# Are nonsingular black holes with super-Planckian hair ruled out by S2 star
data?

M. Cadoni<EMAIL_ADDRESS>Dipartimento di Fisica, Università di
Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy  INFN, Sezione di
Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy  M. De Laurentis
<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”, Università di Napoli
“Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, via Cintia,
I-80126, Napoli, Italy  INFN, Sezione di Napoli, Compl. Univ. di Monte S.
Angelo, Edificio G, via Cintia, I-80126, Napoli, Italy  I. De Martino
<EMAIL_ADDRESS>Departamento de Física Fundamental, Universidad de
Salamanca, Plaza de la Merced, s/n, E-37008 Salamanca, Spain  R. Della Monica
<EMAIL_ADDRESS>Departamento de Física Fundamental, Universidad de
Salamanca, Plaza de la Merced, s/n, E-37008 Salamanca, Spain  M. Oi
<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Cagliari, Cittadella
Universitaria, 09042 Monserrato, Italy  INFN, Sezione di Cagliari, Cittadella
Universitaria, 09042 Monserrato, Italy  A. P. Sanna<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria,
09042 Monserrato, Italy  INFN, Sezione di Cagliari, Cittadella Universitaria,
09042 Monserrato, Italy

###### Abstract

We propose a novel nonsingular black-hole spacetime representing a strong
deformation of the Schwarzschild solution with mass $M$ by an additional hair
$\ell$, which may be hierarchically larger than the Planck scale. Our black-
hole model presents a de Sitter core and $\mathcal{O}(\ell^{2}/r^{2})$ slow-
decaying corrections to the Schwarzschild solution. Our black-hole solutions
are thermodynamically preferred when $0.2\lesssim\ell/GM\lesssim\,0.3$ and are
characterized by strong deviations in the orbits of test particles from the
Schwarzschild case. In particular, we find corrections to the perihelion
precession angle scaling linearly with $\ell$. We test our model using the
available data for the orbits of the S2 star around $\text{SgrA}^{*}$. These
data strongly constrain the value of the hair $\ell$, casting an upper bound
on it of $\sim\,0.47\,GM$, but do not rule out the possible existence of
regular black holes with super-Planckian hair.

_Introduction_.—Since its formulation, General Relativity (GR) has been widely
tested in several different contexts. In particular, one of its most
intriguing predictions is the existence of black holes, whose possible
presence has been tested both directly [1, 2, 3] and indirectly [4, 5].
Although black-hole imaging, gravitational wave and iron-line observations are
compatible with the presence of Kerr black holes [6, 7, 1, 8, 3], there is
still room for small deviations, which could be tested with present and future
experiments. For this reason, there has been an increasing interest in
studying black-hole mimickers. These objects share some properties with GR
solutions but allow for a different phenomenology at the horizon scale, which,
if observed, would represent a smoking gun for deviations from GR.

Regular black holes are among the most fascinating mimickers. Contrary to
classical black holes which present a singularity at their core [9, 10],
indicating the breakdown of the classical theory, these objects are completely
regular everywhere. This point is particularly significant, since we expect a
quantum theory of gravity to resolve the classical singularity problem.
Although there have been some attempts to capture the main properties of the
fundamental theory [11, 12, 13], a clear understanding of its dynamics and of
the mechanism leading to the formation of these regular spacetimes is still
lacking. Consequently, mainly bottom-up approaches have been followed until
now [14, 15, 16, 17, 18, 19, 20, 21, 22], in which one usually modifies GR
solutions to test possible deviations in a phenomenological fashion. This is
particularly suitable, for instance, to study orbits of test particles [23,
24], and it has been used to investigate modifications to the black-hole
shadow [25, 26, 27, 28, 8, 29]. An intriguing feature of nonsingular black-
hole models is the presence of a new length scale (hair) $\ell$, which can be
hierarchically larger than the Planck scale [16]. An important question to be
answered is whether such models with super-Planckian hair can be excluded by
present experimental data.

In this work, following this type of approach, we build a novel regular black-
hole metric, belonging to the general class of models explored in Ref. [16].
The corrections to the Schwarzschild spacetime decrease sufficiently slow to
be experimentally observable also at great distances from the horizon.
Therefore, we test our model with the orbital motion of the S2 star around the
compact radio source Sagittarius A* (SgrA* [2]) in the Galactic Center (GC)
[30, 31].

_The model_.—We start from a static and spherically symmetric spacetime
written in Schwarzschild coordinates $(t,r,\theta,\phi)$ 111Throughout the
paper, we shall use natural units in which $c=\hbar=k_{\text{B}}=1$, unless
otherwise specified.

$\displaystyle\text{d}s^{2}$
$\displaystyle=-f(r)\text{d}t^{2}+\frac{\text{d}r^{2}}{f(r)}+r^{2}\text{d}\Omega^{2}\,,$
(1a) $\displaystyle f(r)$ $\displaystyle=1-\frac{2Gm(r)}{r},\quad
m(r)=4\pi\int_{0}^{r}\text{d}\tilde{r}\,\tilde{r}^{2}\,\rho(\tilde{r})\,,$
(1b)

where $\text{d}\Omega^{2}=\text{d}\theta^{2}+\sin^{2}\theta\,\text{d}\phi^{2}$
is the $2\text{-sphere}$ line element, while $m(r)$ is the Misner-Sharp (MS)
mass, given by a density profile $\rho(r)$. Some profiles of the MS mass (1b)
give a general class of solutions of GR sourced by a anisotropic fluid, with
equation of state $p_{r}=-\rho$, where $p_{r}$ is the radial component of the
fluid pressure [16]. Due to Birkhoff’s theorem, the only vacuum solution of GR
of the form (1a) is the Schwarzschild one, for which $m(r)=M$, being $M$ the
Arnowitt-Deser-Misner mass of the object. This solution is plagued by a
curvature singularity at $r=0$. A widely used approach to eliminate the latter
is imposing a de Sitter (dS)-like behavior of the metric in the core, which
translates to require $f(r)=1-\alpha
r^{2}/\ell^{2}+\mathcal{O}(r^{3}/\ell^{3})$ for $r\ll GM$, where $\alpha$ is
some positive dimensionless constant determining the inner dS length. Here, we
introduced an additional length scale $\ell$, which is responsible for the
smearing of the classical singularity and can be interpreted as an additional
hair of the black hole. Moreover, asymptotic flatness at infinity requires
$f(r)=1-2GM/r+\beta\ell^{2}/r^{2}+\mathcal{O}(\ell^{3}/r^{3})$ for $r\gg GM$,
where $\beta$ is, again, a dimensionless constant. These two minimal
requirements characterize a wide class of regular models, whose general
properties were investigated in Ref. [16].

In this paper we select the particularly simple case belonging to such general
class

$\displaystyle f(r)=1-\frac{2GMr^{2}}{(r+\ell)^{3}}\,,$ (2)

for which $\alpha=2GM/\ell$ and $\beta=6GM/\ell$. The Schwarzschild spacetime
is recovered in the limit $\ell\to 0$.

The main reason for selecting the metric (2) is that it gives
$\mathcal{O}(\ell^{2}/r^{2})$ corrections to the Schwarzschild geometry, which
are the most powerful deformations still compatible with Schwarzschild
asymptotics. We therefore expect strong deviations from the standard GR
phenomenology, having a clear and potentially observable experimental
signature at least when the hair $\ell$ is super-Planckian. Until now, only
models that have at most order $\mathcal{O}(\ell^{4}/r^{4})$ (like, e.g., the
Hayward black hole [15]) or exponentially-suppressed corrections [32] have
been investigated (see also Ref. [16] and references therein). The metric has
also a nice astrophysical analogy. In fact, the density associated with the MS
mass in Eq. 2 reads $4\pi\rho(r)=3M\ell/(r+\ell)^{4}$, whose large $r$
behavior, $\sim r^{-4}$, reproduces that of some density profiles of dark
matter in elliptical and spherical galaxies, like the Hernquist [33] or Jaffe
[34] ones.

The spacetime structure is the same of the general class of models discussed
in Ref. [16]. It has an outer event and an inner Cauchy horizons, respectively
located at $r_{+}$ and $r_{-}$, for which $f(r_{\pm})=0$. The two horizons
coincide at $r_{\text{H,ext}}=2\ell_{\text{c}}$ for the critical value
$\ell=\ell_{\text{c}}\equiv 8GM/27$, and the black hole becomes extremal. The
two horizons remain separate for $\ell<\ell_{\text{c}}$, while they disappear
for $\ell>\ell_{\text{c}}$, which corresponds to a compact horizonless object.

_Black-hole thermodynamics_.—The mass $M$ and the Hawking temperature
$T_{\text{H}}$ of the black-hole model, as functions of the outer horizon
$r_{+}\equiv r_{\text{H}}$ and $\ell$, read

$\displaystyle M=\frac{(\ell+r_{\text{H}})^{3}}{2Gr_{\text{H}}^{2}},\quad
T_{\text{H}}=\frac{r_{\text{H}}-2\ell}{4\pi
r_{\text{H}}(\ell+r_{\text{H}})}\,.$ (3a)

As it is typical, the temperature is zero at extremality. Unlike the
Schwarzschild case, the temperature is nonmonotonic and has a maximum located
at $r_{\text{H,peak}}=\left(2+\sqrt{6}\right)\,\ell$. This implies the
presence of metastable states, i.e., configurations with the same temperature,
but different horizon radii and, therefore, thermodynamic properties. This can
be seen by computing the specific heat

$\displaystyle
C_{\text{H}}=\frac{\text{d}M}{\text{d}T_{\text{H}}}=-\frac{2\pi(r_{\text{H}}-2\ell)(\ell+r_{\text{H}})^{4}}{Gr_{\text{H}}\left(-2\ell^{2}-4\ell
r_{\text{H}}+r_{\text{H}}^{2}\right)}\,.$ (4)

It diverges at $r_{\text{H}}=r_{\text{H,peak}}$, signaling the onset of a
second-order phase transition, separating the models into two branches. For
$r_{\text{H}}>r_{\text{H,peak}}$, i.e.,
$\ell/GM\leq\frac{4}{9}(3-\sqrt{6})\sim 0.245$, $C_{\text{H}}<0$: this is the
branch of thermodynamically unstable holes. For
$r_{\text{H,ext}}<r_{\text{H}}<r_{\text{H,peak}}$, i.e.,
$0.245\,GM\leq\ell\leq\ell_{\text{c}}$, $C_{\text{H}}\geq 0$, we have instead
the branch of stable black holes. This branch includes also the extremal
configuration, for which $C_{\text{H}}=0$. In order to assess which branch is
thermodynamically preferred, we computed the free energy
$\mathcal{F}=M-T_{\text{H}}S$, using both Eq. 3a and the entropy formula in
terms of the integral of the mass function, proposed in Ref. [16]. We found
that the thermodynamically stable black holes, with $\ell\sim\ell_{\text{c}}$,
attain the least free energy and are, thus, thermodynamically preferred with
respect to their unstable counterparts with $\ell\ll GM$.

Orbits of test particles.—The orbits of test particles (geodesics) in the
spacetime (2) can be analyzed using a standard procedure. One writes the
Lagrangian $\mathcal{L}=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$ (the dot means
differentiation with respect to an affine parameter $\lambda$) and uses the
isometries of the metric and the associated conserved quantities (energy
$\mathcal{E}$ and angular momentum). Without loss of generality, we consider,
in our analysis, orbits in the equatorial plane, for which we have

$\displaystyle\dot{r}^{2}+f(r)\left(\epsilon^{2}+\frac{L^{2}}{r^{2}}\right)=\mathcal{E}^{2}.$
(5)

$\epsilon$ is equal to $\pm 1$ or $0$ for timelike and null orbits
respectively, and $L$ denotes the angular momentum about an axis normal to the
invariant plane. The qualitative behavior of both null and timelike
trajectories can be studied by analyzing the effective potential
$V(r)=f(r)\left(\epsilon^{2}+L^{2}/r^{2}\right)$. For null trajectories,
$V(r)$ has a minimum in the region $r_{-}\leq r\leq r_{+}$ and a maximum
located at $r>r_{+}$, the latter corresponding to an unstable circular orbit
(light ring). The light ring radius, equal to $3\,GM$ for the Schwarzschild
spacetime, decreases linearly with $\ell$ for small values of the latter,
going rapidly to zero at $\ell\sim 0.317\,GM$. A similar analysis shows that,
for timelike trajectories, the potential has up to three extrema, depending on
the values of $\ell$ and $L$. A first minimum is located between the outer and
the inner horizon. The other two extrema represent a maximum and a minimum,
which correspond to the marginally bound and the stable circular orbits,
respectively.

Starting from Eq. 5, we can now write an equation for the orbits in the
variable $u=1/r$

$\displaystyle{u^{\prime}}^{2}+f(u)\left(\frac{\epsilon^{2}}{L^{2}}+u^{2}\right)=\frac{\mathcal{E}^{2}}{L^{2}},$
(6)

where a prime indicates differentiation with respect to $\phi$, and
$f(u)=1-2GMu/(1+u\ell)^{3}$. We introduce the dimensionless quantities
$\sigma=(GM/L)^{2}$, $\bar{\ell}=\ell/GM$ and the new variable
$\xi=GMu/\sigma$. When the parameter $\sigma$ is small (e.g.,
$\sigma=1.88\times 10^{-4}$ for the S2 star), we can solve Eq. 6
perturbatively looking for a solution of the form
$\xi=\xi_{0}+\sigma\xi_{1}+\mathcal{O}(\sigma^{2})$.

The zeroth order gives the Newtonian orbits, while at first order we have the
equation

$\displaystyle\xi_{1}^{\prime\prime}+\xi_{1}=3\xi_{0}(\xi_{0}-2\bar{\ell})\,.$
(7)

Neglecting subdominant contributions, and defining $\gamma\equiv
1-\bar{\ell}$, we see that the full solution becomes $\xi\simeq
1+\mathit{e}\cos\left[\left(1-3\gamma\sigma\right)\phi\right]$, where
$\mathit{e}$ is the orbit eccentricity. The orbit precedes of an angle

$\displaystyle\Delta\phi\simeq 6\pi\sigma\gamma\,.$ (8)

Figure 1: Orbital precession for the S2 star for our model as a function of
$\ell$. The black solid line reports the results of our numerical integration
of the geodesic equations in Eq. 6, the blue dashed line represents our first-
order perturbative prediction in Eq. 8. For $\ell=0$ we obtain the
Schwarzschild precession of 12.1 arcmin per orbital period observed in [35].
Moreover, we are able to confirm the perturbative result for small values of
$\ell$ up to a few tens of gravitational radii, where departure from the
linear trend in Eq. 8 is exhibited by the numerical prediction. Finally, we
confirm numerically that for $\ell\geq GM$, the orbital precession becomes
retrograde.

While the usual Schwarzschild result is obtained for $\bar{\ell}\to 0$, we see
that $\Delta\phi$ decreases linearly with the additional hair and it becomes
retrograde for $\bar{\ell}>1$. This is an interesting feature, not present in
the orbits of the Schwarzschild spacetime, that can be used to strongly
constrain the model.

In order to probe the possible existence of the hair $\ell$ in Eq. 2, we have
developed an orbital model for the S2 star in the GC based on the numerical
integration of Eq. 6. In particular, one can recast the energy $\mathcal{E}$
and the angular momentum $L$ in Eq. 6 in terms of the classical Keplerian
elements: the semi-major axis $a$, the eccentricity $e$, the time of
pericenter passage $t_{p}$ and the orbital period $T$ (which can, in fact, be
derived from $M$ and $a$ through Kepler’s third law). A choice of these
parameters uniquely identifies a Keplerian ellipse on the equatorial plane,
that we assume to osculate the real trajectory of the star at a given time. We
hence make use of such ellipse to set the initial conditions at a time $t_{0}$
that, without loss of generality, we fix to be the last time of apocentre
passage for S2 in $\sim$2010.35.

Starting from such initial conditions, we integrate the geodesic equations
numerically by means of a fourth-order Runge-Kutta scheme, over approximately
two orbital periods, covering the time range that spans from 1990 to 2017.
Finally, to compare our synthetic orbit with public data, we need to
reconstruct the observable quantities for S2, i.e., the astrometric sky-
projected position over time for an Earth-based observatory and the
spectroscopically-measured line-of-sight velocity of the star. To this aim, we
perform a geometric projection of the star’s trajectory in the observer
reference frame by means of the Thiele-Innes formulas computed from the three
angular orbital elements: $i$, the orbital inclination; $\Omega$, the
longitude of the ascending node; $\omega$, the argument of the pericenter.
Additionally, for the spectroscopic observables, we take into account the
post-Newtonian time-dilation effects on the light emitted by the star, namely
the special-relativistic transverse and longitudinal Doppler effect and the
general-relativistic gravitational redshift (for more details on how such
quantities can be appropriately accounted for, we refer to previous works on
the subject [36, 24]).

The orbital precession is naturally taken into account on our synthetic orbits
since we directly integrate the fully-relativistic geodesic equations derived
from Eqs. 1a and 2. As a matter of fact, our numerically-integrated orbit
allows us to effectively validate the perturbative results in Eq. 8 by
computing the orbital precession $\Delta\phi$ as the angle spanned by the star
between two subsequent radial turning points. In Fig. 1 we report a comparison
between the numerically computed orbital precession (black solid line) for the
S2 star as a function of the parameter $\ell$, once all the other Keplerian
elements have been fixed to the ones of S2, as derived from the analysis in
[37] (based on a Newtonian orbital model). For $\ell=0$ we obtain the
Schwarzschild precession of 12.1 arcmin per orbital period observed in [35].
The linearly decreasing trend predicted by our perturbative analysis (dashed
blue line) is confirmed up to a few tens of gravitational radii for the
parameter $\ell$, where our numerical predictions start to depart. Moreover,
we are able to confirm numerically the prediction that the orbital precession
becomes retrograde for $\ell\geq GM$.

_Constraining the model with S2 orbital data_.—

Figure 2: Marginalized posterior probability distribution for the parameter
$\ell$ resulting from our MCMC analysis. We are able to constrain the
parameter $\ell$ by providing a 95% confidence level upper limit of
$\ell\lesssim 0.019$ AU, corresponding to $\bar{\ell}\lesssim 0.47$.

We have explored the parameter space of our orbital model for the S2 star,
using the publicly available orbital data for S2. In particular, we have used
near-infrared astrometric positions and radial velocities, coming from 25
years of uninterrupted monitoring of stellar orbits in the GC between $\sim
1992$ and $\sim 2017$, presented in [37]. These data do not cover the last S2
pericenter passage in May 2018 nor its subsequent motion observed by the
GRAVITY Collaboration [35, 38]. Information provided by such portion of the
orbit is crucial in the deed of constraining the gravitational field of SgrA*
[39]. However, as demonstrated in previous works [23, 36], one can consider
orbital data that do not cover the pericenter passage and, then, add as a
single datapoint the precession measurement ($f_{\rm SP}=1.10\pm 0.19$ from
[35], where $f_{\rm SP}=0$ corresponds to a non-preceeding ellipse from
Newtonian gravity and $f_{\rm SP}=1$ corresponds to the GR rate of orbital
precession for the Schwarzschild spacetime).

Besides the knowledge of $\ell$, the full orbital model requires the knowledge
of all the seven Keplerian parameters (which implies leaving the mass of the
central object as a free parameter, as well), along with the observer
galactocentric distance $D$ and 5 reference frame parameters [37, 40]. The
14-dimensional parameter space has been explored through a Markov-Chain Monte
Carlo (MCMC) algorithm implemented in [41]. For the sake of generality, we
have employed uniform priors on all the Keplerian parameters of our orbital
model corresponding to an interval centered on the best-fit values from [37],
with amplitude being 10 times the corresponding observational uncertainty. For
the reference frame parameters, on the other hand, we have taken priors from
the independent analysis in [40]. The interval for the hair $\ell$ has been
set heuristically between 0 and 5 AU, corresponding to over 100 gravitational
radii of the central source. The likelihood adopted for our analysis is the
following

$\displaystyle\log\mathcal{L}=$
$\displaystyle-\frac{1}{2}\sum_{i}\biggl{(}\frac{\textrm{R.A.}_{i}-\textrm{R.A.}_{{\rm
obs},i}}{\sqrt{2}\sigma_{{\rm R.A.},i}}\biggr{)}^{2}+$
$\displaystyle-\frac{1}{2}\sum_{i}\biggl{(}\frac{\textrm{Dec}_{i}-\textrm{Dec}_{{\rm
obs},i}}{\sqrt{2}\sigma_{{\rm Dec},i}}\biggr{)}^{2}+$
$\displaystyle-\frac{1}{2}\sum_{i}\biggl{(}\frac{\textrm{RV}_{i}-\textrm{RV}_{{\rm
obs},i}}{\sqrt{2}\sigma_{{\rm RV},i}}\biggr{)}^{2}+$
$\displaystyle-\frac{1}{2}\biggl{(}\frac{\Delta\phi/\Delta\phi_{\rm
GR}-f_{{\rm SP}}}{\sqrt{2}\sigma_{f_{\rm SP}}}\biggr{)}^{2},$ (9)

where R.A., Dec and RV correspond to the sky-projected right ascension and
declination of S2 and its radial velocity, respectively, while
$\Delta\phi_{\text{GR}}$ is the precession angle predicted for the
Schwarzschild spacetime. The subscript _obs_ represents the observed quantity
at the $i\text{-th}$ epoch, and the $\sigma$’s are the corresponding
observational uncertainties. As done in [23], the factors $\sqrt{2}$ in the
denominators are introduced in order not to double count data points when
considering the last term with the orbital precession (that has been derived
with the same dataset).

The results of our posterior analysis are presented in Table 1 where the
medians and the 68% confidence level intervals for each bounded parameter are
reported. They agree within 1$\sigma$ with previous results in the literature
[37]. We are able to place an upper limit $\ell\lesssim 0.019$ AU
(corresponding to $\ell\lesssim 0.47\,GM$) at 95% confidence level on the
additional hair, whose marginalized posterior distribution is shown in Fig. 2.
Finally, we tested deviations from a Schwarzschild black hole using a mock
catalogue (for more details see [36]) that mirrors future GRAVITY observations
of S2, and we proved the ability of GRAVITY to improve the upper limit on the
hair $\ell$ derived in this work by a factor $\sim 10$.

Parameter (units) |  Best-fit |  Parameter (units) |  Best-fit
---|---|---|---
$D$ (kpc) |  $8.24\pm 0.22$ |  $\omega$ (∘) |  $65.23_{-0.77}^{+0.78}$
$T$ (yr) |  $16.050\pm 0.028$ |  $x_{0}$ (mas) |  $0.26\pm 0.16$
$t_{p}$ (yr) |  $2018.379\pm 0.024$ |  $y_{0}$ (mas) |  $-0.04_{-0.20}^{+0.19}$
$a$ (as) |  $0.1249_{-0.0010}^{+0.0011}$ |  $v_{x,0}$ (mas/yr) |  $0.071_{-0.052}^{+0.053}$
$e$ |  $0.8828\pm 0.0024$ |  $v_{z,0}$ (mas/yr) |  $0.092\pm 0.062$
$i$ (∘) |  $134.42_{-0.49}^{+0.48}$ |  $v_{z,0}$ (km/s) |  $-3.4\pm 4.5$
$\Omega$ (∘) |  $226.75_{-0.82}^{+0.83}$ |  $\bar{\ell}$ |  $\lesssim 0.47$ (95% c.l.)
Table 1: Results of our posterior analysis for the 14 parameters of our
orbital model for the S2 star. In particular, for the bounded parameters, we
derived and reported the 68% confidence interval around the median of the
marginalized distributions. On the other hand, for the parameter $\ell$, our
analysis yields an upper limit which, at 95% confidence level, is given by
$\ell\lesssim 0.019$ AU (corresponding to the dimensionless value
$\bar{\ell}\lesssim 0.47$).

_Discussion and conclusions_.—We proposed a novel regular black-hole spacetime
characterized by an additional hair $\ell$, responsible for the smearing of
the classical singularity. Our metric is designed to have the strongest
allowed corrections at infinity, with respect to the standard Schwarzschild
solution. Due to this property, we have been able to test such geometry with
the orbits of the S2 star around SgrA∗ in the GC. We have found that
corrections to the perihelion precession angle scale linearly with $\ell$. We
have also been able to constrain the parameter $\ell$ with a MCMC analysis,
whose result is the upper bound $\ell\lesssim 0.47\,GM$. This rules out most
horizonless solutions but allows the existence of thermodynamically stable
regular black holes, i.e., models with $0.254\leq\ell/GM\leq 0.296$. Our
results show that regular black holes with super-Planckian hair are not
excluded by the S2 star observational data. The actual proof of the existence
of our nonsingular black holes requires the measurement of $\ell$-dependent
deviations from GR, which could be detected by observations at the light-ring
scale, e.g., with the black hole shadow or gravitational wave experiments.
Nonetheless, we expect our model not to be a good approximation at that scale,
and that a rotating generalization could be necessary.

_Addendum_.—After this manuscript was completed, we were informed that the
metric (2) was first proposed in Ref. [42] and used to investigate the
asymptotic quasinormal modes of regular black holes.

_Acknowledgements_.—We thank S. Zerbini for pointing our attention to Ref.
[42]. RDM acknowledges support from Consejeria de Educación de la Junta de
Castilla y León and from the Fondo Social Europeo. IDM acknowledges support
from Ayuda IJCI2018-036198-I funded by MCIN/AEI/10.13039/501100011033 and: FSE
“El FSE invierte en tu futuro” o “Financiado por la Unión Europea
“NextGenerationEU”/PRTR. IDM is also supported by the project
PID2021-122938NB-I00 funded by the Spanish ”Ministerio de Ciencia e
Innovación” and FEDER “A way of making Europe”, and by the project SA096P20
Junta de Castilla y León. MDL acknowledges the support of Istituto Nazionale
di Fisica Nucleare (INFN) iniziative specifiche QGSKY and TEONGRAV.

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aainstitutetext: Department of Physics, Beijing Normal University, Beijing
100875, Chinabbinstitutetext: Leinweber Center for Theoretical Physics,
University of Michigan, Ann Arbor, MI 48109, USA

#  $sl(2,\mathds{C})\times D$ symmetry and conformal primary basis for
massless fields

Yuan Chen a Mingfeng Li a Kai Shi a Hongbao Zhang a,b Jingchao Zhang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Alternative to the embedding formalism, we provide a group theoretic approach
to the conformal primary basis for the massless field with arbitrary helicity.
To this end, we first point out that $sl(2,\mathds{C})$ isometry gets enhanced
to $sl(2,\mathds{C})\times D$ symmetry for the solution space of the massless
field with arbitrary helicity. Then associated with $sl(2,\mathds{C})\times D$
symmetry, we introduce the novel quadratic Casimirs and relevant tensor/spinor
fields to derive 2 explicit constraints on the bulk dilatation and
$sl(2,\mathds{C})$ Casimirs. With this, we further argue that the candidate
conformal primary basis can be constructed out of the infinite tower of the
descendants of the left and right highest (lowest) conformal primary
wavefunction of $sl(2,\mathds{C})$ Lie algebra, and the corresponding
celestial conformal weights are determined by the bulk scaling dimension
through solving out the exact on-shell conformal primary wavefunctions, where
on top of the two kinds of familiar-looking on-shell conformal primary
wavefunctions, we also obtain another set of independent on-shell conformal
primary wavefunctions for the massless field with helicity $|s|\geq 1$. In
passing, we also develop the relationship between the 4D Lorentz Lie algebra
and 2D conformal Lie algebra from scratch, and present an explicit derivation
for the two important properties associated with the conformal primary
wavefunctions.

##  1 Introduction

Over the last few decades, holographic principle has been standing out as a
guiding principle for us to formulate the quantum theory of gravity. AdS/CFT
correspondence, as one explicit implementation of such a principle, states
that the quantum gravity in an asymptotically Anti-de Sitter spacetime is
encoded fully by the boundary conformal field theory (CFT). On the other hand,
by holography, the only observable in an asymptotically flat spacetime is the
scattering amplitude. However, the scattering amplitude is expressed
conventionally in the momentum representation, which manifests the translation
symmetry but obscures the holographic nature. Given this, a new representation
in terms of the so-called conformal primary basis has been constructed via the
embedding formalism in CFT, whereby the scattering amplitude in the
$(d+2)$-dimensional flat spacetime admits a natural holographic interpretation
of the conformal correlator in the $d$-dimensional celestial sphereBS ; CFS ;
PS ; PSS ; ST ; FT ; DPS ; LZ1 ; LZ2 ; IM . Such a holographic reformulation
of the scattering process in terms of the so-called celestial amplitude
further motivates a recently conjectured duality, called celestial holography,
which proposes that the bulk scattering in the flat spacetime can be dual to a
CFT on the celestial sphereStrominger ; Pasterski1 ; Raclariu ; Pasterski2 ;
PPR .

No matter whether celestial holography turns out to be valid or not, such an
alternative formulation of the scattering process has already shed new light
on our understanding of scattering amplitude, where the conformal primary
basis, as the building block of the whole reformulation, plays a vital role as
it should be the case. In particular, for the case of massless particles, the
conformal primary basis in the new representation turns out to be related to
the familiar plane wave basis in the momentum representation by a Mellin
transformation or further followed by a shadow transformation. The main
purpose of this paper is to offer a group theoretic understanding of the
conformal primary basis for the massless particles in the $4$-dimensional
Minkowski spacetime, which is supposed to serve as an alternative perspective
to the aforementioned embedding formalism. To this end, we shall first develop
the relationship between 4D Lorentz Lie algebra and 2D conformal Lie algebra
from scratch and derive the two important properties associated with the
conformal primary wavefunctions in the subsequent section. Then in Section 3,
with the observation of the bulk dilatation as an emergent symmetry for
massless particles, we will argue that the $sl(2,\mathds{C})\times D$ symmetry
dictated conformal primary basis can serve as an candidate basis for the
massless particle representation of the Poincare Lie algebra, which is further
substantiated by an explicit derivation. In Section 4, we build the specific
correspondence between the 4D bulk scaling dimension and 2D celestial
conformal weights for all the on-shell conformal primary wavefunctions. We
shall conclude our paper with some discussions in the final section.

Notation and conventions follow Chapter $13$ of Wald  unless specified
otherwise. In particular, the signature is $(+,-,-,-)$, and
$\epsilon_{0123}=1$. In addition, a spinor is raised and lowered as
$\phi^{A}=\epsilon^{AB}\phi_{B},\phi_{B}=\epsilon_{AB}\phi^{A}$ with
$\epsilon_{AC}\epsilon^{BC}=\epsilon_{A}{}^{B}=-\epsilon^{B}{}_{A}=\delta_{A}{}^{B}$.

##  2 4D Lorentz group, 2D conformal group, and conformal primary
wavefunctions

In what follows, we shall focus exclusively onto the $4$-dimensional Minkowski
spacetime, where we can take advantage of the spinor machinary to develop the
relationship between 4D Lorentz group and 2D conformal group.

To proceed, we like to take the canonical choice of spinor basis
$o^{A},\iota^{A}$ such that $o_{A}\iota^{A}=1$, with the dual basis given by
$-\iota_{A},o_{A}$. Furthermore, we have
$\epsilon^{AB}=o^{A}\iota^{B}-\iota^{A}o^{B}$ and
$\epsilon_{AB}=o_{A}\iota_{B}-\iota_{A}o_{B}$, which means

$\epsilon^{\Sigma\Gamma}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},\quad\epsilon_{\Sigma\Gamma}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}.$ (1)

The identification between spinors in the above basis and tensors in the
Lorentz coordinates is specified by the soldering form
$\sigma^{\mu}{}_{\Sigma\Gamma^{\prime}}=\frac{1}{\sqrt{2}}(\mathds{1},\mathbf{\sigma})$
with $\mathbf{\sigma}$ the standard Pauli matrices

$\sigma^{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma^{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma^{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (2)

Whence one also has
$\sigma_{\mu}{}^{\Sigma^{\prime}\Gamma}=\frac{1}{\sqrt{2}}(\mathds{1},\sigma^{i})$.
In particular, our Minkowski metric is related to its spinor representation as
follows

$\eta_{\mu\nu}=\sigma_{\mu}{}^{\Sigma^{\prime}\Gamma}\sigma_{\nu}{}^{\Omega^{\prime}\Xi}\epsilon_{\Sigma^{\prime}\Omega^{\prime}}\epsilon_{\Gamma\Xi},$
(3)

where the complex conjugation is indicated implicitly by the primed indices.
With this, the identification $SO(1,3)\simeq SL(2,\mathds{C})/Z_{2}$ can be
expressed as follows

$\Lambda^{\mu}{}_{\nu}x^{\nu}\sigma_{\mu}{}^{\Sigma^{\prime}\Gamma}=x^{\mu}\bar{L}^{\Sigma^{\prime}}{}_{\Omega^{\prime}}\sigma_{\mu}{}^{\Omega^{\prime}\Xi}L^{\Gamma}{}_{\Xi},$
(4)

where $\Lambda\in SO(1,3)$ and $L\in SL(2,\mathds{C})$, satisfying
$L^{\Sigma}{}_{\Gamma}L^{\Omega}{}_{\Xi}\epsilon_{\Sigma\Omega}=\epsilon_{\Gamma\Xi}$.
Accordingly, the corresponding Lie algebras can be related to each other as

$\lambda^{\mu}{}_{\nu}x^{\nu}\sigma_{\mu}{}^{\Sigma^{\prime}\Gamma}=x^{\mu}(\bar{l}^{\Sigma^{\prime}}{}_{\Omega^{\prime}}\sigma_{\mu}{}^{\Omega^{\prime}\Gamma}+\sigma_{\mu}{}^{\Sigma^{\prime}\Xi}l^{\Gamma}{}_{\Xi}).$
(5)

Note that the Pauli matrices serve naturally as the generators for
$sl(2,\mathds{C})$. A straightforward manipulation of the Pauli matrices leads
to the following realization of $sl(2,\mathds{C})$

$\sigma^{1}\rightarrow-2\mathbf{K}_{1},\quad\sigma^{2}\rightarrow
2\mathbf{K}_{2},\quad\sigma^{3}\rightarrow-2\mathbf{K}_{3},\quad
i\sigma^{1}\rightarrow-2\mathbf{J}_{1},\quad i\sigma^{2}\rightarrow
2\mathbf{J}_{2},\quad i\sigma^{3}\rightarrow-2\mathbf{J}_{3}$ (6)

with the Lorentz boosts and rotations defined as111Kindly please refer to
Appendix A for all the generators of the conformal algebra in the
$d$-dimensional Minkowski spacetime.

$\mathbf{K}_{i}=M_{i0},\quad\mathbf{J}_{i}=\frac{1}{2}\epsilon_{0ijk}M^{jk}.$
(7)

However, one can also reach the following two independent realizations of
$sl(2,\mathds{C})$ by the complexified Lorentz generators

$\displaystyle\sigma^{1}\rightarrow-\mathbf{K}_{1}+i\mathbf{J}_{1},\quad\sigma^{2}\rightarrow\mathbf{K}_{2}-i\mathbf{J}_{2},\quad\sigma^{3}\rightarrow-\mathbf{K}_{3}+i\mathbf{J}_{3},$
(8) $\displaystyle i\sigma^{1}\rightarrow
i(-\mathbf{K}_{1}+i\mathbf{J}_{1}),\quad i\sigma^{2}\rightarrow
i(\mathbf{K}_{2}-i\mathbf{J}_{2}),\quad i\sigma^{3}\rightarrow
i(-\mathbf{K}_{3}+i\mathbf{J}_{3})$

from

$\lambda^{\mu}{}_{\nu}x^{\nu}\sigma_{\mu}{}^{\Sigma^{\prime}\Gamma}=x^{\mu}\sigma_{\mu}{}^{\Sigma^{\prime}\Xi}l^{\Gamma}{}_{\Xi},$
(9)

and

$\displaystyle\sigma^{1}\rightarrow-\mathbf{K}_{1}-i\mathbf{J}_{1},\quad\sigma^{2}\rightarrow\mathbf{K}_{2}+i\mathbf{J}_{2},\quad\sigma^{3}\rightarrow-\mathbf{K}_{3}-i\mathbf{J}_{3},$
$\displaystyle i\sigma^{1}\rightarrow-i(-\mathbf{K}_{1}-i\mathbf{J}_{1}),\quad
i\sigma^{2}\rightarrow-i(\mathbf{K}_{2}+i\mathbf{J}_{2}),\quad
i\sigma^{3}\rightarrow-i(-\mathbf{K}_{3}-i\mathbf{J}_{3})$ (10)

from

$\lambda^{\mu}{}_{\nu}x^{\nu}\sigma_{\mu}{}^{\Sigma^{\prime}\Gamma}=x^{\mu}\bar{l}^{\Sigma^{\prime}}{}_{\Omega^{\prime}}\sigma_{\mu}{}^{\Omega^{\prime}\Gamma}.$
(11)

On the other hand, $SL(2,\mathds{C})$ can also be understood as the global
conformal group on the celestial sphere. To this end, let
$\lambda^{\Sigma}=(w,1)$, then its corresponding null vector
$\lambda^{\Sigma}\lambda^{\Sigma^{\prime}}$ is given by
$q^{\mu}=(w\bar{w}+1,w+\bar{w},i(\bar{w}-w),w\bar{w}-1)$, whose spatial
component can be geometrized as a point on a unit celestial sphere as
$\mathbf{q}=(w\bar{w}+1)\hat{\mathbf{q}}$ by performing the stereographic
projection from the north pole of the sphere to the complex plane with
$w=\cot\frac{\theta}{2}e^{i\varphi}$. Similarly, with the choice of
$\lambda^{\Sigma}=(1,\bar{w})$, the corresponding null vector is given by
$q^{\mu}=(1+w\bar{w},w+\bar{w},i(\bar{w}-w),1-w\bar{w})$, which can be
visualized as a point on the unit celestial sphere as
$\mathbf{q}=(1+w\bar{w})\hat{\mathbf{q}}$ instead by performing the
stereographic projection from the south pole to the complex plane with
$w=\tan\frac{\theta}{2}e^{i\varphi}$. In what follows, we prefer to work
exclusively with the first parametrization of our spinor as well as its
corresponding null vector222As to the second parametrization with $\bar{w}$
replaced by $w$, kindly please refer to Appendix B for the relevant results..
Accordingly, $SL(2,\mathds{C})$ acting on our spinor $\lambda^{\Sigma}$ will
induce a global conformal transformation on the celestial sphere as follows

$w^{\prime}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}w=\frac{aw+b}{cw+d},\quad\bar{w}^{\prime}=\begin{pmatrix}\bar{a}&\bar{b}\\\
\bar{c}&\bar{d}\end{pmatrix}\bar{w}=\frac{\bar{a}\bar{w}+\bar{b}}{\bar{c}\bar{w}+\bar{d}}$
(12)

with $ad-bc=1$. Then it is not hard to show that $sl(2,C)$ can also be
realized as follows

$\displaystyle l_{-1}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}=\frac{1}{2}(\sigma^{1}+i\sigma^{2})\rightarrow T_{-1},\quad$
$\displaystyle\bar{l}_{-1}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}=\frac{1}{2}(\overline{\sigma^{1}+i\sigma^{2}})\rightarrow\bar{T}_{-1},$
$\displaystyle l_{1}=\begin{pmatrix}0&0\\\
-1&0\end{pmatrix}=-\frac{1}{2}(\sigma^{1}-i\sigma^{2})\rightarrow T_{1},\quad$
$\displaystyle\bar{l}_{1}=\begin{pmatrix}0&0\\\
-1&0\end{pmatrix}=-\frac{1}{2}(\overline{\sigma^{1}-i\sigma^{2}})\rightarrow\bar{T}_{1}$
$\displaystyle l_{0}=\frac{1}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}=\frac{1}{2}\sigma^{3}\rightarrow T_{0},\quad$
$\displaystyle\bar{l}_{0}=\frac{1}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}=\frac{1}{2}\overline{\sigma^{3}}\rightarrow\bar{T}_{0},$
(13)

where $T_{n}=w^{n+1}\partial_{w}$ and
$\bar{T}_{n}=\bar{w}^{n+1}\partial_{\bar{w}}$ are the vector fields on the
celestial sphere, satisfying the following commutation relations

$[T_{n},T_{m}]=(m-n)T_{n+m},\quad[\bar{T}_{n},\bar{T}_{m}]=(m-n)\bar{T}_{n+m},\quad[T_{n},\bar{T}_{m}]=0.$
(14)

By inspection of Eq. (8), Eq. (2), and Eq. (2), one can obtain the
identification between the 4D Lorentz generators and 2D conformal generators

$\displaystyle T_{-1}\simeq
L_{-1}=\frac{1}{2}(-\mathbf{K}_{1}+\mathbf{J}_{2}+i(\mathbf{K}_{2}+\mathbf{J}_{1})),\quad$
$\displaystyle\bar{T}_{-1}\simeq\bar{L}_{-1}=\frac{1}{2}(-\mathbf{K}_{1}+\mathbf{J}_{2}-i(\mathbf{K}_{2}+\mathbf{J}_{1})),$
$\displaystyle T_{1}\simeq
L_{1}=\frac{1}{2}(\mathbf{K}_{1}+\mathbf{J}_{2}+i(\mathbf{K}_{2}-\mathbf{J}_{1})),\quad$
$\displaystyle\bar{T}_{1}\simeq\bar{L}_{1}=\frac{1}{2}(\mathbf{K}_{1}+\mathbf{J}_{2}-i(\mathbf{K}_{2}-\mathbf{J}_{1})),$
$\displaystyle l_{0}\simeq
L_{0}=\frac{1}{2}(-\mathbf{K}_{3}+i\mathbf{J}_{3}),\quad$
$\displaystyle\bar{l}_{0}\simeq\bar{L}_{0}=\frac{1}{2}(-\mathbf{K}_{3}-i\mathbf{J}_{3}).$
(15)

Accordingly, we have the following commutation relations

$[L_{n},L_{m}]=(m-n)L_{m+n},\quad[\bar{L}_{n},\bar{L}_{m}]=(m-n)\bar{L}_{m+n},\quad[L_{n},\bar{L}_{m}]=0$
(16)

as it should be the case. For the later convenience, we would like to denote
the Lie algebras out of $L_{n}$ and $\bar{L}_{n}$ with $n=0,\pm 1$ as
$sl(2,\mathds{C})_{L}$ and $sl(2,\mathds{C})_{R}$, respectively.

A wavefunction on our Minkowski spacetime and the celestial sphere is called
the conformal primary wavefunction with the $SL(2,\mathds{C})$ conformal
dimension $\Delta$ and spin $J$ if

$\mathcal{O}(x^{\prime\mu}=\Lambda^{\mu}{}_{\nu}x^{\nu};w^{\prime}=\frac{aw+b}{cw+d},\bar{w}^{\prime}=\frac{\bar{a}\bar{w}+\bar{b}}{\bar{c}\bar{w}+\bar{d}})=|\frac{\partial
w^{\prime}}{\partial
w}|^{-\frac{\Delta+J}{2}}|\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}|^{-\frac{\Delta-J}{2}}D(\Lambda)\mathcal{O}(x;w,\bar{w}),$
(17)

where the representation of Lorentz group $D(\Lambda)$ is determined by the
spinor and tensor indices of $\mathcal{O}$ as usual with $|\frac{\partial
w^{\prime}}{\partial w}|=\frac{1}{(cw+d)^{2}}$ and
$|\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}|=\frac{1}{(\bar{c}\bar{w}+\bar{d})^{2}}$.
Whence we have

$D(\Lambda)^{-1}\mathcal{O}(\Lambda
x;w^{\prime},\bar{w}^{\prime})-\mathcal{O}(x;w^{\prime},\bar{w}^{\prime})=|\frac{\partial
w^{\prime}}{\partial
w}|^{-h}|\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}|^{-\bar{h}}\mathcal{O}(x;w,\bar{w})-\mathcal{O}(x;w^{\prime},\bar{w^{\prime}}),$
(18)

where the $SL(2,\mathds{C})$ conformal weights are given by
$(h,\bar{h})=\frac{1}{2}(\Delta+J,\Delta-J)$. This implies

$\displaystyle\mathcal{L}_{L_{n}}\mathcal{O}=-\mathcal{L}_{T_{n}}\mathcal{O}=-(w^{n+1}\partial_{w}+h(n+1)w^{n})\mathcal{O},$
$\displaystyle\mathcal{L}_{\bar{L}_{n}}\mathcal{O}=-\mathcal{L}_{\bar{T}_{n}}\mathcal{O}=-(\bar{w}^{n+1}\partial_{\bar{w}}+\bar{h}(n+1)\bar{w}^{n})\mathcal{O}$
(19)

for $n=0,\pm 1$, which amounts to saying that the Lie derivative of the
Lorentz generators acting on the conformal primary wavefunction can be
understood as the Lie derivative of the corresponding $SL(2,\mathds{C})$
generators acting on it with an additional minus sign. This is essentially the
underlying reason for the definition of the conformal primary wavefunction
through Eq. (17). For our purpose, we would like to list the celestial
conformal weights and the bulk scaling dimension for a few important conformal
primary wavefunctions in Table 1. Furthermore, if a conformal primary wave
function is on-shell, namely satisfies the equation of motion dictated by the
unitary representation of the Poincare group, one can define an operator on
the celestial sphere associated with it as follows

$\hat{\mathcal{O}}(w^{\prime},\bar{w}^{\prime})=(\hat{\Phi}(x^{\prime}),\mathcal{O}(x^{\prime};w^{\prime},\bar{w}^{\prime}))_{\Sigma^{\prime}},$
(20)

where $\hat{\Phi}(x^{\prime})$ is the corresponding bulk quantum field and the
Klein-Gordon inner product $(,)_{\Sigma^{\prime}}$ evaluated on a Cauchy
surface $\Sigma^{\prime}$ is nevertheless independent of the choice of
$\Sigma^{\prime}$. The Lorentz covariance of the Klein-Gordon inner product
implies

$\displaystyle\hat{\mathcal{O}}(w^{\prime},\bar{w}^{\prime})$ $\displaystyle=$
$\displaystyle(D(\Lambda)^{-1}\hat{\Phi}(x^{\prime}),D(\Lambda)^{-1}\mathcal{O}(x^{\prime};w^{\prime},\bar{w}^{\prime}))_{\Sigma}$
(21) $\displaystyle=$
$\displaystyle(U(\Lambda)\hat{\Phi}(x)U(\Lambda)^{-1},|\frac{\partial
w^{\prime}}{\partial
w}|^{-h}|\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}|^{-\bar{h}}\mathcal{O}(x;w,\bar{w}))_{\Sigma}$
$\displaystyle=$ $\displaystyle|\frac{\partial w^{\prime}}{\partial
w}|^{-h}|\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}|^{-\bar{h}}U(\Lambda)\hat{\mathcal{O}}(w,\bar{w})U(\Lambda)^{-1},$

where $U(\Lambda)$ corresponds to the unitary representation of the Lorentz
group in the Fock space. It is noteworthy that due to its presence, the
celestial operator $\hat{\mathcal{O}}$ does not transform under the global
conformal transformation as the ordinary conformal primary operators. But
nevertheless, due to the Lorentz invariance of both the vacuum and S-matrix,
i.e., $U(\Lambda)|0\rangle=|0\rangle$ and $U(\Lambda)^{-1}SU(\Lambda)=S$, we
have

$\displaystyle\langle
0|\hat{\mathcal{O}}_{i}(w^{\prime}_{i},\bar{w}^{\prime}_{i})S\hat{\mathcal{O}}_{j}(w^{\prime}_{j},\bar{w}^{\prime}_{j})|0\rangle$
$\displaystyle=|\frac{\partial w^{\prime}_{i}}{\partial
w_{i}}|^{-h_{i}}|\frac{\partial\bar{w}^{\prime}_{i}}{\partial\bar{w}_{i}}|^{-\bar{h}_{i}}|\frac{\partial
w^{\prime}_{j}}{\partial
w_{j}}|^{-h_{j}}|\frac{\partial\bar{w}^{\prime}_{j}}{\partial\bar{w}_{j}}|^{-\bar{h}_{j}}\langle
0|\hat{\mathcal{O}}_{i}(w_{i},\bar{w}_{i})S\hat{\mathcal{O}}_{j}(w_{j},\bar{w}_{j})|0\rangle,$
(22)

which tells us that the celestial amplitude behaves like the conformal
correlator on the celestial sphere.

| $h$ | $\bar{h}$ | $\Delta$ | $J$ | $\mathcal{D}$
---|---|---|---|---|---
$\lambda^{\Sigma}$ | $-\frac{1}{2}$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$
$\bar{\lambda}^{\Sigma^{\prime}}$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
$q^{\mu}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $0$ | $-1$
$D^{\mu}$ | $0$ | $0$ | $0$ | $0$ | $0$
$\epsilon_{\Sigma\Gamma}$ | $0$ | $0$ | $0$ | $0$ | $1$
$\eta_{\mu\nu}$ | $0$ | $0$ | $0$ | $0$ | $2$
$\frac{1}{q\cdot x}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $-1$
$\frac{\lambda^{\Sigma}}{\sqrt{q\cdot x}}$ | $-\frac{1}{4}$ | $\frac{1}{4}$ | $0$ | $-\frac{1}{2}$ | $-1$
$\frac{D^{\Sigma\Sigma^{\prime}}\bar{\lambda}_{\Sigma^{\prime}}}{\sqrt{q\cdot x}}$ | $\frac{1}{4}$ | $-\frac{1}{4}$ | $0$ | $\frac{1}{2}$ | $0$
Table 1: The celestial conformal weights (conformal dimension and spin) and
the bulk scaling dimension for some conformal primary wavefunctions.

##  3 $sl(2,\mathds{C})\times D$ symmetry and the candidate basis for the
massless particle representation of the Poincare group

The unitary representation of the Poicare group for massless particles is
usually expressed in terms of the simultaneous eigen vectors of the commuting
spatial $3$-momentum or the commuting $4$-momentum with one on-shell condition
$P_{\mu}P^{\mu}=0$. But note that $L^{2},L_{0}$ and $\bar{L}^{2},\bar{L}_{0}$
commuting with one another, where

$L^{2}=L_{0}^{2}-\frac{1}{2}(L_{-1}L_{1}+L_{1}L_{-1}),\quad\bar{L}^{2}=\bar{L}_{0}^{2}-\frac{1}{2}(\bar{L}_{-1}\bar{L}_{1}+\bar{L}_{1}\bar{L}_{-1}),$
(23)

are the corresponding Casimir operators for $sl(2,\mathds{C})_{L}$ and
$sl(2,\mathds{C})_{R}$ Lie algebras, respectively. So it is reasonable to
expect the simultaneous eigen vectors of the above $4$ operators could
constitute the candidate basis as well for the massless particle
representation of the Poincare group. But as alluded above, the number of
degrees of freedom for the $4$-momentum is not $4$ but $3$ due to the on-shell
condition. So there may exist one similar constraint on
$L^{2},L_{0},\bar{L}^{2},\bar{L}_{0}$. As we shall show in this section, this
is the case indeed, where it turns out that the bulk dilation operator come to
play a crucial role. To be more specific, first we note that not only does the
bulk dilatation vector field $D$ together with the Killing vector fields form
a closed Lie algebra, but also commutes with the two Poincare Casimir
operators $P^{2}=P_{\mu}P^{\mu}$ and $W^{2}=W_{\mu}W^{\mu}$ with the Pauli-
Lubanski spin operator defined as
$W_{\mu}=-\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}P^{\nu}M^{\rho\sigma}$ for the
massless particle representation of the Poincare Lie algebra, where
$W_{\mu}=isP_{\mu}$ with $s$ the helicity of the massless particle333More
precisely, $D$ commutes with $s$.. This amounts to saying that the bulk
dilatation emerges as an additional symmetry of the solution space of the
equations of motion dictated by the massless particle representation of the
Poincare group. In addition, $D$ commutes with the Lorentz boosts and
rotations, thus also commutes with $L^{2},L_{0},\bar{L}^{2},\bar{L}_{0}$. As a
result, associated with $sl(2,\mathds{C})\times D$ symmetry, the candidate
basis is formed virtually by the simultaneous eigen vectors of $5$ operators
$D,L^{2},L_{0},\bar{L}^{2},\bar{L}_{0}$ with $2$ constraints on them. In what
follows, we shall present an explicit derivation for this.

###  3.1 Novel quadratic Casimirs and relevant tensor/spinor fields

According to our experience acquired from Zhang ; Zhang2 , we would like to
define a novel quadratic Casimir operator

$\bar{C}^{2}=\bar{L}^{2}-\frac{1}{4}D^{2}.$ (24)

Associated with this Casimir, we further introduce the following two auxiliary
tensor fields

$\displaystyle\bar{H}^{ab}$ $\displaystyle=$
$\displaystyle\bar{L}_{0}^{a}\bar{L}_{0}^{b}-\frac{1}{2}(\bar{L}_{-1}^{a}\bar{L}_{1}^{b}+\bar{L}_{1}^{a}\bar{L}_{-1}^{b})-\frac{1}{4}D^{a}D^{b},$
$\displaystyle\bar{Z}_{abc}$ $\displaystyle=$
$\displaystyle\bar{L}_{0a}\nabla_{b}\bar{L}_{0c}-\frac{1}{2}(\bar{L}_{-1a}\nabla_{b}\bar{L}_{1c}+\bar{L}_{1a}\nabla_{b}\bar{L}_{-1c})-\frac{1}{4}{D}_{a}\nabla_{b}{D}_{c}.$
(25)

Similarly, one can introduce the corresponding Casimir operator and associated
auxiliary fields for the $sl(2,\mathds{C})_{L}$ Lie algebra, which are
obviously related to the right ones by the complex conjugation. A
straightforward calculation gives

$\displaystyle\bar{H}^{ab}=-\frac{1}{4}x^{2}\eta^{ab},\quad\bar{Z}_{abc}=\frac{1}{4}(\eta_{ab}{D}_{c}-\eta_{bc}{D}_{a}-\eta_{ac}{D}_{b}-i\epsilon_{abcd}D^{d}).$
(26)

Whence we further have

$\bar{Z}^{a}{}_{ac}=\frac{1}{2}D_{c},\quad\nabla_{d}\bar{Z}_{abc}=\frac{1}{4}(\eta_{ab}\eta_{cd}-\eta_{bc}\eta_{ad}-\eta_{ac}\eta_{bd}-i\epsilon_{abcd}),$
(27)

where we have used the fact $\nabla_{a}D_{b}=\eta_{ab}$. For the convenience
of the later spinor analysis, we also like to introduce some spinor fields
associated with our novel Casimir as follows

$\displaystyle\alpha_{AA^{\prime}B}{}^{C}(R)$ $\displaystyle=$
$\displaystyle\bar{L}_{0AA^{\prime}}\Gamma_{B}{}^{C}(\bar{L}_{0})-\frac{1}{2}(\bar{L}_{-1AA^{\prime}}\Gamma_{B}{}^{C}(\bar{L}_{1})+\bar{L}_{1AA^{\prime}}\Gamma_{B}{}^{C}(\bar{L}_{-1}))-\frac{1}{4}{D}_{AA^{\prime}}\Gamma_{B}{}^{C}(D)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\bar{Z}_{AA^{\prime}BB^{\prime}}{}^{CB^{\prime}}+\frac{1}{8}D_{AA^{\prime}}\delta_{B}{}^{C}=-\frac{1}{4}D_{AA^{\prime}}\delta_{B}{}^{C}+\frac{1}{8}D_{AA^{\prime}}\delta_{B}{}^{C}=-\frac{1}{8}D_{AA^{\prime}}\delta_{B}{}^{C},$
$\displaystyle\alpha_{AA^{\prime}B}{}^{C}(L)$ $\displaystyle=$ $\displaystyle
L_{0AA^{\prime}}\Gamma_{B}{}^{C}(L_{0})-\frac{1}{2}(L_{-1AA^{\prime}}\Gamma_{B}{}^{C}(L_{1})+L_{1AA^{\prime}}\Gamma_{B}{}^{C}(L_{-1}))-\frac{1}{4}{D}_{AA^{\prime}}\Gamma_{B}{}^{C}(D)$
(28) $\displaystyle=$
$\displaystyle\frac{1}{2}Z_{AA^{\prime}BB^{\prime}}{}^{CB^{\prime}}+\frac{1}{8}D_{AA^{\prime}}\delta_{B}{}^{C}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}(\epsilon_{AB}D^{C}{}_{A^{\prime}}+\epsilon_{A}{}^{C}D_{BA^{\prime}}+D_{AA^{\prime}}\delta_{B}{}^{C})+\frac{1}{8}D_{AA^{\prime}}\delta_{B}{}^{C}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}(\epsilon_{AB}D^{C}{}_{A^{\prime}}+\epsilon_{A}{}^{C}D_{BA^{\prime}})-\frac{1}{8}D_{AA^{\prime}}\delta_{B}{}^{C},$

and

$\displaystyle\gamma_{AD}{}^{BC}(R)=$
$\displaystyle\Gamma_{A}{}^{B}(\bar{L}_{0})\Gamma_{D}{}^{C}(\bar{L}_{0})-\frac{1}{2}(\Gamma_{A}{}^{B}(\bar{L}_{-1})\Gamma_{D}{}^{C}(\bar{L}_{1})+\Gamma_{A}{}^{B}(\bar{L}_{1})\Gamma_{D}{}^{C}(\bar{L}_{-1}))-\frac{1}{4}\Gamma_{A}{}^{B}(D)\Gamma_{D}{}^{C}(D)$
$\displaystyle=\frac{1}{4}(\nabla_{DD^{\prime}}\bar{Z}^{CD^{\prime}}{}_{AB^{\prime}}{}^{BB^{\prime}}+\frac{3}{4}\delta_{A}{}^{B}\delta_{D}{}^{C})=\frac{1}{4}(-\delta_{A}{}^{B}\delta_{D}{}^{C}+\frac{3}{4}\delta_{A}{}^{B}\delta_{D}{}^{C})=-\frac{1}{16}\delta_{A}{}^{B}\delta_{D}{}^{C},$
$\displaystyle\gamma_{AD}{}^{BC}(L)=$
$\displaystyle\Gamma_{A}{}^{B}(L_{0})\Gamma_{D}{}^{C}(L_{0})-\frac{1}{2}(\Gamma_{A}{}^{B}(L_{-1})\Gamma_{D}{}^{C}(L_{1})+\Gamma_{A}{}^{B}(L_{1})\Gamma_{D}{}^{C}(L_{-1}))-\frac{1}{4}\Gamma_{A}{}^{B}(D)\Gamma_{D}{}^{C}(D)$
$\displaystyle=\frac{1}{4}(\nabla_{DD^{\prime}}Z^{CD^{\prime}}{}_{AB^{\prime}}{}^{BB^{\prime}}+\frac{3}{4}\delta_{A}{}^{B}\delta_{D}{}^{C})=\frac{1}{4}(\delta_{A}{}^{C}\delta_{D}{}^{B}+\epsilon_{AD}\epsilon^{CB}-\delta_{A}{}^{B}\delta_{D}{}^{C}+\frac{3}{4}\delta_{A}{}^{B}\delta_{D}{}^{C})$
$\displaystyle=\frac{1}{4}\delta_{A}{}^{C}\delta_{D}{}^{B}+\frac{1}{4}\epsilon_{AD}\epsilon^{CB}-\frac{1}{16}\delta_{A}{}^{B}\delta_{D}{}^{C}$
(29)

where we have used the following identities

$\eta_{AA^{\prime}BB^{\prime}}=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}},\quad\epsilon_{AA^{\prime}BB^{\prime}CC^{\prime}DD^{\prime}}=i(\epsilon_{AB}\epsilon_{CD}\epsilon_{A^{\prime}C^{\prime}}\epsilon_{B^{\prime}D^{\prime}}-\epsilon_{A^{\prime}B^{\prime}}\epsilon_{C^{\prime}D^{\prime}}\epsilon_{AC}\epsilon_{BD})$
(30)

with $\Gamma$ defined later on in Eq. (39). Whence we further have

$\beta_{C}{}^{A}(R)\equiv\gamma_{CD}{}^{DA}(R)=-\frac{1}{16}\delta_{C}{}^{A},\quad\beta_{C}{}^{A}(L)\equiv\gamma_{CD}{}^{DA}(L)=\frac{11}{16}\delta_{C}{}^{A}.$
(31)

###  3.2 Two constraints on $sl(2,\mathds{C})$ Casimirs and dilatation

As a warm-up, let us start with the massless scalar field $\phi$, whose
equation of motion is given by

$\nabla_{a}\nabla^{a}\phi=0.$ (32)

The Lie derivative acting on the scalar field yields

$\mathcal{L}_{X}\mathcal{L}_{Y}\phi=X^{a}\nabla_{a}(Y^{b}\nabla_{b}\phi)=(X^{a}\nabla_{a}Y^{b})\nabla_{b}\phi+X^{a}Y^{b}\nabla_{a}\nabla_{b}\phi,$
(33)

whereby we obtain

$\bar{\mathcal{C}}^{2}\phi=\bar{Z}^{a}{}_{a}{}^{b}\nabla_{b}\phi+\bar{H}^{ab}\nabla_{a}\nabla_{b}\phi=\frac{1}{2}D^{a}\nabla_{a}\phi=\frac{1}{2}\mathcal{D}\phi$
(34)

with $\mathcal{D}=\mathcal{L}_{D}$. Thus we have

$\mathcal{\bar{L}}^{2}\phi=(\frac{1}{4}\mathcal{D}^{2}+\frac{1}{2}\mathcal{D})\phi.$
(35)

It is easy to see that we also have

$\mathcal{L}^{2}\phi=(\frac{1}{4}\mathcal{D}^{2}+\frac{1}{2}\mathcal{D})\phi.$
(36)

Now let us move onto the massless spinor field with the helicity
$s=-\frac{1}{2}$, whose dynamics is governed by the Weyl equation444Kindly
please refer to Appendix C for the relation between the helicity and the
spinor index.

$\nabla_{A^{\prime}A}\phi^{A}=0.$ (37)

Not only is this equation equivalent to
$\nabla_{A^{\prime}}{}^{A}\phi^{B}=\nabla_{A^{\prime}}{}^{(A}\phi^{B)}$, but
also implies $\nabla_{a}\nabla^{a}\phi^{A}=0$. To proceed, let us first recall
the definition of Lie derivative of the spinor field $\phi^{A}{}_{B}$ with
respect to a conformal Killing vector field $\xi^{a}$CD

$\mathcal{L}_{\xi}\phi^{A}{}_{B}=\xi^{a}\nabla_{a}\phi^{A}{}_{B}-\phi^{C}{}_{B}\Gamma_{C}{}^{A}(\xi)+\phi^{A}{}_{C}\Gamma_{B}{}^{C}(\xi)$
(38)

with

$\Gamma_{B}{}^{A}(\xi)=\frac{1}{2}(\nabla_{BB^{\prime}}\xi^{AB^{\prime}}-\frac{1}{4}\nabla_{c}\xi^{c}\delta_{B}{}^{A}).$
(39)

The generalization of this definition to other types of spinor fields is
obvious. It is noteworthy that $\nabla_{a}\Gamma_{B}{}^{A}(\xi)=0$ when
restricted to the dilatation and Killing vector fields in our Minkowski
spacetime. Thus with $X$ and $Y$ such vector fields, we have

$\displaystyle\mathcal{L}_{X}\mathcal{L}_{Y}\phi^{A}$ $\displaystyle=$
$\displaystyle
X^{a}\nabla_{a}(\mathcal{L}_{Y}\phi^{A})-\mathcal{L}_{Y}\phi^{B}\Gamma_{B}{}^{A}(X)$
(40) $\displaystyle=$ $\displaystyle
X^{a}\nabla_{a}(Y^{b}\nabla_{b}\phi^{A}-\phi^{B}\Gamma_{B}{}^{A}(Y))-(Y^{b}\nabla_{b}\phi^{B}-\phi^{C}\Gamma_{C}{}^{B}(Y))\Gamma_{B}{}^{A}(X)$
$\displaystyle=$
$\displaystyle(X^{a}\nabla_{a}Y^{b})\nabla_{b}\phi^{A}+X^{a}Y^{b}\nabla_{a}\nabla_{b}\phi^{A}$
$\displaystyle-\nabla_{a}\phi^{B}(X^{a}\Gamma_{B}{}^{A}(Y)+Y^{a}\Gamma_{B}{}^{A}(X))+\phi^{C}\Gamma_{C}{}^{B}(Y)\Gamma_{B}{}^{A}(X).$

Whence we have

$\displaystyle\mathcal{\bar{C}}^{2}\phi^{A}$ $\displaystyle=$
$\displaystyle\bar{Z}^{a}{}_{a}{}^{b}\nabla_{b}\phi^{A}+\bar{H}^{ab}\nabla_{a}\nabla_{b}\phi^{A}-2\alpha^{CC^{\prime}}{}_{B}{}^{A}(R)\nabla_{CC^{\prime}}\phi^{B}+\phi^{C}\beta_{C}{}^{A}(R)$
(41) $\displaystyle=$
$\displaystyle\frac{1}{2}D^{b}\nabla_{b}\phi^{A}+\frac{1}{4}D^{c}\nabla_{c}\phi^{A}-\frac{1}{16}\phi^{A}=\frac{3}{4}\mathcal{D}\phi^{A}+\frac{5}{16}\phi^{A},$

where we have used
$\mathcal{D}\phi^{A}=D^{a}\nabla_{a}\phi^{A}-\frac{1}{2}\phi^{A}$ in the last
step. Similarly, we also have

$\displaystyle\mathcal{C}^{2}\phi^{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}D^{b}\nabla_{b}\phi^{A}+\frac{1}{4}D^{c}\nabla_{c}\phi^{A}+\frac{1}{2}(\epsilon^{C}{}_{B}D^{AC^{\prime}}+\epsilon^{CA}D_{B}{}^{C^{\prime}})\nabla_{CC^{\prime}}\phi^{B}+\frac{11}{16}\phi^{A}$
(42) $\displaystyle=$
$\displaystyle\frac{3}{4}D^{b}\nabla_{b}\phi^{A}+\frac{1}{2}D_{B}{}^{C^{\prime}}\nabla_{C^{\prime}}{}^{A}\phi^{B}+\frac{11}{16}\phi^{A}=\frac{3}{4}D^{b}\nabla_{b}\phi^{A}+\frac{1}{2}D^{b}\nabla_{b}\phi^{A}+\frac{11}{16}\phi^{A}$
$\displaystyle=$
$\displaystyle\frac{5}{4}\mathcal{D}\phi^{A}+\frac{21}{16}\phi^{A}.$

Thus we end up with the following result

$\mathcal{\bar{L}}^{2}\phi^{A}=(\frac{1}{4}\mathcal{D}^{2}+\frac{3}{4}\mathcal{D}+\frac{5}{16})\phi^{A},\quad\mathcal{L}^{2}\phi^{A}=(\frac{1}{4}\mathcal{D}^{2}+\frac{5}{4}\mathcal{D}+\frac{21}{16})\phi^{A}.$
(43)

Next let us consider the Maxwell equation

$\nabla_{A^{\prime}A}\phi^{AB}=0$ (44)

for the electromagnetic field with $s=-1$, where $\phi^{AB}$ is symmetric with
respect to $A$ and $B$. The Lie derivative acting on $\phi^{AB}$ gives rise to

$\displaystyle\mathcal{L}_{X}\mathcal{L}_{Y}\phi^{AB}$ $\displaystyle=$
$\displaystyle
X^{a}\nabla_{a}(\mathcal{L}_{Y}\phi^{AB})-2\mathcal{L}_{Y}\phi^{C(B}\Gamma_{C}{}^{A)}(X)$
(45) $\displaystyle=$ $\displaystyle
X^{a}\nabla_{a}(Y^{b}\nabla_{b}\phi^{AB}-2\phi^{C(B}\Gamma_{C}{}^{A)}(Y))-2Y^{b}\nabla_{b}\phi^{C(B}\Gamma_{C}{}^{A)}(X)$
$\displaystyle+2\phi^{D(B}\Gamma_{D}{}^{|C|}(Y)\Gamma_{C}{}^{A)}(X)+2\phi^{CD}\Gamma_{D}{}^{(B}(Y)\Gamma_{C}{}^{A)}(X)$
$\displaystyle=(X^{a}\nabla_{a}Y^{b})\nabla_{b}\phi^{AB}+X^{a}Y^{b}\nabla_{a}\nabla_{b}\phi^{AB}$
$\displaystyle-2\nabla_{a}\phi^{C(B}(X^{a}\Gamma_{C}{}^{A)}(Y)+Y^{a}\Gamma_{C}{}^{A)}(X))$
$\displaystyle+2\phi^{D(B}\Gamma_{D}{}^{|C|}(Y)\Gamma_{C}{}^{A)}(X)+2\phi^{CD}\Gamma_{D}{}^{(B}(Y)\Gamma_{C}{}^{A)}(X),$

whereby we have

$\displaystyle\mathcal{\bar{C}}^{2}\phi^{AB}$ $\displaystyle=$
$\displaystyle\frac{1}{2}D^{a}\nabla_{a}\phi^{AB}+\frac{1}{2}D^{a}\nabla_{a}\phi^{AB}-\frac{1}{8}\phi^{AB}-\frac{1}{8}\phi^{AB}$
$\displaystyle=$ $\displaystyle
D^{a}\nabla_{a}\phi^{AB}-\frac{1}{4}\phi^{AB}=\mathcal{D}\phi^{AB}+\frac{3}{4}\phi^{AB}$
$\displaystyle\mathcal{C}^{2}\phi^{AB}$ $\displaystyle=$
$\displaystyle\frac{1}{2}D^{a}\nabla_{a}\phi^{AB}+\frac{1}{2}D^{a}\nabla_{a}\phi^{AB}+D^{a}\nabla_{a}\phi^{AB}+\frac{11}{8}\phi^{AB}+\frac{3}{8}\phi^{AB}$
(46) $\displaystyle=$ $\displaystyle
2D^{a}\nabla_{a}\phi^{AB}+\frac{7}{4}\phi^{AB}=2\mathcal{D}\phi^{AB}+\frac{15}{4}\phi^{AB}.$

Thus we wind up with

$\mathcal{\bar{L}}^{2}\phi^{AB}=(\frac{1}{4}\mathcal{D}^{2}+\mathcal{D}+\frac{3}{4})\phi^{AB},\quad\mathcal{L}^{2}\phi^{AB}=(\frac{1}{4}\mathcal{D}^{2}+2\mathcal{D}+\frac{15}{4})\phi^{AB}.$
(47)

The above spinor analysis has already involved all the necessary ingredients
for one to obtain the corresponding result for the massless spinor field with
other helicities. To be more specific, one can find

$\displaystyle\mathcal{\bar{L}}^{2}\phi^{A\cdots}$ $\displaystyle=$
$\displaystyle(\frac{1}{4}\mathcal{D}^{2}+\frac{n+2}{4}\mathcal{D}+\frac{n^{2}+4n}{16})\phi^{A\cdots}=(\frac{\mathcal{D}}{2}-\frac{s}{2}+1)(\frac{\mathcal{D}}{2}-\frac{s}{2})\phi^{A\cdots},$
$\displaystyle\mathcal{L}^{2}\phi^{A\cdots}$ $\displaystyle=$
$\displaystyle(\frac{1}{4}\mathcal{D}^{2}+\frac{3n+2}{4}\mathcal{D}+\frac{9n^{2}+12n}{16})\phi^{A\cdots}=(\frac{\mathcal{D}}{2}-\frac{3s}{2}+1)(\frac{\mathcal{D}}{2}-\frac{3s}{2})\phi^{A\cdots}$

for the massless spinor field with helicity $s=-\frac{n}{2}$, which is totally
symmetric with respect to $n$ indices and satisfies the equation of motion

$\nabla_{A^{\prime}A}\phi^{A\cdots}=0.$ (49)

Note that the massless spinor field
$\phi^{A^{\prime}B^{\prime}C^{\prime}\cdots}$ with helicity $s=\frac{n}{2}$ is
simply the complex conjugation of the massless spinor field with helicity
$s=-\frac{n}{2}$, so we have

$\displaystyle\mathcal{\bar{L}}^{2}\phi^{A^{\prime}\cdots}$ $\displaystyle=$
$\displaystyle(\frac{1}{4}\mathcal{D}^{2}+\frac{3n+2}{4}\mathcal{D}+\frac{9n^{2}+12n}{16})\phi^{A^{\prime}\cdots}=(\frac{\mathcal{D}}{2}+\frac{3s}{2}+1)(\frac{\mathcal{D}}{2}+\frac{3s}{2})\phi^{A^{\prime}\cdots},$
$\displaystyle\mathcal{L}^{2}\phi^{A^{\prime}\cdots}$ $\displaystyle=$
$\displaystyle(\frac{1}{4}\mathcal{D}^{2}+\frac{n+2}{4}\mathcal{D}+\frac{n^{2}+4n}{16})\phi^{A^{\prime}\cdots}=(\frac{\mathcal{D}}{2}+\frac{s}{2}+1)(\frac{\mathcal{D}}{2}+\frac{s}{2})\phi^{A^{\prime}\cdots}$

for the massless spinor field with helicity $s=\frac{n}{2}$.

On the other hand, by Eq. (2) for a conformal primary wavefunction, we have

$\mathcal{L}^{2}=h(h-1),\quad\bar{\mathcal{L}}^{2}=\bar{h}(\bar{h}-1).$ (51)

So it is reasonable to expect that the candidate basis out of the simultaneous
eigen vectors of $D,L^{2},L_{0},\bar{L}^{2},\bar{L}_{0}$ can be constructed in
terms of the infinite tower of descendants of the left and right highest
(lowest) weight conformal primary wavefunction of $sl(2,\mathds{C})$ Lie
algebra, where the celestial conformal weights are determined by its bulk
scaling dimension555The basis constructed in this way is discrete, compared to
the frequently considered one from the unitary principal series.. Actually, it
has been shown in CMS  that this is the case for the massless scalar field.
Eq. (3.2) and Eq. (3.2) obtained here provide us with an important foundation
to generalize CMS  to the massless field with arbitrary helicity. In the
subsequent section, we shall specify the explicit correspondence between the
2D celestial conformal weights and the 4D bulk scaling dimension for all the
on-shell conformal primary wavefunctions.

##  4 Correspondence between the 4D bulk scaling dimension and 2D celestial
conformal weights

By Eq. (51) and Eq. (3.2), we have the following relationship between the
celestial conformal weight and bulk scaling dimension

$R_{+}:h=\frac{\mathcal{D}}{2}-\frac{3s}{2}+1,\quad\texttt{or}\quad
R_{-}:h=-\frac{\mathcal{D}}{2}+\frac{3s}{2},$ (52)

and

$\bar{R}_{+}:\bar{h}=\frac{\mathcal{D}}{2}-\frac{s}{2}+1,\quad\texttt{or}\quad\bar{R}_{-}:\bar{h}=-\frac{\mathcal{D}}{2}+\frac{s}{2}$
(53)

for the on-shell conformal primary wavefunctions with negative helicity $s$.
However, this relationship demonstrates a certain ambiguity. To fix it, we
like to find the explicit expression for the on-shell conformal primary
wavefunctions. As such, we follow PP ; PPP  to choose
$o^{A}=\frac{1}{\sqrt{q\cdot
x}}\lambda^{A},\iota^{A}=D^{AA^{\prime}}o_{A^{\prime}}$. Accordingly, we have

$\nabla_{A^{\prime}A}o^{B}=-\frac{1}{2}o_{A^{\prime}}o_{A}o^{B},\quad\nabla_{A^{\prime}A}\iota^{B}=\delta_{A}{}^{B}o_{A^{\prime}}-\frac{1}{2}o_{A^{\prime}}o_{A}\iota^{B},$
(54)

which implies

$\nabla_{A^{\prime}A}o^{A}=0,\quad\nabla_{A^{\prime}A}\iota^{A}=\frac{3}{2}o_{A^{\prime}}.$
(55)

Whence we further have

$\displaystyle\nabla_{A^{\prime}A}o^{(AB\cdots}\iota^{CD\cdots)}$
$\displaystyle=$
$\displaystyle\frac{m}{m+n}\nabla_{A^{\prime}A}(o^{A}o^{(B\cdots}\iota^{CD\cdots)}+\frac{n}{m+n}\nabla_{A^{\prime}A}o^{(CD\cdots}\iota^{|A|}\iota^{B\cdots)}$
(56) $\displaystyle=$
$\displaystyle\frac{mn}{m+n}o^{(BC\cdots}\iota^{D\cdots)}o_{A^{\prime}}-\frac{mn-3n-n(n-1)}{2(m+n)}o^{(CD\cdots}\iota^{B\cdots)}o_{A^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{n(n+m+2)}{2(m+n)}o^{(BC\cdots}\iota^{D\cdots)}o_{A^{\prime}},$

where $o^{AB\cdots}$ denotes the spinor field produced by the product of $m$
$o$s and $\iota^{CD\cdots}$ denotes the spinor field produced by the product
of $n$ $\iota$s. With this, we can construct the following linearly
independent on-shell conformal primary wavefunctions for the massless spinor
fields with negative helicity $s$

$\phi^{AB\cdots}=\phi^{\Delta}(x)o^{AB\cdots},\quad\hat{\phi}^{AB\cdots}=\phi^{-s+1}(x)o^{(A\cdots}\iota^{B\cdots)},\quad\tilde{\phi}^{AB\cdots}=(x^{2})^{\Delta+s-1}\phi^{\Delta}(x)\iota^{AB\cdots},$
(57)

where we have used
$D_{AA^{\prime}}D^{AB^{\prime}}=\frac{1}{2}x^{2}\delta_{A^{\prime}}{}^{B^{\prime}}$
with the scalar function defined as $\phi^{\Delta}(x)=\frac{1}{(q\cdot
x)^{\Delta}}$. It is noteworthy that besides the first and third kinds of on-
shell conformal primary wavefunctions, which are familiar to the community and
related to each other by the so-called shadow transformation, we also find the
second kind of on-shell conformal primary wavefunctions for $s\leq-1$.
According to Table.1, we further list the celestial conformal weights and bulk
scaling dimension for the above explicit on-shell conformal primary
wavefunctions in Table 2666It is noteworthy that the second kind of on-shell
conformal primary wavefunctions displays a different correspondence between
the celestial spin and bulk helicity from the first and third ones, whose
celestial spin is related to the bulk helicity simply by $J=\pm s$.. Whence we
obtain a definite relationship between the celestial conformal weights and
bulk scaling dimension for each on-shell conformal primary wavefunction, i.e.,

| $h$ | $\bar{h}$ | $\Delta$ | $J$ | $\mathcal{D}$
---|---|---|---|---|---
$\phi^{AB\cdots}$ | $\frac{\Delta+s}{2}$ | $\frac{\Delta-s}{2}$ | $\Delta$ | $s$ | $-\Delta+2s$
$\hat{\phi}^{AB\cdots}$ | $-s-\frac{o}{2}+\frac{1}{2}$ | $\frac{o}{2}+\frac{1}{2}$ | $-s+1$ | $-s-o$ | $s-o-1$
$\tilde{\phi}^{AB\cdots}$ | $\frac{\Delta-s}{2}$ | $\frac{\Delta+s}{2}$ | $\Delta$ | $-s$ | $\Delta+2s-2$
Table 2: The 2D celestial conformal weights (conformal dimension and spin) and
4D bulk scaling dimension for the on-shell conformal primary wavefunctions
with negative helicity $s$, where $o$ denotes the number of $o^{A}$ in
$\hat{\phi}^{AB\cdots}$.

$\displaystyle R_{-}\quad\texttt{and}\quad\bar{R}_{-}\quad$
$\displaystyle\texttt{ for}\quad\phi^{AB\cdots},$ $\displaystyle
R_{+}\quad\texttt{and}\quad\bar{R}_{-}\quad$
$\displaystyle\texttt{for}\quad\hat{\phi}^{AB\cdots},$ $\displaystyle
R_{+}\quad\texttt{and}\quad\bar{R}_{+}\quad$
$\displaystyle\texttt{for}\quad\tilde{\phi}^{AB\cdots}.$ (58)

As pointed out before, the on-shell massless spinor field with positive
helicity is simply the complex conjugation of that with negative helicity. So
it is not hard to obtain the parallel results for the massless spinor field
with positive helicity, which we shall present below for completeness. Namely,
Eq. (51) together with Eq. (3.2) gives rise to

$R_{+}:\bar{h}=\frac{\mathcal{D}}{2}+\frac{s}{2}+1,\quad\texttt{or}\quad
R_{-}:\bar{h}=-\frac{\mathcal{D}}{2}-\frac{s}{2},$ (59)

and

$\bar{R}_{+}:\bar{h}=\frac{\mathcal{D}}{2}+\frac{3s}{2}+1,\quad\texttt{or}\quad
R_{-}:\bar{h}=-\frac{\mathcal{D}}{2}-\frac{3s}{2}.$ (60)

Such an ambiguity in the correspondence between the celestial conformal
weights and bulk scaling dimension can be resolved by examining the explicit
quantities for each kind of on-shell conformal primary wavefunctions in Table
3. As a result, we have

$\displaystyle R_{-}\quad\texttt{and}\quad\bar{R}_{-}\quad\texttt{
for}\quad\phi^{A^{\prime}B^{\prime}\cdots},$ $\displaystyle
R_{-}\quad\texttt{and}\quad\bar{R}_{+}\quad\texttt{for}\quad\hat{\phi}^{A^{\prime}B^{\prime}\cdots},$
$\displaystyle
R_{+}\quad\texttt{and}\quad\bar{R}_{+}\quad\texttt{for}\quad\tilde{\phi}^{A^{\prime}B^{\prime}\cdots}.$
(61)

| $h$ | $\bar{h}$ | $\Delta$ | $J$ | $\mathcal{D}$
---|---|---|---|---|---
$\phi^{A^{\prime}B^{\prime}\cdots}$ | $\frac{\Delta+s}{2}$ | $\frac{\Delta-s}{2}$ | $\Delta$ | $s$ | $-\Delta-2s$
$\hat{\phi}^{A^{\prime}B^{\prime}\cdots}$ | $\frac{o}{2}+\frac{1}{2}$ | $s-\frac{o}{2}+\frac{1}{2}$ | $s+1$ | $-s+o$ | $-s-o-1$
$\tilde{\phi}^{A^{\prime}B^{\prime}\cdots}$ | $\frac{\Delta-s}{2}$ | $\frac{\Delta+s}{2}$ | $\Delta$ | $-s$ | $\Delta-2s-2$
Table 3: The 2D celestial conformal weights (conformal dimension and spin) and
4D bulk scaling dimension for the on-shell conformal primary wavefunctions
with positive helicity $s$, obtained by taking the complex conjugation of
those with negative helicity $-s$, where $o$ denotes the number of
$o^{A^{\prime}}$ in $\hat{\phi}^{A^{\prime}B^{\prime}\cdots}$.

##  5 Discussion

Although the bulk dilatation does not belong to the isometry Poincare group of
our Minkowski spacetime, it can be regarded as an emergent symmetry of the
solution space of equations of motion for the massless field dictated by the
unitary representation of the Poincare group, reminiscent of the hidden
conformal symmetry of the Kerr black hole discovered in CMS1 . With this in
mind, we have shown that the $sl(2,\mathds{C})\times D$ symmetry dictated
candidate basis for the massless particle representation of the Poincare group
can be constructed out of the infinite tower of the descendants of the left
and right highest (lowest) weight conformal primary wavefunction of
$sl(2,\mathds{C})$ Lie algebra, where the celestial conformal weights are
further determined in an explicit manner by the bulk scaling dimension through
solving out the exact on-shell conformal primary wavefunctions for the
massless field with arbitrary helicity. In particular, on top of the two kinds
of familiar-looking on-shell conformal primary wavefunctions, which are
related to each other by the shadow transformation, we also find another set
of independent on-shell conformal primary wavefunctions for the massless field
with helicity $|s|\geq 1$. In addition, for the massless field with helicity
$|s|\geq 1$, one is also required to introduce the gauge potential to define
the Klein-Gordon inner productPenrose . So we present the exact on-shell
conformal primary wavefunctions as well as the corresponding celestial
conformal weights and bulk scaling dimension in Appendix D for the
electromagnetic potential, which is supposed to be generalized readily to the
massless field with larger helicity.

However, to show that our candidate basis is really the basis for the massless
particle representation of the Poincare group, one is required to generalize
the detailed completeness analysis for the massless scalar field in CMS ; FPR
to the massless field with arbitrary helicity, where the aforementioned new
set of independent on-shell conformal primary wavefunctions may be an
indispensable part for the basis completeness. We hope to address this
important issue elsewhere in the future.

## Acknowledgements

We are grateful to Yichen Feng, Shengyi Liu, Sirui Shuai, Yu Tian, and
Xiaoning Wu for their stimulating discussions. This work is partly supported
by the National Key Research and Development Program of China with Grant No.
2021YFC2203001 and National Natural Science Foundation of China with Grant No.
12075026.

##  Appendix A  Conformal algebra in the $d$-dimensional Minkowski spacetime

For $d$-dimensional Minkowski spacetime with $x^{\mu}$ the Lorentz
coordinates, the global conformal Killing vector fields can be written as

$P_{\mu}=\partial_{\mu},\quad D=x^{\mu}\frac{\partial}{\partial x^{\mu}},\quad
M_{\mu\nu}=x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu},\quad
K_{\mu}=2x_{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial_{\mu}$ (62)

with the non-vanishing commutation relations given by

$\displaystyle[D,P_{\mu}]=-P_{\mu},\quad[P_{\rho},M_{\mu\nu}]=\eta_{\rho\mu}P_{\nu}-\eta_{\rho\nu}P_{\mu},$
$\displaystyle[M_{\mu\nu},M_{\rho\sigma}]=-(\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\mu\sigma}M_{\nu\rho}-\eta_{\nu\rho}M_{\mu\sigma}+\eta_{\nu\sigma}M_{\mu\rho}),$
$\displaystyle[D,K_{\mu}]=K_{\mu},\quad[K_{\mu},P_{\nu}]=-2(\eta_{\mu\nu}D+M_{\mu\nu}).$
(63)

##  Appendix B The formula for the complex coordinate with $w=0$ as the north
pole

Note that the north pole itself corresponds to $w=\infty$ in the complex
coordinate $w$ given by the north pole based stereographic projection, so to
circumvent the potential subtleties associate with the north pole, we prefer
to choose $\lambda^{\Sigma}=(1,w)$. Accordingly,
$q^{\mu}=(1+w\bar{w},w+\bar{w},i(w-\bar{w}),1-w\bar{w})$ with the north pole
located at $w=0$. As a result, Eq. (12) and Eq. (2) will be modified as
follows

$w^{\prime}=\frac{c+dw}{a+bw},\quad\bar{w}^{\prime}=\frac{\bar{c}+\bar{d}\bar{w}}{\bar{a}+\bar{b}\bar{w}},$
(64)

and

$\displaystyle l_{-1}\rightarrow-T_{1},\quad$
$\displaystyle\bar{l}_{-1}\rightarrow-\bar{T}_{1},$ $\displaystyle
l_{1}\rightarrow-T_{-1},\quad$
$\displaystyle\bar{l}_{1}\rightarrow-\bar{T}_{-1},$ $\displaystyle
l_{0}\rightarrow-T_{0},\quad$
$\displaystyle\bar{l}_{0}\rightarrow-\bar{T}_{0}.$ (65)

Similarly, Eq. (17) and Eq. (2) will also get modified in the following way

$\mathcal{O}(x^{\prime\mu}=\Lambda^{\mu}{}_{\nu}x^{\nu};w^{\prime}=\frac{c+dw}{a+bw},\bar{w}^{\prime}=\frac{\bar{c}+\bar{d}\bar{w}}{\bar{a}+\bar{b}\bar{w}})=|\frac{\partial
w^{\prime}}{\partial
w}|^{-\frac{\Delta+J}{2}}|\frac{\partial\bar{w}^{\prime}}{\partial\bar{w}}|^{-\frac{\Delta-J}{2}}D(\Lambda)\mathcal{O}(x;w,\bar{w})$
(66)

with $|\frac{\partial w^{\prime}}{\partial w}|=\frac{1}{(a+bw)^{2}}$ and

$\displaystyle\mathcal{L}_{L_{n}}\mathcal{O}=\mathcal{L}_{T_{-n}}\mathcal{O}=(w^{1-n}\partial_{w}+h(1-n)w^{-n})\mathcal{O},$
$\displaystyle\mathcal{L}_{\bar{L}_{n}}\mathcal{O}=\mathcal{L}_{\bar{T}_{-n}}\mathcal{O}=(\bar{w}^{1-n}\partial_{\bar{w}}+\bar{h}(1-n)\bar{w}^{-n})\mathcal{O}.$
(67)

##  Appendix C $s=-\frac{n}{2}$ for unprimed spinor fields and $s=\frac{n}{2}$
for primed spinor fields

Obviously, the massless scalar field has zero helicity. On the other hand, as
stated in the main body of our paper, unprimed and primed massless spinor
fields have negative and positive helicities, respectively. Here we take the
massless spinor field with one index as an example to show that with our
convention this is the case, i.e.,

$\displaystyle
W^{\mu}\phi^{E}=-\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\mathcal{L}_{P_{\nu}}\mathcal{L}_{M_{\rho\sigma}}\phi^{E}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}\phi^{F}\nabla_{FF^{\prime}}x_{\rho}(\frac{\partial}{\partial
x^{\sigma}})^{EF^{\prime}}$
$\displaystyle=\frac{i}{2}\sigma^{\mu}{}_{AA^{\prime}}(\epsilon^{AB}\epsilon^{CD}\epsilon^{A^{\prime}C^{\prime}}\epsilon^{B^{\prime}D^{\prime}}-\epsilon^{A^{\prime}B^{\prime}}\epsilon^{C^{\prime}D^{\prime}}\epsilon^{AC}\epsilon^{BD})\nabla_{BB^{\prime}}\phi^{F}\epsilon_{CF}\epsilon_{C^{\prime}F^{\prime}}\delta_{D}{}^{E}\delta_{D^{\prime}}{}^{F^{\prime}}$
$\displaystyle=\sigma^{\mu}{}_{AA^{\prime}}(\frac{i}{2}\epsilon^{AB}\epsilon^{A^{\prime}B^{\prime}}\nabla_{BB^{\prime}}\phi^{E}-i\epsilon^{A^{\prime}B^{\prime}}\epsilon^{EB}\nabla_{B^{\prime}B}\phi^{A})$
$\displaystyle=-\frac{i}{2}\eta^{\mu\nu}\partial_{\nu}\phi^{E}=-\frac{i}{2}\mathcal{L}_{P^{\mu}}\phi^{E},$
(68)

which tells us that the unprimed $\phi^{E}$ and the primed $\phi^{E^{\prime}}$
have helicity $-\frac{1}{2}$ and $\frac{1}{2}$, respectively.

##  Appendix D On-shell conformal primary wavefunctions of electromagnetic
potential

For the Killing vector fields or the dilatation field $X$ and $Y$ in Minkowski
spacetime, we have

$\displaystyle\mathcal{L}_{X}\mathcal{L}_{Y}A_{c}=X^{a}\nabla_{a}(Y^{b}\nabla_{b}A_{c}+A_{b}\nabla_{c}Y^{b})+(Y^{b}\nabla_{b}A_{a}+A_{b}\nabla_{a}Y^{b})\nabla_{c}X^{a}=$
$\displaystyle
X^{a}\nabla_{a}Y^{b}\nabla_{b}A_{c}+X^{a}Y^{b}\nabla_{a}\nabla_{b}A_{c}+X^{a}\nabla_{c}Y^{b}\nabla_{a}A_{b}+Y^{b}\nabla_{c}X^{a}\nabla_{b}A_{a}+\nabla_{a}(Y^{b}\nabla_{c}X^{a})A_{b}$

for our electromagnetic potential $A_{c}$ , where we have used the fact the
second derivative of $X$ and $Y$ vanishes. With the gauge condition
$\nabla_{c}A^{c}=D^{c}A_{c}=0$, we further have

$\displaystyle\mathcal{\bar{C}}^{2}A_{c}=\bar{Z}^{a}{}_{a}{}^{b}\nabla_{b}A_{c}+\bar{H}^{ab}\nabla_{a}\nabla_{b}A_{c}+2\bar{Z}^{a}{}_{c}{}^{b}\nabla_{a}A_{b}+\nabla_{a}(\bar{Z}^{b}{}_{c}{}^{a})A_{b}$
$\displaystyle=\frac{1}{2}D^{b}\nabla_{b}A_{c}-\frac{1}{4}x^{2}\Box
A_{c}-\frac{1}{2}\mathcal{D}A_{c}+\frac{i}{2}\nabla_{a}A_{b}\epsilon^{ab}{}_{cd}D^{d}+\frac{1}{2}A_{c}$
$\displaystyle=\mp\frac{1}{2}F_{cd}D^{d}=\pm\frac{1}{2}\mathcal{D}A_{c}$ (70)

for the helicities $s=\pm 1$, which correspond to

$\frac{1}{2}\epsilon_{abcd}F^{cd}=\pm iF_{ab},$ (71)

through the relations
$F^{AA^{\prime}BB^{\prime}}=\phi^{AB}\epsilon^{A^{\prime}B^{\prime}}$ for
$s=-1$ and
$F^{AA^{\prime}BB^{\prime}}=\phi^{A^{\prime}B^{\prime}}\epsilon^{AB}$ for
$s=1$. Likewise, we have

$\mathcal{C}^{2}A_{c}=\mp\frac{1}{2}\mathcal{D}A_{c}$ (72)

for $s=\pm 1$. Thus we end up with

$\mathcal{L}^{2}=\frac{\mathcal{D}}{2}(\frac{\mathcal{D}}{2}-s),\quad\mathcal{\bar{L}}^{2}=\frac{\mathcal{D}}{2}(\frac{\mathcal{D}}{2}+s),$
(73)

which implies the relationship between the bulk scaling dimension and
celestial conformal weight with a certain ambiguity as follows

$\displaystyle
R_{+}:h=\frac{\mathcal{D}}{2}-\frac{s}{2}+\frac{1}{2},\quad\texttt{or}\quad
R_{-}:h=-\frac{\mathcal{D}}{2}+\frac{s}{2}+\frac{1}{2},$
$\displaystyle\bar{R}_{+}:\bar{h}=\frac{\mathcal{D}}{2}+\frac{s}{2}+\frac{1}{2},\quad\texttt{or}\quad\bar{R}_{-}:\bar{h}=-\frac{\mathcal{D}}{2}-\frac{s}{2}+\frac{1}{2}$
(74)

for the on-shell conformal primary wavefunctions of the electromagnetic
potential. Furthermore, with the aforementioned gauge condition as well as the
choice of the null tetrad

$l_{AA^{\prime}}=\iota_{A}\iota_{A^{\prime}},\quad
n_{AA^{\prime}}=o_{A}o_{A^{\prime}},\quad
m_{AA^{\prime}}=\iota_{A}o_{A^{\prime}},\quad\bar{m}_{AA^{\prime}}=o_{A}\iota_{A^{\prime}},$
(75)

the on-shell conformal primary wavefunctions of the electromagnetic potential
can be constructed as follows

$A_{a}=\bar{m}_{a}\phi^{\Delta}(x),\quad\tilde{A}_{a}=m_{a}(x^{2})^{\Delta-1}\phi^{\Delta}(x),$
(76)

which correspond to

$\phi^{AB}=(1-\Delta)\phi^{\Delta}(x)o^{A}o^{B},\quad\tilde{\phi}^{AB}=2(1-\Delta)(x^{2})^{\Delta-2}\phi^{\Delta}(x)\iota^{A}\iota^{B}$
(77)

through the relation
$\phi_{AB}=\nabla_{C^{\prime}(A}A^{C^{\prime}}{}_{B)}=\nabla_{C^{\prime}A}A^{C^{\prime}}{}_{B}$.
Here we have used

$x_{a}=\frac{x^{2}}{2}n_{a}+l_{a},$ (78)

and

$\nabla_{a}\bar{m}_{b}=-\bar{m}_{a}n_{b},\quad\nabla_{a}n_{b}=-n_{a}n_{b}$
(79)

which implies $\nabla_{a}\bar{m}^{a}=0$ and $\Box m_{a}=0$. Obviously, the
above on-shell conformal wavefunctions with $\Delta=1$ correspond to the pure
gauge modes. For $\Delta=1$, one can instead construct the non-gauge modes for
the electromagnetic potential as follows

$A_{a}=\bar{m}_{a}\phi^{1}(x)\ln\frac{q\cdot x}{q\cdot
x_{0}},\quad\tilde{A}_{a}=m_{a}\phi^{1}(x)\ln(\frac{q\cdot x_{0}}{q\cdot
x}x^{2})$ (80)

with $x_{0}=(1,0,0,1)$ the reference point in our Minkowski spacetime, which
give rise to

$\phi^{AB}=\phi^{1}(x)o^{A}o^{B},\quad\tilde{\phi}^{AB}=-2(x^{2})^{-1}\phi^{1}(x)\iota^{A}\iota^{B},$
(81)

respectively. Last, by

$\nabla_{a}l_{b}=n_{a}l_{b}-m_{a}\bar{m}_{b}-\bar{m}_{a}m_{b},$ (82)

one can show that

$\hat{A}_{a}=(-\frac{x^{2}}{2}n_{a}+l_{a})\phi^{2}(x),\quad\hat{A}_{a}=(-\frac{x^{2}}{2}n_{a}+l_{a})\phi^{2}(x)\ln
x^{2}+\phi^{1}(x)(m_{a}\frac{\bar{m}\cdot x_{0}}{q\cdot
x_{0}}-\bar{m}_{a}\frac{m\cdot x_{0}}{q\cdot x_{0}})$ (83)

are also the on-shell conformal primary wavefunctions of the electromagnetic
potential, corresponding to a pure gauge and

$\hat{\phi}^{AB}=4\phi^{2}(x)o^{(A}\iota^{B)},$ (84)

respectively.

| $h$ | $\bar{h}$ | $\Delta$ | $J$ | $\mathcal{D}$
---|---|---|---|---|---
$\bar{m}_{a}\phi^{\Delta}(\Delta=1,\texttt{gauge mode})$ | $\frac{\Delta-1}{2}$ | $\frac{\Delta+1}{2}$ | $\Delta$ | $-1$ | $-\Delta+1$
$m_{a}(x^{2})^{\Delta-1}\phi^{\Delta}(\Delta=1,\texttt{gauge mode})$ | $\frac{\Delta+1}{2}$ | $\frac{\Delta-1}{2}$ | $\Delta$ | $1$ | $\Delta-1$
$\bar{m}_{a}\phi^{1}\ln\frac{q\cdot x}{q\cdot x_{0}}$ | $0$ | $1$ | $1$ | $-1$ | $0(\texttt{up to a gauge})$
$m_{a}\phi^{1}(x)\ln(\frac{q\cdot x_{0}}{q\cdot x}x^{2})$ | $1$ | $0$ | $1$ | $1$ | $0(\texttt{up to a gauge})$
$(-\frac{x^{2}}{2}n_{a}+l_{a})\phi^{2}(x)(\texttt{gauge mode})$ | $1$ | $1$ | $2$ | $0$ | $0$
$(-\frac{x^{2}}{2}n_{a}+l_{a})\phi^{2}(x)\ln x^{2}+$ |  |  |  |  | $0$
$\phi^{1}(x)(m_{a}\frac{\bar{m}\cdot x_{0}}{q\cdot x_{0}}-\bar{m}_{a}\frac{m\cdot x_{0}}{q\cdot x_{0}})$ | $1$ | $1$ | $2$ | $0$ | $(\texttt{up to a gauge})$
Table 4: The 2D celestial conformal weights (conformal dimension and spin) and
4D bulk scaling dimension for the on-shell conformal primary wavefunctions of
the electromagnetic potential with negative helicity $-1$. For the on-shell
celetial conformal primary wavefunctions with positive helicity $1$, which is
the complex conjugation of those with negative helicity $-1$, the
corresponding result can be obtained by simply taking
$h\leftrightarrow\bar{h}$ and $J\rightarrow-J$.

So we have already succeeded in obtaining the on-shell conformal primary
wavefunctions of the electromagnetic potential and its field strength for the
negative helicity. The corresponding celestial conformal weights and bulk
scaling dimension are listed in Table 4. The result for the positive helicity
can be readily obtained by noting that the on-shell conformal primary
wavefunctions for the positive helicity is related to those for the negative
helicity by the complex conjugation. Whence the correpondence between the
celestial conformal weights and bulk scaling dimension can be specified as
follows

$\displaystyle R_{-}\quad\texttt{and}\quad\bar{R}_{-}\quad\texttt{for}\quad
A_{a}(\bar{A}_{a}),$ $\displaystyle
R_{+}\quad\texttt{and}\quad\bar{R}_{-}\quad\texttt{for}\quad\hat{A}_{a},$
$\displaystyle
R_{-}\quad\texttt{and}\quad\bar{R}_{+}\quad\texttt{for}\quad\bar{\hat{A}}_{a},$
$\displaystyle
R_{+}\quad\texttt{and}\quad\bar{R}_{+}\quad\texttt{for}\quad\tilde{A}_{a}(\bar{\tilde{A}}_{a}).$
(85)

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