#  $B$-physics with dynamical domain-wall light quarks and relativistic
$b$-quarks

Ruth S. Van de Water and
Brookhaven National Laboratory, Department of Physics, Upton, NY 11973, USA
E-mail

###### Abstract:

We report on our progress in calculating the $B$-meson decay constants and
$B^{0}$-$\bar{B}^{0}$ mixing parameters using domain-wall light quarks and
relativistic $b$-quarks. We present our computational method and show some
preliminary results obtained on the coarser ($a\approx 0.11$fm) $24^{3}$
lattices. This work is presented on behalf of the RBC and UKQCD
collaborations.

##  1 Introduction

The study of $B$-meson physics on the lattice is of special phenomenological
interest because it allows one to obtain constraints on the CKM unitarity
triangle. In the standard global unitarity triangle fit, the apex of the CKM
unitarity triangle is constrained using lattice input for neutral $B$-meson
mixing. Experimentally, $B_{q}$–$\bar{B}_{q}$ mixing is well measured in terms
of mass differences (oscillation frequencies) $\Delta m_{q}$; in the standard
model it is parameterized by [1]

$\displaystyle\Delta
m_{q}=\frac{G_{F}^{2}m^{2}_{W}}{6\pi^{2}}\eta_{B}S_{0}m_{B_{q}}{f_{B_{q}}^{2}B_{B_{q}}}\lvert
V_{tq}^{*}V_{tb}\rvert^{2},$ (1)

where the index $q$ denotes either a $d$\- or a $s$-quark, $m_{B_{q}}$ is the
mass of the $B_{q}$-meson, and $V_{tq}^{*}$ and $V_{tb}$ are CKM matrix
elements. The Inami-Lim function, $S_{0}$ [2], and the QCD coefficient,
$\eta_{B}$ [1], can be computed perturbatively, whereas the non-perturbative
input is given by $f_{q}^{2}B_{B_{q}}$: the leptonic decay constant
$f_{B_{q}}$ and the $B$-meson bag parameter $B_{B_{q}}$. Computing the ratio
$\Delta m_{s}$/$\Delta m_{d}$ is particularly advantageous because statistical
and systematic uncertainties largely cancel and, moreover, the ratio of CKM
matrix elements becomes accessible

$\displaystyle\frac{\Delta m_{s}}{\Delta
m_{d}}=\frac{m_{B_{s}}}{m_{B_{d}}}\,{\xi^{2}}\,\frac{\lvert
V_{ts}\rvert^{2}}{\lvert V_{td}\rvert^{2}}.$ (2)

The non-perturbative input is now solely contained in the $SU(3)$-breaking
ratio

$\displaystyle\xi$
$\displaystyle=\frac{f_{B_{s}}\sqrt{B_{B_{s}}}}{f_{B_{d}}\sqrt{B_{B_{d}}}}.$
(3)

An alternative way of constraining the CKM triangle has been proposed by
Lunghi and Soni [3] that uses the CKM matrix elements $V_{ub}$ or $V_{cb}$,
thereby avoiding the tension between inclusive and exclusive determinations of
both $V_{ub}$ (3$\sigma$) and $V_{cb}$ (2$\sigma$). This alternative method,
however, requires precise knowledge of the decay constant $f_{B}$ as well as
$BR(B\to\tau\nu)$ and $\Delta m_{s}$. Moreover, Lunghi and Soni point out [4,
5] that already the current data may show signs of physics beyond the standard
model manifesting themselves as deviations of experimental values for
$\sin(2\beta)$ (3.3$\sigma$) and $BR(B\to\tau l\nu)$ (2.8$\sigma$) from the
Standard Model. They argue that the most likely sources for new physics are in
$B_{q}$ mixing and $\sin(2\beta)$.

Therefore, it is timely for the lattice community to determine these
parameters precisely and prove by using statistically and systematically
independent setups that all sources of errors are well under control.
Currently, there are three determinations for $\xi$ using 2+1-flavor gauge
field configurations: an exploratory study by the RBC-UKQCD collaboration [6]
with non-competitive errors and two precise determinations by Fermilab-MILC
[7, 8] and HPQCD [9], which however both rely on the same set of MILC gauge
field configurations. The latter two collaborations also use the same
configurations to obatin the decay constants $f_{B_{d}}$ and $f_{B_{s}}$ and
find good agreement. Here we present the first results of our project to
determine $B$-meson parameters using domain-wall light quarks and relativistic
$b$-quarks.

##  2 Computational setup

Our project is based on using the RBC-UKQCD 2+1-flavor domain wall lattices
with the Iwasaki gauge action for the gluons. Our first results presented in
these proceedings are obtained on the coarser $a\approx 0.11$fm lattices with
a spatial volume of $24^{3}$ in lattice units [10], whereas the full project
will also utilize the finer $a\approx 0.08$fm lattices ($32^{3}$) [11] (for
details see Tab. 1).

|  |  |  |  | approx. | # time
---|---|---|---|---|---|---
L |  $a$(fm) | $m_{l}$ | $m_{s}$ |  $m_{\pi}$(MeV) | # configs. | sources
24 |  $\approx$ 0.11 | 0.005 | 0.040 | 331 | 1640 | 1
24 |  $\approx$ 0.11 | 0.010 | 0.040 | 419 | 1420 | 1
24 |  $\approx$ 0.11 | 0.020 | 0.040 | 558 | 350 | 8
32 |  $\approx$ 0.08 | 0.004 | 0.030 | 307 | 600 | 1
32 |  $\approx$ 0.08 | 0.006 | 0.030 | 366 | 900 | 1
32 |  $\approx$ 0.08 | 0.008 | 0.030 | 418 | 550 | 1
Table 1: Overview of the gauge field ensembles to be used for this project.

For the heavy quarks we use a relativistic formulation derived from the
“Fermilab action” [12] in which we tune the three relevant parameters of the
action non-perturbatively. As demonstraged by Christ, Li and Lin, this
requires only one additional experimental input compared to the formulation of
Fermilab [13]. The relativistic heavy quark (RHQ) action is given by

$\displaystyle
S=\sum_{n,n^{\prime}}\bar{\Psi}_{n}\left\\{\\!m_{0}+\gamma_{0}D_{0}-\\!\frac{aD_{0}^{2}}{2}+\zeta\left[\vec{\gamma}\cdot\vec{D}-\frac{a\left(\vec{D}\right)^{2}}{2}\right]\\!-a\sum_{\mu\nu}\frac{ic_{P}}{4}\sigma_{\mu\nu}F_{\mu\nu}\\!\right\\}_{\\!\\!n,n^{\prime}}\\!\\!\\!\\!\Psi_{n^{\prime}},$
(4)

where the covariant derivative is denoted by $D$, $F_{\mu\nu}$ is the field
strength tensor and we need to tune the three parameters $m_{0}a$, $c_{P}$ and
$\zeta$. Exploratory studies on how to tune these parameters have been
performed by Li and Peng [14, 15, 16]. The idea is to use experimental values
for the spin averaged meson mass ($\overline{m}=(m_{B_{s}}+3m_{B_{s}^{*}})/4$)
and the hyperfine splitting ($\Delta_{m}=m_{B^{*}_{s}}-m_{B_{s}}$) as inputs
together with the constraint from the dispersion relation that the rest mass
$m_{1}$ equal the kinetic mass $m_{2}$. These three quantities are computed
for a set of seven trial parameters determined by making an initial guess for
$m_{0}a$, $c_{P}$, and $\zeta$ and then varying it by a chosen uncertainty
$\pm\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$:

$\displaystyle\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\ \zeta\\\
\end{array}\right],\left[\\!\\!\begin{array}[]{c}m_{0}a-\sigma_{m_{0}a}\\\
c_{P}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a+\sigma_{m_{0}a}\\\
c_{P}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\
c_{P}-\sigma_{c_{P}}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\
c_{P}+\sigma_{c_{P}}\\\ \zeta\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta-\sigma_{\zeta}\\\
\end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta+\sigma_{\zeta}\\\ \end{array}\\!\\!\right]$ (26)

(see Fig. 1). We iterate over the parameters $\\{m_{0}a,\,c_{P},\,\zeta\\}$
until we determine the values that reproduce the known experimental
measurements for $\overline{m},\,\Delta_{m},\,m_{1}/m_{2}$:

$\displaystyle\left[\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta\end{array}\right]^{\text{RHQ}}=J^{-1}\times\left(\left[\begin{array}[]{c}\overline{m}\\\
\Delta_{m}\\\ \frac{m_{1}}{m_{2}}\end{array}\right]^{\text{PDG}}-A\right)$
(33) with $\displaystyle
J=\left[\frac{Y_{3}-Y_{2}}{2\sigma_{m_{0}a}},\,\frac{Y_{5}-Y_{4}}{2\sigma_{c_{P}}},\,\frac{Y_{7}-Y_{6}}{2\sigma_{\zeta}}\right]\qquad\text{and}\quad
A=Y_{1}-J\times\left[m_{0}a,\,c_{P},\,\zeta\right]^{t}.$ (34)

In (34) we use the vectors $Y_{i}$ as shorthand notation for
$[\overline{m},\,\Delta_{m},\,m_{1}/m_{2}]^{t}_{i}$ obtained for the input
parameters as given in Eq. (26), labeled 1-7 from left to right. Eq. (33)
assumes a linear dependence of the meson masses on the parameters of the
action; therefore we must be in a linear regime to reliably extract
$\\{m_{0}a,\,c_{P},\,\zeta\\}$. We stop the iteration once the tuned values
lie within our variation range.

Figure 1: Initial guess for the RHQ parameters and their uncertainties.

In order to finally compute $B$-meson decay constants and mixing parameters as
precisely as possible we repeated the original tuning presented at Lattice
2008 [14] with increased statistics and optimized smeared wavefunction
parameters used for generating the heavy quark propagators. Moreover,
performing the tuning on the same configurations we intend to use for
computing weak matrix elements such as $f_{B}$ will allow for an improved
error analysis in which the correlations among the three RHQ parameters can be
fully taken into account.

Our method for computing the $\Delta B=2$ four-quark operators requires the
light quarks to be generated with a point source and sink, but for the heavy
quarks we are free to explore different smearing choices. In addition to point
sources and sinks, we tried Gaussian smeared sources/sinks and also varied the
radius of the Gaussian smearing. As a first guess we chose the radius of the
Gaussian source to be the rms radius of the $b\bar{b}$\- and $c\bar{c}$ states
[17]. Later we extended the radius and found that $r_{\text{rms}}=0.634$fm
gives the best signal as can be seen in Fig. 2.

Figure 2: $B_{l}$-meson effective masses for different $b$-quark spatial
wavefunctions; in each case the light quark has a point source.

Using this setup we obtain the RHQ parameters on the three $24^{3}$ ensembles
as given in Table 2. In contrast to earlier work [14] the determination uses
only quantities from the heavy-light system. We expect these values to be
close to our final determination. Moreover, we observe that within statistical
uncertainties there is no dependence on the light sea quark mass
($m_{\text{sea}}^{l}$).

$m_{\text{sea}}^{l}$ | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---
0.005 | 8.41(9) | 5.7(2) | 3.1(2)
0.010 | 8.4(1) | 5.8(2) | 3.1(1)
0.020 | 8.4(1) | 5.6(2) | 3.1(1)
Table 2: Preliminary determination of the RHQ parameters on the three $24^{3}$
ensembles with $a\approx 0.11$fm.

##  3 Computation of decay constants

Using these newly determined RHQ parameters we compute as a first non-trivial
test the decay constants of the unitary ($B_{l}$) and strange ($B_{s}$) mesons
via the relation

$\displaystyle f_{B_{q}}=Z_{\Phi}\;\Phi_{B_{q}}\;a^{-3/2}/\sqrt{m_{B_{q}}},$
(35)

where $\Phi_{B}$ is the lattice decay amplitude, $Z_{\Phi}$ is the
renormalization factor. For the $B_{s}$ meson (domain-wall valence quark has
mass of physical $s$-quark) we generated data for the set of seven different
RHQ input parameters. Therefore we can use similar equations to (33) and (34)
in order to determine the decay amplitude $\Phi_{B_{s}}$ at the tuned RHQ
parameters:

$\displaystyle\Phi^{\text{RHQ}}=J_{\Phi}^{(1\times
3)}\times\left[\begin{array}[]{c}m_{0}a\\\ c_{P}\\\
\zeta\end{array}\right]^{\text{RHQ}}+A_{\Phi}$ (39) with
$\displaystyle\displaystyle
J_{\Phi}=\left[\frac{\Phi_{3}-\Phi_{2}}{2\sigma_{m_{0}a}},\,\frac{\Phi_{5}-\Phi_{4}}{2\sigma_{c_{P}}},\,\frac{\Phi_{7}-\Phi_{6}}{2\sigma_{\zeta}}\right]\qquad\text{and}\qquad
A_{\Phi}=\Phi_{1}-J_{\Phi}\times\left[m_{0}a,\,c_{P},\,\zeta\right]^{t}.$ (40)

In Fig. 3 we show the results for $\Phi_{B_{s}}$ obtained on the ensemble with
$m_{\text{sea}}^{l}=0.005$. To test the linearity with respect to the input
parameters $\\{m_{0}a,\,c_{P},\,\zeta\\}$ we used three different sets of
seven RHQ parameters all centered around the same point. The vertical black
line with the gray error band indicates the tuned value of $m_{0}a$, $c_{P}$
or $\zeta$ and allows for a simple estimate of the error in $\Phi_{B_{s}}$ due
to the uncertainty in each of the three parameters. These plots show that the
effect of the uncertainty in $m_{0}a$ and $c_{P}$ is negligible, whereas
$\zeta$ contributes an error of about 1%. Moreover, we see that $\Phi_{B_{s}}$
depends linearly on $m_{0}a$, $c_{P}$ and $\zeta$ in the range of interest.

Figure 3: Dependence of $\Phi_{B_{s}}$ on the three RHQ parameters
$\\{m_{0}a,\,c_{P},\,\zeta\\}$; results are shown for the lightest sea quark
mass $m_{\text{sea}}^{l}=0.005$ on the $24^{3}$ ensembles.

Alternatively, one may simply use the tuned RHQ values to compute the needed
correlation functions. We followed this procedure to calculate the $B_{l}$
meson (domain-wall valence quark mass equals the light sea quark mass) the
decay amplitude $\Phi_{B_{l}}$.

Atfer renormalizing $\Phi_{B}$ multiplicatively at 1-loop [18] we obtain the
decay constants which are given in Tab. 3 and shown in Fig. LABEL:Fig-fB. As
mentioned above, the values for $f_{B_{l}}$ are obtained by simulating
directly at the tuned RHQ parameters, whereas the $f_{B_{s}}$ values are
extracted using Eq. (39). The statistical errors on $f_{B_{s}}$ are therefore
larger because they take into account the statistical uncertainties in the
three RHQ parameters. For a better comparison, we increase the errors in
$f_{B_{l}}$ in the Fig. LABEL:Fig-fB by the error due to the statistical
uncertainty in the RHQ parameters estimated in Fig. 3. Moreover, we emphasize
that this computation is performed without $O(a)$ improvement.

##  4 Conclusion

We presented our first results computing $B$-meson decay constants using
domain-wall light and relativistic $b$-quarks with all parameters of the RHQ
action tuned non-perturbatively. Currently, our results still need to be
${\cal O}(a)$ improved. For $f_{B_{s}}$ we expect only a mild chiral
extrapolation but of course need to perform an extrapolation to the continuum
and estimate other systematic errors. Despite these caveats the small
statistical errors and the fact that our central values lie in the same
ballpark as results of other collaborations indicates the promise of our
method.

$m_{sea}^{l}$ |  $f_{B_{l}}$(MeV) |  $f_{B_{s}}$(Mev)
---|---|---
0.005 | 188(2) | 215(3)
0.010 | 194(2) | 214(4)
0.020 | — | 221(2)
Table 3: Preliminary results for the decay constants using 1-loop
multiplicative renormalization, but without $O(a)$ improvement of the axial-
current operator.

## Acknowledgments

We are thankful to all the members of the RBC and UKQCD collaborations.
Numerical computations for this work utilized USQCD resources and were
performed on the kaon and jpsi clusters at FNAL, in part funded by the Office
of Science of the U.S. Department of Energy. This manuscript has been authored
by an employee of Brookhaven Science Associates, LLC under Contract No. DE-
AC02-98CH10886 with the U.S. Department of Energy.

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# Longitudinally polarized $ZZ$ scattering at the Muon Collider

Tianyi Yang  Department of Physics and State Key Laboratory of Nuclear Physics
and Technology, Peking University, Beijing, 100871, China  Sitian Qian
Department of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing, 100871, China  Zhe Guan  Department of
Physics and State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing, 100871, China  Congqiao Li  Department of Physics and
State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing, 100871, China  Fanqiang Meng  Department of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing,
100871, China  Jie Xiao  Department of Physics and State Key Laboratory of
Nuclear Physics and Technology, Peking University, Beijing, 100871, China
Meng Lu  School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
Qiang Li  Department of Physics and State Key Laboratory of Nuclear Physics
and Technology, Peking University, Beijing, 100871, China

###### Abstract

Measuring longitudinally polarized vector boson scattering in, e.g., the $ZZ$
channel is a promising way to investigate the unitarization scheme from the
Higgs and possible new physics beyond the Standard Model. However, at the LHC,
it demands the end of the HL-LHC lifetime luminosity,
3000$~{}\mathrm{fb^{-1}}$, and advanced data analysis technique to reach the
discovery threshold due to its small production rates. Instead, there could be
great potential at future colliders. In this paper, we perform a Monte Carlo
study and examine the projected sensitivity of longitudinally polarized $ZZ$
scattering at a TeV scale muon collider. We conduct studies at 14 TeV and 6
TeV muon colliders respectively and find that a 5 standard deviation discovery
can be achieved at a 14 TeV muon collider, with 3000$~{}\mathrm{fb^{-1}}$ of
data collected. While a 6 TeV muon collider can already surpass HL-LHC,
reaching 2 standard deviations with around 4$~{}\mathrm{ab^{-1}}$ of data. The
effect from lepton isolation and detector granularity is also discussed, which
may be more obvious at higher energy muon colliders, as the leptons from
longitudinally polarized Z decays tend to be closer.

###### Contents

  1. I Introduction
  2. II Physics processes at the muon collider
  3. III Inclusive and polarized $ZZ$ scattering
  4. IV Simulation and analysis framework
  5. V Analysis results
  6. VI Outlook and conclusions

##  I Introduction

The discovery plb:2012gu ; plb:2012gk  and property measurements higprop  of
the Higgs boson marked a triumph of the Standard Model (SM) of particle
physics and the Large Hadron Collider (LHC). In the next decades, the LHC and
the High-Luminosity LHC (HL-LHC), together with other future colliders in
design, will be further exploring the SM and searching for physics beyond
that. Among the next critical topics, measuring the vector boson scattering
(VBS) processes and especially its longitudinally polarized component (the LL
component) is extremely important yet very demanding.

Vector boson scattering is a type of rare SM processes involving pure
electroweak interactions. It is sensitive to non-Abelian weak gauge boson
interactions and the structure of electroweak symmetry breaking. Typical VBS
signatures at hadron colliders include, for example, large di-jet mass and
large pseudo-rapidity separation. Many VBS studies have been performed at the
LHC, including the discoveries and measurements of $W^{\pm}W^{\pm}$
Sirunyan:2017ret ; Aaboud:2019nmv ; Sirunyan:2020gyx , $W^{\pm}Z$
Sirunyan:2019ksz ; Aaboud:2018ddq ; Sirunyan:2020gyx , $ZZ$ Aad:2020zbq ;
Sirunyan:2020alo , $Z\gamma$ Khachatryan:2017jub ; Aad:2019wpb ; CMS:2019iuv ;
CMS:2021slu , and $W\gamma$ Khachatryan:2016vif ; Sirunyan:2020azs . The topic
of this paper is the VBS channel of $ZZ\rightarrow 4\ell$ ($\ell=$ electron
$e$, or muon $\mu$). While this channel has the advantage of clean final
states, it suffers from a low production cross-section, with a small
branching-ratio of the $Z$ boson decaying to charged leptons.

Measuring the longitudinally polarized component of VBS is a critical next
step for the field, as it is closely related to the important theoretical
property of unitarity restoration through Higgs and possible new physics
Chang:2013aya ; Lee:2018fxj . There have been extensive studies on the LL
fraction measurement through exploiting various kinematic observables and
techniques like Boosted Decision Tree (BDT) or Deep Learning (DL)
Searcy:2015apa ; Lee:2018xtt ; Lee:2019nhm . However, due to the small yields
of this process (typically, the LL fraction is below 10% of the total VBS), it
is quite demanding to distinguish the LL signal from the background. For
example, based on the full simulation of samples with the upgraded CMS
detector at the 14 TeV HL-LHC CMS-PAS-SMP-14-008 ; CMS-PAS-FTR-18-005 ; CMS-
PAS-FTR-18-014 , the expected significance for an integrated luminosity of
3000$~{}\mathrm{fb^{-1}}$ is estimated to be 2.7 and 1.4 standard deviations
($\sigma$), for the LL VBS $W^{\pm}W^{\pm}$ and $ZZ$, respectively. Recently,
the first measurement on the LL VBS has been performed for the
$W^{\pm}W^{\pm}$ channel by the CMS experiment Sirunyan:2020gvn  based on the
full Run 2 data of around 137$~{}\mathrm{fb^{-1}}$. The optimization in the
analysis is based on two kinds of BDT, i.e., one to suppress VBS from non-VBS
and the other to signify LL from non-LL. The observed significance is only
around 1 $\sigma$. However, it symbols the beginning of the experimental
search for the polarized VBS.

##  II Physics processes at the muon collider

A muon–muon collider with the center-of-mass energy at the multi-TeV scale has
received much-revived interest Daniel20  recently, which has several
advantages compared with both hadron–hadron and electron–electron colliders
Mario16 ; Antonio20 ; Dario18 . As massive muons emit much less synchrotron
radiation than electron beams, muons can be accelerated in a circular collider
to higher energies with a much smaller circumference. On the other hand,
because the proton is a composite particle, muon–muon collisions are cleaner
than proton–proton collisions and thus can lead to higher effective center-of-
mass energies. However, due to the short lifetime of the muon, the beam-
induced background (BIB) from muon decays needs to be examined and reduced
properly. Based on a realistic simulation at $\sqrt{s}=1.5$ TeV with BIB
included, Ref. Nazar20  found that the coupling between the Higgs boson and
the b-quark can be measured at percent level with order$~{}\mathrm{ab^{-1}}$
of collected data.

At the TeV scale muon collider, VBS can be an interesting topic and ideal
motivation. As mentioned in Ref. Costantini:2020stv , a TeV scale muon
collider would effectively be a “high-luminosity weak boson collider” and
offer great opportunities to precisely measure electroweak observables and
Higgs coupling.

We target two benchmark scenarios in this study, i.e., a muon–muon collider
with a center-of-mass (c.m.) energy of 14 TeV and 6 TeV, each with order ab-1
of collected data. The physics processes we consider include (P1) $s$-channel
annihilation processes, e.g., $\mu^{+}\mu^{-}\rightarrow
X=n\,t\bar{t}+m\,V+k\,H$, where $n$, $m$ and $k$ are integers that
respectively denote the number of top quark pairs, weak vector bosons $V$, and
Higgs bosons $H$, and (P2) VBS processes including the core interactions as
$VV\to X$. The example diagram of VBS processes at a muon collider is shown in
Fig. 1. Following Ref. Costantini:2020stv , VBS processes can be further
divided into (P2.1) $W^{+}W^{-}$ fusion with two neutrinos in the final states
(symbolized below as $WW$_VBS), (P2.2) $ZZ/Z\gamma/\gamma\gamma$ fusion with
two muons in the final states (symbolized below as $ZZ$_VBS), and (P2.3)
$W^{\pm}Z/W^{\pm}\gamma$ fusion with one muon and one neutrino in the final
states (symbolized below as $WZ$_VBS).

Figure 1: Example diagram of VBS processes at a muon collider

##  III Inclusive and polarized $ZZ$ scattering

In this paper, we focus on the VBS production of $ZZ\rightarrow 4\ell$ at a
muon collider, specifically from P2.1 through $W^{+}W^{-}$ fusion. The
characteristic signal contains four leptons (electrons or muons) accompanied
with the large missing energy $\not{E}$. Accordingly, background processes
(from P1, P2.1, P2.2, and P2.3 as listed above) leading to similar final state
topology are considered in this study.

As mentioned before, we are mainly interested in the longitudinally polarized
component of the VBS $ZZ$ production, i.e., $Z_{L}Z_{L}$ productions through
$W^{+}W^{-}$ fusion. A new option to handle the polarization state from
initial and final particles has been implemented in MadGraph5_aMC@NLO
Alwall:2014hca  from version 2.7.0 BuarqueFranzosi:2019boy . This enables the
scattering and decay simulations to involve polarized, asymptotic states, and
preserves both spin-correlation and off-shell effects, to a good
approximation. However, the polarization definition depends on the reference
frame. Accordingly, MadGraph5_aMC@NLO provides a setting parameter that allows
switching the studied frames. The default is the partonic center-of-mass
frame, which is our choice for this work. Notice in the recent CMS analysis
Sirunyan:2020gvn , the default reference frame and the VBS frame are both
implemented for study, of which the latter has mildly better sensitivity. To
further illustrate the difference between the individual polarization states,
we plot in Fig. 2 the cross-section dependence on the collision energy, for
VBS productions of $ZZ$, $Z_{L}Z_{L}$, the transversely polarized component
$Z_{T}Z_{T}$, as well as $Z_{T}Z_{L}$ through $W^{+}W^{-}$ fusion at a muon
collider.

Figure 2: Cross-section dependence on the center of mass energy of a muon
collider, for the LL and other components of VBS $ZZ$ through $W^{+}W^{-}$
fusion.

##  IV Simulation and analysis framework

Both signal and background events are simulated with MadGraph5_aMC@NLO, then
showered and hadronized by Pythia8 Sjostrand:2014zea . The final state jets
are clustered using FastJet Cacciari:2011ma  with the $k_{\mathrm{T}}$
Cacciari:2008gp  algorithm at a fixed cone size of $R_{\rm jet}=0.5$. We used
Delphes deFavereau:2013fsa  version 3.0 to simulate detector effects with the
default card for the muon collider detector mucard . The simulated backgrounds
are summarized in Table 1.

Table 1: Summary of the background processes at a muon collider considered in this study. SM process type |  Selected background
---|---
P1: $s$-channel |  $ZZ$, $WWZ$
P2.1: $WW$_VBS |  $H$, $HZ(Z)$, $HWW$, $HH$, $WWZ$, $ZZZ$, $Z_{T}Z_{T,L}$, $t\bar{t}Z$
P2.2: $ZZ$_VBS |  $H$, $WW$, $t\bar{t}$, $4e$, $2e2\mu$, $4\mu$
P2.3: $WZ$_VBS |  $WZ$, $WZH$, $WH$, $WWW$, $WZZ$

Events are generated corresponding to a muon collider with collision energies
of 14 (6) TeV and integrated luminosity of 20 (4)$~{}\mathrm{ab^{-1}}$.

We describe the selection criteria as below. First, events must include
exactly four leptons with transverse momentum $p_{\mathrm{T}}>20$ GeV and
absolute pseudo-rapidity $|\eta|<2.5$, and satisfy the lepton flavor and
charge requirements from the $Z$ boson decay, i.e., the leptons should fall
into one of the three flavor types ($4e$, $4\mu$, $2e2\mu$) with exactly two
positive and two negative charges. For the type $2e2\mu$, the di-electron and
di-muon pair must have opposite charges separately. Furthermore, we veto
events with jets cleaned from the four leptons with a distance of $\Delta
R=\sqrt{\Delta^{2}{\eta}+\Delta^{2}{\phi}}>0.5$. We then cluster the selected
four leptons into two reconstructed “$Z$ bosons” (labeled $Z_{1}$ and $Z_{2}$
in the descending order of transverse momenta) using the following algorithm:

  * •

Construct the combinatorics of the leptons
$l_{1}^{+}l_{2}^{+}l_{3}^{-}l_{4}^{-}$ into lepton pair candidates
($l_{1}l_{3},\,l_{2}l_{4}$) and ($l_{1}l_{4},\,l_{2}l_{3}$),

  * •

Calculate $\Delta
M^{2}=(M_{Z_{1}^{\prime}}-M_{Z})^{2}+(M_{Z_{2}^{\prime}}-M_{Z})^{2}$,

  * •

If $\Delta M_{13,24}^{2}>\Delta M_{14,23}^{2}$, choose the lepton pair
($l_{1}l_{4},\,l_{2}l_{3}$), and vice versa,

  * •

For the flavor type $2e2\mu$, adopt the lepton paring $Z_{1}\rightarrow
e^{+}e^{-}$, $Z_{2}\rightarrow\mu^{+}\mu^{-}$.

It is worth mentioning that the four leptons in some background events are not
originated from the two $Z$ bosons. As a result, the reconstructed “$Z$ boson”
in those events may behave differently from the real $Z$ boson, thus can be
exploited to distinguish our signal from the background. Moreover, even for
leptons produced from the decay of a real $Z$ boson, we observe discrepancies
in their distributions among different polarization states. Fig. 3 shows the
invariant mass distributions of $Z_{1}$ and $Z_{2}$ for VBS $ZZ$ through $WW$
fusion and its various polarization fractions, compared with all the
backgrounds. Given the intrinsic shape discrepancies, we train a BDT model for
the better discrimination of the signal and all background components, using
the kinematic observables of the four leptons and two “$Z$ bosons” as input
features. We elaborate on the details in the next section.

Figure 3: The invariant mass distributions of $Z_{1}$ (left) and $Z_{2}$
(right) for VBS $ZZ$ through WW fusion and its various polarization fractions,
compared with all the backgrounds, corresponding to a 14 TeV muon collider
with 20$~{}\mathrm{ab^{-1}}$of data collected. The upper panel compares the
VBS $ZZ$ signal with four background components. The lower panel compares the
kinematics for different polarization fractions of the VBS $ZZ$ process.

##  V Analysis results

For the implementation of the BDT, we shuffle the signal and background events
and define the training and test sets with the event ratio of $2:1$. The BDT
with 200 trees and a maximum depth of 5 is trained. We apply the per-event
weight during the training to account for the cross-section difference among
the processes. The weight is defined by

$n_{L_{X}}=\sigma_{X}L/N_{G_{X}},$ (1)

where $\sigma_{X}$ denotes the cross-section of a process, $L$ denotes the
default target luminosity in this study as 20 (4)$~{}\mathrm{ab^{-1}}$ for a
14 (6) muon collider, and $N_{G_{X}}$ denotes the generated number of events.
The total signal yields are reweighted to match that of the total background
during training for the robustness of the trained model. The input features,
namely reconstructed kinematics of each event used for training, are listed as
follows and summarized in Table 2. In this analysis, up to 37 kinds of event
kinematics are used for the BDT model.

  * •

$(p_{\mathrm{T}},\,\eta,\,\phi)$ of the four leptons (note that due to the
small mass of the leptons, we set the masses to zero in the calculations),

  * •

$(p_{\mathrm{T}},\,\eta,\,\phi,\,m_{\mathrm{inv}})$ of the 2 clustered “$Z$
bosons”,

  * •

$(p_{\mathrm{T}},\,\eta,\,\phi,\,m_{\mathrm{inv}})$ of the summed 4-momenta of
the four leptons,

  * •

$(p_{\mathrm{T}},\,\eta,\,\phi)$ of the missing energy $\not{E}$,

  * •

$(\Delta\eta,\,\Delta\phi,\,\Delta R)$ between the 2 “$Z$ bosons” and the 2
lepton pairs from $Z_{1}$ and $Z_{2}$,

  * •

The flavor types of the 4 leptons $(4e,\,4\mu,\,2e2\mu)$, encoded as
$(1,\,-1,\,0)$ respectively.

Table 2: Summary of features used for the BDT training. Objective |  Features |  Number of features
---|---|---
Each lepton |  $(p_{\mathrm{T}},\,\eta,\,\phi)$ |  12
Each “$Z$ boson” |  $(p_{\mathrm{T}},\,\eta,\,\phi,\,m_{\mathrm{inv}})$ |  8
Four leptons combined |  $(p_{\mathrm{T},4\ell},\,\eta_{4l},\,\phi_{4l},\,m_{4l})$ |  4
$\not{E}$ |  $(p_{\mathrm{T}},\,\eta,\,\phi)$ |  3
Between two Z bosons |  $(\Delta\eta,\,\Delta\phi,\,\Delta R)$ |  3
Between $2\ell$ of $Z_{1}$ |  $(\Delta\eta,\,\Delta\phi,\,\Delta R)$ |  3
Between $2\ell$ of $Z_{2}$ |  $(\Delta\eta,\,\Delta\phi,\,\Delta R)$ |  3
Lepton flavor type |  $(1,\,-1,\,0)$ for $(4e,\,4\mu,\,2e2\mu)$ |  1
Total: |  |  37

The results of the BDT is shown in Fig. 4. In Fig. 4a, we show the feature
importance ranking of the top 10 variables in the BDT model, from which we can
easily conclude that the invariant masses of the two reconstructed “$Z$
bosons” are of vital importance in discriminating the signal and backgrounds.
Fig 4b provides p-values from Kolmogorov-Smirnov test and the BDT score
distributions for the signal and background in the training and test sets, as
proof of no over-training in the BDT model.

(a) Top 10 Kin. Importance

(b) Over Training Test

Figure 4: BDT training results for the LL VBS $ZZ$ analysis at a 14 TeV muon
collider. The left panel 4a shows the importance ranking for the top 10
features; the right panel 4b shows the Kolmogorov-Smirnov test results and
score distributions for signal and background in the training and test sets,
respectively.

The receiver operating characteristic (ROC) curve of the trained model is then
studied from the test set. The significance is also calculated at different
BDT selection thresholds, using the formula

$S=\sqrt{2(s+b)\mathrm{ln}(1+\frac{s}{b})-2s},$ (2)

where $s$ ($b$) represents the weighted signal (background) yields after
requiring the BDT score to be greater than a certain value. The ROC curve and
the significance dependence on the BDT score cut are shown in Fig. 5.

(a) ROC Curve of Trained BDT Model

(b) Thresholds on BDT Score vs. Corresponding Significance

Figure 5: ROC curve and significance dependence on the BDT cut, corresponding
to a 14 TeV muon collider with 20$~{}\mathrm{ab^{-1}}$ of data collected.

It is worth pointing out that, events from some processes with large cross-
sections, such as the $WZH,WH$ and $WZZ$ from $WZ\\_VBS$ (conventions see
Table1), have relatively large weight ($n_{L_{X}}$) after scaling to target
luminosity as the consequence of the intractable complexity in event
generation and the limited CPU resources. Events with big $n_{L_{X}}$ play an
important role when we randomly assign MC events into separate datasets for
training and testing respectively and lead to a large fluctuation on BDT
scores as shown in Fig. 5. Although in most situation tighter cut on BDT score
leads to better significance, fluctuation of achieved significance
corresponding to BDT cut approaching 1 confirms us to adopt a conservative
strategy and pause the scanning of threshold at 0.95. To further validate the
robustness of our results, we split our simulated events to training and
testing sets with 150 random configuration and train 150 same BDT models
respectively. Our target is to extract a reasonable significance with the
collection of all 150 trained models. Results for this validation are
illustrated in Fig. 6.

(a) Results corresponding to different random datasets

(b) Lower envelope of Fig.6a

Figure 6: Training results with the use of 150 different random seeds. The
left plot shows the significance dependence on the BDT cut for all 150
scenarios. It should be noted that we pause the scanning when the backgrounds
yield zero, which results in the straight horizontal line near the end of
scanning. The right plot shows the lower envelope of the significance shape.

In the left panel Fig.6a, we show the training results with the 150 random
dataset configuration. The right plot Fig.6b shows the lower envelope of the
significance as a function of the BDT cut, in these 150 training sessions. We
observe that the optimal cut value is around 0.93, with the corresponding
significance of around 14 $\sigma$. Such a result is the most conservative
estimation of the LL VBS $ZZ$ signal significance for a 14 TeV muon collider.

We also make a comparison between the BDT method and the traditional cut-base
method. The cut-flow results are listed in Table 3, where $\Delta R_{Z2,pm}$
is defined as the $\Delta R$ between the two leptons forming $Z_{2}$. We use
the selections listed in Table 3 to filter all four-lepton events. The
selections are optimized to reach a higher significance. Comparing the maximum
significance reachable by the BDT and the cut-base selection method, we see a
substantial gain from the BDT method in the distinction between signal and
background. Besides, we derive the integrated luminosity where a maximum
significance of 5 $\sigma$ is reached by
$L^{\prime}=(5^{2}\big{/}14^{2})\,L\approx 3000\mbox{$~{}\mathrm{fb^{-1}}$}$.
We conclude that the LL VBS $ZZ$ process can be eventually discovered on a 14
TeV muon collider with 3000$~{}\mathrm{fb^{-1}}$ of collected data, resulting
from a conservative estimation by the BDT method.

Table 3: The cut-flow table and the corresponding significance based on the cut-based method. The selections are optimized to achieve higher significance. cuts |  $s$ |  $b$ |  $S\,[\sigma]$
---|---|---|---
$70\mathrm{GeV}<M_{Z1}$, $M_{Z2}<140\mathrm{GeV}$ |  476.5 |  6592.1 |  5.8
$70\mathrm{GeV}<M_{Z1}$, $M_{Z2}<140\mathrm{GeV}$, $\Delta R_{Z2,pm}<0.4$ |  238.1 |  1165.9 |  6.8
$70\mathrm{GeV}<M_{Z1}$, $M_{Z2}<140\mathrm{GeV}$, $\Delta R_{Z2,pm}<0.4$, |  213.5 |  424.9 |  9.6
$p_{\mathrm{T},4\ell}<300\mathrm{GeV}$ |  |  |
$70\mathrm{GeV}<M_{Z1},M_{Z2}<140\mathrm{GeV}$, $\Delta R_{Z2,pm}<0.4$, |  147.8 |  158.1 |  10.4
$p_{\mathrm{T},4\ell}<300\mathrm{GeV}$, $\not{E}<140\mathrm{GeV}$ |  |  |

We also show the results of our study at a 6 TeV muon collider. With the same
BDT study performed as above, the expected significance for LL VBS $ZZ$ is
found to be around 2 $\sigma$ with 4$~{}\mathrm{ab^{-1}}$, which is much
weaker than the 14 TeV case. However, it is already comparable with the HL-LHC
estimation CMS-PAS-FTR-18-014 . We explain the details of the differences
between a 6 and 14 TeV muon collider as follows. (1) The $N_{L}$ of our signal
after the pre-selection at $\sqrt{s}=6$ TeV is only one-tenth of that at
$\sqrt{s}=14$ TeV. (2) Compared to $\sqrt{s}=14$ TeV, the different
polarization states of the $ZZ$ process in $WW$_VBS at a 6 TeV muon collider
are more similar in their kinematic distributions and thus difficult to be
distinguished. For a better illustration, we plot the distributions of $\Delta
R_{pm,Z2}$ and $p_{\mathrm{T},4\ell}$ at two different collision energies in
Fig. 7. Comparing the left two sub-figures with the right ones, we can see
that in the region where the values of two selected observables approach 0,
the signal process at a 14 TeV muon collider has a more prominent distribution
than at a 6 TeV muon collider. As a result, it becomes more difficult to
ensure the number of the signal remains a considerable size when reducing the
background yields.

(a) $\Delta R_{{Z_{2}},pm}$, $\sqrt{s}=14~{}\text{TeV}$

(b) $\Delta R_{{Z_{2}},pm}$, $\sqrt{s}=6~{}\text{TeV}$

(c) $p_{\mathrm{T},4\ell}$, $\sqrt{s}=14~{}\text{TeV}$

(d) $p_{\mathrm{T},4\ell}$, $\sqrt{s}=6~{}\text{TeV}$

Figure 7: Distributions of $\Delta R_{{Z_{2}},pm}$ (7a & 7b), defined as the
$\Delta R$ between the two leptons forming $Z_{2}$, and $p_{\mathrm{T},4\ell}$
(7c & 7d) from the VBS $ZZ$ process simulated at a 14 TeV (7a & 7c) and a 6
TeV (7b & 7d) muon collider.

Finally, we want to comment on the effects of lepton isolation, which
represents the requirement on the detector granularity of a muon collider. In
above analysis, we follow the Delphes card for muon colliders mucard , where
the lepton isolation cone is set to 0.1 by default. However, from Fig. 7, one
can find LL VBS $ZZ$ peak sharply at low $\Delta R_{{Z_{2}},pm}$ region 111We
have checked that at the Z boson’s rest frame, leptons from Z decay tend to be
perpendicular to the Z boson’s fly direction in the lab frame. However, at the
14 TeV muon collider, Z bosons are quite energetic and the decay products are
boosted to be closer, especially for the longitudinal mode., and if one cuts
on $\Delta R_{{Z_{1,2}},pm}$ to be larger than e.g. 0.2 LClep ;
Yonamine:2011jg , the significance can drop to 6 $\sigma$ as shown in Table 4.
This clearly points out to the needs of detector optimization, in order to
enlarge the gain of measuring LL VBS at a muon collider.

Table 4: The cut-flow table and the corresponding significance when $\Delta R_{{Z_{1,2}},pm}>0.2$. cuts |  $s$ |  $b$ |  $S\,[\sigma]$
---|---|---|---
$\Delta R_{{Z_{1,2}},pm}>0.2$ |  334.3 |  14331.2 |  2.8
$0.2<\Delta R_{{Z_{1}},pm}<0.8$, $0.2<\Delta R_{{Z_{2}},pm}<0.5$ |  108.7 |  1007.6 |  3.4
$0.2<\Delta R_{{Z_{1}},pm}<0.8$, $0.2<\Delta R_{{Z_{2}},pm}<0.5$, |  100.0 |  695.4 |  3.7
$60\mathrm{GeV}<M_{Z1},M_{Z2}<130\mathrm{GeV}$ |  |  |
$0.2<\Delta R_{{Z_{1}},pm}<0.8$, $0.2<\Delta R_{{Z_{2}},pm}<0.5$, |  97.0 |  400.7 |  4.7
$60\mathrm{GeV}<M_{Z1},M_{Z2}<130\mathrm{GeV}$, $p_{\mathrm{T},4\ell}<500\mathrm{GeV}$ |  |  |
$0.2<\Delta R_{{Z_{1}},pm}<0.8$, $0.2<\Delta R_{{Z_{2}},pm}<0.5$, |  61.7 |  90.2 |  5.9
$60\mathrm{GeV}<M_{Z1},M_{Z2}<130\mathrm{GeV}$, $p_{\mathrm{T},4\ell}<500\mathrm{GeV}$, |  |  |
$M_{4l}<3000\mathrm{GeV}$, $\not{E}<180\mathrm{GeV}$ |  |  |

##  VI Outlook and conclusions

Measuring longitudinally polarized vector boson scattering in the $ZZ$ channel
is a promising way to investigate unitarity restoration with the Higgs
mechanism and to search for possible new physics. However, at the LHC, it
demands the end of the HL-LHC lifetime luminosity (i.e.,
3000$~{}\mathrm{fb^{-1}}$) and advanced data analysis technique in order to
reach the discovery threshold, due to the small production rates. We show that
there could be great potential for the discovery of this signal process at
future linear colliders. In this paper, we perform a Monte Carlo study and
examine the projected sensitivity of longitudinally polarized $ZZ$ scattering
at TeV scale muon colliders. With a conservative estimation using the BDT
technique, we find that a 5 standard deviation discovery can be achieved at a
14 TeV muon collider, with 3$~{}\mathrm{ab^{-1}}$ of data collected. The
result outperforms the traditional cut-based analysis strategy. The study is
also performed on a 6 TeV muon collider and shows that 2 standard deviations
can be reached with around 4$~{}\mathrm{ab^{-1}}$ of data. Although less
discriminating power, w.r.t. the 14 TeV muon collider, is observed to
distinguish the longitudinal component of VBS $ZZ$ from the backgrounds, the 2
$\sigma$ significance already surpasses the estimated sensitivity at the HL-
LHC, showing good potentials to reach the first VBS $Z_{L}Z_{L}$ discovery on
a muon collider. The effect from lepton isolation and detector granularity is
also discussed, which shows to be larger at higher energy muon colliders, as
the leptons from longitudinally polarized Z decays tend to be closer.

###### Acknowledgements.

This work is supported in part by the National Natural Science Foundation of
China under Grants No. 12075004 and No. 12061141002, by MOST under grant No.
2018YFA0403900.

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# Binned angular analysis of ${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$
with LHCb

Leon Carus  on behalf of the LHCb collaboration  Physikalisches Institut,
Universität Heidelberg, Im Neuenheimer Feld 226,
69120 Heidelberg, Germany

###### Abstract

The analysis of ${b}\\!\to{s}\ell\ell$ flavor-changing neutral current decays
is a powerful test of the Standard Model (SM). Due to the strong suppression
of these modes in the SM, potential New Physics contributions can have a
significant impact on the measured physics observables. The angular analysis
of ${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays gives access to
optimized angular observables, which are less dependent on hadronic form
factors than for example branching fraction measurements. The angular
observables are extracted in bins of the invariant dimuon mass squared,
$q^{2}$, making the analysis model-independent with respect to any assumptions
about the shape of the signal distribution in $q^{2}$. Previous binned
measurements of the ${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ angular
observables by LHCb were in tension with the SM at the level of $3.3$ standard
deviations ($\sigma$). These proceedings present the ongoing effort to update
the binned angular analysis to include the full Run 1 and Run 2 data sample of
LHCb.

##  1 Introduction

The decay ${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ occurs via a
${b}\\!\to{s}\ell\ell$ flavor-changing neutral current (FCNC) transition.
These transitions are not allowed in the SM at tree level and can therefore
only occur as electroweak penguin decays at the loop level as displayed in
Fig. 1 (left). The combination of loop- and CKM-suppression leads to a very
high sensitivity to New Physics (NP) effects, which can potentially contribute
at the tree level as shown in Fig. 1 (right). Previous analysis of
${b}\\!\to{s}\ell\ell$ decays by LHCb found tensions of data with SM
predictions in angular observables [1, 2, 3] and branching fraction
measurements [4, 5, 6, 7]. The angular analysis of the decay
${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ allows access to optimized
observables, which are less dependent on the theoretical uncertainties of
hadronic form factors [8]. The binned angular analysis of
${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays using the Run 1 and 2016
data sample of LHCb, saw a global tension between the SM prediction and its
measurement of about $3.3\leavevmode\nobreak\ \sigma$ [1]. The results and
predictions for $P_{5}^{\prime}$ and the Wilson Coefficient shifts
$\Delta{\mathcal{R}e(\mathcal{C}_{9})}$ and
$\Delta{\mathcal{R}e(\mathcal{C}_{10})}$ are shown in Fig. 2. A consistent
shift of $P_{5}^{\prime}$ to higher values is visible in the left plot. The
Wilson Coefficients measurement prefers a shift from the SM value to
${\Delta\mathcal{R}e(\mathcal{C}_{9})}\approx-1$.

Figure 1:  Feynman diagram of the SM $b\rightarrow s\mu\mu$ penguin
contribution (left). Example of NP tree-level decay including a lepto-quark.

The estimation of hadronic form factor uncertainties and the effect of $c$
$\overline{c}$-loop contributions present a significant challenge for theory
predictions of the angular observables and are an active field of study. In
contrast to the $q^{2}$-unbinned angular analysis [9, 10, 11], the measurement
of binned angular observables is largely model-independent. These proceedings
discuss the currently ongoing effort to update the binned angular analysis of
${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ adding the LHCb data samples
from 2017 and 2018. This increases the integrated luminosity to a total of 8.4
$\text{\,fb}^{-1}$.

Figure 2:  Measurement of the less form-factor dependent observable
$P_{5}^{\prime}$ (left) by LHCb using Run 1 and 2016 data in black. The SM
predictions [12, 13] are displayed in orange. The right plot shows the results
of the Wilson Coefficient shifts $\Delta{\mathcal{R}e(\mathcal{C}_{9})}$ and
$\Delta{\mathcal{R}e(\mathcal{C}_{10})}$ derived from the measured angular
observables. Both plots show the measurement to be in tension with the SM
prediction.

##  2 Five-dimensional angular fit to
${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$

The decay ${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ can be fully described
by the three decay angles $\theta_{\ell}$, $\theta_{K}$, $\phi$, the invariant
mass of the $K\pi$ system, $m_{K\pi}$, and the squared invariant mass of the
$\mu^{+}\mu^{-}$ system, $q^{2}$. This analysis aims to measure the full set
of angular observables and the branching fraction in bins of $q^{2}$. The
normalized differential decay rate used to fit the data is given by

$\begin{array}[]{rc@{\,}c@{\,}l}\frac{1}{d(\Gamma+\overline{\Gamma})/dq^{2}}\frac{\Gamma}{d\vec{\Omega}\
dq^{2}\
dm_{K\pi}}=&\tfrac{9}{64\pi}&&\biggl{(}\sum_{i\in\mathcal{P}}({S_{i}}\pm{A_{i}})f_{i}(\theta_{l},\theta_{K},\phi){|\mathcal{BW_{P}}(m_{K\pi})|^{2}}\\\
&+&&\sum_{i\in\mathcal{S}}(S_{i}\pm{A_{i}})f_{i}(\theta_{l},\theta_{K},\phi){|\mathcal{L_{S}}(m_{K\pi})|^{2}}\\\
&+&&\sum_{i\in\mathcal{S/P}}({S_{i}}\pm{A_{i}})f_{i}(\theta_{l},\theta_{K},\phi){g_{i}(\mathcal{L_{S}}(m_{K\pi})\mathcal{BW_{P}^{\star}}(m_{K\pi}))}\biggr{)}.\par\end{array}$
(1)

The $m_{{K}\pi}$ dependence of the decay rate is included directly in the
angular fit PDF. The precise description of the $m_{{K}\pi}$ shape is
essential to disentangle spin-1 (P-wave) and spin-0 (S-wave) contributions to
the $K\pi$ system. The P-wave resonance (${K}^{*0}$) peaks in the $m_{{K}\pi}$
distribution and is parameterized using a relativistic Breit-Wigner function
($\mathcal{BW_{P}}$). The P-wave part of the differential decay rate is
described by the first term in Eq. 1, while the second term represents the
S-wave part. The third term arises from the interference between P- and
S-wave. S-wave contributions from ${{K}^{*0}}(1430)$ and $\kappa(800)$ decays
exhibit a broad shape in the $m_{K\pi}$ spectrum and are described by the LASS
model [14] ($\mathcal{L_{S}}$). The function $g_{i}$ in the third term returns
either the $\mathcal{R}e$ or $\mathcal{I}m$ part of its argument depending on
the index i. The five-dimensional angular fit PDF depends on the three decay
angles, $m_{{K}\pi}$, and the invariant mass of the ${B}^{0}$ candidate
$m_{{K}\pi\mu\mu}$. $m_{{K}\pi\mu\mu}$ specifically is used to distinguish
signal candidates, which peak around the ${B}^{0}$ mass, from the remaining
combinatorial background. This type of background arises from randomly
combined tracks in the detector and is distributed exponentially in
$m_{{K}\pi\mu\mu}$. The background shape in $m_{{K}\pi}$ and the decay angles
is modeled using polynomial functions. A five-dimensional maximum likelihood
fit is performed in each $q^{2}$ bin separately to extract the CP-symmetric
($S_{i}$) and CP-asymmetric ($A_{i}$) angular observables and their
uncertainties. The $q^{2}$ spectrum from
$0.06\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ and
$19.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ is analyzed in 8 narrow bins of
an approximate width of $2\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$.
Additionally, the angular observables are extracted in two wide bins from
$1.1\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ to
$6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ and
$15.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ to
$19.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$. A wide range in $m_{{K}\pi}$
is chosen to include both the ${K}^{*0}$ resonance as well as a significant
S-wave component. The spectrum of the decay angles, $q^{2}$ and $m_{{K}\pi}$
in data is distorted by the selection and reconstruction of the decays. In
order to take this acceptance effect into account in the fit to data, it is
modeled using five-dimensional Legendre polynomials. To this end, the
polynomial parameters are extracted from fully reconstructed and selected LHCb
simulation of ${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays using the
method of moments [15]. Separate acceptance functions are calculated for
${B}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ decays in
order to take detection asymmetries into account. Example projections of the
resulting parameterizations are displayed in Fig. 3.

Figure 3:  One-dimensional acceptance projections for $B^{0}\rightarrow
K^{*0}\mu\mu$ decays in $0.4<q^{2}<19.1\leavevmode\nobreak\ GeV^{4}/c^{2}$
determined on LHCb simulation for the year 2016. The plots show the
$cos(\theta_{K})$ (left) and $q^{2}$ (right) projections of the
parameterization along with simulated events.

The parameterization is either evaluated at a fixed point in $q^{2}$ and used
as a multiplicative factor in the probability density function or its inverse
value is used as an acceptance correction weight. The first method is used in
the narrow $q^{2}$ bins, where the variation of the acceptance across the
$q^{2}$ bin is small and the evaluation of the acceptance at a single point is
valid. In bins where this is not the case, a weighted fit is performed.

##  3 Improvements to analysis strategy

Compared to the previous binned measurement [1], the analysis strategy has
been improved in several ways. The description of the $m_{{K}\pi}$ dependence
is now directly included in the angular PDF as described in Eq. 1. Previously,
the $m_{{K}\pi}$ distribution has only been fitted simultaneously to the
angular fit. Along with the wider $m_{{K}\pi}$ selection window compared to
the previous analysis, this significantly reduces the uncertainty on the
observables describing the P/S-wave interference, which will be measured for
the first time. The selection has been improved to increase the signal
efficiency and background rejection by re-optimizing the boosted decision tree
against combinatorial background and the particle identification selection
against misidentified physical backgrounds. As displayed in Eq. 1, the CP-
asymmetries are extracted simultaneously with the CP-symmetries. This is an
improvement compared to previous iterations, where the asymmetries have either
not been extracted or have only been determined separately from the CP-
symmetries. Measuring the asymmetries simultaneously in addition preserves the
correlations between the symmetries and asymmetries. To constrain the total
CP-asymmetry ${\mathcal{A}}^{{C\\!P}}$ an extended term is added to the
likelihood. The information from the extended term is also used to measure the
branching fraction, where the relative signal efficiency can be determined
model-independently using information from the angular observables and the
acceptance.

Overall the sensitivity to the angular observables is expected to increase
significantly compared to the previous iteration of the analysis. The major
part of the improvement originates from the inclusion of 2017/18 LHCb data
samples, which effectively double the number of signal candidates.
Furthermore, the statistical uncertainty is reduced by the re-optimization of
the selection and adding the $m_{{K}\pi}$ dependence directly to the angular
PDF. Figure 4 shows the result of the previous binned angular analysis of
${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays by LHCb using Run 1 and
2016 data for the optimized observable $P_{5}^{\prime}$ (black). Overlayed in
blue the same result with the projected total uncertainty using the full Run 1
and Run 2 data sample is shown.

##  4 Conclusion

In conclusion, the update of the binned angular analysis using the full Run 1
and Run 2 LHCb data sample is expected to significantly improve the precision
of the measurement of the binned angular observables. The observables
describing the angular structure of the S-wave and interference between S- and
P-wave will be published for the first time. Furthermore, this analysis will
publish the most complete set of the angular asymmetries of the decay
${{B}^{0}}\\!\to{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ to date along with a model-
independent measurement of the branching fraction.

Figure 4:  Measurement of the less form-factor dependent observable
$P_{5}^{\prime}$ (left) by LHCb using Run 1 and 2016 data in black. The
projected total uncertainty using the full Run 1 and Run 2 LHCb data sample
assuming identical central values is shown in blue.

## References

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