Properties of a proposed background independent exact renormalization group

Vlad-Mihai Mandric and Tim R. Morris

STAG Research Centre & Department of Physics and Astronomy,
University of Southampton, Highfield, Southampton, SO17 1BJ, U.K.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
We explore the properties of a recently proposed background independent exact
renormalization group approach to gauge theories and gravity. In the process
we also develop the machinery needed to study it rigorously. The proposal
comes with some advantages. It preserves gauge invariance manifestly, avoids
introducing unphysical fields, such as ghosts and Pauli-Villars fields, and
does not require gauge-fixing. However, we show that in the simple case of
$SU(N)$ Yang-Mills it does not completely regularise the longitudinal part of
vertex functions already at one loop, invalidating certain methods for
extracting universal components. Moreover we demonstrate a kind of no-go
theorem: within the proposed structure, whatever choice is made for
covariantisation and cutoff profiles, the two-point vertex flow equation at
one loop cannot be both transverse, as required by gauge invariance, and fully
regularised.

###### Contents

  1. 1 Introduction
  2. 2 A proposal for a background independent exact RG
    1. 2.1 Notation and basic ingredients for regularisation
    2. 2.2 The main idea
    3. 2.3 Regularisation structure
    4. 2.4 Flow equation
  3. 3 Regularisation failure: a sketch
  4. 4 Perturbative expansion
    1. 4.1 Loop expansion
    2. 4.2 Vertex expansion
      1. 4.2.1 Action vertex properties
      2. 4.2.2 Kernel vertex properties
      3. 4.2.3 Two-point action vertices
      4. 4.2.4 Zero-point kernel functions
      5. 4.2.5 Higher-point vertices
      6. 4.2.6 Large momentum behaviour
  5. 5 Regularisation of higher point classical vertices
  6. 6 One loop beta function
    1. 6.1 Classical flow equations
    2. 6.2 Trace terms
    3. 6.3 Two-point vertex at zeroth order in momentum
    4. 6.4 Two-point vertex at second order in momentum
  7. 7 Summary and Conclusions
  8. A Interleave identities for compound kernels
  9. B Why preregularisation is necessary

##  1 Introduction

Understanding the Wilsonian renormalization group (RG) structure of quantum
gravity is surely of importance, see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], and
central to this is the rôle of diffeomorphism invariance. One approach is to
try to generalise the exact RG [11, 12] to gravity, in such a way that it is
manifestly diffeomorphism invariant. On the one hand this would allow
computations to be done whilst keeping exact diffeomorphism invariance at
every stage, i.e. without gauge fixing, and on the other hand, these
computations would be background independent, that is performed without first
choosing the space-time manifold and background metric.

At the classical level this can be done [13]. However in order to compute
quantum corrections, extra ultraviolet regularisation has to be incorporated
into the exact RG so that the integration is properly cut off in some
diffeomorphism invariant way at the effective cutoff scale $\Lambda$. For the
simpler case of gauge invariance in $SU(N)$ Yang-Mills theory this problem was
solved by incorporating gauge invariant PV (Pauli-Villars) fields arising from
a spontaneously broken $SU(N|N)$ gauge theory [14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. It is not clear
whether one can generalise such a scheme to gravity however, in particular it
is not clear what should play the rôle of $SU(N|N)$, although a kind of
supergravity has been suggested in [37].

Recently a different approach to the problem of regularisation has been
pursued [38]. Explicit PV fields are avoided and instead replaced by
functional determinants which, if constructed as squares of simpler
determinants, can be shown to work in the framework of standard perturbation
theory [39, 40, 41]. Furthermore a geometric approach is followed where the
determinants can be regarded as defining a regularised volume element on the
orbit space of the gauge theory [41]. The flow equation is then formulated in
a way that is manifestly invariant under field redefinitions, and applies
equally well to both gravity and gauge theory [38].

Although this new proposal has elegant features, it is not immediately evident
that the regularisation is successfully implemented in the flow equation once
we delve into the details of the relevant Feynman diagrams, first at one loop
and then also at higher loops. In this paper we put the proposal to the test
by constructing explicit expressions for the relevant vertices and then
carefully analyse their UV (ultraviolet) behaviour, as a function of loop
momentum, in the form that they appear in quantum corrections. For this
purpose it is sufficient to focus on Yang-Mills since if the regularisation
fails for Yang-Mills it most certainly fails for quantum gravity (given the
latter’s poor UV behaviour). Working with Yang-Mills also means we can take
over methods used for these investigations in the earlier successful
construction [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36]. As we will see, the proposal of ref. [38]
unfortunately fails to fully regularise already at one loop, but in a rather
subtle way, which in particular invalidates powerful techniques previously
used to extract universal information [23, 50].

The paper is organised as follows. In section 2 we review the proposal, and in
the process, set out our notation and conventions. In sec. 3 we sketch why the
regularisation can fail. In section 4 we build the machinery needed to
rigorously test the structure of the renormalization scheme at the
perturbative level. In sec. 5 we confirm that the higher point classical
vertices incorporate the assumed regularisation. Then in sec. 6 we apply the
techniques extensively to an analysis of the simplest one-loop correction
namely that for the effective action two-point vertex. We show that the
regularisation is sufficient for the momentum independent part, giving a
vanishing result as it should by gauge invariance, only if the
$\Lambda$-derivative of quadratically divergent constant part is discarded.
(Recall that $\Lambda$ is the effective cutoff scale.) The part that is second
order in momentum ought to give the one-loop beta function, if properly
regulated. However we show that the result cannot both be completely
regularised and transverse. It can be taken to be transverse only if the
$\Lambda$-derivative of a linearly divergent part is discarded. Since this
holds for all choices of covariantisation and cutoff profiles, it points to
some inherent limitations in the structure of the proposed flow equation. In
sec. 7 we summarise and draw our conclusions.

##  2 A proposal for a background independent exact RG

###  2.1 Notation and basic ingredients for regularisation

The flow equation proposed in ref. [38] incorporates a geometric approach to
the quantisation of gauge theories [42, 43] and as such it is useful to use
DeWitt notation. On the other hand, as we will see, a detailed understanding
of the UV properties can only be reached by working with explicit expressions
for the vertices. In the case of Yang-Mills theories, the flow equations then
take their simplest form if we regard the gauge fields as valued in the Lie
algebra, i.e. contracted into the generators [14, 15]. We will therefore work
with both notations as appropriate.

We work in Euclidean signature and, therefore, we will keep all gauge group
indices as superscripts and Lorentz indices as subscripts for convenience. For
position and momentum space integrals we introduce the shorthand

$\displaystyle\int d^{D}\\!x$ $\displaystyle=\int_{x}\,,$
$\displaystyle\int\frac{d^{D}\\!p}{(2\pi)^{D}}$ $\displaystyle=\int_{p}\,,$
(2.1)

respectively, where $D$ is the number of dimensions we are working in. We will
frequently switch between position and momentum space representations
throughout the paper and this will help us keep everything clean and concise.
For similar reasons we will adopt the following convention:

$\delta(p)\equiv(2\pi)^{D}\delta^{(D)}(p)\,,$ (2.2)

where $\delta^{(D)}(p)$ is the standard $D$-dimensional Dirac delta function.
Our convention for Fourier transforms is then

$\phi(p)=\int_{x}\\!\phi(x)\,e^{-ipx}\,.$ (2.3)

We will use the DeWitt compact notation and its explicit representation
interchangeably, so for example

$\phi^{a}J_{a}\equiv\sum_{a}\int_{x}A_{\mu}^{a}(x)J_{\mu}^{a}(x)\,.$ (2.4)

Last, but not least, we will adopt the following shorthands for momenta and
scalar functions of momenta, respectively:

$\displaystyle\frac{p}{\Lambda}$ $\displaystyle\equiv\tilde{p}\,,$ (2.5)
$\displaystyle K\bigg{(}\frac{p^{2}}{\Lambda^{2}}\bigg{)}$
$\displaystyle\equiv K_{p}\,,$ (2.6)

where $\Lambda$ is the effective cutoff energy scale.

We work with the gauge group SU($N$). We use DeWitt Latin indices from the
start of the alphabet to label gauge fields, thus $\phi^{a}$, but as already
mentioned, when we need more explicit expressions it will be convenient to
regard the gauge fields $A_{\mu}(x)$ as contracted into the generators:

$\phi^{a}\equiv A_{\mu}(x)=A_{\mu}^{a}(x)T^{a}\,.$ (2.7)

With appropriate definitions for the vertex functions in either language, the
expressions will of course be equal, and the two representations are thus
equivalent. The generators $T^{a}$ are taken to be hermitian, in the
fundamental representation, and orthonormalized as
$tr(T^{a}T^{b})=\frac{1}{2}\delta^{ab}$. A second set of DeWitt Greek indices
from the start of the alphabet is used to label gauge parameters; the map from
the two languages is thus:

$\epsilon^{\alpha}\equiv\omega(x)=\omega^{a}(x)T^{a}\,.$ (2.8)

If we adopt the geometric approach of [42, 43] to the quantisation of gauge
theories, we regard the fields $\phi^{a}$ as coordinates on an infinite
dimensional ‘manifold’ $\Phi$, the space of all possible field configurations.
This has the structure of a fibre bundle, and the fibres are the gauge orbits
$\mathcal{G}$. However, all the physics happens on the quotient space
$\Phi/\mathcal{G}$, where each point belongs to a unique equivalence class
$\\{\phi\\}$, which encompasses all the possible field configurations related
by gauge transformations. The generators of the gauge transformation are of
the following form:

$K_{\alpha}^{a}[\phi]\equiv D_{\mu}\delta(x-y)\,,$ (2.9)

where the covariant derivative is given by
$D_{\mu}:=\partial_{\mu}-iA_{\mu}(x)$, and $A_{\mu}$ is understood to act by
commutation. This means that gauge transformations can be written equivalently
as

$\displaystyle\delta\phi^{a}$
$\displaystyle=K_{\alpha}^{a}[\phi]\>\epsilon^{\alpha}\,,$ (2.10)
$\displaystyle\delta A_{\mu}(x)$
$\displaystyle=D_{\mu}\cdot\omega:=[D_{\mu},\omega]\,.$ (2.11)

The field strength is given by $F_{\mu\nu}:=i[D_{\mu},D_{\nu}]$. We write the
SU($N$) Yang-Mills action in the following way:

$I[\phi]\equiv\frac{1}{4g^{2}}\int_{x}(F^{a}_{\mu\nu})^{2}=\frac{1}{2g^{2}}\>\text{tr}\int_{x}F_{\mu\nu}^{2}\,.$
(2.12)

In momentum space this becomes:

$I[\phi]\equiv\frac{1}{2g^{2}}\int_{p}I_{\mu_{1}\mu_{2}}(p)\>\text{tr}\Big{(}A_{\mu_{1}}(p)A_{\mu_{2}}(-p)\Big{)}\>+\\\
+\frac{1}{3g^{2}}\int_{p_{1}\ldots p_{3}}I_{\mu_{1}\ldots\mu_{3}}(p_{1}\ldots
p_{3})\>\text{tr}\Big{(}A_{\mu_{1}}(p_{1})\ldots
A_{\mu_{3}}(p_{3})\Big{)}\>\delta(p_{1}+p_{2}+p_{3})\>+\\\
+\frac{1}{4g^{2}}\int_{p_{1}\ldots p_{4}}I_{\mu_{1}\dots\mu_{4}}(p_{1}\ldots
p_{4})\>\text{tr}\Big{(}A_{\mu_{1}}(p_{1})\ldots
A_{\mu_{4}}(p_{4})\Big{)}\>\delta(p_{1}+\cdots+p_{4})\,,$ (2.13)

where the vertex functions $I^{(2)},I^{(3)}$ and $I^{(4)}$, are given by:

$\displaystyle I_{\mu_{1}\mu_{2}}(p)$
$\displaystyle=2\,\square_{\mu_{1}\mu_{2}}(p):=2(p^{2}\delta_{\mu_{1}\mu_{2}}-p_{\mu_{1}}p_{\mu_{2}})\,,$
(2.14) $\displaystyle I_{\mu_{1}\ldots\mu_{3}}(p_{1}\ldots p_{3})$
$\displaystyle=2({p_{3}}_{\mu_{2}}\delta_{\mu_{1}\mu_{3}}-{p_{3}}_{\mu_{1}}\delta_{\mu_{2}\mu_{3}})+\text{cycles}\,,$
(2.15) $\displaystyle I_{\mu_{1}\dots\mu_{4}}(p_{1}\ldots p_{4})$
$\displaystyle=(\delta_{\mu_{1}\mu_{4}}\delta_{\mu_{2}\mu_{3}}-\delta_{\mu_{1}\mu_{3}}\delta_{\mu_{2}\mu_{4}})+\text{cycles}\,.$
(2.16)

We note that ‘cycles’ stands for all cyclic permutations of momenta
${p_{1}}_{\mu_{1}}\rightarrow{p_{2}}_{\mu_{2}}\rightarrow\ldots\rightarrow{p_{1}}_{\mu_{1}}$.
The exact preservation of gauge invariance at all stages during the flow,
together with the choice to rescale the coupling in front of the integral in
(2.12), ensures that $A_{\mu}$ cannot run: any wavefunction renormalization by
$Z\neq 1$, $A_{\mu}^{R}=Z^{-1/2}A_{\mu}$, would break gauge invariance:
$\delta A_{\mu}^{R}=Z^{-1/2}\partial_{\mu}\omega-i[A_{\mu}^{R},\omega]$ [44,
14, 15]. Therefore the coupling $g=g(\Lambda)$ is the only quantity that runs.

Functional derivatives are written as

$S,_{a}=\frac{\delta S}{\delta\phi^{a}}\equiv\frac{\delta S}{\delta
A_{\mu}(x)}\,,$ (2.17)

where $\delta/\delta A_{\mu}(x):=2\>T^{a}\>\delta/\delta A_{\mu}^{a}(x)$. The
properties of the latter can be understood as follows [15, 16]. For
convenience here we temporarily write the gauge fields $A_{\mu}(x)$ simply as
$A$, suppressing spacetime dependence and Lorentz indices. Given a well-
behaved gauge invariant function $f(A)$ such that $\delta
f(A)=\text{tr}(\delta A\,X)$, for some $X$, we can exploit the $SU(N)$
completeness relation to see that:

$\frac{\delta f}{\delta A}=X-\frac{1}{N}\text{tr}X\,,$ (2.18)

effectively isolating $X$. This property will be crucial in our endeavour
later on. Following the same reasoning, we have two more useful relations:

$\text{tr}\,Y\frac{\delta f}{\delta
A}=\text{tr}YX-\frac{1}{N}\text{tr}Y\text{tr}X\,,$ (2.19)

and,

$\text{tr}\frac{\delta}{\delta
A}W=\text{tr}Y\text{tr}Z-\frac{1}{N}\text{tr}YZ\,,$ (2.20)

for some $W=T^{a}W^{a}$ such that $\delta W=Y\,\delta A\,Z$. The functional
derivatives of $I[\phi]$ are (in position space):

$\displaystyle\mathcal{E}_{a}=I,_{a}$
$\displaystyle\equiv\frac{1}{g^{2}}D_{\alpha}F_{\mu\alpha}\,,$ (2.21)
$\displaystyle I,_{ab}$
$\displaystyle\equiv\frac{1}{g^{2}}\Big{(}iF_{\mu\nu}+D_{\nu}D_{\mu}-\delta_{\mu\nu}D^{2}\Big{)}\delta(x-y)\,.$
(2.22)

Ultra-local metrics are introduced on $\Phi$ and $\mathcal{G}$, respectively,
such that their corresponding line element (e.g.
$\delta\phi^{a}\gamma_{ab}\delta\phi^{b}$) is made dimensionless by using the
appropriate power of the effective scale $\Lambda$ [38]. They allow us to
raise or lower DeWitt indices, e.g. $\phi^{a}=\gamma^{ab}\phi_{b}$. At this
abstract level the formulation can be developed in a way that applies equally
well to gravity, and indeed ref. [38] treats gravity as a special case. For
gravity however, $\gamma_{ab}$ and $\eta^{\alpha\beta}$ necessarily depend on
$\phi$ (which in this case is the spacetime metric). Then to keep equations
covariant on $\Phi$ requires the use of a connection $\Gamma^{c}_{ab}$ in
function space and covariant derivatives, e.g. replacing $I_{,ab}$ above with
$I_{;ab}=I_{,ab}-\Gamma^{c}_{ab}I_{,c}$. Since we will treat only Yang-Mills
theory in this paper, where one can take the metrics to be $\phi$ independent:

$\displaystyle\gamma_{ab}$
$\displaystyle=\frac{\Lambda^{2}}{g^{2}}\delta_{\mu\nu}\delta(x-y)\,,$
$\displaystyle\gamma^{ab}$
$\displaystyle=\frac{g^{2}}{\Lambda^{2}}\delta_{\mu\nu}\delta(x-y)\,,$ (2.23)
$\displaystyle\eta_{\alpha\beta}$
$\displaystyle=\frac{\Lambda^{4}}{g^{2}}\delta(x-y)\,,$
$\displaystyle\eta^{\alpha\beta}$
$\displaystyle=\frac{g^{2}}{\Lambda^{4}}\delta(x-y)\,,$ (2.24)

we do not need this extra complication and thus we work only in the flat
function space limit.

Again following [38], we will introduce higher covariant derivatives into the
effective action $S$ via functions of Laplace-like differential operators,
which we list below:

$\displaystyle{\Delta^{a}}_{b}$
$\displaystyle=\gamma^{ac}I,_{cb}+K^{a}_{\alpha}\eta^{\alpha\beta}K^{c}_{\beta}\gamma_{cb}$
$\displaystyle\equiv$ $\displaystyle\Delta_{\mu\nu}$
$\displaystyle=\Lambda^{-2}\left(2iF_{\mu\nu}-\delta_{\mu\nu}D^{2}\right)\delta(x-y)\,,$
(2.25) $\displaystyle{\big{(}\Delta_{\parallel}\big{)}^{\alpha}}_{\beta}$
$\displaystyle=\eta^{\alpha\delta}K^{a}_{\delta}\gamma_{ab}K^{b}_{\beta}$
$\displaystyle\equiv$ $\displaystyle\Delta_{\parallel}$
$\displaystyle=-\Lambda^{-2}D^{2}\delta(x-y)\,.$ (2.26)

For two (matrix valued) functions $J_{\mu}(x)$ and $J_{\nu}(x)$ and a momentum
kernel $K_{\mu\nu}(p^{2}/\Lambda^{2})$ we introduce the shorthand:

$\displaystyle J_{\mu}\cdot K_{\mu\nu}\cdot J_{\nu}$
$\displaystyle:=2\,\text{tr}\int_{x}J_{\mu}(x)K_{\mu\nu}\left(-\partial^{2}/\Lambda^{2}\right)J_{\nu}(x)\,,$
(2.27) $\displaystyle
K_{\mu\nu}\left(-\partial^{2}/\Lambda^{2}\right)\delta(x-y)$
$\displaystyle=\int_{p}K_{\mu\nu}(p^{2}/\Lambda^{2})\,e^{ip(x-y)}\,.$ (2.28)

Gauge invariant quantities have all DeWitt indices contracted (i.e. behaves as
a scalar on $\Phi$ and $\mathcal{G}$). One gauge invariant object that plays a
fundamental role in the regularisation is the action $A$ :

$A=I+\frac{1}{2}I,_{a}\big{[}X(\Delta)\big{]}^{ab}I,_{b}\,.$ (2.29)

It is _quasi-local_ , i.e. has an all orders expansion in momenta [15, 16].
The kernel $X(z)$ is given by:

$X(z)=\frac{1-c(z)}{zc(z)}\,,$ (2.30)

where $c(z)$ is a smooth UV cutoff profile such that $c(0)=1$, and
$c(p^{2}/\Lambda^{2})\rightarrow 0$ as $p^{2}/\Lambda^{2}\rightarrow\infty$.

The anomalous dimension is
$\eta:=-g^{2}\Lambda\partial_{\Lambda}(1/g^{2})=\frac{2}{g}\Lambda\partial_{\Lambda}g$.
Given that $\eta\propto\hbar$ (i.e. actually $\hbar g^{2}$), we can write down
a loop expansion for it:

$\eta=\eta_{1}g^{2}+\eta_{2}g^{4}+\cdots\,,$ (2.31)

and thus for $\beta:=\Lambda\partial_{\Lambda}g$ as well
($\eta_{i}=2\beta_{i}$):

$\beta=\beta_{1}g^{3}+\beta_{2}g^{5}+\cdots\,.$ (2.32)

It will prove useful to analyse expressions using the following projection
operators:

$\displaystyle{\Pi_{T}}^{a}{}_{b}$
$\displaystyle:=\delta^{a}{}_{b}-{\Pi_{L}}^{a}{}_{b}$ $\displaystyle\equiv$
$\displaystyle\Pi^{T}_{\mu\nu}$
$\displaystyle:=\delta_{\mu\nu}-\Pi^{L}_{\mu\nu}\,,$ (2.33)
$\displaystyle{\Pi_{L}}^{a}{}_{b}$
$\displaystyle:=K^{a}_{\alpha}\left(\Delta_{\parallel}^{-1}\right)^{\alpha\beta}K^{c}_{\beta}\gamma_{bc}$
$\displaystyle\equiv$ $\displaystyle\Pi^{L}_{\mu\nu}$
$\displaystyle:=D_{\mu}\Big{(}D^{2}\Big{)}^{-1}D_{\nu}\,,$ (2.34)

although they are not directly involved in the regularisation scheme because
of their non-local nature.

###  2.2 The main idea

Before delving into details, it is instructive to outline the main steps
undertaken in ref.[38] to construct a flow equation for the effective action
$S$. Essentially this is a three step procedure, where the first two steps
represent a geometric reformulation of Slavnov’s higher derivative
regularisation scheme [40, 39], while in the last step the freedom one has to
design an exact RG [45, 14, 46, 47] is exploited to avoid fixing the gauge and
to manifestly preserve gauge invariance at all stages during the flow.

First of all we need to incorporate higher covariant derivatives, $S_{UV}$,
into the effective action $S$, to improve the UV behaviour by modifying the
propagators:111As it stands, this equation is almost devoid of meaning since
$\mathcal{S}$ could modify or even cancel terms in $S_{UV}$. At the classical
level, we give it meaning by insisting that $\mathcal{S}^{0}$ contains only
three-point and higher vertices: $\mathcal{S}^{0(n<3)}=0$. , which, in turn,
fixes the on-shell two-point classical effective action. This choice will
allow us to compute classical vertices in terms of quantities already known
[23].

$S=I+S_{UV}+\mathcal{S}\,,$ (2.35)

where $\mathcal{S}$ stands for all the other interactions in the effective
action that are generated by the flow. However higher covariant derivatives do
not regularise all one-loop divergences, and therefore further regularisation
is required [40, 39].

The second step addresses this issue by introducing PV regularisation, which
however is introduced directly through functional determinants in such a way
that these act as _quasi-local_ metrics for $\Phi$, and $\mathcal{G}$; they
are called $G_{ab}[\phi]$ and $H_{\alpha\beta}[\phi]$ respectively. Now, the
partition function $\mathcal{Z}$ takes in general the following form:

$\mathcal{Z}=\int_{\Phi/\mathcal{G}}d\\{\phi\\}\>M[\\{\phi\\}]\>e^{-S[\\{\phi\\}]}\,,$
(2.36)

where $M[\\{\phi\\}]$ is the volume element on the quotient space
$\Phi/\mathcal{G}$. Factoring the integration measure using the functionals
mentioned above,

$d\phi\sqrt{det\>G[\phi]}=d\\{\phi\\}M[\\{\phi\\}]\>d\xi\sqrt{det\>H[\phi]}\,,$
(2.37)

we thus rewrite the partition function (Tr stands for a functional trace of
the kernels):

$\mathcal{Z}\propto\int_{\Phi}d\phi\>\frac{\sqrt{det\>G[\phi]}}{\sqrt{det\>H[\phi]}}\>e^{-S[\phi]}=\int_{\Phi}d\phi\>e^{-S[\phi]+\frac{1}{2}\text{Tr}\ln
G_{ab}[\phi]-\frac{1}{2}\text{Tr}\ln H_{\alpha\beta}[\phi]}$ (2.38)

(up to some actually-infinite constant that we are free to discard without
altering the physics).

The idea then is that $G$ is chosen to cancel one-loop UV divergences arising
from $S$. In the infrared (IR), $G$ and $H$ are both order $\Lambda^{2}$,
corresponding to PV masses of order the cutoff. $G$ thus has a longitudinal
part which will contribute divergences, but $H$ can be used to cancel them.
Finally by choosing $G$ and $H$ each such that they factor into other
operators that are less divergent, we can avoid so-called ‘overlapping’
divergences, ones that correspond at one loop to having external PV legs [48,
40, 49]. Thus at this stage, PV regularisation will require the following:

  * (PV1)

At high momentum the Hessian for fluctuations matches that of $G$ in the
transverse space:

${\Pi_{T}}^{c}{}_{a}\,G_{cd}\,{\Pi_{T}}^{d}{}_{b}\thicksim S_{,ab}\,.$ (2.39)

  * (PV2)

Similarly $H$ matches the longitudinal part of $G$:

$K^{a}_{\alpha}G_{ab}K^{b}_{\beta}\thicksim H_{\alpha\beta}\,.$ (2.40)

  * (PV3)

$G_{ab}[\phi]$ and $H_{\alpha\beta}[\phi]$ must be chosen so that they factor
into operators that are less divergent.

Slavnov’s original scheme does not avoid fixing the gauge [40, 39]. In
ref.[38] gauge fixing is completely avoided by exploiting the freedom [45, 14,
15, 46, 47, 50, 51] one has to design an exact RG [11, 12] flow for the
effective action $S$. This leads to the final flow equation:

$\Lambda D_{\Lambda}S=\frac{1}{2}\text{Tr}\big{[}G^{-1}\Lambda
D_{\Lambda}G-H^{-1}\Lambda D_{\Lambda}H\big{]}\,,$ (2.41)

where $\Lambda D_{\Lambda}:=\Lambda\partial_{\Lambda}+\mathcal{L}_{\Psi}$ is
the total RG derivative, $\mathcal{L}_{\Psi}$ being a Lie derivative on $\Phi$
associated to the change of variables
$\phi^{a}\mapsto\phi^{a}-\delta\Lambda\Psi^{a}$. The total RG derivative acts
in the following way:

$\displaystyle\Lambda D_{\Lambda}S$
$\displaystyle=\Lambda\partial_{\Lambda}S+\Psi^{a}S,_{a}\,,$ (2.42)
$\displaystyle\Lambda D_{\Lambda}H_{\alpha\beta}$
$\displaystyle=\Lambda\partial_{\Lambda}H_{\alpha\beta}+\Psi^{a}H_{\alpha\beta,a}\,,$
(2.43) $\displaystyle\Lambda D_{\Lambda}G_{ab}$
$\displaystyle=\Lambda\partial_{\Lambda}G_{ab}+\Psi^{c}G_{ab,c}+{\Psi^{c}}_{,a}G_{bc}+{\Psi^{c}}_{,b}G_{ac}\>.$
(2.44)

The strategy is that, altogether, the higher covariant derivatives and PV
field determinants should provide a regularised ‘kinetic’ $I+S_{UV}\in S$
part, while a careful choice of $\Psi^{a}$ is meant to ensure that the exact
RG then generates only an $\mathcal{S}\in S$ part that is free of divergences.

###  2.3 Regularisation structure

The method proposed in ref.[38] to construct the regularisation scheme, i.e.
PV operators and higher covariant derivatives, proceeds as follows. First of
all, we need to construct $G_{ab}[\phi]$ and $H_{\alpha\beta}[\phi]$ by taking
into account the requirements outlined in section 2.2. To satisfy (PV3) they
are set to

$\displaystyle G_{ab}$
$\displaystyle={\big{(}C^{-1}\big{)}^{c}}_{a}\>\gamma_{cd}\>{\big{(}C^{-1}\big{)}^{d}}_{b}\,,$
(2.45) $\displaystyle H_{\alpha\beta}$
$\displaystyle={B^{\gamma}}_{\alpha}\>b_{\gamma\delta}\>{B^{\delta}}_{\beta}\,,$
(2.46)

where we require $B^{\alpha}{}_{\beta}$, $b^{\alpha}{}_{\beta}$, and
$\big{(}C^{-1}\big{)}^{a}{}_{b}$ to be constructed such that
$G_{ab}\rightarrow\gamma_{ab}$ and
$H_{\alpha\beta}\rightarrow\eta_{\alpha\beta}$ as $\Lambda\rightarrow\infty$.
This latter requirement ensures that in the continuum limit the _quasi-local_
and _ultra-local_ metrics coincide, and hence encode the same physics [35].
Furthermore, the condition (PV1) implies that $C^{2}S^{(2)}\thicksim 1$, which
essentially means that all divergences coming from $C^{-1}$ vertices should be
regulated by contributions in the effective action involving the effective
propagator. The remaining condition (PV2) together with (2.45), and (2.46)
relates the PV operators to one another in the high momentum limit:

$\displaystyle\big{(}C^{-1}\big{)}^{a}{}_{b}K^{b}_{\alpha}$
$\displaystyle\thicksim K^{a}_{\beta}B^{\beta}{}_{\alpha}\,,$ (2.47)
$\displaystyle b_{\alpha\beta}$
$\displaystyle\thicksim\big{(}\Delta_{\parallel}\big{)}_{\alpha\beta}=K^{a}_{\alpha}\gamma_{ab}K^{b}_{\beta}\,.$
(2.48)

The second part of the regularisation scheme amounts to introducing higher
covariant derivatives into the effective action $S$. Given that $G$ is bi-
linear in $C^{-1}$ we will keep the same structure for $S_{UV}$ [40],

$S_{UV}=\frac{1}{2}A_{,a}\gamma^{ab}A_{,b}\,,$ (2.49)

and thus write the effective action $S$ as

$S=I+\mathcal{S}+\frac{1}{2}A_{,a}\gamma^{ab}A_{,b}\,.$ (2.50)

Here it is instructive to pause and make some comments on the choice of using
$A$ and on the strategy for analysis of the regularisation and flow equation.
If we functionally differentiate (2.29) twice we get:

$A_{,ab}=\Big{(}\Delta\cdot
c^{-1}\big{(}\Delta\big{)}\Big{)}_{ad}{\Pi_{T}}^{d}{}_{b}+O\big{(}\mathcal{E}\big{)}\,.$
(2.51)

Following ref. [38] it is useful to write equations such as this where the
equality is given up to terms that vanish on the equations of motion
$\mathcal{E}$, cf. (2.21). Since $\mathcal{E}$ vanishes if we set $A_{\mu}=0$,
we can read off from the above that the first term provides the effective
inverse propagator that would arise from using $A$ as the action. We see that
it is transverse and that it would lead to an effective propagator $\thicksim
c_{q}/q^{2}$ which is regularised by a cutoff function in the usual way. The
first term above is also solely responsible for the one-loop contribution that
$A$ would provide (through the functional determinant of $A_{,ab}$) when the
result is evaluated on shell. From (2.51) we can also infer the large momentum
behaviour of the vertices $A^{(n)}$. Clearly $A^{(2)}\thicksim
q^{2}c_{q}^{-1}$. In fact, for any integer $n$, $A^{(n)}\thicksim c_{q}^{-1}$
up to some multiplicative power of momentum which can be ignored in comparison
if the cutoff is chosen to be strong enough, for example $c(x)=e^{-x}$ or
$c(x)=(1+x)^{-m}$ for some large $m$. To write this we will use the following
shorthand $A^{(n)}\approx c_{q}^{-1}$. These properties will be more precisely
defined and confirmed in sec. 4.2.6. Until then in the spirit of ref. [38], we
count only powers of cutoffs to analyse the behaviour of different expressions
in the large momentum limit. For example, looking at (2.50) we see that

$S_{,ab}\thicksim A^{(2)}A^{(2)}\approx c_{q}^{-2}\,,$ (2.52)

which means that the (PV1) condition can now be recast as:

$C^{-1}\approx c_{q}^{-1}\,.$ (2.53)

To properly implement regularisation, (PV1-3) ought to be fulfilled or,
equivalently, (2.47), (2.48) and (2.53). A particular choice which satisfies
this is the following :

$\displaystyle{\big{(}C^{-1}\big{)}^{a}}_{b}$
$\displaystyle={\delta^{a}}_{b}+\gamma^{ac}A,_{cb}+K^{a}_{\alpha}Y^{\alpha\beta}K^{c}_{\beta}\gamma_{cb}\,,$
(2.54) $\displaystyle{B^{\alpha}}_{\beta}$
$\displaystyle={\delta^{\alpha}}_{\beta}+Y^{\alpha\gamma}\big{(}\Delta_{\parallel}\big{)}_{\gamma\beta}\,,$
(2.55) $\displaystyle{b^{\alpha}}_{\beta}$
$\displaystyle={t^{\alpha}}_{\beta}+{\big{(}\Delta_{\parallel}\big{)}^{\alpha}}_{\beta}\,,$
(2.56)

where $Y=Y\big{(}\Delta_{\parallel}\big{)}$ is a quasi-local function such
that $Y(z)\rightarrow\infty$ as $z\rightarrow\infty$ at the same rate as
$1/c(z)$ and $Y(0)$ is finite, while $t=t\big{(}\Delta_{\parallel}\big{)}$ is
a quasi-local function such that $t(z)\rightarrow 0$ as $z\rightarrow\infty$
and $t(0)=1$.

###  2.4 Flow equation

For the flow equation to have fixed points it has to be non-linear which
implies that the blocking functional $\Psi[\phi]$ must itself depend on $S$.
Generalising [14, 15, 16, 17, 18, 19, 20, 50, 21, 22, 51, 23, 24, 25, 26, 27,
28, 29, 30, 52, 31, 32, 33, 34, 35, 36], which are themselves generalisations
of the Polchinski equation [53], the blocking functional is written in the
following way:

$\Psi^{a}=-\frac{1}{2}\mathcal{K}^{ab}\Sigma_{,b}+\psi^{a}\,,$ (2.57)

where

$\Sigma:=S-\hat{S}=\mathcal{S}-\frac{1}{2}I_{,a}X^{ab}I_{,b}\,,$ (2.58)

and the “seed” action $\hat{S}$ is given by:

$\hat{S}:=A+S_{UV}=A+\frac{1}{2}A_{,a}\gamma^{ab}A_{,b}\,.$ (2.59)

We will call the quasi-local functionals $\mathcal{K}^{ab}$ and $\psi^{a}$ the
exact RG kernels. We will try to fine-tune them to remove the unwanted
divergences still present in the flow equation. In addition to this, it will
prove useful to fix the classical two-point function [15, 16, 23]. As such, we
require

$S_{0}=I+\frac{1}{2}A_{,a}\gamma^{ab}A_{,b}+\mathcal{S}_{0}\,,$ (2.60)

where $\mathcal{S}_{0}=O\big{(}\mathcal{E}^{3}\big{)}$, to be a solution of
the classical flow equation

$\Lambda D_{\Lambda}S_{0}=0\,.$ (2.61)

This fixes the classical two-point function to be

$S_{0,ab}=I_{,ab}+A_{,ma}\gamma^{mn}A_{,nb}+O\big{(}\mathcal{E}\big{)}\,.$
(2.62)

The only thing left now is to fix the exact RG kernels. The main constraint
comes from the left-hand side of (2.41), i.e. the classical flow equation. One
needs to carefully choose the kernels such that the effective interactions
generated during the flow (i.e. $\mathcal{S}$) do not destroy the
regularisation. In other words we need to ensure that $\mathcal{S}$ diverges
more weakly than $S_{UV}\approx c_{q}^{-2}$. Since we are unable to write a
closed form expression for $\mathcal{S}$ (we can only determine a finite
number of higher order contributions to it by repeated iterations), following
ref. [38] we suppose that its maximum divergence rate is given by:

$\mathcal{S}^{(n)}\thicksim A^{(n)}\approx c_{q}^{-1}\,,$ (2.63)

or, equivalently, using the definition (2.58) of $\Sigma$, that

$\Sigma^{(n)}\approx c_{q}^{-1}\,.$ (2.64)

The strategy now is to fine-tune the exact RG kernels such that they cancel
all the terms which diverge faster than $c_{q}^{-1}$ in the flow equation of
$\mathcal{S}^{(n)}$, and hence prove a posteriori that the assumption (2.63)
is consistent. If we unwrap the left-hand side of (2.41) we get:

$\Lambda
D_{\Lambda}S=\dot{A}_{,a}\gamma^{ab}A_{,b}+\frac{1}{2}A_{,a}\dot{\gamma}^{ab}A_{,b}+A_{,a}\big{(}C^{-1}\big{)}^{a}{}_{b}\psi^{b}-\frac{1}{2}A_{,c}\big{(}C^{-1}\big{)}^{c}{}_{a}\mathcal{K}^{ab}\Sigma_{,b}\\\
+\dot{A}+\dot{\Sigma}-\frac{1}{2}\Sigma_{,a}\mathcal{K}^{ab}\Sigma_{,b}+\Sigma_{,a}\psi^{a}\,,$
(2.65)

where “ $\dot{}$ ” denotes the RG time derivative $\Lambda\partial_{\Lambda}$.
To arrive at this expression we have used the identity:

$\hat{S}_{,b}=A_{,a}\big{(}C^{-1}\big{)}^{a}{}_{b}\,,$ (2.66)

which follows from (2.54) and the gauge invariance of $A$,
$K^{a}_{\alpha}A_{,a}=0$. The first two terms of (2.65) are bi-linear in
$A_{,a}$ and hence diverge as $\approx c_{q}^{-2}$. To eliminate them we use
the third term, setting

$\psi^{a}=-C^{a}{}_{b}\Big{(}\gamma^{bc}\dot{A}_{,c}+\frac{1}{2}\dot{\gamma}^{bc}A_{,c}\Big{)}\,.$
(2.67)

Now looking at the remaining terms, we see that the potentially dangerous ones
are the fourth term which, ignoring $\mathcal{K}^{ab}$ for the moment,
diverges as $\approx c_{q}^{-3}$, and similarly the next to last term, which
diverges as $\approx c_{q}^{-2}$. Cancelling the $C^{-1}$ factor in the former
by writing

$\mathcal{K}^{ab}=C^{a}{}_{c}[\kappa(\Delta)]^{cb}\,,$ (2.68)

the kernel $\kappa^{ab}$ is then fixed uniquely by substituting $S_{0}$ from
(2.60), into (2.65) and requiring cancellation, i.e. requiring the classical
flow (2.61) at $O\big{(}\mathcal{E}^{2}\big{)}$. The result is:

$\kappa^{a}{}_{b}=\bigg{(}2\frac{c(\Delta)-2\Delta
c^{\prime}(\Delta)}{c^{2}(\Delta)+\Delta}\bigg{)}^{a}{}_{b}$ (2.69)

(see sec. IIIB and app. A of ref. [38]). This means that $\kappa^{ab}\approx
c_{q}$ and hence ref. [38] concluded that all the potentially offending terms
in (2.65) are now brought under control. This part of the flow equation (the
classical part) now simplifies to:

$\Lambda
D_{\Lambda}S=-\frac{1}{2}\big{(}\Sigma_{,a}+\hat{S}_{,a}\big{)}C^{a}{}_{c}\kappa^{cb}\Sigma_{,b}+\dot{A}+\dot{\Sigma}+\Sigma_{,a}\psi^{a}\,,$
(2.70)

and indeed a direct substitution of the above estimates suggests that now no
terms diverge faster than $c_{q}^{-1}$. Moreover, although more involved, one
can similarly check that the exact RG kernels (2.67), (2.68) and (2.69) ensure
that the RHS of (2.41) is also apparently properly regularised [38].

##  3 Regularisation failure: a sketch

The problem with the analysis reviewed in the previous section is more easily
grasped if we convert the analysis into the language of Feynman graphs. The
reason this is helpful, is because we can then derive closed formulae for all
the component vertices and propagator-like terms of these graphs, and then
rigorously characterise their asymptotic behaviour in the limit that certain
momenta diverge. In particular in quantum corrections, UV divergences arise
from the limit in which loop momenta $q\to\infty$ while external momenta are
held fixed. The components in the flow equation (2.70) do not necessarily
behave in this limit with the assumed powers of $c_{q}$ because their UV
behaviour depends on the details of how $q$ is routed through the vertices.

By focusing in sec. 6 on the part of the one-loop action that should provide
the beta function, we show that regularisation fails here in a somewhat subtle
way. There are parts that are UV divergent, but can be set to zero once
treated as $\Lambda$ independent. However, gauge transformations still map the
one-loop beta function contribution into those improperly regularised parts.
In fact we show that the contribution cannot then both be gauge invariant and
properly regularised. Actually this latter conclusion holds for all choices of
cutoff functions and some generalisations of the construction. Although the
flow equation is constructed to be covariant by using functions of the
differential operators $\Delta$ and $\Delta_{\parallel}$, other choices of
covariantisation are possible that still reduce to the same functions when the
gauge field $A_{\mu}$ is set to zero (for example by allowing the functions to
depend separately on $F_{\mu\nu}$). Our derivation is independent of these
details.

In fact the first hint of regularisation failure seems to come from how the
regularisation is constructed in the first place. Ref.[38] argues that in the
high momentum limit the leading divergent part of the propagator is its
transverse part. The argument goes as follows. Starting from the gauge
invariance of $S$ (under a gauge transformation
$\phi^{a}\rightarrow\phi^{a}+\delta\phi^{a}$),

$K^{a}_{\alpha}S_{,a}=0\,,$ (3.1)

and differentiating it once we obtain:

$K^{a}_{\alpha}S_{,ab}=-K^{a}_{\alpha,b}S_{,a}\,.$ (3.2)

Regarding this expression as a differential operator one can think of the
large momentum as going from $\alpha$ to $b$. Then this equation tells us that
the behaviour at large momentum is given by $K^{a}_{\alpha,b}$, and since
$K^{a}_{\alpha}$ is a first order differential operator the divergence is only
with one power of momentum. However as an expression about vertices more
generally we see that we cannot exclude the possibility that the loop momentum
threads instead through $S_{,a}$ whose high momentum behaviour can behave as
badly as does $S_{,ab}$.

##  4 Perturbative expansion

In summary, the robustness of the regularisation scheme outlined in sec. 2
needs to be thoroughly tested by analysing carefully the high momentum
behaviour of $n$-point vertices at each order in the loop expansion. In what
follows we will lay out the machinery needed to do this properly, before
returning to these issues in sec. 6.

###  4.1 Loop expansion

The effective action $S$, and in fact any gauge invariant action appearing in
this analysis, has a weak coupling expansion (i.e. loop expansion) in $g^{2}$
(actually in $\hbar g^{2}$):

$S=\frac{1}{g^{2}}S_{0}+S_{1}+g^{2}S_{2}+\cdots\,.$ (4.1)

This means that purely classical actions such as $S_{0}$, but also e.g. $A$,
$S_{UV}$ and $\hat{S}$, carry only a single $1/g^{2}$ prefactor.

The form of the ultralocal metrics $\gamma_{ab}$ and $\eta_{\alpha\beta}$
ensure that the factors of $g$ embedded in the kernels combine with actions to
preserve this property. Therefore we simplify matters from now on by ignoring
the factors of $g$ entirely and trusting that they can be put back in the form
(4.1) above. (Actually, one can also simplify matters by recognising that the
powers of $\Lambda$ appearing through these metrics are as required to make
the equations dimensionally correct, and thus essentially ignore these
metrics.) Note that in the ensuing we will also place the loop index as a
superscript when convenient (so write $S^{0}\equiv S_{0}$ etc. ).

Similarly to (4.1) we can write down a loop expansion for the blocking
functional $\Psi^{a}$:

$\displaystyle\Psi^{a}$ $\displaystyle=\Psi_{0}^{a}+g^{2}\Psi_{1}^{a}+\cdots$
(4.2)
$\displaystyle=\Big{[}-\frac{1}{2}\mathcal{K}^{ab}{\Sigma^{0}}_{,b}+\psi_{0}^{a}\Big{]}+g^{2}\Big{[}-\frac{1}{2}\mathcal{K}^{ab}{\Sigma^{1}}_{,b}+\psi^{a}_{1}\Big{]}+\cdots\,.$
(4.3)

This can be further refined. If we explicitly compute the RG time derivatives
in (2.67) and use the commutativity of RG time and functional derivatives, we
can recast $\psi^{a}$ in the following way:

$\psi^{a}=C^{ab}\theta_{,b}\,,$ (4.4)

where we have introduced the gauge invariant action $\theta$, given by

$\theta:=\frac{\eta}{2}A+I+I_{,m}[W(\Delta)]^{mn}I_{,n}\,,$ (4.5)

with $W(z):=\frac{3}{2}X(z)+zX^{\prime}(z)$, which thus has a loop expansion:

$\displaystyle\theta^{0}=I+I_{,m}W^{mn}I_{,n}\,,$ (4.6)
$\displaystyle\theta^{n}=\frac{\eta_{n}}{2}A\,.$ (4.7)

This enables us to rewrite (4.3) as

$\Psi^{a}=\Big{[}-\frac{1}{2}\mathcal{K}^{ab}{\Sigma^{0}}_{,b}+C^{ab}{\theta^{0}}_{,b}\Big{]}+g^{2}\Big{[}-\frac{1}{2}\mathcal{K}^{ab}{\Sigma^{1}}_{,b}+C^{ab}{\theta^{1}}_{,b}\Big{]}+\cdots\,.$
(4.8)

If we substitute (4.1) and (4.8) into the flow equation (2.41), we will obtain
its loopwise expansion. Thus we find that the flow of the effective action at
the classical level is given by:

$\Lambda\partial_{\Lambda}S^{0}=\frac{1}{2}{S^{0}}_{,a}\mathcal{K}^{ab}{\Sigma^{0}}_{,b}-{S^{0}}_{,a}C^{ab}{\theta^{0}}_{,b}\,,$
(4.9)

at the one-loop level by:

$\displaystyle\Lambda\partial_{\Lambda}S^{1}$
$\displaystyle=\eta_{1}S^{0}+\frac{1}{2}{S^{1}}_{,a}\mathcal{K}^{ab}{\Sigma^{0}}_{,b}+\frac{1}{2}{S^{0}}_{,a}\mathcal{K}^{ab}{\Sigma^{1}}_{,b}-{S^{1}}_{,a}C^{ab}{\theta^{0}}_{,b}-{S^{0}}_{,a}C^{ab}{\theta^{1}}_{,b}$
$\displaystyle+\text{Tr}\Bigg{[}C^{a}{}_{c}\Big{[}\Lambda\partial_{\Lambda}(C^{-1})^{c}{}_{b}\Big{]}+C^{a}{}_{c}(C^{-1})^{c}{}_{b,d}\>\Psi_{0}^{d}+\frac{\delta\Psi_{0}^{a}}{\delta\phi^{b}}+\delta^{a}{}_{b}-(B^{-1})^{\alpha}{}_{\gamma}\big{[}\Lambda\partial_{\Lambda}B^{\gamma}{}_{\beta}\big{]}$
$\displaystyle-(B^{-1})^{\alpha}{}_{\gamma}B^{\gamma}{}_{\beta,a}\Psi_{0}^{a}-\frac{1}{2}(b^{-1})^{\alpha}{}_{\gamma}\big{[}\Lambda\partial_{\Lambda}b^{\gamma}{}_{\beta}\big{]}-\frac{1}{2}(b^{-1})^{\alpha}{}_{\gamma}b^{\gamma}{}_{\beta,a}\Psi_{0}^{a}-2\>\delta^{\alpha}{}_{\beta}\Bigg{]}\,,$

at the two-loop level by:

$\displaystyle\Lambda\partial_{\Lambda}S^{2}$
$\displaystyle=\eta_{2}S^{0}+\frac{1}{2}{S^{2}}_{,a}\mathcal{K}^{ab}{\Sigma^{0}}_{,b}+\frac{1}{2}{S^{1}}_{,a}\mathcal{K}^{ab}{\Sigma^{1}}_{,b}+\frac{1}{2}{S^{0}}_{,a}\mathcal{K}^{ab}{\Sigma^{2}}_{,b}-{S^{2}}_{,a}C^{ab}{\theta^{0}}_{,b}$
$\displaystyle-{S^{1}}_{,a}C^{ab}{\theta^{1}}_{,b}-{S^{0}}_{,a}C^{ab}{\theta^{2}}_{,b}+\text{Tr}\Bigg{[}C^{a}{}_{c}\>(C^{-1})^{c}{}_{b,d}\Psi_{1}^{d}+\frac{\delta\Psi_{1}^{a}}{\delta\phi^{b}}-\frac{\eta_{1}}{2}\delta^{a}{}_{b}+\frac{\eta_{1}}{2}\delta^{\alpha}{}_{\beta}$
$\displaystyle-(B^{-1})^{\alpha}{}_{\gamma}B^{\gamma}{}_{\beta,a}\Psi_{1}^{a}-\frac{1}{2}(b^{-1})^{\alpha}{}_{\gamma}b^{\gamma}{}_{\beta,a}\Psi_{1}^{a}\Bigg{]}\,,$

etc. It will prove useful in the subsequent analysis to rewrite (4.1) in a
different way:

$\displaystyle\Lambda\partial_{\Lambda}S^{1}$
$\displaystyle=\eta_{1}S^{0}+\frac{1}{2}{S^{1}}_{,a}\mathcal{K}^{ab}{\Sigma^{0}}_{,b}+\frac{1}{2}{S^{0}}_{,a}\mathcal{K}^{ab}{\Sigma^{1}}_{,b}-{S^{1}}_{,a}C^{ab}{\theta^{0}}_{,b}-{S^{0}}_{,a}C^{ab}{\theta^{1}}_{,b}$
$\displaystyle+\text{Tr}\Bigg{[}\Lambda\partial_{\Lambda}\big{[}\ln\big{(}C^{-1}\big{)}\big{]}^{a}{}_{b}+\Psi_{0}^{d}\frac{\delta}{\delta\phi^{d}}\big{[}\ln\big{(}C^{-1}\big{)}\big{]}^{a}{}_{b}+\frac{\delta\Psi_{0}^{a}}{\delta\phi^{b}}-\Lambda\partial_{\Lambda}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}$
$\displaystyle-\Psi_{0}^{a}\frac{\delta}{\delta\psi^{a}}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}-\frac{1}{2}\Lambda\partial_{\Lambda}\big{[}\ln
b^{\alpha}{}_{\beta}\big{]}-\frac{1}{2}\Psi_{0}^{a}\frac{\delta}{\delta\psi^{a}}\big{[}\ln
b^{\alpha}{}_{\beta}\big{]}\Bigg{]}\,.$

Here note that we have also discarded the two vacuum contributions
$\delta^{a}{}_{b}$ and $\delta^{\alpha}{}_{\beta}$.

###  4.2 Vertex expansion

In the above equations we have actions ($S^{i},\Sigma^{i},\theta^{i}$) and
kernels (e.g. $\mathcal{K}^{ab},C^{ab},B^{\gamma}_{\beta}$ etc. ). To analyse
their behaviour precisely we break them down into vertices. The above
equations then tell us how these vertices fit together to determine effective
action vertices at each loop order. In this subsection we sketch the form of
these vertices and their properties [15, 16].

####  4.2.1 Action vertex properties

Any gauge invariant action has an expansion in traces and products of traces
[15, 16], which, in position space, takes the following form:

$\displaystyle S=$
$\displaystyle\sum_{n=2}^{\infty}\frac{1}{n}\int_{x_{1}\ldots
x_{n}}S_{\mu_{1}\ldots\mu_{n}}(x_{1}\ldots
x_{n})\text{tr}\Big{(}A_{\mu_{1}}(x_{1})\ldots A_{\mu_{n}}(x_{n})\Big{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2!}\sum_{n,m=2}^{\infty}\frac{1}{nm}\int_{x_{1}\ldots
x_{n}}\int_{y_{1}\ldots
y_{m}}S_{\mu_{1}\ldots\mu_{n},\nu_{1}\ldots\nu_{m}}(x_{1}\ldots
x_{n};y_{1}\ldots y_{m})$
$\displaystyle\text{tr}\Big{(}A_{\mu_{1}}(x_{1})\ldots
A_{\mu_{n}}(x_{n})\Big{)}\text{tr}\Big{(}A_{\nu_{1}}(y_{1})\ldots
A_{\nu_{m}}(y_{m})\Big{)}$ $\displaystyle+$ $\displaystyle\cdots\,.$ (4.13)

From this we can see that single trace vertices are defined as cyclically
symmetric in all indices,

$S_{\mu_{1}\ldots\mu_{n}}(x_{1}\ldots
x_{n})=S_{\mu_{n}\mu_{1}\ldots\mu_{n-1}}(x_{n},x_{1}\ldots x_{n-1})\,,$ (4.14)

whereas the remaining vertices are separately cyclically symmetric for each
string. However, the latter have been included for completeness only and are
of no interest to us in the present paper. Therefore, from now on we will
consider only the single trace vertices. The fact that the action $S$ is real,
together with the hermiticity of the generators $T^{a}$, implies that
$S^{*}_{\mu_{1}\ldots\mu_{n}}(x_{1}\ldots
x_{n})=S_{\mu_{n}\ldots\mu_{1}}(x_{n}\ldots x_{1})$. This is an expression of
charge conjugation invariance. To exploit translation invariance we work in
momentum space where it implies momentum conservation:

$S_{\mu_{1}\ldots\mu_{n}}(p_{1}\ldots
p_{n})\,\delta\Big{(}\sum_{i}p_{i}\Big{)}=\int_{x_{1}\ldots
x_{n}}S_{\mu_{1}\ldots\mu_{n}}(x_{1}\ldots x_{n})\>e^{-i\sum_{i}p_{i}\cdot
x_{i}}\,.$ (4.15)

By convention, we take all momenta pointing in towards the vertex. Note that
the $S_{\mu_{1}\ldots\mu_{n}}(p_{1}\ldots p_{n})$ are well-defined only when
momentum is conserved. For the two-point vertex we write more simply
$S_{\mu\nu}(p)=S_{\mu\nu}(p,-p)$. Lorentz invariance implies that:

$S_{\mu_{1}\ldots\mu_{n}}(p_{1},\ldots,p_{n})=(-1)^{n}S_{\mu_{1}\ldots\mu_{n}}(-p_{1},\ldots,-p_{n})\,.$
(4.16)

In other words, vertices with an even/odd number of legs are even/odd under a
change of sign in all its momentum arguments. Combined with charge conjugation
invariance we thus also have that they are even/odd under reversal of their
arguments

$S_{\mu_{n}\ldots\mu_{1}}(p_{n},\ldots,p_{1})=(-1)^{n}S_{\mu_{1}\ldots\mu_{n}}(p_{1},\ldots,p_{n})\,.$
(4.17)

As an example we note that these symmetries tell us that a three-point action
vertex is totally antisymmetric in its arguments
$({p_{1}}_{\mu_{1}},{p_{2}}_{\mu_{2}},{p_{3}}_{\mu_{3}})$.

The flow equation (2.41) manifestly preserves gauge invariance, namely (2.11),
and this allows us to write a set of ‘trivial’ Ward identities for the
effective action $S$ that relates the longitudinal part of $n$-point vertices
to differences of $(n-1)$-point vertices:

${p_{1}}_{\mu_{1}}S_{\mu_{1}\ldots\mu_{n}}(p_{1}\ldots
p_{n})=S_{\mu_{2}\ldots\mu_{n}}(p_{1}+p_{2},p_{3}\ldots
p_{n})-S_{\mu_{2}\ldots\mu_{n}}(p_{2}\ldots p_{n-1},p_{1}+p_{n})\,.$ (4.18)

Since there are no $(n<2)$-point vertices, the two-point vertex is transverse:

$p_{\mu}S_{\mu\nu}(p)=0\,.$ (4.19)

In the limit of small contracted momentum $p_{1}$, (4.18) yields differential
Ward identities e.g.

$S_{\mu\nu\lambda}(0,p,-p)=\partial^{p}_{\mu}S_{\nu\lambda}(p)\,.$ (4.20)

where we have introduced the shorthand
$\partial^{p}_{\mu}=\frac{\partial}{\partial p_{\mu}}$. This follows
straightforwardly starting from (4.18) after Taylor expanding its right-hand
side:

$\displaystyle\epsilon_{\mu}S_{\mu\nu\lambda}(\epsilon,p,-p-\epsilon)$
$\displaystyle=S_{\nu\lambda}(p+\epsilon)-S_{\nu\lambda}(p)$ (4.21)
$\displaystyle=\epsilon_{\mu}\partial^{p}_{\mu}S_{\nu\lambda}(p)+O(\epsilon^{2})\,,$
(4.22)

recovering (4.20) from $O(\epsilon)$. We can proceed similarly about the four-
point vertex to get:

$S_{\mu\nu\lambda\sigma}(0,0,p,-p)+S_{\nu\mu\lambda\sigma}(0,0,p,-p)=\partial^{p}_{\mu}\partial^{p}_{\nu}S_{\lambda\sigma}(p)\,.$
(4.23)

The $I^{(n)}$ vertices already introduced in eqs. (2.14)–(2.16) satisfy the
above properties.

####  4.2.2 Kernel vertex properties

The regularisation and the flow equation are built up from constructs where
actions are joined together by kernels to form new compound actions.
Generically for two actions, say $R[A]$ and $S[A]$, the construct takes the
form $R_{,a}K^{ab}S_{,b}$ and involves some kernel $K^{ab}[A]$. Since
everything is constructed gauge covariantly, these kernels are themselves
necessarily functionals of $A_{\mu}$. Although the kernels we need are
constructed in a specific way, as reviewed in sec. 2.3, they are special cases
of a more general form of kernel namely any such that provides a
covariantization of some momentum kernel $K^{ab}[A=0]\equiv K_{\mu\nu}(p)$,
cf. (2.27), via gauge fields $A_{\mu}$ that act by commutation. (The latter
property is assured here to the $2$-point level that we will require them, in
essence by constructing the kernels using covariant derivatives $D_{\mu}$, see
sec. 4.2.5 and app. A.) Such kernels have vertices whose properties generalise
those for an action and ensure that the vertices of the compound action
continue to satisfy the correct properties.

Expressions involving such a covariantized kernel $K_{\mu\nu}$ can be
equivalently written in the following way:

$\displaystyle R_{,a}K^{ab}S_{,b}\equiv$
$\displaystyle\frac{1}{\Lambda^{2}}\frac{\delta R}{\delta A_{\mu}}\cdot
K_{\mu\nu}\cdot\frac{\delta S}{\delta A_{\nu}}$ (4.24) $\displaystyle\equiv$
$\displaystyle\frac{1}{2\Lambda^{2}}\text{tr}\int_{x}\int_{y}\frac{\delta
R}{\delta A_{\mu}(x)}K_{\mu\nu}(x,y)\cdot\frac{\delta S}{\delta
A_{\nu}(y)}\,,$ (4.25)

where we suppress the functional dependence on the gauge field. Taylor
expanding with respect to the gauge field in $K^{ab}[A]$ gives a series
expansion in nested commutators [15, 16, 23]:

$R_{,a}{K}^{ab}S_{,b}\equiv\frac{1}{2\Lambda^{2}}\sum_{n=0}\int_{x}\int_{y}\int_{x_{1}\cdots
x_{n}}K_{\mu_{1}\ldots\mu_{n};\mu\nu}(x_{1}\ldots x_{n};x,y)\\\
\text{tr}\Big{[}\frac{\delta R}{\delta A_{\mu}(x)}A_{\mu_{1}}(x_{1})\cdots
A_{\mu_{n}}(x_{n})\cdot\frac{\delta S}{\delta A_{\nu}(y)}\Big{]}\,.$ (4.26)

Now, expanding the commutators, we obtain:

$R_{,a}{K}^{ab}S_{,b}\equiv\frac{1}{2\Lambda^{2}}\sum_{m,n=0}\int_{x}\int_{y}\int_{x_{1}\cdots
x_{n},y_{1}\cdots
y_{m}}K_{\mu_{1}\ldots\mu_{n},\nu_{1}\ldots\nu_{m};\mu\nu}(x_{1}\ldots
x_{n};y_{1}\ldots y_{m};x,y)\\\ \text{tr}\Bigg{[}\frac{\delta R}{\delta
A_{\mu}(x)}A_{\mu_{1}}(x_{1})\cdots A_{\mu_{n}}(x_{n})\frac{\delta S}{\delta
A_{\nu}(y)}A_{\nu_{1}}(y_{1})\cdots A_{\nu_{m}}(y_{m})\Bigg{]}\,.$ (4.27)

In momentum space, kernel vertices take the following form:

$K_{\mu_{1}\ldots\mu_{n},\nu_{1}\ldots\nu_{m};\mu\nu}(p_{1}\ldots
p_{n};q_{1}\ldots q_{m};p,q)\,\delta\Big{(}\sum p_{i}+\sum
q_{i}+p+q\Big{)}=\\\ =\int_{x}\int_{y}\int_{x_{1}\cdots x_{n},y_{1}\cdots
y_{m}}K_{\mu_{1}\ldots\mu_{n},\nu_{1}\ldots\nu_{m};\mu\nu}(x_{1}\ldots
x_{n};y_{1}\ldots y_{m};x,y)e^{-i\Big{(}p\cdot x+q\cdot y+\sum_{i}p_{i}\cdot
x_{i}+\sum_{j}q_{j}\cdot y_{j}\Big{)}}\,.$ (4.28)

The two sets of vertices in (4.26) and (4.27) are related through ‘interleave’
identities (also called ‘coincident line’ identities [15, 16, 23]) which here
just express the fact that covariantisation is via commutation:

$K_{\mu_{1}\ldots\mu_{n},\nu_{1}\ldots\nu_{m};\mu\nu}(p_{1}\ldots
p_{n};q_{1}\ldots
q_{m};p,q)=(-1)^{m}\\!\\!\\!\sum_{\text{interleaves}}K_{\alpha_{1}\ldots\alpha_{n+m};\mu\nu}(k_{1}\ldots
k_{m+n};p,q)\,.$ (4.29)

In the above the sum runs over all the possible arrangements of the combined
sequence of $p_{1}^{\mu_{1}}\ldots p_{n}^{\mu_{n}}$ and $q_{1}^{\nu_{1}}\ldots
q_{m}^{\nu_{m}}$, in which the $p$ momenta remain ordered with one another,
whereas the $q$ momenta order is reversed (via the so-called shuffle product
$\sum k^{\alpha_{1}}_{1}\ldots k^{\alpha_{m+n}}_{m+n}=p_{1}^{\mu_{1}}\ldots
p_{n}^{\mu_{n}}\mathbin{\mathchoice{\kern 0.0pt\hbox to0.0pt{\vrule
height=0.0pt,depth=0.0pt,width=0.0pt\hss}\raise 0.0pt\hbox to0.0pt{\vrule
height=0.0pt,depth=0.0pt,width=0.0pt\hfill\vrule
height=0.0pt,depth=0.0pt,width=0.0pt\hfill\vrule
height=0.0pt,depth=0.0pt,width=0.0pt}\kern 0.0pt}{\kern 0.0pt\hbox
to0.0pt{\vrule height=0.0pt,depth=0.0pt,width=0.0pt\hss}\raise 0.0pt\hbox
to0.0pt{\vrule height=0.0pt,depth=0.0pt,width=0.0pt\hfill\vrule
height=0.0pt,depth=0.0pt,width=0.0pt\hfill\vrule
height=0.0pt,depth=0.0pt,width=0.0pt}\kern 0.0pt}{\kern 0.0pt\hbox
to0.0pt{\vrule height=0.0pt,depth=0.0pt,width=0.0pt\hss}\raise 0.0pt\hbox
to0.0pt{\vrule height=0.0pt,depth=0.0pt,width=0.0pt\hfill\vrule
height=0.0pt,depth=0.0pt,width=0.0pt\hfill\vrule
height=0.0pt,depth=0.0pt,width=0.0pt}\kern 0.0pt}{\kern 0.0pt\hbox
to0.0pt{\vrule height=0.0pt,depth=0.0pt,width=0.0pt\hss}\raise 0.0pt\hbox
to0.0pt{\vrule height=0.0pt,depth=0.0pt,width=0.0pt\hfill\vrule
height=0.0pt,depth=0.0pt,width=0.0pt\hfill\vrule
height=0.0pt,depth=0.0pt,width=0.0pt}\kern 0.0pt}}q_{m}^{\nu_{m}}\ldots
q_{1}^{\nu_{1}}$).

The case where $m=n=0$ (thus leaving a double semi-colon) is of course just
the momentum kernel again:

$K_{,;\mu\nu}(;;p,-p)=K_{\mu\nu}(p)\,.$ (4.30)

We write the $m=0$ case more simply as:

$K_{\mu_{1}\ldots\mu_{n},;\mu\nu}(p_{1}\ldots
p_{n};;p,q)=K_{\mu_{1}\ldots\mu_{n};\mu\nu}(p_{1}\ldots p_{n};p,q)\,,$ (4.31)

then from (4.29) the $n=0$ case is given by

$K_{,\nu_{1}\ldots\nu_{m};\mu\nu}(;q_{1}\ldots
q_{m};p,q)=(-1)^{m}K_{\nu_{m}\ldots\nu_{1};\mu\nu}(q_{m}\ldots q_{1};p,q)\,.$
(4.32)

A useful special case of (4.29) is:

$K_{\mu,\nu;\alpha\beta}(p;q;r,s)=-K_{\mu\nu;\alpha\beta}(p,q;r,s)-K_{\nu\mu;\alpha\beta}(q,p;r,s)\,.$
(4.33)

Kernel vertices obey Ward identities closely similar to (4.18) [15, 16].
Furthermore the contracted momentum can reach the end of the string of
momenta, where it attaches to the outer momentum:

${p_{1}}_{\mu_{1}}K_{\mu_{1}\ldots\mu_{n},\nu_{1}\ldots\nu_{m};\mu\nu}(p_{1}\ldots
p_{n};q_{1}\ldots q_{m};p,q)=\\\
=K_{\mu_{2}\ldots\mu_{n},\nu_{1}\ldots\nu_{m};\mu\nu}(p_{1}+p_{2},p_{3}\ldots
p_{n};q_{1}\ldots q_{m};p,q)\\\
-K_{\mu_{2}\ldots\mu_{n},\nu_{1}\ldots\nu_{m};\mu\nu}(p_{2}\ldots
p_{n};q_{1}\ldots q_{m};p_{1}+p,q)\,.$ (4.34)

Again one can derive differential Ward identities (cf. (4.20), (4.23)). We
list below some of the more useful ones:

$\displaystyle K_{\lambda;\mu\nu}(0;-p,p)$
$\displaystyle=\partial^{p}_{\lambda}K_{\mu\nu}(p)\,,$ (4.35) $\displaystyle
K_{\lambda\sigma;\mu\nu}(-p,0;0,p)$
$\displaystyle=\partial^{p}_{\sigma}K_{\lambda;\mu\nu}(-p;0,p)\,.$ (4.36)

####  4.2.3 Two-point action vertices

Combining these definitions with those of sec. 2 one can derive the
expressions for the vertices that are needed and work through the classical
and quantum corrections systematically. Thus for example, from the definition
(2.29) of the quasi-local action $A$, combined with that of $X$, (2.30), and
the fundamental two-point vertex $I^{(2)}$ in (2.14), or alternatively
directly from (2.51), one gets its two-point vertex:

$A_{\mu_{1}\mu_{2}}(p)=2\,\square_{\mu_{1}\mu_{2}}(p)F_{A}(p)\,,\qquad
F_{A}(p):=\frac{1}{c_{p}}\,.$ (4.37)

Armed with this one finds from (2.59) the two-point vertex for the seed
action:

$\hat{S}_{\mu_{1}\mu_{2}}(p)=2\,\square_{\mu_{1}\mu_{2}}(p)F_{\hat{S}}(p)\,,\qquad
F_{\hat{S}}(p):=\frac{\tilde{p}^{2}+c_{p}}{c_{p}^{2}}\,,$ (4.38)

and from (2.60) the classical effective action $S^{0}$ two-point vertex:

$S^{0}_{\mu_{1}\mu_{2}}(p)=2\,\square_{\mu_{1}\mu_{2}}(p)F(p)\,,\qquad
F(p):=1+\frac{\tilde{p}^{2}}{c_{p}^{2}}\,.$ (4.39)

From the definition (2.58) of $\Sigma$ and $X$ one gets immediately its
classical two-point vertex:

$\Sigma^{0}_{\mu_{1}\mu_{2}}(p)=2\,\square_{\mu_{1}\mu_{2}}(p)F_{\Sigma}(p)\,,\qquad
F_{\Sigma}(p):=1-\frac{1}{c_{p}}\,.$ (4.40)

Finally from its definition (4.5) we get the classical two-point vertex for
the action $\theta$:

$\theta^{0}_{\mu_{1}\mu_{2}}(p)=2\,\square_{\mu_{1}\mu_{2}}(p)F_{\theta}(p)\,,\qquad
F_{\theta}(p):=\frac{c_{p}-2\tilde{p}^{2}c_{p}^{\prime}}{c_{p}^{2}}\,.$ (4.41)

####  4.2.4 Zero-point kernel functions

The zero-point vertices for the simple (i.e. non-compound) kernels $X$, $Y$,
$B$, $b$, $t$, $\kappa$ and $W$, follow straightforwardly from their
definitions in (2.30), (2.55), (2.56), (2.69) and (4.5), by replacing their
argument ($\Delta$ or $\Delta_{\parallel}$ as appropriate) by $\tilde{p}^{2}$,
for example:

$\kappa(p)=2\,\frac{c_{p}-2\tilde{p}^{2}c^{\prime}_{p}}{c_{p}^{2}+\tilde{p}^{2}}\,.$
(4.42)

(The zero-point $X$ kernel was in effect already used to derive the two-point
action vertices above.) Those for the compound kernels follow almost as
straightforwardly. Thus the zero point vertex for
$\big{(}C^{-1}\big{)}^{a}{}_{b}$ follows from its definition (2.54) since the
generator of gauge transformations (2.9) collapses to a partial derivative in
this case. Writing

$C^{-1}_{\mu\nu}(p)=\frac{1}{C_{T}(p)}\frac{\square_{\mu\nu}(p)}{p^{2}}+\frac{1}{C_{L}(p)}\frac{p_{\mu}p_{\nu}}{p^{2}}\,,$
(4.43)

we have that its longitudinal and transverse functions are given by:

$C_{T}(p):=\frac{c_{p}}{c_{p}+\tilde{p}^{2}}\,,\qquad
C_{L}(p):=\frac{1}{1+\tilde{p}^{2}Y_{p}}\,.$ (4.44)

Note that despite appearances in (4.43), the kernel has a Taylor expansion in
$p_{\mu}$ (is quasilocal) because $Y$ is quasilocal and $c_{p}$ is normalised
to $c(0)=1$ (cf. below (2.30)). Inverting gives us the zero-point vertex for
$C^{a}{}_{b}$:

$C_{\mu\nu}(p)=C_{T}(p)\frac{\square_{\mu\nu}(p)}{p^{2}}+C_{L}(p)\frac{p_{\mu}p_{\nu}}{p^{2}}\,,$
(4.45)

which is also quasi-local for the same reasons. Then that for $\mathcal{K}$
follows from (2.68):

$\mathcal{K}_{\mu\nu}(p)=\mathcal{K}_{T}(p)\frac{\square_{\mu\nu}(p)}{p^{2}}+\mathcal{K}_{L}(p)\frac{p_{\mu}p_{\nu}}{p^{2}}\,,\qquad\mathcal{K}_{I}(p):=\kappa(p)C_{I}(p)\quad(I=T,L)\,.$
(4.46)

####  4.2.5 Higher-point vertices

The $n$-point vertices for the simple kernels can be computed by Taylor
expanding in $\Delta$ or $\Delta_{\parallel}$ as appropriate. Keeping only $n$
instances of $A_{\mu_{i}}$ as in (4.26), the momentum dependence can then be
resummed to give the explicit formula. The vertices for those that are simply
functions of $\Delta_{\parallel}=-D^{2}/\Lambda^{2}$ – namely $Y$, $B$, $b$
and $t$ – have vertices that have already been computed this way in sec. 5 of
ref. [16]. We just quote the results:

$\displaystyle Y_{\mu}(p;r,s)$
$\displaystyle=(r-s)_{\mu}\,\frac{Y_{r}-Y_{s}}{p\\!\cdot\\!(r-s)}\,,$ (4.47)
$\displaystyle Y_{\mu\nu}(p,q;r,s)$
$\displaystyle=\delta_{\mu\nu}\frac{Y_{s}-Y_{r}}{s^{2}-r^{2}}-(p+2r)_{\mu}(q+2s)_{\nu}\,\Xi_{Y}(p,q;r,s)\,,$
(4.48) $\displaystyle\Xi_{Y}(p,q;r,s)$
$\displaystyle=\frac{Y_{s+q}}{q\\!\cdot\\!(q+2s)\,p\\!\cdot\\!(p+2r)}+\frac{1}{s^{2}-r^{2}}\left[\frac{Y_{r}}{p\\!\cdot\\!(p+2r)}-\frac{Y_{s}}{q\\!\cdot\\!(q+2s)}\right]\,,$
(4.49)

where we have introduced the shorthand $\Xi_{Y}$, and where for the others one
simply replaces $Y$ by $B$, $b$ or $t$ as appropriate. We note that these
equations need care at special momenta where denominators vanish. For example
the correct equation for $Y_{\mu\nu}(p,-p;r,-r)$ follows from
$\lim_{\epsilon\to 0}Y_{\mu\nu}(p,-p-\epsilon;r+\epsilon,-r)$. Again we refer
to sec. 5 of ref. [16] for details. The vertices for the other simple kernels
$X$, $\kappa$ and $W$ follow from using a similar strategy. We only have to
take into account the presence of $F_{\mu\nu}$ when expanding in a series of
powers of $\Delta_{\mu\nu}$, cf. (2.25). Thus we find:

$\displaystyle X_{\lambda;\mu\nu}(p;r,s)$
$\displaystyle=\left\\{(r-s)_{\lambda}\delta_{\mu\nu}+4\delta_{\lambda[\mu}p_{\nu]}\right\\}\frac{X_{r}-X_{s}}{p\\!\cdot\\!(r-s)}\,,$
(4.50) $\displaystyle X_{\lambda\sigma;\mu\nu}(p,q;r,s)$
$\displaystyle=\left(\delta_{\mu\nu}\delta_{\lambda\sigma}+4\delta_{\lambda[\mu}\delta_{\nu]\sigma}\right)\frac{X_{s}-X_{r}}{s^{2}-r^{2}}+\Bigg{(}16\delta_{\lambda[\mu}p_{\epsilon]}\delta_{\sigma[\epsilon}q_{\nu]}-4\delta_{\lambda[\mu}p_{\nu]}(q+2s)_{\sigma}$
$\displaystyle\quad+4(p+2r)_{\lambda}\delta_{\sigma[\mu}q_{\nu]}-\delta_{\mu\nu}(p+2r)_{\lambda}(q+2s)_{\sigma}\Bigg{)}\Xi_{X}(p,q;r,s)\,.$
(4.51)

Here $T_{[\mu\nu]}:=\frac{1}{2}(T_{\mu\nu}-T_{\nu\mu})$. As above, those for
$\kappa$ and $W$ follow simply by replacing the name.

With the above building blocks and techniques it is a tedious but
straightforward exercise to arrive at explicit expressions for compound kernel
and action $n$-point vertices. For example from (2.29), one finds

$\displaystyle
A_{\mu\nu\lambda}(p,q,-p-q)=I_{\mu\nu\lambda}(p,q,-p-q)+\frac{1}{2\Lambda^{2}}\Bigg{[}I_{\alpha\mu\nu}(-p-q,p,q)X_{p+q}I_{\alpha\lambda}(p+q)$
$\displaystyle+I_{\alpha\nu\lambda}(p,q,-p-q)X_{p}I_{\alpha\mu}(p)+I_{\alpha\lambda\mu}(q,-p-q,p)X_{q}I_{\alpha\nu}(q)+I_{\alpha\mu}(p)X_{\nu;\alpha\beta}(q;p,-p-q)I_{\beta\lambda}(p+q)$
$\displaystyle+I_{\alpha\nu}(q)X_{\lambda;\alpha\beta}(-p-q;q,p)I_{\beta\mu}(p)+I_{\alpha\lambda}(p+q)X_{\mu;\alpha\beta}(p;-p-q,q)I_{\beta\nu}(q)\Bigg{]}\,.$
(4.52)

Using (2.54), this is sufficient to construct the one-point vertex for the
compound kernel $\big{(}C^{-1}\big{)}^{a}{}_{b}$:

$C^{-1}_{\mu;\alpha\beta}(p;r,s)=\frac{1}{2\Lambda^{2}}A_{\mu\beta\alpha}(p,s,r)+\frac{1}{\Lambda^{2}}\left\\{\delta_{\mu\alpha}s_{\beta}Y_{s}-r_{\alpha}\delta_{\mu\beta}Y_{r}-r_{\alpha}s_{\beta}Y_{\mu}(p;r,s)\right\\}\,,$
(4.53)

and from this we also have the $C$ one-point vertex:

$C_{\mu;\alpha\beta}(p;r,s)=-C_{\alpha\gamma}(r)C^{-1}_{\mu;\gamma\delta}(p;r,s)C_{\delta\beta}(s)\,.$
(4.54)

We give further details on the construction of the compound kernels in app. A
where we also prove that up to the two-point level the compound kernels obey
the interleave identities (4.29). Note that, as a consequence of their
symmetric definition as operators, and the interleave identities, all the
kernel one-point vertices so far treated are antisymmetric under
$r_{\alpha}\leftrightarrow s_{\beta}$, for example:

$C^{-1}_{\mu;\alpha\beta}(p;r,s)=C^{-1}_{,\mu;\beta\alpha}(;p;s,r)=-C^{-1}_{\mu;\beta\alpha}(p;s,r)\,.$
(4.55)

The definition (2.68), $\mathcal{K}^{ab}=C^{a}{}_{c}\kappa^{cb}$, is not
symmetric, and thus its one-point vertex does not obey this identity. However,
using (4.54), and (4.50) with $X\mapsto\kappa$, we get an explicit expression
for it also:

$\mathcal{K}_{\mu;\alpha\beta}(p;r,s)=C_{\mu;\alpha\beta}(p;r,s)\,\kappa(s)+C_{\alpha\gamma}(r)\,\kappa_{\mu;\gamma\beta}(p;r,s)\,.$
(4.56)

We should distinguish this from the vertex made from the hermitian conjugate
$\kappa^{ac}C_{c}{}^{b}$:

$\overleftarrow{\mathcal{K}}_{\\!\mu;\alpha\beta}(p;r,s)=\kappa(r)\,C_{\mu;\alpha\beta}(p;r,s)+\kappa_{\mu;\alpha\gamma}(p;r,s)\,C_{\gamma\beta}(s)\,,$
(4.57)

and the symmetrised combination

$\overleftrightarrow{\mathcal{K}}_{\\!\mu;\alpha\beta}(p;r,s)=\frac{1}{2}\left[\mathcal{K}_{\mu;\alpha\beta}(p;r,s)+\overleftarrow{\mathcal{K}}_{\mu;\alpha\beta}(p;r,s)\right]\,,$
(4.58)

coming from $\frac{1}{2}(C^{a}{}_{c}\kappa^{cb}+\kappa^{ac}C_{c}{}^{b})$,
since these versions can also appear. As a consequence of the interleave
identities, $\overleftrightarrow{\mathcal{K}}_{\mu;\alpha\beta}(p;r,s)$ obeys
(4.55), whilst the directed versions satisfy

$\mathcal{K}_{\mu;\alpha\beta}(p;r,s)=-\overleftarrow{\mathcal{K}}_{\\!\mu;\beta\alpha}(p;s,r)\,.$
(4.59)

####  4.2.6 Large momentum behaviour

As we will see in sec. 6, these vertices play a closely similar rôle to those
of Feynman rules. In particular in one-loop diagrams, three-point vertices
will carry a loop momentum $q$ and external momentum $p$ and thus have
arguments $p$, $q$, and $-p-q$. Therefore the large momentum behaviour that
actually determines whether the flow equation is properly regularised is one
where $q\to\infty$ while keeping $p$ fixed. As before, it is sufficient to
characterise the large momentum behaviour in terms of the power of $c_{q}$,
since the proposed regularisation structure works only if the quantum
corrections are regularised overall by some negative power of $c_{q}$.

In fact the three-point action vertices222Recall that three-point action
vertices are totally antisymmetric so the order of the arguments is
irrelevant. $S_{\mu\nu\lambda}(p,q,-p-q)$ diverge at least as rapidly in terms
of $c_{q}$, as their corresponding two-point action vertices
$S_{\mu\lambda}(q)$. This follows from the Ward identity:

$q^{\nu}S_{\mu\nu\lambda}(p,q,-p-q)=S_{\mu\lambda}(p)-S_{\mu\lambda}(p+q)\,.$
(4.60)

Similarly the degree of divergence of a one-point kernel is set by the zero-
point kernel. If $q$ goes from end to end, the one-point kernel behaves with
at least the same power of $c_{q}$ as $K_{q}$:

$p^{\lambda}K_{\lambda;\mu\nu}(p;q,-p-q)=K_{\mu\nu}(q)-K_{\mu\nu}(q+p)\,.$
(4.61)

But if $q$ passes through the side, the large $q$ behaviour depends on whether
$K_{q}$ diverges or decays:

$q^{\lambda}K_{\lambda;\mu\nu}(q;p,-p-q)=K_{\mu\nu}(p)-K_{\mu\nu}(q+p)\,.$
(4.62)

If $K_{q}$ diverges, then the one-point kernel diverges at least as rapidly,
whilst if $K_{q}$ decays then $K_{\lambda;\mu\nu}(q;p,-p-q)$ cannot decay,
because the right hand side of the Ward identity has $K_{\mu\nu}(p)$ which is
independent of $q$.

However, Ward identities only set a lower bound on the power of $c_{q}$,
because there remains the possibility that vertices have a worse behaving
transverse part. We now pin down the precise behaviour as a power of $c_{q}$,
i.e. both longitudinal and transverse parts.333For a detailed analysis of the
large momentum behaviour of arbitrary $n$ higher-point (simple) kernel
vertices see sec. 5 of ref. [16].

We start by analysing $A^{(3)}$. This is displayed in (4.2.5). The fundamental
vertices $I^{(n)}$ in (4.2.5) only provide some power of $q$ at worst, cf.
(2.14) – (2.16), so $A^{(3)}$s large $q$ behaviour is set by its kernels. From
(2.30) we have $X_{q}\approx X_{p+q}\approx c_{q}^{-1}$, and from (4.50) we
see that all the $X$ one-point vertices in (4.2.5) also diverge in this way.
Two terms in $A^{(3)}$ do not diverge with the cutoff, namely the pure $I$
term (i.e. the first term) and the term containing $X_{p}$ on the second line,
but the others dominate, so overall $A^{(3)}\approx c^{-1}_{q}$, as previously
assumed, and matching its two-point vertex (2.51) as per the minimum required
by the Ward identity.

Since from (4.6) the classical action $\theta_{0}$ has the same formula as
(4.2.5), only with the replacement $X\mapsto W$, we see immediately that its
three-point vertex also $\approx c^{-1}_{q}$. In fact, given that the quantum
correction is $\propto A$, this holds for the full $\theta^{(3)}$.

We can borrow the same formula also for the seed action $\hat{S}$. From (2.59)
we replace $I\mapsto A$ and $X\mapsto 1$ on the right hand side of (4.2.5)
(and so delete the three terms involving the $X$ one-point vertex). Since we
have already established that $A_{\mu\nu\lambda}(p,q,-p-q)\approx c^{-1}_{q}$,
we see that indeed $\hat{S}_{\mu\nu\lambda}(p,q,-p-q)\approx c^{-2}_{q}$ again
as previously assumed and enforced by the Ward identity.

It is not hard to see that these estimates hold also for the higher point
$A^{(n)}$, $\theta^{(n)}$ and $\hat{S}^{(n)}$ action vertices, where again two
of their momentum arguments diverge as $\sim\pm q$ with the others held fixed.
For the four-point example, their explicit formulae can be read off from the
right hand side of (6.11). The same is true for higher point kernel vertices
where the zero-point kernel diverges at large momentum. However we will only
use the estimates for the three-point vertices in this paper.

At one loop we need also the classical effective action three-point vertex and
this in turn depends on the kernels $\mathcal{K}$ and $C$ and their one-point
vertices (see eqn. (6.10)). From (4.45) we clearly have
$C_{\alpha\beta}(q)\approx c_{q}$ (recall $Y_{q}\approx c^{-1}_{q}$). Clearly
from (4.53), the $C^{-1}$ one-point vertex $\approx c^{-1}_{q}$ whatever the
arrangement of the arguments $p$, $q$, and $-p-q$, this being the minimum
imposed by the Ward identity. But from (4.54) the arrangement of the arguments
matters for $C$:

$\displaystyle C_{\mu;\alpha\beta}(p;q,-p-q)\approx C_{q}$
$\displaystyle\approx c_{q}\,,$ (4.63) $\displaystyle
C_{\nu;\alpha\beta}(q;p,-p-q)$ $\displaystyle\approx 1\,,$ (4.64)

again precisely as predicted by the Ward identities. One can check using the
explicit formulae from the previous section that the behaviour predicted by
the Ward identities, holds for all the kernel vertices whose zero-point
kernels decay at large momentum, namely $t$, $\kappa$, $C$ and $\mathcal{K}$.
In particular we have also

$\displaystyle\mathcal{K}_{\mu;\alpha\beta}(p;q,-p-q)\approx\mathcal{K}_{q}$
$\displaystyle\approx c^{2}_{q}\,,$ (4.65)
$\displaystyle\mathcal{K}_{\nu;\alpha\beta}(q;p,-p-q)$ $\displaystyle\approx
1\,.$ (4.66)

##  5 Regularisation of higher point classical vertices

As we saw in sec. 2, the space-time trace terms are designed to implement PV
regularisation of the $S_{UV}\approx c^{-2}_{q}$ part. We will see in sec. 6.4
that there are subtleties with this however. The remaining regularisation
works provided $\mathcal{S}^{(n\geq 3)}_{0}$ corrections diverge slower than
$c^{-2}_{q}$. This was the key property set out in eqns. (2.60) and (2.63),
where it was assumed that in fact $\mathcal{S}^{(n)}_{0}\approx c^{-1}_{q}$,
or equivalently $\Sigma^{(n)}_{0}\approx c^{-1}_{q}$. For the three-point we
can now state this more precisely as:

$\Sigma^{0}_{\beta\nu\alpha}(q+p,-p,-q)\approx c^{-1}_{q}\,.$ (5.1)

In this section we confirm in detail that this estimate holds true. It is then
clear that the estimates are correct for all the higher point
$\Sigma^{(n)}_{0}$ vertices also.

In order to compute $\Sigma^{(3)}_{0}$ we need the classical action three-
point vertex. The latter follows from the flow equation $\Lambda
D_{\Lambda}S^{0}=0$. The analysis is simpler if instead we recast it directly
as a flow for $\Sigma_{0}$. Taking the classical limit (i.e. $O(1/g^{2})$ in
the loop expansion) of (2.70), and using the identity (2.66), we have

$\dot{\Sigma}^{0}=-\dot{A}+\frac{1}{2}\Sigma^{0}_{,a}\mathcal{K}^{ab}\Sigma^{0}_{,b}+\frac{1}{2}{A}_{,a}\kappa^{ab}\Sigma^{0}_{,b}-\Sigma^{0}_{,a}C^{ab}\theta^{0}_{,b}\,.$
(5.2)

Expanding to the three-point level:

$\displaystyle\begin{split}\dot{\Sigma}^{0}_{\mu\nu\lambda}(p,q,-p-q)=&-\dot{A}^{0}_{\mu\nu\lambda}(p,q,-p-q)\\\
&+\frac{1}{2\Lambda^{2}}\Sigma^{0}_{\mu\alpha}(p)\mathcal{K}_{T}(p)\Sigma^{0}_{\alpha\nu\lambda}(p,q,-p-q)\\\
&+\frac{1}{2\Lambda^{2}}\Sigma^{0}_{\mu\alpha}(p)\overleftrightarrow{\mathcal{K}}_{\\!\nu;\alpha\beta}(q;p,-p-q)\Sigma^{0}_{\beta\lambda}(p+q)\\\
&+\frac{1}{4\Lambda^{2}}\Sigma^{0}_{\alpha\mu}(p)\kappa(p)A_{\alpha\nu\lambda}(p,q,-p-q)\\\
&+\frac{1}{4\Lambda^{2}}\Sigma^{0}_{\mu\nu\alpha}(p,q,-p-q)\kappa(p+q)A_{\alpha\lambda}(p+q)\\\
&+\frac{1}{4\Lambda^{2}}\Sigma^{0}_{\mu\alpha}(p)\kappa_{\nu;\alpha\beta}(q;p,-p-q)A_{\beta\lambda}(p+q)\\\
&-\frac{1}{4\Lambda^{2}}\Sigma^{0}_{\mu\alpha}(p)\kappa_{\lambda;\alpha\beta}(-p-q;p,q)A_{\beta\nu}(q)\\\
&-\frac{1}{2\Lambda^{2}}\Sigma^{0}_{\alpha\mu}(p)C_{T}(p)\theta^{0}_{\alpha\nu\lambda}(p,q,-p-q)\\\
&-\frac{1}{2\Lambda^{2}}\Sigma^{0}_{\mu\nu\alpha}(p,q,-p-q)C_{T}(p+q)\theta^{0}_{\beta\lambda}(p+q)\\\
&-\frac{1}{2\Lambda^{2}}\Sigma^{0}_{\mu\alpha}(p)C_{\nu;\alpha\beta}(q;p,-p-q)\theta^{0}_{\beta\lambda}(p+q)\\\
&+\frac{1}{2\Lambda^{2}}\Sigma^{0}_{\mu\alpha}(p)C_{\lambda;\alpha\beta}(-p-q;p,q)\theta^{0}_{\beta\nu}(q)\\\
&+\text{cycles}\,.\end{split}$ (5.3)

Here we have used properties of the kernels. First of all, if we contract a
zero-point kernel into a two-point vertex, e.g. as on second line, then only
its transverse part contributes because of the transversality of the two-point
function. In addition to this, we used that $\mathcal{K}^{a}{}_{b}$,
$C^{a}{}_{b}$ and $\kappa^{a}{}_{b}$, kernel one-point vertices obey
interleave identities (4.29).

Those terms that depend on $\Sigma_{0}^{(3)}$ in (5.3) we take to the left of
the equation. Their multiplying factors collect into

$Z_{\mu\nu}(p_{i})=\mathcal{K}_{T}(p_{i})\,\Sigma^{0}_{\mu\nu}(p_{i})$ (5.4)

for each external momentum $p_{i}$, as follows from the identity

$C_{T}(p)\,\theta^{0}_{\mu\nu}(p)=\frac{1}{2}\mathcal{K}_{T}(p)\,S^{0}_{\mu\nu}(p)$
(5.5)

(this can be shown from their explicit form given in secs. 4.2.3, 4.2.4) and
$\kappa(p)A_{\mu\nu}(p)=\mathcal{K}_{T}(p)\hat{S}_{\mu\nu}(p)$, which is the
identity (2.66) at two-point level. Using the results of the previous section
it is straightforward to verify explicitly that the remaining terms, which we
call $T_{\mu\nu\lambda}(p,q,-p-q)$, diverge as
$T_{\mu\nu\lambda}(p,q,-p-q)\approx c^{-1}_{q}$. Indeed, given that in (5.2)
the action vertices diverge as $c^{-1}_{q}$, the only way to produce a
contribution that diverges faster than this would be to have the high momentum
flow, $q$, go through both action factors. But this requires $q$ to flow
through the kernel from end to end, thus reducing the divergence by at least a
factor of $c_{q}$ again ($c_{q}$ for $\kappa$ and $C$, while $\mathcal{K}$
would provide $c^{2}_{q}$). In this way we verify $\Sigma^{(n)}_{0}\approx
c_{q}^{-1}$ for all $n$, i.e. (2.64) at the classical level.

The flow at the three-point level can be written now as

$\Lambda\partial_{\Lambda}\Sigma^{0}_{\mu\nu\lambda}(p,q,-p-q)\\\
-\frac{1}{4\Lambda^{2}}\Big{[}Z_{\mu\alpha}(p)\Sigma^{0}_{\alpha\nu\lambda}(p,q,-p-q)+Z_{\nu\alpha}(q)\Sigma^{0}_{\alpha\lambda\mu}(q,-p-q,p)+Z_{\lambda\alpha}(p+q)\Sigma^{0}_{\alpha\mu\nu}(-p-q,p,q)\Big{]}\\\
=T_{\mu\nu\lambda}(p,q,-p-q)\,.$ (5.6)

This is in a form where it can also be integrated with respect to $\Lambda$.
$T_{\mu\nu\lambda}(p,q,-p-q)$ is a known function which can be constructed
explicitly from the vertices discussed in the previous sections. The terms in
square brackets on the left hand side of (5.6) provide the integrating factor
for the differential equation. Regarding $Z_{\mu\nu}(p)$ as a matrix $Z(p)$,
we define

$\zeta_{\mu\nu}(p)=\left[\exp\int_{\Lambda}^{\infty}\frac{d\Lambda_{1}}{4\Lambda^{3}_{1}}\,Z(p)\right]_{\mu\nu}=\delta_{\mu\nu}+\frac{\Box_{\mu\nu}(p)}{p^{2}}\left(-1+\exp\int_{\Lambda}^{\infty}\frac{d\Lambda_{1}}{2\Lambda_{1}}\,\tilde{p}^{2}\mathcal{K}_{T}(p)F_{\Sigma}(p)\right)\,,$
(5.7)

where for the second equality we use (5.4) and (4.40). The integrated
$\Sigma^{(3)}_{0}$ is then given by

$\Sigma^{0}_{\mu\nu\lambda}(p,q,-p-q)=\\\
-\zeta^{-1}_{\mu\alpha}(p)\,\zeta^{-1}_{\nu\beta}(q)\,\zeta^{-1}_{\lambda\gamma}(-p-q)\int^{\infty}_{\Lambda}\frac{d\Lambda_{1}}{\Lambda_{1}}\,\zeta_{\alpha\alpha^{\prime}}(p)\,\zeta_{\beta\beta^{\prime}}(q)\,\zeta_{\gamma\gamma^{\prime}}(-p-q)\,T_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}(p,q,-p-q)\,,$
(5.8)

where $\zeta^{-1}_{\mu\nu}(p)$ is of course given by the same expression as
(5.7) except for a minus sign in the exponential, and it is understood that
all terms under a $\Lambda_{1}$-integral are evaluated at cutoff scale
$\Lambda_{1}$. Note that the integration constant vanishes because, by gauge
invariance and dimensions, $\Sigma_{0}$ vanishes in the limit
$\Lambda\to\infty$.444As can be seen by Taylor expanding (4.40) for example,
$\Sigma_{0}$ has a minimum of four space-time derivatives and thus has an
overall $1/\Lambda^{2}$ factor. Expanding the exponential in (5.7) we see that
all the corrections that arise from these integrating factors are also
transverse. In this sense the full vertex involves in fact an exponentiation
of the equations of motion. It does not affect the divergence with $c_{q}$
however, since $Z(q)$ vanishes in the large $q$ limit. The integrated version
of the $n\geq 4$-point vertices can be explicitly written down in a similar
way.

##  6 One loop beta function

To see why it is the large $q$ behaviour of the above vertices that is
important, and to provide a test of the formalism, we focus on the simplest
quantum correction: the one-loop contribution to the effective action two-
point vertex. Since gauge invariance is exactly preserved this computation
should, if the flow equation is regularised correctly, also yield the one-loop
beta function (i.e. $\beta_{1}=\eta_{1}/2$). To extract it, we need to define
the coupling constant $g(\Lambda)$ beyond classical level. We do this by
imposing a convenient renormalization condition:

$S=\frac{1}{2g^{2}(\Lambda)}\text{tr}\int_{x}F_{\mu\nu}^{2}+O(\partial^{3})\,,$
(6.1)

which means that the full two-point function of $S$ at $O(p^{2})$ is given by:

$S_{\mu_{1}\mu_{2}}(p)\big{|}_{O(p^{2})}=2\>\square_{\mu_{1}\mu_{2}}(p)\>.$
(6.2)

Extracting the $O(p^{2})$ part of (4.39) we have that:

$S^{0}_{\mu_{1}\mu_{2}}(p)\big{|}_{O(p^{2})}=2\,\square_{\mu_{1}\mu_{2}}(p)=S_{\mu_{1}\mu_{2}}(p)\big{|}_{O(p^{2})}\>.$
(6.3)

This means that the renormalization condition at $O(p^{2})$ is already
saturated at tree level, and, thus, all higher order loop contributions must
vanish,

$\displaystyle S^{n}_{\mu_{1}\mu_{2}}(p)\big{|}_{O(p^{2})}=0$
$\displaystyle,\forall n\geq 1\,.$ (6.4)

Since $\Sigma^{0}=S^{0}-\hat{S}$ and $\Sigma^{n}=S^{n}$, we also obtain that:

$\displaystyle\Sigma^{n}_{\mu_{1}\mu_{2}}(p)\big{|}_{O(p^{2})}$
$\displaystyle=0$ $\displaystyle,\>\forall n\geq 1\>.$ (6.5)

The remaining action $\theta$ behaves differently with respect to $S$ and
$\Sigma$. Its loopwise expansion was already given in (4.7) and from there we
see that at $O(p^{2})$ we have a non-vanishing higher loop contribution from
its two-point vertex:

$\theta^{n}_{\mu_{1}\mu_{2}}(p)\big{|}_{O(p^{2})}=\eta_{n}\square_{\mu_{1}\mu_{2}}(p).$
(6.6)

We computed explicitly all the relevant classical action two-point vertex
functions in sec. 4.2.3.

At this point we are able to recast (4.1) as an algebraic equation for
$\beta_{1}$. If we look at the flow equation of $S^{1}_{\mu\nu}(p)$ and
restrict it to $O(p^{2})$, we obtain:

$-4\beta_{1}\square_{\mu\nu}(p)=\text{Tr}\Bigg{\\{}\Lambda\partial_{\Lambda}\Big{[}\ln\big{(}C^{-1}\big{)}^{a}{}_{b}\Big{]}-\Lambda\partial_{\Lambda}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}-\frac{1}{2}\Lambda\partial_{\Lambda}\big{[}\ln
b^{\alpha}{}_{\beta}\big{]}\\\
+\Psi_{0}^{d}\frac{\delta}{\delta\phi^{d}}\Big{[}\ln\big{(}C^{-1}\big{)}^{a}{}_{b}\Big{]}-\Psi_{0}^{a}\frac{\delta}{\delta\phi^{a}}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}-\frac{1}{2}\Psi_{0}^{a}\frac{\delta}{\delta\phi^{a}}\big{[}\ln
b^{\alpha}{}_{\beta}\big{]}+\frac{\delta\Psi_{0}^{a}}{\delta\phi^{b}}\Bigg{\\}}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{2})\end{subarray}}\,.$ (6.7)

Here we have used the requirement imposed on us by the renormalization
condition (6.1) and the fact that $\Sigma^{0}_{\mu\nu}(p)$ is $O(p^{4})$,
whereas $\theta^{0}_{\mu\nu}(p)$ and $S^{0}_{\mu\nu}(p)$ are $O(p^{2})$. The
only task we are left with is to evaluate the right-hand side of (6.7), and
then $\beta_{1}$ can be extracted from its coefficient.

###  6.1 Classical flow equations

The classical flow equation (4.9) can equivalently be written as follows:

$\Lambda\partial_{\Lambda}S^{0}=\frac{1}{2\Lambda^{2}}\frac{\delta
S^{0}}{\delta
A_{\alpha}}\cdot\mathcal{K}_{\alpha\beta}\cdot\frac{\delta\Sigma^{0}}{\delta
A_{\beta}}-\frac{1}{\Lambda^{2}}\frac{\delta S^{0}}{\delta A_{\alpha}}\cdot
C_{\alpha\beta}\cdot\frac{\delta\theta^{0}}{\delta A_{\beta}}\,.$ (6.8)

The right-hand side of the above expression has two terms with similar
structure, and thus, to write the flow equation for some $n$-point vertex, one
just needs to compute the contributions coming from one term as the
contributions coming from the second one follow from the former by a mere
relabelling. We write below the flow equations for those vertices that are
relevant for the computation of $\beta_{1}$.

The flow equation for $S^{0}_{\mu_{1}\mu_{2}}(p)$ takes the following form:

$\Lambda\partial_{\Lambda}S^{0}_{\mu_{1}\mu_{2}}(p)=\frac{1}{4\Lambda^{2}}S^{0}_{\alpha\mu_{1}}(p)\mathcal{K}_{T}(p)\Sigma^{0}_{\alpha\mu_{2}}(p)-\frac{1}{2\Lambda^{2}}S^{0}_{\alpha\mu_{1}}(p)C_{T}(p)\theta^{0}_{\alpha\mu_{2}}(p)+(\mu_{1}\leftrightarrow\mu_{2})\,.$
(6.9)

By gauge invariance and dimensions, $S^{0}_{\mu\nu}(p)$ (and in fact any two-
point function) must have a structure similar to (4.39). If we substitute this
into the above flow equation and solve for some function $F(p)$, we recover
the same result as stated in (4.39). This is an extra check that the flow
equation (6.9) is derived in a consistent manner.

The flow equation for $S^{0}_{\mu_{1}\mu_{2}\mu_{3}}(p_{1},p_{2},p_{3})$ is
given by:

$\displaystyle\Lambda\partial_{\Lambda}S^{0}_{\mu_{1}\mu_{2}\mu_{3}}(p_{1},p_{2},p_{3})$
$\displaystyle=\frac{1}{4\Lambda^{2}}$ $\displaystyle\Big{[}$ $\displaystyle
S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{\mu_{2};\alpha\beta}(p_{2};p_{1},p_{3})\Sigma^{0}_{\beta\mu_{3}}(p_{3})$
(6.10)
$\displaystyle-S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{\mu_{3};\alpha\beta}(p_{3};p_{1},p_{2})\Sigma^{0}_{\beta\mu_{2}}(p_{2})$
$\displaystyle+S^{0}_{\alpha\mu_{1}\mu_{2}}(p_{3},p_{1},p_{2})\mathcal{K}_{T}(p_{3})\Sigma^{0}_{\alpha\mu_{3}}(p_{3})$
$\displaystyle+S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{T}(p_{1})\Sigma^{0}_{\alpha\mu_{2}\mu_{3}}(p_{1},p_{2},p_{3})\Big{]}$
$\displaystyle-\frac{1}{2\Lambda^{2}}$ $\displaystyle\Big{[}$
$\displaystyle\mathcal{K}\mapsto C\text{ ;
}\Sigma\mapsto\theta\Big{]}+\text{cycles}\,,$

where the terms inside brackets from the last line follow from those written
explicitly inside brackets above them after the relabelling indicated. One can
proceed similarly to derive the flow equation for
$S^{0}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p_{1},p_{2},p_{3},p_{4})$:

$\displaystyle\begin{split}\Lambda\partial_{\Lambda}S^{0}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p_{1},p_{2},p_{3},p_{4})=\frac{1}{4\Lambda^{2}}&\Big{[}S^{0}_{\alpha\mu_{1}\mu_{2}\mu_{3}}(p_{4},p_{1},p_{2},p_{3})\mathcal{K}_{T}(p_{4})\Sigma^{0}_{\alpha\mu_{4}}(p_{4})\\\
&+S^{0}_{\alpha\mu_{1}\mu_{2}}(-p_{1}-p_{2},p_{1},p_{2})\mathcal{K}_{\alpha\beta}(p_{1}+p_{2})\Sigma^{0}_{\beta\mu_{3}\mu_{4}}(-p_{3}-p_{4},p_{3},p_{4})\\\
&+S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{T}(p_{1})\Sigma^{0}_{\alpha\mu_{2}\mu_{3}\mu_{4}}(p_{1},p_{2},p_{3},p_{4})\\\
&+S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{\mu_{2};\alpha\beta}(p_{2};p_{1},p_{3}+p_{4})\Sigma^{0}_{\beta\mu_{3}\mu_{4}}(-p_{3}-p_{4},p_{3},p_{4})\\\
&-S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{\mu_{4};\alpha\beta}(p_{4};p_{1},p_{2}+p_{3})\Sigma^{0}_{\beta\mu_{2}\mu_{3}}(-p_{2}-p_{3},p_{2},p_{3})\\\
&+S^{0}_{\alpha\mu_{1}\mu_{2}}(-p_{1}-p_{2},p_{1},p_{2})\mathcal{K}_{\mu_{3};\alpha\beta}(p_{3};p_{1}+p_{2},p_{4})\Sigma^{0}_{\beta\mu_{4}}(p_{4})\\\
&-S^{0}_{\alpha\mu_{1}\mu_{2}}(-p_{1}-p_{2},p_{1},p_{2})\mathcal{K}_{\mu_{4};\alpha\beta}(p_{4};p_{1}+p_{2},p_{3})\Sigma^{0}_{\beta\mu_{3}}(p_{3})\\\
&+S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{\mu_{2}\mu_{3};\alpha\beta}(p_{2},p_{3};p_{1},p_{4})\Sigma^{0}_{\beta\mu_{4}}(p_{4})\\\
&-S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{\mu_{2}\mu_{4};\alpha\beta}(p_{2},p_{4};p_{1},p_{3})\Sigma^{0}_{\beta\mu_{3}}(p_{3})\\\
&-S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{\mu_{4}\mu_{2};\alpha\beta}(p_{4},p_{2};p_{1},p_{3})\Sigma^{0}_{\beta\mu_{3}}(p_{3})\\\
&+S^{0}_{\alpha\mu_{1}}(p_{1})\mathcal{K}_{\mu_{4}\mu_{3};\alpha\beta}(p_{4},p_{3};p_{1},p_{2})\Sigma^{0}_{\beta\mu_{2}}(p_{2})\Big{]}\\\
-\frac{1}{2\Lambda^{2}}&\Big{[}\mathcal{K}\mapsto C\text{ ;
}\Sigma\mapsto\theta\Big{]}+\text{cycles}\,.\end{split}$ (6.11)

In deriving these, we again exploit the interleave identities (4.29) up to and
including the two-point level, as proven in appendix A.

Although we will not need them in this paper, let us note that these equations
can be integrated to give explicit formulae for the (integrated) classical
effective action vertices. Taking all the $S^{(3)}_{0}$ terms to the left hand
side in (6.10), including the $S^{(3)}_{0}$ part of $\Sigma^{(3)}_{0}$, we
find the same integrating factors $Z_{\mu_{i}\alpha}(p_{i})$ as in (5.4). Thus
the flow for $S^{(3)}_{0}$ can be similarly integrated. This results in an
explicit formula for $S^{(3)}_{0}$ of the same form as (5.8) – except that in
this case there is also an integration constant, namely
$I_{\mu_{1}\mu_{2}\mu_{3}}(p_{1},p_{2},p_{3})$. The latter follows because the
$\Lambda\to\infty$ limit of $S^{(3)}_{0}$ is $I^{(3)}$, as e.g. can be seen
from (6.1) and (2.12). Similarly, taking all $S^{(4)}_{0}$ terms to the left
hand side in (6.11) gives the same integrating factors and allows its flow to
be integrated up to an explicit formula for $S^{(4)}_{0}$. This time on the
right hand side we have instances of the just computed $S^{(3)}_{0}$ vertex,
and now the integration constant is $I^{(4)}$ as in (2.16).

###  6.2 Trace terms

To evaluate the trace terms in (6.7) above one needs to expand the log terms.
The vertex expansion for the $B^{\alpha}{}_{\beta}$ terms, for example, can be
done in the following way:

$\displaystyle\text{Tr}\ln B^{\alpha}{}_{\beta}$
$\displaystyle\equiv\text{Tr}\ln\big{[}\overbrace{B_{r}}^{O(A_{\mu}^{0})}+\overbrace{B_{\mu}(p;r,s)}^{O(A^{1}_{\mu})}+\overbrace{B_{\mu\nu}(p,q;r,s)}^{O(A_{\mu}^{2})}+\ldots\big{]}$
(6.12) $\displaystyle=\int_{r}\ln
B_{r}+\text{Tr}\ln\big{[}1+B_{r}^{-1}B_{\mu}(p;r,s)+B_{r}^{-1}B_{\mu\nu}(p,q;r,s)+\ldots\big{]}$
(6.13)
$\displaystyle=2N\int_{q}\underbrace{B_{q}^{-1}B_{\mu\nu}(p,-p;q,-q)-\frac{1}{2}B_{q}^{-1}B_{\mu}(p;q,-q-p)B^{-1}_{p+q}B_{\nu}(-p;p+q,-q)}_{O(A_{\mu}^{2})}+\ldots\,,$
(6.14)

where $N$ accounts for group combinatorics and the extra factor of two for the
$p_{\mu}\leftrightarrow-p_{\nu}$ symmetry (or, equivalently, for the
definition of the two-point function). Here note that in going from the second
to the third line we have discarded the log term because it is just a vacuum
contribution to the flow equation (or it is differentiated away), and we have
also Taylor expanded the remaining log term using

$\ln(1+x)=x-\frac{x^{2}}{2}+\cdots\,.$ (6.15)

Similar expressions can be written for $\big{(}C^{-1}\big{)}^{a}{}_{b}$ and
$b^{\alpha}{}_{\beta}$ terms, respectively.

To evaluate the remaining trace term $\delta\Psi_{0}^{a}/\delta\phi^{a}$ it is
useful to use its covariant representation:

$\displaystyle\frac{\delta\Psi_{0}^{a}}{\delta\phi^{a}}$
$\displaystyle=\frac{\delta}{\delta\phi^{a}}\bigg{(}-\frac{1}{2}\mathcal{K}^{ab}{\Sigma^{0}}_{,b}+C^{ab}{\theta^{0}}_{,b}\bigg{)}$
(6.16) $\displaystyle\equiv-\frac{1}{2\Lambda^{2}}\frac{\delta}{\delta
A_{\alpha}}\cdot\mathcal{K}_{\alpha\beta}\cdot\frac{\delta\Sigma^{0}}{\delta
A_{\beta}}+\frac{1}{\Lambda^{2}}\frac{\delta}{\delta A_{\alpha}}\cdot
C_{\alpha\beta}\cdot\frac{\delta\theta^{0}}{\delta A_{\beta}}\,.$ (6.17)

This allows us to compute it at the two-point level using (2.20), yielding:

$\displaystyle\begin{split}\frac{\delta\Psi_{0}^{a}}{\delta\phi^{a}}\bigg{|}_{\text{2-pt.function}}&=\frac{1}{2\Lambda^{2}}\int_{q}\Bigg{\\{}2\bigg{(}N-\frac{1}{N}\bigg{)}C_{\alpha\beta}(q)\theta^{0}_{\beta\alpha\mu\nu}(q,-q,p,-p)\\\
&\qquad\qquad\quad-\frac{1}{N}C_{\alpha\beta}(q)\theta^{0}_{\beta\mu\alpha\nu}(q,p,-q,-p)\\\
&\qquad\qquad\quad+2NC_{\mu;\alpha\beta}(p;q,-p-q)\theta^{0}_{\beta\nu\alpha}(q+p,-p,-q)\\\
&\qquad\qquad\quad+2NC_{\mu\nu;\alpha\beta}(p,-p;q,-q)\theta^{0}_{\beta\alpha}(q)\\\
&\qquad\qquad\quad-2NC_{\mu\alpha;\alpha\beta}(p,-q;q,-p)\theta^{0}_{\beta\nu}(p)\Bigg{\\}}\\\
&-\frac{1}{4\Lambda^{2}}\big{[}C\mapsto\mathcal{K}\text{ ;
}\theta\mapsto\Sigma\big{]}\,.\end{split}$ (6.18)

This can be further refined. If we split the kernels into their transverse and
longitudinal parts, respectively, then one can check explicitly that the
$O(1/N)$ longitudinal pieces cancel each other after using Ward identities.
One can also show that, under the $q$ integral and by using Lorentz invariance
($q\rightarrow-q$, $p_{\mu}\leftrightarrow-p_{\nu}$ etc. ) together with
(4.6), (2.59), and (6.11), the following identities hold:

$\displaystyle\theta^{0}_{\alpha\mu\alpha\nu}(q,p,-q,-p)$
$\displaystyle=-2\,\theta^{0}_{\alpha\alpha\mu\nu}(q,-q,p,-p)\,,$ (6.19)
$\displaystyle\Sigma^{0}_{\alpha\mu\alpha\nu}(q,p,-q,-p)$
$\displaystyle=-2\,\Sigma^{0}_{\alpha\alpha\mu\nu}(q,-q,p,-p)\,.$ (6.20)

This means that the $O(1/N)$ transverse pieces vanish as well, and thus (6.18)
becomes:

$\displaystyle\begin{split}\frac{\delta\Psi_{0}^{a}}{\delta\phi^{a}}\bigg{|}_{\text{2-pt.function}}&=\frac{N}{\Lambda^{2}}\int_{q}\Big{\\{}C_{\alpha\beta}(q)\theta^{0}_{\beta\alpha\mu\nu}(q,-q,p,-p)\\\
&\qquad\qquad+C_{\mu;\alpha\beta}(p;q,-p-q)\theta^{0}_{\beta\nu\alpha}(q+p,-p,-q)\\\
&\qquad\qquad+C_{\mu\nu;\alpha\beta}(p,-p;q,-q)\theta^{0}_{\beta\alpha}(q)\\\
&\qquad\qquad-
C_{\mu\alpha;\alpha\beta}(p,-q;q,-p)\theta^{0}_{\beta\nu}(p)\Big{\\}}\\\
&-\frac{N}{2\Lambda^{2}}\big{[}C\mapsto\mathcal{K}\text{ ;
}\theta\mapsto\Sigma\big{]}\,.\end{split}$ (6.21)

###  6.3 Two-point vertex at zeroth order in momentum

If the one-loop two-point vertex is transverse and quasi-local, as it should
be, it can have no momentum independent part, i.e. at $O(p^{0})$ it ought to
vanish. However this property is strictly only true if the flow equation is
properly regulated since, as we will see, it would then follow from the fact
that this contribution can be cast as a momentum-space surface integral at
large $q$, which vanishes if properly regularised. Actually with sufficient
care the construction proposed in ref. [38] does ensure this, up to a
divergent term that can be discarded since it is $\Lambda$ independent but
differentiated by $\Lambda$, because it in fact depends only on two-point
action vertices or zero-point kernels, as we will see.

Thus we turn to computing the two-point $O(p^{0})$ part of the one-loop flow
equation (4.1), or equivalently (4.1). As we have just noted, it ought to
vanish. The first line on the right-hand side is clearly of $O(p^{2})$ and
above and can therefore be discarded. This means that the only contribution
comes from the functional trace terms, and hence, at $O(p^{0})$, the right
hand side of (4.1) becomes:

$\text{Tr}\Bigg{\\{}\Lambda\partial_{\Lambda}\Big{[}\ln\big{(}C^{-1}\big{)}^{a}{}_{b}\Big{]}-\Lambda\partial_{\Lambda}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}-\frac{1}{2}\Lambda\partial_{\Lambda}\big{[}\ln
b^{\alpha}{}_{\beta}\big{]}+\frac{\delta\Psi_{0}^{a}}{\delta\phi^{b}}\\\
+\Psi_{0}^{m}\frac{\delta}{\delta\phi^{m}}\Big{[}\ln\big{(}C^{-1}\big{)}^{a}{}_{b}\Big{]}-\Psi_{0}^{m}\frac{\delta}{\delta\phi^{m}}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}-\frac{1}{2}\Psi_{0}^{m}\frac{\delta}{\delta\phi^{m}}\big{[}\ln
b^{\alpha}{}_{\beta}\big{]}\Bigg{\\}}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{0})\end{subarray}}\,.$ (6.22)

Moreover, one can easily see that the second line above does not contribute at
$O(p^{0})$ because the blocking functional part (i.e. $\Psi_{0}^{m}$) has
either a two-point function residue which is at least of $O(p^{2})$, or a
three-point function residue which is at least $O(p)$ (furthermore it is
multiplied by a term that is antisymmetric in $q$, and hence vanishes under
the $q$ integral). The first three terms can be computed using trace
expansions as described in section 6.2 above. For example, using (6.14) we
will obtain the following:

$\text{Tr}\bigg{\\{}\Lambda\partial_{\Lambda}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}\bigg{\\}}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{0})\end{subarray}}\equiv
2N\Lambda\partial_{\Lambda}\int_{q}B_{q}^{-1}B_{\mu\nu}(0,0;q,-q)-\frac{1}{2}B_{q}^{-1}B_{\mu}(0;q,-q)B^{-1}_{q}B_{\nu}(0;q,-q)\,.$
(6.23)

Here note that quasi-locality ensures that the $p\to 0$ limit is
straightforward. The above expression can be simplified by using differential
Ward identities:

$\displaystyle\text{Tr}\bigg{\\{}\Lambda\partial_{\Lambda}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}\bigg{\\}}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{0})\end{subarray}}$ $\displaystyle\equiv
2N\Lambda\partial_{\Lambda}\int_{q}B_{q}^{-1}\frac{1}{2}\partial_{\mu}^{q}\partial_{\nu}^{q}B_{q}-\frac{1}{2}B_{q}^{-1}\partial_{\mu}^{q}B_{q}\,B^{-1}_{q}\partial_{\nu}^{q}B_{q}$
(6.24)
$\displaystyle=N\Lambda\partial_{\Lambda}\int_{q}\partial_{\mu}^{q}\partial_{\nu}^{q}\big{[}\ln
B_{q}\big{]}\,.$ (6.25)

Similar expressions can be computed for $C$ and $b$ terms, respectively. The
remaining term can be computed starting from (6.21) and following a similar
strategy, yields:

$\displaystyle\begin{split}\frac{\delta\Psi_{0}^{a}}{\delta\phi^{a}}\Bigg{|}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at
}O(p^{0})\end{subarray}}&=\frac{N}{\Lambda^{2}}\int_{q}\Big{\\{}C_{\alpha\beta}(q)\frac{1}{2}\partial_{\mu}^{q}\partial_{\nu}^{q}\theta^{0}_{\beta\alpha}(q)\\\
&\qquad\qquad+\partial_{\mu}^{q}C_{\alpha\beta}(q)\,\partial_{\nu}^{q}\theta^{0}_{\beta\alpha}(q)\\\
&\qquad\qquad+\frac{1}{2}\partial_{\mu}^{q}\partial_{\nu}^{q}C_{\alpha\beta}(q)\,\theta^{0}_{\beta\alpha}(q)\Big{\\}}\\\
&-\frac{N}{2\Lambda^{2}}\big{[}C\mapsto\mathcal{K}\text{ ;
}\theta\mapsto\Sigma\big{]}\,,\end{split}$ (6.26)

which can be recast as:

$\displaystyle\frac{\delta\Psi_{0}^{a}}{\delta\phi^{a}}\Bigg{|}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{0})\end{subarray}}$
$\displaystyle=\frac{N}{4\Lambda^{2}}\int_{q}\partial_{\mu}^{q}\partial_{\nu}^{q}\big{[}2C_{T}(q)\theta^{0}_{\alpha\alpha}(q)-\mathcal{K}_{T}(q)\Sigma^{0}_{\alpha\alpha}(q)\big{]}$
(6.27)
$\displaystyle=\frac{N}{2}(D-1)\int_{q}\partial_{\mu}^{q}\partial_{\nu}^{q}\Big{[}\tilde{q}^{2}\mathcal{K}_{T}(q)F_{\hat{S}}(q)\Big{]}$
(6.28)
$\displaystyle=\frac{N}{2}(D-1)\int_{q}\Lambda\partial_{\Lambda}\partial_{\mu}^{q}\partial_{\nu}^{q}\big{[}\ln\tilde{P}^{\perp}_{q}\big{]}\,,$
(6.29)

where to get the second line we used (5.5). The result can be cast as a total
$\Lambda$-derivative which we do in the final line, where

$\tilde{P}^{\perp}_{q}:=\Lambda^{2}P^{\perp}(q)\,,\qquad
P^{\perp}(q):=\frac{1}{2q^{2}F(q)}\,.$ (6.30)

As we will see in sec. 6.4 this object plays the rôle of an effective action
propagator (in the transverse space). Indeed from (4.39):

$P^{\perp}(q)\,S^{0}_{\alpha\beta}(q)=\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\,.$
(6.31)

In (6.29) one can see that the $\Lambda\partial_{\Lambda}$ sits inside the
integral, whereas in (6.25) it sits outside the momentum integral. In general,
changing the integration and differentiation order is not trivial and to
ensure consistency one must show that the undifferentiated expression is
sufficiently well-behaved in both UV and IR. However as stressed in ref.[38],
the flow equation is consistent only if we consider all trace terms together.
Therefore, we should interpret this as having the $\Lambda\partial_{\Lambda}$
outside the integral in (6.29), as long as we add to it all the remaining
terms similar to and including (6.25). This means that (6.22) becomes:

$\Lambda\partial_{\Lambda}\int_{q}\partial_{\mu}^{q}\partial_{\nu}^{q}\Big{\\{}\big{[}\ln
C\big{]}_{\alpha\alpha}(q)+\ln B_{q}+\frac{1}{2}\ln
b_{q}-\frac{D-1}{2}\ln\tilde{P}^{\perp}_{q}\Big{\\}}\,,$ (6.32)

where $\big{[}\ln C\big{]}_{\alpha\alpha}(q)=(D-1)\ln C_{T}(q)+\ln C_{L}(q)$.
Given that the above integral is an integral of a total derivative, it amounts
to a surface integral at large $q$, as advertised at the beginning of this
section. This surface integral can be discarded, and thus the integral
vanishes, if the term in braces above is UV finite. Using the definitions
(2.54), (2.55) and (2.56), we can write it as follows:

$\displaystyle(D-1)$
$\displaystyle\ln\frac{c_{q}}{c_{q}+\tilde{q}^{2}}+\ln\frac{1}{1+\tilde{q}^{2}Y_{q}}+\ln\Big{(}1+\tilde{q}^{2}Y_{q}\Big{)}+\frac{1}{2}\ln\Big{(}t_{q}+\tilde{q}^{2}\Big{)}+\frac{D-1}{2}\ln\frac{2\tilde{q}^{2}\Big{(}c_{q}^{2}+\tilde{q}^{2}\Big{)}}{c_{q}^{2}}$
$\displaystyle=-(D-1)\ln\bigg{(}\frac{c_{q}}{\tilde{q}^{2}}+1\bigg{)}+\frac{D-1}{2}\ln\bigg{(}\frac{c_{q}^{2}}{\tilde{q}^{2}}+1\bigg{)}+\frac{1}{2}\ln\bigg{(}\frac{t_{q}}{\tilde{q}^{2}}+1\bigg{)}+\frac{1}{2}\ln\tilde{q}^{2}\,.$
(6.33)

All but the last term vanish rapidly for large momentum (from (6.15) and
because $c_{q}$ and $t_{q}$ do). In this calculation, the last term can be
formally discarded because it vanishes under the combined action of $q$ and
$\Lambda$ derivatives. In this sense we are justified in dropping all the
$O(p^{0})$ terms, confirming the arguments outlined in ref.[38] work to
$O(p^{0})$.

However it is unclear whether the part that leads to this unregulated
$\ln\tilde{q}^{2}$ term can always be safely discarded. In particular it is
unclear whether such a part could cause an unrecoverable failure of UV
regularisation at higher loops. If we first compute the $q$ derivatives, the
resulting momentum integral is quadratically UV divergent. By dimensions one
might expect to find an analogous logarithmic UV divergence in the $O(p^{2})$
part. In the next section we confirm this expectation.

###  6.4 Two-point vertex at second order in momentum

The next step is to compute the one-loop beta function by analysing the flow
equation (6.7). The coefficient of the beta function at one loop is a
universal quantity and thus it should be independent of all artefacts of the
regularisation scheme [23]. In the earlier successful construction [14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36] this could be understood as follows. Since the beta function is
dimensionless, the momentum integrals that compute it are also dimensionless.
By trading the higher-point regularisation vertices in these integrals for
$\Lambda$-derivatives of effective action vertices, using the classical flow
equations, one finds that the results combine into terms that either vanish,
because they are $\Lambda$-derivatives of regularised dimensionless integrals
(which thus do not actually depend on $\Lambda$), or terms that survive but
only because when cast in this way a finite result is obtained from a
logarithmic IR divergence, where the effective action is universal – as
determined by the renormalization condition (6.1) [23, 50]. Here we will
follow this route by manipulating the $\delta\Psi_{0}^{a}/\delta\phi^{a}$
term. However, before delving into the details of this, it is better for the
ease of presenting to focus on the other trace terms first.

The remaining trace terms can be partitioned into two groups, namely
$\Lambda\partial_{\Lambda}\ln K$ and
$\Psi^{m}_{0}\frac{\delta}{\delta\phi^{m}}\ln K$, where $K=C^{-1},B,b$. This
means that in order to compute them it will be sufficient to do this for one
value of $K$ chosen for convenience, all the other contributions following
from this one by substitution. It is straightforward to write down the first
$B$ trace terms using (6.14) as follows:

$\displaystyle\text{Tr}\bigg{\\{}\Lambda\partial_{\Lambda}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}\bigg{\\}}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{2})\end{subarray}}\equiv 2N\Lambda\partial_{\Lambda}\int_{q}$
$\displaystyle\bigg{\\{}B_{q}^{-1}B_{\mu\nu}(p,-p;q,-q)$
$\displaystyle-\frac{1}{2}B_{q}^{-1}B_{\mu}(p;q,-q-p)B^{-1}_{p+q}B_{\nu}(-p;p+q,-q)\bigg{\\}}_{O(p^{2})}\,.$
(6.34)

The second $B$ trace term can be written using $\text{Tr
}\Psi^{m}_{0}\frac{\delta}{\delta\phi^{m}}\ln B=\text{Tr
}\Psi^{m}_{0}B_{,m}B^{-1}$, yielding:

$\displaystyle\text{Tr}\bigg{\\{}\Psi_{0}^{m}\frac{\delta}{\delta\phi^{m}}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}\bigg{\\}}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{2})\end{subarray}}\equiv 2N\int_{q}$
$\displaystyle\bigg{\\{}2C_{T}(p)\theta^{0}_{\mu\alpha}(p)B_{\alpha\nu}(p,-p;q,-q)B_{q}^{-1}$
$\displaystyle+C_{T}(p)\theta^{0}_{\mu\alpha}(p)B_{\alpha}(p;-p-q,q)B^{-1}_{\nu}(-p;p+q,-q)$
$\displaystyle+C_{\alpha\beta}(0)\theta^{0}_{\mu\nu\alpha}(p,-p,0)B_{\beta}(0;q,-q)B^{-1}_{q}$
$\displaystyle-\frac{1}{2}\big{[}C\mapsto\mathcal{K}\text{ ;
}\theta\mapsto\Sigma\big{]}\bigg{\\}}_{O(p^{2})}\,.$ (6.35)

The third line is odd in $p$ and thus has no $O(p^{2})$ part. Furthermore the
one-point kernel is antisymmetric in $q$, whereas $B_{q}$ is symmetric, and so
the entire row vanishes under the $q$ integral. A similar argument holds for
the $\mathcal{K}$ sector as well. From (4.6) we get that
$\theta^{0}_{\mu\alpha}(p)\big{|}_{O(p^{2})}=2\,\square_{\mu\alpha}(p)$, and,
given that all the other functions appearing in (6.35) are quasi-local, we are
free to set $p=0$ anywhere else in the $C$ sector. In the $\mathcal{K}$ sector
on the other hand, $\Sigma^{0}_{\mu\alpha}(p)$ is $O(p^{4})$, which means that
it does not contribute to (6.35) at all. Collecting everything together we
have

$\displaystyle\text{Tr}\bigg{\\{}\Psi_{0}^{m}\frac{\delta}{\delta\phi^{m}}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}\bigg{\\}}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{2})\end{subarray}}\equiv 2N\int_{q}$
$\displaystyle\bigg{\\{}2C_{T}(0)2\square_{\mu\alpha}(p)B_{\alpha\nu}(0,0;q,-q)B_{q}^{-1}$
(6.36)
$\displaystyle+C_{T}(0)2\square_{\mu\alpha}(p)B_{\alpha}(0;-q,q)B^{-1}_{\nu}(0;q,-q)\bigg{\\}}_{O(p^{2})}\,,$

which we can recast if we use differential Ward identities for the kernel
vertices:

$\text{Tr}\bigg{\\{}\Psi_{0}^{m}\frac{\delta}{\delta\phi^{m}}\big{[}\ln
B^{\alpha}{}_{\beta}\big{]}\bigg{\\}}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at }O(p^{2})\end{subarray}}\equiv
4N\square_{\mu\alpha}(p)\int_{q}\partial_{\alpha}^{q}\partial_{\nu}^{q}\ln
B_{q}\,.$ (6.37)

The flow equation (6.7) shows that if we add up all trace term contributions
we ought to end up with a transverse expression, i.e.
$\propto\square_{\mu\nu}(p)$. Terms like (6.37) are transverse on the $\nu$
index if we use Lorentz invariance of the $q$ integral. But such arguments
make sense strictly speaking only if the integral is properly regularised. As
already noted in sec. 6.3, trace terms should be considered together, not
individually. However for more involved computations, this is not enough
because the sum would still be ambiguous due to the so-called momentum routing
problem (see app. B). We need in general to make statements about _individual_
terms, and to do so we need to apply some auxiliary regularisation for each
momentum integral to give them a well defined meaning. In the end one can put
all the parts back together again at which point, provided that the trace
terms do actually fully regularise, the result is finite and the “pre-
regularisation” can be safely removed. From now on we will use dimensional
regularisation for this purpose, i.e. compute in $d=4-\varepsilon$ dimensions.
This means that terms similar to (6.37) are indeed transverse on both indices.
If we contract $p_{\mu}$ into (6.34) and use Ward identities we will get
(ignore the $O(p^{2})$ requirement for now):

$\displaystyle
2N\Lambda\partial_{\Lambda}\int_{q}\bigg{\\{}B_{q}^{-1}\Big{(}B_{\mu}(0;q,-q)-B_{\mu}(-p;p+q,-q)\Big{)}$
$\displaystyle\qquad\qquad\quad-\frac{1}{2}B_{q}^{-1}\big{(}B_{q}-B_{p+q}\big{)}B^{-1}_{p+q}B_{\nu}(-p;p+q,-q)\bigg{\\}}\,.$
(6.38)

The first term in the bracket on the first line is odd in $q$, and thus
vanishes when integrated. In the second line we can relabel $q\rightarrow-q-p$
for the first term inside the bracket and use the antisymmetry of the one-
point kernel vertex, $B_{\nu}(-p;p+q,-q)=-B_{\nu}(-p;-q,p+q)$, to arrive at

$\displaystyle
2N\Lambda\partial_{\Lambda}\int_{q}\bigg{\\{}-B_{q}^{-1}B_{\mu}(-p;p+q,-q)+B_{q}^{-1}B_{\nu}(-p;p+q,-q)\bigg{\\}}$
(6.39) $\displaystyle=$ $\displaystyle
2N\Lambda\partial_{\Lambda}\int_{q}\big{\\{}0\big{\\}}\,.$ (6.40)

Lorentz invariance implies that a similar result would have been obtained if
$p_{\nu}$ had been used. Therefore, all terms with a similar structure to
(6.34), when properly regularised, are individually transverse.

The last step is to compute $\delta\Psi_{0}^{a}/\delta\phi^{a}$. At this point
we note that (6.21) is formally transverse prior to any substitution being
done. One can easily check this by contracting it with $p_{\mu}$ (or
equivalently $p_{\nu}$), and then show that the expression vanishes at all
orders, not only at $O(p^{2})$. This means that formally all trace terms in
(6.7) are individually transverse. This is gratifying but unsurprising: it is
a consistency check on the formal preservation of gauge invariance by the flow
equation. As we have just emphasised, for these manipulations to be meaningful
we need to verify this with a gauge invariant pre-regularisation in place. If
the expressions are in fact properly regularised through the PV trace terms,
we should then be able to combine the results into a finite transverse, and in
fact universal, answer.

Thus the real test is to see whether (6.21) remains transverse in dimensional
regularisation after we trade four and three-point $\Sigma^{0}$ and
$\theta^{0}$ vertices for ($\Lambda$ derivatives of) effective action
vertices. If we substitute first (6.11) into (6.21) to trade four-point
$\Sigma^{0}$ and $\theta^{0}$ vertices, and then use (6.10) to trade away the
$\Sigma^{0}$ and $\theta^{0}$ three-point vertices, we obtain eventually the
following:

$\displaystyle\begin{split}\frac{\delta\Psi_{0}^{a}}{\delta\phi^{a}}\bigg{|}_{\begin{subarray}{c}\text{2-pt.function}\\\
\text{at
}O(p^{2})\end{subarray}}=N\int_{q}\Bigg{\\{}&-\Lambda\partial_{\Lambda}\Big{[}P^{\perp}(q)S^{0}_{\alpha\alpha\mu\nu}(q,-q,p,-p)\Big{]}_{O(p^{2})}\\\
&+\frac{1}{2}\Lambda\partial_{\Lambda}\Big{[}P^{\perp}(q)P^{\perp}(p-q)S^{0}_{\alpha\beta\mu}(p-q,q,-p)S^{0}_{\alpha\beta\nu}(p-q,q,-p)\Big{]}_{O(p^{2})}\\\
&+\frac{\square_{\alpha\nu}(p)}{\Lambda^{2}q^{2}F_{q}}\Bigg{[}\frac{\big{(}1-\mathcal{K}_{T}(q)\big{)}\Sigma^{0}_{\alpha\mu}(q)-\big{(}1-2C_{T}(q)\big{)}\theta^{0}_{\alpha\mu}(q)}{q^{2}}\\\
&\qquad\qquad\quad-\frac{1}{2}\partial_{\mu}^{q}\partial_{\alpha}^{q}\big{(}S^{0}_{\beta\beta}(q)-\Sigma^{0}_{\beta\beta}(q)+\theta^{0}_{\beta\beta}(q)\big{)}\\\
&\qquad\qquad\quad+2\frac{q_{\alpha}q_{\mu}}{q^{2}}\big{(}\mathcal{K}_{T}(q)F_{\Sigma}(q)-2C_{T}(q)F_{\theta}(q)-F_{q}\big{)}\\\
&\qquad\qquad\quad+\frac{1}{2\Lambda^{2}q^{2}F_{q}}\partial_{\mu}^{q}\big{(}\square_{\beta\gamma}(q)F_{q}\big{)}\,\partial_{\alpha}^{q}\big{(}S^{0}_{\beta\gamma}(q)-\Sigma^{0}_{\beta\gamma}(q)+\theta^{0}_{\beta\gamma}(q)\big{)}\Bigg{]}\Bigg{\\}}\,,\end{split}$
(6.41)

where $P^{\perp}(q)$ is an effective propagator and was defined already in
(6.30). To arrive at this expression we use the symmetries including
differential Ward identities in a similar way to the trace terms above. As
there, we do not need the explicit formulae except for the zero-point kernels
and two-point action vertices.

All the terms appearing in the above expression are manifestly transverse
except the $\Lambda$-derivative ones. We can evaluate the contribution of the
latter in dimensional regularisation and check whether they are transverse or
not. Again, we are free to move the $\Lambda$ derivative outside the integral
if the resulting momentum integrals are regularised. The first term
(containing the four-point vertex) would then vanish if the UV regularisation
(as also provided by the trace terms) is correctly in place, because the
integral is dimensionless in four dimensions and thus actually independent of
$\Lambda$. This is so because, although $P^{\perp}\sim 1/q^{2}$ for small $q$,
this IR divergence is integrable in four dimensions. Similarly the second term
(containing the three-point effective action vertices) has no surviving UV
contribution if it is properly regulated there, but this time there is a
logarithmic IR divergence which when differentiated with respect to $\Lambda$
gives a finite universal answer dependent only on the renormalization
condition (6.1).

Now we contract the first two terms in (6.41) with $p_{\mu}$ and use Ward
identities to further simplify it, following essentially the same steps that
took us from (6.4) to (6.39). This time we get:

$N\Lambda\partial_{\Lambda}\int_{q}P^{\perp}(q)\Bigg{\\{}S^{0}_{\alpha\alpha\nu}(q,p-q,-p)-P^{\perp}(p-q)S^{0}_{\alpha\beta}(p-q)S^{0}_{\alpha\beta\nu}(q,p-q,-p)\Bigg{\\}}\,.$
(6.42)

Now using the propagator identity (6.31) and Ward identities recursively this
simplifies to

$\Lambda\partial_{\Lambda}\int_{q}\frac{(p+q)_{\beta}}{(p+q)^{2}}\bigg{(}\delta_{\beta\nu}-\frac{q_{\beta}q_{\nu}}{q^{2}}\bigg{)}\,.$
(6.43)

The $\delta_{\beta\nu}$ part is odd in $q$ (one can easily see that this is
the case after expanding the bracket and relabelling $q\rightarrow q-p$), and
hence vanishes, whereas the second term can be recast using
$q\\!\cdot\\!(p+q)=\frac{1}{2}[(p+q)^{2}+q^{2}-p^{2}]$, yielding:

$-\frac{\Lambda}{2}\frac{\partial}{\partial\Lambda}\int_{q}\frac{q_{\nu}}{q^{2}}+\frac{q_{\nu}}{(p+q)^{2}}-\frac{p^{2}q_{\nu}}{(p+q)^{2}q^{2}}\,.$
(6.44)

Again, the first term is odd in $q$. The second term is quadratically
divergent but in dimensional regularisation it vanishes, whereas the third
term is linearly divergent and gives a finite contribution in dimensionsal
regularisation (due to a logarithmic subdivergence):

$\Lambda\partial_{\Lambda}\bigg{(}-\frac{1}{2}\frac{p^{2}p_{\nu}}{(4\pi)^{2}}\ln\Lambda\bigg{)}=-\frac{1}{2}\frac{p^{2}p_{\nu}}{(4\pi)^{2}}\,,$
(6.45)

where we recognise that the regularisation scale is set by $\Lambda$. This
means that the original $\Lambda$ derivative terms from (6.41) give a non-
vanishing longitudinal contribution in dimensional regularisation, which
amounts to

$-\frac{1}{2}\frac{p_{\mu}p_{\nu}}{(4\pi)^{2}}\,,$ (6.46)

and thus they cannot be transverse on their own. In fact, apart from an
overall factor of $\frac{1}{2}$, the effective action terms in (6.41) have
precisely the same structure as found in ref.[23] as part of the one-loop beta
function computation using the manifestly gauge invariant flow equation
developed there. Moreover, the computed longitudinal part (6.46) is precisely
half the one found in ref. [23]. The same arguments in that paper can be used
to extract the full IR contribution from the first two lines of (6.41):

$\frac{19}{6}p^{2}\delta_{\mu\nu}-\frac{11}{3}p_{\mu}p_{\nu}\,,$ (6.47)

and this has the same longitudinal part.

Thus we see that if we assume that (6.7) is sufficiently regularised in the UV
we get a non-transverse answer. As we noted, the above result is arrived at
independently of the detailed form of the regularisation structure, apart from
the expressions for two-point action vertices and zero-point kernels. This
means that the problem cannot be cured by a more careful choice of cutoff
functions or an alternative covariantization, but lies at a deeper structural
level.

Since formally the integrals start out transverse, it follows that there are
unregulated UV divergences which provide a cancelling longitudinal
contribution. In fact if the third term in (6.44) has no UV regularisation we
can (cavalierly) regard it as an infinite constant, independent of $\Lambda$,
which is then annihilated by the $\Lambda$ derivative.

The problem then is that we cannot extract universal information from the
$S_{0}$ terms in (6.41). We can see this explicitly by substituting
$S_{0}=\Sigma_{0}+\hat{S}$. Since $P^{\perp}(q)\approx c^{2}_{q}$, cf. (6.30)
and (4.39), the resulting terms with a $\Sigma_{0}$ three-point or four-point
vertex are UV regularised because $\Sigma^{(3)}_{0}$ and $\Sigma^{(4)}_{0}$
diverge only as $c^{-1}_{q}$, as we confirmed explicitly in sec. 5. As
explained above this means that the $\Sigma^{(4)}_{0}$ contribution actually
vanishes, because the integral is then well defined and dimensionless, and
thus annihilated by $\Lambda\partial_{\Lambda}$. The contributions with a
$\Sigma^{(3)}_{0}$ vertex vanish for the same reason. To see this we only need
to show that there is no longer a logarithmic divergence at small $q$. Since
$\Sigma^{(2)}_{0}$ is $O(p^{4})$, cf. (4.40), we know by gauge invariance (or
equivalently the Ward identities) that $\Sigma^{0}_{\alpha\beta\mu}(p-q,q,-p)$
starts cubic in momenta in a small momentum expansion. In other words,
retaining powers of $p$ up to a maximum of $O(p^{2})$, as $q\to 0$ it vanishes
with at least one power of $q$, and thus indeed there is no longer an IR
logarithmic divergence from the second line of (6.41).

This means that, on substituting $S_{0}=\Sigma_{0}+\hat{S}$, the top two lines
of (6.41) become precisely the same expression but with $S^{(3)}_{0}$ and
$S^{(4)}_{0}$ replaced by $\hat{S}^{(3)}_{0}$ and $\hat{S}^{(4)}_{0}$
respectively. However $\hat{S}$ is freely designed by us as part of the
regularisation scheme so cannot of itself contain the requisite universal
information. Indeed we can go one step further and introduce a transverse
space effective propagator for $\hat{S}_{\alpha\beta}(q)$:

$\hat{P}(q):=\frac{1}{2q^{2}F_{\hat{S}}(q)}\,,$ (6.48)

which satisfies the obvious transverse projector relation, cf. (4.38) and
(6.31). Substituting

$P^{\perp}(q)=\hat{P}(q)\,+\left[P^{\perp}(q)-\hat{P}(q)\right]=\hat{P}(q)\,+\frac{c^{3}_{q}}{2q^{2}}\frac{1-c_{q}}{(c^{2}_{q}+\tilde{q}^{2})(c_{q}+\tilde{q}^{2})}\,,$
(6.49)

for all propagators in the same top two lines, it is apparent that any part
containing the last expression above is again both UV and IR finite (in the
latter case because $c(0)=1$) and therefore vanishes under the $\Lambda$
derivative. Thus the top two lines only contain information determined by our
choice of seed action.

We have shown that if we assume that (6.7) is sufficiently regularised in the
UV we get a universal non-transverse answer from a certain IR contribution.
Since formally the integrals start out transverse, it follows that there are
actually unregulated UV divergences which can formally be viewed as providing
a cancelling longitudinal contribution. Although one can indeed proceed in
this way, it is then no longer possible to prove the result is universal.
Indeed we show that this contribution can be expressed solely in terms of the
seed action, a quantity that we are free to choose.

##  7 Summary and Conclusions

We finish this paper with a brief summary and discussion of our main findings.
In sec. 6.3, we showed that all but one term in the $O(p^{0})$ part of the
two-point function at one-loop level is correctly regularised. Key to this,
was the finding that the $O(p^{0})$ actually only depends on zero-point
kernels and two-point action vertices. If properly regularised, it has to
vanish by gauge invariance. We show that it does vanish. For the properly
regularised terms, this is so because they can be written as a vanishing UV
surface term in momentum space. The one term that is not properly regularised
can however be formally set to zero because it is the $\Lambda$-derivative of
a $\Lambda$-independent quadratically divergent term.

The $O(p^{2})$ part should give us the one-loop beta function, if properly
regularised, and thus be universal (independent of regularisation artefacts).
In sec. 6.4 we recast the result in a way that has previously allowed a
universal answer to be extracted, if properly regularised [23, 50]. We saw
that although the $O(p^{2})$ part is formally transverse, if computed assuming
complete UV regularisation there is actually a non-vanishing longitudinal
part. The problem is caused by unregulated UV divergences, which appears to be
associated to the one found at $O(p^{0})$, and is independent of choice of
cutoff functions or covariantization.

Similarly to the $O(p^{0})$ case, if the incompletely regularised longitudinal
part is regarded formally as an infinite constant, it can then be set to zero,
since it is differentiated by $\Lambda$. The problem then is that the earlier
techniques can no longer be used to extract a universal answer because it is
no longer clear which IR divergent terms should provide a universal answer and
which should be formally set to zero. Indeed we saw explicitly that in the
current case these terms can then be made independent of the classical
effective action. Instead they can be seen to depend only on the seed action,
part of the regularisation structure which we are free to choose (and which is
thus non-universal).

Note that these results are not at variance with ref. [38] where the one-loop
beta function coefficient is successfully computed. It is there extracted from
determinants of zero-point kernels and two-point vertices (the result obtained
in 6.3), using a heat kernel approach. It is thus insensitive to the problems
with UV regularisation that we are highlighting. Indeed the heat-kernel
expression formally depends only on the value of these kernels at vanishing
momentum.

Our results show that the one-loop off-shell contribution is not fully
regularised. At one loop it appears to be possible to proceed formally and
obtain valid answers, but it is unclear whether that would still be possible
at higher loops. Also, the powerful techniques [23, 50] that previously
allowed universal results to be extracted, unfortunately fail in this
formulation. As we have already noted, a repair of this problem (if there is
one which does not introduce explicit PV fields as done previously [14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36]) would seem to require structural changes. Hints lie in the
insufficient regularisation at $O(p^{0})$ seen in 6.3, and in the longitudinal
sector as seen in sec. 6.4 and discussed in more general terms in sec. 3.
Hopefully, following these hints, a way can be found to implement a more
complete regularisation.

## Acknowledgments

We thank Kevin Falls for useful comments on an earlier version of this
manuscript. VMM acknowledges support via an STFC PhD studentship. TRM
acknowledges support from STFC through Consolidated Grant ST/T000775/1.

##  Appendix A Interleave identities for compound kernels

Interleave identites (4.29) hold for those kernels whose covariantization is
carried by gauge fields acting by commutation. This means, for example, that
any well-behaved (or quasi-local) function of $\Delta^{a}{}_{b}$ or
$\Delta_{\parallel}$ fulfills this condition. We exploit these identities for
example in the one-loop beta function calculation where it allows the
cancellation of certain terms. However, the two kernels
$\mathcal{K}^{a}{}_{b}$ and $C^{a}{}_{b}$, are not solely constructed in this
way, and it is not a priori evident that they still obey interleave
identities. It may be that they do not at some higher point level. Fortunately
for us they do obey the interleave identities up to the two-point level, and
this is all that we need in this paper.

In this appendix we prove that the identities do hold up to this level using
the other already established symmetries. At the same time we give more
details on how to construct them. Given the definition of
$\mathcal{K}^{a}{}_{b}$ in (2.68), it is sufficient to check the vertices of
$C^{a}{}_{b}$. Then, since $\kappa^{a}{}_{b}$ is a function of
$\Delta^{a}{}_{b}$, it is guaranteed that a similar behaviour is inherited by
the $\mathcal{K}^{a}{}_{b}$ vertices. From (2.54) we can write
$\big{(}C^{-1}\big{)}^{a}{}_{b}$ in the following way:

$\big{(}C^{-1}\big{)}^{a}{}_{b}=\delta^{a}{}_{b}+R^{a}{}_{b}+z^{a}{}_{b}\,,$
(A.1)

where $R^{ab}:=\gamma^{am}A_{,mn}\gamma^{nb}$ and
$z^{ab}:=K^{a}_{\alpha}Y^{\alpha\beta}K^{b}_{\beta}$. It is easy to see that
$z^{a}{}_{b}$ is made out of gauge fields which act by commutation, and hence
its vertices obey interleave identities. However, checking whether $R$
vertices satisfy these identities or not is non-trivial. From sec. 4.2.2 we
see that for two (matrix valued) functions $J_{\mu}(x)$ and $J_{\nu}(y)$ the
following equivalent statements can be written:

$\displaystyle J_{a}R^{ab}J_{b}$
$\displaystyle\equiv\int_{x}\int_{y}J^{a}_{\mu}(x)R^{ab}_{\mu\nu}(x,y)J^{b}_{\nu}(y)$
(A.2)
$\displaystyle=\frac{2}{\Lambda^{2}}\text{tr}\int_{x}\int_{y}J_{\mu}(x)R_{\mu\nu}(x,y)J_{\nu}(y)$
(A.3)
$\displaystyle=\frac{2}{\Lambda^{2}}\sum_{k=0}^{\infty}\sum_{m=0}^{k}\int_{x}\int_{y}\int_{x_{1}\cdots
x_{k}}R_{\mu_{1}\ldots\mu_{k-m},\mu_{k-m+1}\ldots\mu_{k};\mu\nu}(x_{1}\ldots
x_{k-m};x_{k-m+1}\ldots x_{k};x,y)$ $\displaystyle\hskip
42.67912pt\times\text{tr}\Big{[}J_{\mu}(x)A_{\mu_{1}}(x_{1})\cdots
A_{\mu_{k-m}}(x_{k-m})J_{\nu}(y)A_{\mu_{k-m+1}}(x_{k-m+1})\cdots
A_{\mu_{k}}(x_{k})\Big{]}\,,$ (A.4)

where we note that the sum is half that in (4.27) and we used the definition
of functional derivatives below (2.17) and normalisation of the generators. We
also note that $R^{ab}_{\mu\nu}(x,y)$ is given by555Recall that we factor out
the powers of $g$, cf. sec. 4.1.

$R^{ab}_{\mu\nu}(x,y)=\frac{1}{\Lambda^{4}}\frac{\delta^{2}A}{\delta
A^{b}_{\nu}(y)\delta A^{a}_{\mu}(x)}\,,$ (A.5)

We can look at the functional derivatives of $A$ in more detail if we use the
trace expansion (4.2.1). The first functional derivative takes the following
form:

$\frac{\delta A}{\delta A^{a}_{\mu}(x)}=\sum_{k=2}^{\infty}\int_{x_{1}\ldots
x_{k-1}}A_{\mu\mu_{1}\ldots\mu_{k-1}}(x,x_{1}\ldots
x_{k-1})\text{tr}\Big{[}T^{a}A_{\mu_{1}}(x_{1})\ldots
A_{\mu_{k-1}}(x_{k-1})\Big{]}\,,$ (A.6)

where we consider only the single-trace terms which are the only ones of
interest to us. If we differentiate it a second time, we will obtain:

$\frac{\delta^{2}A}{\delta A^{b}_{\nu}(y)\delta A^{a}_{\mu}(x)}=\\\
=\sum_{k=2}^{\infty}\sum_{m=0}^{k-2}\int_{x_{1}\ldots
x_{k-2}}A_{\mu\mu_{1}\ldots\mu_{k-2-m}\nu\mu_{k-1-m}\ldots\mu_{k-2}}(x,x_{1}\ldots
x_{k-2-m},y,x_{k-1-m}\ldots x_{k-2})\\\
\text{tr}\Big{[}T^{a}A_{\mu_{1}}(x_{1})\ldots
A_{\mu_{k-2-m}}(x_{k-2-m})T^{b}A_{\mu_{k-1-m}}(x_{k-1-m})\ldots
A_{\mu_{k-2}}(x_{k-2})\Big{]}\,.$ (A.7)

If we now relabel $k\mapsto k+2$ and substitute the result into (A.2) using
(A.5), we will obtain the following identity:

$J_{a}R^{ab}J_{b}\equiv\frac{1}{\Lambda^{4}}\sum_{k=0}^{\infty}\sum_{m=0}^{k}\int_{x}\int_{y}\int_{x_{1}\ldots
x_{k}}A_{\mu\mu_{1}\ldots\mu_{k-m}\nu\mu_{k-m+1}\ldots\mu_{k}}(x,x_{1}\ldots
x_{k-m},y,x_{k-m+1}\ldots x_{k})\\\
\text{tr}\Big{[}J_{\mu}(x)A_{\mu_{1}}(x_{1})\ldots
A_{\mu_{k-2-m}}(x_{k-m})J_{\nu}(y)A_{\mu_{k-m+1}}(x_{k-m+1})\ldots
A_{\mu_{k}}(x_{k})\Big{]}\,.$ (A.8)

This means that (A.2) and (A.4) remain equivalent provided that we make the
following identification:

$R_{\mu_{1}\ldots\mu_{k-m},\mu_{k-m+1}\ldots\mu_{k};\mu\nu}(x_{1}\ldots
x_{k-m};x_{k-m+1}\ldots x_{k};x,y)=\\\
=\frac{1}{2\Lambda^{2}}A_{\mu\mu_{1}\ldots\mu_{k-m}\nu\mu_{k-m+1}\ldots\mu_{k}}(x,x_{1}\ldots
x_{k-m},y,x_{k-m+1}\ldots x_{k})\,,$ (A.9)

which take the exact same form in momentum space.

If we now look at the one-point $R$ part of (A.1), we will obtain:

$\displaystyle R_{\mu;\alpha\beta}(p;r,s)$
$\displaystyle=\frac{1}{2\Lambda^{2}}A_{\mu\beta\alpha}(p,s,r)\,,$ (A.10)
$\displaystyle R_{,\mu;\alpha\beta}(;p;r,s)$
$\displaystyle=\frac{1}{2\Lambda^{2}}A_{\mu\alpha\beta}(p,r,s)\,.$ (A.11)

Three-point action vertices are totally anti-symmetric in all their arguments,
and thus the above expressions can be recast as

$R_{\mu;\alpha\beta}(p;r,s)+R_{,\mu;\alpha\beta}(;p;r,s)=0\,,$ (A.12)

which is exactly (4.29) at the one-point level.

At the two-point level the $R$ part of (A.1) becomes:

$\displaystyle R_{\mu\nu;\alpha\beta}(p,q;r,s)$
$\displaystyle=\frac{1}{2\Lambda^{2}}A_{\mu\nu\beta\alpha}(p,q,s,r)\,,$ (A.13)
$\displaystyle R_{\mu,\nu;\alpha\beta}(p;q;r,s)$
$\displaystyle=\frac{1}{2\Lambda^{2}}A_{\mu\beta\nu\alpha}(p,s,q,r)\,,$ (A.14)
$\displaystyle R_{,\mu\nu;\alpha\beta}(;p,q;r,s)$
$\displaystyle=\frac{1}{2\Lambda^{2}}A_{\mu\nu\alpha\beta}(p,q,r,s)\,.$ (A.15)

From (A.13) and (A.15) we can write the following:

$\displaystyle\begin{split}R_{,\mu\nu;\alpha\beta}(;p,q;r,s)-R_{\nu\mu;\alpha\beta}(q,p;r,s)&=\frac{1}{2\Lambda^{2}}A_{\mu\nu\alpha\beta}(p,q,r,s)-\frac{1}{2\Lambda^{2}}A_{\nu\mu\beta\alpha}(q,p,s,r)\\\
&=\frac{1}{2\Lambda^{2}}A_{\mu\nu\alpha\beta}(p,q,r,s)-\frac{1}{2\Lambda^{2}}A_{\mu\nu\alpha\beta}(p,q,r,s)=0\,,\end{split}$
(A.16)

where in going from the first to the second line we have used charge
conjugation invariance. Similarly, working out the explicit form of the four-
point $A$ vertices, one can show that the following expression also holds:

$R_{\mu,\nu;\alpha\beta}(p;q;r,s)+R_{\mu\nu;\alpha\beta}(p,q;r,s)+R_{\nu\mu;\alpha\beta}(q,p;r,s)=0\,.$
(A.17)

These last two identities are exactly the interleave identities (4.29) at the
two-point level for the $R$ part of $C^{-1}$. This means that the full one-
point and two-point vertices of $C^{-1}$ also obey interleave identities.

We are now at the stage where we can inspect the vertices of $C^{a}{}_{b}$
using $C^{a}{}_{m}\big{(}C^{-1}\big{)}^{m}{}_{b}=\delta^{a}{}_{b}$. At
$O(A_{\mu})$, this can be expressed schematically in the following way:

$\text{one-point }C=-\Big{(}\text{zero-point }C\Big{)}\times\bigg{(}\text{one-
point }C^{-1}\bigg{)}\times\Big{(}\text{zero-point }C\Big{)}\,,$ (A.18)

or explicitly as a set of two identities:

$\displaystyle C_{\mu;\alpha\beta}(p;r,s)$
$\displaystyle=-C_{\alpha\gamma}(r)C^{-1}_{\mu;\gamma\delta}(p;r,s)C_{\delta\beta}(s)\,,$
(A.19) $\displaystyle C_{,\mu;\alpha\beta}(p;r,s)$
$\displaystyle=-C_{\alpha\gamma}(r)C^{-1}_{,\mu;\gamma\delta}(p;r,s)C_{\delta\beta}(s)\,.$
(A.20)

If we use (A.12) for the full one-point $C^{-1}$ vertices, we can recast the
expressions above as:

$C_{\mu;\alpha\beta}(p;r,s)+C_{,\mu;\alpha\beta}(;p;r,s)=0\,,$ (A.21)

which is exactly the interleave identity (4.29) for one-point $C$ vertices.

Similarly, at $O(A^{2}_{\mu})$ one can write schematically:

$\text{two-point }C=-\Big{(}\text{zero-point }C\Big{)}\times\bigg{(}\text{two-
point }C^{-1}\bigg{)}\times\Big{(}\text{zero-point }C\Big{)}\\\
+\Big{(}\text{zero-point }C\Big{)}\times\bigg{(}\text{one-point
}C^{-1}\bigg{)}\times\Big{(}\text{zero-point }C\Big{)}\\\
\times\bigg{(}\text{one-point }C^{-1}\bigg{)}\times\Big{(}\text{zero-point
}C\Big{)}\,.$ (A.22)

From the above expression we can infer the explicit expression for the two-
point $C$ vertices:

$\displaystyle\begin{split}C_{\mu,\nu;\alpha\beta}(p;q;r,s)=&-C_{\alpha\alpha_{1}}(r)C^{-1}_{\mu,\nu;\alpha_{1}\beta_{1}}(p;q;r,s)C_{\beta_{1}\beta}(s)\\\
&+C_{\alpha\alpha_{1}}(r)C^{-1}_{\mu;\alpha_{1}\gamma}(p;r,s+q)C_{\gamma\delta}(s+q)C^{-1}_{,\nu;\delta\beta_{1}}(;q;p+r,s)C_{\beta_{1}\beta}(s)\\\
&+C_{\alpha\alpha_{1}}(r)C^{-1}_{,\nu;\alpha_{1}\gamma}(;q;r,s+p)C_{\gamma\delta}(r+q)C^{-1}_{\mu;\delta\beta_{1}}(p;q+r,s)C_{\beta_{1}\beta}(s)\,,\end{split}$
(A.23)
$\displaystyle\begin{split}C_{\mu\nu;\alpha\beta}(p,q;r,s)=&-C_{\alpha\alpha_{1}}(r)C^{-1}_{\mu\nu;\alpha_{1}\beta_{1}}(p,q;r,s)C_{\beta_{1}\beta}(s)\\\
&+C_{\alpha\alpha_{1}}(r)C^{-1}_{\mu;\alpha_{1}\gamma}(p;r,s+q)C_{\gamma\delta}(p+r)C^{-1}_{\nu;\delta\beta_{1}}(q;p+r,s)C_{\beta_{1}\beta}(s)\,,\end{split}$
(A.24)
$\displaystyle\begin{split}C_{,\mu\nu;\alpha\beta}(;p,q;r,s)=&-C_{\alpha\alpha_{1}}(r)C^{-1}_{,\mu\nu;\alpha_{1}\beta_{1}}(;p,q;r,s)C_{\beta_{1}\beta}(s)\\\
&+C_{\alpha\alpha_{1}}(r)C^{-1}_{,\mu;\alpha_{1}\gamma}(;p;r,s+q)C_{\gamma\delta}(p+r)C^{-1}_{,\nu;\delta\beta_{1}}(;q;p+r,s)C_{\beta_{1}\beta}(s)\,.\end{split}$
(A.25)

The interleave identities
$C_{,\mu\nu;\alpha\beta}(;p,q;r,s)-C_{\nu\mu;\alpha\beta}(q,p;r,s)=0$ and
$C_{\mu,\nu;\alpha\beta}(p;q;r,s)+C_{\mu\nu;\alpha\beta}(p,q;r,s)+C_{\nu\mu;\alpha\beta}(q,p;r,s)=0$
follow trivially from (A.23), (A.24), and (A.25) if we use (A.12), (A.16) for
the full $C^{-1}$ vertices. This concludes the proof that the one and two-
point functions of $C^{a}{}_{b}$ and $\mathcal{K}^{a}{}_{b}$, respectively,
obey interleave identities.

##  Appendix B Why preregularisation is necessary

Consider the momentum integral [14]

$\int_{q}\left\\{\frac{1}{q^{2}+\Lambda^{2}}-\frac{1}{(q+p)^{2}+\Lambda^{2}}\right\\}\,.$
(B.1)

Evidently this is zero, since in the second term we can change variables to
$q\mapsto-q-p$ to make it equal and opposite to the first. These sort of
cancelling terms are generic and unavoidable in a system of regularisation
that uses gauge invariant PV regularisation. For example, just such a
relabelling is used in (6.4) to cancel terms against each other. Depending on
how terms are unpacked and combined, they can then give contributions that
appear in a similar way to above. Expanding the second term in (B.1) to
$O(p^{2})$, we obtain

$\int_{q}\left\\{\frac{p^{2}}{(q^{2}+\Lambda^{2})^{2}}-4\frac{(p\\!\cdot\\!q)^{2}}{(q^{2}+\Lambda^{2})^{3}}\right\\}\,.$
(B.2)

In four dimensions by Lorentz invariance (actually rotational invariance since
we are in Euclidean signature) we can replace $(p\\!\cdot\\!q)^{2}$ by
$q^{2}p^{2}/4$, so the above result simplifies to

$\Lambda^{2}p^{2}\int_{q}\frac{1}{(q^{2}+\Lambda^{2})^{3}}\,,$ (B.3)

a manifestly positive convergent answer from a vanishing integral! The problem
is that (B.1) is finite but ambiguous; the result (B.3) can be cast as a total
derivative but with a finite surface term. Generally in situations where PV
regularisation is used, we must first pre-regularise (in a way that is
compatible with gauge invariance) to avoid this so-called momentum routing
problem. This is done, not to subtract divergences with respect to the pre-
regulator, but in order to ensure that the integrals are defined sufficiently
carefully [40, 49, 54]. In this paper we use dimensional regularisation,
working in $d=4-\varepsilon$ dimensions when necessary. Then in (B.2), we
effectively replace $(p\\!\cdot\\!q)^{2}$ by
$q^{2}p^{2}/4+\varepsilon\,q^{2}p^{2}/16$. Using dimensional regularisation it
is straightforward to evaluate the new contribution and verify that it cancels
(B.3), restoring agreement with (B.1).

## References

  * [1] K. S. Stelle.  Renormalization of Higher Derivative Quantum Gravity.  Phys. Rev., D16:953–969, 1977.