_{2p}\Big{)}$
$\displaystyle=c_{n}\,\frac{2^{2n-p}}{2(n-p)}\frac{(2n)!^{2}}{p!^{2}(2n-1-p)!}\Big{(}1-\frac{2n-1-p}{p}\,{\tilde{A}}{\tilde{B}}\Big{)}\,,\quad
p\neq n\,.$ (6.13)

except for $a_{2n}$. To obtain this it is necessary to extend (6.11) to ${\rm
O}(\varepsilon^{3})$.

Using the result (6.12) the expression (6.8) becomes

$\eta^{(2)}=4(n-1)^{2}\frac{n!^{6}}{(2n!)^{3}}\,\frac{n(n-1)(2n-1){\tilde{B}}}{\big{(}n-(n-1){\tilde{A}}{\tilde{B}}\big{)}^{2}}\,.$
(6.14)

The perturbative result (2.18) requires that the final factor, depending on
$A,B$, should be one.

##  7 Critical Exponents with the Derivative Expansion

We now consider small departures from the fixed point of the form

$\displaystyle v(x,t)$ $\displaystyle=v_{*}(x)+e^{\lambda t}f(x)\,,$
$\displaystyle z(x,t)$ $\displaystyle=z_{*}(x)+e^{\lambda t}g(x)\,.$ (7.1)

Substituting this form into (6.3a), (6.3b) we find the eigenvalue equations

$\displaystyle-(D+K)f+v_{*}{\\!}^{\prime}f^{\prime}-\tilde{A}\,g={}$
$\displaystyle\hat{\lambda}f\,,$ (7.2a)
$\displaystyle-(D-L)g+v_{*}{\\!}^{\prime}g^{\prime}+2v_{*}{\\!}^{\prime\prime}g+z_{*}{\\!}^{\prime}f^{\prime}+2z_{*}f^{\prime\prime}-\tilde{B}\,v_{*}{\\!}^{\prime\prime}f^{\prime\prime}={}$
$\displaystyle\hat{\lambda}f\,,$ (7.2b)

with $D$ as in (3.6) and where $\lambda$ is related to $\hat{\lambda}$
according to (this differs from (5.4)).

$\lambda=-{\textstyle\frac{1}{2}}(d-2+\eta)\hat{\lambda}\,.$ (7.3)

Defining the vector

$F=\left(\begin{array}[]{c}f\\\ g\\\ \end{array}\right)\,,$ (7.4)

then (7.2a), (7.2b) may be written more compactly in matrix form as

$\Delta F=\hat{\lambda}F\,.$ (7.5)

The operator $\Delta$ is not hermitian so for later convenience we also
consider the dual equation defining the dual eigenvectors

${\tilde{\Delta}}\tilde{F}=\hat{\lambda}\tilde{F}\,,$ (7.6)

where ${\tilde{\Delta}}$ is the adjoint of $\Delta$ with a scalar product
defined $\int{\rm d}x\,e^{-x^{2}}\,F_{1}{\\!}^{T}F_{2}$.

Just as in Section 5 there are exact eigenfunctions and eigenvalues [28]

$\displaystyle F={}$ $\displaystyle\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)\,,\qquad\qquad\qquad\quad\ \ \hat{\lambda}=-K\,,\qquad\ \
\lambda=d\,,$ (7.7c) $\displaystyle F={}$
$\displaystyle\left(\begin{array}[]{c}v_{*}{\\!}^{\prime}-(K-2)x\\\
z_{*}{\\!}^{\prime}\\\
\end{array}\right)\,,\qquad\hat{\lambda}=-K+1\,,\quad\lambda={\textstyle\frac{1}{2}}(d+2-\eta)\,,$
(7.7f) $\displaystyle F={}$
$\displaystyle\left(\begin{array}[]{c}v_{*}{\\!}^{\prime}\\\
z_{*}{\\!}^{\prime}\\\
\end{array}\right)\,,\qquad\qquad\qquad\quad\,\hat{\lambda}=-1\,,\qquad\quad\,\lambda={\textstyle\frac{1}{2}}(d-2+\eta)\,.$
(7.7i)

The first corresponds to the identity operator and at the $n$-th multicritical
point the second and third to the operators $\phi$ and $\phi^{2n-1}$.

In the $\varepsilon$-expansion

$\displaystyle\Delta=\Delta^{(0)}+\varepsilon\Delta^{(1)}+\dots\,,$ (7.8)

and correspondingly in (7.5)

$\hat{\lambda}=\hat{\lambda}^{(0)}+\varepsilon\hat{\lambda}^{(1)}+\dots\,,\qquad
F=F^{(0)}+\varepsilon F^{(1)}+\dots\,,$ (7.9)

and similarly for $\tilde{F}$. At zeroth order in $\varepsilon$,

$\Delta^{(0)}=\left(\begin{array}[]{cc}-D-2n&-\tilde{A}\\\ 0&-D\\\
\end{array}\right)\,.$ (7.10)

There are two sets of eigenfunctions, which are easily obtained

$F^{(0)}_{k,1}=\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)H_{k}\,,\qquad\tilde{F}^{(0)}_{k,1}=\left(\begin{array}[]{c}1\\\
\frac{\tilde{A}}{2n}\\\ \end{array}\right)H_{k}\,,\qquad k=0,1,2,\dots\,,$
(7.11)

and

$F^{(0)}_{k,2}=\left(\begin{array}[]{c}-\frac{\tilde{A}}{2n}\\\ 1\\\
\end{array}\right)H_{k-2n}\,,\qquad\tilde{F}^{(0)}_{k,2}=\left(\begin{array}[]{c}0\\\
1\\\ \end{array}\right)H_{k-2n}\,,\qquad k=2n,2n+1,\dots\,.$ (7.12)

The eigenvalues in both cases are

$\displaystyle\hat{\lambda}_{k,1}^{(0)}=\hat{\lambda}_{k,2}^{(0)}=k-2n\,,$
(7.13)

which are thus two-fold degenerate for $k\geq 2n$.

In general in terms of the perturbative anomalous dimensions at the fixed
point discussed in Section 2

$\lambda_{k,1}=\frac{2n-k}{n-1}-\omega_{k,1}\,,\qquad\lambda_{k,2}=\frac{2n-k}{n-1}-\omega_{k,2}\,.$
(7.14)

###  7.1 Eigenvalues at ${\rm O}(\varepsilon)$

We now use first-order perturbation theory to compute
$\hat{\lambda}^{(1)}_{k,1}$ and $\hat{\lambda}^{(1)}_{k,2}$. In the expansion
(7.8) at ${\rm O}(\varepsilon)$

$\Delta^{(1)}=\left(\begin{array}[]{cc}-(n-1)^{2}+c_{n}H_{2n}{\\!}^{\prime}\frac{{\rm
d}}{{\rm d}x}&0\\\ -\tilde{B}\,c_{n}H_{2n}{\\!\\!}^{\prime\prime}\frac{{\rm
d}^{2}}{{\rm d}x^{2}}&c_{n}H_{2n}{\\!\\!}^{\prime}\frac{{\rm d}}{{\rm
d}x}+2c_{n}H_{2n}{\\!\\!}^{\prime\prime}\\\ \end{array}\right)\,.$ (7.15)

The ${\rm O}(\varepsilon)$ results are determined in terms of the $2\times 2$
matrix ${\cal M}_{k}$ defined by

${\cal M}_{k,ij}=\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}\tilde{F}^{(0)\,T}_{k,i}\,\Delta^{(1)}F^{(0)}_{k,j}\,.$ (7.16)

For $k<2n$ there is a single eigenfunction so that the perturbation theory
result is just

$\displaystyle N_{k}\hat{\lambda}^{(1)}_{k,1}$ $\displaystyle={\cal M}_{k,11}$
$\displaystyle=-(n-1)^{2}N_{k}+c_{n}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{k}H_{2n}{\\!\\!}^{\prime}H_{k}{\\!}^{\prime}-c_{n}\frac{1}{2n}\tilde{A}\tilde{B}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{k}H_{2n}{\\!\\!}^{\prime\prime}H_{k}{\\!}^{\prime\prime}\,.$
(7.17)

Hence, using the formulae for $N_{k}$, $G_{klm}$ in (3.13), (3.14) we obtain

$\displaystyle\hat{\lambda}^{(1)}_{k,1}={}$
$\displaystyle-(n-1)^{2}+c_{n}\,2^{n+1}\frac{(2n)!k!}{n!^{2}(k-n)!}\big{(}n-(n-1)\tilde{A}\tilde{B}(n-1)\big{)}$
$\displaystyle={}$
$\displaystyle-(n-1)^{2}+2(n-1)^{2}\frac{n!}{(2n)!}\frac{k!}{(k-n)!}\,,$
(7.18)

where $c_{n}$ is determined by (6.12) so that the dependence on
$\tilde{A},\tilde{B}$ disappears. This result is essentially identical to
(5.9) and is in accord with the exact results (7.7c), (7.7f), (7.7i) which
correspond to $k=0,1,2n-1$.

For $k\geq 2n$ there is a two-fold degeneracy, and so we use degenerate
perturbation theory. In particular, for non-trivial first-order eigenfunctions
we require that the first-order perturbations to the eigenvalues solve the
following characteristic equation

$\displaystyle\det\big{[}\lambda^{(1)}_{k}{\cal N}_{k}-{\cal
M}_{k}\big{]}=0\,,$ (7.19)

where the ${\cal N}_{k}$ is the diagonal matrix

${\cal N}_{k}=\left(\begin{array}[]{cc}N_{k}&0\\\ 0&N_{k-2n}\\\
\end{array}\right)\,,$ (7.20)

More explicitly, the elements of $\cal M$ are given by (7.17) and

$\displaystyle{\cal M}_{k,22}$
$\displaystyle=c_{n}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{k-2n}^{\vphantom{T}}\left(H^{\prime}_{2n}H^{\prime}_{k-2n}+2H^{\prime\prime}_{2n}H_{k-2n}+\frac{1}{2n}\tilde{A}\tilde{B}\,H^{\prime\prime}_{2n}H^{\prime\prime}_{k-2n}\right)\,,$
(7.21a) $\displaystyle{\cal M}_{k,21}$
$\displaystyle=-\tilde{B}\,c_{n}\int_{-\infty}^{\infty}\\!\\!\\!\\!{\rm
d}x\,e^{-x^{2}}H_{k-2n}^{\vphantom{T}}H^{\prime\prime}_{2n}H^{\prime\prime}_{k},$
(7.21b) $\displaystyle{\cal M}_{k,12}$
$\displaystyle=c_{n}\frac{1}{2n}\tilde{A}\int_{-\infty}^{\infty}{\rm
d}x\,e^{-x^{2}}H_{k}^{\vphantom{T}}\left(2H^{\prime\prime}_{2n}H_{k-2n}^{\vphantom{T}}+\frac{1}{2n}\tilde{A}\tilde{B}H^{\prime\prime}_{2n}H^{\prime\prime}_{k-2n}\right)$
$\displaystyle=0\,,$ (7.21c)

where in (7.21c) we use that $H_{k}$ is orthogonal to polynomials of degree
$<k$. Hence since the matrix ${\cal M}_{k}$ is lower triangular the
eigenvalues solving (7.19) are just $\hat{\lambda}^{(1)}_{1,k}$ as in (7.18)
and

$\displaystyle\hat{\lambda}^{(1)}_{k,2}={}$ $\displaystyle\frac{{\cal
M}_{k,22}}{N_{k-2n}}=c_{n}\,2^{n+1}\frac{(2n)!(k-2n)!}{n!^{2}(k-3n+1)!}\left(n(k-n+1)+\tilde{A}\tilde{B}(n-1)(k-3n+1)\right)\,.$
(7.22)

From (7.3)
$\lambda=-(\frac{1}{n-1}-{\textstyle\frac{1}{2}}\varepsilon){\hat{\lambda}}+{\rm
O}(\varepsilon^{2})$ so to ${\rm O}(\varepsilon)$ we then have

$\displaystyle\lambda_{k,1}={}$
$\displaystyle\frac{2n-k}{n-1}+\varepsilon\left[\frac{1}{2}k-1-2(n-1)\frac{n!\,k!}{(2n)!\,(k-n)!}\right]\,,$
(7.23a) $\displaystyle\lambda_{k,2}={}$ $\displaystyle\frac{2n-k}{n-1}$
$\displaystyle\\!{}+\varepsilon\left[\frac{1}{2}k-n-2(n-1)\frac{n!\,(k-2n)!}{(2n)!\,(k-3n+1)!}\frac{n(k-n+1)+(n-1)\tilde{A}\tilde{B}(k-3n+1)}{n-(n-1)\tilde{A}\tilde{B}}\right]\,.$
(7.23b)

The result (7.23a) matches the perturbative result in (2.16) but (7.23b)
depends on $AB$ so cannot agree with (2.32) in general.

##  8 Expansion of Exact RG Equation

An alternative approach, which nevertheless has many similarities although
some crucial differences to the derivative expansion, is to expand the full
renormalisation group equation which contains linear and quadratic terms in
terms of translation invariant operators, or scaling fields, which are exact
eigen-solutions for its linearised part. This has been applied in [17] and
[31], [32] to the Wilson RG equation. Here we apply similar methods more
simply to the Polchinski equation for the effective action $S[\varphi,t]$ for
a scalar field $\varphi$. This takes the form, after convenient rescalings by
the cut off scale $\Lambda$, $t=-\ln\Lambda$,

$\displaystyle\frac{\partial}{\partial t}S={}$
$\displaystyle\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,\big{(}{\textstyle\frac{1}{2}}d+1-{\textstyle\frac{1}{2}}\eta+p\cdot\partial_{p}\big{)}\tilde{\varphi}(p)\frac{\delta
S}{\delta\tilde{\varphi}(p)}$ $\displaystyle{}-\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K^{\prime}(p^{2})\bigg{(}\frac{\delta^{2}S}{\delta\tilde{\varphi}(p)\delta\tilde{\varphi}(-p)}-\frac{\delta
S}{\delta\tilde{\varphi}(p)}\,\frac{\delta
S}{\delta\tilde{\varphi}(-p)}\bigg{)}$
$\displaystyle{}-{\textstyle\frac{1}{2}}\eta\,\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K(p^{2})^{-1}p^{2}\tilde{\varphi}(p)\tilde{\varphi}(-p)+C\,,$ (8.1)

where $K(p^{2}/\Lambda^{2})$ is a cut off function and $\tilde{\varphi}$ is
the Fourier transform of $\varphi$,
$\delta\tilde{\varphi}(p)/\delta\tilde{\varphi}(p^{\prime})=(2\pi)^{d}\delta^{d}(p-p^{\prime})$.
$C$ is an additional constant independent of $\varphi$, in general it is
irrelevant and may be neglected222The standard derivation gives
$C=\delta^{d}(0)\int{\rm d}^{d}p\,K^{\prime}(p^{2})K(p^{2})^{-1}p^{2}$ with
$(2\pi)^{d}\delta^{d}(0)$ the overall volume.. Apart from $K(0)=1$ and
sufficient rapid fall off for large $p^{2}$ no restriction on the cut off
function $K(p^{2})$ is imposed here. The Gaussian fixed point corresponds to
$S=0$ and also when the anomalous scale dimension for $\varphi$, $\eta=0$. The
appearance of $\eta$ in (8.1) arises by assuming that $\varphi$ varies with
the cut off $\Lambda$ with an anomalous dimension
${\textstyle\frac{1}{2}}\eta$. As will be apparent later the additional term
proportional to $\eta$ in (8.1) is necessary for consistent RG flow solutions.
The derivative expansion is obtained directly by approximating (8.1) by
assuming $\tilde{\varphi}(p)$ is expanded as
$(2\pi)^{d}(\phi-i\partial\phi\cdot\partial_{p})\delta^{d}(p)$ and requiring
$S[\varphi,t]\to\int{\rm
d}^{d}x\big{(}V(\phi,t)+{\textstyle\frac{1}{2}}Z(\phi,t)(\partial\phi)^{2}\big{)}$.

The starting point of the discussion in this Section requires solutions of

$\big{(}{\cal D}_{1}+{\cal D}_{2}\big{)}{\cal O}=\lambda{\cal O}\,,$ (8.2)

for ${\cal D}_{1}+{\cal D}_{2}$ the differential operator defined by the
linear part of (8.1),

$\displaystyle{\cal D}_{1}={}$ $\displaystyle\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,\big{(}{\textstyle\frac{1}{2}}d+1+p\cdot\partial_{p}\big{)}\tilde{\varphi}(p)\,\frac{\delta}{\delta\tilde{\varphi}(p)}\,,$
$\displaystyle{\cal D}_{2}={}$ $\displaystyle-\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K^{\prime}(p^{2})\frac{\delta^{2}}{\delta\tilde{\varphi}(p)\delta\tilde{\varphi}(-p)}\,.$
(8.3)

The eigenvalue equation

${\cal D}_{1}{\hat{\cal O}}=\lambda{\hat{\cal O}}\,,$ (8.4)

is easily solved in terms of local translational invariant operators by

${\hat{\cal O}}=\frac{1}{(2\pi)^{d(k-1)}}\int\prod_{i=1}^{k}{\rm
d}^{d}p_{i}\,\tilde{\varphi}(p_{i})\
\delta^{d}\big{(}{\textstyle\sum_{i}}p_{i}\big{)}\,O(p_{1},\dots,p_{k})\,,\quad\lambda=d-k({\textstyle\frac{1}{2}}d-1)-r\,,$
(8.5)

where $O(p_{1},\dots,p_{k})$ is a scalar symmetric homogeneous polynomial of
degree $r$. We then obtain corresponding solutions of (8.2) in the form

${\cal O}=e^{-{\cal Y}}{\hat{\cal O}}\,,$ (8.6)

where ${\cal Y}$ is defined by

$[{\cal D}_{1},{\cal Y}]={\cal D}_{2}\,,$ (8.7)

ensuring that ${\cal O}$ as defined by (8.6) has the same eigenvalue as
$\hat{\cal O}$. It is easy to solve (8.7) giving

${\cal Y}={\textstyle\frac{1}{2}}\,\frac{1}{(2\pi)^{d}}\int{\rm d}^{d}p\
G(p)\,\frac{\delta^{2}}{\delta\tilde{\varphi}(p)\delta\tilde{\varphi}(-p)}\,,\qquad
G(p)=\frac{K(p^{2})}{p^{2}}\,,$ (8.8)

so that in (8.6) $e^{-{\cal Y}}$ essentially generates normal ordering.

For comparison with the derivative expansion earlier we consider just
operators with $r=0,2$ for which a convenient basis is

$\displaystyle{\cal O}_{k0}={}$ $\displaystyle e^{-{\cal
Y}}\frac{1}{(2\pi)^{d(k-1)}}\int\prod_{i=1}^{k}{\rm
d}^{d}p_{i}\,\tilde{\varphi}(p_{i})\ \delta^{d}\big{(}p_{(k)}\big{)}\,,\qquad
p_{(k)}={\textstyle\sum_{i=1}^{k}}\,p_{i}\,,$ $\displaystyle{\cal O}_{k2}={}$
$\displaystyle\frac{1}{k(k-1)}\,e^{-{\cal
Y}}\,\frac{1}{(2\pi)^{d(k-1)}}\int\prod_{i=1}^{k}{\rm
d}^{d}p_{i}\,\tilde{\varphi}(p_{i})\
\delta^{d}\big{(}p_{(k)}\big{)}\,\sum_{i=1}^{k}p_{k}{\\!}^{2}\,,\quad
k=2,3,\dots\,.$ (8.9)

The result for ${\cal O}_{k0}$ may be expressed in terms of Hermite polynomial
as a consequence of the identity

$e^{-\frac{1}{4}\,\frac{{\rm d}^{2}}{{\rm d}x^{2}}}(2x)^{n}=H_{n}(x)\,.$
(8.10)

The nonlinear term in (8.1) may be evaluated in terms operators ${\cal O}$ and
${\cal O}^{\prime}$, respectively of degree $l$ and $m$ in $\tilde{\varphi}$
and as given by (8.5) and (8.6), by considering [17]

$\displaystyle e^{{\cal Y}}\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K^{\prime}(p^{2})\,\frac{\delta{\cal
O}}{\delta\tilde{\varphi}(p)}\,\frac{\delta{\cal
O}^{\prime}}{\delta\tilde{\varphi}(-p)}$
$\displaystyle{}=\exp\bigg{(}\frac{1}{(2\pi)^{d}}\\!\int\\!{\rm
d}^{d}p\,G(p)\,\frac{\delta^{2}}{\delta\tilde{\varphi}(p)\delta\tilde{\varphi}^{\prime}(-p)}\bigg{)}\,lm\frac{1}{(2\pi)^{d(l+m-3)}}\\!\int\prod_{i=1}^{l-1}{\rm
d}^{d}p_{i}\,\tilde{\varphi}(p_{i})\\!\int\prod_{j=1}^{m-1}{\rm
d}^{d}q_{j}\,\tilde{\varphi}^{\prime}(q_{j})$
$\displaystyle{}\\!\quad\times\delta^{d}\big{(}p_{(l-1)}+q_{(m-1)}\big{)}\
O\big{(}p_{1},\dots,p_{l-1},-p_{(l-1)}\big{)}\,O^{\prime}\big{(}q_{1},\dots,q_{m-1},-q_{(m-1)}\big{)}\Big{|}_{\tilde{\varphi}^{\prime}=\tilde{\varphi}}\,K^{\prime}\big{(}p_{(l-1)}{\\!\\!}^{2}\big{)}$
$\displaystyle{}=\sum_{n\geq
0}\frac{l!\,m!}{n!(l-1-n)!(m-1-n)!}\,\frac{1}{(2\pi)^{d(l+m-3-2n)}}\int\prod_{i=1}^{l-1-n}\\!\\!\\!{\rm
d}^{d}p_{i}\,\tilde{\varphi}(p_{i})\,\int\prod_{j=1}^{m-1-n}\\!\\!\\!{\rm
d}^{d}q_{j}\,\tilde{\varphi}(q_{j})\,$
$\displaystyle{}\qquad\times\delta^{d}\big{(}p_{(l-1-n)}+q_{(m-1-n)}\big{)}\,F_{n}(p_{1},\dots,p_{l-1-n};q_{1},\dots,q_{m-1-n})\,,$
(8.11)

where, with $p_{(l-1-n)}+q_{(m-1-n)}=0$,

$\displaystyle{}F_{n}(p_{1},\dots,p_{l-1-n};q_{1},\dots,q_{m-1-n})$
$\displaystyle{}=\frac{1}{(2\pi)^{dn}}\int\prod_{h=1}^{n}{\rm
d}^{d}r_{h}\,G(r_{h})\,O\big{(}p_{1},\dots,p_{l-1-n},r_{n},\dots,r_{1},-p_{(l-1-n)}-r_{(n)}\big{)}$
$\displaystyle\quad\quad{}\times
O^{\prime}\big{(}q_{1},\dots,q_{m-1-n},-r_{n},\dots,-r_{1},-q_{(m-1-n)}-r_{(n)}\big{)}\,K^{\prime}\big{(}(p_{(l-1-n)}+r_{(n)})^{2}\big{)}\,.$
(8.12)

For the cases of interest here

$\displaystyle\hskip-14.22636pt{\cal O}={\cal O}_{l0}\,,\quad{\cal
O}^{\prime}={\cal O}_{m0}\,,$
$\displaystyle\hskip-14.22636ptF_{n}(p_{1},\dots,p_{l-1-n};q_{1},\dots,q_{m-1-n})=\rho_{n}\big{(}p_{(l-1-n)}{\\!}^{2}\big{)}\,,$
$\displaystyle\hskip-14.22636pt{\cal O}={\cal O}_{l0}\,,\quad{\cal
O}^{\prime}={\cal O}_{m2}\,,$
$\displaystyle\hskip-14.22636ptF_{n}(p_{1},\dots,p_{l-1-n};q_{1},\dots,q_{m-1-n})={\textstyle{\sum_{j=1}^{m-1-n}\\!q_{j}{\\!}^{2}}}\,\rho_{n}\big{(}p_{(l-1-n)}{\\!}^{2}\big{)}+\tau_{n}\big{(}p_{(l-1-n)}{\\!}^{2}\big{)}\,,$
(8.13)

with

$\displaystyle\rho_{n}(p^{2})={}$
$\displaystyle\frac{1}{(2\pi)^{dn}}\int\prod_{h=1}^{n}{\rm
d}^{d}r_{h}\,G(r_{h})\,K^{\prime}\big{(}(p+r_{(n)})^{2}\big{)}\,,$ (8.14a)
$\displaystyle\tau_{n}(p^{2})={}$
$\displaystyle\frac{1}{(2\pi)^{dn}}\int\prod_{h=1}^{n}{\rm
d}^{d}r_{h}\,G(r_{h})\,\big{(}n\,r_{n}{\\!}^{2}+(p+r_{(n)})^{2}\big{)}\,K^{\prime}\big{(}(p+r_{(n)})^{2}\big{)}\,.$
(8.14b)

With the aid of (8.11) and (8.13) we may then write

$\displaystyle\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K^{\prime}(p^{2})\,\frac{\delta{\cal
O}_{l0}}{\delta\tilde{\varphi}(p)}\frac{\delta{\cal
O}_{m0}}{\delta\tilde{\varphi}(-p)}=\sum_{k}\big{(}{\cal
O}_{k0}\,C_{klm}+{\cal O}_{k2}\,{\tilde{C}}_{klm}+\dots\big{)}\,,$
$\displaystyle\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K^{\prime}(p^{2})\,\frac{\delta{\cal
O}_{l0}}{\delta\tilde{\varphi}(p)}\frac{\delta{\cal
O}_{m2}}{\delta\tilde{\varphi}(-p)}=\sum_{k}\big{(}{\cal
O}_{k0}\,D_{klm}+{\cal O}_{k2}\,{\tilde{D}}_{klm}+\dots\big{)}\,,$ (8.15)

for

$\displaystyle C_{klm}={}$
$\displaystyle\frac{l!\,m!}{(s-1-k)!(s-l)!(s-m)!}\,\rho_{s-1-k}(0)\,,$
$\displaystyle{\tilde{C}}_{klm}={}$
$\displaystyle\frac{l!\,m!}{(s-1-k)!(s-1-l)!(s-1-m)!}\,\rho^{\prime}{\\!}_{s-1-k}(0)\,,$
$\displaystyle D_{klm}={}$
$\displaystyle\frac{l!\,(m-2)!}{(s-1-k)!(s-l)!(s-m)!}\,\tau_{s-1-k}(0)\,,$
$\displaystyle{\tilde{D}}_{klm}={}$
$\displaystyle\frac{l!\,(m-2)!}{(s-1-k)!(s-1-l)!(s-m)!}\,\big{(}(k-1)\rho_{s-1-k}(0)+\tau^{\prime}{\\!}_{s-1-k}(0)\big{)}\,,$
(8.16)

which are non zero so long as
$s={\textstyle\frac{1}{2}}(k+l+m)=k+1,k+2,\dots$.

The truncation of the full RG equation (8.1) corresponding to the derivative
expansion as considered in Section 6 is obtained by writing

$S=\textstyle{\sum_{k}}\big{(}a_{k}\,{\cal O}_{k0}+b_{k}\,{\cal
O}_{k2}\big{)}\,,$ (8.17)

and then reducing (8.1) to

$\displaystyle{\dot{a}}_{k}={}$
$\displaystyle\big{(}d-{\textstyle\frac{1}{2}}(d-2+\eta)k\big{)}\,a_{k}+\textstyle{\sum_{l,m}\big{(}C_{klm}\,a_{l}a_{m}+2D_{klm}\,a_{l}b_{m}}\big{)}\,,$
(8.18a) $\displaystyle{\dot{b}}_{k}={}$
$\displaystyle\big{(}d-2-{\textstyle\frac{1}{2}}(d-2+\eta)k\big{)}\,b_{k}-{\textstyle\frac{1}{2}}\eta\,\delta_{k2}+\textstyle{\sum_{l,m}\big{(}{\tilde{C}}_{klm}\,a_{l}a_{m}+2{\tilde{D}}_{klm}\,a_{l}b_{m}}\big{)}\,.$
(8.18b)

These equations are arbitrary up to the rescalings

$\displaystyle a_{k}\to a_{k}/\alpha_{k}\,,\ b_{k}\to b_{k}/\beta_{k}\,,\quad
C_{klm}\to{}$ $\displaystyle C_{klm}\alpha_{l}\alpha_{m}/\alpha_{k}\,,\quad
D_{klm}\to D_{klm}\alpha_{l}\beta_{m}/\alpha_{k}\,,$
$\displaystyle{\tilde{C}}_{klm}\to{}$
$\displaystyle{\tilde{C}}_{klm}\alpha_{l}\alpha_{m}/\beta_{k}\,,\quad{\tilde{D}}_{klm}\to{\tilde{D}}_{klm}\alpha_{l}\beta_{m}/\beta_{k}\,,$
(8.19)

for any $\alpha_{k},\beta_{k}$ so long as $\beta_{2}=1$ because of the
inhomogeneous $\eta$ term in (8.18b). Deferring further discussion to later we
may impose $b_{2}=0$. Letting $\alpha_{k}=u^{\frac{1}{2}k-1}v$,
$\beta_{k}=u^{\frac{1}{2}k-1}$ then these rescalings correspond to changes in
the cut off dependent functions in (8.16) of the form

$\rho_{n}(0)\to u^{n}v\,\rho_{n}(0)\,,\quad\rho^{\prime}{\\!}_{n}(0)\to
u^{n}v^{2}\,\rho^{\prime}{\\!}_{n}(0)\,,\quad\tau_{n}(0)\to
u^{n}\,\tau_{n}(0)\,,\quad\tau^{\prime}{\\!}_{n}(0)\to
u^{n}v\,\tau^{\prime}{\\!}_{n}(0)\,.$ (8.20)

Critical exponents calculated from (8.18a),(8.18b) must be independent of such
transformations.

The analysis of (8.18a),(8.18b) in the $\varepsilon$-expansion, with $d$ as in
(2.6), is very similar to the previous discussion in the context of the
derivative expansion. At a fixed point a consistent solution is obtained with

$a_{*k}=\varepsilon\,a_{*}\delta_{k\,2n}+\varepsilon^{2}\,a_{*k}{\\!}^{(2)}+{\rm
O}(\varepsilon^{3})\,,\quad b_{*k}=\varepsilon^{2}\,b_{*k}{\\!}^{(2)}+{\rm
O}(\varepsilon^{3})\,,\qquad\eta=\varepsilon^{2}\,\eta^{(2)}+{\rm
O}(\varepsilon^{3})\,,$ (8.21)

where

$\displaystyle a_{*}={}$
$\displaystyle-\frac{n-1}{C_{2n\,2n\,2n}}\,,\qquad\quad\eta^{(2)}=2a_{*}{\\!}^{2}{\tilde{C}}_{2\,2n\,2n}\,,$
$\displaystyle b_{*2p}{\\!}^{(2)}={}$
$\displaystyle\frac{n-1}{2(p-1)}\,a_{*}{\\!}^{2}{\tilde{C}}_{2p\,2n\,2n}\,,\
p\neq 1\,,\qquad
a_{*2p}{\\!}^{(2)}=\frac{n-1}{2(p-n)}\,a_{*}{\\!}^{2}{C}_{2p\,2n\,2n}\,,\
p\neq n\,,$ $\displaystyle a_{*2n}{\\!}^{(2)}={}$
$\displaystyle-\frac{2a_{*}}{n-1}\Big{(}{\textstyle{\sum_{p\neq
n}}}C_{2n\,2n\,2p}\,a_{*2p}{\\!}^{(2)}+{\textstyle{\sum_{p\neq
1}}}D_{2n\,2n\,2p}\,b_{*2p}{\\!}^{(2)}\Big{)}\,.$ (8.22)

Hence with (8.16) this gives

$\eta^{(2)}=4(n-1)^{2}\frac{n!^{6}}{(2n)!^{3}}\,\frac{2n-1}{n}\,\frac{\rho^{\prime}{\\!}_{2n-1}(0)}{\rho_{n-1}(0)^{2}}\,.$
(8.23)

In general $\rho_{n}$, and also $\tau_{n}$, as defined in (8.14a) and (8.14b)
depend on the cut off function but as shown in Appendix C there are certain
universal quantities which are related to logarithmic divergences. In
particular

$\frac{\rho^{\prime}{\\!}_{2n-1}(0)}{\rho_{n-1}(0)^{2}}\bigg{|}_{d=d_{n}}=\frac{n}{2n-1}=\frac{d_{n}}{d_{n}+2}\,,$
(8.24)

so that (8.23) is in exact accord with (2.18).

In the neighbourhood of a fixed point the solutions of (8.18a), (8.18b) are
written as

$a_{k}(t)=a_{*k}+e^{\lambda t}f_{k}\,,\qquad b_{k}(t)=b_{*k}+e^{\lambda
t}g_{k+2(n-1)}\,,$ (8.25)

for $f_{k},g_{k+2(n-1)}$ small. The critical exponents are then determined by
the linear eigenvalue equation

$\frac{2n-k}{n-1}\,\left(\begin{array}[]{cc}f_{k}\\\ g_{k}\\\
\end{array}\right)+\sum_{l}\left(\begin{array}[]{cc}{\cal M}_{11,kl}&{\cal
M}_{12,kl}\\\ {\cal M}_{21,kl}&{\cal M}_{22,kl}\\\
\end{array}\right)\left(\begin{array}[]{cc}f_{l}\\\ g_{l}\\\
\end{array}\right)=\lambda\left(\begin{array}[]{cc}f_{k}\\\ g_{k}\\\
\end{array}\right)\,,$ (8.26)

for

$\displaystyle{\cal M}_{11,kl}={}$
$\displaystyle\big{(}\varepsilon({\textstyle\frac{1}{2}}k-1)-{\textstyle\frac{1}{2}}k\eta\big{)}\delta_{kl}+2{\textstyle{\sum_{m}}}\big{(}C_{klm}\,a_{*m}+D_{klm}\,b_{*m}\big{)}\,,$
$\displaystyle{\cal M}_{22,kl}={}$
$\displaystyle\big{(}\varepsilon({\textstyle\frac{1}{2}}k-n)-({\textstyle\frac{1}{2}}k-n+1)\eta\big{)}\delta_{kl}+2{\textstyle{\sum_{m}}}{\tilde{D}}_{k-2(n-1)\
m\ l-2(n-1)}\,a_{*m}^{\vphantom{g}}\,,$ $\displaystyle{\cal M}_{12,kl}={}$
$\displaystyle 2{\textstyle{\sum_{m}}}D_{km\
l-2(n-1)}\,a_{*m}^{\vphantom{g}}\,,$ $\displaystyle{\cal M}_{21,kl}={}$
$\displaystyle 2{\textstyle{\sum_{m}}}\big{(}{\tilde{C}}_{k-2(n-1)\
lm}\,a_{*m}^{\vphantom{g}}+{\tilde{D}}_{k-2(n-1)\
lm}\,b_{*m}^{\vphantom{g}}\big{)}\,.$ (8.27)

To first order in $\varepsilon$ we may let
$a_{*m}\to\varepsilon\,a_{*}\delta_{m\,2n}$ and $b_{*m}\to 0$. Since

$D_{k\,2n\,k-2(n-1)}=\frac{2n}{k+1-2n}\,\tau_{0}(0)=0\,,$ (8.28)

as from (8.14b) $\tau_{0}(p^{2})=p^{2}K^{\prime}(p^{2})$, then ${\cal
M}_{12,kl}$ is ${\rm O}(\varepsilon^{2})$, whereas otherwise ${\cal
M}_{11,kl},{\cal M}_{21,kl}$ and ${\cal M}_{22,kl}$ are ${\rm
O}(\varepsilon)$. Hence the first order critical exponents from (8.26) are

$\displaystyle\lambda_{1,k}{\\!}^{(1)}={}$
$\displaystyle{\textstyle\frac{1}{2}}k-1+2a_{*}\,C_{kk\,2n}={\textstyle\frac{1}{2}}k-1-2(n-1)\frac{n!\,k!}{(2n)!\,(k-n)!}\,,$
$\displaystyle\lambda_{2,k}{\\!}^{(1)}={}$
$\displaystyle{\textstyle\frac{1}{2}}k-n+2a_{*}\,{\tilde{D}}_{k-2(n-1)\ 2n\
k-(2(n-1)}$ $\displaystyle={}$
$\displaystyle{\textstyle\frac{1}{2}}k-n-2(n-1)\frac{n!\,(k-2n)!}{(2n)!\,(k-3n+1)!}\bigg{(}k-2n+1+n\,\frac{\tau^{\prime}{\\!}_{n-1}(0)}{\rho_{n-1}(0)}\bigg{)}\,.$
(8.29)

Just as in (8.24)

$\tau^{\prime}{\\!}_{n-1}(0)\big{|}_{d=d_{n}}=0\,,$ (8.30)

so that (8.29) is in exact agreement with the perturbative results (2.16) and
(2.32).

##  9 Modified Derivative Expansion

The results of Section 8 can be rewritten in a form which is close to the
derivative expansion. To achieve this it is necessary to make specific choices
of the cut off dependent quantities which appear in (8.16) and which are
arbitrary up to the freedom exhibited in (8.20). It is crucial of course that
the cut-off function independent results in (8.24) and (8.30), as well as
$\tau_{0}(0)=0$, should be satisfied. To this end we choose

$\rho_{n}(0)=-1\,,\qquad\rho^{\prime}{\\!}_{n}(0)=\frac{d}{d+2}\,,\qquad\tau_{n}(0)={\cal
A}n\,,\qquad\tau^{\prime}{\\!}_{n}(0)=0\,,$ (9.1)

where ${\cal A}$ is arbitrary. With these choices, and with (3.13) and (3.14),
then (8.16) gives

$\displaystyle C_{klm}={}$
$\displaystyle-2^{\frac{1}{2}(k-l-m)+1}\,\frac{1}{N_{k}}\,lm\,G_{k\,l-1\,m-1}\,,$
$\displaystyle{\tilde{C}}_{k+2\,lm}={}$
$\displaystyle\frac{d}{d+2}\,2^{\frac{1}{2}(k-l-m)+2}\,\frac{1}{N_{k}}\,l(l-1)m(m-1)\,G_{k\,l-2\,m-2}\,,$
$\displaystyle D_{kl\,m+2}={}$ $\displaystyle{\cal
A}\,2^{\frac{1}{2}(k-l-m)+1}\,\frac{1}{N_{k}}\,l(l-1)G_{k\,l-2\,m}\,,$
$\displaystyle{\tilde{D}}_{k+2\,l\,m+2}={}$
$\displaystyle-2^{\frac{1}{2}(k-l-m)+1}\,\frac{1}{N_{k}}\,\big{(}lm\,G_{k\,l-1\,m-1}+l(l-1)\,G_{k\,l-2\,m}\big{)}\,.$
(9.2)

If we now define

${\tilde{v}}(x)=\sum_{k}a_{k}\,2^{-\frac{1}{2}k}H_{k}(x)\,,\qquad{\tilde{z}}(x)=\sum_{k}b_{k+2}\,2^{-\frac{1}{2}k}H_{k}(x)\,,$
(9.3)

then the truncated RG equations (8.18a) and (8.18b) are equivalent, subject to
requiring non singular solutions for all $x$, to the coupled differential
equations

$\displaystyle{\dot{\tilde{v}}}={}$
$\displaystyle\big{(}d+{\textstyle\frac{1}{2}}(d-2+\eta)D\big{)}\tilde{v}-{\textstyle\frac{1}{2}}\,\tilde{v}^{\prime}{}^{2}+{\cal
A}\,\tilde{v}^{\prime\prime}\tilde{z}\,,$ (9.4a)
$\displaystyle{\dot{\tilde{z}}}={}$
$\displaystyle\big{(}-\eta+{\textstyle\frac{1}{2}}(d-2+\eta)D\big{)}\tilde{z}-\tilde{v}^{\prime}\tilde{z}^{\prime}-\tilde{v}^{\prime\prime}\tilde{z}-{\textstyle\frac{1}{2}}\eta+\frac{d}{4(d+2)}\,\tilde{v}^{\prime\prime}{}^{2}\,.$
(9.4b)

With the redefinitions

$v=\frac{2}{d-2+\eta}\,\tilde{v}\,,\qquad z=2\tilde{z}\,,$ (9.5)

and, with $K,L$ defined as in (6.4), then these become

$\displaystyle\frac{2}{d-2+\eta}\,{\dot{v}}={}$
$\displaystyle(D+K)v-{\textstyle\frac{1}{2}}\,v^{\prime}{}^{2}+{\tilde{\cal
A}}\,v^{\prime\prime}z\,,$ (9.6a)
$\displaystyle\frac{2}{d-2+\eta}\,{\dot{z}}={}$
$\displaystyle(D-L)z-v^{\prime}z^{\prime}-v^{\prime\prime}z-L+{\textstyle\frac{1}{2}}{\tilde{\cal
B}}\,v^{\prime\prime}{}^{2}\,,$ (9.6b)

for

${\tilde{\cal A}}=\frac{2}{d-2+\eta}\,{\cal A}\,,\qquad{\tilde{\cal
B}}={\textstyle\frac{1}{2}}(d-2+\eta)\,\frac{d}{d+2}\,.$ (9.7)

The results (9.6a) and (9.6b) are very similar to (6.3a) and (6.3b), although
there is no linear $z$ term in (9.6a) and the coefficient ${\tilde{\cal B}}$
is determined in (9.6b). As a consequence the coefficient $c_{n}$ in the
leading order solution (3.10) is unchanged from (3.16). Furthermore following
the same discussion as in Sections 6 and 7 gives the correct values for $\eta$
to ${\rm O}(\varepsilon^{2})$. Thus instead of (7.10)

$\Delta^{(0)}=\left(\begin{array}[]{cc}-D-2n&0\\\ 0&-D\\\
\end{array}\right)\,,$ (9.8)

and replacing (7.15)

$\Delta^{(1)}=\left(\begin{array}[]{cc}-(n-1)^{2}+c_{n}H_{2n}{\\!}^{\prime}\frac{{\rm
d}}{{\rm d}x}&-{\tilde{\cal A}}\,c_{n}H_{2n}{\\!\\!}^{\prime\prime}\\\
-\tilde{{\cal B}}\,c_{n}H_{2n}{\\!\\!}^{\prime\prime}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}&c_{n}H_{2n}{\\!\\!}^{\prime}\frac{{\rm d}}{{\rm
d}x}+c_{n}H_{2n}{\\!\\!}^{\prime\prime}\\\ \end{array}\right)\,.$ (9.9)

It is then easy to see that this ensures the correct ${\rm O}(\varepsilon)$
result instead of (7.23b) as well as preserving (7.23a).

In fact it is easy to verify that (7.7c), (7.7f) and (7.7i) still give exact
eigenfunctions and eigenvalues for the linearised perturbations (9.6a) and
(9.6b) about fixed points $v_{*},z_{*}$. These considerations may ensure that
(9.6a) and (9.6b) have a greater chance of predictive success when they are
analysed without using the $\varepsilon$-expansion. Of course setting $z,\eta$
to zero in (9.6a) reduces it to just the LPA.

##  10 Conclusion

The status of the derivative expansion for exact RG flow equations is not
entirely clear. In some respects it may be similar to effective field theories
describing the large distance or low energy aspects of more fundamental
theories. Having identified the relevant degrees of freedom and appropriate
symmetries an effective lagrangian is constructed in terms of all symmetric
scalars formed from the basic fields up to some scale dimension so as to
reproduce physical amplitudes as far as contributions of the form
$(E/\Lambda)^{p}$ for some $p$ where $E$ is a physical energy scale and
$\Lambda$ a cut off [33]. The couplings which appear in the effective
lagrangian can in principle be determined by matching the predictions of the
effective theory with the fundamental theory for some specific physical
amplitude. In a somewhat similar fashion a derivative expansion generates
terms in the differential flow equations whose coefficients appear to depend
on the cut off function and so are essentially arbitrary. A possible
resolution is to match the results to those coming from the
$\varepsilon$-expansion for $\varepsilon\to 0$, although the approximate flow
equations may then be used for general $d$. The results in this paper
essentially show how this can be achieved to ${\rm O}(\varepsilon)$ and to
include thereby the universal aspects of all two vertex Feynman graphs. It
would be interesting although non trivial to extend this to three vertex
graphs. The results in Appendix A show how at this order various
transcendental numbers arise which make achieving this for different
multicritical points simultaneously hard to achieve.

An important issue in using exact RG equations is to determine which solutions
are physically relevant and give results independent of the particular RG
equation or the detailed cut off function as an infra-red fixed point is
approached and we may take $\Lambda\to\infty$. This question becomes more
significant in approximation schemes when the symmetries of the original exact
RG equation are no longer maintained and spurious solutions and critical
exponents may be generated. The Polchinski RG equations for a fixed point
action $S_{*}[\varphi]$ has an exactly marginal operator with zero critical
exponent. This ensures that there is in general a line of physically
equivalent fixed points $S_{*}[\varphi,a]$ depending on a parameter $a$. The
exact marginal operator, which is constructed in detail in Appendix D,
corresponds to an infinitesimal change in the scale of the field $\varphi$
under which the functional integral is invariant (conventionally the kinetic
term in the action may be normalised to one but this is not essential, the
physical couplings need only be redefined appropriately), for a further
discussion see [29]. In the perturbative context the presence of such a
marginal operator was demonstrated after (2.27) and is a property of the ${\rm
O}(\varepsilon)$ results in (2.32) for $k=2n$. The presence of the irrelevant
gauge parameter $a$ is in general necessary for the RG equations to determine
$\eta$.

If the symmetry under rescaling of the fields were to be maintained in a
derivative expansion it would imply that critical exponents should be
independent of $z_{*}(0)$ [34]. In the derivative expansion results obtained
here in (6.5a), (6.5b), or the corresponding equations from (9.6a), (9.6b),
there is a relation in general between $z_{*}(0)$ and $\eta$ so that $\eta$ is
not determined unless $z_{*}(0)$ is fixed. At lowest order in $\varepsilon$ as
in (6.6) the dependence on $z_{*}(0)$ disappears. Imposing $z_{*}(0)=0$ makes
the equations well defined but is not a necessary requirement in general. The
marginal operator constructed in Appendix D involves an integration over
$\tilde{\varphi}(q)$ for all $q$ and so approximations such as the derivative
approximation emphasising low $q$ fail to maintain the exact zero value for
the critical exponent. Nevertheless preserving as far as possible the presence
of a marginal operator is then a potential further constraint on solutions of
exact RG equations in the derivative approximation [32] which may be used to
restrict cut off dependence.

Finally we note that the LPA has desirable features which are absent in any
straightforward fashion in the derivative expansion. As shown in (5.11) the
operator determining critical exponents for the LPA can be recast in self
adjoint form. Related to this is the fact that for the LPA it is possible to
construct a $C$-function from which the RG flow equations can be obtained [35]
and that the equations can be written as a gradient flow [25, 36]. Whether
this is true more generally remains to be demonstrated.

##  Appendix A Further Perturbative Calculations

The results in Section 2 may be extended to the the next order ${\rm
O}(V^{3})$ in a similar fashion and we obtain some results here. The relevant
contributions to $W$ are, with the same notation as (2.4), given by

$\displaystyle W_{2}={}$ $\displaystyle-\frac{1}{6}\sum_{r,s,t\geq
1}\frac{1}{r!s!t!}\int{\rm d}^{d}x_{1}\,{\rm d}^{d}x_{2}\,{\rm
d}^{d}x_{3}\,V^{(r+t)}(\varphi_{1})V^{(r+s)}(\varphi_{2})V^{(s+t)}(\varphi_{3})$
$\displaystyle\hskip 170.71652pt{}\times
G_{0}(x_{12})^{r}G_{0}(x_{23})^{s}G_{0}(x_{31})^{t}$
$\displaystyle{}-\frac{1}{2}\sum_{r,s\geq 2}\frac{1}{r!s!}\int{\rm
d}^{d}x_{1}\,{\rm d}^{d}x_{2}\,{\rm
d}^{d}x_{3}\,V^{(r)}(\varphi_{1})V^{(r+s)}(\varphi_{2})V^{(s)}(\varphi_{3})\,G_{0}(x_{12})^{r}G_{0}(x_{23})^{s}$
$\displaystyle{}+\sum_{r\geq 2}\frac{1}{r!}\int{\rm d}^{d}x_{1}\,{\rm
d}^{d}x_{2}\,V^{(r)}(\varphi_{1})\,G_{0}(x_{12})^{r}\,V^{(r)}_{{\rm{c.t.}1}}(\varphi_{2})\,,$
(A.1)

where $V_{{\rm{c.t.}1}}$ is determined by the first term in (2.8). This
removes subdivergencies arising in (A.1) for $r,s,t=n$ and we restrict
$r,s,t<2n-1$ so that no further subtractions are necessary.

The divergencies coming from the second term in (A.1) are easily obtained
since if ${\cal R}$ is the usual operation defining a finite part, so that
from (2.7a)

${\cal
R}\big{(}G_{0}(x)^{n}\big{)}=G_{0}(x)^{n}-\frac{2}{\varepsilon}\,\frac{1}{(4\pi)^{n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{n-1}\,\delta^{d}(x)\,,$
(A.2)

then in this term the finite part is given by

${\cal R}\big{(}G_{0}(x_{12})^{n}G_{0}(x_{23})^{n}\big{)}={\cal
R}\big{(}G_{0}(x_{12})^{n}\big{)}\,{\cal R}\big{(}G_{0}(x_{23})^{n}\big{)}\,,$
(A.3)

so that the divergent pole terms are given by
$G_{0}(x_{12})^{n}G_{0}(x_{23})^{n}-{\cal
R}\big{(}G_{0}(x_{12})^{n}G_{0}(x_{23})^{n}\big{)}$.

For the first term in (A.1) there is an overall divergence for $d=d_{n}$ when
$r+s+t=2n$. To analyse this we make use of the Mellin-Barnes representation
[37]

$\displaystyle\int{\rm d}^{d}x_{1}\,{\rm d}^{d}x_{2}\,e^{ik_{1}\cdot
x_{1}+k_{2}\cdot x_{2}}\,G_{0}(x_{12})^{r}G_{0}(x_{2})^{s}G_{0}(x_{1})^{t}$
$\displaystyle{}=\frac{1}{(4\pi)^{r+s+t}}\,\frac{\Gamma(\nu)^{r+s+t}}{\Gamma(r\nu)\Gamma(s\nu)\Gamma(t\nu)}\,\frac{1}{\Gamma((r+s+t-1)\nu-1)}\,J_{rst}\bigg{(}\frac{k_{1}^{2}}{k_{3}^{2}},\frac{k_{2}^{2}}{k_{3}^{2}}\bigg{)}\,\bigg{(}\frac{k_{3}^{2}}{4\pi}\bigg{)}^{(r+s+t-2)\nu-2},$
$\displaystyle J_{rst}(u,v)=\frac{1}{(2\pi
i)^{2}}\int_{\gamma-i\infty}^{\gamma+i\infty}\\!\\!\\!\\!\\!\\!\\!\\!{\rm
d}y\int_{\gamma^{\prime}-i\infty}^{\gamma^{\prime}+i\infty}\\!\\!\\!\\!\\!\\!\\!\\!{\rm
d}z\ \Gamma(-y)\,\Gamma(-z)$ $\displaystyle\hskip
85.35826pt{}\times\Gamma\big{(}(r+s-1)\nu-1-y\big{)}\,\Gamma\big{(}(r+t-1)\nu-1-z\big{)}$
$\displaystyle\hskip
85.35826pt{}\times\Gamma\big{(}y+z+1-(r-1)\nu\big{)}\,\Gamma\big{(}y+z+2-(r+s+t-2)\nu\big{)}\,u^{y}v^{z}\,,$
(A.4)

where $k_{3}=-k_{1}-k_{2}$ and $\gamma,\gamma^{\prime}$ are chosen that the
poles in $y,z$ are on the opposite side of the contours from those in $y+z$.
The functions $J_{rst}(u,v)$ satisfy various symmetry relations, in particular

$J_{rst}(u,v)=v^{(r+s+t-2)\nu-2}J_{srt}(u/v,1/v)=u^{(r+s+t-2)\nu-2}J_{tsr}(1/u,v/u)=J_{rts}(v,u)\,,$
(A.5)

which are necessary to ensure that (A.4) is symmetric under permutations of
$k_{1},k_{2},k_{3}$ and also $s,t,r$. For $d=d_{n}-\varepsilon$ as in (2.6),
the poles in $\varepsilon$, reflecting divergences of relevance here, arise
only from the residues of the poles at $y,z=0$. For $r,s,t\neq n,\ r+s+t=2n$
there is then a simple $\varepsilon$-pole arising from
$\Gamma\big{(}2-2(n-1)\nu\big{)}$ since
$\nu=\frac{1}{n-1}-{\textstyle\frac{1}{2}}\varepsilon$ which gives

$\displaystyle G_{0}(x_{12})^{r}G_{0}(x_{23})^{s}G_{0}(x_{31})^{t}$
$\displaystyle\,\big{|}_{r+s+t=2n,r,s,t\neq
n}\sim\frac{1}{\varepsilon}\,\frac{1}{(4\pi)^{2n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-1}K_{rst}\,\delta^{d}(x_{12})\delta^{d}(x_{13})\,,$
$\displaystyle
K_{rst}=\frac{\Gamma\big{(}\frac{n-r}{n-1}\big{)}\,\Gamma\big{(}\frac{n-s}{n-1}\big{)}\,\Gamma\big{(}\frac{n-t}{n-1}\big{)}}{\Gamma\big{(}\frac{r}{n-1}\big{)}\,\Gamma\big{(}\frac{s}{n-1}\big{)}\,\Gamma\big{(}\frac{t}{n-1}\big{)}}\,.$
(A.6)

For $r=n,\ s+t=n$ the $y,z=0$ residues in (A.4) have a double pole in
$\varepsilon$ from $\Gamma\big{(}1-(n-1)\nu\big{)}$ as well as
$\Gamma\big{(}2-(2n-2)\nu\big{)}$. Expanding in $\varepsilon$ gives

$\displaystyle{\cal R}\big{(}G_{0}(x_{12})^{n}\big{)}\,$ $\displaystyle
G_{0}(x_{23})^{s}G_{0}(x_{31})^{t}\,\big{|}_{s+t=n}$ $\displaystyle\sim{}$
$\displaystyle\frac{1}{\varepsilon^{2}}\,\frac{1}{(4\pi)^{2n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\big{(}-2+(n-1)^{2}\varepsilon+(n-1)L_{st}\,\varepsilon\big{)}\,\delta^{d}(x_{12})\delta^{d}(x_{13})\,,$
$\displaystyle
L_{st}=\psi\Big{(}\frac{1}{n-1}\Big{)}-\psi\Big{(}\frac{s}{n-1}\Big{)}-\psi\Big{(}\frac{t}{n-1}\Big{)}+\psi(1)\,,\
s+t=n\,.$ (A.7)

With the aid of (A.6) and (A.7) then the pole terms in (A.1) require

$\displaystyle V_{\rm{c.t.}2}(\phi)={}$
$\displaystyle-\frac{1}{\varepsilon}\,\frac{1}{(4\pi)^{2n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-1}\,\frac{1}{6}\sum_{\genfrac{}{}{0.0pt}{}{r,s,t\geq
1,r,s,t\neq n}{r+s+t=2n}}\frac{K_{rst}}{r!s!t!}\
V^{(r+t)}(\phi)V^{(r+s)}(\phi)V^{(s+t)}(\phi)$
$\displaystyle+\frac{1}{\varepsilon^{2}}\,\frac{1}{(4\pi)^{2n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\,\frac{1}{n!}\sum_{\genfrac{}{}{0.0pt}{}{s,t\geq
1}{s+t=n}}\frac{1}{s!t!}\,\big{(}1-{\textstyle\frac{1}{2}}(n-1)^{2}\varepsilon-{\textstyle\frac{1}{2}}(n-1)L_{st}\big{)}$
$\displaystyle\hskip 184.9429pt{}\times
V^{(n)}(\phi)V^{(n+s)}(\phi)V^{(n+t)}(\phi)$
$\displaystyle+\frac{2}{\varepsilon^{2}}\,\frac{1}{(4\pi)^{2n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\,\frac{1}{n!^{2}}\,V^{(n)}(\phi)^{2}\,V^{(2n)}(\phi)\,.$
(A.8)

The double poles are in accord with standard RG equations from (2.9)

$\bigg{(}\varepsilon-{\hat{\beta}}^{V}\\!\cdot\frac{\partial}{\partial
V}+{\hat{\gamma}}_{\phi}\,\phi\frac{\partial}{\partial\phi}\bigg{)}\big{(}V(\phi)+V_{\rm{c.t.}}(\phi)\big{)}=0\,,$
(A.9)

since we have at this order with (2.10)

$\beta_{2}^{V}(\phi)-\gamma_{\phi,1}\,\phi
V^{\prime}(\phi)+\beta_{1}^{V}\\!\cdot\frac{\partial}{\partial
V}\,V_{\rm{c.t.}1}(\phi)=2(n-1)\varepsilon\,V_{\rm{c.t.}2}(\phi)\,.$ (A.10)

From (A.8) we then obtain

$\displaystyle\beta^{V}_{2}(\phi)={}$
$\displaystyle-\frac{1}{3}(n-1)\,\frac{1}{(4\pi)^{2n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-1}\\!\\!\\!\\!\sum_{\genfrac{}{}{0.0pt}{}{r,s,t\geq
1,r,s,t\neq n}{r+s+t=2n}}\frac{K_{rst}}{r!s!t!}\
V^{(r+t)}(\phi)V^{(r+s)}(\phi)V^{(s+t)}(\phi)$
$\displaystyle-(n-1)^{2}\,\frac{1}{(4\pi)^{2n}}\,\Gamma\Big{(}\frac{1}{n-1}\Big{)}^{2n-2}\,\frac{1}{n!}\sum_{\genfrac{}{}{0.0pt}{}{s,t\geq
1}{s+t=n}}\frac{1}{s!t!}\,\big{(}n-1+L_{st}\big{)}$ $\displaystyle\hskip
199.16928pt{}\times V^{(n)}(\phi)V^{(n+s)}(\phi)V^{(n+t)}(\phi)$
$\displaystyle+\gamma_{\phi,1}\,\phi V^{\prime}(\phi)\,.$ (A.11)

For general $n$ it is not straightforward to analyse (A.11) further so we
content ourselves for the simplest cases of $n=2,3$ which give

$\beta_{2}^{V}(\phi)\big{|}_{n=2}=-{\textstyle\frac{1}{2}}\,\frac{1}{(4\pi)^{4}}\,V^{(2)}(\phi)V^{(3)}(\phi)^{2}+\gamma_{\phi,1}\,\phi
V^{\prime}(\phi)\,,\quad\gamma_{\phi,1}={\frac{1}{12}}\,\frac{g^{2}}{(4\pi)^{4}}\,,$
(A.12)

and

$\displaystyle\beta_{2}^{V}(\phi)\big{|}_{n=3}={}$
$\displaystyle\frac{\pi^{2}}{(4\pi)^{6}}\Big{(}{\textstyle\frac{1}{6}}V^{(2)}(\phi)V^{(5)}(\phi)^{2}-{\textstyle\frac{1}{12}}\pi^{2}V^{(4)}(\phi)^{3}-4V^{(3)}(\phi)V^{(4)}(\phi)V^{(5)}(\phi)\Big{)}$
$\displaystyle{}+\gamma_{\phi,1}\,\phi
V^{\prime}(\phi)\,,\qquad\qquad\gamma_{\phi,1}={\frac{1}{90}}\,\frac{g^{2}\pi^{2}}{(4\pi)^{6}}\,.$
(A.13)

Using (A.12) we may obtain ${\rm O}(\varepsilon^{2})$ corrections to (2.15)
and (2.16) for $n=2$

$\displaystyle\frac{3g_{*}}{(4\pi)^{2}}={}$
$\displaystyle\varepsilon+{\textstyle\frac{2}{3}}\,\varepsilon^{2}-2\eta\,,\qquad\eta={\textstyle\frac{1}{54}}\,\varepsilon^{2}+{\rm
O}(\varepsilon^{3})\,,$ $\displaystyle{\hat{\gamma}}_{k}(g_{*})={}$
$\displaystyle-{\textstyle\frac{1}{2}}(k-2)\,\varepsilon+{\textstyle\frac{1}{2}}k\eta+\textstyle{\frac{1}{6}}k(k-1)(\varepsilon-2\eta)-\textstyle{\frac{1}{18}}k(k-1)(k-3)\,\varepsilon^{2}+{\rm
O}(\varepsilon^{3})\,.$ (A.14)

This agrees with standard results for $k=1,2,3$. Furthermore for $n=3$

$\displaystyle\frac{20}{3}\,\frac{\pi
g_{*}}{(4\pi)^{3}}=2\varepsilon+{\textstyle\frac{9}{20}}\big{(}47+{\textstyle\frac{3}{2}}\pi^{2}\big{)}\,\varepsilon^{2}-3\eta\,,\qquad\eta={\textstyle\frac{1}{500}}\,\varepsilon^{2}+{\rm
O}(\varepsilon^{3})\,,$
$\displaystyle{\hat{\gamma}}_{k}(g_{*})=-{\textstyle\frac{1}{2}}(k-2)\,\varepsilon+{\textstyle\frac{1}{2}}k\eta+\textstyle{\frac{1}{30}}k(k-1)(k-2)\big{(}\varepsilon-{\textstyle\frac{3}{2}}\eta\big{)}$
$\displaystyle\hskip
39.83368pt{}-\textstyle{\frac{3}{100}}k(k-1)(k-5)\big{(}k(k-2)-\textstyle{\frac{1}{24}}k(k-4)+\textstyle{\frac{3}{16}}\pi^{2}(k-2)\big{)}\,\varepsilon^{2}+{\rm
O}(\varepsilon^{3})\,.$ (A.15)

When $k\geq 2n$ the ${\rm O}(\varepsilon^{2})$ results are modified due to
mixing effects. Without computing the ${\rm O}(\varepsilon^{2})$ terms in
${\hat{\gamma}}^{hh}_{k}(g_{*})$ in the matrix (2.27) we have for one
eigenvalue

$\omega_{1,k}={\hat{\gamma}}_{k}(g_{*})+\Delta\omega_{1,k}+{\rm
O}(\varepsilon^{3})\,,\qquad\Delta\omega_{1,k}=\frac{{\gamma}^{gh}_{k,1}(g_{*})\,{\gamma}^{hg}_{k,1}(g_{*})}{{\hat{\gamma}}^{gg}_{k,1}(g_{*})-{\hat{\gamma}}^{hh}_{k,1}(g_{*})}\,.$
(A.16)

This gives

$\displaystyle\Delta\omega_{1,k}\big{|}_{n=2}={}$
$\displaystyle\textstyle{\frac{1}{6}}k(k-1)(k-2)\,\eta\,,\qquad\qquad\ \ k\geq
4\,,$ $\displaystyle\Delta\omega_{1,k}\big{|}_{n=3}={}$
$\displaystyle\textstyle{\frac{1}{30}}k(k-1)(k-3)(k-4)\,\eta\,,\quad\,k\geq
6\,.$ (A.17)

with $\eta$ as in (A.14) and (A.15). We may note that in both cases the
results are consistent with
$\omega_{1,2n}={\hat{\beta}}^{g}{}^{\prime}(g_{*})$.

By considering the residues in (A.4) at $y=1,z=0$ and $y=0,z=1$, and requiring
$r+s+t=3n-1$, we may also determine directly higher order contributions to
$\gamma_{\phi}$ although it is then necessary to include an additional
counterterm for when $r,s,t=2n-1$ in (A.1). Such results are omitted as they
are irrelevant in the context of this paper.

##  Appendix B Verification of Vanishing of a Sum

In the discussion of critical exponents in Section 5 consistency required that
the sum appearing in (5.27)

$S=\sum_{p=0}^{n}(2n-3p)\frac{(2p)!}{p!^{4}(n-p)!^{2}}\,,$ (B.1)

should vanish. Although in the case where it arises here $n$ is even $S=0$ for
any $n$. To show this directly we note that

$S=\frac{1}{n!^{2}}\sum_{p=0}^{n}(2n-3p)\binom{2p}{p}\binom{n}{p}^{2}\,,$
(B.2)

where we may write

$\sum_{r=0}^{p}\binom{p}{r}^{2}=\binom{2p}{p}\,.$ (B.3)

Hence

$\displaystyle S={}$
$\displaystyle\frac{1}{n!^{2}}\sum_{p=0}^{n}\sum_{r=0}^{p}(2n-3p)\binom{p}{r}^{2}\binom{n}{p}^{2}$
$\displaystyle={}$
$\displaystyle\sum_{p=0}^{n}\sum_{r=0}^{p}(2n-3p)\frac{1}{r!^{2}(p-r)!^{2}(n-p)!^{2}}$
$\displaystyle={}$
$\displaystyle\sum_{s=0}^{n}\sum_{r=0}^{n-s}(2n-3s-3r)\frac{1}{r!^{2}s!^{2}(n-s-r)!^{2}}\,,$
(B.4)

where $s=p-r$. Then, setting $t=n-s-r$,

$\displaystyle S=$ $\displaystyle\sum_{\begin{subarray}{c}r,s,t\geq 0\\\
r+s+t=n\end{subarray}}(2t-r-s)\frac{1}{r!^{2}s!^{2}t!^{2}}\,,$ (B.5)

from which it follows using symmetry of the $r,s,t$-sums that $S=0$. This then
implies (8.30).

##  Appendix C Integrals and Cut Off Function Dependence

In the discussion in section 8 the dependence on the cut off function was
reduced to particular integrals such as appeared in (8.14a) and (8.14b). In
general the presence of an arbitrary cut off function $K(p^{2})$, constrained
only by $K(0)=1$ and rapid fall off for large $p^{2}$, ensures that they can
take any value but in special cases the integrals are identical with the
logarithmically divergent part of standard Feynman integrals and so they have
a universal form independent of any particular $K(p^{2})$.

Reinstating the cut off in the propagator which appears in (8.8) so that

$G_{\Lambda}(p)=\frac{K(p^{2}/\Lambda^{2})}{p^{2}}\,,$ (C.1)

then (8.14a) can be written as

$\displaystyle\Lambda^{m(d-2)-2}\rho_{m}\big{(}p^{2}/\Lambda^{2}\big{)}={}$
$\displaystyle\frac{1}{\Lambda^{2}}\,\frac{1}{(2\pi)^{dm}}\\!\int\prod_{h=1}^{m}{\rm
d}^{d}r_{h}\,G_{\Lambda}(r_{h})\,K^{\prime}\big{(}(p+r_{(m)})^{2}/\Lambda^{2}\big{)}$
$\displaystyle={}$
$\displaystyle-\frac{1}{2(m+1)}\,\Lambda\frac{\partial}{\partial\Lambda}\,\frac{1}{(2\pi)^{dm}}\\!\int\prod_{h=1}^{m}{\rm
d}^{d}r_{h}\,G_{\Lambda}(r_{h})\,G_{\Lambda}(p+r_{(m)})\,.$ (C.2)

The logarithmic divergencies present in the product of $m+1$ propagators for
$d=d_{n}$ as in (2.6) when $m(d_{n}-2)=2+2l,\ l=0,1,\dots$ then generate a cut
off independent result for $\rho_{m}^{(l)}(0)$.

To obtain the detailed coefficients, following [17], the momentum space
convolution integrals in (8.14a), (8.14b) are expressed in terms of the
$x$-space propagator

${\tilde{G}}(x)=\frac{1}{(2\pi)^{d}}\int{\rm d}^{d}p\,e^{-ip\cdot x}\,G(p)\,.$
(C.3)

and, using
$(1+{\textstyle\frac{1}{2}}p\cdot\partial_{p})G(p)=K^{\prime}(p^{2})$,

$-{\textstyle\frac{1}{2}}\big{(}d-2+x\cdot\partial_{x}\big{)}\,{\tilde{G}}(x)=\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,e^{-ip\cdot x}K^{\prime}(p^{2})\,.$ (C.4)

Then (8.14a) becomes

$\rho_{m}(p^{2})=-\frac{1}{2}\int{\rm d}^{d}x\,e^{ip\cdot
x}\,{\tilde{G}}(x)^{m}\,\big{(}d-2+x\cdot\partial_{x}\big{)}\,{\tilde{G}}(x)\,.$
(C.5)

This gives for the Taylor expansion coefficients at $p^{2}=0$

$\rho^{(l)}_{m}(0)=-e_{l}\,\frac{1}{2}\int{\rm
d}^{d}x\,(x^{2})^{l}\,{\tilde{G}}(x)^{m}\,\big{(}d-2+x\cdot\partial_{x}\big{)}\,{\tilde{G}}(x)\,,\quad
e_{l}=\frac{(-1)^{l}}{2^{2l}l!({\textstyle\frac{1}{2}}d)_{l}}\,,$ (C.6)

or with $r^{2}=x^{2}$

$\rho^{(l)}_{m}(0)=-e_{l}\,\frac{S_{d}}{2(m+1)}\int_{0}^{\infty}\\!\\!\\!\\!{\rm
d}r\,r^{d+2l-1}\big{(}(m+1)(d-2)+r\partial_{r}\big{)}\,{\tilde{G}}(x)^{m+1}\,,$
(C.7)

for $S_{d}={2\pi^{\frac{1}{2}}}/{\Gamma({\textstyle\frac{1}{2}}d)}$. When
$d+2l=(m+1)(d-2)$ the integrand is a total derivative and using

${\tilde{G}}(x)\sim\frac{1}{(d-2)S_{d}}\,\frac{1}{r^{d-2}}\quad\hbox{as}\quad
r\to\infty\,,$ (C.8)

then there is only a surface term for large $r$ giving

$\rho^{(l)}_{(l+1)(n-1)}(0)\big{|}_{d=d_{n}}=-(-1)^{l}e_{l}\,\frac{n-1}{4(n+l(n-1))}\,\big{(}(d_{n}-2)S_{d_{n}}\big{)}^{-(l+1)(n-1)}\,.$
(C.9)

This result directly implies (8.24). For $l=0,1$ the coefficients obtained in
(C.9) correspond exactly to the pole terms in dimensional regularisation in
(2.7a), (2.7b).

In a similar vein from (8.14b)

$\displaystyle\tau_{m}(p^{2})=\frac{1}{2}\int{\rm d}^{d}x\,e^{ip\cdot
x}\,\Big{(}{}$ $\displaystyle
m\,{\tilde{G}}(x)^{m-1}\partial^{2}{\tilde{G}}(x)\,\big{(}d-2+x\cdot\partial_{x}\big{)}\,{\tilde{G}}(x)$
$\displaystyle{}+{\tilde{G}}(x)^{m}\big{(}d+x\cdot\partial_{x}\big{)}\,\partial^{2}{\tilde{G}}(x)\Big{)}\,,$
(C.10)

and

$\tau^{(l)}_{m}(0)=e_{l}\,\frac{1}{2}S_{d}\int_{0}^{\infty}\\!\\!\\!\\!{\rm
d}r\,r^{d+2l-1}\big{(}m(d-2)+d+r\partial_{r}\big{)}\,{\tilde{G}}(x)^{m}\partial^{2}{\tilde{G}}(x)\,.$
(C.11)

In this case the condition for a the integrand to be a total derivative is
$m(d-2)=2l$ but there is no correspond surface term as
$\partial^{2}{\tilde{G}}(x)$ vanishes more rapidly than $r^{-d}$ as
$r\to\infty$ and therefore

$\tau^{(l)}_{l(n-1)}(0)\big{|}_{d=d_{n}}=0\,.$ (C.12)

##  Appendix D Perturbations of Exact RG Flow Equations

We here discuss perturbations of the exact RG flow equations in (8.1) which
may be written, neglecting $C$, more succinctly in the form

$\frac{\partial}{\partial t}S=({\cal D}_{1}+{\cal
D}_{2})S+S*S-\eta\,\varphi\cdot K^{-1}\varphi\,,$ (D.1)

where

$\displaystyle S*S={}$ $\displaystyle\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K^{\prime}(p^{2})\,\frac{\delta
S}{\delta\tilde{\varphi}(p)}\,\frac{\delta S}{\delta\tilde{\varphi}(-p)}\,,$
$\displaystyle\varphi\cdot K^{-1}\varphi={}$
$\displaystyle{\textstyle\frac{1}{2}}\,\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K(p^{2})^{-1}p^{2}\tilde{\varphi}(p)\tilde{\varphi}(-p)\,,$ (D.2)

and using the definitions (8.3) save that now

${\cal D}_{1}=\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,\big{(}{\textstyle\frac{1}{2}}d+1-{\textstyle\frac{1}{2}}\eta+p\cdot\partial_{p}\big{)}\tilde{\varphi}(p)\,\frac{\delta}{\delta\tilde{\varphi}(p)}\,.$
(D.3)

For a small variation $\delta S$

$\frac{\partial}{\partial t}\delta S=({\cal D}_{1}+{\cal D}_{2}+{\cal
D}_{S})\delta S\,,\qquad{\cal D}_{S}=\frac{2}{(2\pi)^{d}}\int{\rm
d}^{d}p\,K^{\prime}(p^{2})\,\frac{\delta
S}{\delta\tilde{\varphi}(-p)}\,\frac{\delta}{\delta\tilde{\varphi}(p)}\,.$
(D.4)

At a fixed point $S\to S_{*}$ with ${\dot{S}}_{*}=0$. The critical exponents
are then defined by

$\big{(}{\cal D}_{1}+{\cal D}_{2}+{\cal D}_{S_{*}}\big{)}{\cal O}=\lambda{\cal
O}\,,$ (D.5)

for ${\cal O}$ the corresponding eigen-operator.

From (D.1) and the definitions for ${\cal D}_{1},{\cal D}_{2},{\cal
D}_{S_{*}}$ we easily obtain

$\displaystyle\big{(}{\cal D}_{1}+{\cal D}_{2}+{\cal
D}_{S_{*}}\big{)}\tilde{\varphi}(q)={}$
$\displaystyle\big{(}{\textstyle\frac{1}{2}}d+1-{\textstyle\frac{1}{2}}\eta+q\cdot\partial_{q}\big{)}\tilde{\varphi}(q)+2K^{\prime}(q^{2})\,\frac{\delta
S_{*}}{\delta\tilde{\varphi}(-q)}\,,$ $\displaystyle\big{(}{\cal D}_{1}+{\cal
D}_{2}+{\cal D}_{S_{*}}\big{)}\frac{\delta
S_{*}}{\delta\tilde{\varphi}(-q)}={}$
$\displaystyle\big{(}{\textstyle\frac{1}{2}}d-1+{\textstyle\frac{1}{2}}\eta+q\cdot\partial_{q}\big{)}\frac{\delta
S_{*}}{\delta\tilde{\varphi}(-q)}+\eta\,K(q^{2})^{-1}q^{2}\tilde{\varphi}(q)\,.$
(D.6)

Hence there are two exact solutions of (D.5)

$\displaystyle{\cal O}={}$
$\displaystyle\tilde{\varphi}(0)+\frac{2K^{\prime}(0)}{2-\eta}\,\frac{\delta
S_{*}}{\delta\tilde{\varphi}(0)}\,,\qquad\lambda={\textstyle\frac{1}{2}}d+1-{\textstyle\frac{1}{2}}\eta\,,$
(D.7a) $\displaystyle{\cal O}={}$ $\displaystyle\frac{\delta
S_{*}}{\delta\tilde{\varphi}(0)}\,,\hskip
93.89418pt\lambda={\textstyle\frac{1}{2}}d-1+{\textstyle\frac{1}{2}}\eta\,.$
(D.7b)

These are identical with the results obtained in (7.7f) and (7.7i) using the
derivative expansion.

More generally we consider solutions of (D.5) which may be expressed as

${\cal O}_{\Psi}={\cal O}_{\Psi,1}+{\cal O}_{\Psi,2}\,,$ (D.8)

where

${\cal O}_{\Psi,1}=\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}q\,\bigg{(}\Psi(q)\frac{\delta
S_{*}}{\delta\tilde{\varphi}(q)}-\frac{\delta\Psi(q)}{\delta\tilde{\varphi}(q)}\bigg{)}\,,\quad{\cal
O}_{\Psi,2}=\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}q\,K(q^{2})^{-1}q^{2}\,\Psi(q)\tilde{\varphi}(-q)\,.$ (D.9)

For operators of this form a perturbation $\epsilon{\cal O}_{\Psi}$ may be
removed by a redefinition of $\varphi$ in the basic functional integral
$Z=\int{\rm d}[\varphi]\,e^{-\varphi\cdot K^{-1}\varphi-S_{*}[\varphi]}$ so
that $Z$ is invariant. Such operators are termed redundant [17]. The operator
in (D.7b) is of this form by taking $\Psi(q)\to(2\pi)^{d}\delta^{d}(q)$.

For ${\cal O}_{\Psi,1}$ using

${\cal D}_{S}\frac{\delta
S}{\delta\tilde{\varphi}(q)}=\frac{\delta}{\delta\tilde{\varphi}(q)}(S*S)\,,\quad\Big{[}{\cal
D}_{1},\frac{\delta}{\delta\tilde{\varphi}(q)}\Big{]}=\big{(}q\cdot\partial_{q}+{\textstyle\frac{1}{2}}d-1+{\textstyle\frac{1}{2}}\eta\big{)}\frac{\delta}{\delta\tilde{\varphi}(q)}\,.$
(D.10)

we have

$\displaystyle({\cal D}_{1}+{\cal D}_{2}+{\cal D}_{S_{*}}){\cal
O}_{\Psi,1}={}$ $\displaystyle{\cal
O}_{\Psi_{1},1}+\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}q\,\Psi(q)\,\frac{\delta}{\delta\tilde{\varphi}(q)}\big{(}({\cal
D}_{1}+{\cal D}_{2})S_{*}+S_{*}*S_{*}\big{)}$ $\displaystyle\Psi_{1}(q)={}$
$\displaystyle\big{(}{\cal D}_{1}+{\cal D}_{2}+{\cal
D}_{S}-q\cdot\partial_{q}-{\textstyle\frac{1}{2}}d-1+{\textstyle\frac{1}{2}}\eta\big{)}\Psi(q)\,.$
(D.11)

For ${\cal O}_{\Psi,2}$

$\displaystyle({\cal D}_{1}+{\cal D}_{2}+{\cal D}_{S_{*}}){\cal
O}_{\Psi,2}={\cal O}_{\Psi_{2},2}+{\cal O}_{\Psi_{3},1}\,,$
$\displaystyle\Psi_{2}(q)=\Psi_{1}(q)-\eta\Psi(q)+\Psi_{3}(q)\,,\qquad\Psi_{3}(q)=2K(q^{2})^{-1}K^{\prime}(q^{2})q^{2}\Psi(q)\,.$
(D.12)

Hence using the equation for $S_{*}$

$\displaystyle({\cal D}_{1}+{\cal D}_{2}+{\cal D}_{S_{*}}){\cal
O}_{\Psi}={\cal
O}_{\Psi^{\prime}}\,,\qquad\Psi^{\prime}(q)=\Psi_{1}(q)+\Psi_{3}(q)\,.$ (D.13)

The result (D.13) demonstrates that the operators $\\{{\cal O}_{\Psi}\\}$ form
a closed subspace under RG flow near a fixed point. If ${\cal O}(q)$ is a
local operator satisfying the generalisation of (D.5)

$\big{(}{\cal D}_{1}+{\cal D}_{2}+{\cal D}_{S_{*}}\big{)}{\cal
O}(q)=\big{(}q\cdot\partial_{q}+\lambda_{\cal O}\big{)}{\cal O}(q)\,,$ (D.14)

then taking

$\Psi(q)=(q^{2})^{m}K(q^{2}){\cal O}(q)\,,$ (D.15)

gives an eigen-operator ${\cal O}_{\Psi}$ with

$\lambda=\lambda_{\cal
O}-{\textstyle\frac{1}{2}}d-1+{\textstyle\frac{1}{2}}\eta-2m\,.$ (D.16)

If $m$ were arbitrary the eigenvalue could take any value but for locality we
require $m$ to be an integer.

The operator in (D.7a) may be extended to all $q$ by considering

${\cal O}(q)=a(q^{2})\,\tilde{\varphi}(q)+b(q^{2})\,\frac{\delta
S_{*}}{\delta\tilde{\varphi}(-q)}\,,$ (D.17)

where $a(0)=1,\,b(0)=K^{\prime}(0)/(1-{\textstyle\frac{1}{2}}\eta)$. Imposing
(D.14) with $\lambda_{\cal
O}={\textstyle\frac{1}{2}}d+1-{\textstyle\frac{1}{2}}\eta$ gives
$a^{\prime}(x)={\textstyle\frac{1}{2}}\eta K(x)^{-1}b(x)$,
$K^{\prime}(x)a(x)-xb^{\prime}(x)=(1-{\textstyle\frac{1}{2}}\eta)b(x)$ which
have the solutions, assuming $\eta<2$,

$a(x)=\frac{1+xb(x)}{K(x)}\,,\qquad
b(x)=x^{\frac{1}{2}\eta-1}K(x)\int_{0}^{x}\\!u^{-\frac{1}{2}\eta}\,\frac{K^{\prime}(u)}{K(u)^{2}}\
{\rm d}u\,.$ (D.18)

When $\eta=0$, $a(x)=1,\ b(x)=(K(x)-1)/x$.

With these results and using (D.17) and (D.15), with $m=0$, in (D.9) and (D.8)
gives an exactly marginal eigen-operator with $\lambda=0$. Integrating these
marginal deformations generates solutions $S_{*}[\varphi,a]$ for some
parameter $a$ representing a line of equivalent fixed points. The various
formulae may be verified with the Gaussian solution

$S_{*}[\varphi,a]=-{\textstyle\frac{1}{2}}\,\frac{1}{(2\pi)^{d}}\int{\rm
d}^{d}p\,\frac{p^{2}}{K(p^{2})+a}\,\tilde{\varphi}(p)\tilde{\varphi}(-p)\,.$
(D.19)

More generally assuming the eigen-operators corresponding to $\lambda_{k,1}$
${\cal O}_{k,1}$ may be extended to ${\cal O}_{k,1}(q)$ satisfying (D.14) then
this construction determines ${\cal O}_{k+2n-1,2}$ so that from (D.16)

$\lambda_{k,2}=\lambda_{k-2n+1,1}-{\textstyle\frac{1}{2}}d-1+{\textstyle\frac{1}{2}}\eta\,.$
(D.20)

This is compatible with the ${\rm O}(\varepsilon)$ perturbative results and
also the modified derivative expansion calculations described here.

Acknowledgements

J. O’D. is grateful to C. Bervillier and T. Morris for useful correspondence.

## References

  * [1] K.G. Wilson, “Renormalization Group and Critical Phenomena. 1. Renormalization Group and the Kadanoff Scaling Picture,” Phys. Rev. B4 (1971) 3174.
  * [2] F.J. Wegner and A. Houghton, “Renormalization Group Equation for Critical Phenomena,” Phys. Rev. A8 (1972) 401.
  * [3] K.G. Wilson and J.B. Kogut, “The Renormalization Group and the Epsilon Expansion,” Phys. Rept. 12 (1974) 75.
  * [4] J. Polchinski, “Renormalization and Effective Lagrangians,” Nucl. Phys. B231 (1984) 269.
  * [5] C. Bagnuls and C. Bervillier, “Exact Renormalization Group Equations. An Introductory Review,” Phys. Rep. 348 (2001) 91, hep-th/0002034.
  * [6] J. Berges, N. Tetradis and C. Wetterich, “Nonperturbative renormalization flow in quantum field theory and statistical physics,” Phys. Rept. 363 (2002) 223, hep-ph/0005122.
  * [7] J.M. Pawlowski, “Aspects of the Functional Renormalization Group,” hep-th/0512261.
  * [8] B. Delamotte, “An introduction to the nonperturbative renormalization group,” cond-mat/0702365.
  * [9] B. Delamotte, D. Mouhanna and M. Tissier, “Nonperturbative renormalization group approach to frustrated magnets,” Phys. Rev. B69 (2004) 134413, cond-mat/0309101.
  * [10] V.F. Muller, “Perturbative renormalization by flow equations,” Rev. Math. Phys. 15 (2003) 491, hep-th/0208211.
  * [11] A. Hasenfratz and P. Hasenfratz, “Renormalization Group Study of Scalar Field Theories,” Nucl. Phys. B270 (1986) 687.
  * [12] D.F. Litim, “Universality and the Renormalisation Group,” JHEP, 0507 (2005) 005, hep-th/0503096.
  * [13] T.R. Morris, “Equivalence of Local Potential Approximations,” JHEP, JHEP 0507 (2005) 027, hep-th/0503161.
  * [14] K.G. Wilson and M.E. Fisher, “Critical Exponents in 3.99 Dimensions,” Phys. Rev. Lett. 28 (1972) 240.
  * [15] H. Kleinert and V. Schulte-Frohlinde, “Critical Properties of $\phi^{4}$-Theories,” World Scientific, Singapore (2001).
  * [16] K. Pinn, A. Pordt and C. Wieczerkowski, “Computation of Hierarchical Renormalization-Group Fixed Points and Their $\varepsilon$-Expansions,” J. Statistical Physics 77 (1994) 977, hep-lat/9402020.
  * [17] F.J. Wegner in “Phase Transitions and Critical Phenomena, Vol. 6,” ed. C. Domb and M.S. Green, Academic Press, New York (1976).
  * [18] I. Jack and H. Osborn, “Two-loop Background Field Calculations for Arbitrary Background Fields,” Nucl. Phys. B207 (1982) 474.
  * [19] C. Itzykson and J.M. Drouffe, _Statistical Field Theory_ , Vol.1, appendix Chapter 5, (CUP, Cambridge, 1989).
  * [20] G.J. Huish and D.J. Toms, “Renormalization of Interacting Scalar Field Theory in Three-dimensional Curved Spacetime,” Phys. Rev. D49 (1994) 6767.
  * [21] G. Felder, “Renormalization Group in the Local Potential Approximation,” Comm. Math. Phys. 111 (1987) 101.
  * [22] P.C. Lima, “Renormalization Fixed Points in the Local Potential Approximation for $d\geq 3$,” Comm. Math. Phys. 170 (1995) 529.
  * [23] C.S.F. Harvey-Fros, “The Local Potential Approximation of the Renormalization Group,” University of Southampton Ph.D. Thesis (1999), hep-th/0108018.
  * [24] J.F. Nicholl, T.S. Chang and H.E. Stanley, “Approximate Renormalization Group Based on the Wegner-Houghton Differential Generator,” Phys. Rev. Lett. 33 (1974) 540, (E) 1525.
  * [25] P.E. Haagensen et al, “Gradient Flows from an Approximation to the Exact Renormalization Group,” Phys. Lett. B323 (1994) 330, hep-th/9310032.
  * [26] T.R. Morris, “Derivative Expansion of the Exact Renormalization Group,” Phys. Lett. B329 (1994) 241.
  * [27] C. Bervillier, “Status of the derivative expansion in the ERGE,” talk at the 3rd International Conference on the Exact Renormalization Group, Lefkada, Greece (2006), http://www.cc.uoa.gr/%7Epapost/Bervillier.pdf.
  * [28] R.D. Ball, P.E. Haagensen, J.I. Latorre and E. Moreno, “Scheme Independence and the Exact Renormalization Group,” Phys. Lett. B347 (1995) 80, hep-th/9411122.
  * [29] J. Commellas, “Polchinski equation, reparameterization invariance and the derivative expansion,” Nucl. Phys. B509 (1999) 662, hep-th/9705129.
  * [30] G.R. Golner, “Nonperturbative Renormalization Group Calculations for Continuum Spin Systems,” Phys. Rev. B33 (1986) 7863.
  * [31] E.K. Riedel, G.R. Golner and K.E. Newman, “Scaling Field Representation of Wilson’s Exact Renormalization-Group Equation,” Ann. Phys. 161 (1985) 178.
  * [32] K.E. Newman and E.K. Riedel, “Critical exponents by the scaling field method: The isotropic $N$-vector model in three dimensions,” Phys. Rev B30 (1984) 6615.
  * [33] H. Georgi, “Effective Field Theory,” Ann. Rev. Nucl. Part. Sci. 43 (1993) 209.
  * [34] T.R. Morris, “Properties of derivative expansion approximations to the renormalization group,” Int. J. Mod. Phys. B12 (1998) 1343, hep-th/9610012.
  * [35] J. Generowicz, C. Harvey-Fros and T.R. Morris, “C Function Representation of the Local Potential Approximation,” Phys. Lett. B407 (1997) 27, hep-th/9705088.
  * [36] G. Zumbach, “The Renormalization group in the local potential approximation and its applications to the O(n) model,” Nucl. Phys. B413 (1994) 771.
  * [37] A.I. Davydychev and J.B. Tausk, “A magic connection between massive and massless diagrams,” Phys. Rev. D53 (1996) 7381, hep-ph/9504431.