# Theory of wakefields excited by an off–axis drive bunch in a
plasma–dielectric waveguide

K.V. Galaydych<EMAIL_ADDRESS>NSC Kharkiv Institute of Physics and
Technology, 61108 Kharkiv, Ukraine  G.V. Sotnikov  NSC Kharkiv Institute of
Physics and Technology, 61108 Kharkiv, Ukraine  I.N. Onishchenko  NSC Kharkiv
Institute of Physics and Technology, 61108 Kharkiv, Ukraine

###### Abstract

A linear theory of wakefield excitation by a drive electron bunch in a
plasma–dielectric accelerating structure has been constructed for the case of
off–axis bunch injection. The structure under investigation is a round
dielectric–loaded metal waveguide with a channel for charged particles, filled
with homogeneous cold plasma. The developed theory was used to investigate the
spatial distribution of the bunch–excited transverse wakefield components,
which act on the drive electron bunch. The numerical simulations using the
particle-in-cell method have confirmed the predictions of the linear theory in
terms of transverse drive bunch dynamics. The numerical analysis has shown
stable drive bunch transport through the plasma–dielectric accelerating
structure compared with the corresponding dielectric–loaded structure.

##  I INTRODUCTION

Among the advanced acceleration concepts it is possible to highlight four main
schemes, namely, beam–driven plasma wakefield acceleration [1, 2],
laser–driven plasma wakefield acceleration [3], dielectric laser acceleration
[4] and structure–based wakefield acceleration [5]. The dielectric wakefield
accelerating structures belong to the last one and for the present time a
significant progress in their research and development has been made (see, for
example, papers [6, 7, 8]). These structures are in focus of intense studies
at such accelerator centers as ANL, SLAC, BNL, INFN. But despite the efforts
and progress, there are the important tasks and problems that still should be
solved. And solving them may require nonconventional methods and approaches.
As for the conventional accelerators, one of the severe issues of the advanced
accelerators development is the beam breakup (BBU) instability of the drive
bunch. The studies of this instability are of vital importance because this
process is destructive and causes degradation of the drive bunch and the
generated wakefield quality, reduces the drive bunch intensity, and restricts
its propagation length. In turn, all these negative factors can affect the
charged particle acceleration parameters provided by the dielectric wakefield
accelerating structures. The beam breakup problem has been known and studied
for a long time both by theory [9, 10, 11] and by experiment [12, 13], and has
been described in numerous publications. In [14] Lau has presented a
classification of beam breakup instabilities in linacs. The previous studies
[15, 16, 17, 18, 19, 20, 21, 22], devoted to the transverse wakefields, and,
in particular, to the BBU directly in the dielectric wakefield accelerating
structures, point out a great importance of this problem, which is still not
completely overcome at the moment. To control and suppress this instability,
several methods have been proposed. Among them are (I) the
Balakin–Novokhatsky–Smirnov (BNS) damping [23], which consists in variation of
the betatron oscillation frequency along the bunch; (II) the combination of a
profiled quadrupole and FODO focusing system with an energy chirp of bunch
[24, 25]; (III) the excitation of a dielectric resonator by a train of bunches
[26]. In [27, 28, 29] a transverse wakefield reduction using the elliptical
(transversally) drive bunches was investigated theoretically and
experimentally. On the other hand in papers [30, 31] it has been demonstrated
that plasma has the focusing properties, and as a result, this allows one to
focus the drive bunch. The authors of paper [32] have demonstrated that a
combination of a dielectric–loaded structure and plasma provides intrinsic
focusing features and causes the test and drive bunches to be focused. It has
been shown that in this plasma–dielectric structure, the acceleration is
provided by the $TM$ eigenwave of the structure, and the focusing is provided
by the plasma wave. Moreover, further research [33, 34] has shown that using a
nonuniform radial plasma density profile, which takes place at capillary
discharge, improves focusing of both the drive and witness bunches, and also
there exists the optimum vacuum channel radius, at which the focusing is the
best. So, the mentioned features of the plasma–dielectric structure allow one
to consider it as an alternative to a standard quadrupole focusing system, and
as a possible way for the beam breakup mitigation during wakefield excitation
and acceleration. This was the primary motivation of our studies and the
present paper. Here we have centred on the analytical studies of wakefields
generation by the off–axis drive bunch in the plasma–dielectric waveguide. We
have focused our attention on the closed–form expressions for the
bunch–excited electric and magnetic field components in the rigid drive bunch
approximation. The paper is organized as follows. Sec. II gives the statement
of the problem, Sec. III is devoted to the analytical bunch–excited wakefield
studies, Sec. IV demonstrates the results of the numerical studies and Sec. V
gives a short summary.

##  II STATEMENT OF THE PROBLEM

The accelerating structure is a round dielectric–loaded metal waveguide with a
channel for charged particles. This channel is filled with homogeneous cold
plasma. A relativistic drive electron bunch is injected with offset into
plasma in parallel to the structure axis. The wakefield is excited by the
drive bunch passing through the structure. For the case of on–axis injection,
the excited wakefield is described in terms of TM eigenmodes only. However,
due to the drive bunch offset, the excited wakefield is not described only in
terms of TM eigenmodes and has all six components of the electromagnetic
field. A three–dimensional general schematic of the plasma–dielectric
accelerating structure with the drive bunch passing through is shown in Fig.1.

Figure 1: General view of the round plasma–dielectric waveguide. Metal coating
(orange), dielectric (yellow), plasma (blue) and electron bunch (red) are
shown schematically. The off–axis electron bunch moves in parallel to the
axis.

The main goal of the paper is to construct a linear theory of wakefield
excitation in the plasma–dielectric waveguide by the drive electron bunch for
the case of off–axis bunch injection taking into account the high–order
azimuthal modes.

##  III ANALYTICAL STUDIES

The wakefield excited by the point–like charged particle (Green’s function)
should be analyzed first. For the charged particle $q$ moving with the
constant velocity $v$ along the waveguide axis ($z$ direction) with the
transverse coordinates $r_{0}$ and $\varphi_{0}$, the charge and current
densities in the cylindrical coordinates are as follows:

$\begin{split}{\rho}=\frac{q}{v}\frac{{\delta(r-{r_{0}})}}{r}\delta(\varphi-{\varphi_{0}})\delta(t-{t_{0}}-z/v),\\\
{j_{z}}=q\frac{{\delta(r-{r_{0}})}}{r}\delta(\varphi-{\varphi_{0}})\delta(t-{t_{0}}-z/v),\end{split}$
(1)

where $t_{0}$ is the time of particle arrival into the waveguide ($z=0$), and
$\delta$ is the Dirac delta function. As the longitudinal boundedness effect
is neglected, it is supposed that the excited wakefield is defined by the
source only. Therefore, all the electromagnetic field components depend on the
axial coordinate $z$ and time $t$ similar to the functions of $t-z/v$. In this
paper, the transient process of electromagnetic field formation and group
velocity effects associated with the longitudinal boundedness of the
practicable accelerating structures are not taken into account. The
bunch–excited wakefield components, as well as the charge and current
densities, can be described in terms of Fourier transforms by the variables
$t-z/v$ and $\varphi$ as follows:

$\begin{split}{\bf{E}}(r,\varphi,\xi)&=\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}\int\limits_{-\infty}^{+\infty}{d\omega{\bf{E}}_{m}^{\omega}(r,\omega)}}{e^{-i\omega\xi}}{\text{,}}\\\
{\bf{D}}(r,\varphi,\xi)&=\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}\int\limits_{-\infty}^{+\infty}{d\omega\varepsilon(\omega){\bf{E}}_{m}^{\omega}(r,\omega)}}{e^{-i\omega\xi}}{\text{,}}\\\
{\bf{H}}(r,\varphi,\xi)&=\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}\int\limits_{-\infty}^{+\infty}{d\omega{\bf{H}}_{m}^{\omega}(r,\omega)}}{e^{-i\omega\xi}}{\text{,}}\\\
{\bf{B}}(r,\varphi,\xi)&=\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}\int\limits_{-\infty}^{+\infty}{d\omega\mu(\omega){\bf{H}}_{m}^{\omega}(r,\omega)}}{e^{-i\omega\xi}}{\text{,}}\\\
{\rho}(r,\varphi,\xi)&=\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}\int\limits_{-\infty}^{+\infty}{d\omega{\rho}_{m}^{\omega}(r,\omega)}}{e^{-i\omega\xi}}{\text{,}}\\\
{j_{z}}(r,\varphi,\xi)&=\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}\int\limits_{-\infty}^{+\infty}{d\omega{j}_{zm}^{\omega}(r,\omega)}}{e^{-i\omega\xi}}{\text{,}}\end{split}$
(2)

where $\xi=t-z/v$. The set of Maxwell’s equations results in the system of
coupled equations for the Fourier transforms of all the electric and magnetic
field components $E_{rm}^{\omega}$, $E_{\varphi m}^{\omega}$,
$E_{zm}^{\omega}$, $H_{rm}^{\omega}$, $H_{\varphi m}^{\omega}$ and
$H_{zm}^{\omega}$. In turn, the aforementioned system of the coupled equations
leads to the uncoupled wave equations for the Fourier transforms of the
longitudinal electric and magnetic fields $E_{zm}^{\omega}$, and
$H_{zm}^{\omega}$, namely:

$\begin{split}&\frac{1}{r}\frac{\partial}{{\partial r}}\left({r\frac{{\partial
E_{zm}^{\omega}}}{{\partial
r}}}\right)-\frac{{{m^{2}}}}{{{r^{2}}}}E_{zm}^{\omega}-\frac{{{\omega^{2}}}}{{{v^{2}}}}(1-{\beta^{2}}\varepsilon(\omega))E_{zm}^{\omega}=\\\
&\frac{{4\pi\omega
i}}{{{v^{2}}}}\,\frac{{1-{\beta^{2}}\varepsilon(\omega)}}{{\varepsilon(\omega)}}j_{zm}^{\omega},\\\
&\frac{1}{r}\frac{\partial}{{\partial r}}\left({r\frac{{\partial
H_{zm}^{\omega}}}{{\partial
r}}}\right)-\frac{{{m^{2}}}}{{{r^{2}}}}H_{zm}^{\omega}-\frac{{{\omega^{2}}}}{{{v^{2}}}}(1-{\beta^{2}}\varepsilon(\omega))H_{zm}^{\omega}=0.\end{split}$
(3)

All the other four Fourier transforms $E_{rm}^{\omega}$, $E_{\varphi
m}^{\omega}$, $H_{rm}^{\omega}$ and $H_{\varphi m}^{\omega}$ can be described
in terms of $E_{zm}^{\omega}$, and $H_{zm}^{\omega}$.

$\begin{split}E_{rm}^{\omega}&=\frac{v/\omega}{{(1-{\beta^{2}}\varepsilon(\omega))}}\left({\beta\frac{m}{r}H_{zm}^{\omega}-i\frac{{\partial
E_{zm}^{\omega}}}{{\partial r}}}\right),\\\ E_{\varphi
m}^{\omega}&=\frac{{v/\omega}}{{(1-{\beta^{2}}\varepsilon(\omega))}}\left({\frac{m}{r}E_{zm}^{\omega}+i\beta\frac{{\partial
H_{zm}^{\omega}}}{{\partial r}}}\right),\\\
H_{rm}^{\omega}&=\frac{v/\omega}{{(1-{\beta^{2}}\varepsilon(\omega))}}\left({-i\frac{{\partial
H_{zm}^{\omega}}}{{\partial
r}}-\beta\frac{m}{r}\varepsilon(\omega)E_{zm}^{\omega}}\right),\\\ H_{\varphi
m}^{\omega}&=\frac{v/\omega}{{(1-{\beta^{2}}\varepsilon(\omega))}}\left({\frac{m}{r}H_{zm}^{\omega}-i\beta\varepsilon(\omega)\frac{{\partial
E_{zm}^{\omega}}}{{\partial r}}}\right).\end{split}$ (4)

To get the expressions for the Fourier transforms, the boundary conditions for
the waveguide metal coating and plasma–dielectric interface should be added.
These boundary conditions are as follows: (I) for the waveguide wall ($r=b$),
the axial components of the electric field ($E_{zm}^{\omega}$) should be equal
to zero; (II) for the plasma–dielectric interface ($r=a$), the axial
components of the electric field ($E_{zm}^{\omega}$, $E_{\varphi
m}^{\omega}$), axial components of the magnetic induction ($B_{zm}^{\omega}$)
and radial components of the electric induction ($D_{rm}^{\omega}$) should be
continuous. The boundary conditions for the rest of the wakefield components
will be satisfied automatically. In this paper, mathematical details will be
omitted and the results for the axial components of electric and magnetic
fields will be presented for any given azimuthal $m^{th}$ mode in the plasma
and dielectric regions. A detailed description of the procedure for
constructing the solutions for $E_{zm}^{\omega}$ and $H_{zm}^{\omega}$ can be
found, for example, in papers [15, 16, 17, 19]. The results satisfied by the
aforementioned boundary condition are as follows:

$\begin{split}&E_{zm}^{\omega}(r<{r_{0}})=-\frac{{iq}}{\pi}\left({\frac{{{D_{2}}(\omega)}}{{a\omega
D(\omega)}}}\right.\frac{{{I_{m}}({\kappa_{p}}r){I_{m}}({\kappa_{p}}{r_{0}})}}{{I_{m}^{2}({\kappa_{p}}a)}}+\\\
&\left.{\frac{{\kappa_{p}^{2}}}{{\omega{\varepsilon_{p}}(\omega)}}\frac{{{I_{m}}({\kappa_{p}}r)}}{{{I_{m}}({\kappa_{p}}a)}}{\Delta_{m}}({\kappa_{p}}a,{\kappa_{p}}{r_{0}})}\right){e^{-im{\varphi_{0}}+i\omega{t_{0}}}},\\\
&E_{zm}^{\omega}(r>{r_{0}})=-\frac{{iq}}{\pi}\left({\frac{{{D_{2}}(\omega)}}{{a\omega
D(\omega)}}}\right.\frac{{{I_{m}}({\kappa_{p}}r){I_{m}}({\kappa_{p}}{r_{0}})}}{{I_{m}^{2}({\kappa_{p}}a)}}+\\\
&\left.{\frac{{\kappa_{p}^{2}}}{{\omega{\varepsilon_{p}}(\omega)}}\frac{{{I_{m}}({\kappa_{p}}{r_{0}})}}{{{I_{m}}({\kappa_{p}}a)}}{\Delta_{m}}({\kappa_{p}}a,{\kappa_{p}}r)}\right){e^{-im{\varphi_{0}}+i\omega{t_{0}}}},\\\
&E_{zm}^{\omega}(a<r<b)=-\frac{{iq}}{\pi}\frac{{{D_{2}}(\omega)}}{{a\omega
D(\omega)}}\frac{{{I_{m}}({\kappa_{p}}{r_{0}})}}{{{I_{m}}({\kappa_{p}}a)}}\times\\\
&\frac{{{F_{m}}({\kappa_{d}}r,{\kappa_{d}}b)}}{{{F_{m}}({\kappa_{d}}a,{\kappa_{d}}b)}}{e^{-im{\varphi_{0}}+i\omega{t_{0}}}},\end{split}$
(5)

$\begin{split}&H_{zm}^{\omega}(0<r\leq a)=\frac{q}{\pi}\frac{\beta
m}{{a^{2}\omega
D(\omega)}}\frac{{{{(\omega/v)}^{2}}({\varepsilon_{d}}-{\varepsilon_{p}}(\omega))}}{{\kappa_{p}^{2}\kappa_{d}^{2}}}\times\\\
&\frac{{{I_{m}}({\kappa_{p}}r){I_{m}}({\kappa_{p}}{r_{0}})}}{{I_{m}^{2}({\kappa_{p}}a)}}{e^{-im{\varphi_{0}}+i\omega{t_{0}}}},\\\
&H_{zm}^{\omega}(a<r\leq b)=\frac{q}{\pi}\frac{\beta m}{{a^{2}\omega
D(\omega)}}\frac{{{{(\omega/v)}^{2}}({\varepsilon_{d}}-{\varepsilon_{p}}(\omega))}}{{\kappa_{p}^{2}\kappa_{d}^{2}}}\times\\\
&\frac{{{I_{m}}({\kappa_{p}}{r_{0}})}}{{{I_{m}}({\kappa_{p}}a)}}\frac{{{\Phi_{m}}({\kappa_{d}}r,{\kappa_{d}}b)}}{{{\Phi_{m}}({\kappa_{d}}a,{\kappa_{d}}b)}}{e^{-im{\varphi_{0}}+i\omega{t_{0}}}},\end{split}$
(6)

where the following notations are used: $a$ is the inner radius of the
dielectric,
$\kappa_{p}^{2}=(\omega/v)^{2}(1-{\beta^{2}}{\varepsilon_{p}}({\omega}))$,
$\kappa_{d}^{2}=(\omega/v)^{2}(\beta^{2}\varepsilon_{d}-1)$,
${\Delta_{m}}(x,y)={I_{m}}(x){K_{m}}(y)-{K_{m}}(x){I_{m}}(y)$, $I_{m}$ and
$K_{m}$ are the modified Bessel and Macdonald functions of the $m^{th}$ order,
${\varepsilon_{p}}(\omega)=1-\omega_{p}^{2}/{\omega^{2}}$ is the plasma
permittivity, $\omega_{p}$ is the plasma frequency, $\varepsilon_{d}$ is the
dielectric permittivity,
${F_{m}}(x,y)={(-1)^{m}}({J_{m}}(x){Y_{m}}(y)-{Y_{m}}(x){J_{m}}(y))$, $J_{m}$
and $Y_{m}$ are the Bessel and Weber functions of the order $m^{th}$,
${F^{\prime}_{m}}(x,y)={(-1)^{m}}({J^{\prime}_{m}}(x){Y_{m}}(y)-{Y^{\prime}_{m}}(x){J_{m}}(y))$,
${\Phi_{m}}(x,y)={J_{m}}(x){Y^{\prime}_{m}}(y)-{Y_{m}}(x){J^{\prime}_{m}}(y)$,
${\Phi^{\prime}_{m}}(x,y)={J^{\prime}_{m}}(x){Y^{\prime}_{m}}(y)-{Y^{\prime}_{m}}(x){J^{\prime}_{m}}(y)$.
In the expressions (5) and (6) the function $D(\omega)$ is a dispersion
relation and it has the following form

$\begin{split}D(\omega)&=D_{1}(\omega)D_{2}(\omega)-\frac{\beta^{2}m^{2}(\omega/v)^{4}}{a^{2}\kappa_{p}^{4}\kappa_{d}^{4}}(\varepsilon_{d}-\varepsilon_{p})^{2},\\\
D_{1}(\omega)&=\frac{\varepsilon_{p}}{\kappa_{p}}\frac{I^{\prime}_{m}(\kappa_{p}a)}{I_{m}(\kappa_{p}a)}+\frac{\varepsilon_{d}}{\kappa_{d}}\frac{F^{\prime}_{m}(\kappa_{d}a,\kappa_{d}b)}{F_{m}(\kappa_{d}a,\kappa_{d}b)},\\\
D_{2}(\omega)&=\frac{1}{\kappa_{p}}\frac{I^{\prime}_{m}(\kappa_{p}a)}{I_{m}(\kappa_{p}a)}+\frac{1}{\kappa_{d}}\frac{\Phi^{\prime}_{m}(\kappa_{d}a,\kappa_{d}b)}{\Phi_{m}(\kappa_{d}a,\kappa_{d}b)}.\end{split}$
(7)

The frequencies of the plasma–dielectric waveguide eigenmodes, which are in
Cherenkov resonance with the drive bunch, are defined by the roots of the
equation $D(\omega)=0$. For the case of azimuthaly homogenous eigenmodes
($m=0$), these frequencies are defined from the dispersion equation of the
resonant $TM$–modes as follows:

$\begin{split}\frac{{{\varepsilon_{p}}(\omega)}}{{{\kappa_{p}}}}\frac{{{{I^{\prime}}_{0}}({\kappa_{p}}a)}}{{{I_{0}}({\kappa_{p}}a)}}+\frac{{{\varepsilon_{d}}}}{{{\kappa_{d}}}}\frac{{{{F^{\prime}}_{0}}({\kappa_{d}}a,{\kappa_{d}}b)}}{{{F_{0}}({\kappa_{d}}a,{\kappa_{d}}b)}}=0.\end{split}$
(8)

Due to the inverse Fourier transform of the expressions (5) and (6), the
space–time distribution of the wakefield components excited by the point–like
charged particle in the plasma–dielectric waveguide can be obtained. It should
be noted that the expressions (5) and (6) have simple poles
$\varepsilon_{p}(\omega)=0$ and $D(\omega)=0$ only. The wakefield of plasma
wave is defined by the poles $\varepsilon_{p}(\omega)=0$. The wakefield of the
plasma–dielectric eigenmodes, which are synchronous with the drive bunch, is
defined by the poles $D(\omega)=0$, as was mentioned above. The inverse
Fourier transforms with respect to the frequency, using of the calculus of
residues, gives for $E_{zm}$ and $H_{zm}$, the following:

$\begin{split}&{E_{zm}}(r<{r_{0}},\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})=-2qk_{p}^{2}\frac{{{I_{m}}({k_{p}}r)}}{{{I_{m}}({k_{p}}a)}}\times\\\
&{\Delta_{m}}({k_{p}}a,{k_{p}}{r_{0}})\cos{\omega_{p}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}}-\\\
&\sum\limits_{s=1}{\frac{{4q{D_{2}}({\omega_{s}})}}{{a{\omega_{s}}D^{\prime}({\omega_{s}})}}}\frac{{{I_{m}}({\kappa_{ps}}r){I_{m}}({\kappa_{ps}}{r_{0}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}\times\\\
&\cos{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}},\\\
\end{split}$ (9)
$\begin{split}&{E_{zm}}({r_{0}}<r<a,\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})=-2qk_{p}^{2}\frac{{{I_{m}}({k_{p}}{r_{0}})}}{{{I_{m}}({k_{p}}a)}}\times\\\
&{\Delta_{m}}({k_{p}}a,{k_{p}}r)\cos{\omega_{p}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}}-\\\
&\sum\limits_{s=1}{\frac{{4q{D_{2}}({\omega_{s}})}}{{a{\omega_{s}}D^{\prime}({\omega_{s}})}}}\frac{{{I_{m}}({\kappa_{ps}}r){I_{m}}({\kappa_{ps}}{r_{0}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}\times\\\
&\cos{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}},\end{split}$
(10) $\begin{split}&{E_{zm}}(a<r\leq
b,\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})=-\sum\limits_{s=1}{\frac{{4q{D_{2}}({\omega_{s}})}}{{a{\omega_{s}}D^{\prime}({\omega_{s}})}}}\times\\\
&\frac{{{I_{m}}({\kappa_{ps}}{r_{0}})}}{{{I_{m}}({\kappa_{ps}}a)}}\frac{{{F_{m}}({\kappa_{ds}}r,{\kappa_{ds}}b)}}{{{F_{m}}({\kappa_{ds}}a,{\kappa_{ds}}b)}}\cos{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}},\end{split}$
(11) $\begin{split}&H_{zm}(0<r\leq
a,\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})=-\sum\limits_{s=1}{i\frac{{4q\beta
m}}{{{a^{2}}{\omega_{s}}D^{\prime}({\omega_{s}})}}}\times\\\
&\frac{{{I_{m}}({\kappa_{ps}}r){I_{m}}({\kappa_{ps}}{r_{0}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}\frac{{{{({\omega_{s}}/v)}^{2}}}}{{\kappa_{ps}^{2}\kappa_{ds}^{2}}}({\varepsilon_{d}}-{\varepsilon_{p}}({\omega_{s}}))\times\\\
&\cos{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}},\end{split}$
(12) $\begin{split}&H_{zm}(a<r\leq
b,\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})=-\sum\limits_{s=1}{i\frac{{4q\beta
m}}{{{a^{2}}{\omega_{s}}D^{\prime}({\omega_{s}})}}}\times\\\
&\frac{{{I_{m}}({\kappa_{ps}}r)}}{{{I_{m}}({\kappa_{ps}}a)}}\frac{{{\Phi_{m}}({\kappa_{ds}}r,{\kappa_{ds}}b)}}{{{\Phi_{m}}({\kappa_{ds}}a,{\kappa_{ds}}b)}}\frac{{{{({\omega_{s}}/v)}^{2}}}}{{\kappa_{ps}^{2}\kappa_{ds}^{2}}}({\varepsilon_{d}}-{\varepsilon_{p}}({\omega_{s}}))\times\\\
&\cos{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}},\end{split}$
(13)

In this paper, the structure of the longitudinal and radial forces acting on
the bunch particles are of vital interest. Therefore, the expressions for the
axial and radial components of the electric field, as well as for the
azimuthal component of the magnetic field only, are presented as follows:

$\begin{split}&{E_{rm}}(r<{r_{0}},\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})=2qk_{p}^{2}\frac{{I^{\prime}_{m}({k_{p}}r)}}{{{I_{m}}({k_{p}}a)}}\times\\\
&\Delta_{m}({k_{p}}a,{k_{p}}{r_{0}})\sin{\omega_{p}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}}-\\\
&\sum\limits_{s=1}{\frac{{4q}}{{a{\kappa_{ps}}vD^{\prime}({\omega_{s}})}}}\frac{{{I_{m}}({\kappa_{ps}}{r_{0}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}\times\\\
&\left({\frac{{{m^{2}}{\beta^{2}}k_{s}^{2}}}{{a\kappa_{ps}^{2}\kappa_{ds}^{2}}}({\varepsilon_{d}}-{\varepsilon_{p}}({\omega_{s}}))\frac{{{I_{m}}({\kappa_{ps}}r)}}{{{\kappa_{ps}}r}}-{D_{2}}({\omega_{s}})I^{\prime}_{m}({\kappa_{ps}}r)}\right)\times\\\
&\sin{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}},\end{split}$
(14)

$\begin{split}&{E_{rm}}({r_{0}}<r<a,\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})=2qk_{p}^{2}\frac{{{I_{m}}({k_{p}}{r_{0}})}}{{{I_{m}}({k_{p}}a)}}\times\\\
&\Delta^{\prime}_{m}({k_{p}}a,{k_{p}}r)\sin{\omega_{p}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}}-\\\
&\sum\limits_{s=1}{\frac{{4q}}{{a{\kappa_{ps}}vD^{\prime}({\omega_{s}})}}}\frac{{{I_{m}}({\kappa_{ps}}{r_{0}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}\times\\\
&\left({\frac{{{m^{2}}{\beta^{2}}k_{s}^{2}}}{{a\kappa_{ps}^{2}\kappa_{ds}^{2}}}({\varepsilon_{d}}-{\varepsilon_{p}}({\omega_{s}}))\frac{{{I_{m}}({\kappa_{ps}}r)}}{{{\kappa_{ps}}r}}-{D_{2}}({\omega_{s}})I^{\prime}_{m}({\kappa_{ps}}r)}\right)\times\\\
&\sin{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}},\end{split}$
(15)

$\begin{split}&{H_{\varphi m}}(0<r\leq
a,\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})=-\sum\limits_{s=1}{\frac{{4q}}{{a{\kappa_{ps}}cD^{\prime}({\omega_{s}})}}}\times\\\
&\frac{{{I_{m}}({\kappa_{ps}}{r_{0}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}\bigg{(}{\frac{{{m^{2}}k_{s}^{2}}}{{a\kappa_{ps}^{2}\kappa_{ds}^{2}}}({\varepsilon_{d}}-{\varepsilon_{p}}({\omega_{s}}))\frac{{{I_{m}}({\kappa_{ps}}r)}}{{{\kappa_{ps}}r}}-}\\\
&{{\varepsilon_{p}}({\omega_{s}}){D_{2}}({\omega_{s}})I^{\prime}_{m}({\kappa_{ps}}r)}\bigg{)}\sin{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}}\end{split}$
(16)

where ${\kappa_{ps}}={\kappa_{p}}({\omega_{s}})$,
${\kappa_{ds}}={\kappa_{d}}({\omega_{s}})$,
$D^{\prime}({\omega_{s}})=dD({\omega_{s}})/d\omega_{s}$, and $\theta$ is the
Heaviside function. The space–time structure of the wakefield excited by a
point–like charged particle injected with offset in the waveguide under study
is described by the expressions (9)– (16). As it can be seen from the
expressions (9), (10) and (14), (15) for the longitudinal and radial
components of the electric field in the plasma region, the field of plasma
wave is described by the first terms. This field is potential and localized in
the plasma–filled channel for charged particles. The longitudinal and radial
electric fields of the eigenwaves of the plasma–dielectric waveguide, are
described by the other terms, respectively. The azimuthal component of the
magnetic field does not include the plasma wave wakefield. In the process of
wakefield excitation, the longitudinal dynamics of the bunch is determined by
the axial component of the electric field ($F_{z}/q=E_{z}$), and the radial
dynamics is determined by the radial component of the Lorentz force, which is
shown as $F_{r}/q=E_{r}-\beta B_{\varphi}$. In the plasma region expression
for the $F_{r}/q$ can be presented as follows:

$\begin{split}&\frac{{{F_{r}}(r<{r_{0}},\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})}}{q}=2qk_{p}^{2}\frac{{{I^{\prime}_{m}}({k_{p}}r)}}{{{I_{m}}({k_{p}}a)}}\times\\\
&{\Delta_{m}}({k_{p}}a,{k_{p}}{r_{0}})\sin{\omega_{p}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}}+\\\
&\sum\limits_{s=1}{\frac{{4q{\kappa_{ps}}v{D_{2}}({\omega_{s}})}}{{a\omega_{s}^{2}D^{\prime}({\omega_{s}})}}}\frac{{{I_{m}}({\kappa_{ps}}{r_{0}}){I^{\prime}_{m}}({\kappa_{ps}}r)}}{{I_{m}^{2}({\kappa_{ps}}a)}}\times\\\
&\sin{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}},\end{split}$
(17)

$\begin{split}&\frac{{{F_{r}}({r_{0}}<r\leq
a,\varphi,\xi,{r_{0}},{\varphi_{0}},{t_{0}})}}{q}=2qk_{p}^{2}\frac{{{I_{m}}({k_{p}}{r_{0}})}}{{{I_{m}}({k_{p}}a)}}\times\\\
&{\Delta^{\prime}_{m}}({k_{p}}a,{k_{p}}r)\sin{\omega_{p}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}}+\\\
&\sum\limits_{s=1}{\frac{{4q{\kappa_{ps}}v{D_{2}}({\omega_{s}})}}{{a\omega_{s}^{2}D^{\prime}({\omega_{s}})}}}\frac{{{I_{m}}({\kappa_{ps}}{r_{0}})I^{\prime}_{m}({\kappa_{ps}}r)}}{{I_{m}^{2}({\kappa_{ps}}a)}}\times\\\
&\sin{\omega_{s}}(\xi-{t_{0}})\theta(\xi-{t_{0}}){e^{-im{\varphi_{0}}}}\end{split}$
(18)

In order to obtain the wakefield excited by a drive bunch of finite size, it
is necessary to integrate the expressions for the point–like particle over the
arrival time $t_{0}$ and by the transverse coordinates $r_{0}$, and
$\varphi_{0}$. It can be supposed that: (I) the drive bunch has a square
profile of the charge density in both longitudinal and transverse directions
(uniform distribution), (II) the bunch offset $R_{off}$ is greater than the
drive bunch radius $R_{b}$. The resulting expressions for the axial and radial
forces acting on the particles, which are located near the plasma–dielectric
interface ($r=R_{off}+R_{b}$) and accelerating structure axis
($r=R_{off}-R_{b}$) can be presented as follows:

$\begin{split}&\frac{{{F_{z}}(r=R_{off}+R_{b},\varphi,\xi)}}{q}=-\frac{{4{Q_{b}}}}{{{R_{b}}{L_{b}}}}\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}}\times\\\
&\bigg{(}{\frac{{{I_{m}}({k_{p}}{R_{off}}){I_{1}}({k_{p}}{R_{b}})}}{{{I_{m}}({k_{p}}a)}}}{\Delta_{m}}({k_{p}}a,{k_{p}}({R_{off}}+{R_{b}}))\Psi_{\parallel}^{(p)}(\xi)+\\\
&\sum\limits_{s=1}{\frac{{2v{D_{2}}({\omega_{s}})}}{{a{\kappa_{ps}}\omega_{s}^{2}D^{\prime}({\omega_{s}})}}\frac{{{I_{m}}({\kappa_{ps}}{R_{off}}){I_{1}}({\kappa_{ps}}{R_{b}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}}\times\\\
&{{I_{m}}({\kappa_{ps}}({R_{off}}+{R_{b}}))\Psi_{\parallel}^{(s)}(\xi)}\bigg{)}\end{split}$
(19)

$\begin{split}&\frac{{{F_{r}}(r={R_{off}}+{R_{b}},\varphi,\xi)}}{q}=\frac{{4{Q_{b}}}}{{{R_{b}}{L_{b}}}}\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}\times}\\\
&\bigg{(}\frac{{{I_{m}}({k_{p}}{R_{off}}){I_{1}}({k_{p}}{R_{b}})}}{{{I_{m}}({k_{p}}a)}}{\Delta^{\prime}_{m}}({k_{p}}a,{k_{p}}({R_{off}}+{R_{b}}))\Psi_{\bot}^{(p)}(\xi)+\\\
&\sum\limits_{s=1}{\frac{{2{v^{2}}{D_{2}}({\omega_{s}})}}{{a\omega_{s}^{3}D^{\prime}({\omega_{s}})}}\frac{{{I_{m}}({\kappa_{ps}}{R_{off}}){I_{1}}({\kappa_{ps}}{R_{b}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}}\times\\\
&{I^{\prime}_{m}}({\kappa_{ps}}({R_{off}}+{R_{b}}))\Psi_{\bot}^{(s)}(\xi)\bigg{)},\end{split}$
(20)

$\begin{split}&\frac{{{F_{z}}(r={R_{off}}-{R_{b}},\varphi,\xi)}}{q}=-\frac{{4{Q_{b}}}}{{{R_{b}}{L_{b}}}}\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}}\times\\\
&\bigg{(}\frac{{{I_{m}}({k_{p}}({R_{off}}-{R_{b}})){I_{1}}({k_{p}}{R_{b}})}}{{{I_{m}}({k_{p}}a)}}{\Delta_{m}}({k_{p}}a,{k_{p}}{R_{off}})\Psi_{\parallel}^{(p)}(\xi)+\\\
&\sum\limits_{s=1}{\frac{{2v{D_{2}}({\omega_{s}})}}{{a{\kappa_{ps}}\omega_{s}^{2}D^{\prime}({\omega_{s}})}}\frac{{{I_{m}}({\kappa_{ps}}{R_{off}}){I_{1}}({\kappa_{ps}}{R_{b}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}}\times\\\
&{I_{m}}({\kappa_{ps}}({R_{off}}-{R_{b}}))\Psi_{\parallel}^{(s)}(\xi)\bigg{)},\end{split}$
(21)

$\begin{split}&\frac{{{F_{r}}(r={R_{off}}-{R_{b}},\varphi,\xi)}}{q}=\frac{{4{Q_{b}}}}{{{R_{b}}{L_{b}}}}\sum\limits_{m=-\infty}^{+\infty}{{e^{im\varphi}}}\times\\\
&\bigg{(}\frac{{{I^{\prime}_{m}}({k_{p}}({R_{off}}-{R_{b}})){I_{1}}({k_{p}}{R_{b}})}}{{{I_{m}}({k_{p}}a)}}{\Delta_{m}}({k_{p}}a,{k_{p}}{R_{off}})\Psi_{\bot}^{(p)}(\xi)+\\\
&\sum\limits_{s=1}{\frac{{2{v^{2}}{D_{2}}({\omega_{s}})}}{{a\omega_{s}^{3}D^{\prime}({\omega_{s}})}}\frac{{{I_{m}}({\kappa_{ps}}{R_{off}}){I_{1}}({\kappa_{ps}}{R_{b}})}}{{I_{m}^{2}({\kappa_{ps}}a)}}}\times\\\
&{I^{\prime}_{m}}({\kappa_{ps}}({R_{off}}-{R_{b}}))\Psi_{\bot}^{(s)}(\xi)\bigg{)},\end{split}$
(22)

where $Q_{b}$ is the bunch charge, and $L_{b}$ is the bunch length. It is
important to note that for the last integration, the method of waveguide
structure excitation by the off–axis beam, well known in the theory of
gyrotrons, was used. The principle of this integration method is the
transition from the laboratory frame of reference to the frame of reference
with the origin located at the center of the bunch (a detailed description of
the procedure can be found in Appendix). A detailed description of this method
can be found, for example, in paper [35].

The axial structure of the wakefield is described by the functions
$\Psi_{\parallel}^{p,s}(\xi)$ and $\Psi_{\perp}^{p,s}(\xi)$:

$\begin{split}\Psi_{\parallel}^{p,s}(\xi)=&{\theta(\xi)\sin{\omega_{p,s}}\xi}-\\\
&{\theta(\xi-L_{b}/v)\sin{\omega_{p,s}}(\xi-L_{b}/v)},\\\
\Psi_{\perp}^{p,s}(\xi)=&\theta(\xi)(1-\cos{\omega_{p,s}}\xi)-\\\ &\theta(\xi-
L_{b}/v)(1-\cos{\omega_{p,s}}(\xi-L_{b}/v)).\end{split}$ (23)

The analytical expressions (19)–(22) allow carrying out a numerical analysis
of the amplitude distribution of the axial and radial forces acting on the
particles that are on the envelope of the drive bunch, which is injected with
offset.

##  IV NUMERICAL ANALYSIS RESULTS

Based on the derived wakefield excitation theory, a numerical analysis was
performed for the waveguide parameters in the terahertz frequency range; for
the drive bunch we have used parameters of electron bunches accessible at
SLAC. These parameters are presented in Table 1.

Table 1: Parameters of the plasma–dielectric waveguide and the drive electron bunch Parameter | Value
---|---
Inner dielectric radius | $0.5\,mm$
Outer dielectric radius | $0.6\,mm$
Permittivity of the dielectric | $3.75(quartz)$
Plasma density | $4.41\cdot 10^{14}\,cm^{-3}$
Energy of bunch | $5\,GeV$
Charge of bunch | $3\,nC$
Length of bunch | $0.5\,mm$
Radius of bunch | $0.23\,mm$
Bunch offset | $0.24\,mm$

First of all, the numerical methods were used to solve dispersion equations
for resonant $TM$-modes (which are working modes for the accelerator
structures) and hybrid electromagnetic $HEM$-modes (which are parasitic,
because their excitation causes the BBU instability). For the parameters of
the waveguide under study and the electron 5 GeV drive bunch, the resonant
frequencies of the first six eigen $TM$-modes are equal to 0.357 THz, 1.160
THz, 1.898 THz, 2.774 THz, 3.663 THz and 4.558 THz, respectively. The resonant
frequencies of the first six eigen $HEM$-modes are 0.025 THz, 0.356 THz, 0.532
THz, 1.066 THz, 1.387 THz and 1.903 THz, respectively.

The presence of plasma in the channel for charged particles leads to a change
in both the amplitudes of the bunch–excited wakefield components and their
spatial distribution. The axial profiles of the axial and radial forces were
compared for the case of filling the channel for charged particles with plasma
and the case of no plasma in the channel. The results of this comparison are
presented in Fig.2 as the longitudinal distributions of axial and radial
forces that acting on its particles.

Figure 2: Longitudinal profiles of the axial and radial forces at a distance
of 0.47 mm from the waveguide axis for the plasma–filled channel (solid line),
and in the absence of plasma in the channel (dashed line). The drive bunch
(grey rectangle) head position corresponds to $vt-z=0$.

Axial distributions in Fig.2 are shown along a line which is parallel to the
structure axis and located at a distances of 0.47 mm from it, corresponding to
the envelope of the drive bunch closer to the dielectric. Figure 2
illustrates, that the axial force amplitude in the plasma–dielectric structure
is less than in the corresponding dielectric–loaded structure behind the drive
bunch. Filling the channel with plasma also leads to decrease in the spatial
period of the axial force. An analysis showed, that the above changes are
associated with the excitation of the longitudinal electric field of the
plasma wave, as well as with the change in the frequencies of the
plasma–dielectric waveguide eigenwaves. As we are mainly interested in the BBU
of the drive bunch, we start with analysis of the transverse wakefield
amplitudes. As can be seen from Fig.2, the radial force profile shows almost a
harmonic dependence on the longitudinal coordinate, with a period
approximately equal to the plasma wavelength with a low–frequency modulation.
It can be seen that filling with plasma causes a qualitative change in the
longitudinal profile of the radial force. This change is especially noticeable
in the the bunch region, where the radial force has changed its sign. Namely,
in the absence of plasma in the channel for charged particles, the radial
force is defocusing, but in the presence of plasma, this force becomes
focusing. In turn, this should be accompanied by the change in the frequency
composition of the excited wakefield. Therefore, the spectral analysis of the
off–axis bunch–excited wakefield was carried out for the cases of plasma off
and plasma on. The results of the spectral analysis are presented in Fig.3.

Figure 3: Radial force spectrum (at $r=0.47mm$ ) excited by the off–axis drive
electron bunch for the plasma off (top) and plasma on (bottom) cases.

The analysis shows that in the absence of plasma the off–axis drive bunch
leads to significant excitation of $HEM$–modes with the azimuthal index $m=1$
(dipole modes). The analysis has shown that these eigenmodes make a dominant
contribution to the radial force, under the action of which the drive bunch
particles will gradually move to the dielectric. Comparison of spectra
demonstrates that the main reason of difference between the amplitudes and
axial profiles of the radial forces for cases of presence and absence of the
plasma lies in the plasma wave excitation by the drive bunch. The radial
electric field of this wave acts on the bunch particles as a focusing system.
Thus the plasma wave makes a predominant contribution to the radial force. The
low–frequency modulation of the radial force amplitude is due to excitation of
the eigenwaves of the plasma–dielectric accelerating structure. This change in
the excited wakefield spectrum leads, in turn, to a significant (qualitative
and quantitative) change in the transverse dynamics of the drive bunch. The
classical scenario of the BBU instability is as follows. As a result of the
electron bunch injection with offset the bunch tail is deflected during
passing down the accelerating structure. A physical reason of this deflection
is a significant transverse wakefield component excitation. The presence of
plasma in the channel leads to the change of the aforementioned scenario.
Figure 4 presents the distribution of radial forces, which are acting on its
peripheral particles (corresponding, respectively to the drive bunch envelope
closer to the dielectric, and the drive bunch envelope closer to the waveguide
axis).

Figure 4: Axial profiles of the radial force at distances of 0.47 mm (solid
line), and 0.01 mm (dashed line) from the waveguide axis. The drive bunch
(grey rectangle) head position corresponds to $vt-z=0$.

Figure 4 indicates that the radial dynamics of the drive bunch particles
located on the bunch opposite transverse peripheries will differ both
qualitatively and quantitatively. The bunch particles which are closer to the
plasma–dielectric interface will focus in the axial direction during their
propagation along the accelerating structure. The focusing force increases
along the bunch from its head to tail. On the contrary, the bunch particles
which are closer to the waveguide axis will defocus during propagation. The
defocusing force also increases along the bunch from its its head to tail. As
a result, in the process of wakefield excitation the off–axis drive bunch will
not deposit on the dielectric surface. And instead of a gradual increase of
transverse displacement, there will be radial compression of the tail end of
the bunch.

By using the profiles of the transverse force that acts on the drive bunch
particles, the developed theory predicts the transverse dynamics of the drive
bunch. To verify our wakefield amplitude calculations, which are based on
analytical expressions, the particle-in-cell simulations of the drive electron
bunch propagation along the structure under study were carried out, too. The
main goal of the simulations was to verify the derived theory predictions. For
simulations, we have used the same parameters of the waveguide and drive bunch
as those for the analytical calculations (see Table 1). Both the plasma off
and plasma on cases for the off-axis injection were analyzed. We started the
simulations with the case of on–axis injection. Figure 5 demonstrates the
positions of the particles for the time moment when the drive bunch reached
the longitudinal position $z=7.51cm$. The positions of the particles are given
in the vertical–longitudinal plane. It is evident from the figure that the
presence of plasma results in focusing of particles located in the tail of the
drive bunch.

Figure 5: The on–axis injected drive bunch particle distribution on the
vertical–longitudinal plane at the accelerating structure exit for the plasma
on–and plasma off–cases. The vertical axis limits correspond to the full width
of the channel for the charged particles, the horizontal axis limits
correspond to the frame around the bunch.

It should be noted, that the simulated ”model” monoenergetic drive bunch has
zero divergence at the structure input. For the nonzero divergence drive
bunch, the focusing features of the plasma–dielectric structure can partially
compensate the divergence increase in the process of bunch propagation, and
will be investigated further. Figure 6 shows the bunch particles positions on
the vertical–longitudinal plane at the time when the bunch started to deposit
on the dielectric in the plasma off case.

Figure 6: The off–axis injected drive bunch particle distribution on the
vertical–longitudinal plane at the accelerating structure exit for the plasma
on and plasma off cases. The vertical axis limits correspond to the half width
of channel for the charged particles, horizontal axis limits correspond to the
frame around the bunch.

As it is shown, there is a significant qualitative difference in the
transverse bunch dynamics in the plasma off and plasma on cases. As a result
of the initial offset, the transverse deflection of the tail particles is
greater than that of the head ones in the absence of plasma in the channel for
the charged particles. The drive bunch is deflected as a whole while it
propagates through the dielectric–loaded structure, and the BBU instability
does exist in this case, causing undesirable bunch displacement, to occur up
to the total loss of the bunch. The presence of plasma in the channel for the
charged particles leads to the change in the transverse particle dynamics. The
particles in the tail of the off-axis drive bunch are focused (instead of
deflection) due to plasma wave excitation. As a result, the bunch offset does
not tend to increase with time and there is no gradual increase of charge
losses does not occur. The results of numerical simulations show a good
qualitative agreement with the corresponding analytical theory predictions in
terms of the drive bunch envelope particles behavior. More detailed numerical
simulations will be performed during further studies. Thus, it may be
concluded that the presence of plasma in the channel for charged particles
leads to suppression of the transverse instability of the drive bunch. In
turn, from a practical standpoint, this will enable us: (I) to prevent the
charge losses of the drive bunch, which excites the wakefield, (II) to reduce
the requirements for the accuracy of the drive bunch injection into the
accelerating structure, this being particularly important for terahertz
structures having submillimeter transverse dimensions, (III)to increase the
acceleration length of the witness bunch and, consequently, to increase its
output energy. In conclusion, it may be said, that the intrinsic focusing
features of such plasma–dielectric accelerating structure under discussion
allow to improve the drive bunch stability during wakefield excitation.

##  V CONCLUSIONS

An analytical theory of wakefield excitation by the off-axis particle bunch in
the round plasma–dielectric waveguide has been formulated. A comparison
between the longitudinal profiles of radial forces, excited by the off–axis
drive electron bunch, has been carried out for the cases of the absence and
presence of plasma in the channel for charged particles. It has been
demonstrated that in contrast to the dielectric–loaded structure without
plasma filling, the presence of the initial bunch offset in the
plasma–dielectric accelerating structure does not cause to the beam breakup
instability.

###### Acknowledgements.

The study is supported by the National Research Foundation of Ukraine under
the program “Leading and Young Scientists Research Support” (project #
2020.02/0299).

## APPENDIX: INTEGRATION OVER THE PARTICLES

Consider following integral over the arrival time $t_{0}$ and transverse
coordinates $r_{0}$, and $\varphi_{0}$

$\begin{split}\int\int\int
f(r,\varphi,\xi,r_{0},\varphi_{0},t_{0})dt_{0}dr_{0}d\varphi_{0},\end{split}$
(24)

here $f(r,\varphi,\xi,r_{0},\varphi_{0},t_{0})$ is a function, which
represents any of the wakefield components, excited by the point–like particle
(Green’s function). To calculate this integral it is more convenient to
transit from the laboratory frame of reference to the frame of reference with
the origin, located at the center of the drive bunch.

Figure 7: The off–axis injected drive bunch particle positions on
horizontal–vertical plane. Particle transverse coordinates $r_{0}$, and
$\varphi_{0}$ in the laboratory frame of reference, and transverse coordinates
$\rho$ and $\psi$ in the bunch frame of reference are shown schematically.

Particle transverse coordinates $r_{0}$, and $\varphi_{0}$ in the laboratory
frame of reference are expressed in terms of new corresponding transverse
coordinates $\rho$ and $\psi$ in the bunch frame of reference by the
expressions

$\begin{split}&r_{0}(\rho,\psi)=\sqrt{R_{off}^{2}+2R_{off}\rho
cos\psi+\rho^{2}},\\\ &tg\varphi_{0}(\rho,\psi)=\frac{\rho
sin\psi}{R_{off}+\rho cos\psi}\end{split}$ (25)

For the sake of simplicity we present horizontal drive bunch offset only.

Using new variables $\rho$ and $\psi$, one can rewrite the integral (24) as

$\begin{split}\int\int\int&f(r,\varphi,\xi,r_{0},\varphi_{0},t_{0})dt_{0}dr_{0}d\varphi_{0}=\\\
\int\int\int&f(r,\varphi,\xi,r_{0}(\rho,\psi),\varphi_{0}(\rho,\psi),t_{0})\times\\\
&J(r_{0},\varphi_{0};\rho,\psi)dt_{0}d\rho d\psi,\end{split}$ (26)

here $J(r_{0},\varphi_{0};\rho,\psi)$ is the Jacobian determinant.

The Jacobian determinant defines as

$J(r_{0},\varphi_{0};\rho,\psi)=\bigg{(}\frac{\partial
r_{0}}{\partial\rho}\bigg{)}\bigg{(}\frac{\partial\varphi_{0}}{\partial\psi}\bigg{)}-\bigg{(}\frac{\partial
r_{0}}{\partial\psi}\bigg{)}\bigg{(}\frac{\partial\varphi_{0}}{\partial\rho}\bigg{)},$
(27)

and substituting expressions (25) into (27), one obtains

$J(r_{0},\varphi_{0};\rho,\psi)=\frac{\rho}{r_{0}}.$ (28)

Function $f(r,\varphi,\xi,r_{0},\varphi_{0},t_{0})$ contains the combinations
$I_{m}(k_{p}r_{0})e^{-im\varphi_{0}}$ and
$K_{m}(k_{p}r_{0})e^{-im\varphi_{0}}$ (see the expressions (9)– (16)). Using
the Graph theorem for the cylindrical functions [36] it is possible to rewrite
these combinations in terms of new variables $\rho$ and $\psi$, as follows:

$\begin{split}I_{m}(k_{p}r_{0})e^{-im\varphi_{0}}=\sum\limits_{\nu=-\infty}^{+\infty}&(-1)^{\nu}I_{m+\nu}(k_{p}R_{off})I_{\nu}(k_{p}\rho)\times\\\
&e^{-i\nu(\pi-\psi)},\\\
K_{m}(k_{p}r_{0})e^{-im\varphi_{0}}=\sum\limits_{\nu=-\infty}^{+\infty}&K_{m+\nu}(k_{p}R_{off})I_{\nu}(k_{p}\rho)\times\\\
&e^{-i\nu(\pi-\psi)}\end{split}$ (29)

Since we supposed, that the drive bunch has uniform distribution of the charge
density in both longitudinal and transverse directions, the charge of the
point–like particle is

$q=Q_{b}\frac{r_{0}dr_{0}d\varphi_{0}vdt_{0}}{\pi R_{b}^{2}L_{b}}$ (30)

Now, one can accomplish the integration over the arrival time and transverse
coordinates of the particles to get the wakefield, excited by the drive bunch
of finite size.

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# Incorporating the weak mixing angle dependence to reconcile the neutron skin
measurement on $\mathbf{{}^{208}\mathrm{Pb}}$ by PREX-II

M. Atzori Corona  Dipartimento di Fisica, Università degli Studi di Cagliari,
Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042
Monserrato (Cagliari), Italy  Istituto Nazionale di Fisica Nucleare (INFN),
Sezione di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km
0.700, 09042 Monserrato (Cagliari), Italy  M. Cadeddu
<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare (INFN),
Sezione di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km
0.700, 09042 Monserrato (Cagliari), Italy  N. Cargioli
<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di
Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700,
09042 Monserrato (Cagliari), Italy  Istituto Nazionale di Fisica Nucleare
(INFN), Sezione di Cagliari, Complesso Universitario di Monserrato - S.P. per
Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  P. Finelli  Dipartimento
di Fisica e Astronomia, Università degli Studi di Bologna and INFN, Sezione di
Bologna, Via Irnerio 46, I-40126 Bologna, Italy  M. Vorabbi  National Nuclear
Data Center, Bldg. 817, Brookhaven National Laboratory, Upton, NY 11973-5000,
USA

###### Abstract

The only available electroweak measurement of the ${}^{208}\mathrm{Pb}$
neutron skin, $\Delta R_{\rm{np}}$, performed by the PREX-II Collaboration
through polarized electron-lead scattering, shows a mild tension with respect
to both the theoretical nuclear-model predictions and a host of measurements.
However, the dependence on the weak mixing angle should be incorporated in the
calculation, since its low-energy value is experimentally poorly known. We
first repeat the PREX-II analysis confirming their measurement by fixing the
weak mixing angle to its standard model value. Then, we show the explicit
dependence of the PREX-II measurement on the weak mixing angle, obtaining that
it is fully degenerate with the neutron skin. To break this degeneracy, we
exploit the weak mixing angle measurement from atomic parity violation on
lead, obtaining a slightly thinner neutron skin but with about doubled
uncertainties, possibly easing the PREX tension. Relying on the theoretical
prediction, $\Delta R_{\rm{np}}^{\mathrm{th}}\approx 0.13$-$0.19\
\mathrm{fm}$, and using it as a prior in the fit, we find a weak mixing angle
value about $1.2\sigma$ smaller than the standard model prediction. Thus, we
suggest a possible solution of the PREX-II tension by showing that,
considering its underlying dependence on the weak mixing angle, the PREX-II
neutron skin determination could be in agreement with the other available
measurements and predictions if the weak mixing angle at the proper energy
scale is smaller than the standard model prediction.

Introduction.— The neutron skin, $\Delta R_{\rm{np}}\equiv R_{n}-R_{p}$, of a
nucleus quantifies the difference between the neutron and the proton nuclear
distribution radii, $R_{n}$ and $R_{p}$, respectively. Polarized electron-
nucleus scattering happens through both the weak and the electromagnetic
currents, therefore it provides an interesting way to assess the nuclear
structure. Indeed, it is possible to isolate the weak interaction
contribution, which strongly depends on the neutron nuclear density, from the
electromagnetic one, which mainly probes the already highly-tested proton
nuclear density.
Recently, the PREX-II Collaboration released their new measurement of the lead
neutron skin through polarized electron-lead scattering, namely $\Delta
R_{\rm{np}}^{\rm{PREX-II}}(^{208}\rm{Pb})=0.278\pm 0.078\ \mathrm{fm}$ et al.
(2021), after a previous less precise measurement, $\Delta
R_{\rm{np}}^{\rm{PREX-I}}(^{208}\mathrm{Pb})=0.30\pm 0.18\ \mathrm{fm}$
Abrahamyan  _et al._ (2012); Horowitz  _et al._ (2012). The combination of
these results leads to $\Delta
R_{\rm{np}}^{\rm{PREX,comb}}(^{208}\rm{Pb})=0.283\pm 0.071\ \mathrm{fm}$ et
al. (2021) indicating a preference for relatively large values of the skin. In
particular, our analysis is focused on the PREX-II measurement, which is
significantly more precise.
The PREX-II measurement is in tension with the other available determinations,
such as those coming from electric-dipole polarizability Piekarewicz  _et al._
(2012); Roca-Maza  _et al._ (2015); Piekarewicz (2021), antiprotonic atoms
Trzcińska  _et al._ (2001); Zhang  _et al._ (2021); Klos  _et al._ (2007),
proton-nucleus scattering Zenihiro  _et al._ (2010); Starodubsky and Hintz
(1994), coherent pion photoproduction Tarbert  _et al._ (2014) and the
indirect measurements of neutron star observables Essick  _et al._ (2021a);
Fattoyev  _et al._ (2018); Reed  _et al._ (2021); Newton and Crocombe (2021);
Stone (2021); Essick  _et al._ (2021b); Riley  _et al._ (2021); Miller  _et
al._ (2021); Biswas (2021); Abbott  _et al._ (2017, 2018, 2020). All these
non-electroweak measurements are in fair agreement with each other, being also
compatible with the predictions of different energy density functional (EDF)
nuclear models Piekarewicz  _et al._ (2012); Roca-Maza  _et al._ (2015);
Piekarewicz (2021), $\Delta R_{\rm{np}}^{\rm{th}}(^{208}\rm{Pb})=0.13$-$0.19\
\mathrm{fm}$, and the first ab-initio estimate of the lead neutron skin,
$\Delta R_{\rm{np}}^{\textit{ab-initio}}(^{208}\rm{Pb})=0.14$-$0.20\
\mathrm{fm}$ Hu  _et al._ (2021). Although the hadronic probes have been shown
to be affected by uncontrolled theoretical uncertainties Thiel  _et al._
(2019), and the astrophysical constraints are rather indirect, the global
picture that emerges indicates a preference for a thinner lead neutron skin
with respect to PREX, which however is considered to be model independent
being a non-hadronic probe. So far, no solution to such a dilemma has been
found, even if a reasonable compromise seems to be allowed for a specific set
of quantified EDFs Reinhard  _et al._ (2021). Motivated by such a tension, we
attempt to reconcile the PREX observation with all the other determinations by
incorporating the so far neglected dependence on the weak mixing angle,
$\sin^{2}\theta_{W}\equiv s^{2}_{W}$, which is a key parameter of the
electroweak theory Zyla  _et al._ (2020).

PREX-II analysis.— The PREX-II Collaboration measures the lead neutron skin
exploiting 953 MeV polarized electrons scattering at forward angles on
${}^{208}\mathrm{Pb}$ nuclei et al. (2021). The polarization of the incoming
electrons permits to isolate the weak-interaction contribution by building a
parity violating asymmetry, $\mathcal{A}_{\rm{pv}}$, as discussed in Ref. et
al. (2021). Adopting the plane-wave Born approximation, the asymmetry shows
explicitly its dependence on the nuclear weak charge, $\mathcal{Q}_{W}$, and
the weak, $F_{W}$, and charge, $F_{\rm{ch}}$, form factors

$\mathcal{A}_{\rm{pv}}\approx\frac{G_{F}Q^{2}}{4\pi\alpha\sqrt{2}}\frac{\mathcal{Q}_{W}F_{W}(Q^{2})}{ZF_{\rm{ch}}(Q^{2})},$
(1)

where $G_{F}$ is the Fermi constant, $Q$ the momentum transfer, $\alpha$ the
fine-structure constant and $Z$ the atomic number of the target nucleus. In
particular, the weak and charge form factors represent the Fourier transforms
of the corresponding weak and charge densities, $\rho_{W}$ and
$\rho_{\rm{ch}}$.
The charge distribution of a lead nucleus has already been precisely tested
through electromagnetic scattering processes, resulting in a charge
distribution radius of $R_{\rm{ch}}(\mathrm{{}^{208}Pb})=5.503\pm 0.002\
\mathrm{fm}$ De Vries  _et al._ (1987). The charge distribution mainly depends
on the proton nuclear density, so that $R_{\rm{ch}}$ can be translated into a
determination of the proton root-mean-squared radius Ong  _et al._ (2010);
Horowitz  _et al._ (2012), leading to $R_{p}(\mathrm{{}^{208}Pb})=5.449\
\mathrm{fm}$ Horowitz  _et al._ (2012).
However, the distorted-wave Born approximation (DWBA) Überall (1971); Yennie
_et al._ (1954); Ciofi degli Atti (1980), which incorporates the Coulomb
distortion effects on the incoming electron approaching the target nucleus,
must be adopted to properly describe heavy nuclei, such as lead Horowitz
(1998).
From Eq. 1, it is clear that PREX-II determines the weak form factor value at
the experimental mean momentum transfer, $Q_{\mathrm{PREX-II}}\simeq 78\
\mathrm{MeV}$ et al. (2021). After adopting a symmetrized two-parameter Fermi
(2pF) function for the weak nuclear density Piekarewicz  _et al._ (2016), it
is possible to translate the weak form factor measurement into a determination
of the weak nuclear radius, $R_{W}$. In particular, the employed weak nuclear
density reads Piekarewicz  _et al._ (2016)

$\rho_{W}(r,c,a)=\frac{3\ \mathcal{Q}_{W}}{4\pi c\
(c^{2}+\pi^{2}a^{2})}\dfrac{\sinh{(c/a)}}{\cosh{(c/a)}+\cosh{(r/a)}},$ (2)

where $r$ is the radius, $c$ represents the half density radius and $a$ is the
so-called diffuseness, related to the thickness parameter $t$ through
$t=4\,a\,\mathrm{ln}(3)$. We fixed the diffuseness parameter to the EDFs
average value $a=0.605\pm 0.025\ \mathrm{fm}$, as done by the PREX
Collaboration in Ref. et al. (2021). The corresponding weak charge radius
$R_{W}$ can be obtained using Horowitz  _et al._ (2012); Piekarewicz  _et al._
(2016)

$R_{W}^{2}=\frac{1}{\mathcal{Q}_{W}}\int
d^{3}r\,r^{2}\rho_{W}(r,c,a)=\frac{3}{5}c^{2}+\frac{7}{5}\pi^{2}a^{2},$ (3)

which in turn can be translated into the neutron distribution radius by the
relation Horowitz  _et al._ (2012)

$R_{n}^{2}=\frac{\mathcal{Q}_{W}}{\mathcal{Q}_{W}^{n}N}R_{W}^{2}-\frac{\mathcal{Q}_{W}^{p}Z}{\mathcal{Q}_{W}^{n}N}R_{\rm{ch}}^{2}-\langle
r_{p}^{2}\rangle-\frac{Z}{N}\langle
r_{n}^{2}\rangle+\frac{Z+N}{\mathcal{Q}_{W}^{n}N}\langle r_{s}^{2}\rangle,$
(4)

where $N$ is the neutron number, $\langle r_{p(n)}^{2}\rangle$ is the charge
radius of a single proton (neutron), namely, $\langle r_{p}^{2}\rangle=0.7071\
\mathrm{fm}^{2}$ and $\langle r^{2}_{n}\rangle=-0.1161\ \mathrm{fm}^{2}$ Zyla
_et al._ (2020), $\langle r_{s}^{2}\rangle$ the nucleon strangeness radius
($\langle r_{s}^{2}\rangle\simeq-0.013\ \mathrm{fm}^{2}$ Horowitz (2019)) and
$\mathcal{Q}_{W}^{n,p}$ is the weak nucleon charges, defined as twice the
opposite of the couplings of electrons to nucleons, $g_{AV}^{ep}$ and
$g_{AV}^{en}$. Using the standard model (SM) prediction of the weak mixing
angle at low energies, $s^{2\ \rm{SM}}_{W}=0.23857\pm 0.00005$, and taking
into account radiative corrections Zyla  _et al._ (2020); Erler and Su (2013);
Erler  _et al._ (2014), one obtains $g_{AV,\text{SM}}^{ep}=-0.0357$ and
$g_{AV,\text{SM}}^{en}=0.495$, where the coupling to protons is suppressed.
The weak charge of the nucleus, which enters directly both $\rho_{W}$ and
$\mathcal{A}_{\rm{pv}}$, is the weak coupling to the nucleus, and it is
defined as

$\displaystyle\mathcal{Q}_{W}^{\mathrm{th}}(N,Z,s^{2}_{W})\equiv-2[Zg_{AV}^{ep}(s^{2}_{W})+Ng_{AV}^{en}]+\square^{\mathrm{Pb}}_{\gamma
Z}.$ (5)

For ${}^{208}\mathrm{Pb}$, $Z$=82 and $N$=126. Thus, taking into account the
radiative corrections and the total $\gamma$-$Z$ correction, $\square_{\gamma
Z}$, for lead, the weak charge is predicted to be
$\mathcal{Q}^{\mathrm{SM}}_{W}(^{208}\mathrm{Pb})=-117.9(3)$ Erler and
Gorchtein (2021) in the case of polarized electron scattering.
As shown in Eq. 1, PREX-II depends on the weak mixing angle at its
experimental energy scale, where the latter is not well constrained. Indeed,
despite the PREX-II measurement is said to be negligibly dependent on
$s^{2}_{W}$ Donnelly  _et al._ (1989), its dependence should be taken into
account.

Figure 1: Favored regions in the $\sin^{2}\theta_{W}$ vs $\Delta
R_{\rm{np}}(^{208}\mathrm{Pb})$ plane given by our reanalysis of the PREX-II
data. The gray, brown and red shaded areas represent the 1, 2 and 3$\sigma$
confidence level contours. The red horizontal bar shows the PREX-II result et
al. (2021) for $s^{2\ \rm{SM}}_{W}$ (blue line). The green vertical band
corresponds to the theoretical prediction, $\Delta
R^{\rm{th}}_{\rm{np}}(^{208}\mathrm{Pb})$.

To do so, we modified the DREPHA code Klein  _et al._ (1986) to calculate the
asymmetry and cross-section values under the DWBA, giving $\rho_{W}$ as an
input evaluated for a variety of $R_{W}$ and $s^{2}_{W}$ values using the
symmetrized 2pF parametrization Piekarewicz  _et al._ (2016). Then, the
asymmetries have been averaged considering the experimental angular acceptance
$\epsilon(\theta)$, following the PREX-II procedure et al. (2021). We fit the
measured asymmetry $A_{\rm{pv}}^{\rm{meas}}=550\pm 16\ \mathrm{(stat.)}\pm 8\
\mathrm{(syst.)}\ \mathrm{ppb}$ et al. (2021), computing the following
$\chi^{2}$ function

$\chi^{2}_{\rm{PREX-
II}}(s^{2}_{W},R_{n})=\Big{(}\frac{A_{\rm{pv}}(s^{2}_{W},R_{n})-A_{\rm{pv}}^{\rm{meas}}}{\sigma_{A_{\rm{pv}}^{\rm{meas}}}}\Big{)}^{2}\,,$
(6)

where $\sigma_{A_{\rm{pv}}^{\rm{meas}}}$ is the total experimental uncertainty
obtained summing in quadrature the statistical and systematical uncertainties.
We first perform a fit of the neutron skin fixing $s^{2}_{W}=s^{2\
\rm{SM}}_{W}$, leading to $\Delta R_{\rm{np}}(s^{2\ \rm{SM}}_{W})=0.276\pm
0.078\ \mathrm{fm}$, which confirms the PREX-II published result, $\Delta
R_{\rm{np}}^{\rm{PREX-II}}(^{208}\mathrm{Pb})=0.278\pm 0.078\ \mathrm{fm}$ et
al. (2021), via an independent analysis.
Then, leaving $s^{2}_{W}$ free to vary, we obtain the result presented in Fig.
1. The introduction of $s^{2}_{W}$ as a parameter in the fit produces a fully
degenerate diagonal band. Therefore, smaller values of the neutron skin can be
accessed for lower values of $s^{2}_{W}$, so that a neutron skin compatible
with $\Delta R^{\rm{th}}_{\rm{np}}(^{208}\mathrm{Pb})$ can be obtained if
$s^{2}_{W}\approx 0.225$.

Atomic Parity Violation experiment on lead.— To break the degeneracy, we
exploit the atomic parity violation (APV) experiment on ${}^{208}\mathrm{Pb}$,
a weak-interaction process between the electrons and the nucleus, sensitive to
the nuclear weak charge. Indeed, these probes represent the lowest energy
determinations of $s^{2}_{W}$, being the momentum transfer of few MeVs
Bouchiat and Piketty (1983), i.e., $Q_{\mathrm{APV(\rm{Cs})}}\sim 2.4\
\mathrm{MeV}$ for cesium atoms and $Q_{\mathrm{APV(\rm{Pb})}}\sim 8\
\mathrm{MeV}$ for lead atoms. So far, the most precise APV measurement was
performed on cesium and it is extracted from the ratio of the parity violating
amplitude, $E_{\rm APV}$, to the Stark vector transition polarizability,
$\beta$, and by calculating theoretically $E_{\rm APV}$ Zyla  _et al._ (2020);
Dzuba  _et al._ (2012); Bennett and Wieman (1999). The parity nonconserving
nuclear-spin-independent part of the electron-nucleus interaction Hamiltonian
Porsev  _et al._ (2016) depends on $\rho_{W}(\mathbf{r})$. In the original
calculation of $E_{\rm APV}$, a charge-density distribution has been used
instead of $\rho_{W}(\mathbf{r})$ since the charge radius was better
determined. To keep this into account, a so-called neutron-skin correction was
applied to the final calculation using the proper density distribution Pollock
_et al._ (1992); Pollock and Welliver (1999); James and Sandars (1999);
Horowitz  _et al._ (2001); Viatkina  _et al._ (2019). For the amplitude
calculation of APV(Pb), a uniformly charged ball density was assumed Porsev
_et al._ (2016), with $R_{\rm{ch}}(^{208}\mathrm{Pb})=5.5010\,\rm{fm}$ Porsev
_et al._ (2016).
Once $E_{\rm APV}$ is obtained, the quantity $R_{\rm{th}}=\left({{\rm
Im}\,E_{\rm APV}/M_{1}}\right)_{\rm th.}$ can be defined, where $M_{1}$ is the
reduced electric-dipole transition of the magnetic-dipole operator for the
$6p^{2}\,{}^{3}P_{0}\rightarrow 6p^{2}\,{}^{3}P_{1}$ transition relevant for
lead. The theoretical calculated value is $R_{\rm{th}}=-10.6(4)\times
10^{-8}(-\mathcal{Q}^{\mathrm{APV(Pb)}}_{W}/N)$ Porsev  _et al._ (2016) while
experimentally two measurements are available,
$R_{\mathrm{exp}}=-9.86(12)\times 10^{-8}$ Meekhof  _et al._ (1993) and
$R_{\mathrm{exp}}=-9.80(33)\times 10^{-8}$ Phipp  _et al._ (1996), therefore
we consider the experimental average value, $R_{\rm exp}^{\rm av.}$. The lead
weak charge is then obtained through the ratio between $R_{\mathrm{exp}}$ and
$R_{\rm{th}}$, which gives $\mathcal{Q}^{\mathrm{APV(Pb)}}_{W}=-117(5)$ Porsev
_et al._ (2016).
The neutron skin correction can be defined as

$\delta
E^{\mathrm{n.s.}}_{\mathrm{APV}}(R_{n})=\left[(N/\mathcal{Q}^{\mathrm{APV(Pb)}}_{W})\left(1-(q_{n}(R_{n})/q_{p})\right)E_{\mathrm{APV}}\right],$
(7)

where $q_{p(n)}$ are the integrals over the proton and neutron nuclear
densities Pollock  _et al._ (1992); Pollock and Welliver (1999); James and
Sandars (1999); Horowitz  _et al._ (2001); Viatkina  _et al._ (2019),
respectively, determined as discussed in detail in Refs. Cadeddu and Dordei
(2019); Cadeddu  _et al._ (2021a). In the $q_{p}$ calculation, we employ the
charge density used in the original determination Porsev  _et al._ (2016),
while for $q_{n}$ calculation the symmetrized 2pF was adopted.
The weak charge measurement is then obtained through

$\mathcal{Q}_{W}^{\mathrm{APV(Pb)}}(R_{n})=-NR_{\rm exp}^{\rm
av.}\Big{(}\dfrac{M_{1}}{\mathrm{Im}(E_{\rm{APV}}+\delta E_{\rm APV}^{\rm
n.s.}(R_{n}))}\Big{)}_{\rm th.}.$ (8)

This procedure has already been exploited in the case of cesium Cadeddu and
Dordei (2019); Cadeddu  _et al._ (2021a).
The APV measurement, contrary to PREX, is mainly sensitive to $s^{2}_{W}$ and
only feebly on $\Delta R_{\rm np}$. We build the following $\chi^{2}$ function
for APV(Pb)

$\chi^{2}_{\rm{APV}(\mathrm{Pb})}(s^{2}_{W},R_{n})=\Big{(}\dfrac{\mathcal{Q}^{\rm{th,APV}}_{W}(s^{2}_{W})-\mathcal{Q}_{W}^{\rm{APV}(\mathrm{Pb})}(R_{n})}{\sigma^{\rm{APV}(\mathrm{Pb})}(s^{2}_{W},R_{n})}\Big{)}^{2},$
(9)

where $\sigma^{\rm{APV}(\mathrm{Pb})}(s^{2}_{W},R_{n})$ is the total
uncertainty. The weak charge $\mathcal{Q}^{\rm{th},APV}_{W}$ for APV
experiments is slightly different from the one given for polarized electron
scattering in Eq. 5, due to different radiative corrections. In particular,
adopting the description in Refs. Zyla  _et al._ (2020); Cadeddu  _et al._
(2021a), $\mathcal{Q}^{\rm{th},APV}_{W}=-118.79(5)$. We perform a combined fit
of the APV(Pb) and PREX-II measurements by summing the $\chi^{2}$ functions in
Eqs. 6 and 9 to fully exploit their correlations, requiring $s^{2}_{W}$ to be
constant between the corresponding experimental momentum transfers, $8\lesssim
Q\lesssim 78\ \mathrm{MeV}$.

Figure 2: Combined fit results of APV(Pb)+PREX-II (dashed orange and dark red
contours) and APV(Pb)+PREX-II+theory (solid cyan and blue contours, with their
corresponding best fits (orange square and cyan star, respectively), shown in
the $\sin^{2}\theta_{W}$ vs $\Delta R_{\rm{np}}(^{208}\mathrm{Pb})$ plane at
1$\sigma$ and 3$\sigma$ confidence levels. The side panels show the one-
dimensional marginalizations (red line for APV(Pb)+PREX-II, cyan line for
APV(Pb)+PREX-II+theory) for both the fits. The red horizontal bar shows the
PREX-II result et al. (2021) for $s^{2\ \rm{SM}}_{W}$(blue line).

In Fig. 2, we show the combined fit result through the dashed contours. In
particular, the different sensitivities on $s^{2}_{W}$ and $\Delta R_{\rm np}$
of the two experiments allow us to find an elliptical closed region in the
parameter space with best-fit values equal to

$\displaystyle\Delta R_{\rm{np}}^{\rm{Comb}}$ $\displaystyle=$ $\displaystyle
0.262\pm 0.136\ \mathrm{fm},$ (10) $\displaystyle\sin^{2}\theta_{W}(8\lesssim$
$\displaystyle Q$ $\displaystyle\lesssim 78\ \mathrm{MeV})=0.237\pm 0.014.$
(11)

The neutron skin value obtained is slightly smaller than the PREX-II one et
al. (2021), but with about double the uncertainty, so that now it is in
agreement with $\Delta R_{\rm np}^{\rm th}$ as well as with other
nonelectroweak measurements Piekarewicz  _et al._ (2012); Roca-Maza  _et al._
(2015); Piekarewicz (2021); Trzcińska  _et al._ (2001); Zhang  _et al._
(2021); Klos  _et al._ (2007); Zenihiro  _et al._ (2010); Starodubsky and
Hintz (1994); Tarbert  _et al._ (2014); Essick  _et al._ (2021a); Fattoyev
_et al._ (2018); Reed  _et al._ (2021); Newton and Crocombe (2021); Stone
(2021); Essick  _et al._ (2021b); Riley  _et al._ (2021); Miller  _et al._
(2021); Biswas (2021); Abbott  _et al._ (2017, 2018, 2020).
From Fig. 2, it is clear that, to allow for thinner neutron skins, $s^{2}_{W}$
should be lower than its SM value. In Ref. Cadeddu  _et al._ (2021a), this has
already been observed for a combined fit of APV(Cs) and the coherent elastic
neutrino-nucleus scattering data from the COHERENT experiment Akimov  _et al._
(2021). Surprisingly enough, looking at Fig. 3 of Ref. Cadeddu  _et al._
(2021a), it can be retrieved that to obtain a cesium skin value around 0.13
fm, $s^{2}_{W}$ should be smaller than the SM prediction, in agreement with
the results found in this work.
Finally, we combine the two experimental determinations with a Gaussian prior
input on the neutron skin given by $\Delta
R_{\rm{np}}^{\rm{th}}(^{208}\mathrm{Pb})=0.16\pm 0.03\ \mathrm{fm}$. The
result of this combined fit is shown in Fig. 2 through the solid contours.
Clearly, the prior forces the fit to favor smaller values of $s^{2}_{W}$. The
best-fit values correspond to

$\displaystyle\Delta R_{\rm{np}}^{\rm{comb+\rm{th}}}$ $\displaystyle=$
$\displaystyle 0.164\pm 0.029\ \mathrm{fm},$ (12)
$\displaystyle\sin^{2}\theta_{W}(8\lesssim$ $\displaystyle Q$
$\displaystyle\lesssim 78\ \mathrm{MeV})=0.228\pm 0.008.$ (13)

The $s^{2}_{W}$ best fit results in a value lower than the SM predicted one,
and with smaller uncertainty with respect to the PREX-II+APV(Pb) combined fit
result.

Main results and discussion.— The 1$\sigma$ confidence level contours obtained
by the three presented analysis are summarized in Fig. 3 to underline how
thinner skins are allowed for lower values of $s^{2}_{W}$.
Focusing on the implications for $s^{2}_{W}$, in Fig. 4 we summarize the state
of the art of the weak mixing angle measurements in the low-energy regime
($Q\lesssim 200\ \mathrm{MeV}$) through processes involving electrons. The
lowest energy determination belongs to APV(Cs), which is 1$\sigma$ lower than
the SM value Zyla  _et al._ (2020). The APV(Cs) value corresponds to a neutron
skin correction determined from an extrapolation of neutron skin measurements
from antiprotonic data Dzuba  _et al._ (2012); Derevianko (2001, 2000);
Trzcińska  _et al._ (2001), which is compatible with the EDF estimate on
cesium. Let us note that the APV(Cs) result is currently debated in the
community. The theoretical calculations have gone through many reevaluations,
leading to different weak mixing angle determinations. For completeness, see
the work presented in Ref. Porsev  _et al._ (2009) and the recent calculation
performed in Ref. Sahoo  _et al._ (2021). At higher energies ($Q\approx 160\
\mathrm{MeV}$), the combined $Q_{\rm{weak}}$ Androic  _et al._ (2018) and the
E158 Anthony  _et al._ (2005) measurements precisely determine $s^{2}_{W}$ to
be compatible with the SM prediction. The orange square and the light blue
star points are the results obtained in this work for the combined PREX-
II+APV(Pb) fit, see Eq. 11, and the PREX-II+APV(Pb)+theory fit, see Eq. 13,
respectively. The horizontal error bars indicate that we assume $s^{2}_{W}$ to
remain constant between the APV(Pb) and PREX-II experimental energy scales.

Figure 3: Summary of the PREX-only (grey long dashed), combined (orange
dashed) and combined+theory (cyan solid) 1$\sigma$ confidence level contours
in the $\sin^{2}\theta_{W}$ vs $\Delta R_{\rm{np}}(^{208}\mathrm{Pb})$ plane.
The orange square and the cyan star points are the best fits of combined and
combined+theory, respectively. The green vertical band shows $\Delta R_{\rm
np}^{\rm th}$, while the red dot the PREX-II result et al. (2021) for $s^{2\
\rm{SM}}_{W}$(blue line).

Since PREX-II and APV(Pb) are not so precise in measuring the weak mixing
angle, future determinations at the same energy scale are awaited. In
particular, the P2 Becker  _et al._ (2018); Dev  _et al._ (2021) and the
MOLLER Benesch  _et al._ (2014) experiments are going to measure $s^{2}_{W}$
with high precision at energies slightly smaller than the PREX-II one.

Figure 4: Weak mixing angle running with the energy scale $Q$. The SM
prediction (solid blue curve) is compared with some experimental
determinations (black dots) Wood  _et al._ (1997); Dzuba  _et al._ (2012);
Tanabashi  _et al._ (2018); Zyla  _et al._ (2020); Androic  _et al._ (2018);
Anthony  _et al._ (2005), and future measurements (purple dots) Becker  _et
al._ (2018); Dev  _et al._ (2021); Benesch  _et al._ (2014). The orange dashed
and the cyan solid points come from the combined and the combined+theory fits,
respectively. The cyan result is shifted towards lower energies for
illustrative purposes, as indicated by the arrows. The vertical arrows
indicate the momentum transfer for APV(Pb), PREX-II and CREX, while the green
dashed curve represents the modified running of $\sin^{2}\theta_{W}$ in a
scenario involving a new mediator Cadeddu  _et al._ (2021b).

The $s^{2}_{W}$ dependence on the energy scale is fundamental in our
discussion. In fact, the presence of beyond the SM light particles could
significantly modify the running of $s^{2}_{W}$ only at low-energies. The
green dashed curve in Fig. 4 shows an example of a dark $Z$ boson Davoudiasl
_et al._ (2012a, b) of mass around $50\ \mathrm{MeV}$ as discussed in Ref.
Cadeddu  _et al._ (2021b). It shows that $s^{2}_{W}$ can sensibly differ from
the SM at low energies while remaining compatible for $Q\gtrsim 150$ MeV, so
that $Q_{\rm{weak}}$ and PREX-II could be measuring different $s^{2}_{W}$
values.
The PREX twin experiment on calcium, CREX Riordan  _et al._ (2013); Kumar
(2020); Horowitz  _et al._ (2014), is performed at $Q\approx 170\
\mathrm{MeV}$, so near the $Q_{\rm{weak}}$ energy scale. In this regime, the
value of $s^{2}_{W}$ is rather precisely measured to be close to $s^{2\
\rm{SM}}_{W}$, thus, if our interpretation is correct, CREX is expected to
measure a thin calcium neutron skin, compatible with the prediction of coupled
cluster calculations, $\Delta R_{\rm np}^{\rm
cc}(^{48}\mathrm{Ca})=0.12$-$0.15\ \mathrm{fm}$ Hagen  _et al._ (2015).

Conclusions.— In summary, we confirm the PREX-II neutron skin measurement via
an independent analysis at fixed weak mixing angle. Questioning the
assumptions behind this measurement, we show the PREX-II neutron skin
dependence on $s^{2}_{W}$, resulting in a fully degenerate band in the
$s^{2}_{W}$ vs $\Delta R_{\rm{np}}(^{208}\mathrm{Pb})$ plane. Nevertheless,
the fit suggests that a much smaller (or larger) lead neutron skin could be
found if $s^{2}_{W}(Q_{\rm PREX-II})$ is small (or large) enough, so that
neglecting such a dependence the experimental result might be misinterpreted.
A combined fit with APV(Pb) allows us to to break this degeneracy and to
obtain a closed favored region in the parameter space. The best fit is in
great agreement with $s^{2\ \mathrm{SM}}_{W}$ and we find a slightly thinner
neutron skin with respect to the PREX-II result, even if with double the
uncertainty, showing that the PREX-II tension can be eased. We find a smaller
favored region in the parameter space by inserting $\Delta R_{\rm np}^{\rm
th}$ as a prior, obtaining that a $s^{2}_{W}$ value 1.2$\sigma$ smaller than
the SM one is needed to accommodate for smaller neutron skins. Further
$s^{2}_{W}$ measurements are needed, such as the upcoming P2 and MOLLER
experiments Becker  _et al._ (2018); Dev  _et al._ (2021); Benesch  _et al._
(2014), as well as new neutron skin determinations, such as the one by the
MREX experiment Becker  _et al._ (2018), to shed light on this tension.
In conclusion, we discuss the possibility that the upcoming CREX result could
be only feebly affected by the $s^{2}_{W}$ dependence thanks to the higher
experimental energy scale, where $s^{2}_{W}$ is experimentally constrained at
the SM value, reconciling eventual conflicting results between the two twin
experiments.

###### Acknowledgements.

We would like to thank M. Gorchtein and J. Erler for the fruitful discussions
on the lead weak charge. Moreover, the useful discussions with F. Dordei, W.
Bonivento and C. Giunti are kindly acknowledged. The work at Brookhaven
National Laboratory was sponsored by the Office of Nuclear Physics, Office of
Science of the U.S. Department of Energy under Contract No. DE-AC02-98CH10886
with Brookhaven Science Associates, LLC.

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