# Revisiting semileptonic $B^{-}\to p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ decays

C.Q. Geng1,2,3, Chia-Wei Liu2 and Tien-Hsueh Tsai3 1Chongqing University of
Posts & Telecommunications, Chongqing 400065, China
2School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute
for Advanced Study, UCAS, Hangzhou 310024, China
3Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan

###### Abstract

We systematically revisit the baryonic four-body semileptonic decays of
$B^{-}\to{\bf B}\bar{\bf B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ by the
perturbative QCD counting rules with ${\bf B}$ representing octet baryons and
$\ell=e,\mu$. We study the transition form factors of $B^{-}\to{\bf B}\bar{\bf
B}^{\prime}$ in the limit of $(p_{\bf B}+p_{\bar{\bf
B}^{\prime}})^{2}\to\infty$ with the three-body $\bar{B}\to{\bf B}\bar{\bf
B}^{\prime}M$ and $B^{-}\to p\bar{p}\mu^{-}\bar{\nu}_{\mu}$ data along with
$SU(3)_{f}$ flavor symmetry. We calculate the decay branching ratios and
angular asymmetries as well as the differential decay branching fractions of
$B^{-}\to p\bar{p}\ell^{-}\bar{\nu}_{\ell}$. In particular, we find that our
new result of ${\cal B}(B^{-}\to p\bar{p}\ell^{-}\bar{\nu}_{\ell})=(5.21\pm
0.34)\times 10^{-6}$, which is about one order of magnitude lower than the
previous theoretical prediction of $(10.4\pm 2.9)\times 10^{-5}$, agrees well
with both experimental measurements of $(5.8^{+2.6}_{-2.3})\times 10^{-6}$ and
$(5.3\pm 0.4)\times 10^{-6}$ by the Belle and LHCb Collaborations,
respectively. We also evaluate the branching ratios and angular asymmetries in
other channels of $B^{-}\to{\bf B}\bar{\bf B}\ell^{-}\bar{\nu}_{\ell}$, which
can be tested by the ongoing experiments at LHCb and BelleII.

##  I introduction

In 2011, the baryonic four-body semileptonic decay of $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ ($\ell=e$ or $\mu$) was studied with its
decay branching ratio predicted to be $(10.4\pm 2.6\pm 1.2)\times 10^{-5}$ in
Ref. Geng:2011tr . Both $e$ and $\mu$ modes were indeed measured by the Belle
Collaboration Tien:2013nga  in 2014 with ${\cal B}(B^{-}\to
p\bar{p}e^{-}\bar{\nu}_{e})=(8.2^{+3.7}_{-3.2}\pm 0.6)\times 10^{-6}$ and
${\cal B}(B^{-}\to p\bar{p}\mu^{-}\bar{\nu}_{\mu})=(3.1^{+3.1}_{-2.4}\pm
0.7)\times 10^{-6}$ along with the combined value of ${\cal B}(B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell})=(5.8^{+2.4}_{-2.1}\pm 0.9)\times 10^{-6}$,
which is about one oder of magnitude lower than the theoretical prediction.
Recently, the LHCb Collaboration has published the observation of $B^{-}\to
p\bar{p}\mu^{-}\bar{\nu}_{\mu}$ with its decay branching ratio determined to
be $(5.27^{+0.23}_{-0.24}\pm 0.21\pm 0.15)\times 10^{-6}$ Aaij:2019bdu , where
the first and second uncertainties correspond to statistical and systematic,
and the third one is from the branching fraction of the normalization channel,
respectively. Both statistical and systematic uncertainties of the LHCb data
have significant improvements compared with the previous ones by Belle
Tien:2013nga .

These decay modes are useful to determine the value of $|V_{ub}|$ as the works
in the other baryonic modesHsiao:2018zqd  as well as the underlying new CP/T
violating effects. The main difficulty in both extracting $|V_{ub}|$ from
$B^{-}\to p\bar{p}\ell\bar{\nu}_{\ell}$ and constraining the new CP/T
violating effects is how to obtain their hadronic transition amplitude of
$B^{-}\to p\bar{p}$ as it is hard to be calculated via the usual QCD methods,
such as the factorizations and sum rules, which have been widely used in the
mesonic decays of $B^{-}\to\pi^{+}\pi^{-}\ell^{-}\bar{\nu}_{\ell}$ ($B_{\ell
4}$) Feldmann:2018kqr ; Boer:2016iez ; Cheng:2017smj . Nevertheless, these
modes should be considered in the fit for the extraction of $V_{ub}$.
Qualitatively speaking, to reduce the theoretical values for the decay
branching ratios of $B^{-}\to p\bar{p}\ell\bar{\nu}_{\ell}$, a smaller value
of $|V_{ub}|$ is needed besides the form factors. It is similar to the
extractions from the exclusive $B$ and $\Lambda_{b}$ decays, but lower than
that from the inclusive B decays. Clearly, as a baryonic complementary version
of $B_{\ell 4}$ decays, both theoretical and experimental studies of $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ may shed light on the baryonic transition
amplitude of $B^{-}\to p\bar{p}$, uncover the nature of the QCD dynamics, and
improve the measurement of $|V_{ub}|$.

Because of the rareness of four-body $B^{-}\to{\bf B}\bar{\bf
B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ decays with ${\bf B}^{(\prime)}$
representing octet baryons, people have concentrated on the three-body
$\bar{B}\to{\bf B}\bar{\bf B}^{\prime}M$ decays to extract the baryonic
transition from factors in the ${\bf B}\bar{\bf B}^{\prime}$ transitions,
where ${\bf B}(\bar{\bf B}^{\prime})$ and $M$ are octet (anti-)baryons and
pseudoscalar or vector mesons, respectively. There have been several
theoretical analyses on the baryonic three-body $B\to{\bf B}\bar{\bf
B}^{\prime}M$ decays based on the factorization assumptions Cheng:2001tr ;
Cheng:2001ub ; Cheng:2002fp ; Chua:2002yd ; Chua:2002wn ; Geng:2005fh ;
Geng:2005fh ; Chen:2008sw . These baryonic $B$ decays can be basically
classified into current production $\cal C$, transition $\cal T$ and hybrid
${\cal C}+{\cal T}$ types Chen:2008sw , with the quark flow diagrams shown in
Fig. 1.

(a)

(b)

Figure 1: Quark flow diagrams for three-body baryonic B decays $\bar{B}\to{\bf
B}\bar{\bf B}^{\prime}M$ with (a) current and (b) transition types.

Among them, the transition one is the only channel directly related to the
$\bar{B}\to{\bf B}\bar{\bf B}^{\prime}$ baryonic transition amplitudes. In
Ref. Geng:2011tr , perturbative QCD counting rules combined with the available
data of $\bar{B}\to{\bf B}\bar{\bf B}^{\prime}M$ decays at the time were used
to fit the form factors and predict the $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ decays. Although the prediction of $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ motivated its active search, it is clearly
disproved by the experiments of both Belle Tien:2013nga  and LHCb Aaij:2019bdu
. The main reason for such a large prediction is that there were short of
relevant data as well as lack of the understanding of the underlined QCD
dynamics for the baryonic transition of $B^{-}\to p\bar{p}$. In this work, we
would like to reanalyze the semi-leptonic decays of $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ with the same strategy as that in Ref.
Geng:2011tr  with the updated data. In addition, we shall use the flavor
symmetry to extend our results to other $B^{-}\to{\bf B}\bar{\bf
B}\ell^{-}\bar{\nu}_{\ell}$ decays. This work is the first step to know the
property of ${\bar{B}}\to{\bf B}{\bf B}^{\prime}$ transition matrix elements.
After getting a better understanding of these elements, we can use them for
not only improving the measurement of $|V_{ub}|$ but probing or constraining
the new physics effects, such as the T violating triple momentum correlations
due to the rich kinematic structure in the four-body decays of
${\bar{B}}\to{\bf B}{\bf B}^{\prime}\ell\bar{\nu}$.

This paper is organized as follows. In Sec. II, we present our formalism,
which contains the effective Hamiltonians and generalized transition form
factors. In Sec. III, we show our numerical results of the form factors fitted
by three-body $\bar{B}\to{\bf B}\bar{\bf B}^{\prime}M$ processes and the
latest $B^{-}\to p\bar{p}\mu^{-}\bar{\nu}_{\mu}$ result, and present our
predictions of the branching ratios and angular asymmetries in $B^{-}\to{\bf
B}\bar{\bf B}\ell^{-}\bar{\nu}_{\ell}$. We also compare our results of
$p\bar{p}$ invariant mass spectrum in the $B^{-}\to
p\bar{p}\mu^{-}\bar{\nu}_{\mu}$ decay with the one measured by the LHCb. We
give our conclusions in Sec. IV.

##  II Formalism

The effective Hamiltonian for $\bar{B}\to{\bf B}\bar{\bf
B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ at the quark level is given by

$\displaystyle{\cal
H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{ub}\bar{u}\gamma^{\mu}(1-\gamma_{5})b{\bar{\ell}}\gamma_{\mu}(1-\gamma_{5})\nu_{\ell}\,,$
(1)

where $G_{F}$ is the Fermi constant and $V_{ub}$ represents the element of the
CKM matrix. The transition amplitude of $\bar{B}\to{\bf B}\bar{\bf
B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ can be easily factorized into hadronic
and leptonic parts, written as

$\displaystyle{\cal A}(\bar{B}\to{\bf B}\bar{\bf
B}^{\prime}\ell^{-}\bar{\nu}_{\ell})=\frac{G_{F}}{\sqrt{2}}V_{ub}\langle{\bf
B}\bar{\bf
B}^{\prime}|\bar{u}\gamma^{\mu}(1-\gamma_{5})|\bar{B}\rangle\bar{\ell}\gamma_{\mu}(1-\gamma_{5})\nu_{\ell}\,,$
(2)

where $\bar{\ell}$ and $\nu_{\ell}$ are the usual Dirac spinors and
$\langle{\bf B}\bar{\bf
B}^{\prime}|\bar{u}\gamma^{\mu}(1-\gamma_{5})|\bar{B}\rangle$ is the unknown
hadronic transition amplitude. The most general Lorentz invariant forms of the
hadronic transitions for the vector and axial-vector currents can be
parametrized by Geng:2011tr ; Chen:2008sw

$\displaystyle\langle{\bf B}\bar{\bf
B}^{\prime}|\bar{u}\gamma^{\mu}b|\bar{B}\rangle=i\bar{u}(p_{\bf
B})\left[g_{1}\gamma^{\mu}+ig_{2}\sigma^{\mu\nu}p_{\nu}+g_{3}p^{\mu}+g_{4}(p_{\bf
B}+p_{\bar{\bf B}^{\prime}})^{\mu}+g_{5}(p_{\bf B}-p_{\bar{\bf
B}^{\prime}})^{\mu}\right]\gamma_{5}v(p_{\bar{\bf B}^{\prime}}),$
$\displaystyle\langle{\bf B}\bar{\bf
B}^{\prime}|\bar{u}\gamma^{\mu}\gamma_{5}b|\bar{B}\rangle=i\bar{u}(p_{\bf
B})\left[f_{1}\gamma^{\mu}+if_{2}\sigma^{\mu\nu}p_{\nu}+f_{3}p^{\mu}+f_{4}(p_{\bf
B}+p_{\bar{\bf B}^{\prime}})^{\mu}+f_{5}(p_{\bf B}-p_{\bar{\bf
B}^{\prime}})^{\mu}\right]v(p_{\bar{\bf B}^{\prime}}),~{}~{}\,$ (3)

respectively, where $f_{i}$ and $g_{i}$ ($i=1,2,\cdots,5$) are the form
factors and $p^{\mu}=(p_{B^{-}}-p_{\bf B}-p_{\bar{\bf B}^{\prime}})^{\mu}$.
Inspiring from the threshold effects HS , which have been observed in three-
body $\bar{B}\to{\bf B}\bar{\bf B}^{\prime}M$ decays Aaij:2017vnw ;
Aubert:2006qx ; Abe:2002tw , and the pQCD counting rules Brodsky:1980sx ;
Lepage:1980fj ; Lepage:1979za , the momentum dependences of $f_{i}$ and
$g_{i}$ can be assumed to be

$\displaystyle f_{i}=\frac{C_{f_{i}}}{t^{n}},\quad
g_{i}=\frac{C_{g_{i}}}{t^{n}},$ (4)

with $n=3$, where $C_{f_{i},g_{i}}$ are constants determined by the branching
ratios of the input channels. Note that $n$ relates to the number of hard-
gluon propagators as shown in Fig. 2. In the $\bar{B}\to{\bf B}\bar{\bf
B}^{\prime}$ transition, two hard gluons produce the valance quarks in the
${\bf B}\bar{\bf B}^{\prime}$ pair separately as well as one more hard gluon
is needed to speed up the spectator quark in $\bar{B}$ Chen:2008sw . As a
result, we can use $C_{f_{i}}$ and $C_{g_{i}}$ to describe the hadronic form
factors in both transition type three-body and semileptonic four-body decays.

Figure 2: Diagram for the $\bar{B}\to\bar{\bf B}{\bf B}^{\prime}$ transition,
where the curl lines stand for hard-gluons and the symbol of $\otimes$ denotes
the weak vertex, while each hard gluon contributes an $1/t$ in the form
factors.

With the help of $SU(3)$ flavor and $SU(2)$ spin symmetries in $t\to\infty$
and heavy quark limit, $C_{f_{i}}$ and $C_{g_{i}}$ are related by only two
chiral-conserving parameters $C_{RR}$ and $C_{LL}$ and one chiral-flipping
parameter $C_{LR}$. Consequently, we have

$\displaystyle C_{f_{1}}$ $\displaystyle=$ $\displaystyle
m_{\bar{B}}\left(e_{LL}C_{LL}+e_{RR}C_{RR}\right)+\left(m_{\bf B}+m_{\bar{\bf
B}^{\prime}}\right)e_{LR}C_{LR}$ $\displaystyle C_{g_{1}}$ $\displaystyle=$
$\displaystyle m_{\bar{B}}\left(e_{LL}C_{LL}-e_{RR}C_{RR}\right)+\left(m_{\bf
B}-m_{\bar{\bf B}^{\prime}}\right)e_{LR}C_{LR}$ $\displaystyle C_{f_{2}}$
$\displaystyle=$ $\displaystyle-C_{g_{2}}=e_{LR}C_{LR},\quad
C_{f_{i}}=C_{g_{i}}=-e_{LR}C_{LR}\quad\text{for }i=3,4,5\,,$ (5)

where $e_{RR}$, $e_{LL}$ and $e_{LR}$ are the electroweak coefficients
determined by the spin-flavor structure of $\bar{B}$ and ${\bf B}\bar{\bf
B}^{\prime}$, and $m_{{\bf B},\bar{\bf B}^{\prime}}$ correspond to baryon and
anti-baryon masses, respectively. The detail derivations of Eq. 5 are
presented in Appendix. We list the coefficients of relevant channels in Table
1.

Table 1:  Electroweak coefficients of $\bar{B}\to{\bf B}\bar{\bf B}^{\prime}$ under $t\to\infty$ and heavy quark limits. Channel | $e_{RR}$ | $e_{LL}$ | $e_{LR}$
---|---|---|---
$B^{-}\to p\bar{p}$ | $\frac{1}{3}$ | $\frac{5}{3}$ | $-\frac{4}{3}$
$B^{-}\to n\bar{n}$ | $\frac{2}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$
$B^{-}\to\Sigma^{+}\bar{\Sigma}^{+}$ | $\frac{1}{3}$ | $\frac{5}{3}$ | $-\frac{4}{3}$
$B^{-}\to\Sigma^{0}\bar{\Sigma}^{0}$ | $\frac{1}{6}$ | $\frac{5}{6}$ | $-\frac{2}{3}$
$B^{-}\to\Xi^{0}\bar{\Xi}^{0}$ | $\frac{2}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$
$B^{-}\to\Lambda^{0}\bar{\Lambda}^{0}$ | $\frac{1}{2}$ | $\frac{1}{3}$ | $0$
$B^{-}\to\Lambda^{0}\bar{\Sigma}^{0}$ | $-\frac{1}{2\sqrt{3}}$ | $\frac{1}{2\sqrt{3}}$ | $-\frac{1}{\sqrt{3}}$
$B^{-}\to\Sigma^{0}\bar{\Lambda}^{0}$ | $-\frac{1}{2\sqrt{3}}$ | $\frac{1}{2\sqrt{3}}$ | $-\frac{1}{\sqrt{3}}$
$B^{-}\to\Lambda^{0}\bar{p}$ | $0$ | $\sqrt{\frac{2}{3}}$ | $-\sqrt{\frac{2}{3}}$
$\bar{B}^{0}\to\Lambda^{0}\bar{\Lambda}^{0}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$
$\bar{B}^{0}\to p\bar{p}$ | $\frac{2}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$

Following the same formalism in the literature of $B_{\ell 4},D_{\ell 4}$ and
$K_{\ell 4}$ analyses Lee:1992ih ; Pais:1968zza , we examine the
$\bar{B}\to{\bf B}\bar{\bf B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ system in the
$\bar{B}$ rest frame with five kinematic variables,
$s=(p_{\ell}+p_{\bar{\nu}_{\ell}})^{2},~{}t,~{}\theta_{\bf
B},~{}\theta_{\ell}$ and $\phi$, where $\sqrt{s}$ and $\sqrt{t}$ are the
invariant masses of lepton and ${\bf B}\bar{\bf B}^{\prime}$ pairs,
respectively, and three kinematic angles are shown in Fig. 3. The differential
decay width is given by

$\displaystyle d\Gamma=\frac{|\bar{\cal
A}|^{2}}{4(4\pi)^{6}m_{\bar{B}}^{3}}X\beta_{\bf B}\beta_{\bf
L}dsdtd\cos\theta_{\bf B}d\cos\theta_{\ell}d\phi\,,$ (6)

where $|\bar{\cal A}|^{2}$ is the spin-averaged amplitude and $X,\beta_{\bf
B},\beta_{\bf L}$ are given by

$\displaystyle X$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(m_{\bar{B}}^{2}-s-t)^{2}-4st}}{2}$
$\displaystyle\beta_{\bf B}$ $\displaystyle=$
$\displaystyle\frac{1}{t}\lambda^{\frac{1}{2}}(t,m_{\bf B}^{2},m_{\bar{\bf
B}^{\prime}}^{2})$ $\displaystyle\beta_{\bf L}$ $\displaystyle=$
$\displaystyle\frac{1}{s}\lambda^{\frac{1}{2}}(s,m_{\ell}^{2},m_{\bar{\nu}}^{2})\,$
(7)

with $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$.

Figure 3: $\theta_{\bf B},\theta_{\ell}$ and $\phi$ in $\bar{B}\to{\bf
B}\bar{\bf B}\ell^{-}\bar{\nu}_{\ell}$ decays.

We can also define the integrated $\theta_{\bf B}$ and $\theta_{\ell}$
asymmetries of ${\bf B}\bar{\bf B}^{\prime}$ and lepton pairs as follows:

$\displaystyle\langle\alpha_{\theta_{f}}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{\int_{0}^{1}\frac{d\Gamma}{d\cos\theta_{f}}d\cos\theta_{f}-\int_{-1}^{0}\frac{d\Gamma}{d\cos\theta_{f}}d\cos\theta_{f}}{\int_{0}^{1}\frac{d\Gamma}{d\cos\theta_{f}}d\cos\theta_{f}+\int_{-1}^{0}\frac{d\Gamma}{d\cos\theta_{f}}d\cos\theta_{f}}\,,$
(8)

with $f={\bf B}$ and $\ell$, respectively.

##  III Numerical Results

In our numerical analysis, we use the Wolfenstein parameterization for the CKM
matrix with the corresponding parameters, taken to be pdg

$\displaystyle\lambda=0.22650,\quad A=0.790,\quad\rho=0.141,\quad\eta=0.357,$
(9)

leading to $|V_{ub}|=3.8\times 10^{-3}$. To extract the form factors, we use
the factorization assumption and follow the formula in Ref. Chen:2008sw  to
calculate the branching ratios of $\bar{B}\to{\bf B}\bar{\bf B}^{\prime}M$.
The full analysis of three-body kinematics and detail derivations of
$\bar{B}\to{\bf B}\bar{\bf B}^{\prime}M$ factorization amplitudes can be found
in Ref. Chen:2008sw . Based on Refs. Chen:2008sw ; Cheng:1998kd ; Cheng:2001sc
, the effective Wilson coefficients of $a_{2}^{D^{(*)}}$ and $a_{2}^{J^{(*)}}$
should include non-factorizable effects and can be parameterized by the
effective color number $(N_{c}^{eff})$,
$a_{2}^{M}=c_{2}^{M}+c_{1}^{M}/N_{c}^{eff}$, where $N_{c}^{eff}$ will be
fitted . We present the numerical inputs of hadron masses, lifetimes, meson
decay constants, and Wilson coefficients in Table 2 Geng:2011tr ; pdg ;
Buchalla:1995vs .

Table 2: Input values of hadron masses, lifetimes, meson decay constants, and Wilson coefficients, where the masses (decay constants) and lifetimes are in units of GeV and fs, while the Wilson coefficients are dimensionless. $m_{B^{-}}$ | $m_{B^{0}}$ | $m_{D}$ | $m_{D^{*}}$ | $m_{J/\Psi}$ | $m_{p}$ | $m_{n}$ | $m_{\Lambda^{0}}$ | $m_{\Xi^{0}}$ | $m_{\Sigma^{0}}$
---|---|---|---|---|---|---|---|---|---
$5.28$ | $5.28$ | $1.87$ | $2.01$ | $3.10$ | $0.94$ | $0.94$ | $1.12$ | $1.32$ | $1.19$
$m_{\Sigma^{+}}$ | $\tau_{B^{-}}$ | $\tau_{B^{0}}$ | $f_{D}$ | $f_{D^{*}}$ | $f_{J/\Psi}$ | $c_{1}^{D^{(*)}}$ | $c_{2}^{D^{(*)}}$ | $c_{1}^{J/\Psi}$ | $c_{2}^{J/\Psi}$
$1.19$ | $164$ | $152$ | $0.22$ | $0.23$ | $0.41$ | $-0.367$ | $1.169$ | $-0.185$ | $1.082$

By performing the minimum $\chi^{2}$ method with six data points, the free
parameters of $C_{RR,LL,LR}$ in Eq. (5) and the effective color number of
$N_{c}^{eff}$ are fitted to be

$\displaystyle(C_{RR},C_{LL},C_{LR})$ $\displaystyle=$
$\displaystyle(-11.67\pm 1.97,17.78\pm 0.83,6.41\pm 1.62)\text{ GeV}^{4}\,,$
$\displaystyle N_{c}^{eff}$ $\displaystyle=$ $\displaystyle 0.51\pm 0.03\,,$
(10)

respectively, with $\chi^{2}/d.o.f=0.28$. Our fitting results along with the
input data for the transition-type three-body decays of $\bar{B}\to{\bf
B}\bar{\bf B}^{\prime}M$ and $B^{-}\to p\bar{p}\mu^{-}\bar{\nu}_{\mu}$ are
presented in Table. 3.

Table 3: Results for the transition-type decays of $\bar{B}\to{\bf B}\bar{\bf B}^{\prime}M$ and $B^{-}\to p\bar{p}\mu^{-}\bar{\nu}_{e}$ . Channel | Data | Our Results
---|---|---
$10^{5}{\cal B}(B^{-}\to\Lambda^{0}\bar{p}J/\Psi)$ | $1.46\pm 0.12$ | $1.47\pm 0.12$
$10^{5}{\cal B}(B^{0}\to\Lambda^{0}\bar{\Lambda}^{0}D)$ | $1.00\pm 0.30$ | $1.23\pm 0.10$
$10^{5}{\cal B}(B^{0}\to p\bar{p}D)$ | $10.4\pm 0.70$ | $10.42\pm 0.28$
$10^{5}{\cal B}(B^{0}\to p\bar{p}D^{*})$ | $9.9\pm 1.1$ | $9.04\pm 0.49$
$10^{7}{\cal B}(B^{0}\to p\bar{p}J/\Psi)$ | $4.50\pm 0.60$ | $4.83\pm 0.34$
$10^{6}{\cal B}(B^{-}\to p\bar{p}\mu^{-}\bar{\nu}_{\mu})$ | $5.27\pm 0.35$ | $5.21\pm 0.34$

In Table 4, we show our predictions of other four-body $B^{-}\to{\bf
B}\bar{\bf B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ decays. In Tables 3 and 4, we
only consider the errors caused by the data inputs and $\chi^{2}$ fitting, the
other uncertainties are not listed in our results due to the lack of a
comprehensive model to describe $\bar{B}\to{\bf B}{\bf B}^{\prime}$. The most
important source of the theoretical errors is the assumption of the heavy
quark limit as shown around Eq. (19) in Appendix. Without this assumption, we
need to fit totally 10 parameters, which much exceed the number of the current
data points. However, from the kinematical point of view, we expect that the
error from the heavy quark approximation should be the same order as that in
$\Lambda_{c}\to\Lambda$ because of the similar mass ratio of $2m_{\bf
B}/m_{\bar{B}}\simeq m_{\Lambda}/m_{\Lambda_{c}}$ between the two types of the
channels. On the other hand, we are confident with our momentum behaviors of
the form factors given by the QCD counting rules, which have been used to
explain the threshold effects in three-body $\bar{B}\to{\bf B}{\bf
B}^{\prime}M$ decays. Moreover, these momentum behaviors can match the newest
$B^{-}\to p\bar{p}\mu^{+}\nu_{\mu}$ differential decay width measured by the
LHCb. It is interesting to see that the $SU(3)_{f}$ flavor symmetry guarantees
that all observables in
$B^{-}\to\Lambda^{0}\bar{\Sigma}^{0}\ell^{-}\bar{\nu}_{\ell}$ are the same as
those in $B^{-}\to\Sigma^{0}\bar{\Lambda}^{0}\ell^{-}\bar{\nu}_{\ell}$. We
note that the angular distribution asymmetries in $B^{-}\to{\bf B}\bar{\bf
B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ mainly depend on the electroweak
coefficients, which are associated with the spin-flavor structures of the
${\bf B}\bar{\bf B}^{\prime}$ pairs. Interestingly, the angular asymmetries of
$B^{-}\to\Lambda\bar{\Lambda}\ell^{-}\bar{\nu}_{\ell}$ vanish because only the
chiral-conserving interaction participates in $B^{-}\to\Lambda\bar{\Lambda}$.
As a result, the physical observables in
$B^{-}\to\Lambda\bar{\Lambda}\ell^{-}\bar{\nu}_{\ell}$ are sensitive to test
the availability of pQCD counting rules as well as the asymptotic relations in
the limit of $t\to\infty$.

Table 4: Our numerical results of four-body $B^{-}\to{\bf B}\bar{\bf B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ decays, where the errors come from the $\chi^{2}$. Channel | $10^{6}{\cal B}$ | $10^{2}\langle\alpha_{\theta_{\bf B}}\rangle$ | $10^{2}\langle\alpha_{\theta_{\ell}}\rangle$
---|---|---|---
$B^{-}\to p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ | $5.21\pm 0.34$ | $-6.51\pm 1.51$ | $-2.74\pm 0.40$
$B^{-}\to n\bar{n}\ell^{-}\bar{\nu}_{\ell}$ | $0.68\pm 0.10$ | $4.42\pm 1.66$ | $0.41\pm 0.95$
$B^{-}\to\Lambda^{0}\bar{\Lambda}^{0}\ell^{-}\bar{\nu}_{\ell}$ | $0.08\pm 0.01$ | $0.00$ | $0.00$
$B^{-}\to\Sigma^{+}\bar{\Sigma}^{+}\ell^{-}\bar{\nu}_{\ell}$ | $0.24\pm 0.02$ | $-6.91\pm 1.62$ | $-2.83\pm 0.50$
$B^{-}\to\Sigma^{0}\bar{\Sigma}^{0}\ell^{-}\bar{\nu}_{\ell}$ | $0.06\pm 0.01$ | $-6.91\pm 1.62$ | $-2.83\pm 0.49$
$B^{-}\to\Xi^{0}\bar{\Xi}^{0}\ell^{-}\bar{\nu}_{\ell}$ | $0.008\pm 0.001$ | $4.82\pm 1.85$ | $0.28\pm 0.81$
$B^{-}\to\Lambda^{0}\bar{\Sigma}^{0}\ell^{-}\bar{\nu}_{\ell}$ | $0.014\pm 0.004$ | $-5.65\pm 2.05$ | $-7.88\pm 0.64$
$B^{-}\to\Sigma^{0}\bar{\Lambda}^{0}\ell^{-}\bar{\nu}_{\ell}$ | $0.014\pm 0.004$ | $-5.65\pm 2.05$ | $-7.88\pm 0.64$
Table 5: Our results of $B^{-}\to p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ along with those in Ref. Geng:2011tr  and the data. | $10^{6}{\cal B}$ | $10^{2}\langle\alpha_{\theta_{\bf B}}\rangle$ | $10^{2}\langle\alpha_{\theta_{\ell}}\rangle$
---|---|---|---
Our results | $5.21\pm 0.34$ | $-6.51\pm 1.51$ | $-2.74\pm 0.40$
Ref. Geng:2011tr  | $104\pm 29$ | $6\pm 2$ | $59\pm 2$
LHCb ($\ell=\mu$) Aaij:2019bdu  | $5.27^{+0.23}_{-0.24}\pm 0.21\pm 0.15$ | - | -
Belle ($\ell=e$) Tien:2013nga  | $8.2^{+3.7}_{-3.2}\pm 0.6$ | - | -
Belle ($\ell=\mu$) Tien:2013nga  | $3.1^{+3.1}_{-2.4}\pm 0.7$ | - | -
Belle (Combined) Tien:2013nga  | $5.8^{+2.4}_{-2.1}\pm 0.9$ | - | -
Figure 4: Differential branching fraction of $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ as a function of the $p\bar{p}$ invariant
mass ($m(p\bar{p}$), where the red-solid line is our results, and the hollow
dots are the data from the LHCb measurements Aaij:2019bdu .

In Table 5, we summarize our results of $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ along with the previous theoretical ones
Geng:2011tr  as well as the experimental data Aaij:2019bdu ; Tien:2013nga . We
note that the theoretical calculations are insensitive to the lepton mass for
the $\ell=e$ and $\mu$ channels. As seen from Table 5, our new result of
${\cal B}(B^{-}\to p\bar{p}\ell^{-}\bar{\nu}_{\ell})=(5.21\pm 0.34)\times
10^{-6}$ is about one order of magnitude lower than the previous theoretical
prediction of $(10.4\pm 2.9)\times 10^{-5}$ in Ref. Geng:2011tr , but the
effective color number in $\bar{B}\to{\bf B}{\bf B}^{\prime}M$ channel have a
magnificent change. Our fitting result is consistent with the Belle
measurements of ${\cal B}(B^{-}\to
p\bar{p}e^{-}\bar{\nu}_{e})=(8.2^{+3.8}_{-3.3})\times 10^{-6}$ Tien:2013nga
and ${\cal B}(B^{-}\to
p\bar{p}\mu^{-}\bar{\nu}_{\mu})=(3.1^{+3.2}_{-2.5})\times 10^{-6}$
Tien:2013nga  and agrees well with the combined one of
$(5.8^{+2.6}_{-2.3})\times 10^{-6}$ by Belle Tien:2013nga  as well as the
recent $\mu$-channel data of $(5.3\pm 0.4)\times 10^{-6}$ by LHCb Aaij:2019bdu
which is one of our input channels. Clearly, more precise modeling and
explanation are needed to find the QCD origin of the effective color number
being $N_{c}^{eff}=0.51\pm 0.03$, indicating that the non-peturbative effects
in three-body $\bar{B}\to{\bf B}{\bf B}^{\prime}M$ channels is much stronger
than those in two-body $\bar{B}\to M_{1}M_{2}$ ones. In Fig. 4, we plot the
$p\bar{p}$ invariant mass spectrum in $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$. By comparing our results with the LHCb
measurement Aaij:2019bdu , we find that our spectrum is consistent with the
observed one. We further show the differential branching fractions of
$B^{-}\to p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ as functions of $\sqrt{s}\equiv
m_{\ell\bar{\nu}},~{}\cos_{\theta_{\bf B}}$ and $\cos_{\theta_{\ell}}$ in Fig.
5, respectively, which can provide us not only the information of the leptonic
sector but also the the spin-flavor relations in the $t\to\infty$ asymptotic
limit. These differential branching fractions could be tested by the ongoing
experiments.

.

Figure 5: Differential branching fractions of $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ as functions of $\ell\bar{\nu}$ invariant
mass, $\cos\theta_{\bf B}$ and $\cos\theta_{\ell}$, respectively.

##  IV Conclusions

We have systematically revisited the baryonic four-body semileptonic decays of
$B^{-}\to{\bf B}\bar{\bf B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ with
$\ell=e,\mu$. We have reduced the ten form factors in the hadronic transition
of $B^{-}\to{\bf B}\bar{\bf B}^{\prime}$ into three free parameters in the
heavy quark limit and $t\to\infty$. We have performed the minimum $\chi^{2}$
method to fit the three parameters and the effective color number of
$N_{c}^{eff}$ with $\chi^{2}/d.o.f=0.28$ by using five three-body decays of
$\bar{B}\to{\bf B}\bar{\bf B}^{\prime}M$ along with the $B^{-}\to
p\bar{p}\mu^{-}\bar{\nu}_{\mu}$ measurement. We have obtained a consistent
fitting result of ${\cal B}(B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell})=(5.21\pm 0.34)\times 10^{-6}$ as well as
other input channels. Our $B^{-}\to p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ decay
branching ratio is about one order of magnitude lower than the previous
theoretical prediction of $(10.4\pm 2.9)\times 10^{-5}$ in Ref. Geng:2011tr ,
and agrees well with the experimental data of $(5.8^{+2.6}_{-2.3})\times
10^{-6}$ and $(5.3\pm 0.4)\times 10^{-6}$ by Belle Tien:2013nga  and LHCb
Aaij:2019bdu , respectively. In addition, our evaluation of the $m_{p\bar{p}}$
invariant mass spectrum is also consistent with that by the LHCb measurement
Aaij:2019bdu , demonstrating that the threshold effect and the $t^{-3}$
dependence of the form factors from the QCD counting rules is still dominant
in the baryonic four-body semi-leptonic decays, while the other Lorentz
invariant variables, such as $(p_{\bar{B}}+p_{{\bf B}})^{2}$, as well as the
resonant states are highly suppressed. Furthermore, we have plotted the
differential branching fractions with respect to the kinematic variables of
$m_{\ell\bar{\nu}}$ and $\cos\theta_{{\bf B},\ell}$ in $B^{-}\to
p\bar{p}\ell^{-}\bar{\nu}_{\ell}$ to provide the information in the lepton
sector and angular distributions, respectively. We have also used the flavor
symmetry to explore the physical observables in other $B^{-}\to{\bf B}\bar{\bf
B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ decays. In particular, we have found that
that the angular asymmetries of
$B^{-}\to\Lambda\bar{\Lambda}\ell^{-}\bar{\nu}_{\ell}$ vanish due to the
absence of the chiral-flipping interaction ($e_{LR}=0$). On the theoretical
side, the non observed transition modes of the three-body ${\bar{B}}\to{\bf
B}{\bf B}^{\prime}M$ and four-body $\bar{B}\to{\bf B}{\bf
B}^{\prime}\ell\bar{\nu}$ decays can help us to relax the assumption of the
heavy quark limit once they are measured. Otherwise, the lattice simulation
would be currently the most trustworthy theoretical method to reliably extract
the hadronic form factors. On the experimental side, some of our results in
$B^{-}\to{\bf B}\bar{\bf B}^{\prime}\ell^{-}\bar{\nu}_{\ell}$ can be tested by
the ongoing experiments at Belle-II and LHCb. Finally, we remark that the
theoretical determination of the four-body decays of ${\bar{B}}\to{\bf B}{\bf
B}^{\prime}\ell\bar{\nu}$ would provide a valuable opportunity to search for T
violating effects from the triple momentum correlations and improve the
measurement of $|V_{ub}|$ as the works in exclusive $B$ and $\Lambda_{b}$
decays.

## Appendix

Starting with transition matrix elements of $\langle{\bf B}\bar{\bf
B}^{\prime}|J^{\mu}_{V}-J^{\mu}_{A}|\bar{B}\rangle$ with
$J_{V(A)}^{\mu}=\bar{q}\gamma^{\mu}(\gamma^{5})b$, we assume that the
$\bar{B}$ meson state can be approximately expressed by the field operator of
free quarks $|\bar{B}\rangle\sim\bar{b}\gamma^{5}q^{\prime}|0\rangle$.
Therefore, the matrix elements become

$\displaystyle\langle{\bf B}\bar{\bf
B}^{\prime}|J^{\mu}_{V}-J^{\mu}_{A}|\bar{B}(\bar{q}^{\prime}b)\rangle$
$\displaystyle\simeq$ $\displaystyle\langle{\bf B}\bar{\bf
B}^{\prime}|\bar{q}\gamma^{\mu}(1-\gamma^{5})b\bar{b}\gamma^{5}q^{\prime}|0\rangle$
(11) $\displaystyle=$ $\displaystyle\langle{\bf B}\bar{\bf
B}^{\prime}|\bar{q}\gamma^{\mu}(1-\gamma^{5})(\not{p}_{b}+m_{b})\gamma^{5}q^{\prime}|0\rangle$
$\displaystyle=$ $\displaystyle\langle{\bf B}\bar{\bf
B}^{\prime}|\bar{q}\gamma^{\mu}(1-\gamma^{5})(\not{p}_{b}-m_{b})q^{\prime}|0\rangle$
$\displaystyle=$ $\displaystyle 2\langle{\bf B}\bar{\bf
B}^{\prime}|\bar{q}_{L}\gamma^{\mu}\not{p}_{b}q_{R}^{\prime}|0\rangle-2m_{b}\langle{\bf
B}\bar{\bf B}^{\prime}|\bar{q}_{L}\gamma^{\mu}q_{L}^{\prime}|0\rangle$
$\displaystyle=$ $\displaystyle\langle{\bf B}\bar{\bf
B}^{\prime}|J^{\prime\mu}|0\rangle-\langle{\bf B}\bar{\bf
B}^{\prime}|\tilde{J}^{\mu}|0\rangle\,,$

where $J^{\prime\mu}=2\bar{q}_{L}\gamma^{\mu}\not{p}_{b}q_{R}^{\prime}$ and
$\tilde{J}^{\mu}=2m_{b}\bar{q}_{L}\gamma^{\mu}q_{L}^{\prime}$ with
$q_{L(R)}=(1\mp\gamma^{5})/2q$ and $\bar{q}_{L(R)}=\bar{q}(1\pm\gamma^{5})/2$.
Note that, by inserting QCD (gluon-quark-antiquark) vertices in the
corresponding diagrams, the Dirac structure in Eq. (11) could be altered.
However, because of the asymptotic freedom in QCD, we can treat these
alterations from QCD vertices as small perturbations, which are negligible in
the limit of $(p_{\bf B}+p_{{\bf B}^{\prime}})^{2}\to\infty$. In terms of the
crossing-symmetry (c.s.), the final state anti-baryon ($\bar{\bf B}^{\prime}$)
is transformed as the initial baryon (${\bf B}^{\prime}$) in the initial state
with opposite four-momentum $\tilde{p}_{{\bf B}^{\prime}}=-p_{\bar{\bf
B}^{\prime}}$, resulting in

$\displaystyle\langle{\bf B}(p_{\bf B})\bar{\bf B}^{\prime}(p_{\bar{\bf
B}^{\prime}})|J^{\prime\mu}(\tilde{J}^{\mu})|0\rangle\xrightarrow{c.s.}\langle{\bf
B}(p_{\bf B})|J^{\prime\mu}(\tilde{J}^{\mu})|{\bf B}^{\prime}(\tilde{p}_{{\bf
B}^{\prime}})\rangle.$ (12)

According to Refs. Chua:2002yd ; Brodsky:1980sx , the amplitude can be
parameterized as

$\displaystyle\langle{\bf B}|J^{\prime\mu}|{\bf B}^{\prime}\rangle$
$\displaystyle=$ $\displaystyle 2i\bar{u}_{\bf
B}\gamma^{\mu}\not{p}_{b}\left(\frac{1+\gamma^{5}}{2}F^{\prime+}+\frac{1-\gamma^{5}}{2}F^{\prime-}\right)u_{{\bf
B}^{\prime}}$ $\displaystyle=$ $\displaystyle 2i\bar{u}_{\bf
B}^{L}\gamma^{\mu}\not{p}_{b}u_{{\bf
B}^{\prime}}^{R}F^{\prime+}+2i\bar{u}_{\bf
B}^{R}\gamma^{\mu}\not{p}_{b}u_{{\bf B}^{\prime}}^{L}F^{\prime-}$
$\displaystyle\langle{\bf B}|\tilde{J}^{\mu}|{\bf B}^{\prime}\rangle$
$\displaystyle=$ $\displaystyle 2m_{b}i\bar{u}_{\bf
B}\gamma^{\mu}\left(\frac{1+\gamma^{5}}{2}\tilde{F}^{+}+\frac{1-\gamma^{5}}{2}\tilde{F}^{-}\right)u_{{\bf
B}^{\prime}}\,,$ (13) $\displaystyle=$ $\displaystyle 2m_{b}i\bar{u}_{\bf
B}^{R}\gamma^{\mu}u_{{\bf B}^{\prime}}^{R}\tilde{F}^{+}+2m_{b}i\bar{u}_{\bf
B}^{L}\gamma^{\mu}u_{{\bf B}^{\prime}}^{L}\tilde{F}^{-}\,.$

In the asymptotic limit of $(p_{\bf B}-\tilde{p}_{{\bf
B}^{\prime}})^{2}\to\infty$, the helicity of a particle can be approximately
treated as its chirality, so that the amplitudes with a specific chirality can
be written as

$\displaystyle\langle{\bf B},L|J^{\prime\mu}|{\bf B}^{\prime},R\rangle$
$\displaystyle=$ $\displaystyle 2i\bar{u}_{\bf
B}^{L}\gamma^{\mu}\not{p}_{b}\left(e_{LR}F_{LR}\right)u_{{\bf
B}^{\prime}}^{R}$ $\displaystyle\langle{\bf B},R(L)|\tilde{J}^{\mu}|{\bf
B}^{\prime},R(L)\rangle$ $\displaystyle=$ $\displaystyle 2m_{b}i\bar{u}_{\bf
B}^{R(L)}\gamma^{\mu}\left(e_{RR(LL)}F_{RR(LL)}\right)u_{{\bf
B}^{\prime}}^{R(L)}\,$ (14)

where

$\displaystyle e_{LR}$ $\displaystyle=$ $\displaystyle\langle{\bf
B},L|(a^{L}_{q})^{\dagger}a^{R}_{q^{\prime}}|{\bf B}^{\prime},R\rangle,$
$\displaystyle e_{RR}$ $\displaystyle=$ $\displaystyle\langle{\bf
B},R|(a^{L}_{q})^{\dagger}a^{L}_{q^{\prime}}|{\bf B}^{\prime},R\rangle,$
$\displaystyle e_{LL}$ $\displaystyle=$ $\displaystyle\langle{\bf
B},L|(a^{L}_{q})^{\dagger}a^{L}_{q^{\prime}}|{\bf B}^{\prime},L\rangle\,,$
(15)

with the corresponding particle creation and annihilation operators
$(a_{q}^{s})^{\dagger}$ and $a_{q}^{s}$, and other combinations are zero due
to the angular-momentum conservation. From Eqs. (13), (14) and (15), we find
that

$\displaystyle F^{\prime+}=e_{LR}F_{LR},\quad
F^{\prime-}=0,\quad\tilde{F}^{+}=e_{RR}F_{RR},\quad\tilde{F}^{-}=e_{LL}F_{LL}\,.$
(16)

Consequently, the transition amplitude in $(p_{\bf B}-\tilde{p}_{{\bf
B}^{\prime}})^{2}\to\infty$ is given by

$\displaystyle\langle{\bf B}|J^{\prime\mu}-\tilde{J}^{\mu}|{\bf
B}^{\prime}\rangle=2i\bar{u}_{\bf
B}\gamma^{\mu}\left(\not{p}_{b}\frac{1+\gamma^{5}}{2}e_{LR}F_{LR}-m_{b}\frac{1+\gamma^{5}}{2}e_{RR}F_{RR}-m_{b}\frac{1-\gamma^{5}}{2}e_{LL}F_{LL}\right)u_{{\bf
B}^{\prime}}\,.$

After applying the crossing-symmetry again, we get that

$\displaystyle\langle{\bf B}\bar{\bf B}^{\prime}|J^{\mu}_{V}|\bar{B}\rangle$
$\displaystyle=$ $\displaystyle i\bar{u}_{\bf
B}\gamma^{\mu}\left(\not{p}_{b}e_{LR}F_{LR}+m_{b}(e_{LL}F_{LL}-e_{RR}F_{RR})\right)\gamma^{5}v_{\bar{\bf
B}^{\prime}}$ $\displaystyle=$ $\displaystyle i\bar{u}_{\bf
B}\gamma^{\mu}\left(\not{p}_{\bar{B}}e_{LR}F_{LR}+m_{\bar{B}}(e_{LL}F_{LL}-e_{RR}F_{RR})\right)\gamma^{5}v_{\bar{\bf
B}^{\prime}}$ $\displaystyle\langle{\bf B}\bar{\bf
B}^{\prime}|J^{\mu}_{A}|\bar{B}\rangle$ $\displaystyle=$ $\displaystyle
i\bar{u}_{\bf
B}\gamma^{\mu}\left(-\not{p}_{b}e_{LR}F_{LR}+m_{b}(e_{LL}F_{LL}+e_{RR}F_{RR})\right)v_{\bar{\bf
B}^{\prime}}$ (18) $\displaystyle=$ $\displaystyle i\bar{u}_{\bf
B}\gamma^{\mu}\left(-\not{p}_{\bar{B}}e_{LR}F_{LR}+m_{\bar{B}}(e_{LL}F_{LL}+e_{RR}F_{RR})\right)v_{\bar{\bf
B}^{\prime}}\,.$

where we have used the approximations of $p_{b}\simeq p_{\bar{B}}$ and
$m_{b}\simeq m_{\bar{B}}$ in the heavy quark limit. With the help of equation
of motions

$\displaystyle\bar{u}_{\bf B}\not{p}_{\bf B}=\bar{u}_{\bf B}m_{\bf
B},\quad\not{p}_{\bar{\bf B}^{\prime}}v_{\bar{\bf B}^{\prime}}=-m_{\bar{\bf
B}^{\prime}}v_{\bar{\bf B}^{\prime}}\,,$ (19)

and the Dirac algebra,

$\displaystyle\gamma^{\mu}\not{p}_{\bar{B}}$ $\displaystyle=$
$\displaystyle\gamma^{\mu}\not{p}+\gamma^{\mu}(\not{p}_{\bf
B}+\not{p}_{\bar{\bf B}^{\prime}})=p^{\mu}-i\sigma^{\mu\nu}p_{\nu}+2p_{\bf
B}^{\mu}-\not{p}_{\bf B}\gamma^{\mu}+\gamma^{\mu}\not{p}_{\bar{\bf
B}^{\prime}}$ (20) $\displaystyle=$ $\displaystyle-\not{p}_{\bf
B}\gamma^{\mu}+\gamma^{\mu}\not{p}_{\bar{\bf
B}^{\prime}}-i\sigma^{\mu\nu}p_{\nu}+p^{\mu}+(p_{\bf B}-p_{{\bf
B}^{\prime}})^{\mu}+(p_{\bf B}+p_{\bar{\bf B}^{\prime}})^{\mu}\,,$

we finally obtain

$\displaystyle\langle{\bf B}\bar{\bf B}^{\prime}|J^{\mu}_{V}|\bar{B}\rangle$
$\displaystyle=$ $\displaystyle i\bar{u}_{\bf B}\gamma^{\mu}\left[-(m_{\bf
B}-m_{\bar{\bf
B}^{\prime}})e_{LR}F_{LR}+m_{\bar{B}}(e_{LL}F_{LL}-e_{RR}F_{RR})\right.$
$\displaystyle+$
$\displaystyle\left.e_{LR}F_{LR}(-i\sigma^{\mu\nu}p_{\nu}+p^{\mu}+(p_{\bf
B}-p_{{\bf B}^{\prime}})^{\mu}+(p_{\bf B}+p_{\bar{\bf
B}^{\prime}})^{\mu})\right]\gamma^{5}v_{\bar{\bf B}^{\prime}}\,,$
$\displaystyle\langle{\bf B}\bar{\bf B}^{\prime}|J^{\mu}_{A}|\bar{B}\rangle$
$\displaystyle=$ $\displaystyle i\bar{u}_{\bf B}\gamma^{\mu}\left[(m_{\bf
B}+m_{\bar{\bf
B}^{\prime}})e_{LR}F_{LR}+m_{\bar{B}}(e_{LL}F_{LL}-e_{RR}F_{RR})\right.$ (21)
$\displaystyle-$
$\displaystyle\left.e_{LR}F_{LR}(-i\sigma^{\mu\nu}p_{\nu}+p^{\mu}+(p_{\bf
B}-p_{{\bf B}^{\prime}})^{\mu}+(p_{\bf B}+p_{\bar{\bf
B}^{\prime}})^{\mu})\right]v_{\bar{\bf B}^{\prime}}\,,$

which clearly lead to

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle
m_{\bar{B}}\left(e_{LL}F_{LL}+e_{RR}F_{RR}\right)+\left(m_{\bf B}+m_{\bar{\bf
B}^{\prime}}\right)e_{LR}F_{LR}\,,$ $\displaystyle g_{1}$ $\displaystyle=$
$\displaystyle m_{\bar{B}}\left(e_{LL}F_{LL}-e_{RR}F_{RR}\right)-\left(m_{\bf
B}-m_{\bar{\bf B}^{\prime}}\right)e_{LR}F_{LR}\,,$ $\displaystyle f_{2}$
$\displaystyle=$ $\displaystyle-g_{2}=e_{LR}F_{LR},\quad
f_{i}=-g_{i}=-e_{LR}F_{LR}\,,\quad(i=3,4,5)\,.$ (22)

As a result, the constant parts of form factors, $C_{f_{i}(g_{i})}$,
$C_{RR(LL)}$ and $C_{LR}$, in Eq. (22) directly imply the relations in Eq.
(5).

## ACKNOWLEDGMENTS

This work was supported in part by MoST (MoST-107-2119-M-007-013-MY3).

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aainstitutetext: Indian Institute of Technology Kanpur, Kanpur 208016, INDIA.
bbinstitutetext: Institute of Physics, Sachivalaya Marg, Bhubaneshwar 751005,
INDIA.
ccinstitutetext: Homi Bhabha National Institute, Anushakti Nagar, Mumbai
400085, INDIA.
ddinstitutetext: Department of Physics, BITS-Pilani, K K Birla Goa Campus,
Zuarinagar, Goa-403726. INDIA.

# Scattering Amplitudes: Celestial and Carrollian

Arjun Bagchi b,c Shamik Banerjee d Rudranil Basu a and Sudipta Dutta
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>pilani.ac.in<EMAIL_ADDRESS>
###### Abstract

Recent attempts at the construction of holography for asymptotically flat
spacetimes have taken two different routes. Celestial holography, involving a
two dimensional (2d) CFT dual to 4d Minkowski spacetime, has generated novel
results in asymptotic symmetry and scattering amplitudes. A different
formulation, using Carrollian CFTs, has been principally used to provide some
evidence for flat holography in lower dimensions. Understanding of flatspace
scattering has been lacking in the Carroll framework. In this work, using
ideas from Celestial holography, we show that 3d Carrollian CFTs living on the
null boundary of 4d flatspace can potentially compute bulk scattering
amplitudes. 3d Carrollian conformal correlators have two different branches,
one depending on the null time direction and one independent of it. We propose
that it is the time-dependent branch that is related to bulk scattering. We
construct an explicit field theoretic example of a free massless Carrollian
scalar that realises some desired properties.

##  1 Introduction

The Holographic Principle has been one of our primary routes to a theory of
quantum gravity, formulated in terms of a lower dimensional field theory.
Although there has been a great deal of success in understanding holography in
Anti de Sitter spacetime through AdS/CFT, a similar understanding for the
apparently more straight-forward case of asymptotically flat spacetimes is
lacking. There is, however, a consorted recent effort at rectifying this
situation. There are two principal avenues of addressing this problem -
Celestial Holography and Carrollian Holography. Bondi-Metzner-Sachs (BMS)
Bondi:1962px  symmetries, which arise as asymptotic symmetries of flat
spacetimes based on the null boundary, are important to both approaches.

Celestial holography has grown out of the basic observation due to Strominger
Strominger:2013jfa  that in asymptotically flatspace time soft theorems for
$S$-matrix elements can be thought of as Ward identities for asymptotic
symmetries He:2014laa ; Strominger:2014pwa ; Kapec:2014opa ; Kapec:2016jld ;
He:2017fsb ; Ball:2019atb ; Kapec:2017gsg ; Fotopoulos:2019vac ;
Stieberger:2018onx . This correspondence is holographic in nature. The
fundamental claim of celestial holography is that there is a two dimensional
(2d) CFT on the celestial sphere which computes the scattering amplitudes for
processes taking place in the four dimensional asymptotically flat space time.
This computation is facilitated by writing the $S$-matrix in boost eigenstates
deBoer:2003vf ; Pasterski:2016qvg ; Pasterski:2017kqt ; Banerjee:2018gce ;
Banerjee:2018fgd  in which the Ward identities for asymptotic symmetries take
the well known form of Ward identities in a 2d CFT. This CFT is known as the
Celestial CFT 111Please see Appendix A for a brief review of celestial
amplitudes.. This approach to the flat space holography has already produced
many novel results about asymptotic symmetries and scattering amplitudes
Banerjee:2020zlg ; Banerjee:2020vnt ; Guevara:2021abz ; Strominger:2021lvk ;
Himwich:2021dau ; Pasterski:2022joy ; Jiang:2021csc ; Campiglia:2014yka ;
Banerjee:2021dlm ; Donnay:2020guq ; Freidel:2021ytz ; Freidel:2021dfs  in four
dimensions. The reader is pointed to the excellent recent reviews
Strominger:2017zoo ; Pasterski:2021rjz ; Raclariu:2021zjz  for more details on
Celestial holography.

Another school of thought has been the attempt to build duals of
asymptotitcally flat spacetime in terms of a one-dimensional lower field
theory that enjoys BMS symmetry. These field theories are conformal theories
living on the null boundary of spacetime and can be understood as Carroll
contractions of usual relativistic CFTs, which takes the speed of light $c$ to
zero Bagchi:2010zz ; Bagchi:2012cy . We shall call this approach Carroll
holography. The success of this formulation has principally been in the three
dimensional bulk and two dimensional field theories, where various checks have
been performed between the boundary and the bulk, including the matching of
entropy Bagchi:2012xr ; Barnich:2012xq ; Bagchi:2013qva , stress-tensor
correlations Bagchi:2015wna , entanglement entropy Bagchi:2014iea ;
Jiang:2017ecm ; Hijano:2017eii . Some other important advances are
Barnich:2012aw ; Bagchi:2012yk ; Duval:2014lpa ; Hartong:2015usd ;
Bagchi:2016geg ; Bagchi:2021qfe  and higher dimensional explorations include
Bagchi:2016bcd ; Ciambelli:2018wre ; Bagchi:2019xfx . Crucially, the
understanding of scattering processes has been lacking in this formulation.

In this paper, we will provide a bridge between the two formulations. We will
show that using BMS or Conformal Carroll symmetries in a 3d field theory
living on null infinity, one can formulate the scattering problem in 4d
asymptotically flat spacetimes. We will further demonstrate the plausibility
of our proposal by constructing an explicit realisation of Carrollian CFTs in
terms of a 3d massless Carroll scalar with some desired features.

##### Note added:

When this paper was being readied for submission, Donnay:2022aba  appeared on
the arXiv. Although both papers attempt to link Carroll and Celestial
holography, our approaches are complementary.

##  2 BMS and Carroll CFTs

As is now well known, and has been known since the 1960s, the symmetries of
interest in asymptotically flat spacetimes in $d=4$ actually extends beyond
the Poincare group to an infinite dimensional group discovered initially by
Bondi, van der Burgh, Metzner and Sachs Bondi:1962px . The BMS symmetry
algebra of 4d flat spacetime at its null boundary $\mathscr{I}^{\pm}$ is given
by:

$\displaystyle[L_{n},L_{m}]$
$\displaystyle=(n-m)L_{n+m},\quad[\bar{L}_{n},\bar{L}_{m}]=(n-m)\bar{L}_{n+m}$
(1a) $\displaystyle[L_{n},M_{r,s}]$
$\displaystyle=\left(\frac{n+1}{2}-r\right)M_{n+r,s},\quad[\bar{L}_{n},M_{r,s}]=\left(\frac{n+1}{2}-s\right)M_{r,n+s}$
(1b) $\displaystyle[M_{r,s},M_{t,u}]$ $\displaystyle=0.$ (1c)

Here $M_{r,s}$ are the generators of infinite dimensional angle dependent
translations at $\mathscr{I}^{\pm}$ known as supertranslations. The original
BMS group was given by these infinite dimensional supertranslations on top of
the usual Lorentz group denoted here by the generators $\\{L_{0},L_{\pm
1},\bar{L}_{0},\bar{L}_{\pm 1}\\}$. Following Barnich:2009se ; Barnich:2010eb
, there has been an effort to consider the full conformal group on the sphere
at infinity and hence all modes of the $L_{n}$ generators, the so-called
super-rotations222There exists other extensions e.g.Campiglia:2014yka ;
Compere:2018ylh ..

In 2d Celestial CFT superrotation or local conformal transformations on the
celestial sphere are generated by a stress tensor which is the shadow
transform of the subleading soft graviton Cachazo:2014fwa ; Kapec:2014opa ;
Fotopoulos:2019vac . After shadow transformation the subleading soft graviton
theorem Cachazo:2014fwa  becomes the well known Ward identity for stress
tensor in a 2d CFT.

Let us now discuss 3d Carrollian CFT. We are interested in defining a 3d
conformal field theory on $\mathscr{I}^{+}$ which is topologically
$\mathbb{R}_{u}\times\mathbb{S}^{2}$, where $\mathbb{R}_{u}$ is a null line
and $\mathbb{S}^{2}$ is the sphere at infinity. The null line makes the
induced metric of $\mathscr{I}^{+}$ degenerate. Hence the Riemannian structure
is replaced by a so-called Carrollian structures on the intrinsic geometries
of these hypersurfaces. CFTs living on $\mathscr{I}^{\pm}$ are naturally
expected to be invariant under the conformal isometries of these Carrollian
structures. We refer the reader to Appendix B for more details on Carrollian
and conformal Carrollian isometries. Rather intriguingly, conformal Carrollian
symmetries have been shown to be isomorphic to BMS symmetries in one higher
dimension Duval:2014lpa ; Bagchi:2010zz

$\mathfrak{CCarr}_{d}=\mathfrak{bms}_{d+1}.$ (2)

Hence a 3d Carrollian CFT naturally realises the extended infinite-dimensional
BMS4 symmetry. These 3d Carrollian CFTs would be our field theories of
interest, which we will show to be a potential candidate for a holographic
description of scattering amplitudes in 4d asymptotically flat spacetimes.

For these 3d theories, a particular useful representation of vector fields to
consider is Bagchi:2016bcd :

$\displaystyle
L_{n}=-z^{n+1}\partial_{z}-\frac{1}{2}(n+1)z^{n}u\partial_{u}\quad{\bar{L}}_{n}=-{\bar{z}}^{n+1}\partial_{z}-\frac{1}{2}(n+1){\bar{z}}^{n}u\partial_{u}\quad
M_{r,s}=z^{r}{\bar{z}}^{s}\partial_{u}$ (3)

Here $z,{\bar{z}}$ are stereographic coordinates on the sphere. We will label
the Carroll conformal fields $\Phi$ living on $\mathscr{I}^{+}$ with their
weights under $L_{0}$ and ${\bar{L}}_{0}$:

$[L_{0},\Phi(0)]=h\Phi(0),\quad[{\bar{L}}_{0},\Phi(0)]={\bar{h}}\Phi(0).$ (4)

We will assume the existence of Carrollian primary fields living on
$\mathscr{I}^{+}$. The primary conditions are Bagchi:2016bcd ;
Banerjee:2020kaa :

$[L_{n},\Phi(0)]=0,\quad[{\bar{L}}_{n},\Phi(0)]=0,\quad\forall
n>0,\quad[M_{r,s},\Phi(0)]=0,\quad\forall r,s>0.$ (5)

In particular, it is important to stress that the last condition is an
additional requirement on these fields, which is unlike a 2d CFT. Also for the
supertranslations, any one of $r$ or $s$ being greater than zero annihilates
the primary field. The transformation rules of the three dimensional
Carrollian primary fields $\Phi_{h,{\bar{h}}}(u,z,{\bar{z}})$ at an arbitrary
point on $\mathscr{I}^{+}$ under the infinitesimal BMS transformations are
given by

$\displaystyle\delta_{L_{n}}\Phi_{h,{\bar{h}}}(u,z,{\bar{z}})$
$\displaystyle=\epsilon\left[z^{n+1}\partial_{z}+(n+1)z^{n}\left(h+\frac{1}{2}u\partial_{u}\right)\right]\Phi_{h,{\bar{h}}}(u,z,{\bar{z}}),$
(6a) $\displaystyle\delta_{M_{r,s}}\Phi_{h,{\bar{h}}}(u,z,{\bar{z}})$
$\displaystyle=\epsilon
z^{r}{\bar{z}}^{s}\partial_{u}\Phi_{h,{\bar{h}}}(u,z,{\bar{z}}).$ (6b)

There is a similar relation for the antiholomorphic piece.

Let us now discuss how the structure of a Carrolian CFT that we have discussed
above fits into the framework of Celestial Holography.

##  3 Relation to 4d scattering amplitudes via Celestial Holography

As we have stressed above, one of the main reasons for studying Carrollian
CFTs is that its symmetries are the same as the extended BMS algebra. So
potentially Carroll CFTs can be a holographic dual of the quantum theory of
gravity in asymptotically flat spacetime. Now we know, from general
considerations, that the only observables in a quantum theory of gravity in
asymptotically flat space time are the $S$-matrix elements. Therefore, given a
holographic dual, one should be able to compute the spacetime $S$-matrix from
this. Moreover, if the dual is a field theory or at least looks like one then
presumably the $S$-matrix elements should be somehow related to the
correlation functions of the field theory. This is the point of view that we
adopt in this paper.

In the next section, we will focus on the correlation functions of the
Carrollian CFTs. We will find that there are two kinds of correlation
functions or two branches. In one branch, the correlation functions are
independent of the null time direction333The null direction of Carrollian
field theories can be interpreted as the time direction. One of the reasons
for this is that the Carroll limit from the relativistic theories involves a
contraction of the time direction. and have the structure of correlation
functions of a 2d CFT. However, in the other branch the correlators have
explicit (null) time dependence and do not look like those of a 2d CFT. For
example, unlike 2d CFT, the two point function in this branch is ultra-local
in the spatial directions and nonzero even when the scaling dimensions of the
operators are different. Similarly one can show using the 4d Poincare or
global Conformal Carroll invariance of the Carrollian CFT that the time
dependent three point function is zero. This problem can be solved if we treat
$z$ and $\bar{z}$ as independent complex coordinates rather than complex
conjugates of each other. These are reminiscent of the properties of
scattering amplitudes of massless particles in 4d flat spacetime. So what is
the relation of Carroll CFT correlations to scattering amplitudes? In this
paper we propose an answer.

In order to answer this question, we use ideas from Celestial holography. (For
a quick recap of the essential features of Celestial holography, the reader is
pointed to Appendix A). In Celestial holography the dual theory is conjectured
to be a 2d (relativistic) CFT which lives on the celestial sphere. The
important point for our purpose is that the correlation functions of the
celestial CFT are given by the Mellin transform of the 4d scattering
amplitudes deBoer:2003vf ; Pasterski:2016qvg ; Pasterski:2017kqt ;
Banerjee:2018gce ; Banerjee:2018fgd . Let us briefly describe this. For
simplicity let us consider only massless particles.

We parametrize the four momentum of a massless particle as

$p^{\mu}=\omega\left(1+z\bar{z},z+\bar{z},-i(z-\bar{z}),1-z\bar{z}\right),\
p^{\mu}p_{\mu}=0$ (7)

We also introduce a symbol $\epsilon$ which is equal to $\pm 1$ if the
particle is (outgoing) incoming.

Using this parametrization, the Mellin transformation can be written as
Pasterski:2016qvg ; Pasterski:2017kqt ,

$\mathcal{M}\left(\\{z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},\epsilon_{i}\\}\right)=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\omega_{i}^{\Delta_{i}-1}S\left(\\{\epsilon_{i}\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\}\right),\
\Delta\in\mathbb{C},\ \sigma\in\frac{\mathbb{Z}}{2}$ (8)

where $S$ is the $S$-matrix element for $n$ massless particle scattering. Here
we have also defined

$h=\frac{\Delta+\sigma}{2},\ \bar{h}=\frac{\Delta-\sigma}{2}$ (9)

One can show Pasterski:2016qvg ; Pasterski:2017kqt  using the Lorentz
transformation property of $S$-matrix that the object $\mathcal{M}$ on the LHS
indeed transforms like the correlation function of $n$ primary operators of
weight $(h,\bar{h})$ in a 2d CFT 444Note that $\sigma$ is the 4d helicity and
also the 2d spin of the corresponding operator.. After Mellin transformation
the coordinates $(z,\bar{z})$ can be interpreted as the stereographic
coordinates of the celestial sphere and physically represent the direction of
motion of the massless particle. For our purpose however, we will use a
modification Banerjee:2018gce ; Banerjee:2018fgd  of (8) such that the
correlation function $\mathcal{M}$ is now defined on a 3d space with
coordinates $(u,z,\bar{z})$. This space can be interpreted as the (future)
null-infinity with $u$ as the retarded time and $(z,\bar{z})$ as the
stereographic coordinates of the celestial sphere. One can show
Banerjee:2018gce ; Banerjee:2020kaa ; Banerjee:2020zlg ; Banerjee:2018fgd
that under supertranslation,

$u\rightarrow u^{\prime}=u+f(z,\bar{z}),\ z\rightarrow z^{\prime}=z,\
\bar{z}\rightarrow\bar{z}^{\prime}=\bar{z}$ (10)

Similarly under superrotation or local conformal transformations

$u\rightarrow
u^{\prime}=\left(\frac{dw}{dz}\right)^{\frac{1}{2}}\left(\frac{d\bar{w}}{d\bar{z}}\right)^{\frac{1}{2}}u,\
z\rightarrow z^{\prime}=w(z),\
\bar{z}\rightarrow\bar{z}^{\prime}=\bar{w}(\bar{z})$ (11)

Now the modified transformation Banerjee:2018gce ; Banerjee:2018fgd  has the
following form:

$\mathcal{\tilde{M}}\left(\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},\epsilon_{i}\\}\right)=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\omega_{i}^{\Delta_{i}-1}e^{-i\epsilon_{i}\omega_{i}u_{i}}S\left(\\{\epsilon_{i}\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\}\right),\
\Delta\in\mathbb{C}$ (12)

One can show Banerjee:2018gce ; Banerjee:2020kaa ; Banerjee:2020zlg ;
Banerjee:2018fgd  using the celebrated Soft Theorem - Ward Identity
correspondence Strominger:2013jfa ; He:2014laa ; Strominger:2014pwa ;
Kapec:2014opa ; Kapec:2016jld ; He:2017fsb ; Ball:2019atb ; Kapec:2017gsg ;
Fotopoulos:2019vac  that $\tilde{\mathcal{M}}$ transforms covariantly under
the extended BMS4 transformations. In Celestial holography the modified Mellin
transformation (12) is used to compute the graviton celestial amplitudes in
general relativity because the original Mellin transformation integral (8) is
not convergent due to bad UV behaviour of graviton scattering amplitudes in
GR. It turns out that instead when (12) is used the time coordinate $u$ acts
as a UV regulator and as a result $\mathcal{\tilde{M}}$ is finite. For more
details the reader is referred to Banerjee:2020zlg ; Banerjee:2020kaa ;
Banerjee:2019prz .

Now it is useful to write the modified celestial amplitude
$\tilde{\mathcal{M}}$ as a correlation function of fields defined on null
infinity. So following Banerjee:2018gce  we define

$\phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})=\int_{0}^{\infty}d\omega\
\omega^{\Delta-1}e^{-i\epsilon\omega u}a(\epsilon\omega,z,\bar{z},\sigma).$
(13)

where $a(\epsilon\omega,z,\bar{z},\sigma)$ is the momentum space (creation)
annihilation operator of a massless particle with helicity $\sigma$ when
$\left(\epsilon=-1\right)$ $\epsilon=1$. In terms of these fields we can write

$\mathcal{\tilde{M}}\left(\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},\epsilon_{i}\\}\right)=\langle\prod_{i=1}^{n}\phi^{\epsilon_{i}}_{h_{i},\bar{h}_{i}}(u_{i},z_{i},\bar{z}_{i})\rangle.$
(14)

Now, the field $\phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$ transforms
covariantly under the extended BMS4 transformation. Under superrotation
Banerjee:2018gce ; Banerjee:2020kaa ; Banerjee:2020zlg ; Banerjee:2018fgd

$\phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})\rightarrow\left(\frac{dw}{dz}\right)^{h}\left(\frac{d\bar{w}}{d\bar{z}}\right)^{\bar{h}}\phi^{\epsilon}_{h,\bar{h}}(u^{\prime},z^{\prime},\bar{z}^{\prime})$
(15)

where the primed coordinates are defined in (11). Similarly under
supertranslation,

$\phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})\rightarrow\phi^{\epsilon}_{h,\bar{h}}(u+f(z,\bar{z}),z,\bar{z})$
(16)

It is easy to see that for infinitesimal BMS4 transformations (15) and (16)
reduce to the equations (6) written in terms of the primaries of a Carrollian
CFT.

Therefore, it is not unreasonable to wonder whether one can identify the
Carrollian primaries with the primaries
$\phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$ of Celestial Holography. If this is
true then this will open the road towards connecting the Carrollian CFT
correlation functions with bulk scattering amplitudes because the field
$\phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$ is directly related to standard
creation-annihilation operators by (13).

##  4 The Proposal

###  4.1 The central claim

Our central claim in this paper is the following:

It is natural to identify the time-dependent correlation functions of
primaries in a Carrollian CFT with the modified Mellin amplitude

$\mathcal{\tilde{M}}\left(\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},\epsilon_{i}\\}\right)=\prod_{i}\langle\phi^{\epsilon_{i}}_{h_{i},\bar{h}_{i}}(u_{i},z_{i},\bar{z}_{i})\rangle.$

In other words, the time-dependent correlators of a 3d Carrollian CFT compute
the 4d scattering amplitudes in the Mellin basis.

We would like to emphasize that we are not saying that every Carrollian CFT
computes space-time scattering amplitude. But, if a specific Carrollian CFT
does so then it does it in the modified Mellin basis (12).

Now the reader might think that this identification is kinematical because
both the objects transform in the same way under relevant symmetries. While
this is correct, the dynamics enters non-trivially when we choose one of the
branches of the conformal Carrollian correlation functions.

Before we end this section we would like to emphasize few points. First of
all, Celestial holography, as it stands, requires the existence of an infinite
number of conformal primary fields with complex scaling dimensions. So any
Carrollian CFT which can compute 4d scattering amplitudes should also have
this feature.

The second point is regarding the symmetry group of a Carollian CFT. Over the
last few years, study of tree level massless scattering amplitudes using the
framework of celestial holography has revealed a much larger asymptotic
symmetry group than the extended BMS4. For example, the $SL_{2}$ current
algebra at level zero turns out to be a symmetry algebra Banerjee:2020zlg  of
tree level graviton scattering amplitudes. In fact it has been shown that the
$w_{1+\infty}$ is a symmetry algebra Guevara:2021abz ; Strominger:2021lvk ;
Himwich:2021dau ; Pasterski:2022joy ; Jiang:2021csc  for massless scattering
amplitudes. This also holds at the loop level in some special cases. Therefore
the asymptotic symmetry algebra for flat space time is expected to be far more
richer than the extended BMS4 algebra. The current Carrollian framework has to
be extended in order to accommodate these additional symmetries.

###  4.2 Correlation functions in Carrollian CFT and different branches

Having already revealed the main punchline of our paper, we now go ahead and
show the existence of two different branches of correlation functions for 3d
Carroll CFTs.

We are interested in computing the two point vacuum correlation functions of
primary fields in these 3d Carroll CFTs. We will see that just like
relativistic CFTs, it is possible to completely determine (upto constant
factors) the two and three-point functions using symmetry arguments. We demand
that the correlation functions are invariant under the Poincare subalgebra
($\\{M_{l,m},L_{n}\\}$ with $l,m=0,1$ and $n=0,\pm 1$) of the BMS4 algebra.
Consider the two point function

$G(u,z,\bar{z},u^{\prime},z^{\prime},\bar{z}^{\prime})=\langle
0|\Phi(u,z,\bar{z})\Phi^{\prime}(u^{\prime},z^{\prime},\bar{z}^{\prime})|0\rangle.$
(17)

Here $\Phi(u,z,\bar{z})$ and
$\Phi^{\prime}(u^{\prime},z^{\prime},\bar{z}^{\prime})$ are primaries with
weight $(h,h^{\prime})$ and $(\bar{h},\bar{h}^{\prime})$ respectively.
Invariance under Carroll time translations leads to

$\left(\frac{\partial}{\partial u}+\frac{\partial}{\partial
u^{\prime}}\right)G(u,z,\bar{z},u^{\prime},z^{\prime},\bar{z}^{\prime})=0$
(18)

Now considering Carroll boost invariance ($u\to u+bz+\bar{b}{\bar{z}}$)5553d
Carroll boosts are identical to spatial translations in 4d Minkowski
spacetime. For more details, see Appendices A, B. we get,

$\displaystyle\left(z\frac{\partial}{\partial
u}+z^{\prime}\frac{\partial}{\partial
u^{\prime}}\right)G(u,z,\bar{z},u^{\prime},z^{\prime},\bar{z}^{\prime})=0,\,\left(\bar{z}\frac{\partial}{\partial
u}+\bar{z}^{\prime}\frac{\partial}{\partial
u^{\prime}}\right)G(u,z,\bar{z},u^{\prime},z^{\prime},\bar{z}^{\prime})=0.$
(19)

So combining the above equations we get,

$\displaystyle\left(z-z^{\prime}\right)\frac{\partial}{\partial
u}G(u-u^{\prime},z-z^{\prime},\bar{z}-\bar{z}^{\prime})=0,\,\left(\bar{z}-\bar{z}^{\prime}\right)\frac{\partial}{\partial
u}G(u-u^{\prime},z-z^{\prime},\bar{z}-\bar{z}^{\prime})=0.$ (20)

These equations have two independent solutions that give rise to two different
classes of correlators, which has been referred to in the section above666This
was noticed independently in Chen:2021xkw ..

The first class of correlators corresponds to the choice of solution

$\frac{\partial}{\partial
u}G(u-u^{\prime},z-z^{\prime},\bar{z}-\bar{z}^{\prime})=0$ (21)

Using invariances under the subalgebra $\\{L_{0,\pm 1},{\bar{L}}_{0,\pm 1}\\}$
of BMS4, the above gives rise to standard 2d CFT 2-point correlation function
Bagchi:2016bcd :

$G(u,z,\bar{z},u^{\prime},z^{\prime},\bar{z}^{\prime})=\frac{\delta_{h,h^{\prime}}\delta_{{\bar{h}},{\bar{h}}^{\prime}}}{(z-z^{\prime})^{2h}(\bar{z}-\bar{z}^{\prime})^{2\bar{h}}}.$
(22)

For our discussions in this paper, we will not be interested in this
particular branch. The second class of solution corresponds to the choice

$\frac{\partial}{\partial
u}G(u-u^{\prime},z-z^{\prime},\bar{z}-\bar{z}^{\prime})\propto\delta^{2}\left(z-z^{\prime}\right)$
(23)

This gives rise to correlation functions in which we are interested
Banerjee:2018gce . Thus,

$G(u,z,\bar{z},u^{\prime},z^{\prime},\bar{z}^{\prime})=f(u-u^{\prime})\delta^{2}(z-z^{\prime}).$
(24)

We will call this the delta-function branch777In our notation
$\delta^{2}(z)=\delta(x)\delta(y).$. By demanding invariance under the
subalgebra $\\{L_{0,\pm 1},{\bar{L}}_{0,\pm 1}\\}$ of BMS4, it is
straightforward to show

$(\Delta+\Delta^{\prime}-2)f(u-u^{\prime})+(u-u^{\prime})\partial_{u}f(u-u^{\prime})=0,\quad(\sigma+\sigma^{\prime})f(u-u^{\prime})=0.$
(25)

Here $\Delta=(h+\bar{h})$ is the scaling dimension and $\sigma=(h-\bar{h})$ is
spin. The solution of the above equations is

$f(u-u^{\prime})=C\delta_{\sigma+\sigma^{\prime},0}(u-u^{\prime})^{-(\Delta+\Delta^{\prime}-2)},$
(26)

where $C$ is a constant factor. Hence

$G(u,z,\bar{z},u^{\prime},z^{\prime},\bar{z}^{\prime})=\frac{C\,\delta^{2}(z-z^{\prime})}{(u-u^{\prime})^{\Delta+\Delta^{\prime}-2}}\delta_{\sigma+\sigma^{\prime},0}.$
(27)

Once this correlator has this form (27), the equation which imposes $M_{11}$
or the transformation $u\to u+\epsilon z{\bar{z}}$ is trivially satisfied.

Notice that very unlike a relativistic CFT 2 point function, here one does not
have to have equal scaling dimensions for the fields to get a non-zero answer.
Thus this branch cannot be accessed by taking a limit from relativistic CFT
correlation functions.

Let us now discuss how one can obtain the same two point function by modified
Mellin transformation (12) of scattering amplitudes Banerjee:2018gce . Of
course in the case of two point function the scattering amplitude is trivial
and is given by the inner product

$\langle{p_{1},\sigma_{1}}|{p_{2},\sigma_{2}}\rangle=(2\pi)^{3}2E_{p_{1}}\delta^{3}\left(\vec{p}_{1}-\vec{p}_{2}\right)\delta_{\sigma_{1}+\sigma_{2},0}$
(28)

Here the notation is standard except that we label the helicity of an external
particle as if it were an outgoing particle. Now using the parametrization (7)
we can write,

$\langle{p_{1},\sigma_{1}}|{p_{2},\sigma_{2}}\rangle=4\pi^{3}\frac{\delta\left(\omega_{1}-\omega_{2}\right)\delta^{2}\left(z_{1}-z_{2}\right)}{\omega_{1}}\delta_{\sigma_{1}+\sigma_{2},0}$
(29)

Now the Mellin transformed two point function is given by

$\displaystyle\mathcal{\tilde{M}}\left(u_{1},z_{1},\bar{z}_{1},u_{2},z_{2},\bar{z}_{2},h_{1},\bar{h}_{1},h_{2},\bar{h}_{2},\epsilon_{1}=1,\epsilon_{2}=-1\right)$
$\displaystyle=4\pi^{3}\delta_{\sigma_{1}+\sigma_{2},0}\int_{0}^{\infty}d\omega_{1}\int_{0}^{\infty}d\omega_{2}\omega_{1}^{\Delta_{1}-1}\omega_{2}^{\Delta_{2}-1}e^{-i\omega_{1}u_{1}}e^{i\omega_{2}u_{2}}\frac{\delta\left(\omega_{1}-\omega_{2}\right)\delta^{2}\left(z_{1}-z_{2}\right)}{\omega_{1}}$
$\displaystyle=4\pi^{3}\Gamma\left(\Delta_{1}+\Delta_{2}-2\right)\frac{\delta^{2}\left(z_{1}-z_{2}\right)}{\left(i\left(u_{1}-u_{2}\right)\right)^{\Delta_{1}+\Delta_{2}-2}}\delta_{\sigma_{1}+\sigma_{2},0}$
(30)

We can see that, modulo the constant normalization, this has the same
structure as the time dependent two point function of the Carrollian CFT.

More importantly we can see that the presence of the spatial delta function
$\delta^{2}(z_{1}-z_{2})$ in the Carrollian two point function has the (dual)
physical interpretation that the momentum direction of a free particle in the
bulk space time does not change.

In the same way following Banerjee:2018gce  one can show that the time
dependent three point function in the Carrollian CFT is zero. This has the
physical interpretation that in Minkowski signature the scattering amplitude
of three massless particles vanish due to energy momentum conservation.

Therefore, the peculiarities of the time dependent correlation functions of a
Carrollian CFT are precisely what we need to connect to the spacetime
scattering amplitudes of massless particles.

This is the main message of our paper.

##  5 Massless Carrollian scalar field

In this section, our primary goal is to provide a concrete example of a 3d
quantum field theory which is invariant under the BMS4 algebra and give us the
correlation functions in the delta-function branch. We will now focus on a
particular simple example, that of a massless Carroll scalar field.

### The Action

The minimally coupled massless scalar field on Carrollian backgrounds is
described by

$\mathcal{S}=\int
dud^{2}x^{i}\,\tau^{\mu}\tau^{\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi$ (31)

where $\tau^{\mu}$ are vector fields defined in Eq. (60). We shall work with
flat Carroll background by fixing $\tau^{\mu}=(1,0)$ and $g_{ij}=\delta_{ij}$.
On flat Carroll background the action takes the simple form

$\mathcal{S}=\int dud^{2}x^{i}\ \frac{1}{2}(\partial_{u}{\Phi})^{2}$ (32)

The two dimensional version of this action (31) has been extensively used to
study the tensionless limit of string theory where the BMS3 algebra replaces
the two copies of the Virasoro algebra as worldsheet symmetries (see e.g.
Bagchi:2015nca ). We will see that here this simple 3d action carries the
seeds of a potential dual formulation of 4d gravity in asymptotically flat
spacetimes. The above action (31) also arises as the leading piece in the
Carroll expansion (speed of light $c\to 0$) of the action of the free
relativistic massless scalar field theory in (2+1) dimensions.

### Two-point function of the Carroll scalar

Now we turn our attention to the construction of the two point correlation
function of the Carroll scalar. We shall do this in two ways, first by a
Green’s function method and then by canonical analysis. We will see that we
will end up on the delta-function branch elucidated by our general analysis in
the previous sections.

#### Green’s function

We first compute the two point correlation function of the massless Carroll
scalar by computing the Green’s functions. The Green’s function equation for
this theory would be

$\partial^{2}_{u}G(u-u,z^{i}-z^{\prime
i})=\delta^{3}(u-u^{\prime},z^{i}-z^{\prime i}).$ (33)

This equation can be solved in usual way by going to Fourier space where this
takes the form

$\tilde{G}(k_{u},k_{i})=-\frac{1}{k_{u}^{2}}.$ (34)

Transforming back into position space yields

$\displaystyle G(u-u^{\prime},z^{i}-z^{\prime i})$
$\displaystyle=-\int\frac{dk_{u}}{k_{u}^{2}+\mu^{2}}e^{ik_{u}(u-u^{\prime})}\int
d^{2}z\ e^{ik_{i}(z^{i}-z^{\prime i})}$
$\displaystyle=\frac{i}{2}\left[\frac{1}{\mu}-(u-u^{\prime})\right]\delta^{(2)}(z^{i}-z^{\prime
i}).$ (35)

As the equation of motion does not have any spatial derivatives this integral
diverges. We regulate this integral by throwing away the troublesome infinite
piece:

$G(u-u^{\prime},z^{i}-z^{\prime
i})=-\frac{i}{2}(u-u^{\prime})\delta^{2}(z-z^{\prime},\bar{z}-\bar{z}^{\prime}))$
(36)

For scalar fields, the spin $\sigma=0$ and conformal weights of
$\Phi(u,z,\bar{z})$ are (e.g. from the action (32))

$h=\frac{1}{4},\quad\bar{h}=\frac{1}{4}.$ (37)

Hence the answer obtained by the Green’s function method is in perfect
agreement with the 2-point function derived from symmetry arguments in the
earlier section.

#### Canonical approach

We will now rederive the scalar two point correlation function by taking
recourse to cannonical quantisation. To proceed we will put the free scalar
theory on the round sphere and then take radius to infinity limit to recover
our answer in the plane coordinates. The scalar field action on a manifold
with topology $\mathbb{R}\times\mathbb{S}^{2}$ would be

$\mathcal{S}=\int
dud^{2}z\sqrt{q}\left[\frac{1}{2}(\partial_{u}\Phi)^{2}-k^{2}\Phi^{2}\right]$
(38)

Here $k$ is related to the radius of the sphere $R$ by $k=\frac{1}{2R}$. The
metric on the sphere is denoted by $q_{ij}$ and is given by (64). The Euler-
Lagrange equations of motion for this action is

$\ddot{\Phi}+k^{2}\Phi^{2}=0$ (39)

Generic real solutions are given by:

$\Phi(u,z,\bar{z})=\frac{1}{\sqrt{k}}\left(C^{\dagger}(z,\bar{z})e^{iku}+C(z,\bar{z})e^{-iku}\right).$
(40)

The canonical commutation relation between the $C$ fields are and the
Hamiltonian are respectively:

$\displaystyle[C(z,\bar{z}),C^{\dagger}(z^{\prime},\bar{z}^{\prime})]=\frac{1}{2}\delta^{2}(z-z^{\prime}),$
(41) $\displaystyle H=k\int
d^{2}z\sqrt{q}\left(2C^{\dagger}(z,\bar{z})C(z,\bar{z})+\frac{1}{2}\delta^{2}(0)\right).$
(42)

The Hamiltonian has the unphysical zero point energy in the form of the
$\delta^{2}(0)$. The physical part of the Hamiltonian then implies that the
time translation symmetric ground state should be annihilated by $C$:

$C(z,\bar{z})|0\rangle=0,\,\mbox{ for }\,(z,\bar{z})\in\mathbb{S}^{2}.$ (43)

It is therefore straightforward to calculate the 2-point function (keeping in
mind that there are no zero modes):

$\displaystyle G(u,u^{\prime},z^{i},z^{\prime i})=\langle
0|T\Phi(u,z,\bar{z})\Phi(u^{\prime},z^{\prime},\bar{z}^{\prime})|0\rangle.$
(44)

Now, taking $u>u^{\prime}$, we obtain

$\displaystyle G(u,u^{\prime},z^{i},z^{\prime i})=-\frac{1}{2k}[\cos
k(u-u^{\prime})+i\sin
k(u-u^{\prime})]\delta^{2}(z-z^{\prime},\bar{z}-\bar{z}^{\prime})$ (45)

So in the limit $R\to\infty$ , or $k\to 0$ , the two point function becomes

$G(u,u^{\prime},z^{i},z^{\prime
i})=-\left[\frac{1}{2k}+\frac{i}{2}(u-u^{\prime})\right]\delta^{2}(z-z^{\prime},\bar{z}-\bar{z}^{\prime})$
(46)

We read of the scaling dimension of $\Phi$ as $\Delta=1/2$, since this is
spin-less, the physical part of the 2 point function (46) matches exactly with
the one computed using symmetry arguments.

##  6 Discussions

In this paper, we have provided evidence that the correlation functions of 3d
Carrollian CFTs encode scattering amplitudes for 4d asympotically flat
spacetimes, specifically in the Mellin basis. The Carroll correlators have two
distinct branches and one of these, the one with explicit Carroll time
dependence which we called the delta-function branch, was the one relevant for
the connection to flat space scattering.

There are a number of intriguing questions that arise from our considerations
in this paper. Originally, the version of flat space holography that was
envisioned with the connections with Carroll CFTs was one which emerged as a
systematic limit from AdS/CFT. It is clear that the correlation functions that
we focused on in this work cannot emerge as a Carroll limit from standard
relativistic 3d CFT correlators in position space, since e.g. the CFT 2-point
function would vanish for unequal weight primaries and in the time-dependent
Carroll branch, this does not happen. Hence it would seem that the formulation
of Carroll holography we would require for connection to scattering amplitudes
would be disconnected from AdS/CFT. While this makes sense because flat space
and AdS are fundamentally different, how this fits in with e.g. the programme
of attempting to find flat space correlations from AdS/CFT (see e.g.
Giddings:1999jq ; Gary:2009mi ; Raju:2012zr ), remains to be seen.

Our construction and specifically the emergence of two different branches of
correlation functions is also reminiscent of recent advances in the
tensionless regime of string theory where three distinct quantum theories
appear from a single classical theory Bagchi:2020fpr ; Bagchi:2021rfw . Recent
findings of different correlation functions in these theories Sud ;
Hao:2021urq  is an indication that perhaps there is an interesting non-trivial
quantum vacuum structure underlying the Carrollian theories we have discussed
in our work.

It would be of interest to figure out what, if any, is relation of the 2d CFT
branch of the Carroll correlation functions to 4d flatspace physics. We would
also like to understand if it is possible to construct explicit examples of 3d
Carroll CFTs exhibiting 2d CFT correlation functions.

As mentioned before, in order to reproduce flatspace scattering, as in
Celestial CFTs, Carroll CFTs need an infinite number of primary fields with
complex scaling dimension. Moreover, the celestial CFT enjoys a much larger
symmetry Banerjee:2020zlg ; Banerjee:2020vnt ; Guevara:2021abz ;
Strominger:2021lvk ; Himwich:2021dau ; Pasterski:2022joy ; Jiang:2021csc ;
Campiglia:2014yka ; Banerjee:2021dlm ; Donnay:2020guq ; Freidel:2021ytz ;
Freidel:2021dfs  than the extended BMS4 algebra and these additional
symmetries play a central role in holographic computation of scattering
amplitudes. We would ideally like to construct an explicit example of such a
theory to provide a concrete proposal between a gravitational theory in
asymptotically flat spacetimes and a Carroll CFT with additional symmetries
living on its null boundary. But it is obvious that this is presently a
distant goal.

### Acknowledgements

We thank Kinjal Banerjee and Amartya Saha for discussions and Daniel Grumiller
for helpful correspondence.

AB and SB are partially supported by Swarnajayanti Fellowships of the
Department of Science and Technology, India. AB is also partially supported by
Science and Engineering Research Board (SERB) grants SB/SJF/2019-20/08,
CRG/2020/002035. The work of SB is also partially supported by SERB grant
MTR/2019/000937 (Soft-Theorems, S-matrix and Flat-Space Holography). RB is
also supported by the following grants from the SERB: CRG/2020/002035,
SRG/2020/001037. SD is supported by grant number 09/092(0971)/2017-EMR-I from
Council of Scientific and Industrial Research (CSIR).

## APPENDICES

##  Appendix A Brief review of Celestial or Mellin amplitudes for massless
particles

In this appendix, we provide a lightning review of Celestial amplitudes for
massless particles for ready reference. The Celestial or Mellin amplitude for
massless particles in four dimensions is defined as the Mellin transformation
of the $S$-matrix element, given by Pasterski:2016qvg ; Pasterski:2017kqt

$\mathcal{M}_{n}\big{(}\\{z_{i},\bar{z}_{i},h_{i},\bar{h}_{i}\\}\big{)}=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\
\omega_{i}^{\Delta_{i}-1}S_{n}\big{(}\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\}\big{)}$
(47)

where $\sigma_{i}$ denotes the helicity of the $i$-th particle and the on-
shell momenta are parametrized as,

$p_{i}=\omega_{i}(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},-i(z_{i}-\bar{z}_{i}),1-z_{i}\bar{z}_{i}),\quad
p_{i}^{2}=0$ (48)

The scaling dimensions $(h_{i},\bar{h}_{i})$ are defined as,

$h_{i}=\frac{\Delta_{i}+\sigma_{i}}{2},\quad\bar{h}_{i}=\frac{\Delta_{i}-\sigma_{i}}{2}$
(49)

The Lorentz group $SL(2,\mathbb{C})$ acts on the celestial sphere as the group
of global conformal transformations and the Mellin amplitude $\mathcal{M}_{n}$
transforms as,

$\mathcal{M}_{n}\big{(}\\{z_{i},\bar{z}_{i},h_{i},\bar{h}_{i}\\}\big{)}=\prod_{i=1}^{n}\frac{1}{(cz_{i}+d)^{2h_{i}}}\frac{1}{(\bar{c}\bar{z}_{i}+\bar{d})^{2\bar{h}_{i}}}\mathcal{M}_{n}\bigg{(}\frac{az_{i}+b}{cz_{i}+d}\
,\frac{\bar{a}\bar{z}_{i}+\bar{b}}{\bar{c}\bar{z}_{i}+\bar{d}}\
,h_{i},\bar{h}_{i}\bigg{)}$ (50)

This is the familiar transformation law for the correlation function of
primary operators of weight $(h_{i},\bar{h}_{i})$ in a $2$-D CFT under the
global conformal group $SL(2,\mathbb{C})$. In Einstein gravity, the Mellin
amplitude as defined in (47) usually diverges. This divergence can be
regulated by defining a modified Mellin amplitude as Banerjee:2018gce ;
Banerjee:2018fgd ,

$\mathcal{M}_{n}\big{(}\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i}\\}\big{)}=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\
\omega_{i}^{\Delta_{i}-1}e^{-i\sum_{i=1}^{n}\epsilon_{i}\omega_{i}u_{i}}S_{n}\big{(}\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\}\big{)}$
(51)

where $u$ can be thought of as a time coordinate and $\epsilon_{i}=\pm 1$ for
an outgoing (incoming) particle. Under (Lorentz) conformal tranansformation
the modified Mellin amplitude $\mathcal{M}_{n}$ transforms as,

$\mathcal{M}_{n}\big{(}\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i}\\}\big{)}=\prod_{i=1}^{n}\frac{1}{(cz_{i}+d)^{2h_{i}}}\frac{1}{(\bar{c}\bar{z}_{i}+\bar{d})^{2\bar{h}_{i}}}\mathcal{M}_{n}\bigg{(}\frac{u_{i}}{|cz_{i}+d|^{2}}\
,\frac{az_{i}+b}{cz_{i}+d}\
,\frac{\bar{a}\bar{z}_{i}+\bar{b}}{\bar{c}\bar{z}_{i}+\bar{d}}\
,h_{i},\bar{h}_{i}\bigg{)}$ (52)

Under global space-time translation, $u\rightarrow
u+A+Bz+\bar{B}\bar{z}+Cz\bar{z}$, the modified amplitude is invariant, i.e,

$\mathcal{M}_{n}\big{(}\\{u_{i}+A+Bz_{i}+\bar{B}\bar{z}_{i}+Cz_{i}\bar{z}_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i}\\}\big{)}=\mathcal{M}_{n}\big{(}\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i}\\}\big{)}$
(53)

Now in order to make manifest the conformal nature of the dual theory living
on the celestial sphere it is useful to write the (modified) Mellin amplitude
as a correlation function of conformal primary operators. So let us define a
generic conformal primary operator as,

$\phi^{\epsilon}_{h,\bar{h}}(z,\bar{z})=\int_{0}^{\infty}d\omega\
\omega^{\Delta-1}a(\epsilon\omega,z,\bar{z},\sigma)$ (54)

where $\epsilon=\pm 1$ for an annihilation (creation) operator of a massless
particle of helicity $\sigma$. Under (Lorentz) conformal transformation the
conformal primary transforms like a primary operator of scaling dimension
$(h,\bar{h})$

$\phi^{\prime\epsilon}_{h,\bar{h}}(z,\bar{z})=\frac{1}{(cz+d)^{2h}}\frac{1}{(\bar{c}\bar{z}+\bar{d})^{2\bar{h}}}\mathcal{\phi}^{\epsilon}_{h,\bar{h}}\bigg{(}\frac{az+b}{cz+d}\
,\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\bigg{)}$ (55)

Similarly in the presence of the time coordinate $u$ we have,

$\phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})=\int_{0}^{\infty}d\omega\
\omega^{\Delta-1}e^{-i\epsilon\omega u}a(\epsilon\omega,z,\bar{z},\sigma)$
(56)

Under (Lorentz) conformal transformations

$\phi^{\prime\epsilon}_{h,\bar{h}}(u,z,\bar{z})=\frac{1}{(cz+d)^{2h}}\frac{1}{(\bar{c}\bar{z}+\bar{d})^{2\bar{h}}}\mathcal{\phi}^{\epsilon}_{h,\bar{h}}\bigg{(}\frac{u}{|cz+d|^{2}},\frac{az+b}{cz+d}\
,\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\bigg{)}$ (57)

In terms of (54), the Mellin amplitude can be written as the correlation
function of conformal primary operators

$\mathcal{M}_{n}=\bigg{\langle}{\prod_{i=1}^{n}\phi^{\epsilon_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}$
(58)

Similarly using (56), the modified Mellin amplitude can be written as,

$\mathcal{M}_{n}=\bigg{\langle}{\prod_{i=1}^{n}\phi^{\epsilon_{i}}_{h_{i},\bar{h}_{i}}(u_{i},z_{i},\bar{z}_{i})}\bigg{\rangle}$
(59)

##  Appendix B Conformal Carroll symmetry

We will briefly summarize the main points relating to Conformal Carroll
symmetry in this appendix.

### Geometric structures

Throughout the paper, we have been interested in defining quantum field
theories that live on the null boundary $\mathscr{I}^{\pm}$ of asymptotically
flat spacetimes. The null boundary is topologically
$\mathbb{R}_{u}\times\mathbb{S}^{2}$ with $\mathbb{R}_{u}$ being null. The
induced metric of $\mathscr{I}^{\pm}$ is degenerate and Carroll structures
replace usual Riemann structures on it. Carrollian manifolds are endowed with
a degenerate, twice symmetric metric $g_{\mu\nu}$ and its kernel vector field
$\tau^{\mu}$. The isometry algebra of this structure

$\mathcal{L}_{\xi}\tau^{\mu}=0,\quad\mathcal{L}_{\xi}g_{\mu\nu}=0,\quad\tau^{\mu}g_{\mu\nu}=0$
(60)

generates the Carroll algebra, which for flat Carroll manifolds reduces to the
group that is obtained from the Poincare group by sending the speed of light
to zero Bagchi:2016bcd . We shall see the algebra and the contraction below.

We have dealt with CFTs on these Carroll backgrounds. These theories are
naturally expected to be invariant under the conformal isometries of these
Carrollian structures. Conformal Killing equations on these background are

$\displaystyle\mathcal{L}_{\xi}g_{\mu\nu}=\lambda
g_{\mu\nu},\quad\mathcal{L}_{\xi}\tau^{\mu}=-\frac{\lambda}{N}\tau^{\mu}$ (61)

The so called dynamical exponent $N$ encapsulates the relative scaling of
space and time directions. For $N=2$, the space and time dilates
homogeneously. This is the case that we will be interested, as we wish to
connect the conformal Carroll structures to the asymptotic structure of flat
spacetimes, where of course space and time scale in the same way. In (2+1)
dimensions, the set of vector fields that solves the above equation for $N=2$
is

$\xi=\left[\alpha(x^{i})+\frac{u}{2{\sqrt{q}}}\partial_{i}({\sqrt{q}}f^{i}(x^{j}))\right]\partial_{u}+f^{i}(x^{j})\partial_{i}$
(62)

Here $q$ is the determinant of the metric $q_{ij}$ on $\mathbb{S}^{2}$ and
$\alpha(x^{i})$ is an arbitrary function of $x^{i}$, but $f^{i}(x^{i})$ need
to satisfy the following conformal Killing equation on $\mathbb{S}^{2}$:

$\displaystyle\mathcal{L}_{f}q_{ij}=D_{k}f^{k}\,q_{ij}$ (63)

$D$ is the connection compatible with $q_{ij}$. Choosing the stereographic
coordinates $(z,\bar{z})$ on 2 sphere, such that

$ds^{2}=\dfrac{2dz\,d\bar{z}}{(1+z\bar{z})^{2}},$ (64)

the above equation for the components of $f$ is solved by holomorphic and
anti-holomorphic functions, i.e $f^{z}\equiv f^{z}(z)$ and $f^{\bar{z}}\equiv
f^{\bar{z}}(\bar{z})$. The algebra of these vector fields (62) is clearly
infinite dimensional and interestingly closes to form the BMS4 algebra (1)
Duval:2014lpa .

The connection between Carrollian conformal symmetries and BMS symmetries for
arbitrary dimensions was clarified in Duval:2014lpa , following closely
related observations in Bagchi:2010zz .

### Carroll contractions of Conformal algebra

The Carrollian limit of a relativistic CFT is reached by performing an
Inönü–Wigner contraction on the relativistic conformal generators. The
corresponding contraction of the spacetime coordinates for a 3d CFT is
described as

$x_{i}\to x_{i},\qquad t\to\epsilon\,t,\qquad\epsilon\to
0\,\quad\text{with}\quad i=1,2.$ (65)

This is equivalent to taking the speed of light, $c\to 0$. Conformal Carroll
generators are obtained from their relativistic counterparts in this limit:

$\displaystyle H$ $\displaystyle=\partial_{t}$ $\displaystyle B_{i}$
$\displaystyle=-x_{i}\partial_{t}$ $\displaystyle K_{i}$
$\displaystyle=-2x_{j}(t\partial_{t}+x_{i}\partial_{i})+x_{j}x_{j}\partial_{i}$
$\displaystyle K$ $\displaystyle=x_{i}x_{i}\partial_{t}$ (66a) $\displaystyle
D$ $\displaystyle=-(t\partial_{t}+x_{i}\partial_{i})$ $\displaystyle P_{i}$
$\displaystyle=\partial_{i}$ $\displaystyle J_{ij}$
$\displaystyle=-(x_{i}\partial_{j}-x_{j}\partial_{i})\,.$ (66b)

These generate the 3d finite Conformal Carrollian algebra which is $iso(3,1)$
and hence isomorphic to the Poincare group in $d=4$ Bagchi:2016bcd . The
contracted algebra is given by

$\displaystyle[J_{ij},B_{k}]$ $\displaystyle=\delta_{k[i}B_{j]}$
$\displaystyle[J_{ij},P_{k}]$ $\displaystyle=\delta_{k[i}P_{j]}$
$\displaystyle[J_{ij},K_{k}]$ $\displaystyle=\delta_{k[i}K_{j]}$
$\displaystyle[B_{i},P_{j}]$ $\displaystyle=\delta_{ij}H$
$\displaystyle[B_{i},K_{j}]$ $\displaystyle=\delta_{ij}K$ $\displaystyle[D,K]$
$\displaystyle=-K$ $\displaystyle[K,P_{i}]$ $\displaystyle=2B_{i}$
$\displaystyle[K_{i},P_{j}]$ $\displaystyle=-2D\delta_{ij}-2J_{ij}$
$\displaystyle[H,K_{i}]$ $\displaystyle=2B_{i}$ $\displaystyle[D,H]$
$\displaystyle=H$ $\displaystyle[D,P_{i}]$ $\displaystyle=P_{i}$
$\displaystyle[D,K_{i}]$ $\displaystyle=-K_{i}\,.$ (67)

The sub-algebra consisting of the generators $\\{J_{ij},B_{i},P_{i},H\\}$
forms the Carrollian algebra, the $c\to 0$ limit of the 3d Poincaré algebra.
The generators $\\{J_{ij},P_{i},D,K_{i}\\}$ form the conformal algebra of
celestial sphere or equivalently the 4d Lorentz algebra.

### 3d Finite Conformal Carroll = 4d Poincare

One of the most crucial observations that our work is based on is the fact the
3d finite Conformal Carroll algebra is isomorphic to the 4d Poincare algebra.
Here we summarise the relation below:

$\displaystyle M_{00}=H,\quad M_{01}=B_{x}+iB_{y},\quad
M_{10}=B_{x}-iB_{y},\quad M_{11}=K_{0}$ (68a) $\displaystyle
L_{0}=\frac{1}{2}(D+iJ),\quad L_{-1}=\frac{1}{2}(P_{x}+iP_{y})\quad
L_{1}=\frac{1}{2}(K_{x}+iK_{y})$ (68b)
$\displaystyle{\bar{L}}_{0}=\frac{1}{2}(D-iJ)\quad{\bar{L}}_{-1}=\frac{1}{2}(P_{x}-iP_{y})\quad{\bar{L}}_{1}=\frac{1}{2}(K_{x}-iK_{y})$
(68c)

Here the LHS of the equations are the 4d Poincare generators and the RHS are
the finite Conformal Carroll generators. We see that the 4d spacetime
translations $M_{r,s}$ arrange themselves into the Hamiltonian, Carroll
boosts, and the temporal part of the Carroll SCT. The Lorentz generators
$L_{n},{\bar{L}}_{n}$ become the global conformal generators on the sphere, as
expected.

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