LHCP 2013

11institutetext: The Enrico Fermi Institute, The University of Chicago

# The ATLAS Tile Calorimeter Calibration and Performance

Christopher Meyer on behalf of the ATLAS Collaboration 11<EMAIL_ADDRESS>
###### Abstract

A brief summary of the hadronic calorimeter calibration systems and
performance results, in the ATLAS detector at the LHC is given.

##  1 Introduction

The ATLAS Aad:2008zzm  tile calorimeter (TileCal) Aad:2010af  is a sampling,
hadron calorimeter, located at the LHC. The central barrel portion covers
$|\eta|<0.8$, while the extended partitions on either side cover out to
$|\eta|<1.7$. It is composed of alternating layers of plastic scintillating
material and steel (see figure 1), grouped to create cells. In total, 9852
photmultiplier tubes (PMTs) read out energy deposited in the detector. The
calorimeter was designed to have a resolution of $\sigma/E=50\%/\sqrt{E}\oplus
3\%$, and enable a jet energy calibration uncertainty of $<1\%$. The PMT
signal is first shaped into a pulse with FWHM$\sim 50$ ns, then sampled by a
40 MHz analog to digital converter (ADC). To cover the full dynamic range two
gain channels are used, so that smaller signals are amplified before being
sampled by the ADC. This results in precise signal reconstruction over the
full dynamic range. The resulting seven samples of the pulse are fit using the
optimal filter method Usai:2011zz  to determine the amplitude in ADC counts
and timing in ns (see figure 2). Although the digital signal processor which
performs the fit has limited resolution due to the use of fast lookup tables,
the majority of the signals produced by physics show negligible difference
when compared to the full offline reconstruction.

To convert from ADC counts to energy in MeV a series of calibrations is
applied:

  * •

Charge injection system (CIS): provides a calibration $C_{\mathrm{CIS}}$ from
ADC counts to pC.

  * •

Test beam: the initial calibration $C_{\mathrm{testbeam}}$ converting pC to
MeV , derived using test beam results.

  * •

Cesium (${}^{137}\mathrm{Cs}$): provides a relative calibration
$C_{\mathrm{Cs}}$ to account for changes in the scintillating material,
optical fibers, and PMTs since deriving the test beam calibration
Adragna:2009zz .

  * •

Laser: provides a relative calibration $C_{\mathrm{laser}}$ to account for the
drift of the PMTs and optical fibers between ${}^{137}\mathrm{Cs}$ runs.

The total calibration applied to the fitted pulse amplitude $A$ in ADC counts
to derive the measured electromagnetic scale energy $E$ is:

$E[{\mathrm{\ Me\kern-1.00006ptV}}]=C_{\mathrm{testbeam}}\times
C_{\mathrm{Cs}}\times C_{\mathrm{laser}}\times C_{\mathrm{CIS}}\times
A[\mathrm{ADC}]$

Below, the performance of the calibration systems and TileCal as a whole are
discussed in more detail.

Figure 1: One module of the tile calorimeter, showing alternating steel and
scintillating material. Aad:atlaslist  Figure 2: Example pulse shape showing
the 7 sampled points, the fitted pulse shape, and the amplitude, timing, and
pedestal components.

##  2 Calibration Systems

In addition to providing up to date calibration constants for physics signals,
the calibration systems are also staggered such that issues throughout the
physics readout path can be diagnosed. For example, if an issue is found in
the calibration data from the ${}^{137}\mathrm{Cs}$ and Laser systems, but not
in CIS, the problem likely lies in the PMT (see figure 3). This setup is
particularly useful for trouble-shooting unstable high-voltage power supplies
and pathological PMTs. For this reason, as well as providing high quality
calibrations, it is important to keep track of performance for the various
calibration systems, as outlined below.

Figure 3: Overview of the calibration systems in TileCal, and which portions
of the readout electronics they are sensitive to.

###  2.1 Cesium

The cesium system Aad:2010af ; Starchenko:2002ju ; Shalanda:2003rq  circulates
a radioactive ${}^{137}\mathrm{Cs}$ source through 10 km of tubes in TileCal.
Photons with energy of $0.662{\mathrm{\ Me\kern-1.00006ptV}}$ are emitted at a
known rate, and the integrated current is read out from each cell as the
source passes by. This provides a relative calibration $C_{\mathrm{Cs}}$,
which can be combined with the preliminary test beam calibration factor
$C_{\mathrm{testbeam}}$ to convert pC to MeV. Three ${}^{137}\mathrm{Cs}$
sources are circulated through TileCal approximately once per month to measure
the changing conditions of the scintillating material (resulting from
irradiation by physics runs) as well as the PMTs (changes of the PMT vacuum
due to signals flushing out impurities). The results have been cross checked
in situ using muons from cosmic rays as well as physics runs. By using
tracking information to determine the muon momentum, the expected energy
deposited in the calorimeter can be compared with the actual energy measured.

###  2.2 Laser

The laser system Aad:2010af ; Viret:2010zz  sends a pulse of light with a
known amplitude through fiber-optic cables to each PMT. This provides a
relative calibration $C_{\mathrm{laser}}$ with respect to the cesium, tracking
the PMT drift over the period of a month (the time interval between
${}^{137}\mathrm{Cs}$ runs). By definition $C_{\mathrm{laser}}=1$ immediately
following a ${}^{137}\mathrm{Cs}$ run. The evolution of the laser and
${}^{137}\mathrm{Cs}$ calibrations are compared in figure 4 for E1 and E2
cells, located in the crack between the long and extended barrels. Because
these cells have the most direct exposure to the interaction point, they are
expected to show the largest change. Differences are visible because the laser
system is only sensitive to the drift of the PMTs, while the
${}^{137}\mathrm{Cs}$ calibration is also affected by changes in the
scintillating material. For the majority of the cells the PMT and optical
fiber response is much more stable, as shown in figure 5. During the last two
months of 2012 running, the laser calibration changed only $0.5\%$ on average.

Figure 4: Evolution of the ${}^{137}\mathrm{Cs}$ and laser calibration
constants during 2012 data taking. Shown for the E1 and E2 cells, located in
the crack between the long and extended barrel of TileCal Aad:tilepub1 .
Figure 5: Average variation of the laser calibration for low gain channels in
TileCal, shown for the last two months of 2012 data taking Aad:tilepub1 .

###  2.3 Charge Injection System

The charge injection system Aad:2010af ; Anderson:2005ym  injects a pulse of
known charge and records the output in ADC counts. After scanning a range of
input charge, the results are used to derive the $C_{\mathrm{CIS}}$ conversion
factor, converting ADC counts to pC. CIS is generally very stable, with an
average drift of $0.4\%$ during 2012 running (see figure 6). While
instabilities in the electronics can cause shifts of the individual channel
calibrations of up to 1%, these jumps are rare and are generally corrected
within 28 days.

Figure 6: Distribution of fractional change of CIS constants over the 2012 run
period Aad:tilepub1 .

##  3 Performance

In general TileCal provided stable, good quality data throughout Run I at the
LHC. The readout electronics continue to perform well, as seen by the
comparing of a pulse shape from the readout electronics with the reference
pulse shape.The calibration systems obtain stable results, and correct for any
changes due to irradiation before it can affect physics. Good agreement is
observed between data and MC simulation when plotting the mean $E/p$ for
single particle response, shown in figure 7. The high-voltage power supplies
also performed very well, with minimal deviations from the set value. During
long shutdown 1 new radiation-hard low-voltage power supplies are being
installed, which also provide more Gaussian shaped noise with a smaller RMS
(see figure 8). This will reduce the number of tripped modules during physics
runs, as well as further improve the resolution for physics analyses which
rely on TileCal.

Figure 7: Average energy (E) over momentum (p) of single particles measured in
data (black) and MC simulation (red) Aad:tilepub2 . Figure 8: Cell noise in
TileCal readout electronics, comparing the new (red) low voltage power
supplies with the old (blue) Aad:tilepub2 .

##  4 Summary

The calibration systems (${}^{137}\mathrm{Cs}$, Laser, and CIS) provide an up-
to-date status of TileCal performance. In general, the calibrations have shown
to be stable over time. When drifts arise due to changes in the scintillating
material, optical fibers, PMTs, or readout electronics they are quickly caught
and corrected for. TileCal has consistently provided good quality, well
calibrated data for the duration of Run I at the LHC.

## References

  * (1) ATLAS Collaboration, JINST 3, S08003 (2008)
  * (2) ATLAS Collaboration, Eur. Phys. J. C70, 1193 (2010), 1007.5423
  * (3) G. Usai (ATLAS Tile Calorimeter Collaboration), J. Phys. Conf. Ser. 293, 012056 (2011)
  * (4) P. Adragna, C. Alexa, K. Anderson, A. Antonaki, A. Arabidze et al., Nucl. Instrum. Meth. A606, 362 (2009)
  * (5) ATLAS Collaboration, _ATLAS Technical Paper List Of Figures_ , http://twiki.cern.ch/twiki/bin/view/AtlasPublic/AtlasTechnicalPaperListOfFigures (2008)
  * (6) E. Starchenko et al. (ATLAS Collaboration), Nucl. Instrum. Meth. A494, 381 (2002)
  * (7) N. Shalanda et al. (ATLAS Tile Calorimeter Collaboration), Nucl. Instrum. Meth. A508, 276 (2003)
  * (8) S. Viret (LPC ATLAS Collaboration), Nucl. Instrum. Meth. A617, 120 (2010)
  * (9) ATLAS Collaboration, _Approved Tile Calorimeter Plots_ , http://twiki.cern.ch/twiki/bin/view/AtlasPublic/ApprovedPlotsTile (2013)
  * (10) K. Anderson, A. Gupta, F. Merritt, M. Oreglia, J. Pilcher et al., Nucl. Instrum. Meth. A551, 469 (2005)
  * (11) ATLAS Collaboration, _Public Tile Calorimeter Plots for Collision Data_ , http://twiki.cern.ch/twiki/bin/view/AtlasPublic/TileCaloPublicResults (2013)


# A short note on number fields defined by exponential Taylor polynomials

Anuj Jakhar  and  Srinivas Kotyada  Department of Mathematics, Indian
Institute of Technology (IIT) Madras  Department of Mathematics, The Institute
of Mathematical Sciences (IMSc) Chennai<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $n$ be a positive integer and
$f_{n}(x)=1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^{n}}{n!}$ denote the $n$-th
Taylor polynomial of the exponential function. Let $K={\mathbf{Q}}(\theta)$ be
an algebraic number field where $\theta$ is a root of $f_{n}(x)$ and
${\mathbf{Z}}_{K}$ denote the ring of algebraic integers of $K$. In this
paper, we prove that for any prime $p$, $p$ does not divide the index of the
subgroup ${\mathbf{Z}}[\theta]$ in ${\mathbf{Z}}_{K}$ if and only if
$p^{2}\nmid n!$.

###### Key words and phrases:

Ring of algebraic integers; Integral basis and discriminant; Monogenic number
fields.

###### 2010 Mathematics Subject Classification:

11R04; 11R29.

##  1\. Introduction and statements of results

Let $f_{n}(x)=1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^{n}}{n!}$ denote the $n$-th
Taylor polynomial of the exponential function. In 1930, Schur [6] proved that
the Galois group of $f_{n}(x)$ is $A_{n}$, the alternating group on $n$
letters, if $4$ divides $n$ and is $S_{n}$, the symmetric group on $n$
letters, otherwise. In 1987, Coleman [2] gave another proof of this result
using the theory of Newton polygons. He also provided a simple proof of the
irreducibility of $f_{n}(x)$ over the field ${\mathbf{Q}}$ of rational
numbers. Let $K={\mathbf{Q}}(\theta)$ be an algebraic number field and
${\mathbf{Z}}_{K}$ denote the ring of algebraic integers of $K$. In the
present paper, we would like to characterise all the primes dividing the index
of the subgroup ${\mathbf{Z}}[\theta]$ in ${\mathbf{Z}}_{K}$, where $\theta$ a
root of $f_{n}(x)$.

In 1878, Dedekind gave a simple criterion, known as Dedekind Criterion (cf.
[1, Theorem 6.1.4], [3]), which provides a necessary and sufficient condition
for a polynomial $f(x)$ to be satisfied so that $p$ does not divide
$[{\mathbf{Z}}_{K}:Z[\theta]]$, where $\theta$ is a root of $f(x)$.

######  Theorem 1.1.

(Dedekind Criterion) Let $K=\mathbb{Q}(\theta)$ be an algebraic number field
with $f(x)$ as the minimal polynomial of the algebraic integer $\theta$ over
$\mathbb{Q}.$ Let $p$ be a prime and
$\overline{f}(x)=\overline{g}_{1}(x)^{e_{1}}\ldots\overline{g}_{t}(x)^{e_{t}}$
be the factorization of $\overline{f}(x)$ as a product of powers of distinct
irreducible polynomials over $\mathbb{Z}/p\mathbb{Z},$ with each
$g_{i}(x)\in\mathbb{Z}[x]$ monic. Let $M(x)$ denote the polynomial
$\frac{1}{p}(f(x)-g_{1}(x)^{e_{1}}\ldots g_{t}(x)^{e_{t}})$ with coefficients
from $\mathbb{Z}.$ Then $p$ does not divide
$[{\mathbf{Z}}_{K}:\mathbb{Z}[\theta]]$ if and only if for each $i,$ we have
either $e_{i}=1$ or $\bar{g}_{i}(x)$ does not divide $\overline{M}(x).$

Using Dedekind Criterion, Jakhar, Khanduja and Sangwan have given necessary
and sufficient conditions for a prime $p$ to divide the index
$[{\mathbf{Z}}_{K}:{\mathbf{Z}}[\theta]]$ where $\theta$ is a root of an
irreducible polynomial $f(x)=x^{n}+ax^{m}+b\in{\mathbf{Z}}[x]$ over
${\mathbf{Q}}$ (cf. [4], [5]). In this note, we prove the following theorem.

######  Theorem 1.2.

Let $n$ be a positive integer and $p$ be a prime number. Let
$K={\mathbf{Q}}(\theta)$ be an algebraic number field with $\theta$ a root of
$f_{n}(x)=1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^{n}}{n!}$. Then
$p\nmid[{\mathbf{Z}}_{K}:{\mathbf{Z}}[\theta]]$ if and only if $p^{2}\nmid
n!$.

The following corollary is an immediate consequence of the above theorem.

######  Corollary 1.3.

Let $n\geq 4$ be an integer and
$f_{n}(x)=1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^{n}}{n!}$. Let
$K={\mathbf{Q}}(\theta)$ be an algebraic number field with $\theta$ a root of
$f_{n}(x)$, then $2$ divides $[{\mathbf{Z}}_{K}:{\mathbf{Z}}[\theta]]$. In
particular, $\\{1,\theta,\cdots,\theta^{n-1}\\}$ can not be an integral basis
of $K$.

##  2\. Proof of Theorem 1.2.

Let $L={\mathbf{Q}}(\xi)$ with $\xi$ a root of an irreducible polynomial
$g(x)$ and $d_{L}$ denote the discriminant of the field $L$, then it is well
known that the discriminant $D_{g}$ of $g(x)$ and the index
$[{\mathbf{Z}}_{L}:{\mathbf{Z}}[\xi]]$ is connected by the following formula

(2.1) $D_{g}=[{\mathbf{Z}}_{L}:{\mathbf{Z}}[\xi]]^{2}d_{L}.$

It is given in [2] that the discriminant of
$f_{n}(x)=1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^{n}}{n!}$ is given by

(2.2) $D_{f_{n}}=(-1)^{\frac{n(n-1)}{2}}(n!)^{n}.$

###### Proof of Theorem 1.2..

By abuse of language, we take $f_{n}(x)$ as

$f_{n}(x)=x^{n}+nx^{n-1}+\frac{n!}{(n-2)!}x^{n-2}+\cdots+\frac{n!}{2}x^{2}+n!x+n!\in{\mathbf{Z}}[x].$

Keeping in mind (2.2), it is easy to check that the absolute value of the
discriminant of $f_{n}(x)$ is given by $|D_{f_{n}}|=(n!)^{n}.$ In view of
(2.1), we see that if a prime $p$ does not divide $n!$, then $p$ does not
divide $[{\mathbf{Z}}_{K}:{\mathbf{Z}}[\theta]]$. So assume that $p$ is a
divisor of $n!$. Suppose that $i$, $0\leq i\leq n-2$ is the smallest index
such that $p|(n-i)$. We see that

$f_{n}(x)\equiv x^{n}+nx^{n-1}+\cdots+\frac{n!}{(n-i)!}x^{n-i}\equiv
x^{n-i}(x^{i}+nx^{i-1}+\cdots+\frac{n!}{(n-i)!})\mod p.$

Note that $p$ can not divide $i$, because if $p$ divides $i$, then in view of
$p|(n-i)$, we have $p|n$, which contradicts the definition of $i$. So keeping
in mind that $p\nmid i$, the polynomial
$x^{i}+\bar{n}x^{i-1}+\cdots+\overline{\frac{n!}{(n-i)!}}$ belonging to
${\mathbf{Z}}/p{\mathbf{Z}}[x]$ is a separable polynomial. Hence applying
Dedekind Criterion, we see that $p$ does not divide
$[{\mathbf{Z}}_{K}:{\mathbf{Z}}[\theta]]$ if and only if $x$ does not divide
$\overline{M}(x)$, where $M(x)$ is given by

$M(x)=\frac{1}{p}[\frac{n!}{(n-i-1)!}x^{n-i-1}+\cdots+\frac{n!}{2}x^{2}+n!x+n!].$

Thus $p\nmid[{\mathbf{Z}}_{K}:{\mathbf{Z}}[\theta]]$ if and only if $p^{2}$
does not divide $n!$. This completes the proof of the theorem. ∎

## References

  * [1] H. Cohen, A Course in Computational Algebraic Number Theory, Springer- Verlag, Berlin-Heidelberg, 1993.
  * [2] R.F. Coleman, On the Galois groups of the exponential Taylor polynomials, L’Enseignement Mathématique 33 (1987) 183-189.
  * [3] R. Dedekind, Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Götttingen Abh. 23 (1878) 1-23.
  * [4] A. Jakhar, S. K. Khanduja, N. Sangwan, On prime divisors of the index of an algebraic integer, J. Number Theory 166 (2016) 47-61.
  * [5] A. Jakhar, S. K. Khanduja, N. Sangwan, Characterisation of primes dividing the index of a trinomial, Int. J. Number Theory 13 (2017) 2505-2514.
  * [6] I. Schur, Gleichungen ohne Affekt, Gesammelte Abhandlungen 67 (1930) 191-197.


# Exploring the lepton flavor violating decay modes $b\to
s\mu^{\pm}\tau^{\mp}$ in SMEFT approach

Dhiren Panda<EMAIL_ADDRESS>Manas Kumar Mohapatra
<EMAIL_ADDRESS>Rukmani Mohanta<EMAIL_ADDRESS>School of
Physics, University of Hyderabad, Hyderabad-500046, India

###### Abstract

We perform an analysis of the consequences of various new physics operators on
the lepton flavor violating (LFV) decay modes mediated through $b\to
s\ell_{1}\ell_{2}$ transitions. We scrutinize the imprints of the
(pseudo)scalar and axial(vector) operators on the exclusive LFV decay channels
$B_{(s)}\rightarrow(\phi,K^{*},K_{2}^{*})\ell_{1}\ell_{2}$ and
$\Lambda_{b}\rightarrow\Lambda\ell_{1}\ell_{2}$, where $\ell_{1},\ell_{2}$
represent $\mu$ or $\tau$. The new physics parameters are constrained by using
the upper limits of the branching fractions of the $B\to\tau\mu$ and $B\to
K\tau\mu$ processes, assuming the new physics couplings to be real. We then
explore the key observables such as the branching fraction, the forward-
backward asymmetry, and the longitudinal polarisation fraction of the
$B\to(K^{*},\phi,K_{2}^{*})\tau^{\pm}\mu^{\mp}$ decays. In addition, we also
investigate the impact of the new physics couplings on the baryonic
$\Lambda_{b}\to\Lambda\tau^{\pm}\mu^{\mp}$ decay channels mediated by the
$b\to s$ quark level transition. With the experimental prospects at LHCb
upgrade and Belle II, we also predict the upper limits of the above-discussed
observables, which could intrigue the new physics search in these channels.

##  I Introduction

The Standard Model (SM) addresses elementary particles and their interactions,
providing a comprehensive framework that firmly establishes a wide range of
natural phenomena occurring at energies below the electroweak scale. Despite
its remarkable achievements, the pursuit of physics beyond the Standard Model
(BSM) remains crucial. This quest addresses numerous experimental anomalies
and theoretical puzzles, including the prevalence of matter over antimatter in
the universe, the enigma of dark matter and dark energy, as well as the
hierarchy and flavor problems, among others. The potential existence of
physics beyond the Standard Model arises from two distinct avenues. The first
involves indirect effects of new physics (NP) associated with the presence of
heavy new particles, which can modify the Wilson coefficients of the
interaction Hamiltonian within the Standard Model framework. Alternatively,
direct approaches entail detecting new particles through ongoing and upcoming
collider experiments. Among various searches for physics beyond the SM, the
$B$ meson decays have been considered as one of the potential avenues for BSM
search in the indirect approach for many years. In this respect, the flavor-
changing neutral current transition (FCNC) mediated decays, especially the
$(b\to s/d)$ transitions are more captivating for understanding the nature of
rare $b$ quark decays. The LHC at CERN, particularly LHCb LHCb:2022qnv ;
LHCb:2022vje  collaboration recently confirmed the lepton flavor universality
(LFU) violating ratio $R_{K^{(*)}}=\mathcal{B}(B\to
K^{(*)}\mu\mu)/\mathcal{B}(B\to K^{(*)}ee)$ to be consistent with the SM,
which are of the order of unity. On the other hand, there exist various other
observables such as the form factor independent (FFI) observable
$P_{5}^{\prime}$ in $B\to K^{*}\mu^{-}\mu^{+}$ process and the branching
ratios of several decay channels display few sigma deviations from the SM
values. The LHCb LHCb:2013ghj ; LHCb:2015svh  and ATLAS ATLAS:2018gqc
collaborations reported a 3.3$\sigma$ difference from the SM prediction in the
measurement of the celebrated $P_{5}^{\prime}$ observable. The other LFU
violating observables, respecting the $b\to s$ quark level transitions,
$R_{K_{S}^{0}}$ and $R_{K^{*+}}$ LHCb:2021lvy  also show deviations at the
level of $1.4\sigma$ and $1.5\sigma$ compared to the SM predictions,
respectively. On the other hand, the branching fraction of the rare
$B_{s}\to\phi\mu\mu$ decay mode reveals $3.3\sigma$ LHCb:2021zwz  away from
the SM value in $q^{2}\in[1.1,6.0]\leavevmode\nobreak\ \rm GeV^{2}$.

Unlike the lepton flavor-conserving decays, the LFV transitions indicate a
clean probe of NP as they are forbidden in the SM. Various LFV decays in the
charged lepton sector i.e., $\ell_{i}\to\ell_{j}\gamma$,
$\ell_{i}\to\ell_{j}\ell_{k}\bar{\ell}_{k}$ as well as in $B$ meson sector,
mediated through $b\to s\ell_{i}\ell_{j}$ transitions, have been extensively
studied in the literature Lee:2015qra ; Altmannshofer:2015mqa ;
Crivellin:2015mga ; Alonso:2015sja ; Sahoo:2015pzk ; Sahoo:2015wya ;
Mohapatra:2023hhf ; Ali:2023kua . The LFV $B$ meson decays provide an ideal
platform for NP search, particularly the results from LHCb and Belle II
experiments can be used as a guiding principle in this direction. However, so
far we have only the experimental upper limits for these decay modes. The LHCb
experiment reported the upper bound on the branching ratio of the purely
leptonic $B_{s}\to e^{\pm}\mu^{\mp}$ decay channel, which is found to be
$\mathcal{B}(B_{s}\to e^{\pm}\mu^{\mp})<6.3\times 10^{-9}$ at 90% C.L. In
addition, an upper limit of $4.2\times 10^{-5}$ in the branching fraction of
$B\to\tau\mu$ has also been reported by the LHCb experiment LHCb:2019ujz . The
LFV searches in semileptonic $B$ decays, on the other hand, include well-known
decays like $B\to K\ell_{1}\ell_{2}$ processes, where $\ell_{1}$ or
$\ell_{2}=e,\mu,\tau$. Based on Run I data with integrated luminosity 9 ${\rm
fb}^{-1}$, an exclusion limit on $\mathcal{B}(B\to K\mu^{-}e^{+})<7\times
10^{-9}$ is reported by LHCb collaboration LHCb:2019bix . Similar searches
were also performed for $B\to V\ell_{1}\ell_{2}$ ($V=K^{*},\phi$) processes.
The current limit on the branching ratios of $B\to K^{*}\mu e$ and
$B\to\phi\mu e$, set by LHCb, are $6.8\times 10^{-9}$ and $10.1\times
10^{-9}$, respectively. Despite the experimental difficulties due to missing
energy during reconstruction from the environment of $\tau$ lepton present in
the final state of the $B$ meson decays, Belle Belle:2022pcr  experiment put
an upper limit on the branching ratio of $B^{+}\to K^{+}\mu^{\pm}\tau^{\mp}$
decay mode as $3.9\times 10^{-5}$. An analysis is performed using the LHCb
data LHCb:2022wrs  on the $B\to K^{*}\mu^{\pm}\tau^{\mp}$ process. However, no
signal is observed, instead, an upper limit on the branching fraction of $B\to
K^{*}\tau^{+}\mu^{-}$ and $B\to K^{*}\tau^{-}\mu^{+}$ are set to be $1.0\times
10^{-5}$ and $8.2\times 10^{-6}$ LHCb:2022wrs  at the 90$\%$ confidence level.

In this work, we would like to explore the effect of the SMEFT (Standard Model
Effective Field Theory) formalism on the exclusive semileptonic $b\to
s\tau\mu$ decays. Given the current and future experimental prospects, we
mainly focus on the $B\to(K^{*},\phi,K_{2}^{*})\tau^{\pm}\mu^{\mp}$ decays. In
addition, we also investigate the impact of the new physics couplings on the
baryonic $\Lambda_{b}\to\Lambda\tau^{\pm}\mu^{\mp}$ decay channels mediated by
$b\to s$ quark level transition. Our fit anatomy includes the upper limit of
the branching ratios of the leptonic $B\to\tau\mu$ and semileptonic $B\to
K\tau\mu$ processes. Using the constrained values of the NP couplings, we
probe the prominent observables such as the branching fraction, the forward-
backward asymmetry, and the longitudinal polarisation fraction of the above
decay modes in the presence of various SMEFT operators.

The paper is organized in the following manner. In section II, we recapitulate
the theoretical framework of the EFT framework of $b\rightarrow
s\ell_{1}\ell_{2}$ transition. We also discuss on various observable
associated with the $B_{(s)}\to(K^{*},\phi,K_{2}^{*})\ell_{1}\ell_{2}$ and
$\Lambda_{b}\to\Lambda\ell_{1}\ell_{2}$ processes. In section III, we
constrain the new physics parameter space in the presence of various SMEFT
operators. We interpret the outcome followed by numerical analysis of the
$B\to(K^{*},\phi,K_{2}^{*})\tau^{\pm}\mu^{\mp}$ and the baryonic
$\Lambda_{b}\to\Lambda\tau^{\pm}\mu^{\mp}$ decays in the presence of new
physics coefficients. Finally, we conclude our work in section IV.

##  II Theoretical Framework

The most generic effective Hamiltonian describing the $b\to
s\ell_{1}^{-}\ell_{2}^{+}$ decay is given as Das:2019omf

$\begin{split}\mathcal{H}_{eff}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\frac{\alpha_{em}}{4\pi}\sum_{i=9,10,S,P}\big{(}C_{i}^{\ell_{1}\ell_{2}}(\mu)\mathcal{O}_{i}^{\ell_{1}\ell_{2}}(\mu)+C_{i}^{{}^{\prime}\ell_{1}\ell_{2}}(\mu)\mathcal{O^{\prime}}_{i}^{\ell_{1}\ell_{2}}(\mu)\big{)},\end{split}$
(1)

where $G_{F}$ and $V_{tb}V_{ts}^{*}$ are the Fermi coupling constant and CKM
matrix elements respectively. The relevant operators for the process are
expressed as follows,

$\displaystyle\begin{split}&\mathcal{O}^{(\prime)}_{9}=\big{[}\bar{s}\gamma^{\mu}P_{L(R)}b\big{]}\big{[}\ell_{2}\gamma_{\mu}\ell_{1}\big{]}\,,\quad\mathcal{O}^{(\prime)}_{10}=\big{[}\bar{s}\gamma^{\mu}P_{L(R)}b\big{]}\big{[}\ell_{2}\gamma_{\mu}\gamma_{5}\ell_{1}\big{]}\;,\\\
&\mathcal{O}_{S}^{(\prime)}=\big{[}\bar{s}P_{R(L)}b\big{]}\big{[}\ell_{2}\ell_{1}\big{]}\,,\quad\quad\quad\mathcal{O}_{P}^{(\prime)}=\big{[}\bar{s}P_{R(L)}b\big{]}\big{[}\ell_{2}\gamma_{5}\ell_{1}\big{]}\,.\end{split}$
(2)

Here, the operators $\mathcal{O}_{9,10,S,P}$ represent the vector, axial-
vector, scalar, and pseudoscalar operators, respectively. The primed operators
$\mathcal{O}_{i}^{\prime}$ can be obtained by flipping the chirality of the
former operators $\mathcal{O}_{i}$. The $C^{(\prime)}_{9,10,S,P}$ are the
Wilson coefficients that have zero value in the SM and can have a non-zero
value in various new physics scenarios. In SM, the leptons $\ell_{1},\ell_{2}$
correspond to the same flavor which is usually considered as $\ell$.

### The decay observables of $b\to s\ell_{1}\ell_{2}$ transitions

####  II.0.1 Exclusive $\Lambda_{b}\rightarrow\Lambda\ell_{1}\ell_{2}$ decay
channel

To illustrate the kinematics of the
$\Lambda_{b}\rightarrow\Lambda\ell_{1}\ell_{2}$ decay mode, we assume the
baryon $\Lambda_{b}$ is at rest, while the final state particles $\Lambda$ and
the dilepton pair travel along the positive and negative $z$-axis,
respectively. The momenta assigned to the particles $\Lambda_{b}$, $\Lambda$,
$\ell_{1}$, and $\ell_{2}$ are represented by the symbols $p$, $k$, $q_{1}$,
and $q_{2}$ respectively. The spin of the baryon $\Lambda_{b}$ ($\Lambda$) on
to the z-axis in the rest frame is denoted as $s_{p}(s_{k})$. The decay
amplitude of the exclusive $\Lambda_{b}\rightarrow\Lambda\ell_{1}\ell_{2}$
process can be written as Das:2019omf ,

$\begin{split}\mathcal{M}^{\lambda_{1},\lambda_{2}}(s_{p},s_{k})=-\frac{V_{tb}V_{ts}^{*}}{2v^{2}}\frac{\alpha_{em}}{4\pi}\sum_{i=L,R}\Big{[}\sum_{\lambda}\eta_{\lambda}H_{VA,\lambda}^{i,s_{p},s_{k}}L_{i,\lambda}^{\lambda_{2},\lambda_{1}}+H_{SP}^{i,s_{p},s_{k}}L_{i}^{\lambda_{2},\lambda_{1}}\Big{]}.\end{split}$
(3)

In the dilepton rest frame, $q^{\mu}$ represents the four-momentum of the
dilepton pair, and $\theta_{\ell}$ denotes the angle between the $\ell_{1}$
and $z$-axis of the dilepton rest frame. Here the hadronic helicity amplitude
$H_{VA(SP),\lambda}^{i,s_{p},s_{k}}$ correspond to the vector-axial vector
(scalar-pseudoscalar) operators whereas the
$L_{i,\lambda}^{\lambda_{2},\lambda_{1}}$, and
$L_{i}^{\lambda_{2},\lambda_{1}}$ are the leptonic helicity amplitudes.

For the detailed expression of the lepton helicity amplitude, we refer to Ref.
Das:2019omf . Here $i=L,R$ correspond to the chiralities of the lepton
current, and $\lambda=t,\pm 1,0$ represents the helicity state of the virtual
gauge boson that decays into the dilepton pair. The symbols $\lambda_{1,2}$
are the helicities of the leptons. Additionally, the parameters $\eta_{t}$ and
$\eta_{\pm 1,0}$ are assigned a value of 1 and -1, respectively. The
expressions of $H_{VA,\lambda}^{i,s_{p},s_{k}}$ and
$H_{SP,\lambda}^{i,s_{p},s_{k}}$ in terms of Wilson coefficients (WCs) and
form factors (FFs) can be found in Das:2018sms . Alternatively, in the
literature, transversity amplitudes: $A_{\perp(||)1}^{i},\leavevmode\nobreak\
A_{\perp(||)0}^{i}$ and $A_{S\perp(||)},\leavevmode\nobreak\ A_{P\perp(||)}$
are often employed instead of the hadronic helicity amplitudes. The
expressions for these transversity amplitudes can be found in Ref. Das:2019omf
. The amplitudes $L_{i,\lambda}^{\lambda_{2},\lambda_{1}}$ and
$L_{i}^{\lambda_{2},\lambda_{1}}$ are defined as follows:

$\displaystyle\begin{split}&L^{\lambda_{2},\lambda_{1}}_{L(R)}=\langle\bar{\ell}_{2}(\lambda_{2})\ell_{1}(\lambda_{1})|\bar{\ell}_{2}(1\mp\gamma_{5})\ell_{1}|0\rangle\,,\\\
&L^{\lambda_{2},\lambda_{1}}_{L(R),\lambda}=\bar{\epsilon}^{\mu}(\lambda)\langle\bar{\ell}_{2}(\lambda_{2})\ell_{1}(\lambda_{1})|\bar{\ell}_{2}\gamma_{\mu}(1\mp\gamma_{5})\ell_{1}|0\rangle\,.\end{split}$
(4)

Here, $\epsilon^{\mu}$ represents the polarization vector of the virtual gauge
boson that decays into the dilepton pair. The detailed calculations of
$L_{i,\lambda}^{\lambda_{2},\lambda_{1}}$ and
$L_{i}^{\lambda_{2},\lambda_{1}}$ can be found in Ref. Das:2019omf . Based on
these definitions, one can derive the differential branching ratio of
$\Lambda_{b}\rightarrow\Lambda\ell_{1}\ell_{2}$ as follows:

$\begin{split}\frac{d^{2}{\cal
B}}{dq^{2}d\cos\theta_{l}}=\frac{3}{2}(K_{1ss}\sin^{2}\theta_{l}+K_{1cc}\cos^{2}\theta_{l}+K_{1c}\cos\theta_{l})\;.\end{split}$
(5)

The angle $\theta_{\ell}$ can vary within the range
$-\pi\leq\theta_{\ell}\leq\pi$. The long-distance aspect of the decay is
encapsulated within the $\Lambda_{b}\rightarrow\Lambda$ transition matrix
elements, which are parameterized in terms of six $q^{2}$-dependent form
factors, denoted as $f_{t,0,\perp}^{V/A}$ Feldmann:2011xf . For our numerical
analysis, we utilize the form factors obtained from lattice QCD calculations
Detmold:2016pkz . The resulting differential branching ratio is given by:

$\frac{d\mathcal{B}}{dq^{2}}=2K_{1ss}+K_{1cc}\,.$ (6)

Additionally, the FBA (Forward-backward asymmetry) is expressed as,

$A^{\ell}_{\rm FB}=\frac{3}{2}\frac{K_{1c}}{K_{1ss}+K_{1cc}}\,,$ (7)

where the squared dilepton invariant mass ($q^{2}$) varies within the range
$(m_{1}+m_{2})^{2}\leq q^{2}\leq(m_{\Lambda_{b}}-m_{\Lambda})^{2}$.

####  II.0.2 Exclusive $B\to K_{2}^{*}\ell_{1}\ell_{2}$ decay channel

Utilizing the effective Hamiltonian governing $b\to s\ell_{1}\ell_{2}$
transition, provided in Eq. (1), one can derive the transition amplitude for
the $B\to K_{2}^{*}\ell_{1}\ell_{2}$ decay mode. The hadronic matrix elements
of vector and axial-vector currents for $B\to K_{2}^{*}$ transitions can be
parameterized in terms of $q^{2}$-dependent FFs: $V(q^{2})$ and
$A_{0,1,2}(q^{2})$. The expressions in detail are given as follows Wang:2010ni
,

$\displaystyle\langle
K_{2}^{*}(k,\epsilon^{*})|\bar{s}\gamma^{\mu}b|\overline{B}(p)\rangle$
$\displaystyle=$
$\displaystyle-\frac{2V(q^{2})}{m_{B}+m_{K_{2}^{*}}}\epsilon^{\mu\nu\rho\sigma}\epsilon^{*}_{T\nu}p_{\rho}k_{\sigma},$
$\displaystyle\langle
K_{2}^{*}(k,\epsilon^{*})|\bar{s}\gamma^{\mu}\gamma_{5}b|\overline{B}(p)\rangle$
$\displaystyle=$ $\displaystyle
i(m_{B}+m_{K_{2}^{*}})A_{1}(q^{2})\left[\epsilon^{*\mu}_{T}-\frac{\epsilon^{*}_{T}\cdot
q}{q^{2}}q^{\mu}\right]+2im_{K_{2}^{*}}A_{0}(q^{2})\frac{\epsilon^{*}_{T}\cdot
q}{q^{2}}q^{\mu}$ (8) $\displaystyle-iA_{2}(q^{2})\frac{\epsilon^{*}_{T}\cdot
q}{m_{B}+m_{K_{2}^{*}}}\left[(p+k)^{\mu}-\frac{m_{B}^{2}-m_{K_{2}^{*}}^{2}}{q^{2}}q^{\mu}\right],$

where, $p$ ($k$) is the four momentum of $B$ ($K_{2}^{*}$) meson.

We employ the recent values of form factors from the light cone QCD sum rule
(LCSR) approach given in Ref. Aliev:2019ojc . In this method, the form factors
can be expressed as

$F^{B\to
T}=\frac{1}{1-q^{2}/m_{R,F}^{2}}\sum_{n=0}^{1}\alpha_{n}^{F}[z(q^{2})-z(0)]^{n},$
(9)

where
$z(q^{2})=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}}}$,
$t_{\pm}=(m_{B}{\pm}m_{K_{2}^{*}})^{2}$, $t_{0}=t_{+}(1-\sqrt{1-t_{-}/t_{+}})$
and $m_{R,F}$ is the resonance mass corresponding to the form factor. The
associated parameters are provided in Table 1. The resonance masses employed
in our numerical calculations are given as

$\displaystyle m_{R,A_{0}}=5.336\hskip 2.84544pt{\rm GeV},\hskip
14.22636ptm_{R,V}=5.412\hskip 2.84544pt{\rm GeV},\hskip
14.22636ptm_{R,(A_{1},A_{3})}=5.829\hskip 2.84544pt{\rm GeV}.$ (10)

Form factor | $\alpha_{0}$ | $\alpha_{1}$
---|---|---
$V^{B\to K_{2}^{*}}$ | $0.22^{+0.11}_{-0.08}$ | $-0.90^{+0.37}_{-0.50}$
$A^{B\to K_{2}^{*}}_{0}$ | $0.30^{+0.01}_{-0.05}$ | $-1.23^{+0.23}_{-0.23}$
$A^{B\to K_{2}^{*}}_{1}$ | $0.19^{+0.09}_{-0.07}$ | $-0.46^{+0.19}_{-0.25}$
$A^{B\to K_{2}^{*}}_{2}$ | $0.11^{+0.05}_{-0.06}$ | $-0.40^{+0.23}_{-0.16}$
Table 1:  Fit parameters corresponding to $B\to K_{2}^{*}$ form factors in
LCSR method

The differential decay distribution describing three body $B\to
K_{2}^{*}\ell_{1}\ell_{2}$ decay can be expressed as Kumbhakar:2022szr

$\frac{d^{2}\Gamma}{dq^{2}d\cos\theta_{\ell}}=A(q^{2})+B(q^{2})\cos\theta_{\ell}+C(q^{2})\cos^{2}\theta_{\ell}\;,$
(11)

where $\theta_{\ell}$ is the leptonic polar angle which describes the angle
made by lepton $\ell_{1}$ to the dilepton rest frame. The $q^{2}$-dependent
coefficients $A(q^{2}),B(q^{2})$, and $C(q^{2})$ are given in Appendix -B.
Using Eq. (11), the differential decay rate can be given as

$\frac{d\Gamma}{dq^{2}}=2\left(A+\frac{C}{3}\right),$ (12)

and the lepton FBA is found to be

$A_{\rm
FB}(q^{2})=\frac{1}{d\Gamma/dq^{2}}\left(\int_{0}^{1}d\cos\theta_{\ell}\frac{d^{2}\Gamma}{d\cos\theta_{\ell}dq^{2}}-\int_{-1}^{0}d\cos\theta_{\ell}\frac{d^{2}\Gamma}{d\cos\theta_{\ell}dq^{2}}\right)=\frac{B}{2\left(A+\frac{C}{3}\right)}\;.$
(13)

####  II.0.3 Exclusive $B\rightarrow(K^{*},\phi)\ell_{1}^{-}\ell_{2}^{+}$
decay channel

For the analysis of the decay
$B\rightarrow(K^{*},\phi)\ell_{1}^{-}\ell_{2}^{+}$, we adopt the kinematics
and the angular conventions as described in references Korner:1989qb ;
Becirevic:2016zri . The hadronic matrix elements involve a more extensive set
of $q^{2}$-dependent form factors which include

$\displaystyle\langle\bar{K}^{\ast}(k)|\bar{s}\sigma_{\mu\nu}q^{\nu}(1-\gamma_{5})b|\bar{B}(p)\rangle$
$\displaystyle=2i\varepsilon_{\mu\nu\rho\sigma}\varepsilon^{\ast\nu}p^{\rho}k^{\sigma}T_{1}(q^{2})-[(\varepsilon^{\ast}\cdot
q)(2p-q)_{\mu}-\varepsilon_{\mu}^{\ast}(m_{B}^{2}-m_{K^{\ast}}^{2})]T_{2}(q^{2})$
$\displaystyle-(\varepsilon^{\ast}\cdot
q)\Big{[}\frac{q^{2}}{m_{B}^{2}-m_{K^{\ast}}^{2}}(p+k)_{\mu}-q_{\mu}\Big{]}T_{3}(q^{2}),$
$\displaystyle\langle\bar{K}^{\ast}(k)|\bar{s}\gamma^{\mu}(1-\gamma_{5})b|\bar{B}(p)\rangle$
$\displaystyle=\varepsilon_{\mu\nu\rho\sigma}\varepsilon^{\ast\nu}p^{\rho}k^{\sigma}\frac{2V(q^{2})}{m_{B}+m_{K^{\ast}}}-i\Big{[}\varepsilon_{\mu}^{\ast}(m_{B}+m_{K^{\ast}})A_{1}(q^{2})$
$\displaystyle-i(p+k)_{\mu}(\varepsilon^{\ast}\cdot
q)\frac{A_{2}(q^{2})}{m_{B}+m_{K^{\ast}}}-q_{\mu}(\varepsilon^{\ast}\cdot
q)\frac{2m_{K^{\ast}}}{q^{2}}[A_{3}(q^{2})-A_{0}(q^{2})]\Big{]}.$ (14)

The polarisation vector of the $K^{*}(\phi)$ meson is denoted as
$\varepsilon_{\mu}$. The form factor $A_{3}(q^{2})$ related to $A_{1}(q^{2})$
and $A_{2}(q^{2})$ is given by
$2m_{V}A_{3}(q^{2})=(m_{B}+m_{V})A_{1}(q^{2})-(m_{B}-m_{V})A_{2}(q^{2})$.

The $q^{2}$-dependent differential branching ratio, after integrating the full
angular distribution over the angles given in Appendix C, is expressed as

$\begin{split}\frac{d{\cal
B}}{dq^{2}}=\frac{1}{4}[3I_{1}^{c}(q^{2})+6I_{1}^{s}(q^{2})-I_{2}^{c}(q^{2})-2I_{2}^{s}(q^{2})].\end{split}$
(15)

The form factors associated with the transversity amplitude are obtained using
the LCSR method Bharucha:2015bzk  and are expressed as follows:

$F_{i}(q^{2})=\frac{1}{1-q^{2}/m_{R,i}^{2}}\sum_{k}\alpha_{k}^{i}[z(q^{2})-z(0)]^{k},$
(16)

where
$z(q^{2})=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}}}$,
$t_{\pm}=(m_{B}{\pm}m_{V})^{2}$, $t_{0}=t_{+}(1-\sqrt{1-t_{-}/t_{+}})$ and
$m_{R,F}$ is the resonance mass corresponding to the form factor. In our
calculations, we use the resonance masses as follows:

$\displaystyle m_{R,A_{0}}=5.336\hskip 2.84544pt{\rm GeV},\hskip
14.22636ptm_{R,(V,T_{1})}=5.412\hskip 2.84544pt{\rm GeV},\hskip
14.22636ptm_{R,(A_{1},A_{3},T_{2},T_{3})}=5.829\hskip 2.84544pt{\rm GeV}.$
(17)

. We apply the above formalism originally developed for $B\to K^{*}$ decay to
analyze the process $B\to\phi\ell_{1}\ell_{2}$. This is achieved by
straightforwardly substituting the relevant mass and form factors for the
corresponding vector meson $\phi$. The numerical values for the parameters
associated with the form factors with $1\sigma$ uncertainty are given in Table
2.

$F_{i}$ | $\alpha_{0}$ | $\alpha_{1}$ | $\alpha_{2}$
---|---|---|---
$V$ | $0.34\pm 0.04$ | $-1.05\pm 0.24$ | $2.37\pm 1.39$
$A_{0}$ | $0.36\pm 0.05$ | $-1.04\pm 0.27$ | $1.12\pm 1.35$
$A_{1}$ | $0.27\pm 0.03$ | $0.39\pm 0.19$ | $-0.11\pm 0.48$
$A_{3}$ | $0.26\pm 0.03$ | $0.60\pm 0.20$ | $0.12\pm 0.84$
$T_{1}$ | $0.28\pm 0.03$ | $-0.89\pm 0.19$ | 4$1.95\pm 1.10$
$T_{2}$ | $0.28\pm 0.03$ | $0.40\pm 0.18$ | $0.36\pm 0.51$
$T_{3}$ | $0.67\pm 0.08$ | $1.48\pm 0.49$ | $1.92\pm 1.96$

$F_{i}$ | $\alpha_{0}$ | $\alpha_{1}$ | $\alpha_{2}$
---|---|---|---
$V$ | $0.39\pm 0.03$ | $-1.03\pm 0.25$ | $3.50\pm 1.55$
$A_{0}$ | $0.39\pm 0.05$ | $-0.78\pm 0.26$ | $2.41\pm 1.48$
$A_{1}$ | $0.30\pm 0.03$ | $0.48\pm 0.19$ | $0.29\pm 0.65$
$A_{3}$ | $0.25\pm 0.03$ | $0.76\pm 0.20$ | $0.71\pm 0.96$
$T_{1}$ | $0.31\pm 0.03$ | $-0.87\pm 0.19$ | 4$2.75\pm 1.19$
$T_{2}$ | $0.31\pm 0.03$ | $0.58\pm 0.19$ | $0.89\pm 0.71$
$T_{3}$ | $0.68\pm 0.07$ | $2.11\pm 0.46$ | $4.94\pm 2.25$

Table 2: Form factors for $B\to K^{*}\ell_{1}\ell_{2}$ (left panel) and
$B_{s}\to\phi\ell_{1}\ell_{2}$ (right panel).

##  III phenomenological implication

SMEFT-inspired LFV transitions have garnered significant attention. Given the
absence of any new particles observed thus far beyond the electroweak scale,
it is conjectured that the scale of NP is substantially higher compared to the
current running scale of the LHC. In this context, the SMEFT offers a powerful
framework for elucidating LFV decays which encompasses a comprehensive set of
dimension-six operators constructed from the fields of the Standard model. The
corresponding Lagrangian is given as follows Grzadkowski:2010es :

$\begin{split}\mathcal{L}_{SMEFT}&=\mathcal{L}_{SM}-\frac{1}{\Lambda_{\rm
cut}^{2}}\Big{\\{}[C^{(3)}_{lq}]^{ij\alpha\beta}(\bar{Q}^{i}\gamma^{\mu}\sigma^{a}Q^{j})(\bar{L}^{\alpha}\gamma_{\mu}\sigma^{a}L^{\beta})\\\
&+[C^{(1)}_{lq}]^{ij\alpha\beta}(\bar{Q}^{i}\gamma^{\mu}Q^{j})(\bar{L}^{\alpha}\gamma_{\mu}L^{\beta})+[C_{ledq}^{ij\alpha\beta}(\bar{Q}^{i}d_{R}^{j})(\bar{e}_{R}^{\alpha}L^{\beta})]\Big{\\}}.\end{split}$
(18)

Here $Q$ and $L$ represent the left-handed quark and lepton fields, which
transform as doublets under $SU(2)$ whereas $e_{R}$ and $d_{R}$ are the
singlet right-handed charged leptons and down-type quarks, respectively. Here
$\Lambda_{\rm cut}$ is the cut-off scale which can be associated with the mass
of the heavy NP degrees of freedom. The above equation includes the set of all
the dimension six operators contributing to the $b\to s\ell_{1}\ell_{2}$
transitions. It should be noted that none of these above operators contain
tensor currents.

The SMEFT Wilson coefficients can be constrained from low energy processes,
and are related to the Wilson coefficients in Eqn (1) as

$\displaystyle
C_{9}^{\ell_{1}\ell_{2}}=-C_{10}^{\ell_{1}\ell_{2}}=+\frac{v^{2}}{\Lambda_{\rm
cut}^{2}}\frac{\pi}{\alpha_{em}|{V_{tb}V^{*}_{ts}}|}([C^{(3)}_{\ell
q}]^{23\ell_{1}\ell_{2}}+[C^{(1)}_{\ell q}]^{23\ell_{1}\ell_{2}}),$ (19)
$\displaystyle
C_{9}^{{}^{\prime}\ell_{1}\ell_{2}}=-{C^{\prime}}_{10}^{\ell_{1}\ell_{2}}=+\frac{v^{2}}{\Lambda_{\rm
cut}^{2}}\frac{\pi}{\alpha_{em}{V_{tb}V^{*}_{ts}}|}([C_{\ell
d}]^{23\ell_{1}\ell_{2}},$ (20) $\displaystyle
C_{S}^{\ell_{1}\ell_{2}}=-C_{P}^{\ell_{1}\ell_{2}}=+\frac{v^{2}}{\Lambda_{\rm
cut}^{2}}\frac{\pi}{\alpha_{em}{V_{tb}V^{*}_{ts}}|}([C_{leqd}]^{23\ell_{1}\ell_{2}},$
(21) $\displaystyle
C_{S}^{{}^{\prime}\ell_{1}\ell_{2}}=-{C^{\prime}}_{P}^{\ell_{1}\ell_{2}}=+\frac{v^{2}}{\Lambda_{\rm
cut}^{2}}\frac{\pi}{\alpha_{em}{V_{tb}V^{*}_{ts}}|}([C_{\ell
eqd}^{*}]^{32\ell_{1}\ell_{2}}\;.$ (22)

Now, we examine the constraints on various combinations of SMEFT Wilson
coefficients derived from measurements of mesonic LFV decays. We consider the
branching ratios of the decay modes
$\bar{B}_{s}\rightarrow\ell_{1}^{-}\ell_{2}^{+}$ and $B\rightarrow
K\ell_{1}^{-}\ell_{2}^{+}$ and use their experimental upper limits provided in
Table 3 at 90$\%$ confidence level.

Observable | Exp.limit
---|---
$\mathcal{B}(B_{s}\to\mu^{\pm}\tau^{\mp})$ |  $4.2\times 10^{-5}$ LHCb:2019ujz
$\mathcal{B}(B^{+}\to K^{+}\mu^{-}\tau^{+})$ |  $3.9\times 10^{-5}$ LHCb:2020khb
$\mathcal{B}(B^{+}\to K^{+}\mu^{+}\tau^{-})$ |  $4.5\times 10^{-5}$ BaBar:2012azg
Table 3: Experimental upper limits for LFV $B$ decays at 90$\%$ C.L.

The branching fraction of these decay modes are expressed as Becirevic:2016oho
; Gratrex:2015hna

$\displaystyle\mathcal{B}(\bar{B}_{s}\to\ell^{-}_{1}\ell^{+}_{2})=$
$\displaystyle\,\frac{\tau_{B_{s}}}{64\pi^{3}}\frac{\alpha_{\text{em}}^{2}G_{F}^{2}|V_{tb}V_{ts}^{*}|^{2}}{m_{B_{s}}^{3}}f_{B_{s}}^{2}\,\lambda^{1/2}(m_{B_{s}}^{2},m_{\ell_{1}}^{2},m_{\ell_{2}}^{2})$
(23) $\displaystyle\times$
$\displaystyle\left\\{[m_{B_{s}}^{2}-(m_{\ell_{1}}-m_{\ell_{2}})^{2}]\left|(m_{\ell_{1}}+m_{\ell_{2}}){C}_{10-}+\frac{m_{B_{s}}^{2}}{m_{b}+m_{s}}{C}_{P-}\right|^{2}\right.$
$\displaystyle+\left.[m_{B_{s}}^{2}-(m_{\ell_{1}}+m_{\ell_{2}})^{2}]\left|(m_{\ell_{1}}-m_{\ell_{2}})({C}_{9-})+\frac{m_{B_{s}}^{2}}{m_{b}+m_{s}}({C}_{S-})\right|^{2}\right\\},$

$\displaystyle\mathcal{B}(B^{+}\to K^{+}\ell_{1}^{-}\ell_{2}^{+})$
$\displaystyle=10^{-8}\bigg{\\{}c^{S}_{\ell_{1}\ell_{2}}\left|C_{S+}\right|^{2}+c^{P}_{\ell_{1}\ell_{2}}\left|C_{P+}\right|^{2}+c^{9+}_{\ell_{1}\ell_{2}}\left|C_{9+}\right|^{2}$
(24)
$\displaystyle+c^{10+}_{\ell_{1}\ell_{2}}\left|C_{10+}\right|^{2}+c^{S9}_{\ell_{1}\ell_{2}}\,\mathrm{Re}[C_{S+}^{*}C_{9+}]+c^{P10}_{\ell_{1}\ell_{2}}\,\mathrm{Re}[C_{P+}^{*}C_{10+}]\bigg{\\}}\,,$

where we have adopted the notation $C_{X\pm}=C_{X}\pm C_{X}^{\prime}$. To
calculate the branching ratios for these processes, we use the values of
particle masses from PDG ParticleDataGroup:2020ssz , CKM matrix elements from
the UT-fit collaborations utfit , and the decay constant of $B_{s}$ meson as
$f_{B_{s}}$=215 MeV Balasubramamian:2019wgx . The values of the coefficients
$c^{i}_{\ell_{1}\ell_{2}}$ are taken from Bordone:2021usz . For our analysis,
we utilize the Eqs. (23) and (24) to constrain the various combinations of
SMEFT Wilson coefficients. For convenience, we sum over the oppositely charge
lepton decays modes e.g.,
$\mathcal{B}(B_{s}\to\mu^{\pm}\tau^{\mp})=(B_{s}\to\mu^{-}\tau^{+})+(B_{s}\to\mu^{+}\tau^{-})$
and similarly for other channel as well. Here we consider the constraints
coming from the $\mu\tau$ decay channel. It is generally established that
consideration of primed operators is unappealing to fit the $b\to s\ell\ell$
data Alguero:2021anc ; Altmannshofer:2021qrr ; Hurth:2021nsi ;
Ciuchini:2019usw ; Gratrex:2015hna  and hence, the use of only unprimed
operators is well accepted.

In Fig. 1, we illustrate the constrained parameter space governing the SMEFT
Wilson coefficients for the final states involving $\tau\mu$. Here we focus on
the two-dimensional scenario involving the SMEFT new physics couplings
$[C^{(1)}_{\ell q}]^{23\ell_{1}\ell_{2}}$ and
$[C_{leqd}]^{23\ell_{1}\ell_{2}}$ which enables us to elucidate the intricate
relationship between these coefficients within the SMEFT framework.

##  IV Implications OF the Results

Figure 1: Constraints on the NP couplings obtained from the branching ratios
of $B\to\tau\mu$ (top left) and $B\to K^{+}\tau\mu$ (top right). The lower
panel illustrates for the combined measurement of ${\cal B}(B\to\tau\mu$) and
${\cal B}(B\to K^{+}\tau\mu$).

In this section, we illustrate the implications of our results in $b\to
s\ell_{1}\ell_{2}$ transitions. The allowed parameter space is constrained by
using the combined measurements of $B_{s}\to\tau^{\pm}\mu^{\mp}$ and $B\to
K^{+}\tau^{\pm}\mu^{\mp}$, experimental upper limits set at 90$\%$ C.L. is
shown in Fig. 1. Here we focus only on the $\tau\mu$ mode. For our
computation, we set the cut-off scale $\Lambda_{\rm cut}=1$ TeV and use the
benchmark values of the Wilson coefficients as $[C^{(1)}_{\ell
q}]^{23\ell_{1}\ell_{2}}=-0.0484$ and
$[C_{leqd}]^{23\ell_{1}\ell_{2}}=-0.0302$.

Using these values, we analyze the $q^{2}$ dependencies of the key observables
such as the branching fraction, the lepton forward-backward asymmetry, and the
longitudinal polarization fractions in the discussed $b\to s\ell_{1}\ell_{2}$
transitions. The plots presented in Sec. IV utilize the 1$\sigma$ standard
error values of the form factors.

###  IV.1 Impact of SMEFT NP coefficients on
$\Lambda_{b}\to\Lambda\ell_{1}\ell_{2}$ decay observables

  * •

Branching ratio: Considering the $\Lambda_{b}\to\Lambda\tau^{\pm}\mu^{\mp}$
decay process, the $q^{2}$-dependent differential branching ratio is depicted
in the Fig 2. In the high $q^{2}$ region the decay rate distribution indicates
that the contribution of the new Wilson coefficient is quite substantial. The
solid black line in these plots represents the value of the differential
branching ratio considering the central values of the form factors. The band
on both sides of the central solid line indicates $1\sigma$ uncertainty for
the corresponding observable.

Figure 2: Branching ratio (in units of $10^{-6}$) of
$\Lambda_{b}\to\Lambda\tau^{+}\mu^{-}$ (left) and
$\Lambda_{b}\to\Lambda\tau^{-}\mu^{+}$ (right).

From the differential branching ratio plot, it is evident that both decay
modes are expected to have similar order of branching fractions. However, the
predicted value of $\mathcal{B}(\Lambda_{b}\to\Lambda\mu^{-}\tau^{+})$ is
found to be slightly higher than that of
$\mathcal{B}(\Lambda_{b}\to\Lambda\mu^{+}\tau^{-})$.

  * •

Forward-backward asymmetry: Taking into account the lepton forward-backward
asymmetry, the $\Lambda_{b}\to\Lambda\tau^{+}\mu^{-}$ decay exhibits a zero
crossing in the FBA curve, whereas in the case of
$\Lambda_{b}\to\Lambda\tau^{-}\mu^{+}$, no such zero crossing is observed.
Such asymmetry curves are shown in Fig. 3. The zero crossing for the former
$\Lambda_{b}$ decay occurs at $q^{2}$=12 ${\rm GeV}^{2}$. For
$\Lambda_{b}\to\Lambda\tau^{-}\mu^{+}$, the forward backward asymmetry remains
negative throughout all $q^{2}$ regions.

Figure 3: Forward-backward asymmetry of $\Lambda_{b}\to\Lambda\tau^{+}\mu^{-}$
(left) and $\Lambda_{b}\to\Lambda\tau^{-}\mu^{+}$ (right).

The numerical estimation for the branching ratio and $A_{FB}$ are computed
using the central value of the form factors and benchmark points of the SMEFT
Wilson coefficient. This has been shown in Table 4. As mentioned before the
obtained results correspond to the cut-off scale $\Lambda_{cut}$=1 TeV, and
the evolution of the branching ratios of the
$\Lambda_{b}\to\Lambda\mu^{\pm}\tau^{\mp}$ with the cut-off scale are shown in
the Fig. 4.

Figure 4:  Variation of the branching ratio of
$\Lambda_{b}\to\Lambda\tau^{+}\mu^{-}$ (left),
$\Lambda_{b}\to\Lambda\tau^{-}\mu^{+}$ (right) with the cut-off scale.

###  IV.2 Impact of SMEFT NP coefficients on $B\to K_{2}^{*}\ell_{1}\ell_{2}$
decay observables

  * •

Branching ratio: Fig. 5 shows $q^{2}$-dependent differential branching ratios
of the $B\to K_{2}^{*}\mu^{\pm}\tau^{\mp}$ processes. The contribution of
SMEFT NP couplings is significant in the intermediate invariant mass squared
region. The behavior of the differential branching ratios for both decay modes
are of similar. However, the value of ${\cal B}(B\to
K_{2}^{*}\mu^{+}\tau^{-})$ is higher than that for the ${\cal B}(B\to
K_{2}^{*}\mu^{-}\tau^{+})$.

Figure 5: Branching ratio (in units of $10^{-6}$) of $B\to
K_{2}^{*}\mu^{-}\tau^{+}$ (left) and $B\to K_{2}^{*}\mu^{+}\tau^{-}$ (right).

Figure 6: Forward-backward asymmetry of the $B\to K_{2}^{*}\tau^{+}\mu^{-}$
(left) and $B\to K_{2}^{*}\tau^{-}\mu^{+}$(right).

  * •

Forward-backward asymmetry: Fig. 6 depicts the $q^{2}$ dependency of the
forward-backward asymmetry. The two plots illustrate distinct behaviors for
the two different decay modes. For $B\to K_{2}^{*}\mu^{-}\tau^{+}$ decay, the
observable $A_{FB}$ is negative throughout the entire $q^{2}$ region, while
for $B\to K_{2}^{*}\mu^{+}\tau^{-}$ decay, the forward-backward asymmetry
remains positive throughout the $q^{2}$ region.

The numerical estimation is given in Table 4, whereas, the cut-off dependent
branching fraction is depicted in Fig. (7)

Figure 7: Cut off dependent branching ratio of $B\to K_{2}^{*}\tau^{+}\mu^{-}$
(left) and $B\to K_{2}^{*}\tau^{-}\mu^{+}$ (right).

###  IV.3 Impact of SMEFT NP coefficients on
$B\to(K^{*},\phi)\ell_{1}\ell_{2}$ decay observables

  * •

Branching ratio: The numerical values for the expected branching ratios of the
decay process are provided in Table 4. In Fig. 8, we present the
$q^{2}$-dependent differential branching ratio of
$B_{(s)}\to(K^{*},\phi)\tau^{\pm}\mu^{\mp}$ decay processes. Each mode of
$B_{(s)}\to(K^{*},\phi)$ process has almost similar $q^{2}$ dependencies of
the differential branching ratio irrespective of the charge of the heavier
lepton. Therefore, we have shown a single curve for both modes
$\tau^{\pm}\mu^{\mp}$.

Figure 8: Branching ratio (in the units of $10^{-6}$) of $B\to K^{*}\tau\mu$
(left) and $B\to\phi\tau\mu$ (right).

  * •

Forward-backward asymmetry: We see no zero crossing for both of the modes
irrespective of the charge of heavy lepton. For both $B(B_{s})\to
K^{*}(\phi)\tau\mu$ decay, the observable $A_{FB}$ remains positive throughout
the $q^{2}$ region. This is shown in Fig. 9. We also provide the prediction
for the forward-backward asymmetry for the decay process which is provided in
Table 4.

Figure 9: Forward-backward asymmetry of $B\to K^{*}\tau^{\pm}\mu^{\mp}$ (left)
and $B_{s}\to\phi\tau^{\pm}\mu^{\mp}$ (right)

  * •

Longitudinal polarisation asymmetry: The $q^{2}$ dependent lepton longitudinal
polarisation asymmetry of $B\to K^{*}\tau\mu$ and $B\to\phi\tau\mu$ decays are
shown in Fig. 10. The observables have similar behavior irrespective of the
charges of the lepton pair. In addition, the numerical values of $F_{L}$ are
also given in the Table 4.

Figure 10: Longitudinal polarization asymmetry of $B\to K^{*}\tau\mu$(left)and
$B\to\phi\tau\mu$ (right)

Figure 11:  Cutoff dependent branching ratio of $B\to K^{*}\tau\mu$ (left) and
$B\to\phi\tau\mu$ (right).

As mentioned earlier, for our numerical computations, we set the cutoff limit
$\Lambda_{\rm cut}$=1 TeV. The evolution of the branching ratio for the decay
process is depicted in Fig. 11. Similarly the evolution of the $[C^{(1)}_{\ell
q}]^{23\ell_{1}\ell_{2}}$ and $[C_{leqd}]^{23\ell_{1}\ell_{2}}$ can be shown
similarly.

Decay mode Observable | $\mathcal{B}$ | $\mathcal{A}_{FB}$ | $\mathcal{F}_{L}$
---|---|---|---
$\Lambda_{b}\to\Lambda\tau^{-}\mu^{+}$ | $\leq 1.0\times 10^{-5}$ | $\leq 0.030$ | $-$
$\Lambda_{b}\to\Lambda\tau^{+}\mu^{-}$ | $\leq 1.49\times 10^{-5}$ | $\leq-0.18$ | $-$
$B\to K^{*}\tau\mu$ | $\leq 2.805\times 10^{-5}$ | $\leq 0.365$ | $\leq 0.431$
$B_{s}\to\phi\tau\mu$ | $\leq 3.02\times 10^{-5}$ | $\leq 0.243$ | $\leq 0.564$
$B\to K_{2}^{*}\tau^{+}\mu^{-}$ | $\leq 5.856\times 10^{-6}$ | $\leq-0.104$ | $-$
$B\to K_{2}^{*}\tau^{-}\mu^{+}$ | $\leq 9.359\times 10^{-6}$ | $\leq 0.271$ | $-$
Table 4: Predicted upper limits of $\mathcal{B}$ and $\mathcal{A_{FB}}$ for
the $\Lambda_{b}\to\Lambda\tau^{\pm}\mu^{\mp}$,
$B\to(K^{*},\phi)\tau^{\pm}\mu^{\mp}$ and $B\to K_{2}^{*}\tau^{\pm}\mu^{\mp}$
decays, with $\mathcal{F_{L}}$ prediction for
$B\to(K^{*},\phi)\tau^{\pm}\mu^{\mp}$.

##  V Conclusion

The observation of any LFV decays indicates a clear signal of the presence of
new physics beyond the SM, as such decays are strictly forbidden within its
framework. Various extensions of the SM have been proposed to explain the
origin of lepton flavor universality-violating couplings. This in turn
suggests the presence of lepton flavor-violating decays, thus providing
motivating insights to conduct a closer examination of LFV decays. In this
work, we have studied the lepton flavor violating $B_{(s)}$ decays based on
$b\to s\tau\mu$ transitions. For this purpose, we constrained the parameter
space of NP couplings weighted by (pseudo)scalar and axial(vector) operators,
by using the the upper limits of Br($B\to\tau\mu$) and Br($B\to K\tau\mu$) in
the presence of SMEFT Wilson coefficients. Using such constrained NP
parameters, we have investigated the impact on the branching fraction,
forward-backward asymmetry and the lepton polarisation asymmetry of the
exclusive $B_{(s)}\rightarrow(\phi,K^{*},K_{2}^{*})\tau\mu$ and
$\Lambda_{b}\rightarrow\Lambda\tau\mu$ processes. We observed that in the
presence of new SMEFT couplings, these observables receive significant
contributions. Furthermore, we presented the cut-off scale dependency of the
branching ratios for the $b\to s\tau\mu$ decays. Additionally, we made
predictions for the upper limits of the aforementioned observables as
summarized in Table 4. The findings reveal that our predicted values are
substantial and within the reach of current or forthcoming experimental
limits. If detected in the experiments in the future, this would offer a
definitive indication of new physics.

##  VI Acknowledgement

DP would like to acknowledge the support of the Prime Minister’s Research
Fellowship, Government of India. MKM acknowledges IoE PDRF, University of
Hyderabad for the financial support. RM would like to acknowledge the
University of Hyderabad IoE project grant no. RC1-20-012.

##  Appendix A The Angular coefficients of
$\Lambda_{b}\to\Lambda\ell_{1}\ell_{2}$ process

The of the angular coefficients $K_{1ss,1cc,1c}$ can be expressed as follows:

$\displaystyle
K_{1ss,1cc}=K_{1ss,1cc}^{VA}+K_{1ss,1cc}^{SP}+K_{1ss,1cc}^{int},$ (25)

where, the $K_{1ss,1cc}^{VA}$, $K_{1ss,1cc}^{SP}$ and $K_{1ss,1cc}^{int}$ are
given as follows,

$\displaystyle K_{1ss}^{\rm VA}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\bigg{(}2|A_{||0}^{R}|^{2}+|A_{||1}^{R}|^{2}+2|A_{\perp
0}^{R}|^{2}+|A_{||1}^{R}|^{2}+\\{R\leftrightarrow L\\}\bigg{)}$ (26)
$\displaystyle-\frac{m_{+}^{2}+m_{-}^{2}}{4q^{2}}\bigg{[}\bigg{(}|A^{R}_{\|_{0}}|^{2}+|A^{R}_{\perp_{0}}|^{2}+\\{R\leftrightarrow
L\\}\bigg{)}-\bigg{(}|A_{\perp
t}|^{2}+\\{\perp\leftrightarrow\|\\}\bigg{)}\bigg{]}$
$\displaystyle+\frac{m_{+}^{2}-m_{-}^{2}}{4q^{2}}\bigg{[}2\rm
Re\bigg{(}A^{R}_{\perp_{0}}A^{\ast L}_{\perp_{0}}+A^{R}_{\perp_{1}}A^{\ast
L}_{\perp_{1}}+\\{\perp\leftrightarrow\|\\}\bigg{)}\bigg{]}$
$\displaystyle-\frac{m_{+}^{2}m_{-}^{2}}{4q^{4}}\bigg{[}\bigg{(}|A_{\perp
1}^{R}|^{2}+|A_{||1}^{R}|^{2}+\\{R\leftrightarrow
L\\}\bigg{)}+2|A_{\|t}|^{2}+2|A_{\perp t}|^{2}\bigg{]},$ $\displaystyle
K_{1cc}^{\rm VA}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\bigg{(}|A_{\perp
1}^{R}|^{2}+|A_{||1}^{R}|^{2}+\\{R\leftrightarrow
L\\}\bigg{)}+\frac{m_{+}^{2}+m_{-}^{2}}{4q^{2}}$ (27)
$\displaystyle\times\bigg{[}\bigg{(}|A^{R}_{||_{0}}|^{2}-|A^{R}_{||_{1}}|^{2}+|A^{R}_{\perp_{0}}|^{2}-|A^{R}_{\perp_{1}}|^{2}+\\{R\leftrightarrow
L\\}\bigg{)}+\bigg{(}|A_{\perp t}|^{2}+|A_{||t}|^{2}\bigg{)}\bigg{]}$
$\displaystyle+\frac{m_{+}^{2}-m_{-}^{2}}{4q^{2}}\bigg{[}2\rm
Re\bigg{(}A^{R}_{\perp_{0}}A^{*L}_{\perp_{0}}\
+A^{R}_{\perp_{1}}A^{L}_{\perp_{1}}+\\{\perp\leftrightarrow||\\}\bigg{)}\bigg{]}$
$\displaystyle-\frac{m_{+}^{2}m_{-}^{2}}{2q^{4}}\bigg{[}\bigg{(}|A_{\perp
0}^{R}|^{2}+|A_{||0}^{R}|^{2}+\\{R\leftrightarrow
L\\}\bigg{)}+|A_{||t}|^{2}+|A_{\perp t}|^{2}\bigg{]},$ $\displaystyle
K_{1c}^{\rm VA}$ $\displaystyle=$
$\displaystyle-\beta_{\ell}\beta_{\ell}^{\prime}\bigg{(}A^{R}_{\perp_{1}}A^{*R}_{||_{1}}-\\{R\leftrightarrow
L\\}\bigg{)}+\beta_{\ell}\beta_{\ell}^{\prime}\frac{m_{+}m_{-}}{q^{2}}\rm
Re\bigg{(}A_{||0}^{L}A_{||t}^{*}+A_{\perp 0}^{L}A_{\perp t}^{*}\bigg{)}.$ (28)
$\displaystyle K_{1ss}^{\rm SP}=\frac{1}{4}\bigg{(}|A_{\rm
S\perp}|^{2}+|A_{\rm
P\perp}|^{2}+\\{\perp\leftrightarrow\|\\}\bigg{)}-\frac{m_{+}^{2}}{4q^{2}}\big{(}|A_{S||}|^{2}+|A_{S\perp}|^{2}\big{)}-\frac{m_{-}^{2}}{4q^{2}}\big{(}|A_{P||}|^{2}+|A_{P\perp}|^{2}\big{)},$
$\displaystyle K_{1cc}^{\rm SP}=\frac{1}{4}\bigg{(}|A_{\rm
P\perp}|^{2}+|A_{\rm
S\perp}|^{2}+\\{\perp\leftrightarrow\|\\}\bigg{)}-\frac{m_{+}^{2}}{4q^{2}}\big{(}|A_{S||}|^{2}+|A_{S\perp}|^{2}\big{)}-\frac{m_{-}^{2}}{4q^{2}}\big{(}|A_{P||}|^{2}+|A_{P\perp}|^{2}\big{)},$
$\displaystyle K_{1c}^{\rm SP}=0.$ (29) $\displaystyle K_{1ss}^{\rm int}$
$\displaystyle=$ $\displaystyle\frac{m_{+}}{2\sqrt{q^{2}}}\rm
Re\bigg{(}A_{||t}A_{P||}^{*}+A_{\perp
t}A_{P\perp}^{*}\bigg{)}+\frac{m_{-}}{2\sqrt{q^{2}}}Re\bigg{(}A_{||t}A_{S||}^{*}+A_{\perp
t}A_{S\perp}^{*}\bigg{)}$ (30)
$\displaystyle-\frac{m_{+}^{2}m_{-}}{2q^{2}\sqrt{q^{2}}}Re\bigg{(}A_{||t}A_{S||}^{*}+A_{\perp
t}A_{S\perp}^{*}\bigg{)}-\frac{m_{+}m_{-}^{2}}{2q^{2}\sqrt{q^{2}}}Re\bigg{(}A_{||t}A_{P||}^{*}+A_{\perp
t}A_{P\perp}^{*}\bigg{)},$ $\displaystyle K_{1cc}^{\rm int}$ $\displaystyle=$
$\displaystyle\frac{m_{+}}{2\sqrt{q^{2}}}Re\bigg{(}A_{||t}A_{P||}^{*}+A_{\perp
t}A_{P\perp}^{*}\bigg{)}+\frac{m_{-}}{2\sqrt{q^{2}}}Re\bigg{(}A_{||t}A_{S||}^{*}+A_{\perp
t}A_{S\perp}^{*}\bigg{)}$ (31)
$\displaystyle-\frac{m_{+}^{2}m_{-}}{2q^{2}\sqrt{q^{2}}}Re\bigg{(}A_{||t}A_{S||}^{*}+A_{\perp
t}A_{S\perp}^{*}\bigg{)}-\frac{m_{+}m_{-}^{2}}{2q^{2}\sqrt{q^{2}}}Re\bigg{(}A_{||t}A_{P||}^{*}+A_{\perp
t}A_{P\perp}^{*}\bigg{)},$ $\displaystyle K_{1c}^{\rm int}$ $\displaystyle=$
$\displaystyle\frac{\beta_{\ell}\beta_{\ell}^{\prime}}{2\sqrt{q^{2}}}Re\bigg{(}A_{S||}A_{||0}^{L*}+A_{S\perp}A_{\perp
0}^{L*}+A_{S||}A_{||0}^{R*}+A_{|S\perp}A_{\perp 0}^{R*}\bigg{)}$ (32)
$\displaystyle+\frac{\beta_{\ell}\beta_{\ell}^{\prime}}{2\sqrt{q^{2}}}Re\bigg{(}A_{P||}A_{||0}^{L*}+A_{P\perp}A_{\perp
0}^{L*}+-A_{P||}A_{||0}^{R*}-A_{|S\perp}A_{\perp 0}^{R*}\bigg{)}.$

Here, we have defined $m_{\mp}=m_{1}\mp m_{2}$ with $m_{1},m_{2}$ as the
masses of $\ell_{1}$ and $\ell_{2}$, respectively.

##  Appendix B Inputs required for $B\to K_{2}^{*}\ell_{1}\ell_{2}$ decay mode

$\displaystyle C(q^{2})$ $\displaystyle=$
$\displaystyle\frac{3}{8}\beta_{+}^{2}\beta_{-}^{2}\left\\{\left(|A_{L}^{\parallel}|^{2}+|A_{L}^{\perp}|^{2}-2|A_{L}^{0}|^{2}\right)+\left(L\to
R\right)\right\\},$ (33) $\displaystyle B(q^{2})$ $\displaystyle=$
$\displaystyle\frac{3}{2}\beta_{+}\beta_{-}\left\\{\rm
Re\left[A_{L}^{\perp*}A_{L}^{\parallel}-(L\to
R)\right]+\frac{m_{+}m_{-}}{q^{2}}\rm Re\left[A_{L}^{0*}A_{L}^{t}+(L\to
R)\right]\right.$ (34) $\displaystyle\left.+\frac{m_{+}}{\sqrt{q^{2}}}\rm
Re\left[A_{S}^{*}(A_{L}^{0}+A_{R}^{0})\right]-\frac{m_{-}}{\sqrt{q^{2}}}\rm
Re\left[A_{SP}^{*}(A_{L}^{0}-A_{R}^{0})\right]\right\\},$ $\displaystyle
A(q^{2})$ $\displaystyle=$
$\displaystyle\frac{3}{4}\left\\{\frac{1}{4}\left[\left(1+\frac{m_{+}^{2}}{q^{2}}\right)\beta_{-}^{2}+\left(1+\frac{m_{-}^{2}}{q^{2}}\right)\beta_{+}^{2}\right]\left(|A_{L}^{\parallel}|^{2}+|A_{L}^{\perp}|^{2}+(L\to
R)\right)\right.$ (35)
$\displaystyle+\frac{1}{2}\left(\beta_{-}^{2}+\beta_{+}^{2}\right)\left(|A_{L}^{0}|^{2}+|A_{R}^{0}|^{2}\right)+\frac{4m_{1}m_{2}}{q^{2}}\rm
Re\left[A_{L}^{0*}A_{R}^{0}+A_{L}^{\parallel*}A_{R}^{\parallel}+A_{R}^{\perp}A_{L}^{\perp*}-A_{L}^{t}A_{R}^{t*}\right]$
$\displaystyle+\frac{1}{2}\left(\beta_{+}^{2}+\beta_{-}^{2}-2\beta_{+}^{2}\beta_{-}^{2}\right)\left(|A_{L}^{t}|^{2}+|A_{R}^{t}|^{2}\right)+\frac{1}{2}\left(|A_{SP}|^{22}+|A_{S}|^{22}\right)$
$\displaystyle+\frac{2m_{-}}{\sqrt{q^{2}}}\beta_{+}^{2}\rm
Re\left[A_{S}(A_{L}^{t}+A_{R}^{t})^{*}\right]-\frac{2m_{+}}{\sqrt{q^{2}}}\beta_{-}^{2}\rm
Re\left[A_{SP}(A_{L}^{t}-A_{R}^{t})^{*}]\right\\}.$

##  Appendix C Inputs required for $B\to(K^{*},\phi)\ell_{1}\ell_{2}$ process

The full angular distribution for the decay process can be expressed as

$\begin{split}\frac{d^{4}B(B\rightarrow\bar{K}^{*}(\rightarrow
K\pi)\ell_{\alpha}^{-}\ell_{\beta}^{+})}{dq^{2}dcos\theta_{\ell}dcos\theta_{K}d\phi}=\frac{9}{32\pi}I(q^{2},\theta_{\ell},\theta_{K},\phi),\end{split}$
(36)

where

$\displaystyle I(q^{2},\theta_{\ell},\theta_{K},\phi)=$ $\displaystyle
I_{1}^{s}(q^{2})\sin^{2}\theta_{K}+I_{1}^{c}(q^{2})\cos^{2}\theta_{K}+[I_{2}^{s}(q^{2})\sin^{2}\theta_{K}+I_{2}^{c}(q^{2})\cos^{2}\theta_{K}]\cos
2\theta_{\ell}$
$\displaystyle+I_{3}(q^{2})\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\phi+I_{4}(q^{2})\sin 2\theta_{K}\sin 2\theta_{\ell}\cos\phi$
$\displaystyle+I_{5}(q^{2})\sin
2\theta_{K}\sin\theta_{\ell}\cos\phi+[I_{6}^{s}(q^{2})\sin^{2}\theta_{K}+I_{6}^{c}(q^{2})\cos^{2}\theta_{K}]\cos\theta_{\ell}$
$\displaystyle+I_{7}(q^{2})\sin
2\theta_{K}\sin\theta_{\ell}\sin\phi+I_{8}(q^{2})\sin 2\theta_{K}\sin
2\theta_{\ell}\sin\phi$
$\displaystyle+I_{9}(q^{2})\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin 2\phi,$
(37)

with $\theta_{\ell}$ , $\theta_{K}$ and $\phi$ varies as
$-\pi\leq\theta_{\ell},\theta_{K}\leq\pi$ and $0\leq\phi\leq 2\pi$
respectively.
The angular coefficients in terms of transversity amplitude can be expressed
as:

$\displaystyle I_{1}^{s}(q^{2})$
$\displaystyle=\frac{4m_{1}m_{2}}{q^{2}}\mathrm{Re}\left(A_{\parallel}^{L}A_{\parallel}^{R\ast}+A_{\perp}^{L}A_{\perp}^{R\ast}\right)+\biggl{[}|A_{\perp}^{L}|^{2}+|A_{\parallel}^{L}|^{2}+(L\to
R)\biggr{]}\frac{\lambda_{q}+2[q^{4}-(m_{1}^{2}-m_{2}^{2})^{2}]}{4q^{4}},$
$\displaystyle I_{1}^{c}(q^{2})$
$\displaystyle=\frac{8m_{1}m_{2}}{q^{2}}\mathrm{Re}(A_{0}^{L}A_{0}^{R\ast}-A_{t}^{L}A_{t}^{R\ast})+\bigl{[}|A_{0}^{L}|^{2}+|A_{0}^{R}|^{2}\bigr{]}\frac{q^{4}-(m_{1}^{2}-m_{2}^{2})^{2}}{q^{4}}$
$\displaystyle\hskip
99.58464pt-2\frac{(m_{1}^{2}-m_{2}^{2})^{2}-q^{2}(m_{1}^{2}+m_{2}^{2})}{q^{4}}\bigl{(}|A_{t}^{L}|^{2}+|A_{t}^{R}|^{2}\bigr{)},$
$\displaystyle I_{2}^{s}(q^{2})$
$\displaystyle=\frac{\lambda_{q}}{4q^{4}}[|A_{\parallel}^{L}|^{2}+|A_{\perp}^{L}|^{2}+(L\to
R)],$ $\displaystyle I_{2}^{c}(q^{2})$
$\displaystyle=-\frac{\lambda_{q}}{q^{4}}(|A_{0}^{R}|^{2}+|A_{0}^{L}|^{2}),$
$\displaystyle I_{3}(q^{2})$
$\displaystyle=\frac{\lambda_{q}}{2q^{4}}[|A_{\perp}^{L}|^{2}-|A_{\parallel}^{L}|^{2}+(L\to
R)],$ $\displaystyle I_{4}(q^{2})$
$\displaystyle=-\frac{\lambda_{q}}{\sqrt{2}q^{4}}\mathrm{Re}(A_{\parallel}^{L}A_{0}^{L\ast}+(L\to
R)],$ $\displaystyle I_{5}(q^{2})$
$\displaystyle=\frac{\sqrt{2}\lambda_{q}^{1/2}}{q^{2}}\left[\mathrm{Re}(A_{0}^{L}A_{\perp}^{L\ast}-(L\to
R))-\frac{m_{1}^{2}-m_{2}^{2}}{q^{2}}\mathrm{Re}(A_{t}^{L}A_{\parallel}^{L\ast}+(L\to
R))\right],$ $\displaystyle I_{6}^{s}(q^{2})$
$\displaystyle=-\frac{2\lambda_{q}^{1/2}}{q^{2}}[\mathrm{Re}(A_{\parallel}^{L}A_{\perp}^{L\ast}-(L\to
R))],$ $\displaystyle I_{6}^{c}(q^{2})$
$\displaystyle=-\frac{4\lambda_{q}^{1/2}}{q^{2}}\frac{m_{1}^{2}-m_{2}^{2}}{q^{2}}\mathrm{Re}(A_{0}^{L}A_{t}^{L\ast}+(L\to
R)),$ $\displaystyle I_{7}(q^{2})$
$\displaystyle=-\frac{\sqrt{2}\lambda_{q}^{1/2}}{q^{2}}\left[\frac{m_{1}^{2}-m_{2}^{2}}{q^{2}}\mathrm{Im}(A_{\perp}^{L}A_{t}^{L\ast}+(L\to
R))+\mathrm{Im}(A_{0}^{L}A_{\parallel}^{L\ast}-(L\to R))\right],$
$\displaystyle I_{8}(q^{2})$
$\displaystyle=\frac{\lambda_{q}}{\sqrt{2}q^{4}}\mathrm{Im}(A_{0}^{L}A_{\perp}^{L\ast}+(L\to
R)),$ $\displaystyle I_{9}(q^{2})$
$\displaystyle=-\frac{\lambda_{q}}{q^{4}}\mathrm{Im}(A_{\perp}^{R}A_{\parallel}^{R\ast}+A_{\perp}^{L}A_{\parallel}^{L\ast}).$
(38)

The transversity amplitude can be written as follows Becirevic:2016zri ,

$\displaystyle A_{\perp}^{L(R)}=$ $\displaystyle{\cal
N}_{K^{\ast}}\sqrt{2}\lambda_{B}^{1/2}\left[(C_{9}\mp
C_{10})\frac{V(q^{2})}{m_{B}+m_{K^{\ast}}}\right],$ $\displaystyle\hskip
5.69054ptA_{\parallel}^{L(R)}=$ $\displaystyle-{\cal
N}_{K^{\ast}}\sqrt{2}(m_{B}^{2}-m_{K^{\ast}}^{2})\left[(C_{9}\mp
C_{10})\frac{A_{1}(q^{2})}{m_{B}-m_{K^{\ast}}}\right],$ $\displaystyle
A_{0}^{L(R)}=$ $\displaystyle-\frac{{\cal
N}_{K^{\ast}}}{2m_{K^{\ast}}\sqrt{q^{2}}}(C_{9}\mp
C_{10})\left[(m_{B}^{2}-m_{K^{\ast}}^{2}-q^{2})(m_{B}+m_{K^{\ast}})A_{1}(q^{2})-\frac{\lambda_{B}A_{2}(q^{2})}{m_{B}+m_{K^{\ast}}}\right],$
$\displaystyle A_{t}^{L(R)}=$ $\displaystyle-{\cal
N}_{K^{\ast}}\frac{\lambda_{B}^{1/2}}{\sqrt{q^{2}}}\left[(C_{9}\mp
C_{10})+\frac{q^{2}}{m_{b}+m_{s}}\left(\frac{C_{S}}{m_{1}-m_{2}}\mp\frac{C_{P}}{m_{1}+m_{2}}\right)\right]A_{0}(q^{2}).$
(39)

where,

$\displaystyle N_{K^{*}}=$ $\displaystyle
V_{tb}V_{ts}^{*}\bigg{[}\frac{\tau_{B_{d}}G_{F}^{2}\alpha_{em}^{2}\sqrt{\lambda_{B}\lambda_{q}}}{3\times
2^{10}\pi^{5}m_{B}^{3}}\bigg{]}^{\frac{1}{2}},$
$\displaystyle\lambda(a^{2},b^{2},c^{2})=$ $\displaystyle
a^{4}+b^{4}+c^{4}-2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}).$ (40)

In the above equation, $G_{F}$ is the Fermi coupling constant whereas
$\tau_{B_{d}}$ represents the life time of $B_{d}$ meson. The term
proportional to $V_{tb}V_{ts}$ survives in the SM limit.

## References

  * (1) R. Aaij et al., “Test of lepton universality in $b\rightarrow s\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett., vol. 131, no. 5, p. 051803, 2023.
  * (2) R. Aaij et al., “Measurement of lepton universality parameters in $B^{+}\to K^{+}\ell^{+}\ell^{-}$ and $B^{0}\to K^{*0}\ell^{+}\ell^{-}$ decays,” Phys. Rev. D, vol. 108, no. 3, p. 032002, 2023.
  * (3) R. Aaij et al., “Measurement of Form-Factor-Independent Observables in the Decay $B^{0}\to K^{*0}\mu^{+}\mu^{-}$,” Phys. Rev. Lett., vol. 111, p. 191801, 2013.
  * (4) R. Aaij et al., “Angular analysis of the $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay using 3 fb-1 of integrated luminosity,” JHEP, vol. 02, p. 104, 2016.
  * (5) M. Aaboud et al., “Angular analysis of $B^{0}_{d}\rightarrow K^{*}\mu^{+}\mu^{-}$ decays in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS detector,” JHEP, vol. 10, p. 047, 2018.
  * (6) R. Aaij et al., “Tests of lepton universality using $B^{0}\to K^{0}_{S}\ell^{+}\ell^{-}$ and $B^{+}\to K^{*+}\ell^{+}\ell^{-}$ decays,” Phys. Rev. Lett., vol. 128, no. 19, p. 191802, 2022.
  * (7) R. Aaij et al., “Branching Fraction Measurements of the Rare $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ and $B^{0}_{s}\rightarrow f_{2}^{\prime}(1525)\mu^{+}\mu^{-}$\- Decays,” Phys. Rev. Lett., vol. 127, no. 15, p. 151801, 2021.
  * (8) C.-J. Lee and J. Tandean, “Minimal lepton flavor violation implications of the $b\to s$ anomalies,” JHEP, vol. 08, p. 123, 2015.
  * (9) W. Altmannshofer and I. Yavin, “Predictions for lepton flavor universality violation in rare B decays in models with gauged $L_{\mu}-L_{\tau}$,” Phys. Rev. D, vol. 92, no. 7, p. 075022, 2015.
  * (10) A. Crivellin, G. D’Ambrosio, and J. Heeck, “Explaining $h\to\mu^{\pm}\tau^{\mp}$, $B\to K^{*}\mu^{+}\mu^{-}$ and $B\to K\mu^{+}\mu^{-}/B\to Ke^{+}e^{-}$ in a two-Higgs-doublet model with gauged $L_{\mu}-L_{\tau}$,” Phys. Rev. Lett., vol. 114, p. 151801, 2015.
  * (11) R. Alonso, B. Grinstein, and J. Martin Camalich, “Lepton universality violation and lepton flavor conservation in $B$-meson decays,” JHEP, vol. 10, p. 184, 2015.
  * (12) S. Sahoo and R. Mohanta, “Lepton flavor violating B meson decays via a scalar leptoquark,” Phys. Rev. D, vol. 93, no. 11, p. 114001, 2016.
  * (13) S. Sahoo and R. Mohanta, “Scalar leptoquarks and the rare $B$ meson decays,” Phys. Rev. D, vol. 91, no. 9, p. 094019, 2015.
  * (14) M. K. Mohapatra, L. Nayak, R. Dhamija, and A. Giri, “Delving into the $B_{s}\to\ell\ell^{\prime}$, $B_{(s)}\to(K^{(*)},\phi,f_{2}^{\prime},K_{2}^{*})\ell\ell^{\prime}$ processes,” 2 2023.
  * (15) M. I. Ali, U. Chattopadhyay, N. Rajeev, and J. Roy, “SMEFT analysis of charged lepton flavor violating $B$-meson decays,” 12 2023.
  * (16) R. Aaij et al., “Search for the lepton-flavour-violating decays $B^{0}_{s}\to\tau^{\pm}\mu^{\mp}$ and $B^{0}\to\tau^{\pm}\mu^{\mp}$,” Phys. Rev. Lett., vol. 123, no. 21, p. 211801, 2019.
  * (17) R. Aaij et al., “Search for Lepton-Flavor Violating Decays $B^{+}\to K^{+}{\mu}^{\pm}e^{\mp}$,” Phys. Rev. Lett., vol. 123, no. 24, p. 241802, 2019\.
  * (18) S. Watanuki et al., “Search for the Lepton Flavor Violating Decays B+→K+$\tau$$\pm$$\ell$$\mp$ ($\ell$=e, $\mu$) at Belle,” Phys. Rev. Lett., vol. 130, no. 26, p. 261802, 2023.
  * (19) R. Aaij et al., “Search for the lepton-flavour violating decays $B^{0}\to K^{*0}\tau^{\pm}\mu^{\mp}$,” JHEP, vol. 06, p. 143, 2023.
  * (20) D. Das, “Lepton flavor violating $\Lambda_{b}\rightarrow\Lambda\ell_{1}\ell_{2}$ decay,” Eur. Phys. J. C, vol. 79, no. 12, p. 1005, 2019.
  * (21) D. Das, “Model independent New Physics analysis in $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ decay,” Eur. Phys. J. C, vol. 78, no. 3, p. 230, 2018.
  * (22) T. Feldmann and M. W. Y. Yip, “Form factors for $\Lambda_{b}\to\Lambda$ transitions in the soft-collinear effective theory,” Phys. Rev. D, vol. 85, p. 014035, 2012.  [Erratum: Phys.Rev.D 86, 079901 (2012)].
  * (23) W. Detmold and S. Meinel, “$\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$ form factors, differential branching fraction, and angular observables from lattice QCD with relativistic $b$ quarks,” Phys. Rev. D, vol. 93, no. 7, p. 074501, 2016.
  * (24) W. Wang, “B to tensor meson form factors in the perturbative QCD approach,” Phys. Rev. D, vol. 83, p. 014008, 2011.
  * (25) T. M. Aliev, H. Dag, A. Kokulu, and A. Ozpineci, “$B\to T$ transition form factors in light-cone sum rules,” Phys. Rev. D, vol. 100, no. 9, p. 094005, 2019.
  * (26) S. Kumbhakar, R. Sain, and J. Vardani, “Lepton flavor violating decays,” J. Phys. G, vol. 50, no. 9, p. 095003, 2023.
  * (27) J. G. Korner and G. A. Schuler, “Exclusive Semileptonic Heavy Meson Decays Including Lepton Mass Effects,” Z. Phys. C, vol. 46, p. 93, 1990.
  * (28) D. Bečirević, O. Sumensari, and R. Zukanovich Funchal, “Lepton flavor violation in exclusive $b\rightarrow s$ decays,” Eur. Phys. J. C, vol. 76, no. 3, p. 134, 2016.
  * (29) A. Bharucha, D. M. Straub, and R. Zwicky, “$B\to V\ell^{+}\ell^{-}$ in the Standard Model from light-cone sum rules,” JHEP, vol. 08, p. 098, 2016\.
  * (30) B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, “Dimension-Six Terms in the Standard Model Lagrangian,” JHEP, vol. 10, p. 085, 2010.
  * (31) R. Aaij et al., “Search for the lepton flavour violating decay $B^{+}\rightarrow K^{+}\mu^{-}\tau^{+}$ using $B_{s2}^{*0}$ decays,” JHEP, vol. 06, p. 129, 2020.
  * (32) J. P. Lees et al., “A search for the decay modes $B^{+-}\to h^{+-}\tau^{+-}l$,” Phys. Rev. D, vol. 86, p. 012004, 2012.
  * (33) D. Bečirević, N. Košnik, O. Sumensari, and R. Zukanovich Funchal, “Palatable Leptoquark Scenarios for Lepton Flavor Violation in Exclusive $b\to s\ell_{1}\ell_{2}$ modes,” JHEP, vol. 11, p. 035, 2016.
  * (34) J. Gratrex, M. Hopfer, and R. Zwicky, “Generalised helicity formalism, higher moments and the $B\to K_{J_{K}}(\to K\pi)\bar{\ell}_{1}\ell_{2}$ angular distributions,” Phys. Rev. D, vol. 93, no. 5, p. 054008, 2016.
  * (35) P. A. Zyla et al., “Review of Particle Physics,” PTEP, vol. 2020, no. 8, p. 083C01, 2020.
  * (36) UTfit-collaboration. http://www.utfit.org/UTfit/WebHome.
  * (37) R. Balasubramamian and B. Blossier, “Decay constant of $B_{s}$ and $B^{*}_{s}$ mesons from $\mathrm{N_{f}}=2$ lattice QCD,” Eur. Phys. J. C, vol. 80, no. 5, p. 412, 2020.
  * (38) M. Bordone, M. Rahimi, and K. K. Vos, “Lepton flavour violation in rare $\Lambda_{b}$ decays,” Eur. Phys. J. C, vol. 81, no. 8, p. 756, 2021.
  * (39) M. Algueró, B. Capdevila, S. Descotes-Genon, J. Matias, and M. Novoa-Brunet, “$b\rightarrow s\ell^{+}\ell^{-}$ global fits after $R_{K_{S}}$ and $R_{K^{*+}}$,” Eur. Phys. J. C, vol. 82, no. 4, p. 326, 2022.
  * (40) W. Altmannshofer and P. Stangl, “New physics in rare B decays after Moriond 2021,” Eur. Phys. J. C, vol. 81, no. 10, p. 952, 2021.
  * (41) T. Hurth, F. Mahmoudi, D. M. Santos, and S. Neshatpour, “More Indications for Lepton Nonuniversality in $b\to s\ell^{+}\ell^{-}$,” Phys. Lett. B, vol. 824, p. 136838, 2022.
  * (42) M. Ciuchini, A. M. Coutinho, M. Fedele, E. Franco, A. Paul, L. Silvestrini, and M. Valli, “New Physics in $b\to s\ell^{+}\ell^{-}$ confronts new data on Lepton Universality,” Eur. Phys. J. C, vol. 79, no. 8, p. 719, 2019.


* Lykourentzou et al. (2016) Ioanna Lykourentzou, Angeliki Antoniou, Yannick Naudet, and Steven P. Dow. 2016.  Personality Matters: Balancing for Personality Types Leads to Better Outcomes for Crowd Teams. In _Proceedings of the 19th ACM Conference on Computer-Supported Cooperative Work & Social Computing_ _(CSCW ’16)_. Association for Computing Machinery, New York, NY, USA, 260–273.  https://doi.org/10.1145/2818048.2819979
  * Macefield (2007) Ritch Macefield. 2007\.  Usability Studies and the Hawthorne Effect.  _Journal of Usability Studies_ 2, 3 (May 2007), 145 – 154.
  * Margariti et al. (2022) Eleni Margariti, Sean Rintel, Brendan Murphy, and Abigail Sellen. 2022.  Automated mapping of competitive and collaborative overlapping talk in video meetings.. In _CHI Conference on Human Factors in Computing Systems Extended Abstracts_. 8.
  * Mark et al. (2014) Gloria Mark, Shamsi T. Iqbal, Mary Czerwinski, and Paul Johns. 2014.  Bored mondays and focused afternoons: the rhythm of attention and online activity in the workplace. In _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_. ACM, Toronto Ontario Canada, 3025–3034.  https://doi.org/10.1145/2556288.2557204
  * Mejias (2007) Roberto J. Mejias. 2007\.  The Interaction of Process Losses, Process Gains, and Meeting Satisfaction Within Technology-Supported Environments.  _Small Group Research_ 38, 1 (Feb. 2007), 156–194.  https://doi.org/10.1177/1046496406297037
  * Microsoft (2022) Microsoft. 2022\.  _2022 Work Trend Index: Annual Report_.  Technical Report. Microsoft.  https://www.microsoft.com/en-us/worklab/work-trend-index/hybrid-work-is-just-work.
  * Mroz et al. (2018) Joseph E. Mroz, Joseph A. Allen, Dana C. Verhoeven, and Marissa L. Shuffler. 2018.  Do We Really Need Another Meeting? The Science of Workplace Meetings.  _Current Directions in Psychological Science_ 27, 6 (Dec. 2018), 484–491.  https://doi.org/10.1177/0963721418776307
  * Nicol L. Davidson (2013) Nicol L. Davidson. 2013\.  _Trust and Member Inclusion as Communication Factors to Foster Collaboration in Globally Distributed Teams_.  Ph. D. Dissertation.
  * Niebuhr et al. (2021) Oliver Niebuhr, Ronald Böck, and Joseph A. Allen. 2021\.  On the Sound of Successful Meetings: How Speech Prosody Predicts Meeting Performance. In _Companion Publication of the 2021 International Conference on Multimodal Interaction_. ACM, Montreal QC Canada, 240–248.  https://doi.org/10.1145/3461615.3485412
  * Niemantsverdriet and Erickson (2017) Karin Niemantsverdriet and Thomas Erickson. 2017.  Recurring Meetings: An Experiential Account of Repeating Meetings in a Large Organization.  _Proceedings of the ACM on Human-Computer Interaction_ 1, CSCW (Dec. 2017), 1–17.  https://doi.org/10.1145/3134719
  * Nixon and Littlepage (1992) Carol T. Nixon and Glenn E. Littlepage. 1992.  Impact of meeting procedures on meeting effectiveness.  _Journal of Business and Psychology_ 6, 3 (1992), 361–369.  https://doi.org/10.1007/BF01126771
  * Nunamaker et al. (1991) J. F. Nunamaker, Alan R. Dennis, Joseph S. Valacich, Douglas Vogel, and Joey F. George. 1991\.  Electronic meeting systems.  _Commun. ACM_ 34, 7 (July 1991), 40–61.  https://doi.org/10.1145/105783.105793
  * Odermatt et al. (2018) Isabelle Odermatt, Cornelius J. König, Martin Kleinmann, Maria Bachmann, Heiko Röder, and Patricia Schmitz. 2018.  Incivility in Meetings: Predictors and Outcomes.  _Journal of Business and Psychology_ 33, 2 (April 2018), 263–282.  https://doi.org/10.1007/s10869-017-9490-0
  * Oh et al. (2018) Catherine S. Oh, Jeremy N. Bailenson, and Gregory F. Welch. 2018\.  A Systematic Review of Social Presence: Definition, Antecedents, and Implications.  _Frontiers in Robotics and AI_ 5 (Oct. 2018), 114\.  https://doi.org/10.3389/frobt.2018.00114
  * Olson and Olson (2013) Judith S. Olson and Gary M. Olson. 2013.  Working Together Apart: Collaboration over the Internet.  _Synthesis Lectures on Human-Centered Informatics_ 6, 5 (Nov. 2013), 1–151.  https://doi.org/10.2200/S00542ED1V01Y201310HCI020 Publisher: Morgan & Claypool Publishers.
  * Panda et al. (2022) Payod Panda, Molly Jane Nicholas, Mar Gonzalez-Franco, Kori Inkpen, Eyal Ofek, Ross Cutler, Ken Hinckley, and Jaron Lanier. 2022\.  AllTogether: Effect of Avatars in Mixed-Modality Conferencing Environments. In _2022 Symposium on Human-Computer Interaction for Work_. ACM, Durham NH USA, 1–10.  https://doi.org/10.1145/3533406.3539658
  * Pearce and Randel (2004) Jone L. Pearce and Amy E. Randel. 2004.  Expectations of organizational mobility, workplace social inclusion, and employee job performance.  _Journal of Organizational Behavior_ 25, 1 (Feb. 2004), 81–98.  https://doi.org/10.1002/job.232
  * Pendergrass (2019) Angie Pendergrass. 2019\.  Inclusive scientific meetings: Where to start.  https://opensky.ucar.edu/islandora/object/manuscripts%3A983/datastream/PDF/view
  * Podsakoff et al. (2003) Philip M. Podsakoff, Scott B. MacKenzie, Jeong-Yeon Lee, and Nathan P. Podsakoff. 2003.  Common method biases in behavioral research: A critical review of the literature and recommended remedies.  _Journal of Applied Psychology_ 88, 5 (2003), 879–903.  https://doi.org/10.1037/0021-9010.88.5.879
  * Ravid et al. (2020a) Daniel M. Ravid, David L. Tomczak, Jerod C. White, and Tara S. Behrend. 2020a.  EPM 20/20: A Review, Framework, and Research Agenda for Electronic Performance Monitoring.  _Journal of Management_ 46, 1 (Jan. 2020), 100–126.  https://doi.org/10.1177/0149206319869435
  * Ravid et al. (2020b) Daniel M Ravid, Jerod C White, Dave L Tomczak, Ahleah F Miles, and Tara S Behrend. 2020b.  A Meta-Analysis of the Effects of Digital Surveillance of Workers: A Psychology Focused Approach.  _Microsoft New Future of Work Symposium_ (2020).
  * Reece et al. (2022) Andrew Reece, Gus Cooney, Peter Bull, Christine Chung, Bryn Dawson, Casey Fitzpatrick, Tamara Glazer, Dean Knox, Alex Liebscher, and Sebastian Marin. 2022\.  Advancing an Interdisciplinary Science of Conversation: Insights from a Large Multimodal Corpus of Human Speech.  http://arxiv.org/abs/2203.00674 arXiv:2203.00674 [cs].
  * Reinig (2003) Bruce A Reinig. 2003\.  Toward an Understanding of Satisfaction with the Process and Outcomes of Teamwork.  _Journal of Management Information Systems_ 19, 4 (April 2003), 65–83.  https://doi.org/10.1080/07421222.2003.11045750
  * Rice et al. (2020) Darryl B. Rice, Nicole C. J. Young, and Sharon Sheridan. 2020\.  Improving employee emotional and behavioral investments through the trickle-down effect of organizational inclusiveness and the role of moral supervisors.  _Journal of Business and Psychology_ (Jan. 2020).  https://doi.org/10.1007/s10869-019-09675-2
  * Rintel (2010) E. Sean Rintel. 2010\.  Conversational management of network trouble perturbations in personal videoconferencing. In _Proceedings of the 22nd Conference of the Computer-Human Interaction Special Interest Group of Australia on Computer-Human Interaction_. ACM, Brisbane Australia, 304–311.  https://doi.org/10.1145/1952222.1952288
  * Rogelberg et al. (2006) Steven Rogelberg, Desmond Leach, Peter Warr, and Jennifer Burnfield. 2006.  ”Not Another Meeting!” Are Meeting Time Demands Related to Employee Well-Being?  _The Journal of applied psychology_ 91 (Feb. 2006), 83–96.  https://doi.org/10.1037/0021-9010.91.1.83
  * Romano and Nunamaker (2001) N.C. Romano and J.F. Nunamaker. 2001.  Meeting analysis: findings from research and practice. In _Proceedings of the 34th Annual Hawaii International Conference on System Sciences_. IEEE, Maui, HI, USA, 13 pp.  https://doi.org/10.1109/HICSS.2001.926253
  * Saatçi et al. (2019) Banu Saatçi, Roman Rädle, Sean Rintel, Kenton O’Hara, and Clemens Nylandsted Klokmose. 2019\.  Hybrid Meetings in the Modern Workplace: Stories of Success and Failure.  In _Collaboration Technologies and Social Computing_ , Hideyuki Nakanishi, Hironori Egi, Irene-Angelica Chounta, Hideyuki Takada, Satoshi Ichimura, and Ulrich Hoppe (Eds.). Vol. 11677. Springer International Publishing, Cham, 45–61.  https://doi.org/10.1007/978-3-030-28011-6_4 Series Title: Lecture Notes in Computer Science.
  * Samrose et al. (2021) Samiha Samrose, Daniel McDuff, Robert Sim, Jina Suh, Kael Rowan, Javier Hernandez, Sean Rintel, Kevin Moynihan, and Mary Czerwinski. 2021.  MeetingCoach: An Intelligent Dashboard for Supporting Effective & Inclusive Meetings. In _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_. ACM, Yokohama Japan, 1–13.  https://doi.org/10.1145/3411764.3445615
  * Sarkar et al. (2021) Advait Sarkar, Sean Rintel, Damian Borowiec, Rachel Bergmann, Sharon Gillett, Danielle Bragg, Nancy Baym, and Abigail Sellen. 2021.  The promise and peril of parallel chat in video meetings for work. In _Extended Abstracts of the 2021 CHI Conference on Human Factors in Computing Systems_. ACM, Yokohama Japan, 1–8.  https://doi.org/10.1145/3411763.3451793
  * Schwartzman (1989) Helen B. Schwartzman. 1989\.  _The Meeting_.  Springer US, Boston, MA.  https://doi.org/10.1007/978-1-4899-0885-8
  * Sellen (1995) Abigail Sellen. 1995\.  Remote Conversations: The Effects of Mediating Talk With Technology.  _Human-Computer Interaction_ 10, 4 (Dec. 1995), 401–444.  https://doi.org/10.1207/s15327051hci1004_2
  * Smith et al. (2019) Michael G. Smith, Maryam Witte, Sarah Rocha, and Mathias Basner. 2019.  Effectiveness of incentives and follow-up on increasing survey response rates and participation in field studies.  _BMC Medical Research Methodology_ 19, 1 (Dec. 2019), 230.  https://doi.org/10.1186/s12874-019-0868-8
  * Soga et al. (2021) Lebene Richmond Soga, Bernd Vogel, Ana Margarida Graça, and Kofi Osei-Frimpong. 2021.  Web 2.0-enabled team relationships: an actor-network perspective.  _European Journal of Work and Organizational Psychology_ 30, 5 (Sept. 2021), 639–652.  https://doi.org/10.1080/1359432X.2020.1847183
  * Standaert et al. (2016) Willem Standaert, Steve Muylle, and Amit Basu. 2016.  An empirical study of the effectiveness of telepresence as a business meeting mode.  _Information Technology and Management_ 17, 4 (Dec. 2016), 323–339.  https://doi.org/10.1007/s10799-015-0221-9
  * Stray et al. (2016) Viktoria Stray, Dag I.K. Sjøberg, and Tore Dybå. 2016\.  The daily stand-up meeting: A grounded theory study.  _Journal of Systems and Software_ 114 (April 2016), 101–124.  https://doi.org/10.1016/j.jss.2016.01.004
  * Stray et al. (2013) Viktoria Gulliksen Stray, Yngve Lindsjorn, and Dag I.K. Sjoberg. 2013\.  Obstacles to Efficient Daily Meetings in Agile Development Projects: A Case Study. In _2013 ACM / IEEE International Symposium on Empirical Software Engineering and Measurement_. IEEE, Baltimore, Maryland, 95–102.  https://doi.org/10.1109/ESEM.2013.30
  * Tadelis (2016) Steven Tadelis. 2016\.  Reputation and Feedback Systems in Online Platform Markets.  _Annual Review of Economics_ 8, 1 (Oct. 2016), 321–340.  https://doi.org/10.1146/annurev-economics-080315-015325
  * Tang and Isaacs (1992) John C. Tang and Ellen Isaacs. 1992.  Why do users like video?: Studies of multimedia-supported collaboration.  _Computer Supported Cooperative Work (CSCW)_ 1, 3 (Sept. 1992), 163–196.  https://doi.org/10.1007/BF00752437
  * Triana et al. (2012) María del Carmen Triana, Bradley L. Kirkman, and María Fernanda Wagstaff. 2012.  Does the Order of Face-to-Face and Computer-Mediated Communication Matter in Diverse Project Teams? An Investigation of Communication Order Effects on Minority Inclusion and Participation.  _Journal of Business and Psychology_ 27, 1 (March 2012), 57–70.  https://doi.org/10.1007/s10869-011-9232-7
  * Veinott et al. (1999) Elizabeth S. Veinott, Judith Olson, Gary M. Olson, and Xiaolan Fu. 1999.  Video helps remote work: speakers who need to negotiate common ground benefit from seeing each other. In _Proceedings of the SIGCHI conference on Human factors in computing systems the CHI is the limit - CHI ’99_. ACM Press, Pittsburgh, Pennsylvania, United States, 302–309.  https://doi.org/10.1145/302979.303067
  * Wood et al. (2017) Dustin Wood, P. D. Harms, Graham H. Lowman, and Justin A. DeSimone. 2017.  Response Speed and Response Consistency as Mutually Validating Indicators of Data Quality in Online Samples.  _Social Psychological and Personality Science_ 8, 4 (2017), 454–464.  ttps://journals.sagepub.com/doi/abs/10.1177/1948550617703168
  * Woolley et al. (2010) Anita Williams Woolley, Christopher F. Chabris, Alex Pentland, Nada Hashmi, and Thomas W. Malone. 2010\.  Evidence for a Collective Intelligence Factor in the Performance of Human Groups.  _Science_ 330, 6004 (Oct. 2010), 686–688.  https://doi.org/10.1126/science.1193147
  * Yasuoka et al. (2022) Mika Yasuoka, Marko Zivko, Hiroshi Ishiguro, Yuichiro Yoshikawa, and Kazuki Sakai. 2022\.  Effects of Digital Avatar on Perceived Social Presence and Co-presence in Business Meetings Between the Managers and Their Co-workers.  In _Collaboration Technologies and Social Computing_ , Lung-Hsiang Wong, Yugo Hayashi, Cesar A. Collazos, Claudio Alvarez, Gustavo Zurita, and Nelson Baloian (Eds.). Vol. 13632. Springer International Publishing, Cham, 83–97.  https://doi.org/10.1007/978-3-031-20218-6_6 Series Title: Lecture Notes in Computer Science.
  * Zhou et al. (2021) Ke Zhou, Marios Constantinides, Luca Maria Aiello, Sagar Joglekar, and Daniele Quercia. 2021\.  The Role of Different Types of Conversations for Meeting Success.  _IEEE Pervasive Computing_ 20, 4 (Oct. 2021), 35–42.  https://doi.org/10.1109/MPRV.2021.3115879
  * Zhou et al. (2022) Ke Zhou, Marios Constantinides, Sagar Joglekar, and Daniele Quercia. 2022.  Predicting Meeting Success With Nuanced Emotions.  _IEEE Pervasive Computing_ 21, 2 (April 2022), 51–59.  https://doi.org/10.1109/MPRV.2022.3145047

## APPENDIX

##  Appendix A GLM Tables for Side Models

The tables shared in this section show detailed effects and interaction
effects of a subset of attributes on effectiveness and inclusiveness. These
subsets are suggested by the sub-graphs that emerged from graph modeling
fitted on all attributes.

| Coef | Standardized Coef | p-value
---|---|---|---
Intercept | 3.80 | 37.79 | 0.00
Short Call (30min or less) | -0.27 | -2.39 | 0.02
Meeting Size | -0.06 | -8.63 | 0.00
Short Call (30min or less) : Meeting Size | 0.00 | -0.50 | 0.62
Recurring | -0.37 | -3.24 | 0.00
Recurring : Meeting Size | 0.03 | 3.41 | 0.00
Table 7. GLM results in modeling the probability of Effective by Participation | Coef | Standardized Coef | p-value
---|---|---|---
Intercept | -0.40 | -6.90 | 0.00
Meeting Size (8 or less) | 2.00 | 2.00 | 0.02
VideoDuration30% | 0.16 | 1.84 | 0.06
CallDuration | 0.00 | 4.22 | 0.00
Meeting Size (8 or less) : VideoDuration30% | 0.46 | 5.52 | 0.00
VideoDuration30% : CallDuration | 0.00 | -2.05 | 0.04
Meeting Size (8 or less) : VideoDuration30% | 0.00 | -2.00 | 0.05
Table 8. GLM results in modeling the probability of Participation by Video usage Figure 9. Interaction effect between Call Duration, Meeting Size, Video usage, and Participation. Name | Type
---|---
Microphone Failure (Initialization) | bool
Microphone Failure (Mid-Call) | bool
Media Failure | bool
Media Ever Flowed | bool
Reconnect Failure | bool
Reconnect Issue | bool
Call Dropped | bool
Video Duration Percent | float
Audio Only | bool
Video Only | bool
Video or ScreenShare | bool
ScreenShare Only | bool
Audio Participation Rate (based on audio packet decoding counts) | float
Is Rater The Meeting Host | bool
Is Friday | bool
is Monday | bool
Is Country GroupA | bool
Is Country GroupB | bool
Meeting Size | int
Call Duration | float
Predicted Probability of Call Quality Issues | float
Total Time In Meeting In The Same Day | float
Total Calls In The Same Day | int
ScreenShare > 10% | bool
Quality Issues | bool
Reliability Issues | bool
Participation | int
Video Duration > 30% | bool
Recurring | bool
ScreenShare | bool
Small Meeting (8 or less) | bool
Short Call (10min. or less) | bool
Long Call (1hr or more) | bool
Headset | bool
Busy Day (10 or More Calls) | bool
Short Hours in Meetings (Less Than 1hr In Calls On The Same Day) | bool
Table 9. List of variables used for predictive modeling.