# Revisiting the extraction of charge radii of 40Ca and 208Pb with muonic atom
spectroscopy

Hui Hui Xie  College of Physics, Jilin University, Changchun 130012, China
Tomoya Naito  RIKEN Interdisciplinary Theoretical and Mathematical Sciences
Program, Wako 351-0198, Japan  Department of Physics, Graduate School of
Science, The University of Tokyo, Tokyo 113-0033, Japan  Jian Li
<EMAIL_ADDRESS>College of Physics, Jilin University, Changchun 130012,
China  Haozhao Liang<EMAIL_ADDRESS>Department of
Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033,
Japan  RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program,
Wako 351-0198, Japan

###### Abstract

The extractions of nuclear charge radii from muonic atom spectroscopy for 40Ca
and 208Pb are revisited to analyze the model dependencies induced by employing
a Fermi-type charge distribution. For that, the charge densities, together
with the corresponding muonic transition energies, calculated by the covariant
density functional theory are used as a benchmark. The root-mean-square
deviation of transition energies is calculated to quantitatively investigate
the sensitivities of transition energies to the details of the two-parameter
Fermi distribution. It is found that the second and fourth moments of the
charge distribution can be extracted accurately from the muonic atom
spectroscopy without much model dependencies, whereas the obtained two-
parameter Fermi distributions cannot reproduce the details of the benchmarking
charge densities and, in particular, its surface-diffuseness parameter cannot
be determined accurately with the present experimental uncertainties on the
muonic transition energies.

##  I Introduction

Muonic atoms, where a negatively charged muon is captured by a nucleus, have
been widely used to extract nuclear charge radii with high-precision for most
stable isotopes [1, 2, 3, 4, 5, 6], since the technique of X-ray spectroscopy
from $\mu$-mesonic atoms developed in 1953 [7]. In addition, nuclear electric
quadrupole moments can also be accurately deduced from the observed hyperfine
splittings of muonic X-ray transitions [8, 9, 10]. However, there exists a
long-standing problem that limits the extractive accuracy and leads to the
non-negligible theoretical uncertainty, when the experimental transition
energies are analyzed, i.e., there exists a puzzling discrepancy between
theoretical calculations and experimental data for fine-structure splittings
[11, 3].

In the early analysis, the nuclear polarization (NP) correction is supposed to
be the main source of such a discrepancy, due to the existence of challenging
problems in the calculation of the NP effect [11, 3]. Thus, considerable
theoretical efforts have been made in recent decades to perform a rigorous
calculation of NP correction [12, 13, 14, 15, 16]. Employing state-of-the-art
techniques, the most comprehensive calculations of NP corrections are
performed recently by Valuev et al. [16] and it is concluded that the NP
effect is unlikely responsible for the discrepancy. As a result, the other
effects could play roles in explaining the discrepancy. On the other hand,
note that the influence of model dependencies, induced by employing a Fermi-
type charge distribution in the model-dependent analysis, is in general not
examined [3]. Although the influence of model dependencies could be evaded in
a model-independent combined analysis of muonic data and electron-scattering
data as shown in Ref. [11], it is also important to investigate the
contribution of model dependencies to the uncertainty of extracted charge
radii. For this purpose, a detailed analysis of model dependencies with
respect to the Fermi-type charge distribution is performed in the present work
by using the theoretical charge density distribution from the self-consistent
mean-filed calculations as a benchmark.

In recent decades, the covariant (relativistic) density functional theory
(CDFT) has attracted extensive attentions on account of its successful
description of many nuclear phenomena [17, 18, 19, 20, 21, 22]. For instance,
it can well reproduce the isotopic shifts in the Pb [23] and Sn [24] regions,
and naturally give the origins of the pseudospin symmetry in the nucleon
spectrum [25] and the spin symmetry in the anti-nucleon spectrum [26, 27]. In
addition, it can provide a consistent description of currents and time-odd
fields, which play an important role in nuclear magnetic moments [28, 29, 30,
31, 32] and nuclear rotations [33, 34]. Based on the CDFT with the Bogoliubov
transformation in the coordinate representation, the relativistic continuum
Hartree-Bogoliubov (RCHB) theory was developed [35, 36] to provide a proper
treatment of pairing corrections and mean-filed potentials in the presence of
the continuum and has achieved great success in various aspects about exotic
nuclei, such as providing a microscopic and self-consistent description of
halo in 11Li [35], predicting the giant halos phenomena in light and medium-
heavy nuclei [37, 38, 39], and reproducing the interaction cross sections and
charge-changing cross sections in sodium isotopes and other light exotic
nuclei [40, 41].

In this work, nuclear charge densities from the RCHB calculations are used as
a benchmark to revisit the extractive process of charge radii based on two-
parameter Fermi (2pf) distribution for two doubly-magic nuclei, 40Ca and
208Pb. In addition to the second moment, the fourth moment of nuclear charge
distribution has recently attracted great attention for various applications,
such as providing information on nuclear surface properties [42] and
performing a successful study of the neutron-skin thickness [43]. In this
paper, the sensitivities of muonic transition energies to the second and
fourth moments are also investigated.

The paper is organized as follows. In Sec. II, we introduce the RCHB method
for solving the nuclear structure and then construct the nuclear charge
densities. In Sec. III.1, we determine the parameters of 2pf distribution by
fitting the charge densities from RCHB calculations in several ways and
compare the corresponding transition energies with the data of RCHB. In Sec.
III.2, we determine the parameters by fitting the referenced transition
energies, namely the results with the charge densities from RCHB calculations.
In this way, we qualitatively and quantitatively investigate the model
dependencies and the sensitivities of transition energies to the second and
fourth moments. Then, an analysis of the Barrett model is also made for
further comparison. Finally, summary and perspectives are presented in Sec.
IV.

##  II Theoretical framework

###  II.1 Nuclear charge density from RCHB theory

In this study, the RCHB theory constructed with the contact interaction in the
point-coupling representation between nucleons is adopted, where the
conventional finite-range meson-exchange interaction is replaced by the
corresponding local four-point interaction between nucleons. The details of
the RCHB theory with point-coupling density functional can be found in Refs.
[44, 45]. In the following, we introduce the theoretical framework briefly.

Starting from the Lagrangian density, the energy density functional of the
nuclear system can be constructed under the mean-field and no-sea
approximations. By minimizing the energy density functional with respect to
the densities, one obtains the Dirac equation for nucleons within the
relativistic mean-field framework [46]. The relativistic Hartree-Bogoliubov
model provides a unified description of both the mean field and the pairing
correlation.

In the RCHB theory, the proton and neutron densities can be constructed by
quasiparticle wave functions,

$\rho_{\tau}(\bm{r})=\sum_{k\in\tau}V_{k}^{\dagger}(\bm{r})V_{k}(\bm{r})$ (1)

with $\tau\in\\{p,n\\}$. The nuclear charge density then includes the
contributions from the point neutron density, the proton and neutron spin-
orbit densities, and the single-proton and single-neutron charge densities, in
addition to that from the point proton density [47, 48, 49, 50]. The
relativistic nuclear charge density is finally written as

$\rho_{c}(r)=\sum_{\tau}\left[\rho_{c\tau}(r)+W_{c\tau}(r)\right],$ (2)

where

$\displaystyle\rho_{c\tau}(r)=\frac{1}{r}\int_{0}^{\infty}x\rho_{\tau}(x)\left[g_{\tau}(|r-x|)-g_{\tau}(r+x)\right]\mathrm{d}x,$
(3) $\displaystyle
W_{c\tau}(r)=\frac{1}{r}\int_{0}^{\infty}xW_{\tau}(x)\left[f_{2\tau}(|r-x|)-f_{2\tau}(r+x)\right]\mathrm{d}x.$
(4)

Here, $\rho_{\tau}(r)$ is the point nucleon density in Eq. (1) and
$W_{\tau}(r)$ is the spin–orbit density given in Refs. [48, 50]. The functions
$g_{\tau}(x)$ and $f_{2\tau}(x)$ are given by

$\displaystyle
g_{\tau}(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iqx}\,G_{E\tau}(\bm{q}^{2})\,\mathrm{d}q,$
(5) $\displaystyle
f_{2\tau}(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iqx}\,F_{2\tau}(\bm{q}^{2})\,\mathrm{d}q,$
(6)

in which $G_{E\tau}$ and $F_{2\tau}$ denote the electric Sachs and Pauli form
factors of a nucleon, respectively. The following forms [48] are used in this
study,

$\displaystyle G_{Ep}(q^{2})$
$\displaystyle=\frac{1}{\left(1+r_{p}^{2}\bm{q}^{2}/12\right)^{2}},$
$\displaystyle G_{En}(q^{2})$
$\displaystyle=\frac{1}{\left(1+r_{+}^{2}\bm{q}^{2}/12\right)^{2}}-\frac{1}{\left(1+r_{-}^{2}\bm{q}^{2}/12\right)^{2}},$
$\displaystyle F_{2p}(q^{2})$
$\displaystyle=\frac{G_{Ep}}{1+\bm{q}^{2}/4M_{p}^{2}},$ $\displaystyle
F_{2n}(q^{2})$
$\displaystyle=\frac{G_{Ep}-G_{En}/\mu_{n}}{1+\bm{q}^{2}/4M_{n}^{2}},$ (7)

with the proton charge radius $r_{p}=0.8414$ fm [51] and
$r^{2}_{\pm}=r_{\mathrm{av}}^{2}\pm\frac{1}{2}\langle r_{n}^{2}\rangle$, where
$r_{\mathrm{av}}^{2}=0.9$ fm2 is the average of the squared radii for positive
and negative charge distributions, while $\langle r_{n}^{2}\rangle=-0.11$ fm2
[52] is the mean squared charge radius of a neutron. See reference [53] for
more details.

###  II.2 Dirac equation for muonic atom

The Dirac equation for a muon reads

$\left[\bm{\alpha}\cdot\bm{p}+\beta
M_{r}+V(\bm{r})\right]\psi(\bm{r})=\varepsilon\psi(\bm{r}),$ (8)

where the eigenenergy $\varepsilon$ includes both the muonic atom energy
levels $E$ and reduced mass $M_{r}$, i.e.,

$\varepsilon\simeq E+M_{r},\quad\mbox{with}\quad
M_{r}=\frac{M_{A}m_{\mu}}{M_{A}+m_{\mu}}.$ (9)

Note that Eq. (9) holds exactly only at the non-relativistic limit, while it
is a good enough approximation in the present calculations. The mass of muon
$m_{\mu}=206.7682830\,m_{e}$ is adopted [51] and nuclear mass $M_{A}$ is
obtained from Ref. [54]. Since the electrostatic potential $V(r)$ made by the
atomic nucleus is spherically symmetric, the eigenvalue function can be
written as

$\psi_{n\kappa
m}(\bm{r})=\frac{1}{r}\begin{pmatrix}iP_{n\kappa}(r)Y_{jm}^{l}(\theta,\varphi)\\\
Q_{n\kappa}(r)\left(\bm{\sigma}\cdot\bm{\hat{r}}\right)Y_{jm}^{l}(\theta,\varphi)\end{pmatrix},$
(10)

where $P_{n\kappa}(r)$ and $Q_{n\kappa}(r)$ are its large and small
components, respectively.

The radial Dirac equation for one-muon system can be then written in the form
of

$\displaystyle\begin{pmatrix}V(r)&-\left(\frac{\mathrm{d}}{\mathrm{d}r}-\frac{\kappa}{r}\right)\\\
\left(\frac{\mathrm{d}}{\mathrm{d}r}+\frac{\kappa}{r}\right)&V(r)-2M_{r}\end{pmatrix}\begin{pmatrix}P_{n\kappa}(r)\\\
Q_{n\kappa}(r)\end{pmatrix}$ $\displaystyle=E\begin{pmatrix}P_{n\kappa}(r)\\\
Q_{n\kappa}(r)\end{pmatrix},$ (11)

where the mass term $M_{r}$ is subtracted on both sides of the equation.

The electrostatic potential $V(r)$ is obtained via [55]

$V(r)=-4\pi\alpha\left[\int_{0}^{r}\rho_{c}(r^{\prime})\frac{r^{\prime
2}}{r}\,\mathrm{d}r^{\prime}+\int_{r}^{\infty}\rho_{c}(r^{\prime})r^{\prime}\,\mathrm{d}r^{\prime}\right],$
(12)

where $\alpha$ is the fine structure constant. In the present work, the muonic
energy levels are evaluated numerically.

##  III Results and discussion

In order to investigate the influence of model dependency on muonic atom
spectrum, nuclear charge densities of 40Ca and 208Pb obtained from the RCHB
calculations are used as benchmarks. The relativistic density functional PC-
PK1 [44], which provides one of the best density-functional descriptions for
nuclear properties [56, 57, 58, 59, 60, 61, 62, 63], is employed. The box size
$R_{\rm box}=20$ fm, the mesh size $\Delta r=0.1$ fm, and the angular momentum
cutoff $J_{\rm max}=19/2~{}\hbar$ are used in the RCHB calculations. More
numerical details can be found in Ref. [45].

The Dirac equation for the muon is solved using the generalized pseudospectral
(GPS) method, whose powerful performance has been shown in Refs. [64, 65, 66,
67]. The mapping parameters $L=0.01$ and $k=3$ are used in the GPS
calculations.

###  III.1 Fitting the nuclear charge density

In general, the available experimental data of muonic atom spectrum are
analyzed by means of a 2pf distribution in the form of

$\rho_{c}(r)=\frac{\rho_{0}}{1+\exp\left[4\ln 3(r-c)/t\right]},$ (13)

with the half-density-radius $c$ and surface-diffuseness parameters $t$,
together with the normalization

$4\pi\int_{0}^{\infty}\rho_{c}(r)r^{2}\,\mathrm{d}r=Z,$ (14)

where $Z$ represents the proton number. The $n$-th moment is given by

$R_{n}\equiv\langle
r^{n}\rangle=\frac{4\pi}{Z}\int_{0}^{\infty}\rho_{c}(r)r^{n+2}\,\mathrm{d}r.$
(15)

In the following investigation, first of all, the parameters $c$ and $t$ of
2pf distribution are determined by fitting the RCHB charge densities in
several feasible ways, with $\rho_{0}$ determined by the normalization.
Specifically, four 2pf distributions are obtained here, i.e.,
2pf($\sigma_{1}$) is determined by minimizing the loss function $\sigma_{1}$,
2pf($R_{2}$,$\sigma_{1}$) and 2pf($R_{2}$,$\sigma_{2}$) are determined by
minimizing respectively $\sigma_{1}$ and $\sigma_{2}$ with the second moment
$R_{2}$ constraint, and 2pf($R_{2}$,$R_{4}$) is determined by constraining the
second and fourth moments. The loss functions $\sigma_{1}$ and $\sigma_{2}$
are two alternatives that indicate the deviations in shape with the following
forms:

$\sigma_{1}=\int_{0}^{\infty}\left(\rho_{c}^{\mathrm{2pf}}(r)-\rho_{c}^{\mathrm{RCHB}}(r)\right)^{2}\,\mathrm{d}r$
(16)

and

$\sigma_{2}=\int_{0}^{\infty}\left(\rho_{c}^{\mathrm{2pf}}(r)-\rho_{c}^{\mathrm{RCHB}}(r)\right)^{2}r^{2}\,\mathrm{d}r.$
(17)

Figure 1: Comparison of charge densities from the RCHB theory with those of
2pf distributions for (a) 40Ca and (b) 208Pb. The 2pf parameters are
determined by fitting the RCHB charge densities in four different ways, i.e.,
2pf($\sigma_{1}$), 2pf($R_{2}$,$\sigma_{1}$), 2pf($R_{2}$,$\sigma_{2}$), and
2pf($R_{2}$,$R_{4}$). See texts for details.

As shown in Fig. 1, four 2pf distributions are compared with the charge
density from the RCHB calculations. It can be seen from Fig. 1(a) that the
central densities of four 2pf distributions for 40Ca are visibly different.
The curve of 2pf($R_{2}$,$R_{4}$) deviates furthest from the RCHB calculation,
and the curve of 2pf($R_{2},\sigma_{2}$) is the next. In contrast, the curve
of 2pf($\sigma_{1}$) is closest in shape to the RCHB calculation. As for 208Pb
in Fig. 1(b), it shows less differences between four 2pf distributions.
Nevertheless, the curve of 2pf($R_{2}$,$R_{4}$) also shows the most deviation
from the RCHB calculation. On the other hand, in the insets of Fig. 1 the
logarithmic coordinates are taken to emphasize the difference of charge
density at the surface. One can see from the insets that the distributions of
2pf($R_{2}$,$R_{4}$) are the closest ones to the RCHB charge density
distributions among four distributions at the nuclear surface, i.e., $r>6$ fm
for 40Ca and $r>8$ fm for 208Pb. In contrast, the charge distributions
2pf($\sigma_{1}$) are in the worst agreement with the RCHB calculations. With
regard to the rest of two 2pf distributions, the curve of
2pf($R_{2}$,$\sigma_{2}$) is closer to (further away from) the curve of RCHB
than 2pf($R_{2}$,$\sigma_{1}$) for 40Ca (208Pb).

The fitted parameters $c$ and $t$ of four 2pf distributions, together with the
corresponding second and fourth moments and charge radii $r_{c}$, as well as
two loss functions $\sigma_{1}$ and $\sigma_{2}$ are shown in Table 1. Since
the results of 2pf($\sigma_{1}$) are obtained without the second-moment
constraint, there exists a visible difference in the second moment between
2pf($\sigma_{1}$) and the others, i.e., around $0.7$ fm2 for 40Ca and $0.2$
fm2 for 208Pb. Moreover, it can be seen that the parameters determined in
2pf($\sigma_{1}$), 2pf($R_{2}$,$\sigma_{1}$), and 2pf($R_{2}$,$\sigma_{2}$),
without the fourth-moment constraint, generally overestimate the fourth
moments.

Table 1: The parameters $c$ and $t$, the second and fourth moments $R_{2}$ and $R_{4}$, and the charge radii $r_{c}$, as well as the values of two loss functions $\sigma_{1}$ and $\sigma_{2}$, for 40Ca and 208Pb in four 2pf models. Note that the numbers that exactly reproduce the RCHB values are highlighted in bold form. |  40Ca |  208Pb
---|---|---
| 2pf($\sigma_{1}$) | 2pf($R_{2}$,$\sigma_{1}$) | 2pf($R_{2}$,$\sigma_{2}$) | 2pf($R_{2}$,$R_{4}$) | 2pf($\sigma_{1}$) | 2pf($R_{2}$,$\sigma_{1}$) | 2pf($R_{2}$,$\sigma_{2}$) | 2pf($R_{2}$,$R_{4}$)
$c$ (fm) | $3.43431$ | $3.51641$ | $3.63210$ | $3.72839$ | $6.65823$ | $6.66207$ | $6.64228$ | $6.70918$
$t$ (fm) | $2.82022$ | $2.55115$ | $2.41134$ | $2.28476$ | $2.33968$ | $2.26797$ | $2.31614$ | $2.14837$
$R_{2}$ (fm2) | $12.7666$ | 12.0755 | 12.0755 | 12.0755 | 30.5160 | $\bf 30.3103$ | $\bf 30.3103$ | $\bf 30.3103$
$R_{4}$ (fm4) | $256.054$ | $220.280$ | $214.499$ | 209.699 | $1195.877$ | $1174.858$ | $1178.577$ | $\bf 1166.040$
$r_{c}$ (fm) | $3.5730$ | 3.4750 | 3.4750 | 3.4750 | $5.5241$ | $\bf 5.5055$ | $\bf 5.5055$ | $\bf 5.5055$
$\sigma_{1}$ ($10^{-6}$fm-5) | $32.1726$ | $44.8123$ | $90.7370$ | $195.203$ | $9.71232$ | $10.42702$ | $11.3099$ | $15.5466$
$\sigma_{2}$ ($10^{-5}$fm-3) | $19.1357$ | $18.4373$ | $9.36914$ | $16.1918$ | $10.4874$ | $13.9824$ | $12.6133$ | $29.1877$
Table 2: Muonic transition energies (keV) calculated by using RCHB charge density, and the differences between them and the results of four 2pf distributions, for 40Ca and 208Pb. Numbers in parentheses represent the power of ten. Transition | $E_{\mathrm{RCHB}}$ |  40Ca, $\Delta E_{\mathrm{2pf}}-\Delta E_{\mathrm{RCHB}}$ | $E_{\mathrm{RCHB}}$ |  208Pb, $\Delta E_{\mathrm{2pf}}-\Delta E_{\mathrm{RCHB}}$
---|---|---|---|---
(keV) | (40Ca) | 2pf($\sigma_{1}$) | 2pf($R_{2}$,$\sigma_{1}$) | 2pf($R_{2}$,$\sigma_{2}$) | 2pf($R_{2}$,$R_{4}$) | (208Pb) | 2pf($\sigma_{1}$) | 2pf($R_{2}$,$\sigma_{1}$) | 2pf($R_{2}$,$\sigma_{2}$) | 2pf($R_{2}$,$R_{4}$)
$2p_{3/2}$-$1s_{1/2}$ | $778.229$ | $-2.58$ | $1.82(-1)$ | $8.47(-2)$ | $3.69(-3)$ | $5917.293$ | $-8.77$ | $4.84$ | $7.06$ | $-4.51(-1)$
$2p_{1/2}$-$1s_{1/2}$ | $776.766$ | $-2.58$ | $1.82(-1)$ | $8.47(-2)$ | $3.68(-3)$ | $5735.444$ | $-8.24$ | $4.67$ | $6.82$ | $-4.57(-1)$
$3d_{3/2}$-$2p_{1/2}$ | $157.439$ | $-8.03(-3)$ | $-1.06(-3)$ | $-4.46(-4)$ | $7.14(-5)$ | $2619.561$ | $-2.69$ | $-7.98(-4)$ | $-9.02(-2)$ | $2.15(-1)$
$3d_{5/2}$-$2p_{3/2}$ | $156.124$ | $-5.43(-3)$ | $-1.11(-3)$ | $-4.74(-4)$ | $5.95(-5)$ | $2479.793$ | $-2.20$ | $-1.78(-1)$ | $-3.40(-1)$ | $2.12(-1)$
$3d_{3/2}$-$2p_{3/2}$ | $155.976$ | $-5.43(-3)$ | $-1.11(-3)$ | $-4.74(-4)$ | $5.96(-5)$ | $2437.712$ | $-2.16$ | $-1.72(-1)$ | $-3.31(-1)$ | $2.09(-1)$
$5f_{7/2}$-$3d_{5/2}$ | $79.851$ | $-1.96(-6)$ | $-6.14(-7)$ | $-3.05(-7)$ | $-5.66(-8)$ | $1358.683$ | $-5.67(-2)$ | $-1.88(-2)$ | $-2.75(-2)$ | $1.87(-3)$
$5f_{5/2}$-$3d_{5/2}$ | $79.835$ | $-1.95(-6)$ | $-6.05(-7)$ | $-2.96(-7)$ | $-4.78(-8)$ | $1354.023$ | $-5.64(-2)$ | $-1.87(-2)$ | $-2.74(-2)$ | $1.87(-3)$
$4f_{7/2}$-$3d_{5/2}$ | $54.599$ | $-1.96(-6)$ | $-6.07(-7)$ | $-2.99(-7)$ | $-5.04(-8)$ | $932.086$ | $-5.66(-2)$ | $-1.88(-2)$ | $-2.74(-2)$ | $1.87(-3)$
$4f_{5/2}$-$3d_{5/2}$ | $54.568$ | $-1.95(-6)$ | $-6.06(-7)$ | $-2.98(-7)$ | $-4.95(-8)$ | $923.018$ | $-5.63(-2)$ | $-1.87(-2)$ | $-2.73(-2)$ | $1.87(-3)$
$4d_{5/2}$-$3p_{1/2}$ | $55.074$ | $-2.81(-3)$ | $-3.70(-4)$ | $-1.55(-4)$ | $2.53(-5)$ | $914.494$ | $-7.84(-1)$ | $4.73(-2)$ | $4.00(-2)$ | $6.57(-2)$
$4d_{5/2}$-$3p_{3/2}$ | $54.705$ | $-1.91(-3)$ | $-3.90(-4)$ | $-1.66(-4)$ | $2.13(-5)$ | $885.170$ | $-6.94(-1)$ | $-2.38(-2)$ | $-6.41(-2)$ | $7.31(-2)$
$4d_{3/2}$-$3p_{3/2}$ | $54.643$ | $-1.91(-3)$ | $-3.90(-4)$ | $-1.66(-4)$ | $2.13(-5)$ | $867.687$ | $-6.75(-1)$ | $-2.12(-2)$ | $-5.96(-2)$ | $7.15(-2)$

In Table 2, the muonic transition energies calculated by using the RCHB charge
density, and the differences between them and the results calculated by using
the 2pf distributions, for 40Ca and 208Pb, are shown. Twelve muonic transition
energies for 40Ca and 208Pb are given, namely, $2p_{3/2}$-$1s_{1/2}$,
$2p_{1/2}$-$1s_{1/2}$, $3d_{3/2}$-$2p_{1/2}$, $3d_{5/2}$-$2p_{3/2}$,
$3d_{3/2}$-$2p_{3/2}$, $5f_{7/2}$-$3d_{5/2}$, $5f_{5/2}$-$3d_{5/2}$,
$4f_{7/2}$-$3d_{5/2}$, $4f_{5/2}$-$3d_{5/2}$, $4d_{5/2}$-$3p_{1/2}$,
$4d_{5/2}$-$3p_{3/2}$, and $4d_{3/2}$-$3p_{3/2}$, involving fourteen levels.

In comparison, the results of 2pf($\sigma_{1}$) are in the worst agreement
with RCHB calculations and the results of 2pf($R_{2}$,$R_{4}$) are in the best
agreement with RCHB calculations. For example, the 2pf($\sigma_{1}$) gives
$2.58$ keV derivation from RCHB for $2p_{3/2}$-$1s_{1/2}$ in 40Ca. After
considering the second moment constraint $R_{2}$ with different loss function
minimized, 2pf($R_{2}$,$\sigma_{1}$) and 2pf($R_{2}$,$\sigma_{2}$) give the
derivations $-0.182$ and $-0.0847$ keV, respectively. Furthermore, with both
second- and fourth-moment constraints, the derivation from RCHB calculations
decreases to $-3.65\times 10^{-3}$ keV. A similar conclusion for other
transition energies in 40Ca and 208Pb is also observed in Table 2. In view of
the comparison of charge distribution between RCHB calculations and four 2pf
distributions shown in Fig. 1, it is concluded that the surface of nuclear
charge density plays a much more important role in the muonic atom spectrum
than the center. Furthermore, it implies the considerable sensitivities of the
transition energies to the second and fourth moments. A quantitative
discussion is given in the next section.

###  III.2 Fitting the muonic transition energies

In this section, the procedure to extract the charge radii from the muonic
transition energies with the 2pf distribution will be discussed. For the sake
of discussion, the root-mean-square deviation (RMSD) for muonic transition
energies is defined as

$\delta=\left[\frac{1}{N}\sum_{i}\left(\Delta E_{i}^{\mathrm{2pf}}-\Delta
E_{i}^{\mathrm{RCHB}}\right)^{2}\right]^{1/2},$ (18)

where $E_{i}^{\mathrm{2pf}}$ and $E_{i}^{\mathrm{RCHB}}$ are the $i$-th
transition energy calculated with the 2pf charge distributions and the RCHB
charge distributions, respectively, and $N$ is the total number of transitions
considered. For convenience, four transitions with the largest transition
energies, i.e., $2p_{3/2}$-$1s_{1/2}$, $2p_{1/2}$-$1s_{1/2}$,
$3d_{3/2}$-$2p_{1/2}$, and $3d_{5/2}$-$2p_{3/2}$, are considered in Eq. (18).

In panels (a) and (d) of Fig. 2, the trend of $\delta$ with the parameter $t$
changing from $1$ to $3.5$ fm is shown, where the parameter $c$ is determined
by minimizing the RMSD $\delta$ for each $t$. A minimum point of $\delta$ can
be seen in the panels. Here, we use the shorthand writing 2pf($\Delta E$) to
denote the best-fit 2pf distribution which yield a minimum value of $\delta$.
The minimum values of $\delta$ read $\delta\simeq 2.0\times 10^{-5}$ keV with
$t=2.2950$ fm and $c=3.7214$ fm for 40Ca and $\delta\simeq 0.0031$ keV with
$t=2.1821$ fm and $c=6.6977$ fm for 208Pb. In addition, $\delta$ versus the
second moment $\langle r^{2}\rangle$ of the 2pf distribution with the same
parameters as Figs. 2(a) and 2(d) is also displayed in Figs. 2(b) and 2(e).
The second and fourth moments of the best fit are determined as the point of
minimum $\delta$, namely $\langle r^{2}\rangle=12.0778$ fm2, $\langle
r^{4}\rangle=210.145$ fm4 for 40Ca, and $\langle r^{2}\rangle=30.3225$ fm2,
$\langle r^{4}\rangle=1169.38$ fm4 for 208Pb. Meanwhile, the referenced second
moments from the RCHB calculations are also shown in the panels. The relative
deviations in charge radii read
$|\sqrt{12.0778}-\sqrt{12.0755}|/\sqrt{12.0755}\approx 0.01\%$ for 40Ca and
$0.02\%$ for 208Pb. Note that the source of the deviation here is the model
dependency. From the above numbers, it can be seen that the influence of the
model dependency on the extracted charge radii is rather small.

Figure 2: The variation of $\delta$ for 40Ca and 208Pb: (a) and (d) $\delta$
versus parameter $t$ of 2pf distribution, where parameter $c$ are determined
to minimize $\delta$; (b) and (e) $\delta$ versus the second moment of 2pf
distribution, where the black dashed lines indicate the second moment of RCHB
charge distributions; (c) and (f) $\delta$ versus the difference of charge
radii between the 2pf distributions and RCHB calculations, i.e., $\Delta
r_{c}=r_{c}^{\mathrm{2pf}}-r_{c}^{\mathrm{RCHB}}$, where the black dashed
lines represent $\Delta r_{c}=\pm 0.01$ fm.

In practice, the accuracy of the extracted charge radii is limited by the
total uncertainty of transition energies from experimental measurements,
together with various additional corrections, such as vacuum polarization,
relativistic recoil, self-energy, and nuclear polarization corrections [68,
69, 70, 71]. Therefore, the determination of uncertainty of the extracted
charge radii is in general a challenging task. In the present theory-to-theory
benchmarking study, the dependence of the uncertainty in the extracted charge
radii $\Delta r_{c}$ on the uncertainty in the transition energies can be
quantitatively investigated. The RMSD $\delta$ in the present calculations can
be regarded as the total uncertainty of transition energies, and the
difference of charge radii $\Delta
r_{c}=r_{c}^{\mathrm{2pf}}-r_{c}^{\mathrm{RCHB}}$ can be used to indicate the
uncertainty of the extracted charge radii. For this purpose, the variations of
$\delta$ versus $\Delta r_{c}$ for 40Ca and 208Pb are shown in Figs. 2(c) and
2(f), respectively. The points at $\Delta r_{c}=\pm 0.01$ fm are marked and
the corresponding values of $\delta$, i.e., $\delta\approx 0.0017$ keV for
40Ca and $\delta\approx 1$ keV for 208Pb, are illustrated. In fact, it is
quite challenging to make the total uncertainty of transition energies less
than 0.0017 keV for 40Ca. However, the uncertainty less than $1$ keV for 208Pb
is available [3]. As a result, the uncertainty of the extracted charge radii
less than $0.01$ fm for 208Pb can be obtained. It can be deduced that the
charge radii for the heavy nuclei can be extracted from muonic atom
spectroscopy more accurately than for the light nuclei.

Figure 3: The root-mean-square deviation $\delta$ as a function of the fourth
moment $R_{4}$ of 2pf distribution for (a) 40Ca and (b) 208Pb. Different from
Fig. 2, an additional constraint is imposed to keep the second moment
constant, i.e., $R_{2}=12.07784~{}\mathrm{fm^{2}}$ for 40Ca and
$30.32254~{}\mathrm{fm^{2}}$ for 208Pb. The black dashed lines indicate the
data from RCHB calculations.

In order to study the variation of $\delta$ versus the fourth moment, the
influence of the second moment should be avoided. In Fig. 3, the steep
tendency of RMSD $\delta$ with the fourth moment is illustrated, where
parameter $t$ is changed from $1$ to $3.5$ fm, and $c$ is determined according
to the additional constraint of the second moment. The fourth moment of the
RCHB charge distributions is marked and a small deviation from the
corresponding value at the point of minimum $\delta$ can be seen. The relative
deviations of the fourth moment are only around $0.2\%$ and $0.3\%$ for 40Ca
and 208Pb, respectively. Therefore, it can be considered that the prediction
of the fourth moment from the muonic atom spectrum is reliable.

Figure 4: The Barrett equivalent radius $R_{k\alpha}$ for (a) 40Ca and (b)
208Pb. The values of $\alpha$ in Eq. (19) are taken as $0.065$ fm-1 for 40Ca
[2] and $0.1415$ fm-1 for 208Pb [3].

In addition, the analysis using the Barrett model [1] is also performed. The
Barrett equivalent radius $R_{k\alpha}$ is defined by

$\frac{3}{R_{k\alpha}^{3}}\int_{0}^{R_{k\alpha}}r^{k}e^{-\alpha
r}r^{2}\,\mathrm{d}r=\langle r^{k}e^{-\alpha r}\rangle.$ (19)

It can be iteratively calculated from the charge density without resolving the
Dirac equation. The critical sensitivity of transition energies in muonic
atoms to the Barrett equivalent radius has been illustrated [72, 73]. Thus,
the Barrett model provides a particular perspective to compare these 2pf
distributions with the fitted parameters to the referenced charge density from
the RCHB calculations. The comparisons of the Barrett equivalent radius for
four 2pf distributions and the RCHB calculations versus $k$ for 40Ca and 208Pb
are shown in Fig. 4. The values of $\alpha$ are in general determined by the
fitting of the differences of the potential that muon in state $a$ makes and
the potential that muon in state $b$ makes, i.e.,
$f_{ab}(r)=V_{a}(r)-V_{b}(r)$ [72, 3]. The muon potential in the state $a$ is
defined by [1]

$V_{a}(r)=\int_{0}^{\infty}\frac{P^{2}_{a}(r^{\prime})+Q^{2}_{a}(r^{\prime})}{\max(r,r^{\prime})}\,dr^{\prime},$
(20)

where $P_{a}(r)$ and $Q_{a}(r)$ are the large and small components in Eq.
(11), respectively. In the present calculations of the Barrett equivalent
radius, the values of $\alpha$ are taken as $0.065$ fm-1 for 40Ca [2] and
$0.1415$ fm-1 for 208Pb [3]. Different from the practical analysis in
experimental references, e.g., [3], the results of benchmark, i.e., “RCHB”,
are displayed in the form of smooth curves instead of several dots that
correspond to transition energies, because they are evaluated from the RCHB
charge density instead of the transition energies. It can be seen that the
results of 2pf($R_{2}$,$R_{4}$) are in excellent agreement with the results of
RCHB calculations. In contrast, the results of 2pf$(\sigma_{1})$ considerably
deviate from the results of RCHB calculations. As for the rest of two 2pf
distributions, Fig. 4 shows that the result of 2pf($R_{2}$,$\sigma_{2}$) is in
a better (worse) agreement with the result of RCHB than
2pf($R_{2}$,$\sigma_{1}$) for 40Ca (208Pb), which are consistent with the
comparison of transition energies displayed in Table 2.

Furthermore, the results of 2pf($\Delta E$), i.e., the best-fit 2pf
distributions for minimizing the RMSD $\delta$ mentioned above, are also
compared in the Fig. 4. It can be seen that these results are also in
excellent agreements with the results of RCHB calculations. It ensures that
the parameters fitted to the transition energies give a satisfactory
agreement.

Figure 5:  Determination of parameters in 2pf model for 208Pb. (a) The weight
RMSD $\chi$ versus $t$, where $c$ is searched to minimize $\chi$. Two critical
points satisfying condition $\chi\leq 1$ are indicated. (b) Comparison of the
Barrett equivalent radius between the RCHB calculations and the 2pf
distributions with respect to two critical point.  Figure 6:  Determination of
parameters in 2pf model for 208Pb. The elliptic region of 2pf parameters $c$
and $t$ for $\chi\leq 1$ is denoted with the blue shadow area, where the red
star symbol shows the point with the minimum $\chi$ and the green dashed line
shows the determined $c$ values by minimizing $\chi$ for each given $t$. The
contours of the rms charge radii ($0.01$ fm interval) are illustrated with the
black solid lines. Additionally, the red dot-dashed line represents the
contour of the benchmarking charge radius.

In order to perform a more practical investigation, we will then focus on the
weighted RMSD, which is defined as

$\chi=\left[\frac{1}{N}\sum_{i}\frac{\left(\Delta E_{i}^{\mathrm{2pf}}-\Delta
E_{i}^{\mathrm{RCHB}}\right)^{2}}{\left(\eta_{i}\Delta E_{i}^{\rm
RCHB}\right)^{2}}\right]^{1/2},$ (21)

where $\eta_{i}$ represents the relative deviations for $i$-th transition
energy. According to one of the latest experiments for muonic 104Pd [74], the
relative uncertainties of experimental transition energies are approximately
$6.47\times 10^{-5}$, $9.81\times 10^{-5}$, $3.47\times 10^{-4}$, and
$1.20\times 10^{-4}$ for transitions $2p_{3/2}$-$1s_{1/2}$,
$2p_{1/2}$-$1s_{1/2}$, $3d_{3/2}$-$2p_{1/2}$, and $3d_{5/2}$-$2p_{3/2}$,
respectively. Thus, the corresponding $\eta_{i}$ in Eq. (21) are taken as
$10^{-4}$, $10^{-4}$, $3\times 10^{-4}$, and $3\times 10^{-4}$, respectively.

In Fig. 5, the confidence region that satisfies the condition $\chi\leq 1$ for
208Pb is displayed. Parameter $t$ is changed from $1$ to $3.5$ fm with a step
$0.01$ fm, and $c$ is determined to minimize $\chi$ for each $t$. Then two
critical points could be found and the confidence region $2.05\leq t\leq 2.31$
is obtained in Fig. 5(a). This implies that the relative uncertainty of $t$ is
around $(2.31-2.05)/(2.31+2.05)\approx 6\%$, while the relative uncertainty of
$c$ reaches around $0.6\%$.

In Fig. 5(b), the Barrett equivalent radius of the 2pf distributions with
parameters corresponding to two critical points are compared with the results
of RCHB calculations. It is of interest that three curves intersect
approximately at a point around $k\approx 2.25$, and the curve of RCHB is
sandwiched between two 2pf curves with $k$ in the range from $1$ to $4.5$.

Furthermore, in Fig. 6, the elliptic region of 2pf parameters which satisfy
the condition $\chi\leq 1$ is shown. The corresponding charge radii are in the
range from $5.5004$ to $5.5129$ fm. Thus, the uncertainty of the extracted
charge radii is only around $0.005$ fm, and the relative uncertainty is around
$0.11\%$. As a result, it is considered that the extraction of charge radii
from the muonic atom spectrum for 208Pb is reliable.

Meanwhile, there are several interesting observations from the figure. First
of all, it shows the contour of the rms radius in the $c$-$t$ plane. One can
see that the spread of the elliptic region is almost parallel to the contour
of the rms radius. Combined with the comparison of relative uncertainty
between the parameters $c$ and $t$ and the extracted charge radius as
previously discussed, it is confirmed that the transition energy is more
sensitive to the rms radius than to the individual parameter $c$ or $t$. See
also the discussions in Ref. [74]. Second, the contour of the referenced rms
radius, i.e., $r_{c}^{\rm RCHB}$, is shown in Fig. 6. In addition, in the
elliptic region, the point with the minimum $\chi$ and the line with the $c$
determined to minimize $\chi$ for each $t$ is also shown. They correspond to
the red star symbol and the blue curve in Fig. 5(a), respectively. Keeping in
mind that, in this figure, the green dashed line (representing the determined
$c$ values by minimizing $\chi$ for each given $t$) would be identical with
the red dot-dashed line (representing the contour of the benchmarking charge
radius), if there is no model dependency on extracting the charge rms radius.
In contrast, it is seen from the figure that the slopes of these two lines are
slightly different from each other. Precisely, the slopes are $-0.32$ and
$-0.38$ for the former and latter, respectively. Such a difference can be used
as an indicator of the model dependency.

Before ending this section, it is also important to mention that one cannot
constrain the charge radius of 40Ca at all, by taking the same relative
deviations $\eta_{i}$. This is because, to extract the charge radius at the
level of $\Delta r_{c}=\pm 0.01$ fm, the required accuracy of the transition
energies is much higher in the case of 40Ca, as we compared quantitatively in
Figs. 2(c) and 2(f).

##  IV Summary and Perspectives

In this work, the influence of model dependencies on the extraction of charge
radii from the muonic atom spectrum and the sensitivities of muonic transition
energies to the second and fourth moments of charge distribution are
investigated by taking 40Ca and 208Pb as examples. The charge densities from
the RCHB calculations are used as a benchmark. The parameters $c$ and $t$ in
the 2pf model are determined in several ways to fit the referenced charge
densities, and the corresponding transition energies are compared with the
data from RCHB calculations. The sensitivities of transition energies to the
second and fourth moments are illustrated.

In detail, first of all, the variations of RMSD $\delta$ of muonic transition
energies versus parameters $c$ and $t$ are investigated. By changing $t$ from
$1$ to $3.5$ fm and determining $c$ iteratively to minimize $\delta$ for each
$t$, the steep trend of $\delta$ versus the second moment is clearly shown.
Meanwhile, the variations of $\delta$ versus $\Delta r_{c}$ are also shown to
investigate the dependence of the uncertainty in the extracted charge radii on
the uncertainty in the transition energies. On the one hand, it is found that
the total uncertainty of transition energies should be less than $0.0017$ keV
for 40Ca and $1$ keV for 208Pb to make the uncertainty of extracted charge
radii less than $0.01$ fm. This illustrates that the charge radii of heavy
nuclei could be extracted more accurately from muonic atom spectroscopy than
those of light nuclei. On the other hand, when the charge radii are extracted
by minimizing $\delta$, it is found that the relative deviations of extracted
charge radii are around $0.01\%$ for 40Ca and $0.02\%$ for 208Pb, compared
with the referenced charge radii. Therefore, it is considered that the model
dependency only slightly influences the extraction of charge radii. Second,
the variation of $\delta$ versus the fourth moment of 2pf distribution is
displayed with the constraint of the second moment. The sensitivities of
transition energies to the fourth moment can be clearly seen. Finally, the
analysis with the Barrett model is performed. The Barrett equivalent radii
obtained by various 2pf distributions are compared with the referenced
results. It is clearly seen that the 2pf distribution determined by the
transition energies 2pf($\Delta E$) and that with the second- and fourth-
moment constraints 2pf($R_{2}$, $R_{4}$) reproduce the benchmarking Barrett
equivalent radii excellently, whereas those without the fourth-moment
constraint 2pf($R_{2}$, $\sigma_{1}$) and 2pf($R_{2}$, $\sigma_{2}$) provide
less satisfactory results, and that only focusing on the density distribution
alone 2pf($\sigma_{1}$) shows the result considerably deviating from the
benchmark.

As a step further, the analysis with respect to the weighted RMSD $\chi$ with
the relative deviations $\eta_{i}$ taken from the recent experimental data is
performed, by taking 208Pb as an example. The confidence region for parameters
$c$ and $t$, satisfying the condition $\chi\leq 1$, is exhibited in the
$c$-$t$ plane. The small uncertainty of charge radius, i.e., $0.005$ fm,
emphasizes the reliability of the extracted charge radii from the muonic atom
spectrum. Meanwhile, it is considered that the difference of slopes of the two
lines—one represents the determined $c$ values by minimizing $\chi$ for each
given $t$ and the other represents the contour of the benchmarking charge
radius—in the $c$-$t$ plane can be used as an indicator of the model
dependency. However, it is also found that the obtained 2pf distributions
cannot reproduce the details of the benchmarking charge densities. In
particular, its surface-diffuseness parameter $t$ cannot be determined
accurately with the present experimental uncertainties on the muonic
transition energies.

In the present theory-to-theory benchmarking study, we have analyzed the model
dependencies and determined the uncertainty of extracted charge radii from
muonic atom spectroscopy based on the 2pf distribution. It is also expected to
see how much information on the nuclear structure can be extracted from the
muonic atom spectroscopy using a model-independent distribution, such as the
Fourier-Bessel series expansion [75]. This question would be of great interest
in our future studies.

###### Acknowledgements.

This work was supported by the Natural Science Foundation of Jilin Province
(No. 20220101017JC), the National Natural Science Foundation of China (No.
11675063), the Key Laboratory of Nuclear Data Foundation (JCKY2020201C157),
the JSPS Grant-in-Aid for Early-Career Scientists under Grant No. 18K13549,
the JSPS Grant-in-Aid for Scientific Research (S) under Grant No. 20H05648,
the JSPS Grant-in-Aid for Research Activity Start-up under Grant No. 22K20372,
the JSPS Grant-in-Aid for Transformative Research Areas (A) under Grant No.
23H04526, the JSPS Grant-in-Aid for Scientific Research (B) under Grant No.
23H01845, the JSPS Grant-in-Aid for Scientific Research (C) under Grant No.
23K03426, the RIKEN iTHEMS Program, the RIKEN Pioneering Project: Evolution of
Matter in the Universe, and the RIKEN Special Postdoctoral Researchers
Program.

## References

  * Engfer  _et al._ [1974] R. Engfer, H. Schneuwly, J. Vuilleumier, H. Walter, and A. Zehnder, At. Data Nucl. Data Tables 14, 509 (1974).
  * Wohlfahrt  _et al._ [1981] H. D. Wohlfahrt, E. B. Shera, M. V. Hoehn, Y. Yamazaki, and R. M. Steffen, Phys. Rev. C 23, 533 (1981).
  * Bergem  _et al._ [1988] P. Bergem, G. Piller, A. Rueetschi, L. A. Schaller, L. Schellenberg, and H. Schneuwly, Phys. Rev. C 37, 2821 (1988).
  * Schaller  _et al._ [1978] L. Schaller, T. Dubler, K. Kaeser, G. Rinker, B. Robert-Tissot, L. Schellenberg, and H. Schneuwly, Nucl. Phys. A 300, 225 (1978).
  * Yamazaki  _et al._ [1978] Y. Yamazaki, E. B. Shera, M. V. Hoehn, and R. M. Steffen, Phys. Rev. C 18, 1474 (1978).
  * Shera  _et al._ [1989] E. B. Shera, M. V. Hoehn, G. Fricke, and G. Mallot, Phys. Rev. C 39, 195 (1989).
  * Fitch and Rainwater [1953] V. L. Fitch and J. Rainwater, Phys. Rev. 92, 789 (1953).
  * Hitlin  _et al._ [1970] D. Hitlin, S. Bernow, S. Devons, I. Duerdoth, J. W. Kast, E. R. Macagno, J. Rainwater, C. S. Wu, and R. C. Barrett, Phys. Rev. C 1, 1184 (1970).
  * Steffen [1985] R. Steffen, Hyperfine Interactions 24, 223 (1985).
  * Antognini  _et al._ [2020] A. Antognini, N. Berger, T. E. Cocolios, R. Dressler, R. Eichler, A. Eggenberger, P. Indelicato, K. Jungmann, C. H. Keitel, K. Kirch, A. Knecht, N. Michel, J. Nuber, N. S. Oreshkina, A. Ouf, A. Papa, R. Pohl, M. Pospelov, E. Rapisarda, N. Ritjoho, S. Roccia, N. Severijns, A. Skawran, S. M. Vogiatzi, F. Wauters, and L. Willmann, Phys. Rev. C 101, 054313 (2020).
  * Yamazaki  _et al._ [1979] Y. Yamazaki, H. D. Wohlfahrt, E. B. Shera, M. V. Hoehn, and R. M. Steffen, Phys. Rev. Lett. 42, 1470 (1979).
  * Tanaka and Horikawa [1994] Y. Tanaka and Y. Horikawa, Nucl. Phys. A 580, 291 (1994).
  * Haga  _et al._ [2007] A. Haga, Y. Horikawa, and H. Toki, Phys. Rev. C 75, 044315 (2007).
  * Haga  _et al._ [2005a] A. Haga, Y. Horikawa, H. Toki, and Y. Tanaka, Nucl. Instrum. Methods Phys. Res., Sect. 235, 71 (2005a).
  * Haga  _et al._ [2005b] A. Haga, Y. Horikawa, Y. Tanaka, and H. Toki, Nucl. Phys. B 139, 234 (2005b).
  * Valuev  _et al._ [2022] I. A. Valuev, G. Colò, X. Roca-Maza, C. H. Keitel, and N. S. Oreshkina, Phys. Rev. Lett. 128, 203001 (2022).
  * Ring [1996] P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996).
  * Meng  _et al._ [2006] J. Meng, H. Toki, S. G. Zhou, S. Q. Zhang, W. H. Long, and L. S. Geng, Prog. Part. Nucl. Phys. 57, 470 (2006).
  * Meng and Zhou [2015] J. Meng and S. G. Zhou, J. Phys. G: Nucl. Part. Phys. 42, 093101 (2015).
  * Vretenar  _et al._ [2005] D. Vretenar, A. Afanasjev, G. Lalazissis, and P. Ring, Phys. Rep. 409, 101 (2005).
  * Nikšić  _et al._ [2011] T. Nikšić, D. Vretenar, and P. Ring, Prog. Part. Nucl. Phys. 66, 519 (2011).
  * Meng  _et al._ [2013] J. Meng, J. Peng, S.-Q. Zhang, and P.-W. Zhao, Front. Phys. 8, 55 (2013).
  * Sharma  _et al._ [1993] M. M. Sharma, G. A. Lalazissis, and P. Ring, Phys. Lett. B 317, 9 (1993).
  * Naito  _et al._ [2023] T. Naito, T. Oishi, H. Sagawa, and Z. Wang, Phys. Rev. C 107, 054307 (2023).
  * Liang  _et al._ [2015] H. Liang, J. Meng, and S.-G. Zhou, Phys. Rep. 570, 1 (2015).
  * Zhou  _et al._ [2003] S.-G. Zhou, J. Meng, and P. Ring, Phys. Rev. Lett. 91, 262501 (2003).
  * Liang  _et al._ [2010] H. Liang, W. H. Long, J. Meng, and N. Van Giai, Eur. Phys. J. A 44, 119 (2010).
  * Li and Sun [2020] J. Li and W.-J. Sun, Commun. in Theor. Phys. 72, 055301 (2020).
  * Li  _et al._ [2011a] J. Li, J. Meng, P. Ring, J. Yao, and A. Arima, Sci. China Phys. Mech. Astron. 54, 204 (2011a).
  * Li  _et al._ [2011b] J. Li, J. M. Yao, J. Meng, and A. Arima, Prog. Theor. Phys. 125, 1185 (2011b).
  * Wei  _et al._ [2012] J. Wei, J. Li, and J. Meng, Prog. Theor. Phys. Suppl. 196, 400 (2012).
  * Li  _et al._ [2013a] J. Li, J. X. Wei, J. N. Hu, P. Ring, and J. Meng, Phys. Rev. C 88, 064307 (2013a).
  * König and Ring [1993] J. König and P. Ring, Phys. Rev. Lett. 71, 3079 (1993).
  * Zhao  _et al._ [2011] P. W. Zhao, J. Peng, H. Z. Liang, P. Ring, and J. Meng, Phys. Rev. Lett. 107, 122501 (2011).
  * Meng and Ring [1996] J. Meng and P. Ring, Phys. Rev. Lett. 77, 3963 (1996).
  * Meng [1998] J. Meng, Nucl. Phys. A 635, 3 (1998).
  * Meng and Ring [1998] J. Meng and P. Ring, Phys. Rev. Lett. 80, 460 (1998).
  * Meng  _et al._ [2002a] J. Meng, H. Toki, J. Y. Zeng, S. Q. Zhang, and S.-G. Zhou, Phys. Rev. C 65, 041302 (2002a).
  * Zhang  _et al._ [2002] S.-Q. Zhang, J. Meng, S.-G. Zhou, and J.-Y. Zeng, Chinese Phys. Lett. 19, 312 (2002).
  * Meng  _et al._ [1998] J. Meng, I. Tanihata, and S. Yamaji, Phys. Lett. B 419, 1 (1998).
  * Meng  _et al._ [2002b] J. Meng, S.-G. Zhou, and I. Tanihata, Phys. Lett. B 532, 209 (2002b).
  * Reinhard  _et al._ [2020] P.-G. Reinhard, W. Nazarewicz, and R. F. Garcia Ruiz, Phys. Rev. C 101, 021301 (2020).
  * Naito  _et al._ [2021] T. Naito, G. Colò, H. Liang, and X. Roca-Maza, Phys. Rev. C 104, 024316 (2021).
  * Zhao  _et al._ [2010] P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, Phys. Rev. C 82, 054319 (2010).
  * Xia  _et al._ [2018] X. W. Xia, Y. Lim, P. W. Zhao, H. Z. Liang, X. Y. Qu, Y. Chen, H. Liu, L. F. Zhang, S. Q. Zhang, Y. Kim, and J. Meng, At. Data Nucl. Data Tables 121-122, 1 (2018).
  * Meng [2015] J. Meng,  (World Scientific, 2015).
  * Friar and Negele [1975] J. L. Friar and J. W. Negele, Adv. Nucl. Phys. 8, 219 (1975).
  * Kurasawa and Suzuki [2000] H. Kurasawa and T. Suzuki, Phys. Rev. C 62, 054303 (2000).
  * Reinhard and Nazarewicz [2021] P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C 103, 054310 (2021).
  * Kurasawa and Suzuki [2019] H. Kurasawa and T. Suzuki, Prog. Theor. Exp. Phys. 2019, 113D01 (2019).
  * Tiesinga  _et al._ [2021] E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, Rev. Mod. Phys. 93, 025010 (2021).
  * Atac  _et al._ [2021] H. Atac, M. Constantinou, Z.-E. Meziani, M. Paolone, and N. Sparveris, Nat. Commun. 12, 1 (2021).
  * Xie  _et al._ [2023] H. H. Xie, J. Li, L. G. Jiao, and Y. K. Ho, Phys. Rev. A 107, 042807 (2023).
  * Wang  _et al._ [2021] M. Wang, W. Huang, F. Kondev, G. Audi, and S. Naimi, Chinese Phys. C 45, 030003 (2021).
  * Engel [2002] E. Engel,  _Relativistic Electronic Structure Theory, Part 1. Fundamentals_ (Elsevier, Amsterdam, 2002) pp. 524–624.
  * Zhang  _et al._ [2014] Q.-S. Zhang, Z.-M. Niu, Z.-P. Li, J.-M. Yao, and J. Meng, Front. Phys. 9, 529 (2014).
  * Lu  _et al._ [2015] K. Q. Lu, Z. X. Li, Z. P. Li, J. M. Yao, and J. Meng, Phys. Rev. C 91, 027304 (2015).
  * Zhao  _et al._ [2012] P. W. Zhao, L. S. Song, B. Sun, H. Geissel, and J. Meng, Phys. Rev. C 86, 064324 (2012).
  * Yao  _et al._ [2013] J. Yao, H. Mei, and Z. Li, Phys. Lett. B 723, 459 (2013).
  * Li  _et al._ [2013b] Z. Li, B. Song, J. Yao, D. Vretenar, and J. Meng, Phys. Lett. B 726, 866 (2013b).
  * Li  _et al._ [2012] Z. Li, C. Li, J. Xiang, J. Yao, and J. Meng, Phys. Lett. B 717, 470 (2012).
  * Xiang  _et al._ [2013] J. Xiang, Z. P. Li, J. M. Yao, W. H. Long, P. Ring, and J. Meng, Phys. Rev. C 88, 057301 (2013).
  * Wang  _et al._ [2015] Z. H. Wang, J. Xiang, W. H. Long, and Z. P. Li, J. Phys. G 42, 045108 (2015).
  * Canuto  _et al._ [2007] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang,  (Springer Berlin Heidelberg, 2007).
  * Jiao  _et al._ [2021a] L. G. Jiao, Y. Y. He, A. Liu, Y. Z. Zhang, and Y. K. Ho, Phys. Rev. A 104, 022801 (2021a).
  * Xie  _et al._ [2021] H. H. Xie, L. G. Jiao, A. Liu, and Y. K. Ho, Int. J. Quantum Chem. 121, e26653 (2021).
  * Jiao  _et al._ [2021b] L. G. Jiao, H. H. Xie, A. Liu, H. E. Montgomery, and Y. K. Ho, J. Phys. B: At. Mol. Opt. Phys. 54, 175002 (2021b).
  * Borie and Rinker [1982] E. Borie and G. A. Rinker, Rev. Mod. Phys. 54, 67 (1982).
  * Fullerton and Rinker [1976] L. W. Fullerton and G. A. Rinker, Phys. Rev. A 13, 1283 (1976).
  * Cheng  _et al._ [1978] K. T. Cheng, W. D. Sepp, W. R. Johnson, and B. Fricke, Phys. Rev. A 17, 489 (1978).
  * Haga  _et al._ [2002] A. Haga, Y. Horikawa, and Y. Tanaka, Phys. Rev. A 66, 034501 (2002).
  * Barrett [1970] R. Barrett, Phys. Lett. B 33, 388 (1970).
  * Fricke  _et al._ [1995] G. Fricke, C. Bernhardt, K. Heilig, L. Schaller, L. Schellenberg, E. Shera, and C. Dejager, At. Data Nucl. Data Tables 60, 177 (1995).
  * Saito [2021] T. Y. Saito, Ph.D. thesis, Department of Physics, Graduate School of Science, The University of Tokyo (2021).
  * Dreher  _et al._ [1974] B. Dreher, J. Friedrich, K. Merle, H. Rothhaas, and G. Lührs, Nucl. Phys. A 235, 219 (1974).


On the Carrollian Nature of the Light Front

Sucheta Majumdar***Electronic address<EMAIL_ADDRESS>
Aix-Marseille Univ, Université de Toulon, CNRS,

Centre de Physique Théorique, 13288 Marseille, France

Abstract

Motivated by recent advances in non-Lorentzian physics, we revisit the light-
cone formulation of quantum field theories. We discuss some interesting
subalgebras within the light-cone Poincaré algebra, with a key emphasis on the
Carroll, Bargmann, and Galilean kinds. We show that theories on the light
front possess a Hamiltonian of the magnetic Carroll type, thereby proposing a
straightforward method for deriving magnetic Carroll Hamiltonian actions from
Lorentzian field theories.

##  1 Introduction

This article is dedicated to the memory of Lars Brink, an eminent scientist
and an exceptional mentor whose scientific curiosity and wisdom profoundly
influenced my perspective on theoretical physics research. His valuable
guidance and words of encouragement will be deeply missed, and I am forever
indebted to him.

From string theory and supersymmetry to light-cone gravity and higher spins,
the numerous significant contributions Lars made to light-cone physics are a
testament to his keen interest in the subject. As a tribute to his deep
appreciation for life on the light front, this article explores some aspects
of the light-cone formulation that are relevant to Carrollian physics. The
Carroll group, which arises as the ultrarelativistic (speed of light
approaching zero) limit of the Poincaré algebra [1], has been linked to
symmetries of null hypersurfaces, the BMS symmetry of gravity, and flat-space
holography (see [2, 3, 4, 5] and references therein).

The light-cone formulation of field theories is based on Dirac’s front form of
relativistic dynamics [6], where the time evolution of a system occurs along a
lightlike or null direction. Light-cone physics rose to prominence following
the seminal work of Weinberg [7] in the late sixties. Weinberg showed that
certain problematic Feynman diagrams in quantum field theories, such as vacuum
fluctuations, take a simpler form in the light-cone or infinite-momentum
frame, leading to finite or zero contributions. This simplicity was later
attributed to an underlying three-dimensional Galliean invariance, which gives
the four-dimensional theories a nonrelativistic structure [8]. On the other
hand, the light-cone coordinate system comprises two null hypersurfaces that
possess a Carrollian structure. While the Galilean or nonrelativistic nature
of theories in the light-cone formulation has received considerable attention
in the past [8, 9, 10, 11], their Carrollian properties have not been
discussed in much detail. Hence, this article aims to connect some aspects of
the light-cone formulation to Carrollian physics.

We emphasize the double-null nature of the light-cone coordinates, which
corresponds to a choice of time within this framework. The freedom to choose
the time coordinate gives the light-cone Poincaré algebra a unique structure.
As a result, the light-cone Poincaré generators obey various kinematical Lie
algebras in one lower spacetime dimension. Of particular interest are the
subalgebras of Galilei, Carroll, and Bargmann types, and their physical
relevance in the light-cone formulation. Having discussed these subalgebras,
we will focus on the Carrollian features of field theories in the light-cone
formalism. In recent years, Carrollian field theories have emerged as
potential candidates for flat space holography, alongside celestial conformal
field theories. In the standard approaches to Carrollian field theories, the
ultrarelativistic limit of Lorentzian field theories leads to two classes of
theories, namely the electric and the magnetic types. We show that field
theories on the light front have a magnetic Carroll Hamiltonian. Thus, we
propose a straightforward recipe for deriving magnetic Carroll actions without
requiring any rescaling or contraction procedure.

##  2 Choice of light-cone time

Starting with $(d+1)$-dimensional Minkowski spacetime in Cartesian
coordinates, $x^{\mu}=(x^{0},x^{1},...,x^{d})$, we consider two null vectors

$m^{\mu}=\frac{1}{\sqrt{2}}(-1,0,0,...,1)\,,\quad
n^{\mu}=\frac{1}{\sqrt{2}}(-1,0,0,...,-1)\,,$ (2.1)

such that ${n.n}=m.m=0$ and $n.m=-1$. The light fronts are the null planes
obtained by projecting along the two null vectors

$\displaystyle x^{+}=m.x=\frac{1}{\sqrt{2}}(x^{0}+x^{d})\,,$ $\displaystyle
x^{-}=n.x=\frac{1}{\sqrt{2}}(x^{0}-x^{d})\,.$ (2.2)

$m$ and $n$ are normals to the null planes $x^{+}=0$ and $x^{-}=0$
respectively. In the light-cone coordinates, the line element in Minkowski
spacetime reads 222By ‘light-cone’ or ‘light-front’ coordinates, we always
refer to planar null coordinates defined in (2.2), and not the retarded and
advanced time coordinates, $t\pm r$.

$dS^{2}=-2dx^{+}dx^{-}+\delta_{ij}dx^{i}dx^{j}\,,\quad(i,j=1,2,...,d-1)\,.$
(2.3)

An interesting feature to note from the line element is the double-null nature
of the light-cone coordinates, which corresponds to a
$\mathbb{R}^{d-1}\times\mathbb{R}\times\mathbb{R}$ split of the Minkowski
spacetime, as shown in Fig 1. This hints at an apparent ‘duality’ between the
light-cone coordinates $x^{\pm}$ $-$ a choice of time $-$ since either of the
two may be treated as the time coordinate. In most cases, such as low-energy
effective descriptions and scattering amplitudes techniques [12, 13, 14, 15,
16, 17], it suffices to pick one of the light-cone coordinates as time for the
entirety of the analysis while treating the other one as a spatial coordinate.
However, certain physical problems, particularly those pertaining to the
initial value problem and quantization of massless fields on null planes,
require both light-cone directions to be treated as time [18, 19, 20, 21]. In
these instances, the $x^{\pm}$ duality, i.e., the fact that both can play the
role of the light-cone time, is not merely a choice but a necessity for a
complete understanding of the quantization properties of a physical system.

Figure 1: The double-null light-cone coordinate system

In this article, we will shed new light on the $x^{\pm}$ duality from an
algebraic point of view. Following the works of [22, 23], we shall discuss the
concept of two non-Einsteinian times $-$ Galilean (or Newtonian) time and
Carrollian time, adapted to the light-cone framework. Depending on which
light-cone coordinate is taken to be the Newtonian time, one finds two
distinct sets of Galilei, Carroll, and Bargmann subalgebras within the
Poincaré algebra, which are mapped into one another upon exchanging $x^{+}$
with $x^{-}$.

##  3 Aspects of light-cone Poincaré algebra

We begin by reviewing some important features of the light-cone Poincaré
algebra (see [11] for more details). The generators $(P_{\mu},M_{\mu\nu})$ in
light-cone coordinates satisfies the usual Poincaré algebra in $(d+1)$
dimensions

$\displaystyle\left[P_{\mu}\,,P_{\nu}\right]$ $\displaystyle=$ $\displaystyle
0\,,$ (3.1) $\displaystyle\left[M_{\mu\nu}\,,P_{\rho}\right]$ $\displaystyle=$
$\displaystyle\eta_{\nu\rho}P_{\mu}-\eta_{\mu\rho}P_{\nu}\,,$ (3.2)
$\displaystyle\left[M_{\mu\nu}\,,M_{\rho\sigma}\right]$ $\displaystyle=$
$\displaystyle\eta_{\mu\sigma}M_{\nu\rho}+\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\nu\sigma}M_{\mu\rho}\,,$
(3.3)

where the greek indices $\mu,\nu,...$ now run over $(+,-,i)$ and
$\eta_{\mu\nu}$ stands for the light-cone Minkowski metric. For our
convenience, we relabel the generators spanning the light-cone Poincaré
algebra $\mathfrak{p}$ as follows

$\displaystyle P_{+}=E\,,$ $\displaystyle P_{-}=\eta\,,$
$\displaystyle~{}~{}P_{i}\,,$ $\displaystyle M_{+i}=K_{i}\,,$
$\displaystyle~{}~{}M_{-i}=B_{i}\,,$ $\displaystyle~{}~{}M_{ij}\,,\quad
M_{+-}=D\,.$ (3.4)

An element of the light-cone Poincaré algebra $\mathfrak{p}$ is given by

$\displaystyle X_{\mathfrak{p}}$ $\displaystyle=$ $\displaystyle
a^{+}E+a^{-}\eta+a^{i}P_{i}+\beta
D+\gamma^{i}K_{i}+\sigma^{i}B_{i}+\frac{1}{2}\omega^{ij}M_{ij}$ (3.5)
$\displaystyle=$ $\displaystyle(a^{+}-\beta
x^{+}+\gamma_{i}x^{i})\frac{\partial}{\partial x^{+}}+(a^{-}+\beta
x^{-}+\sigma_{i}x^{i})\frac{\partial}{\partial x^{-}}$
$\displaystyle+\left(\omega^{i}{}_{j}x^{j}+\sigma^{i}x^{+}+\gamma^{i}x^{-}+a^{i}\right)\frac{\partial}{\partial
x^{i}}\,,$

where we have relabeled the Lorentz parameters, $\omega^{+-},\omega^{+i}$ and
$\omega^{-i}$, as $\beta,\gamma^{i}$ and $\sigma^{i}$, respectively. The
generator $M_{+-}$, which we denote by $D$, acts as a dilatation operator that
scales the $x^{+}$ and $x^{-}$ directions while keeping $x^{i}$ fixed. Under a
dilatation $D$, a light-cone operator $\mathcal{O}$ transforms as

$[D,\mathcal{O}]=\alpha\mathcal{O}\,,$ (3.6)

where the eigenvalue $\alpha$, often referred to as the “goodness value” [11],
is an integer. We can classify all the light-front Poincaré generators
according to their transformation under $D$, as shown in Table 1.

Table 1: Generators of light-cone Poincaré algebra $\alpha$ | $P_{\mu}$ | $M_{\mu\nu}$ |  | Subgroups $G_{\alpha}$
---|---|---|---|---
1 | $\eta$ | $B_{i}$ |  | $G_{1}$
0 | $P_{i}$ | $M_{ij}\,,$ | $D$ | $G_{0}$
-1 | $E$ | $K_{i}$ |  | $G_{-1}$

Both $G_{+1}$ and $G_{-1}$ are abelian ideals of the light-cone Poincaré
algebra. Additionally, there are two subgroups with a semi-direct product
structure,

$\displaystyle\mathbb{S}_{+}$ $\displaystyle=$ $\displaystyle G_{0}\ltimes
G_{1}~{}=~{}\\{\eta,P_{i},M_{ij},B_{i},D\\}$ $\displaystyle\mathbb{S}_{-}$
$\displaystyle=$ $\displaystyle G_{0}\ltimes
G_{-1}~{}=~{}\\{E,P_{i},M_{ij},K_{i},D\\}\,.$ (3.7)

The subgroups $\mathbb{S}_{\pm}$ are the kinematical groups or stability
groups of the null planes $x^{\pm}=0$. While the subgroup $\mathbb{S}_{+}$ (or
$\mathbb{S}_{-}$) gives the kinematics within the null plane, the remaining
generators comprising the abelian ideal $G_{-1}$ (or $G_{+1}$) form the
dynamical part of the algebra that governs the evolution of the physical
system as a function of null time, $x^{+}$ (or $x^{-}$).

For the sake of completeness, we list all the commutators 333Here, we use the
convention that the first entry in a commutator $[X_{1},X_{2}]$ is taken from
the column and the second one from the row, e.g.$[E,D]=E$ in the first row. of
the light-cone Poincaré algebra $\mathfrak{p}$ in Table 2.

Table 2: Commutators of light-cone Poincaré algebra $\mathfrak{p}$ | $E$ | $\eta$ | $P_{i}$ | $D$ | $M_{ij}$ | $B_{i}$ | $K_{i}$
---|---|---|---|---|---|---|---
$E$ | $0$ | $0$ | $0$ | $E$ | $0$ | $-P_{i}$ | $0$
$\eta$ | $0$ | $0$ | $0$ | $-\eta$ | $0$ | $0$ | $-P_{i}$
$P_{l}$ | $0$ | $0$ | $0$ | $0$ | $\delta_{l[i}P_{j]}$ | $-\delta_{il}\eta$ | $-\delta_{il}E$
$D$ | $-E$ | $\eta$ | $0$ | $0$ | $0$ | $B_{i}$ | $-K_{i}$
$M_{lm}$ | $0$ | $0$ | $-\delta_{i[l}P_{m]}$ | $0$ | $\delta_{[i[m}M_{l]j]}$ | $\delta_{i[m}B_{l]}$ | $\delta_{i[m}K_{l]}$
$B_{l}$ | $P_{l}$ | $0$ | $\delta_{il}\eta$ | $-B_{l}$ | $-\delta_{l[i}B_{j]}$ | $0$ | $M_{li}+\delta_{il}D$
$K_{l}$ | $O$ | $P_{l}$ | $\delta_{il}E$ | $K_{l}$ | $-\delta_{l[i}K_{j]}$ | $-M_{il}-\delta_{il}D$ | $0$

In four spacetime dimensions, the kinematical part of the light-cone Poincaré
algebra is a seven-parameter subgroup as opposed to the usual six-parameter
kinematical subgroups found in the instant form of dynamics. Hence, a key
advantage of working in this formulation is that the light fronts possess the
largest kinematical subgroup among all admissible choices of initial
hypersurfaces for Hamiltonian theories. This feature tremendously simplifies
the unknown dynamical part of the problem, proving to be very effective for
deriving interacting Hamiltonian actions in many instances, such as higher-
spin theories and supersymmetric theories [13, 14].

##  4 Carroll, Bargmann and all that

Minkowski spacetime in light-cone coordinates exhibits a flat Bargmann
structure [22, 23], encompassing both Galilean and Carrollian spacetimes in
one lower dimension. The Galilean spacetime follows from a null dimensional
reduction à la Kaluza-Klein, while the Carrollian spacetime may be viewed as
an embedded null hyperplane [24, 25, 23]. The Bargmannian structure, combined
with the choice of light-cone time, gives rise to a rich array of interesting
kinematical Lie algebras within its Poincaré algebra.

###  4.1 Kinematical Lie algebras

Before exploring the various subalgebras of the light-cone Poincaré algebra,
we briefly review a few basic elements of kinematical Lie algebras (see [26]
for more details). The key ingredients of a $(n+1)$-dimensional kinematical
Lie algebra are the generators

$\mathcal{K}=\\{L_{ab},B_{a},P_{a},H\\}\,,\quad a,b,c,..=1,2,...,n$ (4.1)

where the generators

  * •

$L_{ab}$ span the $\bm{so}(n)$ algebra

$[\bm{L},\bm{L}]=\bm{L}\,,$ (4.2)

  * •

$B_{a}$ and $P_{a}$ transform as vectors under $L_{ab}$

$[\bm{L},\bm{B}]=\bm{B}\,,\quad[\bm{L},\bm{P}]=\bm{P}\,,$ (4.3)

  * •

and, $H$ is a scalar

$[\bm{L},H]=0\,.$ (4.4)

In the commutation relations above, we denote the $\mathfrak{so}(n)$
generators and the vectors with bold letters, while suppressing the
$\mathfrak{so}(n)$ indices on them. The vectors $\bm{B}$ and $\bm{P}$, here,
should not be confused with the light-cone Poincaré generators $B_{i}$ and
$P_{i}$ discussed in the previous section. These $\mathfrak{so}(n)$ vectors
may be identified with the Poincaré generators in some instances, but that is
not always the case.

The kinematical Lie algebras are, then, defined through the non-zero
commutators among $H,\bm{B},$ and $\bm{P}$. In particular, the Galilei,
Carroll and Bargmann algebras are defined as follows

$\displaystyle\text{Galilei}\ \mathfrak{g}:$
$\displaystyle[H,\bm{B}]=-\bm{P}\,,$ (4.5) $\displaystyle\text{Carroll}\
\mathfrak{c}:$ $\displaystyle[\bm{B},\bm{P}]=H\,,$ (4.6)
$\displaystyle\text{Bargmann}\ \mathfrak{b}:$
$\displaystyle[H,\bm{B}]=-\bm{P}\,,\quad[\bm{B},\bm{P}]=Z\,,$ (4.7)

where $Z$ is a central element. Thus, the Carroll and Galilei groups may be
unified into a larger Bargmann group [23, 27].

###  4.2 Subalgebras of light-cone Poincaré algebra

Due to a $\mathbb{R}^{d-1}\times\mathbb{R}\times\mathbb{R}$ split of the
Minkowski spacetime, instead of the usual $\mathbb{R}^{d}\times\mathbb{R}$,
the light-cone Poincaré generators split as

  * •

$M_{ij}$ spanning an $\mathfrak{so}(d-1)$ algebra,

  * •

Three vectors $\\{P_{i}$, $B_{i}$, $K_{i}\\}$ under the $\mathfrak{so}(d-1)$,
and

  * •

Three scalars $\\{E,\eta,D\\}$ under the $\mathfrak{so}(d-1)$.

Among these generators, one can find two subsets that satisfy the conditions
(4.2), (4.3) and (4.4), along with (4.7), and hence, form Bargmann subalgebras
444We shall use the subscript ‘$+$’ or ‘$-$’ to label the algebras in order to
specify which light-cone coordinate, $x^{+}$ or $x^{-}$, is chosen as the
Newtonian time.

$\displaystyle\mathfrak{b}_{+}$ $\displaystyle=$
$\displaystyle\\{E,\eta,M_{ij},B_{i},P_{i}\\}\,,\quad\eta\ \text{central
element}\,,$ (4.8) $\displaystyle\mathfrak{b}_{-}$ $\displaystyle=$
$\displaystyle\\{E,\eta,M_{ij},K_{i},P_{i}\\}\,,\quad E\ \text{central
element}\,.$ (4.9)

By assigning a notion of Newtonian and Carrollian time [23] to the light-cone
coordinates $x^{\pm}$ within each Bargmann group, we can identify a Galilei
and a Carroll subgroup. The important point here is that we have at our
disposal, two vectors, $B_{i}$ and $K_{i}$, that can play the role of the
Galilean (or Carrollian) boosts, and two scalars, $\eta$ and $E$, that
centralize the corresponding Galilean algebra. We elaborate on this point
below.

We can choose one of the null coordinates to be the ‘Newtonian’ time for the
Galilei group. The Hamiltonian (energy) is then given by the momentum
corresponding the Newtonian time. For instance, if $x^{+}$ is taken to be the
Newtonian time, the generator $E=P_{+}$ becomes the Hamiltonian. As a result,
the momentum associated with the other null coordinate $x^{-}$ appears as a
nonrelativistic mass, $\eta=P_{-}$ in the mass-shell condition

$2E\eta-P^{i}P_{i}=0\quad\Rightarrow\quad
H_{\mathfrak{g}}=E=\frac{P^{2}}{2\eta}\,.$ (4.10)

This null coordinate may then be treated as a ‘Carrollian’ time. In the
Bargmann algebra $\mathfrak{b}_{+}$, the generator $\eta$ indeed appears as a
central element. Thus, treating $x^{+}$ as Newtonian time and $x^{-}$ as
Carrollian, we can readily identify the Galilei and Carroll group contained in
$\mathfrak{b}_{+}$ as

$\displaystyle\text{Galilei:}\quad\mathfrak{g}_{+}=\\{E,M_{ij},B_{i},P_{i}\\}\,,$
(4.11)
$\displaystyle\text{Carroll:}\quad\mathfrak{c}_{+}=\\{\eta,M_{ij},B_{i},P_{i}\\}\,.$
(4.12)

The 3-dimensional Galilei subalgebra within the 4-dimensional Poincaré algebra
discussed in the light-cone literature [8, 9, 10], indeed coincides with
$\mathfrak{g}_{+}$. It was also pointed out that $P_{-}$ centralizes this
algebra, but the resulting centrally-extended algebra was not identified as a
Bargmann algebra $\mathfrak{b}_{+}$.

Now, if we instead take $x^{-}$ to be the Newtonian time and $x^{+}$ to be
Carrollian, we find another set of Galilei and Carroll groups contained in the
other Bargmann group $\mathfrak{b}_{-}$ as follows

$\displaystyle\text{Galilei:}\quad\mathfrak{g}_{-}=\\{\eta,M_{ij},K_{i},P_{i}\\}\,,$
(4.13)
$\displaystyle\text{Carroll:}\quad\mathfrak{c}_{-}=\\{E,M_{ij},K_{i},P_{i}\\}\,.$
(4.14)

The Carroll subgroups $\mathfrak{c}_{\pm}$ are the stability groups of the
$d$-dimensional light fronts, $x^{\pm}=\text{constant}$, embedded in a
$(d+1)$-dimensional Minkowski spacetime (see Fig 1). Note that in four
dimensions, the stability groups $\mathbb{S}_{\pm}$, given in (3), are seven-
dimensional only for the light fronts at $x^{\pm}=0$, where the generator
$M_{+-}$ is kinematical. In all other cases, the stability groups of the null
fronts are given by $\mathfrak{c}_{\pm}$, which are six-dimensional.

Interestingly, the Lorentz generators in $\mathfrak{p}$ satisfy another
kinematical Lie algebra characterized by the relations

$[H,\bm{B}]=\bm{B}\,,\quad[H,\bm{P}]=-\bm{P},\quad[\bm{B},\bm{P}]=H+\bm{L}\,,$
(4.15)

with the following identification:
$H=D\,,\,\bm{L}=M_{ij}\,,\,\bm{B}=B_{i}\,,\,\bm{P}=K_{i}\,.$ This Lie algebra
is geometrically associated with a special class of homogenous spaces, named
the ‘Carrollian light cone’ in [28].

Therefore, in $(d+1)$ spacetime dimensions, the light-cone Poincaré algebra
$\mathfrak{p}$, spanned by $dim(\mathfrak{p})=\frac{(d+1)(d+2)}{2}$
generators, contains the $d$-dimensional subalgebras
$(\mathfrak{b}_{\pm},\mathfrak{c}_{\pm},\mathfrak{g}_{\pm})$ with a group
dimension

$\displaystyle dim(\mathfrak{b}_{\pm})=\frac{d(d+1)}{2}+1\,,$ (4.16)
$\displaystyle
dim(\mathfrak{g}_{\pm})=dim(\mathfrak{c}_{\pm})=\frac{d(d+1)}{2}\,.$ (4.17)

We summarize these subalgebras in Table 3, highlighting the central elements
and key commutators that define them, while all other commutation relations
can be found in Table 2.

Table 3: Kinematical Lie subalgebras within light-cone Poincaré algebra Time |  $x^{+}$ Newtonian, $x^{-}$ Carrollian |  $x^{+}$ Carrollian, $x^{-}$ Newtonian
---|---|---
Bargmann $\mathfrak{b}$ | $\mathfrak{b}_{+}=\\{E,\eta,M_{ij},B_{i},P_{i}\\}$ | $\mathfrak{b}_{-}=\\{\eta,E,M_{ij},K_{i},P_{i}\\}$
|  $\eta$ central element |  $E$ central element
Galilei $\mathfrak{g}$ | $\mathfrak{g}_{+}=\\{E,M_{ij},B_{i},P_{i}\\}$ | $\mathfrak{g}_{-}=\\{\eta,M_{ij},K_{i},P_{i}\\}$
| $[E,B_{i}]=-P_{i}$ | $[\eta,K_{i}]=-P_{i}$
|  Hamiltonian $H_{\mathfrak{g}_{+}}=E$ |  Hamiltonian $H_{\mathfrak{g}_{-}}=\eta$
Carroll $\mathfrak{c}$ | $\mathfrak{c}_{+}=\\{\eta,M_{ij},B_{i},P_{i}\\}$ | $\mathfrak{c}_{-}=\\{E,M_{ij},K_{i},P_{i}\\}$
| $[B_{i},P_{j}]=\delta_{ij}\eta$ | $[K_{i},P_{j}]=\delta_{ij}E$
|  Hamiltonian $H_{\mathfrak{c}_{+}}=\eta$ |  Hamiltonian $H_{\mathfrak{c}_{-}}=E$

As alluded to before, the two null coordinates, $x^{+}$ and $x^{-}$, are
completely interchangeable in the light-cone coordinate system. This freedom
to choose the light-cone time is also reflected in these subgroups. On
swapping $x^{+}$ with $x^{-}$, the roles of certain generators in
$\mathfrak{p}$ are exchanged

$\displaystyle x^{+}\longleftrightarrow x^{-}\,,\quad
E\longleftrightarrow\eta\,,\quad K_{i}\longleftrightarrow B_{i}\,,$ (4.18)

which results in the two sets of Bargmann, Galilei and Carroll subgroups being
mapped into each other

$\mathfrak{b}_{+}\longleftrightarrow\mathfrak{b}_{-}\,,\quad\mathfrak{g}_{+}\longleftrightarrow\mathfrak{g}_{-}\,,\quad\mathfrak{c}_{+}\longleftrightarrow\mathfrak{c}_{-}\,.$
(4.19)

The key point is that these subalgebras are inherent to the light-cone
Poincaré algebra and do not arise from a nonrelativistic or ultrarelativistic
limit. In the light-cone framework, these non-Lorentzian symmetries $-$
Carrollian and Galilean $-$ can coexist in the same physical system, albeit as
subalgebras of the Poincaré symmetry in one higher dimension.

##  5 Magnetic Carroll actions on the light front

In this section, we explore the Carrollian nature of field theories on the
light front. In particular, we illustrate that the Hamiltonian density derived
from light-cone actions are of the magnetic Carroll type. For simplicity, we
restrict our discussion to the case of scalar field theory in $(d+1)$
dimensions.

The massless scalar field action

$\mathcal{S}[\phi,\dot{\phi}]=\int
d^{d+1}x\,\mathcal{L}\,,\quad\mathcal{L}=-\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\,,$
(5.1)

in the light-cone coordinates, reads

$\displaystyle\mathcal{S}_{L}=\int
dx^{+}dx^{-}d^{d-1}x\,\left(\partial_{+}\phi\partial_{-}\phi-\frac{1}{2}\partial_{i}\phi\partial^{i}\phi\right)\,.$
(5.2)

We choose $x^{+}$ as the Carrollian time for the light-front Hamiltonian
analysis. This choice aligns with existing light-cone results, where $x^{+}$
is conventionally treated as time. A notable example is the BMS algebra in
light-cone gravity, which can be obtained as a local extension of the Poincaré
algebra [29] as well as a conformal extension of the Carroll algebra
$\mathfrak{c}_{-}$ [30]. One could instead consider a theory with $x^{-}$ as
the Carrollian time, in which case the relevant symmetry group would be
$\mathfrak{c}_{+}$, as described in Table 3.

Treating $x^{+}$ as Carrollian time, we define the conjugate momenta as

$\pi_{\phi}=\frac{\delta\mathcal{L}^{lc}}{\delta(\partial_{+}\phi)}=\partial_{-}\phi\,.$
(5.3)

Since the conjugate momenta involve no time derivates, we get a primary
constraint from the definition of $\pi_{\phi}$

$\chi_{\phi}=\pi_{\phi}-\partial_{-}\phi\,.$ (5.4)

At this point, we choose to strongly impose the second-class constraint,
effectively eliminating the conjugate momenta from the action, as is customary
in the $lc_{2}$ formalism. The canonical Hamiltonian density so obtained
involves the spatial derivatives $\partial_{i}$ only

$\mathcal{H}^{lc}=\pi_{\phi}\partial_{+}\phi-\mathcal{L}^{lc}=\frac{1}{2}\partial_{i}\phi\partial^{i}\phi\,.$
(5.5)

The light-cone Hamiltonian density exactly matches with the Hamiltonian
density obtained in [31] for $d$-dimensional scalar field theory in the
magnetic Carroll case. In fact, the Hamiltonian action,

$\mathcal{S}_{H}=\int
dx^{+}dx^{-}d^{d-1}x(\partial_{+}\phi\partial_{-}\phi-\mathcal{H}^{lc})\,,$
(5.6)

is the same as the Lagrangian action shown above, since the dynamics is now
defined on the reduced phase space endowed with the Poisson (or more
precisely, Dirac) bracket

$\\{\phi(x),\phi(y)\\}=\frac{1}{2}\,\varepsilon(x^{-}-y^{-})\,\delta^{(d-1)}(x-y)\,,$
(5.7)

where $\varepsilon(x^{-}-y^{-})$ is the Heaviside step function.

The crucial point is that upon solving the second-class primary constraint to
eliminate the conjugate momenta, one immediately obtains from the light-cone
action a magnetic Carroll Hamiltonian in one lower dimension without taking
any ultrarelativistic limit.

Alternatively, we could keep the conjugate momenta in the phase space and
impose the constraint by means of a Lagrange multiplier. We will return to
this point in Section 5.2.

###  5.1 Invariance under Carroll transformations

We now focus on the Carroll transformations restricted to a constant $x^{-}$
surface, where the symmetry group of our interest is
$\mathfrak{c}_{-}=\\{M_{ij},P_{i},K_{i},E\\}$

$\displaystyle x^{-}$ $\displaystyle\rightarrow$ $\displaystyle x^{-}\,,$
$\displaystyle x^{+}$ $\displaystyle\rightarrow$ $\displaystyle
x^{+}+\gamma^{i}x_{i}+a^{+}\,,$ $\displaystyle x^{i}$
$\displaystyle\rightarrow$ $\displaystyle
x^{i}+\omega^{i}_{j}x^{j}+\gamma^{i}x^{-}+a^{i}\,.$ (5.8)

One can easily verify that the corresponding field transformations

$\delta_{\mathfrak{c}_{-}}\phi=(\gamma^{i}x_{i}+a^{+})\partial_{+}\phi+(\omega^{i}_{j}x^{j}+\gamma^{i}x^{-}+a^{i})\partial_{i}\phi\,.$
(5.9)

render the light-cone action (5.6) invariant. In the light-cone phase space,
the canonical generators for the Carroll transformations are given in terms of
the energy and momentum densities

$\mathcal{H}(x)=\frac{1}{2}\partial_{i}\phi\,\partial^{i}\phi\,,\quad\mathcal{P}_{i}(x)=\partial_{-}\phi\,\partial_{i}\phi\,.$
(5.10)

The commutation relations among the energy and momentum densities are

$\displaystyle\left[\mathcal{H}(x),\,\mathcal{H}(y)\right]$ $\displaystyle=$
$\displaystyle 0\,,$ (5.11)
$\displaystyle\left[\mathcal{H}(x),\,\mathcal{P}_{i}(y)\right]$
$\displaystyle=$
$\displaystyle\mathcal{H}(y)\,\delta(x^{-}-y^{-})\,\partial_{i}\delta^{d-1}(x-y)\,,$
(5.12) $\displaystyle\left[\mathcal{P}_{i}(x),\,\mathcal{P}_{j}(y)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}\partial_{j}\delta^{d-1}(x-y)\mathcal{P}_{i}(y)+\partial_{i}\delta^{d-1}(x-y)\mathcal{P}_{j}(x)\big{]}.$
(5.13)

From (5.11), we note that the commutator of two Hamiltonian densities vanishes
$-$ a signature of Carrollian dynamics [32, 33]. The canonical generators are
then defined as

$\displaystyle E=\int dx^{-}d^{d-1}x\,\mathcal{H}(x)\,,\quad P_{i}=\int
dx^{-}d^{d-1}x\,\mathcal{P}_{i}(x)\,,$ (5.14) $\displaystyle M_{ij}=\int
dx^{-}d^{d-1}x\,(x_{i}\mathcal{P}_{j}-x_{j}\mathcal{P}_{i})\,,$ (5.15)
$\displaystyle K_{i}=\int
dx^{-}d^{d-1}x\,(x_{+}\mathcal{P}_{i}-x_{i}\mathcal{H})\,.$ (5.16)

The non-vanishing commutators of the Carroll algebra are

$\displaystyle\left[K_{i},\,P_{j}\right]$ $\displaystyle=$
$\displaystyle\delta_{ij}E\,,$ (5.17)
$\displaystyle\left[M_{ij},\,P_{l}\right]$ $\displaystyle=$
$\displaystyle\delta_{lj}P_{i}-\delta_{il}P_{j}\,,$ (5.18)
$\displaystyle\left[M_{ij},\,K_{l}\right]$ $\displaystyle=$
$\displaystyle\delta_{lj}K_{i}-\delta_{il}K_{j}\,,$ (5.19)
$\displaystyle\left[M_{ij},M_{lm}\right]$ $\displaystyle=$
$\displaystyle\delta_{im}M_{jl}-\delta_{il}M_{jm}-\delta_{jm}M_{il}+\delta_{jl}M_{im}\,.$
(5.20)

Therefore, we see that the magnetic Carroll action for scalar field theory
arises naturally in the light-cone formalism, when we solve the second-class
constraints. However, this method does not seem to yield an electric Carroll
action from the light-cone Lagrangian. To this end, we present below an
alternative method for obtaining Carrollian actions on the light front, which
closely resemble the conventional approaches to Carrollian field theories.

###  5.2 Carroll limits of light-cone actions

In the Hamiltonian formulation, the standard way to deal with constraints is
through the introduction of Lagrange multipliers in the action [34, 35]. The
constraints are then implemented in the extended phase space via the equations
of motion for the Lagrange multipliers. We apply this procedure to the primary
constraint obtained from the light-cone Lagrangian (5.2). We augment the
canonical Hamiltonian (5.5) with a term involving a Lagrange multiplier
$\lambda$, resulting in the total Hamiltonian.

$\mathcal{H}_{T}=\mathcal{H}^{lc}+\lambda\chi_{\phi}=\frac{1}{2}\partial_{i}\phi\partial^{i}\phi+\lambda(\pi_{\phi}-\partial_{-}\phi)\,.$
(5.21)

In this extended phase space, the Hamiltonian action takes the form

$\mathcal{S}_{H}[\phi,\pi_{\phi},\lambda]=\int
dx^{-}dx^{+}d^{d-1}x\Big{(}\pi_{\phi}\partial_{+}\phi-\frac{1}{2}\partial_{i}\phi\partial^{i}\phi-\lambda(\pi_{\phi}-\partial_{-}\phi)\Big{)}\,.$
(5.22)

The equation of motions obtained from the action are as follows

$\displaystyle\lambda:$ $\displaystyle\pi_{\phi}-\partial_{-}\phi=0\,,$ (5.23)
$\displaystyle\pi_{\phi}:$ $\displaystyle\partial_{+}\phi-\lambda=0\,,$ (5.24)
$\displaystyle\phi:$
$\displaystyle-\partial_{+}\pi_{\phi}+\partial_{-}\lambda+\frac{1}{2}\partial_{i}\partial^{i}\phi=0\,.$
(5.25)

The constraint $\chi_{\phi}$ is retrieved from the equation of motion for
$\lambda$. The Hamiltonian densities at two points do not commute in this
case, i.e. $[\mathcal{H}_{T}(x),\,\mathcal{H}_{T}(y)]\neq 0$, indicating that
the dynamics is not Carrollian.

We now restrict the theory to a constant $x^{-}$ surface by assuming
$dx^{-}=\rho^{2}$, where $\rho$ is a small constant. In the limit
$\rho\rightarrow 0$, we can devise two distinct rescalings of the canonical
variables $\phi,\pi_{\phi}$ and $\lambda$, that lead to Carroll actions of the
electric and magnetic types. Case I: Magnetic Carroll limit We rescale the
fields in the Hamiltonian action (5.23) as follows

$\phi\rightarrow\frac{\phi}{\rho}\,,\quad\pi_{\phi}\rightarrow\frac{\pi_{\phi}}{\rho}\,,\quad\lambda\rightarrow\lambda\,.$
(5.26)

The volume element, $\int dx^{+}dx^{-}d^{d-1}x$ scales with a factor of
$\rho^{2}$. Thus, as $\rho\rightarrow 0$, we recover the magnetic Carroll
action (5.5)

$\mathcal{S}_{H}=\int
dx^{+}d^{d-1}x\left(\pi_{\phi}\partial_{+}\phi-\mathcal{H}^{mag}\right)\,,$
(5.27)

with the magnetic Carroll Hamiltonian and momentum densities

$\mathcal{H}^{mag}=\frac{1}{2}\partial_{i}\phi\partial^{i}\phi\,,\quad\mathcal{P}_{i}^{mag}=\partial_{-}\phi\partial_{i}\phi\,.$
(5.28)

The canonical generators for the Carroll transformations are the same as in
(5.14), which obey the Carroll algebra (5.17). The invariance of the action
under the Carroll transformations follows accordingly. Case II: Electric
Carroll limit We now consider a different rescaling of the fields with respect
to $\rho$

$\phi\rightarrow\phi\,,\quad\pi_{\phi}\rightarrow\frac{\pi_{\phi}}{\rho^{2}}\,,\quad\lambda\rightarrow\lambda\,.$
(5.29)

As $\rho\rightarrow 0$, the light-cone Hamiltonian action (5.23) becomes

$\mathcal{S}_{H}=\int
dx^{+}d^{d-1}x\left(\pi_{\phi}\partial_{+}\phi-\mathcal{H}^{elec}\right)\,,$
(5.30)

with the electric Carroll Hamiltonian and momentum densities

$\mathcal{H}^{elec}=\lambda\pi_{\phi}\,,\quad\mathcal{P}_{i}^{elec}=\pi_{\phi}\partial_{i}\phi\,.$
(5.31)

The nontrivial Poisson bracket in this case is
$\\{\phi(x),\pi_{\phi}(y)\\}=\delta^{d-1}(x-y)$. The canonical generators
$G_{\mathfrak{c}}$ for the Carroll transformations are defined in the same way
as in (5.14), but with the integrands, $\mathcal{H}^{elec}$ and
$\mathcal{P}_{i}^{elec}$. The Carroll transformation laws for the fields may
be obtained as

$\delta_{G}\phi=\\{\phi\,,G_{\mathfrak{c}}\\}\,,\quad\delta_{G}\pi_{\phi}=\\{\pi_{\phi}\,,G_{\mathfrak{c}}\\}\,.$
(5.32)

The invariance of the light-cone action is established by suitably extending
the transformations to the Lagrange multipliers.

* * *

We have discussed two methods for deriving magnetic Carroll actions from
Lorentzian field theories, which essentially differ in how we impose the
second-class constraints in the Hamiltonian formulation. In the first case, we
solve the constraint to eliminate the conjugate momenta, resulting in a
Hamiltonian with only spatial derivatives. In the second case, the constraint
is implemented through a Lagrange multiplier, and the Carrollian actions are
obtained by suitably rescaling the canonical variables.

Evidently, the magnetic Carroll Hamiltonian appears more naturally in the
light-cone formalism than its electric counterpart. This contrasts with the
standard approaches to Carrollian field theories, where the electric Carroll
action follows from a simple rescaling of the fields, while the magnetic limit
is more subtle. The crucial difference lies in the fact that light-cone
Lagrangians are first-order in time derivatives. Hence, one always obtains
some primary constraints from the definition of the conjugate momenta. Solving
these constraints eliminates the kinetic term in the Legendre transform,
$\mathcal{H}=\pi\dot{\phi}-\mathcal{L}$, leaving a canonical Hamiltonian with
only spatial derivatives, hence rendering it magnetic Carroll.

In a nutshell, if our goal is to find a magnetic Carroll action for a theory
in $d$ dimensions, a straightforward approach would be to begin with the
Lorentzian action in one higher dimension $(d+1)$ in the light-cone
formulation and employ the first method. By treating one of the null
coordinates as the Carrollian time, we can solve the second-class constraints
obtained from the Lagrangian and work with Dirac brackets in the reduced phase
space, thereby arriving at a magnetic Carroll Hamiltonian action.

In order to obtain Carrollian actions for gauge theories, such as light-cone
electromagnetism or gravity, we need an additional ingredient: a choice of
gauge that effectively restricts the vector (or tensor) fields to one of the
light fronts. We wish to address this problem in future work.

## Acknowledgements

We are grateful to Glenn Barnich and Simone Speziale for the initial
discussions that led to this project. We thank Oscar Fuentealba and Marc
Henneaux for fruitful discussions during the thematic program “Carrollian
Physics and Holography” at the Erwin Schrödinger Institute, where a part of
this work was completed. We acknowledge the support of the grant ID# 62312
from the John Templeton Foundation, as part of the project “The Quantum
Information Structure of Spacetime” (QISS).

## References

  * [1] H. Bacry and J. Levy-Leblond, “Possible kinematics,” J. Math. Phys. 9, 1605-1614 (1968).
  * [2] C. Duval, G. W. Gibbons and P. A. Horvathy, “Conformal Carroll groups and BMS symmetry,” Class. Quant. Grav. 31, 092001 (2014) [arXiv:1402.5894].
  * [3] L. Ciambelli, R. G. Leigh, C. Marteau and P. M. Petropoulos, “Carroll Structures, Null Geometry and Conformal Isometries,” Phys. Rev. D 100, no.4, 046010 (2019) [arXiv:1905.02221].
  * [4] L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, “Carrollian Perspective on Celestial Holography,” Phys. Rev. Lett. 129, no.7, 071602 (2022) [arXiv:2202.04702].
  * [5] E. Bergshoeff, J. Figueroa-O’Farrill and J. Gomis, “A non-lorentzian primer,” SciPost Phys. Lect. Notes 69, 1 (2023) [arXiv:2206.12177].
  * [6] P. A. M. Dirac, “Forms of Relativistic Dynamics,” Rev. Mod. Phys. 21, 392-399 (1949).
  * [7] S. Weinberg, “Dynamics at infinite momentum,” Phys. Rev. 150, 1313-1318 (1966).
  * [8] L. Susskind, “Model of self-induced strong interactions,” Phys. Rev. 165, 1535-1546 (1968).
  * [9] J. B. Kogut and D. E. Soper, “Quantum Electrodynamics in the Infinite Momentum Frame,” Phys. Rev. D 1, 2901-2913 (1970).
  * [10] K. Toth, “On the Group - Structure of Theories in the Infinite Momentum Frame,” Nucl. Phys. B 92, 524-540 (1975).
  * [11] H. Leutwyler and J. Stern, “Relativistic Dynamics on a Null Plane,” Annals Phys. 112, 94 (1978).
  * [12] L. Brink, O. Lindgren and B. E. W. Nilsson, “The Ultraviolet Finiteness of the N=4 Yang-Mills Theory,” Phys. Lett. B 123, 323-328 (1983).
  * [13] A. K. H. Bengtsson, I. Bengtsson and L. Brink, “Cubic Interaction Terms for Arbitrary Spin,” Nucl. Phys. B 227, 31-40 (1983).
  * [14] A. K. H. Bengtsson, I. Bengtsson and L. Brink, “Cubic Interaction Terms for Arbitrarily Extended Supermultiplets,” Nucl. Phys. B 227, 41-49 (1983).
  * [15] A. Gorsky and A. Rosly, “From Yang-Mills Lagrangian to MHV diagrams,” JHEP 01, 101 (2006) [arXiv:hep-th/0510111.].
  * [16] P. Mansfield, “The Lagrangian origin of MHV rules,” JHEP 03, 037 (2006) [arXiv:hep-th/0511264].
  * [17] S. Ananth and S. Theisen, “KLT relations from the Einstein-Hilbert Lagrangian,” Phys. Lett. B 652, 128-134 (2007) [arXiv:0706.1778].
  * [18] P. J. Steinhardt, “Problems of Quantization in the Infinite Momentum Frame,” Annals Phys. 128, 425 (1980).
  * [19] G. Mccartor, “Light Cone Quantization for Massless Fields,” Z. Phys. C 41, 271-275 (1988).
  * [20] T. Heinzl and E. Werner, “Light front quantization as an initial boundary value problem,” Z. Phys. C 62, 521-532 (1994) [arXiv:hep-th/9311108].
  * [21] G. Barnich, S. Majumdar, S. Speziale and W. D. Tan, “Lessons from discrete light-cone quantization for physics at null infinity: bosons in two dimensions,” JHEP 05, 326 (2024) [arXiv:2401.14873].
  * [22] C. Duval, G. Burdet, H. P. Kunzle and M. Perrin, “Bargmann Structures and Newton-Cartan Theory,” Phys. Rev. D 31, 1841-1853 (1985).
  * [23] C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, “Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time,” Class. Quant. Grav. 31, 085016 (2014) [arXiv:1402.0657].
  * [24] J. P. Gauntlett, J. Gomis and P. K. Townsend, “Supersymmetry and the physical phase space formulation of spinning particles,” Phys. Lett. B 248, 288-294 (1990).
  * [25] B. Julia and H. Nicolai, “Null Killing vector dimensional reduction and Galilean geometrodynamics,” Nucl. Phys. B 439, 291-326 (1995) [arXiv:hep-th/9412002].
  * [26] J. Figueroa-O’Farrill, “Classification of kinematical Lie algebras,” [arXiv:1711.05676].
  * [27] J. Figueroa-O’Farrill, “Lie algebraic Carroll/Galilei duality,” J. Math. Phys. 64, no.1, 013503 (2023) [arXiv:2210.13924].
  * [28] J. Figueroa-O’Farrill and S. Prohazka, “Spatially isotropic homogeneous spacetimes,” JHEP 01, 229 (2019) [arXiv:1809.01224].
  * [29] S. Ananth, L. Brink and S. Majumdar, “Bondi-Metzner-Sachs algebra as an extension of the Poincaré symmetry in light-cone gravity,” JHEP 07, 129 (2021) [arXiv:2012.07880].
  * [30] S. Ananth, L. Brink and S. Majumdar, “BMS algebra from residual gauge invariance in light-cone gravity,” JHEP 11, 008 (2021) [arXiv:2101.00019].
  * [31] M. Henneaux and P. Salgado-Rebolledo, “Carroll contractions of Lorentz-invariant theories,” JHEP 11, 180 (2021) [arXiv:2109.06708].
  * [32] C. Teitelboim, “How commutators of constraints reflect the space-time structure,” Annals Phys. 79, 542-557 (1973).
  * [33] M. Henneaux, “Geometry of Zero Signature Space-times,” Bull. Soc. Math. Belg. 31, 47-63 (1979) PRINT-79-0606 (PRINCETON).
  * [34] P. G. Bergmann, “Non-Linear Field Theories,” Phys. Rev. 75, 680-685 (1949).
  * [35] P. A. M. Dirac, “Generalized Hamiltonian dynamics,” Can. J. Math. 2, 129-148 (1950).