Preprint INR-TH-2023-016

A New Solution to the Strong CP Problem

S.A. Larin

Institute for Nuclear Research of the Russian Academy of Sciences,

60-th October Anniversary Prospect 7a, Moscow 117312, Russia

Keywords: Quantum Chrom0dynamics, renormalizability, axions, CP violation,
charge conjugation C, space reflection P.

###### Abstract

We suggest a new solution to the strong CP problem. The solution is based on
the proper use of the boundary conditions for the QCD generating functional
integral. We expand the perturbative boundary conditions to both perturbative
and nonperturbative fields integrated in the QCD generating functional
integral. It allows to nullify the CP odd term in the QCD Lagrangian and thus
to solve the strong CP problem. The presently popular solution to the strong
CP problem with introducing axions violates the principle of renormalizability
of Quantum Field Theory, which is very successful phenomenologically. Our
solution obeys the principle of renormalizability of Quantum Field Theory and
does not involve new exotic particles like axions.

##  1 Introduction

The strong CP problem for a long time is considered as in fact an unsolved, or
at least not completely satisfactorily solved, outstanding problem of Quantum
Field Theory and Elementary Particle Physics. For an excellent review of the
subject see [1], where one can find various aspects of this problem. The most
popular presently solution [2], [3] to the strong CP problem introduces new
particles - axions [4],[5]. Axions became so popular that they are even
considered as real candidates for dark matter of the Universe. But presently
only restrictions on their possible properties are established in spite of the
numerous experimental efforts to discover such exotic particles, see e.g.
[6],[7],[8]. Besides, the axion solution of the strong CP problem violates the
fundamental principle of renormalizability of Quantum Field Theory. This basic
principle presently is one of the most phenomenologically successful
principles of Elementary Particles Theory. For example, it ensured in Quantum
Electrodynamics the agreement between the theory and the experiment for
anomalous magnetic moment of the electron within ten decimal points. This
impressive agreement convinces us that renormalizable Quantum Field Theory is
a correct physical theory. Therefore, in our opinion, it seems to be
interesting to find a solution to the strong CP problem which also obeys the
principle of renormalizability of Quantum Field Theory. Besides it is
desirable to find a solution which does not introduce new exotic particles
like axions. This is the goal of the present paper.

To find such a solution we will use in a proper way (a proper way in our
opinion) the boundary conditions in the generating functional of Green
functions of Quantum Chromodynamics. It is well established what kind of
boundary conditions are imposed on the fields of the theory in the functional
integral within perturbative approach. These are known boundary conditions
which produce the correct form of the perturbative propagators of the fields
of the Lagrangian in the considered theory. The derivation of the perturbative
propagators using the boundary conditions of the generating functional of
Green functions of Quantum Chromodynamics can be found in [9]. We will assume
that the same boundary conditions are valid for all fields of the theory which
are integrated in the functional integral. That is we will suppose that
perturbation theory calibrates the whole nonperturbative functional integral.

Our solution will obey the principle of renormalizability of Quantum Field
Theory and will not involve new exotic particles like axions.

The CP problem is the question of why the strong interaction does not violate
the charge-parity (CP) symmetry, which is the combination of charge
conjugation (C) and parity (P) symmetries. The CP symmetry states that the
laws of physics should be the same if a particle is replaced by its
antiparticle and its spatial coordinates are inverted (P). However, the weak
interaction is known to violate the CP symmetry, and there is no fundamental
reason why the strong interaction should not do the same.

The CP problem is also related to the origin of the matter-antimatter
asymmetry in the universe, which is another unsolved mystery in physics. The
CP violation involves scalar fields that couple to the quarks and induce a
complex phase in the quark mass matrix. This phase could affect the properties
of the neutron stars and black holes in X-ray binaries, such as their mass,
radius, magnetic field, and spin, see e.g. [10]. Another possible connection
is that some models of CP violation involve new particles that have spin1/2
and interact with the standard model particles via a new force [11]. These
particles could affect the X-ray spectrum or the gravitational waves emitted
by the system.

The CP violation in the early universe could have generated primordial
magnetic fields that were amplified by the collapse of stars into neutron
stars. These fields could then explain the existence of magnetars powered by
extremely strong magnetic fields, see e.g. [12]. However, this scenario is
highly speculative and requires more theoretical and observational support.

##  2 Materials and Methods

In the present work we will deal with the Quantum Chromodynamics (QCD)
generating functional of Green functions, which will be the basic object of
our considerations

$Z(J)=\int d\Phi~{}~{}exp\left(i\int
d^{4}x\left(L_{QCD}+J_{k}\cdot\Phi_{k}\right)\right),$ (1)

where $d\Phi$ denotes the integration measure of the functional integral
$Z(J)$ over all fields $\Phi_{k}$ of the theory, gluons and quarks. $J_{k}$
are sources of the fields. The symbol $J$ in $Z(J)$ denotes the full set of
sources $J_{k}$ of the fields.

Within perturbation theory the QCD Lagrangian $L_{QCD}$ is invariant, in
particular, under the combined symmetry transformations CP, where C is the
charge conjugation operator and P is the space reflection. More precisely the
QCD Lagrangian within perturbation theory is invariant under both the charge
conjugation C and the space reflection C.

The essence of the CP problem is that in full nonperturbative QCD one can add
to the QCD Lagrangian the CP odd gauge invariant term which seems to be not
forbidden from the first principles

$\Delta
L_{\theta}=\frac{\theta}{32\pi^{2}}G_{\mu\nu}^{a}\tilde{G}_{\mu\nu}^{a}.$ (2)

It is invariant under the charge conjugation C and is not invariant under the
space reflection P, hence it is also non invariant under the combined CP
transformation. But this term is forbidden by experiments with the rather high
precision as we will see it below.

The dual field strength tensor $\tilde{G}_{\mu\nu}^{a}$ in (2) is defined in
the standard way

$\tilde{G}_{\mu\nu}^{a}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}G_{\rho\sigma}^{a}.$
(3)

The $\theta$-term in (2) is purely nonperturbative since it is invisible in
perturbation theory because it can be rewritten as a total derivative

$\Delta L_{\theta}=\theta\partial_{\mu}K_{\mu}.$ (4)

Here $K_{\mu}$ is the known Chern-Simons current

$K_{\mu}=\frac{1}{16\pi^{2}}\epsilon_{\mu\nu\rho\sigma}\left(A_{\nu}^{a}\partial_{\rho}A_{\sigma}^{a}+\frac{1}{3}f^{abc}A_{\nu}^{a}A_{\rho}^{b}A_{\sigma}^{c}\right).$
(5)

The $\theta$-term can be discarded within perturbation theory. It can be
easily seen in the Euclidean space since the fields of the theory decrease in
the Euclidean space at the time infinities and the total derivative (4) does
not contribute to the QCD action. But with the discovery of instantons [13] it
was realized that the field configurations with the instanton boundary
conditions give nonzero nonperturbative contributions to the action. In
particular, the one instanton contribution looks like

$\Delta S_{\theta}=\int d^{4}x\Delta L_{\theta}=\theta.$ (6)

The key notion here is the famous topological charge which has the following
form

$\mathcal{V}=\int d^{4}x\partial_{\mu}K_{\mu}=\int
d^{3}xK_{0}(\vec{x},t)|^{t=+\infty}_{t=-\infty}=\mathcal{K}(t\rightarrow+\infty)-\mathcal{K}(t\rightarrow-\infty),$
(7)

where $\mathcal{K}$ is the Pontryagin number. The topological charge is zero
for perturbative fields, i.e. in perturbation theory. But instanton fields,
for example in the $A_{0}=0$ gauge, interpolate between the zero gluon fields
$A_{i}(\vec{x},t=-\infty)=0,~{}~{}i=1,2,3$ and the nonzero gluon fields
$A_{i}(\vec{x},t=+\infty)=U^{+}\partial_{i}U,~{}~{}i=1,2,3$. Here the matrix
$U$ is the Polyakov hedgehog

$U(\vec{x})=exp\left(-\frac{i\pi\vec{x}\cdot\vec{\sigma}}{\sqrt{\vec{x}^{2}+\rho^{2}}}\right).$
(8)

For this instanton configuration one has that the Pontryagin number and
correspondingly the topological charge are equal to unity

$\mathcal{V}=\mathcal{K}(t=+\infty)=1,$ (9)

Thus the $\theta$-term gives the nonzero nonperturbative contribution to the
QCD action.

In the full QCD, with quarks, there is also contributions to the CP odd part
of the QCD Lagrangian from the imaginary phases of the quark mass matrix. The
phases can be rotated away by the chiral transformations of quark fields. But
there is the famous axial anomaly [14],[15]. It generates non invariance of
the measure of the Feynman functional integral under chiral transformations
[16]. Therefore the phases of the quark mass matrix arise before the
$G\tilde{G}$ term in the Lagrangian. Hence the parameter which determines the
value of the CP violation is in fact

$\theta+arg(det\mathcal{M}),$ (10)

where $\mathcal{M}$ is the quark mass matrix.

Below we shall use the same symbol $\theta$ for this parameter to simplify
notations assuming that it already includes the effects of the quark mass
matrix.

Probably the most essential phenomenological effect of the $\theta$-term is a
nonzero electric dipole moment of the neutron $d_{n}$. The electric dipole
moment is given by the effective interaction Lagrangian

$L_{nEDM}=\frac{d_{n}}{2}\bar{n}i\gamma_{5}\sigma_{\mu\nu}nF^{\mu\nu},$ (11)

where $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the photon
field strength tensor, $n$ stands for the neutron field ,
$\sigma_{\mu\nu}=\frac{1}{21}[\gamma_{mu}\gamma_{nu}]$ is the standard
antisymmetric product of Dirac gamma matrices.

The $\theta$-term generates the following electric dipole moment of the
neutron $d_{n}$

$<n(p_{f})\gamma(k)|eJ_{\mu}^{em}A^{\mu}i\int d^{4}x\Delta
L_{\theta}|n(p_{i})>=d_{n}\bar{n}(p_{f})\gamma_{5}\sigma_{\mu\nu}n(p_{i})k^{\mu}\epsilon^{\mu}(k).$
(12)

Here $J_{\mu}$ is the quark electromagnetic current.
$k^{\mu}=p_{f}^{\mu}-p_{i}^{\mu}$, where $p_{i}^{\mu}$ is the incoming
momentum of the neutron and $p_{f}^{\mu}$ is the outgoing momentum of the
neutron. $\epsilon^{\mu}(k)$ is the photon polarization.

The matrix element in the left hand side of eq.(12) is zero in perturbative
theory and is calculated within purely nonperturbative QCD. There are several
nonperturbative methods by which the electric dipole moment of the neutron
$d_{n}$ was estimated, for the detailed overview see [1] and references
therein. Here we give the short summary of the results of the corresponding
nonperturbative approaches. The performed bag model calculations produced the
following result $d_{n}\approx\theta 2.7\cdot 10^{-16}$ e$\cdot$cm. Shortly
after this result the chiral logarithms approach was used to obtain the
following estimate $d_{n}\approx\theta 5.2\cdot 10^{-16}$e$\cdot$cm. The
approach of chiral perturbation theory was further developed to produce the
slightly less result $d_{n}\approx\theta 3.3\cdot 10^{-16}$e$\cdot$cm. At last
but not at least the calculations based on the QCD sum rules method gave the
following again slightly less estimate $d_{n}\approx\theta 1.2\cdot
10^{-16}$e$\cdot$cm. All these results have considerable uncertainties of the
order of $50$ per cent because of essential difficulties of nonperturbative
QCD calculations.

But anyway the average theoretical value for $d_{n}$ can be confidently
estimated within the same of the order of $50$ per cent uncertainty as

$d_{n,theor}\approx\theta\cdot 10^{-16}$e$\cdot$cm.

This should be compared with the most recent experimental value [17] for the
electric dipole moment of the neutron $d_{n}$ which is

$d_{n}=(0.0\pm 1.1)\times 10^{-26}$e$\cdot$cm.

Thus one gets an extremely strong restriction on the value of the $\theta$
coupling

$|\theta|\leq 10^{-10}.$ (13)

The explanation of this practically zero value of the coupling $\theta$ is the
essence of a solution to the strong CP problem.

The presently popular solution to the problem is the famous axion solution. It
assumes the addition to the QCD Lagrangian the term with the new axion field
$a(x)$, which in fact reduces to the shift of the coupling $\theta$ in the QCD
Lagrangian $\theta\rightarrow a(x)/f_{a}+\theta$. So the corresponding term
$\Delta L_{\theta}$ of eq.(2) in the QCD Lagrangian becomes as follows

$\Delta
L_{\theta}\rightarrow\left(\frac{a(x)}{f_{a}}+\theta\right)\frac{1}{32\pi^{2}}G_{\mu\nu}^{b}\tilde{G}_{\mu\nu}^{b}.$
(14)

After the spontaneous symmetry breaking of the global Peccei Quinn symmetry
[2],[3] one calculates the effective potential for the axion field $a(x)$.
Then one finds that when the axion rests at the minimum of this potential, the
CP violating term (14) nullifies. This is the known axion solution to the
strong CP problem.

There are different types of axions suggested in the literature. Let us
consider some of them. As the scalar Higgs fields produce vacuum expectation
values the electroweak local symmetry group is broken spontaneously. This
develops masses of the gauge W and Z intermediate vector bosons. At the same
time the global Peccei Quinn U(1) group is also spontaneously broken.This
spontaneous breaking of the U(1) global symmetry leads to the appearance of
the massless Goldstone boson, which is called the Weinberg-Wilczek (WW) axion
in this case [4],[5]. In the Standard Model including two Higgs doublets this
axion is presented as the following superposition

$a=1/v(v_{\phi}Im\phi_{0}-v_{\chi}Im\chi_{0})$,

here $\phi_{0}$ and $\chi_{0}$ are neutral components of the two Higgs
doublets. Besides, $v=\sqrt{v_{\phi}^{2}+v_{\chi}^{2}}\approx$ 250 GeV, where
$v_{\phi}$ and $v_{\chi}$ denote vacuum expectation values of the fields
$\phi$ and $\chi$ correspondingly. Within the considered approximation the WW
axion is massless. But as it was mentioned above the nonperturbative effects
of Quantum Chromodynamics (for example instantons) can generate the potential
for the WW axion. In this way the WW axion obtains the nonzero mass value
which is estimated according to [4],[5] as following

$m_{a}\approx f_{\pi}m_{\pi}/v\approx$ 100 KeV.

Besides, the decay constant of the WW axion is $1/v$. Hence it is clear that
the mass and the decay constant of the WW axion is connected to the breaking
scale $v$ of electroweak symmetry. This constraint turns out to be too strong
and correspondingly the WW axion turns out to be excluded by the existing
experimental data.

If the scale of the breaking of the Peccei Quinn symmetry is much more than
the electroweak scale $v$, then according to the above formula the axion is
essentially lighter and the decay constant of the axion is much smaller. This
type of the light axion could be in agreement with the existing experimental
data.

A solution with the light ’invisible’ axion was first suggested in [18], [19]
(the so called KSVZ axion). In the Ref. [19] this type of the axion is named
the phantom axion. It should be underlined that in order to uncouple the
’phantom’ from the electroweak scale it is necessary to decouple the proper
scalar fields from the standard quarks and couple these fields to very heavy
hypothetical fermions carrying color.

To be more precise one should introduce a complex scalar $\Phi$ which is
coupled to the hypothetical quark field $Q$, the electroweak singlet in the
fundamental representation of the SU(3) color group.

Then the modulus of the scalar $\Phi$ is supposed to produce the large vacuum
expectation value $f/\sqrt{2}$ and the argument of the field $\Phi$ is just
the axion field $a$ up to normalization:

$a(x)=f\alpha(x),~{}~{}~{}\alpha(x)=Arg\Phi(x),~{}~{}~{}~{}f>>\Lambda.$

Further the low energy coupling of this axion to the gluons is as follows

$\Delta
L=\frac{1}{f}a\frac{1}{32\pi^{2}}G_{\mu\nu}^{a}\tilde{G}_{\mu\nu}^{a}.$

In this way the Lagrangian of QCD depends on the expression
$\theta+\alpha(x)$.

More generally, it is possible to introduce more than one quark field $Q$ or
to introduce these fields in a higher representation of the SU(3) color group.
In this case the coupling of the axion to gluons obtains an integer number
$N$:

$\Delta
L=\frac{1}{f}aN\frac{1}{32\pi^{2}}G_{\mu\nu}^{a}\tilde{G}_{\mu\nu}^{a}.$

This multiplier $N$ (should not be confused with the color number $N_{c}$) is
usually called as the axion index. Hence, in general, the Lagrangian of
Quantum Chromodynamics depends on the sum $\theta+N\alpha(x)$. It can be
assumed that the nonperturbative QCD effects produce the potential for
$\theta+N\alpha(x)$. The latter sum is minimized at the point
$\theta+N\alpha_{vac}=0$. Hence the strong CP problem is solved in this way.

One more way to produce an ’invisible’ axion is suggested in [20],[21] (the so
called ZDFS axion). In this case one keeps the Peccei Quinn symmetry of the
two doublet Standard Model but splits the scales of the Peccei Quinn and
electroweak breaking. For this purpose the Lagrangian of the Standard Model is
extended, one adds the scalar Standard Model singlet field $\Sigma$

Then one note that this Lagrangian is in variant under the axial
transformations

$q_{L}\rightarrow e^{i\alpha}q_{L},~{}~{}q_{R}\rightarrow
e^{i\alpha}q_{R},~{}~{}\phi\rightarrow e^{2i\alpha}\phi,~{}~{}\chi\rightarrow
e^{-2i\alpha}\chi,~{}~{}\Sigma\rightarrow e^{2i\alpha}\Sigma.$

After spontaneous breaking of this axial symmetry the Goldstone particle, the
axion, appears as the superposition

$a=\frac{1}{V}(v_{\phi}Im\phi_{0}-v_{\chi}Im\chi_{0}+v_{\Sigma}Im\Sigma.$

Here $V=\sqrt{v_{\phi}^{2}+v_{\chi}^{2}+v_{\Sigma}^{2}}$,
$v_{\phi},~{}v_{chi}$ and $v_{\Sigma}$ represent vacuum expectation values of
the fields $\phi,~{}\chi$ and $\Sigma$ correspondingly. The vacuum expectation
value of the field $\Sigma$ is not necessarily connected to the scale of the
electroweak symmetry breaking. Actually it can be chosen as large as the Grand
Unification scale. In this case the considered axion is very light and its
decay constant is small.

But first of all, the term with the axion field $a(x)$ in (14) has the
dynamical dimension (that is the dimensions of the fields plus the dimensions
of the derivatives of the fields) five instead of four necessary for
renormalizability. Hence the term with the axion field $a(x)$ violates
renormalizability of the Lagrangian. As we already have mentioned in the
introduction, renormalizability of the Lagrangian is a rather important
principle of Quantum Field Theory. This principle turned out to be very
successful phenomenologically as it is demonstrated for example by the famous
case of the anomalous magnetic moment of the electron. Therefore it is quite
important to preserve the principle of renormalizability when solving the
strong CP problem. Secondly, the axion is not found experimentally in spite of
numerous experimental attempts, as we also have underlined in the
introduction.

Therefore we find it necessary to suggest a new solution to the strong CP
problem which preserves renormalizability of the theory and does not involve
new exotic particles like axions.

Let us now again consider the QCD generating functional (1). It is well known
that this integral is not defined yet completely if only the QCD Lagrangian is
defined with the corresponding gauge condition. At least within perturbation
theory one should impose on the Lagrangian fields the proper boundary
conditions.

In perturbation theory one has the well known boundary conditions. For example
for the gluon fields one has the following conditions

$A_{\mu}^{a}(\vec{x},t\to-\infty)\to A_{\mu,in}^{a}(x),$ (15)
$A_{\mu}^{a}(\vec{x},t\to+\infty)\to A_{\mu}^{a,out}(x).$

Here the incoming asymptotic gluon fields $A_{\mu,in}^{a}(x)$ contain only the
positive frequency part and the outgoing gluon fields $A_{\mu,}^{a,out}(x)$
contain only the negative frequency part:

$A_{\mu,in}^{a}(x)=\frac{1}{(2\pi)^{3/2}}\int
d^{3}k~{}e^{i(\vec{k}\vec{x}-\omega t)}v_{\mu}^{i}(k)a_{i}(k)/\sqrt{2w},\\\ $
(16) $A_{\mu,}^{a,out}(x)=\frac{1}{(2\pi)^{3/2}}\int
d^{3}k~{}e^{i(\vec{k}\vec{x}+\omega t)}v_{\mu}^{i}(k)a^{*}_{i}(k)/\sqrt{2w},$

where $\omega=\sqrt{\vec{k}^{2}}$ and $v_{\mu}^{i}(k)$ are polarization
vectors of the gluons. Here the sums over the gluon polarizations $i=1,2$ are
assumed.

These are the known Feynman boundary conditions. They are necessary to obtain
the correct form of the perturbative propagators of the fields of the type
$1/(k^{2}+i\epsilon$) with the correct plus $i\epsilon$ prescription. Thus
with this boundary conditions the gluon fields (the quark fields also)
oscillate at the time infinities. After transition to the Euclidean space by
means of the Wick rotation $t\rightarrow ix_{4}$ the fields decrease at the
time infinities, so in the Euclidean space it is easy to see that in
perturbation theory total derivatives in the Lagrangian are zero.

Hence one can write perturbative boundary conditions (15) for all fields
$\Phi_{i}$ of the QCD Lagrangian symbolically as follows

$\Phi(t\to\pm\infty)\to\Phi_{in}^{out}(x).$ (17)

This helpful notation will be used below to formulate the QCD generating
functional integral as a compact formula.

##  3 Results

Let us now exactly formulate our solution to the strong CP problem. As it is
was already underlined above the solution is based on the proper use of the
boundary conditions for all Lagrangian fields in the QCD generating functional
integral. So let us now again consider the QCD generating functional integral
(1.

In our opinion, it is necessary and natural to generalize the boundary
conditions (17) for perturbative fields to all fields $\Phi_{i}$ of the QCD
Lagrangian which are integrated in the generating functional integral (1).
Then all Lagrangian fields will decrease in the Euclidean space at the time
infinities. Hence such a definition of the boundary conditions nullifies all
total derivatives in the Lagrangian both for the perturbative contributions
and the nonperturbative contributions. Thus the CP odd term in the QCD
Lagrangian will be nullified and it solves the strong CP problem. Besides this
definition allows to formulate exactly the complete (the perturbative part
plus the nonperturbative part) QCD generating functional integral as one
compact mathematical formula:

$Z(J)=\int_{\Phi(t\to\pm\infty)\to\Phi_{in}^{out}}d\Phi~{}exp\left(i\int
d^{4}x\left(L_{QCD}+J_{k}\cdot\Phi_{k}\right)\right).$ (18)

##  4 Discussions

The strong CP problem for a long time is in fact considered as the still
unsolved (or not completely adequately solved) prominent problem of Quantum
Field Theory and Elementary Particle Physics. For an excellent review of the
CP problem and related topics see the fist reference of the present paper,
where one can discover different aspects of the subject. The presently most
popular axion solution [2], [3] to the strong CP problem introduces new exotic
elementary particles - axions. Axions are now so popular that they are
presently considered as the real candidates for Dark Matter of the Universe. A
lot of advanced experiments are performed to discover some sorts of these
axions. But presently only some kinds of restrictions on their possible
properties are obtained in spite of the numerous huge experimental efforts to
find such exotic elementary particles, see for example [6],[7],[8]. Besides,
the axion solution of the strong CP problem is in contradiction with the
fundamental, in our opinion, principle of renormalizability of Quantum Field
Theory. This basic, in our opinion, principle presently is one of the most
experimentally successful principles of Quantum Field Theory and Elementary
Particles Physics. For example, this principle produced for the famous
anomalous magnetic moment of the electron within renormalized Quantum
Electrodynamics the outstanding agreement between the theoretical value and
the experimental value within ten decimal points. This prominent agreement
between the theory and the experiment convinces us that renormalizable Quantum
Electrodynamics and more generally renormalizable Quantum Field Theory are the
proper physical theories.

Therefore, in our opinion, it seems to be important to have a solution to the
strong CP problem which is in agreement with the principle of
renormalizability of Quantum Field Theory. Besides we suppose that it is
desirable to find a solution to the strong CP problem which does not involve
new exotic elementary particles like axions or something similar. Therefor the
goal of the present paper is to find the solution which satisfies these two
conditions (renormalizability and the absence of new exotic particles).

To find such a solution we have used in this paper in a proper way the
boundary conditions for the Lagrangian fields in the generating functional of
the Green functions of Quantum Chromodynamics (a proper way in our opinion).
It is well understood what kind of boundary conditions should be imposed on
the fields of the theory in the generating functional integral within
perturbation approach. These are the known boundary conditions which generate
the necessary form of the perturbative propagators of the fields of the
Lagrangian of Quantum Chromodynamics. These boundary conditions produce the
correct ’$+i\epsilon$’ prescription for the perturbative propagators of the
fields. We have suggested that the same boundary conditions should be valid
for all Lagrangian fields integrated in the QCD generating functional
integral, i.e. for both perturbative and nonperturbative contributions. That
is we have supposed that perturbation theory calibrates the whole
nonperturbative functional integral, i.e. it calibrates both the perturbative
and the nonperturbative parts of the generating functional integral.

Our solution satisfies two important criteria. The solution does obey the
principle of renormalizability of Quantum Field Theory and does not involve
new exotic particles like axions.

Let us make now necessary here remarks concerning the famous $U(1)$ problem.
The essence of this problem is that the mass of the flavour singlet pseudo-
scalar $\eta^{\prime}$ meson $m_{\eta^{\prime}}\approx 958MeV$ is surprisingly
heavier than the masses of the flavour octet pseudo-scalar mesons. One can
argue that it is not possible to extend the discussed above perturbatve
boundary conditions for the fields, which are well established within
perturbation theory, to all fields which are integrated in the QCD generating
functional integral. Such an extension excludes from the generating functional
integral for example the instanton contributions not obeying the perturbative
boundary conditions. In particular there is the well known statement [22],[23]
that instantons solve the $U(1)$ problem. Hence it seems that they should not
be excluded from the theory. But one can note that there is also the well
known solution [24],[25] to the $U(1)$ problem using the axial anomaly which
was suggested before the discovery of instantons. Therefore one can argue that
the $U(1)$ problem can be solved without involving the instantons. Thus the
extension of the perturbative boundary conditions to all fields of the
generating functional integral is well allowed.

##  5 Conclusions

We have suggested a new solution to the strong CP problem. To find such a
solution we use in a new way the boundary conditions in the Quantum
Chromodynamics generating functional of Green functions. It is well
established what kind of boundary conditions are imposed on the fields of the
QCD Lagrangian in the functional integral within perturbation theory. We
assume that the same boundary conditions are valid for all Lagrangian fields
of the functional integral, i.e. for both perturbative and nonperturbative
fields. This allows to nullify total derivatives in the QCD action, in
particular the CP odd term which can be presented as the total derivative.
Hence it solves the strong CP problem. Thus we suppose that perturbation
theory calibrate the complete nonperturbative functional integral.

Maybe it is worthwhile also to mention that our solution does not violate any
symmetries of Quantum Chromodynamics. We would like to underline once more
that our solution to the strong CP problem obeys the principle of
renormalizability of Quantum Field Theory and does not involve new exotic
particles like axions.

##  6 Acknowledgments

The author is grateful to the collaborators of the Theory Division of the
Institute for Nuclear Research for valuable discussions.

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# CLIP-Embed-KD: Computationally Efficient Knowledge Distillation Using
Embeddings as Teachers

Lakshmi Nair
Georgia Institute of Technology
Atlanta, GA, USA

###### Abstract

Contrastive Language-Image Pre-training (CLIP) has been shown to improve zero-
shot generalization capabilities of language and vision models. In this paper,
we extend CLIP for efficient knowledge distillation, by utilizing embeddings
as teachers. Typical knowledge distillation frameworks require running forward
passes through a teacher model, which is often prohibitive in the case of
billion or trillion parameter teachers. In these cases, using only the
embeddings of the teacher models to guide the distillation can yield
significant computational savings. Our preliminary findings show that CLIP-
based knowledge distillation with embeddings can outperform full scale
knowledge distillation using $9\times$ less memory and $8\times$ less training
time.

##  1 Introduction

---
(a) CLIP-Teacher-KD

---
(b) CLIP-Embed-KD

Figure 1: CLIP-Teacher-KD uses the teacher model to generate embeddings for
every training sample. CLIP-Embed-KD uses pre-computed teacher embeddings thus
avoiding the need to run forward passes through the teacher model for every
sample.

Contrastive Language-Image Pre-training (CLIP) [8], introduced in 2021, uses
natural language supervision to efficiently learn visual concepts. CLIP gained
significant popularity owing to its improved zero-shot transfer robustness
compared to regularly pre-trained ImageNet models. CLIP involves pre-training
an image encoder and text encoder to predict image-text pairings within a
dataset. At its core, CLIP uses a constrastive objective function that first
computes the scaled pair-wise cosine similarity between image embeddings (of
an image encoder) and text embeddings (of a text encoder) to generate output
logits. During training, the logits are compared to labels that match input
images to their corresponding ground truth text embeddings, using a cross
entropy loss. Aligning the image and text modalities in this way enables CLIP
models to achieve robust zero-shot performances across classification tasks.
Originally applied to visual classification tasks on images, several follow up
papers have since successfully extended the application of CLIP to other data
modalities such as videos [10], point clouds [12], and multimodal contexts
[1].

In this brief paper, we investigate the application of CLIP to knowledge
distillation (KD). Knowledge distillation is the process of transferring (or
distilling) the knowledge of a larger teacher model into a smaller, more
compressed, student model [6]. This is achieved by comparing the outputs of
the teacher and student models using an appropriate loss function (called
distillation loss). Existing knowledge distillation approaches can be
categorized as response-based, feature-based, or instance-based [5]. Response-
based KD compares outputs of the final classification layer (i.e., logits) of
the teacher and student models for each training sample. Feature-based KD
compares the outputs of the final and intermediate layers (i.e., feature maps)
of the student and teacher models. Lastly, instance-based KD compares the
similarity between intermediate feature maps of pairs of layers in the teacher
and student models. However, all distillation approaches require performing
several forward passes through both the teacher and student models for
comparing the corresponding outputs. This can be prohibitive when the teacher
models are billion parameters in size as with recent models [2], and we wish
to run the KD training on limited computational resources (e.g., mobile
devices).

Intuitively, aligning the teacher and student feature maps can be likened to
the alignment of text and image embeddings with CLIP. Prior work on knowledge
distillation with CLIP uses CLIP-trained vision and text encoder models, with
token selection and confidence weighting applied to a distillation loss that
is formulated as the L1 difference between the base model tokens and teacher
tokens [9]. Other approaches have applied knowledge distillation on models
that are pre-trained with CLIP, for performing video-language retrieval task
using KL-divergence loss [7]. In [11], the authors compare CLIP-based
knowledge distillation approaches through different loss functions including
mean-squared error (MSE) or KL-divergence. In contrast to existing methods, we
explore the question: Can pre-computed embeddings obtained from the teacher
model be used to train the student model in knowledge distillation? We
investigate two methods in this context.

  * •

CLIP-Teacher-KD: We investigate the direct application of CLIP’s contrastive
pre-training objective in the distillation loss of a teacher and student,
along with the student’s typical training loss (similar to feature-based KD).

  * •

CLIP-Embed-KD: We extend the above to investigate if pre-computed teacher
embeddings can be used in place of the teacher model in order to effectively
align the student. This eliminates the need to run repeated forward passes of
the teacher, except for the initial pre-computation step.

Our goal with this preliminary work is not to present state-of-the-art KD
results, rather to present initial findings on the potential of extending CLIP
with teacher embeddings for more computationally efficient knowledge
distillation.

##  2 Approach

###  2.1 Knowledge distillation

Knowledge distillation (KD) involves three components: a) teacher model which
is typically a large model exhibiting the behavior that we would like to
distill into a smaller model; b) the smaller student model; c) a distillation
loss that captures the difference between the teacher and student outputs with
the goal of minimizing their differences:

$\displaystyle\mathcal{L_{D}}(z_{s},z_{t},\hat{y})=\alpha_{1}\mathcal{L_{CE}}(z_{s},\hat{y})+\alpha_{2}\mathcal{L_{KD}}(z_{s},z_{t})$
(1)

Here, $\mathcal{L_{D}}$ is the overall distillation loss over the output
logits of the student, $z_{s}$ and the output logits of the teacher, $z_{t}$.
The term $\mathcal{L_{CE}}$ is the typical cross-entropy supervision loss of
the student logits and ground truth labels $\hat{y}$. Additionally,
$\mathcal{L_{KD}}$ represents the distillation term that captures the
difference between teacher and student logits, and is commonly the KL-
divergence between $z_{s}$ and $z_{t}$. The scalars $\alpha_{1}$, $\alpha_{2}$
represent a weighting of the two losses. Minimizing $\mathcal{L_{D}}$ aligns
the student and teacher model behaviors while minimizing the student loss with
respect to the ground truth $\hat{y}$. Follow up research has proposed several
variations on the loss functions [5]. However, in all cases of KD, even if the
teacher parameters are frozen (i.e., no backward pass is needed), computing
the logits $z_{t}$ for the loss function requires forward passes through the
teacher model which is often much larger than the student model [5]. Thus, a
significant proportion of the computational resources in KD are allocated
towards the operation of the teacher.

Our preliminary work is motivated to address this issue and we present our
approach in two parts: CLIP-Teacher-KD evaluates the performance of using
CLIP’s pre-training objective within distillation, but requires running
repeated forward passes through the teacher model. We subsequently adapt this
to CLIP-Embed-KD which uses pre-computed embeddings in place of the full
teacher model.

###  2.2 CLIP-Teacher-KD

In CLIP-Teacher-KD, we first pass the input training data through the student
and teacher models and extract the output embeddings for the [CLS] token. The
[CLS] token (often used in models like BERT [3] and ViT [4]) represents a
summarized feature map which is extracted and used for the final
classification. The [CLS] teacher embeddings are projected through a learnable
projection layer into the embedding dimensions of the student model. We
further normalize the resultant embeddings and compute the dot product of the
normalized teacher and student embeddings. We pass the resultant dot product
with ground truth labels into a cross entropy loss ($\mathcal{L}_{clip}$). We
use an identity matrix as the ground truth (since teacher embedding $i$ should
correspond to student embedding $i$). We also generate the final output logits
of the student model through its classifier, which consumes the [CLS] token
embeddings and maps them to class probabilities using softmax. The
distillation loss ($\mathcal{L_{D}}$) is a combination of the cross entropy
loss of the student logits and ground truth labels, and $\mathcal{L}_{clip}$:

$\displaystyle E_{s}=f_{student}(x);E_{t}=f_{teacher}(x);$ (2)
$\displaystyle\hat{E}_{t}=E_{t}\cdot W^{T}_{proj}$ (3)
$\displaystyle\mathcal{L}_{clip}=\mathcal{L_{CE}}(\lVert
E_{s}\rVert\cdot\lVert\hat{E}_{t}\rVert^{T},G)$ (4)
$\displaystyle\mathcal{L_{D}}(E_{s},\hat{E}_{t},z_{s},\hat{y})=\alpha_{1}\mathcal{L_{CE}}(z_{s},\hat{y})+\alpha_{2}\mathcal{L}_{clip}(E_{s},\hat{E}_{t})$
(5)

Here, $E_{s}\in\mathbb{R}^{B\times D_{s}}$ and $E_{t}\in\mathbb{R}^{B\times
D_{t}}$ denote student and teacher embeddings of the [CLS] token,
respectively. $B$ denotes the batch size and $G$ is the identity matrix
$\mathbb{I}^{B\times B}$. The learnable projection
$W_{proj}\in\mathbb{R}^{D_{s}\times D_{t}}$ projects teacher embeddings into
the student’s embedding dimension. We use $\alpha_{1}=0.5$ and
$\alpha_{2}=(1-\alpha_{1})=0.5$ in our experiments.

###  2.3 CLIP-Embed-KD

In this approach, we adapt CLIP-Teacher-KD to use teacher embeddings in place
of the full teacher model, to improve the computational efficiency of
knowledge distillation by avoiding the need to run repeated forward passes
through the teacher model for each incoming training sample. We begin by
randomly sampling $N$ samples of data for each class $c$ in the dataset. We
obtain teacher embeddings of the [CLS] tokens for the collected data and
compute a cumulative representation of each class embedding, by simply
averaging the embeddings of each class along the embedding dimension. This
gives an “averaged” embedding for each class representing the teacher
embeddings $E_{t}$. In our future work, we seek to explore alternate ways of
computing more faithful representations of these teacher embeddings.

The averaged embeddings are $E_{t}\in\mathbb{R}^{N_{c}\times D_{t}}$, where
$N_{c}$ is number of classes. As with CLIP-Teacher-KD, we project the averaged
embeddings using a learnable projection layer to the embedding dimensions of
the student. When computing $\mathcal{L}_{clip}$ with the averaged embeddings,
the ground truth is a $B\times N_{c}$ matrix, representing a one-hot encoding
of the labels for each sample in the batch. That is, each row of the matrix is
a one-hot encoding of the label for the corresponding sample. The dot product
of $E_{s}$ and the transposed averaged teacher embeddings result in an output
of dimensions $B\times N_{c}$. The output is passed into a cross entropy loss
function with the ground truth matrix to compute $\mathcal{L}_{clip}$. The
remainder of the loss function is same as in Equation (5). We make our code
publicly available at: https://github.com/lnairGT/CLIP-Distillation/.

Figure 2: Training and validation curves of base student model. Figure 3:
Training and validation curves of large student model.

##  3 Experiments and Results

Student | Layers | Embed dim | Heads | MLP
---|---|---|---|---
Base | 6 | 256 | 8 | 1024
Large | 10 | 512 | 8 | 2048
Teacher | Layers | Embed dim | Heads | MLP
Base-16/32 | 12 | 768 | 12 | 3072
Large-16/32 | 24 | 1024 | 16 | 4096
Table 1: Architecture specification of the models.

In this section, we evaluate CLIP-Teacher-KD and CLIP-Embed-KD on image
classification with CIFAR100. Our goal is not to achieve state-of-the-art
performances, rather to investigate the value of using CLIP loss in knowledge
distillation through the following research questions:

a) How does using the CLIP distillation loss from Equation (5) compare to the
regular KD loss in Equation (1)?

b) How does CLIP-Embed-KD compare to CLIP-Teacher-KD in terms of computational
efficiency and accuracy?

We use Vision Transformers (ViT) [4] and the teacher architectures (Table 1)
use ImageNet pre-trained checkpoints available on HuggingFace, with variants
of google/vit-base-patch16-224-in21k. We use two teacher and student sizes:
base and large. The teacher models use patch sizes $16$, $32$ and student
models use patch size $4$. We use batch size of $64$, and image sizes
$32\times 32$ and $64\times 64$. The teacher models always use an image size
of $224\times 224$ to generate their outputs (logits or embeddings). For CLIP-
Embed-KD we compute the averaged embeddings over 100 samples per class. We
train all student models for 200 epochs with a learning rate of 0.0001, cosine
annealing and weight decay of 0.1.

Teacher | CLIP-Embed-KD | CLIP-Teacher-KD
---|---|---
Base-16 | 49.32 | 51.68
Large-16 | 49.67 | 51.92
Base-32 | 49.34 ($17\times\downarrow$ mem) | 52.61
Large-32 |  50.33 ($59\times\downarrow$ mem) | 51.95
Table 2: Performance on base student with image $32\times 32$. Student | Teacher | Image sz | CLIP-Embed-KD
---|---|---|---
Base |  | 64 | 53.28
Large | Large-16 | 32 | 52.92
Large |  | 64 | 54.36
Base |  | 64 | 53.40
Large | Large-32 | 32 | 52.77
Large |  | 64 | 54.92
Table 3: Scaling CLIP-Embed-KD to larger models and images. Figure 4: Training
resource utilization (with Large-32 teacher) of CLIP-Embed-KD, CLIP-Teacher-KD
[student-size][image-size]: CLIP-Embed-KD scales well (to larger models and
larger images) to outperform CLIP-Teacher-KD for much less memory. Figure 5:
Accuracy vs. $\alpha_{2}$ ($\alpha_{1}=1-\alpha_{2}$). Non-zero weighting of
both losses yield the best performance ($\alpha_{2}=1$ causes overfitting).

###  3.1 CLIP distillation loss vs. regular KD loss

Figures 2 and 3 compare the training loss and validation accuracy curves of
regular KD that uses equation (1), with CLIP-Teacher-KD and CLIP-Embed-KD on
small and large student models with a fixed ViT-base teacher of patch size 16.
Here, we see that using CLIP’s contrastive objective outperforms the regular
KD loss function from equation (1). CLIP-Teacher-KD slightly outperforms CLIP-
Embed-KD. CLIP-Teacher-KD and CLIP-Embed-KD are much closer in performance for
the larger student model (Figure 3). We further analyze the differences
between the two approaches.

###  3.2 CLIP-Teacher-KD vs. CLIP-Embed-KD

Table 2 compares the accuracy of CLIP-Teacher-KD vs. CLIP-Embed-KD for
different teacher sizes and patch sizes, for the base student model. We note
that for CLIP-Embed-KD, final accuracy of the student seems to have a small
improvement when using the large teacher models over the base ones. This
pattern is not explicit with CLIP-Teacher-KD where the base teacher model has
a slightly improved student accuracy compared to the large teacher at patch
size 32. Since CLIP-Embed-KD uses pre-computed teacher embeddings to train the
student, larger teacher sizes potentially contribute to improved quality of
the average embeddings.

CLIP-Embed-KD is computationally more resource efficient than CLIP-Teacher-KD,
and can scale better to outperform CLIP-Teacher-KD. In Table 2, CLIP-Embed-KD
achieves roughly about $\approx$2% lesser accuracy than CLIP-Teacher-KD, since
the averaged embeddings result in some loss of information per sample compared
to CLIP-Teacher-KD. However, CLIP-Teacher-KD uses more memory as teacher size
grows, whereas, CLIP-Embed-KD uses fixed memory for embeddings alone, leading
to better scaling behavior ($17\times$, $59\times$ less memory used). In
Figure 4, we see that CLIP-Embed-KD can get higher accuracy with larger
student models and larger image sizes while staying at a significantly lower
computational budget in comparison to CLIP-Teacher-KD. Eliminating the need to
store the teacher model and run repeated forward passes through it, allows the
freed resources to be better utilized for training larger student models
instead. Even with slightly larger students, the memory used is much lesser
than the teacher, and CLIP-Embed-KD outperforms CLIP-Teacher-KD in accuracy.

Validation accuracy for different $\alpha_{2}$ values are in Figure 5 (for
large student; large-32 teacher; and image size 32). We see that non-zero
weighting of both losses yield the best performance. Using only
$\mathcal{L}_{clip}$ (i.e., $\alpha_{2}=1$) leads to overfitting. Using only
the typical supervised cross-entropy loss (i.e., $\alpha_{2}=0$) performs
worse than when $\mathcal{L}_{clip}$ is included.

##  4 Future Work

Our preliminary work demonstrates the potential of CLIP-Embed-KD for
computationally efficient distillation. In the future, we seek to evaluate
alternative teacher embedding representations that can be used in CLIP-Embed-
KD to further improve the performance, and evaluate our approach on billion-
trillion parameter NLP models and diverse datasets.

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# Casimir boundaries, monopoles, and deconfinement transition in 3+1
dimensional compact electrodynamics

M. N. Chernodub  Institut Denis Poisson UMR 7013, Université de Tours, 37200
Tours, France  V. A. Goy  Pacific Quantum Center, Far Eastern Federal
University, 690950 Vladivostok, Russia  A. V. Molochkov  Pacific Quantum
Center, Far Eastern Federal University, 690950 Vladivostok, Russia  A. S.
Tanashkin  Pacific Quantum Center, Far Eastern Federal University, 690950
Vladivostok, Russia

(August 28, 2024)

###### Abstract

Compact U(1) gauge theory in 3+1 dimensions possesses the confining phase,
characterized by a linear raise of the potential between particles with
opposite electric charges at sufficiently large inter-particle separation. The
confinement is generated by condensation of Abelian monopoles at strong gauge
coupling. We study the properties of monopoles and deconfining order parameter
in zero-temperature theory in the presence of ideally conducting parallel
metallic boundaries (plates) usually associated with the Casimir effect. Using
first-principle numerical simulations in compact U(1) lattice gauge theory, we
show that as the distance between the plates diminishes, the vacuum in between
the plates experiences a deconfining transition. The phase diagram in the
space of the gauge coupling and the inter-plane distance is obtained.

##  I Introduction

Physical objects affect fluctuations of quantum fields and modify dispersion
relations of quantum fluctuations in the vacuum around them. This phenomenon
is the essence of the Casimir effect [1], which predicts that the energy of
vacuum fluctuations is modified by the presence of physical bodies [2, 3].
Moreover, the shift in the energy of virtual particles has a fundamental
physical consequence as the Casimir effect results in a tiny force between
neutral objects [4] that can be detected experimentally [5, 6, 7]. The Casimir
effect is one of the demonstrations of the importance of vacuum fluctuations
and the physical significance of the mysterious vacuum energy.

Most straightforwardly, the Casimir effect reveals itself in non-interacting
field theories. However, even in the absence of interactions, the shift in
vacuum energy sets a complex analytical problem apart from a few simplest
geometries of physical bodies. Therefore, the Casimir effect is usually
studied using, for example, the analytical proximity force approximation [8],
or utilizing numerical tools [9] that include the world line approaches [10]
and first-principle methods of lattice gauge theory [11, 12, 13, 14, 15, 16,
17, 18].

In the presence of (self-) interactions of fields, for example, in quantum
electrodynamics (QED), the calculations of the Casimir interaction become much
more involved. However, in the experimentally relevant cases, the effect of
interactions of fundamental fields is very small due to the weakness of the
QED coupling constant. The electron-photon coupling affects the Casimir-Polder
force in the second-order of the perturbation theory, thus making this
contribution undetectable with current experimental techniques [19].

In strongly coupled field theories, interactions can substantially change the
magnitude of the Casimir force and, unexpectedly, modify the vacuum structure
of the theories themselves. In theories with dynamical matter fields, the
presence of reflective boundaries can affect vacuum condensates and generate
new (and modify existing) phase transitions such as the chiral phase
transition in the four-fermion effective field theory [20, 21, 22].
Interactions can also change the sign of the Casimir–Polder force in fermionic
systems with condensates [23], as well as in the $\mathbb{C}P^{N-1}$ model
[24] (see also [25]).

The Casimir geometry can also affect non-perturbative phenomena associated
with the gauge (vector) fields. For example, vacua of compact Abelian U(1)
gauge theory and non-Abelian SU(2) gauge theory in two spatial dimensions lose
the confining property at sufficiently small separations between,
respectively, perfectly metallic and chromometallic plates [17, 18]. In
compact Abelian theory, the deconfining transition is associated with the
binding transition of the Abelian monopoles, which emerges in the monopoles
plasma due to the presence of the boundaries [17].

We study the effect produced by two closely-spaced, perfectly-conducting
boundaries on the phase structure of 3+1 dimensional compact U(1) gauge
theory. The study may be relevant to the MIT bag model, which treats the
hadron as a (spherical) deconfinement region separated from the confining
exterior by a (reflective) wall [26, 27]. Thermodynamically, the wall is
supported from the collapse by a Casimir pressure of this finite-volume system
[28, 29, 30]. We also investigate the simplest statistical properties of the
Abelian monopoles in the space between the plates since these topological
objects are responsible for the confinement of charge in compact U(1) gauge
theory.

The structure of this paper is as follows. In Section II we describe the
lattice model, the monopoles, and the definition of the perfectly metallic
Casimir boundary conditions. Then, in Section III we show how the Casimir
plates affect the monopoles. Next, we use the monopole properties to determine
the phase diagram of the model. The deconfinement order parameter in the
inter-plate space are discussed in Section IV. The last section is devoted to
our conclusions.

##  II Compact electrodynamics on the lattice, Casimir plates and monopoles

###  II.1 The model

We study a 3+1 dimensional compact U(1) gauge theory in lattice regularization
suitable for numerical simulations. The calculations in thermal equilibrium
are performed, after Wick rotation, in four Euclidean spacetime dimensions.
Below we briefly discuss the model, topological defects, and the Casimir
boundary conditions, following the discussion in 2+1 dimensional case [15, 16]
closely.

The compact U(1) gauge model describes the dynamics of the lattice gauge
(photon) field $\theta_{x,\mu}\in[-\pi,+\pi)$ which is defined on the
elementary links $l=\\{x,\mu\\}$ set by its starting point $x$ and the
direction $\mu$. In the continuum limit, $a\to 0$, the lattice field
$\theta_{x\mu}=aA_{\mu}(x)$ is related to the continuum gauge field
$A_{\mu}(x)$ and the lattice spacing $a$. The lattice analogue of the field-
strength tensor $F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
is played by the plaquette angle,

$\displaystyle\theta_{P_{x,\mu\nu}}=\theta_{x,\mu}+\theta_{x+\hat{\mu},\nu}-\theta_{x+\hat{\nu},\mu}-\theta_{x,\nu}\,,$
(1)

constructed from the link fields (link angles) $\theta_{x,\mu}$. Each
plaquette $P\equiv P_{x,\mu\nu}$ is set by the position $x$ of one of its
corners and by two vectors of the plaquette plane, $\mu<\nu$, with the axes
labeled by the indices $\mu,\nu=1,\dots,4$. The indices $\mu=1,2,3$ correspond
to spatial directions while $\mu=4$ marks the imaginary Euclidean time.

In the continuum limit, the plaquette angle (1) reduces to its continuum
analogue $\theta_{P_{x,\mu\nu}}=a^{2}F_{\mu\nu}(x)+O(a^{4})$ for small
(perturbative) fluctuations of the photon field. In addition to the
perturbative fluctuations, the model also possesses the topological
configurations of the gauge fields, the Abelian monopoles, which correspond to
large variations of the lattice gauge field $\theta_{x,\mu}\sim 1$. These
configurations are singular in the continuum limit.

The action of the lattice model,

$\displaystyle S[\theta]=\beta\sum_{P}\left(1-\cos\theta_{P}\right)\,,$ (2)

is given by the sum over all elementary lattice plaquettes $P$. For
configurations without monopoles, the lattice action (2) becomes the standard
photon action if one associates the lattice coupling constant $\beta=4/e^{2}$
with the electric charge $e$. In the presence of the Abelian monopole
singularities, the continuum action becomes more complicated as it includes
singular Dirac sheets attached to the worldlines of the Abelian monopoles. The
continuum formulation of the compact QED has been briefly discussed in one of
our previous papers [16].

The model (2) is also called the “compact” model because the Abelian gauge
group of the theory corresponds to a compact manifold, $S^{1}$. The action is
invariant under discrete shifts of the plaquette variable,
$\theta_{P}\to\theta_{P}+2\pi n_{P}$ with an integer $n_{P}\in{\mathrm{Z}}$.
This invariance implies that two lattice field strengths $\theta_{P}$ and
$\theta_{P}^{\prime}=\theta_{P}+2\pi n$ with $n\in{\mathbb{Z}}$ are physically
equivalent to each other, thus reducing the physical gauge group to a circle,
${\mathbb{R}}/{\mathbb{Z}}\sim S^{1}$. The same symmetry is also applied to
the gauge field itself, $\theta_{x,\mu}\to\theta_{x,\mu}+2\pi k_{x,\mu}$ with
$k_{x,\mu}\in{\mathrm{Z}}$.

###  II.2 Magnetic monopoles and electric confinement

The compactness of the model naturally leads to the appearance of singular
configurations of the gauge field, the Abelian monopoles. In the continuum
limit, the mentioned $2\pi$ shifts, which leave invariant the lattice action
(2), correspond to the physically unobservable displacements of the singular
Dirac sheets (i.e., the worldlines of the Dirac strings attached to the
Abelian monopoles). The end-points of the open Dirac strings correspond to the
trajectories of the Abelian monopoles, which are physical, gauge-invariant
topological defects.

The monopoles are particle-like objects in the (3+1)-dimensional compact
electrodynamics. On the lattice, the monopole current $j_{x,\mu}$ can be
determined via a finite-difference divergence of the physical part

$\displaystyle{\bar{\theta}}_{P}=\theta_{P}+2\pi k_{P}\in[-\pi,\pi)\,,\qquad
k_{P}\in{\mathrm{Z}}\,,$ (3)

of the lattice field-strength tensor $\theta_{P}$. The monopole trajectory
corresponds to a collection of three-dimensional cubes $C_{x,\mu}$ which
contain a nonzero magnetic charge, $j_{x,\mu}\neq 0$:

$\displaystyle j_{x,\mu}=\frac{1}{2\pi}\sum_{P\in\partial
C_{x,\mu}}{\bar{\theta}}_{P}\in{\mathbb{Z}}\,.$ (4)

Here the sum goes over all elementary sides $P$ of the cube $C_{x,\mu}$, and
the index $\mu$ specifies the local direction of the monopole current. The
index $\mu$ is normal to the three axes that form the 3d cube $C_{x,\mu}$. For
example, if $j_{x,4}\neq 0$, then the corresponding 3-cube is a spatial cube
which contains a static segment of the monopole trajectory.

In the continuum limit, the sum in Eq. (4) reduces to the divergence of the
magnetic field, thus signaling violation of the Bianchi identities for
singular field configurations,
$\epsilon^{\mu\nu\alpha\beta}\partial_{\nu}F_{\alpha\beta}\neq 0$. The
monopole trajectory (4) forms a closed loop defined at the dual hypercubic
lattice (for a review, see Ref. [31]).

It is convenient to define the global monopole density,

$\displaystyle\rho=\frac{1}{{\mathrm{Vol}}_{4}}\sum_{x,\mu}|j_{x,\mu}|\,,$ (5)

where the sum is performed over the volume Vol4 of the four-dimensional
hypercubic lattice. We will also calculate the density (5) and the related
quantities in restricted volumes (in between the Casimir plates).

In lattice gauge theories, the Abelian monopoles have been intensively probed
for their possible role in the phenomenon of the charge confinement, which is,
presumably, closely related to the color confinement in non-Abelian gauge
theories such as QCD [32, 33, 31]. The monopole condensation in an Abelian
gauge theory leads to linear confinement of the electric charges because the
monopole condensate squeezes the electric flux emanating from the electric
charges into a thin electric tube. The tube plays the role of a confining
string. Since the string is a linear object with a constant energy density
$\sigma$ per string length, the increasing distance $R$ between the particle-
antiparticle pair leads to a linearly rising potential $V(R)\simeq\sigma R$ at
large distances. The dimensionful parameter $\sigma$ has the sense of string
tension.

This confining mechanism is similar (and dual) to the formation of the
Abrikosov vortices in superconductors, where the electrically charged
condensate of electron Cooper pairs squeezes the magnetic flux into thin
vortices. If a monopole–anti-monopole pair were immersed into the
superconductor, it would be confined due to the appearance of the Abrikosov
vortex stretched between the constituents of the pair. The mechanism of the
charge confinement based on the monopole condensation is often called the dual
superconductor mechanism [32, 33].

The dual superconductor mechanism was shown to work in the four-dimensional
compact electrodynamics, which possesses the straightforward and unambiguous
definition (4) of the Abelian monopoles [34]. The onset of the monopole
condensation, related to percolation of monopole trajectories [35], is a well-
defined phenomenon in this model. More complicated, non-Abelian Yang-Mills
theories were also shown to possess the dual superconductivity phenomenon in
low-temperature, confining phase [35, 36, 37, 38, 39, 40].

###  II.3 Casimir plates for compact gauge fields

In (3+1) dimensions, the Casimir problem is defined, in general, for three-
dimensional physical materials possessing two-dimensional surfaces. If the
surfaces are made of an ideal metal, then two tangential (to the surface at
each point) components of the electric field and a normal component of the
magnetic field vanish. These boundary conditions can be written in a covariant
form:

$\displaystyle\varepsilon^{\mu\nu\lambda\sigma}F_{\nu\lambda}(x)v_{\sigma}(x)=0\,,\qquad\mu=1,\dots
4\,,$ (6)

where

$\displaystyle v_{\mu}(x)=\varepsilon_{\mu\nu\lambda\sigma}\int
d^{3}\xi\,\frac{\partial{\bar{x}}^{\nu}}{\partial\xi_{1}}\frac{\partial{\bar{x}}^{\lambda}}{\partial\xi_{2}}\frac{\partial{\bar{x}}^{\sigma}}{\partial\xi_{3}}\,\delta^{(4)}\bigl{(}x-{\bar{x}}({\vec{\xi}}\,)\bigr{)},\quad$
(7)

is the dual 3-volume element of the world sheet of the surface. The latter is
described by the vector function
${\bar{x}}^{\mu}={\bar{x}}^{\mu}(\vec{\xi}\,)$ parameterized by the three
dimensional vector $\vec{\xi}=(\xi_{1},\xi_{2},\xi_{3})$.

Figure 1: The geometric setup of two parallel Casimir plates.

In our paper, we consider two static flat plates normal to the $x_{1}$ axis
set at the positions $x_{1}=l_{1}$ and $x_{1}=l_{2}$ as shown in Fig. 1. For
each plate, the local volume element of the corresponding world volume (7) is
given by the following formula

$\displaystyle\nu_{\mu}(x)=\delta_{\mu,1}\delta(x_{1}-l_{a})\,,\qquad
a=1,2\,,$ (8)

where the parameter $a$ labels the surfaces. To derive the above formula, one
can use the parameterization of the $a$th surface as follows:
${\bar{x}}^{\mu}_{a}=(l_{a},\xi_{1},\xi_{2},\xi_{3})$. Consequently, the
covariant conditions (6), reduce to the three conditions that include the
normal component of the magnetic field and two tangential components of the
(Euclidean) electric field, respectively:

$\displaystyle B_{1}$ $\displaystyle\equiv$ $\displaystyle
F^{23}(x){\biggl{|}}_{x_{1}=l_{a}}=0\,,$ (9) $\displaystyle E_{2}$
$\displaystyle\equiv$ $\displaystyle F^{24}(x){\biggl{|}}_{x_{1}=l_{a}}=0\,,$
(10) $\displaystyle\ E_{3}$ $\displaystyle\equiv$ $\displaystyle
F^{34}(x){\biggl{|}}_{x_{1}=l_{a}}=0\,.$ (11)

The definition of the lattice field strength tensor (1) and its physical part
(3), imply that the lattice Casimir conditions (11) read as follows [15, 16]:

$\displaystyle\cos\theta_{x,\mu\nu}{\biggl{|}}_{x_{1}=l_{a}}=1,\qquad(\mu,\nu)=(23,24,34)\,,$
(12)

for all possible $(x_{2},x_{3},x_{4})$ and fixed $x_{1}=l_{a}$ with $a=1,2$.

The simplest way to implement the boundary condition (12) in the path integral
formalism is to add a set of Lagrange multipliers to the standard action (2):

$\displaystyle
S_{\varepsilon}[\theta]=\sum_{P}\beta_{P}(\varepsilon)\cos\theta_{P}\,,$ (13)

To this end, we modify the lattice plaquette couplings
$\beta_{P}\to\beta_{P}(\varepsilon)$. The inhomogeneous coupling

$\displaystyle\beta_{P}(\varepsilon)=\beta\bigl{[}1+(\varepsilon-1)\,\delta_{P,{\mathcal{V}}}\bigr{]}\,.$
(14)

is a function of the dielectric permittivity $\varepsilon$ of the Casimir
plates. Here we used the notation $\mathcal{V}$ to denote the collection of
plaquettes $P_{x,\mu\nu}$ belonging to the world volume of the plates. The
plates are effectively absent if $\varepsilon=1$ while in the limit
$\varepsilon\to\infty$, the components of the physical lattice field-strength
tensor (3) vanish at the word-volume of the plates and, consequently, the
lattice condition (12) is satisfied. In this limit, the plates become
perfectly metallic [2].

We perform our simulations on a $24^{4}$ lattice corresponding to a zero-
temperature model. To generate and update gauge field configurations, we used
the Monte-Carlo heatbath algorithm [41, 42]. For each point, set by gauge
coupling constant $\beta$ and distance between plates $R$, we generated
$7.5\times 10^{5}$ trajectories. The first $10^{5}$ configurations are omitted
to achieve thermalization.

##  III Monopoles and Casimir plates

###  III.1 Monopole density in the absence of the plates

It is well known that in the strong coupling region of the theory,
$\beta\lesssim 1$, the monopole trajectory forms a dense percolating cluster
[35]. This cluster corresponds to the monopole condensate, which, according to
the dual superconducting scenario, produces the confinement of oppositely
charged electric test particles. The property of percolation implies that any
of two points in space have a nonzero probability of being connected by a
monopole trajectory. Furthermore, the infrared nature of the monopole
condensate implies that at infinitely large separation between the points, the
nonzero percolation probability stays constant as the distance between the
points increases.

In the weak coupling regime, $\beta\gtrsim 1$, the percolation cluster breaks
down, thus signaling that the monopole condensate disappears. The confinement
property is, consequently, lost. The confining strongly-coupled regime and the
weakly coupled deconfining phase are separated by the first-order phase
transition [43, 44]. While we do not study the percolation properties of the
monopole cluster in our paper, the transition point can be deduced from the
behavior of the much simpler quantity, the monopole density (5).

(a)

(b)

(c)

Figure 2: (a) Monopole density $\rho$, (b) its susceptibility and (c) the
Binder cummulant (16) for ${\cal O}=\rho$ vs lattice coupling $\beta$ in the
absence of plates. The vertical line marks the position of the phase
transition calculated from these observables.

Before performing calculations of Casimir effects, we consider case of the
homogeneous lattice in the absence of metallic plates. In Fig. 2, we show the
monopole density $\rho$, the corresponding susceptibility,

$\displaystyle\chi_{\rho}=\langle\rho^{2}\rangle-\langle\rho\rangle^{2}\,,$
(15)

and the Binder cummulant (with $\rho=\rho$),

$\displaystyle
B_{\rho}=\frac{\langle\rho^{4}\rangle}{\langle\rho^{2}\rangle^{2}}-3\,,$ (16)

as the functions of the lattice coupling $\beta$. The quantities (15) and (16)
characterize the fluctuations of the monopole density $\rho$.

The monopole density diminishes with the increase of the coupling constant
$\beta$. The position of the deconfining phase transition,

$\displaystyle\beta_{c}=1.010(1),\qquad\mbox{(no plates)}\,,$ (17)

is well visible from the discontinuity in the monopole density, Fig. 2(a). The
discontinuity – which is pertinent to the first-order phase transition –
coincides with high accuracy with the positions in the peaks of the
susceptibility of the monopole density, Fig. 2(b) and the corresponding Binder
cumulant, Fig. 2(c). In the vicinity of the phase transition, we take finely
spaced values of $\beta$ with the step $\delta\beta=0.001$ to achieve
reasonable accuracy of our simulations in evaluation of the (pseudo)critical
values of $\beta_{c}$ calculated as a maximal value of the derivative of the
monopole density with respect to $\beta$, and as a position of the peaks of
the susceptibility and the Binder cumulant of monopole density. The position
of the pseudocritical point (17) coincides, within the error bars, with the
infinite-volume result of Ref. [43].

###  III.2 Monopole properties with Casimir plates

Perfectly conducting Casimir plates are introduced via the inhomogeneous
coupling (14) which serves as the Lagrange multiplier that reduces physical
fluctuations of the gauge field at the plates. In practice, we take the
sufficiently large value of the dielectric constant, $\varepsilon=10^{3}$,
which corresponds to the asymptotically large coupling constant at the plates,
$\beta_{P}\to\infty$. The coupling constant in bulk (outside and inside the
plates) is fixed to take a homogeneous value $\beta$. We consider the
separations $R\equiv|l_{1}-l_{2}|=1,\dots,8$ between the plates.

|
---|---
(a) | (b)
Figure 3: Typical examples of monopole configurations in (a) the confining
phase ($\beta=0.8$) and (b) the deconfining phase ($\beta=0.9$) for the plates
separated by the distance $R=3$. The monopoles and anti-monopoles are
represented by the red and blue dots, respectively. The plates, positioned
vertically in the middle of the lattice, are not shown.

We immediately notice that closely separated plates affect the monopoles
between them. The effect is readily visible in the examples of the typical
monopole configurations shown in Fig. 3 for two values of the bulk coupling
constant $\beta$. The plates tend to diminish the monopole density in the
volume between them compared to the monopole density outside the plates. The
suppression effect is enhanced for larger values of the coupling constant
$\beta$ (at weaker coupling), as one can see from comparisons of Figs. 3(a)
and (b). The suppression of the monopole density suggests that the confining
property should be weakened in between the plates, and therefore the
confinement-deconfinement phase transition should occur at stronger values of
the coupling constant (smaller $\beta$’s). This observation will be confirmed
below.

Figure 4: The ratio $\rho_{\mathrm{ins}}/\rho^{\mathrm{np}}_{\mathrm{ins}}$ of
the monopole density $\rho_{\mathrm{ins}}$ inside the Casimir plates to the
monopole density in the absence of the plates,
$\rho_{\mathrm{ins}}^{\mathrm{np}}$, vs. the inter-plate separation $R$ for a
fixed set of the lattice coupling $\beta$.

The ratio of the monopole density in between the plates $\rho_{\mathrm{ins}}$
and the monopole density at the same $\beta$ in the absence of the plates,
$\rho^{\mathrm{np}}_{\mathrm{all}}$, are shown in Fig. 4 for various
separations $R$. All couplings $\beta$ shown in the plot, the shrinking plates
lead to the diminishing monopole density. At weaker coupling (larger $\beta$),
the relative monopole density is affected stronger than at stronger coupling
(smaller $\beta$). The relative monopole density,
$\rho_{\mathrm{ins}}/\rho^{\mathrm{np}}_{\mathrm{ins}}$, has an inflection
point at certain $R=R^{*}$ at fixed $\beta$. This point moves towards smaller
values of $R$ as the lattice coupling $\beta$ decreases. The latter fact
indicates that the model may have a $\beta$-dependent transition which moves
towards smaller $R$ as the coupling $\beta$ gets larger.

|  |
---|---|---
|  |
|  |
Figure 5: The monopole density (the left panel), its susceptibility (the
middle panel) and the Binder cumulant (the right panel) plotted as the
function of $\beta$ for three separations between the plates (from top to
bottom): $R={2,4,8}$. The corresponding critical coupling constants,
$\beta_{c}=\beta_{c}(R)$, are shown in the insets.

In Fig. 5 we show the monopole density in between the plates
$\rho_{\mathrm{ins}}$, its susceptibility, and the corresponding Binder
cumulant for three values of the inter-plate distance $R$. One can immediately
make a few qualitative observations from these figures.

Firstly, we notice that all these quantities behave similarly to the case
without plates (Fig. 2) implying affinity of these transitions. Secondly, we
notice that for any fixed $R$, the monopole density, its susceptibility, and
the Binder cumulant experience the singularities at the same value of the
coupling constant $\beta$, highlighting the presence of a genuine
thermodynamic instability. Thirdly, the positions of these singularities,
$\beta_{c}=\beta_{c}(R)$, shift towards the strong coupling region as the
distance between $R$ the plates diminishes. In other words, the closer the
plates, the weaker the monopole component of the vacuum.

Figure 6: The phase diagram of the vacuum of the compact U(1) gauge theory in
between the perfectly metallic plates separated by the distance $R$. The solid
line represents the best fit (18) of the critical coupling $\beta_{c}$ which
corresponds to the confinement-deconfinement phase transition at the inter-
plate distance $R$. The $R\to\infty$ limit is shown by the dashed horizontal
line.

The dependence of the critical coupling $\beta_{c}$ of the inter-plate
distance $R$ is shown in Fig. 6. As we discussed above, the smaller $R$ the
smaller $\beta_{c}$. The dependence can be fitted by the function

$\displaystyle\beta_{c}^{\mathrm{fit}}(R)=\beta^{\infty}_{c}-\alpha\exp{\left[-(R^{2}/R_{0}^{2})^{\nu}\right]}\,,$
(18)

where the best fit parameter $\beta_{c}=1.0071(16)$ gives us the critical
coupling at the infinitely separated plates. This value is close to the
transition point in the absence of the plates (17), highlighting the self-
consistency of our approach. The other best fit parameters in Eq. (18) are as
follows: $\alpha=3.7(6)$, $R_{0}=0.28(7)$ and $\nu=0.257(16)$. The best fit
function is shown in Fig. 6 by the solid line.

The critical coupling (18) separates the confinement phase ($\beta<\beta_{c}$)
and the deconfinement phase ($\beta>\beta_{c}$). The phase transition line is
a rising function of the inter-plate distance $R$. The critical coupling
vanishes, $\beta_{c}^{\mathrm{fit}}(R_{c})=0$, at the critical distance
$R_{c}=0.47(7)$. Formally, at the separations smaller than critical value,
$R<R_{c}$, the theory cannot reside in the confining phase. Since the inter-
plate separation is a positive integer number in the discretized theory,
$R=1,2,\dots$, this asymptotic critical value cannot be reached on the
lattice.

The loss of the confinement property of the vacuum between the metallic plates
in 3+1 dimensions, revealed by Fig. 6, can be understood using the analogy
with the similar effect in the 2+1 dimensional theory [16]. Namely, the
monopoles and anti-monopoles interact with each other via the long-range
massless photon exchange. In the absence of the plates, this interaction, in
3+1 dimensions, decays as $|x|^{-2}$ as the 4-dimensional distance $x$ between
the objects increases. In the presence of the plates, the system experiences
the dimensional reduction from four- to three-dimensional spacetime. In the
latter, the interaction between the (anti-)monopoles strengthens and decays
slower, as $|x|^{-1}$. These two factors lead to the breaking of the infrared
monopole clusters into smaller clusters and, consequently, to the
disappearance of the monopole condensate.

In the 2+1 dimensional model, the same effect leads to the pairing of
monopoles and anti-monopoles into the magnetically neutral monopole pairs (the
lower-dimensional counterparts of the small clusters) and to the decay of the
Coulomb monopole gas (the lower-dimensional analogue of the monopole
condensate). The neutral pairs (the small clusters) cannot support the
confinement and confining property is lost between the sufficiently close
plates [16, 17].

##  IV Deconfinement order parameter

We determined the nature of confinement and deconfinement phases in the whole
phase diagram of Fig. 6 using the simple fact that these phases at finite
separation $R$ are smoothly connected to the known phases in the $R\to\infty$
limit. In this section, we quantify this assertion by calculating the
deconfinement order parameter, the expectation of the Polyakov loop, in
between the plates.

Usually, the Polyakov loop is determined at a finite temperature where the
extension of the lattice in the imaginary time direction is finite. The same
order parameter can also be used on the zero-temperature lattice with a finite
extension in the temporal direction, $N_{T}$.

In the Abelian gauge theory, the Polyakov loop $P_{\bm{x}}$ at the spatial
space point ${\bm{x}}$ is given by a cyclic product of temporal link
variables:

$P_{\bm{x}}=\prod_{x_{4}=0}^{N_{T}-1}e^{i\theta_{{\bm{x}},x_{4};\mu=4}}\,.$
(19)

Due to the property of the temporal cyclicity, this quantity does not depend
on the time slice where it is defined.

The expectation value of this gauge-invariant quantity, $P=\left\langle
P_{\bm{x}}\right\rangle$ serves as a deconfinement order parameter: at an
infinite-volume lattice, $P\neq 0$ in the deconfinement phase and $P=0$ in the
confinement phase. At finite lattice (as in our case), the expectation value
of the Polyakov loop is nonzero in both phases, being small (large) in the
confinement (deconfinement) phase.

We compute the expectation value of the modulus of the operator (19) averaged
over a set of lattice points in a fixed timeslice of the 3-dimensional volume
V3:

$|P|=\left|\frac{1}{V_{3}}\sum\limits_{{\bm{x}}\in V_{3}}P_{\bm{x}}\right|\,.$
(20)

The expectation value of the Polyakov order parameter (20) in the absence of
the plate is shown in Fig. 7. We show the region around the phase transition
point (17), where the change in the behavior is well seen.

Figure 7: The Polyakov loop as the function of lattice coupling $\beta$ in the
absence of plates.

The order parameter evaluated in the space between the plates is presented in
Fig. 8. We show this quantity, as a function of the inter-plate separation
$R$, for the same set of $\beta$ values as used in Fig. 4 for the inter-plate
monopole density. The shrinking plates induce the deconfining phase resulting
in the increase of the Polyakov loop (Fig. 8) in agreement with the
diminishing monopole density (shown in Fig. 4). The effect appears to work at
all values of the coupling constant $\beta$. The similar tendency is seen in
Fig. 9 which shows the same quantity vs. $\beta$ at a set of fixed inter-plate
separations $R$.

Figure 8: The expectation value of the Polyakov loop in the space between the
Casimir plates at the separation $R$ at a set of fixed coupling constants
$\beta$. Figure 9: The Polyakov loop inside the plates vs $\beta$ at fixed
$R$.

These results support the phase diagram of Fig. 6.

##  V Conclusions

Using the first-principle numerical simulations, we show that the vacuum of
confining gauge theory, the compact U(1) gauge model in 3+1 dimensions, is
affected by closely spaced perfectly conducting parallel plates. The non-
perturbative Casimir effect modifies the vacuum structure and leads to the
Casimir-induced deconfining phase transition in between the plates. Our main
result, the phase diagram in the plane “coupling constant”–“distance between
the plates” is given in Fig. 6.

###### Acknowledgements.

The numerical calculations were performed in 2021 at the computing cluster
Vostok-1 of the Far Eastern Federal University. The work of VAG, AVM, and AST
was supported by Grant No. 0657-2020-0015 of the Ministry of Science and
Higher Education of Russia.

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