# Vortex Structures and Momentum Sharing
in Dynamic Sauter–Schwinger Process

A. Bechler<EMAIL_ADDRESS>F. Cajiao Vélez  K. Krajewska  J. Z.
Kamiński  Institute of Theoretical Physics, Faculty of Physics, University of
Warsaw, Pasteura 5, 02-093 Warszawa, Poland

(August 28, 2024)

###### Abstract

Vortex pattern formation in electron-positron pair creation from vacuum by a
time-dependent electric field of linear polarization is analyzed. It is
demonstrated that in such scenario the momentum distributions of created
particles exhibit vortex-antivortex pairs. Their sensitivity to the laser
field parameters such as the field frequency and intensity is also studied.
Specifically, it is shown that with increasing field frequency accross the
one-photon threshold additional vortex-antivortex pairs appear. Their location
in the momentum space is consistent with a general threshold behavior of
probability distributions of created electrons (positrons). Namely, while for
small field frequencies the particles tend to be created along the field
polarization direction, for large enough frequencies they are predominantly
generated in the perpendicular direction. Such change in longitudinal and
transverse momentum sharing of created particles occurs accross the one-photon
threshold.

electron-positron pair creation, vortices, threshold effects, Dirac-
Heisenberg-Wigner formalism

##  I Introduction

Nonlinear response of quantum vacuum to macroscopic electromagnetic fields,
leading to creation of electron-positron ($e^{-}e^{+}$) pairs, has been
predicted by Sauter Sauter , Heisenberg and Euler Heisenberg-Euler , and
Schwinger Schwinger . Since then, various authors have made significant
contributions to our current understanding of this process, which we will
refer to as the Sauter-Schwinger process. Specifically, Białynicki-Birula,
Górnicki and Rafelski have established a new framework for treating the
quantum vacuum in electromagnetic fields IBB1  (see, also, Refs. IBB102 ;
IBB103  and the PhD thesis of Ł. Rudnicki LR2012 ). This is by means of what
they called the Dirac-Heisenberg-Wigner (DHW) function, which describes the
$e^{-}e^{+}$ densities in phase space. Later on, the method was largely
explored for the case of spatially homogeneous electric fields (see, e.g., HAG
; IBB2 ; archiwum ; Blinne ; Blinne1 ; PhDthesis ; Xie1 ; Xie2 ; IZBB ). For
instance, the quantum kinetic approach was recovered in that case HAG  and
various analytical results for exactly solvable fields were derived HAG ; IBB2
; archiwum . Recently, the spontaneous formation of time-crystal structures in
the $e^{-}e^{+}$ pair creation was discovered by Białynicki-Birula and
Białynicka-Birula in Ref. IZBB . Other applications of the DHW formalism in
the context of pair creation concern the case of parallel spatially
homogeneous electric and magnetic fields Sheng , the standing electric wave
Heben ; Berenyi ; Naseri ; Mohamedsedik , and inhomogeneous electric and
magnetic fields in one spatial direction Alkofer1 ; Kohlfu1 ; Kohlfu2 . The
latter limitation follows entirely from performing capabilities of current
computers, as the DHW method is very general and can be used in arbitrary
dimensions. It is also important to emphasize that the DHW method is not
limited to describing pair creation from vacuum. For instance, it was argued
that DHW is very useful for practical plasma applications such as studies of
Langmuir waves in a high-density plasma plasmareview .

Another area of research where prof. I. Białynicki-Birula has largely
contributed to is related to quantum vortices. It follows from the
hydrodynamical formulation of quantum mechanics that the probability fluid can
inherently possess vortices Cieplak . They are defined as phase singularities
of the wave function and their strenght is measured in terms of a topological
charge Cieplak ; BS ; Taylor . As it was discussed in BS ; Taylor , the
vortices form isolated lines, that either emerge from a single point forming a
closed loop or can be created as a pair of lines with opposite topological
charges. Those mechanisms of creation and subsequent annihilation of vortex-
antivortex pairs were confirmed recently in a series of papers focused on
vortex structures in strong-field ionization Pe2 ; Pe1 ; Pe3 ; Felipe ;
Majczak . Specifically, it was demonstrated that the vortex structures are
very sensitive to the laser field parameters, and so they can be easily
steered by the field. While the aforementioned papers deal with quantum
vortices in nonrelativistic quantum mechanics, their notion can be also
extended to relativistic quantum theory as has been proposed by Białynicki-
Birula and Białynicka-Birula in BBvortices . See also, the construction of
knotted vortex states, or hopfionslike states, in relativistic quantum
mechanics BBhopfions  by the same authors.

Note that electron-positron pair creation and ionization are formally similar,
as they are both threshold-related phenomena which can be driven by external
dynamically changing fields. For this reason, one might expect similar effects
being exhibited in both processes. Keeping this in mind, in the current paper
we investigate whether vortex structures similar to Pe2 ; Pe1 ; Pe3 ; Felipe ;
Majczak  can be observed in the probability amplitude of $e^{-}e^{+}$ pair
creation in the presence of a linearly polarized time-dependent electric
field. Our emphasis will be on threshold behavior of those patterns, which can
be studied, for instance, by changing the frequency of the driving field. As
we will show, this is in agreement with the longitudinal and the transverse
momentum sharing of created particles across the threshold which has been
studied in Ref. KK012104 . At this point we would like to mention that other
structures, known as spiral vortex patterns, were found in strong-field
ionization SAF ; Wollen  and later on in pair production Xie1 ; Xie2  for
certain combinations of circularly polarized electric field pulses. As
demonstrated however in Ref. Pe3 , in the case of ionization such spirals in
momentum distributions of photoelectrons does not necessarily carry a nonzero
topological charge which distinguishes them from vortices analyzed in Refs.
Pe2 ; Pe1 ; Pe3 ; Felipe ; Majczak . The same is expected to hold for pair
creation.

Our paper is organized as follows. Based on the original derivation presented
in Ref. IBB1 , we introduce the DHW formalism in Sect. II. Bispinorial
decomposition of the DHW-function for a spatially homogeneous electric field
is presented in Sect. III and the final equations for a linear polarization
are given in Sect. IV. Section V is devoted to vortex patterns in $e^{-}e^{+}$
pair creation and their sensitivity to external field parameters, especially
when passing across the threshold. Another threshold-related effect is
discussed in Sect. VI, where we demonstrate how the particles momentum is
redistributed across the threshold of pair creation. Our final remarks are
given in Sect. VII.

##  II The DHW-function for fermion field

The DHW-function for the fermion field is defined as IBB1

$\displaystyle W_{\alpha\beta}(\bm{x},\bm{p},t)=-\frac{1}{2}\int
d^{3}se^{-i\bm{p}\cdot\bm{s}}\langle
0|\mathcal{U}(\bm{s},\bm{x},t)[\Psi_{\alpha}(\bm{x}+\bm{s}/2,t),\Psi^{\dagger}_{\beta}(\bm{x}-\bm{s}/2,t)]|0\rangle,$
(1)

where the factor $\mathcal{U}(\bm{s},\bm{x},t)$ contains line integral of the
vector potential in temporal gauge $A^{0}=0$,

$\displaystyle\mathcal{U}(\bm{s},\bm{x},t)=\exp\left[-ie\int_{-1/2}^{1/2}d\xi\bm{s}\cdot\bm{A}(\bm{x}+\xi\bm{s},t)\right],$
(2)

and assures gauge invariance of the DHW-function, whereas
$\Psi_{\alpha},\,\Psi_{\beta}$ are the fermion field operators in Heisenberg
picture. We use here the version of DHW-function with the vacuum expectation
value HAG ; in general, however any pure or mixed state can be used IBB1 . The
DHW-function is a $4\times 4$ Hermitian matrix and as such can be decomposed
in terms of 16 Hermitian matrices $\Gamma_{a}$ with real coefficients
depending, in general, on $\bm{x}$, $\bm{p}$ and time. Matrices
$\Gamma_{a},\,a=0,1,2,...,15$ can be constructed as Kronecker products of two
sets of Pauli matrices (including the identity matrix), $(I_{2},\rho_{j})$ and
$(I_{2},\sigma_{j})$ IBB1 . The correspondence looks as follows

$\displaystyle\begin{split}&\Gamma_{0}=I_{4},\,\Gamma_{j}=\rho_{j}\otimes
I_{2},\,\Gamma_{j+3}=I_{2}\otimes\sigma_{j},\,\Gamma_{j+6}=\rho_{1}\otimes\sigma_{j},\\\
&\Gamma_{j+9}=\rho_{2}\otimes\sigma_{j},\,\Gamma_{j+12}=\rho_{3}\otimes\sigma_{j},\end{split}$
(3)

where the index $j=1,2,3$. In terms of standard $\gamma$-matrices,

$\displaystyle\begin{split}&\Gamma_{0}=I_{4},\,\Gamma_{1}=\gamma_{5},\,\Gamma_{2}=-i\gamma^{0}\gamma_{5},\,\Gamma_{3}=\gamma^{0},\,\Gamma_{j+3}=\Sigma^{j},\,\\\
&\Gamma_{j+6}=\alpha^{j},\,\Gamma_{j+9}=-i\gamma^{j},\,\Gamma_{j+12}=\gamma^{0}\Sigma^{j},\end{split}$
(4)

where $\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$, and
$\Sigma^{j}=\gamma^{5}\alpha^{j}$ are the $4\times 4$ spin matrices. With the
use of Eq. (4) expansion of the DHW-function can be written in the form IBB1

$\displaystyle\begin{split}W(\bm{x},\bm{p},t)=&\frac{1}{4}(f_{0}+\gamma_{5}f_{1}-i\gamma^{0}\gamma_{5}f_{2}+\gamma^{0}f_{3}+\bm{\Sigma}\cdot\bm{g}_{0}+\bm{\alpha}\cdot\bm{g}_{1}\\\
&-i\bm{\gamma}\cdot\bm{g}_{2}+\gamma^{0}\bm{\Sigma}\cdot\bm{g}_{3}).\end{split}$
(5)

The expansion coefficients are the same as $f_{0},\,f_{1},\,f_{2},\,f_{3}$ and
$\bm{g}_{0},\,\bm{g}_{1},\,\bm{g}_{2},\,\bm{g}_{3}$ used in IBB1 .

Equations fulfilled by the expansion coefficients can be found by calculating
their time derivatives using Dirac equation for the fermion field operators.
In deriving these equations one usually adopts the Hartree-, or mean-EB
approximation neglecting quantum fluctuations of the electromagnetic field
IBB1 ; HAG . This is equivalent to the replacements

$\displaystyle\langle
0|\hat{F}^{\mu\nu}(\bm{x},t)\mathcal{U}(\bm{s},\bm{x},t)[\Psi(\bm{x}_{1},t),\Psi^{\dagger}(\bm{x}_{2},t)]|0\rangle\rightarrow
F^{\mu\nu}(\bm{x},t)\langle
0|\mathcal{U}(\bm{s},\bm{x},t)[\Psi(\bm{x}_{1},t),\Psi^{\dagger}(\bm{x}_{2},t)]|0\rangle,$

i.e., the operator of quantum electromagnetic field is replaced by classical
C-number field. Application of the Dirac equation for fermion field operators
in general case of space- and time-dependent electromagnetic field results in
a complicated system of 16 integro-differential equations for the expansion
coefficients of the DHW-function. These equations simplify significantly for
spatially homogeneous electric field vanishing for $t\rightarrow\pm\infty$,
which is the subject of main interest in the present paper. Initial conditions
are determined by free vacuum value of DHW-function. It follows then from Eq.
(1) with zero electromagnetic field and free Dirac field operators that only
the coefficients $f_{3}$ and $\bm{g}_{1}$ survive and their vacuum values are

$\displaystyle
f^{vac}_{3}=-\frac{2mc^{2}}{E_{\bm{p}}},\quad\bm{g}^{vac}_{1}=-\frac{2c\bm{p}}{E_{\bm{p}}},$
(6)

where $E_{\bm{p}}=\sqrt{c^{2}\bm{p}^{2}+m^{2}c^{4}}$ is the free particle
energy. In the case of spatially homogeneous electric field only the
coefficients $\bm{g}_{0}$ and $\bm{g}_{2}$ couple to the vacuum values (6), so
that it is sufficient to consider 10 equations for
$f_{3},\,\bm{g}_{0},\,\bm{g}_{1},\,\bm{g}_{2}$. They have the form IBB2 ;
archiwum

$\displaystyle[\partial_{t}+e\bm{\mathcal{E}}(t)\cdot\nabla_{\bm{p}}]W(\bm{p},t)=\frac{c}{\hbar}M(\bm{p})W(\bm{p},t),$
(7)

where $W$ denotes 10-dimensional vector

$\displaystyle W=[f_{3},\,\bm{g}_{0},\,\bm{g}_{1},\,\bm{g}_{2}],$ (8)

and the $10\times 10$ matrix $M$ has the following block structure

$\displaystyle
M(\bm{p})=\left[\begin{array}[]{cccc}0&\bm{0}^{T}&\bm{0}^{T}&2\bm{p}^{T}\\\
\bm{0}&\mathds{O}_{3}&2\bm{p}\times&\mathds{O}_{3}\\\
\bm{0}&2\bm{p}\times&\mathds{O}_{3}&-2mcI_{3}\\\
-2\bm{p}&\mathds{O}_{3}&2mcI_{3}&\mathds{O}_{3}\end{array}\right],$ (13)

where $\bm{0}$ and $\bm{p}$ are 3-dimensional null and momentum column
vectors, $\mathds{O}_{3}$ \- the $3\times 3$ null matrix and $I_{3}$ is the
3-dimensional identity matrix. Notation $\bm{p}\times$ means that when acting
on a 3-dimensional vector to the right it gives its vector product with
$\bm{p}$. Explicitly,

$\displaystyle\bm{p}\times=\left[\begin{array}[]{ccc}0&-p_{3}&p_{2}\\\
p_{3}&0&-p_{1}\\\ -p_{2}&p_{1}&0\end{array}\right].$ (17)

In closing this Section, we note that physical interpretation of the DHW-
functions can be found in Ref. IBB1 . In particular, the phase space energy
density is given by IBB1 ; Blinne ,

$\displaystyle\varepsilon(t,\bm{r},\bm{p})=c\bm{p}\cdot\bm{g}_{1}(t,\bm{r},\bm{p})+mc^{2}f_{3}(t,\bm{r},\bm{p}).$
(18)

The one particle distribution function, which will be used in Sect. V for
numerical analysis of momentum distributions, is defined as Blinne

$\displaystyle
f(t,\bm{r},\bm{p})=\frac{\varepsilon(t,\bm{r},\bm{p})-\varepsilon_{vac}}{2E_{p}}=\frac{\varepsilon(t,\bm{r},\bm{p})}{2E_{p}}+1,$
(19)

where $\varepsilon_{vac}$ was expressed by vacuum DHW-functions (6). It is
also worth noting that the DHW formalism is very general as it allows one to
account for an arbitrary electromagnetic field. However, for a spatially
homogeneous electric field, other approaches can be conveniently applied; one
of which being developed next.

##  III Bispinorial representation of the DHW functions for spatially
homogeneous electric field

We consider Dirac equation in the spatially homogeneous electric field
$\bm{\mathcal{E}}(t)=-\partial_{t}\bm{\mathcal{A}}(t)$, with the vector
potential vanishing both for $t\rightarrow-\infty$ and $t\rightarrow\infty$.
Due to translational invariance of the problem spatial dependence of the wave
function is of plane wave type,

$\displaystyle\Psi(t,\bm{x})=\exp\left(\frac{i}{\hbar}\bm{p}\cdot\bm{x}\right)\Phi_{\bm{p}r}(t),$
(20)

where time-dependent bispinor $\Phi_{\bm{p}r}(t)$ is labeled by asymptotic
momentum $\bm{p}$ and spin index $r$, and fulfills the equation

$\displaystyle i\hbar\partial_{t}\Phi_{\bm{p}r}(t)=H_{D}(t)\Phi_{\bm{p}r}(t),$
(21)

where the time-dependent Hamiltonian reads

$\displaystyle
H_{D}(t)=c\bm{\alpha}\cdot[\bm{p}-e\bm{\mathcal{A}}(t)]+\gamma^{0}mc^{2}.$
(22)

To make contact with the DHW-functions we construct 16 expressions bilinear in
the bispinor $\Phi_{\bm{p}r}(t)$

$\displaystyle
S_{a}(\bm{p},t)=\sum_{r}\Phi^{\dagger}_{\bm{p}r}(t)\Gamma_{a}\Phi_{\bm{p}r}(t).$
(23)

Using the Dirac equation (21) and its Hermitian conjugate one finds equations
fulfilled by the functions $S_{a}$,

$\displaystyle\partial_{t}S_{a}=\frac{i}{\hbar}\sum_{r}\Phi^{\dagger}_{\bm{p}r}[H_{D}(t),\Gamma_{a}]\Phi_{\bm{p}r}.$
(24)

The Dirac Hamiltonian $H_{D}(t)$ can be written in terms of $\Gamma$-matrices
as

$\displaystyle
H_{D}(t)=c\Gamma_{j+6}[p^{j}-e\mathcal{A}^{j}(t)]+mc^{2}\Gamma_{3},$ (25)

where summation convention for the Cartesian index $j$ was used. The $\Gamma$
matrices fulfill commutation relations

$\displaystyle[\Gamma_{a},\,\Gamma_{b}]=i\sum_{c=0}^{15}f^{c}\,_{ab}\Gamma_{c},$
(26)

where $f^{c}\,_{ab}$ are real structure constants of the algebra of $\Gamma$
matrices. Substituting Eq. (25) into Eq. (24) and using Eq. (26) gives

$\displaystyle\partial_{t}S_{a}=-\frac{c}{\hbar}(p^{j}-e\mathcal{A}^{j})\sum_{b=0}^{15}f^{b}\,_{j+6,a}S_{b}-\frac{mc^{2}}{\hbar}\sum_{b=0}^{15}f^{b}\,_{3a}S_{b}.$
(27)

Nonvanishing structure constants are (indices $i,j,k$ take the values $1,2,3$)

$\displaystyle f^{k}\,_{ij}=2\epsilon_{ijk},$ $\displaystyle
f^{i+12}\,_{1,\,i+9}=2,\,f^{i+9}\,_{1,\,i+12}=-2,\,f^{i+12}\,_{2,\,i+6}=-2,\,f^{i+6}\,_{2,\,i+12}=2,$
$\displaystyle f^{i+9}\,_{3,\,i+6}=2,\,f^{i+6}\,_{3,\,i+9}=-2,$ $\displaystyle
f^{k+3}\,_{i+3,j+3}=f^{k+3}\,_{i+6,j+6}=f^{k+3}\,_{i+9,j+9}=f^{k+3}\,_{i+12,j+12}=2\epsilon_{ijk},$
(28) $\displaystyle
f^{k+6}\,_{i+3,j+6}=f^{k+9}\,_{i+3,j+9}=f^{k+12}\,_{i+3,j+12}=2\epsilon_{ijk},$
$\displaystyle
f^{1}\,_{i+9,j+12}=2\delta_{ij},\,f^{2}\,_{i+6,j+12}=-2\delta_{ij},\,f^{3}\,_{i+6,j+9}=2\delta_{ij},$

plus the structure constants obtained from anti-symmetry relation
$f^{a}\,_{c\,b}=-f^{a}\,_{b\,c}$. It is now straightforward, though a little
tedious, to derive 16 equations fulfilled by the functions $S_{a}$

$\displaystyle\partial_{t}S_{0}=0,$ (29a)
$\displaystyle\partial_{t}S_{1}=-2\frac{mc^{2}}{\hbar}S_{2},$ (29b)
$\displaystyle\partial_{t}S_{2}=-2\frac{c}{\hbar}(p^{j}-e\mathcal{A}^{j})S_{j+12}+2\frac{mc^{2}}{\hbar}S_{1},$
(29c)
$\displaystyle\partial_{t}S_{3}=2\frac{c}{\hbar}(p^{j}-e\mathcal{A}^{j})S_{j+9},$
(29d)
$\displaystyle\partial_{t}S_{k+3}=2\frac{c}{\hbar}\epsilon_{kjl}(p^{j}-e\mathcal{A}^{j})S_{l+6},$
(29e)
$\displaystyle\partial_{t}S_{k+6}=2\frac{c}{\hbar}\epsilon_{kjl}(p^{j}-e\mathcal{A}^{j})S_{l+3}-2\frac{mc^{2}}{\hbar}S_{k+9},$
(29f)
$\displaystyle\partial_{t}S_{k+9}=-2\frac{c}{\hbar}(p^{k}-e\mathcal{A}^{k})S_{3}+2\frac{mc^{2}}{\hbar}S_{k+6},$
(29g)
$\displaystyle\partial_{t}S_{k+12}=2\frac{c}{\hbar}(p^{k}-e\mathcal{A}^{k})S_{2}.$
(29h)

Note that equations containing 6 functions
$S_{0},\,S_{1},\,S_{2},\,S_{13},\,S_{14},\,S_{15}$ do not couple to remaining
ten equations for
$S_{3},\,S_{4},\,S_{5},\,S_{6},\,S_{7},\,S_{8},\,S_{9},\,S_{10},\,S_{11},\,S_{12}$.
Denoting

$\displaystyle S_{3}=h_{3},\quad(S_{4},S_{5},S_{6})=\bm{h}_{0},\quad$
$\displaystyle(S_{7},S_{8},S_{9})=\bm{h}_{1},\quad(S_{10},S_{11},S_{12})=\bm{h}_{2},$
(30)

we see that equations (29) for ten-dimensional vector
$V=[h_{3},\,\bm{h}_{0},\,\bm{h}_{1},\,\bm{h}_{2}]$ can be written in the
matrix form as

$\displaystyle\partial_{t}V=\frac{c}{\hbar}M(\bm{p}(t))V,$ (31)

where

$\displaystyle\bm{p}(t)=\bm{p}-e\bm{\mathcal{A}}(t),$ (32)

and the matrix $M$ is given by (13). The same system of ordinary differential
equations follows from Eq. (7) after applying the method of characteristics to
first order partial differential equations IBB1 ; HAG ; archiwum ; Blinne .
Therefore two vectors $W$ and $V$ obey the same system of ordinary
differential equations. In order to identify fully $V$ and $W$ one needs to
show that they fulfill also the same initial conditions, which for $W$ are
given by Eq. (6) and $\bm{g}_{0}^{vac}=0=\bm{g}_{2}^{vac}$.

The Dirac wave function pertaining to pair creation process should fulfill
Feynman boundary conditions: for $t\rightarrow-\infty$ it contains only
solutions of free Dirac equation with negative energy, whereas for
$t\rightarrow\infty$ it is a combination of positive and negative energy parts
with the negative energy contribution equal to the wave function of the
created positron. Extensive discussion of boundary conditions fulfilled by
solutions of Dirac equation in classical electromagnetic field can be found in
IBB3 . It can be also shown that Feynman boundary conditions are “forced” by
LSZ-reduction formulae for the S-matrix element of pair creation. For
$t\rightarrow-\infty$ we have therefore

$\displaystyle\Phi_{\bm{p}s}=\exp\left(\frac{i}{\hbar}E_{\bm{p}}t\right)w^{(-)}_{-\bm{p}s}.$
(33)

Substituting Eq. (33) into Eq. (23) one can show that coefficients (30)
fulfill the following initial conditions for $t\rightarrow-\infty$

$\displaystyle\bm{h}_{0}^{0}=0=\bm{h}_{2}^{0},\quad
h^{0}_{3}=-\frac{2mc^{2}}{E_{\bm{p}}},\quad\bm{h}^{0}_{1}=-\frac{2c\bm{p}}{E_{\bm{p}}},$
(34)

corresponding exactly to vacuum initial conditions for the vector $W$.

The bispinorial approach to the dynamic Sauter-Schwinger pair production by
spatially homogeneous electric fields has been developed in this Section.
Importantly, it has been proven to be equivalent to the DHW formalism
described in Sect. II. Therefore, similar to the DHW method, it has advantaged
over other approaches. Specifically, it allows to treat an arbitrarily
polarized time-dependent electric field. Having said that, we turn to the case
of a linear polarization, for which other well-established theories exist and
can be tested against (see, for instance, Refs. KMK ; KK2019  and references
therein).

##  IV Linearly polarized field and analogy with two level atom

In general, the vector $W$ (or, equivalently $V$) can be expressed as a
combination of ten orthonormal basis vectors $\mathds{E}_{a}$

$\displaystyle W=-2\sum_{a=1}^{10}u_{a}\mathds{E}_{a}.$ (35)

With choice of $-\frac{1}{2}W^{vac}$ as one of the basis elements one can show
that for linearly polarized field $\bm{\mathcal{A}}(t)=\mathcal{A}(t)\bm{n}$
three vectors

$\displaystyle\mathds{E}_{1}=\frac{c}{E_{\bm{p}}\epsilon_{\perp}}\left[\begin{array}[]{c}-mc^{2}(\bm{n}\cdot\bm{p})\\\
\bm{0}\\\ \dfrac{E^{2}_{\bm{p}}\bm{n}}{c}-c(\bm{n}\cdot\bm{p})\bm{p}\\\
\bm{0}\end{array}\right],\,\mathds{E}_{2}=\frac{c}{\epsilon_{\perp}}\left[\begin{array}[]{c}0\\\
\bm{p}\times\bm{n}\\\ \bm{0}\\\
mc\bm{n}\end{array}\right],\,\mathds{E}_{3}=\frac{1}{E_{\bm{p}}}\left[\begin{array}[]{c}mc^{2}\\\
\bm{0}\\\ c\bm{p}\\\ \bm{0}\end{array}\right],$ (48)

where $\epsilon_{\perp}=\sqrt{c^{2}\bm{p}_{\perp}^{2}+m^{2}c^{4}}$, form a set
closed under the action of $\bm{n}\cdot\nabla_{\bm{p}}$ and $M$ in Eq. (7).
Note that $\mathds{E}_{3}=-(1/2)W^{vac}$. Choosing $\bm{n}$ in the
$z$-direction (${\bm{n}}={\bm{e}}_{3}$) we have

$\displaystyle\begin{split}&\frac{\partial}{\partial
p_{3}}\mathds{E}_{1}=-\frac{c\epsilon_{\perp}}{E_{\bm{p}}^{2}}\mathds{E}_{3},\quad\frac{\partial}{\partial
p_{3}}\mathds{E}_{2}=0,\quad\frac{\partial}{\partial
p_{3}}\mathds{E}_{3}=\frac{c\epsilon_{\perp}}{E_{\bm{p}}^{2}}\mathds{E}_{1},\quad\\\
&M\mathds{E}_{1}=\frac{2E_{\bm{p}}}{c}\mathds{E}_{3},\quad
M\mathds{E}_{2}=-\frac{2E_{\bm{p}}}{c}\mathds{E}_{1},\quad
M\mathds{E}_{3}=0.\end{split}$ (49)

Solution of Eq. (7) can be expressed as

$\displaystyle
W(\bm{p},t)=-2\sum_{a=1}^{3}u_{a}(\bm{p},t)\mathds{E}_{a}(\bm{p}(t)).$ (50)

Substituting Eq. (50) to Eq. (31), and denoting
$[u_{1},\,u_{2},\,u_{3}]=\bm{u}$ we obtain the precession-type equation for
$\bm{u}$

$\displaystyle\partial_{t}\bm{u}=\bm{a}\times\bm{u},$ (51)

with the vector $\bm{a}$ given by

$\displaystyle\bm{a}=[0,\,-2\Omega_{\bm{p}}(t),\,2\omega_{\bm{p}}(t)],$ (52)

where

$\displaystyle\omega_{\bm{p}}(t)=\frac{E_{{\bm{p}}(t)}}{\hbar}=\frac{1}{\hbar}\sqrt{c^{2}\bm{p}_{\perp}^{2}+c^{2}[p_{3}^{2}-e\mathcal{A}(t)]^{2}+m^{2}c^{4}},\quad\Omega_{\bm{p}}(t)=\frac{ce\epsilon_{\perp}\mathcal{E}(t)}{2E_{{\bm{p}}(t)}^{2}},$
(53)

and where the temporal dependence of the electric field is given by ${\cal
E}(t)=-\dot{\cal A}(t)$. Note that the initial condition for $\bm{u}$ has the
form $\bm{u}^{vac}=[0,\,0,\,1]$.

Three equations (51) can be reduced to a system of two equations by expressing
$\bm{u}$ in the form of spinorial decomposition, analogous to that used in
IBB2 ; archiwum

$\displaystyle\bm{u}=\chi^{\dagger}\bm{\sigma}\chi,$ (54)

where $\chi$ is a two-component spinor and $\bm{\sigma}$ are Pauli matrices.
Substitution of Eq. (54) to (51) leads to the equation for $\chi$ which has
the same structure as the Schrödinger equation describing time evolution of a
two-level atom. This equation has been derived in the context of pair-creation
by different method previously (see, e.g., Refs. IBB2 ; KMK  and references
therein),

$\displaystyle i\partial_{t}\left[\begin{array}[]{c}c_{\bm{p}}^{(1)}(t)\\\
c_{\bm{p}}^{(2)}(t)\end{array}\right]=\left[\begin{array}[]{cc}\omega_{\bm{p}}(t)&i\Omega_{\bm{p}}(t)\\\
-i\Omega_{\bm{p}}(t)&-\omega_{\bm{p}}(t)\end{array}\right]\left[\begin{array}[]{c}c_{\bm{p}}^{(1)}(t)\\\
c_{\bm{p}}^{(2)}(t)\end{array}\right],$ (61)

where $c_{\bm{p}}^{(1)}(t)$ and $c_{\bm{p}}^{(2)}(t)$ are, respectively, upper
and lower components of $\chi$. Initial conditions read

$\displaystyle c_{\bm{p}}^{(1)}|_{t\rightarrow-\infty}=1,\quad
c_{\bm{p}}^{(2)}|_{t\rightarrow-\infty}=0.$ (62)

Third component of $\bm{u}$ is equal to
$|c^{(1)}_{\bm{p}}|^{2}-|c^{(2)}_{\bm{p}}|^{2}$ and for two level atom
corresponds to ”population inversion” (with opposite sign). Before action of
the electric field $u_{3}=1$, which corresponds to the vacuum state with no
pairs. During action of the electric field $e^{+}e^{-}$ pairs are created so
that $|c^{(1)}_{\bm{p}}|^{2}<1$ and $|c^{(2)}_{\bm{p}}|^{2}>0$ with
$|c^{(1)}_{\bm{p}}(t)|^{2}+|c^{(2)}_{\bm{p}}(t)|^{2}=1$. Hence,
$|c^{(2)}_{\bm{p}}|^{2}$ for $t\rightarrow\infty$ can be interpreted as
momentum distribution of created fermionic pairs $f(\bm{p})$ (19). Explicitly,

$\displaystyle f(\bm{p})=1-u_{3}=2|c^{(2)}_{\bm{p}}|^{2}.$ (63)

Let us note in closing this section that a similar to (61) system of equations
can be derived for bosons by applying other, then those based on the Wigner
formalism, methods of QED (see, e.g., Ref. KK2019  and references therein).
However, in this case the time-evolution is pseudounitary.

From now on we use units in which $\hbar=1$. Moreover, $m$ and $e$ will refer
to the electron rest mass and charge, respectively.

##  V Threshold effects and  vortices

In our further investigations we choose the electric field $\mathcal{E}(t)$
such that

$\mathcal{E}(t)=\begin{cases}\mathcal{E}_{0}\sin^{4}\bigl{(}\frac{1}{2N}\phi\bigr{)}\cos(\phi),&\phi\in[0,2\pi
N],\cr 0,&\phi\notin[0,2\pi N],\end{cases}$ (64)

where $\phi=\omega t$ and $N=3$. The integer $N$ determines the number of
cycles within the electric field pulse and for $N\geqslant 3$ the condition

$\int_{-\infty}^{\infty}\mathrm{d}t\,\mathcal{E}(t)=0$ (65)

is satisfied. Due to this property the vector potential function,

$\mathcal{A}(t)=-\int_{-\infty}^{t}\mathrm{d}\tau\,\mathcal{E}(\tau),$ (66)

can be chosen such that it vanishes both in the remote past and in the far
future,

$\lim_{t\rightarrow\pm\infty}\mathcal{A}(t)=0.$ (67)

The shapes of both functions for $\mathcal{E}_{0}=0.1\mathcal{E}_{S}$ and
$\omega=mc^{2}$ are presented in Fig. 1, where
$\mathcal{E}_{S}=m^{2}c^{3}/|e|$ is the Sauter-Schwinger electric field
strength Sauter ; Sauter2 ; Schwinger . For the electron momentum vector, we
will separate its parallel and perpendicular components as measured with
respect to the direction of the electric field oscillations ${\bm{e}}_{3}$
such that $\bm{p}=p_{\bot}\bm{e}_{\bot}+p_{\|}\bm{e}_{3}$, where
$\bm{e}_{\bot}$ is a unit vector perpendicular to $\bm{e}_{3}$.

Figure 1: Time-dependent electric field strength $\mathcal{E}(t)$ for
$\mathcal{E}_{0}=0.1\mathcal{E}_{S}$ and $N=3$, as defined by Eq. (64), and
the corresponding vector potential function $\mathcal{A}(t)$. Contrary to the
electric field, the amplitude of the vector potential depends on the frequency
$\omega$.  Figure 2: Momentum distributions of electrons created from the QED
vacuum by the electric field, illustrated in Fig. 1. In the upper row, the
distributions $|c_{\bm{p}}^{(2)}|^{1/2}$ (the power $1/2$ is chosen for visual
purposes) are presented for three chosen frequencies $\omega$ (equivalent to
photon energies). In the lower row, the corresponding phases of
$c_{\bm{p}}^{(2)}$ are demonstrated.

As was mentioned in Introduction, the process of creating electron-positron
pairs in QED has many analogies with ionization of atoms, in which the role of
the time-dependent electric field is played by a strong laser pulse in the
dipole approximation. For this case the concept of photons as quanta of energy
absorbed or emitted by the system is commonly used. One can then talk about
multiphoton ionization and an energy threshold for that process. Moreover,
such a threshold is dynamically increased as the electric field becomes
stronger, which leads to the so-called threshold effects and channel closing
in ionization KFS2006 . It turns out that in the case of the dynamic Sauter-
Schwinger process this heuristic picture can also be applied in order to
describe qualitative changes in momentum distributions of the created
particles (as, for instance, in the coherent energy combs studied in Refs. KMK
; Dunne ). This can be done even for the very short pulses considered here.

Figure 3: The same as in Fig. 2, but for lager electric field amplitude
$\mathcal{E}_{0}=0.5\mathcal{E}_{S}$ and lager frequencies.

Another interesting effect, which appears as a result of a time-dependent
electric field interacting with QED vacuum, is the creation or annihilation of
vortex lines in the electron momentum distributions. The properties of vortex
lines and their entanglement were thoroughly analyzed in Refs. BS ; Taylor .
In both photoionization and photodetachment, the creation and annihilation of
vortex lines were studied for linearly Pe1  and circularly Pe2 ; Pe3 ; Felipe
; Majczak  polarized fields. It has been shown how the time-reversal symmetry
of the laser pulse leads to the annihilation of vortex-antivortex pairs and
the creation of spirals in momentum distributions Majczak . Note that such
spirals have been predicted theoretically in SAF  and confirmed experimentally
in Wollen . Moreover, the application of the DHW-function formalism allowed to
show that similar spiral structures also appear in pair creation by a train of
two circularly polarized electric field pulses of opposite helicity Xie1 ;
Xie2 . Therefore, the question arises: Can vortices be expected in momentum
distributions of created pairs for linearly polarized electric field pulses?

To address this question, in Fig. 2 we present the momentum distributions of
electrons created by a linearly polarized electric field pulse of different
frequencies, which have been selected close to the ‘two-photon’ threshold of
pair creation. For $\omega=0.99mc^{2}$ (i.e., just before opening the ‘two-
photon’ channel) we observe two singular points for which the amplitude
$c_{\bm{p}}^{(2)}$ vanishes and the phase ${\rm arg}[c_{\bm{p}}^{(2)}]$ cannot
be uniquely defined. Because of the axial symmetry of the problem, it can be
concluded that these two points belong to the same vortex line. In the current
case, the latter is represented by a circle in the three-dimensional momentum
space. In fact, one can even define the orientation of this closed line by
exploiting the analogies with the circuit along which the electric current
flows and generates, according to the Amperé’s law, the vortex-type magnetic
field. To this end, let us define the ‘magnetic field’ $\bm{B}(\bm{p})$ such
that

$\bm{B}(\bm{p})=\bm{\nabla}_{\bm{p}}\bigl{(}{\rm
arg}[c_{\bm{p}}^{(2)}]\bigr{)}.$ (68)

Its circulation around a singular point is $\pm 2\pi$, hence the ‘electric
current’ becomes $I=\pm 2\pi$, if we put the magnetic permeability
$\mu_{0}=1$. In particular, for $\omega=0.99mc^{2}$ (left panel in Fig. 2), we
have ‘$-$’ for $(p_{\|}=0,p_{\bot}>0)$ and the current flows behind the plane,
whereas for $(p_{\|}=0,p_{\bot}<0)$ we have ‘$+$’ and the current flows
towards the reader. Thus the orientation of the vortex line can be uniquely
attributed to the direction of the ‘electric current’. As the frequency
$\omega$ increases, we observe the appearance of a new vortex line. The case
of $\omega=mc^{2}$ (which is the threshold frequency for the two-photon pair
creation) corresponds to a transition in which, for $\bm{p}=\bm{0}$, the
radius of the new vortex circle is close to zero (middle column in Fig. 2).
After exceeding this value (the case of $\omega=1.01mc^{2}$, i.e., the right
column in Fig. 2), the second circular vortex line appears, with an
orientation opposite to the previous one (bottom panels). While increasing the
frequency $\omega$, the radii of both circular vortex lines also grow. This,
in turn, results in the merging of the two well-defined lobes of high
probability into a single structure, the maximum of which is found at zero
momentum. This situation is discussed in Sect. VI.

In Fig. 3 we demonstrate the same phenomenon, but for a larger amplitude of
the electric field. The only significant difference is that now the threshold
frequency for the two-photon pair creation is shifted upwards and its value is
between $1.1mc^{2}$ (one vortex line) and $1.2mc^{2}$ (two vortex lines). A
plausible interpretation of this fact can be based on analogies with the
multiphoton ionization in which, for larger intensity of the electromagnetic
field, the so-called ponderomotive shift of the threshold energy is observed
KFS2006 . Similar effects, but in the context of photodetachment by circularly
polarized laser pulses, have been discussed in Ref. Felipe .

Figure 4: Momentum distributions of electrons created by oscillating electric
field for different frequencies and for the electric field amplitude
$\mathcal{E}_{0}=0.5\mathcal{E}_{S}$. Starting from the one-photon threshold
frequency, $\omega=2mc^{2}$, a qualitative change in the shapes of the high-
probability structures is observed.

##  VI Longitudinal and transverse momentum sharing

The Schwinger formula for probability rate of pair production per unit volume
in the case of a constant (or slowly-changing-in-time) electric field can be
derived using the tunneling formalism LR . According to this formula, an
increment of the perpendicular momentum of the particles,
$|{\bm{p}}_{\perp}|$, is accompanied by a rapidly vanishing creation rate.
However, for rapidly changing fields the tunneling theory is no longer
applicable. This is supported by the analysis presented above, as the momentum
distributions for pair production are not elongated in the direction of the
electric field. As it has been shown in Ref. KK012104 , for sufficiently high
frequencies $\omega$ particles prefer to be created in the direction
perpendicular to the electric field. This counterintuitive phenomenon is
illustrated in Fig. 4. For low frequencies, $\omega\leqslant 0.8mc^{2}$, the
distributions are concentrated around the axis of vanishing transverse
momentum. However, as the frequency increases, the distributions begin to
concentrate around zero momentum. This happens until the one-photon threshold
is reached. A further increase of frequency causes the position of the high-
probability regions in the distribution to migrate towards the direction
perpendicular to the electric field (i.e., towards larger $p_{\perp}$).
Furthermore, at $\omega=4mc^{2}$ the high-probability zone, in the three-
dimensional space, takes the shape of a torus centered around the
${\bm{p}}_{\perp}={\bm{0}}$. This means that under such conditions particles
prefer to be ejected in the direction perpendicular to the electric field
vector. As shown in Ref. KK012104 , the distribution for pair creation, when
integrated over particles momenta, starts to saturate (or even decreases) with
the increasing frequency, leading to the seemingly unexpected stabilization
phenomenon. In fact, the stabilization effects appear to be quite common in
the strong-field QED, as discussed for instance in Refs. KKE ; LR2 ; ER21 ;
Kemp1 ; Kemp2 .

In summary, although we have concentrated our discussion on the fermionic
distribution function $f(\bf{p})$ and the phase of the momentum amplitude
$c^{(2)}_{\bm{p}}$, the other components of the DHW functions (8) can be also
determined by applying Eqs. (54), (50) and (48). This topic is, however,
beyond the scope of the present work and is going to be considered in due
course.

##  VII Conclusions

In this paper, we have formulated the bispinorial approach to $e^{-}e^{+}$
pair production in spatially homogeneous electric fields. The method has
turned out to be equivalent to the DHW formalism, that was introduced in Ref.
IBB1 . We have shown that the Sauter-Schwinger pair production by a linearly
polarized time-dependent electric field reduces formally to solving a two-
level model in compliance with Ref. KMK  (see also, references therein). The
advantage of this approach is that one gains access to probability amplitudes
and, therefore, to their phases. The latter allow to uniquely identify
vortices and antivortices in momentum distributions of created pairs. As it
has been demonstrated in our paper, for a linearly polarized pulsed electric
field they appear in pairs. We have also analyzed the vortex patterns while
increasing the field frequency across the two-photon threshold. While we have
observed new vortex-antivortex pair, the general features of momentum
distributions also change across the threshold. Specifically, we have seen
that below the one-photon threshold the particles are created most efficiently
along the polarization direction of the electric field, whereas above the
threshold – in the perpendicular direction. This shows different
characteristics of the Sauter-Schwinger process while passing from low- to
high-frequency regimes of electric-field–vacuum interactions.

## Acknowledgements

The work of (F.C.V., K.K., and J.Z.K.) was supported by the National Science
Centre (Poland) under Grant No. 2018/31/B/ST2/01251.

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# Non-local partner to the cosmological constant

John F. Donoghue1<EMAIL_ADDRESS>1Department of Physics,
University of Massachusetts, Amherst, MA 01003, USA

###### Abstract

I show that quantum corrections due to a massive particle generates a non-
local term in the gravitational effective action which is of zeroth order in
the derivative expansion, much like the cosmological constant. It carries a
fixed coefficient which is very much larger than the cosmological constant,
and which cannot be fine-tuned. The interaction is active at scales above the
particle’s mass. This is of the form $m^{4}(\frac{1}{\Box}R)_{x}``\langle
x|\log(\Box+m^{2})|y\rangle^{\prime\prime}(\frac{1}{\Box}R)_{y}$, and I
discuss the meaning of $``\langle x|\log(\Box+m^{2})|y\rangle^{\prime\prime}$
and other aspects of its interpretation.

††preprint: ACFI-T22-01

##  I Local and non-local interactions

The foundation of General Relativity is general covariance - the invariance of
the theory under local changes in the coordinates. One can write an action
which respects this symmetry using the curvatures

$S=\int d^{4}x\sqrt{-g}\left[-\Lambda+\frac{1}{16\pi
G}R+c_{1}R^{2}+c_{2}R_{\mu\nu}R^{\mu\nu}+~{}...\right]$ (1)

Here $\Lambda$ is the cosmological constant (of mass dimension $m^{4}$),
$R_{\mu\nu}$ and $R=g^{\mu\nu}R_{\mu\nu}$ are the Ricci tensor and scalar
curvature, $G$ is Newton’s constant (with mass dimension $1/G\sim m^{2}$) and
$c_{1},~{}c_{2}$ are dimensionless constants. This action has been ordered in
the derivative expansion, with the cosmological constant being of zeroth
order, the scalar curvature having two derivatives and the curvature-squared
terms involving four derivatives. The ellipses indicate terms with yet more
derivatives. In applications the derivatives turn into factors of energy and
momentum, and therefore the terms with two or more powers of the curvature are
negligible at low energy. Using the first two terms in this action, we obtain
Einstein’s equations with a cosmological constant. Rather famously, the
cosmological constant is remarkably small, taking the value $\Lambda\simeq
10^{-47}~{}{\rm GeV}^{4}$ if the present accelerated expansion of the universe
is due the presence of $\Lambda$.

If we had a complete theory of quantum gravity, we could presumably predict
the coefficients in the gravitational action from the fundamental parameters
in that theory. In place of that knowledge, all that we really know is that
the unknown physics from high energy will described by local terms in the
action. This follows from the uncertainty principle which tells us that heavy
fields do not propagate far when viewed at low energy. When one treats quantum
effects of matter fields, or of the gravitational field itself, one needs to
renormalize the parameters in this action. The divergences also are local
because they occur at high energy.

However, we can still make predictions at low energy without knowing the
complete quantum theory of gravityDonoghue:1994dn ; Donoghue:2017pgk . At low
energy, gravitons and other light particles can propagate a long distance,
which distinguishes them from the local high energy effects. Knowing the low
energy degrees of freedom and their couplings are sufficient to calculate
these effects. In momentum space, where the calculations are most often
performed, this is distinguished by non-analytic behavior, such as
$\sqrt{q^{2}},~{}\log q^{2}$, which cannot be Taylor expanded in powers of
$q^{2}$ and as such cannot be represented by derivative operators in a local
Lagrangian. For example, in the quantum correction to the Newtonian potential
due to gravition loops, one calculates the logarithmic non-analytic term

${\cal M}=\frac{4\pi
GMm}{q^{2}}\left[1+\frac{41}{20\pi}Gq^{2}\log{-q^{2}}\right]$ (2)

which turns into a long-distance correction when Fourier transformed Bjerrum-
Bohr:2002gqz ; Khriplovich:2002bt

$V(r)=-\frac{GMm}{r}\left[1+\frac{41}{10\pi}\frac{G}{r^{2}}\right]$ (3)

The one-loop logarithm yields the leading quantum effect.

The non-analytic and non-local terms can be described by a non-local effective
action. The most well-known are in the curvature squared terms, where
Barvinsky and Vilkovisky and collaborators Barvinsky:1990up ; Barvinsky:1993en
; Avramidi:1990ap ; Barvinsky:1985an  have included the logarithmic non-
analytic terms using the generic form

${\cal
L}=c_{1}(\mu)R^{2}+c_{2}(\mu)R_{\mu\nu}R^{\mu\nu}+d_{1}R\log(\frac{\Box}{\mu^{2}})R+d_{2}R_{\mu\nu}\log(\frac{\Box}{\mu^{2}})R^{\mu\nu}$
(4)

with $d_{1,2}$ being known dimensionless constants calculated at one loop
order. Here $\log\Box$ represents generalization of the Fourier transform of
$\log q^{2}$ \- it will be described more fully below. Although this is
written in a form which appears local, it really represents a non-local action
because $\langle x|\log\Box|y\rangle$ is non-local. In particular the
correspondence is

${\cal L}=R\log(\frac{\Box}{\mu^{2}})R~{}~{}~{}\Rightarrow~{}~{}~{}S=\int
d^{4}x\sqrt{-g(x)}~{}d^{4}y\sqrt{-g(y)}~{}~{}R(x)~{}\langle
x|\log(\frac{\Box}{\mu^{2}})|y\rangle~{}R(y)$ (5)

In the derivative expansion, these terms are of fourth order in the
derivatives.

The basic result of this paper is that when dealing with loops of massive
particles, there are also non-local effects at zeroth order and second order
in the derivative expansion. That at zeroth order is

$\displaystyle{\cal L}=$
$\displaystyle\frac{m^{4}}{40\pi^{2}}\left[\left(\frac{1}{\Box}R_{\lambda\sigma}\right)\left[\log((\Box+m^{2})/m^{2})\right]\left(\frac{1}{\Box}R^{\lambda\sigma}\right)-\frac{1}{8}\left(\frac{1}{\Box}R\right)\left[{\log((\Box+m^{2})/m^{2})}\right]\left(\frac{1}{\Box}R\right)\right]$
(6)

from the loop of a massive scalar field. A massive fermion yields $-2$ times
this result. The two derivatives in the curvatures are canceled by the
$1/\Box$ factors, so that overall this is zeroth order in the derivative
expansion. The structure of this is explained below, but in the sense of the
derivative expansion this is a non-local partner to the cosmological constant.
It comes with a coefficient which cannot be adjusted.

In most settings the fact that there was a non-local component to the
interaction would not be remarkable, as these effects are expected in usual
quantum corrections. However, in the case of the cosmological constant there
is a special feature. The experimental value of the local cosmological
constant is very much smaller than expected, while the non-local partner
enters at normal size and cannot be removed. Therefore, it has an enormous
numerical advantage over the tiny cosmological constant. For example
considering the top quark in the loop, we have $m_{t}^{4}/\Lambda\sim
10^{56}$. However, we will see that this interaction is only applicable at
scales above the particle’s mass, as the effect of the massive field becomes
local below that scale.

##  II Origin of non-local interactions

One-loop diagrams can all be reduced to functions of momentum times the basic
scalar tadpole, bubble, triangle and box diagrams Passarino:1978jh . The UV
divergent diagrams are the tadpole and bubble. The tadpole has no external
momentum running through it, and in dimensional regularization has the form
near $d=4$

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle-i\mu^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{[k^{2}-m^{2}]}\
\ $ (7) $\displaystyle=$
$\displaystyle\frac{m^{2}}{16\pi^{2}}\left[\frac{1}{\epsilon}-\gamma+\log(4\pi)+\log(\mu^{2}/m^{2})+1\right]$

with $\epsilon=(4-d)/2$. This is always independent of any external momenta.
The massive bubble does depend on the external momentum and has the form

$\displaystyle I_{2}(q)$ $\displaystyle=$
$\displaystyle-i\mu^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{[k^{2}-m^{2}][(k-q)^{2}-m^{2}]}\
\ $ (8) $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left[\frac{1}{\epsilon}-\gamma+\log(4\pi)+\log(\mu^{2}/m^{2})-J(q^{2})\right]$

with

$J(q^{2})=\int_{0}^{1}dx~{}\log\left[\frac{m^{2}-x(1-x)q^{2}}{m^{2}}\right]\ \
.$ (9)

When the mass is zero or the momentum become large, one obtains the $\log
q^{2}$ which has been referenced above.

The logarithm in the Barvinsky-Vilkovisky effective action comes from the
bubble diagram. While they derive this using the background field method and a
non-local version of the the heat kernel Barvinsky:1985an , is easy to
reproduce it in a perturbative expansion also Codello:2012kq ;
Donoghue:2014yha ; Donoghue:2015nba ; Donoghue:2015xla . One expands
$g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x)$ and calculates the bubble diagram
involving two $h_{\mu\nu}$. One finds

${\cal M}\sim
h_{\mu\nu}h_{\alpha\beta}\left(q^{\mu}q^{\nu}q^{\alpha}q^{\beta}+...\right)\left[\frac{1}{\epsilon}+...-\log(-q^{2})\right]\
\ .$ (10)

This can be matched to the possible terms in a covariant effective
action.There is a unique connection between the tensor structure of the
momentum factors and the leading terms in an expansion in the curvatures,
resulting in the identification of the renormalization of the curvature-
squared terms and also the $R\log\Box R$ terms.

In flat space we have have the clear identification of the meaning of
$\log\Box$. It is a function defined by the Fourier transform of $\log-q^{2}$,

$\displaystyle L_{0}(x-y)$ $\displaystyle=$ $\displaystyle\langle
x|\log\Box/\mu^{2}|y\rangle$ (11) $\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}e^{-iq\cdot(x-y)}\log(-q^{2}-i\epsilon)/\mu^{2}\
\ .$

For curved spacetime we need a covariant generalization of this. This is not
uniquely defined - there are several options which could reduce to the flat
space version in the appropriate limit. This issue is discussed in the
appendix.

In the Barvinsky Vilkovisky formalism, there are also non-local terms
following from the one-loop triangle diagram, which involves three factors of
the gravitational field Barvinsky:1993en . These provide the triple graviton
portions of the $R^{2}$ and $R\log\Box R$ effects previously identified, as
well as some new terms. When these latter are written in covariant form they
yield contributions which are proportional to actions such as
$R^{2}(1/\Box)R$. Here $1/\Box$ is a propagator

$\langle x|\frac{1}{\Box}|y\rangle=G(x,y)$ (12)

which has covariant versions DeWitt:1975ys ; DeWitt:1967ub ; DeWitt:1967uc .
Such interactions are said to be of “higher order in the curvature”. However
they are of the same order in the derivative expansion as are the local
$R^{2}$ terms and the nonlocal $R\log\Box R$ terms, i.e. all are of fourth
order in the derivative expansion.

In practice, the Barvinsky Vilkovisky expansion in the curvature is a weak
field expansion rather than a derivative expansion. It however has the great
advantage that all factors are written in covariant form, so that it manifests
the coordinate invariance of general relativity.

##  III Renormalization of the cosmological constant

What about the lower order terms in the derivative expansion? The coefficients
of the curvature-squared terms are dimensionless. Starting with massless
fields one can generate these as their coefficients are just pure numbers and
do not require any dimensionful parameters. However, to generate terms with
fewer derivatives, one needs a extra mass parameter because the coefficients
of such term must carry a dimensionful factor. With loops of a massive field,
one generates the renormalization of the scalar curvature term at order
$m^{2}$, and the cosmological constant at order $m^{4}$ when using dimensional
regularization, i.e.

$\displaystyle\delta\sqrt{-g}\frac{1}{16\pi G}R$ $\displaystyle\sim$
$\displaystyle\sqrt{-g}\frac{m^{2}R}{\epsilon}+{\rm non-local}$
$\displaystyle\delta\sqrt{-g}\Lambda$ $\displaystyle\sim$
$\displaystyle\sqrt{-g}\frac{m^{4}}{\epsilon}+{\rm non-local}\ \ .$ (13)

Our goal is to calculate the non-local actions in these cases. First we need
to understand the renormalization of the local terms.

Again we can identify the effect by an expansion about Minkowski space. For
reference, the weak field expansion of the cosmological term is

$\sqrt{-g}\Lambda=\Lambda(1+\frac{1}{2}h^{\sigma}_{\sigma}+\frac{1}{8}(h^{\sigma}_{\sigma})^{2}-\frac{1}{4}h_{\sigma\lambda}h^{\sigma\lambda}+...)$
(14)

We can then identify changes in the cosmological constant through the
couplings to the gravitational field $h_{\mu\nu}$, with interactions which do
not involve derivatives.

For a minimally coupled scalar field, the gravitational couplings come from
the Lagrangian

$\displaystyle\sqrt{-g}{\cal L}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-g}}{2}\left[g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-m^{2}\phi^{2}\right]$
(15) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(1+\frac{h}{2}+\frac{h^{2}}{8}-\frac{1}{2}h^{\alpha\beta}h_{\alpha\beta}\right)\left(\partial_{\mu}\phi\partial^{\mu}\phi-m^{2}\phi^{2}\right)-h^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right.$
$\displaystyle\left.~{}~{}~{}+\left(h^{\mu\beta}h^{~{}\nu}_{\beta}-\frac{1}{2}hh^{\mu\nu}\right)\partial_{\mu}\phi\partial_{\nu}\phi\right]$

Here $h=h^{\lambda}_{\lambda}$.

Figure 1: The tadpole diagrams (a) and (b) and the bubble diagram (c). The
solid line is the scalar field and the dashed line is the metric field.

The one loop correction linear in the gravitational field comes from the
tadpole diagram of Fig 1a, and has the form

$\displaystyle-i{\cal M}_{a}$ $\displaystyle=$ $\displaystyle-
ih^{\mu\nu}\int\frac{d^{4}k}{(2\pi)^{4}}~{}k_{\mu}k_{\nu}\frac{i}{k^{2}-m^{2}+i\epsilon}$
(16) $\displaystyle=$ $\displaystyle-
ih^{\mu\nu}\mu^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}~{}\frac{\eta_{\mu\nu}}{d}k^{2}\frac{i}{k^{2}-m^{2}+i\epsilon}$

where I have dropped a dimensionless integral because I am using dimensionless
regularization111The effect of this dimensionless integral also vanishes when
using a cutoff because there is an extra interaction from the measure required
when using cutoff regularization Donoghue:2020hoh  .. This integral is readily
evaluated and yields

$-i{\cal
M}_{a}=ih\frac{m^{4}}{64\pi^{2}}\left[\frac{1}{\bar{\epsilon}}+\log\frac{\mu^{2}}{m^{2}}+\frac{3}{2}\right]$
(17)

with the shorthand notation

$\frac{1}{\bar{\epsilon}}=\frac{1}{\epsilon}-\gamma+\log(4\pi)\ \ .$ (18)

The second order terms in the gravitational field are similarly
straightforward. That of Figure 1b is given by

$\displaystyle-i{\cal M}_{b}$ $\displaystyle=$ $\displaystyle
i\left(h^{\mu\beta}h_{\beta}^{~{}\nu}-\frac{1}{2}hh^{\mu\nu}\right)\int\frac{d^{4}k}{(2\pi)^{4}}~{}k_{\mu}k_{\nu}\frac{i}{k^{2}-m^{2}+i\epsilon}$
(19) $\displaystyle=$
$\displaystyle-i\frac{m^{4}}{64\pi^{2}}\left(h^{\mu\beta}h_{\beta}^{~{}\nu}-\frac{1}{2}hh^{\mu\nu}\right)\eta_{\mu\nu}\left[\frac{1}{\bar{\epsilon}}+\log\frac{\mu^{2}}{m^{2}}+\frac{3}{2}\right]\
\ .$

For the bubble diagram of Figure 1c, for our purposes in this section we can
evaluate it at zero momentum because the cosmological constant does not
involve momentum dependence. This yields

$\displaystyle-i{\cal M}_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\left[h^{\mu\nu}k_{\mu}k_{\nu}-\frac{1}{2}h(k^{2}-m^{2})\right]\left[h^{\alpha\beta}k_{\alpha}k_{\beta}-\frac{1}{2}h(k^{2}-m^{2})\right]}{(k^{2}-m^{2}+i\epsilon)^{2}}$
(20) $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left[h^{\mu\nu}h^{\alpha\beta}\frac{k_{\mu}k_{\nu}k_{\alpha}k_{\beta}}{(k^{2}-m^{2}+i\epsilon)^{2}}-hh^{\mu\nu}\frac{k_{\mu}k_{\nu}}{k^{2}-m^{2}+i\epsilon}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left[\frac{h^{\mu\nu}h^{\alpha\beta}}{d(d+2)}(\eta_{\mu\nu}\eta_{\alpha\beta}+\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha})\left(\frac{m^{4}}{(k^{2}-m^{2}+i\epsilon)^{2}}+\frac{2m^{2}}{k^{2}-m^{2}+i\epsilon}\right)\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{hh^{\mu\nu}}{d}\frac{m^{2}}{k^{2}-m^{2}+i\epsilon}\right]$
$\displaystyle=$ $\displaystyle
i\frac{m^{4}}{128\pi^{2}}\left(h^{\mu\nu}h_{\mu\nu}-\frac{1}{2}h^{2}\right)\left[\frac{1}{\bar{\epsilon}}+\log\frac{\mu^{2}}{m^{2}}+\frac{3}{2}\right]\
\ .$

In this case, it is non-trivial that the finite part (the $3/2$ within the
final square bracket) comes out identical to the finite part in the previous
two diagrams. However, this is required for general covariance. Summing the
three diagrams one gets the appropriate combination for the expansion of
$\sqrt{-g}$, i.e.

$-i{\cal
M}=i\frac{m^{4}}{32\pi^{2}}\left(\frac{1}{2}h+\frac{1}{8}h^{2}-\frac{1}{4}h^{\mu\nu}{\mu\nu}\right)\left[\frac{1}{\bar{\epsilon}}+\log\frac{\mu^{2}}{m^{2}}+\frac{3}{2}\right]\
\ .$ (21)

This lets us read off the one-loop contribution to the cosmological constant

$\delta\Lambda=-\frac{m^{4}}{32\pi^{2}}\left[\frac{1}{\epsilon}-\gamma+\log(4\pi)+\log\frac{\mu^{2}}{m^{2}}+\frac{3}{2}\right]\
\ .$ (22)

This quantum effect is one contribution to the physical renormalized value of
the cosmological constant. All physical predictions are expressed in terms of
the measured value after renormalization. As is well known, it is striking
that the measured value is so much smaller than all the known mass scales. The
present calculation reinforces that mystery, but does not help explaining it.

One can also readily calculate the renormalization of the Einstein term,
proportional to $R$, by this method. In this case one must include the
external momentum dependence. The result is proportional to $m^{2}$, and does
not add anything particular to our discussion, so I do not describe it
further.

##  IV The non-local partner

While the ingredients calculated in the previous section disappear into the
renormalized value of $\Lambda$, non-local effects are finite and physical.
Having worked through the renormalization calculation, we can readily see that
such effects must occur. The fact that the scalar bubble diagram contributes
to the renormalization, as seen in Eq. 20, and that this diagram has
logarithmic momentum dependence, as seen in Eq. 8, mean that there will be
logarithmic non-localities when one includes non-zero momentum/spatial
dependence.

For this calculation, one needs to include the external momentum dependence in
the bubble diagram, which will be denoted by the momentum $q^{\mu}$. The
reduction to scalar integrals is more extensive in this case. It is however
useful to show one such integral which displays an important facet of the
final result. Consider the integral

$\displaystyle I_{\mu\nu\alpha\beta}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\mu}k_{\nu}k_{\alpha}k_{\beta}}{[k^{2}-m^{2}+i\epsilon][(k+q)^{2}-m^{2}+i\epsilon]}$
(23) $\displaystyle=$ $\displaystyle
F\left(\eta_{\mu\nu}\eta_{\alpha\beta}+\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\alpha\nu}\right).$
$\displaystyle+$ $\displaystyle
G\left(\eta_{\mu\nu}q_{\alpha}q_{\beta}+\rm{5~{}perms.}\right)$
$\displaystyle+$ $\displaystyle Hq_{\mu}q_{\nu}q_{\alpha}q_{\beta}\ \ .$

The $m^{4}$ portions of the coefficient functions $F,~{}G,~{}H$ are given by

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\frac{m^{4}}{8}\left[\frac{1}{\bar{\epsilon}}+\log\frac{\mu^{2}}{m^{2}}+\frac{3}{2}-\frac{8}{15}J(q^{2})\right]+{\cal
O}(m^{2})$ $\displaystyle G$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\frac{m^{4}}{15q^{2}}J(q^{2})+{\cal
O}(m^{2})$ $\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\frac{-m^{4}}{5q^{4}}J(q^{2})+{\cal
O}(m^{2})$ (24)

where $J(q^{2})$ is the logarithm of Eq. 9. The feature worth noticing is the
inverse powers of $q^{2}$ in the $G$ and $H$ formfactors. Combined with the
numerator factors in the integral definition of Eq. 23, these mean that all of
these terms are of zeroth order when counting powers of the momenta.

Consider a matrix element with two external gravitons having polarization
vectors $\epsilon_{\mu\nu}$ and $\epsilon^{*}_{\alpha\beta}$ and define

$Q_{\mu\nu}=q_{\mu}q_{\nu}-\eta_{\mu\nu}q^{2}$ (25)

with $q$ being the momentum carried by the gravitons. With this notation, the
residual matrix element at order $m^{4}$ and $m^{2}$ after renormalization has
been performed is

${\cal
M}_{\mu\nu\alpha\beta}=\frac{1}{160\pi^{2}q^{4}}\left(Q_{\mu\nu}Q_{\alpha\beta}+Q_{\mu\alpha}Q_{\nu\beta}+Q_{\mu\beta}Q_{\nu\alpha}\right)\left[m^{4}J(q^{2})+\frac{1}{6}m^{2}q^{2}-3m^{2}q^{2}J(q^{2})\right]$
(26)

There are also terms of order $m^{0}$ , which contribute to the Barvinsky
Vilkovisky action in ways which are described above. What is most important to
note is the overall factor of $1/q^{4}$. Combined with the factors of $q$ in
the $Q_{\mu\nu}$, this makes the overall pre-factor zeroth order in the
momentum expansion.

In position space this interaction can only be represented by a non-local
effective action. This can be first displayed using the gravitational field
$h_{\mu\nu}$ as a preliminary step. The gravitational field is not gauge
invariant by itself, and in this case the choice of harmonic gauge,
$\partial_{\mu}h^{\mu\nu}=\frac{1}{2}\partial^{\nu}h$ will get rid of the
pesky inverse powers of $q^{2}$. In this gauge, we have

$\displaystyle\epsilon^{\mu\nu}Q_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}q^{2}\epsilon^{\lambda}_{~{}\lambda}$
$\displaystyle\epsilon^{\mu\nu}Q_{\mu\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}q^{\nu}q_{\alpha}\epsilon^{\lambda}_{~{}\lambda}-q^{2}\epsilon^{\nu}_{\alpha}$
(27)

In terms of the field, and again using the quasi-local notation, we find

${\cal
L}=\frac{m^{4}}{160\pi^{2}}\left[h_{\mu\nu}\log[(\Box+m^{2})/m^{2}]h^{\mu\nu}.-\frac{1}{8}h\log[(\Box+m^{2})/m^{2}]h\right]$
(28)

where we have defined the notation $\log[(\Box+m^{2}/m^{2})]$ by

$\langle
x|\log[(\Box+m^{2}/m^{2})|y\rangle=\int\frac{d^{4}q}{(2\pi)^{4}}e^{iq\cdot(x-y)}J(q^{2})\
\ .$ (29)

More discussion of this function is found in the appendix.

In turning this into a covariant expression, we need to provide a
correspondence between the possible curvatures and their matrix elements. At
second order in the fields the different products of curvatures have matrix
elements.

$\displaystyle\langle RR\rangle$ $\displaystyle=$ $\displaystyle
2Q_{\mu\nu}Q_{\alpha\beta}$ $\displaystyle\langle
R_{\lambda\sigma}R^{\lambda\sigma}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[2Q_{\mu\nu}Q_{\alpha\beta}+Q_{\mu\alpha}Q_{\nu\beta}+Q_{\mu\beta}Q_{\nu\alpha}\right]$
(30)

When using the curvatures, we still have the inverse powers of $q^{2}$ in the
matrix element so we use

$\langle(\frac{1}{\Box}R)(\frac{1}{\Box}R)\rangle=\frac{2}{q^{4}}Q_{\mu\nu}Q_{\alpha\beta}$
(31)

This leads to the non-local action

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{m^{4}}{40\pi^{2}}\left[\left(\frac{1}{\Box}R_{\lambda\sigma}\right)\log((\Box+m^{2})/m^{2})\left(\frac{1}{\Box}R^{\lambda\sigma}\right)-\frac{1}{8}\left(\frac{1}{\Box}R\right)\log((\Box+m^{2})/m^{2})\left(\frac{1}{\Box}R\right)\right]$
(32) $\displaystyle+$
$\displaystyle\frac{m^{2}}{240\pi^{2}}\left[R_{\lambda\sigma}\frac{1}{\Box}R^{\lambda\sigma}-\frac{1}{8}R\frac{1}{\Box}R\right]$

Here I have included the effect of the second term in the square brackets of
Eq. 26 even though it is second order in the derivative expansion, as it will
be important for the discussion of decoupling and locality which will be
discussed soon. The first line is the non-local partner of the cosmological
constant.

##  V Decoupling

At first sight, this operator would seem to violate the principle of
decoupling Appelquist:1974tg . The effects of very heavy masses should be
local, when applied at energies much lower that the mass. The factors of
$1/\Box$ seem to violate this as they appear non-local down to low energies.
However, this is not the case. The power counting is different at low and high
energy because

$\displaystyle J(q^{2})$ $\displaystyle\sim$
$\displaystyle\log\frac{-q^{2}}{m^{2}}-2~{}~{}~{}~{}~{}~{}~{}~{}q^{2}\gg
m^{2}.\ $ (33) $\displaystyle\sim$
$\displaystyle-\frac{q^{2}}{6m^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q^{2}\ll
m^{2}$

This implies that at low energy there is a cancelation between the terms in
Eq. 26. The overall factor in the square brackets of Eq. 26 is of order
$q^{4}$ for $q^{2}\ll m^{2}$ and results in a local operator.

This implies that the non-local operator is only operative at scales above the
particle mass. What is meant by the phrase “scales” depends on the context. In
the original computation, this clearly referred to the momentum. However in
general relativistic applications, it would refer rather to spatial or
temporal derivatives. In cosmological settings, the Hubble parameter
$H=\dot{a}/a$ would likely play the role of the energy scale in applications.

The presence of the integral $J(q^{2})$ has been previously noted in various
settings in gravitational physicsDalvit:1994gf ; Gorbar:2002pw ; Burns:2014bva
including appropriate comments about non-localities and decoupling. These
works provide possible avenues for application of the effective action
described above.

##  VI Discussion

I have calculated the non-local effects due to quantum loops of a massive
particle, and found an effect which is zeroth order in the derivative
expansion, much like the cosmological constant. Non-local effects are
ubiquitous when loops involve particles whose mass is smaller than the
relevant energy scales. In this case the result is special because it is very
many orders of magnitude larger than the local effect (the cosmological
constant), although the non-locality is active only at high energy scales222A
related special case is the inflaton potential, where the local component can
be fine-tuned to be almost flat. However, Miao and WoodardMiao:2015oba  have
pointed out that couplings of the inflaton to fields needed for reheating
generate a non-local component which cannot be fine-tuned.. I have used the
Barvinsky-Vilkovisky technique to express this in covariant form involving the
curvatures.

The cosmological constant is not a running coupling constant in the usual
sense. Its value in applications does not depend on the energy scales
involved. One might be tempted to identify the unphysical dimensional
regularization parameter $\mu$, which appear in the renormalization procedure
of Eq. 22, as a running parameter. In mass-independent renormalization
schemes, or when the mass is negliable, this $\mu$ dependence is sometimes
used as a surrogate for the energy dependence, because in those theories the
scale dependence tracks the energy dependence. However, here it just totally
disappears into the renormalized parameter and does not track any energy
dependence, as can be seen most obviously in the tadpole loop. Energy
dependence in the coupling could be described in position space by a non-local
effective action333For an example with QED, see Donoghue:2015nba . In the case
of the cosmological constant, the non-local effect has a different structure
than the local effect. The non-local partner is the closest thing that we have
to a running effect at zeroth-order in the derivative expansion.

The triangle diagram can bring in further non-local effects. Some of these
will be part of the interaction calculated in this paper, when the curvature
products are expanded to order $h^{3}$. There likely will also be new
operators which can be best expressed as the product of three curvatures. Some
of these can also be of zeroth-order in the derivative expansion. The result
of this paper is an approximation to a much more complicated general result.
This expansion in the curvatures is a weak field expansion. It is also
possible that repeating this procedure around other fixed background
spacetimes may yield forms which are more useful for those backgrounds.

There is a subset of the literature dealing with non-local actions in gravity,
which can be traced through the references in Barvinsky:2003kg ; Deser:2007jk
; Deser:2019lmm ; Maggiore:2014sia ; Maggiore:2016gpx . Many of these are
speculative suggestions for non-local actions. In contrast, the Barvinsky
Vilkovisky program and the present work are derived from the quantum effects
of standard local theories and are not speculative.

Techniques in applying non-local actions are less well-developed than those of
local Lagrangians, and further work here would be useful for applications.

## Acknowledgements

This work has been partially supported by the US National Science Foundation
under grant NSF-PHY-21-12800. I thank Basem El-Menoufi, Roberto Percacci and
Richard Woodard for discussions.

##  VII Appendix - Covariant non-local functions

In flat space the meaning of $\log\Box$ is not controversial - its matrix
element is the Fourier transform of $\log(-q^{2}-i\epsilon)$ (Eq. 11).
Notationally it appears as an operator, but it is applied as just a function.
When we generalize to curved spacetime, we seek a covariant definition which
reduces to the flat space version in the appropriate limit. Such a
generalization will involve the metric as well as the endpoints. Unfortunately
there is not a unique choice - I display three possibilities below and there
are undoubtably more. The non-local operators $\log\Box$ and
$\log(\Box+\frac{1}{6}R)$ also are discussed in connection with the conformal
anomaly Deser:1976yx ; Deser:1993yx ; Deser:1999zv  as these provide a
representation for the anomaly.

Given this non-uniqueness, how can one proceed? A first point is that within
the framework of an expansion in the curvatures differences in the definitions
can be corrected for by adjusting higher order terms in the curvature
expansion. If you are working at second order in the curvature, the
differences between the various covariant operators appears at third order in
the curvature. At leading order, one may in principle use any covariant
definition.

However, it is still possible that some forms are better than others for
particular applications. For example, a given form may introduce spurious
infrared divergences which should not be present. This has been seen in at
least one application Donoghue:2015nba . Beyond this caveat, not much is
known.

Here are three possible definitions of covariant $\log\Box$:

1) Barvinsky Vilkovisky and collaborators use a definition with a single
propagator,

$\langle x|\log\Box/\mu^{2}|y>=\int_{0}^{\infty}dm^{2}\langle
x|\left[\frac{1}{\Box-m^{2}}-\frac{1}{\mu^{2}-m^{2}}\right]|y\rangle\ \ .$
(34)

Here

$\langle x|\frac{1}{\Box-m^{2}}|y\rangle$ (35)

is the covariant propagator for a massive particle DeWitt:1975ys .

2) Osborn and Erdmenger have proposed using two massless propagators, and the
subtraction of the divergence which occurs in this combination

$\langle x|\log\Box/\mu^{2}|y>=\left[\langle
x|\frac{1}{\Box-m^{2}}|y\rangle\right]^{2}-\frac{1}{\sqrt{-g}}\delta^{4}(x-y)\left[\frac{1}{\bar{\epsilon}}+\log\mu^{2}/m^{2}+1\right]\
\ .$ (36)

The flat space version of this is the scalar bubble diagram, so this
definition has the good feature that it close to what one does when
calculating Feynman diagrams. One proves this relation by Fourier transforming
both sides.

3) Perhaps the simplest generalization uses the proper time parameterization.
In this case one uses the flat space representation

$\displaystyle L_{0}(x-y)$ $\displaystyle=$ $\displaystyle\langle
x|\log\Box/\mu^{2}|y\rangle$ (37) $\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}e^{-iq\cdot(x-y)}\log(-q^{2}-i\epsilon)/\mu^{2}$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}~{}\int_{0}^{\infty}\frac{ds}{s}\left[e^{i(q^{2}+i\epsilon)s}-e^{i\mu^{2}s}\right]e^{-iq\cdot(x-y)}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{3}}e^{-i\frac{(x-y)^{2}-i\epsilon}{4s}}`+~{}{\rm
local}$

This can be made covariant most simply by replacing the space time separation
by the geodetic distance, i.e. the proper distance along the geodesic
connecting the points $\frac{1}{2}(x-y)^{2}\to\sigma$ so that one can use

$L(\sigma)=\frac{1}{16\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{3}}e^{-i\frac{\sigma-i\epsilon}{2s}}$
(38)

That these definitions are different can be verified by expanding them about
flat spacetime in an expansion in the gravitational field.

There is another aspect to the use of non-local functions, which is the fact
that tensor quantities transform differently at different spacetime locations
because general coordinate invariance is a local symmetry. In
$R_{\mu\nu}(x)\langle
x|\log\Box|y\rangle^{\mu\nu\alpha\beta}R_{\alpha\beta}(y)$ the curvature
tensors transform the differently at $x$ and $y$. The treatment of this using
Wilson lines is reserved for a future publication nonlocalBasem .

The function

$J(x,y)=\langle x|\log(\Box+m^{2})/m^{2}|y\rangle$ (39)

is a covariant generalization of

$J_{0}(x-y)=\int\frac{d^{4}q}{(2\pi)^{4}}e^{iq\cdot(x-y)}J(q^{2})=\int\frac{d^{4}q}{(2\pi)^{4}}e^{iq\cdot(x-y)}\int_{0}^{1}dx~{}\log\left[\frac{m^{2}-x(1-x)q^{2}}{m^{2}}\right]\
\ .\ .$ (40)

Again, this is not uniquely defined. Because it is to be applied only above
the energy scale $m$, it is probably most useful to use an approximation valid
for $q^{2}\gg m^{2}$, namely $J(q^{2})\sim\log(-q^{2}/m^{2})-2$, in which case
the non-local part of $J(x,y)$ is approximated by $\log\Box$.

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