$\Lambda_{b}\rightarrow\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$ as probe of CP-
violating New Physics

Diganta Das a, Jaydeb Das b, Girish Kumar c and Niladribihari Sahoo d

a Center for Computational Natural Sciences and Bioinformatics, International
Institute of Information Technology, Hyderabad 500 032, India

b Department of Physics and Astrophysics, University of Delhi, Delhi 110007,
India

c Department of Physics, National Taiwan University, Taipei 10617, Taiwan

d School of Physics and Astronomy, University of Birmingham, Birmingham B15
2TT, UK

###### Abstract

We investigate the possible sizes of all the CP-violating asymmetries offered
by the angular distribution of rare decay $\Lambda_{b}\rightarrow\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-}$ in the Standard Model and new physics scenarios
motivated by the recent $b\to s\ell^{+}\ell^{-}$ anomalies. We work in a
model-independent effective theory framework and discuss the sensitivity of CP
asymmetries to new ${O}_{9,10}$ operators and their chirality flipped
counterparts. We find that the size of many of the CP asymmetries can be at
the level of few percent in new physics scenarios consistent with current
$b\to s\ell^{+}\ell^{-}$ data at level of $1\sigma$. We emphasize that
measurements of these CP asymmetries can be used to discriminate different new
physics scenarios in $b\to s\ell^{+}\ell^{-}$.

##  1 Introduction

A major motivation for going beyond the Standard Model (SM) is to find new
sources of charge-parity violation (CPV) required for the explanation of
baryonic asymmetry of the Universe (BAU). The SM possesses two CPV sources:
the Cabibbo-Kobayashi-Maskawa (CKM) phase (related to weak interactions), and
a strong CP phase that is severely constrained by the upper limit of neutron
electric dipole moment measurement [1]. So far, all experimental observations
of CPV have been in the quark sector and are consistent with the CKM
mechanism. However, it is well known that the SM fails to satisfy the
Sakharov’s conditions [2] needed for explaining the BAU. Therefore,
investigations of experimentally accessible CP-violating observables offering
excellent sensitivity to physics beyond the SM are highly motivated.

At the luminosity frontier of new physics (NP) searches, the physical
processes with underlying quark current $b\to s\ell^{+}\ell^{-}$ transition
have been of particular interest in recent times. The measurements of various
observables related to $B\to K^{\ast}\mu^{+}\mu^{-}$ [3, 4, 5],
$B_{s}\to\phi\mu^{+}\mu^{-}$ [6, 7], show deviations from the SM expectation.
Another set of deviations are observed in the measurements of lepton flavor
universality (LFU) ratios $R_{K^{(\ast)}}={\cal B}(B\to
K^{(\ast)}\mu^{+}\mu^{-})/{\cal B}(B\to K^{(\ast)}e^{+}e^{-})$ [8, 9]. The
reported results [10, 11, 12, 13] indicate of a possible breakdown of LFU, an
innate characteristic of the SM, and therefore are of profound importance for
indirect probe of NP. Indeed, the global likelihood analysis of $b\to
s\ell^{+}\ell^{-}$ performed by many groups [14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29] reveal large preference for NP hypothesis over
the SM. The anomalies, collectively known as neutral-current $B$-anomalies,
will be tested rigorously in the upcoming measurements with more data, and if
confirmed, would be the indisputable evidence of NP in $b\to
s\ell^{+}\ell^{-}$ transitions.

The aforementioned anomalies, if confirmed, say nothing about the CP nature of
the underlying NP as the concerned $b\to s\ell^{+}\ell^{-}$ observables are CP
averaged. However, $b\to s\ell^{+}\ell^{-}$ decays also offer a multitude of
observables which are highly sensitive to the CP nature of NP, aided by the
fact that in the SM the $b\to s\ell^{+}\ell^{-}$ transitions are double
Cabibbo-suppressed [30]. A combined measurements of CP-violating and CP-
conserving observables therefore provide a more powerful method to understand
the CP properties of NP. In literature, several works have investigated the CP
asymmetries in $B\to K^{\ast}\mu^{+}\mu^{-}$ angular distributions to probe NP
[31, 32, 30, 33, 34, 35, 36, 37, 38], and as illustrated in Ref. [35], the CP
asymmetries are capable of distinguishing between NP models that address the
$b\to s\ell^{+}\ell^{-}$ anomalies.

If the observed deviation in the $b\to s\ell^{+}\ell^{-}$ are indeed due to
unambiguous short-distance NP then it would in principle also affect all
semileptonic processes with the same underlying current. One important example
is the baryonic decay $\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$
for an unpolarized $\Lambda_{b}$, which is the topic of this paper. There are
several benefits of studying this decay. The angular distribution of
$\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$, similar to its mesonic
counterpart $B\to K^{\ast}(\to K\pi)\ell^{+}\ell^{-}$, offers a large number
of CP-conserving as well as CP-violating angular asymmetries that provide
complimentary information about NP in $b\to s\ell^{+}\ell^{-}$ [39]. The
secondary decay $K^{\ast}\to K\pi$ in mesonic mode $B\to K^{\ast}(\to
K\pi)\ell^{+}\ell^{-}$ is a strong decay and therefore it conserves parity. On
the other hand, the decay $\Lambda\to p\pi$ in the baryonic mode is a parity
violating weak decay; this characteristic will play a key role in constructing
several CP-violating asymmetries in $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-}$, as discussed later in this paper. Furthermore, as
pointed out in Ref. [40], $\Lambda_{b}\to\Lambda$ form factors, in comparison
to $B\to K^{\ast}$, are more suitable to be computed with higher precision
using lattice QCD due to stability of $\Lambda$ under strong interactions.

The first observation of $\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$
was reported by the CDF [41]. The recent angular analysis of
unpolarized#1#1#1A full angular analysis of $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-}$ with polarized $\Lambda_{b}$ has been performed by
LHCb in Ref. [42]. Predictions in the SM and beyond for polarized
$\Lambda_{b}$ decay $\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$ can
be found in Refs. [43, 44]. $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\ell^{+}\ell^{-}$ by the LHCb indicate a branching ratio that is
smaller than the SM expectation [45], a pattern also observed in the $B\to
K^{(\ast)}\mu^{+}\mu^{-}$, $B_{s}\to\phi\mu^{+}\mu^{-}$ modes. There are
extensive theoretical works on model-independent study of
$\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$ in the SM and beyond
[46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 39, 59, 40, 60, 61, 43,
62, 63, 64, 65, 66, 67, 68]. These works mostly focus on CP-conserving angular
observables. To the best of our knowledge, angular observables which
discriminate the decay with its CP-conjugated mode
$\bar{\Lambda}_{b}\to\bar{\Lambda}(\to\bar{p}\pi^{+})\ell^{+}\ell^{-}$ and
their role in the probe of NP in $b\to s\ell^{+}\ell^{-}$ have not yet been
discussed yet. This paper investigates the prospects of CP asymmetries
associated with the baryonic mode and assess their sensitivity to CP-violating
NP. We give description of all possible CP-violating asymmetries that are at
disposal from $\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}$ angular
distribution. Focusing on the muonic mode, we identify sensitivity of these
asymmetries to various NP WCs and show that the measurement of CP asymmetries
in $\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}$ can be used to
distinguish various NP solutions that are favored by the current $b\to
s\mu^{+}\mu^{-}$ data.

The paper is organized as follows. We begin the next section with a
description of the model-independent effective framework relevant for study of
$b\to s\ell^{+}\ell^{-}$ transitions. In section 3 we define the full angular
distribution of both ${\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}}$ and
its CP-conjugated mode, and discuss the subtleties the CP properties of the
secondary decay of $\Lambda\,(\bar{\Lambda})$ particles bring in defining the
corresponding angular coefficients. In section 4 we define all the CP-
violating asymmetries of the $\Lambda_{b}$ decay angular distribution and
present our main numerical results. Finally we offer our conclusions in
section 5.

##  2 $b\to s\ell^{+}\ell^{-}$ effective Lagrangian

In the SM, the $b\to s\ell^{+}\ell^{-}$ decays arise at the loop level only.
The Lagrangian relevant at the scale $\mu=m_{b}$ is given by [33],

$\displaystyle{\cal L}_{\rm
eff}=\frac{4G_{F}}{\sqrt{2}}\lambda_{t}\Big{[}C_{1}O_{1}^{c}+C_{2}O_{2}^{c}+\sum_{i=3}^{6}C_{i}O_{i}+\sum_{i=7}^{10}\left(C_{i}O_{i}+C_{i}^{\prime}O_{i}^{\prime}\right)]$
$\displaystyle+(\lambda_{u}/\lambda_{t})\\{C_{1}(O_{1}^{c}-O_{1}^{u})+C_{2}(O_{2}^{c}-O_{2}^{u})\\}\Big{]}\,,$
(1)

where $G_{F}$ is the Fermi’s constant, and $\lambda_{i}=V_{ib}V_{is}^{\ast}$,
where $V_{ij}$ denotes the CKM matrix element. The WCs $C_{i}$ contain
information about short-distance physics associated with local operators
$O_{i}$. For the discussion of $b\to s\ell^{+}\ell^{-}$ transition,
$O_{7,9,10}$ are the dominant operators which we list below. Denoting the
chiral projectors as $P_{L,R}=(1\mp\gamma_{5})/2$, one has,

$\displaystyle O_{7}$
$\displaystyle=\frac{e}{16\pi^{2}}m_{b}[\bar{s}\sigma_{\mu\nu}P_{R}b]F^{\mu\nu},$
(2) $\displaystyle O_{9}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}\left[\bar{s}\,\gamma_{\mu}P_{L}b\right]\left[\bar{\ell}\,\gamma^{\mu}\ell\right],\quad
O_{10}=\frac{e^{2}}{16\pi^{2}}\left[\bar{s}\,\gamma_{\mu}P_{L}b\right]\left[\bar{\ell}\,\gamma^{\mu}\gamma_{5}\ell\right],$
(3)

with $C_{7}(m_{b})\simeq-0.3$, and $C_{9}(m_{b})\approx-C_{10}(m_{b})\simeq
4.2$ in the SM. Note that the primed operators are the chirality flipped
counterparts of the SM operators, and in the SM their contributions
$(C_{i}^{\prime})$ are vanishing. The operators $O_{1-6}$ are 4-quark
operators related to decays $b\to s\bar{q}q$, and $O_{8}$ is dipole operator
related to radiative decay $b\to sg$. Their explicit form can be found, for
example, in Refs. [69, 70, 71]. These operators contribute to the $b\to
s\ell^{+}\ell^{-}$ through quark loops. It is customary to include their
contribution in the effective WCs $C_{7(9)}^{\rm eff}$ of operators
$O_{7(9)}$. These effective coefficients at next-to-next-to-leading
logarithmic are given as,

$\displaystyle C_{7}^{\rm eff}$
$\displaystyle=C_{7}-\frac{1}{3}\left(C_{3}+\frac{4}{3}C_{4}+20C_{5}+\frac{80}{3}C_{6}\right)-\frac{\alpha_{s}}{4\pi}\left((C_{1}-6C_{2})F_{1,c}^{7}(q^{2})+C_{8}F_{8}^{7}(q^{2})\right),$
(4) $\displaystyle C_{9}^{\rm eff}$
$\displaystyle=C_{9}+\frac{4}{3}C_{3}+\frac{64}{9}C_{5}+\frac{64}{27}C_{6}+h(q^{2},0)\left(-\frac{1}{2}C_{3}-\frac{2}{3}C_{4}-8C_{5}-\frac{32}{3}C_{6}\right)$
$\displaystyle+h(q^{2},m_{b})\left(-\frac{7}{2}C_{3}-\frac{2}{3}C_{4}-38C_{5}-\frac{32}{3}C_{6}\right)+h(q^{2},m_{c})\left(\frac{4}{3}C_{1}+C_{2}+6C_{3}+60C_{5}\right)$
$\displaystyle-\frac{\alpha_{s}}{4\pi}\left(C_{1}F_{1,c}^{9}(q^{2})+C_{2}F_{2,c}^{9}(q^{2})+C_{8}F_{8}^{9}(q^{2})\right)+\frac{\lambda_{u}}{\lambda_{t}}\left(\frac{4}{3}C_{1}+C_{2}\right)\left(h(q^{2},m_{c})-h(q^{2},0)\right).$
(5)

The functions $h(a,b)$ and $F_{8}^{7,9}(q^{2})$ are given in Ref. [72], and
the functions $F_{i,c}^{7,9}(q^{2})$ $(i=1,2)$ are provided in Refs. [73, 74].
Note that the quark masses appearing in Eqs. (4) and (5) are in pole scheme,
and the corresponding values ($m_{b}^{\rm pole}=4.74174$ GeV, $m_{c}^{\rm
pole}=1.5953$ GeV) are taken from Ref. [40]. The numerical values of SM Wilson
coefficients contributing to $b\to s\ell^{+}\ell^{-}$ are given in Table 1.

$C_{1}$ | $C_{2}$ | $C_{3}$ | $C_{4}$ | $C_{5}$ | $C_{6}$ | $C_{7}$ | $C_{8}$ | $C_{9}$ | $C_{10}$
---|---|---|---|---|---|---|---|---|---
$-0.2877$ | $1.0101$ | $-0.0060$ | $-0.0860$ | $0.0004$ | $0.0011$ | $-0.3361$ | $-0.1821$ | $4.2745$ | $-4.1602$
Table 1: Values of $b\to s\ell^{+}\ell^{-}$ WCs in the SM at $\mu=4.2$ GeV
taken from Ref. [40].

There are $(\bar{s}b)(\bar{\ell}\ell)$ operators with scalar and tensor
structure which may arise in NP; but scalar operators are highly constrained
from $B_{s}\to\mu^{+}\mu^{-}$ data [75, 76]. Furthermore, since the current
global fits to $b\to s\mu^{+}\mu^{-}$ data strongly prefer NP in WCs of left-
handed (axial)vector operators in Eq. (3), we will also neglect NP tensor
operators for the simplicity of the analysis.

##  3 Angular Distribution

Assuming the $\Lambda_{b}$ to be unpolarized, the 4-fold differential angular
distributions for ${\Lambda_{b}\to\Lambda(\to p\pi^{-})\ell^{+}\ell^{-}}$ is
given by [39],

$\displaystyle\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}q^{2}\mathrm{d}\cos\theta_{\ell}\mathrm{d}\cos\theta_{\Lambda}\mathrm{d}\phi}=\frac{3}{8\pi}[$
$\displaystyle\left(K_{1ss}\sin^{2}\theta_{\ell}+K_{1cc}\cos^{2}\theta_{\ell}+K_{1c}\cos\theta_{\ell}\right)$
$\displaystyle+\left(K_{2ss}\sin^{2}\theta_{\ell}+K_{2cc}\cos^{2}\theta_{\ell}+K_{2c}\cos\theta_{\ell}\right)\cos\theta_{\Lambda}$
$\displaystyle+\left(K_{3sc}\sin\theta_{\ell}\cos\theta_{\ell}+K_{3s}\sin\theta_{\ell}\right)\sin\theta_{\Lambda}\sin\phi$
$\displaystyle\left.+\left(K_{4sc}\sin\theta_{\ell}\cos\theta_{\ell}+K_{4s}\sin\theta_{\ell}\right)\sin\theta_{\Lambda}\cos\phi\right].$
(6)

The distribution is completely described by four variables: the invariant
lepton mass squared ($q^{2}$) and three Euler angles $\theta_{\ell}$,
$\theta_{\Lambda}$, and $\phi$. In the rest frame of $\Lambda_{b}$, the
daughter baryon is assumed to travel along the $+z$ axis. The
$\theta_{\Lambda}$ is the angle made by the proton with the $+z$ axis in the
rest frame of the $\Lambda$, $\theta_{\ell}$ is the angle made by the
$\ell^{-}$ with respect to the $+z$ axis in the rest frame of the lepton pair,
and $\phi$ defines angle in the rest frame of $\Lambda_{b}$ between planes
containing $p\pi^{-}$ and the lepton pair. Denoting mass of $\Lambda_{b}$,
$\Lambda$, and charged lepton $\ell$ as $m_{\Lambda_{b}}$, $m_{\Lambda}$, and
$m_{\ell}$, respectively, the physical region of the decay process is defined
by the following values:

$\displaystyle
q^{2}\in[4m_{\ell}^{2},(m_{\Lambda_{b}}-m_{\Lambda})^{2}],\quad\cos\theta_{\Lambda}\in[-1,1],\quad\cos\theta_{\ell}\in[-1,1],\quad\phi\in[0,2\pi].$
(7)

In Eq. (3), the angular coefficients $K_{i}$ are functions of $q^{2}$. These
are conveniently described in terms of $\Lambda_{b}\to\Lambda$ transversity
amplitudes, $A^{L,R}_{i}(q^{2})$ as follows,

$\displaystyle K_{1ss}$
$\displaystyle=\frac{1}{4}\big{(}2|{A^{R}_{\|_{0}}}|^{2}+|{A^{R}_{\|_{1}}}|^{2}+2|{A^{R}_{\perp_{0}}}|^{2}+|{A^{R}_{\perp_{1}}}|^{2}+\\{R\leftrightarrow
L\\}\big{)},$ (8) $\displaystyle K_{1cc}$
$\displaystyle=\frac{1}{2}\big{(}|{A^{R}_{\|_{1}}}|^{2}+|{A^{R}_{\perp_{1}}}|^{2}+\\{R\leftrightarrow
L\\}\big{)},$ (9) $\displaystyle K_{1c}$ $\displaystyle=-{\rm
Re}\big{(}A^{R}_{\perp_{1}}A^{\ast R}_{\|_{1}}-\\{R\leftrightarrow
L\\}\big{)},$ (10) $\displaystyle K_{2ss}$
$\displaystyle=\frac{\alpha_{\Lambda}}{2}{\rm
Re}\big{(}2A^{R}_{\perp_{0}}A^{\ast R}_{\|_{0}}+A^{R}_{\perp_{1}}A^{\ast
R}_{\|_{1}}+\\{R\leftrightarrow L\\}\big{)},$ (11) $\displaystyle K_{2cc}$
$\displaystyle=\alpha_{\Lambda}{\rm Re}\big{(}A^{R}_{\perp_{1}}A^{\ast
R}_{\|_{1}}+A^{L}_{\perp_{1}}A^{\ast L}_{\|_{1}}\big{)},$ (12) $\displaystyle
K_{2c}$ $\displaystyle=-\frac{\alpha_{\Lambda}}{2}{\rm
Re}\big{(}|{A^{R}_{\perp_{1}}}|^{2}+|{A^{R}_{\|_{1}}}|^{2}-\\{R\leftrightarrow
L\\}\big{)},$ (13) $\displaystyle K_{3sc}$
$\displaystyle=\frac{\alpha_{\Lambda}}{\sqrt{2}}{\rm
Im}\big{(}{A^{R}_{\perp_{1}}}{A^{\ast R}_{\perp_{0}}}-{A^{R}_{\|_{1}}}{A^{\ast
R}_{\|_{0}}}+\\{R\leftrightarrow L\\}\big{)},$ (14) $\displaystyle K_{3s}$
$\displaystyle=\frac{\alpha_{\Lambda}}{\sqrt{2}}{\rm
Im}\big{(}A^{R}_{\perp_{1}}A^{\ast R}_{\|_{0}}-A^{R}_{\|_{1}}A^{\ast
R}_{\perp_{0}}-\\{R\leftrightarrow L\\}\big{)},$ (15) $\displaystyle K_{4sc}$
$\displaystyle=\frac{\alpha_{\Lambda}}{\sqrt{2}}{\rm
Re}\big{(}A^{R}_{\perp_{1}}A^{\ast R}_{\|_{0}}-A^{R}_{\|_{1}}A^{\ast
R}_{\perp_{0}}+\\{R\leftrightarrow L\\}\big{)},$ (16) $\displaystyle K_{4s}$
$\displaystyle=\frac{\alpha_{\Lambda}}{\sqrt{2}}{\rm
Re}\big{(}A^{R}_{\perp_{1}}A^{\ast R}_{\perp_{0}}-A^{R}_{\|_{1}}A^{\ast
R}_{\|_{0}}-\\{R\leftrightarrow L\\}\big{)},$ (17)

where the $q^{2}$ dependence of the transversity amplitudes is implied. Since
we will be focussing on muonic mode only, we have ignored the lepton mass in
writing the coefficients $K_{i}$ above. The expressions for $K_{i}$ including
lepton mass effects can be found in Ref. [64]. The transversity amplitudes
$A^{L,R}_{i}(q^{2})$ in terms of the $\Lambda_{b}\to\Lambda$ form factors
$f_{i}(q^{2})$ (see section 4) and $b\to s\ell^{+}\ell^{-}$ Wilson
coefficients are given by [64]

$\displaystyle A^{L,(R)}_{\perp_{1}}$
$\displaystyle=-\sqrt{2}N\bigg{(}f^{V}_{\perp}\sqrt{2s_{-}}C^{L,(R)}_{+}+\frac{2m_{b}}{q^{2}}f^{T}_{\perp}({m_{\Lambda_{b}}}+{m_{\Lambda}})\sqrt{2s_{-}}C_{7}^{\rm
eff}\bigg{)}\,,$ (18) $\displaystyle A^{L,(R)}_{\|_{1}}$
$\displaystyle=\sqrt{2}N\bigg{(}f^{A}_{\perp}\sqrt{2s_{+}}C^{L,(R)}_{-}+\frac{2m_{b}}{q^{2}}f^{T5}_{\perp}({m_{\Lambda_{b}}}-{m_{\Lambda}})\sqrt{2s_{+}}C_{7}^{\rm
eff}\bigg{)}\,,$ (19) $\displaystyle A^{L,(R)}_{\perp_{0}}$
$\displaystyle=\sqrt{2}N\bigg{(}f^{V}_{0}({m_{\Lambda_{b}}}+{m_{\Lambda}})\sqrt{\frac{s_{-}}{q^{2}}}C^{L,(R)}_{+}+\frac{2m_{b}}{q^{2}}f^{T}_{0}\sqrt{q^{2}s_{-}}C_{7}^{\rm
eff}\bigg{)}\,,$ (20) $\displaystyle A^{L,(R)}_{\|_{0}}$
$\displaystyle=-\sqrt{2}N\bigg{(}f^{A}_{0}({m_{\Lambda_{b}}}-{m_{\Lambda}})\sqrt{\frac{s_{+}}{q^{2}}}C^{L,(R)}_{-}+\frac{2m_{b}}{q^{2}}f^{T5}_{0}\sqrt{q^{2}s_{+}}C_{7}^{\rm
eff}\bigg{)}\,,$ (21)

where $s_{\pm}=({m_{\Lambda_{b}}}\pm{m_{\Lambda}})^{2}-q^{2}$, and
$C_{\pm}^{L(R)}$ are combinations of Wilson coefficients,

$\displaystyle C_{+}^{L(R)}=(C_{9}^{\rm eff}\mp C_{10})+(C_{9}^{\prime}\mp
C_{10}^{\prime})\,,\quad C_{-}^{L(R)}=(C_{9}^{\rm eff}\mp
C_{10})-(C_{9}^{\prime}\mp C_{10}^{\prime})\,,$ (22)

and the normalization constant $N$, a function of $q^{2}$, is given by,

$N(q^{2})=\left[\frac{G_{F}^{2}|V_{tb}V_{ts}^{\ast}|^{2}\alpha_{e}^{2}}{3.2^{11}m^{3}_{\Lambda_{b}}\pi^{5}}{{q^{2}{\lambda^{1/2}({m^{2}_{\Lambda_{b}}},{m^{2}_{\Lambda}},q^{2})}}{}\beta_{\ell}}\right]^{1/2},\quad\beta_{\ell}=\sqrt{1-\frac{4m_{\ell}^{2}}{q^{2}}}\,,$
(23)

with $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+bc+ca)$.

In the expressions of $K_{i}$, the parity violating decay parameter
$\alpha_{\Lambda}$ arises through the secondary decay $\Lambda\to p\pi$. The
corresponding hadronic matrix element is given by [39],

$\displaystyle\langle
p(k_{1},s_{p})\pi^{-}(k_{2})|(\bar{d}u)_{V-A}(\bar{u}s)_{V-A}|\Lambda(k,s_{\Lambda})\rangle=[\bar{u}(k_{1},s_{p})(\xi\,\gamma_{5}+\omega)u(k,s_{\Lambda})]\,,$
(24)

which depends on only two parameters, $\xi$ and $\omega$. These can be
determined from experimental data on $\Lambda\to p\pi$. The decay parameter
$\alpha_{\Lambda}$ then is given by [39],

$\displaystyle\alpha_{\Lambda}=\frac{-2\,\text{Re}(\omega\,\xi)}{\sqrt{r_{-}/r_{+}}|\xi|^{2}+\sqrt{r_{+}/r_{-}}|\omega|^{2}}\,.$
(25)

where $r_{\pm}=(m_{\Lambda}\pm m_{p})^{2}-m_{\pi}^{2}$.

To write down the corresponding distribution for CP-conjugated mode
$\bar{\Lambda}_{b}\to\bar{\Lambda}(\to\bar{p}\pi^{+})\ell^{+}\ell^{-}$, we
take the following definition for Euler angles: In the rest frame of the
$\bar{\Lambda_{b}}$, the $\bar{\Lambda}$ is assumed to travel along the $+z$
axis, and $\theta_{\Lambda}$ is the angle between $\bar{\Lambda}$ and anti-
proton in the $(\bar{p}\pi^{+})$ rest frame. The lepton angle $\theta_{\ell}$
is the angle between $\bar{\Lambda}$ and $\ell^{-}$#2#2#2 Note that direction
of $\ell^{-}$ is taken as reference for both decay and CP-conjugated mode.
This convention is similar to the one used for mesonic counterpart decay in
Ref. [33]. in the dilepton rest frame, and $\phi$ is the angle between planes
of $(\bar{p}\pi^{+})$ and lepton pair. With the above convention, the decay
distribution for
$\bar{\Lambda}_{b}\to\bar{\Lambda}(\to\bar{p}\pi^{+})\ell^{+}\ell^{-}$ is
simply obtained from Eq. (3) after the transformation
($\theta_{\Lambda}\to\theta_{\Lambda},$ $\theta_{\ell}=\theta_{\ell}-\pi$, and
$\phi\to-\phi$). This is equivalent to replacing the functions $K_{i}$s in Eq.
(3) with $\bar{K}_{i}$s in the following way,

$\displaystyle K_{1cc,1ss,2cc,2ss,4sc,3s}\to$
$\displaystyle+\bar{K}_{1cc,1ss,2cc,2ss,4sc,3s}\,,$ (26) $\displaystyle
K_{1c,2c,4s,3sc}\to$ $\displaystyle-\bar{K}_{1c,2c,4s,3sc}\,,$ (27)

where $\bar{K}_{i}$ equals $K_{i}$ except for the weak phases conjugated.
Additionally, in all but three coefficients $K_{1ss}$, $K_{1cc}$, and
$K_{1c}$, we replace $\Lambda\to p\pi$ decay parameter $\alpha_{\Lambda}$ by
$\bar{\alpha}_{\Lambda}$ that corresponds to the CP-conjugated decay
$\bar{\Lambda}\to\bar{p}\pi^{+}$. This replacement follows from the fact that
under a CP transformation the Dirac-field bilinears transform as
$\bar{\psi}_{1}\psi_{2}\xrightarrow{CP}\bar{\psi}_{2}\psi_{1}$,
$\bar{\psi}_{1}\gamma_{5}\psi_{2}\xrightarrow{CP}-\bar{\psi}_{2}\gamma_{5}\psi_{1}$.
Therefore, in the expression of $\bar{\Lambda}\to\bar{p}\pi^{+}$ hadronic
matrix element obtained through a CP transformation of equation (24), the
$\xi$ term picks up a minus sign but $\omega$ term does not. The definition of
$\bar{\alpha}_{\Lambda}$, defined similar to Eq. (25), then implies
$\bar{\alpha}_{\Lambda}=-\alpha_{\Lambda}$ under strict $CP$-symmetry.
Experimentally measured values $\alpha_{\Lambda}=0.7519\pm 0.0010\pm 0.0024$
and $\bar{\alpha}_{\Lambda}=-0.7559\pm 0.0036\pm 0.0030$ by BESIII
collaboration [77] agree well with theory.

##  4 CP-Violating Asymmetries and Results

Let $\Gamma(H\to f)$ be the decay rate of $H\to f$ (where $H$ indicates the
initial state $\Lambda_{b}$ and $f$ indicates the final state) and
$\bar{\Gamma}(\bar{H}\to\bar{f})$ be the decay rate of the CP-conjugate mode
$\bar{H}\to\bar{f}$. As one needs two interfering decay amplitudes to observe
a direct CPV, we write the amplitudes as

$\displaystyle A[H\to f]$
$\displaystyle=|A_{1}|e^{i\phi^{w}_{1}}e^{i\phi^{s}_{1}}+|A_{2}|e^{i\phi^{w}_{2}}e^{i\phi^{s}_{2}}\,,$
(28) $\displaystyle A[\bar{H}\to\bar{f}]$
$\displaystyle=|A_{1}|e^{-i\phi^{w}_{1}}e^{i\phi^{s}_{1}}+|A_{2}|e^{-i\phi^{w}_{2}}e^{i\phi^{s}_{2}}\,,$
(29)

where $\phi^{w}_{1,2}$ are the weak phases, $\phi^{s}_{1,2}$ are the strong
phases (arising due to final state interactions), and $|A_{1,2}|$ are the
moduli of the interfering matrix elements. The decay rate asymmetry that
signals the presence of CPV is

$\Gamma(H\to
f)-\bar{\Gamma}(\bar{H}\to\bar{f})\propto|A_{1}||A_{2}|\sin(\phi_{1}^{s}-\phi_{2}^{s})\sin(\phi_{1}^{w}-\phi_{2}^{w})\,.$
(30)

As evident from this expression, a nonvanishing CP asymmetry requires that
both the relative strong and weak phases of the amplitudes must be
nonvanishing. The SM has a finite strong phase emanating from the imaginary
part of $C_{9}^{\rm eff}$, which is generated by the $\bar{q}q$ loops
($q=c,u$) in the current-current operators. However, the weak phase, coming
from the CKM elements in the last term of Eq. (5), is double Cabibbo-
suppressed and small. Therefore, the CPV in $b\to s\ell^{+}\ell^{-}$
transitions in the SM is expected to be very small.

As discussed in the Introduction, the measurements of several observables
associated with the $b\to s\ell^{+}\ell^{-}$ current are in tension with the
SM, and the global fits to data show large preference to NP hypothesis over
the SM. Except the CP asymmetry $A_{\rm CP}^{K^{(\ast)}}$ [78], most of the
measured $b\to s\ell^{+}\ell^{-}$ observables, including the ones that show
the tensions with the SM, are CP averaged and therefore are not sensitive to
the complex phases of NP. Therefore whether the NP in question is real or
complex is not clear at present. To answer this question, one needs to study
CP-violating observable in $b\to s\ell^{+}\ell^{-}$ transition. To this
purpose, we construct several CP-violating observables in
${\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}}$ decay and investigate their
sensitivity to complex NP Wilson coefficients. We follow the results of Ref.
[79] which performed global fit analysis of $b\to s\mu^{+}\mu^{-}$ data to
complex Wilson coefficients. In particular, we consider the following three NP
scenarios for $b\to s\mu^{+}\mu^{-}$ WCs :

  * •

Case I: $C_{9}^{\text{NP}}=-1.0-0.9i$ (NP in left-handed vector operator only)

  * •

Case II: $C_{9}^{\text{NP}}=-C_{10}^{\text{NP}}=-0.7-1.4i$ ($SU(2)_{L}$
invariant NP)

  * •

Case III: $C_{9}^{\prime}=-C_{10}^{\prime}=0.04+0.16i$ (Right-handed NP)

The Case I and Case II have the largest pulls (more than $6\sigma$) and
describe the data best. The Case III although cannot explain tensions in $b\to
s$ data, we include it in our analysis to assess the sensitivity of CP
asymmetries to right-handed NP.

In order to make numerical predictions of observables, one also needs
$\Lambda_{b}\to\Lambda$ form factors for which we use lattice QCD results of
Ref.[40]. The $\Lambda_{b}\to\Lambda$ hadronic matrix elements is parametrized
in terms of ten form factors#3#3#3Note that the form factors labels used in
Ref. [40] are different from labels we use in this paper. These are related as
$f^{V}_{t,0,\perp}=f_{0,+,\perp}$, $f^{A}_{t,0,\perp}=g_{0,+,\perp}$,
$f^{T}_{0,\perp}=h_{+,\perp}$ and $f^{T5}_{0,\perp}=\tilde{h}_{+,\perp}$.
$f^{V}_{t,0,\perp}$, $f^{A}_{t,0,\perp}$, $f^{T}_{0,\perp}$,
$f^{T5}_{0,\perp}$, which are function of $q^{2}$. The lattice calculations
are fitted to two $z$-parametrizations: the “nominal” fit and “higher-order”
fit. Defining the parameter $z(q^{2},t_{+})$ as,

$z(q^{2},t_{+})=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}}}\,,$
(31)

where $t_{0}=({m_{\Lambda_{b}}}-{m_{\Lambda}})^{2}$ and
$t_{+}=(m_{B}+m_{K})^{2}$, in the so called “nominal” fit the form factor
parametrization is given as,

$f(q^{2})=\frac{1}{1-q^{2}/(m^{f}_{\rm
pole})^{2}}\big{[}a^{f}_{0}+a_{1}^{f}z(q^{2},t_{+})\big{]}\,,$ (32)

while in “higher-order” fit, the parametrization is given as,

$f(q^{2})=\frac{1}{1-q^{2}/(m^{f}_{\rm
pole})^{2}}\big{[}a^{f}_{0}+a_{1}^{f}z(q^{2},t_{+})+a_{2}^{f}(z(q^{2},t_{+}))^{2}\big{]}\,,$
(33)

where values of the coefficients $a_{i}^{f}$ and the correlations among them
are taken from Ref. [40].

In our numerical analysis, apart from already discussed form factors and the
Wilson coefficients, we use results of CKMfitter Group [80] for values of CKM
elements, while particle masses and their lifetime values are taken from
Particle Data Group [81].

With the numerical inputs at our disposal, we make bin-wise predictions of
different observables that we describe in the next section. To be precise, the
prediction of an observable ${\cal O}(q^{2})$ in a given bin $q^{2}\in[a,b]$
is

$\displaystyle\langle{\cal O}\rangle=\frac{1}{|b-a|}{\int_{a}^{b}dq^{2}\,{\cal
O}}(q^{2})\,.$ (34)

For observable involving ratio of two quantities, the binned prediction is
obtained after integrating the numerator and denominator separately and then
taking their ratio. The two regions of $q^{2}$ where we make the predictions
are $q^{2}\in[0.1,6]$ and $q^{2}\in[15,20]$. To avoid the charmonium
resonances, we refrain from making any prediction in the $6-15~{}{\rm
GeV}^{2}$ range.

Our numerical determinations are subject to uncertainties coming from
different inputs including the form factors. Regarding form factors
uncertainties, as described in Ref. [40], we use the “nominal” fit of Eq. (32)
and for estimation of systematic uncertainties “higher-order” fit of Eq. (33)
is used. The total uncertainty is obtained after adding statistical and
systematic uncertainties in quadrature. In the appendix A we have collected
bin wise predictions of all observables that we describe next.

###  4.1 CP asymmetry in decay rate

The CP asymmetry in decay rate is defined as,

$\displaystyle{\cal
A}_{CP}=\frac{d\Gamma/dq^{2}-d\bar{\Gamma}/dq^{2}}{d\Gamma/dq^{2}+d\bar{\Gamma}/dq^{2}}\,,$
(35)

where $d\Gamma/dq^{2}$ and $d\bar{\Gamma}/dq^{2}$ are decay rates of the decay
${\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}}$ and its CP-conjugate
decay, respectively, which in terms of angular coefficients are defined as,

$\displaystyle\frac{d\Gamma}{dq^{2}}=2K_{1ss}+K_{1cc},\quad\frac{d\bar{\Gamma}}{dq^{2}}=2\bar{K}_{1ss}+\bar{K}_{1cc}\,.$
(36)

Figure 1: Predictions for direct CP asymmetry in decay rate (${\cal A}_{CP}$)
of ${\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}}$.

In Fig. 1, we show the ${\cal A}_{CP}$ as function of $q^{2}$ (left plot) in
the SM and NP scenarios. The corresponding binned predictions are shown in the
plot to the right. We find that in the SM the ${\cal A}_{CP}$ is, as expected,
very small and is $\sim{\cal O}(10^{-3})$. In NP case, we note the following:

  * •

${\cal A}_{CP}$ is sensitive to left-handed NP at high $q^{2}$ as seen from
Fig. 1 where NP Cases I and II show large deviation from the SM. We find that
in case II, which has WCs with larger imaginary part, ${\cal A}_{CP}$ can be
up to $6\%$.

  * •

We find that ${\cal A}_{CP}$ is not sensitivity to right-handed currents (NP
case III).

  * •

At low $q^{2}$, we find ${\cal A}_{CP}$ is not sensitive to any of NP
scenario.

###  4.2 CP asymmetry in longitudinal polarization

We define the CP asymmetry in longitudinal polarization fraction as the
difference between longitudinal polarization of ${\Lambda_{b}\to\Lambda(\to
p\pi^{-})\mu^{+}\mu^{-}}$ (denoted as $F_{L}(q^{2})$) and its CP-conjugate
decay (denoted as $\bar{F}_{L}(q^{2})$), normalized by sum of corresponding
decay rates,

$\displaystyle{\cal
A}_{F_{L}}=\frac{F_{L}(q^{2})-\bar{F}_{L}(q^{2})}{d\Gamma/dq^{2}+d\bar{\Gamma}/dq^{2}},$
(37)

where $F_{L}(q^{2})$ and $\bar{F}_{L}(q^{2})$ in terms of angular coefficients
are given as,

$\displaystyle
F_{L}(q^{2})=2K_{1ss}-K_{1cc},\quad\bar{F}_{L}(q^{2})=2\bar{K}_{1ss}-\bar{K}_{1cc}\,.$
(38)

Figure 2: Predictions for CP asymmetry in longitudinal polarization of
${\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}}$. The color scheme is same as in Fig.
1.

Since angular coefficients appearing in ${\cal A}_{f_{L}}$ are same as in
${\cal A}_{CP}$, one expects ${\cal A}_{f_{L}}$ to be sensitive to NP. In Fig.
2, we show results for asymmetry ${\cal A}_{f_{L}}$, and indeed we find that
observable is sensitive to left-handed NP (NP case I and II) while very weakly
sensitive to right-handed NP (NP case III). The NP effects are higher in large
$q^{2}$ region and ${\cal A}_{f_{L}}$ can be $\lesssim 3\%$. In low $q^{2}$
region, ${\cal A}_{f_{L}}$ remains well below $1\%$.

###  4.3 CP asymmetry in forward-backward asymmetries

Figure 3: Predictions for CP asymmetries ${\cal A}_{1c}$, ${\cal A}_{2c}$,
${\cal A}_{2cc}$, and ${\cal A}_{2ss}$. The color scheme is same as in Fig. 1.

The decay ${\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}}$ offers three
types of forward-backward (FB) asymmetries [39]: FB asymmetry ($a_{\rm
FB}^{\ell}$) with respect to (w.r.t) leptonic angle $\theta_{\ell}$, FB
asymmetry ($a_{\rm FB}^{\Lambda}$) w.r.t to hadronic angle $\theta_{\Lambda}$,
and FB asymmetry ($a_{\rm FB}^{\ell\Lambda}$) w.r.t combination of
$\theta_{\ell}$ and $\theta_{\Lambda}$, respectively. In terms of angular
coefficients, these are given as,

$\displaystyle a^{\ell}_{\rm FB}=\frac{3}{2}K_{1c},\quad a^{\Lambda}_{\rm
FB}=\frac{1}{2}(2K_{2ss}+K_{2cc}),\quad a^{\ell\Lambda}_{\rm
FB}=\frac{3}{4}K_{2c}\,.$ (39)

One can then take the difference between measurement of the FB asymmetry in
the decay ${\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}}$ and its CP-
conjugate to define new CP asymmetries. These CP asymmetries can be accessed
by measuring the CP asymmetries associated with angular coefficients $K_{1c}$,
$K_{2c}$, $K_{2ss}$ and $K_{2cc}$. To this end, we define following four CP
asymmetries:

$\displaystyle{\cal A}_{1c}$
$\displaystyle=\frac{K_{1c}(q^{2})-\bar{K}_{1c}(q^{2})}{d\Gamma/dq^{2}+d\bar{\Gamma}/dq^{2}},$
(40) $\displaystyle{\cal A}_{j}$
$\displaystyle=\frac{K_{j}(q^{2})+\bar{K}_{j}(q^{2})}{d\Gamma/dq^{2}+d\bar{\Gamma}/dq^{2}},\quad\text{for}\
j=2c,2ss,2cc.$ (41)

${\cal A}_{1c}$ and ${\cal A}_{2c}$ are equivalent to CP asymmetries in
$a^{\ell}_{\rm FB}$ and $a^{\ell\Lambda}_{\rm FB}$ up to a normalization
constant, while CP asymmetry in $a^{\Lambda}_{\rm FB}$ can be determined from
combined measurements of ${\cal A}_{2ss}$ and ${\cal A}_{2cc}$. Also, note
that ${\cal A}_{1c}$ involves difference of $K_{1c}$ and $\bar{K}_{1c}$, the
other CP asymmetries involves sum of corresponding coefficients. This is
because the angular coefficients, in Eqs. (8)-(17) are proportional to the
decay parameter $\alpha_{\Lambda}$ (or $\bar{\alpha}_{\Lambda}$ in case of CP-
conjugate mode). As discussed in the previous section,
$\alpha_{\Lambda}\simeq-\bar{\alpha}_{\Lambda}$ experimentally, therefore it
is the combination $K_{j}+\bar{K}_{j}$ ($j=2c,2ss,2cc,3s,3sc,4s,4sc$) that
vanishes in case of purely real WCs.

In Fig. 3, we show results for ${\cal A}_{1c}$, ${\cal A}_{2c}$, ${\cal
A}_{2cc}$, and ${\cal A}_{2ss}$. We note the following:

  * •

CP asymmetry ${\cal A}_{1c}$ is sensitive to $C_{10}$, while no sensitivity to
vector and right-handed current is found. We find that in NP case II which has
nonzero $C_{10}^{\rm NP}$, the value of ${\cal A}_{1c}$ can be at $\sim 1-2\%$
level in large $q^{2}$ region. In other NP cases considered, ${\cal A}_{1c}$
remain indistinguishable from the SM.

  * •

For the asymmetry ${\cal A}_{2c}$, we observe similar NP sensitivity as in
${\cal A}_{1c}$ except that the sign of the asymmetry is opposite to that of
${\cal A}_{1c}$. We also note that ${\cal A}_{2c}$ is largest ($\sim 1.5\%$)
at the kinematic end point $q^{2}\sim 20~{}{\rm GeV}^{2}$.

  * •

In contrast, the asymmetries ${\cal A}_{2ss}$ and ${\cal A}_{2ss}$ are
sensitive to $C_{9}$, while no sensitivity to axial vector and right-handed
currents is found. Both asymmetries are found to be $\sim 1\%$ in large
$q^{2}$ region, with ${\cal A}_{2ss}$ being somewhat slighly larger.

###  4.4 More CP asymmetries

One can define four more CP asymmetries associated with angular coefficients
$K_{3s}$, $K_{3sc}$, $K_{4s}$, and $K_{4sc}$, respectively,

$\displaystyle{\cal A}_{j}$
$\displaystyle=\frac{K_{j}(q^{2})+\bar{K}_{j}(q^{2})}{d\Gamma/dq^{2}+d\bar{\Gamma}/dq^{2}},\quad\text{for}\
j=3s,3sc,4s,4sc.$ (42)

Figure 4: Predictions for CP asymmetries ${\cal A}_{3s}$, ${\cal A}_{3sc}$,
${\cal A}_{4s}$, and ${\cal A}_{4sc}$. The color scheme is same as in Fig. 1.

In Fig. 4 we show the results for these CP asymmetries and make the following
observations:

  * •

We find ${\cal A}_{3s}$ to be sensitive to right-handed currents (NP case III)
in both low and high $q^{2}$ regions, where it can be up to $\sim 2\%$. It is
also mildly sensitive to $C_{10}$ in the low $q^{2}$ region, but has large
theoretical uncertainties, as can be seen from low $q^{2}$ bins. Another point
worth mentioning is that there is zero-crossing in low $q^{2}$ region of
${\cal A}_{3s}$ for right-handed NP scenario (Case III), while no such
behavior is seen in left-handed NP scenarios.

  * •

In case of ${\cal A}_{3sc}$, we find the CP asymmetry to be sensitive to
$C_{9}$ as well as right-handed NP case, with its size being $\sim 1-3\%$ in
low $q^{2}$ region. At large $q^{2}$, CP asymmetry remains negligible.

  * •

For ${\cal A}_{4s}$, we find that the asymmetry is sensitive to $C_{10}$ at
large $q^{2}$. The CP asymmetry remain SM-like in cases with NP in $C_{9}$ or
right-handed NP. In NP cases II (which has nonzero $C_{10}$), ${\cal A}_{4s}$
can be $\sim 1\%$.

  * •

On the other hand, for ${\cal A}_{4sc}$ we find that the observable is most
sensitive to $C_{9}$ (NP case I and II) and ${\cal A}_{4sc}$ can be up to
$\sim 2\%$ at large $q^{2}$ regions.

##  5 Conclusion

The recent experimental data on the $b\to s\ell^{+}\ell^{-}$ transitions hints
towards the presence of NP. However, the CP properties of the possible
underlying NP is unknown. One concrete way to answer this question is provided
by the measurements of CP-violating asymmetries associated with the $b\to
s\ell^{+}\ell^{-}$ transitions. In the SM, the CPV in the $b\to
s\ell^{+}\ell^{-}$ transition is very small, but can be enhanced in many NP
models. Thus, measurements of sizable $b\to s\ell^{+}\ell^{-}$ CP asymmetries
are highly motivated. With this in mind, in this paper, we have investigated
in a model-independent fashion the prospects of probing CP-violating NP in
${\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}}$ decay. To this end, we
list all the CP asymmetries offered by the angular distribution of an
unpolarized $\Lambda_{b}$ decay ${\Lambda_{b}\to\Lambda(\to
p\pi^{-})\mu^{+}\mu^{-}}$. We then present their determinations in the SM and
several NP scenarios motivated by the present global fits to $b\to
s\ell^{+}\ell^{-}$ data. We find that several of the CP asymmetries, depending
on their NP sensitivity and $q^{2}$ region, can be enhanced to a few percent
level. More importantly, we find that the measurements of these CP asymmetries
can provide new methods to not only probe but also potentially distinguish NP
cases discussed in the paper. Therefore, the CP asymmetries in
${\Lambda_{b}\to\Lambda(\to p\pi^{-})\mu^{+}\mu^{-}}$ provide new avenues to
cross-check the SM and, in conjunction with CP asymmetries of $B\to
K^{\ast}\ell^{+}\ell^{-}$, can play useful role in searching NP in $b\to
s\ell^{+}\ell^{-}$ transitions.

Acknowledgments DD would like to thank the DST, Govt. of India for the INSPIRE
Faculty Award (grant no IFA16-PH170). DD also thanks the Institute for
Theoretical Physics III, University of Stuttgart for kind hospitality during
various stages of the work. JD acknowledges the Council of Scientific and
Industrial Research (CSIR), Government of India, for the SRF fellowship grant
with File No. 09/045(1511)/ 2017-EMR-I. JD also would like to acknowledge
Research Grant No. SERB/CRG/004889/SGBKC/2022/04 of the SERB, India, for
partial financial support. The work of GK is supported by NSTC
111-2639-M-002-002-ASP of Taiwan. NS would like to acknowledge support from
the UK Science and Technology Facilities Council (STFC).

##  Appendix A Prediction of CP asymmetries

###  A.1 ${\cal A}_{CP}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | 0.399$\pm$0.053 | 0.398$\pm$0.103 | 0.386$\pm$0.138 | 0.405$\pm$0.052
$[1,2]$ | 0.197$\pm$0.049 | 0.165$\pm$0.172 | 0.130$\pm$0.245 | 0.209$\pm$0.050
$[2,4]$ | 0.098$\pm$0.024 | 0.034$\pm$0.167 | -0.013$\pm$0.272 | 0.112$\pm$0.028
$[4,6]$ | 0.066$\pm$0.017 | -0.013$\pm$0.159 | -0.058$\pm$0.263 | 0.079$\pm$0.020
$[1.1,6]$ | 0.098$\pm$0.022 | 0.033$\pm$0.166 | -0.011$\pm$0.258 | 0.111$\pm$0.025
$[15,16]$ | -0.201$\pm$0.023 | -3.461$\pm$0.239 | -5.423$\pm$0.370 | -0.335$\pm$0.028
$[16,18]$ | -0.229$\pm$0.020 | -3.504$\pm$0.208 | -5.472$\pm$0.332 | -0.433$\pm$0.025
$[18,20]$ | -0.250$\pm$0.019 | -3.559$\pm$0.178 | -5.540$\pm$0.285 | -0.590$\pm$0.019
$[15,20]$ | -0.231$\pm$0.019 | -3.514$\pm$0.198 | -5.486$\pm$0.318 | -0.467$\pm$0.023

###  A.2 ${\cal A}_{f_{L}}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | 0.033$\pm$0.003 | -0.086$\pm$0.046 | -0.156$\pm$0.071 | 0.042$\pm$0.007
$[1,2]$ | 0.039$\pm$0.011 | -0.254$\pm$0.057 | -0.426$\pm$0.092 | 0.058$\pm$0.018
$[2,4]$ | 0.068$\pm$0.012 | -0.257$\pm$0.053 | -0.445$\pm$0.093 | 0.087$\pm$0.018
$[4,6]$ | 0.084$\pm$0.011 | -0.196$\pm$0.056 | -0.352$\pm$0.097 | 0.097$\pm$0.015
$[1.1,6]$ | 0.071$\pm$0.012 | -0.229$\pm$0.054 | -0.400$\pm$0.085 | 0.087$\pm$0.016
$[15,16]$ | -0.084$\pm$0.009 | -1.558$\pm$0.104 | -2.442$\pm$0.161 | -0.123$\pm$0.015
$[16,18]$ | -0.090$\pm$0.008 | -1.458$\pm$0.087 | -2.278$\pm$0.144 | -0.155$\pm$0.012
$[18,20]$ | -0.090$\pm$0.007 | -1.320$\pm$0.061 | -2.055$\pm$0.103 | -0.202$\pm$0.008
$[15,20]$ | -0.089$\pm$0.007 | -1.431$\pm$0.082 | -2.234$\pm$0.127 | -0.165$\pm$0.012

###  A.3 ${\cal A}_{1c}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | 0.020$\pm$0.004 | 0.021$\pm$0.004 | -0.067$\pm$0.020 | 0.022$\pm$0.004
$[1,2]$ | 0.044$\pm$0.010 | 0.048$\pm$0.010 | -0.146$\pm$0.054 | 0.046$\pm$0.011
$[2,4]$ | 0.044$\pm$0.011 | 0.050$\pm$0.012 | -0.151$\pm$0.076 | 0.046$\pm$0.011
$[4,6]$ | 0.03$\pm$0.008 | 0.035$\pm$0.010 | -0.124$\pm$0.081 | 0.032$\pm$0.010
$[1.1,6]$ | 0.037$\pm$0.009 | 0.043$\pm$0.009 | -0.138$\pm$0.073 | 0.040$\pm$0.010
$[15,16]$ | 0.08$\pm$0.008 | 0.098$\pm$0.009 | 1.420$\pm$0.132 | 0.080$\pm$0.008
$[16,18]$ | 0.083$\pm$0.007 | 0.101$\pm$0.009 | 1.412$\pm$0.109 | 0.082$\pm$0.007
$[18,20]$ | 0.069$\pm$0.006 | 0.085$\pm$0.008 | 1.178$\pm$0.087 | 0.069$\pm$0.006
$[15,20]$ | 0.078$\pm$0.007 | 0.095$\pm$0.009 | 1.331$\pm$0.101 | 0.077$\pm$0.007

###  A.4 ${\cal A}_{2c}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | -0.004$\pm$0.016 | -0.004$\pm$0.018 | 0.057$\pm$0.022 | -0.006$\pm$0.016
$[1,2]$ | -0.015$\pm$0.028 | -0.015$\pm$0.035 | 0.125$\pm$0.050 | -0.018$\pm$0.026
$[2,4]$ | -0.025$\pm$0.014 | -0.023$\pm$0.023 | 0.121$\pm$0.070 | -0.026$\pm$0.015
$[4,6]$ | -0.031$\pm$0.016 | -0.027$\pm$0.009 | 0.085$\pm$0.069 | -0.031$\pm$0.016
$[1.1,6]$ | -0.026$\pm$0.009 | -0.023$\pm$0.016 | 0.105$\pm$0.059 | -0.027$\pm$0.010
$[15,16]$ | -0.118$\pm$0.085 | -0.125$\pm$0.074 | -1.197$\pm$0.133 | -0.153$\pm$0.088
$[16,18]$ | -0.128$\pm$0.090 | -0.137$\pm$0.081 | -1.280$\pm$0.132 | -0.18$\pm$0.093
$[18,20]$ | -0.138$\pm$0.097 | -0.149$\pm$0.089 | -1.392$\pm$0.120 | -0.224$\pm$0.100
$[15,20]$ | -0.129$\pm$0.088 | -0.139$\pm$0.082 | -1.302$\pm$0.120 | -0.19$\pm$0.093

###  A.5 ${\cal A}_{2cc}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | -0.076$\pm$0.098 | -0.116$\pm$0.105 | -0.138$\pm$0.114 | -0.074$\pm$0.093
$[1,2]$ | -0.034$\pm$0.043 | -0.125$\pm$0.063 | -0.178$\pm$0.076 | -0.031$\pm$0.041
$[2,4]$ | 0.003$\pm$0.024 | -0.091$\pm$0.052 | -0.146$\pm$0.067 | 0.005$\pm$0.024
$[4,6]$ | 0.025$\pm$0.029 | -0.049$\pm$0.050 | -0.092$\pm$0.074 | 0.026$\pm$0.029
$[1.1,6]$ | 0.008$\pm$0.028 | -0.077$\pm$0.052 | -0.127$\pm$0.078 | 0.010$\pm$0.028
$[15,16]$ | 0.092$\pm$0.079 | 0.721$\pm$0.100 | 1.100$\pm$0.124 | 0.092$\pm$0.079
$[16,18]$ | 0.096$\pm$0.077 | 0.721$\pm$0.096 | 1.096$\pm$0.116 | 0.096$\pm$0.082
$[18,20]$ | 0.083$\pm$0.067 | 0.604$\pm$0.081 | 0.917$\pm$0.087 | 0.082$\pm$0.065
$[15,20]$ | 0.091$\pm$0.074 | 0.680$\pm$0.091 | 1.034$\pm$0.103 | 0.090$\pm$0.075

###  A.6 ${\cal A}_{2ss}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | -0.014$\pm$0.103 | 0.002$\pm$0.107 | 0.015$\pm$0.111 | -0.013$\pm$0.108
$[1,2]$ | 0.040$\pm$0.138 | 0.089$\pm$0.134 | 0.123$\pm$0.136 | 0.043$\pm$0.125
$[2,4]$ | 0.060$\pm$0.131 | 0.122$\pm$0.144 | 0.162$\pm$0.149 | 0.062$\pm$0.136
$[4,6]$ | 0.062$\pm$0.150 | 0.122$\pm$0.142 | 0.158$\pm$0.155 | 0.064$\pm$0.124
$[1.1,6]$ | 0.058$\pm$0.129 | 0.117$\pm$0.147 | 0.154$\pm$0.155 | 0.060$\pm$0.134
$[15,16]$ | 0.122$\pm$0.114 | 0.982$\pm$0.122 | 1.500$\pm$0.146 | 0.123$\pm$0.113
$[16,18]$ | 0.119$\pm$0.098 | 0.919$\pm$0.118 | 1.399$\pm$0.136 | 0.121$\pm$0.100
$[18,20]$ | 0.096$\pm$0.080 | 0.717$\pm$0.092 | 1.088$\pm$0.105 | 0.097$\pm$0.083
$[15,20]$ | 0.112$\pm$0.095 | 0.861$\pm$0.103 | 1.311$\pm$0.120 | 0.113$\pm$0.095

###  A.7 ${\cal A}_{3s}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | 0.021$\pm$0.023 | 0.022$\pm$0.024 | -0.742$\pm$0.948 | 0.392$\pm$0.060
$[1,2]$ | 0.026$\pm$0.027 | 0.029$\pm$0.029 | -1.008$\pm$1.154 | -0.194$\pm$0.223
$[2,4]$ | 0.020$\pm$0.019 | 0.022$\pm$0.023 | -0.803$\pm$0.875 | -1.059$\pm$0.269
$[4,6]$ | 0.013$\pm$0.012 | 0.015$\pm$0.014 | -0.535$\pm$0.561 | -1.686$\pm$0.198
$[1.1,6]$ | 0.017$\pm$0.018 | 0.020$\pm$0.020 | -0.713$\pm$0.800 | -1.223$\pm$0.245
$[15,16]$ | 0.002$\pm$0.002 | 0.002$\pm$0.002 | -0.047$\pm$0.086 | -2.047$\pm$0.039
$[16,18]$ | 0.001$\pm$0.001 | 0.001$\pm$0.002 | -0.024$\pm$0.065 | -1.917$\pm$0.044
$[18,20]$ | 0.0$\pm$0.001 | 0.001$\pm$0.001 | -0.003$\pm$0.047 | -1.494$\pm$0.058
$[15,20]$ | 0.001$\pm$0.001 | 0.001$\pm$0.001 | -0.021$\pm$0.061 | -1.796$\pm$0.049

###  A.8 ${\cal A}_{3sc}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | -0.023$\pm$0.032 | 0.733$\pm$0.888 | 1.169$\pm$1.413 | -0.618$\pm$0.041
$[1,2]$ | -0.030$\pm$0.039 | 0.953$\pm$1.079 | 1.550$\pm$1.774 | -0.802$\pm$0.067
$[2,4]$ | -0.022$\pm$0.025 | 0.681$\pm$0.691 | 1.114$\pm$1.140 | -0.600$\pm$0.085
$[4,6]$ | -0.013$\pm$0.013 | 0.397$\pm$0.35 | 0.647$\pm$0.569 | -0.384$\pm$0.060
$[1.1,6]$ | -0.019$\pm$0.022 | 0.594$\pm$0.611 | 0.970$\pm$0.991 | -0.530$\pm$0.071
$[15,16]$ | -0.001$\pm$0.002 | 0.024$\pm$0.033 | 0.038$\pm$0.054 | -0.046$\pm$0.006
$[16,18]$ | -0.001$\pm$0.001 | 0.014$\pm$0.025 | 0.022$\pm$0.040 | -0.031$\pm$0.004
$[18,20]$ | -0.0$\pm$0.001 | 0.005$\pm$0.012 | 0.008$\pm$0.019 | -0.013$\pm$0.002
$[15,20]$ | -0.0$\pm$0.001 | 0.013$\pm$0.022 | 0.021$\pm$0.036 | -0.028$\pm$0.004

###  A.9 ${\cal A}_{4s}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | -0.004$\pm$0.015 | -0.004$\pm$0.018 | 0.036$\pm$0.057 | -0.029$\pm$0.017
$[1,2]$ | -0.010$\pm$0.017 | -0.008$\pm$0.015 | 0.022$\pm$0.051 | -0.037$\pm$0.019
$[2,4]$ | -0.011$\pm$0.023 | -0.008$\pm$0.018 | -0.008$\pm$0.044 | -0.027$\pm$0.026
$[4,6]$ | -0.008$\pm$0.028 | -0.006$\pm$0.020 | -0.022$\pm$0.037 | -0.015$\pm$0.032
$[1.1,6]$ | -0.010$\pm$0.022 | -0.007$\pm$0.017 | -0.010$\pm$0.043 | -0.023$\pm$0.024
$[15,16]$ | 0.039$\pm$0.03 | 0.042$\pm$0.028 | 0.414$\pm$0.087 | 0.165$\pm$0.030
$[16,18]$ | 0.06$\pm$0.046 | 0.065$\pm$0.042 | 0.621$\pm$0.089 | 0.185$\pm$0.045
$[18,20]$ | 0.100$\pm$0.074 | 0.107$\pm$0.063 | 1.017$\pm$0.093 | 0.225$\pm$0.075
$[15,20]$ | 0.070$\pm$0.050 | 0.075$\pm$0.048 | 0.717$\pm$0.087 | 0.195$\pm$0.050

###  A.10 ${\cal A}_{4sc}$ (in units of $10^{-2}$)

Bin | SM | Case I | Case II | Case III
---|---|---|---|---
$[0.1,1]$ | 0.003$\pm$0.013 | -0.017$\pm$0.025 | -0.028$\pm$0.035 | 0.028$\pm$0.015
$[1,2]$ | 0.006$\pm$0.017 | -0.020$\pm$0.033 | -0.036$\pm$0.052 | 0.040$\pm$0.019
$[2,4]$ | 0.006$\pm$0.019 | -0.011$\pm$0.029 | -0.022$\pm$0.046 | 0.033$\pm$0.018
$[4,6]$ | 0.005$\pm$0.015 | -0.002$\pm$0.027 | -0.006$\pm$0.037 | 0.023$\pm$0.018
$[1.1,6]$ | 0.005$\pm$0.015 | -0.009$\pm$0.028 | -0.017$\pm$0.041 | 0.029$\pm$0.016
$[15,16]$ | 0.010$\pm$0.011 | 0.09$\pm$0.034 | 0.137$\pm$0.056 | 0.015$\pm$0.011
$[16,18]$ | 0.011$\pm$0.010 | 0.091$\pm$0.028 | 0.138$\pm$0.042 | 0.014$\pm$0.011
$[18,20]$ | 0.009$\pm$0.009 | 0.076$\pm$0.019 | 0.116$\pm$0.028 | 0.011$\pm$0.009
$[15,20]$ | 0.010$\pm$0.011 | 0.085$\pm$0.026 | 0.130$\pm$0.038 | 0.013$\pm$0.011

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