# New physics search via angular distribution of $B\to D^{*}\ell\nu_{\ell}$
decay in the light of the new lattice data

Tejhas Kapoor<EMAIL_ADDRESS>Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France  Zhuo-Ran Huang  Department of
Physics, College of Physics, Mechanical and Electrical Engineering, Jishou
University, Jishou 416000, China  Emi Kou  Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France

###### Abstract

In this article, we investigate the angular distribution of the ${B}\to
D^{*}\ell{\nu}_{\ell}\;$process to search for New Physics signals. The Belle
collaboration has analysed it to constraint $V_{cb}$ and the $B\to D^{*}$ form
factors, under the assumption of the Standard Model. With the newly released
lattice QCD data, we can perform a simultaneous fit of the form factors,
$V_{cb}$ as well as of furthermore new physics parameters. We use the untagged
Belle data and the lattice data to constrain right-handed new physics. In
addition, we also generate unbinned pseudo-dataset and perform a sensitivity
study on more general new physics models, along with the lattice data.

##  1 Introduction

The ${B}\to D^{*}\ell{\nu}_{\ell}\;$decay has been paid a great attention in
recent years. The first reason for this is the $V_{cb}$ puzzle: there has been
a longstanding tension between the values of $|V_{cb}|$ determined using the
inclusive measurements and that determined by the exclusive $b\rightarrow
c\ell\nu$ decays. The inclusive measurement uses the semileptonic decays of
the type $B\rightarrow X_{c}\ell\nu$, where $X_{c}$ are all possible hadronic
states including charm quark. The current world average is $|V_{cb}|=(42.2\pm
0.8)\times 10^{-3}$ [1]. On the other hand, the exclusive measurement of
$|V_{cb}|$ has the following world average:
${\color[rgb]{0,0,0}|V_{cb}|=}(38.46\pm 0.40\pm 0.55)\times 10^{-3}$, where
the first uncertainty combines the statistical and systematic uncertainties
from the experimental measurement and the second is theoretical (lattice QCD
calculation and electroweak correction) [2].

The second reason  is the so-called $\mathcal{R}(D^{(*)})$ anomaly: to test
the lepton flavour universality in the $b\rightarrow c\ell\nu$ decays, a ratio
is defined:

$\displaystyle\mathcal{R}(D^{(*)})=\frac{\mathcal{B}(B\rightarrow
D^{(*)}\tau\nu)}{\mathcal{B}(B\rightarrow
D^{(*)}\ell\nu)};\quad\ell=\\{e,\mu\\}$ (1)

The combined results from BaBar [3, 4], Belle [5, 6, 7, 8], LHCb [9, 10] and
Belle II [11] exceed the SM predictions [2] by about $3.3\sigma$. Theoretical
efforts to explain such anomalies branch into model-independent and model-
dependent analysis. For the model-dependent analysis, several classes of
models have been proposed to provide a resolution to such anomalies. These
classes of models can be divided into three major categories: extended higgs
sector models [12, 13, 14, 15, 16, 17, 18, 19, 20, 21], heavy charged vector
boson models [22, 23, 24, 25, 26, 27] and leptoquark models [28, 29, 30, 31,
32, 33], with the latter being the most popular [34]. Given that the ratio
$\mathcal{R}(D^{(*)})$ is measured to be higher than its SM prediction, the
model-dependent approaches usually try to enhance the $B\rightarrow
D^{(*)}\tau\nu$ rate (say, by introducing new heavy particles), rather than
suppress the $B\rightarrow D^{(*)}\ell\nu$ rate, as the latter have stringent
constraints on new physics coupling to electrons and muons [35].

In this work, we explore the potential of the angular distribution analysis of
the ${B}\to D^{*}\ell{\nu}_{\ell}\;$process ($\ell=e,\mu$) to search for
signals beyond the SM, following the model-independent approach. We will work
with a low-energy effective Hamiltonian containing all dimension six, four-
fermion operators, assuming no right-handed neutrinos. We expect the angular
distribution to be a powerful tool to distinguish the different Dirac
structure that these operators have: (left)right-handed, (pseudo)scalar and
tensor. These contributions can arise in several NP scenarios. For example,
the right-handed contribution can arise from $W_{L}-W_{R}$ mixing left-right
symmetric models [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. On the
other hand, the scalar contribution is generated in models with extra charged
scalars [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60], while the tensor
operators are often generated by a Fierz transformation on scalar operators
[34]. These NP effects can be both lepton flavour universal and lepton flavour
violating, thus, having strong implication on the $R(D^{(*)})$ anomaly.

Belle has used the angular distribution data of ${B}\to
D^{*}\ell{\nu}_{\ell}\;$decay to fit simultaneously $|V_{cb}|$ and the form
factors [61, 62, 63]. The angular distribution is written in terms of 4
kinematical variables (one invariant mass and three angles). The analysis uses
four, one-dimensional binned distributions (corresponding to each kinematical
variable), in which the other three variables are integrated away. Since
$|V_{cb}|$ is always multiplied with a scaling factor $\mathcal{F}(w)\eta_{\rm
EW}$, the fit parameter is actually $|V_{cb}|\mathcal{F}(1)\eta_{\rm EW}$,
where $\eta_{\rm EW}$ is the electroweak correction, and
$\mathcal{F}(1)=h_{A_{1}}(1)$ is a form factor at zero-recoil. Therefore, the
value of $\mathcal{F}(1)$ is required to obtain $|V_{cb}|$. It should be noted
that addition of NP effects increases the number of parameters, and
experimental data alone cannot fit form factors, $V_{cb}$ and NP parameters
simultaneously.

The recent lattice QCD results from Fermilab-MILC[64], JLQCD[65] and HPQCD
collaborations [66] provided a breakthrough, by publishing the results on the
form factors for this decay. Previously, only $\mathcal{F}(1)$ was known at
zero recoil; but now we have results on four form factors
$(f,g,\mathcal{F}_{1},\mathcal{F}_{2})$ in the low-recoil region. In the light
of these new results, we present a simultaneous fit of Belle and lattice QCD
data. Now, thanks to the lattice data, we can even fit the NP parameters,
allowing us to constrain different types of NP. In addition to using the Belle
binned dataset, we will also demonstrate the fit using an unbinned belle
pseudo dataset [67] – the latter will become possible once a larger amount of
data becomes available at the Belle II experiment [68].

The organisation of the article is as follows: In Section 2 we introduce the
effective Hamiltonian and the different NP operators relevant for the ${B}\to
D^{*}\ell{\nu}_{\ell}\;$decay. In Section 3, we write the complete angular
decay distribution for the ${B}\to D^{*}\ell{\nu}_{\ell}\;$process in terms of
the NP Wilson coefficients, hadronic and leptonic amplitudes. In Section 4, we
describe the new lattice QCD data in the BGL form factor parameteriation, and
show the form factor fit results to the higher recoil region using the new
lattice data. In Section 5, we use the untagged Belle binned data along with
the lattice data to fit form factors, $V_{cb}$ and right-handed NP Wilson
coefficient, and present the fit results. Finally, in Section 6, we
demonstrate the fit procedure with unbinned Belle and lattice data and perform
a sensitivity study with different NP operators and we conclude in Sedction 7

##  2 New physics model

In this article, we consider the new physics particle which contributes to the
semi-leptonic ${B}\to D^{*}\ell{\nu}_{\ell}\;$decay with $\ell=e,\mu$. We
start with the most general effective Hamiltonian, containing all dimension
six, four-fermion operators, assuming no right-handed neutrinos:

$\mathcal{H}_{\rm eff}=\frac{4G_{F}}{\sqrt{2}}V_{cb}\sum_{\ell=e,\mu}[C_{\rm
V_{L}}^{\ell}O^{\ell}_{\rm V_{L}}+C_{\rm V_{R}}^{\ell}O^{\ell}_{\rm
V_{R}}+{C_{\rm S}^{\ell}O^{\ell}_{\rm S}}+C_{\rm P}^{\ell}O^{\ell}_{\rm
P}+C_{\rm T}^{\ell}O^{\ell}_{\rm T}]$ (2)

where ${\rm V_{L},V_{R},S,P,T}$ indicate the left-handed, right-handed,
scalar, pseudoscalar, and tensor terms, respectively. The corresponding
operators are given as

$\displaystyle\begin{aligned} O^{\ell}_{\rm
V_{L}}&=(\bar{c}_{L}\gamma^{\mu}b_{L})(\bar{{\ell}}_{L}\gamma_{\mu}\nu_{{\ell}L})\\\
O^{\ell}_{\rm
V_{R}}&=(\bar{c}_{R}\gamma^{\mu}b_{R})(\bar{{\ell}}_{L}\gamma_{\mu}\nu_{{\ell}L})\\\
O^{\ell}_{\rm S}&=(\bar{c}b)(\bar{{\ell}}_{R}\nu_{{\ell}L})\\\ O^{\ell}_{\rm
P}&=(\bar{c}\gamma^{5}b)(\bar{{\ell}}_{R}\nu_{{\ell}L})\\\ O^{\ell}_{\rm
T}&=(\bar{c}_{R}\sigma^{\mu\nu}b_{L})(\bar{{\ell}}_{R}\sigma_{\mu\nu}\nu_{{\ell}L})\end{aligned}$
(3)

where $\psi_{L/R}=\frac{1\mp\gamma_{5}}{2}\psi$ with
$\psi=\\{c,b,\ell,\nu\\}$. The contribution from the scalar operator
$O^{\ell}_{\rm S}$ does not appear in the following. We ignore the superscript
$\ell$ in the following discussion. In the SM, $C_{\rm V_{L}}=1$, while the
rest of the Wilson coefficients are zero.

##  3 Angular distribution of the ${B}\to D^{*}\ell{\nu}_{\ell}\;$decay

Using the Hamiltonian given in Eq. (2), we compute the the angular
distribution of the ${B}\to D^{*}\ell{\nu}_{\ell}\;$decay (shown in Fig. 1):

$\displaystyle\begin{aligned} \frac{\text{d}\Gamma(\bar{B}\to D^{*}(\to
D\pi)\,\ell^{-}\,\bar{\nu}_{\ell})}{\text{d}q^{2}\,\text{d}\cos\theta_{V}\,\text{d}\cos\theta_{\ell}\,\text{d}\chi}=&\frac{3}{8(4\pi)^{4}}\frac{(q^{2}-m_{\ell}^{2})^{2}|\vec{p}_{D^{*}}|^{2}}{M_{B}^{2}q^{2}}G_{F}^{2}|V_{cb}|^{2}|\eta_{\rm
EW}|^{2}\mathcal{B}(D^{*}\rightarrow D\pi)\\\
&\sum_{\lambda_{\ell},\lambda_{\nu_{\ell}}}|\mathcal{M}_{\rm
V_{L}}+\mathcal{M}_{\rm V_{R}}+\mathcal{M}_{\rm P}+\mathcal{M}_{\rm
T}|^{2}\end{aligned}$ (4)

Figure 1: Decay angles for the ${B}\to D^{*}\ell{\nu}_{\ell}\;$decay, where
$\theta_{V}$ is the angle between the direction of $D$ meson and the direction
opposite to that of $B$ meson, in the $D^{*}$ meson rest frame. Similarly,
$\theta_{\ell}$ is the angle between the direction of lepton and the direction
opposite to that of $B$ meson, in the virtual $W$ rest frame. $\chi$ is the
angle between the two decay planes.

where $q^{2}$ is the invariant mass of the lepton-neutrino pair. Following the
method given in [69], we compute the amplitudes $\mathcal{M}_{\rm
V_{L},V_{R},P,T}$. In this method, we introduce the $W$ boson polarisation and
separate the amplitudes into the hadronic and leptonic part as:

$\displaystyle\begin{aligned} \mathcal{M}_{\rm V_{L}}&=\,C_{\rm
V_{L}}\sum_{\lambda_{D^{*}}=\pm,0}P(\lambda_{D^{*}})\sum_{\lambda=t,\pm,0}g_{\lambda\lambda}H^{\lambda_{D^{*}}}_{\rm
V_{L},\lambda}L_{\lambda}^{\rm VA}\\\ \mathcal{M}_{\rm V_{R}}&=\,C_{\rm
V_{R}}\sum_{\lambda_{D^{*}}=\pm,0}P(\lambda_{D^{*}})\sum_{\lambda=t,\pm,0}g_{\lambda\lambda}H^{\lambda_{D^{*}}}_{\rm
V_{R},\lambda}L_{\lambda}^{\rm VA}\\\ \mathcal{M}_{\rm P}&=\,C_{\rm
P}\sum_{\lambda_{D^{*}}=\pm,0}P(\lambda_{D^{*}})H^{\lambda_{D^{*}}}_{\rm
P}L^{\rm SP}\\\ \mathcal{M}_{\rm T}&=\,C_{\rm
T}\sum_{\lambda_{D^{*}}=\pm,0}P(\lambda_{D^{*}})\sum_{\lambda=t,\pm,0}\sum_{\lambda^{\prime}=t,\pm,0}g_{\lambda\lambda}g_{\lambda^{\prime}\lambda^{\prime}}H^{\lambda_{D^{*}}}_{\rm
T,\lambda\lambda^{\prime}}L^{\rm T}_{\lambda\lambda^{\prime}}\end{aligned}$
(5)

where $P(\lambda_{D^{*}})=\langle
D\pi|D^{*}\rangle=2g\epsilon_{D^{*}}^{*\alpha}(\lambda_{D^{*}})p_{D,\alpha}$
($g$ is the $D^{*}$-$D$-$\pi$ coupling), $\epsilon_{D^{*}}$ is the
polarisation vector of $D^{*}$ meson and $\lambda_{D^{*}}\in\\{+,-,0\\}$ is
the polarisation of $D^{*}$ meson. The polarisation vectors are given in [69].

The hadronic amplitudes are given as

$\displaystyle\begin{aligned} H_{\rm
V_{L},\lambda}^{\lambda_{D^{*}}}(q^{2})&=\bar{\epsilon}^{*}_{\mu}(\lambda)\langle
D^{*}(p_{D^{*}},\epsilon_{D^{*}}\left(\lambda_{D^{*}})\right)|\bar{c}\gamma^{\mu}(1-\gamma^{5})b|\bar{B}(p_{B})\rangle\,,\\\
H_{\rm
V_{R},\lambda}^{\lambda_{D^{*}}}(q^{2})&=\bar{\epsilon}^{*}_{\mu}(\lambda)\langle
D^{*}(p_{D^{*}},\epsilon_{D^{*}}\left(\lambda_{D^{*}})\right)|\bar{c}\gamma^{\mu}(1+\gamma^{5})b|\bar{B}(p_{B})\rangle\,,\\\
H_{\rm P}^{\lambda_{D^{*}}}(q^{2})&=\langle
D^{*}(p_{D^{*}},\epsilon_{D^{*}}\left(\lambda_{D^{*}})\right)|\bar{c}\gamma^{5}b|\bar{B}(p_{B})\rangle\,,\\\
H_{\rm
T,\lambda\lambda^{\prime}}^{\lambda_{D^{*}}}(q^{2})&=i\bar{\epsilon}^{*}_{\mu}(\lambda)\bar{\epsilon}^{*}_{\nu}(\lambda^{\prime})\langle
D^{*}(p_{D^{*}},\epsilon\left(\lambda_{D^{*}})\right)|\bar{c}\sigma^{\mu\nu}(1-\gamma^{5})b|\bar{B}(p_{B})\rangle,\end{aligned}$
(6)

and the leptonic amplitudes are given as

$\displaystyle\begin{aligned} L^{\rm
V_{L/R}}_{\lambda}&=\bar{\epsilon}_{\mu}(\lambda)\bar{u}_{\ell}\gamma^{\mu}(1\mp\gamma^{5})v_{\bar{\nu}_{\ell}}\\\
L^{\rm P}&=\bar{u}_{\ell}(1-\gamma^{5})v_{\bar{\nu}_{\ell}}\\\ L^{\rm
T}_{\lambda,\lambda^{\prime}}&=-i\bar{\epsilon}_{\mu}(\lambda)\bar{\epsilon}_{\nu}(\lambda^{\prime})\bar{u}_{\ell}\sigma^{\mu\nu}(1-\gamma^{5})v_{\bar{\nu}_{\ell}}\end{aligned}$
(7)

Both are written by $\bar{\epsilon}$, the polarisation vector of $W$ boson,
and $\lambda^{(^{\prime})}\in\\{+,-,0,t\\}$, its polarisation. Note that
$\bar{\epsilon}(t)$ is not orthogonal but proportional to $W$-momentum. The
polarisation vectors are given in [69].

Then, using the completeness relation of the polarisation vectors of $W$
boson, we square the hadronic and leptonic parts of the amplitudes separately.
This allows us to evaluate each part in a convenient frame: we choose the $W$
rest frame for the leptonic part and $B$ rest frame for the hadronic part.

The hadronic part is a complex function of form factors and we find

$\displaystyle\begin{aligned} H_{\rm V}^{\pm}(q^{2})&\equiv H_{\rm
V_{L},\pm}^{\pm}(q^{2})=-H_{\rm V_{R},\mp}^{\mp}(q^{2})\\\
&=(M_{B}+M_{D^{*}})A_{1}(q^{2})\mp\frac{\sqrt{\lambda_{D^{*}}(q^{2})}}{M_{B}+M_{D^{*}}}V(q^{2})\\\
H_{\rm V}^{0}(q^{2})&\equiv H_{\rm V_{L},0}^{0}(q^{2})=-H_{\rm
V_{R},0}^{0}(q^{2})\\\
&=\frac{M_{B}+M_{D^{*}}}{2M_{D^{*}}\sqrt{q^{2}}}\left[-(M_{B}^{2}-M_{D^{*}}^{2}-q^{2})A_{1}(q^{2})+\frac{\lambda_{D^{*}}(q^{2})}{(M_{B}+M_{D^{*}})^{2}}A_{2}(q^{2})\right]\\\
H_{\rm P}(q^{2})&\equiv H_{\rm P}^{0}(q^{2})\\\
&=-\frac{\sqrt{\lambda_{D^{*}}(q^{2})}}{M_{B}+m_{c}}A_{0}(q^{2})\\\ H_{\rm
T}^{\pm}(q^{2})&\equiv\pm H_{\rm{T},\pm t}^{\pm}(q^{2})\\\
&=\frac{1}{\sqrt{q^{2}}}\bigg{[}\sqrt{\lambda_{D^{*}}(q^{2})}T_{1}(q^{2})\pm(M_{B}^{2}-M_{D^{*}}^{2})T_{2}(q^{2})\bigg{]}\\\
H_{\rm T}^{0}(q^{2})&\equiv
H^{0}_{\rm{T},+-}(q^{2})=H^{0}_{\rm{T},0t}(q^{2})\\\
&=\frac{1}{2M_{D^{*}}}\bigg{[}(M_{B}^{2}+3M_{D^{*}}^{2}-q^{2})T_{2}(q^{2})-\frac{\lambda_{D^{*}}(q^{2})}{(M_{B}^{2}-M_{D^{*}}^{2})}T_{3}(q^{2})\bigg{]}\\\
\end{aligned}$ (8)

These expressions agree with the results in [70] while we find an overall sign
difference in $H_{T,0}$ with respect to [28].

Finally the angular distribution of $\bar{B}\rightarrow D^{*}(\rightarrow
D\pi)\ell^{-}\bar{\nu}_{\ell}$ is found to be

$\displaystyle\begin{aligned} \frac{\text{d}\Gamma(\bar{B}\rightarrow
D^{*}(\rightarrow
D\pi)\ell^{-}\bar{\nu}_{\ell})}{\text{d}w\text{d}\cos\theta_{V}\text{d}\cos\theta_{\ell}\text{d}\chi}=&\frac{6M_{B}M_{D^{*}}^{2}}{8(4\pi)^{4}}\sqrt{w^{2}-1}(1-2\,w\,r+r^{2})\,G_{F}^{2}\,\left|V_{cb}\right|^{2}|\eta_{\rm
EW}|^{2}\,\times\\\ &\mathcal{B}(D^{*}\to
D\pi)\Big{\\{}J_{1s}\sin^{2}\theta_{V}+J_{1c}\cos^{2}\theta_{V}\\\
&+(J_{2s}\sin^{2}\theta_{V}+J_{2c}\cos^{2}\theta_{V})\cos 2\theta_{\ell}\\\
&+J_{3}\sin^{2}\theta_{V}\sin^{2}\theta_{\ell}\cos 2\chi\\\ &+J_{4}\sin
2\theta_{V}\sin 2\theta_{\ell}\cos\chi+J_{5}\sin
2\theta_{V}\sin\theta_{\ell}\cos\chi\\\
&+(J_{6s}\sin^{2}\theta_{V}+J_{6c}\cos^{2}\theta_{V})\cos\theta_{\ell}\\\
&+J_{7}\sin 2\theta_{V}\sin\theta_{\ell}\sin\chi+J_{8}\sin 2\theta_{V}\sin
2\theta_{\ell}\sin\chi\\\ &+J_{9}\sin^{2}\theta_{V}\sin^{2}\theta_{\ell}\sin
2\chi\Big{\\}}\end{aligned}$ (9)

where $w\equiv\frac{M_{B}^{2}+M_{D^{*}}^{2}-q^{2}}{2M_{B}M_{D^{*}}}$ with
$q^{2}\equiv(p_{B}-p_{D^{*}})^{2}$ and $r\equiv\frac{M_{D^{*}}}{M_{B}}$. The
angular coefficients $J_{i}(i=1\sim 9)$ are given in Tables 1 and 2.

J | $|C_{\rm V_{L}}|^{2}$ | $|C_{\rm V_{R}}|^{2}$ | $\textrm{Re}[C_{\rm V_{L}}C_{\rm V_{R}}^{*}]$ | $\textrm{Im}[C_{\rm V_{L}}C_{\rm V_{R}}^{*}]$
---|---|---|---|---
$J_{1s}$ | $\frac{3}{2}[(H_{\rm V}^{+})^{2}+(H_{\rm V}^{-})^{2}]$ | $\frac{3}{2}[(H_{\rm V}^{+})^{2}+(H_{\rm V}^{-})^{2}]$ | $-6H_{\rm V}^{+}H_{\rm V}^{-}$ | 0
$J_{1c}$ | $2(H_{\rm V}^{0})^{2}$ | $2(H_{\rm V}^{0})^{2}$ | $-4(H_{\rm V}^{0})^{2}$ | 0
$J_{2s}$ | $\frac{1}{2}[(H_{\rm V}^{+})^{2}+(H_{\rm V}^{-})^{2}]$ | $\frac{1}{2}[(H_{\rm V}^{+})^{2}+(H_{\rm V}^{-})^{2}]$ | $-2H_{\rm V}^{+}H_{\rm V}^{-}$ | 0
$J_{2c}$ | $-2(H_{\rm V}^{0})^{2}$ | $-2(H_{\rm V}^{0})^{2}$ | $4(H_{\rm V}^{0})^{2}$ | 0
$J_{3}$ | $-2H_{\rm V}^{+}H_{\rm V}^{-}$ | $-2H_{\rm V}^{+}H_{\rm V}^{-}$ | $2[(H_{\rm V}^{+})^{2}+(H_{\rm V}^{-})^{2}]$ | 0
$J_{4}$ | $(H_{\rm V}^{+}H_{\rm V}^{0}+H_{\rm V}^{-}H_{\rm V}^{0})$ | $(H_{\rm V}^{+}H_{\rm V}^{0}+H_{\rm V}^{-}H_{\rm V}^{0})$ | $-2(H_{\rm V}^{+}H_{\rm V}^{0}+H_{\rm V}^{-}H_{\rm V}^{0})$ | 0
$J_{5}$ | $-2(H_{\rm V}^{+}H_{\rm V}^{0}-H_{\rm V}^{-}H_{\rm V}^{0})$ | $2(H_{\rm V}^{+}H_{\rm V}^{0}-H_{\rm V}^{-}H_{\rm V}^{0})$ | 0 | 0
$J_{6s}$ | $-2[(H_{\rm V}^{+})^{2}-(H_{\rm V}^{-})^{2}]$ | $2[(H_{\rm V}^{+})^{2}-(H_{\rm V}^{-})^{2}]$ | 0 | 0
$J_{6c}$ | 0 | 0 | 0 | 0
$J_{7}$ | 0 | 0 | 0 | 0
$J_{8}$ | 0 | 0 | 0 | $2(H_{\rm V}^{+}H_{\rm V}^{0}+H_{\rm V}^{-}H_{\rm V}^{0})$
$J_{9}$ | 0 | 0 | 0 | $-2[(H_{\rm V}^{+})^{2}-(H_{\rm V}^{-})^{2}]$
Table 1: The angular coefficients ($J$-functions) appearing in the differential decay rate in Eq. (9). $H_{\rm V}^{\pm,0}$ are the hadronic amplitudes, defined in Eq. (8). The columns contain the left- and right-handed contributions and their interference terms. The complete $J_{i}$-function is sum of all terms in its row, multiplied with the corresponding Wilson coefficient term in the corresponding column (written in the topmost row). J | $|C_{\rm P}|^{2}$ | $|C_{\rm T}|^{2}$ | $\textrm{Re}[C_{\rm P}C_{\rm T}^{*}]$ | $\textrm{Im}[C_{\rm P}C_{\rm T}^{*}]$
---|---|---|---|---
$J_{1s}$ | 0 | $8[(H_{\rm T}^{+})^{2}+(H_{\rm T}^{-})^{2}]$ | 0 | 0
$J_{1c}$ | $2H_{\rm P}^{2}$ | $32(H_{\rm T}^{0})^{2}$ | 0 | 0
$J_{2s}$ | 0 | $-8[(H_{\rm T}^{+})^{2}+(H_{\rm T}^{-})^{2}]$ | 0 | 0
$J_{2c}$ | 0 | $32(H_{\rm T}^{0})^{2}$ | 0 | 0
$J_{3}$ | 0 | $-32H_{\rm T}^{+}H_{\rm T}^{-}$ | 0 | 0
$J_{4}$ | 0 | $16(H_{\rm T}^{-}H_{\rm T}^{0}-H_{\rm T}^{+}H_{T}^{0})$ | 0 | 0
$J_{5}$ | 0 | 0 | $-8H_{\rm P}(H_{\rm T}^{+}-H_{\rm T}^{-})$ | 0
$J_{6s}$ | 0 | 0 | 0 | 0
$J_{6c}$ | 0 | 0 | $-32H_{\rm P}H_{\rm T}^{0}$ | 0
$J_{7}$ | 0 | 0 | 0 | $8H_{\rm P}(H_{\rm T}^{+}+H_{\rm T}^{-})$
$J_{8}$ | 0 | 0 | 0 | 0
$J_{9}$ | 0 | 0 | 0 | 0
Table 2: The angular coefficients ($J$-functions) appearing in the
differential decay rate in Eq. (9). $H_{\rm T}^{\pm,0}$ and $H_{\rm P}$ are
the hadronic amplitudes, defined in Eq. (8). The columns contain the
pseudocalar and tensor contributions and their interference terms. The
complete $J_{i}$-function is sum of all terms in its row, multiplied with the
corresponding Wilson coefficient term in the corresponding column (written in
the topmost row).

At the limit of massless leptons, the interferences occur only between left-
and right-handed currents, and between psudoscalar and tensor contributions,
which means that the SM (i.e. $C_{\rm V_{L}}=1$) interferes only with the
right-handed NP. Since the SM does not produce any of $J_{6c,7,8,9}$, a
nonzero measurement of these angular coefficients indicates immediately an
appearance of NP (null-test). In particular, the $J_{7,8,9}$ are CP violating
observable, which can be induced only by complex Wilson coefficients.

As given in Eq. (8), the helicity amplitudes $(H_{\rm V}^{\pm/0},H_{\rm
P},H_{\rm T}^{\pm/0})$ are the functions of the form factors. Thus, the
desired NP information which is engraved in the Wilson coefficients can be
obtained only by knowing them. As we discuss in the next section, the vector
and the axial vector form factors have recently been computed by the lattice
QCD and we will use those information to investigate the NP effects.

##  4 The new lattice data on the form factors

As mentioned in the introduction, the Belle experiment has attempted to fit
the form factors and the Standard Model parameters, i.e. $V_{cb}$
simultaneously. However, it is important to mention that it has always
required one lattice input, such as $\mathcal{F}(1)$. Furthermore, an
inclusion of the NP effects increase the number of parameters and the
experiment cannot fit all of them, as demonstrated in our previous paper [67]:
for example, in order to fit the right-handed contribution, we needed to
include another form factor $h_{V}(1)$ as an input. Now that the lattice QCD
results are available, one can perform the NP fit in a more systematic way, by
including the lattice data together with the experimental data. In the
following sections, we will illustrate how to perform such fit.

The three lattice QCD collaborations have recently published the results on
the $B\to D^{*}$ vector and axial-vector form factors [64, 65, 66]. Previously
only the zero-recoil results were available [71]; however, the new papers have
computed the form factors on a few points around the low recoil region. As
shown in the previous section, our study requires not only these two types of
form factors but also the pseudoscalar and tensor ones. These form factors
could in principle, be obtained from the lattice simulation as well (see e.g.
[66]). On the other hand, as introduced in [28], they can be related to the
vector and axial-vector form factors. We derive these relations in Appendix A,
which agree with the results in [28]. Thus, in the following, we relate the
pseudoscalar and tensor form factors to the vector and axial-vector form
factors, for which the lattice results are available.

In the lattice results, the form factors are parameterised using the so-called
BGL parameterisation [72]. Thus, in this article, we use the same
parameterisation. The form factors are related to them as follows:

$\displaystyle\begin{aligned} g&=\frac{2}{M_{B}+M_{D^{*}}}V\\\
f&=(M_{B}+M_{D^{*}})A_{1}\\\
\mathcal{F}_{1}&=\frac{1}{2M_{D^{*}}}\left[(M_{B}^{2}-M_{D^{*}}^{2}-q^{2})(M_{B}+M_{D^{*}})A_{1}-\frac{4M_{B}^{2}|{\vec{p}}_{D^{*}}|^{2}}{M_{B}+M_{D^{*}}}A_{2}\right]\\\
\mathcal{F}_{2}&=2A_{0}\end{aligned}$ (10)

where the $q^{2}$ (or $w$) dependence of the form factors is implicit. By
using these definitions, the helicity amplitudes can be now written in a very
simple manner:

$\displaystyle\begin{aligned} H_{V}^{\pm}(w)&=f\mp
gM_{B}|{\vec{p}_{D^{*}}}|\\\
H_{V}^{0}(w)&=\frac{\mathcal{F}_{1}}{\sqrt{q^{2}}}\\\
H_{P}(w)&=-\frac{\mathcal{F}_{2}M_{B}|{\vec{p}_{D^{*}}}|}{M_{B}+m_{c}}\\\
H_{T}^{\pm}(w)&=\frac{\pm
f(m_{b}-m_{c})+gM_{B}|{\vec{p}_{D^{*}}}|(m_{b}+m_{c})}{\sqrt{q^{2}}}\\\
H_{T}^{0}(w)&=\frac{-(m_{b}-m_{c})(-\mathcal{F}_{1}(M_{B}^{2}-M_{D^{*}}^{2})+2\mathcal{F}_{2}M_{B}^{2}|{\vec{p}_{D^{*}}}|^{2})}{(M_{B}^{2}-M_{D^{*}}^{2}){q^{2}}}\end{aligned}$
(11)

where $|{\vec{p}_{D^{*}}}|=M_{D^{*}}\sqrt{w^{2}-1}$ and
$\sqrt{q^{2}}=M_{B}\sqrt{1-2wr+r^{2}}$. $m_{b(c)}$ is the $b(c)$-quark mass.
Note that the pseudoscalar and tensor form factors are reduced to the above
four form factors thanks to the relations given in Appendix A.

The momentum dependence of these form factors are given in the $z-$ expansion:

$\displaystyle\begin{aligned}
g(z)&=\frac{1}{P_{1^{-}}(z)\phi_{g}(z)}\sum_{n=0}^{\infty}a^{g}_{n}z_{n},\quad\quad\;f(z)=\frac{1}{P_{1^{+}}(z)\phi_{f}(z)}\sum_{n=0}^{\infty}a^{f}_{n}z^{n},\\\
\mathcal{F}_{1}(z)&=\frac{1}{P_{1^{+}}(z)\phi_{\mathcal{F}_{1}}(z)}\sum_{n=0}^{\infty}a^{\mathcal{F}_{1}}_{n}z_{n},\quad\mathcal{F}_{2}(z)=\frac{1}{P_{0^{-}}(z)\phi_{\mathcal{F}_{2}}(z)}\sum_{n=0}^{\infty}a^{\mathcal{F}_{2}}_{n}z_{n},\end{aligned}$
(12)

where

$z\equiv\frac{\sqrt{w+1}-\sqrt{2}}{\sqrt{w+1}+\sqrt{2}}$ (13)

Note that the expansion coefficients must satisfy the unitarity conditions:

$\sum_{n=0}^{\infty}(a^{g}_{n})^{2}<1,\quad\sum_{n=0}^{\infty}(a^{f}_{n})^{2}+(a^{\mathcal{F}_{1}}_{n})^{2}<1,\quad\sum_{n=0}^{\infty}(a^{\mathcal{F}_{2}}_{n})^{2}<1$
(14)

From Eq. (10), we can see that the form factors are not completely
independent.

Type |  $B_{c}^{(*)}$ mass (GeV) | $\chi^{T,L}_{1^{\pm}}$
---|---|---
$f$, $\mathcal{F}_{1}$ | 6.739, 6.750, 7.145, 7.150 | $3.894\times 10^{-4}\ {\rm GeV}^{-2}$
$g$ | 6.329, 6.920, 7.020, 7.280 | $5.131\times 10^{-4}\ {\rm GeV}^{-2}$
$\mathcal{F}_{2}$ | 6.275, 6.842, 7.250 | $1.9421\times 10^{-2}$
Table 3: The input parameters for the BGL form factors given in Eq. (12). We
use the values applied in [64]

At the zero-recoil limit, i.e. $|{\vec{p}_{D^{*}}}|=0$ $(w_{\rm min}=1)$, we
find that both $f$ and $\mathcal{F}_{1}$ are written only by the $A_{1}$ form
factor and are related as

$\mathcal{F}_{1}(0)=(M_{B}-M_{D^{*}})f(0)$ (15)

On the other hand, at the maximum recoil, $q^{2}=0$ $(w_{\rm
max}=\frac{M_{B}^{2}+M_{D^{*}}^{2}}{2M_{B}M_{D^{*}}})$, we have a condition,
$A_{0}=\frac{M_{B}+M_{D^{*}}}{2M_{D^{*}}}A_{1}-\frac{M_{B}-M_{D^{*}}}{2M_{D^{*}}}A_{2}$,
to avoid the pole. This leads to

$\mathcal{F}_{1}(z_{\rm
max})=\frac{(M_{B}^{2}-M_{D^{*}}^{2})}{2}\mathcal{F}_{2}(z_{\rm max})$ (16)

In order to take into account these relations, we will eliminate the lowest
order $\mathcal{F}_{1}$ constant, i.e $a^{\mathcal{F}_{1}}_{0}$ and the
highest order $\mathcal{F}_{2}$ constant $a^{\mathcal{F}_{2}}_{j_{\rm max}}$
by solving these equations.

The functions $P_{1^{-},1^{+},0^{-}}$ and
$\phi_{g,f,\mathcal{F}_{1},\mathcal{F}_{2}}$, whose expression can be found in
the original article [72], contain several input parameters. We follow the
reference [64] (originally obtained in [73]) and use the values given in Table
3 as input parameters111These values are very different from the one used in
[62] which uses the values from [72], with $n_{I}=2.6$, $M_{B}=5.280$GeV and
$M_{D^{*}}=2.010$GeV.

###  4.1 The lattice QCD data and their extrapolations to higher $w$ regions

Form factor | JLQCD | FM | JLQCD (prior) | FM (prior)
---|---|---|---|---
$a^{f}_{0}$ | $0.01197(19)$ | $0.01209(19)$ | $0.01197(19)$ | $0.01208(19)$
$a^{f}_{1}$ | $0.020(11)$ | $-0.012(20)$ | $0.021(10)$ | $0.001(15)$
$a^{f}_{2}$ | $0.00(49)$ | $0.8(14)$ | $-0.08(43)$ | $0.04(79)$
$a^{g}_{0}$ | $0.0294(18)$ | $0.0329(12)$ | $0.0293(18)$ | $0.0328(12)$
$a^{g}_{1}$ | $-0.057(51)$ | $-0.15(10)$ | $-0.057(37)$ | $-0.136(55)$
$a^{g}_{2}$ | $1.0(31)$ | $0.9(55)$ | $0.72(95)$ | $0.27(98)$
$a^{\mathcal{F}_{1}}_{1}$ | $0.0010(42)$ | $-0.0080(38)$ | $0.0012(41)$ | $-0.006(3)$
$a^{\mathcal{F}_{1}}_{2}$ | $0.04(21)$ | $0.14(23)$ | $0.03(20)$ | $0.01(15)$
$a^{\mathcal{F}_{2}}_{0}$ | $0.0484(17)$ | $0.0500(15)$ | $0.0485(17)$ | $0.0499(15)$
$a^{\mathcal{F}_{2}}_{1}$ | $-0.055(89)$ | $-0.377(89)$ | $-0.054(88)$ | $-0.324(77)$
Table 4: The BGL form factor fitted values with the lattice data from JLQCD
and Fermilab-MILC (FM) collaborations. The 2nd-3rd and 4th-5th columns contain
results of fit respectively without and with the unitarity condition, using
the prior given in Eq. (19).

The lattice QCD can compute the values of the form factors in the low $w$
region. Fermilab-MILC (FM) and JLQCD collaborations give the values of the
four form factors, $F\equiv(f,g,\mathcal{F}_{1},\mathcal{F}_{2})$, at these
three points: $w_{\rm FM}=(1.03.1.10,1.17)$ for Fermilab-MILC collaboration
and $w_{\rm JLQCD}=(1.025.1.060,1.100)$ for JLQCD 222Hereafter, we focus only
on the results provided by these two collaboration as the result from [66] is
given in terms of the HQET form factors and requires an extra transformation..
They also provides the $12\times 12$ covariance matrix for them. Using this
lattice data, we fit the BGL parameters, $\vec{a}_{\rm
BGL}=(a^{f}_{n},a^{g}_{n},a^{\mathcal{F}_{1}}_{n},a^{\mathcal{F}_{2}}_{n})$,
given in Eq. (12). The $\chi^{2}$ can be symbolically written as

$\chi^{2}_{\rm latt}(\vec{a}_{\rm BGL})=(\vec{F}_{\rm BGL}(w_{\rm
latt})-\vec{F}_{\rm latt}(w_{\rm latt}))V_{\rm latt}^{-1}(\vec{F}_{\rm
BGL}(w_{\rm latt})-\vec{F}_{\rm latt}(w_{\rm latt}))^{T},$ (17)

where ”latt“ is either FM or JLQCD. The $\vec{F}_{\rm latt}(w_{\rm latt})$ are
the form factors evaluated by the lattice and $V_{\rm latt}$ is the covariance
matrix mentioned above. The $\vec{F}_{\rm BGL}(w_{\rm latt})$ are the form
factor expanded in terms of the BGL parameters $\vec{a}_{\rm BGL}$ and
evaluated, using Eq. (12), at $w_{\rm latt}$. We keep terms up to $n=2$ in the
following. Using the two relations in Eqs. (15) and (16), we eliminate
$a^{\mathcal{F}_{1}}_{0}$ and $a^{\mathcal{F}_{2}}_{2}$ and we fit 10
parameters. The fit results for JLQCD and Fermilab-MILC are given in Table 4,
in the second and third coumns, respectively.

|
---|---
|
Figure 2: The extrapolated $w$-dependence of the form factors using the BGL
parameteristaion: the green and orange bounds are obtained by using the
lattice QCD inputs at the low $w$ region by the JLQCD and the Fermilab-MILC
collaborations, respectively. No unitarity condition is imposed in these
plots.

Previously, the only lattice QCD result available was for $\mathcal{F}$ at
zero-recoil:

$\mathcal{F}(1)\equiv\frac{1}{2\sqrt{M_{B}M_{D^{*}}}}\frac{a^{f}_{0}}{P_{1^{+}}(0)\phi_{f}(0)}$
(18)

with the value being $\mathcal{F}(1)=0.906\pm 0.004\pm 0.012$ [71]. With the
new lattice results, the fitted value of $a^{f}_{0}$ above gives:

$\mathcal{F}^{\rm JLQCD}(1)=0.887\pm 0.014,\quad\mathcal{F}^{\rm
FM}(1)=0.896\pm 0.014$

Using these values, we can extrapolate the form factors to higher value of
$w$. The extrapolated plots are shown in Fig. 2. As mentioned earlier, the
expansion parameters have constraints coming from the unitarity, i.e. Eq.
(14). One can impose these limit by adding some prior like

$\chi^{2}_{\rm unitarity-bound}=a_{\rm BGL}^{2}$ (19)

The results obtained with this prior are given in Table 4 in the fourth and
fifth column for JLQCD and Fermilab-MILC, respectively. The extrapolated plots
are shown in Fig. 3. Even after taking into account the unitarity bound, the
lattice QCD data itself does not constrain well the higher $w$ region. As we
will see in the next section, the experimental data has a much better handle
on this.

|
---|---
|
Figure 3: The extrapolated $w$-dependence of the form factors using the BGL
parameteristaion: the green and orange bounds are obtained by using the
lattice QCD inputs at the low $w$ region by the JLQCD and the Fermilab-MILC
collaborations, respectively. The unitarity condition is imposed by using a
prior given in Eq. (19).

##  5 The Belle binned results in the light of lattice data

The BaBar and Belle collaborations used their data and obtained the form
factors along with the CKM matrix element $V_{cb}$ in [61, 62, 63, 74, 75,
76]. They use the binned distribution of one of the four kinematical
variables, $w,\cos\theta_{l},\cos\theta_{V}$ or $\chi$. It is a one
dimensional distribution where the other 3 variables are integrated each time:

$\displaystyle\begin{aligned}
d\Gamma(w)&=\frac{8K}{9}\pi\Big{\\{}6J_{1s}+3J_{1c}-2J_{2s}-J_{2c}\Big{\\}}\\\
d\Gamma(\cos\theta_{V})&=-\frac{4K}{3}\pi\Big{\\{}{-\frac{8}{3}J_{1s}}+({\frac{8}{3}J_{1s}-4J_{1c}})\cos^{2}\theta_{V}\Big{\\}}\\\
d\Gamma(\cos\theta_{\ell})&=\frac{4K}{3}\pi\Big{\\{}({\frac{4}{3}J_{1s}+2J_{1c}})+{2J_{6s}}\cos\theta_{\ell}+({\frac{4}{3}J_{1s}-2J_{1c}})\cos^{2}\theta_{\ell}\Big{\\}}\\\
d\Gamma(\chi)&=\frac{4K}{9}\Big{\\{}({\frac{16}{3}J_{1s}+4J_{1c}})+{4J_{3}}\cos(2{\chi})+{4J_{9}}\sin(2{\chi})\Big{\\}}\end{aligned}$
(20)

where
$K=\frac{3M_{B}M_{D^{*}}^{2}}{4(4\pi)^{4}}\sqrt{w^{2}-1}(1-2\,w\,r+r^{2})\,G_{F}^{2}\,{\color[rgb]{0,0,0}|\eta_{\rm
EW}|^{2}}\left|V_{cb}\right|^{2}\mathcal{B}(D^{*}\to D\pi)$. They divided each
distribution into 10 bins and thus, the number of events for the total of 40
bins are published together with the detector response matrix and the
selection efficiency, the systematic uncertainties and the $40\times 40$
statistical uncertainty correlation matrix. The correlation among the bins are
obtained using the SM theoretical formula.

In this section, we first reproduce the result of Belle [62], where the form
factors as well as the $|V_{cb}|$ are extracted from the data within the SM
framework. As mentioned earlier, the $|V_{cb}|$ is an overall factor and can
not be extracted without an input of at least one form factor. In Section 5.1,
we first obtain the tilde parameters, $\tilde{a}_{\rm BGL}\equiv a_{\rm
BGL}|V_{cb}|\eta_{EW}$, which is independent of the form factor. Then, in
Section 5.2, we use the lattice results introduced in the Section 4, and fit
the form factors and $|V_{cb}|$ simultaneously. In Section 5.3, we extend our
model from SM to the one including right-handed current, parameterised by
$C_{\rm V_{R}}$ and perform the fit of form factor, $|V_{cb}|$ and $C_{\rm
V_{R}}$. There is a caveat on the last analysis: the correlation of the Belle
result is obtained based on the SM theory formula. Thus, the result of Section
5.3 remains as just a demonstration and we suggest the Belle collaboration to
obtain the correlation including right-handed model to perform the fit.

###  5.1 Form factor determination with Belle data

In this section we obtain the tilde parameters from the Belle data published
in [62]. The $\chi^{2}$ to minimise can be written as

$\displaystyle\chi^{2}_{\rm Belle}(\vec{\tilde{a}}_{\rm
BGL})=\sum_{i,j}(N_{i}^{\rm obs}-N_{i}^{\rm exp})C^{-1}_{ij}(N_{j}^{\rm
obs}-N_{j}^{\rm exp})$ (21)

where $N_{i}^{\rm obs}$ is the observed number of events, $N^{\rm exp}_{i}$ is
the expected number of events and $C_{ij}$ is the covariance matrix. To avoid
the problem with the highly correlated experimental uncertainties, we follow
the procedure given in [77] and perform a Toy Monte Carlo (MC): the MC samples
are generated by using the Cholesky decomposition method for both statistical
and systematic covariance matrices $C_{\rm stat,syst}$

$N^{{\rm obs}\prime}=N^{\rm obs}+Lu,\quad C=LL^{T}$ (22)

where $u$ is the random value taken from a Gaussian distribution of mean 0 and
variance 1. We iterate the fit a few thousands times with this MC sample and
average the result to obtain the best fit values for the parameters as well as
the error matrix.

Form factor | Value
---|---
$\tilde{a}^{f}_{0}$ | $0.000459(9)(20)$
$\tilde{a}^{f}_{1}$ | $0.0017(13)(13)$
$\tilde{a}^{f}_{2}$ | $-0.013(43)(32)$
$\tilde{a}^{g}_{0}$ | $0.0016(6)(5)$
$\tilde{a}^{g}_{1}$ | $-0.019(24)(21)$
$\tilde{a}^{g}_{2}$ | $-0.02(17)(17)$
$\tilde{a}^{\mathcal{F}_{1}}_{1}$ | $-0.0003(1)(1)$
$\tilde{a}^{\mathcal{F}_{1}}_{2}$ | $-0.0028(19)(16)$
Table 5: The BGL parameters fitted by using the Belle data published in [62]. The tilde notation is defined as $\tilde{a}_{\rm BGL}\equiv a_{\rm BGL}|V_{cb}|\eta_{\rm EW}$. The first error is statistical and the second is systematic. |
---|---
Figure 4: The $w$-dependence of the form factors using the fitted BGL
parameters with the Belle data [62]. Note that the originally obtained BGL
parameters are in the tilde form, i.e. $\tilde{a}_{\rm BGL}\equiv a_{\rm
BGL}|V_{cb}|\eta_{\rm EW}$. In this figure, we use a factor $|V_{cb}|\eta_{\rm
EW}=0.039$ to relate it to the BGL parameter, $a_{\rm BGL}$ so as to be able
to compare to the $w$-dependence of the form factors in the other figures.

The obtained results are given in Table 4, where the first error is
statistical and the second one is systematic. Since the distribution doesn’t
depend on the $\mathcal{F}_{2}$, we fit only eight parameters here. The result
agrees qualitatively well with the results given in [62]. Using this results,
we may draw the $f,g,F_{1}$ form factors in terms of $w$. However, as the
obtained result from the fit is $\tilde{a}_{\rm BGL}$, we have to scale it by
a factor $|V_{cb}|\eta_{\rm EW}$ to obtain ${a}_{\rm BGL}$. Just for the sake
of comparing the $w$ dependence with Fig. 2, we chose here a scaling factor as
0.039 and the result is given in Fig. 4.

###  5.2 SM fit of the Belle data with the new lattice data

Next, we will perform the fit including the new lattice data given in Section
4. The new $\chi^{2}$ to minimise is the sum of Eq.(17) and Eq.(21), i.e.

$\chi^{2}_{\rm Belle+latt}(\vec{a}_{\rm BGL},|V_{cb}|)=\chi^{2}_{\rm
Belle}(\vec{a}_{\rm BGL},|V_{cb}|)+\chi^{2}_{\rm latt}(\vec{a}_{\rm BGL})$
(23)

Now, we can determine both the form factor parameters $\vec{a}_{\rm BGL}$ and
$|V_{cb}|$. It is important to mention that in this case, we also have to
generate the MC sample for the lattice data, by fluctuating them within their
error, to perform the fit. The obtained fit results are given in Table 6.
Using the BGL parameters, we draw the $w$ dependence of the form factors,
$f,g,F_{1}$ and $F_{2}$ (without prior) in Figs. 5 and 6. We observe that the
obtained bounds on $f$, and especially on $g$ form factors becomes very narrow
in whole region of $w$, as compared to Figs. 2 and 3. This seems to come from
the fact that the Belle data constrains these parameters in the middle range
of $w$ as seen in Fig. 4. Together with the lattice data constraint in the
lower $w$ region, these two form factors become well constrained in the whole
range of $w$.

Concerning $|V_{cb}|$, with the $\eta_{\rm EW}=1.0066$, we find

$\displaystyle\begin{aligned} |V_{cb}|=0.0394\pm 0.0011\quad\quad\quad&{\rm
JLQCD}\\\ |V_{cb}|=0.0386\pm 0.0009\quad\quad\quad&{\rm Fermilab-
MILC}\end{aligned}$ (24)

As expected, the $|V_{cb}|$ is strongly correlated to the value of
$a^{f}_{0}$, which is depicted in Fig. 7.

Fit parameters | JLQCD | Fermilab-MILC
---|---|---
$a^{f}_{0}$ | $0.01167(23)(16)$ | $0.01219(17)(19)$
$a^{f}_{1}$ | $0.0428(70)(69)$ | $0.0233(54)(105)$
$a^{f}_{2}$ | $-0.47(20)(26)$ | $-0.16(12)(17)$
$a^{g}_{0}$ | $0.02876(93)(64)$ | $0.03229(99)(96)$
$a^{g}_{1}$ | $-0.061(33)(25)$ | $-0.166(35)(37)$
$a^{g}_{2}$ | $-0.0165(80)(87)$ | $-0.03(24)(22)$
$a^{F_{1}}_{1}$ | $0.0089(21)(21)$ | $0.0030(13)(18)$
$a^{F_{1}}_{2}$ | $-0.109(44)(37)$ | $0.002(26)(25)$
$a^{F_{2}}_{0}$ | $0.0515(18)(18)$ | $0.0519(12)(13)$
$a^{F_{2}}_{1}$ | $-0.03(12)(13)$ | $-0.175(55)(67)$
$|V_{cb}|\eta_{\rm EW}$ | $0.03969(86)(67)$ | $0.03892(52)(69)$
Table 6: The BGL form factors and $|V_{cb}|\eta_{\rm EW}$ obtained by a simultaneous fit of the Belle data [62] and the lattice data by JLQCD [65] (second column) and Fermilab-MILC [64] (third column) collaborations. The first error is statistical and the second is systematic. |
---|---
|
Figure 5: The $w$-dependence of the form factors using the fitted BGL parameters with the Belle data [62] and the JLQCD lattice data [65]. |
---|---
|
Figure 6: The $w$-dependence of the form factors using the fitted BGL parameters with the Belle data [62] and the Fermilab-MILC lattice data [64].  Belle+JLQCD |  Belle+Fermilab-MILC
---|---
|
Figure 7: Correlation plots between $a^{f}_{0}$ and $|V_{cb}|\eta_{\rm EW}$,
from combined fit of Belle data [62] and lattice data. The first plot is with
JLQCD data [65] and the second plot is with Fermilab-MILC data [64].

###  5.3 Test of new physics by fitting the Belle and lattice data: example of
right-handed model

In this section, we suggest the simultaneous fit of Belle and lattice data to
test the new physics model with right-handed current as an example. Strictly
speaking, we cannot use the currently available Belle data to perform this fit
since its correlation assumes the SM as mentioned earlier. For this reason,
our fit result is not completely reliable though, it may gives a useful
instruction for the future analysis for Belle II.

With the right-handed contribution, the $J_{i}$ functions which describe the
differential decay rate in Eq. (9) include a new parameter $C_{\rm V_{R}}$ as
given in Table 1. In general $C_{\rm V_{R}}$ can be a complex number, but we
assume it to be real parameter for simplicity in this section. Now the
$\chi^{2}$ to fit is given as

$\displaystyle\chi^{2}_{\rm Belle+latt}(\vec{a}_{\rm BGL},|V_{cb}|,C_{\rm
V_{R}})=\chi^{2}_{\rm Belle}(\vec{a}_{\rm BGL},|V_{cb}|,C_{\rm
V_{R}})+\chi^{2}_{\rm latt}(\vec{a}_{\rm BGL})$ (25)

where the $C_{\rm V_{R}}$ is included in the $N_{i}^{\rm exp}$ in
$\chi^{2}_{\rm Belle}$. The obtained fit results are given in Table 7. Here,
we observe a significant difference between results with lattice inputs of
JLQCD and Fermilab-MILC: $C_{\rm V_{R}}$ is zero consistent for JLQCD while
the Fermilab-MILC shows a negative 5% shift. At the same time, the $|V_{cb}|$
value decreases for the latter. Although it is not a significant effect, we
attempt to understand better this phenomena. Looking at the error matrix, we
find three large entries between $a_{0}^{f}-|V_{cb}|$, $a_{0}^{g}-C_{\rm
V_{R}}$ and $|V_{cb}|-C_{\rm V_{R}}$, which we show in Figs. 8 and 9. Indeed,
the central value of $a_{0}^{g}$ for two lattice inputs disagree as seen in,
for example, Table 4, and that seems to be the reason for different $C_{\rm
V_{R}}$ obtained in the fit. As the correlation between $|V_{cb}|$ and $C_{\rm
V_{R}}$ is positive, the negative $C_{\rm V_{R}}$ obtained for Fermilab-MILC
lowers the value of $|V_{cb}|$. As mentioned in the introduction, the SM
$|V_{cb}|$ is expected to be higher than the nominal fit, this contradicts to
the global picture. It is premature to give any conclusion, since i) we need
the Belle collaboration to provide their correlation assuming right-handed
current, ii) the two lattice collaboration do not agree on the vector form
factor, namely $a_{0}^{g}$. Nevertheless, we urge both experimental and
lattice collaborations to clarify these issues.

Fit parameters | JLQCD | Fermilab-MILC
---|---|---
$a^{f}_{0}$ | $0.01200(20)(18)$ | $0.01212(19)(17)$
$a^{f}_{1}$ | $0.0193(34)(69)$ | $0.0193(29)(74)$
$a^{f}_{2}$ | $-0.036(69)(51)$ | $-0.066(25)(26)$
$a^{g}_{0}$ | $0.0288(19)(20)$ | $0.0335(12)(12)$
$a^{g}_{1}$ | $-0.070(34)(40)$ | $-0.123(46)(49)$
$a^{g}_{2}$ | $-0.0051(71)(59)$ | $-0.0060(18)(20)$
$a^{F_{1}}_{1}$ | $0.0051(13)(19)$ | $0.0026(11)(15)$
$a^{F_{1}}_{2}$ | $-0.049(27)(27)$ | $0.004(23)(22)$
$a^{F_{2}}_{0}$ | $0.0481(18)(18)$ | $0.0522(12)(12)$
$a^{F_{2}}_{1}$ | $0.030(60)(67)$ | $-0.193(50)(52)$
$C_{\rm V_{R}}$ | $0.008(47)(51)$ | $-0.052(27)(28)$
$|V_{cb}|\eta_{\rm EW}$ | $0.0399(21)(22)$ | $0.0374(11)(12)$
Table 7: The BGL form factors, $C_{\rm V_{R}}$ and $|V_{cb}|\eta_{\rm EW}$ obtained by a simultaneous fit of the Belle data [62] and the lattice data by JLQCD [65] (second column) and Fermilab-MILC [64] (third column) collaborations. The first error is statistical and the second is systematic. |
---|---
Figure 8: Correlation plots between $a^{f}_{0}$, $|V_{cb}|\eta_{\rm EW}$ and $C_{\rm V_{R}}$, from combined fit of Belle data [62] and JLQCD lattice data [65]. $C_{V_{R}}$ is assumed to be real. |
---|---
Figure 9: Correlation plots between $a^{f}_{0}$, $a^{g}_{0}$,
$|V_{cb}|\eta_{\rm EW}$ and $C_{\rm V_{R}}$, from combined fit of Belle data
[62] and Fermilab-MILC lattice data [64]. $C_{\rm V_{R}}$ is assumed to be
real.

##  6 Unbinned angular analysis of ${B}\to D^{*}\ell{\nu}_{\ell}\;$

In this section, following the method introduced in [67], we investigate the
sensitivity of the new physics in the unbinned method. The Belle analysis is
limited to the binned analysis as of today, while once higher statistics will
be available, a possibility of the unbinned analysis will open. The unbinned
analysis, with full kinematical information, unlike the binned method which
uses only integrated one dimension information, should provide a more detailed
information. In this section, we investigate the unbinned method to explore
what new information we can obtain comparing to the binned method which is
studied in the previous section.

As unbinned data is not available from Belle, we synthesise the data first: we
use all the fitted parameters by Belle experiment in [62] (form factors and
$V_{cb}$) as truth values, and reproduce the differential decay rate of
${B}\to D^{*}\ell{\nu}_{\ell}\;$decay. Then, to compensate for the limited
statistics, we first separate the data in $w$ bins of 10 equal intervals.
Then, for each $w$ bin, we investigate the fully unbinned distribution of the
3 angles ($\cos\theta_{v},\cos\theta_{\ell},\chi$). This is the same strategy
applied for the $P_{i}^{(\prime)}$ measurements at LHCb in [78].

Comparing to our study in [67], we have three new inputs for this work; i) we
use new lattice data introduced in Section 4, ii) the new physics model
includes not only right-handed current but also psudoscalar and tensor terms,
iii) we include the branching ratio to the fit. The point iii) is important in
order to determine the new physics parameters (i.e. $C_{\rm V_{R},P,T}$) and
$V_{cb}$ simultaneously. In our previous work, we attempted to use the number
of signal of ${B}\to D^{*}\ell{\nu}_{\ell}\;$process for this purpose, but to
compare its theoretical predictions to the experimental value, it requires
experimental efficiency, which was not so easy to obtain. Using the branching
ratio provides a much simpler solution.

###  6.1 Unbinned maximum likelihood method

Let us now illustrate how to perform the unbinned fit with maximum likelihood
method. We start by obtaining the normalised Probability Density Function
(PDF) in terms of the 11 independent angular observables $J_{i}$. For this, we
first give the expression of decay rate integrated over all the angles

$\frac{d\Gamma}{dw}=\frac{3M_{B}M_{D^{*}}^{2}}{4(4\pi)^{2}}G_{F}^{2}|V_{cb}|^{2}|\eta_{\rm
EW}|^{2}\mathcal{B}(D^{*}\rightarrow
D\pi)\frac{8\pi}{9}\left(6J^{\prime}_{1s}+3J^{\prime}_{1c}-2J^{\prime}_{2s}-J^{\prime}_{2c}\right)$
(26)

where $J^{\prime}_{i}\equiv J_{i}\sqrt{w^{2}-1}(1-2wr+r^{2})$. Then, we
integrate over each $w$-bin and define the integrated decay rate as

$\displaystyle\langle\Gamma\rangle_{w-\rm bin}$
$\displaystyle=\frac{3M_{B}M_{D^{*}}^{2}}{4(4\pi)^{2}}G_{F}^{2}|V_{cb}|^{2}|\eta_{\rm
EW}|^{2}\mathcal{B}(D^{*}\rightarrow D\pi)$ (27)
$\displaystyle\times\frac{8\pi}{9}\left(6\langle J^{\prime}_{1s}\rangle_{w-\rm
bin}+3\langle J^{\prime}_{1c}\rangle_{w-\rm bin}-2\langle
J^{\prime}_{2s}\rangle_{w-\rm bin}-\langle J^{\prime}_{2c}\rangle_{w-\rm
bin}\right)$

where

$\langle J_{i}\rangle_{w-{\rm bin}}\equiv\int_{w-{\rm bin}}J_{i}dw$

Hereafter, the index ”$w$-bin” is implicit. The normalised PDF for each
$w$-bin now can be written by the new normalised angular coefficients
$\vec{\langle g\rangle}=\langle g_{i}\rangle$ as

$\displaystyle\begin{aligned}
\hat{f}_{\langle\vec{g}\rangle}(\cos\theta_{V},\cos\theta_{\ell},\chi)&=\frac{9}{8\pi}\Big{\\{}\frac{1}{6}(1-3\langle
g_{1c}\rangle+2\langle g_{2s}\rangle+\langle
g_{2c}\rangle)\sin^{2}\theta_{V}+\langle g_{1c}\rangle\cos^{2}\theta_{V}\\\
&+(\langle g_{2s}\rangle\sin^{2}\theta_{V}+\langle
g_{2c}\rangle\cos^{2}\theta_{V})\cos 2\theta_{l}\\\ &+\langle
g_{3}\rangle\sin^{2}\theta_{V}\sin^{2}\theta_{l}\cos 2\chi_{l}+\langle
g_{4}\rangle\sin 2\theta_{V}\sin 2\theta_{l}\cos\chi_{l}\\\ &+\langle
g_{5}\rangle\sin 2\theta_{V}\sin\theta_{l}\cos\chi_{l}+(\langle
g_{6s}\rangle\sin^{2}\theta_{V}+\langle
g_{6c}\rangle\cos^{2}\theta_{V})\cos\theta_{l}\\\ &+\langle g_{7}\rangle\sin
2\theta_{V}\sin\theta_{l}\sin\chi_{l}+\langle g_{8}\rangle\sin 2\theta_{V}\sin
2\theta_{l}\sin\chi_{l}\\\ &+\langle
g_{9}\rangle\sin^{2}\theta_{V}\sin^{2}\theta_{l}\sin
2\chi_{l}\Big{\\}}\end{aligned}$ (28)

where

$\langle g_{i}\rangle\equiv\frac{\langle J^{\prime}_{i}\rangle}{6\langle
J^{\prime}_{1s}\rangle+3\langle J^{\prime}_{1c}\rangle-2\langle
J^{\prime}_{2s}\rangle-\langle J^{\prime}_{2c}\rangle}$ (29)

Then, we can write the log-likelihood to determine the angular coefficients
$\langle\vec{g}\rangle$ from the data:

$\mathcal{L}(\langle\vec{g}\rangle)=\sum_{i=1}^{N}ln\hat{f}_{\langle\vec{g}\rangle}(e_{i})$
(30)

where $e_{i}$ are the experimental events in
($\cos\theta_{V},\cos\theta_{\ell},\chi$) and $N$ are the number of events for
each $w$-bin. The best fit value for $\langle g_{i}\rangle$ is obtained by
maximising the likelihood

$\left.\frac{\partial\mathcal{L}}{\partial\langle
g_{i}\rangle}\right|_{\langle g_{i}\rangle=\langle g_{i}\rangle_{\rm fit}}=0$
(31)

The inverse of the covariance matrix can be obtained by

$V^{-1}_{ij}=\left.\frac{\partial^{2}\mathcal{L}}{\partial\langle
g_{i}\rangle\partial\langle g_{j}\rangle}\right|_{\langle g_{i}\rangle=\langle
g_{i}\rangle_{\rm fit}}$ (32)

Just to be clear, $V_{ij}$ is the $11\times 11$ matrix for each $w$-bin (so we
need 10 of these matrix if we have 10 bins in $w$).

###  6.2 Fitting the theory parameters with the pseudo-data together with the
lattice inputs

In this work, instead of performing a Toy Monte Carlo with generated events,
we obtain directly $\langle g_{i}\rangle_{\rm fit}$ as well as $V_{ij}$ from
the normalised PDF $\hat{f}_{\langle\vec{g}\rangle}$ with truth inputs (i.e.
the fitted form factors and $V_{cb}$ in [62] as mentioned earlier). The
detailed procedure is given in the Appendix B.

Now, we can fit the theory parameters to the ’experimentally’ obtained angular
coefficients and their correlations by using the following $\chi^{2}$:

$\displaystyle\chi^{2}_{\rm angle}(\vec{a}_{\rm BGL},C_{\rm NP})=$ (33)
$\displaystyle\sum_{w-{\rm bin}=1}^{10}\left[N_{w-{\rm
bin}}\hat{V}^{-1}_{ij}\left(\langle g_{i}^{\rm exp}\rangle-\langle g_{i}^{\rm
th}(\vec{a}_{\rm BGL},C_{\rm NP})\rangle\right)\left(\langle g_{j}^{\rm
exp}\rangle-\langle g_{j}^{\rm th}(\vec{a}_{\rm BGL},C_{\rm
NP})\rangle\right)\right]_{w-{\rm bin}}$

where $\hat{V}$ is the scaled covariance matrix, $N_{w-{\rm bin}}$ are the
number of events in the $w$-bin and $\langle g_{i}^{\rm exp}\rangle$ is the
best fit value of the angular coefficients (see Appendix B for details) while
$\langle g_{i}^{\rm th}(\vec{a}_{\rm BGL},C_{\rm NP})\rangle$ are the
theoretical expressions, which depend on the form factors and NP Wilson
coefficients. We assume about the same amount of the signal events as in [62].
Note that as $\chi^{2}_{\rm angle}$ term is made from the normalised angular
coefficients, the overall constants such as $V_{cb}$ gets cancelled.

In order to constrain from the total number of events, we use the world
average of the branching ratio, as mentioned earlier:

$\mathcal{B}^{\rm exp}(B\to D^{*}\ell\nu)=0.0495\pm 0.0011$

Then, the $\chi^{2}$ from the branching ratio measurement can be written as,

$\chi^{2}_{\mathcal{B}}(\vec{a}_{\rm BGL},V_{cb},C_{\rm
NP})=\left(\frac{\mathcal{B}^{\rm th}(\vec{a}_{\rm BGL},V_{cb},C_{\rm
NP})-0.0495}{0.0011}\right)^{2}$ (34)

where $\mathcal{B}^{\rm th}(\vec{a}_{\rm BGL},V_{cb},C_{\rm NP})$ is the
theoretical prediction of the branching ratio including new physics
contribution.

Finally adding the lattice constraint, the total $\chi^{2}$ to fit is

$\displaystyle\chi^{2}_{\rm unbinned}(\vec{a}_{\rm BGL},V_{cb},C_{\rm
NP})=\chi^{2}_{\rm angle}(\vec{a}_{\rm BGL},C_{\rm NP})+\chi^{2}_{\rm
latt}(\vec{a}_{\rm BGL})+\chi^{2}_{\mathcal{B}}(\vec{a}_{\rm
BGL},V_{cb},C_{\rm NP})$ (35)

In the following, we show the fit results. We assume single non-zero new
physics coupling at a time (i.e. $C_{\rm V_{R}}\neq 0,C_{\rm P}\neq 0$ or
$C_{\rm T}\neq 0$ and also assume that they are real number.

####  6.2.1 Right-handed model: $C_{\rm V_{R}}\neq 0$

The fitted result is given in Table 8. Let us first compare this unbinned
analysis result with the binned one given in Table 7. As our pseudodata does
not have systematic error, we can compare only the statistical one (i.e. the
first parenthesis in Table 7). As expected, the statistical errors are better
for the unbinned analysis though, hardly a factor of 2 for $C_{\rm V_{R}}$.
Figs. 10 and 11 show the correlations, which are also very similar to the
binned study in Figs. 8 and Fig. 9. The $C_{\rm V_{R}}$ is more significantly
different from zero in this case.

Fit parameters | JLQCD | Fermilab-MILC
---|---|---
$a^{f}_{0}$ | $0.01294(20)$ | $0.01327(20)$
$a^{f}_{1}$ | $0.018(11)$ | $-0.005(13)$
$a^{f}_{2}$ | $-0.21(53)$ | $-0.28(61)$
$a^{g}_{0}$ | $0.0255(15)$ | $0.0289(11)$
$a^{g}_{1}$ | $-0.045(35)$ | $-0.130(49)$
$a^{g}_{2}$ | $0.4(11)$ | $1.0(14)$
$a^{\mathcal{F}_{1}}_{1}$ | $0.0061(22)$ | $-0.0001(23)$
$a^{\mathcal{F}_{1}}_{2}$ | $-0.095(94)$ | $-0.05(10)$
$a^{\mathcal{F}_{2}}_{0}$ | $0.0483(14)$ | $0.0537(13)$
$a^{\mathcal{F}_{2}}_{1}$ | $-0.004(50)$ | $-0.276(62)$
$C_{\rm V_{R}}$ | $-0.023(34)$ | $-0.070(23)$
$|V_{cb}|\eta_{\rm EW}$ | $0.0390(21)$ | $0.0385(20)$
Table 8: The BGL form factors, $C_{\rm V_{R}}$ and $|V_{cb}|\eta_{\rm EW}$ obtained by a simultaneous fit of the pseudo-data and the lattice data by JLQCD [65] (second column) and Fermilab-MILC [64] (third column) collaborations. The errors are statistical only. |
---|---
Figure 10: Correlation plots between $a^{g}_{0}$, $|V_{cb}|\eta_{\rm EW}$ and $C_{\rm V_{R}}$, from combined fit of the pseudo-data and JLQCD lattice data [65]. $C_{\rm V_{R}}$ is assumed to be real.  |
---|---
Figure 11: Correlation plots between $a^{g}_{0}$, $|V_{cb}|\eta_{\rm EW}$ and
$C_{\rm V_{R}}$, from combined fit of the pseudo-data and Fermilab-MILC
lattice data [64] $C_{\rm V_{R}}$ is assumed to be real.

####  6.2.2 Pseudoscalar model: $C_{\rm P}\neq 0$

As we can see in the Tables 1 and 2, the $C_{\rm P}$ parameter enters only to
the angular observable $J_{1c}$ in squared form, $|C_{\rm P}|^{2}$. There is
no interference with the SM term i.e. $C_{\rm V_{L}}$, as such terms are
proportional to the lepton mass, which is neglected. As a result the
sensitivity to this parameter is quite poor. The fitted result is given in
Table 9 where we find that $|C_{\rm P}|$ is constrained only at 20-30%
precision. As mentioned earlier, the psuedoscalar form factor is obtained by
relating it to the axial vector form factor. This could potentially induce an
error from the uncertainties of charm and quark masses. However, we find that
this error is negligible here: we tested this by inflating the errors in the
quark masses by order of 1 GeV. The results on $|C_{\rm P}|$ with JLQCD and
Fermilab-MILC inputs are slightly different. As shown in Figs. 12 and 13,
there is a small correlation between $C_{\rm P}$ and $a_{1}^{\mathcal{F}_{1}}$
whose values are different for these two inputs and this might be the reason
of this difference. Also shown in Figs. 12 and 13, the correlation between
$|C_{\rm P}|$ and $|V_{cb}|$ is very small. Thus, a shift in $|C_{\rm P}|$
would not influence the value of $V_{cb}$ as seen in the the case of $C_{\rm
V_{R}}\neq 0$.

Fit parameters | JLQCD | Fermilab-MILC
---|---|---
$a^{f}_{0}$ | $0.01294(20)$ | $0.01336(20)$
$a^{f}_{1}$ | $0.018(11)$ | $-0.003(13)$
$a^{f}_{2}$ | $-0.20(53)$ | $-0.47(61)$
$a^{g}_{0}$ | $0.02456(60)$ | $0.02626(70)$
$a^{g}_{1}$ | $-0.057(29)$ | $-0.154(45)$
$a^{g}_{2}$ | $0.6(11)$ | $0.9(12)$
$a^{\mathcal{F}_{1}}_{1}$ | $0.0062(22)$ | $-0.0003(23)$
$a^{\mathcal{F}_{1}}_{2}$ | $-0.095(94)$ | $-0.08(10)$
$a^{\mathcal{F}_{2}}_{0}$ | $0.0484(14)$ | $0.0530(13)$
$a^{\mathcal{F}_{2}}_{1}$ | $-0.002(50)$ | $-0.271(61)$
$|C_{\rm P}|$ | $-0.08(33)$ | $-0.17(16)$
$|V_{cb}|\eta_{\rm EW}$ | $0.0398(17)$ | $0.0413(21)$
Table 9: The BGL form factors, $|C_{\rm P}|$ and $|V_{cb}|\eta_{\rm EW}$ obtained by a simultaneous fit of the pseudo-data and the lattice data by JLQCD [65] (second column) and Fermilab-MILC [64] (third column) collaborations. The errors are statistical only. |
---|---
Figure 12: Correlation plots between $a^{\mathcal{F}_{1}}_{1}$, $|V_{cb}|\eta_{\rm EW}$ and $|C_{\rm P}|$, from combined fit of the pseudo-data and JLQCD lattice data [65]. |
---|---
Figure 13: Correlation plots between $a^{\mathcal{F}_{1}}_{1}$,
$|V_{cb}|\eta_{\rm EW}$ and $|C_{\rm P}|$, from combined fit of the pseudo-
data and Fermilab-MILC lattice data [64].

####  6.2.3 Tensor model: $C_{\rm T}\neq 0$

In the case of $C_{\rm T}\neq 0$ model, several $J_{i}$ functions depend on
$C_{\rm T}$, though, similar to $C_{\rm P}\neq 0$ model, the interference to
the SM term cancels due to the small lepton masses (see Tables 1 and 2). The
fitted results are given in Table 10 where we find that $|C_{\rm T}|$ is much
more constrained than $|C_{\rm P}|$, at 2-3% level. The result does not change
after taking into account the uncertainties from charm and quark masses, which
appear in the formula relating the tensor and vector form factors. In this
case, we observe almost no correlation between $|C_{\rm T}|$ and form factors
and the correlation between $|C_{\rm T}|$ and $V_{cb}$ is also small as well
(see Figs. 14).

Fit parameters | JLQCD | Fermilab-MILC
---|---|---
$a^{f}_{0}$ | $0.01294(20)$ | $0.01336(20)$
$a^{f}_{1}$ | $0.018(11)$ | $-0.003(13)$
$a^{f}_{2}$ | $-0.21(54)$ | $-0.48(61)$
$a^{g}_{0}$ | $0.02456(60)$ | $0.02626(70)$
$a^{g}_{1}$ | $-0.057(29)$ | $-0.154(45)$
$a^{g}_{2}$ | $0.6(11)$ | $0.9(12)$
$a^{\mathcal{F}_{1}}_{1}$ | $-0.0062(22)$ | $-0.0003(23)$
$a^{\mathcal{F}_{1}}_{2}$ | $-0.095(94)$ | $-0.08(10)$
$a^{\mathcal{F}_{2}}_{0}$ | $0.0484(14)$ | $0.0530(13)$
$a^{\mathcal{F}_{2}}_{1}$ | $-0.002(50)$ | $-0.271(61)$
$|C_{\rm T}|$ | $0.001(26)$ | $0.002(17)$
$|V_{cb}|\eta_{\rm EW}$ | $0.0398(17)$ | $0.0413(21)$
Table 10: The BGL form factors, $|C_{\rm T}|$ and $|V_{cb}|\eta_{\rm EW}$ obtained by a simultaneous fit of the pseudo-data and the lattice data by JLQCD [65] (second column) and Fermilab-MILC [64] (third column) collaborations. The errors are statistical only. |
---|---
Figure 14: Correlation plots between $|V_{cb}|\eta_{\rm EW}$ and $|C_{\rm
T}|$, from combined fit of the pseudo-data and JLQCD lattice data (left) [65]
and Fermilab-MILC lattice data (right) [64].

##  7 Conclusions

In this article, we investigated the potential of the angular analysis of
${B}\to D^{*}\ell{\nu}_{\ell}\;$decay to search for new physics in the light
of the new lattice QCD data on the form factors. Previously, such an analysis
was not possible, as the form factors, the SM CKM parameter $V_{cb}$ and the
NP parameters could not be fitted simultaneously from the experimental data.
Starting with an effective Hamiltonian containing different NP operators
(left/right-handed, pseudoscalar and tensor), we first computed the complete
angular distribution of ${B}\to D^{*}\ell{\nu}_{\ell}\;$decay, which confirms
the previous computation [70]. While the lattice data provide only the vector
and axial-vector form factors, the pseudoscalar and tensor form factors can be
derived from the equation of motion, up to the uncertainties in the charm and
bottom quark masses. We use the lattice data from two collaborations,
Fermilab-MILC and JLQCD, which provide the four form factors at three
different $w$ points in the low recoil region. We first extrapolated the form
factors to the higher $w$ region based on the BGL parametrisation and showed
that the errors on them at high $w$ region are large even after taking into
account the unitary bound.

Next, we used the public Belle data from [62] and fit it together with the
lattice data. The result shows that the errors on the form factors in both low
and high $w$ region reduces significantly. The SM parameter, $V_{cb}$, appears
as an overall factor and it can not be fitted from the experimental data only.
We have fitted $V_{cb}$ with the new lattice data and found $|V_{cb}|=(39.4\pm
1.1)\times 10^{-3}$ (JLQCD) and $|V_{cb}|=(38.6\pm 0.9)\times 10^{-3}$
(Fermilab-MILC), which agrees with previous studies. In order to evaluate
whether Belle data (one dimensional and binned) can further constrain NP
parameters, we next introduced the right-handed current, $C_{\rm V_{R}}$.
Strictly speaking, this fit is not possible as of today as Belle data assume
SM correlations. Nevertheless, we performed the fit with $C_{\rm V_{R}}\neq 0$
for demonstration. We found a surprisingly high sensitivity of this binned
Belle data to $C_{\rm V_{R}}$, at the level of 4-7%. We also found that
$C_{\rm V_{R}}$ is strongly correlated to the vector form factor parameter
$a_{0}^{g}$ and the current slight discrepancy between JLQCD and Fermilab-MILC
on this parameter leads to different conclusions: the latter shows some hint
of a negative shift of $C_{\rm V_{R}}$. On the other hand, as $C_{\rm V_{R}}$
is positively correlated to the $|V_{cb}|$ value, this lowers $V_{cb}$, which
is not favourable from the global fit point of view. In any case, our result
demonstrated that the Belle (II) binned fit can provide valuable information
on NP and we urge Belle (II) collaboration to perform such fits.

In the last section, we have investigated the possibility of the unbinned fit.
Using synthesised data based on the same Belle dataset together with lattice
inputs, we have performed a sensitivity study on the right-handed,
pseudoscalar and tensor types of NP models. For the right-handed case, we find
very similar results as the binned analysis, with the sensitivity slightly
higher. The pseudoscalar and tensor models are expected to be less constrained
as their interference with SM is proportional to the final state lepton mass
and it is suppressed in the $\ell=e,\mu$ cases. Nevertheless, we find that the
pseudoscalar and tensor couplings, $|C_{\rm P,T}|$, are constrained,
respectively, at 20-30% and 2-3% levels. For these cases, different from the
right-handed case, the correlation between the NP couplings and $|V_{cb}|$ is
very small. Thus, a nonzero observation of these couplings would not conflict
with the global fit result of $|V_{cb}|$.

## Acknowledgements

We would like to acknowledge T. Kaneko, Ph. Urquijo, D. Ferlewicz and E.
Waheed for their collaboration at the the early stage of this project. We
thank F. Le Diberder for his advice and careful reading of the manuscript. We
also thank R. Mizuk for his helpful comments. This work was supported by TYL-
FJPPL.

##  Appendix A Relationship between pseudoscalar/tensor and vector form
factors

We use equations of motion in order to express pseudoscalar and tensor form
factors in terms of vector form factors [28].

Pseudoscalar form factors:

Using the equation of motion

$i\partial_{\mu}(\bar{c}\gamma^{\mu}\gamma^{5}b)=-(m_{b}+m_{c})\bar{c}\gamma^{5}b$
(36)

the pseudo scalar matrix event becomes

$\displaystyle\langle
D^{*}(p_{D^{*}},\epsilon_{2})|\bar{c}\gamma^{5}b|\bar{B}(p_{B})\rangle$
$\displaystyle=\frac{-1}{m_{b}+m_{c}}q_{\mu}\langle
D^{*}(p_{D^{*}},\epsilon_{2})|\bar{c}\gamma^{\mu}\gamma^{5}b|\bar{B}(p_{B})\rangle$
(37)

where $m_{b,c}$ are $b$\- and $c$\- quark masses, respectively, and
$q^{\mu}=p_{B}^{\mu}-p_{D^{*}}^{\mu}$. In our numerical analysis we use
$m_{b}=4.18^{+0.03}_{-0.02}$ GeV and $m_{c}=1.27\pm 0.02$ GeV [1].

Tensor form factors
The guiding equation of motion is given by

$\partial_{\mu}(\bar{c}\sigma^{\mu\nu}b)=-(m_{b}+m_{c})\bar{c}\gamma^{\nu}b$
(38)

which gives us the equations

$-iq_{\mu}\langle
D^{*}(p_{D^{*}},\epsilon_{2})|\bar{c}\sigma^{\mu\nu}b|\bar{B}(p_{B})\rangle=-(m_{b}+m_{c})\langle
D^{*}(p_{D^{*}},\epsilon_{2})|\bar{c}\sigma^{\mu\nu}b|\bar{B}(p_{B})\rangle$
(39)

This equation gives the same relation for $+$ and $-$ polarization of $D^{*}$,
and both sides go to zero for longitudinal (0) polarization. Therefore, we
have only two equations in total, and we need two more to determine all the
tensor form factors (including the form factor for $B\rightarrow D$ decay).
For this, we have the pseudo tensor equation of motion given by

$\partial_{\mu}(\bar{c}\sigma^{\mu\nu}\gamma^{5}b)=(m_{b}-m_{c})\bar{c}\gamma^{\nu}\gamma^{5}b$
(40)

where the pseudo tensor matrix elements can be related to the tensor matrix
elements by the relation
$\bar{c}\sigma_{\mu\nu}\gamma^{5}b=-\frac{i}{2}\epsilon_{\mu\nu\alpha\beta}\bar{c}\sigma^{\alpha\beta}b$
with the convention $\epsilon_{0123}=1$. This relation gives us the following
equation

$-\frac{q_{\mu}}{2}\epsilon^{\mu\nu\alpha\beta}\langle
D^{*}(p_{D^{*}},\epsilon_{2})|\bar{c}\sigma_{\alpha\beta}b|\bar{B}(p_{B})\rangle=(m_{b}-m_{c})\langle
D^{*}(p_{D^{*}},\epsilon_{2})|\bar{c}\gamma^{\nu}\gamma^{5}b|\bar{B}(p_{B})\rangle$
(41)

where we get two equations - one from $\epsilon_{2}(\pm)$ and other from
$\epsilon_{2}(0)$. Now these four equations can be solved to write the
expressions of the four tensor form factors ($h_{T,T_{1},T_{2},T_{3}}$) in
terms of vector form factors ($h_{+,-,V,A_{1},A_{2},A_{3}}$), which are given
as follows [28]:

$\displaystyle\begin{aligned}
h_{T}(w)&=\frac{m_{b}+m_{c}}{m_{B}+m_{D}}\Big{[}h_{+}(w)-\frac{1+r_{D}}{1-r_{D}}h_{-}(w)\Big{]}\\\
h_{T_{1}}(w)&=\frac{1}{2(1+r_{D^{*}}^{2}-2r_{D^{*}}w)}\Big{[}\frac{m_{b}-m_{c}}{m_{B}-m_{D^{*}}}(1-r_{D^{*}})^{2}(w+1)h_{A_{1}}(w)\\\
&-\frac{m_{b}+m_{c}}{m_{B}+m_{D^{*}}}(1+r_{D^{*}})^{2}(w-1)h_{V}(w)\Big{]}\\\
h_{T_{2}}(w)&=\frac{(1-r_{D^{*}}^{2})(w+1)}{2(1+r_{D^{*}}^{2}-2r_{D^{*}}w)}\Big{[}\frac{m_{b}-m_{c}}{m_{B}-m_{D^{*}}}h_{A_{1}}(w)-\frac{m_{b}+m_{c}}{m_{B}+m_{D^{*}}}h_{V}(w)\Big{]}\\\
h_{T_{3}}(w)&=\frac{-1}{2(1+r_{D^{*}})(1+r_{D^{*}}^{2}-2r_{D^{*}}w)}\Big{[}2\frac{m_{b}-m_{c}}{m_{B}-m_{D^{*}}}r_{D^{*}}(w+1)h_{A_{1}}(w)\\\
&+\frac{m_{b}-m_{c}}{m_{B}-m_{D^{*}}}(1+r_{D^{*}}^{2}-2r_{D^{*}}w)(h_{A_{3}}(w)-r_{D^{*}}h_{A_{2}}(w))\\\
&-\frac{m_{b}+m_{c}}{m_{B}+m_{D^{*}}}(1+r_{D^{*}})^{2}h_{V}(w)\Big{]}\end{aligned}$
(42)

In addition, the BGL form factors are related to the HQET form factors as
follows:

$\displaystyle g$ $\displaystyle=\frac{h_{V}}{M_{B}\sqrt{r}}$ (43)
$\displaystyle f$ $\displaystyle=M_{B}\sqrt{r}(1+w)h_{A_{1}}$
$\displaystyle\mathcal{F}_{1}$
$\displaystyle=M_{B}^{2}\sqrt{r}(1+w)\left[(w-r)h_{A_{1}}-(w-1)(rh_{A_{2}}+h_{A_{3}})\right]$
$\displaystyle\mathcal{F}_{2}$
$\displaystyle=\frac{1}{\sqrt{r}}\left[(1+w)h_{A_{1}}+(rw-1)h_{A_{2}}+(r-w)h_{A_{3}}\right]$

We can solve Eq. (43) and Eq. (42) to write the tensor form factors in terms
of BGL parameters, which gives us the tensor helicity amplitudes given in Eq.
(11).

##  Appendix B Statistical procedure

This method of generating the covariance matrices from the truth values is
based on the fact that the statistical error scales as $\sqrt{N_{\rm event}}$,
where $N_{\rm event}$ are the number of events.

Starting with the normalied PDF $\hat{f}_{\langle\vec{g}\rangle}$ given in Eq.
(28), the covariance matrix for $\langle\vec{g}\rangle$ functions is given by

$\displaystyle V_{ij}^{-1}=N_{\rm event}\int\
\left.\left(\frac{\partial\hat{f}(x)_{\langle\vec{g}_{i}\rangle}}{\partial\langle\vec{g}_{i}\rangle}\frac{\partial\hat{f}(x)_{\langle\vec{g}_{j}\rangle}}{\partial\langle\vec{g}_{j}\rangle}\frac{1}{\hat{f}(x)_{\langle\vec{g}_{i}\rangle}}\right)\right|_{\langle\vec{g}_{i}\rangle=\langle\vec{g}_{i}\rangle^{\rm
exp}}\ dx$ (44)

where

  * •

$\langle\vec{g}_{i}\rangle$ is the vector of observables.

  * •

$\langle\vec{g}_{i}\rangle^{\rm exp}$ is the vector of truth value of $w$-bin
integrated $g_{i}$ functions, which we obtain from the experimentally measured
form factors in [62].

  * •

$dx$ represents integration over the complete phase space, i.e. the three
angles that describe the decay.

  * •

$N_{\rm event}$ is the number of events in the corresponding $w$-bin: we have
10 $w$-bins, and thus, we have 10 such covariance matrices for each bin, the
dimension of each being $11\times 11$ (corresponding to 11 independent
$\langle{g}_{i}\rangle$ functions).

We can define a rescaled covariance matrix $\hat{V}$ as

$\displaystyle\hat{V}=N_{\rm event}V$ (45)

Using this pseudo dataset, we can perform a maximum likelihood estimate with
our model assumptions (right-handed, pseudoscalar, tensor etc.), using the
$\chi^{2}$ given in Eq. (33).

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# A Computer Vision Method for Estimating Velocity from Jumps

Soumyadip Roy1, Chaitanya Roygaga2, Nathaniel Blanchard1, Aparna Bharati2
1 Colorado State University, CO, USA 2 Lehigh University, PA, USA

###### Abstract

Athletes routinely undergo fitness evaluations to evaluate their training
progress. Typically, these evaluations require a trained professional who
utilizes specialized equipment like force plates. For the assessment, athletes
perform drop and squat jumps, and key variables are measured, e.g. velocity,
flight time, and time to stabilization, to name a few. However, amateur
athletes may not have access to professionals or equipment that can provide
these assessments. Here, we investigate the feasibility of estimating key
variables using video recordings. We focus on jump velocity as a starting
point because it is highly correlated with other key variables and is
important for determining posture and lower-limb capacity. We find that
velocity can be estimated with a high degree of precision across a range of
athletes, with an average R-value of 0.71 (SD = 0.06).

##  1 Introduction

Athletes benefit from feedback from professional evaluations to maximize
performance and minimize injury risk. Often, these evaluations require
athletes to perform key movements that facilitate evaluation, e.g. drop or
squat jumps, with the use of specialized equipment like force plates. However,
not all athletes have access to this equipment, nor to trained professionals
to help them understand the variables measured by these movements and how they
should adjust their training in response. Here, we establish that key
variables from these jumps can be estimated using from videos of athletes
performing both squat and drop jumps.

We use force plate data as the ground truth for our variables. The accuracy of
force plates has been well established. Garcia-Ramos _et al_. [4] show that
the force plate variables like velocity and power are highly reliable when
evaluating a countermovement jump for an athlete’s performance. Velocity is an
important variable when determining posture [3] or to represent the maximum
lower-limb capacity of an athlete [10], which is helpful in creating
individual and prioritized training profiles for athletes.

Figure 1: Overview of our pipeline. The Swin Transformer block takes as input
the video frames and gives out flattened features further processed through
two dense layers for an estimated velocity score.

The usefulness of force plates is limited to laboratory settings [6], but it
is substantially more feasible to record a video outside of such settings and
utilize a computer vision model to estimate these key variables. However, for
this to be realistic, the computer vision model has to be sufficiently
reliable. Prior work has examined the feasibility of identifying when an
athlete incorrectly performs a drop or squat jump [9], but thus far no works
have estimated the key variables themselves.

Prior work has also focused on alternate solutions for athlete performance
evaluation, useful in the absence of standard force plate data. Johnson _et
al_. [6] provide a method to analyze athlete ground reaction forces and
moments (GRF/Ms) in real time using eight marker motion capture trajectories,
allowing for a field-based study in identifying early injury risk; they [7]
also use the motion capture data to estimate Knee Joint Movements (KJM).
Goldacre _et al_. [5] utilized preprocessed marker motion capture data and
compared a Least Squares Estimator method for estimating GRF with a
convolutional neural network (CNN) estimator and found the CNN-best method was
more reliable and consistent. All of these works showcase the feasibility of
estimating a key metric, GRF, but none have considered other essential metrics
like velocity. Further, none of these works have attempted to estimate
variables using raw video data. Here, we train a custom Swin Transformer
architecture to estimate key metrics like velocity using the dataset from
Blanchard _et al_. [1].

We propose an automatic force plate variable estimation tool that uses athlete
motion videos — performing evaluative jumps such as countermovement and drop
jumps — as input, which can be run on any handy smart device such as a mobile
phone. The tool is trained on the videos directly with minimal preprocessing
steps, allowing faster inference during its execution. In the absence of a
force plate, our method can provide a good approximation of the desired force
plate variables, which can further be used to investigate the performance of
an athlete.

In summary, this work has several novel contributions:

  1. 1.

This is the first work to showcase that key variables for assessing athlete
performance can be estimated using video alone.

  2. 2.

We create a custom variant on the Swin Transformer architecture (see Figure 1)
and achieve state-of-the-art performance across 86 athletes.

  3. 3.

Our solution is generalized across two different kinds of jumps: the squat
jump and the drop jump. We did not create separate models for each kind of
jump.

##  2 Methods

###  2.1 Data

Blanchard _et al_. [1] collected a dataset which consisted of 582 RGB videos
of 89 athletes performing multiple countermovement and drop jumps [1]. Each
jump was recorded from three different angles: a left, right, and center
perspective. Jumps were performed on force plate data, which recorded a
multitude of variables associated with the jump (181 for countermovement jumps
and 58 for drop jumps). Only 86 athletes’ force plate data (545 videos) were
available, so we only utilized this subset of the data.

We divided the dataset into three separate folds. No participant from one fold
was present in any other fold. The first two folds included 29 unique
participants each, and the third fold included the remaining 28 participants.
There were 180, 178, and 161 videos in each fold, respectively.

We focused on training a model to estimate concentric peak velocity. Figure 2
displays the distribution of concentric peak velocity (hereafter, velocity)
across the three folds of our dataset. In our experiments, models used for
evaluation were trained on each individual view, with an additional model
trained on the combined input of the three views.

Figure 2: Distribution of velocity across data folds.

###  2.2 Model Architectures

Estimating velocity, as well as other metrics from the force plate data, is a
regression problem. We experimented with three different methods to evaluate
the feasibility of estimating velocity.

Baseline Models We evaluated two different baseline methods for estimating the
participant’s velocity using video data in order to measure athletic
performance without a force plate setup. First, we used a ResNet-50
architecture to extract features from all video frames. For our first
baseline, we trained a support vector machine (SVM) using these features. For
our second, we added a regression dense layer to the ResNet-50.

In the first baseline method, we use the SVM regression method for velocity
estimation, with the kernel set to ’rbf’, degree to 3, coef0 to 0.0, tol to
0.0001, C to 1.0 and epsilon to 0.01. The second baseline method uses a dense
layer for final score estimation.

Swin Transformer Architecture We hypothesized a transformer architecture [2]
would be better suited for video data than our baseline methods. We modified
the Swin Transformer [8] to extract features from video frames and utilized a
dense layer to predict velocity. The first layer of the Swin Transformer
divided the image into smaller patches with equal window sizes using the
window multi-attention head (W-MSA), as shown in Equation 1. The second layer
repeated this, but shifted the starting position using shifted window multi-
attention head (SW-MSA), as shown in Equation 2. This change in how window
patches are made across layers produced cross-connections that let the model
identify the relevant image features. As illustrated in Equations 3 and 4,
self attention was conducted in the local window patches before the window
patches were sent via a multi-layer perceptron (MLP). Finally, the adjacent
patches were combined. The two successive layers were part of a Swin
Transformer block; we used four such Swin Transformer blocks to generate the
features with respect to each video frame.

$\hat{z}^{l}=\mathrm{W}\\\
-\mathrm{MSA}\left(\mathrm{LN}\left(\mathrm{z}^{l-1}\right)\right)+\mathrm{z}^{l-1}$
(1)

$\hat{z}^{l+1}=\mathrm{SW-
MSA}\left(\mathrm{LN}\left(\mathrm{z}^{l}\right)\right)+\mathrm{z}^{l}$ (2)

$z^{l}=\mathrm{MLP}\left(\mathrm{LN}\left(\mathrm{\hat{z}}^{l}\right)\right)+\mathrm{\hat{z}}^{l}$
(3)

$z^{l+1}=\mathrm{MLP}\left(\mathrm{LN}\left(\mathrm{\hat{z}}^{l+1}\right)\right)+\mathrm{\hat{z}}^{l+1}$
(4)

where $\hat{z}^{l}$ are the output features of window-based multihead self-
attention layers ($\mathrm{W}-\mathrm{MSA}$) and shifted window-based
multihead self-attention layers ($\mathrm{SW}-\mathrm{MSA}$) for block $l$
respectively.

The architecture with the Swin Transformer backbone is as shown in Figure 1.
The inputs to the model are video frames from each view with respect to a
single participant. Each of the video frames are processed through the Swin
Transformer Block [8].

The mean of the frames from each view after flattening these features are
calculated. Each view is passed through a dense layer before concatenating
them all together. The combined features are then processed through another
dense layer to obtain the final features.

###  2.3 Training and Evaluation Details

Velocity scores followed a Gaussian distribution, with the majority of data
ranging between 0.4 and 0.6, as seen in Figure 2. We exploited this
distribution during training by using a balanced sampler to create mini
batches, thus allowing the model to accurately learn the embedding space of
jumps across all velocities.

All of our models were trained on a Titan V GPU with an Intel(R) Xeon(R)
E5-1650 v4 processor and 16GB RAM. We used L1 loss for training and kept a
fixed learning rate of 1e-3. The models were trained for 100 epochs using the
Adam Optimizer.

All models were trained and evaluated following a 3-fold cross validation
paradigm. No participants present in the training data were present in the
test data.

We evaluated models using two metrics: Pearson’s R and Mean Absolute Error
(MAE). All reported results are the mean and standard deviation of each metric
across all folds.

##  3 Results

Table 1: Our baseline when compared to SVM and Dense Layers for estimation on the Video and Force plate data. Method |  MAE ($\downarrow$) |  R ($\uparrow$)
---|---|---
Baseline method with SVM |  0.19 ±0.017 |  0.12 ±0.06
Baseline method with Dense Layer |  0.28 ±0.017 |  0.16 ±0.05
Our method (Combined view) |  0.21 ±0.015 |  0.71 ±0.06

Table 1 shows the performance of our baseline estimators and our transformer
method. Although these methods appear similar when considering only MAE, our
transformer-based model performs substantially better when evaluated by
Pearson’s R. In Figure 3, we showcase how our estimated velocity compares with
the actual velocity for each fold — our models slightly underestimated
velocity, but they were extremely consistent with their estimations across
folds. The low R values of the baseline methods indicate the estimations from
those models were extremely inconsistent.

(a) Fold 1

(b) Fold 2

(c) Fold 3

Figure 3: Actual vs estimated velocity scores for the Transformer pipeline
trained on a combined view input.

###  3.1 Ablation

We investigated how influential each component of the data was for our
method’s prediction. Table 2 showcases the results of this experiment — video
from the center perspective was more important for accurately estimating
velocity, exhibiting an R value 0.21 higher than either the left or right
perspective (the left and right perspectives exhibited similar performances).
This contrasts the findings of [9], which found the left and right
perspectives were more influential for estimating when a jump was performed
incorrectly. This indicates that different perspectives may be essential to
fully estimating all components of a jump. Further, data from all three
perspectives culminated in the best model — combining all perspectives
increased the R value by 0.15 over the center view alone.

Table 2: Experiments to determine the importance of combining videos captured from different views. The transformer pipeline trained on the combination of the three views improves the correlation score (R), which shows that the model is able to distinguish between a low and a high velocity jump when provided with video input from multiple angles. But the model trained on the Left view provides the least error in estimation, followed by the Right view, which shows that the model can still learn accurate estimations from videos from a single view but with a much lower capacity to distinguish between the higher and lower velocity jumps. Video capture view |  MAE ($\downarrow$) |  R ($\uparrow$)
---|---|---
Left |  0.16 ±0.016 |  0.35 ±0.06
Right |  0.20 ±0.014 |  0.34 ±0.04
Center |  0.19 ±0.014 |  0.56 ±0.06
Combined View |  0.21 ±0.015 |  0.71 ±0.06

##  4 Conclusion and Future Work

Velocity is a key evaluation metric for evaluating an athlete’s performance.
To our knowledge, this is the first work that estimates velocity from RGB
video alone. Our novel solution is generalized across participants and two
kinds of evaluative jumps: the squat jump and the drop jump. This work also
provides insights into what kinds of data are important for estimating key
metrics like velocity. In the future, we plan to expand our architecture to
estimate a multitude of other essential variables for athlete performance
following a multi-task regression paradigm.

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