UFIFT-QG-21-06 , CCTP-2021-05

Summing Inflationary Logarithms in Nonlinear Sigma Models

S. P. Miao1⋆, N. C. Tsamis2† and R. P. Woodard3‡

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

2 Institute of Theoretical Physics & Computational Physics,
Department of Physics, University of Crete,
GR-710 03 Heraklion, HELLAS

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

ABSTRACT

We consider two nonlinear sigma models on de Sitter background which involve
the same derivative interactions as quantum gravity but without the gauge
issue. The first model contains only a single field, which can be reduced to a
free theory by a local field redefinition; the second contains two fields and
cannot be so reduced. Loop corrections in both models produce large temporal
and spatial logarithms which cause perturbation theory to break down at late
times and large distances. Many of these logarithms derive from the “tail”
part of the propagator and can be summed using a variant of Starobinsky’s
stochastic formalism involving a curvature-dependent effective potential. The
remaining logarithms derive from the ultraviolet and can be summed using a
variant of the renormalization group based on a special class of curvature-
dependent renormalizations. Explicit results are derived at 1-loop and 2-loop
orders.

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

In their pioneering work on perturbation theory in nontrivial geometries,
DeWitt and Brehme devoted special attention to the behavior of propagators and
Green’s functions near coincidence [1]. They noted that while the leading
singularity is a universal function of the invariant separation, there is a
less singular part which depends on the geometry and on the properties of the
field. They called this sub-dominant singularity the “tail” term.

The tail terms of certain fields become maximally strong during the
accelerated expansion of inflation. For many purposes the background geometry
of inflation can be taken to be de Sitter,

$ds^{2}=a^{2}(\eta)\Bigl{[}-d\eta^{2}+d\vec{x}\\!\cdot\\!d\vec{x}\Bigr{]}=-dt^{2}+a^{2}d\vec{x}\\!\cdot\\!d\vec{x}\quad,\quad
a(\eta)=-\frac{1}{H\eta}=e^{Ht}\;.$ (1)

On $D=4$ dimensional de Sitter background the propagator of a massless,
minimally coupled scalar, in Bunch-Davies vacuum, is,

$i\Delta(x;x^{\prime})\Bigl{|}_{D=4}=\frac{1}{4\pi^{2}}\Biggl{\\{}\frac{1}{aa^{\prime}\Delta
x^{2}}-\frac{H^{2}}{2}\ln\Bigl{[}\frac{1}{4}H^{2}\Delta
x^{2}\Bigr{]}\Biggr{\\}}\;,$ (2)

where the Poincaré interval is,

$\Delta
x^{2}(x;x^{\prime})\equiv\Bigl{\|}\vec{x}\\!-\\!\vec{x}^{\prime}\Bigr{\|}^{2}-\Bigl{(}|\eta\\!-\\!\eta^{\prime}|\\!-\\!i\epsilon\Bigr{)}^{2}\;.$
(3)

The tail term of expression (2) is the part involving the logarithm.

A curious feature of the massless, minimally coupled scalar tail is that its
coincidence limit grows with time [2, 3, 4],

$i\Delta(x;x)=\Bigl{(}{\rm Divergent\
constant}\Bigr{)}+\frac{H^{2}}{4\pi^{2}}\ln(a)\;.$ (4)

Expression (4) was the first example of a general sort of secular effect
encountered in loop corrections involving interactions between nearly massless
and minimally coupled scalars [5, 6, 7, 8, 9]. These secular logarithms
attracted much attention during the opening decade of the 21st century because
they have the potential to enhance loop corrections to the power spectrum [10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

A fascinating aspect of the secular logarithms encountered in loop corrections
to scalar potential models,

$\mathcal{L}=-\frac{1}{2}\partial_{\mu}\Phi\partial_{\nu}\Phi
g^{\mu\nu}\sqrt{-g}-V(\Phi)\sqrt{-g}\;,$ (5)

is that the steady growth of $\ln(a)=Ht$ must eventually overwhelm even the
smallest loop-counting parameter. One cannot conclude from this that loop
corrections ever become large, just that the standard loop expansion breaks
down. Some sort of nonperturbative resummation is required to determine what
actually happens.

Starobinsky quite early developed a stochastic formalism which not only
predicts the leading logarithms of scalar potential models at each order in
perturbation theory [25], but also gives the late time form in those cases for
which a static limit is approached [26]. Starobinsky’s formalism is based on
replacing the full field operator $\Phi(t,\vec{x})$ with a stochastic field
$\varphi(t,\vec{x})$ which commutes with itself
$[\varphi(t,\vec{x}),\varphi(t^{\prime},\vec{x}^{\prime})]=0$, and whose
correlators are completely free of ultraviolet divergences. This stochastic
field $\varphi(t,\vec{x})$ is constructed from the same free creation and
annihilation operators that appear in $\Phi(t,\vec{x})$ in such a way that the
two fields produce the same leading logarithms at each order in perturbation
theory. The Heisenberg field equation for $\Phi$ gives rise to a Langevin
equation for $\varphi$ (which we express in co-moving coordinates),

$\frac{\delta
S[\Phi]}{\delta\Phi(x)}=\partial_{\mu}\Bigl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}\Phi\Bigr{]}-V^{\prime}(\Phi)\sqrt{-g}\longrightarrow
3Ha^{3}\Bigl{[}\dot{\varphi}-\dot{\varphi}_{0}\Bigr{]}-V^{\prime}(\varphi)a^{3}\;.$
(6)

Here $\varphi_{0}(t,\vec{x})$ is a truncation of the Yang-Feldman free field
with the ultraviolet excised and the mode function taken to its limiting
infrared form,

$\varphi_{0}(t,\vec{x})\equiv\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\theta\Bigl{(}aH\\!-\\!k\Bigr{)}\frac{\theta(k\\!-\\!H)H}{\sqrt{2k^{3}}}\Bigl{\\{}\alpha_{\vec{k}}e^{i\vec{k}\cdot\vec{x}}+\alpha^{\dagger}_{\vec{k}}e^{-i\vec{k}\cdot\vec{x}}\Bigr{\\}}\;,$
(7)

One derives (6) by first integrating the exact field equation to reach the
Yang-Feldman form. One then notes that reaching leading logarithm order
requires each free field to contribute an infrared logarithm, so there will be
no change to correlators, at leading logarithm order, if the full free field
mode sum is replaced by (7). Differentiating this truncated Yang-Feldman
equation gives Starobinsky’s Langevin equation [27].

The problem of summing up large logarithms in flat space scattering amplitudes
seems similar, and that has prompted particle theorists to try applying
renormalization group methods to understanding the evolution of cosmological
correlators [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. However, the
problems with this approach become obvious upon closer examination of the
analogy on which it is based. The renormalization group of flat space
describes how correlators change when the positions of field operators are
adiabatically expanded (or compressed) by some constant:

${\rm Renormalization\ Group:}\qquad x^{\mu}\longrightarrow A\times
x^{\mu}\;.$ (8)

What we really want to know in cosmology is how correlators change when
infinitesimal intervals are expanded by the time-dependent scale factor:

${\rm Cosmological\ Evolution:}\qquad dx^{\mu}\longrightarrow a(\eta)\times
dx^{\mu}\;.$ (9)

It is not clear how to relate the two processes, and simple correspondences
such as $A\longrightarrow a(\eta)$, or the renormalization scale
$\mu\longrightarrow H$, can easily be shown to fail by direct computation
[40]. Another crucial obstacle is that the leading logarithms of scalar
potential models arise entirely from the infrared, without regard to
renormalization. And the fact is that, despite years of heroic effort by
talented physicists [41, 42], no one has yet been able to devise a version of
the renormalization group which gives complete agreement for the leading
logarithms of scalar potential models [43].

Massless, minimally coupled scalars also engender large logarithms when they
interact with fermions [44] and with photons [45, 46, 47, 48]. In both cases
the other fields do not themselves generate large logarithms, but their
dynamics modify the ways in which these logarithms manifest. Such
modifications derive as much from the ultraviolet as from the infrared, so no
simple truncation procedure captures the correct result. However, integrating
out the other fields produces a scalar potential model whose large logarithms
are correctly captured by the stochastic formalism [44, 49].

On a general cosmological background it turns out that dynamical gravitons
obey the same equation as the massless, minimally coupled [50], so loop
corrections from inflationary gravitons should also induce large logarithms.
Of course the computations are much more difficult but a number of 1PI (one-
particle-irreducible) 1-point and 2-point functions have been evaluated at
1-loop and 2-loop orders in pure gravity [51, 52, 53, 54] and in gravity plus
various matter theories [55, 56, 57, 58, 59, 60, 61, 62]. When the 1PI 2-point
functions are used to quantum-correct the linearized effective field equations
one often (but not always) finds large logarithmic corrections to mode
functions and to exchange potentials [63, 64, 65, 66, 67, 68, 69]. These are
very challenging calculations, and it has been suggested that some of them may
be gauge artifacts [70, 71, 72, 73, 74, 75, 76, 77]. A procedure has been
developed to purge gauge dependence from 1PI 2-point functions [78, 79], and
its implementation is being undertaken as rapidly as the formidable
computational challenges permit [80].

Assuming the large logarithms of inflationary gravitons are real, the question
is how they can be re-summed. The derivative couplings of gravity pose an
obstacle to a completely stochastic explanation of these results because
derivatives preclude every free field from inducing a large logarithm, which
was an essential part of the proof that the stochastic formalism works for
scalar potential models [27]. Further, direct studies have shown that some of
the logarithms cannot be explained using the stochastic formalism [81], nor
are all of the logarithms due to the tail part of the graviton propagator
[82]. What we need is a simple format in which the complications of derivative
interactions can be sorted out without intricate computations which require a
year or more to complete.

Nonlinear sigma models would seem to provide a natural paradigm for derivative
interactions. These models consist of normal scalar kinetic terms which are
multiplied by functions of undifferentiated scalars, giving rise to the same
sort of derivative interactions as quantum gravity but without the
distractions of tensor indices and gauge fixing. Early work focused on
deriving a completely stochastic representation of the large logarithms
induced by these models [27], and that approach has been extensively pursued
by Kitamoto and Kitazawa [83, 84, 85]. We have thought it good to revisit this
problem after the realization that no completely stochastic approach can
capture all the large logarithms induced inflationary gravitons [81, 82]. The
point of this paper is to demonstrate that the large logarithmics of nonlinear
sigma models on de Sitter can be explained by combining a variant of
Starobinsky’s stochastic formalism with a variant of the renormalization
group.

This paper consists of six sections, of which the first is nearly done. In
section 2 we introduce the two nonlinear sigma models that will be studied.
Section 3 works out 1-loop corrections to the mode functions and exchange
potentials of the first model, as well as to 1-loop and 2-loop expectation
values of the field and its square. The same things are computed for the
second model in section 4. Section 5 collects the various large logarithms
exposed by all this work. We then demonstrate that many of these large
logarithms arise from stochastic effects associated with a curvature-dependent
effective potential induced by the kinetic terms. The remaining large
logarithms follow from employing the Callan-Symanzik equation to a special
class of counterterms that can be viewed as curvature-dependent
renormalizations of the bare theories. Our conclusions comprise section 6,
particularly the lessons for quantum gravity.

##  2 Two Nonlinear Sigma Models

This section introduces the two models upon which this study is based. The
first is a single field model which gives a free theory by a local field
redefinition; the second is a model based on two fields which is fundamentally
interacting. For each model we give the bare Lagrangian and the first two
variations of the action. We also present the Feynman rules and some important
identities for the coincidence limits of the propagator.

###  2.1 Single Field Model

The Lagrangian of the first model we will study is,

$\mathcal{L}=-\frac{1}{2}f^{2}(\Phi)\partial_{\mu}\Phi\partial_{\nu}\Phi
g^{\mu\nu}\sqrt{-g}\;.$ (10)

A nonlinear sigma model based on a single field can be reduced to free
theories by local field redefinitions. For the case of (10) the free field
$\Psi(x)$ obeys,

$d\Psi\equiv
f(\Phi)d\Phi\qquad\Longrightarrow\qquad\mathcal{L}=-\frac{1}{2}\partial_{\mu}\Psi\partial_{\nu}\Psi
g^{\mu\nu}\sqrt{-g}\;.$ (11)

Of course the existence of such a local field redefinition means that the flat
space $S$-matrix is unity but interactions can still cause interesting changes
to the kinematics of free fields, and to the evolution of the $\Phi$
background. Quantifying these changes at 1-loop and 2-loop orders will teach
us much.

We must select the function $f(\Phi)$ in order to define a specific model. The
simplest choice involves a single, dimensionful coupling constant $\lambda$,

$f(\Phi)=1\\!+\\!\frac{\lambda}{2}\Phi\quad\Longrightarrow\quad\Psi[\Phi]=\Phi\\!+\\!\frac{\lambda}{4}\Phi^{2}\quad\Longleftrightarrow\quad\Phi[\Psi]=\frac{2}{\lambda}\Bigl{[}\sqrt{1\\!+\\!\lambda\Psi}\\!-\\!1\Bigr{]}\;.$
(12)

With this choice of $f(\Phi)$ the Heisenberg field equation is,

$\frac{\delta
S[\Phi]}{\delta\Phi(x)}=\Bigl{(}1\\!+\\!\frac{1}{2}\lambda\Phi\Bigr{)}\partial_{\mu}\Bigl{[}(1\\!+\\!\frac{1}{2}\lambda\Phi\Bigr{)}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\Bigr{]}\;.$
(13)

We will also sometimes need the second variation,

$\displaystyle\frac{\delta^{2}S[\Phi]}{\delta\Phi(x)\delta\Phi(x^{\prime})}=\lambda\delta^{D}(x\\!-\\!x^{\prime})\partial_{\mu}\Bigl{[}\Bigl{(}1\\!+\\!\frac{1}{2}\lambda\Phi\Bigr{)}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\Bigr{]}$
(14)
$\displaystyle-\frac{1}{4}\lambda^{2}\delta^{D}(x\\!-\\!x^{\prime})\sqrt{-g}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi+\partial_{\mu}\Bigl{[}\Bigl{(}1\\!+\\!\frac{1}{2}\lambda\Phi\Bigr{)}^{2}\\!\\!\sqrt{-g}g^{\mu\nu}\partial_{\nu}\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}.\qquad$

In $D$ spacetime dimensions the propagators of both the $\Phi$ and the $\Psi$
fields obey the same equation,

$\partial^{\mu}\Bigl{[}a^{D-2}\partial_{\mu}i\Delta(x;x^{\prime})\Bigr{]}\equiv\mathcal{D}i\Delta(x;x^{\prime})=i\delta^{D}(x\\!-\\!x^{\prime})\;.$
(15)

The solution is [5, 6],

$i\Delta(x;x^{\prime})=F\Bigl{(}y(x;x^{\prime})\Bigr{)}+k\ln(aa^{\prime})\qquad,\qquad
k\equiv\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\;,$
(16)

where the de Sitter length function is $y(x;x^{\prime})\equiv
aa^{\prime}H^{2}\Delta x^{2}(x;x^{\prime})$ and the first derivative of the
function $F(y)$ is,

$\displaystyle
F^{\prime}(y)=-\frac{H^{D-2}}{4(4\pi)^{\frac{D}{2}}}\Biggl{\\{}\Gamma\Bigl{(}\frac{D}{2}\Bigr{)}\Bigl{(}\frac{4}{y}\Bigr{)}^{\frac{D}{2}}+\Gamma\Bigl{(}\frac{D}{2}\\!+\\!1\Bigr{)}\Bigl{(}\frac{4}{y}\Bigr{)}^{\frac{D}{2}-1}$
(17) $\displaystyle\hskip
56.9055pt+\sum_{n=0}^{\infty}\Biggl{[}\frac{\Gamma(n\\!+\\!\frac{D}{2}\\!+\\!2)}{\Gamma(n\\!+\\!3)}\Bigl{(}\frac{y}{4}\Bigr{)}^{n-\frac{D}{2}+2}-\frac{\Gamma(n\\!+\\!D)}{\Gamma(n\\!+\\!\frac{D}{2}\\!+\\!1)}\Bigl{(}\frac{y}{4}\Bigr{)}^{n}\Biggr{]}\Biggr{\\}}.\qquad$

The coincidence limits of the propagator and its first two derivatives are,

$\displaystyle i\Delta(x;x)=k\Bigl{[}-\pi{\rm
cot}\Bigl{(}\frac{D\pi}{2}\Bigr{)}+2\ln(a)\Bigr{]}$ ,
$\displaystyle\partial_{\mu}i\Delta(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}=kHa\delta^{0}_{~{}\mu}\;,\qquad$
(18)
$\displaystyle\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}=-\Bigl{(}\frac{D\\!-\\!1}{D}\Bigr{)}kH^{2}g_{\mu\nu}$
, $\displaystyle\partial_{\mu}i\Delta(x;x)=2kHa\delta^{0}_{~{}\mu}\;.\qquad$
(19)

###  2.2 Two Field Model

The simplest truly interacting nonlinear sigma model would seem to be,

$\mathcal{L}=-\frac{1}{2}\partial_{\mu}A\partial_{\nu}Ag^{\mu\nu}\sqrt{-g}-\frac{1}{2}\Bigl{(}1\\!+\\!\frac{1}{2}\lambda
A\Bigr{)}^{2}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;.$ (20)

The first variations of its action are,

$\displaystyle\frac{\delta S[A,B]}{\delta A(x)}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Bigl{[}\sqrt{-g}g^{\mu\nu}\partial_{\nu}A\Bigr{]}-\frac{1}{2}\lambda\Bigl{(}1\\!+\\!\frac{1}{2}\lambda
A\Bigr{)}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;,\qquad$ (21)
$\displaystyle\frac{\delta S[A,B]}{\delta B(x)}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Bigl{[}\Bigl{(}1\\!+\\!\frac{1}{2}\lambda
A\Bigr{)}^{2}\sqrt{-g}g^{\mu\nu}\partial_{\nu}B\Bigr{]}\;.\qquad$ (22)

And the second variations work out to be,

$\displaystyle\frac{\delta^{2}S[A,B]}{\delta A(x)\delta A(x^{\prime})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\partial_{\mu}\Bigl{[}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}$
(23) $\displaystyle\hskip
99.58464pt-\frac{1}{4}\lambda^{2}\delta^{D}(x\\!-\\!x^{\prime})\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;,\qquad$
$\displaystyle\frac{\delta^{2}S[A,B]}{\delta B(x)\delta B(x^{\prime})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\partial_{\mu}\Bigl{[}\Bigl{(}1\\!+\\!\frac{1}{2}\lambda
A\Bigr{)}^{2}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}\;.\qquad$
(24)

The propagators of both $A$ and $B$ are the same as $i\Delta(x;x^{\prime})$
given in expression (16). The other Feynman rules for the bare action are the
$\lambda A\partial B\partial B$ and $\lambda^{2}A^{2}\partial B\partial B$
vertices. All are depicted in Figure 1.

Figure 1: The Feynman rules of the $A$-$B$ model. $A$ lines are solid whereas
$B$ lines are dashed, and both propagators have the functional form (16).

##  3 Large Logarithms in the Single Field Model

In this section we calculate 1-loop and 2-loop corrections to a variety of
quantities in the single field model. We begin with the 1-loop self-mass,
which is then used to compute 1-loop corrections to the plane wave mode
function and the response to a point source. The section closes with a
evaluation of the expectation values of the field and its square at 1-loop and
2-loop orders.

###  3.1 The Self-Mass

The 1PI 2-point function can be expressed as,

$-iM^{2}(x;x^{\prime})\equiv\Biggl{\langle}\Omega\Biggl{|}\frac{i\delta
S[\Phi]}{\delta\Phi(x)}\frac{i\delta
S[\Phi]}{\delta\Phi(x^{\prime})}+\frac{i\delta^{2}S[\Phi]}{\delta\Phi(x)\delta\Phi(x^{\prime})}\Biggr{|}\Omega\Biggr{\rangle}\;.$
(25)

Substituting (13) into the first term and using (12) gives,

$\displaystyle-iM^{2}_{\Phi
3}(x;x^{\prime})=\Bigl{(}\frac{i\lambda}{2}\Bigr{)}^{2}\Biggl{\\{}\frac{1}{2}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}$
(26)
$\displaystyle-\mathcal{D}\Bigl{[}{a^{\prime}}^{D-2}{\partial^{\prime}}^{\rho}i\Delta(x;x^{\prime})\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\Bigr{]}\\!-\\!\mathcal{D}^{\prime}\Bigl{[}a^{D-2}\partial^{\mu}i\Delta(x;x^{\prime})\partial_{\mu}i\Delta(x;x^{\prime})\Bigr{]}$
$\displaystyle\hskip
136.5733pt+2(aa^{\prime})^{D-2}\partial^{\mu}{\partial^{\prime}}^{\rho}i\Delta(x;x^{\prime})\partial_{\mu}\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\Biggr{\\}}.\qquad$

The second term in (25) comes from the second variation (14),

$\displaystyle-iM^{2}_{\Phi
4}(x;x^{\prime})=\frac{i\lambda^{2}}{4}\Biggl{\\{}-\delta^{D}(x\\!-\\!x^{\prime})\,a^{D-2}\partial^{\prime}_{\mu}\partial^{\mu}i\Delta(x;x^{\prime})$
(27) $\displaystyle\hskip
170.71652pt+\partial^{\mu}\Bigl{[}i\Delta(x;x)a^{D-2}\partial_{\mu}\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}\Biggr{\\}}.\qquad$

The 3-point contribution (26) can be reduced by a series of partial
integrations whose general form will occur repeatedly. We will present them
this once in detail and not again,

$\displaystyle
a^{D-2}\partial^{\mu}i\Delta(x;x^{\prime})\partial_{\mu}i\Delta(x;x^{\prime})=\partial^{\mu}\Bigl{[}a^{D-2}i\Delta(x;x^{\prime})\partial_{\mu}i\Delta(x;x^{\prime})\Bigr{]}$
(28) $\displaystyle\hskip
28.45274pt-i\Delta(x;x^{\prime})\mathcal{D}i\Delta(x;x^{\prime})=\frac{1}{2}\mathcal{D}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\\!-i\Delta(x;x)i\delta^{D}(x\\!-\\!x^{\prime})\;,\qquad$
$\displaystyle
2(aa^{\prime})^{D-2}\partial^{\mu}{\partial^{\prime}}^{\rho}i\Delta(x;x^{\prime})\partial_{\mu}\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})=2\partial^{\mu}\Bigl{[}(aa^{\prime})^{D-2}{\partial^{\prime}}^{\rho}i\Delta(x;x^{\prime})$
(31) $\displaystyle\hskip
71.13188pt\times\partial_{\mu}\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\Bigr{]}-2{a^{\prime}}^{D-2}{\partial^{\prime}}^{\rho}i\Delta(x;x^{\prime})\partial^{\prime}_{\rho}\mathcal{D}i\Delta(x;x^{\prime})\;,\qquad$
$\displaystyle=\mathcal{D}\Bigl{[}{a^{\prime}}^{D-2}{\partial^{\prime}}^{\rho}i\Delta(x;x^{\prime})\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\Bigr{]}-2{a^{\prime}}^{D-2}{\partial^{\prime}}^{\rho}i\Delta(x;x^{\prime})\partial^{\prime}_{\rho}i\delta^{D}(x\\!-\\!x^{\prime})\;,\qquad$
$\displaystyle=\frac{1}{2}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\\!\\!\\!-\\!\mathcal{D}\Bigl{[}i\Delta(x;x)i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}\\!-\\!2kHa^{D-1}\partial_{0}i\delta^{D}(x\\!-\\!x^{\prime})\;.\qquad$

Reducing the four parts of (26) gives,

$-iM^{2}_{\Phi
3}(x;x^{\prime})=\frac{\lambda^{2}}{4}\Biggl{\\{}-i\Delta(x;x)\mathcal{D}\Bigl{[}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}+2kHa^{D-1}\partial_{0}i\delta^{D}(x\\!-\\!x^{\prime})\Biggr{\\}}.$
(32)

Applying the similar reductions to (27) allows the 4-point contribution to be
expressed as,

$\displaystyle-iM^{2}_{\Phi
4}(x;x^{\prime})=\frac{\lambda^{2}}{4}\Biggl{\\{}i\Delta(x;x)\mathcal{D}\Bigl{[}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}$
(33) $\displaystyle\hskip
68.28644pt-2kHa^{D-1}\partial_{0}i\delta^{D}(x\\!-\\!x^{\prime})+(D\\!-\\!1)kH^{2}a^{D}i\delta^{D}(x\\!-\\!x^{\prime})\Biggr{\\}}.\qquad$

When (32) and (33) are added, all the divergences cancel and we can take the
unregulated limit for the final result,

$-iM^{2}_{\Phi}(x;x^{\prime})=\frac{3\lambda^{2}H^{4}a^{4}}{32\pi^{2}}\,i\delta^{4}(x\\!-\\!x^{\prime})\;.$
(34)

Equation (34) corresponds to a tachyonic mass of
$m^{2}_{\Phi}=-3\lambda^{2}H^{4}/32\pi^{2}$. Note that the unit S-matrix
implied by (12) does not preclude interactions from changing the free field
kinematics. We will see that evolution can also occur, and that composite
operators still require field strength renormalization.

###  3.2 1-Loop Mode Function and Exchange Potential

The self-mass supplies the quantum correction to the linearized effective
field equation for $\Phi(x)$,

$\mathcal{D}\Phi(x)-\int\\!\\!d^{4}x^{\prime}M^{2}(x;x^{\prime})\Phi(x^{\prime})=\Bigl{[}\mathcal{D}+\frac{3\lambda^{2}H^{4}a^{4}}{32\pi^{2}}+O(\lambda^{4})\Bigr{]}\Phi(x)=J(x)\;,$
(35)

where $\mathcal{D}\equiv\partial^{\mu}a^{2}\partial_{\mu}$ is the kinetic
operator which was introduced in equation (15). We will study 1-loop
corrections to the kinematics of free scalar fields (with $J(x)=0$) and to the
response to a point source (with $J(x)=Ka\delta^{3}(\vec{x})$). It will be
useful to consider the scale factor $a$ as the time variable,

$\mathcal{D}\equiv
a^{2}\Bigl{[}-\partial_{0}^{2}-2aH\partial_{0}+\nabla^{2}\Bigr{]}=a^{4}H^{2}\Bigl{[}-a^{2}\partial_{a}^{2}-4a\partial_{a}+\frac{\nabla^{2}}{a^{2}H^{2}}\Bigr{]}\;.$
(36)

####  3.2.1 Mode Function

Scalar radiation takes the form,

$J(x)=0\qquad\Longrightarrow\qquad\Phi(x)=u_{\Phi}(\eta,k)e^{i\vec{k}\cdot\vec{x}}\;,$
(37)

where the mode function $u_{\Phi}(\eta,k)$ obeys,

$\Bigl{[}a^{2}\partial_{a}^{2}+4a\partial_{a}+\frac{k^{2}}{a^{2}H^{2}}-\frac{3\lambda^{2}H^{2}}{32\pi^{2}}+O(\lambda^{4})\Bigr{]}u_{\Phi}(\eta,k)=0\;.$
(38)

The canonically normalized solution for Bunch-Davies vacuum is,

$u_{\Phi}(\eta,k)=i\sqrt{\frac{\pi}{4Ha^{3}}}\,H^{(1)}_{\nu}\Bigl{(}\frac{k}{aH}\Bigr{)}\qquad,\qquad\nu\equiv\sqrt{\frac{9}{4}\\!-\\!\frac{m^{2}_{\Phi}}{H^{2}}}\;.$
(39)

The form at late times is,

$u_{\Phi}(\eta,k)\longrightarrow\frac{\Gamma(\nu)}{\sqrt{4\pi
Ha^{3}}}\Bigl{(}\frac{2aH}{k}\Bigr{)}^{\nu}\Bigl{\\{}1+\frac{1}{\nu\\!-\\!1}\Bigl{(}\frac{k}{2aH}\Bigr{)}^{2}+\dots\Bigr{\\}}\;.$
(40)

Because $\nu=\frac{3}{2}-\frac{m^{2}}{3H^{2}}+O(\lambda^{4})$ is greater than
$\frac{3}{2}$ for tachyonic masses, we see that the mode function (40)
experiences slow growth at late times.

####  3.2.2 Exchange Potential

The exchange potential is the response to a point source,

$J(x)=Ka\delta^{3}(\vec{x})\qquad\Longrightarrow\qquad\Phi(x)=P_{\Phi}(\eta,r)\;,$
(41)

where $r\equiv\|\vec{x}\|$ and the potential obeys,

$a^{4}H^{2}\Bigl{[}-a^{2}\partial_{a}^{2}-4a\partial_{a}+\frac{\nabla^{2}}{a^{2}H^{2}}+\frac{3\lambda^{2}H^{2}}{32\pi^{2}}+O(\lambda^{4})\Bigr{]}P_{\Phi}(\eta,r)=Ka\delta^{3}(\vec{x})\;.$
(42)

The order $\lambda^{0}$ solution and its late time limit are [86],

$\displaystyle P_{0}(\eta,r)$ $\displaystyle=$
$\displaystyle\frac{KH}{4\pi}\Bigl{\\{}\ln(Hr)+\ln\Bigl{(}1\\!+\\!\frac{1}{aHr}\Bigr{)}-\frac{1}{aHr}\Bigr{\\}}\;,$
(43) $\displaystyle\longrightarrow$
$\displaystyle\frac{KH}{4\pi}\Bigl{\\{}\ln(Hr)-\frac{1}{2(aHr)^{2}}+\dots\Bigr{\\}}\;.$
(44)

The simplest way of solving equation (42) is by using the retarded Green’s
function for a massive, minimally coupled scalar,

$\displaystyle m^{2}\neq 0\Longrightarrow G_{\rm
ret}(x;x^{\prime})=-\frac{1}{4\pi}\Biggl{\\{}\frac{\delta(\Delta\eta\\!-\\!\Delta
r)}{aa^{\prime}\Delta r}$ (45) $\displaystyle\hskip
68.28644pt-\frac{H^{2}\theta(\Delta\eta\\!-\\!\Delta
r)}{2\Gamma(\frac{1}{2}\\!+\\!\nu)\Gamma(\frac{1}{2}\\!-\\!\nu)}\sum_{n=0}^{\infty}\frac{\Gamma(\frac{3}{2}\\!+\\!\nu\\!+\\!n)\Gamma(\frac{3}{2}\\!-\\!\nu\\!+\\!n)}{n!(n\\!+\\!1)!}\Bigl{(}\frac{y}{4}\Bigr{)}^{n}\Biggr{\\}},\qquad$

where $\nu^{2}\equiv\frac{9}{4}-\frac{m^{2}}{H^{2}}$ and $y\equiv
aa^{\prime}H^{2}\Delta x^{2}$ is the de Sitter length function. (Note that
expression (45) takes the form predicted by DeWitt and Brehme [1] with a
universal light-cone singularity plus a mass-dependent tail term.) For small
$m^{2}/H^{2}$ we can expand $G_{\rm ret}(x;x^{\prime})$,

$\displaystyle G_{\rm
ret}(x;x^{\prime})=-\frac{1}{4\pi}\Biggl{\\{}\frac{\delta(\Delta\eta\\!-\\!\Delta
r)}{aa^{\prime}\Delta r}+H^{2}\theta(\Delta\eta\\!-\\!\Delta r)$ (46)
$\displaystyle\hskip 76.82234pt-m^{2}\theta(\Delta\eta\\!-\\!\Delta
r)\Biggl{[}\frac{1}{3}\ln\Bigl{(}1\\!-\\!\frac{y}{4}\Bigr{)}\\!+\\!\frac{1}{2}\\!+\\!\frac{\frac{y}{6}}{y\\!-\\!4}\Biggr{]}+O(m^{4})\Biggr{\\}}.\qquad$

Integrating against the source and taking the late time limit gives,

$\displaystyle P_{\Phi}(\eta,r)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\int\\!\\!d^{4}x^{\prime}G_{\rm
ret}(x;x^{\prime})\\!\times\\!Ka^{\prime}\delta^{3}(\vec{x}^{\prime})\;,$ (47)
$\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle\frac{KH}{4\pi}\Biggl{\\{}\ln(Hr)+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)\ln(Hr)+O(\lambda^{4})\Biggr{\\}}\;.$
(48)

###  3.3 The Expectation Values of $\Phi(x)$ and $\Phi^{2}(x)$

The field definition (12) makes it simple to evaluate expectation values of
$\Phi(x)$ and its square. The first power is,

$\Phi[\Psi]=\Psi-\frac{1}{4}\lambda\Psi^{2}+\frac{1}{8}\lambda^{2}\Psi^{3}-\frac{5}{24}\lambda^{3}\Psi^{4}+O\Bigl{(}\lambda^{4}\Psi^{5}\Bigr{)}\;.$
(49)

Taking the expectation value gives,

$\Bigl{\langle}\Omega\Bigl{|}\Phi(x)\Bigr{|}\Omega\Bigr{\rangle}=-\frac{1}{4}\lambda
i\Delta(x;x)-\frac{15}{64}\lambda^{3}\Bigl{[}i\Delta(x;x)\Bigr{]}^{2}+O\Bigl{(}\lambda^{5}\Bigl{[}i\Delta(x;x)\Bigr{]}^{3}\Bigr{)}\;.$
(50)

Expression (18) shows that the coincident propagator is time dependent, so we
see that $\langle\Omega|\Phi(x)|\Omega\rangle$ evolves, in spite of vanishing
flat space scattering amplitudes.

The expansion of $\Phi^{2}$ is,

$\Phi^{2}(x)=\Psi^{2}(x)-\frac{1}{2}\lambda\Psi^{3}(x)+\frac{5}{16}\lambda^{2}\Psi^{4}(x)+O\Bigl{(}\lambda^{3}\Psi^{5}\Bigr{)}\;.$
(51)

Taking its expectation value gives,

$\Bigl{\langle}\Omega\Bigl{|}\Phi^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}=i\Delta(x;x)+\frac{15}{16}\lambda^{2}\Bigl{[}i\Delta(x;x)\Bigr{]}^{2}+O\Bigl{(}\lambda^{4}\Bigl{[}i\Delta(x;x)\Bigr{]}^{3}\Bigr{)}\;.$
(52)

$\Phi^{2}(x)$ is a composite operator and requires renormalization with
counterterms of the form,

$\delta\Phi^{2}=K_{\Phi 1}R+K_{\Phi 2}R\Phi^{2}+K_{\Phi
3}R^{2}+O(\lambda^{4})\;.$ (53)

Comparison with the primitive expression (52) implies,

$\displaystyle K_{\Phi 1}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\mu^{D-4}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi{\rm
cot}(\frac{D\pi}{2})}{D(D\\!-\\!1)}\;,$ (54) $\displaystyle K_{\Phi 2}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{15\lambda^{2}\mu^{D-4}}{8(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi{\rm
cot}(\frac{D\pi}{2})}{D(D\\!-\\!1)}\;,$ (55) $\displaystyle K_{\Phi 3}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{15\lambda^{2}\mu^{D-4}}{16(4\pi)^{D}}\Bigl{[}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi{\rm
cot}(\frac{D\pi}{2})}{D(D\\!-\\!1)}\Biggr{]}^{2}\;.$ (56)

Using these values in (53), adding $\delta\Phi^{2}$ to (52), and taking the
unregulated limit gives the fully renormalized result,

$\Bigl{\langle}\Omega\Bigl{|}\Phi^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}_{\rm
ren}=\frac{H^{2}}{4\pi^{2}}\ln\Bigl{(}\frac{\mu
a}{H}\Bigr{)}+\frac{15\lambda^{2}H^{4}}{256\pi^{4}}\ln^{2}\Bigl{(}\frac{\mu
a}{H}\Bigr{)}+O(\lambda^{4})\;.$ (57)

##  4 Large Logarithms in the Two Field Model

The task of this section is computing the same things for the two field model
(20) that we previously did for the single field model (10). The order of
presentation is the same as in the previous section, although our labor is
complicated by the inability to remove interactions by a local field
redefinition. We must also digress to explain the Schwinger-Keldysh formalism
when solving the effective field equations.

###  4.1 1-Loop Self-Masses for $A$ and $B$

The four counterterms we require for renormalizing the self-masses at 1-loop
are,

$\displaystyle\Delta\mathcal{L}=-\frac{1}{2}C_{A1}\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptA\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptA\sqrt{-g}-\frac{1}{2}C_{A2}\,R\partial_{\mu}A\partial_{\nu}Ag^{\mu\nu}\sqrt{-g}$
(58) $\displaystyle\hskip 99.58464pt-\frac{1}{2}C_{B1}\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptB\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptB\sqrt{-g}-\frac{1}{2}C_{B2}\,R\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;.\qquad$

The three diagrams which contribute to the $A$ self-mass at 1-loop are shown
in Figure 2.

Figure 2: Diagrams which represent the 1-loop contributions to the $A$-field
self-mass $-iM^{2}_{A}(x;x^{\prime})$. Recall that $A$ lines are solid,
whereas $B$ lines are dashed.

From left to right, their analytic expressions are,

$\displaystyle-iM^{2}_{A3}(x;x^{\prime})$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{(-i\lambda)^{2}}{2}(aa^{\prime})^{D-2}\partial^{\mu}{\partial^{\prime}}^{\rho}i\Delta(x;x^{\prime})\partial_{\mu}\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\;,\qquad$
(59) $\displaystyle-iM^{2}_{A4}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\lambda^{2}}{4}i\delta^{D}(x\\!-\\!x^{\prime})\,a^{D-2}\partial^{\mu}\partial^{\prime}_{\mu}i\Delta(x;x^{\prime})\;,\qquad$
(60) $\displaystyle-iM^{2}_{Ac}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle-
C_{A1}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{(aa^{\prime})^{\frac{D}{2}}}\Bigr{]}+C_{A2}\partial^{\mu}\Bigl{[}Ra^{D-2}\partial_{\mu}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}\;.\qquad$
(61)

After the same sort of reductions employed in the single-field model, the two
primitive diagrams take the forms,

$\displaystyle-iM^{2}_{A3}(x;x^{\prime})$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{(-i\lambda)^{2}}{2}\Biggl{\\{}\frac{1}{4}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}-\frac{1}{2}\mathcal{D}\Bigl{[}i\Delta(x;x)i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}$
(62) $\displaystyle\hskip 147.95424pt-
kHa^{D-1}\partial_{0}i\delta^{D}(x\\!-\\!x^{\prime})\Biggr{\\}},\qquad$
$\displaystyle-iM^{2}_{A4}(x;x^{\prime})$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{i\lambda^{2}}{4}\delta^{D}(x\\!-\\!x^{\prime})\\!\times\\!-(D-1)kH^{2}a^{D}\;.\qquad$
(63)

The square of the propagator in expression (62) is logarithmically divergent
so we need only retain dimensional regularization for the leading term and can
take $D=4$ for the rest,

$\displaystyle\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}=\frac{\Gamma^{2}(\frac{D}{2}\\!-\\!1)}{16\pi^{D}}\frac{1}{(aa^{\prime}\Delta
x^{2})^{D-2}}$ (64) $\displaystyle\hskip
62.59596pt-\frac{H^{2}}{16\pi^{4}}\frac{\ln(\frac{1}{4}H^{2}\Delta
x^{2})}{aa^{\prime}\Delta
x^{2}}+\frac{H^{4}}{64\pi^{4}}\ln^{2}\Bigl{(}\frac{1}{4}H^{2}\Delta
x^{2}\Bigr{)}+O(D\\!-\\!4)\;.\qquad$

The fundamental logarithmic divergence is $1/\Delta x^{2D-4}$. We localize
this by first extracting a d‘Alembertian, then adding zero in the form of the
massless propagator equation in flat space [5, 6],

$\displaystyle\frac{1}{\Delta
x^{2D-4}}=\frac{\partial^{2}}{2(D\\!-\\!3)(D\\!-\\!4)}\Bigl{[}\frac{1}{\Delta
x^{2D-6}}\Bigr{]}=\frac{\mu^{D-4}}{2(D\\!-\\!3)(D\\!-\\!4)}\frac{4\pi^{\frac{D}{2}}i\delta^{D}(x\\!-\\!x^{\prime})}{\Gamma(\frac{D}{2}\\!-\\!1)}$
(66) $\displaystyle\hskip
142.26378pt+\frac{\partial^{2}}{2(D\\!-\\!3)(D\\!-\\!4)}\Bigl{[}\frac{1}{\Delta
x^{2D-6}}-\frac{\mu^{D-4}}{\Delta x^{D-2}}\Bigr{]}\;,\qquad$
$\displaystyle\hskip
11.38092pt=\frac{\mu^{D-4}}{2(D\\!-\\!3)(D\\!-\\!4)}\frac{4\pi^{\frac{D}{2}}i\delta^{D}(x\\!-\\!x^{\prime})}{\Gamma(\frac{D}{2}\\!-\\!1)}-\frac{\partial^{2}}{4}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Bigr{]}+O(D\\!-\\!4)\;.\qquad$

Here $\mu$ is the mass scale of dimensional regularization.

Comparison with expression (61) gives the two $A$-type counterterms,

$C_{A1}=-\frac{\lambda^{2}\mu^{D-4}}{32\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)}\quad,\quad
C_{A2}=\frac{\lambda^{2}\mu^{D-4}}{4(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi{\rm
cot}(\frac{D\pi}{2})}{D(D\\!-\\!1)}\;.$ (67)

Combining $-iM^{2}_{A3}(x;x^{\prime})$ and $-iM^{2}_{A4}(x;x^{\prime})$ with
$-iM^{2}_{3c}(x;x^{\prime})$ and taking the unregulated limit gives the
renormalized self-mass at one loop,

$\displaystyle-
iM^{2}_{A}(x;x^{\prime})=-\frac{3\lambda^{2}H^{4}a^{4}}{32\pi^{2}}\,i\delta^{4}(x\\!-\\!x^{\prime})+\frac{\lambda^{2}H^{2}\partial^{\mu}}{16\pi^{2}}\Bigl{[}\ln\Bigl{(}\frac{\mu
a}{H}\Bigr{)}a^{2}\partial_{\mu}i\delta^{4}(x\\!-\\!x^{\prime})\Bigr{]}$ (68)
$\displaystyle\hskip
56.9055pt+\frac{\lambda^{2}\mathcal{D}\mathcal{D}^{\prime}}{512\pi^{4}}\Biggl{\\{}\frac{\ln(aa^{\prime})4\pi^{2}i\delta^{4}(x\\!-\\!x^{\prime})}{(aa^{\prime})^{2}}+\frac{\partial^{2}}{(aa^{\prime})^{2}}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Bigr{]}$ $\displaystyle\hskip
128.0374pt+\frac{4H^{2}}{aa^{\prime}}\frac{\ln(\frac{1}{4}H^{2}\Delta
x^{2})}{\Delta x^{2}}-H^{4}\ln^{2}\Bigl{(}\frac{1}{4}H^{2}\Delta
x^{2}\Bigr{)}\Biggr{\\}}.\qquad$

Note that the first term represents a positive mass-squared of the same
magnitude as the tachyonic mass we saw in expression (34). It is accompanied
by many other contributions which signal that this system is not reducible to
a free field.

Figure 3 depicts the three diagrams which contribute to the self-mass of $B$
at 1-loop.

Figure 3: Diagrams which represent the 1-loop contributions to the $B$-field
self-mass $-iM^{2}_{B}(x;x^{\prime})$. Recall that $A$ lines are solid,
whereas $B$ lines are dashed.

The analytic expressions for these three diagrams are, from left to right,

$\displaystyle-iM^{2}_{B3}(x;x^{\prime})$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle(i\lambda)^{2}\partial^{\mu}{\partial^{\prime}}^{\rho}\Bigl{[}i\Delta(x;x^{\prime})(aa^{\prime})^{D-2}\partial_{\mu}\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\Bigr{]}\;,\qquad$
(69) $\displaystyle-iM^{2}_{B4}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\lambda^{2}}{4}\partial^{\mu}\Bigl{[}i\Delta(x;x)a^{D-2}\partial_{\mu}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}\;,\qquad$
(70) $\displaystyle-iM^{2}_{Bc}(x;x^{\prime})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle-
C_{B1}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{(aa^{\prime})^{\frac{D}{2}}}\Bigr{]}+C_{B2}\partial^{\mu}\Bigl{[}Ra^{D-2}\partial_{\mu}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}\;.\qquad$
(71)

After some familiar reductions the two primitive diagrams take the form,

$\displaystyle-iM^{2}_{B3}(x;x^{\prime})$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{(i\lambda)^{2}}{2}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}$
(72) $\displaystyle\hskip
42.67912pt-(i\lambda)^{2}\partial^{\mu}{\partial^{\prime}}^{\rho}\Bigl{[}(aa^{\prime})^{D-2}\partial_{\mu}i\Delta(x;x^{\prime})\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\Bigr{]}\;,\qquad$
$\displaystyle-iM^{2}_{B4}(x;x^{\prime})$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\lambda^{2}}{4}k\pi{\rm
cot}\Bigl{(}\frac{D\pi}{2}\Bigr{)}\mathcal{D}\Bigl{[}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}$
(73) $\displaystyle\hskip
51.21504pt+\frac{\lambda^{2}H^{2}\partial^{\mu}}{16\pi^{2}}\Bigl{[}\ln(a)a^{2}\partial_{\mu}i\delta^{4}(x\\!-\\!x^{\prime})\Bigr{]}+O(D\\!-\\!4)\;.\qquad$

Expression (73) is already fully reduced, and the reduction of the first term
of (72) is identical to that of (64), but the second term requires new
analysis. Because this contribution is quadratically divergent we must retain
dimensional regularization for the first two terms in the power series of the
propagator,

$\partial_{\mu}i\Delta(x;x^{\prime})=-\frac{\Gamma(\frac{D}{2})}{2\pi^{\frac{D}{2}}(aa^{\prime})^{\frac{D}{2}-1}}\Biggl{\\{}\frac{\Delta
x_{\mu}}{\Delta
x^{D}}+\frac{[\frac{1}{2}aH\delta^{0}_{~{}\mu}\\!+\\!\frac{D}{8}aa^{\prime}H^{2}\Delta
x_{\mu}]}{\Delta x^{D-2}}+\dots\Biggr{\\}}.$ (74)

Taking the product of two such differentiated propagators, extracting
derivatives and taking $D=4$ in integrable terms gives,

$\displaystyle-(i\lambda)^{2}\partial^{\mu}{\partial^{\prime}}^{\rho}\Bigl{[}(aa^{\prime})^{D-2}\partial_{\mu}i\Delta(x;x^{\prime})\partial^{\prime}_{\rho}i\Delta(x;x^{\prime})\Bigr{]}$
(75)
$\displaystyle=-\frac{(i\lambda)^{2}\Gamma^{2}(\frac{D}{2})}{4\pi^{D}}\Biggl{\\{}\frac{\mathcal{D}\mathcal{D}^{\prime}}{4(D\\!-\\!2)^{2}}\Bigl{[}\frac{1}{(aa^{\prime}\Delta
x^{2})^{D-2}}\Bigr{]}-\frac{DH^{2}\partial\\!\cdot\\!\partial^{\prime}}{8(D\\!-\\!2)}\Bigl{[}\frac{aa^{\prime}}{\Delta
x^{2D-4}}\Bigr{]}$ $\displaystyle\hskip
28.45274pt+\frac{1}{8}\mathcal{D}\mathcal{D}^{\prime}\Bigl{[}\frac{H^{2}}{aa^{\prime}\Delta
x^{2}}\Bigr{]}\\!+\\!\frac{a^{2}{a^{\prime}}^{2}H^{4}\nabla^{2}}{8}\Bigl{[}\frac{1}{\Delta
x^{2}}\Bigr{]}\\!-\\!\frac{H^{4}}{16}\mathcal{D}\mathcal{D}^{\prime}\ln(H^{2}\Delta
x^{2})\Biggr{\\}}.\qquad$

Comparison with expression (71) gives the two $B$-type counterterms,

$\displaystyle C_{B1}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\lambda^{2}\mu^{D-4}}{16\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)}\;,$
(76) $\displaystyle C_{B2}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\lambda^{2}\mu^{D-4}}{4(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi{\rm
cot}(\frac{D\pi}{2})}{D(D\\!-\\!1)}-\frac{\lambda^{2}\mu^{D-4}}{32\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)}\Bigl{(}\frac{D\\!-\\!2}{D\\!-\\!1}\Bigr{)}\;.\qquad$
(77)

Note that $C_{B2}$ cancels divergences in $-iM^{2}_{B4}(x;x^{\prime})$ — the
left hand contribution to (77) — and in $-iM^{2}_{B3}(x;x^{\prime})$ — the
right hand term of (77). Combining the two primitive diagrams with the
counterterm and taking the unregulated limit gives,

$\displaystyle-
iM^{2}_{B}(x;x^{\prime})=\frac{\lambda^{2}\mathcal{D}\mathcal{D}^{\prime}}{64\pi^{2}}\Bigl{[}\frac{\ln(aa^{\prime})i\delta^{4}(x\\!-\\!x^{\prime})}{(aa^{\prime})^{2}}\Bigr{]}\\!-\\!\frac{\lambda^{2}H^{2}\partial^{\mu}}{16\pi^{2}}\Bigl{[}\ln\Bigl{(}\\!\frac{Ha}{\mu}\\!\Bigr{)}a^{2}\partial_{\mu}i\delta^{4}(x\\!-\\!x^{\prime})\Bigr{]}$
(78)
$\displaystyle+\frac{\lambda^{2}\mathcal{D}\mathcal{D}^{\prime}}{256\pi^{4}}\Biggl{\\{}\frac{\partial^{2}}{(aa^{\prime})^{2}}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta
x^{2}}\Bigr{]}+\frac{H^{2}\partial^{2}}{aa^{\prime}}\Bigl{[}\ln^{2}\Bigl{(}\frac{H^{2}\Delta
x^{2}}{4}\Bigr{)}\Bigr{]}-2H^{4}\Bigl{[}\ln^{2}\Bigl{(}\frac{H^{2}\Delta
x^{2}}{4}\Bigr{)}$ $\displaystyle+2\ln\Bigl{(}\\!\frac{H^{2}\Delta
x^{2}}{4}\\!\Bigr{)}\\!\Bigr{]}\\!\\!\Biggr{\\}}\\!+\\!\frac{\lambda^{2}H^{2}\partial\\!\cdot\\!\partial^{\prime}}{64\pi^{4}}\Biggl{\\{}\\!aa^{\prime}\partial^{2}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta
x^{2}}\Bigr{]}\\!\\!\Biggr{\\}}\\!+\\!\frac{\lambda^{2}H^{4}(aa^{\prime})^{2}\nabla^{2}}{32\pi^{4}}\frac{1}{\Delta
x^{2}}.\qquad$

###  4.2 1-Loop Mode Functions & Exchange Potentials

####  4.2.1 Schwinger-Keldysh Effective Field Equations

The linearized effective field equations for $A$ reads,

$\mathcal{D}A(x)=J(x)+\int\\!\\!d^{4}x^{\prime}\,M^{2}_{A}(x;x^{\prime})A(x^{\prime})\;.$
(79)

The $B$ equation is the same with $A(x)$ replaced by $B(x)$ and
$M^{2}_{A}(x;x^{\prime})$ replaced by $M^{2}_{B}(x;x^{\prime})$. Setting the
source $J(x)=0$ describes the propagation of scalar radiation, while the
choice $J(x)=a(\eta)\delta^{3}(\vec{x})$ gives the scalar exchange potential.

Using our in-out results (68) and (78) in equation (79) would result in two
problems:

  * •

The fact that the self-masses are complex precludes real solutions; and

  * •

The fact that the self-masses $M^{2}_{A,B}(x;x^{\prime})$ are nonzero for
${x^{\prime}}^{\mu}$ outside the past light-cone of $x^{\mu}$ leads to a
response before its cause.

Both of these embarrassments can be avoided by employing the self-masses of
the Schwinger-Keldysh formalism [87, 88, 89, 90]. There are many fine
references on the Schwinger-Keldysh effective field equations [91, 92, 93] but
we only need the simple rules for converting the in-out self-masses (68) and
(78) to in-in self-masses [94]:

  * •

The Dirac delta function terms are not changed;

  * •

For every term involving the Poincaré interval $\Delta x^{2}(x;x^{\prime})$,
defined in expression (3), one must subtract the very same function of,

$\Delta
x^{2}_{+-}(x;x^{\prime})\equiv\Bigl{\|}\vec{x}-\vec{x}^{\prime}\Bigr{\|}^{2}-\Bigl{(}\eta-\eta^{\prime}+i\epsilon\Bigr{)}^{2}\;.$
(80)

In converting the in-out self-mass to its in-in cognate it is desirable to
extract d‘Alembertians from inverse powers of $1/\Delta x^{2}$ to reach powers
of logarithms,

$\frac{1}{\Delta x^{2}}=\frac{\partial^{2}}{4}\Bigl{[}\ln(\mu^{2}\Delta
x^{2})\Bigr{]}\quad,\quad\frac{\ln(\mu^{2}\Delta x^{2})}{\Delta
x^{2}}=\frac{\partial^{2}}{8}\Bigl{[}\ln^{2}(\mu^{2}\Delta
x^{2})\\!-\\!2\ln(\mu^{2}\Delta x^{2})\Bigr{]}\;.$ (81)

Differences of powers of logarithms of $\Delta x^{2}$ and $\Delta x^{2}_{+-}$
give a form that makes the reality and causality of
$-iM^{2}_{A,B}(x;x^{\prime})$ manifest,

$\displaystyle\ln(\mu^{2}\Delta x^{2})-\ln(\mu^{2}\Delta x^{2}_{+-})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle 2\pi
i\theta(\Delta\eta\\!-\\!\Delta r)\;,$ (82)
$\displaystyle\ln^{2}(\mu^{2}\Delta x^{2})-\ln^{2}(\mu^{2}\Delta x^{2}_{+-})$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle 4\pi
i\theta(\Delta\eta\\!-\\!\Delta r)\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})]\;.$ (83)

where $\Delta\eta\equiv\eta-\eta^{\prime}$ and $\Delta
r\equiv\|\vec{x}-\vec{x}^{\prime}\|$.

Finally, we must adapt (79) to the fact that only one loop results for the
self-masses are available. This means we can only solve the equation
perturbatively in powers of $\lambda^{2}$,

$A(x)\equiv A_{0}(x)+\lambda^{2}A_{1}(x)+O(\lambda^{4})\quad,\quad B(x)\equiv
B_{0}(x)+\lambda^{2}B_{1}(x)+O(\lambda^{4})\;.$ (84)

The zeroth order solutions obey,

$\mathcal{D}A_{0}(x)=J_{A}(x)\quad,\quad\mathcal{D}B_{0}(x)=J_{B}(x)\;.$ (85)

At 1-loop order we have,

$\displaystyle\mathcal{D}A_{1}(x)=\frac{3H^{4}a^{4}A_{0}}{32\pi^{2}}-\frac{H^{2}\partial^{\mu}}{16\pi^{2}}\Bigl{[}\ln\Bigl{(}\frac{\mu
a}{H}\Bigr{)}a^{2}\partial_{\mu}A_{0}\Bigr{]}-\frac{\mathcal{D}}{64\pi^{2}}\Bigl{[}\frac{\ln(a)\mathcal{D}A_{0}}{a^{4}}\Bigr{]}$
(86)
$\displaystyle-\frac{\mathcal{D}}{512\pi^{3}}\int\\!\\!d^{4}x^{\prime}\Biggl{\\{}\frac{\partial^{4}}{2a^{2}{a^{\prime}}^{2}}\Biggl{[}\theta(\Delta\eta\\!-\\!\Delta
r)\Bigl{(}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})]\\!-\\!1\Bigr{)}\Biggr{]}$ $\displaystyle\hskip
42.67912pt+\frac{2H^{2}\partial^{2}}{aa^{\prime}}\Biggl{[}\theta(\Delta\eta\\!-\\!\Delta
r)\Bigl{(}\ln\Bigl{[}\frac{1}{4}H^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})\Bigr{]}\\!-\\!1\Bigr{)}\Biggr{]}$ $\displaystyle\hskip
85.35826pt-4H^{4}\theta(\Delta\eta\\!-\\!\Delta
r)\ln\Bigl{[}\frac{1}{4}H^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})\Bigr{]}\Biggr{\\}}\mathcal{D}^{\prime}A_{0}(x^{\prime})\;,\qquad$
$\displaystyle\mathcal{D}B_{1}(x)=-\frac{\mathcal{D}}{32\pi^{2}}\Bigl{[}\frac{\ln(a)\mathcal{D}B_{0}}{a^{4}}\Bigr{]}+\frac{H^{2}\partial^{\mu}}{16\pi^{2}}\Bigl{[}\ln\Bigl{(}\frac{Ha}{\mu}\Bigr{)}a^{2}\partial_{\mu}B_{0}\Bigr{]}$
(87)
$\displaystyle-\frac{\mathcal{D}}{256\pi^{3}}\int\\!\\!d^{4}x^{\prime}\Biggl{\\{}\frac{\partial^{4}}{2a^{2}{a^{\prime}}^{2}}\Biggl{[}\theta(\Delta\eta\\!-\\!\Delta
r)\Bigl{(}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})]\\!-\\!1\Bigr{)}\Biggr{]}$ $\displaystyle\hskip
56.9055pt+\frac{4H^{2}\partial^{2}}{aa^{\prime}}\Bigl{(}\theta(\Delta\eta\\!-\\!\Delta
r)\ln\Bigl{[}\frac{1}{4}H^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})\Bigr{]}\Bigr{)}$ $\displaystyle\hskip
85.35826pt-8H^{4}\theta(\Delta\eta\\!-\\!\Delta
r)\Bigl{(}\ln\Bigl{[}\frac{1}{4}H^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})\Bigr{]}\\!+\\!1\Bigr{)}\Biggr{\\}}\mathcal{D}^{\prime}B_{0}(x^{\prime})$
$\displaystyle-\frac{H^{2}}{128\pi^{3}}\int\\!\\!d^{4}x^{\prime}\partial\\!\cdot\\!\partial^{\prime}\Biggl{\\{}aa^{\prime}\partial^{4}\Biggl{[}\theta(\Delta\eta\\!-\\!\Delta
r)\Bigl{[}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!\Delta
x^{2})]\\!-\\!1\Bigr{]}\Biggr{]}\Biggr{\\}}B_{0}(x^{\prime})$
$\displaystyle\hskip
113.81102pt-\frac{H^{4}}{64\pi^{3}}\int\\!\\!d^{4}x^{\prime}a^{2}{a^{\prime}}^{2}\nabla^{2}\partial^{2}\Bigl{[}\theta(\Delta\eta\\!-\\!\Delta
r)\Bigr{]}B_{0}(x^{\prime})\;.\qquad$

####  4.2.2 1-Loop Corrected Mode Functions

Scalar radiation corresponds to $J_{A,B}(x)=0$ and has zeroth order solution,

$A_{0}(x)=B_{0}(x)=u_{0}(\eta,k)e^{i\vec{k}\cdot\vec{x}}\qquad\Longrightarrow\qquad
u_{0}(\eta,k)=\frac{H}{\sqrt{2k^{3}}}\Bigl{[}1+ik\eta\Bigr{]}e^{-ik\eta}\;.$
(88)

Because $\mathcal{D}A_{0}(x)=0=\mathcal{D}B_{0}(x)$, very few terms of the
1-loop field equations (86-87) survive,

$\displaystyle\mathcal{D}A_{1}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{3H^{4}a^{4}A_{0}}{32\pi^{2}}+\frac{H^{3}a^{3}\partial_{0}A_{0}}{16\pi^{2}}\;,$
(89) $\displaystyle\mathcal{D}B_{1}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{H^{3}a^{3}\partial_{0}B_{0}}{16\pi^{2}}+\frac{H^{4}k^{2}a^{2}}{64\pi^{3}}\\!\int\\!\\!d^{4}x^{\prime}\,\theta(\Delta\eta\\!-\\!\Delta
r){\partial^{\prime}}^{2}\Bigl{[}{a^{\prime}}^{2}B_{0}(x^{\prime})\Bigr{]}\noindent$
(91)
$\displaystyle-\frac{H^{2}\partial^{\mu}}{128\pi^{3}}\Biggl{\\{}a\partial^{2}\\!\\!\\!\int\\!\\!d^{4}x^{\prime}\,\theta(\Delta\eta\\!-\\!\Delta
r)\Bigl{[}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})]\\!-\\!1\Bigr{]}{\partial^{\prime}}^{2}\Bigl{[}a^{\prime}\partial^{\prime}_{\mu}B_{0}(x^{\prime})\Bigr{]}\Biggr{\\}}.\qquad$

1-loop corrections to the $A$ mode function are dominated by the first term on
the right hand side of expression (89) which corresponds to a mass,

$m^{2}_{A}=\frac{3\lambda^{2}H^{4}}{32\pi^{2}}+O(\lambda^{4})\quad\Longrightarrow\quad\nu\equiv\sqrt{\frac{9}{4}\\!-\\!\frac{m_{A}^{2}}{H^{2}}}=\frac{3}{2}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}+O(\lambda^{4})\;.$
(92)

Substituting (92) into our previous result (40) for the late time form of a
massive scalar mode function, and expanding in powers of $\lambda^{2}$ gives,

$\displaystyle
i\sqrt{\frac{\pi}{4Ha^{3}}}\,H^{(1)}_{\nu}\Bigl{(}\frac{k}{aH}\Bigr{)}\longrightarrow\frac{H}{\sqrt{2k^{3}}}\Biggl{\\{}1+\frac{k^{2}}{2a^{2}H^{2}}+O\Bigl{(}\frac{k^{4}}{a^{4}H^{4}}\Bigr{)}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\Biggl{[}\ln\Bigl{(}\frac{2aH}{k}\Bigr{)}$
(93) $\displaystyle\hskip
8.5359pt+\psi\Bigl{(}\frac{3}{2}\Bigr{)}+\Bigl{[}\ln\Bigl{(}\frac{2aH}{k}\Bigr{)}\\!+\\!\psi\Bigl{(}\frac{3}{2}\Bigr{)}\\!-\\!2\Bigr{]}\frac{k^{2}}{2a^{2}H^{2}}+O\Bigr{(}\frac{k^{4}}{a^{4}H^{4}}\Bigr{)}\Biggr{]}+O(\lambda^{4})\Biggr{\\}}.\qquad$

In contrast, the final term in (89) is down by two factors of $a$,

$\partial_{0}u_{0}(\eta,k)=\frac{H}{\sqrt{2k^{3}}}\\!\times\\!-\frac{k^{2}}{aH}\exp\Bigl{[}\frac{ik}{aH}\Bigr{]}\;,$
(94)

and corrects the $A$ mode function by terms which fall off like $1/a^{2}$.

The first term on the right hand side of (91) is opposite to the final term of
(89), and is similarly irrelevant. To evaluate the nonlocal contribution to
(91) we first note,

${\partial^{\prime}}^{2}\Bigl{[}{a^{\prime}}^{2}B_{0}(x^{\prime})\Bigr{]}=12{a^{\prime}}^{4}H^{2}B_{0}(x^{\prime})+2{a^{\prime}}^{3}H\partial_{0}^{\prime}B_{0}(x^{\prime})\longrightarrow
12{a^{\prime}}^{4}H^{2}\\!\times\\!\frac{H}{\sqrt{2k^{3}}}\;.$ (95)

Hence the leading late time form of the right hand side of (91) is,

$\displaystyle\frac{H^{4}k^{2}a^{2}}{64\pi^{3}}\int\\!\\!d^{4}x^{\prime}\theta(\Delta\eta\\!-\\!\Delta
r){\partial^{\prime}}^{2}\Bigl{[}{a^{\prime}}^{2}B_{0}(x^{\prime})\Bigr{]}$
(97) $\displaystyle\hskip
42.67912pt\longrightarrow\frac{H^{4}k^{2}a^{2}}{64\pi^{3}}\int\\!\\!d^{4}x^{\prime}\theta(\Delta\eta\\!-\\!\Delta
r)\\!\times\\!12{a^{\prime}}^{4}H^{2}u_{0}(0,k)\;,$ $\displaystyle\hskip
42.67912pt=\frac{H^{6}k^{2}a^{2}}{4\pi^{2}}\,u_{0}(0,k)\\!\\!\int_{\eta_{i}}^{\eta}\\!\\!d\eta^{\prime}{a^{\prime}}^{4}\Delta\eta^{3}\longrightarrow\frac{H^{2}k^{2}a^{2}\ln(a)}{4\pi^{2}}\,u_{0}(0,k)\;.\qquad$

The nonlocal source in (91) is therefore only enhanced over the minuscule
local contribution by a factor of $\ln(a)$. The net effect is no large
logarithms in 1-loop corrections to $u_{B}(\eta,k)$, just a slightly slower
approach to the constant late time limit of the tree order result,

$u_{B}(\eta,k)\longrightarrow\Biggl{\\{}1+\frac{\lambda^{2}H^{2}}{8\pi^{2}}\Bigl{(}\frac{k}{aH}\Bigr{)}^{2}\ln(a)+O(\lambda^{4})\Biggr{\\}}u_{0}(\eta,k)\;.$
(98)

####  4.2.3 1-Loop Corrected Exchange Potentials

We define the exchange potential as the response to a point source
$J(\eta,\vec{x})=Ka\delta^{3}(\vec{x})$. These potentials are functions of
$\eta$ and $r\equiv\|\vec{x}\|$. The order $\lambda^{0}$ solutions for $A$ and
$B$ are the same [86],

$\mathcal{D}P_{0}(\eta,r)=Ka\delta^{3}(\vec{x})\Longrightarrow
P_{0}(\eta,r)=\frac{KH}{4\pi}\Bigl{\\{}\ln\Bigl{(}Hr\\!+\\!\frac{1}{a}\Bigr{)}-\frac{1}{aHr}\Bigr{\\}}\;.$
(99)

Other derivatives of $P_{0}(\eta,r)$ are,

$\displaystyle\partial_{0}P_{0}(\eta,r)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{KH^{2}}{4\pi}\Bigl{[}\frac{1}{Hr}-\frac{1}{Hr\\!+\\!\frac{1}{a}}\Bigr{]}\;\;,\;\;\partial_{0}^{2}P_{0}(\eta,r)=-\frac{KH^{3}}{4\pi}\frac{1}{(Hr\\!+\\!\frac{1}{a})^{2}}\;,\qquad$
(100) $\displaystyle\nabla^{2}P_{0}(\eta,r)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{K\delta^{3}(\vec{x})}{a}+\frac{KH^{3}}{4\pi}\Bigl{[}\frac{2a}{Hr}-\frac{2a}{Hr\\!+\\!\frac{1}{a}}-\frac{1}{(Hr\\!+\\!\frac{1}{a})^{2}}\Bigr{]}\;.\qquad$
(101)

Recall also that the late time limit of $Hr\gg\frac{1}{a}$ is constant in time
but not in space,

$P_{0}(\eta,r)\longrightarrow\frac{KH}{4\pi}\Bigl{\\{}\ln(Hr)+\frac{1}{2a^{2}H^{2}r^{2}}+\dots\Bigr{\\}}\;.$
(102)

The exchange potential for $A$ takes the form,

$P_{A}(\eta,r)=P_{0}(\eta,r)+P_{A1}(\eta,r)+O(\lambda^{4})\;.$ (103)

From equations (86) and (99) we see that $\mathcal{D}P_{A1}(\eta,r)$ is,

$\displaystyle a^{4}H^{2}m_{A}^{2}P_{0}-\frac{\lambda^{2}H^{2}\ln(\frac{\mu
a}{H})Ka\delta^{3}(\vec{x})}{16\pi^{2}}+\frac{\lambda^{2}H^{3}a^{3}\partial_{0}P_{0}}{16\pi^{2}}-\frac{\lambda^{2}\mathcal{D}}{64\pi^{2}}\Bigl{[}\frac{\ln(a)K\delta^{3}(\vec{x})}{a^{3}}\Bigr{]}$
(104)
$\displaystyle-\frac{\lambda^{2}K\mathcal{D}}{512\pi^{3}}\\!\\!\int\\!\\!d^{4}\eta^{\prime}\Biggl{\\{}\frac{\partial^{4}}{2a^{2}a^{\prime}}\Biggl{[}\theta(\Delta\eta\\!-\\!r)\Bigl{(}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\\!-\\!1\Bigr{)}\Biggr{]}+\frac{2H^{2}\partial^{2}}{a}\Biggl{[}\theta(\Delta\eta\\!-\\!r)$
$\displaystyle\times\Bigl{(}\ln\Bigl{[}\frac{1}{4}H^{2}(\Delta\eta^{2}\\!-\\!r^{2})\Bigr{]}\\!-\\!1\Bigr{)}\Biggr{]}-4a^{\prime}H^{4}\theta(\Delta\eta\\!-\\!r)\ln\Bigl{[}\frac{1}{4}H^{2}(\Delta\eta^{2}\\!-\\!r^{2})\Bigr{]}\Biggr{\\}}.\qquad$

It turns out that only the first two contributions to (104) make significant
contributions to $P_{A1}(\eta,r)$ at late times. Of course the term
proportional to $m_{A}^{2}=\frac{3\lambda^{2}H^{2}}{32\pi^{2}}=-m^{2}_{\Phi}$
makes the opposite contribution from the tachyonic mass of $\Phi$ that we
worked out in expression (48),

$\mathcal{D}\Delta P_{A1}=a^{4}H^{2}m_{A}^{2}P_{0}\Longrightarrow\Delta
P_{A1}(\eta,r)\longrightarrow-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)\\!\times\\!\frac{KH}{4\pi}\ln(Hr)\;.$
(105)

To work out the result from the second term in (104) we simply integrate
against the $\lambda^{0}$ retarded Green’s function (46),

$\displaystyle\mathcal{D}\Delta P_{A1}=-\frac{\lambda^{2}H^{2}\ln(\frac{\mu
a}{H})Ka\delta^{3}(\vec{x})}{16\pi^{2}}$ (108)
$\displaystyle\Longrightarrow\Delta
P_{A1}(\eta,r)=\int_{\eta_{i}}^{0}\\!\\!d\eta^{\prime}\Bigl{\\{}\frac{\delta(\Delta\eta\\!-\\!r)}{aa^{\prime}r}+H^{2}\theta(\Delta\eta\\!-\\!r)\Bigr{\\}}\frac{\lambda^{2}KH^{2}a^{\prime}\ln(\frac{\mu
a^{\prime}}{H})}{64\pi^{3}},\qquad$ $\displaystyle\hskip
65.44142pt=\frac{\lambda^{2}KH^{3}}{64\pi^{3}}\Bigl{\\{}\frac{1}{aHr}\ln\Bigl{(}\frac{\mu
a(\eta\\!-\\!r)}{H}\Bigr{)}+\frac{1}{2}\ln^{2}\Bigl{(}\frac{\mu
a(\eta\\!-\\!r)}{H}\Bigr{)}\Bigr{\\}},\qquad$ $\displaystyle\hskip
65.44142pt\longrightarrow\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)\times\frac{KH}{4\pi}\ln(Hr)\;.\qquad$

The final step is facilitated by noting that $a(\eta-r)=1/(Hr+\frac{1}{a})$.
Combining (105) and (108) gives the leading late time correction at 1-loop
order,

$P_{A1}(\eta,r)\longrightarrow\Bigl{\\{}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)\;.$
(109)

The $B$ exchange potential can be written as,

$P_{B}(\eta,r)=P_{0}(\eta,r)+P_{B1}(\eta,r)+O(\lambda^{4})\;.$ (110)

Substituting the tree order solution (99) in the generic 1-loop equation (87)
implies that the 1-loop correction obeys,

$\displaystyle\mathcal{D}P_{B1}(x)=\frac{\lambda^{2}H^{2}\ln(\frac{\mu
a}{H})Ka\delta^{3}(\vec{x})}{16\pi^{2}}-\frac{\lambda^{2}H^{3}a^{3}\partial_{0}P_{0}}{16\pi^{2}}-\frac{\lambda^{2}\mathcal{D}}{32\pi^{2}}\Bigl{[}\frac{\ln(a)K\delta^{3}(\vec{x})}{a^{3}}\Bigr{]}$
(111)
$\displaystyle-\frac{\lambda^{2}K\mathcal{D}}{256\pi^{3}}\\!\int\\!\\!d\eta^{\prime}\Biggl{\\{}\frac{\partial^{4}}{2a^{2}a^{\prime}}\Biggl{[}\theta(\Delta\eta\\!-\\!r)\Bigl{(}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\\!-\\!1\Bigr{)}\Biggr{]}+\frac{4H^{2}\partial^{2}}{a}\Bigl{(}\theta(\Delta\eta\\!-\\!r)$
$\displaystyle\hskip
14.22636pt\times\ln\Bigl{[}\frac{1}{4}H^{2}(\Delta\eta^{2}\\!-\\!r^{2})\Bigr{]}\Bigr{)}-8a^{\prime}H^{4}\theta(\Delta\eta\\!-\\!r)\Bigl{(}\ln\Bigl{[}\frac{1}{4}H^{2}(\Delta\eta^{2}\\!-\\!r^{2})\Bigr{]}\\!+\\!1\Bigr{)}\Biggr{\\}}$
$\displaystyle+\frac{\lambda^{2}H^{2}\partial^{\mu}}{128\pi^{3}}\Biggl{\\{}a\partial^{4}\\!\\!\\!\int\\!\\!d^{4}x^{\prime}a^{\prime}\theta(\Delta\eta\\!-\\!\Delta
r)\Bigl{[}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!\Delta
x^{2})]\\!-\\!1\Bigr{]}\partial^{\prime}_{\mu}P_{0}(x^{\prime})\Biggr{\\}}$
$\displaystyle\hskip
113.81102pt-\frac{\lambda^{2}a^{2}H^{4}\partial^{2}}{64\pi^{3}}\int\\!\\!d^{4}x^{\prime}{a^{\prime}}^{2}\theta(\Delta\eta\\!-\\!\Delta
r)\nabla^{2}P_{0}(x^{\prime})\;.\qquad$

Many of the contributions in (111) are similar to those of (104), and it turns
out that only the first one induces large logarithms at late times. Because
the sign of this first term is opposite to its cousin in (104) we need only
reverse the sign of (108) to obtain the leading late time contribution,

$P_{B1}(\eta,r)\longrightarrow-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)\times\frac{KH}{4\pi}\ln(Hr)\;.$
(112)

###  4.3 Expectation Values at 1 and 2 Loops

Computing expectation values is much more difficult without a local field
redefinition like (12) which expresses the full field in terms of a free
field. However, we struggle through most of the same computations for the two
field model (20) that we did for the single field model (10). We first
evaluate primitive results in dimensional regularization, then renormalize and
take the unregulated limit.

####  4.3.1 Primitive Result for $\langle A(x)\rangle$

Figure 4 shows the 1-loop contributions to the expectation value of $A(x)$.

Figure 4: Diagram which represents the 1-loop contribution to the expectation
value of $A(x)$. Recall that $A$ lines are solid, whereas $B$ lines are
dashed. Square vertices are fixed, whereas circular vertices are integrated.

The initial expression of this diagram is,

$A_{1}=-\frac{i}{2}\lambda\\!\\!\int\\!\\!d^{D}x^{\prime}\,i\Delta(x;x^{\prime})\sqrt{-g(x^{\prime})}g^{\mu\nu}(x^{\prime})\\!\times\\!\partial^{\prime}_{\mu}\partial^{\prime\prime}_{\nu}i\Delta(x^{\prime};x^{\prime\prime})\Bigr{|}_{x^{\prime\prime}=x^{\prime}}\;.$
(113)

Employing relation (19) to evaluate the doubly differentiated, coincident
propagator, and then interpreting the diagram in the Schwinger-Keldysh sense
gives,

$A_{1}=\frac{i}{2}\lambda(D\\!-\\!1)kH^{2}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}\Bigl{[}i\Delta_{++}(x;x^{\prime})\\!-\\!i\Delta_{+-}(x;x^{\prime})\Bigr{]}\;.$
(114)

Expression (114) is ultraviolet finite so we can set $D=4$ and use expressions
(2) and (81) to conclude,

$\Bigl{\langle}\Omega\Bigl{|}A(x)\Bigr{|}\Omega\Bigr{\rangle}=\frac{\lambda
H^{2}}{16\pi^{2}}\Bigl{[}\ln(a)-\frac{1}{3}+\frac{1}{3a^{3}}\Bigr{]}+O(\lambda^{3})\;.$
(115)

####  4.3.2 Primitive Results for $\langle[A(x)-\langle
A(x)\rangle]^{2}\rangle$

The expectation value of $A^{2}(x)$ contains a disconnected part that is the
square of (115),

$\Bigl{\langle}\Omega\Bigl{|}A^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}=\Bigl{\langle}\Omega\Bigl{|}A(x)\Bigr{|}\Omega\Bigr{\rangle}^{2}+\Bigl{\langle}\Omega\Bigl{|}\Bigl{[}A(x)-\langle\Omega|A(x)|\Omega\rangle\Bigr{]}^{2}\Bigr{|}\Omega\Bigr{\rangle}\;.$
(116)

Figure 5 shows the 1-loop and 2-loop contributions to the connected part.

Figure 5: Diagrams representing those 1-loop and 2-loop contributions to the
expectation value of $A^{2}(x)$ which are not already included in the square
of the expectation value of $A(x)$. Recall that $A$ lines are solid, whereas
$B$ lines are dashed. Square vertices are fixed, whereas circular vertices are
integrated.

Of course the 1-loop part is just the coincident propagator (18). Our initial
expressions for the three 2-loop contributions are,

$\displaystyle A^{2}_{2a}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{(-i\lambda)^{2}}{2}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}g^{\mu\nu}(x^{\prime})\,i\Delta(x;x^{\prime})$
(117)
$\displaystyle\times\\!\\!\int\\!\\!d^{D}x^{\prime\prime}\sqrt{-g(x^{\prime\prime})}g^{\rho\sigma}(x^{\prime\prime})\,i\Delta(x;x^{\prime\prime})\,\partial^{\prime}_{\mu}\partial^{\prime\prime}_{\rho}i\Delta(x^{\prime};x^{\prime\prime})\partial^{\prime}_{\nu}\partial^{\prime\prime}_{\sigma}i\Delta(x^{\prime};x^{\prime\prime})\;,\qquad$
$\displaystyle A^{2}_{2b}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{i\lambda^{2}}{4}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}g^{\mu\nu}(x^{\prime})\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\partial^{\prime}_{\mu}\partial^{\prime\prime}_{\nu}i\Delta(x^{\prime};x^{\prime\prime})\Bigl{|}_{x^{\prime\prime}=x^{\prime}}\;,\qquad$
(118) $\displaystyle A^{2}_{2c}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}\Biggl{\\{}-iC_{A1}\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0pt^{\prime}i\Delta(x;x^{\prime})\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0pt^{\prime}i\Delta(x;x^{\prime})$ (119) $\displaystyle\hskip 113.81102pt-
iC_{A2}\,Rg^{\mu\nu}(x^{\prime})\partial^{\prime}_{\mu}i\Delta(x;x^{\prime})\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\Biggr{\\}}.\qquad$

We reduce expression (117) with (31) and an invocation of the Schwinger-
Keldysh formalism,

$\displaystyle A^{2}_{2a}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\lambda^{2}}{8}\Bigl{[}i\Delta(x;x)\Bigr{]}^{2}$ (120)
$\displaystyle-\frac{i\lambda^{2}(D\\!-\\!1)kH^{2}}{4}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}\Biggl{\\{}\\!\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\\!\\!\\!-\\!\Bigl{[}i\Delta_{+-}(x;x^{\prime})\Bigr{]}^{2}\\!\Biggr{\\}}.\qquad$

The 4-point contribution (118) follows from (19) and another application of
the Schwinger-Keldysh formalsim,

$A^{2}_{2b}=\frac{i\lambda^{2}(D\\!-\\!1)kH^{2}}{4}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}\Biggl{\\{}\\!\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\\!\\!\\!-\\!\Bigl{[}i\Delta_{+-}(x;x^{\prime})\Bigr{]}^{2}\\!\Biggr{\\}}.$
(121)

And the counterterm insertion (119) follows from the propagator equation and
an application of (28),

$A^{2}_{2c}=-C_{A2}\,R\,i\Delta(x;x)\;.$ (122)

####  4.3.3 Primitive Results for $\langle B^{2}(x)\rangle$

The expectation value of $B(x)$ vanishes to all orders by virtue of the shift
symmetry of (20). Hence there is no disconnected part to the expectation value
of $B^{2}(x)$. The diagrams which contribute to it are depicted in Figure 6

Figure 6: Diagrams which represent the 1-loop and 2-loop contributions to the
expectation value of $B^{2}(x)$. Recall that $A$ lines are solid, whereas $B$
lines are dashed. Square vertices are fixed, whereas circular vertices are
integrated.

The associated analytic expressions are,

$\displaystyle B^{2}_{2a}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle(-i\lambda)^{2}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}g^{\mu\nu}(x^{\prime})\,\partial^{\prime}_{\mu}i\Delta(x;x^{\prime})$
(123)
$\displaystyle\times\\!\\!\int\\!\\!d^{D}x^{\prime\prime}\sqrt{-g(x^{\prime\prime})}g^{\rho\sigma}(x^{\prime\prime})\,\partial^{\prime\prime}_{\rho}i\Delta(x;x^{\prime\prime})\,i\Delta(x^{\prime};x^{\prime\prime})\partial^{\prime}_{\nu}\partial^{\prime\prime}_{\sigma}i\Delta(x^{\prime};x^{\prime\prime})\;,\qquad$
$\displaystyle B^{2}_{2b}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{i\lambda^{2}}{4}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}g^{\mu\nu}(x^{\prime})\partial^{\prime}_{\mu}i\Delta(x;x^{\prime})\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\,i\Delta(x^{\prime};x^{\prime})\;,\qquad$
(124) $\displaystyle B^{2}_{2c}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}\Biggl{\\{}-iC_{B1}\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0pt^{\prime}i\Delta(x;x^{\prime})\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0pt^{\prime}i\Delta(x;x^{\prime})$ (125) $\displaystyle\hskip 113.81102pt-
iC_{B2}\,Rg^{\mu\nu}(x^{\prime})\partial^{\prime}_{\mu}i\Delta(x;x^{\prime})\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\Biggr{\\}}.\qquad$

Familiar partial integration procedures reduce (123-125) to the form,

$\displaystyle B^{2}_{2a}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{3\lambda^{2}}{4}\Bigl{[}i\Delta(x;x)\Bigr{]}^{2}$ (126)
$\displaystyle-\frac{i\lambda^{2}(D\\!-\\!1)kH^{2}}{2}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}\Biggl{\\{}\\!\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\\!\\!\\!-\\!\Bigl{[}i\Delta_{+-}(x;x^{\prime})\Bigr{]}^{2}\\!\Biggr{\\}},\qquad$
$\displaystyle B^{2}_{2b}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\lambda^{2}}{4}\Bigl{[}i\Delta(x;x)\Bigr{]}^{2}$ (127)
$\displaystyle+\frac{i\lambda^{2}(D\\!-\\!1)kH^{2}}{4}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}\Biggl{\\{}\\!\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\\!\\!\\!-\\!\Bigl{[}i\Delta_{+-}(x;x^{\prime})\Bigr{]}^{2}\\!\Biggr{\\}},\qquad$
$\displaystyle B^{2}_{2c}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle-
C_{B2}\,R\,i\Delta(x;x)\;.$ (128)

The coincident propagator was given in (18), so we need only employ the
Schwinger-Keldysh formalism to show that,

$\displaystyle-\frac{i\lambda^{2}(D\\!-\\!1)kH^{2}}{4}\\!\\!\int\\!\\!d^{D}x^{\prime}\sqrt{-g(x^{\prime})}\Biggl{\\{}\\!\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\\!\\!\\!-\\!\Bigl{[}i\Delta_{+-}(x;x^{\prime})\Bigr{]}^{2}\\!\Biggr{\\}}$
(129)
$\displaystyle=\frac{\lambda^{2}H^{2D-4}}{(4\pi)^{D}}\frac{\Gamma(D)}{\Gamma(\frac{D}{2})}\Biggl{\\{}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)}\\!-\\!\frac{\ln^{2}(a)}{3}\\!-\\!\frac{2}{3}\ln(a)\\!+\\!\frac{79}{54}+O(a^{-1})\Biggr{\\}}.\qquad$

Note that we have simplified ultraviolet finite contributions to (129) by
taking $D=4$.

####  4.3.4 Renormalized Results

The expectation value of $A(x)$ is ultraviolet finite and requires no
renormalization. The square of $A(x)$ is a composite operator and
renormalizing it at 1-loop and 2-loop orders requires three counterterms,

$\delta A^{2}=K_{A1}R+K_{A2}RA^{2}+K_{A3}R^{2}+\dots$ (130)

Comparison with expressions (120-122) reveals that the three constants are,

$\displaystyle K_{A1}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\mu^{D-4}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi{\rm
cot}(\frac{D\pi}{2})}{D(D\\!-\\!1)}\;,$ (131) $\displaystyle K_{A2}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle 0\;,$ (132) $\displaystyle
K_{A3}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\lambda^{2}\mu^{2D-8}}{8(4\pi)^{D}}\frac{\Gamma^{2}(D\\!-\\!1)}{\Gamma^{2}(\frac{D}{2})}\frac{\pi^{2}{\rm
cot}^{2}(\frac{D\pi}{2})}{D^{2}(D\\!-\\!1)^{2}}\;.$ (133)

Putting everything together gives the final renormalized result,

$\displaystyle\Bigl{\langle}\Omega\Bigl{|}A^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}_{\rm
ren}=\frac{H^{2}}{4\pi^{2}}\ln\Bigl{(}\frac{\mu
a}{H}\Bigr{)}-\frac{\lambda^{2}H^{4}}{128\pi^{4}}\ln^{2}\Bigl{(}\frac{\mu
a}{H}\Bigr{)}$ (134) $\displaystyle\hskip
113.81102pt\frac{\lambda^{2}H^{4}}{256\pi^{4}}\Biggl{[}\ln(a)\\!-\\!\frac{1}{3}\\!+\\!\frac{1}{3a^{3}}+O(\lambda^{2})\Biggr{]}^{2}+O(\lambda^{4})\;.\qquad$

The square of $B(x)$ is also a composite operator and requires similar
counterterms,

$\delta B^{2}=K_{B1}R+K_{B2}RB^{2}+K_{B3}R^{2}+\dots$ (135)

Comparison with expressions (126-128) determines the constants to be,

$\displaystyle K_{B1}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\mu^{D-4}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi{\rm
cot}(\frac{D\pi}{2})}{D(D\\!-\\!1)}\;,$ (136) $\displaystyle K_{B2}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{5\lambda^{2}\mu^{D-4}}{4(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\pi{\rm
cot}(\frac{D\pi}{2})}{D(D\\!-\\!1)}-\frac{\lambda^{2}\mu^{D-4}}{32\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)}\Bigl{(}\frac{D\\!-\\!2}{D\\!-\\!1}\Bigr{)},\qquad$
(137) $\displaystyle K_{B3}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\lambda^{2}\mu^{2D-8}}{2(4\pi)^{D}}\frac{\Gamma^{2}(D\\!-\\!1)}{\Gamma^{2}(\frac{D}{2})}\frac{\pi^{2}{\rm
cot}^{2}(\frac{D\pi}{2})}{D^{2}(D\\!-\\!1)^{2}}$ (138) $\displaystyle\hskip
14.22636pt-\frac{\lambda^{2}\mu^{2D-8}}{(4\pi)^{D}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{2(D\\!-\\!3)(D\\!-\\!4)}\frac{1}{D^{2}(D\\!-\\!1)}-\frac{\lambda^{2}}{128\pi^{4}}\frac{79}{2592}\;.\qquad$

Substituting these values in (135), adding that to the sum of (126-128), and
taking the unregulated limit gives,

$\displaystyle\Bigl{\langle}\Omega\Bigl{|}B^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}_{\rm
ren}=\frac{H^{2}}{4\pi^{2}}\ln\Bigl{(}\frac{\mu
a}{H}\Bigr{)}+\frac{\lambda^{2}H^{4}}{128\pi^{4}}\Biggl{\\{}3\ln^{2}(a)\\!-\\!2\ln(a)$
(139) $\displaystyle\hskip
85.35826pt+8\ln\Bigl{(}\frac{\mu}{H}\Bigr{)}\ln(a)\\!+\\!4\ln^{2}\Bigl{(}\frac{\mu}{H}\Bigr{)}\\!-\\!3\ln\Bigl{(}\frac{\mu}{H}\Bigr{)}\Biggr{\\}}+O(\lambda^{4})\;.\qquad$

##  5 Describing the Large Logarithms

The purpose of this section is to explain the various large logarithms derived
in the previous two sections. We begin by summarizing them. We then show how
to infer stochastic effects based on effective potentials for $\Phi(x)$ and
$A(x)$. The remaining large logarithms can be explained using a variant of the
renormalization group which is based on a special class of counterterms that
can be viewed as curvature-dependent renormalizations of parameters in the
bare theories. The section closes with a color-coded summary of the various
logarithms, in which stochastic effects are red and renormalization group
effects are green.

###  5.1 Summary

The large logarithms of the single scalar model (10) reside in expressions
(40), (48), (50) and (57). Table 1 summarizes these results.

Quantity | Leading Logarithms
---|---
$u_{\Phi}(\eta,k)$ | $\Bigl{\\{}1+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)+O(\lambda^{4})\Bigr{\\}}\times\frac{H}{\sqrt{2k^{3}}}$
$P_{\Phi}(\eta,r)$ | $\Bigl{\\{}1+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)+O(\lambda^{4})\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)$
$\langle\Omega|\Phi(x)|\Omega\rangle$ | $-\Bigl{\\{}1+\frac{15\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Bigr{\\}}\times\frac{\lambda H^{2}}{16\pi^{2}}\ln(a)$
$\langle\Omega|\Phi^{2}(x)|\Omega\rangle_{\rm ren}$ | $\Bigl{\\{}1+\frac{15\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Bigr{\\}}\times\frac{H^{2}}{4\pi^{2}}\ln(a)$
Table 1: Leading logarithms in the single scalar model (10).

The large logarithms of the two scalar model (20) reside in expressions (93),
(98), (109), (112), (115), (134) and (139). Of course the expectation value of
$B(x)$ vanishes to all orders. Table 2 summarizes these results.

Quantity | Leading Logarithms
---|---
$u_{A}(\eta,k)$ | $\Bigl{\\{}1-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)+O(\lambda^{4})\Bigr{\\}}\times\frac{H}{\sqrt{2k^{3}}}$
$u_{B}(\eta,k)$ | $\Bigl{\\{}1+0+O(\lambda^{4})\Bigr{\\}}\times\frac{H}{\sqrt{2k^{3}}}$
$P_{A}(\eta,r)$ | $\Bigl{\\{}1-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)+O(\lambda^{4})\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)$
$P_{B}(\eta,r)$ | $\Bigl{\\{}1-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)+O(\lambda^{4})\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)$
$\langle\Omega|A(x)|\Omega\rangle$ | $\Bigl{\\{}1+O(\lambda^{2})\Bigr{\\}}\times\frac{\lambda H^{2}}{16\pi^{2}}\ln(a)$
$\langle\Omega|A^{2}(x)|\Omega\rangle_{\rm ren}$ | $\Bigl{\\{}1-\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Bigr{\\}}\times\frac{H^{2}}{4\pi^{2}}\ln(a)$
$\langle\Omega|B(x)|\Omega\rangle$ | $0$
$\langle\Omega|B^{2}(x)|\Omega\rangle_{\rm ren}$ | $\Bigl{\\{}1+\frac{3\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)+O(\lambda^{4})\Bigr{\\}}\times\frac{H^{2}}{4\pi^{2}}\ln(a)$
Table 2: Leading logarithms in the two scalar model (20).

###  5.2 1-Loop Effective Potentials

Stochastic effects in both models can understood using the effective
potential. One derives this by setting the undifferentiated fields equal to a
constant and then integrating the differentiated fields out of the field
equations.

####  5.2.1 Single Field Model

The key to evaluating $V_{\rm eff}(\Phi)$ is that the $\Phi$ propagator in the
presence of a constant field configuration $\Phi(x)=\Phi_{0}$ is a field
strength renormalization of the free propagator,

$\Bigl{\langle}\Omega\Bigl{|}T\Bigl{[}\Phi(x)\Phi(x^{\prime})\Bigr{]}\Bigr{|}\Omega\Bigr{\rangle}_{\Phi_{0}}=\frac{i\Delta(x;x^{\prime})}{(1\\!+\\!\frac{1}{2}\lambda\Phi_{0})^{2}}\;.$
(140)

The 1-loop effective potential follows from taking the expectation value of
the action’s first variation (13) in the presence of constant
$\Phi(x)=\Phi_{0}$,

$\displaystyle-V^{\prime}_{\rm eff}(\Phi_{0})a^{D}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\Bigl{(}1\\!+\\!\frac{1}{2}\lambda\Phi_{0}\Bigr{)}\partial^{\mu}\Bigl{[}\frac{1}{4}\lambda
a^{D-2}\partial_{\mu}\Bigl{\langle}\Omega\Bigl{|}\Phi^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}_{\Phi_{0}}\Bigr{]}\;,\qquad$
(141) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\frac{1}{2}\lambda(D\\!-\\!1)kH^{2}a^{D}}{1\\!+\\!\frac{1}{2}\lambda\Phi_{0}}\;.\qquad$
(142)

Expression (142) is ultraviolet finite and corresponds to a 1-loop effective
potential of,

$V_{\rm
eff}(\Phi)=\frac{3H^{4}}{8\pi^{2}}\ln\Bigl{|}1+\frac{1}{2}\lambda\Phi\Bigr{|}\;.$
(143)

The effective potential (143) explains the tachyonic mass (34) we found after
the lengthy computation of the 1-loop self-mass,

$m^{2}_{\Phi}\equiv\frac{\partial^{2}V_{\rm
eff}(\Phi)}{\partial\Phi^{2}}\Bigl{|}_{\Phi=0}=-\frac{3\lambda^{2}H^{4}}{32\pi^{2}}\;.$
(144)

Recall expression (40) for the late time limit of the massive mode function,

$u_{\Phi}(\eta,k)\longrightarrow\frac{\Gamma(\nu)}{\sqrt{4\pi
Ha^{3}}}\Bigl{(}\frac{2aH}{k}\Bigr{)}^{\nu}\qquad,\qquad\nu=\sqrt{\frac{9}{4}-\frac{m^{2}_{\Phi}}{H^{2}}}=\frac{3}{2}-\frac{m^{2}_{\Phi}}{3H^{2}}+\dots$
(145)

Substituting (144) into (145) and expanding for small $\lambda$ gives
quantitative agreement with the entry for the mode function in Table 1,

$u_{\Phi}(\eta,k)\longrightarrow\Biggl{\\{}1+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\Bigl{[}\ln\Bigl{(}\frac{2aH}{k}\Bigr{)}+\psi\Bigl{(}\frac{3}{2}\Bigr{)}\Bigr{]}+O(\lambda^{4})\Biggr{\\}}\frac{H}{\sqrt{2k^{3}}}\;.$
(146)

From expression (48) we see that the stochastically generated mass (144) also
explains the entry for the exchange potential.

The effective potential also explains the tendency for the expectation value
of $\Phi(x)$ to become more and more negative as per expression (50). To see
this we write the homogeneous evolution equation (in co-moving time) that
follows from adding the variation of the effective potential to the classical
variation (13),

$-\Bigl{(}1\\!+\\!\frac{\lambda}{2}\Phi\Bigr{)}\frac{d}{dt}\Biggl{[}\Bigl{(}1\\!+\\!\frac{\lambda}{2}\Phi\Bigr{)}a^{3}\dot{\Phi}\Biggr{]}-V^{\prime}_{\rm
eff}(\Phi)a^{3}=0\;.$ (147)

Because the evolution of $\Phi$ is much slower than that of the scale factor
$a=e^{Ht}$, the largest contribution to the first term of (147) is from the
external derivative acting on the factor of $a^{3}$. At this point the
equation can be integrated,

$3H\Bigl{(}1\\!+\\!\frac{\lambda}{2}\Phi\Bigr{)}^{2}\dot{\Phi}\simeq-\frac{3\lambda
H^{4}}{16\pi^{2}}\frac{1}{1\\!+\\!\frac{1}{2}\lambda\Phi}\Longrightarrow\frac{1}{2\lambda}\Bigl{[}\Bigl{(}1\\!+\\!\frac{\lambda}{2}\Phi\Bigr{)}^{4}-1\Bigr{]}\simeq-\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)\;.$ (148)

Inverting to solve for $\Phi$ gives,

$\Phi=\frac{2}{\lambda}\Biggl{\\{}\Bigl{[}1-\frac{\lambda^{2}H^{2}}{8\pi^{2}}\ln(a)\Bigr{]}^{\frac{1}{4}}-1\Biggr{\\}}=-\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)\Biggl{\\{}1+\frac{3\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Biggr{\\}}.$
(149)

The fact that the order $\lambda^{3}$ contribution (149) disagrees with Table
1 is due to not having included fluctuations around the homogeneous solution
driven by the stochastically truncated free field,

$\varphi_{0}(t,\vec{x})\equiv\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\theta\Bigl{(}aH\\!-\\!k\Bigr{)}\frac{\theta(k\\!-\\!H)H}{\sqrt{2k^{3}}}\Bigl{\\{}\alpha_{\vec{k}}e^{i\vec{k}\cdot\vec{x}}+\alpha^{\dagger}_{\vec{k}}e^{-i\vec{k}\cdot\vec{x}}\Bigr{\\}}\;,$
(150)

where $\alpha^{\dagger}_{\vec{k}}$ and $\alpha_{\vec{k}}$ are canonically
normalized creation and annihilation operators,

$\Bigl{[}\alpha_{\vec{k}},\alpha^{\dagger}_{\vec{p}}\Bigr{]}=(2\pi)^{3}\delta^{3}(\vec{k}\\!-\\!\vec{p})\;.$
(151)

If we use the symbol $\varphi(t,\vec{x})$ to distinguish the full ultraviolet
finite stochastic field from $\Phi$, then the Langevin equation associated
with (148) is,

$\dot{\varphi}=\dot{\varphi}_{0}-\frac{\lambda
H^{3}}{16\pi^{2}}\frac{1}{(1\\!+\\!\frac{1}{2}\lambda\Psi)^{3}}\;.$ (152)

It is simple to generate a perturbative solution which includes stochastic
fluctuations around (149),

$\displaystyle\varphi=\varphi_{0}-\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)+\frac{3\lambda^{2}H^{3}}{32\pi^{2}}\int_{0}^{t}\\!\\!dt^{\prime}\varphi_{0}$
(153) $\displaystyle\hskip
113.81102pt-\frac{3\lambda^{3}H^{4}}{1024\pi^{4}}\ln^{2}(a)-\frac{3\lambda^{3}H^{3}}{32\pi^{2}}\int_{0}^{t}\\!\\!dt^{\prime}\varphi_{0}^{2}+O(\lambda^{4})\;.\qquad$

The expectation value of (153) reproduces the entry for
$\langle\Omega|\Phi(x)|\Omega\rangle$ in Table 1,

$\displaystyle\Bigl{\langle}\Omega\Bigl{|}\varphi(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle 0-\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)+0$ (154) $\displaystyle\hskip
11.38092pt-\frac{3\lambda^{3}H^{4}}{1024\pi^{4}}\ln^{2}(a)-\frac{3\lambda^{3}H^{3}}{32\pi^{2}}\int_{0}^{t}\\!\\!dt^{\prime}\frac{H^{2}}{4\pi^{2}}\ln(a^{\prime})+O(\lambda^{5})\;,\qquad$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle-\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)\Biggl{\\{}1+\frac{15\lambda^{2}H^{2}}{64\pi^{4}}\ln^{2}(a)+O(\lambda^{4})\Biggr{\\}}.\qquad$
(155)

Note that stochastic fluctuations cause the expectation value of the field to
roll down its potential more rapidly than the result (149) because a downward
fluctuation is more probable than an upward one.

####  5.2.2 Two Field Model

The same techniques can be applied to the two scalar model (20). The exact
shift symmetry of $B$ precludes there being any effective potential for the
field $B$, but $A$ has one. We can compute it by noting that the expectation
value of $B$ in the presence of constant $A(x)=A_{0}$ is a field strength
renormalization,

$\Bigl{\langle}\Omega\Bigl{|}T\Bigl{[}B(x)B(x^{\prime})\Bigr{]}\Bigr{|}\Omega\Bigr{\rangle}_{A_{0}}=\frac{i\Delta(x;x^{\prime})}{(1\\!+\\!\frac{1}{2}\lambda
A_{0})^{2}}\;.$ (156)

The 1-loop effective potential follows from taking the expectation value of
the action’s first A variation (21),

$\displaystyle-V^{\prime}_{\rm eff}(A_{0})a^{D}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{1}{2}\lambda\Bigl{(}1\\!+\\!\frac{1}{2}\lambda
A_{0}\Bigr{)}a^{D-2}\Bigl{\langle}\Omega\Bigl{|}\partial^{\mu}B(x)\partial_{\mu}B(x)\Bigr{|}\Omega\Bigr{\rangle}_{A_{0}}\;,\qquad$
(157) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle+\frac{\frac{1}{2}\lambda(D\\!-\\!1)kH^{2}a^{D}}{1\\!+\\!\frac{1}{2}\lambda
A_{0}}\;.$ (158)

Taking the unregulated limit and integrating gives the 1-loop effective
potential,

$V_{\rm eff}(A)=-\frac{3H^{4}}{8\pi^{2}}\ln\Bigl{|}1+\frac{1}{2}\lambda
A\Bigr{|}\;.$ (159)

The $A$ effective potential (159) explains the positive mass-squared we found
in expression (68) after a lengthy computation,

$m^{2}_{A}=\frac{\partial^{2}V_{\rm eff}(A)}{\partial
A^{2}}\Bigl{|}_{A=0}=\frac{3\lambda^{2}H^{4}}{32\pi^{2}}\;.$ (160)

This is opposite of the tachyonic mass for $\Phi$ (that is,
$m^{2}_{A}=-m^{2}_{\Phi}$), which explains the factors of
$-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)$ in the entries for $u_{A}(\eta,k)$
and $P_{A}(\eta,r)$ in Table 2. The effective potential for $A$ also explains
the tendency for $\langle\Omega|A(x)|\Omega\rangle$ to grow without bound.
Specializing the $A$ field equation (21) to homogeneous evolution in co-moving
coordinates, adding the effective potential, and neglecting derivatives of $A$
with respect to derivatives of the scale factor $a$ gives,

$-\frac{d}{dt}\Bigl{(}a^{3}\dot{A}\Bigr{)}-V_{\rm
eff}^{\prime}(A)a^{3}=0\qquad\Longrightarrow\qquad
3H\dot{A}\simeq\frac{3\lambda
H^{4}}{16\pi^{2}}\frac{1}{1\\!+\\!\frac{1}{2}\lambda A}\;.$ (161)

Equation (161) can be solved exactly,

$A\simeq\frac{2}{\lambda}\Biggl{[}\sqrt{1+\frac{\lambda^{2}H^{2}}{16\pi^{2}}\ln(a)}-1\Biggr{]}=\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)\Biggl{\\{}1-\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Biggr{\\}}.$
(162)

Gaining quantitative agreement with $\langle\Omega|A(x)|\Omega\rangle$
requires the inclusion of stochastic jitter from the truncated free field
$\mathcal{A}_{0}(t,\vec{x})$,

$\mathcal{A}_{0}(t,\vec{x})\equiv\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\theta\Bigl{(}aH\\!-\\!k\Bigr{)}\frac{\theta(k\\!-\\!H)H}{\sqrt{2k^{3}}}\Bigl{\\{}\alpha_{\vec{k}}e^{i\vec{k}\cdot\vec{x}}+\alpha^{\dagger}_{\vec{k}}e^{-i\vec{k}\cdot\vec{x}}\Bigr{\\}}\;.$
(163)

The Langevin equation associated with (161) is,

$\dot{\mathcal{A}}=\dot{\mathcal{A}}_{0}+\frac{\lambda
H^{3}}{16\pi^{2}}\frac{1}{1\\!+\\!\frac{1}{2}\lambda\mathcal{A}}\;.$ (164)

Iteration of (164) generates a solution which includes the ${\rm C}\hbox
to0.0pt{\hss\vrule height=7.1pt,width=1.0pt,depth=-0.4pt\phantom{t}}$-number
solution (162) plus stochastic jitter involving $\mathcal{A}_{0}$,

$\displaystyle\mathcal{A}=\mathcal{A}_{0}+\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)-\frac{\lambda^{2}H^{3}}{32\pi^{2}}\int_{0}^{t}\\!\\!dt^{\prime}\mathcal{A}_{0}$
(165) $\displaystyle\hskip
113.81102pt-\frac{\lambda^{3}H^{4}}{1024\pi^{4}}\ln^{2}(a)+\frac{\lambda^{3}H^{3}}{64\pi^{2}}\int_{0}^{t}\\!\\!dt^{\prime}\mathcal{A}^{2}_{0}+O(\lambda^{4})\;.\qquad$

The expectation value of (165) agrees exactly with the entry for
$\langle\Omega|A(x)|\Omega\rangle$ in Table 2,

$\displaystyle\Bigl{\langle}\Omega\Bigl{|}\mathcal{A}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle 0+\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)+0$ (166) $\displaystyle\hskip
14.22636pt-\frac{\lambda^{3}H^{4}}{1024\pi^{4}}\ln^{2}(a)+\frac{\lambda^{3}H^{3}}{64\pi^{2}}\int_{0}^{t}\\!\\!dt^{\prime}\frac{H^{2}}{4\pi^{2}}\ln(a^{\prime})+O(\lambda^{5})\;,\qquad$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)\Biggl{\\{}1+\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Biggr{\\}}.$
(167)

Note again that stochastic jitter again increases the rate at which the field
rolls down its potential.

The expectation value of $\mathcal{A}^{2}(t,\vec{x})$ is also straightforward,

$\displaystyle\Bigl{\langle}\Omega\Bigl{|}\mathcal{A}^{2}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\Biggl{\langle}\Omega\Biggl{|}\mathcal{A}_{0}^{2}+\frac{\lambda
H^{2}}{8\pi^{2}}\ln(a)\mathcal{A}_{0}$ (168) $\displaystyle\hskip
28.45274pt+\frac{\lambda^{2}H^{4}}{256\pi^{4}}\ln^{2}(a)-\frac{\lambda^{2}H^{3}}{16\pi^{2}}\mathcal{A}_{0}\int_{0}^{t}\\!\\!dt^{\prime}\mathcal{A}_{0}+O(\lambda^{3})\Biggr{\rangle},\qquad$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{H^{2}}{4\pi^{2}}\ln(a)\Biggl{\\{}1-\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Biggr{\\}}.$
(169)

Expression (169) is in perfect agreement with the entry for
$\langle\Omega|A^{2}(x)|\Omega\rangle$ in Table 2. This means that the leading
logarithms of the field $A(x)$ and its square are purely stochastic — at least
to this order.

###  5.3 Curvature-Dependent Renormalizations

Let us start with the renormalized expectation of $\Phi^{2}(x)$. Recall from
relation (53) that the renormalized composite operator is,

$\Phi^{2}_{\rm ren}\equiv\Phi^{2}+K_{\Phi 1}R+K_{\Phi 2}R\Phi^{2}+K_{\Phi
3}R^{2}+O(\lambda^{4})\;.$ (170)

Some of the 1-loop and 2-loop counterterms in expression (170) have no flat
space analogs, but the $K_{\Phi 2}R\Phi^{2}$ counterterm can be regarded as
part of a curvature-dependent field strength renormalization,
$\Phi^{2}=\sqrt{Z_{\Phi^{2}}}\times\Phi^{2}_{\rm ren}$ with,

$Z_{\Phi^{2}}=1-2K_{\Phi 2}\\!\times\\!R+O(\lambda^{4})\;.$ (171)

The associated $\gamma$ function is,

$\gamma_{\Phi^{2}}\equiv\frac{\partial\ln(Z_{\Phi^{2}})}{\partial\ln(\mu^{2})}=-\frac{15\lambda^{2}H^{2}}{32\pi^{2}}+O(\lambda^{4})\;.$
(172)

Expression (57) shows that the renormalized expectation value of $\Phi^{2}(x)$
actually depends on the product $\mu a/H$. Hence we can replace the
$\mu\frac{\partial}{\partial\mu}$ term in the Callan-Symanzik equation with
$a\frac{\partial}{\partial a}$,

$\Bigl{[}a\frac{\partial}{\partial
a}+\beta\frac{\partial}{\partial\lambda}+\gamma_{\Phi^{2}}\Bigr{]}\Bigl{\langle}\Omega\Bigl{|}\Phi^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}_{\rm
ren}=0\;.$ (173)

The $\beta$ function for this model is of order $\lambda^{3}$, so we see that
equation (173) perfectly explains the order $\lambda^{2}$ (2-loop)
contribution to $\langle\Omega|\Phi^{2}(x)|\Omega\rangle_{\rm ren}$

$a\frac{\partial}{\partial
a}\Bigl{\\{}\frac{15\lambda^{2}H^{4}}{256\pi^{4}}\ln^{2}(a)\Bigr{\\}}-\frac{15\lambda^{2}H^{2}}{32\pi^{2}}\times\frac{H^{2}}{4\pi^{2}}\ln(a)=0$
(174)

Note that the order $\lambda^{0}$ (1-loop) contribution does not obey the
Callan-Symanzik equation (173); the $\frac{H^{2}}{4\pi^{2}}\ln(a)$
contribution is a stochastic effect which is not explained by the
renormalization group.

Expressions (130) and (132) show that the composite operator $A^{2}(x)$ does
not require a curvature-dependent field strength renormalization at this
order,

$Z_{A^{2}}=1-2K_{A2}\times
R+O(\lambda^{4})=1+O(\lambda^{4})\quad\Longrightarrow\quad\gamma_{A^{2}}=0+O(\lambda^{4})\;.$
(175)

Hence the Callan-Symanzik equation does not constrain
$\langle\Omega|A^{2}(x)|\Omega\rangle_{\rm ren}$ at this order, and we saw
from expression (169) that stochastic effects completely explain the 1-loop
and 2-loop contributions for this entry in Table 2.

The composite operator $B^{2}(x)$ experiences a curvature-dependent field
strength renormalization we can read off from expression (135) and (137),

$Z_{B^{2}}=1-2K_{B2}\times
R+O(\lambda^{4})\qquad\Longrightarrow\qquad\gamma_{B^{2}}=-\frac{3\lambda^{2}H^{2}}{16\pi^{2}}+O(\lambda^{4}H^{4})\;.$
(176)

The Callan-Symanzik equation for $\langle\Omega|B^{2}(x)|\Omega\rangle_{\rm
ren}$ is,

$\Bigl{[}a\frac{\partial}{\partial
a}+\beta\frac{\partial}{\partial\lambda}+\gamma_{B^{2}}\Bigr{]}\Bigl{\langle}\Omega\Bigl{|}B^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}_{\rm
ren}=0\;.$ (177)

Because the $\beta$ function of the two field model is of order $\lambda^{3}$,
the equation precisely predicts the
$\frac{3\lambda^{2}H^{2}}{32\pi^{2}}\times\frac{H^{2}}{4\pi^{2}}\ln(a)$
contribution reported in Table 2. Note again that the Callan-Symanzik equation
does not predict the logarithm at order $\lambda^{0}$ which is a stochastic
effect.

Expression (58) shows that we can also think of the 1PI 2-point functions for
$A$ and $B$ experiencing curvature-dependent field strength renormalizations
whose associated $\gamma$ functions can be computed from (67) and (77),

$\displaystyle Z_{A}=1+C_{A2}\\!\times\\!R+O(\lambda^{4})$
$\displaystyle\quad,\quad$ $\displaystyle
Z_{B}=1+C_{B2}\\!\times\\!R+O(\lambda^{4})\;,$ (178)
$\displaystyle\gamma_{A}=+\frac{\lambda^{2}H^{2}}{32\pi^{2}}+O(\lambda^{4})$
$\displaystyle\quad,\quad$
$\displaystyle\gamma_{B}=-\frac{\lambda^{2}H^{2}}{32\pi^{2}}+O(\lambda^{4})\;.$
(179)

The tree order mode functions both approach constants at late times, so the
1-loop corrections are unconstrained by the Callan-Symanzik equation. However,
the tree order exchange potentials approach $\frac{KH}{4\pi}\ln(Hr)$ at late
times. We must therefore interpret the $\mu\frac{\partial}{\partial\mu}$ term
as $r\frac{\partial}{\partial r}$. The 1-loop corrections are integrals of the
1PI 2-point functions, so the Callan-Symanzik equations read,

$\Bigl{[}r\frac{\partial}{\partial
r}+\beta\frac{\partial}{\partial\lambda}-2\gamma_{A}\Bigr{]}P_{A}(\eta,r)=0=\Bigl{[}r\frac{\partial}{\partial
r}+\beta\frac{\partial}{\partial\lambda}-2\gamma_{B}\Bigr{]}P_{B}(\eta,r)\;.$
(180)

With the $\gamma$ functions (179), these equations predict the
$\pm\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)\times\frac{KH}{4\pi}\ln(Hr)$
contributions for $P_{A}(\eta,r)$ and $P_{B}(\eta,r)$ in Table 2. Note that
they do not predict the
$-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)\times\frac{KH}{4\pi}\ln(Hr)$
contribution to $P_{A}(\eta,r)$. This is a stochastic effect from the mass
generated by the effective potential $V_{\rm eff}(A)$ of expression (159).

###  5.4 Color-Coded Tables

So many different logarithms occurred that we have thought it good to provide
color-coded versions of Tables 1 and 2 to distinguish stochastic effects (in
red) from those explained by the renormalization group (in green).

Quantity | Leading Logarithms
---|---
$u_{\Phi}(\eta,k)$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times\frac{H}{\sqrt{2k^{3}}}$
$P_{\Phi}(\eta,r)$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)$
$\langle\Omega|\Phi(x)|\Omega\rangle$ | $-\Bigl{\\{}1{\color[rgb]{1,0,0}+\frac{15\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times{\color[rgb]{1,0,0}\frac{\lambda H^{2}}{16\pi^{2}}\ln(a)}$
$\langle\Omega|\Phi^{2}(x)|\Omega\rangle_{\rm ren}$ | $\Bigl{\\{}1{\color[rgb]{0,1,0}+\frac{15\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times{\color[rgb]{1,0,0}\frac{H^{2}}{4\pi^{2}}\ln(a)}$
Table 3: Color-coded explanations of single scalar logarithms from Table 1.
Red denotes stochastic logarithms and green those explained by the
renormalization group.

Note that mass effects are considered stochastic because $m^{2}_{\Phi}$ and
$m^{2}_{A}$ were induced by the effective potentials $V_{\rm eff}(\Phi)$ and
$V_{\rm eff}(A)$ which give rise to the Langevin equations (152) and (164).

Quantity | Leading Logarithms
---|---
$u_{A}(\eta,k)$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times\frac{H}{\sqrt{2k^{3}}}$
$u_{B}(\eta,k)$ | $\Bigl{\\{}1+0+O(\lambda^{4})\Bigr{\\}}\times\frac{H}{\sqrt{2k^{3}}}$
$P_{A}(\eta,r)$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)}{\color[rgb]{0,1,0}+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)}+O(\lambda^{4})\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)$
$P_{B}(\eta,r)$ | $\Bigl{\\{}1{\color[rgb]{0,1,0}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)}+O(\lambda^{4})\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)$
$\langle\Omega|A(x)|\Omega\rangle$ | $\Bigl{\\{}1+O(\lambda^{2})\Bigr{\\}}\times{\color[rgb]{1,0,0}\frac{\lambda H^{2}}{16\pi^{2}}\ln(a)}$
$\langle\Omega|A^{2}(x)|\Omega\rangle_{\rm ren}$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}-\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times{\color[rgb]{1,0,0}\frac{H^{2}}{4\pi^{2}}\ln(a)}$
$\langle\Omega|B(x)|\Omega\rangle$ | $0$
$\langle\Omega|B^{2}(x)|\Omega\rangle_{\rm ren}$ | $\Bigl{\\{}1{\color[rgb]{0,1,0}+\frac{3\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times{\color[rgb]{1,0,0}\frac{H^{2}}{4\pi^{2}}\ln(a)}$
Table 4: Color-coded explanations of two scalar logarithms from Table 2. Red
denotes stochastic logarithms and green those explained by the renormalization
group.

##  6 Epilogue

Proponents of the renormalization group have long contended with supporters of
the stochastic formalism in attempting to explain and re-sum the large
logarithms which arise when making perturbative computations during inflation.
Although the stochastic formalism provides a complete description for scalar
potential models, the outcome for nonlinear sigma models is more nuanced. Many
of their logarithms can be explained by a variant of the stochastic formalism
which is based on using the effective potential to infer a scalar potential
model. Unlike the cases of Yukawa theory [44] and Scalar Quantum
Electrodynamics [49], the effective potentials of nonlinear sigma models
derive from kinetic terms and would vanish in flat space background. The
remaining logarithms can be explained by a variant of the renormalization
group based on regarding certain higher-derivative counterterms as curvature-
dependent renormalizations of couplings in the bare theory.

We considered a single field model (10), which can be reduced to a free theory
by a field redefinition,111The absence of flat space scattering in no way
precludes interactions from changing the kinematics of free fields or the
evolution of the background. and a two field model (20) which cannot be. For
the mode functions and exchange potentials of each model we derived tree order
and 1-loop results; for the expectation values of the fields and the squares
we derived 1-loop and 2-loop results. Tables 3 and 4 give a color-coded
summary of which logarithms have a stochastic explanation and which ones
derive from the renormalization group. Note that many of the entries derive
partially from one technique and partly from the other. This is particularly
evident for 1-loop corrections to the exchange potential for $A$ in the two
field model (20).

The need to combine ultraviolet and stochastic techniques can be seen in the
passage from the exact field equation (21) for $A(x)$ to its stochastic
realization (164). The exact Heisenberg operator equation is,

$\partial_{\mu}\Bigl{(}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}A\Bigr{)}-\frac{\lambda}{2}\Bigl{(}1\\!+\\!\frac{\lambda}{2}A\Bigr{)}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}=0\;.$
(181)

The stochastic realization of the first term in (181) is straightforward [27],

$\partial_{\mu}\Bigl{(}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}A\Bigr{)}\longrightarrow-3H\Bigl{(}\dot{\mathcal{A}}-\dot{\mathcal{A}}_{0}\Bigr{)}a^{3}\;.$
(182)

However, there is no completely stochastic derivation of the stochastic
realization of the second term,

$-\frac{\lambda}{2}\Bigl{(}1\\!+\\!\frac{\lambda}{2}A\Bigr{)}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\longrightarrow+\frac{3\lambda
H^{4}}{16\pi^{2}}\frac{a^{3}}{1\\!+\\!\frac{1}{2}\lambda\mathcal{A}}\;,$ (183)

because it depends on the ultraviolet sector of $B(x)$ to produce the correct
stochastic result,

$\partial_{\mu}B(x)\partial_{\nu}B(x)\longrightarrow-\frac{(\frac{D-1}{D})kH^{2}g_{\mu\nu}(x)}{[1\\!+\\!\frac{1}{2}\lambda\mathcal{A}(x)]^{2}}\;.$
(184)

Any stochastic truncation of $B(x)$ would result in ultraviolet finite fields
whose expectation values could never reproduce the indefinite signature so
evident in expression (184). Note also that it can be the same field whose
ultraviolet must be integrated out in derivative terms to give the appropriate
Langevin equation. This is evident for the single field model (10) in the
passage from the exact field equation (13),

$\Bigl{(}1\\!+\\!\frac{\lambda}{2}\Phi\Bigr{)}^{2}\partial_{\mu}\Bigl{(}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}\Phi\Bigr{)}+\frac{\lambda}{2}\Bigl{(}1\\!+\\!\frac{\lambda}{2}\Phi\Bigr{)}\partial_{\mu}\Phi\partial_{\nu}\Phi
g^{\mu\nu}\sqrt{-g}=0\;,$ (185)

to its stochastic realization (152),

$-\Bigl{(}1\\!+\\!\frac{\lambda}{2}\varphi\Bigr{)}^{2}\\!\times\\!3Ha^{3}\Bigl{(}\dot{\varphi}-\dot{\varphi}_{0}\Bigr{)}-\frac{3\lambda
H^{4}}{16\pi^{2}}\frac{a^{3}}{1\\!+\\!\frac{1}{2}\lambda\varphi}=0\;.$ (186)

Quantum field theories in an expanding universe have instantaneous energy
eigenstates, but the expansion of the universe prevents these eigenstates from
evolving onto one another. So what was the minimum energy state at one instant
is not generally minimum energy later on. Bunch-Davies vacuum corresponds to
the state that was minimum energy in the distant past. Although Starobinsky’s
stochastic formalism, and hence also our variant of it, was derived assuming
quantum fields in Bunch-Davies vacuum, it ought to apply broadly to states
which are perturbatively nearby. On the other hand, this is not true for
states which are highly excited from Bunch-Davies vacuum. Indeed, by making
suitable Bogoliubov transformations one can change the scalar and tensor power
spectra by potentially momentum-dependent factors which range from zero to
infinity! That same ambiguity must also afflict the stochastic formalism, as
it does all the other predictions of inflationary cosmology. We suspect that
the process by which the states of originally trans-Planckian wave numbers are
red-shifted to the point where quantum general relativity can be used as an
effective field theory leaves these states near Bunch-Davies vacuum. However,
that is a conjecture which can and should be studied.

We should comment on the peculiar notion of applying the renormalization group
to nonrenormalizable theories such as (10) and (20). At the order we have
worked, no renormalization was required for the 1PI 2-point functions of the
single field model (10), but we did need four counterterms (58) to renormalize
the self-masses for $A$ and $B$,

$\displaystyle\Delta\mathcal{L}=-\frac{1}{2}C_{A1}\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptA\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptA\sqrt{-g}-\frac{1}{2}C_{A2}\,R\partial_{\mu}A\partial_{\nu}Ag^{\mu\nu}\sqrt{-g}$
(187) $\displaystyle\hskip 96.73918pt-\frac{1}{2}C_{B1}\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptB\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptB\sqrt{-g}-\frac{1}{2}C_{B2}\,R\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;.\qquad$

Each of these counterterms involves higher derivatives, but there is an
important distinction between when those derivatives act on $A$ and $B$ and
when they act on the metric. The terms proportional to $C_{A1}$ and $C_{B1}$
involve higher derivatives of the fields $A$ and $B$ and play no role in the
generation of large inflationary logarithms. However, the terms proportional
to $C_{A2}$ and $C_{B2}$ can be viewed as curvature-dependent field strength
renormalizations of $A$ and $B$, respectively. It is the flow of these
couplings which serves to capture the green-colored logarithms in Tables 3 and
4. It is also worth noting that the basis for stochastic effects, the
effective potentials (143) and (159), are also curvature-dependent and would
vanish in the flat space limit.

This project suggests a number of extensions. The most urgent of these is
working out the curvature-dependent coupling constant renormalizations that
would allow us to determine the renormalization group flows. We would like to
determine the late time behavior and also whether or not renormalization group
improvement of the effective potentials matters at leading logarithm order. It
would also be interesting to compute the order $\lambda^{3}$ (two loop)
contribution to the expectation value of $A(x)$ to see if it agrees with the
stochastic prediction in expression (167). And we would like to know whether
or not the renormalization group can be used to explain the sub-dominant
logarithms one sometimes encounters in the rate at which the mode functions
freeze in,

$\displaystyle u_{0}(\eta,k)$ $\displaystyle\longrightarrow$
$\displaystyle\frac{H}{\sqrt{2k^{3}}}\Bigl{\\{}1+\frac{k^{2}}{2a^{2}H^{2}}+\dots\Bigr{\\}}\;,$
(188) $\displaystyle u_{1}(\eta,k)$ $\displaystyle\longrightarrow$
$\displaystyle\frac{H}{\sqrt{2k^{3}}}\Bigl{\\{}0+\frac{\\#\lambda^{2}k^{2}\ln(a)}{a^{2}}+\dots\Bigr{\\}}\;.$
(189)

A final spin-off is understanding the painfully accumulated collection of
large logarithms induced by inflationary gravitons in the mode functions and
exchange potentials of various matter theories [63, 65, 66, 95] and gravity
itself [96, 69].

Of course the primary motivation for studying nonlinear sigma models was to
understand the derivative interactions of quantum gravity, without the
plethora of indices and the miasma of confusion associated with gauge fixing.
We particularly wish to understand the viability of back-reaction to slow the
expansion rate in $\Lambda$-driven inflation [51, 97]. It is therefore worth
summarizing what this project suggests for quantum gravity:

  * •

The large logarithms induced by inflationary gravitons can likely be explained
using a combination of curvature-dependent effective potentials and curvature-
dependent renormalization group flows;

  * •

There seems to be no obstacle to inferring an effective potential by
integrating the ultraviolet out of the invariant Lagrangian [98],

$\displaystyle\mathcal{L}_{\rm
inv}=a^{D-2}\sqrt{-\widetilde{g}}\,\widetilde{g}^{\alpha\beta}\widetilde{g}^{\gamma\delta}\widetilde{g}^{\epsilon\zeta}\Bigl{[}\frac{1}{2}h_{\alpha\gamma,\epsilon}h_{\zeta\delta,\beta}\\!-\\!\frac{1}{2}h_{\alpha\beta,\gamma}h_{\delta\epsilon,\zeta}\\!+\\!\frac{1}{4}h_{\alpha\beta,\gamma}h_{\epsilon\zeta,\delta}$
(190) $\displaystyle\hskip
28.45274pt-\frac{1}{4}h_{\alpha\gamma,\epsilon}h_{\beta\delta,\zeta}\Bigr{]}+\frac{1}{2}(D\\!-\\!2)a^{D-1}H\sqrt{-\widetilde{g}}\,\widetilde{g}^{\alpha\beta}\widetilde{g}^{\gamma\delta}h_{\alpha\beta,\gamma}h_{\delta
0}\;,\qquad$

where $\widetilde{g}_{\mu\nu}\equiv\eta_{\mu\nu}+\kappa h_{\mu\nu}$ is
considered to be constant ($\kappa^{2}\equiv 16\pi G$), at which point we can
follow back-reaction by solving for the homogeneous evolution with the
effective potential, the same way we did with equations (148-149) for the
single field model and with equations (161-162) for the two field model;

  * •

Of the two invariant 1-loop counterterms,

$\Delta\mathcal{L}=\alpha_{1}R^{2}\sqrt{-g}+\alpha_{2}C^{\alpha\beta\gamma\delta}C_{\alpha\beta\gamma\delta}\sqrt{-g}\;,$
(191)

the one proportional to $\alpha_{2}$ likely plays no role in producing large
logarithms while the one proportional to $\alpha_{1}$ can be viewed as a
curvature-dependent renormalization of Newton’s constant, and its flow has the
potential to explain the unnaturally large value of $\alpha_{1}$ in
Starobinsky’s original model of inflation [99];

  * •

As long as the curvature remains nonzero there is no reason to assume that
evolution approaches a static limit, and the two field model provides an
explicit example of significant evolution persisting to arbitrarily late
times, cf. expression (162);

  * •

It seems inevitable that significant modifications to the expansion rate and
to the force of gravity will persist to late times; and

  * •

Curvature-dependent effective potentials and renormalization group flows pose
a challenge when back-reaction causes the curvature to change, but they also
provide a natural mechanism through which the effects of inflationary
gravitons can become dormant during radiation domination (with $R=0$) and then
reassert themselves at late times, after the transition to matter domination.

Acknowledgements

This work was partially supported by Taiwan MOST grants 109-2112-M-006-002 and
110-2112-M-006-026; by the European Union’s Horizon 2020 Programme under grant
agreement 669288-SM-GRAV-ERC-2014-ADG; by NSF grant 1912484; and by the
Institute for Fundamental Theory at the University of Florida.

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