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}{}\pgfsys@moveto{52.63777pt}{-2.84544pt}\pgfsys@lineto{52.63777pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.92009pt}{-10.68399pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$y_{1},s_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{44.06331pt}{-10.68399pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{74.6649pt}{-1.25055pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$x_{0}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.30379pt}{-2.10318pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$x,t$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (70)

As in Eq. (49), red solid lines denote bare transition probabilities. The blue
dashed line connecting two internal vertices, Eq. (23), represents the
correlation kernel $C_{2}(s_{2},s_{1})$. The two vertical bars inserted to the
right of each such vertex represent the gradient operator acting on the target
point of the incoming transition probability. As usual for Feynman diagrams,
external fields depend on fixed parameters ($x_{0},x,t$), whilst internal
fields depend on variables to be integrated over (e.g.,
$y_{1},s_{1},y_{2},s_{2}$). In addition to providing a better overview of
terms arising in the perturbative expansion, diagrams also act as graphical
cues illustrating how the driving noise induces memory into the evolution of
$x_{t}$.

###  C.2 The driving noise averaged visit probability

Following Eq. (26), we are able to evaluate driving noise averaged
observables. In order to derive the central results of Eqs. (28) and (29), it
remains to evaluate the driving noise averaged visit probability

$\displaystyle\mathcal{Q}(x,t)$
$\displaystyle=\left\langle\psi(x,t)\widetilde{\varphi}(0,0)\mathcal{Z}_{y}[g\int\widetilde{\varphi}\nabla\varphi]\right\rangle_{\mathcal{S}}$
(71)
$\displaystyle=\Bigg{\langle}\psi(x,t)\varphi(x_{0},0)\Bigg{(}1+g^{2}\int\\!\\!\\!\\!\\!\int{\rm
d}{y_{1,2}}\,{\rm
d}{s_{1,2}}\,\widetilde{\varphi}_{1}\partial_{y_{1}}\varphi_{1}C_{2}(s_{1}-s_{2})\widetilde{\varphi}_{2}\partial_{y_{2}}\varphi_{2}\Bigg{)}\Bigg{\rangle}_{\mathcal{S}}+\mathcal{O}(g^{3})$
(72)

where we use $\varphi_{i}=\varphi(y_{i},s_{i})$ for brevity. Each term
appearing in the $\gamma$-perturbative expansion of the trace function (cf.
(18)) is additionally corrected to order $g^{2}$ by replacing two internal
$\varphi\widetilde{\varphi}$ propagators by two “$y_{t}$-driven propagators”
$\widetilde{\varphi}\partial_{x}\varphi$ and connecting them with the two-
point correlator $C_{2}$ of $y_{t}$. As can be seen easiest diagrammatically,
all possible corrections fall into one of the four categories shown in Fig. 2.
They are classified according to whether both $g$-vertices couple to the same
or different propagator of a transition or a return. It is simplest to compute
the four contributions in frequency rather than direct time as the loops
factorise. The calculation itself is given in App. E.

##  Appendix D Renormalisation of transmutation rate

In this appendix, we derive the result for the renormalisation of the coupling
$\tau$ which is stated in Eq. (19). To simplify the computation, we perform
the calculation in $x,\omega$ variables, i.e., in real space and Fourier
transformed time. The renormalisation is given by (cf. Eq. (18))

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which is a diagrammatic representation of the terms arising in the path-
integrated average of the visit probability, Eq. (12), when expanding in
$\gamma$,

$n_{0}^{-1}\left\langle\psi(x,\omega_{1})\widetilde{\varphi}(x_{0},\omega_{0})\right\rangle_{\mathcal{S}}=n_{0}^{-1}\gamma\tau\int\\!\\!\\!\\!\\!\int{\rm
d}{z}\,{\rm d}\bar{}\hskip
1.00006pt{\omega^{\prime}}\left\langle\psi(x,\omega_{1})\widetilde{\psi}(z,\omega^{\prime})\varphi(z,\omega^{\prime})\widetilde{\varphi}(x_{0},\omega_{0})\right\rangle_{\mathcal{S};\gamma=0}\\\
-n_{0}^{-1}\gamma^{2}\lambda\sigma\int\\!\\!\\!\\!\\!\int{\rm d}{z_{1}}\,{\rm
d}{z_{2}}\,{\rm d}\bar{}\hskip 1.00006pt{\omega^{\prime}_{1}}{\rm
d}\bar{}\hskip 1.00006pt{\omega^{\prime}_{2}}{\rm d}\bar{}\hskip
1.00006pt{\omega_{1}{{}^{\prime\prime}}}{\rm d}\bar{}\hskip
1.00006pt{\omega_{2}{{}^{\prime\prime}}}\\\
\times\left\langle\psi(x,\omega_{1})\widetilde{\psi}(z_{1},-\omega^{\prime}_{1}-\omega^{\prime\prime}_{1})\psi(z_{1},\omega_{1}{{}^{\prime\prime}})\varphi(z_{1},\omega^{\prime}_{1})\widetilde{\psi}(z_{2},\omega_{2}{{}^{\prime\prime}})\widetilde{\varphi}(z_{2},\omega^{\prime}_{2})\varphi(z_{2},-\omega^{\prime}_{2}-\omega^{\prime\prime}_{2})\widetilde{\varphi}(x_{0},\omega_{0})\right\rangle_{\mathcal{S};\gamma=0}+\cdots.$
(74)

Crucially, the averages
$\left\langle\bullet\right\rangle_{\mathcal{S};\gamma=0}$ are taken over
Gaussian random variables since for $\gamma=0$ the path action (Eq. (11)) is
bilinear. Thus, Wick’s theorem (e.g., [58]) applies and all averages in Eq.
(74) decompose into products of two-point functions. The only such Gaussian
two-point functions which do not vanish are

$\displaystyle\left\langle\varphi(z_{1},\omega_{1})\widetilde{\varphi}(z_{0},\omega_{0})\right\rangle_{\mathcal{S};\gamma=0}$
$\displaystyle={T}\left(z_{0},z_{1};\omega_{1}\right){\delta}\bar{}\hskip
1.00006pt(\omega_{0}+\omega_{1})=\int{\rm
d}{t}\,e^{i\omega_{1}(t-t_{0})}{T}\left(z_{0},z_{1};t\right){\delta}\bar{}\hskip
1.00006pt(\omega_{0}+\omega_{1})$ (75)
$\displaystyle\left\langle\psi(z_{1},\omega_{1})\widetilde{\psi}(z_{0},\omega_{0})\right\rangle_{\mathcal{S};\gamma=0}$
$\displaystyle=\frac{\delta(z_{1}-z_{0}){\delta}\bar{}\hskip
1.00006pt(\omega_{0}+\omega_{1})}{-i\omega_{1}+\varepsilon}=\int\\!\\!\\!\\!\\!\int{\rm
d}{t_{0}}\,{\rm
d}{t}\,e^{i\omega_{1}t+i\omega_{0}t_{0}}\Theta(t-t_{0})\delta(z_{1}-z_{0})e^{-\varepsilon(t-t_{0})}$
(76)

The second correlator intuitively characterises the behaviour of the trace
which, once deposited at $z_{0}$ at time $t_{0}$, remains there for an
infinitely long time, as $\varepsilon\to 0$. Equipped with these correlators,
the non-vanishing contributions to the averages appearing in Eq. (74) are,
following Wick’s theorem,

$\displaystyle n_{0}^{-1}\gamma\tau\int\\!\\!\\!\\!\\!\int{\rm d}{z}\,{\rm
d}\bar{}\hskip
1.00006pt{\omega^{\prime}}\left\langle\psi(x,\omega_{1})\widetilde{\psi}(z,\omega^{\prime})\varphi(z,\omega^{\prime})\widetilde{\varphi}(x_{0},\omega_{0})\right\rangle_{\mathcal{S};\gamma=0}$
$\displaystyle=n_{0}^{-1}\gamma\tau\int\\!\\!\\!\\!\\!\int{\rm d}{z}\,{\rm
d}\bar{}\hskip
1.00006pt{\omega^{\prime}}\frac{\delta(x-z)\delta(\omega_{1}+\omega^{\prime})}{-i\omega_{1}+\varepsilon}T(x_{0},z,\omega^{\prime}){\delta}\bar{}\hskip
1.00006pt(\omega^{\prime}+\omega_{0})$
$\displaystyle=n_{0}^{-1}\gamma\tau\frac{{T}\left(x_{0},x,-\omega_{1}\right)}{-i\omega_{1}+\varepsilon}{\delta}\bar{}\hskip
1.00006pt(\omega_{0}-\omega_{1}),$ (77)

and to second order,

$\displaystyle-n_{0}^{-1}\gamma^{2}\lambda\sigma\int\\!\\!\\!\\!\\!\int{\rm
d}{z_{1}}\,{\rm d}{z_{2}}\,{\rm d}\bar{}\hskip
1.00006pt{\omega^{\prime}_{1}}{\rm d}\bar{}\hskip
1.00006pt{\omega^{\prime}_{2}}{\rm d}\bar{}\hskip
1.00006pt{\omega{{}^{\prime\prime}}_{1}}{\rm d}\bar{}\hskip
1.00006pt{\omega_{2}{{}^{\prime\prime}}}$ (78)
$\displaystyle\qquad\times\left\langle\psi(x,\omega_{1})\widetilde{\psi}(z_{1},-\omega^{\prime}_{1}-\omega^{\prime\prime}_{1})\psi(z_{1},\omega_{1}{{}^{\prime\prime}})\varphi(z_{1},\omega^{\prime}_{1})\widetilde{\psi}(z_{2},\omega_{2}{{}^{\prime\prime}})\widetilde{\varphi}(z_{2},\omega^{\prime}_{2})\varphi(z_{2},-\omega^{\prime}_{2}-\omega^{\prime\prime}_{2})\widetilde{\varphi}(x_{0},\omega_{0})\right\rangle_{\mathcal{S};\gamma=0}$
$\displaystyle\qquad\qquad=-n_{0}^{-1}\gamma^{2}\lambda\sigma\frac{1}{-i\omega_{1}+\varepsilon}\int{\rm
d}\bar{}\hskip
1.00006pt{\omega^{\prime\prime}_{2}}\frac{{R}\left(x,x,\omega_{1}-\omega^{\prime\prime}_{2}\right)}{-i\omega^{\prime\prime}_{1}+\varepsilon}{T}\left(x_{0},x,\omega_{0}\right){\delta}\bar{}\hskip
1.00006pt(\omega_{0}+\omega_{1})$ (79)
$\displaystyle\qquad\qquad=-n_{0}^{-1}\gamma^{2}\lambda\sigma{R}\left(x,\omega_{1}+i\varepsilon\right)\frac{{T}\left(x_{0},x,\omega_{0}\right)}{-i\omega_{1}+\varepsilon}{\delta}\bar{}\hskip
1.00006pt(\omega_{0}+\omega_{1}),$ (80)

where in the first equality we used the definition of the return probability
to abbreviate

$\displaystyle\int\\!\\!\\!\\!\\!\int{\rm d}{z_{1}}\,{\rm
d}{z_{2}}\,{\delta}\bar{}\hskip
1.00006pt(x-z_{1})\delta(z_{1}-z_{2}){T}\left(z_{1},z_{2},\omega\right)={R}\left(x,\omega\right),$
(81)

and in the second equality used Cauchy’s residue formula to solve the integral
by evaluating the residue of the simple pole at
$\omega^{\prime\prime}_{2}=-i\varepsilon$.

Since both correlators in Eq. (75) are proportional to ${\delta}\bar{}\hskip
1.00006pt(\omega_{0}+\omega_{1})$, all higher order expansion terms factorise,
after integrating over the internal frequencies, into a product over amputated
one-loop bubble diagrams (i.e., interpreted here as a function of external
parameters $z_{1},\omega_{1}$ and $z_{2},\omega_{2}$, respectively) which by
analogous reasoning to the calculation above evaluate as

$z_{1},\omega_{1}$$z_{2},\omega_{2}$
$\displaystyle=\gamma^{2}\lambda\sigma{R}\left(z_{1},\omega_{1}+i\varepsilon\right)\delta(\omega_{1}+\omega_{2})\delta(z_{1}-z_{2}).$
(82)

The bubble diagram graphically encodes the probability of a particle
depositing a trace and then returning to it, in other words a “time-ordered”
return probability: The green wriggly line may be understood as the trace
which once placed remains immobile, meanwhile the red solid line represents
the diffusing, and returning, walker. Likewise, the higher order diagrams in
Eq. (18) may be interpreted as repeated returns to $x$.

Returning to Eq. (18), the renormalised $\tau$ coupling, $\tau_{R}$, is the
effective factor satisfying

$\displaystyle\left\langle\psi(x,\omega_{1})\widetilde{\varphi}(x_{0},\omega_{0})\right\rangle_{\mathcal{S}}=\frac{1}{-i\omega_{1}+\varepsilon}\tau_{R}(\omega_{1}){T}\left(x_{0},x,-\omega_{0}\right){\delta}\bar{}\hskip
1.00006pt(\omega_{0}+\omega_{1})$ (83)

and collecting the factors generated by the terms in Eq. (74), and evaluated
using Eq. (82), one obtains

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}\pgfsys<EMAIL_ADDRESS>(84)
$\displaystyle=\left[\gamma\tau-\gamma^{2}\lambda\sigma{R}\left(x,\omega_{1}+i\varepsilon\right)+\gamma^{3}\lambda\sigma\kappa\left({R}\left(x,\omega_{1}+i\varepsilon\right)\right)^{2}-\gamma^{4}c\sigma\kappa^{2}\left({R}\left(x,\omega_{1}+i\varepsilon\right)\right)^{3}+\cdots\right]{\delta}\bar{}\hskip
1.00006pt(\omega_{0}+\omega_{1})$ (85)

This series can be resummed using the geometric series, in field theory often
referred to as Dyson summation [58]. Rearranging the sum gives

$\displaystyle\tau_{R}(\omega_{1})$
$\displaystyle=\left[\gamma\tau+\gamma\frac{\lambda\sigma}{\kappa}\sum_{r=1}^{\infty}\left(-\gamma\kappa{R}\left(x,\omega_{1}+i\varepsilon\right)\right)^{r}\right]$
(86)
$\displaystyle=\left[\gamma\tau+\gamma\frac{\lambda\sigma}{\kappa}\sum_{r=0}^{\infty}\left(-\gamma\kappa{R}\left(x,\omega_{1}+i\varepsilon\right)\right)^{r}-\gamma\frac{c\sigma}{\kappa}\right]$
(87)
$\displaystyle=\left[\frac{\gamma\tau}{1+\gamma\kappa{R}\left(x,\omega_{1}+i\varepsilon\right)}+\gamma\left(\underbrace{\tau-\frac{c\sigma}{\kappa}}_{=0}\right)\right]$
(88)
$\displaystyle=\frac{\gamma\tau}{1+\gamma\kappa{R}\left(x,\omega_{1}+i\varepsilon\right)}$
(89)

where we made use of the bare values given in Eq. (14), i.e., replaced
$\sigma$ with $\tau$ and used $\lambda=\kappa$ at bare level. This vertex
interpolates the physical pictures for $\gamma=0$, where no deposition takes
place ($\tau_{R}=0$), and $\gamma\to\infty$, where every newly visited site
gets marked immediately by a deposited trace. For $\gamma\to\infty$, the
coupling tends to

$\displaystyle\lim_{\gamma\to\infty}\tau_{R}(\omega_{1})=\frac{n_{0}}{{R}\left(x,\omega_{1}+i\varepsilon\right)},$
(90)

using $n_{0}=\tau/\kappa$.

##  Appendix E Derivation of four non-Markovian correction terms to trace
function

When expanding the nonlinear action in Eq. (72) to all orders in $\gamma$ and
to $\mathcal{O}(g^{2})$, the nonvanishing contribution of joint perturbative
order $\gamma^{n}g^{2}$ is obtained by inserting the (Fourier-transformed)
$g^{2}$-decoration

(91) $\displaystyle=\int\\!\\!\\!\\!\\!\int{\rm d}\bar{}\hskip
1.00006pt^{d}{\tilde{\omega}_{1}}\ldots{\rm d}\bar{}\hskip
1.00006pt^{d}{\tilde{\omega}_{4}}{\rm d}{y_{1}}\,...{\rm
d}{y_{4}}\,\widetilde{\varphi}(y_{1},\tilde{\omega}_{1})\partial_{y_{2}}\varphi(y_{2},\tilde{\omega}_{2})C_{2}(\tilde{\omega}_{3}+\tilde{\omega}_{4})\widetilde{\varphi}(y_{3},\tilde{\omega}_{3})\partial_{y_{4}}\varphi(y_{4},\tilde{\omega}_{4})\times{\delta}\bar{}\hskip
1.00006pt(\tilde{\omega}_{1}+\tilde{\omega}_{2}+\tilde{\omega}_{3}+\tilde{\omega}_{4})$

into the expansion of the trace function
$\left\langle\psi\widetilde{\varphi}\right\rangle$ as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathcal{Q}(x_{0},x,\omega){\delta}\bar{}\hskip
1.00006pt(\omega+\omega^{\prime})$ (92) $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\gamma^{2+n}\int{\rm d}\bar{}\hskip
1.00006pt{\omega_{1}}...{\rm d}\bar{}\hskip 1.00006pt{\omega_{n}}{\rm
d}\bar{}\hskip 1.00006pt{\tilde{\omega}_{1}}...{\rm d}\bar{}\hskip
1.00006pt{\tilde{\omega}_{4}}{\rm d}{z_{1}}\,...{\rm d}{z_{n}}\,{\rm
d}{y_{1}}\,...{\rm d}{y_{4}}\,$
$\displaystyle\left\langle\psi(x,\omega)\left(-\lambda\widetilde{\psi}(z_{1},\omega_{1})\varphi(z_{1},\omega_{1})\psi(z_{1},\omega_{1})\right)\left(-\kappa\widetilde{\varphi}(z_{2},\omega_{2})\widetilde{\psi}(z_{2},\omega_{2})\varphi(z_{2},\omega_{2})\psi(z_{2},\omega_{2})\right)\right.$
$\displaystyle...\left(-\kappa\widetilde{\varphi}(z_{n-1},\omega_{n-1})\widetilde{\psi}(z_{n-1},\omega_{n-1})\varphi(z_{n-1},\omega_{n-1})\psi(z_{n-1},\omega_{n-1})\right)\left(\sigma\widetilde{\varphi}(z_{n},\omega_{n})\widetilde{\psi}(z_{n},\omega_{n})\varphi(z_{n},\omega_{n})\right)\varphi(x_{0},\omega^{\prime})$
$\displaystyle\times\left.g^{2}\widetilde{\varphi}(y_{1},\tilde{\omega}_{1})\partial_{y_{2}}\varphi(y_{2},\tilde{\omega}_{2})C_{2}(\tilde{\omega}_{3}+\tilde{\omega}_{4})\widetilde{\varphi}(y_{3},\tilde{\omega}_{3})\partial_{y_{4}}\varphi(y_{4},\tilde{\omega}_{4})\times{\delta}\bar{}\hskip
1.00006pt(\tilde{\omega}_{1}+\tilde{\omega}_{2}+\tilde{\omega}_{3}+\tilde{\omega}_{4})\right\rangle.$

As usual, we employ Wick’s theorem to evaluate this average over Gaussian
random variables: When expanding the average
$n_{0}^{-1}\left\langle\psi(x,\omega)\widetilde{\varphi}(x_{0},\omega^{\prime})\right\rangle$
in powers of $\gamma$ and $g$, as shown in Eq. (92), the resulting
coefficients are averages over finite products of
$\varphi_{1},...\varphi_{j_{1}},\widetilde{\varphi}_{1}...\varphi_{j_{2}},\psi_{1}...\psi_{j_{3}}$
and $\widetilde{\psi}_{1}..\widetilde{\psi}_{j_{4}}$ fields (where
$\varphi_{i}=\varphi(z_{i},\omega_{i})$ etc.) with respect to the Gaussian
measure (defined by the action in Eq. (11) for $\gamma=0$). Each of those
coefficients is evaluated using Wick’s theorem, i.e.,

$\displaystyle\left\langle\psi_{1}...\psi_{j_{1}}\widetilde{\psi}_{1}...\widetilde{\psi}_{j_{2}}\varphi_{1}...\varphi_{j_{3}}\widetilde{\varphi}_{1}...\widetilde{\varphi}_{j_{4}}\right\rangle_{\mathcal{S};\gamma=0}=\sum_{\text{pairings}}\prod_{(k_{m},\ell_{m})}\left\langle\phi_{k_{m}}\phi_{\ell_{m}}\right\rangle_{\mathcal{S};\gamma=0}$
(93)

where the sum runs over all possible pairwise pairings of the
$j_{1}+j_{2}+j_{3}+j_{4}$ indices, and we use $\phi$ in lieu of
$\varphi,\widetilde{\varphi},\psi,\widetilde{\psi}$ to alleviate notation.
Although the number of possible combinations of such pairings is very large,
the right hand side of Eq. (93) drastically simplifies because most of the
pairwise averages vanish under the Gaussian average. Again, as in the case of
$g=0$ (cf. App. B), the only non-vanishing Gaussian correlators are those of
the form
$\left\langle\varphi\widetilde{\varphi}\right\rangle,\left\langle\psi\widetilde{\psi}\right\rangle$
as given by Eqs. (75), (76). Thus, the sum over averages in Eq. (92)
simplifies into a sum over integrals over products of these two Gaussian
propagators. The integrals run over $n$ internal spatial variables
$z_{1},...,z_{n}$ which stem from the expansion in $\gamma^{n}$, and four
internal spatial variables $y_{1},...,y_{4}$ which stem from the expansion in
$g^{2}$ (and thus are related to the non-Markovian correction). Analagously,
the integration also runs over $n$ frequencies $\omega_{1},...,\omega_{n}$
stemming from the $\gamma^{n}$-expansion and four frequencies
$\tilde{\omega}_{1},...,\tilde{\omega}_{4}$ from the $g^{2}$-expansion. The
integral over the internal space variables $z_{1},...,z_{n}$, simplifies
significantly, since the corresponding
$\left\langle\psi\widetilde{\psi}\right\rangle$ propagators (cf. Eq. (76)) are
proportional to spatial $\delta$-functions. This results in all internal space
coordinates to identify as $z_{1}=...=z_{n}=x$.

The integral over the internal spatial coordinates $y_{1},...,y_{4}$ appearing
in the four $\varphi$ and $\widetilde{\varphi}$ fields in Eq. (92), however,
is less trivial as it involves the correlators (“propagators”) of
$\left\langle\varphi\widetilde{\varphi}\right\rangle$. Likewise, the
integration over $\tilde{\omega}_{1},...,\tilde{\omega}_{4}$, further involves
the correlator of the driving noise,
$C_{2}(\tilde{\omega}_{3}+\tilde{\omega}_{4})$. Hence, the integrals stemming
from the $g^{2}$-expansion need to be dealt with more carefully: The four
fields
$\widetilde{\varphi}(y_{1},\tilde{\omega}_{1})\partial_{y_{2}}\widetilde{\varphi}(y_{2},\tilde{\omega}_{2})\widetilde{\varphi}(y_{3},\tilde{\omega}_{4})\partial_{y_{4}}\varphi(y_{4},\tilde{\omega}_{4})$
appearing in every term of Eq. (92) have to be paired up (according to Eq.
(93)) with another creation field $\widetilde{\varphi}(z_{i})$ or annihilation
field $\widetilde{\varphi}(z_{i})$ appearing in Eq. (92), respectively, in
order to have a non-vanishing contribution (cf. Eq. (75)). Up to permutation
of indices, each of these possible combinations (which we refer to as “ _Wick
pairings_ ”, Eq. (93)) can fundamentally be grouped into four different ways
that are best understood diagrammatically: In any case, each of the two
$\varphi(y_{i})\partial_{y_{j}}\widetilde{\varphi}(y_{j})$ attaches via Wick-
pairing to a $\varphi(z_{k})\widetilde{\varphi}(z_{\ell})$ field appearing in
the terms of Eq. (92). Concurrently,
$\widetilde{\varphi}(z_{k}),\varphi(z_{\ell})$, Wick-pair with two other
corresponding fields giving rise to connected
$\left\langle\varphi\widetilde{\varphi}\right\rangle$ \- “propagators” and so
on. Thereby, the Wick-pairing of
$\varphi(y_{i})\partial_{y_{j}}\widetilde{\varphi}(y_{j})$, diagrammatically
speaking, splits an existing propagator in the $\gamma^{n}$-expansion into two
or, as shown below, in three.

In the expansion to order $\gamma^{n}$, there is one propagator from $x_{0}$
to $x$ (_transition_) and $n$ propagators from $x$ to $x$ (_return_)
represented by loop diagrams. The $g$-vertex can thus occur in four different
ways, Fig. 2, to be distinguished by whether and how the $g$-vertex enters
into the initial _transition propagator_ , , in $\leavevmode\hbox
to23.86pt{\vbox to3.85pt{\pgfpicture\makeatletter\hbox{\hskip
11.88092pt\lower-1.92264pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{11.38092pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}} }
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.45,0}\pgfsys@color@rgb@stroke{0}{0.45}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to26.71pt{\vbox to10.12pt{\pgfpicture\makeatletter\hbox{\hskip
9.0359pt\lower-8.19748pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{17.07182pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.45,0}\pgfsys@color@rgb@stroke{0}{0.45}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.625pt}{1.42264pt}{-0.9375pt}{1.42264pt}{-1.5625pt}\pgfsys@curveto{1.56621pt}{-1.99161pt}{1.0316pt}{-2.58171pt}{0.47015pt}{-3.1993pt}\pgfsys@curveto{-0.09128pt}{-3.81683pt}{-0.6259pt}{-4.40697pt}{-0.48233pt}{-4.83609pt}\pgfsys@curveto{-0.14554pt}{-5.13828pt}{0.60837pt}{-4.88197pt}{1.39813pt}{-4.61195pt}\pgfsys@curveto{2.18785pt}{-4.34193pt}{2.94179pt}{-4.08563pt}{3.2786pt}{-4.38782pt}\pgfsys@curveto{3.72086pt}{-4.48352pt}{3.9552pt}{-5.24454pt}{4.19943pt}{-6.04265pt}\pgfsys@curveto{4.44366pt}{-6.84073pt}{4.67801pt}{-7.60178pt}{5.12027pt}{-7.69748pt}\pgfsys@curveto{5.5519pt}{-7.56166pt}{5.71553pt}{-6.7824pt}{5.88562pt}{-5.96527pt}\pgfsys@curveto{6.05571pt}{-5.1482pt}{6.21933pt}{-4.36888pt}{6.65097pt}{-4.23306pt}\pgfsys@lineto{9.67383pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\ldots$, Eq. (73), or
into any of the _return propagators_ . First, we consider
$\mathcal{Q}^{(1)}_{\mathrm{I}}$, I) in Fig. 2, the case of the correlation
coupling appearing twice within the same return propagator leading to the
diagrammatic sum

$\displaystyle\mathcal{Q}^{(1)}_{\mathrm{I}}(x_{0},x,\omega)$
$\displaystyle=n_{0}^{-1}\lim_{\gamma\to\infty}\left[-\gamma^{2}\leavevmode\hbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)^{s}\times\leavevmode\hbox
to15.43pt{\vbox to1.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right]$ (95)
$\displaystyle=-\frac{1}{-i\omega}\frac{{T}\left(x_{0},x\right)}{\left({R}\left(x,\omega\right)\right)^{2}}\int\\!\\!\\!\\!\\!\int{\rm
d}{y_{1}}\,{\rm d}{y_{2}}\,{\rm d}\bar{}\hskip
1.00006pt{\tilde{\omega}}\left(\partial_{y_{1}}{T}\left(x,y_{1},\omega\right)\right)\left(\partial_{y_{2}}{T}\left(y_{1},y_{2},\omega-\tilde{\omega}\right)\right){T}\left(y_{2},x,\omega\right)C_{2}(\tilde{\omega})$
(96)

In the last line, we replaced the geometric sums in Eq. (95) by the expression
found for $\tau_{R}$ in Eq. (89). To be precise, this is not a matter of
trivially renormalising $\tau$ to $\tau_{R}$ , as the sums appearing in Eq.
(95) are in fact renormalisations of $\lambda$ and $\sigma$, respectively,

$\displaystyle-\lambda_{R}$
$\displaystyle=(-\lambda)\gamma\sum_{r=0}^{\infty}\left(\gamma\leavevmode\hbox
to16.24pt{\vbox to10.8pt{\pgfpicture\makeatletter\hbox{\hskip
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}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.45,0}\pgfsys@color@rgb@stroke{0}{0.45}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-0.625pt}{-12.80374pt}{-0.9375pt}{-12.80374pt}{-1.5625pt}\pgfsys@curveto{-12.70557pt}{-2.00421pt}{-13.29863pt}{-2.53554pt}{-13.92122pt}{-3.0914pt}\pgfsys@curveto{-14.54381pt}{-3.64723pt}{-15.13689pt}{-4.17859pt}{-15.0387pt}{-4.6203pt}\pgfsys@curveto{-14.79375pt}{-5.00076pt}{-13.99893pt}{-4.95285pt}{-13.16588pt}{-4.90118pt}\pgfsys@curveto{-12.33289pt}{-4.84952pt}{-11.53802pt}{-4.8016pt}{-11.29306pt}{-5.18207pt}\pgfsys@curveto{-10.93256pt}{-5.45554pt}{-11.04143pt}{-6.24434pt}{-11.15697pt}{-7.07095pt}\pgfsys@curveto{-11.27252pt}{-7.89752pt}{-11.3814pt}{-8.68636pt}{-11.02087pt}{-8.95984pt}\pgfsys@curveto{-10.58812pt}{-9.09203pt}{-10.01234pt}{-8.54199pt}{-9.40982pt}{-7.96443pt}\pgfsys@curveto{-8.80734pt}{-7.38687pt}{-8.23154pt}{-6.83684pt}{-7.79877pt}{-6.96902pt}\pgfsys@curveto{-7.34694pt}{-6.94449pt}{-6.91986pt}{-7.61655pt}{-6.47343pt}{-8.32178pt}\pgfsys@curveto{-6.02704pt}{-9.027pt}{-5.59995pt}{-9.69907pt}{-5.1481pt}{-9.67453pt}\pgfsys@curveto{-4.73259pt}{-9.49539pt}{-4.6493pt}{-8.70348pt}{-4.56346pt}{-7.87328pt}\pgfsys@curveto{-4.47763pt}{-7.04309pt}{-4.39435pt}{-6.25117pt}{-3.97882pt}{-6.07202pt}\pgfsys@curveto{-3.65028pt}{-5.76088pt}{-2.88976pt}{-5.99678pt}{-2.093pt}{-6.24542pt}\pgfsys@curveto{-1.2963pt}{-6.49408pt}{-0.53574pt}{-6.72998pt}{-0.20718pt}{-6.41882pt}\pgfsys@lineto{0.00002pt}{-0.00002pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)^{r}=\gamma(-\lambda)+\gamma(-\lambda)\sum_{r=1}^{\infty}\left(-\gamma\kappa{R}\left(x,\omega\right)\right)^{r}\stackrel{{\scriptstyle\gamma\to\infty}}{{\longrightarrow}}-\frac{1}{{R}\left(x,\omega\right)}$
(97) $\displaystyle\sigma_{R}$
$\displaystyle=\sigma\gamma\sum_{r=0}^{\infty}\left(\gamma\leavevmode\hbox
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}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-0.625pt}{-12.80374pt}{-0.9375pt}{-12.80374pt}{-1.5625pt}\pgfsys@curveto{-12.70557pt}{-2.00421pt}{-13.29863pt}{-2.53554pt}{-13.92122pt}{-3.0914pt}\pgfsys@curveto{-14.54381pt}{-3.64723pt}{-15.13689pt}{-4.17859pt}{-15.0387pt}{-4.6203pt}\pgfsys@curveto{-14.79375pt}{-5.00076pt}{-13.99893pt}{-4.95285pt}{-13.16588pt}{-4.90118pt}\pgfsys@curveto{-12.33289pt}{-4.84952pt}{-11.53802pt}{-4.8016pt}{-11.29306pt}{-5.18207pt}\pgfsys@curveto{-10.93256pt}{-5.45554pt}{-11.04143pt}{-6.24434pt}{-11.15697pt}{-7.07095pt}\pgfsys@curveto{-11.27252pt}{-7.89752pt}{-11.3814pt}{-8.68636pt}{-11.02087pt}{-8.95984pt}\pgfsys@curveto{-10.58812pt}{-9.09203pt}{-10.01234pt}{-8.54199pt}{-9.40982pt}{-7.96443pt}\pgfsys@curveto{-8.80734pt}{-7.38687pt}{-8.23154pt}{-6.83684pt}{-7.79877pt}{-6.96902pt}\pgfsys@curveto{-7.34694pt}{-6.94449pt}{-6.91986pt}{-7.61655pt}{-6.47343pt}{-8.32178pt}\pgfsys@curveto{-6.02704pt}{-9.027pt}{-5.59995pt}{-9.69907pt}{-5.1481pt}{-9.67453pt}\pgfsys@curveto{-4.73259pt}{-9.49539pt}{-4.6493pt}{-8.70348pt}{-4.56346pt}{-7.87328pt}\pgfsys@curveto{-4.47763pt}{-7.04309pt}{-4.39435pt}{-6.25117pt}{-3.97882pt}{-6.07202pt}\pgfsys@curveto{-3.65028pt}{-5.76088pt}{-2.88976pt}{-5.99678pt}{-2.093pt}{-6.24542pt}\pgfsys@curveto{-1.2963pt}{-6.49408pt}{-0.53574pt}{-6.72998pt}{-0.20718pt}{-6.41882pt}\pgfsys@lineto{0.00002pt}{-0.00002pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)^{r}=\gamma\sigma+\gamma\sigma\sum_{r=1}^{\infty}\left(-\gamma\kappa{R}\left(x,\omega\right)\right)^{r}\stackrel{{\scriptstyle\gamma\to\infty}}{{\longrightarrow}}\frac{n_{0}}{{R}\left(x,\omega\right)},$
(98)

where again we made use of the definition $n_{0}=\sigma\kappa^{-1}$. However,
comparing to Eq. (89), they renormalise identically, as does $\kappa$

$\displaystyle-\kappa_{R}$
$\displaystyle=(-\kappa)\gamma\sum_{r=0}^{\infty}\left(\gamma\leavevmode\hbox
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}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-14.22638pt}{-0.625pt}{-12.80374pt}{-0.9375pt}{-12.80374pt}{-1.5625pt}\pgfsys@curveto{-12.70557pt}{-2.00421pt}{-13.29863pt}{-2.53554pt}{-13.92122pt}{-3.0914pt}\pgfsys@curveto{-14.54381pt}{-3.64723pt}{-15.13689pt}{-4.17859pt}{-15.0387pt}{-4.6203pt}\pgfsys@curveto{-14.79375pt}{-5.00076pt}{-13.99893pt}{-4.95285pt}{-13.16588pt}{-4.90118pt}\pgfsys@curveto{-12.33289pt}{-4.84952pt}{-11.53802pt}{-4.8016pt}{-11.29306pt}{-5.18207pt}\pgfsys@curveto{-10.93256pt}{-5.45554pt}{-11.04143pt}{-6.24434pt}{-11.15697pt}{-7.07095pt}\pgfsys@curveto{-11.27252pt}{-7.89752pt}{-11.3814pt}{-8.68636pt}{-11.02087pt}{-8.95984pt}\pgfsys@curveto{-10.58812pt}{-9.09203pt}{-10.01234pt}{-8.54199pt}{-9.40982pt}{-7.96443pt}\pgfsys@curveto{-8.80734pt}{-7.38687pt}{-8.23154pt}{-6.83684pt}{-7.79877pt}{-6.96902pt}\pgfsys@curveto{-7.34694pt}{-6.94449pt}{-6.91986pt}{-7.61655pt}{-6.47343pt}{-8.32178pt}\pgfsys@curveto{-6.02704pt}{-9.027pt}{-5.59995pt}{-9.69907pt}{-5.1481pt}{-9.67453pt}\pgfsys@curveto{-4.73259pt}{-9.49539pt}{-4.6493pt}{-8.70348pt}{-4.56346pt}{-7.87328pt}\pgfsys@curveto{-4.47763pt}{-7.04309pt}{-4.39435pt}{-6.25117pt}{-3.97882pt}{-6.07202pt}\pgfsys@curveto{-3.65028pt}{-5.76088pt}{-2.88976pt}{-5.99678pt}{-2.093pt}{-6.24542pt}\pgfsys@curveto{-1.2963pt}{-6.49408pt}{-0.53574pt}{-6.72998pt}{-0.20718pt}{-6.41882pt}\pgfsys@lineto{0.00002pt}{-0.00002pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)^{r}=\gamma(-\kappa)\sum_{r=0}^{\infty}\left(-\gamma\kappa{R}\left(x,\omega\right)\right)^{r}\stackrel{{\scriptstyle\gamma\to\infty}}{{\longrightarrow}}-\frac{1}{{R}\left(x,\omega\right)}$
(99)

to be used below.

Another identity entering into expressing $\mathcal{Q}^{(1)}_{\mathrm{I}}$,
Eq. (94), as Eq. (96), is the key-ingredient of
$\mathcal{Q}^{(1)}_{\mathrm{I}}$,

$(x,\omega)$$(x,\omega)$$\tilde{\omega}$$\partial_{y_{1}}$$\partial_{y_{2}}$
$\displaystyle=\int\\!\\!\\!\\!\\!\int{\rm d}{y_{1}}\,{\rm d}{y_{2}}\,{\rm
d}\bar{}\hskip
1.00006pt{\tilde{\omega}}\left(\partial_{y_{1}}{T}\left(x,y_{1},\omega\right)\right)\left(\partial_{y_{2}}{T}\left(y_{1},y_{2},\omega-\tilde{\omega}\right)\right){T}\left(y_{2},x,\omega\right)C_{2}(\tilde{\omega})$
(100)

Considering $\mathcal{Q}^{(1)}_{\mathrm{II}}$, shown as II) in Fig. 2, next,
the two $g$-vertices may be inserted into two different return propagators  of
a diagram in the expansion Eq. (73),

$\displaystyle\mathcal{Q}^{(1)}_{\mathrm{II}}(x_{0},x,\omega)$
$\displaystyle=n_{0}^{-1}\lim_{\gamma\to\infty}\left[\gamma^{3}\leavevmode\hbox
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