CCTP-2024-7

UFIFT-QG-24-04

Summing Gravitational Effects from Loops of Inflationary Scalars

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 develop a procedure for re-summing the large logarithms induced in gravity
by loops of inflationary scalars. We first show how the scalar can be
integrated out of the field equations in the presence of constant graviton
field. We then extend this result to a fully conserved form which explains the
need for a finite renormalization of the cosmological constant which was
previously inferred from explicit computation. A variant of the
renormalization group turns out to explain the large logarithmic corrections
revealed by explicit computation in the electric field strength of
gravitational radiation and in the potentials which characterize the response
to a point mass. The implications for graviton loops are discussed.

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

⋆ e-mail<EMAIL_ADDRESS>
† e-mail<EMAIL_ADDRESS>
‡ e-mail<EMAIL_ADDRESS>
##  1 Prologue

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

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

The accelerated expansion ($H>0$ with $0\leq\epsilon<1$) of primordial
inflation rips virtual particles out of the vacuum [1]. The phenomenon is
largest for particles such as massless, minimally coupled (MMC) scalars and
gravitons, which are both massless and not conformally invariant [2, 3]. This
is what is thought to have caused the primordial spectra of gravitons [4] and
scalars [5].

Because more and more quanta are created as inflation progresses, correlators
which involve interacting MMC scalars and gravitons often show secular growth
in the form of powers of $\ln[a(t)]$ [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. For
example, the 2-loop dimensionally regulated and fully renormalized expectation
value of the stress tensor of an MMC scalar with a
$\frac{\lambda}{4!}\phi^{4}$ self-interaction, on de Sitter background
($\epsilon=0$) takes the perfect fluid form $\langle
T_{\mu\nu}\rangle=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$ with energy density and
pressure [6, 16, 17],

$\displaystyle\rho(t)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\lambda
H^{4}}{2^{7}\pi^{4}}\\!\times\\!\ln^{2}(a)+O(\lambda^{2})\;,\qquad$ (2)
$\displaystyle p(t)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\lambda
H^{4}}{2^{7}\pi^{4}}\Bigl{\\{}-\ln^{2}(a)-\tfrac{2}{3}\ln(a)\Bigr{\\}}+O(\lambda^{2})\;.\qquad$
(3)

In the correlators of this theory each factor of $\lambda$ can be associated
with as many as two factors of $\ln(a)$. Contributions which saturate this
bound are known as leading logarithm, whereas those which have fewer factors
of $\ln(a)$ are known as subleading. In expression (3) the factor of
$-\ln^{2}(a)$ is a leading logarithm whereas the factor of
$-\frac{2}{3}\ln(a)$ is subleading.

During a prolonged period of inflation factors of $\ln[a(t)]$ can grow so
large that they overwhelm even the smallest coupling constant. Developing a
technique to sum up the series of leading logarithms may eventually be as
important for cosmology as the renormalization group summation of leading
momentum logarithms was to flat space quantum field theory. The late Alexei
Starobinsky solved this problem for scalar potential models [18],

$\mathcal{L}=-\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi
g^{\mu\nu}\sqrt{-g}-\frac{\lambda}{4!}\phi^{4}\sqrt{-g}\;.$ (4)

Starobinsky’s technique can be proven to reproduce each order’s leading
logarithms [19] and, when $V(\phi)$ is bounded below, it can be summed up to
give the late time limits of cosmological correlators [20]. However,
straightforward application of the method fails for more general theories [8,
21, 22].

To understand the problem with more general theories it is necessary to
distinguish between “Active” fields which induce large logarithms, such as MMC
scalars and gravitons, and “Passive” fields which do not, such as conformally
coupled scalars, fermions, photons and even differentiated Active fields. One
can sum the leading logarithms of theories which contain Passive fields
without derivative interactions by integrating out the Passive fields,
assuming the Active fields are constant in space and time. This results in a
scalar potential model which can then be treated using Starobinsky’s formalism
[8, 21].

The large logarithms of scalar potential models derive from what DeWitt and
Brehme termed the “tail” part [23] of the MMC scalar propagator, 111In this de
Sitter background example (5), the tail is the logarithmic term.

$D=4\quad\Longrightarrow\quad
i\Delta(x;x^{\prime})=\frac{1}{4\pi^{2}}\frac{1}{aa^{\prime}(x\\!-\\!x^{\prime})^{2}}-\frac{H^{2}}{8\pi^{2}}\ln\Bigl{[}H^{2}(x\\!-\\!x^{\prime})^{2}\Bigr{]}\;.$
(5)

The situation when derivative interactions are present is more complicated
because renormalization provides a new mechanism for generating large
logarithms. When using dimensional regularization it turns out that
$D$-dependent factors of $a(t)$ do not occur in divergent primitive
contributions, which are incompletely canceled by counterterms,

$\displaystyle\Bigl{(}{\rm
Primitive:}\;\;\frac{(2H)^{D-4}}{D\\!-\\!4}\Bigr{)}-\Bigl{(}{\rm
Counterterm:}\;\;\frac{(\mu a)^{D-4}}{D\\!-\\!4}\Bigr{)}$ (6)
$\displaystyle\hskip 213.39566pt=-\ln\Bigl{(}\frac{\mu
a}{2H}\Bigr{)}+O(D\\!-\\!4)\;.\qquad$

These sorts of large logarithms are correctly described by a variant of the
renormalization group [24].

Derivative interactions are also more complicated in the way one integrates
out Passive fields in the present of a constant Active background. When
derivative interactions are absent the coupling to a constant Active field is
typically a mass term, as in Yukawa [8] or scalar quantum electrodynamics in
Lorenz gauge [21]. This leads to a conventional Coleman-Weinberg potential
[25], of course modified by functions of the dimensionless ratio of
$\frac{\phi}{H}$. Derivative interactions lead to a new type of effective
potential in which the field strength of a Passive field depends upon constant
Active fields [24]. In this paper we consider a third type of effective
potential in which constant Active fields shift the Hubble parameter on which
the Passive field propagators depend. 222Hubble-induced effective potentials
can also occur in theories for which matter fields assume a non-trivial
background [26].

The model we shall study is how a loop of MMC scalars on de Sitter background
induce changes in gravity,

$\mathcal{L}=-\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi
g^{\mu\nu}\sqrt{-g}+\frac{[R\\!-\\!(D\\!-\\!2)\Lambda]\sqrt{-g}}{16\pi G}\;,$
(7)

where $G$ is Newton’s constant and $\Lambda=(D-1)H^{2}$ is the cosmological
constant. The scalar loop contribution to the graviton self-energy was made
about a decade ago [27], and used to solve the effective field equations for
gravitational radiation [28, 29] and for the gravitational response to a point
mass [30]. Unfortunately, the original computation fails to be conserved
because it lacks a finite renormalization of the cosmological constant, the
necessity for which was only realized recently [31]. This problem was
compounded by representing the non-conserved graviton self-energy using a sum
of structure functions which are automatically conserved [29]. When the
computation was re-done, including the finite renormalization, and making no
assumptions about structure functions, a somewhat different result was found
for the $\ln[a(t)]$ correction to the potentials [32]. The effects on
gravitational radiation were also computed to enough accuracy to reveal a
logarithmic change in the electric component of the Weyl tensor. The purpose
of this paper is to explain both logarithmic corrections using a variant of
the renormalization group. We will also explain how to integrate the scalar
out of the gravitational field equation for constant graviton background, and
show that the induced stress tensor implies precisely the finite
renormalization of the cosmological constant which is needed to make the
graviton self-energy conserved [31].

This paper consists of six sections. In Section 2 we review non-linear sigma
models, which manifest large logarithms from renormalization (6) and also
stochastic logarithms from effective potentials derived from Active fields
modifying the field strengths of Passive fields. Section 3 shows that
integrating the scalar out of (7) in the presence of a spacetime constant
graviton background induces a new type of effective stress tensor by modifying
the Hubble constant on which the scalar propagator depends. We show that this
result explains the finite renormalization required in the exact calculation.
Because the induced stress tensor is only valid at leading logarithm order, it
is not conserved when the graviton field is allowed to depend on space and
time. In Section 4 we show how the stress tensor can be extended to give a
fully conserved form. A variant of the renormalization group is used in
Section 5 to explain the large logarithms found for gravitational radiation
and for the response to a point mass. Section 6 gives our conclusions and also
discusses the prospects for extending this analysis to loops of gravitons.

##  2 Non-linear Sigma Models

Non-linear sigma models have the same $\,h\partial h\partial h\,$ derivative
interactions as gravity, and induce the same factors of $\ln[a(t)]$ on de
Sitter background, but without the complex index structure or the gauge issue.
They have therefore received much attention as a simple venue for sorting out
the complexities of derivative interactions [19, 33, 34, 35, 24, 36, 37]. A
simple example consists of two scalar fields $A(x)$ and $B(x)$ with the
following Lagrangian,

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

The first variations of its action provide the equations of motion,

$\displaystyle\frac{\delta S[A,B]}{\delta A(x)}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\partial_{\mu}\Bigl{[}\sqrt{-g}g^{\mu\nu}\partial_{\nu}A\Bigr{]}-\frac{1}{2}\lambda\Bigl{(}1\\!+\\!\frac{1}{2}\lambda
A\Bigr{)}\partial_{\mu}B\;\partial_{\nu}B\;g^{\mu\nu}\sqrt{-g}=0\;,\qquad$ (9)
$\displaystyle\frac{\delta S[A,B]}{\delta B(x)}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\partial_{\mu}\Bigl{[}\Bigl{(}1\\!+\\!\frac{1}{2}\lambda
A\Bigr{)}^{2}\sqrt{-g}\;g^{\mu\nu}\partial_{\nu}B\Bigr{]}=0\;.$ (10)

The above theory is not renormalizable and hence requires an ever-increasing
number of counterterms.
* Perturbative Results

The quantities computed were the expectation values of $A,\,B$ and their
squares, the mode functions $u_{A}(\eta,k)\,\&\,u_{B}(\eta,k)$, and the
exchange potentials $P_{A}(\eta,r)\;\&\;P_{B}(\eta,k)$. The latter two are
obtained from the scalar self-mass $-iM^{2}(x;x^{\prime})$ which supplies the
quantum corrections to the linearized effective field equation for a scalar
$\Phi(x)$,

$\mathcal{D}\Phi(x)-\int\\!\\!d^{4}x^{\prime}M^{2}(x;x^{\prime})\Phi(x^{\prime})=J(x)\;,$
(11)

where $\,{\mathcal{D}}\equiv\partial^{\mu}a^{2}\partial_{\mu}\,$ is the
kinetic operator and $J(x)$ the source. The quantum corrections to the
propagation of scalar radiation are imprinted in the mode function correction
which is obtained when $J(x)=0$. The choice
$J(x)=a(\eta)\delta^{3}(\mathbf{x})$ gives the scalar exchange potential.

The above perturbative calculations of self-masses and VEV’s were performed in
the $``in-in"$ formalism which is causal and which allows only real self-
masses [38, 39, 40, 41, 42, 43, 44, 45, 46]. The regulation technique is
dimensional regularization which preserves coordinate invariance. As a result
the Lagrangian (8) requires two counterterms per scalar field to renormalize
the self-masses at 1-loop,

$\displaystyle\Delta\mathcal{L}_{-iM^{2}}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{1}{2}C_{\scriptscriptstyle A1}\,\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptA\,\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptA\,\sqrt{-g}-\frac{1}{2}C_{\scriptscriptstyle
A2}\,R\,\partial_{\mu}A\,\partial_{\nu}A\,g^{\mu\nu}\sqrt{-g}$ (12)
$\displaystyle-\frac{1}{2}C_{\scriptscriptstyle B1}\,\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptB\,\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptB\,\sqrt{-g}-\frac{1}{2}C_{\scriptscriptstyle
B2}\,R\,\partial_{\mu}B\,\partial_{\nu}B\,g^{\mu\nu}\sqrt{-g}\;.$

Moreover, the VEV’s of the squares of the scalar fields require composite
operator renormalization which at 1-loop and 2-loop orders implies the
following counterterms, 333No renormalizations are needed for the VEV’s of
$A(x)$ which is ultraviolet finite to this order and the VEV of $B(x)$
vanishes to all orders by virtue of the shift symmetry of (8).

$\displaystyle A_{\rm ren}^{2}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle
A^{2}+K_{\scriptscriptstyle A1}R+K_{\scriptscriptstyle
A2}R\,A^{2}+K_{\scriptscriptstyle A3}R^{2}+O(\lambda^{4})\;,$ (13)
$\displaystyle B_{\rm ren}^{2}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle
B^{2}+K_{\scriptscriptstyle B1}R+K_{\scriptscriptstyle
B2}R\,B^{2}+K_{\scriptscriptstyle B3}R^{2}+O(\lambda^{4})\;.$ (14)

Finally, to renormalize the 3-point vertex of the theory at 1-loop order, the
counterterm Lagrangian is,

$\displaystyle\Delta\mathcal{L}_{ABB}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{1}{2}C_{\scriptscriptstyle ABB1}\,\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptA\,\partial_{\mu}B\,\partial_{\nu}B\,g^{\mu\nu}\sqrt{-g}-\frac{1}{2}C_{\scriptscriptstyle
ABB2}\,\partial_{\mu}A\,\partial_{\nu}B\,\kern 1.0pt\vbox{\hrule
height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptB\,g^{\mu\nu}\sqrt{-g}$
(15) $\displaystyle-\frac{1}{2}C_{\scriptscriptstyle ABB3}\,A\,\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0ptB\,\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0ptB\,\sqrt{-g}-\frac{1}{2}C_{\scriptscriptstyle
ABB4}\,R\,A\,\partial_{\mu}B\,\partial_{\nu}B\,g^{\mu\nu}\sqrt{-g}\;.\qquad$

In (15) the last term is the curvature-dependent coupling constant
renormalization. It turns out that at 1-loop there is no such renormalization
and the associated 1-loop $\beta$-function vanishes [24, 36],

$\displaystyle\delta\lambda$ $\displaystyle\\!\\!\equiv\\!\\!$
$\displaystyle\lambda_{\rm ren}-\lambda=C_{\scriptscriptstyle ABB4}\times
R+O(\lambda^{5})=0\,\lambda^{3}+O(\lambda^{5})\;\Rightarrow$ (16)
$\displaystyle\beta$ $\displaystyle\\!\\!\equiv\\!\\!$
$\displaystyle\frac{\partial\delta\lambda}{\partial\ln(\mu)}=0\,\lambda^{3}+O(\lambda^{5})\;\;.$
(17)

All the coefficients in (12-15) are determined so that they absorb the
primitive divergences of the perturbative diagrams. The leading logarithm
renormalized results are most conveniently presented in the form of Table 1
[24, 36]:

Quantity | Leading Logarithms
---|---
$u_{A}(\eta,k)$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times\frac{H}{\sqrt{2k^{3}}}$
$u_{B}(\eta,k)$ | $\Bigl{\\{}1+0+O(\lambda^{4})\Bigr{\\}}\times\frac{H}{\sqrt{2k^{3}}}$
$P_{A}(\eta,r)$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)}{\color[rgb]{0,1,0}+\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)}+O(\lambda^{4})\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)$
$P_{B}(\eta,r)$ | $\Bigl{\\{}1{\color[rgb]{0,1,0}-\frac{\lambda^{2}H^{2}}{32\pi^{2}}\ln(Hr)}+O(\lambda^{4})\Bigr{\\}}\times\frac{KH}{4\pi}\ln(Hr)$
$\langle\Omega|A(x)|\Omega\rangle$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}+\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times{\color[rgb]{1,0,0}\frac{\lambda H^{2}}{16\pi^{2}}\ln(a)}$
$\langle\Omega|A^{2}(x)|\Omega\rangle_{\rm ren}$ | $\Bigl{\\{}1{\color[rgb]{1,0,0}-\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times{\color[rgb]{1,0,0}\frac{H^{2}}{4\pi^{2}}\ln(a)}$
$\langle\Omega|B(x)|\Omega\rangle$ | $0$
$\langle\Omega|B^{2}(x)|\Omega\rangle_{\rm ren}$ | $\Bigl{\\{}1{\color[rgb]{0,1,0}+\frac{3\lambda^{2}H^{2}}{32\pi^{2}}\ln(a)}+O(\lambda^{4})\Bigr{\\}}\times{\color[rgb]{1,0,0}\frac{H^{2}}{4\pi^{2}}\ln(a)}$
Table 1: Leading logarithm renormalized perturbative results for the AB model
to leading logarithm. Red denotes leading logarithms explained by the
stochastic formalism while green denotes those explained by the
renormalization group.

* Re-summation Techniques: Stochastic

For the infrared secular contributions due to the ever-increasing number of
degrees of freedom with super-horizon wavelengths $\,k<a(t)H$, the stochastic
method can re-sum the leading contributions and provide the late time
evolution; this was explicitly displayed by Starobinsky and Yokoyama [20] for
a scalar $\Phi(x)$ in de Sitter spacetime with a non-derivative self-
interacting potential $V(x)$ and a static late time evolution limit. The basic
idea is to replace $\Phi(x)$ with a stochastic field $\varphi(x)$ which
commutes with itself $[\varphi(x),\varphi(x^{\prime})]=0$, and whose
correlators are completely free of ultraviolet divergences. The stochastic
field $\varphi(x)$ is constructed from the same free creation and annihilation
operators that appear in $\Phi(x)$ in such a way that the two fields produce
the same leading logarithms at each order in perturbation theory. The
requirement to concentrate on these infrared modes, simplifies the equation of
motion to a Langevin equation obeyed by the stochastic field $\varphi(x)$,
444For the purposes of this analysis we have converted from conformal time
$\eta$ to co-moving time $t$.

${\dot{\varphi}}(x)-{\dot{\varphi}}_{o}(x)=-\frac{1}{3H}V^{\prime}({\varphi})\;\;,$
(18)

where $\varphi_{o}$ is the stochastic jitter given by the infrared truncated
free field mode sum emanating from $\Phi(x)$.

- The extension of the method to scalars with derivative self-interactions can be seen by noting, for instance in the equation of motion (9) for $A(x)$, that a constant $A(x)={\overline{A}}$ field is a field strength renormalization of $B(x)$. It follows that we can quantify the effect of the undifferentiated $A(x)$ to all orders by simply integrating out the differentiated $B(x)$ fields in (9) for constant $A(x)={\overline{A}}$. Taking the VEV of (9) gives the first order equation for the 1-loop effective potential,

$\displaystyle-V^{\prime}_{\rm eff}({\overline{A}})\,a^{D}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{1}{2}\lambda\Bigl{(}1\\!+\\!\frac{1}{2}\lambda{\overline{A}}\Bigr{)}a^{D-2}\Bigl{\langle}\Omega\Bigl{|}\partial^{\mu}B(x)\,\partial_{\mu}B(x)\Bigr{|}\Omega\Bigr{\rangle}_{\overline{A}}\;,$
(19) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle+\frac{\frac{1}{2}\lambda(D\\!-\\!1)kH^{2}a^{D}}{1\\!+\\!\frac{1}{2}\lambda{\overline{A}}}\;,$
(20)

which upon taking the unregulated $D\\!=\\!4$ limit and integrating gives the
1-loop effective potential,

$V_{\rm eff}(A)=-\frac{3H^{4}}{8\pi^{2}}\ln\Bigl{|}1+\frac{1}{2}\lambda
A\Bigr{|}\;.$ (21)

By integrating the derivative interactions out of the field equations in the
presence of a constant scalar background, we obtained a curvature-dependent
effective potential $V_{\rm eff}$ for which the standard stochastic procedure
applies: one merely replaces $V$ in the non-derivative re-summation rule with
$V_{\rm eff}$.

- When the generic field equation (9) of $A(x)$ is restricted to de Sitter spacetime we get, 555Since the scale factor varies much faster than the field during inflation, it is preferable to have derivatives act on the scale factor instead of the field. Thus, the single time derivative of the Hubble friction term dominates over the second time and space derivatives terms.

$-\frac{d}{dt}\Bigl{(}a^{3}\dot{A}\Bigr{)}-V_{\rm
eff}^{\prime}(A)a^{3}=0\qquad\Longrightarrow\qquad\dot{A}=\frac{\lambda
H^{3}}{16\pi^{2}}\frac{1}{1\\!+\\!\frac{1}{2}\lambda A}\;.$ (22)

Equation (22) can be solved exactly, and for the initial condition $A_{\rm
in}=0$ the solution is,

$A_{\rm
cl}=\frac{2}{\lambda}\Biggl{[}\sqrt{1+\frac{\lambda^{2}H^{2}}{16\pi^{2}}\ln(a)}-1\Biggr{]}\;.$
(23)

The Langevin equation (18) for the associated stochastic field
${\mathcal{A}}(x)$,

$\dot{\mathcal{A}}-\dot{\mathcal{A}}_{o}=\frac{\lambda
H^{3}}{16\pi^{2}}\frac{1}{1\\!+\\!\frac{1}{2}\lambda\mathcal{A}}\;,$ (24)

introduces the stochastic jitter coming from the infrared truncated free field
$\mathcal{A}_{o}$. The general solution consists of the $c$-number solution
(23) plus a series in powers of ${\mathcal{A}}_{o}$ coming from iterating
(24),

$\mathcal{A}=\mathcal{A}_{\rm
cl}+\mathcal{A}_{o}-\frac{\lambda^{2}H^{3}}{32\pi^{2}}\int_{0}^{t}dt^{\prime}\mathcal{A}_{o}+\frac{\lambda^{3}H^{3}}{64\pi^{2}}\int_{0}^{t}dt^{\prime}\mathcal{A}_{o}^{2}+O(\lambda^{4})\;.$
(25)

* Re-summation Techniques: Renormalization Group

For the leading logarithms coming from the ultraviolet sector, the standard
QFT re-summation methodology of the renormalization group needs, as well, an
extension due to the presence of a curved spacetime: counterterms can be
regarded as curvature-dependent renormalizations of the bare parameters
present in the original theory. Consider, for instance, the renormalization of
the composite operator $A^{2}(x)$. Of the general set of counterterms (13),
the $\,K_{\scriptscriptstyle A2}R\,A^{2}\,$ is a field strength
renormalization,

$A^{2}=\sqrt{Z_{\scriptscriptstyle A^{2}}}\\!\times\\!A_{\rm
ren}^{2}\quad,\quad Z_{\scriptscriptstyle A^{2}}=1-2K_{\scriptscriptstyle
A2}\\!\times\\!R+O(\lambda^{4})\;,$ (26)

which is curvature dependent.

- The re-summation of the leading logarithms from the renormalization group is well known in QFT: the 1-loop approximation to the running coupling re-sums the leading logarithmic behaviour of all the loop diagrams. 666Furthermore, the running coupling obtained by solving the renormalization group equations with the $\beta$-function approximated with its $\ell$-loop expression re-sums not only the leading logarithm arising at $n$-loop order, but also the first $\ell$-1 subleading logarithms. Recall that the $N$-point function $G_{N}(x_{1},\dots,x_{N};\lambda;\mu)$ at a scale $\mu$ can be expressed in terms of its value at a scale $\mu_{0}$ via the running coupling constant $\overline{\lambda}(\mu_{0})$,

$\displaystyle G_{N}(x_{1},\dots,x_{N};\lambda;\mu)$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle
G_{N}(x_{1},\dots,x_{N};\overline{\lambda}(\mu);\mu_{0})$ (27)
$\displaystyle\hskip
54.06006pt\times\exp\\!\left[-N\int_{\mu_{0}}^{\mu}\frac{d\mu^{\prime}}{\mu^{\prime}}\,\gamma\big{(}\overline{\lambda}(\mu^{\prime})\big{)}\right]\;,$

where $\gamma(\overline{\lambda}(\mu))$ is the $\gamma$-function of the
associated field.

Since for the AB model the 1-loop $\beta$-function vanishes, so that
$\overline{\lambda}(\mu)=\lambda$, the leading $\ln(\mu)$ re-summation becomes
a power law,

$G_{N}(x_{1},\dots,x_{N};\lambda;\mu)=G_{N}(x_{1},\dots,x_{N};\lambda;\mu_{0})\times\left[\frac{\mu^{\prime}}{\mu}\right]^{N\,\gamma(\lambda)}\;.$
(28)

* Agreement: Stochastic Analysis

To substantiate that the above re-summations do indeed produce the leading
logarithms, we must compare their predictions against the perturbative results
of Table 1.

- Starting from the stochastic re-summation, we expand the stochastic solution (25) in powers of $\lambda$ and take its VEV, 777The infrared truncated free field $\mathcal{A}_{o}$ has the canonically commuting creation and annihilation operators of $A$ so that: $\langle\Omega|\mathcal{A}_{o}(t,\vec{x})|\Omega\rangle=0$ and $\langle\Omega|\mathcal{A}_{o}^{2}(t,\vec{x})|\Omega\rangle=\frac{H^{2}\ln(a)}{4\pi^{2}}.$

$\Bigl{\langle}\Omega\Bigl{|}\mathcal{A}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}=\frac{\lambda
H^{2}}{16\pi^{2}}\ln(a)\Biggl{\\{}1+\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Biggr{\\}}\;,$
(29)

as well as the VEV of its square,

$\Bigl{\langle}\Omega\Bigl{|}\mathcal{A}^{2}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}=\frac{H^{2}}{4\pi^{2}}\ln(a)\Biggl{\\{}1-\frac{\lambda^{2}H^{2}}{64\pi^{2}}\ln(a)+O(\lambda^{4})\Biggr{\\}}\;.$
(30)

Comparing (29) and (30) with their perturbative counterparts
$\langle\Omega|A(x)|\Omega\rangle$ and $\langle\Omega|A^{2}(x)|\Omega\rangle$
in Table 1 shows perfect agreement, a highly non-trivial achievement indeed.
* Agreement: Renormalization Group

A similar non-trivial agreement is maintained when we consider the
renormalization group analysis and its re-summation of the leading logarithms.
For instance, the Callan-Symanzik equation for the composite operator
$A^{2}(x)$ has the form [24],

$\Bigl{[}a\frac{\partial}{\partial
a}+\beta\frac{\partial}{\partial\lambda}+\gamma_{\scriptscriptstyle
A^{2}}\Bigr{]}\Bigl{\langle}\Omega\Bigl{|}A^{2}(x)\Bigr{|}\Omega\Bigr{\rangle}_{\rm
ren}=0\;.$ (31)

The $\beta$ and $\gamma$ functions are,

$\beta\equiv\frac{\partial\delta\lambda}{\partial\ln(\mu)}=0+O(\lambda^{5})\quad,\quad\gamma_{\scriptscriptstyle
A^{2}}\equiv\frac{\partial\ln(Z_{\scriptscriptstyle
A^{2}})}{\partial\ln(\mu^{2})}=0+O(\lambda^{4})\;.$ (32)

Therefore, there should no leading logarithms coming from this analysis to
this order (2-loop) and indeed the corresponding perturbative result seen in
Table 1 for $\langle\Omega|A^{2}(x)|\Omega\rangle_{\rm ren}$ confirm this;
both leading logarithms are stochastic effects as described above. 888The same
is true for the renormalized VEV of $A(x)$.

- The curvature-dependent field strength renormalization of the composite operator $B^{2}(x)$ is [24],

$Z_{\scriptscriptstyle B^{2}}=1-2K_{\scriptscriptstyle
B2}\\!\times\\!R+O(\lambda^{4})\quad\Longrightarrow\quad\gamma_{\scriptscriptstyle
B^{2}}=-\frac{3\lambda^{2}H^{2}}{16\pi^{2}}+O(\lambda^{4}H^{4})\;,$ (33)

so that its associated Callan-Symanzik equation predicts the leading logarithm
contribution to this order to be $+\frac{3\lambda^{2}H^{2}}{16\pi^{2}}$ which
is identical to what is seen in Table 1 for
$\langle\Omega|B^{2}(x)|\Omega\rangle_{\rm ren}$.
* Comments

- It turns out that all entries in Table 1 are reproduced by one or the other of the re-summation techniques. Yet another example, is the $\lambda^{0}$ order term $\frac{H^{2}}{4\pi^{2}}\ln(a)$ that appears in the VEV’s of Table 1: it comes from the coincidence limit of the scalar propagator and is a stochastic effect; it could not be explained by the renormalization group since it does not obey the Callan-Symanzik equation.

- Among the physical conclusions from the above analysis, we could highlight the behaviour of the scalar field $A(x)$. It provides an example of a field which, (i) rolls down its potential at a faster rate due to its stochastic jitter, (ii) develops a positive mass from a stochastically induced effective potential, and (iii) has a time evolution that does not approach a static limit but grows and persists to arbitrarily late times.

##  3 Constant Graviton Induced Stress Tensor

On $D=4$ dimensional de Sitter the cosmological constant is $\Lambda=3H^{2}$.
The gravitational field equation associated with our model (7) is,

$R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+3H^{2}g_{\mu\nu}=8\pi
G\left[\delta^{\rho}_{~{}\mu}\delta^{\sigma}_{~{}\nu}-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\right]\partial_{\rho}\phi\,\partial_{\sigma}\phi\;.$
(34)

We seek to integrate out the scalar fields on the right hand side of (34). If
this could be done for an arbitrary metric field $g_{\mu\nu}(x)$ it would give
us the exact 1-loop effective field equation, however, this is not possible.
What we can do instead is to integrate out the scalars for constant graviton
field, which should suffice at leading logarithm order. As we shall see, this
constant background is just de Sitter spacetime with a different cosmological
constant.
- Consider the general class of conformally rescaled backgrounds with constant $H$ and arbitrary $\widetilde{g}_{\mu\nu}(x)$,

$g_{\mu\nu}(x)\equiv a^{2}\,\widetilde{g}_{\mu\nu}(x)\equiv
a^{2}\Big{[}\eta_{\mu\nu}+\kappa h_{\mu\nu}(x)\Big{]}\quad,\quad
a=-H{\eta}^{-1}\;.$ (35)

Here and henceforth $\kappa^{2}\equiv 16\pi G$ is the loop-counting parameter
of quantum gravity. For geometries (35) we find, 999We define:
$\widetilde{\Gamma}^{\rho}_{~{}\mu\nu}\equiv\frac{1}{2}\widetilde{g}^{\rho\sigma}\Bigl{(}\widetilde{g}_{\sigma\mu,\nu}+\widetilde{g}_{\nu\sigma,\mu}-\widetilde{g}_{\mu\nu,\sigma}\Bigr{)}$.

$\displaystyle\Gamma^{\rho}_{~{}\mu\nu}=aH\Bigl{(}\delta^{\rho}_{~{}\mu}\delta^{0}_{~{}\nu}+\delta^{\rho}_{~{}\nu}\delta^{0}_{~{}\mu}-\widetilde{g}^{0\rho}\,\widetilde{g}_{\mu\nu}\Bigr{)}+\widetilde{\Gamma}^{\rho}_{~{}\mu\nu}\;,$
(36) $\displaystyle
R^{\rho}_{~{}\sigma\mu\nu}=-H^{2}\,\widetilde{g}^{00}\Bigl{(}\delta^{\rho}_{~{}\mu}\,g_{\sigma\nu}-\delta^{\rho}_{~{}\nu}\,g_{\sigma\mu}\Bigr{)}$
$\displaystyle+\,aH\Bigl{[}\delta^{\rho}_{~{}\mu}\widetilde{\Gamma}^{0}_{~{}\nu\sigma}-\delta^{\rho}_{~{}\nu}\widetilde{\Gamma}^{0}_{~{}\mu\sigma}+\widetilde{g}^{\rho\alpha}\Bigl{(}\widetilde{\Gamma}^{0}_{~{}\mu\alpha}\,\widetilde{g}_{\nu\sigma}-\widetilde{\Gamma}^{0}_{~{}\nu\alpha}\,\widetilde{g}_{\mu\sigma}\Bigr{)}\Bigr{]}+\widetilde{R}^{\rho}_{~{}\sigma\mu\nu}\;,$
(37) $\displaystyle
R_{\mu\nu}=-3H^{2}\,\widetilde{g}^{00}g_{\mu\nu}+aH\Bigl{[}2\widetilde{\Gamma}^{0}_{~{}\mu\nu}+\widetilde{g}_{\mu\nu}\,\widetilde{g}^{\alpha\beta}\,\widetilde{\Gamma}^{0}_{~{}\alpha\beta}\Bigr{]}+\widetilde{R}_{\mu\nu}\;,$
(38) $\displaystyle
R=-12H^{2}\,\widetilde{g}^{00}+6\frac{H}{a}\,\widetilde{g}^{\alpha\beta}\,\widetilde{\Gamma}^{0}_{~{}\alpha\beta}+\frac{1}{a^{2}}\,\widetilde{R}\;.$
(39)

Thus, the left hand side of (34) becomes,

$\displaystyle
R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+3H^{2}g_{\mu\nu}=3H^{2}\Bigl{(}1+\widetilde{g}^{00}\Bigr{)}g_{\mu\nu}+2aH\Bigl{[}\widetilde{\Gamma}^{0}_{~{}\mu\nu}\\!-\\!\widetilde{g}_{\mu\nu}\,\widetilde{g}^{\alpha\beta}\,\widetilde{\Gamma}^{0}_{~{}\alpha\beta}\Bigl{]}$
$\displaystyle\hskip
76.82234pt+\widetilde{R}_{\mu\nu}-\frac{1}{2}\widetilde{g}_{\mu\nu}\widetilde{R}\;.$
(40)

- When we focus on geometries with $h_{\mu\nu}(x)$ constant (which implies constant $\widetilde{g}_{\mu\nu}(x)$), so that only terms without derivatives acting on the field are selected, the curvature tensor (37) simplifies,

$R^{\rho}_{~{}\sigma\mu\nu}\Big{|}_{\widetilde{g}_{\mu\nu}=c}=-H^{2}\,\widetilde{g}^{00}\Bigl{(}\delta^{\rho}_{~{}\mu}\,g_{\sigma\nu}-\delta^{\rho}_{~{}\nu}\,g_{\sigma\mu}\Bigr{)}\;,$
(41)

and we recognize a de Sitter geometry albeit with a different cosmological
constant,

$\widetilde{g}_{\mu\nu,\rho}=0\quad\Longrightarrow\quad
H^{2}\;\longrightarrow\;-\widetilde{g}^{00}H^{2}\;.$ (42)

- To integrate out the differentiated scalars present in the right hand side of (34),

$\partial_{\mu}\phi(x)\,\partial_{\nu}\phi(x)\;\;\Longrightarrow\;\;\Bigl{\langle}\Omega\,\Bigl{|}\partial_{\mu}\phi(x)\,\partial_{\nu}\phi(x)\Bigr{|}\,\Omega\Bigr{\rangle}\;,$
(43)

the above expectation value is to be evaluated in the scalar vacuum for the
background (35) with $\widetilde{g}_{\mu\nu}(x)$ a spacetime constant.

What is needed for the computation of (43) is the coincidence limit of the
mixed derivative of the scalar propagator in de Sitter spacetime, a quantity
that has been evaluated using dimensional regularization (and the old Hubble
parameter $H$) [6, 16],

$\displaystyle\lim_{x^{\prime}\to
x}\partial_{\mu}\partial^{\prime}_{\nu}\,i\Delta_{dS}(x;x^{\prime})=-\frac{H^{D}}{(4\pi)^{\frac{D}{2}}}\,\frac{\Gamma(D)}{2\Gamma(\frac{D}{2}\\!+\\!1)}\times
g_{\mu\nu}(x)\Big{|}_{\widetilde{g}_{\mu\nu}=c}$ $\displaystyle\hskip
22.76228pt\longrightarrow\;-\frac{3H^{4}}{32\pi^{2}}\times
g_{\mu\nu}(x)\Big{|}_{\widetilde{g}_{\mu\nu}=c}\;.$ (44)

The finite $D\\!=\\!4$ remainder in (44) emerges after the automatic
subtraction of the quartic and quadratic divergences of the doubly
differentiated propagator.

By effecting the replacement (42) in the evaluation (44), the right hand side
(43) equals,

$\Bigl{\langle}\Omega\,\Bigl{|}\partial_{\mu}\phi(x)\,\partial_{\nu}\phi(x)\Bigr{|}\;\Omega\Bigr{\rangle}_{g_{\mu\nu}=a^{2}\widetilde{g}_{\mu\nu}}=-\frac{3\,[-\widetilde{g}^{00}H^{2}]^{2}}{32\pi^{2}}\\!\times\\!g_{\mu\nu}(x)\;,$
(45)

so that,

$\displaystyle 8\pi
G\,\Bigl{\langle}\Omega\,\Bigl{|}\partial_{\mu}\phi\,\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\partial_{\rho}\phi\,\partial_{\sigma}\phi\Bigr{|}\,\Omega\Bigr{\rangle}_{g_{\mu\nu}=a^{2}\widetilde{g}_{\mu\nu}}$
$\displaystyle\hskip
167.87108pt=\frac{3\kappa^{2}[-\widetilde{g}^{00}H^{2}]^{2}}{64\pi^{2}}\times
g_{\mu\nu}(x)\,.$ (46)

Note that (46) represents a negative contribution to the cosmological constant
which can become arbitrarily large if $\,-\widetilde{g}^{00}$ achieves
arbitrarily large values.
- Substituting (40) and (46) into the gravitational field equations (34) we arrive at the leading logarithm form,

$\displaystyle\widetilde{R}_{\mu\nu}-\frac{1}{2}\widetilde{g}_{\mu\nu}\widetilde{R}+2aH\Bigl{[}\widetilde{\Gamma}^{0}_{~{}\mu\nu}\\!-\\!\widetilde{g}_{\mu\nu}\,\widetilde{g}^{\alpha\beta}\,\widetilde{\Gamma}^{0}_{~{}\alpha\beta}\Bigl{]}$
$\displaystyle\hskip
82.51282pt=-3H^{2}\left[1+\widetilde{g}^{00}-\frac{\kappa^{2}H^{2}}{64\pi^{2}}(\widetilde{g}^{00})^{2}\right]g_{\mu\nu}\;,$
(47)

displaying the scalar-induced gravitational stress-energy tensor $T^{\rm
ind}_{\mu\nu}\,$,

$T^{\rm
ind}_{\mu\nu}=\frac{3}{32\pi^{2}}\Big{[}-\widetilde{g}^{00}H^{2}\Big{]}^{2}g_{\mu\nu}\;,$
(48)

always with the understanding that $\,\widetilde{g}_{\mu\nu}(x)\\!=\\!{\rm
constant}$.

For $\widetilde{g}^{00}\\!=\\!-1$ the induced stress tensor (48) takes the
form of a negative addition to the cosmological constant of,

$\Delta\Lambda=-\frac{3\kappa^{2}H^{4}}{64\pi^{2}}\;.$ (49)

If we want the quantity “$H$” to represent the true Hubble parameter, at least
initially, (49) must be absorbed by making a finite renormalization of the
cosmological constant. This is precisely the finite renormalization which was
previously recognized as being necessary to make the scalar loop contribution
to the graviton self-energy conserved — see equations (33) and (38) of [31].

##  4 Generally Conserved Extension

The induced stress tensor (48) is not conserved when $\widetilde{g}_{\mu\nu}$
is allowed to depend on space and time. It is important to understand that
this is not an error but rather the inevitable consequence of working at
leading logarithm order. Consider, for example, the $\lambda\phi^{4}$ energy
density and pressure (2-3). Under the symmetries of homogeneity and isotropy
conservation reduces to $\dot{\rho}=-3H(\rho+p)$, and relations (2-3) do obey
this,

$\displaystyle\dot{\rho}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\lambda H^{4}}{2^{7}\pi^{2}}\times
2H\ln(a)+O(\lambda^{2})\;,$ (50) $\displaystyle-3H(\rho+p)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle-3H\Biggl{\\{}\frac{\lambda
H^{4}}{2^{7}\pi^{2}}\Bigl{[}+\ln^{2}(a)-\ln^{2}(a)-\tfrac{2}{3}\ln(a)\Bigr{]}+O(\lambda^{2})\Biggr{\\}}\;.\qquad$
(51)

However, the right hand side (51) only agrees with the left (50) by virtue of
the sub-leading factor of $-\frac{2}{3}\ln(a)$ in the pressure (3). With just
the $\,\pm\ln^{2}(a)$ leading logarithm contributions, conservation is
violated.

Although the violation of conservation of the induced stress tensor (48) has a
good explanation, it does present us with a problem. The gravitational field
equations (34) consist of 10 relations, for a general metric, of which 4 are
automatically implied by conservation. If one adds the non-conserved stress
tensor (48) this will no longer be true. All 10 of the equations cannot be
used, because the 0th order equations are still conserved. So which 6 of the
10 equations should we solve, and why? Note that this sort of issue could not
arise with the very similar effective potentials which are induced in non-
linear sigma models; it is specific to gravity and it will have to be
confronted as well when graviton loops are considered.

We believe that the answer is to extend the leading logarithm stress tensor
(48) to a generally conserved form. The expressions (38)and (39) for the Ricci
tensor and scalar suggest that we could regard $[-\widetilde{g}^{00}H^{2}]$ as
$\frac{R}{12}$ and try to extend the stress-energy tensor (48) so that it
emerges from a more general Lagrangian,

$\mathcal{L}_{T}=f(R)\sqrt{-g}\;,$ (52)

which gives the conserved source,

$\frac{\kappa^{2}}{\sqrt{-g}}\frac{\delta S_{T}[g]}{\delta
g^{\mu\nu}(x)}=-\frac{1}{2}\kappa^{2}\,g_{\mu\nu}f(R)+\kappa^{2}\Bigl{[}R_{\mu\nu}+g_{\mu\nu}\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0pt-
D_{\mu}D_{\nu}\Bigr{]}f^{\prime}(R)\;,$ (53)

where $\,\kern 1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip
3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule
height=0.6pt}\kern
1.0pt\equiv\frac{1}{\sqrt{-g}}\partial_{\mu}[\sqrt{-g}g^{\mu\nu}\partial_{\nu}]$
is the scalar covariant d’Alembertian and $D_{\mu}$ is the covariant
derivative operator. By substituting (38) and (39) into (53), by discarding
the sub-leading terms involving $\widetilde{\Gamma}^{0}_{~{}\alpha\beta}$ and
$\widetilde{R}$, and by defining $X\equiv-12\,\widetilde{g}^{00}H^{2}$,
equation (53) reduces to a 1st order differential equation for the function
$f(X)$,

$\frac{X^{2}}{2^{9}\cdot
3\cdot\pi^{2}}=-f(X)+\frac{1}{2}Xf^{\prime}(X)=\frac{X^{3}}{2}\Big{(}\frac{f(X)}{X^{2}}\Big{)}^{\prime}\;,$
(54)

which has the solution,

$f(X)=\frac{X^{2}\ln(X)}{2^{8}\cdot 3\cdot\pi^{2}}\;,$ (55)

whose 1st and 2nd derivatives equal,

$f^{\prime}(X)=\frac{2X\ln(X)+X}{2^{8}\cdot 3\cdot\pi^{2}}\quad,\quad
f^{\prime\prime}(X)=\frac{2\ln(X)+3}{2^{8}\cdot 3\cdot\pi^{2}}\;.$ (56)

- Furthermore, the two derivative terms present in (53) are,

$\displaystyle g_{\mu\nu}\kern 1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule
width=1.2pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule
width=0.6pt}\hrule height=0.6pt}\kern 1.0pt$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\widetilde{g}_{\mu\nu}\Bigl{[}\widetilde{g}^{\rho\sigma}\partial_{\rho}\partial_{\sigma}+2aH\,\widetilde{g}^{0\rho}\partial_{\rho}-\widetilde{g}^{\rho\sigma}\widetilde{\Gamma}^{\alpha}_{~{}\rho\sigma}\partial_{\alpha}\Bigr{]}\;,$
(57) $\displaystyle-D_{\mu}D_{\nu}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\partial_{\mu}\partial_{\nu}+aH\Bigl{[}\delta^{0}_{~{}\mu}\partial_{\nu}+\delta^{0}_{~{}\nu}\partial_{\mu}-\widetilde{g}_{\mu\nu}\widetilde{g}^{0\rho}\partial_{\rho}\Bigr{]}+\widetilde{\Gamma}^{\rho}_{~{}\mu\nu}\partial_{\rho}\;.$
(58)

It will turn out to be rather useful to extract the metric from the Ricci
tensor (38),

$\displaystyle R_{\mu\nu}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{1}{4}g_{\mu\nu}R+2aH\Bigl{[}\widetilde{\Gamma}^{0}_{~{}\mu\nu}\\!-\\!\frac{1}{4}\widetilde{g}_{\mu\nu}\,\widetilde{g}^{\alpha\beta}\,\widetilde{\Gamma}^{0}_{~{}\alpha\beta}\Bigr{]}+\Bigl{[}\widetilde{R}_{\mu\nu}\\!-\\!\frac{1}{4}\widetilde{g}_{\mu\nu}\widetilde{R}\Bigr{]}$
(59) $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\frac{1}{4}g_{\mu\nu}R+\Delta R_{\mu\nu}\;,$

and to define the dimensionless Ricci scalar $\mathcal{R}$ from (39),

$R=12H^{2}\Big{[}\\!-\widetilde{g}^{00}+\frac{\widetilde{g}^{\alpha\beta}\,\widetilde{\Gamma}^{0}_{~{}\alpha\beta}}{2aH}+\frac{\widetilde{R}}{12a^{2}H^{2}}\Big{]}\equiv
12H^{2}\times{\mathcal{R}}\;.$ (60)

In view of (59), (60) and (55), (56) the three terms on the right hand side of
(53) become,

$\displaystyle-\frac{1}{2}\kappa^{2}\,g_{\mu\nu}\,f(R)\;=\;-6\epsilon
H^{2}\mathcal{R}^{2}\ln(\mathcal{R})\,g_{\mu\nu}\;,$ (61)
$\displaystyle\kappa^{2}R_{\mu\nu}f^{\prime}(R)\;=\;3\epsilon
H^{2}\mathcal{R}^{2}\Bigl{[}2\ln(\mathcal{R})+1\Bigr{]}g_{\mu\nu}+\epsilon\mathcal{R}\Bigl{[}2\ln(\mathcal{R})+1\Bigr{]}\Delta
R_{\mu\nu}\;,\qquad$ (62) $\displaystyle\kappa^{2}\Bigl{[}g_{\mu\nu}\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0pt\\!-\\!D_{\mu}D_{\nu}\Bigr{]}f^{\prime}(R)\;=\;\epsilon\Bigl{[}g_{\mu\nu}\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern
1.0pt\\!-\\!D_{\mu}D_{\nu}\Bigr{]}\Bigl{[}2\mathcal{R}\ln(\mathcal{R})+\mathcal{R}\Bigr{]}\;,$
(63)

where $\epsilon\equiv\frac{\kappa^{2}H^{2}}{64\pi^{2}}$.

Summing (61-63) gives,

$\displaystyle\frac{\kappa^{2}}{\sqrt{-g}}\frac{\delta S_{T}[g]}{\delta
g^{\mu\nu}(x)}=3\epsilon
H^{2}\mathcal{R}^{2}\,g_{\mu\nu}+\epsilon\mathcal{R}\Bigl{[}2\ln(\mathcal{R})+1\Bigr{]}\Delta
R_{\mu\nu}$ $\displaystyle\hskip 93.89418pt+\epsilon\Bigl{[}g_{\mu\nu}\kern
1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip 3.0pt\vbox{\vskip
6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule height=0.6pt}\kern 1.0pt-
D_{\mu}D_{\nu}\Bigr{]}\Bigl{[}2\mathcal{R}\ln(\mathcal{R})+\mathcal{R}\Bigr{]}\;.\qquad$
(64)

The spacetime geometries of cosmological interest are homogeneous,

$\widetilde{g}_{00}=-\frac{1}{x(t)}\qquad,\qquad\widetilde{g}_{0i}=0\qquad,\qquad\widetilde{g}_{ij}=\delta_{ij}\;.$
(65)

The quantities of interest have the following form,

$\displaystyle\widetilde{\Gamma}^{0}_{~{}00}=-\frac{a\dot{x}}{2x}\qquad\Longrightarrow\qquad\widetilde{g}^{\alpha\beta}\Gamma^{0}_{~{}\alpha\beta}=\frac{1}{2}a\dot{x}\;,$
(66) $\displaystyle\widetilde{R}_{\mu\nu}=0\qquad,\qquad\Delta
R_{\mu\nu}=-a^{2}H\dot{x}\Bigl{[}\,\frac{1}{x}\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}+\frac{1}{4}\widetilde{g}_{\mu\nu}\Bigr{]}\;,\qquad$
(67) $\displaystyle\mathcal{R}=x+\frac{\dot{x}}{4H}\;,$ (68)

so that the relevant derivative operators become,

$g_{\mu\nu}\kern 1.0pt\vbox{\hrule height=1.2pt\hbox{\vrule width=1.2pt\hskip
3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule width=0.6pt}\hrule
height=0.6pt}\kern 1.0pt-
D_{\mu}D_{\nu}=-g_{\mu\nu}\Bigl{[}x\partial_{t}+2xH+\frac{1}{2}\dot{x}\Bigr{]}\partial_{t}-\frac{a^{2}}{x}\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}\Bigl{[}x\partial_{t}-xH+\frac{1}{2}\dot{x}\Bigr{]}\partial_{t}\;.$
(69)

Consequently, the two non-zero components in this background are,

$\displaystyle\frac{\kappa^{2}}{\sqrt{-g}}\frac{\delta S_{T}[g]}{\delta
g^{00}(x)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\epsilon\,g_{00}\times A\;,$ (71) $\displaystyle
A\;\equiv\;3H^{2}\mathcal{R}^{2}+3H\Bigl{(}\frac{1}{4}\dot{x}-x\partial_{t}\Bigr{)}\Bigl{[}2\mathcal{R}\ln(\mathcal{R})+\mathcal{R}\Bigr{]}\;.$
$\displaystyle\frac{\kappa^{2}}{\sqrt{-g}}\frac{\delta S_{T}[g]}{\delta
g^{ij}(x)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\epsilon\,g_{ij}\times B\;,$ (73) $\displaystyle
B\;\equiv\;3H^{2}\mathcal{R}^{2}-\Bigl{(}\frac{1}{4}H\dot{x}+2Hx\partial_{t}+\frac{1}{2}\dot{x}\partial_{t}+x\partial_{t}^{2}\Bigr{)}\Bigl{[}2\mathcal{R}\ln(\mathcal{R})+\mathcal{R}\Bigr{]}\;,\qquad$

and the conservation equation is satisfied,

$\dot{A}=3H(-A+B)\;.$ (74)

Moreover, the extended source (64) has the proper correspondence limit (48)
because when $\widetilde{g}_{\mu\nu}$ is constant only the first term in the
right hand side of (64) survives. It should also be noted that the extension
we analyzed is by no means unique.

##  5 Renormalization Group Analysis

Analysis of the exact 1-loop effective field equations produced three results
[32]. First, the electric component of the Weyl tensor for plane wave
gravitational radiation
($h_{ij}(t,\vec{x})=\epsilon_{ij}(\vec{k},\lambda)u(t,k)e^{i\vec{k}\cdot\vec{x}}$)
receives a logarithmic correction,

$C_{0i0j}(t,\vec{x})=C^{(0)}_{0i0j}(t,\vec{x})\Bigl{\\{}1-\tfrac{3GH^{2}}{10\pi}\\!\times\\!\ln(a)+O(G^{2})\Bigr{\\}}\;.$
(75)

A similar logarithmic correction affects the gravitational potentials,

$ds^{2}=-\Bigl{[}1\\!-\\!2\Psi(t,r)\Bigr{]}dt^{2}+a^{2}(t)\Bigl{[}1-2\Phi(t,r)\Bigr{]}d\vec{x}\\!\cdot\\!d\vec{x}\;,$
(76)

generated in response to a static point mass $M$,

$\displaystyle\Psi(t,r)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tfrac{GM}{ar}\Bigl{\\{}1+\tfrac{G}{20\pi
a^{2}r^{2}}-\tfrac{3GH^{2}}{10\pi}\\!\times\\!\ln(aHr)+O(G^{2})\Bigr{\\}}\;,$
(77) $\displaystyle\Phi(t,r)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\tfrac{GM}{ar}\Bigl{\\{}1-\tfrac{G}{60\pi
a^{2}r^{2}}-\tfrac{3GH^{2}}{10\pi}\\!\times\\!\Bigl{[}\ln(aHr)\\!+\\!1\Bigr{]}+O(G^{2})\Bigr{\\}}\;.$
(78)

The factors of $\frac{G}{a^{2}r^{2}}$ in (77-78) represent de Sitter versions
of corrections which have long been known on flat space background [47, 48,
49]. They are not leading logarithm contributions and have no interest for us.

The three leading logarithm corrections (75) and (77-78) are not explained by
the induced stress tensor we considered in the previous section. They closely
resemble the logarithmic corrections previously found to the electric
components of the field strength tensor for plane wave electromagnetic
radiation [13], and to the Coulomb potential for a static point charge [12].
Those two results can be explained by a variant of the renormalization group
[50], and it turns out that a similar explanation works for the gravitational
logarithms (75) and (77-78).

The model (7) was the first quantum gravitational system ever studied using
dimensional regularization [51]. It has long been known that this model
requires two counterterms at 1-loop order,

$\Delta\mathcal{L}=c_{1}R^{2}\sqrt{-g}+c_{2}C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}\sqrt{-g}\;.$
(79)

The coefficients $c_{1}$ and $c_{2}$ which were used to derive the 1-loop
results (75) and (77-78) are [27, 32],

$\displaystyle c_{1}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\mu^{D-4}\Gamma(\frac{D}{2})}{2^{8}\pi^{\frac{D}{2}}}\frac{(D\\!-\\!2)}{(D\\!-\\!1)^{2}(D\\!-\\!3)(D\\!-\\!4)}\;,$
(80) $\displaystyle c_{2}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\mu^{D-4}\Gamma(\frac{D}{2})}{2^{8}\pi^{\frac{D}{2}}}\frac{2}{(D\\!+\\!1)(D\\!-\\!1)(D\\!-\\!3)^{2}(D\\!-\\!4)}\;,$
(81)

where $\mu$ is the scale of dimensional regularization.

Like the non-linear sigma models considered in Section 3, our model (7) is
nonrenormalizable. Recall from Section 3 that it was necessary to sort the
resulting BPHZ (Bogoliubov, Parasiuk [52], Hepp [53] and Zimmermann [54, 55]
counterterms into those which can be viewed as renormalizing parameters of the
bare theory and those that provide irrelevant higher derivative contributions
which do not induce large logarithms. Experience with the large
electromagnetic logarithms induced by graviton loops [50] shows that this
sometimes requires dissecting the original counterterms. For example,
renormalizing the 1-loop graviton contribution to the vacuum polarization on
de Sitter background requires three counterterms [56, 57], 101010The non-
covariant $\Delta C$ counterterm arises from the unavoidable breaking of de
Sitter invariance in the graviton propagator.

$\displaystyle\Delta\mathcal{L}=\Delta
C\,H^{2}F_{ij}F_{k\ell}g^{ik}g^{j\ell}\sqrt{-g}+\overline{C}H^{2}F_{\mu\nu}F_{\rho\sigma}g^{\mu\rho}g^{\nu\sigma}\sqrt{-g}$
(82)
$\displaystyle+C_{4}\,D_{\alpha}F_{\mu\nu}\,D_{\beta}F_{\rho\sigma}\,g^{\alpha\beta}g^{\mu\rho}g^{\nu\sigma}\sqrt{-g}\;.\qquad$

However, the curvature-dependent field strength renormalization which explains
the large electromagnetic logarithms is [50],

$\delta Z=-4\Bigl{[}\overline{C}-(3D\\!-\\!8)C_{4}\Bigr{]}H^{2}\;.$ (83)

The analogous decomposition of the gravitational counterterms (79) begins by
writing the Eddington ($R^{2}$) counterterm as the sum of three terms,

$R^{2}=\Bigl{(}R\\!-\\!D\Lambda\Bigr{)}^{2}+2D\Lambda\Bigl{[}R\\!-\\!(D\\!-\\!2)\Lambda\Bigr{]}+D(D\\!-\\!4)\Lambda^{2}\;.$
(84)

(i) The first term in (84) involves products of second derivatives of the
graviton field, and explicit computation shows that it engenders no large
logarithms [32].
(ii) However, the second term can be viewed either as a renormalization of
Newton’s constant in the original model (7), or else as a renormalization of
the graviton field strength. Because we have no information about higher order
couplings it is convenient to adopt the latter interpretation.
(iii) The third term is finite.

The interpretation adopted for the second term in (84) applies as well to the
special term arising, at quadratic order, from the term proportional to
$\,\partial^{2}_{0}h_{\nu\sigma}$ in the Weyl counterterm of (79),

$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}=-\tfrac{2\kappa}{a^{4}}\,\partial_{\mu}\partial_{\rho}h_{\nu\sigma}\,\widetilde{C}^{\mu\nu\rho\sigma}+O(\kappa^{3}h^{3})\;\longrightarrow\;-\tfrac{2\kappa}{a^{4}}\,\partial_{0}^{2}h_{ij}\,\widetilde{C}_{0i0j}\;,$
(85)

where $\widetilde{C}^{\mu\nu\rho\sigma}$ is the conformally rescaled Weyl
tensor. Direct calculation [32] shows that the two time derivatives can act on
scale factors to eventually contribute to the large logarithms (75) and
(77-78).

Hence the total field strength renormalization is,

$\delta
Z=D\Bigl{[}2(D\\!-\\!1)c_{1}-c_{2}\Bigr{]}\kappa^{2}H^{2}\quad\Longrightarrow\quad\gamma\equiv\frac{\partial\ln(1\\!+\\!\delta
Z)}{\partial\ln(\mu^{2})}=-\frac{3GH^{2}}{20\pi}\;.$ (86)

The Callan-Symanzik equation for a 2-point Green’s function reads,

$\Bigl{[}\frac{\partial}{\partial\ln(\mu)}+\beta_{G}\,\frac{\partial}{\partial
G}+2\gamma\Bigr{]}G^{(2)}=0\;.$ (87)

The $\beta$-function for $G$ vanishes at the order we are working. From
expression (6) we see that factors of the renormalization scale $\mu$ come in
the form $\ln(\mu a)$, so that we can replace the derivative with respect to
$\ln(\mu)$ by a derivative with respect to $\ln(a)$. The Weyl tensor (75) can
be considered as the 2-point correlator between the operator $C_{0i0j}$ and
the single particle creation operator; similarly, the potentials (77-78) can
be considered as 2-point correlators. It follows that the three large
logarithms in (75) and (77-78) can all be explained using the renormalization
group. If we ignore possible running of Newton’s constant it is even possible
to sum up the effects to conclude,

$\displaystyle C_{0i0j}(t,\vec{x})$ $\displaystyle\\!\\!\longrightarrow\\!\\!$
$\displaystyle
C^{(0)}_{0i0j}(t,\vec{x})\\!\times\\!\Bigl{[}a(t)\Bigr{]}^{-\frac{3GH^{2}}{10\pi}}\;,$
(88) $\displaystyle\Psi(t,r)$ $\displaystyle\\!\\!\longrightarrow\\!\\!$
$\displaystyle\frac{GM}{a(t)r}\\!\times\\!\Bigl{[}a(t)Hr\Bigr{]}^{-\frac{3GH^{2}}{10\pi}}\;,$
(89) $\displaystyle\Phi(t,r)$ $\displaystyle\\!\\!\longrightarrow\\!\\!$
$\displaystyle-\frac{GM}{a(t)r}\\!\times\\!\Bigl{[}a(t)Hr\Bigr{]}^{-\frac{3GH^{2}}{10\pi}}\;.$
(90)

##  6 Epilogue

The continuous production of MMC scalars and gravitons during inflation causes
loop corrections which would be constant in flat space to acquire secular
growth factors in the form of powers of the logarithm of the scale factor [6,
7, 8, 9, 10, 11, 12, 13, 14, 15]. Over a prolonged period of inflation these
factors must overwhelm even the smallest coupling constant. At this point
perturbation theory breaks down and one must employ some non-perturbative re-
summation. Starobinsky’s stochastic formalism [18, 20] sums the series of
leading logarithms scalar potential models (4) but it must be extended to
correctly describe the leading logarithms of more general models. In these
models one must distinguish between Active fields which engender secular
growth factors and Passive fields which do not. One form of extending
Starobinsky’s formalism is to integrate out Passive fields in the presence of
a spacetime constant Active field background. The result is an effective
potential which can then be treated using Starobinsky’s formalism. With the
work done in this paper we now know of three ways in which these effective
potentials can be induced:

  * •

Through a constant Active field giving rise to a Passive field mass, as in
Yukawa [8];

  * •

Through a constant Active field changing a Passive field strength, as in
nonlinear sigma models [24]; and

  * •

Through a constant Active field changing the Hubble parameter on which a
Passive field propagator depends, as occurs in our model (7).

In Section 3 we considered an MMC scalar coupled to gravity (7) and integrated
the scalar out of the gravitational field equation in the presence of a
constant graviton background. It turns out that this is equivalent to merely
changing the de Sitter Hubble constant according to the rule (42). The
resulting stress tensor (48) precisely explains the finite renormalization of
the cosmological constant which was previously noted as being necessary to
make the graviton self-energy conserved [31].

In Section 4 we noted that the induced stress tensor (48) is not conserved
when one goes beyond leading logarithm order to permit to graviton field to
vary in space and time. This is not an error, but rather the inevitable
consequence of working at leading logarithm order. However, it does pose a
problem in solving the gravitational field equations. We showed that (48) can
be extended to a form (64) which is generally conserved.

For theories which possess derivative interactions, such as non-linear sigma
models and gravity, a second extension must be made of Starobinsky’s formalism
to include the large logarithms induced by the incomplete cancellation between
primitive divergences and counterterms (6). These logarithms are described by
a variant of the renormalization group in which some portion of the BPHZ
counterterms can be regarded as renormalizing couplings or field strengths of
the original theory. That procedure was described in Section 2 for non-linear
sigma models, and we applied it in Section 5 to the large logarithms (75 and
(77-78) which occur in our model (7). The two BPHZ counterterms (79) can be
regarded as giving rise to a graviton field strength renormalization (86)
which explains all three of the large logarithms. If we ignore possible
running of Newton’s constant, it is even possible to give fully re-sumed
results (88-90). We remark that it should be possible to determine any running
of $G$ by computing the 1PI 3-point function, the same as has recently been
done for non-linear sigma models [36].

The point of making this study was to facilitate the development of a leading
logarithm re-summation technique for graviton loops, so it is worth commenting
on what will generalize and what may differ. First, the procedure of
integrating out differentiated graviton fields for constant
$\widetilde{g}_{\mu\nu}$ should involve the same replacement (42) as for our
scalar model (7). However, the more complicated form of the graviton
propagator [58, 59] makes it likely that the induced stress tensor will
contain a term proportional to $\delta^{0}_{\mu}\delta^{0}_{\nu}$ in addition
to one proportional to $g_{\mu\nu}$. Like the scalar-induced stress tensor
(48), it is inevitable that the graviton-induced stress tensor will not be
conserved when one permits the graviton field to depend on space and time. We
plan to seek a fully conserved extension, but we do not expect that a local
one such as (64) can be found. We also expect that 1-loop corrections to the
graviton mode function and to the response to a point mass will involve
stochastic fluctuations in these extended, effective field equations, in
addition to renormalization group effects. Finally, we expect the need for
more counterterms than just the two covariant possibilities (79), owing to the
same de Sitter breaking which produced the noncovariant $\Delta C$ counterterm
(82) for electromagnetism.

Acknowledgements

This work was partially supported by Taiwan NSTC grants 111-2112-M-006-038 and
112-2112-M-006-017, by NSF grant PHY-2207514 and by the Institute for
Fundamental Theory at the University of Florida.

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