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appendix.lyx
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#LyX 2.2 created this file. For more info see http://www.lyx.org/
\lyxformat 508
\begin_document
\begin_header
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theorems-ams
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\end_header
\begin_body
\begin_layout Chapter
\start_of_appendix
The auxiliary vertex and its renormalisation
\begin_inset CommandInset label
LatexCommand label
name "chap:Auxiliary-vertex-and-renorm"
\end_inset
\end_layout
\begin_layout Standard
As described in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:SI-3PIEA-Three-loop-truncation"
\end_inset
the renormalisation of the three loop 3PIEA requires the definition of
an auxiliary vertex
\begin_inset Formula $V_{abc}^{\mu}$
\end_inset
with the same asymptotic behaviour as the full self-consistent solution
at large momentum.
This auxiliary vertex can be found in terms of a six point kernel
\begin_inset Formula $\mathcal{K}_{abcdef}$
\end_inset
which obeys the integral equation
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:K-6pt-kernel-eom"
\end_inset
, reproduced here
\begin_inset Formula
\begin{equation}
\mathcal{K}_{abcdef}=\delta_{ad}\delta_{be}\delta_{cf}+\frac{1}{3!}\sum_{\pi}\left(-\frac{3i\hbar}{2}\right)\delta_{\pi\left(a\right)h}W_{\pi\left(b\right)\pi\left(c\right)kg}\Delta_{ki}^{\mu}\Delta_{gj}^{\mu}\mathcal{K}_{hijdef}.
\end{equation}
\end_inset
This can be written graphically as
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{fmffile}{BS-kernel-K-eom}
\end_layout
\begin_layout Plain Layout
\backslash
begin{equation}
\end_layout
\begin_layout Plain Layout
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40)
\backslash
fmfstraight
\backslash
fmfleftn{i}{3}
\backslash
fmfrightn{o}{3}
\backslash
fmfpolyn{shaded}{K}{6}
\backslash
fmf{plain}{i1,K5}
\backslash
fmf{plain}{i2,K4}
\backslash
fmf{plain}{i3,K3}
\backslash
fmf{plain}{o1,K6}
\backslash
fmf{plain}{o2,K1}
\backslash
fmf{plain}{o3,K2}
\backslash
end{fmfgraph}} =
\backslash
parbox{15mm}{
\backslash
begin{fmfgraph}(40,40)
\backslash
fmfstraight
\backslash
fmfleftn{i}{3}
\backslash
fmfrightn{o}{3}
\backslash
fmf{plain}{i1,o1}
\backslash
fmf{plain}{i2,o2}
\backslash
fmf{plain}{i3,o3}
\backslash
end{fmfgraph}} +
\backslash
parbox{27mm}{
\backslash
begin{fmfgraph}(75,40)
\backslash
fmfstraight
\backslash
fmfleftn{i}{3}
\backslash
fmfrightn{o}{3}
\backslash
fmfpolyn{full}{Pi}{6}
\backslash
fmf{plain}{i1,Pi5}
\backslash
fmf{plain}{i2,Pi4}
\backslash
fmf{plain}{i3,Pi3}
\backslash
fmfpolyn{shaded}{K}{6}
\backslash
fmf{plain,tension=0.5}{K5,Pi6}
\backslash
fmf{phantom,tension=0.5}{Pi1,K4}
\backslash
fmf{phantom,tension=0.5}{Pi2,K3}
\backslash
fmf{plain}{o1,K6}
\backslash
fmf{plain}{o2,K1}
\backslash
fmf{plain}{o3,K2}
\backslash
fmffreeze
\backslash
fmf{plain}{v,Pi1}
\backslash
fmf{plain}{v,Pi2}
\backslash
fmf{plain}{v,K3}
\backslash
fmf{plain}{v,K4}
\backslash
fmfdot{v}
\backslash
end{fmfgraph}},
\end_layout
\begin_layout Plain Layout
\backslash
end{equation}
\end_layout
\begin_layout Plain Layout
\backslash
end{fmffile}
\end_layout
\end_inset
where the shaded hexagonal blob is
\begin_inset Formula $\mathcal{K}$
\end_inset
and the solid hexagonal blob is the sum over all permutations of the lines
coming in from the left and connecting to those on the right.
\end_layout
\begin_layout Standard
Solving
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:K-6pt-kernel-eom"
\end_inset
by iteration generates an infinite number of terms, one of which is
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{fmffile}{BS-kernel-K-der-stab-stab-der}
\end_layout
\begin_layout Plain Layout
\backslash
begin{equation}
\end_layout
\begin_layout Plain Layout
\backslash
parbox{40mm}{
\end_layout
\begin_layout Plain Layout
\backslash
begin{fmfgraph*}(100,40)
\end_layout
\begin_layout Plain Layout
\backslash
fmfstraight
\end_layout
\begin_layout Plain Layout
\backslash
fmfleftn{i}{3}
\end_layout
\begin_layout Plain Layout
\backslash
fmflabel{$a$}{i3}
\end_layout
\begin_layout Plain Layout
\backslash
fmflabel{$b$}{i2}
\end_layout
\begin_layout Plain Layout
\backslash
fmflabel{$c$}{i1}
\end_layout
\begin_layout Plain Layout
\backslash
fmfrightn{o}{3}
\end_layout
\begin_layout Plain Layout
\backslash
fmflabel{$d$}{o3}
\end_layout
\begin_layout Plain Layout
\backslash
fmflabel{$e$}{o2}
\end_layout
\begin_layout Plain Layout
\backslash
fmflabel{$f$}{o1}
\end_layout
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\backslash
fmftopn{t}{6}
\end_layout
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\backslash
fmfbottomn{b}{6}
\end_layout
\begin_layout Plain Layout
\backslash
fmf{phantom}{t2,v1}
\backslash
fmf{phantom,tension=2}{v1,b2}
\end_layout
\begin_layout Plain Layout
\backslash
fmf{phantom}{t3,v2}
\backslash
fmf{phantom,tension=2}{v2,b3}
\end_layout
\begin_layout Plain Layout
\backslash
fmf{phantom}{t4,v3}
\backslash
fmf{phantom,tension=2}{v3,b4}
\end_layout
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\backslash
fmf{phantom}{t5,v4}
\backslash
fmf{phantom,tension=2}{v4,b5}
\end_layout
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\backslash
fmffreeze
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\backslash
fmf{plain}{i3,v1}
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\backslash
fmf{plain}{i2,t2,t3,t4,v4}
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\backslash
fmf{plain}{i1,v1}
\end_layout
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\backslash
fmf{plain,left=1}{v1,v2,v1}
\end_layout
\begin_layout Plain Layout
\backslash
fmf{plain,left=1}{v2,v3,v2}
\end_layout
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\backslash
fmf{plain}{v3,v4}
\end_layout
\begin_layout Plain Layout
\backslash
fmf{plain}{o3,v3}
\end_layout
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\backslash
fmf{plain}{o2,v4}
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\backslash
fmf{plain}{o1,v4}
\end_layout
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\backslash
fmfdot{v1}
\backslash
fmfdot{v2}
\backslash
fmfdot{v3}
\backslash
fmfdot{v4}
\end_layout
\begin_layout Plain Layout
\backslash
end{fmfgraph*}}.
\end_layout
\begin_layout Plain Layout
\backslash
end{equation}
\end_layout
\begin_layout Plain Layout
\backslash
end{fmffile}
\end_layout
\end_inset
Each contribution is in one-to-one correspondence with the sequence of permutati
ons
\begin_inset Formula $\pi_{1}\pi_{2}\cdots\pi_{n}\cdots$
\end_inset
of the propagator lines (read from left to right in relation to the diagram).
The permutations fall into two classes:
\begin_inset Quotes eld
\end_inset
stabilisers,
\begin_inset Quotes erd
\end_inset
for which
\begin_inset Formula $\pi\left(a\right)=a$
\end_inset
, and
\begin_inset Quotes eld
\end_inset
derangements,
\begin_inset Quotes erd
\end_inset
for which
\begin_inset Formula $\pi\left(a\right)=b$
\end_inset
or
\begin_inset Formula $c$
\end_inset
.
Any sequence of permutations is of the form of an alternating sequence
of runs of (possibly zero) stabilisers, separated by derangements.
\end_layout
\begin_layout Standard
Consider a run of
\begin_inset Formula $n$
\end_inset
stabilisers,
\begin_inset Formula $\cdots\pi_{a}\left(\pi_{1}\pi_{2}\cdots\pi_{n}\right)\pi_{b}\cdots$
\end_inset
, where
\begin_inset Formula $\pi_{a}$
\end_inset
and
\begin_inset Formula $\pi_{b}$
\end_inset
are derangements and
\begin_inset Formula $\pi_{1}$
\end_inset
through
\begin_inset Formula $\pi_{n}$
\end_inset
are all stabilisers.
The case for
\begin_inset Formula $n=2$
\end_inset
is shown above.
Each stabiliser creates a logarithmically divergent loop on the bottom
two lines
\begin_inset Formula $\sim-\lambda\mathcal{I}^{\mu}$
\end_inset
.
Derangements on the other hand, if they create loops at all, create loops
with
\begin_inset Formula $>2$
\end_inset
propagators, and hence are convergent.
Thus all divergences in
\begin_inset Formula $\mathcal{K}_{abcdef}$
\end_inset
can be removed by rendering a single primitive divergence finite.
Note that the whole series
\begin_inset Formula $\sum_{n=0}^{\infty}\cdots\pi_{a}\left(\prod_{i=1}^{n}\pi_{i}\right)\pi_{b}$
\end_inset
, where again
\begin_inset Formula $\pi_{a,b}$
\end_inset
are derangements and
\begin_inset Formula $\left\{ \pi_{i}\right\} $
\end_inset
are stabilisers, can be summed because the series is geometric.
The result is that the six point kernel can be determined by an equation
like
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:K-6pt-kernel-eom"
\end_inset
, except that the sum over all permutations is replaced by a sum over derangemen
ts only, and the bare vertex
\begin_inset Formula $W$
\end_inset
is replaced by a resummed four point kernel
\begin_inset Formula $\mathcal{K}_{abcd}^{\left(4\right)}\sim\lambda/\left(1+\lambda\mathcal{I}^{\mu}\right)$
\end_inset
.
Substituting the solution for
\begin_inset Formula $\mathcal{K}$
\end_inset
in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:aux-vertex-soln"
\end_inset
gives the solution for
\begin_inset Formula $V_{abc}^{\mu}$
\end_inset
.
\end_layout
\begin_layout Standard
This expression for
\begin_inset Formula $V_{abc}^{\mu}$
\end_inset
can be dramatically simplified in
\begin_inset Formula $3$
\end_inset
or
\begin_inset Formula $1+2$
\end_inset
dimensions because
\begin_inset Formula $\mathcal{I}^{\mu}$
\end_inset
is finite and the geometric sum in
\begin_inset Formula $\mathcal{K}_{abcd}^{\left(4\right)}$
\end_inset
converges.
Indeed
\begin_inset Formula $\mathcal{K}_{abcd}^{\left(4\right)}\left(p_{1},p_{2},p_{3},p_{1}+p_{2}-p_{3}\right)\sim\lambda/\left[1+\lambda/\left(p_{1}+p_{2}\right)^{4-d}\right]\to\lambda$
\end_inset
as
\begin_inset Formula $p_{1,2,3,4}\to\infty$
\end_inset
.
Further, every loop integral in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:aux-vertex-soln"
\end_inset
likewise converges, and every loop yields a factor of
\begin_inset Formula $\sim1/p^{4-d}$
\end_inset
.
Thus the dominant behaviour as
\begin_inset Formula $p\to\infty$
\end_inset
is just the tree level behaviour and the auxiliary vertex can be eliminated
completely.
\end_layout
\begin_layout Standard
However, in
\begin_inset Formula $4$
\end_inset
or
\begin_inset Formula $1+3$
\end_inset
dimensions
\begin_inset Formula $V_{abc}^{\mu}$
\end_inset
apparently cannot be simplified further.
First
\begin_inset Formula $\mathcal{K}_{abcd}^{\left(4\right)}$
\end_inset
must be renormalised, then the bubble appearing in the non-trivial terms
in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:aux-vertex-soln"
\end_inset
(or the equivalent integral equation) must be renormalised, then the resulting
series must be summed (or the equivalent integral equation solved), noting
that on the basis of power counting every term is apparently equally important.
On this basis ones expect that no compact analytic expression for
\begin_inset Formula $V_{abc}^{\mu}$
\end_inset
, or even its asymptotic behaviour, exists and that the renormalisation
must be accomplished as part of the self-consistent numerical solution
of the full equations of motion.
\end_layout
\begin_layout Standard
This style of argument can be quickly generalised to many other theories,
such as gauge theories, where the diagrammatic expansion has a similar
combinatorial structure to scalar
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset
theory, showing up the well known problem of the renormalisation of
\begin_inset Formula $n$
\end_inset
PIEA for
\begin_inset Formula $n\geq3$
\end_inset
in four dimensions.
The discussion here certainly does not solve this problem, which remains
open, to the author's knowledge, though hopefully this discussion may be
helpful.
\end_layout
\begin_layout Chapter
Deriving counter-terms for three loop truncations
\begin_inset CommandInset label
LatexCommand label
name "chap:Deriving-Counter-terms-for-3-Loop-Truncations"
\end_inset
\end_layout
\begin_layout Standard
Here the renormalisation of the three loop truncation of the SI-3PIEA is
carried out in
\begin_inset Formula $1+2$
\end_inset
dimensions as discussed in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:SI-3PIEA-Three-loop-truncation"
\end_inset
.
The effective action is as in the two loop truncations in Appendix
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Appendix-Renormalisation-Details"
\end_inset
except for a new counter-term
\begin_inset Formula $\delta\lambda\to\delta\lambda_{C}$
\end_inset
for the the
\begin_inset Formula $\Phi_{2}$
\end_inset
term and the addition of the three loop diagrams
\begin_inset Formula
\begin{align}
\Phi_{3} & =Z_{V}^{4}Z_{\Delta}^{6}\left[\left(N-1\right)\frac{i\hbar^{3}}{3!}V_{N}\left(\bar{V}\right)^{3}\left(\Delta_{H}\right)^{3}\left(\Delta_{G}\right)^{3}+\frac{i\hbar^{3}}{4!}\left(V_{N}\right)^{4}\left(\Delta_{H}\right)^{6}\right.\nonumber \\
& \left.+\left(N-1\right)\frac{i\hbar^{3}}{8}\left(\bar{V}\right)^{4}\Delta_{H}\Delta_{H}\left(\Delta_{G}\right)^{4}\right],\\
\Phi_{4} & =\frac{i\hbar^{3}\left(\lambda+\delta\lambda\right)}{24}Z_{V}^{2}Z_{\Delta}^{5}\left[2\left(N-1\right)\bar{V}V_{N}\left(\Delta_{H}\right)^{3}\Delta_{G}\Delta_{G}\right.+\left(N^{2}-1\right)\bar{V}\bar{V}\Delta_{H}\left(\Delta_{G}\right)^{4}\nonumber \\
& \left.+3V_{N}V_{N}\left(\Delta_{H}\right)^{5}+2^{2}\left(N-1\right)\bar{V}\bar{V}\left(\Delta_{G}\right)^{3}\Delta_{H}\Delta_{H}\right],\\
\Phi_{5} & =\frac{i\hbar^{3}\left(\lambda+\delta\lambda\right)^{2}}{144}Z_{\Delta}^{4}\left\{ \left[\left(N-1\right)\Delta_{G}\Delta_{G}+\Delta_{H}\Delta_{H}\right]^{2}+2\left(N-1\right)\left(\Delta_{G}\right)^{4}+2\left(\Delta_{H}\right)^{4}\right\} .
\end{align}
\end_inset
\end_layout
\begin_layout Standard
The equations of motion following from
\begin_inset Formula $\Gamma^{\left(3\right)}$
\end_inset
are then
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
\Delta_{G}^{-1} & =-\left(ZZ_{\Delta}\partial_{\mu}\partial^{\mu}+m^{2}+\delta m_{1}^{2}+Z_{\Delta}\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\right)\nonumber \\
& -\frac{\hbar}{6}\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]Z_{\Delta}^{2}\mathcal{T}_{G}-\frac{\hbar}{6}\left(\lambda+\delta\lambda_{2}^{A}\right)Z_{\Delta}^{2}\mathcal{T}_{H}\nonumber \\
& -i\hbar Z_{V}^{2}Z_{\Delta}^{3}\left[-2\frac{\left(\lambda+\delta\lambda_{C}\right)Z_{V}^{-1}v}{3}-\bar{V}\right]\Delta_{H}\Delta_{G}\bar{V}\nonumber \\
& +\hbar^{2}Z_{V}^{4}Z_{\Delta}^{6}\left[V_{N}\left(\bar{V}\right)^{3}\left(\Delta_{H}\right)^{3}\left(\Delta_{G}\right)^{2}+\left(\bar{V}\right)^{4}\Delta_{H}\Delta_{H}\left(\Delta_{G}\right)^{3}\right]\nonumber \\
& +\frac{\hbar^{2}\left(\lambda+\delta\lambda\right)}{3}Z_{V}^{2}Z_{\Delta}^{5}\left[\bar{V}V_{N}\left(\Delta_{H}\right)^{3}\Delta_{G}+\left(N+1\right)\bar{V}\bar{V}\Delta_{H}\left(\Delta_{G}\right)^{3}+3\bar{V}\bar{V}\left(\Delta_{G}\right)^{2}\Delta_{H}\Delta_{H}\right]\nonumber \\
& +\frac{\hbar^{2}\left(\lambda+\delta\lambda\right)^{2}}{18}Z_{\Delta}^{4}\left[\left(N+1\right)\left(\Delta_{G}\right)^{3}+\Delta_{H}\Delta_{H}\Delta_{G}\right],\label{eq:three-loop-Delta_G-eom}
\end{align}
\end_inset
for the Goldstone propagator,
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
\bar{V} & =-\frac{\left(\lambda+\delta\lambda_{C}\right)v}{3}Z_{V}^{-1}+i\hbar Z_{V}^{2}Z_{\Delta}^{3}\left[V_{N}\left(\bar{V}\right)^{2}\left(\Delta_{H}\right)^{2}\Delta_{G}+\left(\bar{V}\right)^{3}\Delta_{H}\left(\Delta_{G}\right)^{2}\right]\nonumber \\
& +\frac{i\hbar\left(\lambda+\delta\lambda\right)}{6}Z_{\Delta}^{2}\left[V_{N}\left(\Delta_{H}\right)^{2}+\left(N+1\right)\bar{V}\left(\Delta_{G}\right)^{2}+4\bar{V}\Delta_{G}\Delta_{H}\right],\label{eq:three-loop-Vbar-eom}
\end{align}
\end_inset
for the Higgs-Goldstone-Goldstone vertex,
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
V_{N} & =-\left(\lambda+\delta\lambda_{C}\right)vZ_{V}^{-1}+i\hbar Z_{V}^{2}Z_{\Delta}^{3}\left[\left(N-1\right)\left(\bar{V}\right)^{3}\left(\Delta_{G}\right)^{3}+\left(V_{N}\right)^{3}\left(\Delta_{H}\right)^{3}\right]\nonumber \\
& +\frac{i\hbar\left(\lambda+\delta\lambda\right)}{2}Z_{\Delta}^{2}\left[\left(N-1\right)\bar{V}\Delta_{G}\Delta_{G}+3V_{N}\left(\Delta_{H}\right)^{2}\right],\label{eq:three-loop-V_N-eom}
\end{align}
\end_inset
for the triple Higgs vertex, and finally
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
0 & =\Delta_{G}^{-1}\left(p=0\right)v,\label{eq:WI-1-three-loop}\\
0 & =Z_{V}Z_{\Delta}\bar{V}\left(p,-p,0\right)v+\Delta_{G}^{-1}\left(p\right)-\Delta_{H}^{-1}\left(p\right),\label{eq:WI2-three-loop}
\end{align}
\end_inset
for the Ward identities.
\end_layout
\begin_layout Standard
Note that the only divergent integrals in these equations are the linearly
divergent tadpole integrals
\begin_inset Formula $\mathcal{T}_{G/H}$
\end_inset
and the logarithmically divergent
\noun on
BBALL
\noun default
integrals (last line of
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:three-loop-Delta_G-eom"
\end_inset
).
By power counting one finds that the third, fourth, and fifth lines of
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:three-loop-Delta_G-eom"
\end_inset
produce finite self-energy contributions with leading asymptotics
\begin_inset Formula $\sim p^{-1}$
\end_inset
,
\begin_inset Formula $p^{-4}$
\end_inset
, and
\begin_inset Formula $p^{-2}$
\end_inset
respectively.
One can separate finite and divergent parts of
\begin_inset Formula $\Delta_{G}^{-1}$
\end_inset
as
\begin_inset Formula
\begin{equation}
\Delta_{G}^{-1}=-\left(\partial_{\mu}\partial^{\mu}+m^{2}+\frac{\lambda}{6}v^{2}\right)-\left[\Sigma_{G}^{0}\left(p\right)-\Sigma_{G}^{0}\left(m_{G}\right)\right]-\Sigma_{G}^{\infty}\left(p\right),
\end{equation}
\end_inset
where
\begin_inset Formula
\begin{align}
-\Sigma_{G}^{0}\left(p\right) & =-\frac{\hbar}{6}\left(N+1\right)\lambda\left(\mathcal{T}_{G}-\mathcal{T}^{\mu}\right)-\frac{\hbar}{6}\lambda\left(\mathcal{T}_{H}-\mathcal{T}^{\mu}\right)\nonumber \\
& -i\hbar\left[-2\frac{\left(\lambda+\delta\lambda_{C}\right)Z_{V}^{-1}v}{3}-\bar{V}\right]\Delta_{H}\Delta_{G}\bar{V}\nonumber \\
& +\hbar^{2}\left[V_{N}\left(\bar{V}\right)^{3}\left(\Delta_{H}\right)^{3}\left(\Delta_{G}\right)^{2}+\left(\bar{V}\right)^{4}\Delta_{H}\Delta_{H}\left(\Delta_{G}\right)^{3}\right]\nonumber \\
& +\frac{\hbar^{2}\left(\lambda+\delta\lambda\right)Z_{\Delta}^{2}}{3}\left[\bar{V}V_{N}\left(\Delta_{H}\right)^{3}\Delta_{G}+\left(N+1\right)\bar{V}\bar{V}\Delta_{H}\left(\Delta_{G}\right)^{3}+3\bar{V}\bar{V}\left(\Delta_{G}\right)^{2}\Delta_{H}\Delta_{H}\right]\nonumber \\
& +\frac{\hbar^{2}\left(\lambda+\delta\lambda\right)^{2}Z_{\Delta}^{4}}{18}\left[\left(N+1\right)\left(\Delta_{G}\right)^{3}+\Delta_{H}\Delta_{H}\Delta_{G}-\left(N+2\right)\mathcal{B}^{\mu}\right],
\end{align}
\end_inset
and
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
-\Sigma_{G}^{\infty}\left(p\right) & =-\Sigma_{G}^{0}\left(m_{G}\right)-\left(\left(ZZ_{\Delta}-1\right)\partial_{\mu}\partial^{\mu}+\delta m_{1}^{2}+\frac{\delta\lambda_{1}^{A}}{6}v^{2}+\left(Z_{\Delta}-1\right)\frac{\lambda+\delta\lambda_{1}^{A}}{6}v^{2}\right)\nonumber \\
& -\frac{\hbar}{6}\left(N+1\right)\lambda\mathcal{T}^{\mu}-\frac{\hbar}{6}\left[\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\mathcal{T}_{G}\nonumber \\
& -\frac{\hbar}{6}\left[\left(N+1\right)\lambda+\left(N-1\right)\delta\lambda_{2}^{A}+2\delta\lambda_{2}^{B}\right]\left(Z_{\Delta}^{2}-1\right)\mathcal{T}_{G}\nonumber \\
& -\frac{\hbar}{6}\lambda\mathcal{T}^{\mu}-\frac{\hbar}{6}\delta\lambda_{2}^{A}\mathcal{T}_{H}-\frac{\hbar}{6}\left(\lambda+\delta\lambda_{2}^{A}\right)\left(Z_{\Delta}^{2}-1\right)\mathcal{T}_{H}\nonumber \\
& -i\hbar\left(Z_{V}^{2}Z_{\Delta}^{3}-1\right)\left[-2\frac{\left(\lambda+\delta\lambda_{C}\right)Z_{V}^{-1}v}{3}-\bar{V}\right]\Delta_{H}\Delta_{G}\bar{V}\nonumber \\
& +\hbar^{2}\left(Z_{V}^{4}Z_{\Delta}^{6}-1\right)\left[V_{N}\left(\bar{V}\right)^{3}\left(\Delta_{H}\right)^{3}\left(\Delta_{G}\right)^{2}+\left(\bar{V}\right)^{4}\Delta_{H}\Delta_{H}\left(\Delta_{G}\right)^{3}\right]\nonumber \\
& +\frac{\hbar^{2}\left(\lambda+\delta\lambda\right)Z_{\Delta}^{2}}{3}\left(Z_{V}^{2}Z_{\Delta}^{3}-1\right)\nonumber \\
& \times\left[\bar{V}V_{N}\left(\Delta_{H}\right)^{3}\Delta_{G}+\left(N+1\right)\bar{V}\bar{V}\Delta_{H}\left(\Delta_{G}\right)^{3}+3\bar{V}\bar{V}\left(\Delta_{G}\right)^{2}\Delta_{H}\Delta_{H}\right]\nonumber \\
& +\left(N+2\right)\frac{\hbar^{2}\left(\lambda+\delta\lambda\right)^{2}Z_{\Delta}^{4}}{18}\mathcal{B}^{\mu},
\end{align}
\end_inset
are the finite and divergent parts respectively and we introduced the
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integral
\begin_inset Formula $\mathcal{B}^{\mu}=\int_{qp}\Delta^{\mu}\left(q\right)\Delta^{\mu}\left(p\right)\Delta^{\mu}\left(p+q\right)$
\end_inset
.
This split has already assumed that
\begin_inset Formula $\left(\lambda+\delta\lambda_{C}\right)Z_{V}^{-1}$
\end_inset
and
\begin_inset Formula $\left(\lambda+\delta\lambda\right)Z_{\Delta}^{2}$
\end_inset
are finite, which will be demonstrated to be consistent shortly.
Renormalisation requires
\begin_inset Formula $\Sigma_{G}^{\infty}\left(p\right)=0$
\end_inset
.
Note the explicit subtraction of
\begin_inset Formula $\Sigma_{G}^{0}\left(m_{G}\right)$
\end_inset
in order to fulfil the mass shell condition.
Doing the same now for
\begin_inset Formula $\Delta_{H}^{-1}$
\end_inset
gives the pole condition
\begin_inset Formula
\begin{equation}