UFIFT-QG-23-07

Large Inflationary Logarithms in a Nontrivial Nonlinear Sigma Model

C. Litos†, R. P. Woodard∗ and B. Yesilyurt‡

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

ABSTRACT

Loops of inflationary gravitons are known to induce large temporal and spatial
logarithms which can cause perturbation theory to break down. Nonlinear sigma
models possess the same kind of derivative interactions and induce the same
sorts of large logarithms, without the complicated index structure and
potential gauge problem. Previous studies have examined models with zero field
space curvature which can be reduced to free field theories by local,
invertible field redefinitions. Here we study a model which cannot be so
reduced and still shows the same sorts of large logarithms. We compute the
evolution of the background at 1-loop and 2-loop orders, and we find the
1-loop $\beta$ and $\gamma$ functions.

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

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

The background geometry of cosmology is characterized by scale factor $a(t)$,
Hubble parameter $H(t)$, and first slow roll parameter $\epsilon(t)$,

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

The accelerated expansion of inflation ($H(t)>0$ with $0\leq\epsilon(t)<1$)
rips virtual scalars and gravitons out of the vacuum. This is the basis for
the primordial spectra of scalars [1] and tensors [2]. At some level these
quanta must interact among themselves and with other particles. Gravitons are
especially interesting because their couplings are universal and because their
tensor structure allows them to mediate effects which scalars cannot. For
example, on de Sitter background ($\epsilon(t)=0$) one loop gravitons modify
the plane wave mode function $u(t,k)$ of gravitational radiation [3] and the
gravitational response $\Psi(t,r)$ to a point mass [4] to,

$\displaystyle u(t,k)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
u_{0}(t,k)\Biggl{\\{}1+\frac{16GH^{2}}{3\pi}\ln^{2}(a)+O(G^{2})\Biggr{\\}}\;,\qquad$
(2) $\displaystyle\Psi(t,r)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{GM}{ar}\Biggl{\\{}1+\frac{103G}{15\pi
a^{2}r^{2}}-\frac{8GH^{2}}{\pi}\ln^{3}(a)+O(G^{2})\Biggr{\\}}\;.\qquad$ (3)

Similar results have been obtained for the corrections of inflationary
gravitons to fermions [5], to electrodynamics [6, 7], and to massless,
minimally coupled scalars [8].

A fascinating aspect of these corrections is that the steady production of
inflationary gravitons endows them with a secular growth which must eventually
overwhelm the loop-counting parameter $GH^{2}$ provided inflation persists
long enough. One must develop a nonperturbative resummation technique in order
to evolve past that point. Nonlinear sigma models provide a useful testing
ground for developing such a technique because they possess the same sorts of
derivative interactions as gravitons, and induce the same sorts of large
logarithms, without the complicating tensor structures or the potential for
gauge dependence [9, 10, 11, 12]. A successful technique has been devised
through combining a variant of Starobinsky’s stochastic formalism [13, 14]
with a variant of the renormalization group [15, 16]. Even better, the
technique can be generalized from de Sitter to an arbitrary cosmological
background (1) which has undergone primordial inflation [17]. Applying this
technique shows that the large scales of primordial inflation can be
transmitted to late time [18].

The obvious next step is generalizing the resummation technique from nonlinear
sigma models to quantum gravity. This seems to be entirely possible, and works
for the one graviton loop correction on which it has been checked [8].
However, our purpose here is to clear up a worry concerning the sorts of
nonlinear sigma models on which the resummation technique has so far been
applied. Specifically, both of those models can be reduced to free theories by
means of local invertible field redefinitions, which means that their flat
space S-matrices are unity by Borchers Theorem [19]. That in no way precludes
interesting evolutions for the scalar backgrounds, and for the 1-particle
states, and it was these evolutions which suggested and confirmed the
resummation technique. One might still worry that the technique only works for
models which can be reduced to free theories. The purpose of this paper is to
demonstrate that the resummation technique applies for a model whose flat
space $S$-matrix is nontrivial.

This paper contains five sections, of which this Introduction is the first. In
Section 2 we describe the old and new models, and we give the Feynman rules.
Section 3 computes the time-evolving background at 1-loop and 2-loop orders.
It also computes and renormalizes the 1PI (one-particle-irreducible) 2-point
and 3-point functions at 1-loop order. The resummation technique is applied in
Section 4. Our conclusions comprise Section 5.

##  2 The Model

The purpose of this section is to define the model and give those of its
Feynman rules which are required for our work. We begin by presenting the old
models and explaining why their $S$-matrices are trivial. Then a new model is
proposed by making a small variation which preserves the lowest order
interactions.

The resummation technique was developed based on work with two nonlinear sigma
models [15]. One of these was based on a single field. All models of this sort
can be made free by a local, invertible field redefinition,

$\mathcal{L}=-\frac{1}{2}f^{2}(\Phi)\partial_{\mu}\Phi\partial_{\nu}\Phi
g^{\mu\nu}\sqrt{-g}\quad,\quad d\Psi\equiv
f(\Phi)d\Phi\quad\Longrightarrow\quad\mathcal{L}=-\frac{1}{2}\partial_{\mu}\Psi\partial_{\nu}\Psi
g^{\mu\nu}\sqrt{-g}\;.$ (4)

A second model was based on two fields,

$\mathcal{L}_{\rm
old}=-\frac{1}{2}\partial_{\mu}A\partial_{\nu}Ag^{\mu\nu}\sqrt{-g}-\frac{1}{2}\Bigl{(}1+\frac{\lambda}{2}A\Bigr{)}^{2}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;.$
(5)

Although it was not initially realized, this model can be also reduced to a
theory of two free scalars by making a local, invertible field
redefinition,111We thank Arkady Tseytlin for this observation.

$\displaystyle X$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\frac{2}{\lambda}\Bigl{(}1+\frac{\lambda}{2}A\Bigr{)}\cos\Bigl{(}\frac{\lambda}{2}B\Bigr{)}\;,$
(6) $\displaystyle Y$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\frac{2}{\lambda}\Bigl{(}1+\frac{\lambda}{2}A\Bigr{)}\sin\Bigl{(}\frac{\lambda}{2}B\Bigr{)}\;.$
(7)

Hence the flat space S-matrices of (4) and (5) are both unity by Borchers
Theorem [19]. That in no way precludes these models from manifesting
interesting evolutions of their backgrounds and of the single-particle
kinematics.

To be certain that the resummation technique does not rely on having a trivial
S-matrix we devised a slight modification of the 2-field model (5) whose field
space curvature implies that it cannot be reduced to a fee theory,

$\mathcal{L}_{\rm
new}=-\frac{1}{2}\partial_{\mu}A\partial_{\nu}Ag^{\mu\nu}\sqrt{-g}-\frac{1}{2}\Bigl{(}1+\frac{\lambda}{4}A\Bigr{)}^{4}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;.$
(8)

The 3-point coupling in this model is identical to that of the old model (5),
and the 4-point coupling has the same field content with the old numerical
coefficient of $\frac{1}{8}$ replaced by $\frac{3}{16}$. There are additional
5-point and 6-point interactions which make only simple contributions to the
diagrams we evaluate in section 3,

$\frac{1}{2}\Bigl{(}1\\!+\\!\frac{\lambda}{4}A\Bigr{)}^{4}(\partial
B)^{2}-\frac{1}{2}(\partial B)^{2}=\frac{\lambda}{2}A(\partial
B)^{2}+\frac{3\lambda^{2}}{16}A^{2}(\partial
B)^{2}+\frac{\lambda^{3}}{32}A^{3}(\partial
B)^{2}+\frac{\lambda^{4}}{512}A^{4}(\partial B)^{2}.$ (9)

A diagrammatic representation of the Feynman rules is shown in Figure 1.

Figure 1: Primitive interactions of the bare Lagrangian (8). A lines are solid
and $B$ lines are dashed.

The new model (8) is no more renormalizable than the old one (5). Hence
divergences must be subtracted, order-by-order, using BPHZ counterterms
(Bogoliubov-Parasiuk-Hepp-Zimmermann [20, 21, 22, 23]). The 1-loop
counterterms we require are shown in Figure 2.

Figure 2: Diagrammatic representation of the 1-loop counterterms we require.
The first two diagrams renormalize the $A$ and $B$ self-masses while the
rightmost diagram renormalizes the vertex function. Recall that $A$ lines are
solid and $B$ lines are dashed.

The first and second diagrams renormalize the $A$ and $B$ self-masses and
correspond to the counterterm,

$\displaystyle\Delta\mathcal{L}^{(2)}=-\frac{1}{2}C_{1A^{2}}\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptA\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptA\sqrt{-g}-\frac{1}{2}C_{2A^{2}}R\,\partial_{\mu}A\partial_{\nu}Ag^{\mu\nu}\sqrt{-g}$
(10) $\displaystyle\hskip 170.71652pt-\frac{1}{2}C_{1B^{2}}\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptB\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptB\sqrt{-g}-\frac{1}{2}C_{2B^{2}}R\,\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;.\qquad$

The 3rd diagram is required to renormalize the 3-point vertex and corresponds
to,

$\displaystyle\delta\mathcal{L}^{(3)}=-\frac{1}{2}C_{1AB^{2}}\Box
A\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}-C_{2AB^{2}}\partial_{\mu}A\partial_{\nu}B\Box
Bg^{\mu\nu}\sqrt{-g}$ (11) $\displaystyle\hskip
142.26378pt-\frac{1}{2}C_{3AB^{2}}A\Box B\Box
B\sqrt{-g}-\frac{1}{2}C_{4AB^{2}}RA\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\
.\qquad$

Here $R=D(D-1)H^{2}$ is the Ricci scalar. Section 3 will determine the various
coefficients in expressions (10-11) as functions of $\lambda$, $D$ and the
dimensional regularization scale $\mu$.

In $D$ spacetime dimensions, the propagators of both fields obey the equation,

$\partial^{\mu}\left[a^{D-2}\partial_{\mu}i\Delta(x;x^{\prime})\right]\equiv\mathcal{D}i\Delta(x;x^{\prime})=i\delta^{D}(x-x^{\prime}).$
(12)

The solution consists of a de Sitter invariant part plus a de Sitter breaking
logarithm [24, 25],

$i\Delta(x;x^{\prime})=F\Bigl{(}\mathcal{Y}(x;x^{\prime})\Bigr{)}+k\ln(aa^{\prime})\qquad,\qquad
k\equiv\frac{H^{D-2}}{(4\pi)^{D/2}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}.$
(13)

Here $\mathcal{Y}(x;x^{\prime})\equiv aa^{\prime}H^{2}\Delta
x^{2}(x;x^{\prime})\equiv aa^{\prime}H^{2}(x-x^{\prime})^{2}$ is the de-Sitter
length function, and the first derivative of $F(\mathcal{Y}(x;x^{\prime}))$
obeys,

$\displaystyle
F^{\prime}(\mathcal{Y})=-\frac{H^{D-2}}{4(4\pi)^{D/2}}\Biggl{\\{}\Gamma\Bigl{(}\frac{D}{2}\Bigr{)}\left(\frac{4}{\mathcal{Y}}\right)^{\frac{D}{2}}+\Gamma\Bigl{(}\frac{D}{2}\\!+\\!1\Bigr{)}\left(\frac{4}{\mathcal{Y}}\right)^{\frac{D}{2}-1}$
(14) $\displaystyle\hskip
128.0374pt+\sum_{n=0}^{\infty}\left[\frac{\Gamma(n\\!+\\!\frac{D}{2}\\!+\\!2)}{\Gamma(n\\!+\\!3)}\left(\frac{\mathcal{Y}}{4}\right)^{n-\frac{D}{2}+2}-\frac{\Gamma(n\\!+\\!D)}{\Gamma(n\\!+\\!\frac{D}{2}\\!+\\!1)}\left(\frac{\mathcal{Y}}{4}\right)^{n}\right]\Biggr{\\}}.\qquad$

In dimensional regularization the coincidence limits of the propagator and its
derivatives are,

$\displaystyle
i\Delta(x;x)=k\left[-\pi\cot\left(\frac{D\pi}{2}\right)+2\ln(a)\right]\qquad$
,
$\displaystyle\qquad\partial_{\mu}i\Delta(x;x^{\prime})\Bigl{|}_{x=x^{\prime}}=kHa\delta^{0}_{\
\mu}\;,\qquad$ (15)
$\displaystyle\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}=-\left(\frac{D-1}{D}\right)kH^{2}g_{\mu\nu}\qquad$
, $\displaystyle\qquad\partial_{\mu}i\Delta(x;x)=2kHa\delta^{0}_{\ \mu}\;.$
(16)

We close by commenting on notation. Because the de Sitter metric
$g_{\mu\nu}=a^{2}\eta_{\mu\nu}$ is conformal to the Minkowski metric
$\eta_{\mu\nu}$, we adopt a notation where $\partial_{\mu}$ stands for
$\frac{\partial}{\partial x^{\mu}}$, no matter what sort of tensor it acts
upon. Further, we raise and lower its indices using the Minkowski metric,
$\partial^{\mu}\equiv\eta^{\mu\nu}\partial_{\nu}$. And we define
$\partial^{2}\equiv\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$. To save space
we sometimes write coordinate arguments of the metric and its scale factor
using a subscript or a superscript, as in
$\sqrt{-g(x)}\equiv\sqrt{-g_{x}}\equiv a_{x}^{D}$ and $g^{\mu\nu}(y)\equiv
g^{\mu\nu}_{y}\equiv a^{-2}_{y}\eta^{\mu\nu}$. The same notation applies to
derivatives, as in $\frac{\partial}{\partial
z^{\mu}}\equiv\partial^{z}_{\mu}$.

##  3 Explicit Results

The purpose of this section is to carry out the same explicit calculations for
(8) that were done for the old model (5) [15, 16]. We begin with the 1-loop
and 2-loop expectation values of $A(x)$. Next, the self-masses of $A$ and $B$
are computed and renormalized at 1-loop order. Finally, we evaluate the 1-loop
vertex function.

###  3.1 The 1-Loop and 2-Loop Background

We start with the 1-loop expectation value of $A(x)$ whose diagrammatic
representation is shown in Figure 3.

Figure 3: 1-loop contribution to the expectation value of $A(x)$.

Because the 3-point couplings of the old and new models agree, this diagram is
unchanged from the old result [15],

$\boxed{\left\langle\Omega\left|A(x)\right|\Omega\right\rangle=\frac{\lambda
H^{2}}{16\pi^{2}}\left[\ln(a)-\frac{1}{3}+\frac{1}{3a^{3}}\right]+\mathcal{O}(\lambda^{3})}.$
(17)

The 2-loop contributions are shown in Figure 4.

Figure 4: Contributions to the expectation value of $A(x)$ at 2-loop order.

The first 7 diagrams (which we label $A_{2a}$ through $A_{2g}$) were
calculated in [15]. We need only include a relative factor of $\frac{3}{2}$
for each 4-point coupling to express their final contributions, at leading
logarithm order, as multiples of the factor
$S\equiv\lambda^{3}H^{4}\ln^{2}(a)/2^{10}\pi^{4}$,

$\displaystyle A_{2a}\longrightarrow-3\cdot S\quad,\quad
A_{2b}\longrightarrow+8\cdot S\quad$ , $\displaystyle\quad
A_{2c}\longrightarrow 0\cdot S\quad,\quad A_{2d}\longrightarrow-6\cdot
S\;,\qquad$ (18) $\displaystyle A_{2e}\longrightarrow+\frac{3}{2}\cdot
S\quad,\quad A_{2f}\longrightarrow-2\cdot S\quad$ , $\displaystyle\quad
A_{2g}\longrightarrow 0\cdot S\;.\qquad$ (19)

Only the last diagram, $A_{2h}$ is really new. It has a symmetry factor of
$\frac{1}{4}$ and its initial expression is,

$A_{2h}=\frac{-i}{4}\cdot\frac{3\lambda^{3}}{8}\cdot\int d^{D}y\
\sqrt{-g(y)}\,g^{\mu\nu}(y)\cdot i\Delta(x;y)\cdot
i\Delta(y;y)\cdot\partial_{\mu}^{z}\partial_{\nu}^{y}i\Delta(y;z)\Bigl{|}_{y=z}\;.$
(20)

The two coincidence limits can be read from (15-16) to give,

$A_{2h}=\frac{3i\lambda^{3}}{2^{5}}(D-1)kH^{2}I_{2},$ (21)

where $I_{2}$, and its leading logarithm result, was given in [18],

$I_{2}\equiv\int
d^{D}y\sqrt{-g(y)}i\Delta(x;y)i\Delta(y;y)\longrightarrow-\frac{i}{24\pi^{2}}\times\ln^{2}(a).$
(22)

This is finite and does not require renormalization, so we can set $D=4$,

$A_{2h}\longrightarrow\frac{3}{2}\cdot S.$ (23)

Adding this to the leading logarithm results for the first seven diagrams
gives,

$\displaystyle A_{2a}\longrightarrow-3\cdot S\quad,\quad
A_{2b}\longrightarrow+8\cdot S\quad$ , $\displaystyle\quad
A_{2c}\longrightarrow 0\cdot S\quad,\quad A_{2d}\longrightarrow-6\cdot
S\;,\qquad$ (24) $\displaystyle A_{2e}\longrightarrow+\frac{3}{2}\cdot
S\quad,\quad A_{2f}\longrightarrow-2\cdot S\quad$ , $\displaystyle\quad
A_{2g}\longrightarrow 0\cdot S\quad,\quad
A_{2h}\longrightarrow+\frac{3}{2}\cdot S\;.\qquad$ (25)

At leading logarithm order the eight diagrams of Figure 4 sum to zero, so our
result for the expectation value of $A$ is,

$\boxed{\left\langle\Omega\left|A(x)\right|\Omega\right\rangle=\frac{\lambda
H^{2}\ln(a)}{16\pi^{2}}\Bigl{[}1+0\Bigr{]}+\mathcal{O}(\lambda^{4})}\;.$ (26)

###  3.2 The 1-Loop Self-Masses

We now move on to the 1-loop self-masses. Absorbing the divergences of these
reveals curvature-dependent field strength renormalizations $Z_{A}$ and
$Z_{B}$ from the terms proportional to $C_{2A^{2}}$ and $C_{2B^{2}}$ in
expression (10). These give the $\gamma$ functions that will be used in the
Renormalization Group (RG) analysis of Section 4.

The 1-loop contributions to $-iM^{2}_{A}(x;x^{\prime})$ are shown in Figure 5.

Figure 5: 1-loop contributions to the $A$ self mass
$-iM_{A}^{2}(x;x^{\prime})$.

The first diagram is unchanged from the old model, and the second diagram is
just $\frac{3}{2}$ times the previous result [5],

$\displaystyle-iM_{A_{3}}^{2}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{(-i\lambda)^{2}}{2}\Biggl{\\{}\frac{1}{4}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}$
(27) $\displaystyle\hskip
85.35826pt-\frac{1}{2}\mathcal{D}\Bigl{[}i\Delta(x;x)i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}-kHa^{D-1}\partial_{0}i\delta^{D}(x\\!-\\!x^{\prime})\Biggr{\\}},\qquad$
$\displaystyle-iM^{2}_{A_{4}}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{i3\lambda^{2}}{8}\delta^{D}(x\\!-\\!x^{\prime})\cdot-(D\\!-\\!1)kH^{2}a^{D}\;.$
(28)

The counterterm is $i$ times the second variation of (10),

$-iM^{2}_{Ac}(x;x^{\prime})=-C_{1A^{2}}\mathcal{D}\mathcal{D}^{\prime}\left[\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{(aa^{\prime})^{\frac{D}{2}}}\right]+C_{2A^{2}}\partial^{\mu}\Bigl{[}Ra^{D-2}\partial_{\mu}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}.$
(29)

Because only (27) diverges in dimensional regularization, and this diagram is
identical to the old model, so too are the coefficients $C_{A_{1}}$ and
$C_{A_{2}}$,

$C_{1A^{2}}=-\frac{\lambda^{2}\mu^{D-4}}{32\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)}\quad,\quad
C_{2A^{2}}=\frac{\lambda^{2}\mu^{D-4}}{4(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi\cot(\frac{D\pi}{2})}{D(D\\!-\\!1)}.$
(30)

We move to the $B$ self-mass, whose 1-loop contributions are shown in Figure
6.

Figure 6: 1-loop contributions to the $B$ self-mass
$-iM_{B}^{2}(x;x^{\prime})$.

As before, the 3-point diagram is unchanged from the old model and the 4-point
diagram is just the result from the old model times $\frac{3}{2}$,

$-iM^{2}_{B_{3}}=\frac{(i\lambda)^{2}}{2}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}-(i\lambda)^{2}\partial^{\mu}\partial^{\prime\rho}\Bigl{[}(aa^{\prime})^{D-2}\partial_{\mu}i\Delta(x;x^{\prime})\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\Bigr{]}$
(31)
$-iM^{2}_{B_{4}}=-\frac{3\lambda^{2}k\pi\cot(\frac{D\pi}{2})\mathcal{D}}{8}\left[i\delta^{D}(x\\!-\\!x^{\prime})\right]+\frac{3\lambda^{2}H^{2}\partial^{\mu}}{32\pi^{2}}\Bigl{[}\ln(a)a^{2}\partial_{\mu}i\delta^{4}(x\\!-\\!x^{\prime})\Bigr{]}+\mathcal{O}(D\\!-\\!4).$
(32)

The counterterm is $i$ times the second variation of (10) with respect to $B$,

$-iM^{2}_{B_{c}}(x;x^{\prime})=-C_{1B^{2}}\mathcal{D}\mathcal{D}^{\prime}\left[\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{(aa^{\prime})^{\frac{D}{2}}}\right]+C_{2B^{2}}\partial^{\mu}\Bigl{[}Ra^{D-2}\partial_{\mu}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}.$
(33)

Multiplying the divergences from (32) by $\frac{3}{2}$ and combining with
those from (31) gives,

$C_{1B^{2}}=-\frac{\lambda^{2}\mu^{D-4}}{16\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)},$
(34)
$C_{2B^{2}}=\frac{3\lambda^{2}\mu^{D-4}}{8(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi\cot(\frac{D\pi}{2})}{D(D\\!-\\!1)}-\frac{\lambda^{2}\mu^{D-4}}{32\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)}\frac{D\\!-\\!2}{D\\!-\\!1}.$
(35)

###  3.3 The 1-Loop Vertex Function

In this subsection we first isolate the primitive divergences of the 3-point
vertex $-iV(x;y;z)$ at 1-loop order. These divergences are removed by the
counterterm (11), which determines the coefficients $C_{1AB^{2}}$,
$C_{2AB^{2}}$, $C_{3AB^{2}}$ and $C_{4AB^{2}}$. We regard the $C_{4AB^{2}}$
counterterm as a curvature-dependent renormalization of the bare 3-point
coupling and infer the corresponding beta function as,

$\delta\lambda=C_{4AB^{2}}\\!\times\\!R+O(\lambda^{5})\qquad\Longrightarrow\qquad\beta\equiv\frac{\partial\delta\lambda}{\partial\ln(\mu)}\;.$
(36)

The tree order vertex can be inferred from the Feynman rules,

$-iV_{0}(x;y;z)=-i\lambda\sqrt{-g(x)}\,g^{\mu\nu}_{x}\partial_{\mu}\delta^{D}(x-y)\partial_{\nu}\delta^{D}(x\\!-\\!z).$
(37)

The various 1-loop contributions are shown in Figure 7.

Figure 7: 1-loop contributions to the $ABB$ vertex $-iV(x;y;z)$.

Because the leftmost diagram involves only 3-point couplings, it is unchanged
from the old model [16],

$\displaystyle-
iV_{1a}(x;y;z)=\frac{i\lambda^{3}}{2}\mathcal{D}_{x}\partial_{\sigma}^{y}\partial_{\beta}^{z}\Biggl{\\{}\sqrt{-g_{y}}\,g_{y}^{\rho\sigma}\partial_{\rho}^{y}i\Delta(x;y)\sqrt{-g_{z}}\,g_{z}^{\alpha\beta}\partial_{\alpha}^{z}i\Delta(x;z)i\Delta(y;z)\Biggr{\\}}$
(38) $\displaystyle\hskip
156.49014pt+\frac{\lambda^{3}}{4}\mathcal{D}_{y}\mathcal{D}_{z}\Biggl{\\{}\Bigl{[}i\Delta(y;z)\Bigr{]}^{2}\Bigl{[}\delta^{D}(x\\!-\\!y)+\delta^{D}(x\\!-\\!z)\Bigr{]}\Biggr{\\}}$
$\displaystyle\hskip
34.14322pt-\frac{\lambda^{3}}{2}\partial_{\sigma}^{y}\partial_{\beta}^{z}\Biggl{\\{}\sqrt{-g_{y}}\,g_{y}^{\rho\sigma}\partial_{\rho}^{y}i\Delta(y;z)\sqrt{-g_{z}}\,g_{z}^{\alpha\beta}\partial_{\alpha}^{z}i\Delta(y;z)\Bigl{[}\delta^{D}(x\\!-\\!y)+\delta^{D}(x\\!-\\!z)\Bigr{]}\Biggr{\\}}.\qquad$

The 2nd and 3rd diagrams involve a single 4-point coupling and are therefore
$\frac{3}{2}$ times the results of the old model [16],

$\displaystyle-
iV_{1b}(x;y;z)=\frac{3\lambda^{3}}{8}\mathcal{D}_{y}\Biggl{\\{}\sqrt{-g_{x}}\,g_{x}^{\mu\nu}\partial_{\mu}^{x}\Bigl{[}i\Delta(x;y)\Bigr{]}^{2}\partial_{\nu}\delta^{D}(x\\!-\\!z)\Biggr{\\}}$
(39) $\displaystyle\hskip
85.35826pt+\frac{3\lambda^{3}}{4}\partial_{\nu}^{z}\partial_{\beta}^{y}\Biggl{\\{}\sqrt{-g_{z}}\,g_{z}^{\mu\nu}\partial_{\mu}^{z}i\Delta(z;y)\sqrt{-g_{y}}\,g_{y}^{\alpha\beta}\partial_{\alpha}^{y}i\Delta(z;y)\delta^{D}(x\\!-\\!z)\Biggr{\\}},\qquad$
$\displaystyle-
iV_{1c}(x;y;z)=\frac{3\lambda^{3}}{8}\mathcal{D}_{z}\Biggl{\\{}\sqrt{-g_{x}}\,g_{x}^{\mu\nu}\partial_{\mu}^{x}[i\Delta(x;z)]^{2}\partial_{\nu}\delta^{D}(x\\!-\\!y)\Biggr{\\}}$
(40) $\displaystyle\hskip
85.35826pt+\frac{3\lambda^{3}}{4}\partial_{\sigma}^{y}\partial_{\beta}^{z}\Biggl{\\{}\sqrt{-g_{y}}\,g_{y}^{\rho\sigma}\partial_{\rho}^{y}i\Delta(y;z)\sqrt{-g_{z}}\,g_{z}^{\alpha\beta}\partial_{\alpha}^{z}i\Delta(y;z)\delta^{D}(x\\!-\\!y)\Biggr{\\}}.\qquad$

Before considering the 4th diagram, we combine and reduce expressions (38-40).
Because the last two differ from the old model, the reduction is more
complicated. Adding all three terms and performing some judicious partial
integrations gives,

$\displaystyle-
iV_{1abc}=\frac{i\lambda^{3}}{2}\mathcal{D}_{x}\partial^{\rho}_{y}\partial^{\alpha}_{z}\Bigl{\\{}(a_{y}a_{z})^{D-2}i\Delta(y;z)\partial_{\rho}^{y}i\Delta(x;y)\partial_{\alpha}^{z}i\Delta(x;z)\Bigr{\\}}$
(41)
$\displaystyle+\frac{3\lambda^{3}}{8}\mathcal{D}_{y}\partial_{z}^{\alpha}\Bigl{\\{}a_{z}^{D-2}\Bigl{[}i\Delta(y;z)\Bigr{]}^{2}\partial_{\alpha}^{z}\delta^{D}(x\\!-\\!z)\Bigr{\\}}+\frac{3\lambda^{3}}{8}\mathcal{D}_{z}\partial_{y}^{\alpha}\Bigl{\\{}a_{y}^{D-2}\Bigl{[}i\Delta(y;z)\Bigr{]}^{2}\partial_{\alpha}^{y}\delta^{D}(x\\!-\\!y)\Bigr{\\}}$
$\displaystyle\hskip
113.81102pt-\frac{\lambda^{3}}{8}\mathcal{D}_{y}\mathcal{D}_{z}\Bigl{\\{}\Bigl{[}i\Delta(y;z)\Bigr{]}^{2}\Bigl{[}\delta^{D}(x\\!-\\!y)+\delta^{D}(x\\!-\\!z)\Bigr{]}\Bigr{\\}}$
$\displaystyle\hskip
85.35826pt+\frac{\lambda^{3}}{4}\partial^{\rho}_{y}\partial^{\alpha}_{z}\Bigl{\\{}(a_{y}a_{z})^{D-2}\partial_{\rho}^{y}i\Delta(y;z)\partial_{\alpha}^{z}i\Delta(y;z)\Bigl{[}\delta^{D}(x\\!-\\!y)+\delta^{D}(x\\!-\\!z)\Bigr{]}\Bigr{\\}}.\qquad$

After considerable manipulations explained in the Appendix A expression the
divergent part of (41) can be brought to the form,

$-iV_{1abc}(x;y;z)\longrightarrow-\frac{i\lambda^{3}\Gamma(\frac{D}{2}\\!+\\!1)\mu^{D-4}H^{2}a_{x}^{2}}{32\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\,\partial^{\mu}\delta^{D}(x\\!-\\!y)\partial_{\mu}\delta^{D}(x\\!-\\!z)-i\widetilde{V}(x;y;z),$
(42)

where $-i\widetilde{V}(x;y;z)$ consists of higher derivative divergences,

$\displaystyle-i\widetilde{V}(x;y;z)=\frac{i\lambda^{3}\mu^{D-4}}{16\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{D(D\\!-\\!3)(D\\!-\\!4)}\mathcal{D}_{x}\left[\frac{\partial_{\mu}\delta^{D}(x\\!-\\!y)\partial^{\mu}\delta^{D}(x\\!-\\!z)}{a_{x}^{D-2}}\right]$
(43)
$\displaystyle+\frac{5i\lambda^{3}\mu^{D-4}\Gamma(\frac{D}{2}\\!-\\!1)}{128\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\Biggl{\\{}\mathcal{D}_{y}\partial_{x}^{\mu}\left[\frac{\delta^{D}(x\\!-\\!y)\partial_{\mu}\delta^{D}(x\\!-\\!z)}{a_{x}^{D-2}}\right]+\mathcal{D}_{z}\partial_{x}^{\mu}\left[\frac{\partial_{\mu}\delta^{D}(x\\!-\\!y)\\!\cdot\\!\delta^{D}(x\\!-\\!z)}{a_{x}^{D-2}}\right]\Biggr{\\}}$
$\displaystyle\hskip
156.49014pt-\frac{\lambda^{3}\Gamma(\frac{D}{2}\\!-\\!1)\mu^{D-4}}{64\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\mathcal{D}_{y}\mathcal{D}_{z}\left[\frac{i\delta^{D}(x\\!-\\!z)\delta^{D}(x\\!-\\!y)}{a_{x}^{2D-4}}\right].\qquad$

We call the new diagram $-iV_{1d}(x;y;z)$ and its contribution is,

$-iV_{1d}=-i\frac{3\lambda^{3}}{16}\sqrt{-g(x)}\,g^{\mu\nu}(x)\partial_{\mu}\delta^{D}(x\\!-\\!y)\partial_{\nu}\delta^{D}(x\\!-\\!z)\cdot
i\Delta(x;x)\;.$ (44)

This is just the bare vertex (37) times the simple factor
$\frac{3\lambda^{2}}{16}i\Delta(x;x)$. Recall that the coincident propagator
is given in equation (15),

$\displaystyle-iV_{1d}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-i\frac{3\lambda^{3}}{16}a_{x}^{D-2}\,\partial^{\mu}\delta^{D}(x\\!-\\!y)\partial_{\mu}\delta^{D}(x\\!-\\!z)k\left[-\pi\cot\Bigl{(}\frac{D\pi}{2}\Bigr{)}+2\ln(a)\right]\;,\qquad$
(45) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-i\frac{3\lambda^{3}k}{16}\\!\times\\!-\pi\cot\Bigl{(}\frac{D\pi}{2}\Bigr{)}\\!\times\\!a_{x}^{2}\,\partial^{\mu}\delta^{D}(x\\!-\\!y)\partial_{\mu}\delta^{D}(x\\!-\\!z)+O(D\\!-\\!4)\;.\qquad$
(46)

Adding the primitive divergences from (42) to this gives,

$\displaystyle-iV_{1{\rm
pdiv}}(x;y;z)=-i\frac{\lambda^{3}H^{2}a_{x}^{2}}{8}\,\partial^{\mu}\delta^{D}(x\\!-\\!y)\partial_{\mu}\delta^{D}(x\\!-\\!z)$
(47) $\displaystyle\hskip
113.81102pt\times\left[\frac{3k}{2H^{2}}\\!\times\\!-\pi\cot\Bigl{(}\frac{D\pi}{2}\Bigr{)}+\frac{\mu^{D-4}\Gamma(\frac{D}{2}\\!+\\!1)}{4\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\right]-i\tilde{V}(x;y;z).\qquad$

The primitive divergences (47) are absorbed by the third variation of (11),

$\displaystyle\frac{i\delta^{3}S_{AB^{2}}[A,B]}{\delta A(x)\delta B(y)\delta
B(z)}=-iC_{1AB^{2}}\mathcal{D}_{x}\left[\frac{\partial^{\mu}\delta^{D}(x\\!-\\!y)\partial_{\mu}\delta^{D}(x\\!-\\!z)}{a_{x}^{2}}\right]$
(48)
$\displaystyle+iC_{2AB^{2}}\mathcal{D}_{y}\partial^{\mu}_{x}\left[\frac{\delta^{D}(x\\!-\\!y)\partial_{\mu}\delta^{D}(x\\!-\\!z)}{a_{x}^{2}}\right]+iC_{2AB^{2}}\mathcal{D}_{z}\partial^{\mu}_{x}\left[\frac{\partial_{\mu}\delta^{D}(x\\!-\\!y)\\!\times\\!\delta^{D}(x\\!-\\!z)}{a_{x}^{2}}\right]$
$\displaystyle\hskip 48.36958pt-
iC_{3AB^{2}}\mathcal{D}_{y}\mathcal{D}_{z}\left[\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x\\!-\\!z)}{a_{x}^{D}}\right]-iC_{4AB^{2}}Ra_{x}^{2}\,\partial^{\mu}\delta^{D}(x\\!-\\!y)\partial_{\mu}\delta^{D}(x\\!-\\!z)\;.\qquad$

Adding (48) to (47) and demanding regularity as $D\to 4$ implies,

$\displaystyle C_{1AB^{2}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\lambda^{3}\mu^{D-4}\,\Gamma(\frac{D}{2}\\!-\\!1)}{16\pi^{\frac{D}{2}}D(D\\!-\\!3)(D\\!-\\!4)}\;,\qquad$
(49) $\displaystyle C_{2AB^{2}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{5\lambda^{3}\mu^{D-4}\,\Gamma(\frac{D}{2}\\!-\\!1)}{128\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\;,\qquad$
(50) $\displaystyle C_{3AB^{2}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\lambda^{3}\mu^{D-4}\,\Gamma(\frac{D}{2}\\!-\\!1)}{64\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\;,\qquad$
(51) $\displaystyle C_{4AB^{2}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\lambda^{3}\mu^{D-4}}{32\pi^{\frac{D}{2}}D(D\\!-\\!1)}\left[\frac{3\pi\cot(\frac{D\pi}{2})\Gamma(D\\!-\\!1)}{8\Gamma(\frac{D}{2})}-\frac{\Gamma(\frac{D}{2}\\!+\\!1)}{(D\\!-\\!3)(D\\!-\\!4)}\right]\;.\qquad$
(52)

Because we have suppressed the finite contributions the renormalized result
will not be given but let us take note of the fact that logarithms of $\mu$
come in the form $\ln(\frac{\mu a}{H})$. We can regard $C_{4AB^{2}}\times
R=\delta\lambda$ as a curvature-dependent renormalization. The associated
1-loop beta function (36) is,

$\boxed{\beta=\frac{\partial\delta\lambda}{\partial\ln(\mu)}=-\frac{\lambda^{3}H^{2}}{64\pi^{2}}+O(\lambda^{5})}\;.$
(53)

##  4 Summing the Logarithms

Here we demonstrate that the results of the previous section for the new model
(8) can be explained using the same combination of Starobinsky’s stochastic
formalism and the renormalization group that worked for the old model (5). We
begin by deriving the curvature-dependent effective potential and showing that
it explains the evolution of the background. We next apply our explicit
results for the 1-loop counterterms to work out the curvature-dependent
renormalization group corrections.

###  4.1 Curvature-Dependent Effective Potential

The shift symmetry of the field $B(x)$ evident in (8) prevents it from
developing an effective potential. However, $A(x)$ does acquire one from
integrating out the differentiated $B$ fields from the $A$ field equation in
the presence of a constant $A(x)=A_{0}$ background,

$\frac{\delta S[A,B]}{\delta
A(x)}=\partial_{\mu}\Bigl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}A\Bigr{]}-\frac{\lambda}{2}\Bigl{(}1\\!+\\!\frac{\lambda}{4}A\Bigr{)}^{3}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}=0\;.$
(54)

One can see from the Lagrangian (8) that a constant $A(x)=A_{0}$ background
just renormalizes the $B$ field strength,

$\Bigl{\langle}\Omega\Bigl{|}T\Bigl{[}B(x)B(x^{\prime})\Bigr{]}\Bigr{|}\Omega\Bigr{\rangle}_{A=A_{0}}=\frac{i\Delta(x;x^{\prime})}{(1+\frac{\lambda}{4}A_{0})^{4}}\;.$
(55)

Hence the $A$ equation can be recognized as that of a scalar potential model,

$\displaystyle\frac{\delta S[A,B]}{\delta
A(x)}\longrightarrow\partial_{\mu}\Bigl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}A\Bigr{]}-\frac{\lambda}{2}\Bigl{(}1\\!+\\!\frac{\lambda}{4}A\Bigr{)}^{3}\times\frac{\sqrt{-g}\,g^{\mu\nu}\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})|_{x^{\prime}=x}}{(1\\!+\\!\frac{\lambda}{4}A)^{4}}\;,$
(57)
$\displaystyle=\partial_{\mu}\Bigl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}A\Bigr{]}+\frac{(\frac{D-1}{2})\lambda
kH^{2}\sqrt{-g}}{1\\!+\\!\frac{\lambda}{4}A}\equiv\partial_{\mu}\Bigl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}A\Bigr{]}-V^{\prime}_{\rm
eff}(A)\sqrt{-g}\;.\qquad$

Note that we have employed expression (16) to evaluate the coincidence limit
of the doubly differentiated propagator. This is free of divergences, so we
can set $D=4$ to find,

$V^{\prime}_{\text{eff}}(A)=-\frac{3\lambda
H^{4}}{16\pi^{2}}\Bigl{(}1+\frac{\lambda}{4}A\Bigr{)}^{-1}\qquad\Longrightarrow\qquad
V_{\rm
eff}(A)=-\frac{3H^{4}}{4\pi^{2}}\ln\Bigl{(}1\\!+\\!\frac{\lambda}{4}A\Bigr{)}\;.$
(58)

We see that the new model’s curvature-dependent effective potential (58) is
almost the same as the old model’s result of
$-\frac{3H^{4}}{8\pi^{2}}\ln(1+\frac{\lambda}{2}A)$ [15].

Starobinsky long ago showed how to sum the leading logarithms of a scalar
potential model like (57) [13, 14]. The leading logarithms contained in
correlators of the quantum field $A(t,\vec{x})$ turn out to agree, to all
orders [9], with those of stochastic random field $\mathcal{A}(t,\vec{x})$
which obeys the Langevin equation,

$3H\Bigl{[}\dot{\mathcal{A}}-\dot{\mathcal{A}}_{0}\Bigr{]}=V^{\prime}_{\rm
eff}(\mathcal{A})\;.$ (59)

The “stochastic jitter” in this equation is supplied by the time derivative of
the infrared-truncated, free field expansion of $A(t,\vec{x})$,

$\mathcal{A}_{0}(t,\vec{x})\equiv\int_{H}^{aH}\\!\\!\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\frac{H}{\sqrt{2k^{3}}}\Bigl{\\{}\alpha(\vec{k})e^{i\vec{k}\cdot\vec{x}}+\alpha^{\dagger}(\vec{k})e^{-i\vec{k}\cdot\vec{x}}\Bigr{\\}}\quad,\quad\Bigl{[}\alpha(\vec{k}),\alpha^{\dagger}(\vec{k}^{\prime})\Bigr{]}=(2\pi)^{3}\delta^{3}(\vec{k}\\!-\\!\vec{k}^{\prime})\;.$
(60)

If we turn off the stochastic jitter then equation (59) is simple to solve,
adopting the initial condition $\mathcal{A}(0,\vec{x})=0$ and noting that
$\ln(a)=Ht$,

$\mathcal{A}(t,\vec{x})\Bigl{|}_{\mathcal{A}_{0}=0}=\frac{4}{\lambda}\left[\sqrt{1\\!+\\!\frac{\lambda^{2}H^{2}\ln(a)}{32\pi^{2}}}-1\right]=\frac{\lambda
H^{2}\ln(a)}{16\pi^{2}}-\frac{\lambda^{3}H^{4}\ln^{2}(a)}{2^{11}\pi^{4}}+O(\lambda^{5})\;.$
(61)

Because it is easier to fluctuate down the potential than up, we expect that
the effect of adding stochastic jitter is to accelerate the evolution of
$\mathcal{A}$ down its potential. Solving (59) perturbatively gives,

$\mathcal{A}=\mathcal{A}_{0}+\frac{\lambda
H^{2}\ln(a)}{16\pi^{2}}-\frac{\lambda^{2}H^{3}}{2^{6}\pi^{2}}\\!\\!\int_{0}^{t}\\!\\!\\!dt^{\prime}\mathcal{A}_{0}(t^{\prime},\vec{x})-\frac{\lambda^{3}H^{4}\ln^{2}(a)}{2^{11}\pi^{4}}+\frac{\lambda^{3}H^{3}}{2^{8}\pi^{2}}\\!\\!\int_{0}^{t}\\!\\!\\!dt^{\prime}\mathcal{A}^{2}_{0}(t^{\prime},\vec{x})+O(\lambda^{4})\;.$
(62)

By taking the expectation value of the previous equation, and using the mode
sum (60) to conclude,

$\Bigl{\langle}\Omega\Bigl{|}\mathcal{A}_{0}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}=0\qquad,\qquad\Bigl{\langle}\Omega\Bigl{|}\mathcal{A}_{0}^{2}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}=\frac{H^{2}\ln(a)}{4\pi^{2}}\;,$
(63)

we find,

$\Bigl{\langle}\Omega\Bigl{|}\mathcal{A}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}=\frac{\lambda
H^{2}\ln(a)}{16\pi^{2}}+O(\lambda^{5})\;.$ (64)

This is exactly the result (26) we got by explicit computation, which is
impressive confirmation of the stochastic prediction.

###  4.2 Curvature-Dependent Renormalization Group

In a theory with derivative interactions not all of the large logarithms
derive from stochastic effects. Some of them arise instead from the incomplete
cancellation between primitive divergences — which have no $D$-dependent
powers of the scale factor — and counterterms — which inherit a $a^{D}$ from
the measure factor $\sqrt{-g}$,

$\Bigl{(}\frac{H^{D-4}}{D\\!-\\!4}\Bigr{)}-\Bigl{(}\frac{\mu^{D-4}a^{D-4}}{D\\!-\\!4}\Bigr{)}=-\ln\Bigl{(}\frac{\mu
a}{H}\Bigr{)}+O(D\\!-\\!4)\;.$ (65)

These logarithms can be recovered using the renormalization group, which
follows factors of $\ln(\mu)$. Because we are only interested in cases where
the factors of $\ln(a)$ are not suppressed by inverse powers of the scale
factor, the counterterms of concern are those which can be viewed as
curvature-dependent renormalizations of bare parameters [15]. Of the four
counterterms (11) required to renormalize the vertex function at 1-loop order
we have already seen that the contribution proportional to $C_{4AB^{2}}$ can
be viewed as a curvature-dependent renormalization of the bare vertex, and we
used this to compute the associated beta function (36). Of the four
counterterms (10) required to renormalize the 1-loop self-masses it is the
contributions proportional to $C_{2A^{2}}$ and $C_{2B^{2}}$ which can be
regarded as curvature-dependent field strength renormalizations,

$Z_{A}\equiv 1+C_{2A^{2}}\\!\times\\!R+O(\lambda^{4})\qquad,\qquad Z_{B}\equiv
1+C_{2B^{2}}\\!\times\\!R+O(\lambda^{4})\;.$ (66)

Our explicit results (30) and (35) give the associated gamma functions,

$\boxed{\gamma_{A}\equiv\frac{\partial\ln(Z_{A})}{\partial\ln(\mu^{2})}=\frac{\lambda^{2}H^{2}}{32\pi^{2}}+O(\lambda^{4})\qquad,\qquad\gamma_{B}\equiv\frac{\partial\ln(Z_{B})}{\partial\ln(\mu^{2})}=-\frac{\lambda^{2}H^{2}}{64\pi^{2}}+O(\lambda^{4})\;.}$
(67)

We are ready to investigate the Callan-Symanzik equations for the n-point
Green’s functions $G_{n}(x_{1};x_{2};\ldots;x_{n};\lambda;\mu)$ of the field
$A$,2221PI $n$-point functions obey a similar equation with the last term
replaced by $-n\gamma_{A}$.

$\left[\mu\frac{\partial}{\partial\mu}+\beta\frac{\partial}{\partial\lambda}+n\gamma_{A}\right]G_{n}\Bigl{(}x_{1};x_{2};\ldots;x_{n};\lambda;\mu\Bigr{)}=0\;.$
(68)

This equation can be solved using the method of characteristics. We first find
a running coupling constant $\bar{\lambda}(\mu)$ which obeys the differential
equation and initial condition,

$\mu\frac{d\bar{\lambda}}{d\mu}=-\beta\Bigl{(}\bar{\lambda}(\mu)\Bigr{)}\quad,\quad\bar{\lambda}(\mu_{0})=\lambda\qquad\Longrightarrow\qquad\beta(\lambda)\frac{\partial\bar{\lambda}}{\partial\lambda}=\beta(\bar{\lambda})\;.$
(69)

We can then write the solution as,

$G_{n}\Bigl{(}x_{1};x_{2};\ldots;x_{n};\lambda;\mu\Bigr{)}=G_{n}\Bigl{(}x_{1};x_{2};\ldots;x_{n};\bar{\lambda}(\mu);\mu_{0}\Bigr{)}\\!\times\\!\exp\left[-n\\!\\!\int_{\mu_{0}}^{\mu}\\!\\!\frac{d\mu^{\prime}}{\mu^{\prime}}\gamma\Bigl{(}\bar{\lambda}(\mu^{\prime})\Bigr{)}\right]\;.$
(70)

Inserting the $\beta$ function (53) into (69), and ignoring higher loop
corrections yields,

$\bar{\lambda}(\mu)=\frac{\lambda}{\sqrt{1\\!-\\!\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(\frac{\mu}{\mu_{0}})}}\;.$
(71)

Substituting (71) and (67) into (70), and again ignoring higher loop
corrections, gives,

$G_{n}\Bigl{(}x_{1};x_{2};\ldots
x_{n};\lambda;\mu\Bigr{)}=G_{n}\Bigl{(}x_{1};x_{2};\ldots;x_{n};\bar{\lambda}(\mu);\mu_{0}\Bigr{)}\\!\times\\!\Bigl{[}1\\!-\\!\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln\Bigl{(}\frac{\mu}{\mu_{0}}\Bigr{)}\Bigr{]}^{n}\;.$
(72)

Similar results could be derived for Green’s functions which also, or even
exclusively, involve the field $B(x)$.

Having a negative beta function traditionally means that the theory runs
towards weak coupling in the ultraviolet because logarithms of the scale $\mu$
are associated with inverse factors of some characteristic momentum in the
process. In cosmology we are interested in how things behave at late times,
and we note from expression (65) that the scale $\mu$ is associated with the
scale factor $a(t)$ in the form $\ln[\mu a/H]$. It should therefore be that
having a negative beta function means the theory evolves towards strong
coupling at late times.

##  5 Conclusion

In this work, we examined a nontrivial nonlinear sigma model (defined in
section 2) whose loop corrections on de Sitter background show the same large
logarithms as have been reported from inflationary gravitons [3, 4, 5, 6, 7,
8]. Previous work [15, 16] on nonlinear sigma models which can be reduced to
free theories has shown that these large logarithms can be resummed by
combining a variant of Starobinsky’s stochastic formalism [13, 14] — based on
curvature-dependent effective potentials — with a variant of the
renormalization group — based on the subset of counterterms which can be
viewed as curvature-dependent renormalizations of bare parameters. Our
analysis confirms that these techniques continue to apply for models whose
S-matrix is nontrivial. In section 3.1, we explicitly computed the evolution
of the background (26) at 1-loop and 2-loop orders. In section 4.1 we showed
that the resummation techniques correctly predict these results.

One significant difference associated with a nontrivial S-matrix is that the
beta function does not vanish. The 1-loop beta function (53) was derived in
section 3.3. The 1-loop gamma functions (67) were derived in section 3.2, and
combined with the beta function in section 4.2, to solve the Callan-Symanzik
equation for $n$-point Green’s functions (72). Because the beta function (53)
is negative, the running coupling constant $\bar{\lambda}(\mu)$ grows with the
scale $\mu$ as shown in equation (71). From (65) we see that logarithms of
$\mu$ are associated with the scale factor $a(t)$ in the form $\ln[\mu
a(t)/H]$. This implies that the model becomes strongly coupled at late times.

An interesting higher loop phenomenon is that there can be mixing of large
stochastic logarithms with large logarithms from the renormalization group. We
speculate that the correct way to include these is to use the renormalization
group to improve the curvature-dependent effective potential and then use
that, without any extra RG corrections. From (67) we observe that, because
$\gamma_{A}\sim\lambda^{2}$, an RG-improvement to the effective potential (58)
will include lowest order corrections of the form $\lambda^{3}A$. Those would
not engender any changes in the background evolution at leading logarithm
order.

We should comment on how this work can be extended to slow-roll inflation
models, in particular to ultra-slow-roll inflation in which a nearly flat
region of the potential causes the scalar to almost stop rolling [26]. One
must be clear that neither of the nonlinear sigma model scalars $A(x)$ and
$B(x)$ serves as the inflaton; they are just spectators to inflation driven by
some other scalar field.333To make $A$ or $B$ the inflaton would require the
introduction of a potential for them, which is a very substantial modification
of the model. In this case exact calculations are no longer possible because
we lack the propagators for a general inflationary background. However, the
great thing about the resummation technique described in section 4 is that it
can be implemented on a general inflationary background. The stochastic
formalism described in section 4.1 relies on a curvature-dependent effective
potential whose effective force (57) derives from the coincidence limit of the
doubly differentiated free scalar propagator
$\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})$ in the
appropriate background. Although we do not have exact expressions for this
quantity, very good analytic approximations have been developed [17], and one
can use them to solve the resulting Langevin equation (59) numerically for any
cosmological background which has experienced primordial inflation [18]. If
the analytic approximations should happen to become unreliable for a
particular expansion history they can easily be improved [27]. The situation
for the curvature-dependent renormalization group described in section 4.2 is
even better because the coefficients of the counterterms are universal,
independent of the background geometry.

The infrared modes which engender the large logarithms which we have studied
are fascinating, but experience with gravity [28, 29, 30, 31, 32] shows that
one must employ a resummation technique which goes beyond linearized order in
order to understand what happens when perturbation theory breaks down.
Although our 1-loop and 2-loop results are obviously perturbative, we
emphasize that they were made merely to check the validity of the resummation
technique of section 4, which is fully nonperturbative. One can see that the
technique includes nonlinear effects from the form (58) of the curvature-
dependent effective potential. Note also that its field dependence of
$\ln(1+\tfrac{\lambda}{4}A)$ applies to any cosmological background; only the
multiplicative coefficient of $-\tfrac{3H^{4}}{4\pi^{2}}$ changes when the
geometry is no longer de Sitter. The solution (61) which results from ignoring
stochastic jitter is similarly nonperturbative — and it should be generally
true that jitter merely serves to accelerate the rate at which the $A$
background rolls down the effective potential. Finally, the Renormalization
Group formalism sums logarithms which result from renormalization (65) to all
orders in perturbation theory.

Acknowledgements

This work was partially supported by NSF grant PHY-2207514 and by DOE grant
DE-SC0022148 and by the Institute for Fundamental Theory at the University of
Florida.

##  Appendix A Reduction of the 1-loop vertex function

The aim of this appendix is to reduce the last line of equation (41) for
$-iV_{1abc}(x;y;z)$,

$I(x;y;z)\equiv\frac{\lambda^{3}}{4}\partial^{\rho}_{y}\partial^{\alpha}_{z}\Bigl{\\{}(a_{y}a_{z})^{D-2}\partial_{\rho}^{y}i\Delta(y;z)\partial_{\alpha}^{z}i\Delta(y;z)\Bigl{[}\delta^{D}(x\\!-\\!y)+\delta^{D}(x\\!-\\!z)\Bigr{]}\Bigr{\\}}\;.$
(73)

Note that $\partial_{\rho}^{y}i\Delta(y;z)\partial_{\alpha}^{z}i\Delta(y;z)$
is quadratically divergent, so extracting just the divergent part of
$I(x;y;z)$ requires only the two terms given on the first line of expansion
(14) for each propagator,

$\displaystyle\partial^{y}_{\rho}i\Delta(y;z)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\Gamma(\frac{D}{2})}{2\pi^{\frac{D}{2}}(a_{y}a_{z})^{\frac{D}{2}-1}}\left[\frac{(y\\!-\\!z)_{\rho}}{(y\\!-\\!z)^{D}}+\frac{a_{y}H\delta^{0}_{~{}\rho}}{2(y\\!-\\!z)^{D-2}}+\frac{Da_{y}a_{z}H^{2}(y\\!-\\!z)_{\rho}}{8(y\\!-\\!z)^{D-2}}+\ldots\right]\;,\qquad$
(74) $\displaystyle\partial^{z}_{\alpha}i\Delta(y;z)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\Gamma(\frac{D}{2})}{2\pi^{\frac{D}{2}}(a_{y}a_{z})^{\frac{D}{2}-1}}\left[-\frac{(y\\!-\\!z)_{\alpha}}{(y\\!-\\!z)^{D}}+\frac{a_{z}H\delta^{0}_{~{}\alpha}}{2(y\\!-\\!z)^{D-2}}-\frac{Da_{y}a_{z}H^{2}(y\\!-\\!z)_{\alpha}}{8(y\\!-\\!z)^{D-2}}+\ldots\right]\;.\qquad$
(75)

Multiplying (74) and (75), and retaining only potentially divergent terms
yields,

$\displaystyle\partial^{y}_{\rho}i\Delta(y;z)\\!\times\\!\partial^{z}_{\alpha}i\Delta(y;z)=\frac{\Gamma^{2}(\frac{D}{2})}{4\pi^{D}(a_{y}a_{z})^{D-2}}\Biggl{\\{}{\color[rgb]{1,0,0}-\frac{(y\\!-\\!z)_{\rho}(y\\!-\\!z)_{\alpha}}{(y\\!-\\!z)^{2D}}+\frac{a_{z}H\delta_{~{}\alpha}^{0}(y\\!-\\!z)_{\rho}}{2(y\\!-\\!z)^{2D-2}}\color[rgb]{0,0,0}}$
(76) $\displaystyle\hskip
71.13188pt{\color[rgb]{1,0,0}-\frac{a_{y}H\delta^{0}_{~{}\rho}(y\\!-\\!z)_{\alpha}}{2(y\\!-\\!z)^{2D-2}}+\frac{a_{y}a_{z}H^{2}\delta_{~{}\rho}^{0}\delta_{~{}\alpha}^{0}}{4(y\\!-\\!z)^{2D-4}}\color[rgb]{0,0,0}}{\color[rgb]{0,0,1}-\frac{Da_{y}a_{z}H^{2}(y\\!-\\!z)_{\rho}(y\\!-\\!z)_{\alpha}}{4(y\\!-\\!z)^{2D-2}}\color[rgb]{0,0,0}}+\dots\Biggr{\\}}\;.\qquad$

The terms highlighted in red and blue require separate reductions which we
give below.

We first extract two derivatives from the red terms of expression (76),

$\displaystyle\frac{\Gamma^{2}(\frac{D}{2})}{16\pi^{D}(a_{y}a_{z})^{D-2}}\Biggl{\\{}-\frac{(y\\!-\\!z)_{\rho}(y\\!-\\!z)_{\alpha}}{(y\\!-\\!z)^{2D}}+\frac{a_{z}H\delta_{~{}\alpha}^{0}(y\\!-\\!z)_{\rho}}{2(y\\!-\\!z)^{2D-2}}-\frac{a_{y}H\delta^{0}_{~{}\rho}(y\\!-\\!z)_{\alpha}}{2(y\\!-\\!z)^{2D-2}}+\frac{a_{y}a_{z}H^{2}\delta_{~{}\rho}^{0}\delta_{~{}\alpha}^{0}}{4(y\\!-\\!z)^{2D-4}}\Biggr{\\}}$
(77) $\displaystyle\hskip
19.91684pt=\frac{\Gamma^{2}(\frac{D}{2}\\!-\\!1)}{64\pi^{D}}\Biggr{\\{}\partial_{\rho}^{y}\partial_{\alpha}^{z}\left[\frac{1}{(a_{y}a_{z})^{D-2}(y\\!-\\!z)^{2D-4}}\right]+\frac{[\partial_{\rho}^{y}\partial_{\alpha}^{y}\\!-\\!\eta_{\rho\alpha}\partial^{2}_{y}]}{(D\\!-\\!1)(a_{y}a_{z})^{D-2}}\left[\frac{1}{(y\\!-\\!z)^{2D-4}}\right]\Biggr{\\}}.\qquad$

The denominator $(y-z)^{2D-4}$ is logarithmically divergent so we can extract
a local divergence from it by adding the flat space propagator equation [24,
25],

$\frac{1}{(y\\!-\\!z)^{2D-4}}=\frac{\mu^{D-4}}{2(D\\!-\\!3)(D\\!-\\!4)}\frac{4\pi^{\frac{D}{2}}\,i\delta^{D}(y\\!-\\!z)}{\Gamma(\frac{D}{2}\\!-\\!1)}-\frac{\partial^{2}_{y}}{4}\Biggl{[}\frac{\ln[\mu^{2}(y\\!-\\!z)^{2}]}{(y\\!-\\!z)^{2}}\Biggr{]}+O(D\\!-\\!4)\;.$
(78)

Substituting (78) in (77), and retaining only the divergences, reduces the red
terms of expression (76) to,

$\displaystyle\frac{\Gamma^{2}(\frac{D}{2})}{16\pi^{D}(a_{y}a_{z})^{D-2}}\Biggl{\\{}-\frac{(y\\!-\\!z)_{\rho}(y\\!-\\!z)_{\alpha}}{(y\\!-\\!z)^{2D}}+\frac{a_{z}H\delta_{~{}\alpha}^{0}(y\\!-\\!z)_{\rho}}{2(y\\!-\\!z)^{2D-2}}-\frac{a_{y}H\delta^{0}_{~{}\rho}(y\\!-\\!z)_{\alpha}}{2(y\\!-\\!z)^{2D-2}}+\frac{a_{y}a_{z}H^{2}\delta_{~{}\rho}^{0}\delta_{~{}\alpha}^{0}}{4(y\\!-\\!z)^{2D-4}}\Biggr{\\}}$
(79)
$\displaystyle=\frac{\Gamma(\frac{D}{2}\\!-\\!1)\mu^{D-4}}{32\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\Biggl{\\{}\partial^{y}_{\rho}\partial^{z}_{\alpha}\left[\frac{i\delta^{D}(y\\!-\\!z)}{(a_{y}a_{z})^{D-2}}\right]+\frac{[\partial_{\rho}^{y}\partial_{\alpha}^{y}\\!-\\!\eta_{\rho\alpha}\partial^{2}_{y}]\,i\delta^{D}(y\\!-\\!z)}{(D\\!-\\!1)(a_{y}a_{z})^{D-2}}\Biggr{\\}}+\Bigl{(}{\rm
Finite}\Bigr{)}.\qquad$

We now extract derivatives from the blue term in (76),

$\displaystyle-\frac{\Gamma(\frac{D}{2})\Gamma(\frac{D}{2}\\!+\\!1)H^{2}}{8\pi^{D}(a_{y}a_{z})^{D-3}}\\!\times\\!\frac{(y\\!-\\!z)_{\rho}(y\\!-\\!z)_{\alpha}}{(y\\!-\\!z)^{2D-2}}$
(80) $\displaystyle\hskip
71.13188pt=-\frac{\Gamma(\frac{D}{2}\\!-\\!1)\Gamma(\frac{D}{2}\\!+\\!1)H^{2}}{32\pi^{D}(a_{y}a_{z})^{D-3}}\\!\times\\!\left[\frac{\partial_{\rho}^{y}\partial_{\alpha}^{y}}{2(D\\!-\\!3)}+\frac{\eta_{\rho\alpha}\partial^{2}_{y}}{2(D\\!-\\!3)(D\\!-\\!4)}\right]\frac{1}{(y-z)^{2D-6}}.\qquad$

Of the two terms inside the square brackets at the end of (80), only the one
proportional to $\eta_{\rho\alpha}\partial^{2}_{y}$ is divergent. Using the
same reduction as (78) we therefore reduce the blue term of expression (76)
to,

$-\frac{\Gamma(\frac{D}{2})\Gamma(\frac{D}{2}\\!+\\!1)H^{2}}{8\pi^{D}(a_{y}a_{z})^{D-3}}\\!\times\\!\frac{(y\\!-\\!z)_{\rho}(y\\!-\\!z)_{\alpha}}{(y\\!-\\!z)^{2D-2}}=-\frac{\Gamma(\frac{D}{2}\\!+\\!1)}{16\pi^{\frac{D}{2}}}\frac{\mu^{D-4}\eta_{\rho\alpha}H^{2}i\delta^{D}(y\\!-\\!z)}{(D\\!-\\!3)(D\\!-\\!4)(a_{y}a_{z})^{D-3}}+\Bigl{(}{\rm
Finite}\Bigr{)}.$ (81)

We are now ready to employ expressions (79) and (81) in (73). Note that the
transverse projection operator in the second term of (79) vanishes upon
canceling the scale factors and exploiting the $i\delta^{D}(y-z)$ to reflect
derivatives where necessary,

$\displaystyle\frac{\lambda^{3}}{4}\partial^{\rho}_{y}\partial^{\alpha}_{z}\Biggl{\\{}(a_{y}a_{z})^{D-2}\\!\times\\!\frac{\Gamma(\frac{D}{2}\\!-\\!1)\mu^{D-4}}{32\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\frac{[\partial^{y}_{\rho}\partial^{y}_{\alpha}\\!-\\!\eta_{\rho\alpha}\partial^{2}_{y}]i\delta^{D}(y\\!-\\!z)}{(D\\!-\\!1)(a_{y}a_{z})^{D-2}}\Bigl{[}\delta^{D}(x\\!-\\!y)+\delta^{D}(x\\!-\\!z)\Bigr{]}\Biggr{\\}}$
(82)
$\displaystyle=\frac{\lambda^{3}\mu^{D-4}\Gamma(\frac{D}{2}\\!-\\!1)\partial^{\rho}_{y}\partial^{\alpha}_{z}}{128\pi^{\frac{D}{2}}(D\\!-\\!1)(D\\!-\\!3)(D\\!-\\!4)}\Biggl{\\{}\Bigl{[}\partial^{z}_{\rho}\partial^{z}_{\alpha}\\!-\\!\eta_{\rho\alpha}\partial^{2}_{z}\Bigr{]}i\delta^{D}(y\\!-\\!z)\\!\times\\!\delta^{D}(x\\!-\\!y)$
$\displaystyle\hskip
199.16928pt+\Bigl{[}\partial^{y}_{\rho}\partial^{y}_{\alpha}\\!-\\!\eta_{\rho\alpha}\partial^{2}_{y}\Bigr{]}i\delta^{D}(y\\!-\\!z)\\!\times\\!\delta^{D}(x\\!-\\!z)\Biggr{\\}}=0\;.\qquad$

The divergent part of $I(x;y;z)$ comes from the first term of (79) and from
(80). After some judicious partial integrations it can be written as,

$\displaystyle I_{\rm
div}=\frac{i\lambda^{3}\mu^{D-4}\Gamma(\frac{D}{2}\\!-\\!1)}{128\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\Biggl{\\{}2\mathcal{D}_{y}\mathcal{D}_{z}\left[\frac{\delta^{D}(x\\!-\\!y)\delta^{D}(x\\!-\\!z)}{(a_{y}a_{z})^{D-2}}\right]-\mathcal{D}_{y}\partial^{\alpha}_{z}\left[\frac{\delta^{D}(y\\!-\\!z)\partial^{z}_{\alpha}\delta^{D}(x\\!-\\!z)}{a_{y}^{D-2}}\right]$
(83) $\displaystyle\hskip
14.22636pt-\mathcal{D}_{z}\partial^{\rho}_{y}\\!\\!\left[\frac{\delta^{D}(y\\!-\\!z)\partial^{y}_{\rho}\delta^{D}(x\\!-\\!y)}{a_{z}^{D-2}}\right]\\!\\!\Biggr{\\}}-\frac{i\lambda^{3}H^{2}a_{x}^{2}\mu^{D-4}\Gamma(\frac{D}{2}\\!+\\!1)}{32\pi^{\frac{D}{2}}(D\\!-\\!3)(D\\!-\\!4)}\,\partial^{\mu}_{x}\delta^{D}(x\\!-\\!y)\partial^{x}_{\mu}\delta^{D}(x\\!-\\!z)\;.\qquad$

##  Appendix B Parameters and Variables

The aim of this appendix is to list all the parameters and variables used in
the paper.

In the first section, we introduce scale factor $a(t)$, Hubble parameter
$H(t)$ and first slow roll parameter $\epsilon(t)$ in (1). In the same
section, variables $t$, $r$ and $k$ are comoving time, comoving coordiante
distance to a point source and plane wave number, respectively, given in (2)
and (3). Moreover, $G$ denotes Newton’s constant of gravitation.
In the second section, we introduce our model. $A$ and $B$ denotes two scalar
fields, and the model is given in (8). $\lambda$ denotes the dimensionful
coupling constant of our model. In (10) and (11), the coefficients
$C_{1A^{2}}$, $C_{2A^{2}}$, $C_{1B^{2}}$, $C_{2A^{2}}$,
$C_{1AB^{2}}$,$C_{2AB^{2}}$, $C_{3AB^{2}}$ and $C_{4AB^{2}}$ are counterterms.
Also, $D$ denotes the spacetime dimension and $\mu$ is the scale of the
dimensional regularization. In the same section, we define the kinetic
operator $\mathcal{D}$ in (12), and provide the scalar propagator
$i\Delta(x:x^{\prime})$ for fields $A$ and $B$ in (13). It should be noted
that $k$ does not denote the wave number in the rest of the paper, but defined
as a function of Hubble parameter in (13). The notation adopted in the paper
is summarised in the last paragraph of this section.
In the third section, the contribution of each diagram in Figure (4) to
expectation value of $A$ is labeled $A_{2a}$ through $A_{2h}$, and the results
are given in (24) and (25). Furthermore, we provide the explicit form of
3-point and 4-point contributions to 1PI 2-loop functions
$-iM^{2}_{A}(x;x^{\prime})$ and $-iM^{2}_{B}(x;x^{\prime})$ in (27) through
(33). In the same section, $\beta$ denotes the beta function for the field
$A$, and it is defined in (36).
Lastly in the fourth section, $\mathcal{A}$ denotes stochastic random field.
In (66) and (67), we provide the expressions for field strength
renormalizations and gamma functions. They are denoted by $Z$ and $\gamma$,
respectively. $G(x_{1};x_{2};...;x_{n};\lambda;\mu)$ denotes the n-point
Green’s function, first time appearing in (68).

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