# Conserved charges for
rational electromagnetic knots

Lukas Hantzko, Kaushlendra Kumar111Corresponding author:
<EMAIL_ADDRESS>and Gabriel Picanço Costa
Institut für Theoretische Physik
Leibniz Universität Hannover
Appelstraße 2, 30167 Hannover, Germany

( )

###### Abstract

We revisit a newfound construction of rational electromagnetic knots based on
the conformal correspondence between Minkowski space and a finite
$S^{3}$-cylinder. We present here a more direct approach for this conformal
correspondence based on Carter–Penrose transformation that avoids a detour to
de Sitter space. The Maxwell equations can be analytically solved on the
cylinder in terms of $S^{3}$ harmonics $Y_{j;m,n}$, which can then be
transformed into Minkowski coordinates using the conformal map. The resultant
“knot basis” electromagnetic field configurations have non-trivial topology in
that their field lines form closed knots. We consider finite, complex linear
combinations of these knot-basis solutions for a fixed spin $j$ and compute
all the 15 conserved Noether charges associated with the conformal group. We
find that the scalar charges either vanish or are proportional to the energy.
For the non-vanishing vector charges, we find a nice geometric structure that
facilitates computation of their spherical components as well. We present
analytic results for all charges for up to $j{=}1$. We demonstrate possible
applications of our findings through some known previous results.

##  1 Introduction and summary

Topologically non-trivial solutions to source-free Maxwell’s equations on
Minkowski space $\mathds{R}^{1,3}$ were first discovered by Rañada [1] based
on the Hopf fibration. Other methods for constructing such linked and knotted
rational solutions involving Bateman’s complex Euler potentials, conformal
inversion and Penrose twistors have been found since then (see [2] for a
review). Key features characterizing these solutions are helicity (conserved
under conformal transformations) and Noether charges associated with the
conformal group $SO(2,4)$ of $\mathds{R}^{1,3}$ [3, 4]. Quite interestingly,
some simple EM knot configurations, including the one with figure-8 topology,
have been produced in the laboratory using laser beams with knotted
polarization singularities [5]. With such a discovery, this line of research
no longer remains just a theoretical endeavor; more complicated knot
configurations discussed here, or elsewhere, could be realized experimentally
in the near future. Trajectories of charged particles in the background of
these knotted electromagnetic fields have also been studied recently [6].

It was shown in [7] that a complete family of such rational knot solutions on
$\mathds{R}^{1,3}$, labelled by the so called left-right $S^{3}$ harmonics
$Y_{j;m,n}$, can be constructed; any finite-action rational Maxwell solution
can be expanded in terms of these basis solutions. The construction relies on
the conformal correspondence between the de Sitter space dS4 and
$\mathds{R}^{1,3}$ via a finite Lorentzian $S^{3}$-cylinder over conformal
time $\tau$. Here, we establish this conformal correspondence between the
cylinder and $\mathds{R}^{1,3}$ more directly using Carter–Penrose
transformation. These “knot basis” solutions are first obtained on the
cylinder by employing the $SO(4)$ isometry of $S^{3}$ coupled with the right-
multiplication action of $SU(2)$ on $S^{3}$ (being the group manifold of
$SU(2)$) and are later transferred to $\mathds{R}^{1,3}$ via the conformal
map. In another paper [8], co-authored by one of us, general linear
combination of such “knot basis” solutions in terms of arbitrary complex
coefficients $\Lambda_{j,m,n}$ was considered and, for a fixed $j$, its
helicity was found to be proportional to the conserved energy.

Here, we reconsider these generic solutions for a fixed $j$ and proceed to
compute the remaining charges associated with the conformal group, namely
momenta, angular momenta, boosts, dilatation and special conformal
transformation (SCT) charges. The charge densities are evaluated on de Sitter
space at $\tau=0$ (without loss of generality, since they are conserved) where
considerable simplifications occur demonstrating the usefulness of this “de
Sitter method”. It is found that the dilatation charge vanishes while the
scalar SCT charge $V_{0}$ is proportional to the energy $E$. Furthermore, the
boost charges $K_{i}$ vanish and the vector SCT charges $V_{i}$ are
proportional to the momentum charges $P_{i}$. Interestingly, for the vector
charge densities viz. momenta $p_{i}$, angular momenta $l_{i}$ and vector SCT
$v_{i}$ we find that a one-form constructed with them on the spatial slice
$\mathds{R}^{3}\hookrightarrow\mathds{R}^{1,3}$, e.g. $p_{i}\mathrm{d}x^{i}$
is proportional to a similar one on de Sitter space. This correspondence
allows us to compute additional charges $(p_{r},p_{\theta},p_{\phi})$ by the
action of such one-forms on spherical vector fields
$(\mbox{$\partial$}_{r},\mbox{$\partial$}_{\theta},\mbox{$\partial$}_{\phi})$.
At $j=0$ it turns out that there are only four independent nonzero charges:
the energy $E$ and the momenta $P_{i}$. The situation for higher spin $j$ is
more complicated, but some of the components of the charges in spherical
coordinates are found to vanish for arbitrary $j$. The action of the
$\mathfrak{so(3)}$ generators $\mathcal{D}_{a}$ on the indices of
$\Lambda_{j,m,n}$ can easily be obtained for a fixed $j$ owing to the $SO(4)$
isometry. This allows for an action of these generators on the charges. For
(the Cartesian components of) the vector charges this action is found to
inherit the original $\mathfrak{so(3)}$ Lie algebraic structure, as expected.
We also compute the correct coefficients $\Lambda_{0;0,n}$ corresponding to
two very interesting generalizations of the Hopfian solution obtained via
Bateman’s construction in [4], which allow us to validate our generic formulae
of these charges. Further works in this direction could be to relate these new
knot solutions and their conserved charges to other known field configurations
in [4], such as plane waves or to study Fourier analysis of general electro-
magnetic knot configurations to understand single mode behaviour.

The paper is organized as follows. In Section 2, we first establish the
conformal map between Minkowski space and the Lorentzian $S^{3}$-cylinder
using Carter–Penrose transformation and review the necessary computational
tools related to $S^{3}$, including its harmonics. We then construct the
rational Maxwell solutions on the cylinder in terms of the $S^{3}$ harmonics
and illustrate some knotted field configurations using figures. Section 3, at
first, reviews the Noether current and associated charges arising from the
action of the conformal group on free Maxwell theory and presents a
computational strategy. The $15$ conformal charges are then computed in four
subsections and the action of $\mathfrak{so(3)}$ generators $\mathcal{D}_{a}$
on some of the vector charges is analyzed. Finally, in Section 4 we validate
the formulae of the charges computed in Section 3 using other known results.

##  2 Construction of rational electromagnetic knots

We present below the construction of rational electromagnetic knots by
completely skipping the detour to de Sitter space in contrast with [7], where
these solutions were first presented. For a clear presentation, we split the
construction in three parts.

###  2.1 Minkowski to cylinder: Carter–Penrose transformation

The metric on the Minkowski space $\mathds{R}^{1,3}$ in polar coordinates is
given by

$\mathrm{d}s_{\text{Mink}}^{2}\ =\
-\mathrm{d}t^{2}+\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\Omega_{2}^{2}\
;\qquad\mathrm{d}\Omega_{2}^{2}\ =\
\mathrm{d}\theta^{2}+\sin^{2}\theta\,\mathrm{d}\phi^{2},$ (2.1)

Figure 1:  Penrose diagram of Minkowski space $\mathds{R}^{1,3}$. Each point
hides a two-sphere $S^{2}\ni\\{\theta,\phi\\}$. Blue curves indicate
$t{=}\textrm{const}$ slices while brown curves depict the world hypersurface
of $r{=}\textrm{const}$ spheres. The lightcone of the Minkowski-space origin
is drawn in red.

where $-\infty<t<\infty,\ 0\leq r<\infty,\ 0\leq\theta\leq\pi$, and
$0\leq\phi\leq 2\pi$. We first employ the lightcone coordinates $(u,v)$ to
transform the metric in the following way

$\displaystyle u\ :=\ t-r\ ,\quad v\ :=\ t+r\ ;\quad-\infty<u\leq v<\infty\ $
(2.2) $\displaystyle\implies\mathrm{d}s_{\text{Mink}}^{2}\ =\
-\mathrm{d}v\,\mathrm{d}u+{\textstyle\frac{1}{4}}(v-u)^{2}\,\mathrm{d}\Omega_{2}^{2}\
.$

In the second step, we compactify the spacetime with the help of the
coordinate (U,V) as follows:

$\displaystyle U\ :=\ \arctan u\ ,\quad V\ :=\ \arctan v\
;\quad-{\textstyle\frac{\pi}{2}}<U\leq V<{\textstyle\frac{\pi}{2}}\ $ (2.3)
$\displaystyle\implies\mathrm{d}s_{\text{Mink}}^{2}\ =\
\frac{1}{4\cos^{2}V\cos^{2}U}\left[-4\,\mathrm{d}V\,\mathrm{d}U+\sin^{2}(V-U)\,\mathrm{d}\Omega_{2}^{2}\right]\
.$

Finally, we rotate the coordinate system back using $(\tau,\chi)$ to obtain
the desired form of the metric:

$\displaystyle\tau\ :=\ V+U\ ,\quad\chi\ :=\ \pi+U-V\ ;\quad 0<\chi\leq\pi\
,|\tau|<\chi\ $ (2.4) $\displaystyle\implies\mathrm{d}s_{\text{Mink}}^{2}\ =\
\gamma^{-2}\left[-\mathrm{d}\tau^{2}+\mathrm{d}\chi^{2}+\sin^{2}\chi\,\mathrm{d}\Omega_{2}^{2}\right]\
;\quad\gamma\ =\ \cos\tau-\cos\chi\ .$

We realize that the Minkowski metric is conformally equivalent to the metric
on a Lorentzian cylinder $\mathcal{I}\times S^{3}\ ;\ \mathcal{I}=(-\pi,\pi)$
with a conformal factor that can be recasted in terms of Minkowski coordinates
using the above transformations (2.2-2.4):

$\mathrm{d}s_{\text{cyl}}^{2}\ :=\
-\mathrm{d}\tau^{2}+\mathrm{d}\Omega_{3}^{2}\ =\
\gamma^{2}\,\mathrm{d}s_{\text{Mink}}^{2}\
;\quad\gamma=\frac{2\ell^{2}}{\sqrt{4\,t^{2}\,\ell^{2}+(r^{2}-t^{2}+\ell^{2})^{2}}}\
,$ (2.5)

where we have made $\gamma$ dimensionless using a global scale factor $\ell$,
chosen to be the de Sitter radius so as to make the connection with previous
works apparent. We depict in Figure 1 the causal structure of the Minkowski
spacetime (preserved under a conformal transformation) using the Penrose
diagram. Note that the spatial infinity $i^{0}$ (where $t=0$ and
$r\rightarrow\infty$) and the future/past temporal infinities $i^{\pm}$ (where
$r=0$ and $t\rightarrow\pm\infty$) are not included in the figure.
Furthermore, the boundaries of the Penrose diagram are (a) $r=0$ line, (b)
past null infinity $\mathscr{I}^{-}$ corresponding to $r-t\rightarrow\infty$,
$r+t=$ constant and (c) future null infinity $\mathscr{I}^{+}$ corresponding
to $r+t\rightarrow\infty$, $r-t=$ constant. The natural embedding coordinates
$\omega_{{}_{A}}$ with $A=1,2,3,4$ of $S^{3}\hookrightarrow\mathds{R}^{4}$
satisfying $\omega_{{}_{A}}^{2}=1$ has a natural parametrization in terms of
hyperspherical coordinates $(\chi,\theta,\phi)$ of (2.4):

$\omega_{a}\ =\ \sin\chi\;\hat{\omega}_{a}\ ,\ \omega_{4}\ =\ \cos\chi\
,\quad\textrm{with}\quad\left(\hat{\omega}_{1},\,\hat{\omega}_{2},\,\hat{\omega}_{3}\right)\
=\ \left(\sin\theta\cos\phi,\,\sin\theta\sin\phi,\,\cos\theta\right)\ .$ (2.6)

It is not difficult to obtain the following useful relations

$\gamma\,t\ =\ \ell\,\sin\tau\quad\quad\textrm{and}\quad\quad\gamma\,r\ =\
\ell\,\sin\chi\ ,$ (2.7)

which can be used to find the Jacobian of transformation between the
coordinates $y^{m}\in\\{\tau,\chi,\theta,\phi\\}$ and
$x^{\mu}\in\\{t,r,\theta,\phi\\}$:

$\bigl{(}J^{m}_{\ \ \mu}\bigr{)}\ :=\
\frac{\mbox{$\partial$}(\tau,\chi,\theta,\phi)}{\mbox{$\partial$}(t,r,\theta,\phi)}\
=\ \frac{1}{\ell}\biggl{(}\begin{matrix}p&-q\\\\[4.0pt]
q&-p\end{matrix}\biggr{)}\oplus\mathds{1}_{2}\
\quad\textrm{with}\quad\biggl{\\{}\begin{array}[]{l}p\ =\
{\textstyle\frac{\gamma^{2}}{2\ell^{2}}}\,(r^{2}{+}t^{2}{+}\ell^{2})\ =\
1{-}\cos\tau\cos\chi\\\\[4.0pt] q\ =\
{\textstyle\frac{\gamma^{2}}{\ell^{2}}}\,t\,r\ =\
\sin\tau\sin\chi\end{array}\biggr{\\}}\ .$ (2.8)

###  2.2 Structure of three-sphere and its harmonics

The presence of $S^{3}$ in the metric (2.5) is an added advantage, which can
be exploited to write a $SO(4)$-invariant vector potential and obtain the
corresponding electromagnetic fields by solving Maxwell’s equations on the
Lorentzian cylinder. These quantities can be later exported to the Minkowski
spacetime owing to the conformal invariance of the vacuum Maxwell equations in
four dimensions. To this end, we start with the group $SO(4)$, which is
isomorphic to two copies of $SU(2)$ (up to a $\mathds{Z}_{2}$ grading). Each
of these $SU(2)$ has a group action that generates a left (right)
multiplication (a.k.a. translation) on $S^{3}$. This can easily be checked
using the map

$g:\;S^{3}\rightarrow SU(2);\leavevmode\nobreak\ \leavevmode\nobreak\
(\omega_{1},\ \omega_{2},\ \omega_{3},\
\omega_{4})\,\mapsto\,-i\begin{pmatrix}\beta&\alpha^{*}\\\
\alpha&-\beta^{*}\end{pmatrix}\leavevmode\nobreak\
\textrm{with}\leavevmode\nobreak\ \ \alpha:=\omega_{1}+i\omega_{2}\
\leavevmode\nobreak\ \textrm{and}\leavevmode\nobreak\ \
\beta:=\omega_{3}+i\omega_{4}\ .$ (2.9)

This parametrization of $g$ ensures that the identity element
$e=\mathds{1}_{2}$ of the group $SU(2)$ can be obtained from $(0,0,0,1)$, i.e.
the North pole of $S^{3}$. It is well known that $S^{3}$ is the group manifold
of $SU(2)$. Keeping this in mind, we consider the Cartan one-form

$\Omega_{l}(g)\,:=\,g^{-1}\,\mathrm{d}g\ =\ e^{a}\,T_{a},\leavevmode\nobreak\
\leavevmode\nobreak\ \textrm{with}\leavevmode\nobreak\ \leavevmode\nobreak\
T_{a}=-i\sigma_{a}$ (2.10)

being the $SU(2)$ generators. The left-invariant one-form222Named so because
they remain invariant under the dragging induced by the left SU(2)
multiplication. $e^{a}$ can, alternatively, be expressed using the so-called
self-dual ’t Hooft symbol $\eta^{a}_{{}_{\ BC}}$:

$e^{a}\ =\ -\eta^{a}_{{}_{\
BC}}\,\omega_{{}_{B}}\,\mathrm{d}\omega_{{}_{C}}\quad\textrm{with}\quad\eta^{a}_{\
bc}\ =\ \varepsilon_{abc}\quad\textrm{and}\quad\eta^{a}_{\ b4}\ =\
-\eta^{a}_{\ 4b}\ =\ \delta^{a}_{b}\ .$ (2.11)

They satisfy the following useful identities

$\delta_{ab}\,e^{a}\,e^{b}\ =\
\mathrm{d}\Omega_{3}^{2}\quad\quad\textrm{and}\quad\quad\mathrm{d}e^{a}+\varepsilon_{abc}\,e^{b}\wedge
e^{c}\ =\ 0.$ (2.12)

The left-invariant vector fields $L_{a}$ generating the right translations are
dual to $e^{a}$ and are given by

$L_{a}\ =\ -\eta^{a}_{{}_{\
BC}}\,\omega_{{}_{B}}\,{\textstyle\frac{\partial}{\partial\omega_{{}_{C}}}}\qquad\Rightarrow\qquad\left[L_{a},L_{b}\right]\
=\ 2\,\varepsilon_{abc}\,L_{c}\ .$ (2.13)

In a similar way, the right-invariant vector fields $R_{a}$ generating the
left translations are given by

$R_{a}\ =\ -\tilde{\eta}^{a}_{{}_{\
BC}}\,\omega_{{}_{B}}\,{\textstyle\frac{\partial}{\partial\omega_{{}_{C}}}}\qquad\Rightarrow\qquad\left[R_{a},R_{b}\right]=2\,\varepsilon_{abc}\,R_{c}\
,$ (2.14)

where the anti-self-dual ’t Hooft symbols $\tilde{\eta}^{a}_{{}_{\ BC}}$ are
obtained from (2.11) by flipping the $B,C\\!=4$ sign. The stability subgroup
$SO(3)\subset SO(4)$ under the group action on the coset space
$SO(4)/SO(3)\simeq S^{3}$ is generated by

$\mathcal{D}_{a}\ :=\
L_{a}+R_{a}=-2\,\varepsilon_{abc}\,\omega_{b}\,{\textstyle\frac{\partial}{\partial\omega_{c}}}\quad\implies\quad[\mathcal{D}_{a},\mathcal{D}_{b}]=2\,\varepsilon_{abc}\,\mathcal{D}_{c}\
.$ (2.15)

Furthermore, the vector fields $L_{a}$ and $R_{a}$ act on the one-forms
$e^{a}$ via their Lie derivative333One can use, e.g., the Cartan formula:
$\mathcal{L}_{X}=\mathrm{d}\circ\iota_{X}+\iota_{X}\circ\mathrm{d}$ where
$\mathcal{L}_{X}$ is the Lie derivative w.r.t $X$, $\mathrm{d}$ is the
exterior derivative and $\iota_{X}$ is the interior product. yielding

$L_{a}\,e^{b}\ =\ 2\,\varepsilon_{abc}\,e^{c}\quad\quad\textrm{and}\quad\quad
R_{a}\,e^{b}\ =\ 0\ .$ (2.16)

We can now write the differential $\mathrm{d}$ of the functions $f\in
C^{\infty}\left(\mathcal{I}\times S^{3}\right)$ using $L_{a}$ as

$\mathrm{d}f\ =\ \mathrm{d}\tau\;\partial_{\tau}f\ +\ e^{a}\,L_{a}f\ .$ (2.17)

Furthermore, the restriction of such functions on $S^{3}$ has a natural
expansion in terms of left-right harmonics $Y_{j;m,n}$ with
$2j\in\mathds{N}_{0}$ and $m,n\in\\{-j,-j+1,...,j\\}$ that are eigenfunctions
of Casimirs444This is same as the scalar Laplacian on $S^{3}$:
$L^{2}=R^{2}=\bigtriangleup_{3}$. $L^{2}:=L_{a}L_{a},\ R^{2}:=R_{a}R_{a}$, and
operators $L_{3}$, $R_{3}$:

$\displaystyle\begin{split}-{\textstyle\frac{1}{4}}L^{2}\,Y_{j;m,n}&\ =\
-{\textstyle\frac{1}{4}}R^{2}\,Y_{j;m,n}\ =\ j(j{+}1)\,Y_{j;m,n}\ ,\\\
{\textstyle\frac{\mathrm{i}}{2}}R_{3}\,Y_{j;m,n}&\ =\ m\,Y_{j;m,n}\
,\leavevmode\nobreak\ {\textstyle\frac{\mathrm{i}}{2}}L_{3}\,Y_{j;m,n}\ =\
n\,Y_{j;m,n}\ .\end{split}$ (2.18)

The action of the corresponding ladder operators

$L_{\pm}\ =\
(L_{1}\pm\mathrm{i}L_{2})/\sqrt{2}\quad\quad\textrm{and}\quad\quad R_{\pm}\ =\
(R_{1}\pm\mathrm{i}R_{2})/\sqrt{2}$ (2.19)

on the harmonics $Y_{j;m,n}$ is, therefore, given by

${\textstyle\frac{\mathrm{i}}{2}}\,R_{\pm}\,Y_{j;m,n}\ =\
\sqrt{(j{\mp}m)(j{\pm}m{+}1)/2}\,Y_{j;m\pm
1,n}\quad\\!\quad\textrm{and}\quad\\!\quad{\textstyle\frac{\mathrm{i}}{2}}\,L_{\pm}\,Y_{j;m,n}\
=\ \sqrt{(j{\mp}n)(j{\pm}n{+}1)/2}\,Y_{j;m,n\pm 1}\ .$ (2.20)

The normalized harmonics $Y_{j;m,n}$ can be expanded in terms of functions
$\alpha,\beta$ and their complex conjugates as

$Y_{j;m,n}(\omega)\ =\
\sqrt{\frac{2j+1}{2\pi^{2}}}\,\sum_{k=0}^{2j}\,(-1)^{m+n+k}\frac{\sqrt{(j+m)!(j-m)!(j+n)!(j-n)!}}{(n+m+k)!(j-n-k)!(j-m-k)!k!}\,\alpha^{n+m+k}\,\bar{\alpha}^{k}\,\beta^{j-m-k}\,\bar{\beta}^{j-n-k}\
.$ (2.21)

They satisfy the orthonormality condition

$\int\mathrm{d}^{3}\Omega_{3}\,Y_{j;m,n}\,\bar{Y}_{j^{\prime};m^{\prime},n^{\prime}}\
=\
\delta_{j,j^{\prime}}\,\delta_{m,m^{\prime}}\,\delta_{n,n^{\prime}}\quad\textrm{with}\quad\mathrm{d}^{3}\Omega_{3}\
=\ \sin^{2}\chi\sin\theta\,\mathrm{d}\chi\,\mathrm{d}\theta\,\mathrm{d}\phi\
.$ (2.22)

There also exist the so-called adjoint harmonics $\tilde{Y}_{j,l,M}$ [8] that
are related with the left-right harmonics $Y_{j;m,n}$ through the
Clebsch–Gordan coefficients $C_{m,n}^{l,M}$ as

$Y_{j;m,n}\ =\
\sum\limits_{l=0}^{2j}\sum\limits_{M=-l}^{l}\,C_{m,n}^{l,M}\,{\widetilde{Y}}_{j;l,M}\
.$ (2.23)

These adjoint harmonics can be written in terms of the standard $S^{2}$
spherical harmonics $Y_{l,M}$ and the associated Legendre polynomials of the
first kind $P_{a}^{b}$:

${\widetilde{Y}}_{j;l,M}(\chi,\theta,\phi)\ =\
R_{j,l}(\chi)\,Y_{l,M}(\theta,\phi)\quad\textrm{with}\quad R_{j,l}(\chi)\ =\
\mathrm{i}^{2j+l}\,\sqrt{{\textstyle\frac{2j+1}{\sin\chi}}{\textstyle\frac{(2j+l+1)!}{(2j-l)!}}}\,P_{2j+\frac{1}{2}}^{-l-\frac{1}{2}}(\cos\chi)\
.$ (2.24)

We can write down the cylinder one-forms in Minkowski coordinates using
relations (2.6-2.8) in (2.11)

$\displaystyle\mathrm{d}\tau\,=:\,e^{\tau}$ $\displaystyle\ =\
{\textstyle\frac{\gamma^{2}}{\ell^{3}}}\bigl{(}{\textstyle\frac{1}{2}}(t^{2}{+}r^{2}{+}\ell^{2})\,\mathrm{d}t-t\,r\,\mathrm{d}r\bigr{)}$
(2.25) $\displaystyle e^{a}$ $\displaystyle\ =\
{\textstyle\frac{\gamma^{2}}{\ell^{3}}}\bigl{(}\hat{\omega}^{a}\bigl{[}r\,t\,\mathrm{d}t-{\textstyle\frac{1}{2}}(t^{2}{+}r^{2}{+}\ell^{2})\,\mathrm{d}r\bigr{]}-{\textstyle\frac{1}{2}}(t^{2}{-}r^{2}{+}\ell^{2})\,r\,\mathrm{d}\hat{\omega}^{a}-\ell\,r^{2}\varepsilon^{ajk}\hat{\omega}^{j}\mathrm{d}\hat{\omega}^{k}\bigr{)}\
.$

We further note down the expressions for the vector fields $L_{a}$ in terms of
$S^{3}$ angles by using (2.6) in (2.13) for later purposes:

$\displaystyle L_{1}$ $\displaystyle\ =\
\sin\theta\cos\phi\,\mbox{$\partial$}_{\chi}\ +\
(\cot\chi\cos\theta\cos\phi+\sin\phi)\,\mbox{$\partial$}_{\theta}\ -\
(\cot\chi\csc\theta\sin\phi-\cot\theta\cos\phi)\,\mbox{$\partial$}_{\phi}\ ,$
(2.26) $\displaystyle L_{2}$ $\displaystyle\ =\
\sin\theta\sin\phi\,\mbox{$\partial$}_{\chi}\ +\
(\cot\chi\cos\theta\sin\phi-\cos\phi)\,\mbox{$\partial$}_{\theta}\ +\
(\cot\chi\csc\theta\cos\phi+\cot\theta\sin\phi)\,\mbox{$\partial$}_{\phi}\ ,$
$\displaystyle L_{3}$ $\displaystyle\ =\ \cos\theta\,\mbox{$\partial$}_{\chi}\
-\ \cot\chi\sin\theta\,\mbox{$\partial$}_{\theta}\ -\
\mbox{$\partial$}_{\phi}\ .$

###  2.3 Rational solutions to vacuum Maxwell equations

We start with the following ansatz [8] for a real-valued Maxwell gauge
potential $\mathcal{A}$ on the Lorentzian cylinder

${\cal A}\ =\ {\cal A}_{\tau}(\tau,\omega)\,e^{\tau}\ +\ \sum_{a=1}^{3}{\cal
A}_{a}(\tau,\omega)\,e^{a}\quad\quad\textrm{with}\quad\omega\equiv\\{\omega_{{}_{A}}\\}\
,$ (2.27)

which can be extended to the Minkowski space since the Maxwell’s theory is
conformally invariant here. We further impose the following “temporal+Coulomb”
gauge on this space

${\cal A}_{\tau}(\tau,\omega)=0\quad\quad\textrm{and}\quad\quad R_{a}\,{\cal
A}_{a}(\tau,\omega)=0\ .$ (2.28)

Note that the first part in (2.28) is not the usual temporal gauge
$A_{t}\\!=0$. In fact, we can make use of the inverse Jacobian (2.8) while
promoting the gauge potential to the Minkowski space ${\cal A}\\!={\cal
A}_{a}\,e^{a}\\!=A_{\mu}\,dx^{\mu}\\!=A$ to get

$0={\cal A}_{\tau}={\cal
A}\left(\partial_{\tau}\right)=A\left({\textstyle\frac{\ell^{2}}{\gamma^{2}}}\left(p\,\partial_{t}+q\,\partial_{r}\right)\right)\implies\left(r^{2}+t^{2}+\ell^{2}\right)A_{t}+2\,r\,t\,A_{r}=0\
.$ (2.29)

The field strength $\cal{F}$ is easily obtained with this gauge potential
using (2.12):

${\cal F}\ =\ \mathrm{d}{\cal A}\ =\ \mbox{$\partial$}_{\tau}{\cal
A}_{a}\,e^{\tau}{\wedge}e^{a}+\bigl{(}{\textstyle\frac{1}{2}}R_{[b}{\cal
A}_{c]}-{\cal A}_{a}\,\varepsilon^{a}_{\ bc}\bigr{)}\,e^{b}{\wedge}e^{c}\ .$
(2.30)

The corresponding vacuum Maxwell equations $\mathrm{d}*{\cal F}=0$ read

$\mbox{$\partial$}_{\tau}^{2}{\cal A}_{a}\ =\ (R^{2}-4)\,{\cal
A}_{a}+2\,\varepsilon_{abc}R_{b}{\cal A}_{c}\ .$ (2.31)

Solutions to these coupled linear wave equations on $S^{3}$ were obtained in
[7] as an infinite tower of basis solutions labelled by $j$ and divided into
two categories: type-I with frequency $\Omega_{j}\\!=2(j{+}1)$ and type-II
with frequency $\Omega_{j}\\!=2j$. We shall confine our discussions in this
paper to type-I solutions. The corresponding results for type-II solutions can
easily be obtained using a duality transformation (see [7]) where the basis
solutions of type-I at level $j$ are related with the basis solutions of type-
II at level $j{+}1$ via the exchange $m\leftrightarrow n$. We take a general
complex linear combination of these basis solutions and obtain (see [8]) the
real-valued functions ${\cal A}_{a}(\tau,\omega)$ as

${\cal A}_{a}(\tau,\omega)\ =\ \sum_{2j=0}^{\infty}Z_{a}^{j}(\omega)\
\mathrm{e}^{2(j+1)\mathrm{i}\tau}\ +\
\sum_{2j=0}^{\infty}\bar{Z}_{a}^{j}(\omega)\
\mathrm{e}^{-2(j+1)\mathrm{i}\tau}\ ,$ (2.32)

where the complex angular functions $Z_{a}^{j}$ are expanded in terms of
$(2j{+}1)(2j{+}3)$ arbitrary complex coefficients $\Lambda_{j;m,n}$ and basis
solutions $Z_{*}^{j;m,n}(\omega)\leavevmode\nobreak\
\textrm{with}\leavevmode\nobreak\ *\in\\{+,3,-\\}$ as

$Z_{*}^{j}(\omega)\ =\ \sum_{m=-j}^{j}\sum_{n=-j-1}^{j+1}\Lambda_{j;m,n}\
Z_{*}^{j;m,n}(\omega)\quad\quad\textrm{with}\quad\quad
Z_{\pm}^{j}\,:=\,{\textstyle\frac{1}{\sqrt{2}}}\left(Z_{1}^{j}\pm\mathrm{i}Z_{2}^{j}\right)\
.$ (2.33)

The complex basis functions $Z_{*}^{j;m,n}$ are given in terms of harmonics
$Y_{j;m,n}$ as

$\left.\begin{aligned} Z_{+}^{j;m,n}(\omega)&\ =\
\sqrt{(j{-}n)(j{-}n{+}1)/2}\leavevmode\nobreak\ Y_{j;m,n+1}(\omega)\\\
Z_{3}^{j;m,n}(\omega)\,&\ =\ \sqrt{(j{+}1)^{2}-n^{2}}\leavevmode\nobreak\
Y_{j;m,n}(\omega)\\\ Z_{-}^{j;m,n}(\omega)&\ =\
-\sqrt{(j{+}n)(j{+}n{+}1)/2}\leavevmode\nobreak\
Y_{j;m,n-1}(\omega)\end{aligned}\right\\}\quad\textrm{with}\quad\biggl{\\{}\begin{array}[]{l}m\
=\ -j,-j+1,...,j\\\\[4.0pt] n\ =\ -j-1,-j,...,j+1\end{array}\biggr{\\}}\ .$
(2.34)

The expected harmonic expansion for $Z_{a}^{j}(\omega)$ is obtained by using
(2.33) in (2.34) while inverting the basis from $(+,3,-)$ to $(1,2,3)$:

$Z_{a}^{j}(\omega)\ =\ \sum_{m,n=-j}^{j}X_{a}^{j;m,n}\
Y_{j;m,n}(\omega)\quad\quad\textrm{with}\quad\quad X_{a}^{j;m,n}\ =\
\sum_{\tilde{n}=-j-1}^{j+1}C^{n,\tilde{n}}_{j,a}\,\Lambda_{j;m,\tilde{n}}\ ,$
(2.35)

where the real coefficients $C^{n,\tilde{n}}_{j,a}$ are determined from
(2.34).

The useful expression for the “sphere-frame” electric and magnetic fields,

${\cal F}\ =\ {\cal
E}_{a}\,e^{a}{\wedge}e^{\tau}+{\textstyle\frac{1}{2}}\,{\cal
B}_{a}\,\varepsilon^{a}_{\ bc}\,e^{b}{\wedge}e^{c}\ $ (2.36)

for a fixed spin $j$ is obtained by eliminating $R_{[b}A_{c]}$ in (2.30) using
(2.31) and employing

$\mbox{$\partial$}_{\tau}^{2}{\cal A}^{(j)}=-\Omega_{j}^{2}\,{\cal
A}^{(j)}\quad\quad\textrm{and}\quad\quad R^{2}\,{\cal A}^{(j)}\ =\
-4j(j{+}1)\,{\cal A}^{(j)}$ (2.37)

to get

${\cal E}_{a}^{(j)}\ =\ -\mbox{$\partial$}_{\tau}{\cal
A}_{a}^{(j)}\quad\quad\textrm{and}\quad\quad{\cal B}_{a}^{(j)}\ =\
-\Omega_{j}\,{\cal A}_{a}^{(j)}\ .$ (2.38)

Figure 2:  Sample electric (red) and magnetic (green) field lines at $t{=}0$.
Left: $(j;m,n)=(0;0,-1)$ configuration, Centre:
$(j;m,n)=({\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}})$
configuration, Right: $(j;m,n)=(1;0,1)$ configuration. More self-knotted field
lines start appearing with additional seed points in the simulation; the size
of these new knots increases with the increase in the seed point’s distance
from the origin.

It may be noted here that, due to the compactness of the Lorentzian cylinder,
the sphere-frame energy and action are always finite. One can obtain, by a
cumbersome calculation, the electric/magnetic fields in Minkowski space by
using the results (2.25) in (2.36) and realizing that

${\cal F}\ =\ F\ =\
E_{a}\,\mathrm{d}x^{a}\wedge\mathrm{d}t+{\textstyle\frac{1}{2}}\,B_{a}\,\varepsilon^{a}_{\
bc}\,\mathrm{d}x^{b}\wedge\mathrm{d}x^{c}\ .$ (2.39)

The actual expression for electric/magnetic fields are rather involved and it
bodes well to combine them as the Riemann–Silberstein vector S:

$\textbf{S}\ =\ \textbf{E}+\mathrm{i}\textbf{B}\ .$ (2.40)

The expression of S becomes more and more complicated with increasing
$j$.555Expression of the RS vector for $(j;m,n)=(1;0,0)$ has been recorded in
[8]. The (self) knotted field lines corresponding to such electromagnetic
configurations also increase in complexity with $j$. We illustrate this point
by producing field lines for $j=0,1/2$ and $1$ in Figure 2.

##  3 Conformal group and Noether charges

It is well known [9] that free Maxwell theory on $\mathds{R}^{1,3}$ arising
from the action

$S\left[A_{\mu}\right]\ =\ \int d^{4}x\,\mathcal{L}\ ;\qquad\mathcal{L}\ =\
-{\textstyle\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}$ (3.1)

is invariant under the conformal group $SO(2,4)$. Furthermore, the above
action is also invariant under the gauge transformations:
$A_{\mu}(x)\rightarrow A_{\mu}(x)+\mbox{$\partial$}_{\mu}\lambda(x)$ for some
scalar field $\lambda(x)$. The conformal group is generated by transformations
$x^{\mu}\rightarrow x^{\mu}+\xi^{\mu}(x)$, where the vector fields $\xi^{\mu}$
obey the conformal Killing equations:

$\xi_{\mu,\nu}\,+\,\xi_{\nu,\mu}\ =\
{\textstyle\frac{1}{2}}\,\eta_{\mu\nu}\,\xi^{\alpha}_{\leavevmode\nobreak\
,\alpha}\quad\textrm{with}\quad\\{\eta_{\mu\nu}\\}\ =\
\textrm{diag}(-1,1,1,1)\ .$ (3.2)

The conserved Noether current $J^{\mu}$ is obtained by equating the “on-shell
variation” of the action where the variations $\delta A_{\mu}$ are arbitrary
and the fields $A_{\mu}$ satisfies the Euler–Langrange equations with its
“symmetry variation” where the fields $A_{\mu}$ are arbitrary but variations
$\delta A_{\mu}$ satisfy the symmetry condition. The correct variation $\delta
A_{\mu}$ is obtained by imposing the gauge invariance on the Lie derivative of
$A_{\mu}$ w.r.t. the vector field $\xi^{\mu}$:

$\mathcal{L}_{\xi^{\alpha}}A_{\mu}:=A^{\prime}_{\mu}(x)-A_{\mu}(x)=-\xi^{\alpha}\,\mbox{$\partial$}_{\alpha}A_{\mu}-A_{\alpha}\,\mbox{$\partial$}_{\mu}\xi^{\alpha}\leavevmode\nobreak\
\longrightarrow\leavevmode\nobreak\ \delta A_{\mu}\ =\ F_{\mu\nu}\xi^{\nu}\ .$
(3.3)

Finally, the conserved current is obtained as

$J^{\mu}\ =\
\frac{\mbox{$\partial$}\mathcal{L}}{\mbox{$\partial$}A_{\rho,\mu}}\delta
A_{\rho}+\xi^{\mu}\mathcal{L}\ =\
\xi^{\rho}\left(F^{\mu\alpha}F_{\rho\alpha}-{\textstyle\frac{1}{4}}\delta^{\mu}_{\rho}F^{2}\right)\quad\textrm{with}\quad
F^{2}=F_{\beta\gamma}F^{\beta\gamma}\ ,$ (3.4)

which satisfies the continuity equation and gives the conserved (in time)
charge $Q$666For source-free fields, the current J is assumed to vanish when
the surface $\mbox{$\partial$}V$ is taken to infinity.:

$\mbox{$\partial$}_{\mu}\,J^{\mu}=0\quad\implies\quad\frac{\mathrm{d}Q}{\mathrm{d}t}\
=\ -\int_{\mbox{$\partial$}V}\mathrm{d}^{2}\textbf{s}\cdot\textbf{J}\ =\ 0\
;\quad Q\ =\ \int_{V}\mathrm{d}^{3}x\,J^{0}\ .$ (3.5)

Before proceeding further, a couple of remarks pertaining to the subsequent
calculations are in order:

  * •

All the charges $Q$ are computed at $t\\!=0$ owing to the simple $J^{0}$
expressions on this time-slice. To that end, we record the following useful
identities at $t=\tau=0$:

$\displaystyle e^{a}_{i}$ $\displaystyle\ =\
{\textstyle\frac{1}{\ell}}\left(\gamma\,\omega_{4}\,\delta^{a}_{i}-\omega_{a}\,\omega_{i}+\epsilon_{aic}\,\gamma\,\omega_{c}\right)\
,\quad e^{a}_{i}\,e^{b}_{i}\ =\
{\textstyle\frac{\gamma^{2}}{\ell^{2}}}\delta^{ab}$ (3.6)
$\displaystyle\gamma$ $\displaystyle\ =\ 1-\omega_{4}\ ,\leavevmode\nobreak\
\mathrm{d}^{3}x\ =\
{\textstyle\frac{\ell^{3}}{\gamma^{3}}}\mathrm{d}^{3}\Omega_{3}\
\leavevmode\nobreak\ ;\leavevmode\nobreak\
\mathrm{d}^{3}\Omega_{3}:=e^{1}\wedge e^{2}\wedge e^{3}\ .$

Moreover, the electromagnetic fields at $t\\!=0$ are given in terms of tetrads
$e^{\tau}=e^{\tau}_{\mu}\,\mathrm{d}x^{\mu},\
e^{a}=e^{a}_{\mu}\,\mathrm{d}x^{\mu}$:

$E_{i}\ =\
e^{\tau}_{0}\,e^{a}_{i}\,\mathcal{E}_{a}\quad\quad\textrm{and}\quad\quad
B_{i}\ =\
{\textstyle\frac{1}{2}}\,\epsilon_{ijk}\,\epsilon_{abc}\,e^{b}_{j}\,e^{c}_{k}\,\mathcal{B}_{a}\
$ (3.7)

using the two-form (2.36) and its Minkowski counterpart (2.39).

  * •

Furthermore, these charges $Q$ are computed for a fixed spin-$j$ and, thus, we
will suppress the index $j$ from now onwards, unless necessary. Note that the
sphere-frame EM fields for fixed $j$ can be obtained by using the expansion
(2.32) in (2.38) as

${\cal A}_{a}\ =\
Z_{a}(\omega)\,\mathrm{e}^{\Omega\,\mathrm{i}\tau}+\bar{Z}_{a}(\omega)\,\mathrm{e}^{-\Omega\,\mathrm{i}\tau}\qquad\Rightarrow\qquad\biggl{\\{}\begin{array}[]{l}{\cal
E}_{a}\ =\
-\mathrm{i}\,\Omega\,Z_{a}\,\mathrm{e}^{\Omega\,\mathrm{i}\tau}+\mathrm{i}\,\Omega\,\bar{Z}_{a}\,\mathrm{e}^{-\Omega\,\mathrm{i}\tau}\\\\[2.0pt]
{\cal B}_{a}\ =\
\,-\,\Omega\,Z_{a}\,\mathrm{e}^{\Omega\,\mathrm{i}\tau}\,-\,\Omega\,\bar{Z}_{a}\,\mathrm{e}^{-\Omega\,\mathrm{i}\tau}\end{array}\biggr{\\}}$
(3.8)

with $\bar{Z}_{a}$ denoting the complex conjugate of $Z_{a}$.

  * •

The simplifications below for the charge density $J^{0}$ are carried out using
the harmonic expansion (2.35) while using the results (3.6) in (3.7).

  * •

We frequently use below the well-known fact that an odd
$\\{\omega_{{}_{A}}\\}$ integral over $S^{3}$ vanishes because of the opposite
contributions coming from the antipodal points on the sphere. In particular,
it can be checked that the following integral vanishes777Note that the power
of $\omega_{{}_{A}}$ in $Y_{j;m,n}\bar{Y}_{j;m^{\prime},n^{\prime}}$ is always
even. One way to check this is by employing the toroidal coordinates:
$\omega_{1}=\cos\eta\cos\zeta_{1},\omega_{2}=\cos\eta\sin\zeta_{1},\omega_{3}=\sin\eta\cos\zeta_{2},\omega_{4}=\sin\eta\sin\zeta_{2}$
with $\eta\in(0,{\textstyle\frac{\pi}{2}})$ and
$\zeta_{1},\zeta_{2}\in(0,2\pi)$ in (2.21). The resultant selection rules
coming from $\zeta_{1},\zeta_{2}$ integral would yield
$m-m^{\prime}\in{\textstyle\frac{2k+1}{2}}$ with $k\in\mathds{N}_{0}$, which
is not feasible for fixed $j$.:

$\int\mathrm{d}^{3}\Omega_{3}\,(\omega_{1})^{a}\,(\omega_{2})^{b}\,(\omega_{3})^{c}\,(\omega_{4})^{d}\,Y_{j;m,n}\,\overline{Y}_{j;m^{\prime},n^{\prime}}\
=\ 0\quad\textrm{for}\quad a+b+c+d\ \in\ 2\mathds{N}_{0}+1\ .$ (3.9)

Having made these remarks, we now proceed to compute the charges $Q$ for
various conformal transformations $\xi^{\mu}$ obeying (3.2) in following four
categories.

###  3.1 Translations

An easily seen solution to (3.2) is the set of four constant translations

$\xi^{\mu}=\epsilon^{\mu}\ ,$ (3.10)

which also partly generates the Poincaré group and give rise to the usual
stress-energy tensor of electrodynamics

$J^{\mu}\ =\ T^{\mu}_{\ \;\nu}\ =\
F^{\mu\alpha}F_{\nu\alpha}-{\textstyle\frac{1}{4}}\delta^{\mu}_{\nu}F^{2}\ ,$
(3.11)

corresponding to the $\mu$-component of the translation for an arbitrary
$\epsilon^{\nu}$. The corresponding charges are the energy $E$ and the
momentum P.

Energy. The expression of the energy density $e:=T^{00}$ simplifies to

$e\,:=\,{\textstyle\frac{1}{2}}\left(E_{i}^{2}+B_{i}^{2}\right)\ =\
({\textstyle\frac{\gamma}{\ell}})^{4}\,\rho\quad\quad\textrm{with}\quad\quad\rho\
=\ {\textstyle\frac{1}{2}}\left({\cal E}_{a}^{2}+{\cal B}_{a}^{2}\right)\ ,$
(3.12)

which, in turn, simplifies the expression for the energy $E$ to

$E\ =\ \int_{V}\mathrm{d}^{3}x\;e\ =\
{\textstyle\frac{2}{\ell}}\,\Omega^{2}\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,(1-\omega_{4})\,Z_{a}\,\bar{Z}_{a}\
.$ (3.13)

The resultant expression for $E$ in terms of complex parameters
$\Lambda_{m,n}$ and its conjugate was already computed in [8]. The
$\omega_{4}$ part of the above integral vanishes as discussed in an earlier
remark and we obtain

$E\ =\ \frac{8}{\ell}\,(j+1)^{3}(2j+1)\,\sum_{m,n}\,|\Lambda_{m,n}|^{2}\ .$
(3.14)

Momentum. For the momentum densities
$p_{i}=T^{0i}=(\textbf{E}\times\textbf{B})_{i}$ we obtain an interesting
correspondence relating the one-form $p:=p_{i}\,\mathrm{d}x^{i}$ on Minkowski
space with a similar one on de Sitter space:

$p\ =\ ({\textstyle\frac{\gamma}{\ell}})^{3}\,\mathcal{P}_{a}\,e^{a}\ =:\
({\textstyle\frac{\gamma}{\ell}})^{3}\,\mathcal{P}\quad\quad\textrm{with}\quad\quad\mathcal{P}_{a}\
:=\ \varepsilon_{abc}\,\mathcal{E}_{b}\,\mathcal{B}_{c}\ .$ (3.15)

A straightforward calculation then yields the expression of momenta $P_{i}$:

$P_{i}\ =\ \int_{V}\mathrm{d}^{3}x\,p_{i}\ =\
\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,\mathcal{P}_{a}\,e^{a}_{i}\ =\
2\mathrm{i}\ \Omega^{2}\ \varepsilon_{abc}\
\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,e^{a}_{i}\,Z_{b}\,\bar{Z}_{c}$ (3.16)

with $e^{a}_{i}$ given by (3.6). The results for $j=0$ are

$\displaystyle P_{1}^{(j=0)}$ $\displaystyle\ =\
-{\textstyle\frac{\sqrt{2}}{\ell}}\left(\left(\bar{\Lambda}_{0,-1}+\bar{\Lambda}_{0,1}\right)\Lambda_{0,0}+\bar{\Lambda}_{0,0}\left(\Lambda_{0,-1}+\Lambda_{0,1}\right)\right)\
,$ (3.17) $\displaystyle P_{2}^{(j=0)}$ $\displaystyle\ =\
{\textstyle\frac{\mathrm{i}\sqrt{2}}{\ell}}\left(\left(-\bar{\Lambda}_{0,-1}+\bar{\Lambda}_{0,1}\right)\Lambda_{0,0}+\bar{\Lambda}_{0,0}\left(\Lambda_{0,-1}-\Lambda_{0,1}\right)\right)\
,$ $\displaystyle P_{3}^{(j=0)}$ $\displaystyle\ =\
{\textstyle\frac{2}{\ell}}\left(|\Lambda_{0,-1}|^{2}-|\Lambda_{0,1}|^{2}\right)\
.$

As a consistency requirement, we check that the vector charges $P_{i}$ are
rotated according to the algebra of $\mathcal{D}_{a}$ (2.15) (see appendix A):

$\mathcal{D}_{a}\,P_{b}\ =\ 2\,\varepsilon_{abc}\,P_{c}\ .$ (3.18)

We also note down $P_{3}$ for $j=1/2\ \leavevmode\nobreak\
\textrm{and}\leavevmode\nobreak\ 1$ in Table 1.

|  $j=1/2$ |  $j=1$
---|---|---
$P_{3}$ |  $\begin{aligned} {\textstyle\frac{9}{\ell}}\Big{(}&|\Lambda_{-1/2,-3/2}|^{2}-|\Lambda_{-1/2,1/2}|^{2}\\\ &-2|\Lambda_{-1/2,3/2}|^{2}+2|\Lambda_{1/2,-3/2}|^{2}\\\ &+|\Lambda_{1/2,-1/2}|^{2}-|\Lambda_{1/2,3/2}|^{2}\Big{)}\end{aligned}$ |  $\begin{aligned} {\textstyle\frac{24}{\ell}}\Big{(}&|\Lambda_{-1,-2}|^{2}-|\Lambda_{-1,0}|^{2}-2|\Lambda_{-1,1}|^{2}-3|\Lambda_{-1,2}|^{2}\\\ &+2|\Lambda_{0,-2}|^{2}+|\Lambda_{0,-1}|^{2}-|\Lambda_{0,1}|^{2}-2|\Lambda_{0,2}|^{2}\\\ &+3|\Lambda_{1,-2}|^{2}+2|\Lambda_{1,-1}|^{2}+|\Lambda_{1,0}|^{2}-|\Lambda_{1,2}|^{2}\Big{)}\end{aligned}$
$P_{\phi}$ |  $\begin{aligned} 9\Big{(}&2|\Lambda_{-1/2,-3/2}|^{2}+|\Lambda_{-1/2,-1/2}|^{2}\\\ &-|\Lambda_{-1/2,3/2}|^{2}+|\Lambda_{1/2,-3/2}|^{2}\\\ &-|\Lambda_{1/2,1/2}|^{2}-2|\Lambda_{1/2,3/2}|^{2}\Big{)}\end{aligned}$ |  $\begin{aligned} 24\Big{(}&3|\Lambda_{-1,-2}|^{2}+2|\Lambda_{-1,-1}|^{2}+|\Lambda_{-1,0}|^{2}-|\Lambda_{-1,2}|^{2}\\\ &+2|\Lambda_{0,-2}|^{2}+|\Lambda_{0,-1}|^{2}-|\Lambda_{0,1}|^{2}-2|\Lambda_{0,2}|^{2}\\\ &+|\Lambda_{1,-2}|^{2}-|\Lambda_{1,0}|^{2}-2|\Lambda_{1,1}|^{2}-3|\Lambda_{1,2}|^{2}\Big{)}\end{aligned}$
$L_{3}$ |  $\begin{aligned} 9\ell\,\Big{(}&-2|\Lambda_{-1/2,-3/2}|^{2}-|\Lambda_{-1/2,-1/2}|^{2}\\\ &+|\Lambda_{-1/2,3/2}|^{2}-|\Lambda_{1/2,-3/2}|^{2}\\\ &+|\Lambda_{1/2,1/2}|^{2}+2|\Lambda_{1/2,3/2}|^{2}\Big{)}\end{aligned}$ |  $\begin{aligned} -24\ell\,\Big{(}&3|\Lambda_{-1,-2}|^{2}+2|\Lambda_{-1,-1}|^{2}+|\Lambda_{-1,0}|^{2}-|\Lambda_{-1,2}|^{2}\\\ &+2|\Lambda_{0,-2}|^{2}+|\Lambda_{0,-1}|^{2}-|\Lambda_{0,1}|^{2}-2|\Lambda_{0,2}|^{2}\\\ &+|\Lambda_{1,-2}|^{2}-|\Lambda_{1,0}|^{2}-2|\Lambda_{1,1}|^{2}-3|\Lambda_{1,2}|^{2}\Big{)}\end{aligned}$
Table 1: Expressions of $P_{3}$, $P_{\phi}$ and $L_{3}$ for
$j=1/2\leavevmode\nobreak\ \textrm{and}\leavevmode\nobreak\ 1$.

One can compute the corresponding $P_{1}$ and $P_{2}$ for $j=1/2$ and $j=1$ by
employing the action of an appropriate $\mathcal{D}_{a}$ of Table 3.

We can additionally compute the spherical components of the momentum
$(P_{r},P_{\theta},P_{\phi})$ by letting the one-form $e^{a}$ in (3.15) act on
the vector fields
$(\mbox{$\partial$}_{r},\mbox{$\partial$}_{\theta},\mbox{$\partial$}_{\phi})$.
In practice, we first write
$\mbox{$\partial$}_{r}=-{\textstyle\frac{\gamma}{\ell}}\mbox{$\partial$}_{\chi}$
using (2.8) at $t=0$ and then invert the vector fields
$(\mbox{$\partial$}_{r},\mbox{$\partial$}_{\theta},\mbox{$\partial$}_{\phi})$
in terms of the left invariant vector fields $(L_{1},L_{2},L_{3})$ using
(2.26). Finally, using the duality relation $e^{a}(L_{b})=\delta^{a}_{b}$ we
obtain

$\displaystyle P_{r}$ $\displaystyle\ =\
-\frac{1}{\ell}\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,(1-\cos\chi)\,\left(\sin\theta\cos\phi\,\mathcal{P}_{1}+\sin\theta\sin\phi\,\mathcal{P}_{2}+\cos\theta\,\mathcal{P}_{3}\right)\
,$ (3.19) $\displaystyle P_{\theta}$ $\displaystyle\ =\
\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,\sin\chi\cos\chi\left((\cos\theta\cos\phi+\tan\chi\sin\phi)\,\mathcal{P}_{1}+(\cos\theta\sin\phi-\tan\chi\cos\phi)\mathcal{P}_{2}-\sin\theta\,\mathcal{P}_{3}\right),$
$\displaystyle P_{\phi}$ $\displaystyle\ =\
\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,\sin^{2}\chi\sin\theta\left((\cos\theta\cos\phi-\cot\chi\sin\phi)\,\mathcal{P}_{1}+(\cos\theta\sin\phi+\cot\chi\cos\phi)\,\mathcal{P}_{2}-\sin\theta\,\mathcal{P}_{3}\right)\
.$

From the expression of $P_{r}$ we see that the integrand over $S^{2}$, i.e.
$\hat{\omega}_{a}\mathcal{P}_{a}$ (2.6) is an odd function,888The terms of
$\mathcal{P}_{a}$ using (2.24) are proportional to
$Y_{l,M}Y_{l^{\prime},M^{\prime}}$, which is even in $\hat{\omega}_{a}$ for a
fixed $j$. which makes $P_{r}$ vanish. We also find with explicit calculations
(verified for up to $j=1$) that $P_{\theta}$ vanishes. For $j=0$ we find that
$P_{\phi}$ are proportional to $P_{3}$:

$P_{\phi}^{(j=0)}\ =\ \ell\,P_{3}^{(j=0)}\ =\
2\left(|\Lambda_{0,-1}|^{2}-|\Lambda_{0,1}|^{2}\right)\ .$ (3.20)

The expressions of $P_{\phi}$ for $j=1/2$ and $1$ has been recorded in the
Table 1.

###  3.2 Lorentz transformations

Another solution of (3.2) is given by

$\xi^{\mu}(x)\ =\ \epsilon^{\mu}_{\
\;\nu}\,x^{\nu}\quad\textrm{where}\quad\epsilon_{\mu\nu}\ =\
-\epsilon_{\nu\mu}\ ,$ (3.21)

which correspond to the six generators of the Lorentz group $SO(1,3)$. These
six together with the above four translations generates the full Poincaré
group. The corresponding six charges for these transformations are grouped
into the boost K and the angular momentum L.

Boost. The conserved charge densities arising from $J^{0}$ (3.4) corresponding
to $\epsilon_{0i}$ are the boost densities $\textbf{k}=\rho\
\textbf{x}-\textbf{p}\ t$, which simplify for $t=0$ to

$k_{i}\ =\
({\textstyle\frac{\gamma}{\ell}})^{3}\,\mathcal{K}_{i}\quad\quad\textrm{with}\quad\quad\mathcal{K}_{i}\
=\ \rho\,\omega_{i}\ .$ (3.22)

The corresponding charges $K_{i}$ vanish because of the odd integrand as
discussed in an earlier remark:

$K_{i}\ =\ \int_{V}\mathrm{d}^{3}x\,k_{i}\ =\
\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,\mathcal{K}_{i}\ =\
2\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,\omega_{i}\,Z_{a}^{j}\,\bar{Z}_{a}^{j}\
=\ 0\ .$ (3.23)

Angular momentum. The other three conserved charge densities $J^{0}$
corresponding to $\epsilon_{ij}$ in (3.4) are the angular momentum densities
$\textbf{l}=\textbf{p}\times\textbf{x}$, which take a simple form just like in
momentum (3.15):

$l_{i}\,\mathrm{d}x^{i}\ =\
({\textstyle\frac{\gamma}{\ell}})^{2}\,\mathcal{L}_{a}\,e^{a}\quad\quad\textrm{with}\quad\quad\mathcal{L}_{a}\
=\ \varepsilon_{abc}\,\mathcal{P}_{b}\,\omega_{c}\ .$ (3.24)

The expressions for the charges $L_{i}$ simplify to

$L_{i}\ =\ \int_{V}\mathrm{d}^{3}x\,l_{i}\ =\
\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,({\textstyle\frac{\ell}{\gamma}})\,\mathcal{L}_{a}\,e^{a}_{i}\
=\ 2\mathrm{i}\ell\ \Omega^{2}\
\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,{\textstyle\frac{1}{\gamma}}\,e^{a}_{i}\left(\bar{Z}_{a}\,\omega_{b}Z_{b}-Z_{a}\,\omega_{b}\bar{Z}_{b}\right)\
.$ (3.25)

Explicit calculations show that for $j=0$ the angular momenta $L_{i}$ are
proportional to the momenta $P_{i}$:

$L_{i}^{(j=0)}\ =\ -\ell^{2}\,P_{i}^{(j=0)}\ .$ (3.26)

This is, however, not true for higher spin $j$. The angular momenta $L_{i}$
has the same rotation behaviour as for the momenta $P_{i}$ (3.18). We,
therefore, note down the results of $L_{3}$ for $j=1/2$ and $1$ in Table 1,
from which the corresponding expressions of $L_{1}$ and $L_{2}$ can be
obtained using Table 3.

We can again compute the spherical components of the angular momentum
$(L_{r},L_{\theta},L_{\phi})$ using the relations (3.19) by replacing
$\mathcal{P}$ with $\mathcal{L}$ in it. We realize that the $S^{2}$ integrand
for $L_{r}$ would only have terms like
$\hat{\omega}_{a}\hat{\omega}_{b}\mathcal{P}_{c}$, which are all odd
functions999Here again the terms of $\mathcal{P}_{c}$, which goes like
$Y_{l,M}Y_{l^{\prime},M^{\prime}}$ (2.24), are all even functions of $\phi$
for a fixed $j$. of $\phi$ and, therefore, the $\phi$ integral over the domain
$(0,2\pi)$ would make $L_{r}$ vanish. We also find, with explicit
computations, that the charges $L_{\theta}$ and $L_{\phi}$ for $j=0$ are
proportional to $P_{3}$:

$L_{\theta}^{(j=0)}\ =\
{\textstyle\frac{4}{3}}\,\ell^{2}\,P_{3}^{(j=0)}\quad\quad\textrm{and}\quad\quad
L_{\phi}^{(j=0)}\ =\ -{\textstyle\frac{1}{3}}\,\ell^{2}\,P_{3}^{(j=0)}\ .$
(3.27)

As a non trivial example, we collect these charges for $j=1/2$ below:

$\displaystyle L_{\theta}^{\left(j=1/2\right)}\ =\
{\textstyle\frac{12}{5}}\ell\,\Big{(}$ $\displaystyle
9|\Lambda_{-1/2,-3/2}|^{2}+6|\Lambda_{-1/2,-1/2}|^{2}+|\Lambda_{-1/2,1/2}|^{2}-6|\Lambda_{-1/2,3/2}|^{2}$
(3.28)
$\displaystyle\qquad\qquad\qquad\qquad+6|\Lambda_{1/2,-3/2}|^{2}-|\Lambda_{1/2,-1/2}|^{2}-6|\Lambda_{1/2,1/2}|^{2}-9|\Lambda_{1/2,3/2}|^{2}\Big{)}\
,$ $\displaystyle L_{\phi}^{\left(j=1/2\right)}\ =\
-{\textstyle\frac{3}{5}}\ell\,\Big{(}$ $\displaystyle
6|\Lambda_{-1/2,-3/2}|^{2}-|\Lambda_{-1/2,-1/2}|^{2}-6|\Lambda_{-1/2,1/2}|^{2}-9|\Lambda_{-1/2,3/2}|^{2}+9|\Lambda_{1/2,-3/2}|^{2}$
(3.29)
$\displaystyle\qquad\quad+6|\Lambda_{1/2,-1/2}|^{2}+|\Lambda_{1/2,1/2}|^{2}-6|\Lambda_{1/2,3/2}|^{2}+\sqrt{3}\big{(}\bar{\Lambda}_{1/2,-3/2}\Lambda_{-1/2,-1/2}$
$\displaystyle\qquad\qquad\qquad-\bar{\Lambda}_{1/2,1/2}\Lambda_{-1/2,3/2}+\bar{\Lambda}_{-1/2,-1/2}\Lambda_{1/2,-3/2}-\bar{\Lambda}_{-1/2,3/2}\Lambda_{1/2,1/2}\big{)}\Big{)}\
.$

###  3.3 Dilatation

It is easy to verify that a constant rescaling by $\lambda$:

$\xi^{\mu}\ =\ \lambda\,x^{\mu}$ (3.30)

is also a solution of (3.2). The charge density corresponding to this single
generator of the conformal group is $\textbf{p}\cdot\textbf{x}-e\,t$, which
for $t\\!=0$ simplifies to

$p_{i}\,x_{i}\ =\
({\textstyle\frac{\gamma}{\ell}})^{3}\,\mathcal{P}_{a}\,\omega_{a}\ .$ (3.31)

The corresponding charge $D$ vanishes because of the odd integrand:

$D\ =\ \int_{V}\mathrm{d}^{3}x\,p_{i}\,x_{i}\ =\
2\mathrm{i}\,\Omega^{2}\,\varepsilon_{abc}\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,\omega_{a}\,Z_{b}\,\bar{Z}_{c}\
=\ 0\ .$ (3.32)

###  3.4 Special conformal transformations

A fairly straightforward calculation shows that the following not so obvious
transformation

$\xi^{\mu}\ =\ 2\,x^{\mu}\,b_{\nu}x^{\nu}\ -\ b^{\mu}\,x^{\nu}x_{\nu}$ (3.33)

also satisfies (3.2). The four generators corresponding to $b_{\mu}$ give rise
to four different charges $V_{0}$ and V.

Scalar SCT. The charge density $J^{0}$ corresponding to $b_{0}$ is
$v_{0}=(\textbf{x}^{2}+t^{2})\,e-2t\,\textbf{p}\cdot\textbf{x}$, which for
$t\\!=0$ simplifies to

$v_{0}\ =\ \textbf{x}^{2}\,e\ =\
({\textstyle\frac{\gamma}{\ell}})^{2}\rho\,\omega_{a}^{2}\ .$ (3.34)

The expression for the corresponding charge $V_{0}$ takes the following simple
form:

$V_{0}\ =\ \int_{V}\mathrm{d}^{3}x\,v_{0}\ =\
\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,({\textstyle\frac{\ell}{\gamma}})\,(1-\omega_{4}^{2})\,\rho\
=\
2\ell\,\Omega^{2}\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,(1+\omega_{4})\,Z_{a}\,\bar{Z}_{a}\
.$ (3.35)

Here again the $\omega_{4}$ term of the integral, being odd, vanishes and
yields

$V_{0}\ =\ \ell^{2}\,E\ =\
8\,\ell\,(j+1)^{3}(2j+1)\,\sum_{m,n}\,|\Lambda_{m,n}|^{2}\ .$ (3.36)

Vector SCT. The charge densities $J^{0}$ that correspond to $b_{i}$ read:
$\textbf{v}=2\textbf{x}(\textbf{x}\cdot\textbf{p})-2t\,\textbf{x}\,e-(\textbf{x}^{2}-t^{2})\textbf{p}$.
This simplify at $t\\!=0$ and take a structure similar to the momentum
densities $p_{i}$ (3.15):

$v_{i}\,dx^{i}\ =\
({\textstyle\frac{\gamma}{\ell}})\,\mathcal{V}_{a}\,e^{a}\quad\textrm{with}\quad\mathcal{V}_{a}\
=\ 2\omega_{a}\,(\mathcal{P}_{b}\,\omega_{b})-\omega_{b}^{2}\,\mathcal{P}_{a}\
.$ (3.37)

The expressions for the charges $V_{i}$ then simplify to

$V_{i}\ =\ \int_{V}\mathrm{d}^{3}x\,v_{i}\ =\
2\mathrm{i}\,\ell^{2}\,\Omega^{2}\int_{S^{3}}\mathrm{d}^{3}\Omega_{3}\,\gamma^{-2}e^{a}_{i}\left(2\varepsilon_{bcd}\,\omega_{a}\,\omega_{b}\,Z_{c}\,\bar{Z}_{d}-\varepsilon_{abc}\,(1-\omega_{4}^{2})\,Z_{b}\,\bar{Z}_{c}\right)\
.$ (3.38)

With explicit computation we observe that the charges $V_{i}$ are proportional
to the momenta $P_{i}$ (verified explicitly for up to $j=1$)

$V_{i}\ =\ \ell^{2}\,P_{i}\ .$ (3.39)

As before, we can compute the spherical components
$(V_{r},V_{\theta},V_{\phi})$ by using the expressions (3.19) and replacing
$\mathcal{P}$ with $\mathcal{V}$ in it. We notice that the charge $V_{r}$
vanishes owing to the odd $S^{2}$ integrand101010Observe that the terms in
$\mathcal{V}_{a}$ (3.37) are all even in $\hat{\omega}_{a}$. just like in the
case of $P_{r}$. However, unlike $P_{\theta}$ here the charge $V_{\theta}$ is
non-vanishing. Explicit calculations show that for $j=0$ the charges
$V_{\theta}$ and $V_{\phi}$ are proportional to the momentum $P_{3}$:

$V_{\theta}^{(j=0)}\ =\
-{\textstyle\frac{4}{3}}\ell^{3}\,P_{3}^{(j=0)}\quad\quad\textrm{and}\quad\quad
V_{\phi}^{(j=0)}\ =\ -{\textstyle\frac{5}{3}}\ell^{3}\,P_{3}^{(j=0)}\ .$
(3.40)

Additionally, we record below the charges $V_{\theta}$ and $V_{\phi}$ for the
non-trivial case of $j=1/2$

$\displaystyle V_{\theta}^{\left(j=1/2\right)}\ =\
-{\textstyle\frac{6}{5}}\ell^{2}\,\Big{(}$ $\displaystyle
9|\Lambda_{-1/2,-3/2}|^{2}+|\Lambda_{-1/2,-1/2}|^{2}-9|\Lambda_{-1/2,1/2}|^{2}-21|\Lambda_{-1/2,3/2}|^{2}$
(3.41)
$\displaystyle\qquad\qquad\qquad\qquad+21|\Lambda_{1/2,-3/2}|^{2}+9|\Lambda_{1/2,-1/2}|^{2}-|\Lambda_{1/2,1/2}|^{2}-9|\Lambda_{1/2,3/2}|^{2}\Big{)}\
,$ $\displaystyle V_{\phi}^{\left(j=1/2\right)}\ =\
-{\textstyle\frac{3}{5}}\ell^{2}\,\Big{(}$ $\displaystyle
42|\Lambda_{-1/2,-3/2}|^{2}+33|\Lambda_{-1/2,-1/2}|^{2}+8|\Lambda_{-1/2,1/2}|^{2}-33|\Lambda_{-1/2,3/2}|^{2}+33|\Lambda_{1/2,-3/2}|^{2}$
(3.42)
$\displaystyle\qquad-8|\Lambda_{1/2,-1/2}|^{2}-33|\Lambda_{1/2,1/2}|^{2}-42|\Lambda_{1/2,3/2}|^{2}+8\sqrt{3}\big{(}-\bar{\Lambda}_{1/2,-3/2}\Lambda_{-1/2,-1/2}$
$\displaystyle\qquad\qquad\quad\quad+\bar{\Lambda}_{1/2,1/2}\Lambda_{-1/2,3/2}-\bar{\Lambda}_{-1/2,-1/2}\Lambda_{1/2,-3/2}+\bar{\Lambda}_{-1/2,3/2}\Lambda_{1/2,1/2}\big{)}\Big{)}\
.$

One can compute these charges for higher spin $j$ by following the same
strategy.

##  4 Applications

The method of constructing rational electromagnetic fields presented in this
paper has the added advantage that it produces a complete set labelled by
$(j,m,n)$; any electromagnetic field configuration having finite energy can,
in principle, be obtained from an expansion like in (2.32-2.34), albeit with a
varying $j$. The operational difficulty involved in this procedure has to do
with the fact that this set is infinite as
$j\in{\textstyle\frac{\mathds{N}}{2}}$. There are, however, many important
cases where only a finite number of knot-basis solutions (sometimes only with
a fixed $j$) need to be combined to get the desired EM field configuration.
One such very important case is that of Hopfian solution given by Ranãda,
which was already covered in [7]. Below we analyse two very interesting
generalizations of Hopfian solution presented in [4] in the context of present
construction. It is imperative to note here that while the scope of
construction of a new solution from the known ones as presented in [4] is
limited, the same is not true for the method presented in this paper, which by
design can produce arbitrary number of new field configurations. Some of these
possible new field configurations obtained from the $j\\!=\\!0$ sector
(possibly from $j\\!=\\!1/2\leavevmode\nobreak\
\textrm{or}\leavevmode\nobreak\ 1$ as well) could find experimental
application like, e.g. in [5].

Figure 3:  Sample electric (red) and magnetic (green) field lines at $t{=}0$.
Left: time-translated Hopfian with $c=60$, Right: Rotated Hopfian with
$\theta=1$.

Bateman’s construction, employed in [4], hinges on a judicious ansatz for the
Riemann–Silberstein vector (2.40) satisfying Maxwell’s equations:

$\textbf{S}\ =\
\nabla\alpha\times\nabla\beta\quad\implies\quad\nabla\cdot\textbf{S}\ =\
0\leavevmode\nobreak\ \ \&\leavevmode\nobreak\ \
\mathrm{i}\,\partial_{t}\textbf{S}\ =\ \nabla\times\textbf{S}\ ,$ (4.1)

using a pair of complex functions $(\alpha,\beta)$. An interesting
generalization of the Hopfian solution is obtained in equations (3.16-3.17) of
[4] using (complex) time-translation (TT) to obtain the following
$(\alpha,\beta)$ pair (up to a normalization)

$(\alpha,\beta)_{TT}\ =\
\left(\frac{A-1-\mathrm{i}z}{A+\mathrm{i}(t+\mathrm{i}c)},\frac{x-\mathrm{i}y}{A+\mathrm{i}(t+\mathrm{i}c)}\right)\
;\quad A\ =\
{\textstyle\frac{1}{2}}\left(x^{2}+y^{2}+z^{2}-(t+\mathrm{i}c)^{2}+1\right)\
,$ (4.2)

where $c$ is a constant real parameter. The corresponding EM field
configuration is obtained, in our case, by choosing $\ell=1-c$ and only the
following $j=0$ complex coefficients in (2.33)

$\left(\Lambda_{0;0,1}\ ,\leavevmode\nobreak\ \Lambda_{0;0,0}\
,\leavevmode\nobreak\ \Lambda_{0;0,-1}\right)_{TT}\ =\ \left(0\
,\leavevmode\nobreak\ 0\ ,\leavevmode\nobreak\
-\mathrm{i}\frac{\pi}{2\ell^{2}}\right)\ .$ (4.3)

A sample electromagnetic knot configuration of this modified Hopfian is
illustrated in Figure 3. Another interesting generalization of the Hopfian is
constructed in equations (3.20-3.21) of [4] using a (complex) rotation (R) to
get the following $(\alpha,\beta)$ pair (again, up to a normalization)

$(\alpha,\beta)_{R}\ =\
\left(\frac{A-1+\mathrm{i}(z\cos{\mathrm{i}\theta}+x\sin{\mathrm{i}\theta})}{A+\mathrm{i}t},\frac{x\cos{\mathrm{i}\theta}-z\sin{\mathrm{i}\theta}-\mathrm{i}y}{A+\mathrm{i}t}\right)\
;\ A\ =\ {\textstyle\frac{1}{2}}\left(x^{2}+y^{2}+z^{2}-t^{2}+1\right)\ .$
(4.4)

To get this particular EM field configuration, we need to set $\ell=1$ and use
the following combination of $j=0$ coefficients in (2.33)

$\left(\Lambda_{0;0,1}\ ,\leavevmode\nobreak\ \Lambda_{0;0,0}\
,\leavevmode\nobreak\ \Lambda_{0;0,-1}\right)_{R}\ =\
\left(\mathrm{i}\frac{\pi}{4}(\cosh{\theta}-1)\ ,\leavevmode\nobreak\
-\frac{\pi}{2\sqrt{2}}\sinh{\theta}\ ,\leavevmode\nobreak\
-\mathrm{i}\frac{\pi}{4}(\cosh\theta+1)\right)\ .$ (4.5)

We illustrate the EM field lines for a sample $\theta$ value of this modified
Hopfian in Figure 3. We note down the conformal charges corresponding to these
solutions in Table 2 by plugging the $j=0$ coefficients (4.3) and (4.5) in the
appropriate formulae of the previous section. The results match with the ones
given in [4] up to a rescaling of the energy, which can be achieved by an
appropriate choice of normalization.

|  Time-translated Hopfian |  Rotated Hopfian
---|---|---
Energy (E) |  $\frac{2\pi^{2}}{(1-c)^{5}}=:E_{TT}$ |  $2\pi^{2}\cosh^{2}\theta=:E_{R}$
Momentum (P) |  $\left(0\ ,\leavevmode\nobreak\ 0\ ,\leavevmode\nobreak\ \frac{1}{4}\right)E_{TT}$ |  $\left(0\ ,\leavevmode\nobreak\ -\frac{1}{4}\tanh{\theta}\ ,\leavevmode\nobreak\ \frac{1}{4}\operatorname{sech}\theta\right)E_{R}$
Boost (K) |  $\left(0\ ,\leavevmode\nobreak\ 0\ ,\leavevmode\nobreak\ 0\right)$ |  $\left(0\ ,\leavevmode\nobreak\ 0\ ,\leavevmode\nobreak\ 0\right)$
Ang. momentum (L) |  $\left(0\ ,\leavevmode\nobreak\ 0\ ,\leavevmode\nobreak\ -\frac{1}{4}(1-c)^{2}\right)E_{TT}$ |  $\left(0\ ,\leavevmode\nobreak\ \frac{1}{4}\tanh{\theta}\ ,\leavevmode\nobreak\ -\frac{1}{4}\operatorname{sech}\theta\right)E_{R}$
Dilatation (D) |  $0$ |  $0$
Scalar SCT ($V_{0}$) |  $(1-c)^{2}E_{TT}$ |  $E_{R}$
Vector SCT (V) |  $\left(0\ ,\leavevmode\nobreak\ 0\ ,\leavevmode\nobreak\ \frac{1}{4}(1-c)^{2}\right)E_{TT}$ |  $\left(0\ ,\leavevmode\nobreak\ -\frac{1}{4}\tanh{\theta}\ ,\leavevmode\nobreak\ \frac{1}{4}\operatorname{sech}\theta\right)E_{R}$
Table 2: Conformal charges for the time-translated and rotated Hopfian
configurations.

## Acknowledgements

KK is grateful to Deutscher Akademischer Austauschdienst (DAAD) for the
doctoral research grant 57381412. We thank Olaf Lechtenfeld for several
insightful discussions and valuable suggestions.

##  Appendix A Rotation of indices

By construction the gauge potential ${\cal A}$ is $SO(4)$ invariant, which
means it is also invariant under the action of $SO(3)$ generators
$\mathcal{D}_{a}$ (2.15). For a complex-valued ${\cal A}={\cal A}_{a}\,e^{a}$
expanded using (2.32) and (2.35) at a fixed $j$ this means

$0\ =\ \mathcal{D}_{a}({\cal A})\ =\
\sum_{m,n,\tilde{n}}C^{n,\tilde{n}}_{j,b}\,\mathrm{e}^{\mathrm{i}\Omega\tau}\,\left(\mathcal{D}_{a}(\Lambda_{m,\tilde{n}})\,e^{b}\,Y_{j;m,n}+\Lambda_{m,\tilde{n}}\,\mathcal{D}_{a}(e^{b})\,Y_{j;m,n}+\Lambda_{m,\tilde{n}}\,e^{b}\,\mathcal{D}_{a}(Y_{j;m,n})\right)$
(A.1)

where $\mathcal{D}_{a}(e^{b})$ are determined from (2.16) while
$\mathcal{D}_{a}(Y_{j;m,n})$ are determined from (2.18-2.20). By collecting
the coefficients of various linearly independent $e^{b}$ and $Y_{j;m,n}$ terms
in the above expansion for a fixed $\mathcal{D}_{a}$ one gets a set of coupled
linear equations for $\mathcal{D}_{a}(\Lambda_{m,\tilde{n}})$, which can be
easily solved. The action of the generators $\mathcal{D}_{a}$ on
$\Lambda_{m,\tilde{n}}$ for $j=0,\leavevmode\nobreak\ 1/2\leavevmode\nobreak\
\textrm{and}\leavevmode\nobreak\ 1$ is given in Table 3.

|  $\mathcal{D}_{1}$ |  $\mathcal{D}_{2}$ |  $\mathcal{D}_{3}$
---|---|---|---
$\left.j=0:\right.\begin{aligned} &\Lambda_{0,-1}\mapsto\\\ &\Lambda_{0,0}\mapsto\\\ &\Lambda_{0,1}\mapsto\end{aligned}$ |  $\begin{aligned} &\sqrt{2}\mathrm{i}\Lambda_{0,0}\\\ &\sqrt{2}\mathrm{i}(\Lambda_{0,1}+\Lambda_{0,-1})\\\ &\sqrt{2}\mathrm{i}\Lambda_{0,0}\end{aligned}$ |  $\begin{aligned} &-\sqrt{2}\Lambda_{0,0}\\\ &\sqrt{2}(\Lambda_{0,-1}-\Lambda_{0,1})\\\ &\sqrt{2}\Lambda_{0,0}\end{aligned}$ |  $\begin{aligned} &-\sqrt{2}\mathrm{i}\Lambda_{0,-1}\\\ &\quad 0\\\ &\sqrt{2}\mathrm{i}\Lambda_{0,1}\end{aligned}$
$\left.\begin{aligned} &\quad j={\textstyle\frac{1}{2}}:\\\ &\underbrace{\textrm{Notation}}\\\ &\pm{\textstyle\frac{1}{2}}\equiv\pm\\\ &\pm{\textstyle\frac{3}{2}}\equiv\uparrow\downarrow\end{aligned}\right\\}\begin{aligned} &\Lambda_{-,\downarrow}\mapsto\\\\[1.99997pt] &\Lambda_{-,-}\mapsto\\\\[1.99997pt] &\Lambda_{-,+}\mapsto\\\\[1.99997pt] &\Lambda_{-,\uparrow}\mapsto\\\\[1.99997pt] &\Lambda_{+,\downarrow}\mapsto\\\\[1.99997pt] &\Lambda_{+,-}\mapsto\\\\[1.99997pt] &\Lambda_{+,+}\mapsto\\\\[1.99997pt] &\Lambda_{+,\uparrow}\mapsto\end{aligned}$ |  $\begin{aligned} &\mathrm{i}(\sqrt{3}\Lambda_{-,-}+\Lambda_{+,\downarrow})\\\ &\mathrm{i}(\sqrt{3}\Lambda_{-,\downarrow}+2\Lambda_{-,+}+\Lambda_{+,-})\\\ &\mathrm{i}(2\Lambda_{-,-}+\sqrt{3}\Lambda_{-,\uparrow}+\Lambda_{+,+})\\\ &\mathrm{i}(\sqrt{3}\Lambda_{-,+}+\Lambda_{+,\uparrow})\\\ &\mathrm{i}(\Lambda_{-,\downarrow}+\sqrt{3}\Lambda_{+,-})\\\ &\mathrm{i}(\Lambda_{-,-}+\sqrt{3}\Lambda_{+,\downarrow}+2\Lambda_{+,+})\\\ &\mathrm{i}(\Lambda_{-,+}+2\Lambda_{+,-}+\sqrt{3}\Lambda_{+,\uparrow})\\\ &\mathrm{i}(\Lambda_{-,\uparrow}+\sqrt{3}\Lambda_{+,+})\\\ \end{aligned}$ |  $\begin{aligned} &-\sqrt{3}\Lambda_{-,-}-\Lambda_{+,\downarrow}\\\ &\sqrt{3}\Lambda_{-,\downarrow}-2\Lambda_{-,+}-\Lambda_{+,-}\\\ &2\Lambda_{-,-}-\sqrt{3}\Lambda_{-,\uparrow}-\Lambda_{+,+}\\\ &\sqrt{3}\Lambda_{-,+}-\Lambda_{+,\uparrow}\\\ &\Lambda_{-,\downarrow}-\sqrt{3}\Lambda_{+,-}\\\ &\Lambda_{-,-}+\sqrt{3}\Lambda_{+,\downarrow}-2\Lambda_{+,+}\\\ &\Lambda_{-,+}+2\Lambda_{+,-}-\sqrt{3}\Lambda_{+,\uparrow}\\\ &\Lambda_{-,\uparrow}+\sqrt{3}\Lambda_{+,+}\\\ \end{aligned}$ |  $\begin{aligned} &-4\mathrm{i}\Lambda_{-,\downarrow}\\\\[1.99997pt] &-2\mathrm{i}\Lambda_{-,-}\\\\[1.99997pt] &0\\\\[1.99997pt] &2\mathrm{i}\Lambda_{-,\uparrow}\\\\[1.99997pt] &-2\mathrm{i}\Lambda_{+,\downarrow}\\\\[1.99997pt] &0\\\\[1.99997pt] &2\mathrm{i}\Lambda_{+,+}\\\\[1.99997pt] &4\mathrm{i}\Lambda_{+,\uparrow}\end{aligned}$
$\left.\begin{aligned} &\quad j=1:\\\ &\underbrace{\textrm{Notation}}\\\ &\pm 1\equiv\pm\\\ &\pm 2\equiv\uparrow\downarrow\end{aligned}\right\\}\begin{aligned} &\Lambda_{-,\downarrow}\mapsto\\\\[1.99997pt] &\Lambda_{-,-}\mapsto\\\\[1.99997pt] &\Lambda_{-,0}\mapsto\\\\[1.99997pt] &\Lambda_{-,+}\mapsto\\\\[1.99997pt] &\Lambda_{-,\uparrow}\mapsto\\\\[1.99997pt] &\Lambda_{0,\downarrow}\mapsto\\\\[1.99997pt] &\Lambda_{0,-}\mapsto\\\\[1.99997pt] &\Lambda_{0,0}\mapsto\\\\[1.99997pt] &\Lambda_{0,+}\mapsto\\\\[1.99997pt] &\Lambda_{0,\uparrow}\mapsto\\\\[1.99997pt] &\Lambda_{+,\downarrow}\mapsto\\\\[1.99997pt] &\Lambda_{+,-}\mapsto\\\\[1.99997pt] &\Lambda_{+,0}\mapsto\\\\[1.99997pt] &\Lambda_{+,+}\mapsto\\\\[1.99997pt] &\Lambda_{+,\uparrow}\mapsto\end{aligned}$ |  $\begin{aligned} &\mathrm{i}(2\Lambda_{-,-}+\sqrt{2}\Lambda_{0,\downarrow})\\\ &\mathrm{i}(2\Lambda_{-,\downarrow}+\sqrt{6}\Lambda_{-,0}+\sqrt{2}\Lambda_{0,-})\\\ &\mathrm{i}(\sqrt{6}\Lambda_{-,-}+\sqrt{6}\Lambda_{-,+}+\sqrt{2}\Lambda_{0,0})\\\ &\mathrm{i}(\sqrt{6}\Lambda_{-,0}+2\Lambda_{-,\uparrow}+\sqrt{2}\Lambda_{0,+})\\\ &\mathrm{i}(2\Lambda_{-,+}+\sqrt{2}\Lambda_{+,\uparrow})\\\ &\mathrm{i}(\sqrt{2}\Lambda_{-,\downarrow}+2\Lambda_{0,-}+\sqrt{2}\Lambda_{+,\downarrow})\\\ &\mathrm{i}\sqrt{2}(\Lambda_{-,-}+\sqrt{2}\Lambda_{0,\downarrow}+\sqrt{3}\Lambda_{0,0}+\Lambda_{+,-})\\\ &\mathrm{i}\sqrt{2}(\Lambda_{-,0}+\sqrt{3}\Lambda_{0,-}+\sqrt{3}\Lambda_{0,+}+\Lambda_{+,0})\\\ &\mathrm{i}\sqrt{2}(\Lambda_{-,+}+\sqrt{3}\Lambda_{0,0}+\sqrt{2}\Lambda_{0,\uparrow}+\Lambda_{+,+})\\\ &\mathrm{i}(\sqrt{2}\Lambda_{-,\uparrow}+2\Lambda_{0,+}+\sqrt{2}\Lambda_{+,\uparrow})\\\ &\mathrm{i}(\sqrt{2}\Lambda_{0,\downarrow}+2\Lambda_{+,-})\\\ &\mathrm{i}(\sqrt{2}\Lambda_{0,-}+2\Lambda_{+,\downarrow}+\sqrt{6}\Lambda_{+,0})\\\ &\mathrm{i}(\sqrt{2}\Lambda_{0,0}+\sqrt{6}\Lambda_{+,-}+\sqrt{6}\Lambda_{+,+})\\\ &\mathrm{i}(\sqrt{2}\Lambda_{0,+}+\sqrt{6}\Lambda_{+,0}+2\Lambda_{+,\uparrow})\\\ &\mathrm{i}(\sqrt{2}\Lambda_{0,\uparrow}+2\Lambda_{+,+})\\\ \end{aligned}$ |  $\begin{aligned} &-2\Lambda_{-,-}-\sqrt{2}\Lambda_{0,\downarrow}\\\ &2\Lambda_{-,\downarrow}-\sqrt{6}\Lambda_{-,0}-\sqrt{2}\Lambda_{0,-}\\\ &\sqrt{6}\Lambda_{-,-}-\sqrt{6}\Lambda_{-,+}-\sqrt{2}\Lambda_{0,0}\\\ &\sqrt{6}\Lambda_{-,0}-2\Lambda_{-,\uparrow}-\sqrt{2}\Lambda_{0,+}\\\ &2\Lambda_{-,+}-\sqrt{2}\Lambda_{0,\uparrow}\\\ &\sqrt{2}\Lambda_{-,\downarrow}-2\Lambda_{0,-}-\sqrt{2}\Lambda_{+,\downarrow}\\\ &\sqrt{2}(\Lambda_{-,-}+\sqrt{2}\Lambda_{0,\downarrow}-\sqrt{3}\Lambda_{0,0}-\Lambda_{+,-})\\\ &\sqrt{2}(\Lambda_{-,0}+\sqrt{3}\Lambda_{0,-}-\sqrt{3}\Lambda_{0,+}-\Lambda_{+,0})\\\ &\sqrt{2}(\Lambda_{-,+}+\sqrt{3}\Lambda_{0,0}-\sqrt{2}\Lambda_{0,\uparrow}-\Lambda_{+,+})\\\ &\sqrt{2}\Lambda_{-,\uparrow}+2\Lambda_{0,+}-\sqrt{2}\Lambda_{+,\uparrow}\\\ &\sqrt{2}\Lambda_{0,\downarrow}+-2\Lambda_{+,-}\\\ &\sqrt{2}\Lambda_{0,-}+2\Lambda_{+,\downarrow}-\sqrt{6}\Lambda_{+,0}\\\ &\sqrt{2}\Lambda_{0,0}+\sqrt{6}\Lambda_{+,-}-\sqrt{6}\Lambda_{+,+}\\\ &\sqrt{2}\Lambda_{0,+}+\sqrt{6}\Lambda_{+,0}-2\Lambda_{+,\uparrow}\\\ &\sqrt{2}\Lambda_{0,\uparrow}+2\Lambda_{+,+}\\\ \end{aligned}$ |  $\begin{aligned} &-6\mathrm{i}\Lambda_{-,\downarrow}\\\\[1.99997pt] &-4\mathrm{i}\Lambda_{-,-}\\\\[1.99997pt] &-2\mathrm{i}\Lambda_{-,0}\\\\[1.99997pt] &0\\\\[1.99997pt] &2\mathrm{i}\Lambda_{-,\uparrow}\\\\[1.99997pt] &-4\mathrm{i}\Lambda_{0,\downarrow}\\\\[1.99997pt] &-2\mathrm{i}\Lambda_{0,-}\\\\[1.99997pt] &0\\\\[1.99997pt] &2\mathrm{i}\Lambda_{0,+}\\\\[1.99997pt] &4\mathrm{i}\Lambda_{0,\uparrow}\\\\[1.99997pt] &-2\mathrm{i}\Lambda_{+,\downarrow}\\\\[1.99997pt] &0\\\\[1.99997pt] &2\mathrm{i}\Lambda_{+,0}\\\\[1.99997pt] &4\mathrm{i}\Lambda_{+,+}\\\\[1.99997pt] &6\mathrm{i}\Lambda_{+,\uparrow}\end{aligned}$

Table 3: Action of $\mathcal{D}_{a}$ on $\Lambda_{m,\tilde{n}}$ for
$j=0,\leavevmode\nobreak\ 1/2\leavevmode\nobreak\
\textrm{and}\leavevmode\nobreak\ 1$.

## References

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  * [4] C. Hoyos, N. Sircar and J. Sonnenschein. J. Phys. A: Math. Theor. 48 (2015) 255204,  arXiv:1502.01382 [hep-th].
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  * [6] K. Kumar, O. Lechtenfeld, G. Picanço Costa, Trajectories of charged particles in knotted electromagnetic fields, arXiv:2202.00169[math-ph].
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# The Axion Quality and Fifth Force

Yue Zhang  Department of Physics, Carleton University, Ottawa, ON K1S 5B6,
Canada

###### Abstract

We investigate what it takes for the axion to address the strong CP problem in
the presence of explicit Peccei-Quinn (PQ) symmetry breaking effects besides
the strong interaction. In cases where the PQ-Higgsing scalar field directly
couples to the Standard Model sector, it is pointed out that existing fifth
force experiments can set better constraints on the axion quality over neutron
electric dipole moment.

The QCD axion, a light pseudo-Goldstone boson from spontaneous breaking of the
Peccei-Quinn (PQ) symmetry Peccei and Quinn (1977a, b); Wilczek (1978);
Weinberg (1978), provides a dynamical and elegant solution to the strong CP
problem of the Standard Model. With an appropriate choice of its decay
constant, axion can also comprise all the dark matter that fills our universe
Preskill  _et al._ (1983); Abbott and Sikivie (1983); Dine and Fischler
(1983). Currently there is an extensive experimental program in search for
evidence of such an appealing new physics candidate Sikivie (2021); Irastorza
and Redondo (2018).

A key ingredient for the axion solution to work is the quality of the PQ
global symmetry Kamionkowski and March-Russell (1992); Holman  _et al._
(1992); Barr and Seckel (1992); Dine (2022). In general, explicit PQ symmetry
violation besides strong interaction could cause the physical $\bar{\Theta}$
parameter to deviate from zero, which is tightly constrained by neutron
electric dipole moment (EDM) measurement Baker  _et al._ (2006) to be less
than $\sim 10^{-10}$. While the strong CP problem asks why $\bar{\Theta}$ is
so small, the axion turns the question into why such PQ breaking effects are
so feeble. Indeed, to satisfy the EDM constraint, a Planck-scale suppressed
operator made of the PQ-Higgsing scalar must be of unusually high dimensions
Dine (2022). It inspired a number of proposals of building models with a high-
quality PQ symmetry Georgi  _et al._ (1981); Dias  _et al._ (2004); Cheung
(2010); Di Luzio  _et al._ (2017); Duerr  _et al._ (2018); Lillard and Tait
(2018); Alvey and Escudero (2021).

Global symmetries are not meant to be exact. Such a viewpoint has been widely
appreciated regarding the baryon and lepton number symmetries of nature, in
which case various symmetry breaking phenomena have been suggested and under
scrutiny Reines  _et al._ (1954); Kuzmin (1970); Georgi and Glashow (1974);
Furry (1939); Ng and Kamal (1978); Missimer  _et al._ (1994). In this note, we
apply a similar philosophy to the PQ symmetry. Rather than respecting it to
high degrees, we are most interested in the physical consequences of general
quality-violating effects on the axion, beyond EDM. In particular, we will
consider direct couplings of the PQ field to the Standard Model particles,
where the lightness of axion allows the fifth force experiments to play an
important role.

The leading-order axion potential arises from the strong interaction

$V(a)=\frac{1}{2}m_{a}^{2}(a+\bar{\Theta}f_{a})^{2},\quad\quad
m_{a}=\frac{\sqrt{m_{u}m_{d}}}{m_{u}+m_{d}}\frac{m_{\pi}F_{\pi}}{f_{a}}\ .$
(1)

Without other terms, the axion condensate $a/f_{a}=-\bar{\Theta}$ minimizes
the potential and the solution to strong CP problem remains intact. In this
work, we will consider small perturbations to this minimum from additional
explicit PQ breaking effects. It suffices to truncate the potential to
quadratic order rather than resorting to the more general form Grilli di
Cortona  _et al._ (2016).

Next, we turn on additional explicit PQ breaking operators, by directly
coupling the PQ field to known particles. Because only low-energy experimental
constraints will be explored, we simply work in the broken electroweak
symmetry phase.

Case 1: coupling to electron.

We first explore explicit PQ-breaking effects from the effective operator

$\delta\mathcal{L}=-\frac{m_{e}}{\Lambda_{e}}e^{i\delta}\phi\,\bar{e}_{L}e_{R}+{\rm
h.c.}\ ,$ (2)

where the scalar field $\phi$ that Higgses the PQ symmetry takes the low-
energy form

$\phi=\frac{f_{a}}{\sqrt{2}}e^{ia/f_{a}}\ .$ (3)

We first explore contribution to the axion potential from the above
interaction. By closing the electron fields in a loop, Eq. (2) could give
radiative correction to another operator $\phi H^{\dagger}H$ and in turn an
axion potential that is quadratically sensitive to ultraviolet energy scale.
Given the ignorance of detailed high-scale physics, the coefficient of $\phi
H^{\dagger}H$ is not calculable but only can be determined by experiments. In
the spirit of the “finite naturalness” argument Farina  _et al._ (2013);
Bardeen (1995), we do not include such contributions in this analysis, but
rather focus on those involving only known physical scales, which is the
electron mass here. The finite radiative correction corresponds to a Coleman-
Weinberg potential Coleman and Weinberg (1973),

$\delta
V(a)=\frac{|M_{e}(a)|^{4}}{64\pi^{2}}\left(\ln\frac{|M_{e}(a)|^{2}}{\mu^{2}}-\frac{3}{2}\right)\
,$ (4)

where $\mu$ is the renormalization scale, and the axion field dependent
electron mass takes the form

$M_{e}(a)=m_{e}\left(1+\frac{f_{a}}{\sqrt{2}\Lambda_{e}}e^{i(a/f_{a}+\delta)}\right)\
.$ (5)

The total potential $V+\delta V(a)$ is minimized with a nonzero axion
condensate that deviates from $-\bar{\Theta}f_{a}$. In the limit
$f_{a}\ll\Lambda_{e}$,

$\frac{a}{f_{a}}+\bar{\Theta}\simeq\frac{(m_{u}+m_{d})^{2}}{m_{u}m_{d}}\frac{m_{e}^{4}}{16\sqrt{2}\pi^{2}m_{\pi}^{2}F_{\pi}^{2}}\frac{f_{a}\sin(\delta-\bar{\Theta})}{\Lambda_{e}}\left(\ln\frac{m_{e}^{2}}{\mu^{2}}-1\right)\
.$ (6)

In general, the right-hand side does not vanish because there is no reason for
the phase factor $\delta$ to be close to $\bar{\Theta}$. The neutron EDM
constrains $a/f_{a}+\bar{\Theta}<10^{-10}$. By approximating the factor
$[\ln(m_{e}^{2}/\mu^{2})-1]$ as unity, EDM sets a lower bound on the axion
quality parameter,

$\frac{\Lambda_{e}}{\left|\sin(\delta-\bar{\Theta})\right|}>5.3\times
10^{10}\,{\rm GeV}\left(\frac{10^{-5}\,\rm eV}{m_{a}}\right)\ ,$ (7)

where we have used $m_{e}=0.511\,$MeV, $m_{u}=2.16\,$MeV, $m_{d}=4.67\,$MeV,
$m_{\pi}=135\,$MeV, $F_{\pi}=92.2\,$MeV Zyla  _et al._ (2020). This bound
corresponds to the purple curve in FIG. 1 (left).

We proceed to consider more effects due to the operator Eq. (2), by observing
that it contains an axion coupling to the CP-even bilinear fermion operator.
Taylor expanding it to linear order in the axion field, we get

$\delta\mathcal{L}\simeq\frac{m_{e}}{\sqrt{2}\Lambda_{e}}\left[\sin(\delta-\bar{\Theta})\bar{e}e-\cos(\delta-\bar{\Theta})\bar{e}i\gamma_{5}e\right]a\
.$ (8)

Here $a$ should be understood as the excitation on top of the axion condensate
found above, which at leading order is approximately $-\bar{\Theta}f_{a}$. Eq.
(8) implies a coherent coupling of axion to many atoms at low momentum
transfers. With a very small mass, the axion can mediate a fifth force between
two macroscopic objects. Their long-range potential is modified from pure
gravity by a factor $\left(1+\alpha e^{-m_{a}r}\right)$, where

$\alpha=\frac{Z_{1}Z_{2}}{A_{1}A_{2}}\frac{m_{e}^{2}\sin^{2}(\delta-\bar{\Theta})}{8\pi
Gu^{2}\Lambda_{e}^{2}}\ ,$ (9)

and $G$ is the gravitational constant, $u=931.5\,$MeV is the atomic mass unit,
$Z_{1,2}$ and $A_{1,2}$ denote the atomic charge and atomic weight of the
gravitating objects, respectively. A nonzero $\alpha$ modifies the
gravitational inverse-square law and is constrained by torsion balance
experiments Adelberger  _et al._ (2009) as a function of the mediator mass.
The fifth force constraint is shown by the blue curve in FIG. 1 (left).

For an even lighter axion, Eq. (8) is also constrained by the test of
equivalent principle (EP). Testing point objects made of different materials
can experience different acceleration toward a common source. This effect is
characterized by a parameter similar to $\alpha$,

$\widetilde{\alpha}=\frac{m_{e}^{2}\sin^{2}(\delta-\bar{\Theta})}{8\pi
Gu^{2}\Lambda_{e}^{2}}\ .$ (10)

Experimental constraint on $\widetilde{\alpha}$ Adelberger  _et al._ (2009) is
shown by the yellow curve in FIG. 1 (left).

Moreover, the scalar coupling $a\bar{e}e$ is tightly constrained by
astrophysics due to excessive cooling to red giant stars. The limit derived in
Hardy and Lasenby (2017) corresponds to the green curve in the figure. We note
this constraint weakens substantially Raffelt and Weiss (1995) due to the lack
of plasmon effect if $\delta=\bar{\Theta}$ and only the pseudoscalar coupling
$a\bar{e}i\gamma_{5}e$ is present.

Remarkably, the effects of operator Eq. (19) in EDM, violations of $1/r^{2}$
law, and EP all vanish in the limit where the phase factor $\delta$ approaches
to $\bar{\Theta}$ or $\Lambda_{e}$ goes to infinity. All of them constrain the
axion quality violation through the same parameter
$\Lambda_{e}/|\sin(\delta-\bar{\Theta})|$. FIG. 1 (left) states that the
leading constraints on the PQ field coupling to electron are never from the
EDM experiment throughout the entire mass range of axion between $10^{-12}$ eV
and eV scale. Laboratory tests of the equivalence principle, inverse-square
law of gravity, and stellar cooling set much better constraints. In this case,
they are more sensitive probes of the axion quality over EDM.

As additional remarks, the two interactions in Eq. (8) also contribute to the
electron EDM at one-loop level. However, given the above constraints on
$\Lambda_{e}$, we find the contribution to be negligible compared to the
latest limit from the ACME experiment Andreev  _et al._ (2018). The physical
electron mass in the presence of the PQ breaking operator is $|M_{e}(a)|$ with
$a$ set by the minimum value Eq. (6), instead of $m_{e}$. Experimentally, the
electron mass has been measured to very high precision Sturm  _et al._ (2014).
In the case of axion being the dark matter around us, its coherent oscillation
could lead to potentially interesting time dependence in the mass of electron.

Figure 1: Experimental lower limits on the axion quality parameter
$\Lambda_{e,g}/|\sin(\delta-\bar{\Theta})|$ in the two cases considered in
this work, including neutron EDM (purple), test of gravitational inverse-
square law (blue), test of the equivalence principle (yellow), and red giant
cooling (green), when applicable.

Case 2: coupling to gluon.

As the second exercise, we directly couple the PQ-field to the CP-even gluon
operator,

$\delta\mathcal{L}=-\frac{\alpha_{s}}{\Lambda_{g}}e^{i\delta}\phi\,G_{\mu\nu}^{a}G^{a\mu\nu}+{\rm
h.c.}\ .$ (11)

The vacuum condensate of the gluon operator takes the value Shifman  _et al._
(1979)

$\left\langle
0\left|\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G^{a\mu\nu}\right|0\right\rangle=0.012\,{\rm
GeV}^{4}\ .$ (12)

Through Eq. (11), it generates an axion potential

$\delta V(a)=\frac{\sqrt{2}f_{a}}{\Lambda_{g}}\left\langle
0\left|\alpha_{s}G_{\mu\nu}^{a}G^{a\mu\nu}\right|0\right\rangle\cos\left(\frac{a}{f_{a}}+\delta\right)\
.$ (13)

Minimizing $V+\delta V(a)$ leads to a nonzero axion condensate. For
$f_{a}\ll\Lambda_{g}$, we get

$\frac{a}{f_{a}}+\bar{\Theta}\simeq\frac{(m_{u}+m_{d})^{2}}{m_{u}m_{d}}\frac{\left\langle
0\left|\alpha_{s}G_{\mu\nu}^{a}G^{a\mu\nu}\right|0\right\rangle}{m_{\pi}^{2}F_{\pi}^{2}}\frac{\sqrt{2}f_{a}\sin(\delta-\bar{\Theta})}{\Lambda_{g}}\
.$ (14)

The neutron EDM constraint, $a/f_{a}+\bar{\Theta}<10^{-10}$, translates into

$\frac{\Lambda_{g}}{\left|\sin(\delta-\bar{\Theta})\right|}>9.2\times
10^{24}\,{\rm GeV}\left(\frac{10^{-5}\,\rm eV}{m_{a}}\right)\ ,$ (15)

as shown by the purple curve in FIG. 1 (right). This constraint is
substantially stronger compared to Eq. (7) due to a parametric enhancement
factor, of order $16\pi^{2}(\Lambda_{\rm QCD}/m_{e})^{4}$.

The gluon operator in Eq. (11) also has a non-zero matrix element at the
nucleon level. We use the relation between nucleon mass and the trace anomaly
of energy-momentum tensor Shifman  _et al._ (1978); Donoghue  _et al._ (2014),

$m_{N}\bar{\psi}_{N}\psi_{N}=-\frac{9}{8\pi}\left\langle
N\left|\alpha_{s}G_{\mu\nu}^{a}G^{a\mu\nu}\right|N\right\rangle+\cdots\ ,$
(16)

where $N=p,n$, $m_{N}\simeq 937\,$MeV, and $\dots$ represent contributions
proportional to light quark masses. Under the approximation that the light
quark contributions are negligible Giedt  _et al._ (2009), the coupling of a
single axion excitation to the CP-even bilinear nucleon operator is

$-\frac{8\sqrt{2}\pi
m_{N}\sin(\delta-\bar{\Theta})}{9\Lambda_{g}}(\bar{p}p+\bar{n}n)\,a\ .$ (17)

Again, a light axion can mediate long-range fifth force, where the $\alpha$
and $\tilde{\alpha}$ parameters (the counterpart of Eqs. (9) and (10)) are

$\alpha=\frac{32\pi
m_{N}^{2}\sin^{2}(\delta-\bar{\Theta})}{81Gu^{2}\Lambda_{g}^{2}}\
,\quad\quad\tilde{\alpha}=\frac{64\pi
m_{N}^{2}\sin^{2}(\delta-\bar{\Theta})}{81Gu^{2}\Lambda_{g}^{2}}\ .$ (18)

The corresponding constraints from $1/r^{2}$ law and EP tests Adelberger  _et
al._ (2009) are depicted by the blue and yellow curves in FIG. 1 (right),
respectively.

Like the previous case, both constraints from EDM and the fifth force
experiments are controlled by the same axion quality parameter
$\Lambda_{g}/|\sin(\delta-\bar{\Theta})|$, although here the neutron EDM still
provides the leading constraint on the axion quality. We find a similar result
of comparison holds if the dominant low energy operator is instead $\phi
H^{\dagger}H$.

Case 3: coupling to quark.

Finally, we comment on the coupling of PQ field to the quark mass term. Taking
the up-quark as example,

$\delta\mathcal{L}=-\frac{m_{u}}{\Lambda_{u}}e^{i\delta}\phi\,\bar{u}_{L}u_{R}+{\rm
h.c.}\ .$ (19)

This term modifies the quark mass

$m_{u}\to
M_{u}(a)=m_{u}\left(1+\frac{f_{a}}{\sqrt{2}\Lambda_{u}}e^{i(a/f_{a}+\delta)}\right)\
.$ (20)

To derive the impact on the axion potential, one feeds the $a$-dependent quark
mass to the low-energy chiral Lagrangian. The argument of the complex factor
in the bracket of Eq. (20) simply acts as an extra contribution to the
$\bar{\Theta}$ parameter. The condition for minimizing the axion potential
becomes

$\frac{a}{f_{a}}+\bar{\Theta}+{\rm
arg}\left(1+\frac{f_{a}}{\sqrt{2}\Lambda_{u}}e^{i(a/f_{a}+\delta)}\right)=0\
,$ (21)

which still has a solution as long as $f_{a}\ll\Lambda_{u}$. In this case, the
operator Eq. (19) does not lead to extra terms in the axion potential, nor a
contribution to EDM. On the other hand, it is still constrained by the torsion
balance experiments due to the axion coupling to CP-even quark operator if
$\delta\neq\bar{\Theta}$.

We conclude by stressing that the Peccei-Quinn symmetry for solving the strong
CP problem is a global symmetry and in general allowed to be explicitly broken
by high-scale physics. The quality of this symmetry and the resulting axion
solution could be subject to various experimental probes, not limited to the
EDM measurement. In this work, we discuss a class of examples where the PQ
field has a direct coupling to the Standard Model sector, and deformation of
the axion potential is accompanied with strongly modified charge-parity nature
of the axion. We show that tests of a fifth force mediated by the light axion
can set leading constraint on the higher dimensional operator in certain
cases. The findings of this work suggest it could be fruitful to confront the
axion quality to a broader landscape of experiments and theoretical frameworks
of explicitly broken PQ symmetry.

This work is supported by the Arthur B. McDonald Canadian Astroparticle
Physics Research Institute. The author thanks the organizers of the Pollica
Summer Workshop supported by the Regione Campania, Università degli Studi di
Salerno, Università degli Studi di Napoli “Federico II”, i dipartimenti di
Fisica “Ettore Pancini” and “E R Caianiello”, and Istituto Nazionale di Fisica
Nucleare; and to the townspeople of Pollica, IT, for their generosity and
hospitality.

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