# On the instability of a uniform electric field in pure non-Abelian Yang-
Mills theory

Carlos Cardona, Tanmay Vachaspati  ∗Physics Department, Arizona State
University, Tempe, AZ 85287, USA.

###### Abstract

We study the Schwinger process in a uniform non-Abelian electric field using a
dynamical approach in which we evolve an initial quantum state for gluonic
excitations. We evaluate the spectral energy density and number density in the
excitations as functions of time. The total energy density has an ultraviolet
divergence which we argue gets tamed due to asymptotic freedom, leading to
$g^{4}E^{4}t^{4}$ growth, where $g$ is the coupling and $E$ the electric field
strength. We also find an infrared divergence in the number density of
excitations whose resolution requires an effect such as confinement.

##  I Introduction

The Schwinger effect Schwinger (1951), whereby non-perturbative quantum
effects in a background electric field lead to electron-positron pair
production, has received much attention (for example, see the reviews Holstein
(1999); Dunne (2005)). Heuristically, the electric field pulls apart the
electron-positron pairs that are fluctuating in and out of the vacuum. For
weak electric field strength $E\ll\pi m_{e}^{2}/e$, where $e$ and $m_{e}$ are
the positron’s electric charge and mass, the Schwinger process can be thought
of as a quantum tunneling event and is exponentially suppressed in the
dimensionless combination of variables $\pi m_{e}^{2}/eE$. The rate of
creation of number density of electrons is,

${\dot{n}}\propto e^{2}E^{2}\exp(-\pi m_{e}^{2}/eE)$ (1)

Schwinger’s original computation deals with the probability of vacuum
persistence111For a nice account on the relation between the vacuum
persistence probability and rate of pair creation see Cohen and McGady (2008)
for spin-0 and spin-1/2 matter fields and has since been generalized to
numerous other systems besides electromagnetism. Pair creation has also been
studied for massless charges, equivalently for supercritical electric fields
($eE\gg\pi m_{e}^{2}$) in 1+1 dimensions Chu and Vachaspati (2010); Gold et
al. (2020). In these cases, the exponential suppression of the original
Schwinger effect is not present, and other techniques have to be employed as
pair creation is no longer a tunneling process that is exponentially
suppressed. For example, the massless Schwinger model can be solved
completely, including backreaction on the electric field, and results in a
$1/\sqrt{t}$ decay of the electric field strength Chu and Vachaspati (2010).

In the present work we are interested in non-Abelian gauge theory in the
background of a homogeneous (color) electric field and the consequent
Schwinger pair creation of “gluons”222We will refer to the model as “pure QCD”
even though we will consider the simpler SU(2) gauge group. The “pure” means
that we will only consider gauge fields and not include any other fields.. The
process has been investigated before using effective action techniques to
calculate vacuum persistence amplitudes Yildiz and Cox (1980); Ambjorn and
Hughes (1982a, b); Cooper and Nayak (2006); Nayak and van Nieuwenhuizen
(2005); Ragsdale and Singleton (2017); Cooper et al. (2008); Matinyan and
Savvidy (1978); Brown and Weisberger (1979); Nair and Yelnikov (2010) with the
result that there is a constant rate of particle number density production,
still given by (1) with $m_{e}=0$. The result is surprising to us since the
Schwinger process can be viewed as a tunneling process and one might expect
that the WKB Brezin and Itzykson (1970); Kim and Page (2002) and other
approximations used to obtain (1) would break down for massless gauge fields.
For this reason we wish to re-examine the problem using a different approach.

We take a dynamical approach to the problem. At the initial time, we consider
a color electric field background and quantum excitations in their non-
interacting ground state. We then evolve the system, just as one would do for
small quantum excitations in a time-dependent background. We use the method of
Bogoliubov transformations Bogolyubov (1958); Birrell and Davies (1984), re-
cast as a “classical-quantum correspondence” (CQC) whereby quantum evolution
is described in terms of the classical evolution of a related classical system
Ambjorn et al. (1983); Vachaspati and Zahariade (2018, 2019); Bojowald and
Ding (2021). The method is explained in Appendix A. Our approach simply
evolves an initial state in contrast to other methods that compute (non-
interacting) vacuum persistence amplitudes.

Our results indeed differ from (1) with $m_{e}$ set to zero. We find that the
energy density in excitations, ${\cal E}$, grows with time as ${\cal E}\propto
g^{4}E^{4}t^{4}$ where $g$ is the non-Abelian coupling constant (see also
Akhmedov and Popov (2015); Akhmedov et al. (2014)). We also examine the number
density of particles in the leading adiabatic approximation Dabrowski and
Dunne (2016) and find that it is not well-defined as there is a zero frequency
mode at all times for which the particle number density diverges. The total
number density, found by integrating over all excitation modes, also diverges.

In Sec. II we set up the basic framework, identifying the background field and
the small excitations. Here we also discuss some limitations of our setup.
Sec. III diagonalizes the Hamiltonian by expanding the quantum fields in
modes. We find it convenient to discretize the modes for numerical analysis
(Sec. IV). The quantum analysis is reduced to a classical analysis in Sec. V,
following which we numerically evaluate the spectral energy density and the
total energy in quantum excitations as a function of time in Sec. VI. The
number density of quantum excitations is discussed in Sec. VII and shown to
diverge for all times due to the presence of a zero frequency mode. We
conclude in Sec. IX.

##  II Set up

We will consider a pure $SU(2)$ gauge theory with Lagrangian density,

$L=-\frac{1}{4}(W^{a}_{\mu\nu})^{2},\ \ a=1,2,3$ (2)

with the field strength defined in the usual way

$W^{a}_{\mu\nu}=\partial_{\mu}W^{a}_{\nu}-\partial_{\nu}W^{a}_{\mu}+g\epsilon^{abc}W^{b}_{\mu}W^{c}_{\nu}$
(3)

We have to make certain approximations to proceed with our analysis. Our main
approximation is that we expand the fields about a fixed background electric
field to quadratic order in the Lagrangian and ignore higher order
interactions.

Let us write

$W^{a}_{\mu}=A^{a}_{\mu}+Q^{a}_{\mu}$ (4)

where $A^{a}_{\mu}$ is a classical background and $Q^{a}_{\mu}$ denotes
quantum excitations on top of the classical background. We will work in
temporal gauge, so $W^{a}_{0}=0$, and take

$A^{a}_{\mu}=-A(t)\,\delta^{a3}\delta_{\mu 3}$ (5)

Then there is an externally imposed classical electric field but no magnetic
field,

$E^{a}_{i}=-A^{a}_{0i}={\dot{A}}(t)\,\delta^{a3}\delta_{i3},\ \ B^{a}_{i}=0.$
(6)

We will evolve the quantum variables, $Q^{a}_{i}$, assuming that they are in
their non-interacting ground state at $t=0$. We insert (4) in (2), then use
the background (5), and expand to quadratic order in the $Q^{a}_{\mu}$ to
obtain,

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\dot{Q}^{1}_{i})^{2}-\frac{1}{4}(\partial_{i}Q^{1}_{j}-\partial_{j}Q^{1}_{i}-gA(Q^{2}_{i}\delta^{3}_{j}-\delta^{3}_{i}Q^{2}_{j}))^{2}$
(7) $\displaystyle+$
$\displaystyle\frac{1}{2}(\dot{Q}^{2}_{i})^{2}-\frac{1}{4}(\partial_{i}Q^{2}_{j}-\partial_{j}Q^{2}_{i}+gA(Q^{1}_{i}\delta^{3}_{j}-\delta^{3}_{i}Q^{1}_{j}))^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}({\dot{Q}}^{3}_{3}-{\dot{A}})^{2}-\frac{1}{4}(\partial_{i}Q^{3}_{j}-\partial_{j}Q^{3}_{i})^{2}+{\cal
O}\left((Q^{a}_{i})^{3}\right)$

The classical electric field is externally imposed, i.e. there are external
sources that produce and maintain the electric field $E^{3}_{3}$ which is
assumed to be constant. Therefore, we take333We will also consider an
adiabatically turned on and off electric field in Sec. VIII.

$A(t)=Et,\ \ {\dot{A}}=E$ (8)

Then the variables $Q^{3}_{i}$ decouple from the other quantum variables. We
can calculate the rate of particle production in a fixed external field by
considering the truncated Lagrangian,

$L^{\prime}=\frac{1}{2}(\dot{Q}^{1}_{i})^{2}+\frac{1}{2}(\dot{Q}^{2}_{i})^{2}-\frac{1}{4}(Q^{1}_{ij})^{2}-\frac{1}{4}(Q^{2}_{ij})^{2}$
(9)

where,

$\displaystyle Q^{1}_{ij}$ $\displaystyle\equiv$
$\displaystyle\partial_{i}Q^{1}_{j}-\partial_{j}Q^{1}_{i}-gEt(Q^{2}_{i}\delta^{3}_{j}-\delta^{3}_{i}Q^{2}_{j}))$
$\displaystyle Q^{2}_{ij}$ $\displaystyle\equiv$
$\displaystyle\partial_{i}Q^{2}_{j}-\partial_{j}Q^{2}_{i}+gEt(Q^{1}_{i}\delta^{3}_{j}-\delta^{3}_{i}Q^{1}_{j})$
(10)

At this level of approximation, the $Q^{1}_{i}$ and $Q^{2}_{i}$ fluctuations
do not backreact on the background electric field. The backreaction will only
appear due to the cubic and higher order terms in the $Q^{a}_{i}$ in (7).

We will expand the excitations in momentum modes in the next section. There
are quantum issues at both ends of the spectrum. For modes with low energy,
the coupling constant is strong and confinement should play a role. The lowest
energy excitations will be massive glueballs, not massless gluons444Indeed the
assumed background uniform electric field itself ignores confinement. The
spirit of the present work is that we work as if there is no confinement and
study the consequences.. Modes with very high energy are in the regime of
asymptotic freedom as the coupling constant becomes small. Inclusion of these
effects in our calculations is beyond our reach and we shall proceed based on
(9) as if it is the full story and see if there are any inconsistencies.
Indeed we will encounter two inconsistencies in this approach. In Sec. VI.1 we
will encounter an ultraviolet divergence that we argue will be resolved by
properly accounting for asymptotic freedom. We will also encounter a
divergence in the number density of excitations at low energy at all times
whose interpretation will change radically once we take confinement into
account.

##  III Expansion in modes

In this section we expand the fields in modes and diagonalize the Lagrangian.
The calculations are straightforward if tedious; the end result for the
diagonalized Lagrangian is given in (34) and (35).

The gauge fields are expanded in the physical transverse modes as,

$Q^{1}_{\mu}=\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{\sqrt{2E_{p}}}\sum_{r=0}^{3}\left[a^{r}_{\bf p}\epsilon_{{\bf
p},\mu}^{r}e^{i{\bf p}\cdot{\bf x}}+h.c.\right]$ (11)

where $d\bar{}\hskip 1.00006pt^{3}p\equiv d^{3}p/(2\pi)^{3}$, $E_{p}=|{\bf
p}|$, and reality of $Q^{1}$ implies $a^{r*}_{\bf p}=a^{r}_{-{\bf p}}$.
Similarly,

$Q^{2}_{\mu}=\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{\sqrt{2E_{p}}}\sum_{r=0}^{3}\left[b^{r}_{\bf p}\epsilon_{{\bf
p},\mu}^{r}e^{i{\bf p}\cdot{\bf x}}+h.c.\right]$ (12)

with $b^{r*}_{\bf p}=b^{r}_{-{\bf p}}$.

The magnetic fields are

$\displaystyle({\bf B}^{1})_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon_{ijk}Q^{1}_{jk}=\nabla\times{\bf
Q}^{1}-gEt\,{\bf Q}^{2}\times{\hat{z}}$ (13) $\displaystyle({\bf B}^{2})_{i}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon_{ijk}Q^{2}_{jk}=\nabla\times{\bf
Q}^{2}+gEt\,{\bf Q}^{1}\times{\hat{z}}$ (14)

Therefore

$\displaystyle{\bf B}^{1}$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{\sqrt{2E_{p}}}\sum_{r=0}^{3}\left[(a^{r}_{\bf p}i{\bf
p}+gEt\,b^{r}_{\bf p}{\hat{z}})\times{\bm{\epsilon}}^{r}_{\bf p}e^{i{\bf
p}\cdot{\bf x}}+h.c.\right]$ (15) $\displaystyle\equiv$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{\sqrt{2E_{p}}}\sum_{r=0}^{3}\left[{\bf a}^{r}_{\bf p}e^{i{\bf
p}\cdot{\bf x}}+h.c.\right]$

where ${\bf a}^{r}_{\bf p}\equiv(a^{r}_{\bf p}i{\bf p}+gEt\,b^{r}_{\bf
p}{\hat{z}})\times{\bm{\epsilon}}^{r}_{\bf p}$. And

$\displaystyle{\bf B}^{2}$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{\sqrt{2E_{p}}}\sum_{r=0}^{3}\left[(b^{r}_{\bf p}i{\bf
p}-gEt\,a^{r}_{\bf p}{\hat{z}})\times{\bm{\epsilon}}^{r}_{\bf p}e^{i{\bf
p}\cdot{\bf x}}+h.c.\right]$ (16) $\displaystyle\equiv$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{\sqrt{2E_{p}}}\sum_{r=0}^{3}\left[{\bf b}^{r}_{\bf p}e^{i{\bf
p}\cdot{\bf x}}+h.c.\right]$

where ${\bf b}^{r}_{\bf p}\equiv(b^{r}_{\bf p}i{\bf p}-gEt\,a^{r}_{\bf
p}{\hat{z}})\times{\bm{\epsilon}}^{r}_{\bf p}$.

These expressions give,

$\displaystyle{\cal E}_{B1}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\int d^{3}x({\bf B}^{1})^{2}$ (17) $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{2E_{p}}\sum_{r,s=0}^{3}\left[{\bf a}^{r}_{\bf p}\cdot{\bf
a}^{s{\dagger}}_{\bf p}+h.c.\right]$

Note that there are two polarizations and $\\{{\hat{\epsilon}}^{1}_{\bf
p},{\hat{\epsilon}}^{2}_{\bf p},{\hat{p}}\\}$ form a right-handed orthonormal
basis. For example,

$\displaystyle{\hat{\epsilon}}^{1}_{\bf p}$ $\displaystyle=$
$\displaystyle(-\cos\theta\cos\phi,-\cos\theta\sin\phi,\sin\theta)$ (18)
$\displaystyle{\hat{\epsilon}}^{2}_{\bf p}$ $\displaystyle=$
$\displaystyle(-\sin\phi,\cos\phi,0)$ (19) $\displaystyle{\hat{p}}$
$\displaystyle=$
$\displaystyle(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ (20)

We check

$({\bf p}\times{\hat{\epsilon}}^{r}_{\bf p})\cdot({\bf
p}\times{\hat{\epsilon}}^{s}_{\bf p})=p^{2}\delta^{rs}$ (21)
$({\hat{z}}\times{\hat{\epsilon}}^{r}_{\bf p})\cdot({\bf
p}\times{\hat{\epsilon}}^{s}_{\bf p})=p_{z}\delta^{rs}=p\cos\theta\delta^{rs}$
(22) $({\hat{z}}\times{\hat{\epsilon}}^{r}_{\bf
p})\cdot({\hat{z}}\times{\hat{\epsilon}}^{s}_{\bf
p})=\delta^{rs}-({\hat{\epsilon}}^{r}_{\bf
p}\cdot{\hat{z}})\,({\hat{\epsilon}}^{s}_{\bf p}\cdot{\hat{z}})$ (23)

and with the choice of vectors in (18)-(20),

$({\hat{\epsilon}}^{r}_{\bf p}\cdot{\hat{z}})\,({\hat{\epsilon}}^{s}_{\bf
p}\cdot{\hat{z}})=\sin^{2}\theta\,\delta^{r1}\delta^{s1}$ (24)

Therefore

$\displaystyle{\cal E}_{B1}$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\sum_{r=1}^{2}\biggl{[}p^{2}|a^{r}_{\bf
p}|^{2}+g^{2}E^{2}t^{2}|b^{r}_{\bf p}|^{2}(1-\sin^{2}\theta\delta_{r1})$ (25)
$\displaystyle\hskip 56.9055pt-igEtp_{z}(a^{r{\dagger}}_{\bf p}b^{r}_{\bf
p}-a^{r}_{\bf p}b^{r{\dagger}}_{\bf p})\biggr{]}$

Similarly

$\displaystyle{\cal E}_{B2}$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\sum_{r=1}^{2}\biggl{[}p^{2}|b^{r}_{\bf
p}|^{2}+g^{2}E^{2}t^{2}|a^{r}_{\bf p}|^{2}(1-\sin^{2}\theta\delta_{r1})$ (26)
$\displaystyle\hskip 56.9055pt-igEtp_{z}(a^{r{\dagger}}_{\bf p}b^{r}_{\bf
p}-a^{r}_{\bf p}b^{r{\dagger}}_{\bf p})\biggr{]}$

$\displaystyle{\cal E}_{B1+2}$ $\displaystyle=$ $\displaystyle{\cal
E}_{B1}+{\cal E}_{B2}$ (27) $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}(p^{2}+g^{2}E^{2}t^{2}\cos^{2}\theta)(|a^{1}_{\bf
p}|^{2}+|b^{1}_{\bf p}|^{2})$ $\displaystyle\hskip
28.45274pt-i2gEtp\cos\theta(a^{1{\dagger}}_{\bf p}b^{1}_{\bf p}-a^{1}_{\bf
p}b^{1{\dagger}}_{\bf p})$ $\displaystyle\hskip
28.45274pt+(p^{2}+g^{2}E^{2}t^{2})(|a^{2}_{\bf p}|^{2}+|b^{2}_{\bf p}|^{2})$
$\displaystyle\hskip 28.45274pt-i2gEtp\cos\theta(a^{2{\dagger}}_{\bf
p}b^{2}_{\bf p}-a^{2}_{\bf p}b^{2{\dagger}}_{\bf p})\biggr{]}$

Next let

$a^{r}_{\bf p}=\alpha^{r}_{\bf p}+i\beta^{r}_{\bf p},\ \ b^{r}_{\bf
p}=\gamma^{r}_{\bf p}+i\delta^{r}_{\bf p}$ (28)

where $\alpha^{r}_{\bf p}$, $\beta^{r}_{\bf p}$, $\gamma^{r}_{\bf p}$ and
$\delta^{r}_{\bf p}$ are real. The reality conditions, $a^{r*}_{\bf
p}=a^{r}_{-{\bf p}}$ and $b^{r*}_{\bf p}=b^{r}_{-{\bf p}}$, imply

$\alpha^{r}_{\bf p}=\alpha^{r}_{-{\bf p}},\ \ \beta^{r}_{\bf
p}=-\beta^{r}_{-{\bf p}},\ \ \gamma^{r}_{\bf p}=\gamma^{r}_{-{\bf p}},\ \
\delta^{r}_{\bf p}=-\delta^{r}_{-{\bf p}}$ (29)

For convenience, define

$P=gEt,\ \ P_{z}=P\cos\theta$ (30)

Then

$\displaystyle{\cal E}_{B1+2}$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}(p^{2}+P_{z}^{2})\\{(\alpha^{1}_{\bf
p})^{2}+(\delta^{1}_{\bf p})^{2}\\}+4pP_{z}\alpha^{1}_{\bf p}\delta^{1}_{\bf
p}$ $\displaystyle\hskip 14.22636pt+(p^{2}+P_{z}^{2})\\{(\beta^{1}_{\bf
p})^{2}+(\gamma^{1}_{\bf p})^{2}\\}-4pP_{z}\beta^{1}_{\bf p}\gamma^{1}_{\bf
p}$ $\displaystyle\hskip 14.22636pt+(p^{2}+P^{2})\\{(\alpha^{2}_{\bf
p})^{2}+(\delta^{2}_{\bf p})^{2}\\}+4pP_{z}\alpha^{2}_{\bf p}\delta^{2}_{\bf
p}$ $\displaystyle\hskip 14.22636pt+(p^{2}+P^{2})\\{(\beta^{2}_{\bf
p})^{2}+(\gamma^{2}_{\bf p})^{2}\\}-4pP_{z}\beta^{2}_{\bf p}\gamma^{2}_{\bf
p}\biggr{]}$

The energy has separated into 4 disjoint sectors: ($\alpha^{1}_{\bf
p},\delta^{1}_{\bf p}$), ($\beta^{1}_{\bf p},\gamma^{1}_{\bf p}$),
($\alpha^{2}_{\bf p},\delta^{2}_{\bf p}$) and ($\alpha^{2}_{\bf
p},\delta^{2}_{\bf p}$). The ($\alpha^{1}_{\bf p},\delta^{1}_{\bf p}$) sector
is equivalent to the ($\beta^{1}_{\bf p},\gamma^{1}_{\bf p}$) sector under
$P_{z}\to-P_{z}$ which is the same as $g\to-g$. Similarly for the
($\alpha^{2}_{\bf p},\delta^{2}_{\bf p}$) and ($\beta^{2}_{\bf
p},\gamma^{2}_{\bf p}$) sectors. The ($\alpha^{1}_{\bf p},\delta^{1}_{\bf p}$)
and ($\alpha^{2}_{\bf p},\delta^{2}_{\bf p}$) sectors look very similar but
they differ in their first terms: the former has $P_{z}$ while the latter has
$P$.

The energy in the electric field is

$\displaystyle{\cal E}_{E1+2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{3}x\left[(\partial_{t}{\bf W}^{1})^{2}+(\partial_{t}{\bf
W}^{2})^{2}\right]$ $\displaystyle=$ $\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\sum_{r=1}^{2}\left[({\dot{\alpha}}^{r}_{\bf
p})^{2}+({\dot{\beta}}^{r}_{\bf p})^{2}+({\dot{\gamma}}^{r}_{\bf
p})^{2}+({\dot{\delta}}^{r}_{\bf p})^{2}\right]$

and we only need to solve for the dynamics of the two quantum systems,

$\displaystyle L_{1}$ $\displaystyle=$ $\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}({\dot{\alpha}}^{1}_{\bf
p})^{2}+({\dot{\delta}}^{1}_{\bf p})^{2}-(p^{2}+P_{z}^{2})\\{(\alpha^{1}_{\bf
p})^{2}+(\delta^{1}_{\bf p})^{2}\\}$ (31) $\displaystyle\hskip
113.81102pt-4pP_{z}\alpha^{1}_{\bf p}\delta^{1}_{\bf p}\biggr{]}$
$\displaystyle L_{2}$ $\displaystyle=$ $\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}({\dot{\alpha}}^{2}_{\bf
p})^{2}+({\dot{\delta}}^{2}_{\bf p})^{2}-(p^{2}+P^{2})\\{(\alpha^{2}_{\bf
p})^{2}+(\delta^{2}_{\bf p})^{2}\\}$ (32) $\displaystyle\hskip
113.81102pt-4pP_{z}\alpha^{2}_{\bf p}\delta^{2}_{\bf p}\biggr{]}$

As noted above, the Lagrangians for ($\beta^{r}_{\bf p},\gamma^{r}_{\bf p}$)
are related to $L_{1}$ and $L_{2}$ by $g\to-g$. Also recall that there is
time-dependence in these Lagrangians because $P_{z}$ and $P$ grow in
proportion to $t$ as defined in (30).

The Lagrangians can be diagonalized by using linear combinations,

$\phi_{{\bf p},\pm}=\frac{\alpha^{1}_{\bf p}\pm\delta^{1}_{\bf p}}{\sqrt{2}},\
\ \psi_{{\bf p},\pm}=\frac{\alpha^{2}_{\bf p}\pm\delta^{2}_{\bf p}}{\sqrt{2}}$
(33)

Then

$\displaystyle L_{1}$ $\displaystyle=$ $\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}{\dot{\phi}}_{{\bf
p},+}^{2}-(p+P_{z})^{2}\phi_{{\bf p},+}^{2}$ (34) $\displaystyle\hskip
42.67912pt+{\dot{\phi}}_{{\bf p},-}^{2}-(p-P_{z})^{2}\phi_{{\bf
p},-}^{2}\biggr{]}$ $\displaystyle L_{2}$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}{\dot{\psi}}_{{\bf
p},+}^{2}-\\{(p_{z}+P)^{2}+p_{\perp}^{2}\\}\psi_{{\bf p},+}^{2}$ (35)
$\displaystyle\hskip 28.45274pt+{\dot{\psi}}_{{\bf
p},-}^{2}-\\{(p_{z}-P)^{2}+p_{\perp}^{2}\\}\psi_{{\bf p},-}^{2}\biggr{]}$

where $P=gEt$, $p_{\perp}=p\sin\theta$ and $p_{z}=p\cos\theta$.

Similarly we can easily write down $L_{3}$ and $L_{4}$ for the
$(\beta^{r}_{\bf p},\gamma^{r}_{\bf p})$ sector since these are related to
$L_{1}$ and $L_{2}$ by $g\to-g$. The full Lagrangian is given by the sum of
$L_{1},\ldots,L_{4}$.

##  IV Discretization

We now discretize the integrations in (34) and (35). For example,

$\displaystyle L_{1}$ $\displaystyle=$ $\displaystyle\sum_{{\bf
p}}\frac{(\Delta p)^{3}}{E_{p}}\biggl{[}{\dot{\phi}}_{{\bf
p},+}^{2}-(p+P_{z})^{2}\phi_{{\bf p},+}^{2}$ (36) $\displaystyle\hskip
7.11317pt+{\dot{\phi}}_{{\bf p},-}^{2}-(p-P_{z})^{2}\phi_{{\bf
p},-}^{2}\biggr{]}=\sum_{{\bf p}}L_{1{\bf p}}$

and similarly for $L_{2}$. The volume element in momentum space is

$(\Delta p)^{3}=\left(\frac{2\pi}{L}\right)^{3}=\frac{(2\pi)^{3}}{V}$ (37)

where $L$ is the size of the (compactified) spatial domain and $V$ its volume.
The Hamiltonian for each mode can now be written as that for a simple harmonic
oscillator,

$\displaystyle H_{1{\bf p}}$ $\displaystyle=$ $\displaystyle\frac{\pi_{{\bf
p}+}^{2}}{2m_{\bf p}}+\frac{m_{\bf p}}{2}(p+P_{z})^{2}|\phi_{{\bf p}+}|^{2}$
(38) $\displaystyle+\frac{\pi_{{\bf p}-}^{2}}{2m_{\bf p}}+\frac{m_{\bf
p}}{2}(p-P_{z})^{2}|\phi_{{\bf p}-}|^{2}$ $\displaystyle H_{2{\bf p}}$
$\displaystyle=$ $\displaystyle\frac{\Pi_{{\bf p}+}^{2}}{2m_{\bf
p}}+\frac{m_{\bf p}}{2}\\{(p_{z}+P)^{2}+p_{\perp}^{2}\\}|\psi_{{\bf p}+}|^{2}$
(39) $\displaystyle+\frac{\Pi_{{\bf p}-}^{2}}{2m_{\bf p}}+\frac{m_{\bf
p}}{2}\\{(p_{z}-P)^{2}+p_{\perp}^{2}\\}|\psi_{{\bf p}-}|^{2}$

where $\pi_{{\bf p}\pm}$ and $\Pi_{{\bf p}\pm}$ are conjugate momenta to
$\phi_{{\bf p}\pm}$ and $\psi_{{\bf p}\pm}$, and

$m_{{\bf p}}=2\frac{(\Delta
p)^{3}}{E_{p}}=\frac{2}{p}\left(\frac{2\pi}{L}\right)^{3}$ (40)

Note that $m_{\bf p}$ has dimensions of mass squared (not mass).

As noted at the end of Sec. III, we will also have $H_{3{\bf p}}$ and
$H_{4{\bf p}}$ corresponding to the $(\beta^{r}_{\bf p},\gamma^{r}_{\bf p})$
sector.

The Hamiltonians in (38) and (39) are those of simple harmonic oscillators
with time dependent frequencies,

$\omega_{\phi,{\bf p},\pm}=+\sqrt{(p\pm P_{z})^{2}}$ (41) $\omega_{\psi,{\bf
p},\pm}=+\sqrt{(p_{z}\pm P)^{2}+p_{\perp}^{2}}$ (42)

The $+$ signs in the pre-factor are to emphasize that we are taking the
positive square root. The structure of the frequencies is easy to understand
because the $p_{z}\pm P$ follows from the covariant derivative acting on the
excitations. The different forms of $\omega_{\phi,{\bf p},\pm}$ and
$\omega_{\psi,{\bf p},\pm}$ arise since the variables $\phi_{{\bf p},\pm}$ are
associated with ${\hat{\epsilon}}_{\bf p}^{r=1}$, while $\psi_{{\bf p},\pm}$
are associated with ${\hat{\epsilon}}_{\bf p}^{r=2}$.

An important point for us is that there are certain modes for which the
frequency vanishes. For example, $\omega_{\phi,{\bf p},-}=|p-P_{z}|$ and, at
any time, the frequency vanishes for $p=P_{z}=gEt\cos\theta$. Similarly for
$\omega_{\psi,{\bf p},-}$. The frequencies $\omega_{\phi,{\bf p},+}$ and
$\omega_{\psi,{\bf p},+}$ do not vanish for $t>0$. We will return to this
point in our discussion of the particle number density in Sec. VII.

##  V Classical-Quantum Correspondence

To obtain particle production of the fields $\phi_{{\bf p},\pm}$, $\psi_{{\bf
p},\pm}$ we will use the CQC. (See Appendix A for a summary of the CQC.) Then
the variables $\phi_{{\bf p},\pm}$ and $\psi_{{\bf p},\pm}$ are complexified
and we solve the classical equations of motion

$\displaystyle{\ddot{\phi}}_{{\bf p},\pm}+(P_{z}\pm p)^{2}\phi_{{\bf p},\pm}$
$\displaystyle=$ $\displaystyle 0$ (43) $\displaystyle{\ddot{\psi}}_{{\bf
p},\pm}+\\{(P\pm p_{z})^{2}+p_{\perp}^{2}\\}\psi_{{\bf p},\pm}$
$\displaystyle=$ $\displaystyle 0$ (44)

with initial conditions,

$\displaystyle\phi_{{\bf p},\pm}(t=0)=-\frac{i}{\sqrt{2m_{{\bf
p}}p}}=\psi_{{\bf p},\pm}(t=0)$ (45) $\displaystyle{\dot{\phi}}_{{\bf
p},\pm}(t=0)=\sqrt{\frac{p}{2m_{{\bf p}}}}={\dot{\psi}}_{{\bf p},\pm}(t=0)$
(46)

(Note that the simple harmonic oscillator frequencies at $t=0$ are simply $p$
because $P(t=0)=0$.) Using (40) we write

$\displaystyle\phi_{{\bf
p},\pm}(0)=-\frac{i}{2}\left(\frac{L}{2\pi}\right)^{3/2}=\psi_{{\bf
p},\pm}(0)$ (47) $\displaystyle{\dot{\phi}}_{{\bf
p},\pm}(0)=\frac{p}{2}\left(\frac{L}{2\pi}\right)^{3/2}={\dot{\psi}}_{{\bf
p},\pm}(0)$ (48)

To calculate the energy density in excitations, we simply need to evaluate the
classical energy in the complexified $\phi_{{\bf p},\pm}$ and $\psi_{{\bf
p},\pm}$ as we describe next.

##  VI Energy density production

The energy in the complexified variables $\phi_{{\bf p},\pm}$ $\psi_{{\bf
p},\pm}$ follows from (34) and (35),

$\displaystyle{\cal E}_{1}$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}|{\dot{\phi}}_{{\bf
p},+}|^{2}+(p+P_{z})^{2}|\phi_{{\bf p},+}|^{2}$ (49) $\displaystyle\hskip
42.67912pt+|{\dot{\phi}}_{{\bf p},-}|^{2}+(p-P_{z})^{2}|\phi_{{\bf
p},-}|^{2}\biggr{]}$ $\displaystyle{\cal E}_{2}$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}|{\dot{\psi}}_{{\bf
p},+}|^{2}+\\{(p_{z}+P)^{2}+p_{\perp}^{2}\\}|\psi_{{\bf p},+}|^{2}$ (50)
$\displaystyle\hskip 14.22636pt+|{\dot{\psi}}_{{\bf
p},-}|^{2}+\\{(p_{z}-P)^{2}+p_{\perp}^{2}\\}|\psi_{{\bf p},-}|^{2}\biggr{]}$

The energies in the $(\beta^{r}_{\bf p},\gamma^{r}_{\bf p})$ sector give
identical expressions and we will include these in the end in the total energy
by multiplying by a factor of 2.

The energy expressions in (49) and (50) include the ground state energy – the
$\omega/2$ of the simple harmonic oscillator – whereas we are interested in
the energy of the excitations only. As described in Appendix A, the ground
state energy can be discarded by writing the energies as,

$\displaystyle:{\cal E}_{1}:$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}\left|{\dot{\phi}}_{{\bf
p},+}-i\omega_{\phi,{\bf p},+}\,\phi_{{\bf p},+}\right|^{2}$ (51)
$\displaystyle\hskip 42.67912pt+\left|{\dot{\phi}}_{{\bf
p},-}-i\omega_{\phi,{\bf p},-}\,\phi_{{\bf p},-}\right|^{2}\biggr{]}$
$\displaystyle:{\cal E}_{2}:$ $\displaystyle=$
$\displaystyle\int\frac{d\bar{}\hskip
1.00006pt^{3}p}{E_{p}}\biggl{[}\left|{\dot{\psi}}_{{\bf
p},+}-i\omega_{\psi,{\bf p},+}\,\psi_{{\bf p},+}\right|^{2}$ (52)
$\displaystyle\hskip 42.67912pt+\left|{\dot{\psi}}_{{\bf
p},-}-i\omega_{\psi,{\bf p},-}\,\psi_{{\bf p},-}\right|^{2}\biggr{]}$

The total energy in the excitations is

$\displaystyle:{\cal E}:$ $\displaystyle=$ $\displaystyle 2(\,:{\cal
E}_{1}:+:{\cal E}_{2}:\,)$ (53) $\displaystyle=$ $\displaystyle
8\pi\int_{0}^{\infty}dp\,p\int_{0}^{1}du\,$
$\displaystyle\hskip-56.9055pt\times\biggl{[}\left|{\dot{\phi}}_{{\bf
p},+}-i\omega_{\phi,{\bf p},+}\,\phi_{{\bf
p},+}\right|^{2}+\left|{\dot{\phi}}_{{\bf p},-}-i\omega_{\phi,{\bf
p},-}\,\phi_{{\bf p},-}\right|^{2}$
$\displaystyle\hskip-56.9055pt+\left|{\dot{\psi}}_{{\bf
p},+}-i\omega_{\psi,{\bf p},+}\,\psi_{{\bf
p},+}\right|^{2}+\left|{\dot{\psi}}_{{\bf p},-}-i\omega_{\psi,{\bf
p},-}\,\psi_{{\bf p},-}\right|^{2}\biggr{]}$

where the factor of 2 in the first line accounts for the excitations in the
$(\beta^{r}_{\bf p},\gamma^{r}_{\bf p})$ sector, $u\equiv\cos\theta$, and we
have used the symmetry under $u\to-u$ to restrict $u$ to the interval $[0,1]$.
We remark that the expressions occurring in the integrands of (51) and (52),
such as ${\dot{\phi}}_{{\bf p},+}-i\omega_{\phi,{\bf p},+}\,\phi_{{\bf p},+}$,
are the usual Bogolyubov $\beta$ coefficients up to a factor of
$1/\sqrt{\omega}$ where $\omega$ stands for the frequency.

Now all that is required is to solve the equations of motion in (43) and (44),
insert the solutions in the energy expressions above, and perform the
integrations. The first step is formally accomplished since the solutions to
the equations of motion can be written in terms of parabolic cylindrical
functions. However, we have found it more practical to solve the differential
equations numerically followed by numerical integration555Part of the
numerical work has been performed in Python and the code can be seen at the
following link
https://github.com/cargicar/Vacuum_inestability_Non_Abelian_YM/blob/96625bc7e89863901ba59439d2427def826cae0b/NASchwingerEffect4_carlos_kernel.ipynb.

###  VI.1 Numerical evaluation of the energy

We have numerically solved the differential equations in (43) and (44) for
$u\in[0,1]$ and $p\in[0,p_{c}]$ where $p_{c}\gg gEt$ is a momentum cutoff. As
we solve the differential equations, we also numerically evaluate the energy
in (53). In the numerical computations we choose units so that $gE=1$.

Figure 1:  $\rho(x,t)(t_{f}/t)^{3}$ vs. $x=p(t_{f}/t)$ for $t_{f}=20$ and
$t=t_{f}/2,2t_{f}/3$, and $t_{f}$. There are three curves in the plot but they
all overlap. The peak in $\rho(p,t)$ is located at $p\approx gEt/2$.

We define the spectral energy density, $\rho(p,t)$, via

$\frac{:{\cal E}:}{L^{3}}=\int_{0}^{\infty}dp\,\rho(p,t)$ (54)

or explicitly,

$\displaystyle\rho(p)$ $\displaystyle=$ $\displaystyle\frac{8\pi
p}{L^{3}}\int_{0}^{1}du\,\sum_{s=\pm}\biggl{[}\left|{\dot{\phi}}_{{\bf
p},s}-i\omega_{\phi,{\bf p},s}\,\phi_{{\bf p},s}\right|^{2}$ (55)
$\displaystyle\hskip 56.9055pt+\left|{\dot{\psi}}_{{\bf
p},s}-i\omega_{\psi,{\bf p},s}\,\psi_{{\bf p},s}\right|^{2}\biggr{]}$

In Fig. 1 we plot the spectral energy density rescaled by $t^{-3}$, i.e.
$t^{-3}\rho(p,t)$, as a function of $p/t$ at three different times. It is
clear that the peak of the spectrum $\rho$ grows as $t^{3}$ and the width
grows as $t$, implying that the total energy in excitations grows as $t^{4}$.

The linear growth of the width of $\rho(p,t)$ follows from the form of the
oscillation frequencies in (41) and (42). For very large values of $p$, the
time dependence of the frequencies can be ignored. Then the solutions for
$\phi_{{\bf p},\pm}$ and $\psi_{{\bf p},\pm}$ are simply trigonometric
functions for which there is no contribution to the energy. Hence there is no
particle production for $p\gg gEt$; there is only particle production for
$p\lesssim gEt$ and so the width in $p$ contributing to particle production
grows linearly in $t$. To understand the growth of the peak of $\rho(p,t)$
that goes as $t^{3}$, we note that the peak is located at $p\approx gEt/2$.
Thus the $p$ prefactor in (55) contributes one factor of $t$. In the
integrand, the initial conditions for the variables ${\dot{\phi}}_{{\bf
p},\pm}$ and ${\dot{\psi}}_{{\bf p},\pm}$ are proportional to $p$ as in (48)
and since the $p$ that contributes to the energy integral grows proportional
to $t$, and the variables enter quadratically in the energy integral, the peak
of the spectral energy density grows as $t^{3}$.

The $t^{4}$ growth of the total energy density is further confirmed in Fig. 2
where we plot the total energy density divided by $t^{4}$ vs. time.

Figure 2:  Energy density rescaled by $t^{4}$, $:{{\cal E}}:/(L^{3}t^{4})$,
vs. time in the $\phi$ excitations (blue curve), $\psi$ excitations (red
curve), and in total (black curve). The flat curves at late times confirm that
$:{\cal E}:\propto t^{4}$ as also indicated in Fig. 1.

To obtain the total energy density, $:{\cal E}:$, we have integrated over
$p\in[0,p_{c}]$ where $p_{c}\gg gEt$ is a cutoff. To study the dependence of
our result on the cutoff, we zoom into the large $p$ behavior of $\rho(p,t)$,
shown in Fig. 3. This gives $\rho(p,t)\propto 1/p$ at large $p$ and the total
energy diverges logarithmically as the cutoff $p_{c}$ is taken to infinity.
However this ultraviolet divergence will be controlled once asymptotic freedom
is taken into account. To see this in more detail, note that $\rho(p,t)$ in
Fig. 1 has a dominant peak structure followed by the $1/p$ fall off, which
after integration over $p$, lead to an asymptotic contribution to the energy
density given by $g^{4}\,\log(p_{c}/M)$, where $M$ is a renormalization scale.
However the asymptotic value of the coupling constant at the cut-off scale
evolves from its value $g_{M}$ at the renormalization scale as Peskin and
Schroeder (1995),

$g^{2}={g_{M}^{2}\over 1+g_{M}^{2}\log(p_{c}/M)}\,,$ (56)

implying,

$\lim_{p_{c}\to\infty}g^{4}\,\log(p_{c}/M)\to 0\,.$ (57)

Hence the $1/p$ tail contribution to the integration vanishes in the
$p_{c}\to\infty$ limit once we take the running of $g^{2}$, i.e. asymptotic
freedom, into account.

Figure 3:  Log-log plot showing the asymptotic behavior of $\rho(p,t)$ vs. $p$
for $t=2t_{f}/3$ with $t_{f}=20$. The dashed line shows a $1/p$ fall off.

##  VII Number density production

The number density of produced particles, $n$, is ambiguous at intermediate
times though there is no ambiguity in the infinite time limit if the electric
field is switched off Dabrowski and Dunne (2016). Here we adopt the definition
that the number density is given by the energy density for each mode divided
by the frequency of that mode. Then using (53) we have,

$\displaystyle n$ $\displaystyle=$ $\displaystyle
8\pi\int_{0}^{\infty}dp\,p\int_{0}^{1}du\,\sum_{s=\pm}\biggl{[}\frac{\left|{\dot{\phi}}_{{\bf
p},s}-i\omega_{\phi,{\bf p},s}\,\phi_{{\bf p},s}\right|^{2}}{\omega_{\phi,{\bf
p},s}}$ (58) $\displaystyle\hskip 71.13188pt+\frac{\left|{\dot{\psi}}_{{\bf
p},s}-i\omega_{\psi,{\bf p},s}\,\psi_{{\bf p},s}\right|^{2}}{\omega_{\psi,{\bf
p},s}}\biggr{]}$

The expressions for the frequencies are given in (41) and (42). The issue is
that at any given time, there is a momentum mode for which the frequency
vanishes. Thus the integrand in (58) is singular at all times.

To determine if the singularity is integrable, we focus our attention on the
$\phi_{{\bf p},-}$ sector for which the number density is

$n_{\phi,-}=8\pi\int_{0}^{\infty}dp\,p\int_{0}^{1}du\,\frac{\left|{\dot{\phi}}_{{\bf
p},-}-i|p-gEtu|\,\phi_{{\bf p},-}\right|^{2}}{|p-gEtu|}$ (59)

First consider the numerator of the integrand. For $p=gEtu$ it is simply
$|{\dot{\phi}}_{{\bf p},-}|^{2}$. Non-vanishing energy in the $\phi_{{\bf
p},-}$ excitations implies that $|{\dot{\phi}}_{{\bf p},-}|^{2}\neq 0$. So the
singularity structure of the integral is

$n_{\phi,-}=8\pi\langle p|{\dot{\phi}}_{{\bf
p},-}|^{2}\rangle\int_{0}^{\infty}dq\int_{0}^{1}du\,\frac{1}{|q-u|}$ (60)

where $q\equiv p/gEt$ and $\langle\cdot\rangle$ denotes an effective value
along the singular curve, $p=gEtu$ (or $q=u$), in the integration plane. By
transforming integration variables to $x_{\pm}=q\pm u$ it is clear that the
integral is logarithmically divergent due to the singularity along $x_{-}=0$.

We note that the divergence in particle number density arises at low energy
where the frequencies vanish. It is worth emphasizing that this is a
particular behavior of the massless theory and will not be present in massive
cases such as QED. In the full interacting theory we can expect this infrared
divergence to be resolved due to confinement effects. At such low frequencies,
the “soft” gluons are confined and are only present in massive glueball
states. In our analysis, as discussed in Sec. II, we are examining where the
road goes when we ignore confinement effects. It is interesting that the
calculation without confinement leads to a divergent number density of
excitations, suggesting a self-inconsistency.

##  VIII Adiabatic case

Often in the Bogoliubov method, the background is taken to turn on
adiabatically, survive for a certain time period, and then slowly turned off.
Then the asymptotic vacua are unambiguously defined and the total energy
density of particles produced is evaluated at $t\rightarrow\infty$.

We have also treated the adiabatic case. Now our choice for $A(t)$ is,

$A(t)=E\tau\left[\tanh\left(\frac{t-t_{E}}{\tau}\right)-\tanh\left(\frac{-t_{E}}{\tau}\right)\right]$
(61)

where $t_{E}$ is some large time at which the electric field is maximum and
$\tau$ is the duration for which the field is turned on. The electric field is
given by,

${\dot{A}}=E\,{\rm sech}^{2}\left(\frac{t-t_{E}}{\tau}\right)$ (62)

Our analysis from the previous sections remains unchanged except that the
frequencies in (63) and (64) now become

$\omega_{\phi,{\bf p},\pm}=+\sqrt{(p\pm gA(t)u)^{2}}$ (63) $\omega_{\psi,{\bf
p},\pm}=+\sqrt{(p_{z}\pm gA(t))^{2}+p_{\perp}^{2}}$ (64)

We have repeated the numerical analysis in this case and show the results for
the energy density versus $\tau$ in Fig. 4. Once again we find $:{\cal
E}:\propto\tau^{4}$.

Figure 4:  Log-log plot of $:{\cal E}:$ vs. $\tau$ for the adiabatic case.

We can also consider the number density of particles in the adiabatic case.
From Eqs. (63) we see that the mode with $p=gA(\infty)u$ has vanishing
frequency $\omega_{\phi,{\bf p},-}$ as $t\to\infty$. Similarly the mode with
$p_{z}=gA(\infty)$, $p_{\perp}=0$ has vanishing frequency $\omega_{\psi,{\bf
p},-}$ as $t\to\infty$. Therefore the singularity discussed in Sec. VII is
present even with the short duration electric field of (62).

##  IX Conclusions

We have considered the fate of a uniform color electric field in a pure gauge
Yang-Mills theory with SU(2) gauge group666The gauge group is not important
for our analysis since SU(N) models have SU(2) subgroups.. Our approach
assumes initial conditions with a uniform electric field and quantum
excitations in their non-interacting ground state. We then truncate the
excitations to quadratic order in the action and evolve the system using the
CQC.

We find that there is rapid particle (gluon) production for which we are able
to characterize the energy spectrum and its time dependence as shown in Fig.
1. Production occurs for modes in a range $p\in[0,gEt]$ with peak production
at $p\approx gEt/2$. The amount of energy produced in an interval $dp$ grows
as $t^{3}$. We can understand this growth in terms of phase space factors that
give one power of $t$, and the square of the amplitude of vacuum fluctuations
that are proportional to $p^{2}$, and hence grow as $t^{2}$. The $t^{4}$
dependence we find is in sharp contrast to usual Schwinger pair production, as
in (1), for which the energy grows linearly with time. However, in contrast to
the usual Schwinger pair productions of massive particles, gauge excitations
are massless and this may be sufficient to explain the different production
rates.

To the order in which we perform our calculations, the coupling constant does
not run with energy scale. However, we encounter an ultraviolet divergence in
our calculation of the energy produced in excitations. We have argued that the
divergence would get controlled if we were to properly account for the running
of the coupling constant at high energies (asymptotic freedom). In calculating
the particle number density we also encounter an infrared divergence that we
argue will be absent in the confining theory. In other words, if we ignore
confinement, our analysis implies that the particle number density diverges,
which we interpret as an indication of a lack of self-consistency of the
unconfined assumption.

Our analysis leaves open several directions of interest such as the
backreaction of particle production on the background color electric field. If
we simply use energy conservation as a guide, the background electric field
has an energy density proportional to $E^{2}$ whereas the energy density in
excitations grows as $g^{4}E^{4}t^{4}$. Equating these two gives the decay
time for the electric field to be $\tau\sim(g\sqrt{E})^{-1}$. However, we
cannot exclude the possibility that the electric field will be antiscreened as
argued for non-Abelian gauge theories (see for example Sec. 16.7 of Peskin and
Schroeder (1995)), in which case the electric field strength would actually
increase with time. We hope to investigate this issue by a more detailed
analysis in the future.

###### Acknowledgements.

We thank Jan Ambjørn, Mainak Mukhopadhyay, Subodh Patil, Doug Singleton and
George Zahariade for feedback. This work was supported by the U.S. Department
of Energy, Office of High Energy Physics, under Award No. DE-SC0019470 and CG
by the National Science Foundation Award No. PHY-2012195 at Arizona State
University.

##  Appendix A Classical-Quantum Correspondence

The CQC is equivalent to the Bogoliubov transformation method for calculating
particle production in the case of a time dependent but spatially homogeneous
background. Once the model is discretized, each mode is equivalent to a simple
harmonic oscillator as in Sec. IV. So we only need to demonstrate the method
for a simple harmonic oscillator.

Consider the Hamiltonian for a simple harmonic oscillator of mass $m$ – which
can also be time dependent in the general case – with an arbitrary time
dependent frequency $\omega(t)$,

$H=\frac{p^{2}}{2m}+\frac{m}{2}\omega^{2}(t)x^{2}$ (65)

Then the quantum operators $x$ and $p$ can be written as

$x=z^{*}a_{0}+za_{0}^{\dagger},\ \
p=m({\dot{z}}^{*}a_{0}+{\dot{z}}a_{0}^{\dagger})$ (66)

where $a_{0}$ and $a_{0}^{\dagger}$ are the initial annihilation and creation
operators defined in terms of the initial position operator $x_{0}$ and
momentum operator $p_{0}$,

$a_{0}=\frac{p_{0}-im\omega_{0}x_{0}}{\sqrt{2m\omega_{0}}},\ \
a_{0}^{\dagger}=\frac{p_{0}+im\omega_{0}x_{0}}{\sqrt{2m\omega_{0}}}$ (67)

where $\omega_{0}$ is the frequency at the initial time, and $z(t)$ is a
complex-valued c-number function of time. Using the Heisenberg equations for
$x$ and $p$ we find that $z(t)$ must satisfy

${\ddot{z}}+\omega^{2}z=0$ (68)

The initial conditions follow from the above relations and are,

$z_{0}=\frac{-i}{\sqrt{2m\omega_{0}}},\ \
{\dot{z}}_{0}=\sqrt{\frac{\omega_{0}}{2m}}$ (69)

The Wronskian is constant and fixed by the initial conditions,

$m(z^{*}{\dot{z}}-z{\dot{z}}^{*})=i$ (70)

Other quantum operators can be re-written in terms of $z(t)$, $x_{0}$ and
$p_{0}$. Expectation values are written entirely in terms of $z(t)$. In
particular, the energy is

$\langle H\rangle=\frac{1}{2}m|{\dot{z}}|^{2}+\frac{1}{2}m\omega^{2}|z|^{2}$
(71)

This can also be written as

$\langle H\rangle=\frac{m}{2}|{\dot{z}}-i\omega
z|^{2}-\frac{i}{2}m\omega(z^{*}{\dot{z}}-z{\dot{z}}^{*})$ (72)

The second term is the Wronskian and hence is a constant determined by the
initial conditions. In the quantum simple harmonic oscillator this term
corresponds to the ground state energy $\omega/2$. So the energy in the
excitations is given by

$:{\cal E}:=\frac{m}{2}|{\dot{z}}-i\omega z|^{2}$ (73)

which is what we use in (51) and (52).

## References

  * Schwinger (1951) J. S. Schwinger, Phys. Rev. 82, 664 (1951).
  * Holstein (1999) B. R. Holstein, American Journal of Physics 67, 499 (1999), eprint https://doi.org/10.1119/1.19313, URL https://doi.org/10.1119/1.19313.
  * Dunne (2005) G. V. Dunne, From Fields to Strings: Circumnavigating Theoretical Physics p. 445–522 (2005), URL http://dx.doi.org/10.1142/9789812775344_0014.
  * Cohen and McGady (2008) T. D. Cohen and D. A. McGady, Phys. Rev. D 78, 036008 (2008), eprint 0807.1117.
  * Chu and Vachaspati (2010) Y.-Z. Chu and T. Vachaspati, Phys. Rev. D 81, 085020 (2010), eprint 1001.2559.
  * Gold et al. (2020) G. Gold, D. A. Mcgady, S. P. Patil, and V. Vardanyan (2020), eprint 2012.15824.
  * Yildiz and Cox (1980) A. Yildiz and P. H. Cox, Phys. Rev. D 21, 1095 (1980).
  * Ambjorn and Hughes (1982a) J. Ambjorn and R. J. Hughes, Nucl. Phys. B 197, 113 (1982a).
  * Ambjorn and Hughes (1982b) J. Ambjorn and R. J. Hughes, Phys. Lett. B 113, 305 (1982b).
  * Cooper and Nayak (2006) F. Cooper and G. C. Nayak, Phys. Rev. D 73, 065005 (2006), eprint hep-ph/0511053.
  * Nayak and van Nieuwenhuizen (2005) G. C. Nayak and P. van Nieuwenhuizen, Phys. Rev. D 71, 125001 (2005), eprint hep-ph/0504070.
  * Ragsdale and Singleton (2017) M. Ragsdale and D. Singleton, J. Phys. Conf. Ser. 883, 012014 (2017), eprint 1708.09753.
  * Cooper et al. (2008) F. Cooper, J. F. Dawson, and B. Mihaila, in _Conference on Nonequilibrium Phenomena in Cosmology and Particle Physics_ (2008), eprint 0806.1249.
  * Matinyan and Savvidy (1978) S. G. Matinyan and G. K. Savvidy, Nucl. Phys. B 134, 539 (1978).
  * Brown and Weisberger (1979) L. S. Brown and W. I. Weisberger, Nucl. Phys. B 157, 285 (1979), [Erratum: Nucl.Phys.B 172, 544 (1980)].
  * Nair and Yelnikov (2010) V. P. Nair and A. Yelnikov, Phys. Rev. D 82, 125005 (2010), eprint 1005.2582.
  * Brezin and Itzykson (1970) E. Brezin and C. Itzykson, Phys. Rev. D 2, 1191 (1970).
  * Kim and Page (2002) S. P. Kim and D. N. Page, Phys. Rev. D 65, 105002 (2002), eprint hep-th/0005078.
  * Bogolyubov (1958) N. N. Bogolyubov, Sov. Phys. JETP 7, 51 (1958).
  * Birrell and Davies (1984) N. D. Birrell and P. C. W. Davies, _Quantum Fields in Curved Space_ , Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, UK, 1984), ISBN 978-0-521-27858-4, 978-0-521-27858-4.
  * Ambjorn et al. (1983) J. Ambjorn, R. J. Hughes, and N. K. Nielsen, Annals Phys. 150, 92 (1983).
  * Vachaspati and Zahariade (2018) T. Vachaspati and G. Zahariade, Phys. Rev. D 98, 065002 (2018), eprint 1806.05196.
  * Vachaspati and Zahariade (2019) T. Vachaspati and G. Zahariade, JCAP 09, 015 (2019), eprint 1807.10282.
  * Bojowald and Ding (2021) M. Bojowald and D. Ding, JCAP 03, 083 (2021), eprint 2011.03018.
  * Akhmedov and Popov (2015) E. T. Akhmedov and F. K. Popov, JHEP 09, 085 (2015), eprint 1412.1554.
  * Akhmedov et al. (2014) E. T. Akhmedov, N. Astrakhantsev, and F. K. Popov, JHEP 09, 071 (2014), eprint 1405.5285.
  * Dabrowski and Dunne (2016) R. Dabrowski and G. V. Dunne, Phys. Rev. D 94, 065005 (2016), eprint 1606.00902.
  * Peskin and Schroeder (1995) M. E. Peskin and D. V. Schroeder, _An Introduction to quantum field theory_ (Addison-Wesley, Reading, USA, 1995), ISBN 978-0-201-50397-5.