UFIFT-QG-24-01

Gauge Independent Logarithms from Inflationary Gravitons

D. Glavan1∗, S. P. Miao2⋆, T. Prokopec3† and R. P. Woodard4‡

1 CEICO, Institute of Physics of the Czech Academy of Sciences (FZU),
Na Slovance 1999/2, 182 21 Prague 8, CZECH REPUBLIC

2 Department of Physics, National Cheng Kung University,
No. 1 University Road, Tainan City 70101, TAIWAN

3 Institute for Theoretical Physics, Spinoza Institute & EMME$\Phi$,
Utrecht University, Postbus 80.195, 3508 TD Utrecht, THE NETHERLANDS

4 Department of Physics, University of Florida,
Gainesville, FL 32611, UNITED STATES

ABSTRACT

PACS numbers: 04.50.Kd, 95.35.+d, 98.62.-g

Dependence on the graviton gauge enters the conventional effective field
equations because they fail to account for quantum gravitational correlations
with the source which excites the effective field and with the observer who
measures it. Including these correlations has been shown to eliminate gauge
dependence in flat space background. We generalize the technique to de Sitter
background for the case of the 1-loop graviton corrections to the exchange
potential of a massless, minimally coupled scalar.

∗ e-mail<EMAIL_ADDRESS>
⋆ e-mail<EMAIL_ADDRESS>
† e-mail<EMAIL_ADDRESS>
‡ e-mail<EMAIL_ADDRESS>
##  1 Introduction

The background geometry of cosmology with scale factor $a(t)$ is,

$ds^{2}=-dt^{2}+a^{2}(t)d\vec{x}\\!\cdot\\!d\vec{x}\qquad\Longrightarrow
H(t)\equiv\frac{\dot{a}}{a}\quad,\quad\epsilon(t)\equiv-\frac{\dot{H}}{H^{2}}\;.$
(1)

The Hubble parameter $H(t)$ measures the expansion rate while the first slow
roll parameter $\epsilon(t)$ measures the rate at which $H(t)$ changes.
Inflation is characterized by positive $H(t)$ and $0\leq\epsilon(t)<1$, which
means that both the first and second time derivatives of $a(t)$ are positive.
The most highly accelerated geometry is de Sitter, with $\epsilon(t)=0$ and
$H$ constant, which serves as a paradigm for primordial inflation.

The accelerated expansion of inflation can rip massless, not conformally
invariant virtual quanta out of the vacuum. That process is responsible for
the primordial power spectra of scalars [1] and gravitons [2], but these
quanta must, at some level, interact with themselves and with other particles
to affect kinematics and long-range forces. One studies these changes on a
field by computing its 1PI (one-particle-irreducible) 2-point function and
then using that to quantum-correct the linearized field equation. For a scalar
field the 1PI 2-point function is known as the self-mass
$-iM^{2}(x;x^{\prime})$. If the scalar is massless and minimally coupled (MMC)
then its quantum-corrected, linearized field equation is,

$\partial_{\mu}\Bigl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}\varphi(x)\Bigr{]}-\int\\!\\!d^{4}x^{\prime}\,M^{2}(x;x^{\prime})\varphi(x^{\prime})=J(x)\;.$
(2)

Here $g_{\mu\nu}(x)$ is the background metric and $J(x)$ is the source. Zero
source solutions tell one about modifications to the kinematics of single
particle solutions whereas setting $J$ to a static point source determines the
exchange potential.

Effects mediated by inflationary scalars are generally stronger and simpler to
compute, but they suffer from being model dependent. In contrast, graviton-
mediated effects are weaker, and much more difficult to work out, but are
completely generic because general relativity is the unique low energy
effective field theory of gravity [3, 4, 5, 6, 7, 8]. Despite the complexity
of working with gravitons on an inflationary background, six cases have been
found of significant effects on de Sitter:

  * •

Growth of the fermion field strength [9];

  * •

Temporal and spatial growth of the electric force [10];

  * •

Growth of the of the photon field strength [11];

  * •

Growth of the graviton field strength [12];

  * •

Spatial suppression of the force carried by an MMC scalar [13]; and

  * •

Temporal suppression of the gravitational force [14].

A standard concern when studying vector-mediated or tensor-mediated
corrections to effective field equations is how to distinguish physical
effects from gauge artifacts such as the acausality apparently implied by the
instantaneous Coulomb potential. Exhibiting the gauge dependence of 1PI
$n$-point functions is relatively simple in flat space [15], but very
difficult for inflationary backgrounds. The simplest gauge [16, 17] happens to
break de Sitter invariance, which has been a point of contention for decades
[18, 19, 20, 21, 22, 23, 24, 25, 26]. However, other gauges are so difficult
to use that, of the ten graviton loops so far evaluated [27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 13],111An independent computation of graviton corrections
to the self-mass of a massless, conformally coupled scalar [37, 38] also used
the simplest gauge, but disagrees with [36]. only one was performed using a
different gauge. This one was of the graviton contributions to the vacuum
polarization [34] in a 1-parameter family of de Sitter invariant gauges [39].
When it was used to check the growth of the photon field strength, the same
time dependence was found as in the simplest gauge [11], but with a different
multiplicative factor [40]. One can therefore conclude that there is nothing
particularly misleading about the simplest gauge, but that 1PI functions on de
Sitter harbor the same sort of gauge dependence as they do on flat space.

It was recently shown that the usual effective field equations are gauge
dependent because they ignore quantum gravitational correlations with the
source which excites the effective field and with the observer who measures it
[41]. When these correlations are restored on flat space background, the
effective field equations for both an MMC scalar [41] and for electromagnetism
[42] become gauge independent. One restores source and observer correlations
by writing down the position-space diagrams which contribute to $t$-channel
scattering between two massive source particles. These diagrams consist of
4-point, 3-point and 2-point Green’s functions. A series of identities due to
Donoghue and collaborators [43, 3, 44, 45] permits one to extract the
$t$-channel poles from the 3-point and 4-point diagrams, which reduces them to
2-point form. Then the propagator equation is used to regard these 2-point
forms as corrections to the 1PI 2-point function, and hence to the effective
field equation. Our goal here is to generalize this technique to de Sitter for
graviton corrections to an MMC scalar.

This paper contains seven sections. In section 2 we review the Feynman rules,
including the massive source-observer field, along with gravity and the MMC
scalar. We also generalize the Donoghue identities from flat space to de
Sitter. It turns out that five classes of diagrams contribute to $t$-channel
scattering, in addition to the self-mass. These diagrams are presented in
section 3. Section 4 demonstrates an important cancellation which occurs for
any background geometry and in any gauge. The remaining diagrams are reduced
in section 5, and incorporated as corrections to the renormalized self-mass.
Section 6 gives our conclusions, including the crucial issue of whether or not
the 1-loop correction to the scalar exchange potential contains a large
logarithm.

##  2 Feynman Rules and Reduction Strategy

The purpose of this section is to present the basics of our computation. We
begin with the full action which describes the massive source-observer field
$\Psi$, in addition to gravity and the massless, minimally coupled scalar
whose exchange potential it modifies. The necessary Feynman rules are
presented, then the general strategy is described. The section closes with
some useful derivative identities.

###  2.1 Propagators and Vertices

We model the source and observer as a massive, minimally coupled scalar $\Psi$
which has a cubic coupling to the massless, minimally coupled scalar
$\varphi$. The bare Lagrangian is,

$\displaystyle\mathcal{L}=\frac{[R\\!-\\!(D\\!-\\!2)\Lambda]\sqrt{-g}}{16\pi
G}-\frac{1}{2}\partial_{\mu}\varphi\partial_{\nu}\varphi g^{\mu\nu}\sqrt{-g}$
(3) $\displaystyle\hskip
122.34692pt-\frac{1}{2}\partial_{\mu}\Psi\partial_{\nu}\Psi
g^{\mu\nu}\sqrt{-g}-\frac{1}{2}\Bigl{(}m^{2}\\!+\\!\lambda\varphi\Bigr{)}\Psi^{2}\sqrt{-g}\;.\qquad$

Here $G$ is Newton’s constant, $\Lambda$ is the positive cosmological constant
and $D$ is the dimension of spacetime, left arbitrary to facilitate the use of
dimensional regularization. It is convenient to change the time variable from
co-moving time $t$ to conformal time $\eta$, with $d\eta\equiv dt/a(t)$. This
makes the de Sitter background conformal to flat space,

$ds^{2}=a^{2}\Bigl{[}-d\eta^{2}+d\vec{x}\\!\cdot\\!d\vec{x}\Bigr{]}\qquad,\qquad
a=-\frac{1}{H\eta}\quad,\quad H\equiv\sqrt{\frac{\Lambda}{D\\!-\\!1}}\;.$ (4)

We adopt the convention that the indices of partial derivatives are raised and
lowered with the Minkowski metric,
$\partial^{\mu}\equiv\eta^{\mu\nu}\partial_{\nu}$.

Our analysis employs four scalar propagators. The source and observer ($\Psi$)
propagator $i\Delta_{m}(x;x^{\prime})$ has a mass $m\gg H$ which is assumed
much larger than the Hubble parameter and spatial momenta. The massless scalar
($\varphi$) propagator $i\Delta_{A}(x;x^{\prime})$ has zero mass, while the
graviton propagator requires three scalar propagators
$i\Delta_{I}(x;x^{\prime})$ whose masses are,

$M^{2}_{A}=0\qquad,\qquad M^{2}_{B}=(D\\!-\\!2)H^{2}\qquad,\qquad
M^{2}_{C}=2(D\\!-\\!3)H^{2}\;.$ (5)

All four propagators obey equations involving the differential operator
$\mathcal{D}\equiv\partial_{\mu}[\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}]=\partial^{\mu}a^{D-2}\partial_{\mu}$,

$\Bigl{(}\mathcal{D}\\!-\\!a^{D}m^{2}\Bigr{)}i\Delta_{m}(x;x^{\prime})=i\delta^{D}(x\\!-\\!x^{\prime})=\Bigl{(}\mathcal{D}\\!-\\!a^{D}M^{2}_{I}\Bigr{)}i\Delta_{I}(x;x^{\prime})\;.$
(6)

The three propagators $i\Delta_{I}(x;x^{\prime})$ can be usefully expanded as
differences of a series in $D$-dependent powers of $aa^{\prime}\Delta
x^{2}\equiv
aa^{\prime}(x-x^{\prime})^{\mu}(x-x^{\prime})^{\nu}\eta_{\mu\nu}+i\epsilon$
and a series of integer powers of the same quantity,

$\displaystyle
i\Delta_{A}=\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{4\pi^{\frac{D}{2}}}\Biggl{\\{}\frac{1}{[aa^{\prime}\Delta
x^{2}]^{\frac{D}{2}-1}}+\frac{D(D\\!-\\!2)}{8(D\\!-\\!4)}\frac{H^{2}}{[aa^{\prime}\Delta
x^{2}]^{\frac{D}{2}-2}}+\ldots\Biggr{\\}}$ (7) $\displaystyle\hskip
39.83368pt-\frac{(\frac{H}{2})^{D-4}}{4\pi^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{2\Gamma(\frac{D}{2})}\Biggl{\\{}0+\tfrac{1}{2}H^{2}\Bigl{[}\pi\cot\Bigl{(}\frac{D\pi}{2}\Bigr{)}\\!-\\!\ln(aa^{\prime})\Bigr{]}+\ldots\Biggr{\\}},\qquad$
$\displaystyle
i\Delta_{B}=\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{4\pi^{\frac{D}{2}}}\Biggl{\\{}\frac{1}{[aa^{\prime}\Delta
x^{2}]^{\frac{D}{2}-1}}+\Bigl{(}\frac{D\\!-\\!2}{8}\Bigr{)}\frac{H^{2}}{[aa^{\prime}\Delta
x^{2}]^{\frac{D}{2}-2}}+\ldots\Biggr{\\}}$ (8) $\displaystyle\hskip
133.72786pt-\frac{(\frac{H}{2})^{D-4}}{4\pi^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{2\Gamma(\frac{D}{2})}\Biggl{\\{}0+\frac{H^{2}}{2(D\\!-\\!2)}+\ldots\Biggr{\\}},\qquad$
$\displaystyle
i\Delta_{C}=\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{4\pi^{\frac{D}{2}}}\Biggl{\\{}\frac{1}{[aa^{\prime}\Delta
x^{2}]^{\frac{D}{2}-1}}+\Bigl{(}\frac{D\\!-\\!6}{8}\Bigr{)}\frac{H^{2}}{[aa^{\prime}\Delta
x^{2}]^{\frac{D}{2}-2}}+\ldots\Biggr{\\}}$ (9) $\displaystyle\hskip
99.58464pt-\frac{(\frac{H}{2})^{D-4}}{4\pi^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{2\Gamma(\frac{D}{2})}\Biggl{\\{}0-\frac{H^{2}}{2(D\\!-\\!2)(D\\!-\\!3)}+\ldots\Biggr{\\}}.\qquad$

The neglected terms in expressions (7-9) are not only less singular at
coincidence (${x^{\prime}}^{\mu}=x^{\mu}\Longrightarrow\Delta x^{2}=0$), they
also vanish in $D=4$ dimensions. Indeed, even summing the order
$[aa^{\prime}\Delta x^{2}]^{2-\frac{D}{2}}$ and $[aa^{\prime}\Delta
x^{2}]^{0}$ terms of $i\Delta_{B}$ and $i\Delta_{C}$ vanishes in $D=4$. This
fact has great significance for our calculation.

The great thing about the simplest gauge is that the graviton propagator (in
conformal coordinates) consists of a sum of spacetime constant tensor factors
multiplying the three scalar propagators $i\Delta_{I}(x;x^{\prime})$ [16, 17],

$\displaystyle
i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;x^{\prime})=\Bigl{[}2\overline{\eta}_{\mu(\rho}\overline{\eta}_{\sigma)\nu}\\!-\\!\frac{2\overline{\eta}_{\mu\nu}\overline{\eta}_{\rho\sigma}}{D\\!-\\!3}\Bigr{]}i\Delta_{A}(x;x^{\prime})-4\delta^{0}_{~{}(\mu}\overline{\eta}_{\nu)(\rho}\delta^{0}_{~{}\sigma)}i\Delta_{B}(x;x^{\prime})$
(10)
$\displaystyle+\frac{2}{(D\\!-\\!3)(D\\!-\\!2)}\Bigl{[}\overline{\eta}_{\mu\nu}\\!+\\!(D\\!-\\!3)\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}\Bigr{]}\Bigl{[}\overline{\eta}_{\rho\sigma}\\!+\\!(D\\!-\\!3)\delta^{0}_{~{}\rho}\delta^{0}_{~{}\sigma}\Bigr{]}i\Delta_{C}(x;x^{\prime})\;.\qquad$

(Parenthesized indices are symmetrized.) The three tensor factors involve the
Minkowski metric $\eta_{\mu\nu}$, the Kronecker delta function
$\delta^{0}_{~{}\mu}$, and a combination
$\overline{\eta}_{\mu\nu}\equiv\eta_{\mu\nu}+\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}$
which gives the purely spatial part of the Minkowski metric. The only
contraction we require is,

$\eta^{\rho\sigma}\\!\times\\!i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}\Bigr{]}(x;x^{\prime})=-\frac{4\eta_{\mu\nu}}{D\\!-\\!2}\,i\Delta_{C}-\frac{4\overline{\eta}_{\mu\nu}}{D\\!-\\!3}\Bigl{(}i\Delta_{A}\\!-\\!i\Delta_{C}\Bigr{)}\;.$
(11)

Note that summing the three tensor factors in expression (10) produces the
tensor factor of the de Donder gauge propagator in flat space,

$\displaystyle\Bigl{[}2\overline{\eta}_{\mu(\rho}\overline{\eta}_{\sigma)\nu}\\!-\\!\frac{2\overline{\eta}_{\mu\nu}\overline{\eta}_{\rho\sigma}}{D\\!-\\!3}\Bigr{]}-4\delta^{0}_{~{}(\mu}\overline{\eta}_{\nu)(\rho}\delta^{0}_{~{}\sigma)}+\frac{2}{(D\\!-\\!3)(D\\!-\\!2)}\Bigl{[}\overline{\eta}_{\mu\nu}\\!+\\!(D\\!-\\!3)\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}\Bigr{]}$
(12) $\displaystyle\hskip
110.96556pt\times\Bigl{[}\overline{\eta}_{\rho\sigma}\\!+\\!(D\\!-\\!3)\delta^{0}_{~{}\rho}\delta^{0}_{~{}\sigma}\Bigr{]}=2\eta_{\mu(\rho}\eta_{\sigma)\nu}-\frac{2\eta_{\mu\nu}\eta_{\rho\sigma}}{D\\!-\\!2}\;.\qquad$

This suggests the advantages of singling out a particular scalar propagator —
call it $i\Delta_{J}$ — and then expanding the other two propagators around
it,

$i\Delta_{I}=i\Delta_{J}+(i\Delta_{I}\\!-\\!i\Delta_{J})\;.$ (13)

Substituting (13) into (10) will then produce $i\Delta_{J}$ times the flat
space tensor (12), plus a series of differences of propagators. These
differences are less singular at coincidence because expressions (7-9) show
that all three propagators have the same leading singularity
$\Gamma(\frac{D}{2}-1)/4\pi^{\frac{D}{2}}[aa^{\prime}\Delta
x^{2}]^{\frac{D}{2}-1}$. The best choice for our calculation is $J=C$, which
gives,

$\displaystyle
i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;x^{\prime})=\Bigl{[}2\eta_{\mu(\rho}\eta_{\sigma)\nu}\\!-\\!\frac{2\eta_{\mu\nu}\eta_{\rho\sigma}}{D\\!-\\!2}\Bigr{]}i\Delta_{C}+4\eta_{\mu)(\rho}\overline{\eta}_{\sigma)(\nu}\Bigl{(}i\Delta_{B}\\!-\\!i\Delta_{C}\Bigr{)}$
(14) $\displaystyle\hskip
54.06006pt+2\overline{\eta}_{\mu(\rho}\overline{\eta}_{\sigma)\nu}\Bigl{(}i\Delta_{A}\\!-\\!2i\Delta_{B}+i\Delta_{C}\Bigr{)}-\frac{2\overline{\eta}_{\mu\nu}\overline{\eta}_{\rho\sigma}}{D\\!-\\!3}\Bigl{(}i\Delta_{A}\\!-\\!i\Delta_{C}\Bigr{)}\;.\qquad$

This form is especially effective for us because the 4 indices of the graviton
propagator are typically contracted into derivatives from vertices and any
spatial derivatives are assumed small compared to the source observer mass
$m$. In the 3-point and 4-point contributions these vertex derivatives must
supply at least two factors of $m$ to cancel inverse factors from the Donoghue
Identities. Hence we can neglect all the terms on the last line of (14) which
have only spatial indices. It turns out that even the last term on the first
line of (14) drops out because it is proportional to the difference
$(i\Delta_{B}-i\Delta_{C})$, which lacks the leading singularity and actually
vanishes in $D=4$ dimensions.

Figure 1: Vertices A-E from left to right, given in expressions (15-19). Solid
lines represent the massless, minimally coupled scalar, dashed lines represent
the massive source-observer field, and wavy lines represent the graviton.

Figure 1 depicts the vertices we require. From left to right they are,

$\displaystyle{\rm Vertex\ A}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-i\lambda a_{x}^{D}\;,\qquad$ (15) $\displaystyle{\rm Vertex\
B}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle-i\kappa
a_{y}^{D-2}\Bigl{[}-\partial^{\mu}_{1}\partial^{\nu}_{2}+\tfrac{1}{2}\eta^{\mu\nu}(\partial_{1}\\!\cdot\\!\partial_{2}\\!+\\!a_{y}^{2}m^{2})\Bigr{]}\;,\qquad$
(16) $\displaystyle{\rm Vertex\ C}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-i\kappa
a_{z}^{D-2}\Bigl{[}-\partial^{\mu}_{1}\partial^{\nu}_{2}+\tfrac{1}{2}\eta^{\mu\nu}\partial_{1}\\!\cdot\\!\partial_{2}\Bigr{]}\;,\qquad$
(17) $\displaystyle{\rm Vertex\ D}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\tfrac{i}{2}\kappa\lambda a_{x}^{D}\eta^{\mu\nu}\;,\qquad$ (18)
$\displaystyle{\rm Vertex\ E}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-i\kappa^{2}\lambda
a_{x}^{D}\Bigl{[}\tfrac{1}{8}\eta^{\mu\nu}\eta^{\rho\sigma}-\tfrac{1}{4}\eta^{\mu\rho}\eta^{\sigma\nu}\Bigr{]}\;.\qquad$
(19)

The subscripts on scale factors indicate the coordinate argument, for example,
$a_{x}\equiv-\frac{1}{Hx^{0}}$ (with $x^{0}<0$). Subscripts on derivatives
indicate which leg is differentiated, and we have not bothered to symmetrize
indices because they are contracted into the symmetric graviton propagator.

###  2.2 Reduction Strategy

As discussed in section 1, the gauge dependence of a 1PI 2-point function can
be removed by first writing down (in position space) all the diagrams which
contribute to the $t$-channel scattering process $\Psi_{1}(x)$ \+
$\Psi_{3}(x^{\prime})\longrightarrow\Psi_{2}(y)+\Psi_{4}(y^{\prime})$. This
consists of a collection of 2-point, 3-point and 4-point Green’s functions,

$-iV(x;x^{\prime})\;\;,\;\;-iV(x;x^{\prime};y)\;\;,\;\;-iV(x;x^{\prime};y^{\prime})\;\;,\;\;-iV(x;x^{\prime};y;y^{\prime})\;.$
(20)

The 2-point and 3-point Green’s functions must be multiplied by appropriate
delta functions to fix $y^{\mu}$ and $y^{\prime\mu}$, for example,

$-iV(x;x^{\prime})\longrightarrow\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-iV(x;x^{\prime})\;.$
(21)

One then extracts the $t$-channel poles from 3-point and 4-point contributions
using identities derived by Donoghue and collaborators [43, 3, 44, 45] and
easy to check from general results [46, 47]. In flat position space one can
reduce 3-point Green’s functions to 2-point form using the relation [41],

$i\Delta_{m}(x;y)i\Delta(x;x^{\prime})i\Delta(y;x^{\prime})\longrightarrow\frac{i\delta^{D}(x\\!-\\!y)}{2m^{2}}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\;,$
(22)

where $i\Delta(x;x^{\prime})$ denotes the massless scalar propagator in flat
space. Reducing the flat 4-point Green’s functions to 2-point form requires
two such identities [45, 44],

$\displaystyle
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta(x;x^{\prime})i\Delta(y;y^{\prime})\longrightarrow-\frac{i}{m^{2}}\Bigl{[}1-\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}\\!-\\!m^{2}}{3m^{2}}\Bigr{)}\Bigr{]}$
$\displaystyle\hskip
79.6678pt\times\Biggl{\\{}\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\int\\!\\!d^{D}z\Bigl{[}i\Delta(x;z)\Bigr{]}^{2}i\Delta(z;x^{\prime})\Biggr{\\}}\;,\qquad$
(23) $\displaystyle
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta(x;y^{\prime})i\Delta(x^{\prime};y)\longrightarrow+\frac{i}{m^{2}}\Bigl{[}1+\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}\\!+\\!m^{2}}{3m^{2}}\Bigr{)}\Bigr{]}$
$\displaystyle\hskip
79.6678pt\times\Biggl{\\{}\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\int\\!\\!d^{D}z\Bigl{[}i\Delta(x;z)\Bigr{]}^{2}i\Delta(z;x^{\prime})\Biggr{\\}}\;,\qquad$
(24)

where the over-lined derivatives are understood to act on the external wave
functions. Note that the integrals at the end of expressions (23) and (24) is
symmetric under interchange of $x^{\mu}$ and $x^{\prime\mu}$.

Figure 2: The four classes of diagrams ($\alpha$ through $\delta$, from left
to right) produced after reducing diagrams with source and observer
correlations to 2-point form. The shaded circles are
$-i\lambda^{2}(a_{x}a_{x^{\prime}})^{D}f_{\alpha}(x;x^{\prime})$, $-i\lambda
a_{x}^{D}f_{\beta}(x;z^{\prime})$, $-i\lambda
a_{x^{\prime}}^{D}f_{\gamma}(z;x^{\prime})$ and $-if_{\delta}(z;z^{\prime})$,
respectively.

Figure 2 shows the generic 2-point forms produced by these reductions. We can
regard each of these 2-point contributions as a correction to the 1PI 2-point
function by using the propagator equation
$i\delta^{D}(x-z)=\mathcal{D}_{z}i\Delta_{A}(x;z)$ and then partially
integrating. We will give all the steps for the leftmost diagram,

$\displaystyle{\rm
Class}\,\alpha\equiv\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!-if_{\alpha}(x;x^{\prime})\;,$
(28) $\displaystyle=-i\lambda
a_{x}^{D}\\!\\!\int\\!\\!d^{D}z\,i\delta^{D}(x\\!-\\!z)\\!\times\\!-i\lambda
a_{x^{\prime}}^{D}\\!\\!\int\\!\\!d^{D}z^{\prime}\,i\delta^{D}(x^{\prime}\\!-\\!z^{\prime})\\!\times\\!-if_{\alpha}(z;z^{\prime})\;,\qquad$
$\displaystyle=-i\lambda
a_{x}^{D}\\!\\!\int\\!\\!d^{D}z\,\mathcal{D}_{z}i\Delta_{A}(x;z)\\!\times\\!-i\lambda
a_{x^{\prime}}^{D}\\!\\!\int\\!\\!d^{D}z^{\prime}\,\mathcal{D}_{z^{\prime}}i\Delta_{A}(x^{\prime};z^{\prime})\\!\times\\!-if_{\alpha}(z;z^{\prime})\;,\qquad$
$\displaystyle=-i\lambda
a_{x}^{D}\\!\\!\int\\!\\!d^{D}z\,i\Delta_{A}(x;z)\\!\times\\!-i\lambda
a_{x^{\prime}}^{D}\\!\\!\int\\!\\!d^{D}z^{\prime}\,i\Delta_{A}(x^{\prime};z^{\prime})\\!\times\\!-i\mathcal{D}_{z}\mathcal{D}_{z^{\prime}}f_{\alpha}(z;z^{\prime})\;.\qquad$

We merely give the results for the remaining diagrams,

$\displaystyle{\rm Class}\,\beta\equiv-i\lambda
a_{x^{\prime}}^{D}\\!\\!\int\\!\\!d^{D}z^{\prime}\,i\Delta_{A}(x^{\prime};z^{\prime})\\!\times-i\lambda
a_{x}^{D}f_{\beta}(x;z^{\prime})\;,$ (30) $\displaystyle=-i\lambda
a_{x}^{D}\\!\\!\int\\!\\!d^{D}z\,i\Delta_{A}(x;z)\\!\times\\!-i\lambda
a_{x^{\prime}}^{D}\\!\\!\int\\!\\!d^{D}z^{\prime}\,i\Delta_{A}(x^{\prime};z^{\prime})\\!\times\\!-i\mathcal{D}_{z}f_{\beta}(z;z^{\prime})\;\,\qquad$
$\displaystyle{\rm Class}\,\gamma\equiv-i\lambda
a_{x}^{D}\\!\\!\int\\!\\!d^{D}z\,i\Delta_{A}(x;z)\\!\times-i\lambda
a_{x^{\prime}}^{D}f_{\gamma}(z;x^{\prime})\;,$ (32) $\displaystyle=-i\lambda
a_{x}^{D}\\!\\!\int\\!\\!d^{D}z\,i\Delta_{A}(x;z)\\!\times\\!-i\lambda
a_{x^{\prime}}^{D}\\!\\!\int\\!\\!d^{D}z^{\prime}\,i\Delta_{A}(x^{\prime};z^{\prime})\\!\times\\!-i\mathcal{D}_{z^{\prime}}f_{\gamma}(z;z^{\prime})\;,\qquad$
$\displaystyle{\rm Class}\,\delta\equiv-i\lambda
a_{x}^{D}\\!\\!\int\\!\\!d^{D}z\,i\Delta_{A}(x;z)\\!\times\\!-i\lambda
a_{x^{\prime}}^{D}\\!\\!\int\\!\\!d^{D}z^{\prime}\,i\Delta_{A}(x^{\prime};z^{\prime})\\!\times\\!-if_{\delta}(z;z^{\prime})\;.$
(33)

The gauge invariant self-mass is the sum of all four classes,

$M^{2}_{\rm
inv}(x;x^{\prime})=\mathcal{D}_{x}\mathcal{D}_{x^{\prime}}f_{\alpha}(x;x^{\prime})+\mathcal{D}_{x}f_{\beta}(x;x^{\prime})+\mathcal{D}_{x^{\prime}}f_{\gamma}(x;x^{\prime})+f_{\delta}(x;x^{\prime})\;.$
(34)

It remains to generalize the Donoghue Identities (22-24) from flat space to de
Sitter. This is a matter of preserving general coordinate invariance and
taking account of the scalar propagator $i\Delta_{A}$ possibly differing from
the scalar propagators $i\Delta_{I}$ in the graviton propagator. For the
3-point identity (22) the required generalization is,

$i\Delta_{m}(x;y)i\Delta_{A}(x;x^{\prime})i\Delta_{I}(y;x^{\prime})\longrightarrow\frac{i\delta^{D}(x\\!-\\!y)}{2m^{2}a_{x}^{D}}\,i\Delta_{A}(x;x^{\prime})i\Delta_{I}(x;x^{\prime})\;.$
(35)

We also need a 3-point identity with a single derivative,

$i\Delta_{m}(x;y)i\Delta_{A}(x;z)\partial_{y}^{\mu}i\Delta_{C}(y;z)\longrightarrow(\overline{\partial}_{y}^{\mu}\\!+\\!\overline{\partial}_{z}^{\mu})\frac{i\delta^{D}(x\\!-\\!y)}{2m^{2}a_{x}^{D}}\,i\Delta_{A}(x;z)i\Delta_{C}(x;z)\;.$
(36)

The 4-point identities (23-24) have the complication of contracting
derivatives at different points. This can be made invariant using the parallel
transport matrix $[\mbox{}^{\mu}g^{\nu}](x;x^{\prime})$ whose $3+1$
decomposition in de Sitter conformal coordinates is,

$\displaystyle[\mbox{}^{\mu}g^{\nu}](x;x^{\prime})$ $\displaystyle\equiv$
$\displaystyle\left(\begin{matrix}[\mbox{}^{0}g^{0}]&[\mbox{}^{0}g^{n}]\\\
[\mbox{}^{m}g^{0}]&[\mbox{}^{m}g^{n}]\\\ \end{matrix}\right)\;,$ (37)
$\displaystyle=$
$\displaystyle\frac{\eta^{\mu\nu}}{a_{x}a_{x^{\prime}}}+\frac{2}{4\\!-\\!y}\left(\begin{matrix}-H^{2}\Delta
r^{2}&-(\frac{1}{a_{x}}\\!+\\!\frac{1}{a_{x^{\prime}}})H\Delta r^{n}\\\
(\frac{1}{a_{x}}\\!+\\!\frac{1}{a_{x^{\prime}}})H\Delta r^{m}&H^{2}\Delta
r^{m}\Delta r^{n}\\\ \end{matrix}\right)\;,\qquad$ (38)

where $\Delta r^{i}\equiv(x-x^{\prime})^{i}$ and $y\equiv
a_{x}a_{x^{\prime}}H^{2}\Delta x^{2}$. Note that
$[\mbox{}^{\mu}g^{\nu}](x;x^{\prime})\simeq\eta^{\mu\nu}/a_{x}a_{x^{\prime}}$
near spatial coincidence. The appropriate generalizations of the 4-point
Donoghue Identities (23-24) are,

$\displaystyle
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i\Delta_{C}(y;y^{\prime})$
$\displaystyle\longrightarrow-\frac{i}{m^{2}}\Bigl{[}1-\Bigl{(}\frac{\overline{\partial}^{x}_{\mu}\overline{\partial}^{x^{\prime}}_{\nu}[\mbox{}^{\mu}g^{\nu}](x;x^{\prime})\\!-\\!m^{2}}{3m^{2}}\Bigr{)}\Bigr{]}\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})}{(a_{x}a_{x^{\prime}})^{D}}$
$\displaystyle\hskip
28.45274pt\times\frac{1}{2}\Biggl{\\{}\\!\int\\!\\!d^{D}z\,a_{z}^{D}i\Delta_{A}(x;z)i\Delta_{C}(x;z)i\Delta_{A}(z;x^{\prime})+\Bigl{(}x^{\mu}\longleftrightarrow{x^{\prime}}^{\mu}\Bigr{)}\\!\Biggr{\\}},\qquad$
(39) $\displaystyle
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;y^{\prime})i\Delta_{C}(x^{\prime};y)$
$\displaystyle\longrightarrow+\frac{i}{m^{2}}\Bigl{[}1+\Bigl{(}\frac{\overline{\partial}^{x}_{\mu}\overline{\partial}^{y^{\prime}}_{\nu}[\mbox{}^{\mu}g^{\nu}](x;y^{\prime})\\!+\\!m^{2}}{3m^{2}}\Bigr{)}\Bigr{]}\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})}{(a_{x}a_{x^{\prime}})^{D}}$
$\displaystyle\hskip
28.45274pt\times\frac{1}{2}\Biggl{\\{}\\!\int\\!\\!d^{D}z\,a_{z}^{D}i\Delta_{A}(x;z)i\Delta_{C}(x;z)i\Delta_{A}(z;y^{\prime})+\Bigl{(}x^{\mu}\longleftrightarrow{y^{\prime}}^{\mu}\Bigr{)}\\!\Biggr{\\}}.\qquad$
(40)

###  2.3 Derivative Identities

Momentum conservation at a 3-point vertex (15-17) is complicated by the
factors of $a^{D-2}$ (for derivative vertices) and $a^{D}$ (for non-derivative
vertices). If we label derivatives of the three propagators (or external wave
functions) by “$\partial_{1}^{\mu}$”, “$\partial_{2}^{\mu}$” and
“$\partial_{3}^{\mu}$” then we might use “$\partial_{4}^{\mu}$ to stand for
differentiation of the scale factors. With this understanding, momentum
conservation reads,

$\Bigl{(}\partial_{1}+\partial_{2}+\partial_{3}+\partial_{4}\Bigr{)}^{\mu}=0\;.$
(41)

We often require contractions of two derivatives
$\partial_{1}\cdot\partial_{2}\equiv\eta_{\mu\nu}\partial_{1}^{\mu}\partial_{2}^{\nu}$.
When the vertex contains a factor of $a^{D-2}$ such contractions can be
expressed in terms of the scalar covariant d’Alembertian,

$\mathcal{D}_{i}\equiv\eta_{\mu\nu}\partial_{i}^{\mu}a^{D-2}\partial_{i}^{\nu}=a^{D-2}\partial_{i}\\!\cdot\\!(\partial_{i}\\!+\\!\partial_{4})\;.$
(42)

The derivation is,

$\displaystyle\partial_{1}\\!\cdot\\!\partial_{2}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{1}{2}(\partial_{1}\\!+\\!\partial_{2})^{2}-\tfrac{1}{2}\partial_{1}^{2}-\tfrac{1}{2}\partial_{2}^{2}\;,$
(43) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{1}{2}(\partial_{3}\\!+\\!\partial_{4})^{2}-\tfrac{1}{2}\partial_{1}^{2}-\tfrac{1}{2}\partial_{2}^{2}\;,$
(44) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{1}{2}\partial_{3}\\!\cdot\\!(\partial_{3}\\!+\\!\partial_{4})+\tfrac{1}{2}\partial_{4}\\!\cdot\\!(\partial_{3}\\!+\\!\partial_{4})-\tfrac{1}{2}\partial_{1}^{2}-\tfrac{1}{2}\partial_{2}^{2}\;,\qquad$
(45) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{1}{2a^{D-2}}\Bigl{[}\mathcal{D}_{3}-\mathcal{D}_{1}-\mathcal{D}_{2}\Bigr{]}\;.$
(46)

A related identity of great utility is,

$a^{D-2}\partial_{1}\\!\cdot\\!\partial_{2}+a^{D}m^{2}=\frac{1}{2}\mathcal{D}_{3}-\frac{1}{2}\Bigl{(}\mathcal{D}_{1}\\!-\\!a^{D}m^{2}\Bigr{)}-\frac{1}{2}\Bigl{(}\mathcal{D}_{2}\\!-\\!a^{D}m^{2}\Bigr{)}\;.$
(47)

##  3 The Five New Diagrams

Figure 3: Diagrams $0a$ (left) and $0b$ (right) included in the uncorrected,
self-mass.

The scattering process
$\Psi_{1}(x)+\Psi_{3}(x^{\prime})\longrightarrow\Psi_{2}(y)+\Psi_{4}(y^{\prime})$
receives a $t$-channel contribution from the 1-loop self-mass
$-iM_{0}^{2}(x;x^{\prime})$ as shown in Figure 3. The analytic expression is,

$-iV_{0}(x;x^{\prime})=-\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\\!\int\\!\\!d^{D}z\,i\Delta_{A}(x;z)\\!\\!\int\\!\\!d^{D}z^{\prime}\,i\Delta_{A}(x^{\prime};z^{\prime})\\!\times\\!-iM_{0}^{2}(z;z^{\prime})\;.$
(48)

The purpose of this section is to give the five other classes of diagrams
which make $t$-channel contributions to the same 1-loop scattering amplitude.
We use subscripts to distinguish scale factors at different spacetime points,
as in $a_{x}$ and $a_{x^{\prime}}$ in expression (48). The same convention
applies to derivative operators, whose indices are raised and lowered using
the Minkowski metric, as in
$\partial^{\mu}_{y}\equiv\eta^{\mu\nu}\frac{\partial}{\partial y^{\nu}}$.
Because there are sometimes too many propagators to distinguish how
derivatives act by order, we employ a special convention for a
$h_{\mu\nu}\Psi_{\rm int}\Psi_{\rm ext}$ vertex: a bar indicates that the
derivative acts on the external $\Psi$ leg, a tilde indicates that it acts on
the graviton leg, and a derivative without either distinction acts on the
internal $\Psi$ leg.

###  3.1 Vertex-Vertex

Figure 4: Diagrams $1a$ (left), $1b$ (middle) and $1c$ (right) in which the
vertex is corrected.

The first of the new diagrams derive from correlators of gravitons which
emerge from the vertex. Figure 4 shows the three possibilities. The first case
is where a single graviton emerges from each vertex,

$-iV_{1a}(x;x^{\prime})=\Bigl{(}-\tfrac{i}{2}\kappa\lambda
a_{x}^{D}\eta^{\mu\nu}\Bigr{)}i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;x^{\prime})\Bigl{(}-\tfrac{i}{2}\kappa\lambda
a_{x^{\prime}}^{D}\eta^{\rho\sigma}\Bigr{)}i\Delta_{A}(x;x^{\prime})\;.$ (49)

The 2nd and 3rd are where two gravitons emerge from the same vertex,

$\displaystyle-iV_{1b}(x;x^{\prime})$ (50)
$\displaystyle=\Bigl{[}-i\kappa^{2}\lambda
a_{x}^{D}\Big{(}\tfrac{1}{8}\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\tfrac{1}{4}\eta^{\mu\rho}\eta^{\nu\sigma}\Bigr{)}\Bigr{]}i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;x)\Bigl{(}-i\lambda
a_{x^{\prime}}^{D}\Bigr{)}i\Delta_{A}(x;x^{\prime})\;,\qquad$ $\displaystyle-
iV_{1c}(x;x^{\prime})$ (51) $\displaystyle=\Bigl{(}-i\lambda
a_{x}^{D}\Bigr{)}i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x^{\prime};x^{\prime})\Bigl{[}-i\kappa^{2}\lambda
a_{x^{\prime}}^{D}\Big{(}\tfrac{1}{8}\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\tfrac{1}{4}\eta^{\mu\rho}\eta^{\nu\sigma}\Bigr{)}\Bigr{]}i\Delta_{A}(x;x^{\prime})\;.\qquad$

###  3.2 Source(Observer)-Vertex

Figure 5: Diagrams $2a$ (left), $2b$ (2nd), $2c$ (3rd) and $2d$ (right)
including a correlation from the propagation of the source or observer with
the vertex.

Figure 5 shows the second class of new diagrams: those which involve a
graviton from the propagation of the source or observer correlated with a
graviton from the vertex on the opposite side. (Having the vertex is on the
same side produces an irrelevant external leg correction.) The leftmost
diagram is,

$\displaystyle-iV_{2a}(x;x^{\prime};y)=\Bigl{[}-i\kappa
a_{y}^{D-2}\Bigl{(}-\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}+\tfrac{1}{2}\eta^{\mu\nu}[\overline{\partial}_{y}\\!\cdot\\!\partial_{y}\\!+\\!a_{y}^{2}m^{2}]\Bigr{)}\Bigr{]}i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;x^{\prime})$
(53) $\displaystyle\hskip 99.58464pt\times\Bigl{(}-\tfrac{i}{2}\kappa\lambda
a_{x^{\prime}}^{D}\eta^{\rho\sigma}\Bigr{)}i\Delta_{m}(y;x)\Bigl{(}-i\lambda
a_{x}^{D}\Bigr{)}i\Delta_{A}(x;x^{\prime})\;,\qquad$
$\displaystyle=\frac{i}{2}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}a_{y}^{D-2}\Bigl{[}-\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}+\tfrac{1}{2}\eta^{\mu\nu}(\overline{\partial}_{y}\\!\cdot\\!\partial_{y}\\!+\\!a_{y}^{2}m^{2})\Bigr{]}$
$\displaystyle\hskip 156.49014pt\times
i[\mbox{}_{\mu\nu}\Delta^{\rho}_{~{}\rho}](y;x^{\prime})i\Delta_{m}(x;y)i\Delta_{A}(x;x^{\prime})\;.\qquad$

We merely give final results for the other three diagrams,

$\displaystyle-
iV_{2b}(x;x^{\prime};y)=\frac{i}{2}\kappa^{2}\lambda^{2}(a_{y}a_{x^{\prime}})^{D}a_{x}^{D-2}\Bigl{[}-\overline{\partial}_{x}^{\mu}\partial_{x}^{\nu}+\tfrac{1}{2}\eta^{\mu\nu}(\overline{\partial}_{x}\\!\cdot\\!\partial_{x}\\!+\\!a_{x}^{2}m^{2})\Bigr{]}$
(54) $\displaystyle\hskip 150.79968pt\times
i[\mbox{}_{\mu\nu}\Delta^{\rho}_{~{}\rho}](x;x^{\prime})i\Delta_{m}(x;y)i\Delta_{A}(y;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{2c}(x;x^{\prime};y^{\prime})=\frac{i}{2}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}a_{y^{\prime}}^{D-2}\Bigl{[}-\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}+\tfrac{1}{2}\eta^{\rho\sigma}(\overline{\partial}_{y^{\prime}}\\!\cdot\\!\partial_{y^{\prime}}\\!+\\!a_{y^{\prime}}^{2}m^{2})\Bigr{]}$
(55) $\displaystyle\hskip 150.79968pt\times
i[\mbox{}^{\mu}_{~{}\mu}\Delta_{\rho\sigma}](x;y^{\prime})i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{2d}(x;x^{\prime};y^{\prime})=\frac{i}{2}\kappa^{2}\lambda^{2}(a_{x}a_{y^{\prime}})^{D}a_{x^{\prime}}^{D-2}\Bigl{[}-\overline{\partial}_{x^{\prime}}^{\rho}\partial_{x^{\prime}}^{\sigma}+\tfrac{1}{2}\eta^{\rho\sigma}(\overline{\partial}_{x^{\prime}}\\!\cdot\\!\partial_{x^{\prime}}\\!+\\!a_{x^{\prime}}^{2}m^{2})\Bigr{]}$
(56) $\displaystyle\hskip 150.79968pt\times
i[\mbox{}^{\mu}_{~{}\mu}\Delta_{\rho\sigma}](x;x^{\prime})i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;y^{\prime})\;,\qquad$

###  3.3 Vertex-Exchange

Figure 6: Diagrams $3a$ (left) and $3b$ (right) included a correlation between
a vertex and the propagation of the exchange scalar.

Another class of corrections is where a graviton from a vertex correlates with
a graviton from the propagation of the exchange scalar. Figure 6 shows these
diagrams. Their analytic expressions are,

$\displaystyle-
iV_{3a}(x;x^{\prime})=\tfrac{i}{2}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}-\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}+\tfrac{1}{2}\eta^{\rho\sigma}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}$
(57) $\displaystyle\hskip 156.49014pt\times
i[\mbox{}^{\mu}_{~{}\mu}\Delta_{\rho\sigma}](x;z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{3b}(x;x^{\prime})=\tfrac{i}{2}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}-\overline{\partial}_{z}^{\mu}\partial_{z}^{\nu}+\tfrac{1}{2}\eta^{\mu\nu}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}$
(58) $\displaystyle\hskip 156.49014pt\times
i[\mbox{}_{\mu\nu}\Delta^{\rho}_{~{}\rho}](z;x^{\prime})i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;.\qquad$

Our notation in (57-58) is that $\overline{\partial}_{z}$ differentiates the
scalar propagator external to the loop, while $\partial_{z}$ acts on the
scalar propagator inside the loop.

###  3.4 Source-Observer

Figure 7: Diagrams $4a$ (left), $4b$ (2nd), $4c$ (3rd) and $4d$ (right)
including a correlation between the propagation of the source and the
propagation of the observer.

Figure 7 shows diagrams which result from a graviton in the propagation of the
source correlating with a graviton from the observer. (Source-source and
Observer-observer correlations either give external state factors or else make
vertex corrections which are not enhanced.) The leftmost diagram involves two
A vertices (15) and two B vertices (16),

$\displaystyle-iV_{4a}(x;x^{\prime};y;y^{\prime})=\Bigl{[}-i\kappa
a_{y}^{D-2}\Bigl{(}-\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}+\tfrac{1}{2}\eta^{\mu\nu}[\overline{\partial}_{y}\\!\cdot\\!\partial_{y}\\!+\\!a_{y}^{2}m^{2}]\Bigr{)}\Bigr{]}i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;y^{\prime})$
(60) $\displaystyle\hskip 71.13188pt\times\Bigl{[}-i\kappa
a_{y^{\prime}}^{D-2}\Bigl{(}-\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}+\tfrac{1}{2}\eta^{\rho\sigma}[\overline{\partial}_{y^{\prime}}\\!\cdot\\!\partial_{y^{\prime}}\\!+\\!a_{y^{\prime}}^{2}m^{2}]\Bigr{)}\Bigr{]}$
$\displaystyle\hskip 71.13188pt\times i\Delta_{m}(x;y)\Bigl{(}-i\lambda
a_{x}^{D}\Bigr{)}i\Delta_{A}(x;x^{\prime})\Bigl{(}-i\lambda
a_{x^{\prime}}^{D}\Bigr{)}i\Delta_{m}(x^{\prime};y^{\prime})\;,\qquad$
$\displaystyle=\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}(a_{y}a_{y^{\prime}})^{D-2}\Bigl{[}\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}\\!-\\!\tfrac{1}{2}\eta^{\mu\nu}(\overline{\partial}_{y}\\!\cdot\\!\partial_{y}\\!+\\!a_{y}^{2}m^{2})\Bigr{]}\Bigl{[}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}$
$\displaystyle-\tfrac{1}{2}\eta^{\rho\sigma}(\overline{\partial}_{y^{\prime}}\\!\cdot\\!\partial_{y^{\prime}}\\!+\\!a_{y^{\prime}}^{2}m^{2})\Bigr{]}i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;y^{\prime})\;.\qquad$

For the remaining three diagrams we give only the final forms,

$\displaystyle-
iV_{4b}(x;x^{\prime};y;y^{\prime})=\kappa^{2}\lambda^{2}(a_{y}a_{y^{\prime}})^{D}(a_{x}a_{x^{\prime}})^{D-2}\Bigl{[}\overline{\partial}_{x}^{\mu}\partial_{x}^{\nu}\\!-\\!\tfrac{\eta^{\mu\nu}}{2}(\overline{\partial}_{x}\\!\\!\cdot\\!\partial_{x}\\!+\\!a_{x}^{2}m^{2})\\!\Bigr{]}\\!\Bigl{[}\overline{\partial}_{x^{\prime}}^{\rho}\partial_{x^{\prime}}^{\sigma}$
(61)
$\displaystyle-\tfrac{\eta^{\rho\sigma}}{2}(\overline{\partial}_{x^{\prime}}\\!\\!\cdot\\!\partial_{x^{\prime}}\\!+\\!a_{x^{\prime}}^{2}m^{2})\Bigr{]}i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(y;y^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{4c}(x;x^{\prime};y;y^{\prime})=\kappa^{2}\lambda^{2}(a_{y}a_{x^{\prime}})^{D}(a_{x}a_{y^{\prime}})^{D-2}\Bigl{[}\overline{\partial}_{x}^{\mu}\partial_{x}^{\nu}\\!-\\!\tfrac{\eta^{\mu\nu}}{2}(\overline{\partial}_{x}\\!\\!\cdot\\!\partial_{x}\\!+\\!a_{x}^{2}m^{2})\\!\Bigr{]}\\!\Bigl{[}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}$
(62)
$\displaystyle-\tfrac{\eta^{\rho\sigma}}{2}(\overline{\partial}_{y^{\prime}}\\!\\!\cdot\\!\partial_{y^{\prime}}\\!+\\!a_{y^{\prime}}^{2}m^{2})\Bigr{]}i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(y;x^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;y^{\prime})\;,\qquad$
$\displaystyle-
iV_{4d}(x;x^{\prime};y;y^{\prime})=\kappa^{2}\lambda^{2}(a_{x}a_{y^{\prime}})^{D}(a_{y}a_{x^{\prime}})^{D-2}\Bigl{[}\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}\\!-\\!\tfrac{\eta^{\mu\nu}}{2}(\overline{\partial}_{y}\\!\\!\cdot\\!\partial_{y}\\!+\\!a_{y}^{2}m^{2})\\!\Bigr{]}\\!\Bigl{[}\overline{\partial}_{x^{\prime}}^{\rho}\partial_{x^{\prime}}^{\sigma}$
(63)
$\displaystyle-\tfrac{\eta^{\rho\sigma}}{2}(\overline{\partial}_{x^{\prime}}\\!\\!\cdot\\!\partial_{x^{\prime}}\\!+\\!a_{x^{\prime}}^{2}m^{2})\Bigr{]}i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;y^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;x^{\prime})\;.\qquad$

###  3.5 Source(Observer)-Exchange

Figure 8: Diagrams $5a$ (left), $5b$ (2nd), $5c$ (3rd) and $5d$ (right)
including a correlation between the propagation of the source or observer and
the propagation of the exchange scalar.

The final class of contributions consists of gravitons from the propagation of
the source or observer correlation with gravitons from the exchange scalar.
Figure 8 shows these diagrams, which each involves two $\varphi\Psi^{2}$
vertices (15), a single $h_{\mu\nu}\Psi^{2}$ vertex (16) and a single
$h_{\mu\nu}\varphi^{2}$ vertex (17). The leftmost diagram is,

$\displaystyle-iV_{5a}(x;x^{\prime};y)=\Bigl{[}-i\kappa
a_{y}^{D-2}\Bigl{(}-\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}+\tfrac{1}{2}\eta^{\mu\nu}[\overline{\partial}_{y}\\!\\!\cdot\\!\partial_{y}\\!+\\!a_{y}^{2}m^{2}]\Bigr{)}\Bigr{]}i\Delta_{m}(y;x)$
(65) $\displaystyle\hskip
85.35826pt\times\\!\int\\!\\!d^{D}z\,i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;z)\Bigl{[}-i\kappa
a_{z}^{D-2}\Bigl{(}-\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}+\tfrac{1}{2}\eta^{\rho\sigma}\overline{\partial}_{z}\\!\\!\cdot\\!\partial_{z}\Bigr{)}\Bigr{]}$
$\displaystyle\hskip 128.0374pt\times i\Delta_{A}(z;x)\Bigl{(}-i\lambda
a_{x}^{D}\Bigr{)}i\Delta_{A}(z;x^{\prime})\Bigl{(}-i\lambda
a_{x^{\prime}}^{D}\Bigr{)}\;,\qquad$
$\displaystyle=\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\\!\int\\!\\!d^{D}z(a_{y}a_{z})^{D-2}\Bigl{[}\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}-\tfrac{\eta^{\mu\nu}}{2}(\overline{\partial}_{y}\\!\\!\cdot\\!\partial_{y}\\!+\\!a_{y}^{2}m^{2})\Bigr{]}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}$
$\displaystyle\hskip
68.28644pt-\tfrac{\eta^{\rho\sigma}}{2}\overline{\partial}_{z}\\!\\!\cdot\\!\partial_{z}\Bigr{]}i\Delta_{m}(x;y)i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;.\qquad$

Recall that $\overline{\partial}_{z}$ indicates differentiation of the scalar
propagator external to the loop, where $\partial_{z}$ acts on the scalar
propagator inside the loop. For the remaining three diagrams we give only the
final forms,

$\displaystyle-
iV_{5b}(x;x^{\prime};y)=\kappa^{2}\lambda^{2}(a_{y}a_{x^{\prime}})^{D}\\!\\!\int\\!\\!d^{D}z(a_{x}a_{z})^{D-2}\Bigl{[}\overline{\partial}_{x}^{\mu}\partial_{x}^{\nu}-\tfrac{\eta^{\mu\nu}}{2}(\overline{\partial}_{x}\\!\\!\cdot\\!\partial_{x}\\!+\\!a_{x}^{2}m^{2})\Bigr{]}$
(66) $\displaystyle\hskip
14.22636pt\times\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}\\!-\\!\tfrac{\eta^{\rho\sigma}}{2}\overline{\partial}_{z}\\!\\!\cdot\\!\partial_{z}\Bigr{]}i\Delta_{m}(x;y)i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;z)i\Delta_{A}(y;z)i\Delta_{A}(z;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{5c}(x;x^{\prime};y^{\prime})=\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\\!\int\\!\\!d^{D}z(a_{y^{\prime}}a_{z})^{D-2}\Bigl{[}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}-\tfrac{\eta^{\rho\sigma}}{2}(\overline{\partial}_{y^{\prime}}\\!\\!\cdot\\!\partial_{y^{\prime}}\\!+\\!a_{y^{\prime}}^{2}m^{2})\Bigr{]}$
(67) $\displaystyle\hskip
14.22636pt\times\Bigl{[}\overline{\partial}_{z}^{\mu}\partial_{z}^{\nu}\\!-\\!\tfrac{\eta^{\mu\nu}}{2}\overline{\partial}_{z}\\!\\!\cdot\\!\partial_{z}\Bigr{]}i\Delta_{m}(x^{\prime};y^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](z;y^{\prime})i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{5d}(x;x^{\prime};y^{\prime})=\kappa^{2}\lambda^{2}(a_{x}a_{y^{\prime}})^{D}\\!\\!\int\\!\\!d^{D}z(a_{x^{\prime}}a_{z})^{D-2}\Bigl{[}\overline{\partial}_{x^{\prime}}^{\rho}\partial_{x^{\prime}}^{\sigma}-\tfrac{\eta^{\rho\sigma}}{2}(\overline{\partial}_{x^{\prime}}\\!\\!\cdot\\!\partial_{x^{\prime}}\\!+\\!a_{x^{\prime}}^{2}m^{2})\Bigr{]}$
(68) $\displaystyle\hskip
14.22636pt\times\Bigl{[}\overline{\partial}_{z}^{\mu}\partial_{z}^{\nu}\\!-\\!\tfrac{\eta^{\mu\nu}}{2}\overline{\partial}_{z}\\!\\!\cdot\\!\partial_{z}\Bigr{]}i\Delta_{m}(x^{\prime};y^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](z;x^{\prime})i\Delta_{A}(x;z)i\Delta_{A}(z;y^{\prime})\;.\qquad$

##  4 Consolidation

When the derivative identity (47) is used on the $h_{\mu\nu}\Psi^{2}$ vertex
(16) at an external point, and the external state condition
$\overline{\mathcal{D}}=a^{D}m^{2}$ is employed, the vertex simplies to,

$\displaystyle-i\kappa
a^{D-2}\Bigl{(}-\overline{\partial}^{\mu}\partial^{\nu}+\tfrac{1}{2}\eta^{\mu\nu}\Bigl{[}\overline{\partial}\\!\cdot\\!\partial+a^{2}m^{2}\Bigr{]}\Bigr{)}$
(69) $\displaystyle\hskip
99.58464pt=-i\kappa\Bigl{(}-a^{D-2}\overline{\partial}^{\mu}\partial^{\nu}+\tfrac{1}{4}\eta^{\mu\nu}\Bigl{[}\widetilde{\mathcal{D}}-(\mathcal{D}\\!-\\!a^{D}m^{2})\Bigr{]}\Bigr{)}\;,\qquad$

Recall that $\overline{\partial}^{\mu}$ acts on the external $\Psi$ leg,
$\partial^{\mu}$ acts on the internal $\Psi$ line and
$\widetilde{\partial}^{\mu}$ differentiates the graviton propagator. Of course
the factor of $(\mathcal{D}-a^{D}m^{2})$ acts on the internal $\Psi$
propagator to give a delta function, which effectively reduces the number of
distinct points of that portion of the diagram. This part of the
Source(Observer)-Exchange ($V_{5}$) diagrams turns out to cancel the Vertex-
Exchange ($V_{3}$) corrections. Similarly, these parts of the Source-Observer
($V_{4}$) diagrams cancel the main Vertex-Vertex ($V_{1a}$) correction and all
of the Source(Observer)-Vertex ($V_{2}$) corrections. The purpose of this
section is to demonstrate these cancellations, which are especially powerful
because they pertain for any gauge. We close by listing the residual diagrams
which remain.

###  4.1 The 3-5 Complex

It is convenient to label the three terms inside the large parentheses of
expression (69) as “$i$”, “$ii$” and “$iii$”,

$\displaystyle-a^{D-2}\overline{\partial}^{\mu}\partial^{\nu}$
$\displaystyle\Longleftrightarrow$ $\displaystyle i\;,$ (70)
$\displaystyle\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}$
$\displaystyle\Longleftrightarrow$ $\displaystyle ii\;,$ (71)
$\displaystyle-\frac{1}{4}\eta^{\mu\nu}(\mathcal{D}\\!-\\!a^{D}m^{2})$
$\displaystyle\Longleftrightarrow$ $\displaystyle iii\;.$ (72)

We can decompose $-iV_{5}(x;x^{\prime};y)$ accordingly,

$\displaystyle-iV_{5a_{i}}(x;x^{\prime};y)$
$\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!a_{y}^{D-2}\overline{\partial}^{\mu}\partial_{y}^{\nu}\\!\times\\!\\!\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}\\!-\\!\tfrac{1}{2}\eta^{\mu\nu}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}$
(73) $\displaystyle\hskip 34.14322pt\times
i\Delta_{m}(x;y)i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;,\qquad$
$\displaystyle-iV_{5a_{ii}}(x;x^{\prime};y)$
$\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!-\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}_{y}}\\!\times\\!\\!\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}\\!-\\!\tfrac{1}{2}\eta^{\mu\nu}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}$
(74) $\displaystyle\hskip 34.14322pt\times
i\Delta_{m}(x;y)i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;,\qquad$
$\displaystyle-iV_{5a_{iii}}(x;x^{\prime};y)$
$\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!\tfrac{1}{4}\eta^{\mu\nu}(\mathcal{D}_{y}\\!-\\!a^{D}m^{2})\\!\times\\!\\!\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}$
(75)
$\displaystyle-\tfrac{1}{2}\eta^{\mu\nu}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}\\!\times\\!i\Delta_{m}(x;y)i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;.\qquad$

The same conventions pertain for diagrams $-iV_{5b}(x;x^{\prime};y)$,
$-iV_{5c}(x;x^{\prime};y^{\prime})$ and $-iV_{5d}(x;x^{\prime};y^{\prime})$.
We use a bar to indicate the sum of the “$i$” and “$ii$” terms,

$-iV_{\overline{5}}(x;x^{\prime};y)\equiv-
iV_{5_{i}}(x;x^{\prime};y)-iV_{5_{ii}}(x;x^{\prime};y)\;.$ (76)

Note that $-iV_{5a_{iii}}(x;x^{\prime};y)$ can be simplified by acting the
$\Psi$ kinetic operator $(\mathcal{D}_{y}-a^{D}m^{2})$ on the $\Psi$
propagator $i\Delta_{m}(x;y)$,

$\displaystyle-iV_{5a_{iii}}(x;x^{\prime};y)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\delta^{D}(x\\!-\\!y)\\!\times\\!\tfrac{i}{4}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}-\tfrac{1}{2}\eta^{\mu\nu}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}$
(77) $\displaystyle\hskip 85.35826pt\times
i[\mbox{}^{\mu}_{~{}\mu}\Delta_{\rho\sigma}](x;z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;.\qquad$

One gets the same result for $-iV_{5b_{iii}}(x;x^{\prime};y)$. Taken together,
they cancel $\delta^{D}(x-y)\times-iV_{3a}(x;x^{\prime})$,

$\displaystyle\delta^{D}(x\\!-\\!y)\\!\times\\!-iV_{3a}(x;x^{\prime})=\delta^{D}(x\\!-\\!y)\\!\times\\!-\tfrac{i}{2}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}$
(78) $\displaystyle\hskip
96.73918pt-\tfrac{1}{2}\eta^{\rho\sigma}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}\\!\times\\!i[\mbox{}^{\mu}_{~{}\mu}\Delta_{\rho\sigma}](x;z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;.\qquad$

Hence we can write,

$-iV_{5a_{iii}}(x;x^{\prime};y)-iV_{5b_{iii}}(x;x^{\prime};y)+\delta^{D}(x\\!-\\!y)\\!\times\\!-iV_{3a}(x;x^{\prime})=0\;.$
(79)

Similarly, one can show,

$-iV_{5c_{iii}}(x;x^{\prime};y^{\prime})-iV_{5d_{iii}}(x;x^{\prime};y^{\prime})+\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-iV_{3b}(x;x^{\prime})=0\;.$
(80)

It follows that the Vertex-Exchange diagrams are completely canceled by the
“$iii$” parts of the Source(Observer)-Exchange diagrams.

###  4.2 The 1a-2-4 Complex

The Source-Observer diagrams ($V_{4}$) each have two $h_{\mu\nu}\Psi^{2}$
vertices (16), each one of which can be decomposed into terms “$i$”, “$ii$”
and “$iii$” as in (70-72). For $V_{4a}$ we first expand on the left hand
vertex, keeping the right hand vertex together,

$V_{4a}=V_{\overline{4a}}+V_{4a_{iii}}\;.$ (81)

We then expand on the right hand vertex,

$V_{\overline{4a}}=V_{\overline{\overline{4a}}}+V_{\overline{4a}_{iii}}\qquad,\qquad
V_{4a_{iii}}=V_{\overline{4a_{iii}}}+V_{(4a_{iii})_{iii}}\;,$ (82)

where the resulting four terms are,

$\displaystyle-iV_{\overline{\overline{4a}}}$
$\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\Bigl{[}a_{y}^{D-2}\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}\Bigl{[}a_{y^{\prime}}^{D-2}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}\\!-\\!\tfrac{1}{4}\eta^{\rho\sigma}\widetilde{D}_{y^{\prime}}\Bigr{]}$
(83) $\displaystyle\hskip 48.36958pt\times
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;y^{\prime})\;,\qquad$
$\displaystyle-iV_{\overline{4a}_{iii}}$
$\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\tfrac{1}{4}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\Bigl{[}a_{y}^{D-2}\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}(\mathcal{D}_{y^{\prime}}\\!-\\!a_{y^{\prime}}^{D}m^{2})$
(84) $\displaystyle\hskip 48.36958pt\times
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i[\mbox{}_{\mu\nu}\Delta^{\rho}_{~{}\rho}](y;y^{\prime})\;,\qquad$
$\displaystyle-iV_{\overline{4a_{iii}}}$
$\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\tfrac{1}{4}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}(\mathcal{D}_{y}\\!-\\!a_{y}^{D}m^{2})\Bigl{[}a_{y^{\prime}}^{D-2}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}\\!-\\!\tfrac{1}{4}\eta^{\rho\sigma}\widetilde{D}_{y^{\prime}}\Bigr{]}$
(85) $\displaystyle\hskip 48.36958pt\times
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i[\mbox{}^{\mu}_{~{}\mu}\Delta_{\rho\sigma}](y;y^{\prime})\;,\qquad$
$\displaystyle-iV_{(4a_{iii})_{iii}}$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\tfrac{1}{16}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}(\mathcal{D}_{y}\\!-\\!a_{y}^{D}m^{2})(\mathcal{D}_{y^{\prime}}\\!-\\!a_{y^{\prime}}^{D}m^{2})$
(86) $\displaystyle\hskip 48.36958pt\times
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i[\mbox{}^{\mu}_{~{}\mu}\Delta^{\rho}_{~{}\rho}](y;y^{\prime})\;.\qquad$

For $V_{4b}$ we also expand first on the left hand vertex and then on the
right hand vertex. However, it is best to expand $V_{4c}$ and $V_{4d}$ first
on the right hand vertex and then on the left hand one. This convention makes
for the simplest correspondence with the Source(Observer)-Vertex diagrams
($V_{2}$).

The Source(Observer)-Vertex diagrams (53-56) contain only one
$h_{\mu\nu}\Psi^{2}$ vertex. We expand this as usual, for example,

$V_{2a}=V_{\overline{2a}}+V_{2a_{iii}}\;.$ (87)

To be explicit, the two terms are,

$\displaystyle-iV_{\overline{2a}}$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle-\tfrac{i}{2}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\Bigl{[}a_{y}^{D-2}\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}$
(88) $\displaystyle\hskip 113.81102pt\times
i\Delta_{m}(x;y)i\Delta_{A}(x;x^{\prime})i[\mbox{}_{\mu\nu}\Delta^{\rho}_{~{}\rho}](y;x^{\prime})\;,\qquad$
$\displaystyle-iV_{2a_{iii}}$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle-\tfrac{i}{8}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}(\mathcal{D}_{y}\\!-\\!a_{y}^{D}m^{2})$
(89) $\displaystyle\hskip 113.81102pt\times
i\Delta_{m}(x;y)i\Delta_{A}(x;x^{\prime})i[\mbox{}^{\mu}_{\mu}\Delta^{\rho}_{~{}\rho}](y;x^{\prime})\;.\qquad$

Let us first act the $\Psi$ kinetic operators on the $\Psi$ propagators in
expressions (86) and (89) to conclude,

$\displaystyle-iV_{(4a_{iii})_{iii}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})$
(90) $\displaystyle\hskip
65.44142pt\\!\times\\!-\tfrac{1}{16}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}i\Delta_{A}(x;x^{\prime})i[\mbox{}^{\mu}_{~{}\mu}\Delta^{\rho}_{~{}\rho}](x;x^{\prime})\;,\qquad$
$\displaystyle-iV_{2a_{iii}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\delta^{D}(x\\!-\\!y)\\!\times\\!\tfrac{1}{8}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}i\Delta_{A}(x;x^{\prime})i[\mbox{}^{\mu}_{~{}\mu}\Delta^{\rho}_{~{}\rho}](x;x^{\prime})\;.\qquad$
(91)

Adding the two contributions (with appropriate delta functions) gives
$-\tfrac{1}{4}$ times the $-iV_{1a}$ contribution (49),

$\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-iV_{2a_{iii}}-iV_{(4a_{iii})_{iii}}=-\frac{1}{4}\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-iV_{1a}\;.$
(92)

The same relation pertains as well for the “$b$”, “$c$” and “$d$”
contributions, with the result that $-iV_{1a}$ is completely canceled.

Now act the $\Psi$ kinetic operator of expression (84),

$\displaystyle-iV_{\overline{4a}_{iii}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!\tfrac{i}{4}\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\Bigl{[}a_{y}^{D-2}\overline{\partial}_{y}^{\mu}\partial_{y}^{\nu}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}$
(93) $\displaystyle\hskip 122.34692pt\times
i\Delta_{m}(x;y)i\Delta_{A}(x;x^{\prime})i[\mbox{}_{\mu\nu}\Delta^{\rho}_{~{}\rho}](y;y^{\prime})\;.\qquad$

One gets exactly the same thing from $-iV_{\overline{4c_{iii}}}$, and their
sum cancels $-iV_{\overline{2a}}$,

$-iV_{\overline{4a}_{iii}}-iV_{\overline{4d_{iii}}}=-\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-iV_{\overline{2a}}\;.$
(94)

A number of similar relations pertain,

$\displaystyle-iV_{\overline{4b}_{iii}}-iV_{\overline{4c_{iii}}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-iV_{\overline{2b}}\;,\qquad$
(95) $\displaystyle-iV_{\overline{4c}_{iii}}-iV_{\overline{4a_{iii}}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\delta^{D}(x\\!-\\!y)\\!\times\\!-iV_{\overline{2c}}\;,\qquad$
(96) $\displaystyle-iV_{\overline{4d}_{iii}}-iV_{\overline{4b_{iii}}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\delta^{D}(x\\!-\\!y)\\!\times\\!-iV_{\overline{2d}}\;.\qquad$
(97)

Hence the $-iV_{2}$ diagrams are completely canceled.

###  4.3 Surviving Residual Diagrams

After all these consolidations there are just three classes of “new” diagrams
which remain. The simplest are the two surviving $i=1$ diagrams (50-51),

$\displaystyle-
iV_{1b}(x;x^{\prime})=-\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}$ (98)
$\displaystyle\hskip
85.35826pt\times\Bigl{\\{}\tfrac{1}{8}i[\mbox{}^{\alpha}_{~{}\alpha}\Delta^{\beta}_{~{}\beta}](x;x)-\tfrac{1}{4}i[\mbox{}^{\alpha\beta}\Delta_{\alpha\beta}](x;x)\Bigr{\\}}i\Delta_{A}(x;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{1c}(x;x^{\prime})=-\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}$ (99)
$\displaystyle\hskip 85.35826pt\times
i\Delta_{A}(x;x^{\prime})\Bigl{\\{}\tfrac{1}{8}i[\mbox{}^{\alpha}_{~{}\alpha}\Delta^{\beta}_{~{}\beta}](x^{\prime};x^{\prime})-\tfrac{1}{4}i[\mbox{}^{\alpha\beta}\Delta_{\alpha\beta}](x^{\prime};x^{\prime})\Bigr{\\}}\;.\qquad$

Next come the four residual $i=4$ diagrams,

$\displaystyle-
iV_{\overline{\overline{4a}}}(x;x^{\prime};y;y^{\prime})=\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\Bigl{[}a_{y}^{D-2}\overline{\partial}^{\mu}_{y}\partial^{\nu}_{y}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}\Bigl{[}a_{y^{\prime}}^{D-2}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}\\!-\\!\tfrac{1}{4}\eta^{\rho\sigma}\widetilde{\mathcal{D}}_{y^{\prime}}\Bigr{]}$
(100) $\displaystyle\hskip 102.43008pt\times
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;y^{\prime})\;,\qquad$
$\displaystyle-
iV_{\overline{\overline{4b}}}(x;x^{\prime};y;y^{\prime})=\kappa^{2}\lambda^{2}(a_{y}a_{y^{\prime}})^{D}\Bigl{[}a_{x}^{D-2}\overline{\partial}^{\mu}_{x}\partial^{\nu}_{x}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{x}\Bigr{]}\Bigl{[}a_{x^{\prime}}^{D-2}\overline{\partial}_{x^{\prime}}^{\rho}\partial_{x^{\prime}}^{\sigma}\\!-\\!\tfrac{1}{4}\eta^{\rho\sigma}\widetilde{\mathcal{D}}_{x^{\prime}}\Bigr{]}$
(101) $\displaystyle\hskip 102.43008pt\times
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(y;y^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{\overline{\overline{4c}}}(x;x^{\prime};y;y^{\prime})=\kappa^{2}\lambda^{2}(a_{y}a_{x^{\prime}})^{D}\Bigl{[}a_{x}^{D-2}\overline{\partial}^{\mu}_{x}\partial^{\nu}_{x}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{x}\Bigr{]}\Bigl{[}a_{y^{\prime}}^{D-2}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}\\!-\\!\tfrac{1}{4}\eta^{\rho\sigma}\widetilde{\mathcal{D}}_{y^{\prime}}\Bigr{]}$
(102) $\displaystyle\hskip 102.43008pt\times
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(y;x^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;y^{\prime})\;,\qquad$
$\displaystyle-
iV_{\overline{\overline{4d}}}(x;x^{\prime};y;y^{\prime})=\kappa^{2}\lambda^{2}(a_{x}a_{y^{\prime}})^{D}\Bigl{[}a_{y}^{D-2}\overline{\partial}^{\mu}_{y}\partial^{\nu}_{y}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}\Bigl{[}a_{x^{\prime}}^{D-2}\overline{\partial}_{x^{\prime}}^{\rho}\partial_{x^{\prime}}^{\sigma}\\!-\\!\tfrac{1}{4}\eta^{\rho\sigma}\widetilde{\mathcal{D}}_{x^{\prime}}\Bigr{]}$
(103) $\displaystyle\hskip 102.43008pt\times
i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;y^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;x^{\prime})\;.\qquad$

Finally, there are the four $i=5$ residual diagrams,

$\displaystyle-
iV_{\overline{5a}}(x;x^{\prime};y)=\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\Bigl{[}a_{y}^{D-2}\overline{\partial}^{\mu}_{y}\partial^{\nu}_{y}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}$
(104) $\displaystyle\hskip
36.98866pt-\tfrac{1}{2}\eta^{\rho\sigma}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}\\!\times\\!i\Delta_{m}(x;y)i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y;z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{\overline{5b}}(x;x^{\prime};y)=\kappa^{2}\lambda^{2}(a_{y}a_{x^{\prime}})^{D}\Bigl{[}a_{x}^{D-2}\overline{\partial}^{\mu}_{x}\partial^{\nu}_{x}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{x}\Bigr{]}\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}$
(105) $\displaystyle\hskip
36.98866pt-\tfrac{1}{2}\eta^{\rho\sigma}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}\\!\times\\!i\Delta_{m}(x;y)i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x;z)i\Delta_{A}(y;z)i\Delta_{A}(z;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{\overline{5c}}(x;x^{\prime};y)=\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\Bigl{[}a_{y^{\prime}}^{D-2}\overline{\partial}^{\mu}_{y^{\prime}}\partial^{\nu}_{y^{\prime}}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y^{\prime}}\Bigr{]}\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}$
(106) $\displaystyle\hskip
36.98866pt-\tfrac{1}{2}\eta^{\rho\sigma}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}\\!\times\\!i\Delta_{m}(x^{\prime};y^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](y^{\prime};z)i\Delta_{A}(x;z)i\Delta_{A}(z;x^{\prime})\;,\qquad$
$\displaystyle-
iV_{\overline{5d}}(x;x^{\prime};y)=\kappa^{2}\lambda^{2}(a_{x}a_{y^{\prime}})^{D}\Bigl{[}a_{x^{\prime}}^{D-2}\overline{\partial}^{\mu}_{x^{\prime}}\partial^{\nu}_{x^{\prime}}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{x^{\prime}}\Bigr{]}\\!\int\\!\\!d^{D}z\,a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}$
(107) $\displaystyle\hskip
36.98866pt-\tfrac{1}{2}\eta^{\rho\sigma}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}\\!\times\\!i\Delta_{m}(x^{\prime};y^{\prime})i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}](x^{\prime};z)i\Delta_{A}(x;z)i\Delta_{A}(z;y^{\prime})\;.\qquad$

It is important to note that the reductions to these residual forms are exact,
and independent of the graviton gauge.

##  5 Gauge Independent Self-Mass

The purpose of this section is to combine the $t$-channel contributions from
the five new diagrams with $-iM_{0}(x;x^{\prime})$ to produce a gauge
independent self-mass. We begin by discussing the general reduction strategy
for the $\overline{5}$ and $\overline{\overline{4}}$ contributions. Then each
of these is worked out. The section closes by adding the $1b$ and $1c$
contributions and renormalizing the final result.

###  5.1 General Reduction Strategy

A number of simplifications concern the graviton propagator. First, because
the propagator’s indices are typically contracted into derivatives, only the
temporal components of which can be significant for the infrared, $t$-channel
exchange contributions of interest to us, we need retain just the first term
in expression (14),

$i[\mbox{}_{\mu\nu}\Delta_{\rho\sigma}]\longrightarrow\Bigl{[}\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{2}{D\\!-\\!2}\eta_{\mu\nu}\eta_{\rho\sigma}\Bigr{]}\\!\times\\!i\Delta_{C}\;.$
(108)

Second, in acting $\widetilde{D}$ on $i\Delta_{C}$ in 4-point contributions we
can ignore the delta function because that term will have one fewer massless
propagator,

$\widetilde{\mathcal{D}}i\Delta_{C}(x;x^{\prime})=i\delta^{D}(x\\!-\\!x^{\prime})+2(D\\!-\\!3)H^{2}i\Delta_{C}(x;x^{\prime})\longrightarrow
2(D\\!-\\!3)H^{2}i\Delta_{C}(x;x^{\prime})\;.$ (109)

Finally, single derivatives of $i\Delta_{C}$ can be approximately reflected
from one coordinate to the other using the relation,222Recall that
$\partial^{\mu}_{a_{x}}=-(D-2)Ha_{x}\delta^{\mu}_{~{}0}$ indicates the
derivative acting on the factor of $a^{D-2}_{x}$ which accompanies vertices
from the kinetic energy.

$\Bigl{(}\widetilde{\partial}^{\mu}_{x}+\frac{1}{2}\partial^{\mu}_{a_{x}}\Bigr{)}i\Delta_{C}(x;x^{\prime})\simeq-\Bigl{(}\widetilde{\partial}^{\mu}_{x^{\prime}}+\frac{1}{2}\partial^{\mu}_{a_{x^{\prime}}}\Bigr{)}i\Delta_{C}(x;x^{\prime})\;.$
(110)

This approximation is exact on the first term of the
$i\Delta_{C}(x;x^{\prime})$ expansion; and the higher terms should not matter
because they vanish for $D=4$.

Our basic strategy is to follow the same reductions as in flat space [41],
collecting de Sitter obstructions as they occur. However, we need not retain
terms with more than two factors of $H$ because any additional factors of $H$
will necessarily come in the dimensionless form of $H/m$ which vanishes when
the source and observer masses become large.

One major change from the flat space reduction concerns the external state
factors for the source and observer. In flat space these are simple phase
factors; in de Sitter they are spatial plane waves times Hankel functions with
imaginary arguments. Their series expansions in powers of $k^{2}/a^{2}$ are
more useful for our purposes,

$u_{m}(x;\vec{k})=\frac{e^{-i\mu t+i\vec{k}\cdot\vec{x}}}{\sqrt{2\mu
a^{D-1}}}\sum_{n=0}^{\infty}\frac{\Gamma(1\\!+\\!\frac{i\mu}{H})}{n!\,\Gamma(1\\!+\\!n\\!+\\!\frac{i\mu}{H})}\Bigl{(}\frac{-k^{2}}{4H^{2}a^{2}}\Bigr{)}^{n}\;,\;\mu^{2}\equiv
m^{2}\\!-\\!(\tfrac{D-1}{2})^{2}H^{2}\;.$ (111)

Note that this expression employs the co-moving time $t=\ln(a)/H$, rather than
the conformal time. We can expand external first derivatives as,

$\partial_{0}u_{m}(x;\vec{k})=-ia\Bigl{[}\mu-i(\tfrac{D-1}{2})H+\tfrac{k^{2}}{2ma^{2}}+\ldots\Bigr{]}u_{m}\quad,\quad\partial_{i}u_{m}(x;\vec{k})=ik_{i}u_{m}\;.$
(112)

We sometimes also need second time derivatives,

$\partial_{0}^{2}u_{m}(x;\vec{k})=-a^{2}\Bigl{[}m^{2}-i(D\\!-\\!2)\mu
H-\tfrac{1}{2}(D\\!-\\!1)(D\\!-\\!2)H^{2}+\tfrac{k^{2}}{a^{2}}+\ldots\Bigr{]}u_{m}(x;\vec{k})\;.$
(113)

We regard the two in-coming wave functions as,

$u_{m}(x;\vec{k}_{1})\qquad,\qquad u_{m}(x^{\prime};\vec{k}_{3})\;.$ (114)

The corresponding out-going wave functions are,

$u_{m}^{*}(y;\vec{k}_{2})\qquad,\qquad u_{m}^{*}(y^{\prime};\vec{k}_{4})\;.$
(115)

And spatial momentum conservation implies,

$\vec{k}_{1}+\vec{k}_{3}=\vec{k}_{2}+\vec{k}_{4}\qquad\Longrightarrow\qquad\vec{k}_{1}-\vec{k}_{2}=\vec{k}_{4}-\vec{k}_{3}\equiv\vec{q}\;.$
(116)

###  5.2 $V_{\overline{5}}(x;x^{\prime};y;y^{\prime})$ Diagrams

Here we give the reduction for the residual diagrams of type $\overline{5}$
listed in (104-107). The details are given for part $\overline{5a}$, and the
remaining three parts are inferred from symmetry.

The initial step of the reduction is contracting the simplified graviton
propagator (108) into the tensor structures and derivatives of the two
3-vertices in (104),

$\displaystyle\Bigl{[}a_{y}^{D-2}\overline{\partial}^{\mu}_{y}\partial^{\nu}_{y}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}a_{z}^{D-2}\Bigl{[}\overline{\partial}_{z}^{\rho}\partial_{z}^{\sigma}-\tfrac{1}{2}\eta^{\rho\sigma}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\Bigr{]}\Bigl{[}\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{2\eta_{\mu\nu}\eta_{\rho\sigma}}{D\\!-\\!2}\Bigr{]}$
$\displaystyle\hskip
28.45274pt=a_{z}^{D-2}\Bigl{[}a_{y}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\partial_{y}\\!\cdot\\!\partial_{z}+a_{y}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\partial_{z}\partial_{y}\\!\cdot\\!\overline{\partial}_{z}-\tfrac{1}{2}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\widetilde{\mathcal{D}}_{y}\Bigr{]}\,.$
(117)

The derivatives in the resulting three terms above are then partially
integrated using (41), and reflected over the graviton propagator using (110)
so as to extract as many of them from the triangular loop. Not all derivatives
can be extracted this way. The ones remaining inside we arrange when possible
to be contracted with another derivative at the same vertex. This procedure
gives the following for the first term in (117),

$\displaystyle(a_{y}a_{z})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\partial_{y}\\!\cdot\\!\partial_{z}\longrightarrow-\tfrac{1}{2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\bigr{(}a_{z}^{D-2}\mathcal{D}_{y}+a_{y}^{D-2}\mathcal{D}_{z}\bigr{)}$
(118) $\displaystyle\hskip
14.22636pt+\tfrac{1}{2}(a_{y}a_{z})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigl{[}\overline{\partial}_{y}\\!\cdot\\!(\overline{\partial}_{y}\\!+\\!\partial_{a_{y}})+\overline{\partial}_{z}\\!\cdot\\!(\overline{\partial}_{z}\\!+\\!\partial_{a_{z}})+2\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigr{]}\qquad$
(119) $\displaystyle\hskip
14.22636pt+\tfrac{1}{4}(D\\!-\\!2)(a_{y}a_{z})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigl{\\{}H(a_{y}\\!-\\!a_{z})\Bigl{[}2\bigl{(}{\overline{\partial}_{y^{0}}}-\overline{\partial}_{z^{0}}\bigr{)}$
$\displaystyle\hskip
56.9055pt+\widetilde{\partial}_{y^{0}}-\widetilde{\partial}_{z^{0}}+(D\\!-\\!2)H(a_{y}\\!-\\!a_{z})\Bigr{]}+H^{2}(a_{y}^{2}\\!+\\!a_{z}^{2})\Bigr{\\}}\,.\qquad$
(120)

Note that we recognized scalar d’Alembertians (42) where that was not impeded
by other derivatives or powers of the scale factor. They are defined to act
directly onto the corresponding propagator or external leg. The first two
lines in the expression above contain terms that are de Sitter generalizations
of flat space contributions, that we refer to as flat space descendants, while
the remaining lines contain only pure de Sitter terms.

The second term in (117), upon reorganizing the derivatives, doubles the
contribution of the first term, and provides additional contributions,

$\displaystyle(a_{y}a_{z})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\partial_{z}\partial_{y}\\!\cdot\\!\overline{\partial}_{z}\longrightarrow(a_{y}a_{z})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\partial_{y}\\!\cdot\\!\partial_{z}$
(121)
$\displaystyle+\tfrac{1}{2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\bigl{(}a_{z}^{D-2}\widetilde{\mathcal{D}}_{y}+a_{y}^{D-2}\widetilde{\mathcal{D}}_{z}\bigr{)}+\tfrac{1}{4}\bigl{(}\mathcal{D}_{y}-\overline{\mathcal{D}}_{y}-\widetilde{\mathcal{D}}_{y}\bigr{)}\bigl{(}\mathcal{D}_{z}-\overline{\mathcal{D}}_{z}-\widetilde{\mathcal{D}}_{z}\bigr{)}\qquad$
(122)
$\displaystyle+\tfrac{1}{4}(D\\!-\\!2)(a_{y}a_{z})^{D-2}\Bigl{\\{}H(a_{y}\\!-\\!a_{z})\Bigl{[}2\overline{\partial}_{y}\\!\cdot\\!\widetilde{\partial}_{y}\overline{\partial}_{z^{0}}-2\overline{\partial}_{z}\\!\cdot\\!\widetilde{\partial}_{z}\overline{\partial}_{y^{0}}$
(123)
$\displaystyle-\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\bigl{(}\widetilde{\partial}_{y^{0}}\\!-\\!\widetilde{\partial}_{z^{0}}\bigr{)}-(D\\!-\\!2)H(a_{y}\\!-\\!a_{z})\overline{\nabla}_{y}\\!\cdot\\!\overline{\nabla}_{z}\Bigr{]}-H^{2}(a_{y}^{2}\\!+\\!a_{z}^{2})\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigr{\\}}.\qquad$
(124)

The latter can again be recognized as flat space descendants in line (122),
and as pure de Sitter contributions in lines (123) and (124). Finally, the
third term in (117) produces only flat space descendants,

$\displaystyle-\tfrac{1}{2}a_{z}^{D-2}\overline{\partial}_{z}\\!\cdot\\!\partial_{z}\widetilde{\mathcal{D}}_{y}\longrightarrow-\tfrac{1}{4}\widetilde{\mathcal{D}}_{y}\bigl{(}\widetilde{\mathcal{D}}_{z}-\overline{\mathcal{D}}_{z}-\mathcal{D}_{z}\bigr{)}\,.$
(125)

Further reduction of terms in (118-125) is accomplished by acting the scalar
kinetic operators where possible, applying 3-point Donoghue identities (35)
and (36), and finally acting the remaining derivatives onto external legs. We
consider these reductions separately for flat space descendants and for pure
de Sitter contributions.

####  5.2.1 Reduction of flat space descendants

Flat space descendant contributions comprises twice the terms in lines (118)
and (119), and terms in lines (122) and (125), that added together read,

$\displaystyle(a_{y}a_{z})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigl{[}2\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}+\overline{\partial}_{y}\\!\cdot\\!(\overline{\partial}_{y}\\!+\\!\partial_{a_{y}})+\overline{\partial}_{z}\\!\cdot\\!(\overline{\partial}_{z}\\!+\\!\partial_{a_{z}})\Bigr{]}$
(126)
$\displaystyle-\tfrac{1}{2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigr{[}2a_{z}^{D-2}\mathcal{D}_{y}+2a_{y}^{D-2}\mathcal{D}_{z}-a_{z}^{D-2}\widetilde{\mathcal{D}}_{y}-a_{y}^{D-2}\widetilde{\mathcal{D}}_{z}\Bigr{]}$
(127)
$\displaystyle+\tfrac{1}{4}\bigl{(}\mathcal{D}_{y}-\overline{\mathcal{D}}_{y}\bigr{)}\bigl{(}\mathcal{D}_{z}-\overline{\mathcal{D}}_{z}-\widetilde{\mathcal{D}}_{z}\bigr{)}+\tfrac{1}{2}\widetilde{\mathcal{D}}_{y}\overline{\mathcal{D}}_{z}\,.$
(128)

Simplest to reduce are terms in the last line (128), whose contribution we
label by “$(A)$”, that consist of products of scalar d’Alembertians. Only
terms containing combination
$(\mathcal{D}_{y}\\!-\\!\overline{\mathcal{D}}_{y})$ that contracts the
massive propagator contribute in the end,

$(A):\quad\tfrac{1}{4}\bigl{(}\mathcal{D}_{y}\\!-\\!\overline{\mathcal{D}}_{y}\bigr{)}\bigl{(}\mathcal{D}_{z}\\!-\\!\overline{\mathcal{D}}_{z}\\!-\\!\widetilde{\mathcal{D}}_{z}\bigr{)}i\Delta_{m}(x;y)\longrightarrow\tfrac{1}{4}i\delta^{D}(x\\!-\\!y)\bigl{(}\mathcal{D}_{z}\\!-\\!\overline{\mathcal{D}}_{z}\\!-\\!\widetilde{\mathcal{D}}_{z}\bigr{)}\,,\
\ $ (129)

while the remaining ones vanish in the large mass limit.

The terms in line (127) that have two external derivatives are best considered
together with the latter two terms in first line (126) that contain two
external would-be scalar d’Alembertians. We label this contribution by
“$(B)$”. It is only the middle term in (126) and the first term in (127) that
contain sufficient number of derivatives acting on either external mode
functions or on the massive propagator to generate a nonvanishing
contribution. This contribution is obtained by first acting the scalar
d’Alembertian, then applying the 3-point Donoghue identity (35), followed by
acting derivatives on external mode functions using (112),

$\displaystyle(B):\qquad
a_{z}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigl{[}a_{y}^{D-2}\overline{\partial}_{y}\\!\cdot\\!(\overline{\partial}_{y}\\!+\\!\partial_{a_{y}})-\mathcal{D}_{y}\Bigr{]}i\Delta_{m}(x;y)$
(130) $\displaystyle\longrightarrow
a_{z}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigl{\\{}a_{y}^{D-2}\Bigl{[}\overline{\partial}_{y}\\!\cdot\\!(\overline{\partial}_{y}\\!+\\!\partial_{a_{y}})-m^{2}a_{y}^{2}\Bigr{]}i\Delta_{m}(x;y)-i\delta^{D}(x\\!-\\!y)\Bigr{\\}}\qquad$
(131) $\displaystyle\longrightarrow
a_{z}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigl{[}\tfrac{1}{2m^{2}a_{y}^{2}}\overline{\partial}_{y}\\!\cdot\\!(\overline{\partial}_{y}\\!+\\!\partial_{a_{y}})-\tfrac{3}{2}\Bigr{]}i\delta^{D}(x\\!-\\!y)$
(132) $\displaystyle\longrightarrow
a_{z}^{D-2}\Bigl{[}-\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}-Ha_{y}\overline{\partial}_{z^{0}}\Bigr{]}i\delta^{D}(x\\!-\\!y)\,.$
(133)

In the last step above one needs to account for all the terms generated by
multiple derivatives acting on external mode functions,

$\overline{\partial}_{z}\\!\cdot\\!\overline{\partial}_{y}\,\overline{\partial}_{y}\\!\cdot\\!(\overline{\partial}_{y}\\!+\\!\partial_{a_{y}})\longrightarrow
m^{2}a_{y}^{2}\Bigl{[}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}-2Ha_{y}\overline{\partial}_{z^{0}}-\tfrac{(D-2)H^{2}}{m^{2}}\overline{\partial}_{y^{0}}\overline{\partial}_{z^{0}}\Bigr{]}\,,$
(134)

and keep only ones contributing in the large mass limit.

The last flat space descendant contribution, that we label by “$(C)$”, is the
first term in line (126) with all four derivatives extracted. Reducing it is
straightforward and again follows from applying the 3-point Donoghue identity,
followed by acting the derivatives on the external mode function,

$\displaystyle(C):\quad
2(a_{y}a_{z})^{D-2}(\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z})^{2}i\Delta_{m}(x;y)$
$\displaystyle\longrightarrow
a_{z}^{D-2}(\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z})^{2}\frac{i\delta^{D}(x\\!-\\!y)}{m^{2}a_{y}^{2}}\longrightarrow-
a_{z}^{D-2}(\overline{\partial}_{z^{0}})^{2}i\delta^{D}(x\\!-\\!y)\,.\quad$
(135)

####  5.2.2 Reduction of pure de Sitter terms

Pure de Sitter contributions to $\overline{5a}$ residual diagram consist of
twice the contributions in (120), and lines (123) and (124), which added
together read,

$\displaystyle\tfrac{1}{4}(D\\!-\\!2)(a_{y}a_{z})^{D-2}\times\Bigl{\\{}H(a_{y}\\!-\\!a_{z})\Bigl{[}4\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\bigl{(}{\overline{\partial}_{y^{0}}}-\overline{\partial}_{z^{0}}\bigr{)}$
(136)
$\displaystyle+\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\bigl{(}\widetilde{\partial}_{y^{0}}-\widetilde{\partial}_{z^{0}}\bigr{)}+2\overline{\partial}_{y}\\!\cdot\\!\widetilde{\partial}_{y}\,\overline{\partial}_{z^{0}}-2\overline{\partial}_{z}\\!\cdot\\!\widetilde{\partial}_{z}\,\overline{\partial}_{y^{0}}$
(137)
$\displaystyle+(D\\!-\\!2)H(a_{y}\\!-\\!a_{z})\bigl{(}2\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}-\overline{\nabla}_{y}\\!\cdot\\!\overline{\nabla}_{z}\bigr{)}\Bigr{]}+H^{2}(a_{y}^{2}\\!+\\!a_{z}^{2})\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\Bigr{\\}}\,.\qquad$
(138)

Terms in the first line (136) with all derivatives extracted, whose
contribution we label by “$(D)$”, are immediately reduced by the Donoghue
3-point identity followed by acting derivatives on external legs. Only the
first term in (136) manages to contribute,

$\displaystyle(D):\qquad(D\\!-\\!2)H(a_{y}\\!-\\!a_{z})(a_{y}a_{z})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}{\overline{\partial}_{y^{0}}}i\Delta_{m}(x;y)$
(139) $\displaystyle\hskip
14.22636pt\longrightarrow(D\\!-\\!2)H(a_{y}\\!-\\!a_{z})a_{z}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}{\overline{\partial}_{y^{0}}}\frac{i\delta^{D}(x\\!-\\!y)}{2m^{2}a_{y}^{2}}\qquad$
(140) $\displaystyle\hskip
14.22636pt\longrightarrow\tfrac{1}{2}(D\\!-\\!2)H(a_{y}\\!-\\!a_{z})a_{z}^{D-2}\overline{\partial}_{z^{0}}i\delta^{D}(x\\!-\\!y)\,.$
(141)

The second line (137) contains terms with single internal derivatives that
cannot be extracted from the triangular loop. Reduction of such terms requires
the derivative 3-point Donoghue identity (36). For the first two terms, their
contribution labeled by $(E)$, we can apply this identity immediately after
reflecting the internal derivative to act on the $y$ coordinate of the
graviton propagator. Acting derivatives on external legs then concludes the
reduction,

$\displaystyle(E):\quad\tfrac{1}{2}(D\\!-\\!2)(a_{y}a_{z})^{D-2}H(a_{y}\\!-\\!a_{z})\Bigl{[}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\widetilde{\partial}_{y^{0}}\\!+\\!\overline{\partial}_{y}\\!\cdot\\!\widetilde{\partial}_{y}\,\overline{\partial}_{z^{0}}\Bigr{]}i\Delta_{m}(x;y)\qquad$
(142) $\displaystyle\hskip
25.6073pt\longrightarrow\tfrac{1}{2}(D\\!-\\!2)a_{z}^{D-2}H(a_{y}\\!-\\!a_{z})\\!\Bigl{[}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{z}\,\overline{\partial}_{y^{0}}+\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y}\,\overline{\partial}_{z^{0}}\Bigr{]}\\!\frac{i\delta^{D}(x\\!-\\!y)}{2m^{2}a_{y}^{2}}\qquad\
$ (143) $\displaystyle\hskip
25.6073pt\longrightarrow\tfrac{1}{2}(D\\!-\\!2)a_{z}^{D-2}H(a_{y}\\!-\\!a_{z})\overline{\partial}_{z^{0}}i\delta^{D}(x\\!-\\!y)\,.$
(144)

For the remaining term in line (137), labeled by “$(F)$”, the derivative
Donoghue identity should not be applied directly in order not to lose the
subleading terms. We first write out the derivatives contracted at the same
vertex in terms of d’Alembertian operators keeping track of correction terms
up to order $H^{2}$,

$\displaystyle(F):\quad-\tfrac{1}{2}(D\\!-\\!2)(a_{y}a_{z})^{D-2}H(a_{y}\\!-\\!a_{z})\overline{\partial}_{z}\\!\cdot\\!\widetilde{\partial}_{z}\,\overline{\partial}_{y^{0}}i\Delta_{m}(x;y)\qquad$
(145) $\displaystyle\hskip
14.22636pt\longrightarrow-\tfrac{1}{4}(D\\!-\\!2)a_{y}^{D-2}H(a_{y}\\!-\\!a_{z})\bigl{(}\mathcal{D}_{z}-\overline{\mathcal{D}}_{z}-\widetilde{\mathcal{D}}_{z}\bigr{)}\overline{\partial}_{y^{0}}i\Delta_{m}(x;y)\qquad$
(146) $\displaystyle\hskip
48.36958pt-\tfrac{1}{4}(D\\!-\\!2)(a_{y}a_{z})^{D-2}H^{2}a_{z}^{2}\widetilde{\partial}_{y^{0}}\overline{\partial}_{y^{0}}i\Delta_{m}(x;y)\,.$
(147)

None of the terms in line (146) can contribute, but the term in line (147)
produces a nonvanishing contribution after applying the derivative 3-point
Donoghue identity and acting the external leg derivatives,

$(F):\quad\
\longrightarrow-\tfrac{1}{4}(D\\!-\\!2)H^{2}a_{z}^{D}(\overline{\partial}_{y^{0}})^{2}\frac{i\delta^{D}(x\\!-\\!y)}{2m^{2}a_{y}^{2}}\longrightarrow\tfrac{1}{4}(D\\!-\\!2)H^{2}a_{z}^{D}i\delta^{D}(x\\!-\\!y)\,.\
\ $ (148)

Lastly, the remaining terms, given in line (138), do not have enough external
derivatives to contribute.

####  5.2.3 Self-mass contributions

The reduced diagram $\overline{5a}$ collects $(A)$–$(F)$ contributions from
(129), (133), (135), (141), (144), and (148). It is natural to consider it
together with the reduced diagram $\overline{5b}$, that is obtained from
$\overline{5a}$ by reflecting $x\\!\leftrightarrow\\!y$ in the final results
of reductions in sections 5.2.1 and 5.2.2. Together they make up diagrams of
the same topology,

$\displaystyle\bigl{[}-iV_{\overline{5a+b}}\bigr{]}^{\rm
3pt}_{(I)}=(-i\lambda)^{2}\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})(a_{x}a_{x^{\prime}})^{D}$
$\displaystyle\hskip 14.22636pt\times
i\kappa^{2}\\!\\!\int\\!d^{D\\!}z\,\Theta_{I}\Bigl{[}a_{z}^{D-2}i\Delta_{C}(x;z)i\Delta_{A}(x;z)\Bigr{]}\times
i\Delta_{A}(z;x^{\prime})\,,$ (149)

where contributions $(A)$–$(F)$ are accounted for by derivative operators that
act to their right,

$\displaystyle\Theta_{(A)}=\tfrac{1}{2}\mathcal{D}_{z}a_{z}^{2-D}\\!+\\!(D\\!-\\!3)H^{2}a_{z}^{2}\,,\
\ \
\Theta_{(B)}=\partial_{x}\\!\cdot\\!\partial_{z}\\!-\\!(D\\!+\\!2)Ha_{x}\partial_{0}^{z}\,,$
(150) $\displaystyle\Theta_{(C)}=2(\partial_{0}^{z})^{2}\,,\ \ \
\Theta_{(D+E)}\\!=\\!2(D\\!-\\!2)\partial_{0}^{z}H(a_{x}\\!-\\!a_{z})\,,\ \ \
\Theta_{(F)}=-\tfrac{1}{2}(D\\!-\\!2)H^{2}a_{z}^{2}\,.$

In addition $(A)$ contributions generate diagrams of different topology when
scalar d’Alembertians in (129) contract either the graviton or the massless
scalar propagator,

$\displaystyle\bigl{[}-iV_{\overline{5a+b}}\bigr{]}^{\rm
4pt}_{(A)}=(-i\lambda)^{2}\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})(a_{x}a_{x^{\prime}})^{D}$
$\displaystyle\hskip
42.67912pt\times\tfrac{1}{2}\kappa^{2}\bigl{[}i\Delta_{C}(x;x)-i\Delta_{A}(x;x)\bigr{]}\times
i\Delta_{A}(x;x^{\prime})\,.$ (151)

Some of the differential operators in (150) can be further rewritten owing to
the topology of diagram (149) and the identity,

$\displaystyle
a_{x}^{D-n_{x}}\partial_{0}^{z}\Bigl{[}a_{z}^{D-n_{z}}i\Delta_{C}(x;z)i\Delta_{A}(x;z)\Bigr{]}$
$\displaystyle=a_{x}^{D-n_{x}}\bigl{(}-\partial_{0}^{x}+\partial_{0}^{x}+\partial_{0}^{z}\bigr{)}\Bigl{[}a_{z}^{D-n_{z}}i\Delta_{C}(x;z)i\Delta_{A}(x;z)\Bigr{]}$
$\displaystyle\longrightarrow
a_{x}^{D-n_{x}}\Bigl{[}(1\\!-\\!n_{x})Ha_{x}+\partial_{0}^{x}+\partial_{0}^{z}\Bigr{]}\Bigl{[}a_{z}^{D-n_{z}}i\Delta_{C}(x;z)i\Delta_{A}(x;z)\Bigr{]}\,,\qquad$
(152)

that is derived by partial integration and accounting for action of time
derivatives on external legs,

$\partial_{0}\bigl{[}u_{m}(x,\vec{k})u_{m}^{*}(x,\vec{k}^{\prime})\bigr{]}\xrightarrow{m^{2}\to\infty}-(D\\!-\\!1)Ha_{x}\times
u_{m}(x,\vec{k})u_{m}^{*}(x,\vec{k}^{\prime})\,.$ (153)

It is $(B)$ and $(C)$ contributions that benefit from reorganizing
derivatives,

$\displaystyle\Theta_{(B)}\\!\longrightarrow\\!-\mathcal{D}_{z}a_{z}^{2-D}\\!\\!-\\!\bigl{(}\partial_{0}^{z}\\!+\\!2Ha_{x}\bigr{)}\bigl{(}\partial_{0}^{x}\\!+\\!\partial_{0}^{z}\\!+\\!2Ha_{x}\bigr{)}\\!-\\!(D\\!-\\!2)\partial_{0}^{z}H(a_{x}\\!-\\!a_{z})\,,\quad$
$\displaystyle\Theta_{(C)}\\!\longrightarrow\\!2\bigl{(}\partial_{0}^{z}\\!-\\!Ha_{x}\bigr{)}\bigl{(}\partial_{0}^{x}\\!+\\!\partial_{0}^{z}\\!+\\!2Ha_{x}\bigr{)}\,.\qquad$
(154)

The utility of this form is that only $\mathcal{D}_{z}a_{z}^{2-D}$ term will
contain divergences, while all the other terms will be manifestly finite.
Collecting all the contributions $(A)$–$(F)$ to diagrams
$\overline{5a}\\!+\\!\overline{5b}$ finally produces a contribution to the
$\beta$-type diagram from Fig. 2,

$\displaystyle\bigl{[}-if_{\beta}(x;x^{\prime})\bigr{]}_{\overline{5}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{\kappa^{2}}{2}\Bigl{[}-\mathcal{D}_{x^{\prime}}a_{x^{\prime}}^{2-D}+2\bigl{(}\partial_{0}^{x^{\prime}}\\!-\\!4Ha_{x}\bigr{)}\bigl{(}\partial_{0}^{x}\\!+\\!\partial_{0}^{x^{\prime}}\\!+\\!2Ha_{x}\bigr{)}$
(155)
$\displaystyle+2(D\\!-\\!2)\partial_{0}^{x^{\prime}}(a_{x}\\!-\\!a_{x^{\prime}})+(D\\!-\\!4)H^{2}a_{x^{\prime}}^{2}\Bigr{]}\Bigl{[}a_{x^{\prime}}^{D-2}i\Delta_{C}(x;x^{\prime})i\Delta_{A}(x;x^{\prime})\Bigr{]}$
$\displaystyle-\tfrac{\kappa^{2}}{2}i\delta^{D}(x\\!-\\!x^{\prime})\Bigl{[}i\Delta_{C}(x^{\prime};x^{\prime})-i\Delta_{A}(x^{\prime};x^{\prime})\Bigr{]}\,.$

The two remaining diagrams $\overline{5c}\\!+\\!\overline{5d}$ are then
inferred by reflecting $x\\!\leftrightarrow\\!x^{\prime}$ in the $\beta$-type
contribution above, yielding a $\gamma$-type contribution from Fig. 2,

$\displaystyle\bigl{[}-if_{\gamma}(x;x^{\prime})\bigr{]}_{\overline{5}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{\kappa^{2}}{2}\Bigl{[}-\mathcal{D}_{x}a_{x}^{2-D}+2\bigl{(}\partial_{0}^{x}\\!-\\!4Ha_{x^{\prime}}\bigr{)}\bigl{(}\partial_{0}^{x}\\!+\\!\partial_{0}^{x^{\prime}}\\!+\\!2Ha_{x^{\prime}}\bigr{)}$
(156)
$\displaystyle-2(D\\!-\\!2)\partial_{0}^{x}(a_{x}\\!-\\!a_{x^{\prime}})+(D\\!-\\!4)H^{2}a_{x}^{2}\Bigr{]}\Bigl{[}a_{x}^{D-2}i\Delta_{C}(x;x^{\prime})i\Delta_{A}(x;x^{\prime})\Bigr{]}$
$\displaystyle-\tfrac{\kappa^{2}}{2}i\delta^{D}(x\\!-\\!x^{\prime})\Bigl{[}i\Delta_{C}(x;x)-i\Delta_{A}(x;x)\Bigr{]}\,.$

###  5.3 $V_{\overline{\overline{4}}}(x;x^{\prime};y;y^{\prime})$ Diagrams

Each of the residual diagrams (100-103) contains a product of truncated
vertices from a single graviton coupled to the massive scalar kinetic term.
For the case of $-iV_{\overline{\overline{4a}}}$ these are,

$\displaystyle\Bigl{[}a_{y}^{D-2}\overline{\partial}^{\mu}_{y}\partial^{\nu}_{y}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}\Bigl{[}a_{y^{\prime}}^{D-2}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}\\!-\\!\tfrac{1}{4}\eta^{\rho\sigma}\widetilde{\mathcal{D}}_{y^{\prime}}\Bigr{]}=(a_{y}a_{y^{\prime}})^{D-2}\overline{\partial}^{\mu}_{y}\partial^{\nu}_{y}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}$
(157) $\displaystyle\hskip
42.67912pt-\tfrac{1}{4}a_{y}^{D-2}\overline{\partial}^{\mu}_{y}\partial^{\nu}_{y}\widetilde{\mathcal{D}}_{y^{\prime}}\eta^{\rho\sigma}-\tfrac{1}{4}a_{y^{\prime}}^{D-2}\widetilde{\mathcal{D}}_{y}\eta^{\mu\nu}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}+\tfrac{1}{16}\widetilde{\mathcal{D}}_{y}\widetilde{\mathcal{D}}_{y^{\prime}}\eta^{\mu\nu}\eta^{\rho\sigma}\;.\qquad$

With the simplification (108) these factors will be contracted into the flat
space tensor,

$\displaystyle\Bigl{[}a_{y}^{D-2}\overline{\partial}^{\mu}_{y}\partial^{\nu}_{y}\\!-\\!\tfrac{1}{4}\eta^{\mu\nu}\widetilde{\mathcal{D}}_{y}\Bigr{]}\Bigl{[}a_{y^{\prime}}^{D-2}\overline{\partial}_{y^{\prime}}^{\rho}\partial_{y^{\prime}}^{\sigma}\\!-\\!\tfrac{1}{4}\eta^{\rho\sigma}\widetilde{\mathcal{D}}_{y^{\prime}}\Bigr{]}\Bigl{[}\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{2}{D\\!-\\!2}\eta_{\mu\nu}\eta_{\rho\sigma}\Bigr{]}$
(158)
$\displaystyle=(a_{y}a_{y^{\prime}})^{D-2}\Bigl{[}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}\partial_{y}\\!\cdot\\!\partial_{y^{\prime}}+\overline{\partial}_{y}\\!\cdot\\!\partial_{y^{\prime}}\overline{\partial}_{y^{\prime}}\\!\cdot\\!\partial_{y}\Bigr{]}-\frac{2}{D\\!-\\!2}\Bigl{[}m^{2}a_{y}^{D}+\tfrac{1}{2}(\mathcal{D}_{y}\\!-\\!m^{2}a_{y}^{D})\Bigr{]}$
$\displaystyle\hskip
142.26378pt\times\Bigl{[}m^{2}a_{y^{\prime}}^{D}+\tfrac{1}{2}(\mathcal{D}_{y^{\prime}}\\!-\\!m^{2}a_{y^{\prime}}^{D})\Bigr{]}-\tfrac{1}{4}\widetilde{\mathcal{D}}_{y}\widetilde{\mathcal{D}}_{y^{\prime}}\;.\qquad$

Note that the last term can be dropped. We label the remaining terms “$A$”,
“$B$” and “$C$”, respectively.

The “$C$” terms proportional to $-2/(D-2)$ in expression (158) can already be
recognized as making 2-point, 3-point and 4-point contributions,

$\displaystyle{\rm 2\\!-\\!Point}$
$\displaystyle\\!\\!\\!\Longrightarrow\\!\\!\\!$
$\displaystyle-\tfrac{1}{2(D-2)}\\!\times\\!(\mathcal{D}_{y}\\!-\\!m^{2}a_{y}^{D})\\!\times\\!(\mathcal{D}_{y^{\prime}}\\!-\\!m^{2}a_{y^{\prime}}^{D})\;,\qquad$
(159) $\displaystyle{\rm 3\\!-\\!Point}$
$\displaystyle\\!\\!\\!\Longrightarrow\\!\\!\\!$
$\displaystyle-\tfrac{1}{D-2}\\!\times\\!(\mathcal{D}_{y}\\!-\\!m^{2}a_{y}^{D})\\!\\!\times\\!m^{2}a_{y^{\prime}}^{D}\\!-\\!\tfrac{1}{D-2}\\!\times\\!m^{2}a_{y}^{D}\\!\\!\times\\!(\mathcal{D}_{y^{\prime}}\\!-\\!m^{2}a_{y^{\prime}}^{D})\;,\qquad$
(160) $\displaystyle{\rm 4\\!-\\!Point}$
$\displaystyle\\!\\!\\!\Longrightarrow\\!\\!\\!$
$\displaystyle-\tfrac{2}{D-2}\\!\times\\!m^{2}a_{y}^{D}\\!\times\\!m^{2}a_{y^{\prime}}^{D}\;.\qquad$
(161)

To reach the four classes of 2-point forms described in Figure 2 we first
recall that each term in $-iV_{\overline{\overline{4a}}}$ shares a common
factor of,

$\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i\Delta_{C}(y;y^{\prime})\;.$
(162)

One now acts the massive scalar kinetic operators and invokes the appropriate
Donoghue Identity. For example, the 2-point contribution (159) gives

$\displaystyle-\tfrac{\kappa^{2}\lambda^{2}}{2(D-2)}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!i\delta^{D}(x\\!-\\!y)\\!\times\\!i\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!i\Delta_{A}(x;x^{\prime})i\Delta_{C}(y;y^{\prime})$
(163)
$\displaystyle=\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-i\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!\tfrac{i\kappa^{2}}{2(D-2)}i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;.\qquad$

The final factor of (163) contributes to $f_{\alpha}(x;x^{\prime})$. The
3-point contribution (160) requires the 3-point Donoghue Identity (35) after
acting the scalar kinetic operators,

$\displaystyle-\tfrac{\kappa^{2}\lambda^{2}}{D-2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!i\delta^{D}(x\\!-\\!y)\\!\times\\!m^{2}a_{y^{\prime}}^{D}\\!\times\\!i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i\Delta_{C}(y;y^{\prime})$
(165) $\displaystyle\hskip
14.22636pt-\tfrac{\kappa^{2}\lambda^{2}}{D-2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!i\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!m^{2}a_{y}^{D}\\!\times\\!i\Delta_{m}(x;y)i\Delta_{A}(x;x^{\prime})i\Delta_{C}(y;y^{\prime})$
$\displaystyle\longrightarrow-\tfrac{\kappa^{2}\lambda^{2}}{D-2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!i\delta^{D}(x\\!-\\!y)\\!\times\\!i\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;,\qquad$
$\displaystyle=\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-i\lambda^{2}(a_{x}a_{x^{\prime}})^{D}\\!\times\\!\tfrac{i\kappa^{2}}{D-2}\,i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;.\qquad$

This is a contribution to $f_{\alpha}(x;x^{\prime})$ of twice the size as
(163).

The 4-point contribution (161) has no scalar kinetic operators but it requires
the 4-point Donoghue Identity (39). The first line contains the factor,

$\Bigl{[}-\frac{i}{m^{2}}+\frac{i}{3m^{4}}\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}-m^{2}\Bigr{)}\Bigr{]}\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})}{(a_{x}a_{x^{\prime}})^{D}}\;.$
(166)

The external state derivatives can be expanded using (112),

$\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}=m^{2}-i(D\\!-\\!1)mH-\tfrac{1}{2}(D\\!-\\!1)^{2}H^{2}+\tfrac{1}{2}\Bigl{\|}\frac{\vec{k}_{1}}{a_{x}}-\frac{\vec{k}_{3}}{a_{x^{\prime}}}\Bigr{\|}^{2}+O\Bigl{(}\frac{1}{m}\Bigr{)}\;.$
(167)

It follows that the 4-point contribution (161) reduces to,

$\displaystyle-\tfrac{2m^{4}(a_{y}a_{y^{\prime}})^{D}}{D-2}\\!\times\\!\kappa^{2}\lambda^{2}(a_{x}a_{x^{\prime}})^{D}i\Delta_{m}(x;y)i\Delta_{m}(x^{\prime};y^{\prime})i\Delta_{A}(x;x^{\prime})i\Delta_{C}(y;y^{\prime})$
(168)
$\displaystyle\longrightarrow\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})\\!\times\\!-i\lambda^{2}(a_{x}a_{x^{\prime}})^{D}$
$\displaystyle\times-\tfrac{\kappa^{2}}{3(D-2)}\Bigl{[}3m^{2}+i(D\\!-\\!1)mH+\tfrac{1}{2}(D\\!-\\!1)^{2}H^{2}-\tfrac{1}{2}\|\tfrac{\vec{k}_{1}}{a_{x}}\\!-\\!\tfrac{\vec{k}_{3}}{a_{x^{\prime}}}\|^{2}\Bigr{]}$
$\displaystyle\hskip
28.45274pt\times\Biggl{\\{}\int\\!\\!d^{D}z\,a_{z}^{D}i\Delta_{A}(x;z)i\Delta_{C}(x;z)i\Delta_{A}(z;x^{\prime})+\Bigl{(}x^{\mu}\longleftrightarrow{x^{\prime}}^{\mu}\Bigr{)}\Biggr{\\}}.\qquad$

The 4-point “$C$” contribution from $-iV_{\overline{\overline{4b}}}$ is almost
the same as (168) except the square-bracketed term in the 3rd line is,

$-iV_{\overline{\overline{4b}}}\Longrightarrow
3m^{2}-i(D\\!-\\!1)mH+\tfrac{1}{2}(D\\!-\\!1)^{2}H^{2}-\tfrac{1}{2}\|\tfrac{\vec{k}_{2}}{a_{x}}\\!-\\!\tfrac{\vec{k}_{4}}{a_{x^{\prime}}}\|^{2}\;.$
(169)

The other contributions are simpler,

$-iV_{\overline{\overline{4c}}}\Longrightarrow-3m^{2}+\tfrac{1}{2}\|\tfrac{\vec{k}_{1}}{a_{x}}\\!-\\!\tfrac{\vec{k}_{4}}{a_{x^{\prime}}}\|^{2}\qquad,\qquad-
iV_{\overline{\overline{4d}}}\Longrightarrow-3m^{2}+\tfrac{1}{2}\|\tfrac{\vec{k}_{2}}{a_{x}}\\!-\\!\tfrac{\vec{k}_{3}}{a_{x^{\prime}}}\|^{2}\;.$
(170)

Adding (169) and (170) to the 3rd line of (168), and using spatial momentum
conservation (116) gives,

$-iV_{\overline{\overline{4a,b,c,d}}}\Longrightarrow(D\\!-\\!1)^{2}H^{2}-\frac{\|\vec{q}\,\|^{2}}{a_{x}a_{x^{\prime}}}\;.$
(171)

This corresponds to Group $C$ contributions to the Class $\beta$ and $\gamma$
diagrams of Figure 2,

$\displaystyle-if_{\overline{\overline{4C}}\beta}(x;x^{\prime})$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\tfrac{\kappa^{2}a_{x^{\prime}}^{D}}{3(D-2)}\Bigl{[}\tfrac{q^{2}}{a_{x}a_{x^{\prime}}}-(D\\!-\\!1)^{2}H^{2}\Bigr{]}i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;,\qquad$
(172) $\displaystyle-if_{\overline{\overline{4C}}\gamma}(x;x^{\prime})$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\tfrac{\kappa^{2}a_{x}^{D}}{3(D-2)}\Bigl{[}\tfrac{q^{2}}{a_{x}a_{x^{\prime}}}-(D\\!-\\!1)^{2}H^{2}\Bigr{]}i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;.\qquad$
(173)

In contrast, the $C$ contributions to the Class $\alpha$ diagram of Figure 2
is just four times the sum of the final factors of expressions (163) and
(165),

$-if_{\overline{\overline{4C}}\alpha}(x;x^{\prime})=\tfrac{6\kappa^{2}}{D-2}\,i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;.$
(174)

There is no contribution to the Class $\delta$ diagram.

The “$A$” and “$B$” terms in expression (158) require extensive reduction
using momentum conservation (41), the graviton reflection identity (110) and
action of the external derivatives (112). This is simplest for the two factors
of the “$B$” term
$\overline{\partial}_{y}\cdot\partial_{y^{\prime}}\times\overline{\partial}_{y^{\prime}}\cdot\partial_{y}$.
The first factor gives,

$\displaystyle\overline{\partial}_{y}\\!\cdot\\!\partial_{y^{\prime}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\overline{\partial}_{y}\\!\cdot\\!\Bigl{(}\overline{\partial}_{y^{\prime}}+\widetilde{\partial}_{y^{\prime}}+\partial_{a_{y^{\prime}}}\Bigr{)}\;,\qquad$
(175) $\displaystyle\\!\\!\\!\simeq\\!\\!\\!$
$\displaystyle-\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}+\overline{\partial}_{y}\\!\cdot\\!\Bigl{(}\widetilde{\partial}_{y}+\tfrac{1}{2}\partial_{a_{y}}-\tfrac{1}{2}\partial_{a_{y^{\prime}}}\Bigr{)}\;,\qquad$
(177) $\displaystyle\\!\\!\\!\simeq\\!\\!\\!$
$\displaystyle-\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}+\tfrac{1}{2}\Bigl{[}\partial_{y}\\!\cdot\\!(\partial_{y}\\!+\\!\partial_{a_{y}})-m^{2}a_{y}^{2}\Bigr{]}-\tfrac{1}{2}\widetilde{\partial}_{y}\\!\cdot\\!(\widetilde{\partial}_{y}\\!+\\!\partial_{a_{y}})$
$\displaystyle\hskip
184.9429pt-(\tfrac{D-2}{2})a_{y}a_{y^{\prime}}H^{2}\Delta\eta_{yy^{\prime}}\overline{\partial}_{y^{0}},\qquad$

where $\Delta\eta_{yy^{\prime}}\equiv y^{0}-y^{\prime 0}$. The second factor
produces a similar result,

$\displaystyle\overline{\partial}_{y^{\prime}}\\!\cdot\\!\partial_{y}\simeq-\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}+\tfrac{1}{2}\Bigl{[}\partial_{y^{\prime}}\\!\cdot\\!(\partial_{y^{\prime}}\\!+\\!\partial_{a_{y^{\prime}}})-m^{2}a_{y^{\prime}}^{2}\Bigr{]}-\tfrac{1}{2}\widetilde{\partial}_{y^{\prime}}\\!\cdot\\!(\widetilde{\partial}_{y^{\prime}}\\!+\\!\partial_{a_{y^{\prime}}})$
(178) $\displaystyle\hskip
213.39566pt+(\tfrac{D-2}{2})a_{y}a_{y^{\prime}}H^{2}\Delta\eta_{yy^{\prime}}\overline{\partial}_{{y^{\prime}}^{0}}.\qquad$

If we use
$\widetilde{\mathcal{D}}_{y}i\Delta_{C}(y;y^{\prime})=\widetilde{\mathcal{D}}_{y^{\prime}}i\Delta_{C}(y;y^{\prime})\longrightarrow
2(D-3)i\Delta_{C}(y;y^{\prime})$ the “$B$” term in (158) gives,

$\displaystyle(a_{y}a_{y^{\prime}})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\partial_{y^{\prime}}\overline{\partial}_{y^{\prime}}\\!\cdot\\!\partial_{y}\longrightarrow\Biggl{\\{}a_{y}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}+(\tfrac{D-2}{2})a^{D-1}_{y}a_{y^{\prime}}H^{2}\Delta\eta_{yy^{\prime}}\overline{\partial}_{y^{0}}$
(179)
$\displaystyle+(D\\!-\\!3)a_{y}^{D}H^{2}\\!-\\!\tfrac{1}{2}\Bigl{[}\mathcal{D}_{y}\\!-\\!m^{2}a_{y}^{D}\Bigr{]}\Biggr{\\}}\Biggl{\\{}a_{y^{\prime}}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}\\!-\\!(\tfrac{D-2}{2})a_{y}a^{D-1}_{y^{\prime}}H^{2}\Delta\eta_{yy^{\prime}}\overline{\partial}_{{y^{\prime}}^{0}}$
$\displaystyle\hskip
170.71652pt+(D\\!-\\!3)a_{y^{\prime}}^{D}H^{2}\\!-\\!\tfrac{1}{2}\Bigl{[}\mathcal{D}_{y^{\prime}}\\!-\\!m^{2}a_{y^{\prime}}^{D}\Bigr{]}\Biggr{\\}}.\qquad$

The 2-point contribution from (179) is easy to read off,

${\rm
2\\!-\\!Point}\Longrightarrow\tfrac{1}{4}\\!\times\\!(\mathcal{D}_{y}\\!-\\!m^{2}a_{y}^{D})\\!\times\\!(\mathcal{D}_{y^{\prime}}\\!-\\!m^{2}a_{y^{\prime}}^{D})\;.$
(180)

This differs from the “$C$” contribution (159) only by the replacement of
$-\frac{1}{2(D-2)}$ by $\frac{1}{4}$, so we can immediately conclude,

$-if_{\overline{\overline{4B}}\alpha}(x;x^{\prime})=-\kappa^{2}i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;.$
(181)

Because the 3-point Donoghue Identity (35) goes like $1/m^{2}$, which can only
be compensated by two external derivatives. For
$-iV_{\overline{\overline{4a}}}$ the only 3-point contribution which survives
for large $m$ is,

$\overline{\overline{4aB}}\Longrightarrow-\tfrac{1}{2}a_{y^{\prime}}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}(\mathcal{D}_{y}\\!-\\!m^{2}a_{y}^{D})-\tfrac{1}{2}a_{y}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}(\mathcal{D}_{y^{\prime}}\\!-\\!m^{2}a_{y^{\prime}}^{D})\;.$
(182)

The other three variations give,

$\displaystyle\overline{\overline{4bB}}$
$\displaystyle\\!\\!\\!\Longrightarrow\\!\\!\\!$
$\displaystyle-\tfrac{1}{2}a_{x^{\prime}}^{D-2}\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}(\mathcal{D}_{x}\\!-\\!m^{2}a_{x}^{D})-\tfrac{1}{2}a_{x}^{D-2}\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}(\mathcal{D}_{x^{\prime}}\\!-\\!m^{2}a_{x^{\prime}}^{D})\;,\qquad$
(183) $\displaystyle\overline{\overline{4cB}}$
$\displaystyle\\!\\!\\!\Longrightarrow\\!\\!\\!$
$\displaystyle-\tfrac{1}{2}a_{y^{\prime}}^{D-2}\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}(\mathcal{D}_{x}\\!-\\!m^{2}a_{x}^{D})-\tfrac{1}{2}a_{x}^{D-2}\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}(\mathcal{D}_{y^{\prime}}\\!-\\!m^{2}a_{y^{\prime}}^{D})\;,\qquad$
(184) $\displaystyle\overline{\overline{4dB}}$
$\displaystyle\\!\\!\\!\Longrightarrow\\!\\!\\!$
$\displaystyle-\tfrac{1}{2}a_{x^{\prime}}^{D-2}\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}(\mathcal{D}_{y}\\!-\\!m^{2}a_{y}^{D})-\tfrac{1}{2}a_{y}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{x^{\prime}}(\mathcal{D}_{x^{\prime}}\\!-\\!m^{2}a_{x^{\prime}}^{D})\;,\qquad$
(185)

Because the external derivatives in (182-183) are either both incoming or
outgoing they cancel against the mixed incoming-outgoing derivatives of
(184-185) when account is taken of the factors of
$\delta^{D}(x-y)\delta^{D}(x^{\prime}-y^{\prime})$,

$\displaystyle\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}\longrightarrow+m^{2}a_{y}a_{y^{\prime}}+O(m)$
,
$\displaystyle\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}\longrightarrow+m^{2}a_{x}a_{x^{\prime}}+O(m)\;,\qquad$
(186)
$\displaystyle\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}\longrightarrow-m^{2}a_{x}a_{y^{\prime}}+O(m)$
,
$\displaystyle\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{x^{\prime}}\longrightarrow-m^{2}a_{y}a_{x^{\prime}}+O(m)\;.\qquad$
(187)

The 4-point “$B$” contributions are more complicated because they can involve
as many as four external derivatives, and because the 4-point Donoghue
Identities (39-40) contain factors of both $1/m^{2}$ and $1/m^{4}$, as well as
up to two external derivatives. It is therefore best to make a preliminary
reduction based on each external derivative potentially contributing a factor
of $m$,

$\displaystyle{\rm
4\\!-\\!Point}\Longrightarrow(a_{y}a_{y^{\prime}})^{D}\Biggl{\\{}\\!\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}^{2}\\!\\!+H^{2}\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}\Bigl{[}(\tfrac{D-2}{2})\Delta\eta_{yy^{\prime}}(\overline{\partial}_{y^{0}}\\!-\\!\overline{\partial}_{{y^{\prime}}^{0}})$
(188) $\displaystyle\hskip
110.96556pt+(D\\!-\\!3)(\tfrac{a_{y}}{a_{y^{\prime}}}\\!+\\!\tfrac{a_{y^{\prime}}}{a_{y}})\Bigr{]}\\!-\\!(\tfrac{D-2}{2})^{2}H^{4}\Delta\eta^{2}_{yy^{\prime}}\overline{\partial}_{y^{0}}\overline{\partial}_{{y^{\prime}}^{0}}\\!\Biggr{\\}}.\qquad$

The appropriate factors for the three other permutations are given by making
the following coordinate replacements in (188),

$\displaystyle-iV_{\overline{\overline{4b}}}$ $\displaystyle\Longrightarrow$
$\displaystyle x^{\mu}\longleftrightarrow y^{\mu}\qquad{\rm
and}\qquad{x^{\prime}}^{\mu}\longleftrightarrow{y^{\prime}}^{\mu}\;,$ (189)
$\displaystyle-iV_{\overline{\overline{4c}}}$ $\displaystyle\Longrightarrow$
$\displaystyle{x^{\prime}}^{\mu}\longleftrightarrow{y^{\prime}}^{\mu}\;,$
(190) $\displaystyle-iV_{\overline{\overline{4d}}}$
$\displaystyle\Longrightarrow$ $\displaystyle x^{\mu}\longleftrightarrow
y^{\mu}\;.$ (191)

Each of these factors must be multiplied into a factor from the appropriate
4-point Donoghue Identity,

$\displaystyle-iV_{\overline{\overline{4a}}}$ $\displaystyle\Longrightarrow$
$\displaystyle\Bigl{[}-\frac{i}{m^{2}}+\frac{i}{3m^{4}}\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}-m^{2}\Bigr{)}\Bigr{]}\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})}{(a_{x}a_{x^{\prime}})^{D}}\;,$
(192) $\displaystyle-iV_{\overline{\overline{4b}}}$
$\displaystyle\Longrightarrow$
$\displaystyle\Bigl{[}-\frac{i}{m^{2}}+\frac{i}{3m^{4}}\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}-m^{2}\Bigr{)}\Bigr{]}\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})}{(a_{y}a_{y^{\prime}})^{D}}\;,$
(193) $\displaystyle-iV_{\overline{\overline{4c}}}$
$\displaystyle\Longrightarrow$
$\displaystyle\Bigl{[}\frac{i}{m^{2}}+\frac{i}{3m^{4}}\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{y}a_{x^{\prime}}}+m^{2}\Bigr{)}\Bigr{]}\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})}{(a_{y}a_{x^{\prime}})^{D}}\;,$
(194) $\displaystyle-iV_{\overline{\overline{4d}}}$
$\displaystyle\Longrightarrow$
$\displaystyle\Bigl{[}\frac{i}{m^{2}}+\frac{i}{3m^{4}}\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{x}a_{y^{\prime}}}+m^{2}\Bigr{)}\Bigr{]}\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x^{\prime}\\!-\\!y^{\prime})}{(a_{x}a_{y^{\prime}})^{D}}\;.$
(195)

Then all four permutations are summed and one discards terms which vanish in
the large $m$ limit.

It is best to proceed sequentially with each of the four terms in (188), and
its permutations, times the two terms from expressions (192-195). The product
of the first factors gives,

$\displaystyle\frac{i}{m^{2}}\Biggl{\\{}-\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}^{2}-\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}\Bigr{)}^{2}+\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{x}a_{y^{\prime}}}\Bigr{)}^{2}+\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{y}a_{x^{\prime}}}\Bigr{)}^{2}\Biggr{\\}}$
(196) $\displaystyle\hskip 199.16928pt\longrightarrow
i\Bigl{[}4(D\\!-\\!2)^{2}H^{2}-\tfrac{2q^{2}}{a_{x}a_{x^{\prime}}}\Bigr{]}.\qquad$

The product of the first factor from (188) and its permutations with the final
factor of (192-195) is,

$\displaystyle\frac{i}{3m^{4}}\Biggl{\\{}\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}^{2}\Bigl{[}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}\\!-\\!m^{2}\Bigr{]}+\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}\Bigr{)}^{2}\Bigl{[}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\\!-\\!m^{2}\Bigr{]}$
(197) $\displaystyle\hskip
85.35826pt+\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{x}a_{y^{\prime}}}\Bigr{)}^{2}\Bigl{[}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{y}a_{x^{\prime}}}\\!+\\!m^{2}\Bigr{]}+\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{y}a_{x^{\prime}}}\Bigr{)}^{2}\Bigl{[}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{x}a_{y^{\prime}}}\\!+\\!m^{2}\Bigr{]}\Biggr{\\}}$
$\displaystyle\longrightarrow\frac{i}{3}\Bigl{[}4(D\\!-\\!1)(D\\!-\\!2)H^{2}-(D\\!-\\!1)^{2}H^{2}+\tfrac{q^{2}}{a_{x}a_{x^{\prime}}}\Bigr{]}\;.\qquad$

The product of the second factor of (188) and its permutations with the first
factor of (192-195) is,

$\displaystyle
i(\tfrac{D-2}{2})\frac{H^{2}}{m^{2}}\Biggl{\\{}-\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}\Delta\eta_{yy^{\prime}}(\overline{\partial}_{y^{0}}\\!-\\!\overline{\partial}_{{y^{\prime}}^{0}})-\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}\Bigr{)}\Delta\eta_{xx^{\prime}}(\overline{\partial}_{x^{0}}\\!-\\!\overline{\partial}_{{x^{\prime}}^{0}})$
(198) $\displaystyle\hskip
42.67912pt+\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{y}a_{x^{\prime}}}\Bigr{)}\Delta\eta_{yx^{\prime}}(\overline{\partial}_{y^{0}}\\!-\\!\overline{\partial}_{{x^{\prime}}^{0}})+\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{x}a_{y^{\prime}}}\Bigr{)}\Delta\eta_{xy^{\prime}}(\overline{\partial}_{x^{0}}\\!-\\!\overline{\partial}_{{y^{\prime}}^{0}})\Biggr{\\}}$
$\displaystyle\hskip 156.49014pt\longrightarrow
i\tfrac{3}{2}(D\\!-\\!1)(D\\!-\\!2)a_{x}a_{x^{\prime}}H^{4}\Delta\eta^{2}_{xx^{\prime}}\;.\qquad$

For the second factor of (188) times the second factor of (192-195) is is only
the $a$ and $b$ permutations which survive the large $m$ limit,

$\displaystyle
i(\tfrac{D-2}{2})\frac{H^{2}}{3m^{4}}\Biggl{\\{}\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}\Delta\eta_{yy^{\prime}}(\overline{\partial}_{y^{0}}\\!-\\!\overline{\partial}_{{y^{\prime}}^{0}})\Bigl{[}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}\\!-\\!m^{2}\Bigr{]}$
(199) $\displaystyle\hskip
85.35826pt+\Bigl{(}\frac{\overline{\partial}_{x}\\!\cdot\\!\overline{\partial}_{x^{\prime}}}{a_{x}a_{x^{\prime}}}\Bigr{)}\Delta\eta_{xx^{\prime}}(\overline{\partial}_{x^{0}}\\!-\\!\overline{\partial}_{{x^{\prime}}^{0}})\Bigl{[}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\\!-\\!m^{2}\Bigr{]}\Biggr{\\}}$
$\displaystyle\hskip
156.49014pt\longrightarrow-\tfrac{2}{3}i(D\\!-\\!1)(D\\!-\\!2)a_{x}a_{x^{\prime}}H^{4}\Delta\eta_{xx^{\prime}}^{2}.\qquad$

The 3rd and 4th factors of (188), and their permutations, contain only two
external derivatives so only the first factor of (192-195) survives the large
$m$ limit,

$\displaystyle\frac{iH^{2}}{m^{2}}\Biggl{\\{}-\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}(D\\!-\\!3)(\tfrac{a_{y}}{a_{y^{\prime}}}\\!+\\!\tfrac{a_{y^{\prime}}}{a_{y}})-(\tfrac{D-2}{2})^{2}\overline{\partial}_{y^{0}}\overline{\partial}_{{y^{\prime}}^{0}}H^{2}\Delta\eta_{yy^{\prime}}^{2}+\ldots\Biggr{\\}}$
(201) $\displaystyle\hskip 68.28644pt\longrightarrow
i4H^{2}\Bigl{[}-(D\\!-\\!3)(\tfrac{a_{x}}{a_{x^{\prime}}}\\!+\\!\tfrac{a_{x^{\prime}}}{a_{x}})+(\tfrac{D-2}{2})^{2}a_{x}a_{x^{\prime}}H^{2}\Delta\eta_{xx^{\prime}}^{2}\Bigr{]},\qquad$
$\displaystyle=i4H^{2}\Bigl{[}-2(D\\!-\\!3)+\tfrac{1}{4}(D^{2}\\!-\\!8D\\!+\\!28)a_{x}a_{x^{\prime}}H^{2}\Delta\eta_{xx^{\prime}}^{2}\Bigr{]}.\qquad$

The factors of $a_{x}a_{x^{\prime}}H^{2}\Delta\eta_{xx^{\prime}}^{2}$ in
expressions (198), (199) and (201) take us beyond the 4-point Donoghue
Identities (39-40). These factors must actually be internal to the $d^{D}z$
integrations on the last lines of expressions (39-40),

$\displaystyle\int\\!\\!d^{D}z\,a_{z}^{D}\\!\times\\!a_{x}a_{z}H^{2}\Delta\eta_{xz}^{2}\\!\times\\!i\Delta_{A}(x;z)i\Delta_{C}(x;z)i\Delta_{A}(z;x^{\prime})$
(202) $\displaystyle\hskip
39.83368pt+\int\\!\\!d^{D}z\,a_{z}^{D}i\Delta_{A}(x;z)\\!\times\\!a_{z}a_{x^{\prime}}H^{2}\Delta\eta_{zx^{\prime}}^{2}\\!\times\\!i\Delta_{A}(z;x^{\prime})i\Delta_{C}(z;x^{\prime}).\qquad$

The same comments apply to the factors of $q^{2}/a_{x}a_{x^{\prime}}$ which
appear in expressions (196) and (197). These factors must give internal
derivatives on the last lines of (39-40),

$\displaystyle\int\\!\\!d^{D}z\,\mathcal{D}_{z}\Bigl{[}i\Delta_{A}(x;z)i\Delta_{C}(x;z)\Bigr{]}\\!\times\\!i\Delta_{A}(z;x^{\prime})$
(203) $\displaystyle\hskip
113.81102pt+\int\\!\\!d^{D}z\,i\Delta_{A}(x;z)\\!\times\\!\mathcal{D}_{z}\Bigl{[}i\Delta_{A}(z;x^{\prime})i\Delta_{C}(z;x^{\prime})\Bigr{]}.\qquad$

With these understandings we can sum the “$B$” contributions from expressions
(196), (197), (198), (199) and (201) to give,

$\displaystyle-if_{\overline{\overline{4B}}\beta}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\kappa^{2}\Bigl{[}-\tfrac{5}{6}\mathcal{D}_{x^{\prime}}+F_{1}(D)a_{x^{\prime}}^{D}H^{2}$
(204) $\displaystyle\hskip
42.67912pt+F_{2}(D)a_{x}a^{D+1}_{x^{\prime}}H^{4}\Delta\eta_{xx^{\prime}}^{2}\Bigr{]}i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime}),\qquad$
$\displaystyle-if_{\overline{\overline{4B}}\gamma}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\kappa^{2}\Bigl{[}-\tfrac{5}{6}\mathcal{D}_{x}+F_{1}(D)a_{x}^{D}H^{2}$
(205) $\displaystyle\hskip
42.67912pt+F_{2}(D)a^{D+1}_{x}a_{x^{\prime}}H^{4}\Delta\eta_{xx^{\prime}}^{2}\Bigr{]}i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime}).\qquad$

The functions $F_{1}(D)$ and $F_{2}(D)$ are,

$\displaystyle F_{1}(D)$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle
2(D\\!-\\!2)^{2}+\tfrac{2}{3}(D\\!-\\!1)(D\\!-\\!2)-\tfrac{1}{6}(D\\!-\\!1)^{2}-4(D\\!-\\!3)\;,\qquad$
(206) $\displaystyle F_{2}(D)$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\tfrac{1}{8}(D\\!-\\!2)^{2}+\tfrac{5}{12}(D\\!-\\!1)(D\\!-\\!2)+D\\!-\\!3\;.$
(207)

The “$A$” term of (158) seems simpler than “$B$” because its first factor of
$\overline{\partial}_{y}\cdot\overline{\partial}_{y^{\prime}}$ is already
reduced. However, the reduction of the second factor is complex,

$\displaystyle\partial_{y}\\!\cdot\\!\partial_{y^{\prime}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{1}{2}(\partial_{y}\\!+\\!\partial_{y^{\prime}})^{2}-\tfrac{1}{2}\partial_{y}^{2}-\tfrac{1}{2}\partial_{y^{\prime}}^{2}\;,$
(208) $\displaystyle\\!\\!\\!\simeq\\!\\!\\!$
$\displaystyle\tfrac{1}{2}\Bigl{[}\overline{\partial}_{y}\\!+\\!\tfrac{1}{2}\partial_{a_{y}}\\!+\\!\overline{\partial}_{y^{\prime}}\\!+\\!\tfrac{1}{2}\partial_{a_{y^{\prime}}}\Bigr{]}^{2}-\tfrac{1}{2}\partial_{y}^{2}-\tfrac{1}{2}\partial_{y^{\prime}}^{2}\;,$
(210) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}-\tfrac{1}{2}\Bigl{[}\partial_{y}\\!\cdot\\!(\partial_{y}\\!+\\!\partial_{a_{y}})-m^{2}a_{y}^{2}\Bigr{]}-\tfrac{1}{2}\Bigl{[}\partial_{y^{\prime}}\\!\cdot\\!(\partial_{y^{\prime}}\\!+\\!\partial_{a_{y^{\prime}}})-m^{2}a_{y^{\prime}}^{2}\Bigr{]}$
$\displaystyle\hskip
56.9055pt+\tfrac{1}{2}(\overline{\partial}_{y^{\prime}}\\!+\\!\partial_{y})\\!\cdot\\!\partial_{a_{y}}+\tfrac{1}{2}(\overline{\partial}_{y}\\!+\\!\partial_{y^{\prime}})\\!\cdot\\!\partial_{a_{y^{\prime}}}+\tfrac{1}{8}(\partial_{a_{y}}\\!+\\!\partial_{a_{y^{\prime}}})^{2}.\qquad$

The terms on the last line can be usefully rearranged using momentum
conservation,

$\displaystyle\tfrac{1}{2}(\overline{\partial}_{y^{\prime}}\\!+\\!\partial_{y})\\!\cdot\\!\partial_{a_{y}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{1}{2}(\overline{\partial}_{y^{\prime}}\\!-\\!\overline{\partial}_{y})\\!\cdot\\!\partial_{a_{y}}-\tfrac{1}{2}(\widetilde{\partial}_{y}\\!+\\!\tfrac{1}{2}\partial_{a_{y}})\\!\cdot\\!\partial_{a_{y}}-\tfrac{1}{4}\partial^{2}_{a_{y}}\;,\qquad$
(211)
$\displaystyle\tfrac{1}{2}(\overline{\partial}_{y}\\!+\\!\partial_{y^{\prime}})\\!\cdot\\!\partial_{a_{y^{\prime}}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{1}{2}(\overline{\partial}_{y}\\!-\\!\overline{\partial}_{y^{\prime}})\\!\cdot\\!\partial_{a_{y^{\prime}}}-\tfrac{1}{2}(\widetilde{\partial}_{y^{\prime}}\\!+\\!\tfrac{1}{2}\partial_{a_{y^{\prime}}})\\!\cdot\\!\partial_{a_{y^{\prime}}}-\tfrac{1}{4}\partial^{2}_{a_{y^{\prime}}}\;.\qquad$
(212)

Substituting (211-212) in (210) and using the reflection identity gives,

$\displaystyle\partial_{y}\\!\cdot\\!\partial_{y^{\prime}}\simeq\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}-\tfrac{1}{2}\Bigl{[}\partial_{y}\\!\cdot\\!(\partial_{y}\\!+\\!\partial_{a_{y}})-m^{2}a_{y}^{2}\Bigr{]}-\tfrac{1}{2}\Bigl{[}\partial_{y^{\prime}}\\!\cdot\\!(\partial_{y^{\prime}}\\!+\\!\partial_{a_{y^{\prime}}})-m^{2}a_{y^{\prime}}^{2}\Bigr{]}$
(213)
$\displaystyle-\tfrac{1}{2}(\overline{\partial}_{y}\\!-\\!\overline{\partial}_{y^{\prime}})\\!\cdot\\!(\partial_{a_{y}}\\!-\\!\partial_{a_{y^{\prime}}})-\tfrac{1}{2}(\widetilde{\partial}_{y}\\!+\\!\tfrac{1}{2}\partial_{a_{y}})\\!\cdot\\!(\partial_{a_{y}}\\!-\\!\partial_{a_{y^{\prime}}})-\tfrac{1}{8}(\partial_{a_{y}}\\!-\\!\partial_{a_{y^{\prime}}})^{2}\;.\qquad$

Expression (213) can be further simplified using the relation,

$\partial_{a_{y}}^{\mu}\\!-\\!\partial_{a_{y^{\prime}}}^{\mu}=-(D\\!-\\!2)a_{y}a_{y^{\prime}}H^{2}\Delta\eta_{yy^{\prime}}^{2}\;.$
(214)

Multiplying by
$(a_{y}a_{y^{\prime}})^{D-2}\overline{\partial}_{y}\cdot\overline{\partial}_{y^{\prime}}$
gives the total “$A$” factor,

$\displaystyle(a_{y}a_{y^{\prime}})^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}\,\partial_{y}\\!\cdot\\!\partial_{y^{\prime}}\simeq(a_{y}a_{y^{\prime}})^{D}\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}^{2}-\tfrac{1}{2}a_{y^{\prime}}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}\Bigl{[}\widetilde{D}_{y}\\!-\\!m^{2}a_{y}^{D}\Bigr{]}$
(215)
$\displaystyle-\tfrac{1}{2}a_{y}^{D-2}\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}\Bigl{[}\widetilde{D}_{y^{\prime}}\\!-\\!m^{2}a_{y^{\prime}}^{D}\Bigr{]}+(\tfrac{D-2}{2})H^{2}(a_{y}a_{y^{\prime}})^{D}\Bigl{(}\frac{\overline{\partial}_{y}\\!\cdot\\!\overline{\partial}_{y^{\prime}}}{a_{y}a_{y^{\prime}}}\Bigr{)}\Biggl{\\{}\Delta\eta_{yy^{\prime}}(\overline{\partial}_{y^{0}}\\!-\\!\overline{\partial}_{{y^{\prime}}^{0}})$
$\displaystyle\hskip
71.13188pt+\Delta\eta_{yy^{\prime}}\Bigl{[}\widetilde{\partial}_{y^{0}}\\!+\\!(\tfrac{D-2}{2})a_{y}H\Bigr{]}\\!+\\!\tfrac{1}{2}\\!+\\!(\tfrac{D-1}{4})a_{y}a_{y^{\prime}}H^{2}\Delta\eta_{yy^{\prime}}^{2}\Biggr{\\}}.\qquad$

There are no 2-point contributions and, like the “$B$” case, the 3-point
contributions cancel. Most of the 4-point contributions are similar to those
of (188) and can be read off from expressions (196), (197), (198), (199) and
(201). The only exception is penultimate term, which involves a derivative
$\widetilde{\partial}_{y^{0}}+(\tfrac{D-2}{2})a_{y}H\equiv\widetilde{D}_{y}$
of the $C$-type propagator. We understand this as acting on the internal
propagator in the same sense as the scale factors. The final result is,

$\displaystyle-
if_{\overline{\overline{4A}}\beta}(x;x^{\prime})=\kappa^{2}\Bigl{[}-\tfrac{5}{6}\mathcal{D}_{x}+F_{3}(D)a_{x^{\prime}}^{D}H^{2}+F_{4}(D)a_{x}a^{D+1}_{x^{\prime}}H^{4}\Delta\eta_{xx^{\prime}}^{2}\Bigr{]}$
(216) $\displaystyle\times
i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})+(D\\!-\\!2)\kappa^{2}H^{2}a_{x^{\prime}}^{D}i\Delta_{A}(x;x^{\prime})\Delta\eta_{xx^{\prime}}\widetilde{D}_{x}i\Delta_{C}(x;x^{\prime}),\qquad$
$\displaystyle-
if_{\overline{\overline{4A}}\gamma}(x;x^{\prime})=\kappa^{2}\Bigl{[}-\tfrac{5}{6}\mathcal{D}_{x}+F_{4}(D)a_{x}^{D}H^{2}+F_{4}(D)a^{D+1}_{x}a_{x^{\prime}}H^{4}\Delta\eta_{xx^{\prime}}^{2}\Bigr{]}$
(217) $\displaystyle\times
i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})+(D\\!-\\!2)\kappa^{2}H^{2}a_{x}^{D}i\Delta_{A}(x;x^{\prime})\Delta\eta_{xx^{\prime}}\widetilde{D}_{x}i\Delta_{C}(x;x^{\prime}).\qquad$

The functions $F_{3}(D)$ and $F_{4}(D)$ are,

$\displaystyle F_{3}(D)$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle
2(D\\!-\\!2)^{2}+\tfrac{2}{3}(D\\!-\\!1)(D\\!-\\!2)-\tfrac{1}{6}(D\\!-\\!1)^{2}-\tfrac{1}{2}(D\\!-\\!2)\;,\qquad$
(218) $\displaystyle F_{4}(D)$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\tfrac{2}{3}(D\\!-\\!1)(D\\!-\\!2)\;.$ (219)

It is time to sum up all the contributions from
$-iV_{\overline{\overline{4}}}$ to the four classes of 2-point forms in Figure
2. The ones of Class $\alpha$ come from expressions (174) and (181),

$-if_{\overline{\overline{4}}\alpha}(x;x^{\prime})=-(\tfrac{D-8}{D-2})\kappa^{2}i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;.$
(220)

Class $\beta$ diagrams derive from expressions (172), (204) and (216),

$\displaystyle-
if_{\overline{\overline{4}}\beta}(x;x^{\prime})=\kappa^{2}\Bigl{[}-\tfrac{(5D-11)}{3(D-2)}\mathcal{D}_{x^{\prime}}+G_{1}(D)a_{x^{\prime}}^{D}H^{2}+G_{2}(D)a_{x}a^{D+1}_{x^{\prime}}H^{4}\Delta\eta_{xx^{\prime}}^{2}\Bigr{]}$
(221) $\displaystyle\times
i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})+(D\\!-\\!2)\kappa^{2}H^{2}a_{x^{\prime}}^{D}i\Delta_{A}(x;x^{\prime})\Delta\eta_{xx^{\prime}}\widetilde{D}_{x}i\Delta_{C}(x;x^{\prime}),\qquad$

where the functions $G_{1}(D)$ and $G_{2}(D)$ are,

$\displaystyle G_{1}(D)$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle-\tfrac{(D-1)^{2}}{3(D-2)}+4(D-2)^{2}+\tfrac{4}{3}(D\\!-\\!1)(D\\!-\\!2)$
(222) $\displaystyle\hskip
128.0374pt-\tfrac{1}{3}(D\\!-\\!1)^{2}-\tfrac{1}{2}(D\\!-\\!2)-4(D\\!-\\!3)\;,\qquad$
$\displaystyle G_{2}(D)$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\tfrac{1}{8}(D\\!-\\!2)^{2}+\tfrac{13}{12}(D\\!-\\!1)(D\\!-\\!2)+D\\!-\\!3\;.\qquad$
(223)

Expressions (173), (205) and (217) give the Class $\gamma$ diagrams,

$\displaystyle-
if_{\overline{\overline{4}}\gamma}(x;x^{\prime})=\kappa^{2}\Bigl{[}-\tfrac{(5D-11)}{3(D-2)}\mathcal{D}_{x}+G_{1}(D)a_{x}^{D}H^{2}+G_{2}(D)a^{D+1}_{x}a_{x^{\prime}}H^{4}\Delta\eta_{xx^{\prime}}^{2}\Bigr{]}$
(224) $\displaystyle\times
i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})+(D\\!-\\!2)\kappa^{2}H^{2}a_{x}^{D}i\Delta_{A}(x;x^{\prime})\Delta\eta_{xx^{\prime}}\widetilde{D}_{x}i\Delta_{C}(x;x^{\prime}),\qquad$

There are no Class $\delta$ diagrams from $-iV_{\overline{\overline{4}}}$.

###  5.4 Renormalization

Recall the four classes of diagrams described in Figure 2 which contribute to
the gauge invariant self-mass,

$M^{2}_{\rm
inv}(x;x^{\prime})=\mathcal{D}_{x}\mathcal{D}_{x^{\prime}}f_{\alpha}(x;x^{\prime})+\mathcal{D}_{x}f_{\beta}(x;x^{\prime})+\mathcal{D}_{x^{\prime}}f_{\gamma}(x;x^{\prime})+f_{\delta}(x;x^{\prime})\;.$
(225)

The only contribution to Class $\delta$ is from the original, gauge-dependent
result $-if_{\delta}(x;x^{\prime})=-iM_{0}(x;x^{\prime})$. In section 5.3 we
derived the $-iV_{\overline{\overline{4}}}$ contributions to Classes $\alpha$,
$\beta$ and $\gamma$ in expressions (220), (221) and (224), respectively.
Section 5.2 gives the Class $\beta$ and $\gamma$ contributions from
$-iV_{\overline{5}}$ in expressions (155) and (156), respectively. (There are
no Class $\alpha$ contributions from $-iV_{\overline{5}}$.) Finally, there are
Class $\beta$ and $\gamma$ contributions from $-iV_{1b}(x;x^{\prime})$ and
$-iV_{1c}(x;x^{\prime})$, respectively,

$\displaystyle-if_{1\beta}(x;x^{\prime})$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle
i\kappa^{2}\Bigl{[}\tfrac{D}{2}(\tfrac{D-1}{2})i\Delta_{A}+(\tfrac{D-1}{2})i\Delta_{B}+\tfrac{1}{2}i\Delta_{C}\Bigr{]}\delta^{D}(x\\!-\\!x^{\prime})\;,\qquad$
(226) $\displaystyle-if_{1\gamma}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
i\kappa^{2}\Bigl{[}\tfrac{D}{2}(\tfrac{D-1}{2})i\Delta_{A}+(\tfrac{D-1}{2})i\Delta_{B}+\tfrac{1}{2}i\Delta_{C}\Bigr{]}\delta^{D}(x\\!-\\!x^{\prime})\;.\qquad$
(227)

To carry out renormalization we must first localize the primitive divergences
using expressions (7-9) for the three propagators. The simplest case is the
coincidence limits which appear in (226-227),

$\displaystyle\tfrac{D}{2}(\tfrac{D-1}{2})i\Delta_{A}+(\tfrac{D-1}{2})i\Delta_{B}+\tfrac{1}{2}i\Delta_{C}=-\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\Bigl{[}\pi{\rm
cot}(\tfrac{D\pi}{2})\\!-\\!2\ln(a)\Bigr{]}$ (228) $\displaystyle\hskip
42.67912pt\times\tfrac{D}{2}(\tfrac{D-1}{2})-\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!2)}{\Gamma(\frac{D}{2})}\\!\times\\!(\tfrac{D-1}{2})+\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!3)}{\Gamma(\frac{D}{2})}\\!\times\\!\tfrac{1}{2}\;.\qquad$

There are also divergences which appear in two products of noncoincident
propagators,

$i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\qquad,\qquad
i\Delta_{A}(x;x^{\prime})\Delta\eta_{xx^{\prime}}\widetilde{D}_{x}i\Delta_{C}(x;x^{\prime})\;,$
(229)

where we recall,

$\widetilde{D}_{x}\equiv\frac{\partial}{\partial
x^{0}}+(\tfrac{D-2}{2})a_{x}H\;.$ (230)

There are no divergences in the product,

$a_{x}a_{x^{\prime}}H^{2}\Delta\eta_{xx^{\prime}}^{2}i\Delta_{A}(x;x^{\prime})i\Delta_{C}(x;x^{\prime})\;.$
(231)

The two products are reduced by first expanding, retaining only the
potentially divergent terms in $D$ dimensions and setting $D=4$ for the
others,

$i\Delta_{A}i\Delta_{C}=\frac{\Gamma^{2}(\frac{D}{2}\\!-\\!1)}{16\pi^{D}}\Biggl{\\{}\frac{1}{[aa^{\prime}\Delta
x^{2}]^{D-2}}-\frac{H^{2}\ln(\frac{1}{4}H^{2}\Delta x^{2})}{2aa^{\prime}\Delta
x^{2}}+O(D\\!-\\!4)\Biggr{\\}}.$ (232)

The next step is to localize the ultraviolet divergence by extracting a
derivative and then adding zero in the form of the flat space propagator
equation [48, 49],

$\displaystyle\frac{1}{\Delta
x^{2D-4}}=\frac{\partial^{2}}{2(D\\!-\\!3)(D\\!-\\!4)}\Biggl{[}\frac{1}{\Delta
x^{2D-6}}\Biggr{]}\;,$ (235)
$\displaystyle=\frac{\mu^{D-4}}{2(D\\!-\\!3)(D\\!-\\!4)}\frac{4\pi^{\frac{D}{2}}i\delta^{D}(x\\!-\\!x^{\prime})}{\Gamma(\frac{D}{2}\\!-\\!1)}$
$\displaystyle\hskip
142.26378pt+\frac{\partial^{2}}{2(D\\!-\\!3)(D\\!-\\!4)}\Biggl{[}\frac{1}{\Delta
x^{2D-6}}-\frac{\mu^{D-4}}{\Delta x^{D-2}}\Biggr{]},\qquad$
$\displaystyle=\frac{\mu^{D-4}}{2(D\\!-\\!3)(D\\!-\\!4)}\frac{4\pi^{\frac{D}{2}}i\delta^{D}(x\\!-\\!x^{\prime})}{\Gamma(\frac{D}{2}\\!-\\!1)}-\frac{\partial^{2}}{4}\Biggl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Biggr{]}+O(D\\!-\\!4)\;.\qquad$

The final result for the first product is,

$\displaystyle
i\Delta_{A}i\Delta_{C}\longrightarrow\frac{\mu^{D-4}\Gamma(\frac{D}{2}\\!-\\!1)}{8(D\\!-\\!3)(D\\!-\\!4)\pi^{\frac{D}{2}}}\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{(aa^{\prime})^{D-2}}$
(236) $\displaystyle\hskip
113.81102pt-\frac{\partial^{2}}{64\pi^{4}(aa^{\prime})^{2}}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Bigr{]}-\frac{H^{2}\ln(\frac{1}{4}H^{2}\Delta
x^{2})}{32\pi^{4}aa^{\prime}\Delta x^{2}}.\qquad$

Using expression (230) in the second product of (229) gives,

$\displaystyle
i\Delta_{A}\\!\\!\times\\!\\!\Delta\eta\widetilde{D}_{x}i\Delta_{C}=\frac{\Gamma^{2}(\frac{D}{2}\\!-\\!1)}{16\pi^{D}}\Biggl{\\{}\\!\\!\frac{(D\\!-\\!2)\Delta\eta^{2}}{(aa^{\prime})^{D-2}\Delta
x^{2D-2}}\\!-\\!\frac{H^{2}\Delta\eta^{2}\ln(\frac{1}{4}H^{2}\Delta
x^{2})}{aa^{\prime}\Delta x^{4}}\\!\\!\Biggr{\\}},$ (239)
$\displaystyle=\frac{\Gamma^{2}(\frac{D}{2}\\!-\\!1)}{16\pi^{D}}\Biggl{\\{}-\frac{1}{2(aa^{\prime})^{D-2}\Delta
x^{2D-4}}+\frac{H^{2}[\ln(\frac{1}{4}H^{2}\Delta
x^{2})\\!+\\!1]}{2aa^{\prime}\Delta x^{2}}$ $\displaystyle\hskip
45.52458pt+\frac{\partial_{0}^{2}}{4(aa^{\prime})^{2}}\Bigl{[}\frac{1}{\Delta
x^{2}}\Bigr{]}+\frac{H^{2}\partial_{0}^{2}}{8aa^{\prime}}\Bigl{[}\ln^{2}(\tfrac{1}{4}H^{2}\Delta
x^{2})\\!+\\!2\ln(\tfrac{1}{4}H^{2}\Delta x^{2})\Bigr{]}\Biggr{\\}},\qquad$
$\displaystyle\longrightarrow-\frac{\mu^{D-4}\Gamma(\frac{D}{2}\\!-\\!1)}{16(D\\!-\\!3)(D\\!-\\!4)\pi^{\frac{D}{2}}}\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{(aa^{\prime})^{D-2}}$
$\displaystyle\hskip
28.45274pt+\frac{\partial^{2}}{128\pi^{4}(aa^{\prime})^{2}}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Bigr{]}+\frac{H^{2}[\ln(\frac{1}{4}H^{2}\Delta
x^{2})\\!+\\!1]}{32\pi^{4}aa^{\prime}\Delta x^{2}}$ $\displaystyle\hskip
14.22636pt+\frac{\partial_{0}^{2}}{64\pi^{4}(aa^{\prime})^{2}}\Bigl{[}\frac{1}{\Delta
x^{2}}\Bigr{]}+\frac{H^{2}\partial_{0}^{2}}{128\pi^{4}aa^{\prime}}\Bigl{[}\ln^{2}(\tfrac{1}{4}H^{2}\Delta
x^{2})\\!+\\!2\ln(\tfrac{1}{4}H^{2}\Delta x^{2})\Bigr{]}.\qquad$

The 1-loop divergences of the invariant self-mass are canceled by three
counterterms,

$\Delta\mathcal{L}=-\tfrac{1}{2}\alpha_{1}\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0pt\phi\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0pt\phi\sqrt{-g}-\tfrac{1}{2}\alpha_{2}R\partial_{\mu}\phi\partial_{\nu}\phi
g^{\mu\nu}\sqrt{-g}-\tfrac{1}{2}\alpha_{3}R\partial_{0}\phi\partial_{0}\phi
g^{00}\sqrt{-g}\;,$ (240)

where the Ricci scalar in de Sitter is $R=D(D-1)H^{2}$. The associated