aainstitutetext: University of Oxford, United Kingdombbinstitutetext:
Département de Physique, École Normale Supérieure, Paris,
Franceccinstitutetext: Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France

# Model-independent method for measuring the angular coefficients of $B^{0}\to
D^{*-}\tau^{+}\nu_{\tau}$ decays

Donal Hill,111Corresponding author. a Malcolm John b Wenqi Ke c Anton
Poluektov<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Reconstruction of the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ angular distribution
is complicated by the strongly biasing effect of losing the neutrino
information from both the $B$ and $\tau$ decays. In this work, a novel method
for making unbiased measurements of the angular coefficients while preserving
the model independence of the angular technique is demonstrated. The twelve
angular functions that describe the signal decay, in addition to background
terms, are modelled in a multidimensional fit, using template probability
density functions that encapsulate all resolution and acceptance effects.
Sensitivities at the LHCb and Belle II experiments are estimated, and sources
of systematic uncertainty are discussed, notably in the extrapolation to a
measurement of $R(D^{*})$.

##  1 Introduction

An indication of new physics (NP) is emerging though the evidence that $b\to
c\tau\nu_{\tau}$ transitions proceed at a higher rate than would be expected
from a comparison to the same process with the lighter leptons. The ratio of
branching fractions

$R(D^{(*)})=\frac{\mathcal{B}(B\to D^{(*)}\tau\nu)}{\mathcal{B}(B\to
D^{(*)}l\nu)}\hskip 22.76228ptl\in\\{e,\mu\\}$

is measured above the Standard Model (SM) expectation in results from the
$B$-factories Lees:2012xj ; Lees:2013uzd ; Huschle:2015rga ; Hirose:2016wfn ;
Hirose:2017dxl ; Abdesselam:2019dgh  and LHCb Aaij:2015yra ;
PhysRevD.97.072013 . The current global averages of experimental results agree
with the SM predictions HFLAV16  at only $\sim 3$ standard deviations:

$\displaystyle R(D^{*})^{\rm exp}=0.295\pm 0.014\,,$ $\displaystyle R(D)^{\rm
exp}=0.340\pm 0.030\,,$ $\displaystyle R(D^{*})^{\rm SM}=0.258\pm 0.005\,,$
$\displaystyle R(D)^{\rm SM}=0.299\pm 0.003\,.$

In order to further characterise the underlying physics in $b\to
c\tau\nu_{\tau}$ transitions, it is necessary to study the kinematics of
semitauonic $B$ decays in addition to their rates. Many polarisation and
asymmetry observables have been shown to discriminate between the SM and NP
scenarios Biancofiore:2013ki ; Duraisamy2013 ; PhysRevD.85.094025 ; Tanaka1995
; PhysRevD.82.034027 ; PhysRevD.87.054002 ; PhysRevD.86.034027 ;
PhysRevD.92.114022 ; Becirevic:2016hea ; PhysRevD.94.094021 ;
PhysRevD.95.115038 ; Bardhan2017 ; PhysRevD.95.036021 ; PhysRevD.95.093006 ;
PhysRevD.99.035015 ; Colangelo:2018cnj . One such example is the $D^{*}$
longitudinal polarisation fraction, which has recently been measured to be
$F_{L}^{D^{*}}=0.60\pm 0.09$ Abdesselam:2019wbt . Several calculations of the
SM expectation exist, which centre around 0.45 Tanaka:2012nw ;
PhysRevD.92.114022 ; PhysRevD.95.115038 ; PhysRevD.98.095018 ;
Bhattacharya:2018kig ; Becirevic:2019tpx ; this tension constitutes another
potential indication of deviation from the SM.

Complete information on the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ decay
kinematics is ultimately obtained from the full angular decay rate
Becirevic:2019tpx

$\begin{split}\frac{d^{4}\Gamma}{dq^{2}\,d(\cos{\theta_{D}})\,d(\cos{\theta_{L}})\,d\chi}&\propto
I_{1c}\cos^{2}{\theta_{D}}+I_{1s}\sin^{2}{\theta_{D}}\\\
&+[I_{2c}\cos^{2}{\theta_{D}}+I_{2s}\sin^{2}{\theta_{D}}]\cos 2\theta_{L}\\\
&+[I_{6c}\cos^{2}{\theta_{D}}+I_{6s}\sin^{2}{\theta_{D}}]\cos\theta_{L}\\\
&+[I_{3}\cos 2\chi+I_{9}\sin 2\chi]\sin^{2}{\theta_{L}}\sin^{2}{\theta_{D}}\\\
&+[I_{4}\cos\chi+I_{8}\sin\chi]\sin 2\theta_{L}\sin 2\theta_{D}\\\
&+[I_{5}\cos\chi+I_{7}\sin\chi]\sin\theta_{L}\sin 2\theta_{D}\,,\end{split}$
(1)

where the angles (${\theta_{D}}$, ${\theta_{L}}$, $\chi$) parameterise the
spin-0 $B^{0}$ meson decay topology, and are defined in App. A. This
expression involves a sum of twelve independent angular functions, each of
which is multiplied by a coefficient $I_{X}$
($X\in\\{1c,1s,2c,2s,3,4,5,6c,6s,7,8,9\\}$) that encapsulates the dependence
on the square of the dilepton invariant mass, $q^{2}$, form factors, and the
fundamental couplings. The angular distribution can reveal the influence of NP
even if $R(D^{*})$ becomes fully compatible with the SM.

Angular analysis is well established in the study of rare dimuon decays such
as $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ Alok:2011gv ; Aaij2013 . The principal
advantage of the technique is that the coefficients contain all form factor
dependence, so there is no experimental uncertainty due to a choice of form
factor scheme. Combinations of the angular coefficients can also reduce
dependence on the form factors in subsequent phenomenological interpretations.
The difficulty that arises in applying angular analysis methods to semitauonic
decays is the missing information due to the lost neutrinos in both the $B$
and $\tau$ decays, which strongly sculpts the angular distribution and makes a
parametric fit to data impossible.

In this paper, a novel approach is presented that uses a multidimensional
template fit in the angular basis to measure the $I_{X}$ coefficients in a
model-independent manner without statistical biases. The technique assumes and
requires excellent agreement between data and simulated samples for the
construction of the templates, which must describe all reconstruction,
resolution and migration effects. In this case the fit reduces to a linear sum
of twelve independent templates, preserving the inherent model independence of
the angular method. Eleven $I_{X}$ parameters are measured (one is fixed by
$\sum I_{X}=1$), and the model dependence is confined to one overall signal
yield. The expected resolutions and covariance matrix of the $I_{X}$
parameters are determined under realistic experimental conditions emulating
both hadron collider and $B$-factory scenarios; assessment of the $I_{X}$
sensitivity to NP is beyond the scope of the work.

For this demonstration, the angular coefficients are measured integrated over
all values of $q^{2}$. This is not required and parallel angular fits in
several $q^{2}$ bins could follow a similar procedure. NP may also be sought
by measuring $I_{X}$ and $\bar{I}_{X}$ (the ${C\\!P}$ conjugate) values
separately with the same template fit by tagging according to the $\tau$
lepton charge.

It is also noted that the methods developed in this work are applicable to the
light lepton $B^{0}\to D^{*-}l^{+}\nu_{l}$ modes. These decays suffer from
lower background levels, and superior angular resolution due to the stable
charged lepton coming directly from a well-defined $B$ decay vertex. A
measurement of the $I_{X}$ coefficients of these modes is well motivated to
validate form factor schemes and provide a null test of the SM.

###  1.1 Monte Carlo simulation

Monte Carlo (MC) signal samples of $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ decays
are generated using the RapidSim package Cowan:2016tnm . RapidSim is a fast MC
generator for simulating heavy-quark hadron decays. It uses TGenPhaseSpace
Brun:1997pa  to generate $b$-quark hadron decays and FONLL Cacciari:2001td  to
give the $b$ quark the correct production kinematics for the Large Hadron
Collider (LHC). Exclusive decays of the $D^{*-}$ meson decay to ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\pi^{-}$ are generated, with the
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ meson decaying to
$K^{+}\pi^{-}$. The three-prong
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu}_{\tau}$ decay, rather than the more
abundant muonic $\tau$ decay, is the focus of this study. This is because the
presence of a $\tau$ decay vertex results in lower backgrounds, and with only
two neutrinos in the final state, this mode has the best decay angle
resolution.

Signal $B^{0}$ mesons are decayed using EvtGen Lange:2001uf . The ISGW2
PhysRevD.39.799  model distributes $B^{0}$ decays according to Eq. (1), with
SM values for each $I_{X}$ coefficient. The VSS model is used to model the
vector $D^{*-}$ decay, and TAUOLA Chrzaszcz:2016fte  (model number 5) produces
the correct kinematic and invariant mass structure for the three-prong $\tau$
decay. To emulate the effects of detector resolution, the track momenta and
decay vertex coordinates are smeared according to RapidSim LHCb resolution
presets Cowan:2016tnm . The missing momentum due to the presence of two final
state neutrinos is modelled by ignoring both particles in any calculations of
reconstructed quantities. Detector acceptance effects are modelled by
restricting generated $B^{0}$ mesons to the momentum range $[0,100]$ GeV$/c$
and the pseudorapidity range $[1,6]$, which is similar to the geometrical
acceptance of the LHCb detector.

##  2 Kinematic reconstruction

Due to the lost neutrinos in the final state and the absence of a constraint
from the initial state, neither the $\tau$ nor $B^{0}$ momentum can be fully
reconstructed at a hadron collider. The best calculation of the decay angles
uses estimates of the $\tau$ and $B^{0}$ momentum that are determined from the
topology of the decay. As the $\tau$ lepton mass is well known
PhysRevD.98.030001 , its momentum can be estimated up to a two-fold ambiguity
from its line of flight between the reconstructed $B^{0}$ and $\tau$ vertices.
The $\tau$ momentum magnitude in the laboratory frame is

$|\vec{p}_{\tau}|=\frac{(m_{3\pi}^{2}+m_{\tau}^{2})|\vec{p}_{3\pi}|\cos\theta_{\tau,3\pi}\pm
E_{3\pi}\sqrt{(m_{\tau}^{2}-m_{3\pi}^{2})^{2}-4m_{\tau}^{2}|\vec{p}_{3\pi}|^{2}\sin^{2}\theta_{\tau,3\pi}}}{2(E_{3\pi}^{2}-|\vec{p}_{3\pi}|^{2}\cos^{2}\theta_{\tau,3\pi})}\,,$
(2)

where $m_{3\pi}$, $|\vec{p}_{3\pi}|$, and $E_{3\pi}$ are the reconstructed
invariant mass, momentum, and energy of the three-prong system, $m_{\tau}$ is
the known $\tau$ mass, and $\theta_{\tau,3\pi}$ is the angle between the
three-prong momentum vector and the flight vector of the $\tau$. Eq. (2) has a
single solution when $\theta_{\tau,3\pi}$ takes the maximum allowed value such
that the square-root term is zero, i.e.

$\theta_{\tau,3\pi}^{\text{max}}=\arcsin\bigg{(}\frac{m_{\tau}^{2}-m_{3\pi}^{2}}{2m_{\tau}|\vec{p}_{3\pi}|}\bigg{)}\,.$
(3)

Combined with the $\tau$ line of flight, this provides an estimate of the
$\tau$ momentum components with minimal bias. In a similar fashion, the
$B^{0}$ momentum is estimated using

$|\vec{p}_{B^{0}}|=\frac{(m_{Y}^{2}+m_{B^{0}}^{2})|\vec{p}_{Y}|\cos\theta_{B^{0},Y}\pm
E_{Y}\sqrt{(m_{B^{0}}^{2}-m_{Y}^{2})^{2}-4m_{B^{0}}^{2}|\vec{p}_{Y}|^{2}\sin^{2}\theta_{B^{0},Y}}}{2(E_{Y}^{2}-|\vec{p}_{Y}|^{2}\cos^{2}\theta_{B^{0},Y})}\,,$
(4)

where $Y$ represents the $D^{*-}\tau^{+}$ system as reconstructed using Eq.
(2). The maximum opening angle between the $B^{0}$ flight vector and the
momentum vector of $Y$ is

$\theta_{B^{0},Y}^{\text{max}}=\arcsin\bigg{(}\frac{m_{B^{0}}^{2}-m_{Y}^{2}}{2m_{B^{0}}|\vec{p}_{Y}|}\bigg{)}\,,$
(5)

which is used in Eq. (4) to provide a single estimate for the $B^{0}$ momentum
magnitude. The $\tau$ and $B^{0}$ momentum bias and resolution are tabulated
in Tab. 1.

Particle | Mean $p_{\text{True}}$ [GeV/$c$] | Res. $\mu$ | Res. $\sigma$
---|---|---|---
$B^{0}$ | 150.6 | 0.09 | 0.19
$\tau$ | 61.9 | 0.01 | 0.25
Table 1: Summary of $\tau$ and $B^{0}$ momentum bias and resolution, defined
as the mean and width of $(p_{\text{Reco}}-p_{\text{True}})/p_{\text{True}}$,
respectively.

Since the four-vectors of the $\tau$ and $B^{0}$ are necessary inputs for the
calculation of the decay angles, they too suffer substantial resolution
effects. This is shown in Fig. 1, where the true and reconstructed angular
distributions of $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ are compared. The angular
resolutions are quantified in Tab. 2, which shows that $\cos\theta_{D}$ is the
most well-reconstructed quantity. As a result of the large resolution effects
on $\cos\theta_{L}$ and $\chi$, considerable migration of events occurs within
the angular phase space. This is illustrated in Fig. 2, where the two-
dimensional projections of the truth-level and reconstructed angular
distributions are shown. The density difference within each bin is also shown,
where red (blue) regions indicate increases (decreases) in density caused by
the reconstruction. The overall effect of the event migration is to reduce the
density variation across the phase space, but a bias in $\cos\theta_{L}$
towards more positive values is also evident.

Due to the reconstruction-induced event migration, a parametric fit to the
reconstructed decay angles using Eq. (1) cannot be used to measure the $I_{X}$
coefficients. Any attempt to correct the reconstruction biases leads to a
dependence on the model used in the Monte Carlo from which the correction is
derived. Instead, it is demonstrated that the $I_{X}$ coefficients can be
measured with a binned fit using multidimensional histogram templates, where
the angular degradation and other detector effects are included directly in
each of the twelve templates that describe the signal probability density
function (PDF).

Figure 1: (Top) True (red) and reconstructed (blue) angular distributions from
100,000 generated $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ events. (Bottom)
Distributions of reconstructed angular variables versus true, where darker
colours indicate a higher density of events.

Figure 2: Two-dimensional projections of the truth-level (top) and reconstructed (middle) angular distributions; darker colours indicate regions of higher density. The density difference (bottom) in each bin indicates where the density increases (red) and decreases (blue) as a result of the reconstruction. Angle | Res. $\mu$ | Res. $\sigma$
---|---|---
$\cos\theta_{D}$ | 0.00 | 0.23
$\cos\theta_{L}$ | 0.15 | 0.65
$\chi$ | -0.01 rad | 2.24 rad
Table 2: Angular variable resolution mean ($\mu$) and width ($\sigma$)
determined using generated $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ events. The
resolution is defined as $a_{Reco}-a_{True}$, where
$a\in\\{\cos\theta_{D},\cos\theta_{L},\chi\\}$.

##  3 Building and using templates

The decay rate defined in Eq. (1) involves a sum of twelve independent angular
functions in the $(\cos\theta_{D},\cos\theta_{L},\chi)_{\text{True}}$ variable
space. Because of the bias and resolution effects that arise in the
reconstruction of semitauonic decays, a template fit in the
$(\cos\theta_{D},\cos\theta_{L},\chi)_{\text{Reco}}$ variable space must be
used. The twelve angular functions become twelve multidimensional template
histograms each scaled by an $I_{X}$ coefficient. In this way, the fit is
performed with the reconstruction effects included directly within the PDF.

The template histograms are created by first filling twelve density histograms
$D_{I_{X}}$. Each density histogram contains a large number of bins across
$(\cos\theta_{D},\cos\theta_{L},\chi)_{\text{True}}$ space; $30\times 30\times
30$ uniform bins are used here. The $D_{I_{X}}$ histograms are filled
according to the angular function associated with each $I_{X}$ in Eq. (1). The
twelve $D_{I_{X}}$ histograms are divided by a density histogram, $M$, of the
total signal model given by Eq. (1). By taking the ratio
$R_{I_{X}}={D_{I_{X}}}/{M}$, the model used in the simulation cancels and
ensures that the $R_{I_{X}}$ histograms are model independent. The $R_{I_{X}}$
histograms are then used to assign weights to simulated signals events based
on their $(\cos\theta_{D},\cos\theta_{L},\chi)_{\text{True}}$ value. A
simulated event falling within the true angular bin $i$ will be assigned
twelve weights, $w_{I_{X}}=R_{I_{X}}(i)$.

Subsequently, the per-event weights $w_{I_{X}}$ are applied when constructing
histogram templates, $h_{I_{X}}$, in the reconstructed angular variables
$(\cos\theta_{D},\cos\theta_{L},\chi)_{\text{Reco}}$. The result of this
procedure is illustrated in Fig. 3, where the $h_{I_{1c}}$ and $h_{I_{2s}}$
templates created using a sample of one million generated events are shown.
The large size of the sample ensures that the template statistical uncertainty
in each bin is negligible. The corresponding density histograms $D_{I_{1c}}$
and $D_{I_{2s}}$ are also shown, to illustrate the sculpting effect of the
reconstruction. Note that the $h_{I_{X}}$ templates may not be positive in all
bins, but the sum of all twelve templates in any given bin is always positive,
and is proportional to the total decay rate in that bin.

Figure 3: Projections of the $D_{I_{X}}$ density histograms (red) and the
$h_{I_{X}}$ templates (blue) for $X=1c$ (top) and $X=2s$ (bottom). The
differences between $D_{I_{X}}$ and $h_{I_{X}}$ are caused by the bias and
resolution effects introduced by the reconstruction.

###  3.1 Signal-only template fit

The twelve $h_{I_{X}}$ templates are normalised to have unit integral, then
multiplied by their corresponding $I_{X}$ coefficients to build the total
signal PDF

$\begin{split}P((\cos\theta_{D},\cos\theta_{L},\chi)_{\text{Reco}})&=\frac{1}{3}(4-6I_{1s}+I_{2c}+2I_{2s})h_{1_{1c}}\\\
&+I_{1s}h_{I_{1s}}+I_{2c}h_{I_{2c}}+I_{2s}h_{I_{2s}}+I_{6c}h_{I_{6c}}+I_{6s}h_{I_{6s}}\\\
&+I_{3}h_{I_{3}}+I_{9}h_{I_{9}}+I_{4}h_{I_{4}}+I_{8}h_{I_{8}}+I_{7}h_{I_{7}}+I_{5}h_{I_{5}}\,.\end{split}$
(6)

This expression is analogous to Eq. (1), but where the angular functions are
replaced with the $h_{I_{X}}$ templates. The normalisation condition from Ref.
Becirevic:2019tpx  is imposed,

$\Gamma=\frac{1}{4}(3I_{1c}+6I_{1s}-I_{2c}-2I_{2s})=1\,,$ (7)

which constrains the value of $I_{1c}$ from the other $I_{X}$ coefficients.
Importantly, the form of each $h_{I_{X}}$ template remains the same regardless
of the underlying physics model; only the values of the $I_{X}$ coefficients
are modified by the presence of NP.

Using this PDF, a binned maximum likelihood fit to the reconstructed decay
angle distribution is demonstrated using the TensorFlowAnalysis package TFA ,
which provides an interface between TensorFlow tensorflow2015-whitepaper  and
the MINUIT James:1975dr  minimisation package. For this demonstration,
$N_{\text{sig}}=100,000$ signal events are generated across $n_{\text{bins}}$
bins in each of the three angular variables, where $n_{\text{bins}}=16$ is
chosen to ensure that there are approximately 25 signal events in each bin. An
alternative binning scheme based on placing approximately 50 events into each
bin has been tested and shows consistent behaviour. The $I_{X}$ values found
by the binned fit are summarised in Tab. 3 (a).

(a)

(a)  Coefficient | Value
---|---
$I_{1c}$ | $\phantom{-}0.53\pm 0.01$
$I_{1s}$ | $\phantom{-}0.40\pm 0.00$
$I_{2c}$ | $-0.17\pm 0.02$
$I_{2s}$ | $\phantom{-}0.08\pm 0.01$
$I_{3}$ | $-0.10\pm 0.01$
$I_{4}$ | $-0.14\pm 0.02$
$I_{5}$ | $\phantom{-}0.28\pm 0.01$
$I_{6c}$ | $\phantom{-}0.28\pm 0.02$
$I_{6s}$ | $-0.24\pm 0.01$
$I_{7}$ | $-0.01\pm 0.01$
$I_{8}$ | $\phantom{-}0.01\pm 0.02$
$I_{9}$ | $-0.01\pm 0.02$

(b)

(b)

Coefficient | Value
---|---
$I_{1c}$ | $\phantom{-}0.52$
$I_{1s}$ | $\phantom{-}0.40$
$I_{2c}$ | $-0.16$
$I_{2s}$ | $\phantom{-}0.06$
$I_{3}$ | $-0.12$
$I_{4}$ | $-0.14$
$I_{5}$ | $\phantom{-}0.28$
$I_{6c}$ | $\phantom{-}0.32$
$I_{6s}$ | $-0.25$
$I_{7}$ | $-0.01$
$I_{8}$ | $\phantom{-}0.00$
$I_{9}$ | $-0.00$

Table 3: Results of the (a) binned template fit to the reconstructed angles,
and (b) unbinned parametric fit to the truth-level angles. In both cases only
the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ signal sample is used. The
uncertainties quoted in (a) are statistical, and arise from the use of a
finite number of bins. The uncertainties in (b) are due only to finite sample
size, and are negligibly small compared to those of (a); they are thus omitted
for this comparison.

###  3.2 Validating with a truth-level fit

To validate the binned template fit results, a second sample of $100,000$
signal decays is generated without acceptance cuts, resolution smearing, or
the effect of the missing neutrino in the four-vector calculation. Using the
true angles and Eq. (1) as the PDF, an unbinned parametric fit is performed
with the normalisation condition of Eq. (7) again imposed. The total fit
projections are shown in Figure 4 and the fitted $I_{X}$ values are recorded
in Table 3 (b). The level of agreement between the truth-level parametric fit
and the binned fit to the reconstructed angular variables is excellent. By
correctly describing reconstruction biases and resolution effects using
templates, the binned fit correctly measures the angular coefficients.

Figure 4: Normalised one-dimensional projections of the unbinned parametric
fit to the truth-level angular distribution in a sample of $100,000$ $B^{0}\to
D^{*-}\tau^{+}\nu_{\tau}$ decays. The black points are the signal sample, and
the red solid line represents the best fit using Eq. (1), where the $I_{X}$
coefficient values listed in Tab. 3 (b) are found.

It is desirable to show that the template method can correctly recover SM
$I_{X}$ values under an alternative form factor scheme. To demonstrate this,
the signal-only template fit is repeated with per-event weights applied to the
generated signal sample in order to align it with the CLN form factor scheme
Caprini:1997mu . The reweighting is performed using the HAMMER package
Duell:2016maj ; Ligeti:2016npd . Both the parametric unbinned fit and the
binned template fit are rerun; the $h_{I_{X}}$ templates are unaltered as they
do not contain any form factor dependence. The $I_{X}$ coefficients resulting
from the fits are displayed in Fig. 5, and are seen to agree well with the SM
values calculated using the CLN scheme in Ref. Becirevic:2019tpx .
Furthermore, the longitudinal $D^{*-}$ polarisation fraction calculated from
this signal-only template fit,

$F_{L}^{D^{*}}=\frac{3I_{1c}-I_{2c}}{3I_{1c}-I_{2c}+6I_{1s}-2I_{2s}}=0.446\pm
0.004\,,$ (8)

aligns satisfactorily with the SM expectation noted in Sec. 1.

Figure 5: Comparison of the signal-only truth-level unbinned parametric fit
(orange points), the binned template fit (navy points), and the
$q^{2}$-integrated SM $I_{X}$ values calculated in Ref. Becirevic:2019tpx
(red points). The fitted MC samples are re-weighted to follow the CLN form
factor scheme, which has also been used in the calculation of the cited SM
values.

###  3.3 On binning in ${\boldmath q^{2}}$

The weighting procedure can be applied inclusively or in a number of discrete
$q^{2}_{\rm True}$ bins. Assuming simulation accurately models all $q^{2}_{\rm
True}\to q^{2}_{\rm Reco}$ sculpting and bin migration, the model independence
of the $I_{X}$ measurements is preserved. Specifically, with $N$ bins in
$q^{2}_{\rm True}$, the event weights in the template definition become
$w_{I_{X}}=R_{I_{X}}(i,j)$, where $i$ again identifies the true angular bin
and $j$ tags the true $q^{2}_{\rm True}$ bin. Because an event from the
$j^{\rm th}$ $q^{2}_{\rm True}$ bin may migrate to any $k^{\rm th}$
$q^{2}_{\rm Reco}$ bin, $12N^{2}$ templates are required. These
$h_{j,I_{X}}^{k}$ templates are built in the reconstructed angular variables
space $(\cos\theta_{D},\cos\theta_{L},\chi)_{\text{Reco}}$ for each
$q^{2}_{\rm Reco}$ bin by applying the weights.222The $q^{2}_{\rm Reco}$ bins
are not required to align with the $q^{2}_{\rm True}$ bins. Since the
reconstruction may measure $q^{2}_{\rm Reco}$ values outside the possible
$q^{2}_{\rm True}$ range, different ranges may be necessary. Also, the number
of $q^{2}_{\rm True}$ bins and $q^{2}_{\rm Reco}$ bins may differ, though they
are both taken to be $N$ in this explanation. Although there are $12N^{2}$
templates, there are still only $11N$ freely varying angular coefficients and
$N$ signal fractions, $A_{j}$. This is because the apportioning across the
$q^{2}_{\rm Reco}$ bins is fixed from simulation via the weighting procedure.
In a generalisation of Eq. 6, the PDF is defined summed over $q^{2}_{\rm
Reco}$ bins and $q^{2}_{\rm True}$ bins,

$\begin{split}P\left((\cos\theta_{D},\cos\theta_{L},\chi)_{\text{Reco}};q^{2}_{\rm
Reco}\right)&=\sum_{j}A_{j}\sum_{k}\bigg{\\{}\frac{1}{3}\left(4-6y_{j,1s}+y_{j,2c}+2y_{j,2s}\right)h^{k}_{j,1_{1c}}\\\
&+y_{j,1s}h^{k}_{j,I_{1s}}+y_{j,2c}h^{k}_{j,I_{2c}}+y_{j,2s}h^{k}_{j,I_{2s}}\\\
&+y_{j,6c}h^{k}_{j,I_{6c}}+y_{,j6s}h^{k}_{j,I_{6s}}+y_{j,3}h^{k}_{j,I_{3}}+y_{j,9}h^{k}_{j,I_{9}}\\\
&+y_{j,4}h^{k}_{j,I_{4}}+y_{j,8}h^{k}_{j,I_{8}}+y_{j,7}h^{k}_{j,I_{7}}+y_{j,5}h^{k}_{j,I_{5}}\bigg{\\}}\,.\end{split}$
(9)

The coefficients are $y^{k}_{j,X}=f^{k}_{j,X}I_{j,X}$, where the fractions
$f^{k}_{j,X}$ with $\sum_{k}f^{k}_{j,X}=1$ quantify the bin migration
probabilities are derived from simulation through the weighting procedure.

It is stressed that because the $q^{2}_{\rm True}$ binning is included in the
event weights $R_{I_{X}}(i,j)={D_{j,I_{X}}}/{M_{j}}$, the dependence on the
model used in the simulation is removed in a multi-$q^{2}$-bin implementation
in the same way that it is removed in the inclusive implementation documented
in this work. However, as $12N^{2}$ templates are required just for the
signal, large simulated samples would been needed in such an analysis as well
as large data sets.

##  4 Dealing with backgrounds

The template fit is studied in a more realistic manner by considering
backgrounds to the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ signal. A large number
of backgrounds are generated using RapidSim, matching the list of backgrounds
considered in recent experimental work PhysRevD.97.072013 . There are three
categories to consider: (1) prompt $B\to D^{*-}\pi^{+}\pi^{+}\pi^{-}(X)$
backgrounds that are reduced by requiring a displaced three-prong vertex, (2)
the dominant double-charm $B\to D^{*-}D_{(s)}^{+}(X)$ background, and (3)
$B\to D^{*-}(X)\,\tau^{+}\nu_{\tau}$ feed-down.

Several high branching fraction $B$ decays produce three charged pions at the
$B$ decay vertex, with one or more additional particles ($X$) missed in the
reconstruction. The prompt backgrounds considered for this study are recorded
in Tab. 5 (App. B). These prompt $B\to D^{*-}\pi^{+}\pi^{+}\pi^{-}(X)$
backgrounds are reduced by applying a flight requirement to the $3\pi$ system.
In Ref. PhysRevD.97.072013 , $\tau$ candidate vertices are required to be
displaced from the $B$ vertex along the $z$-axis with a separation of
$4\sigma_{z}$, where the vertex uncertainties of the $B$ and $\tau$ candidates
determine the standard deviation $\sigma_{z}$. Vertex fits are not performed
in RapidSim so the effect of the flight requirement must be approximated.
Assuming a combined $B$ and $\tau$ vertex uncertainty of 1 mm, the flight
requirement is emulated by requiring that all $3\pi$ vertices are displaced 4
mm from their corresponding $B$ decay vertex. This is applied to all RapidSim
samples, including the signal sample; the efficiencies on signal decays and
prompt $B\to D^{*-}\pi^{+}\pi^{+}\pi^{-}(X)$ decays are similar to those
reported in Ref. PhysRevD.97.072013 .

The largest source of background remaining after the $\tau$ flight requirement
arises from double-charm decays of the type $B\to D^{*-}D_{s}^{+}(X)$. The
$D_{s}^{+}$ meson flies before decaying to final states that include three
charged pions, mimicking the
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu}_{\tau}$ signature. The $B\to
D^{*-}D_{s}^{+}(X)$ modes generated for this study are recorded in Tab. 6
(App. B) and the $D_{s}^{+}$ modes are listed in Tab. 7 (App. B). A realistic
sample is created for each double-charm $B$ mode by weighting contributions
according to measured branching fractions PhysRevD.98.030001 . To create a
single $B\to D^{*-}D_{s}^{+}(X)$ sample, the subsamples are summed according
to the proportions measured in Ref. PhysRevD.97.072013 . Similarly, $B\to
D^{*-}D^{+}(X)$ and $B\to D^{*-}D^{0}(X)$ decays are also present in the
background. The $D^{0}$ and $D^{+}$ mesons also fly, and often produce three
or more charged particles in their decay. The $B\to D^{*-}D^{(0,+)}(X)$ modes
generated for this study are recorded in Tab. 8 (App. B), and the $D^{(0,+)}$
modes are listed in Tab. 9 (App. B).

The $B\to D^{**}\tau\nu_{\tau}$ decay is identical to signal aside from the
fact that the charm meson is produced in a higher state of angular momentum.
Given this similarity, the feed-down background must be included as a small
fixed fraction relative to signal, and treated as a systematic pollution in
any phenomenological interpretation. Two contributions are included, namely
$B^{+}\to D_{1}(2420)^{0}\tau^{+}\nu_{\tau}$ and $B^{+}\to
D_{2}^{*}(2460)^{0}\tau^{+}\nu_{\tau}$ decays, where the excited charm mesons
decay to $D^{*-}\pi^{+}$ and the additional pion is not reconstructed. The
EvtGen models used are given in Tab. 10 (App. B). No $B\to
D^{**}\tau\nu_{\tau}$ modes have been observed, so the $D_{1}(2420)^{0}$ and
$D_{2}^{*}(2460)^{0}$ samples are summed in equal proportion.

###  4.1 Multivariate classifier

The reconstructed angular distributions for each background category are shown
in Fig. 6 along with the total signal template. All events are required to
fall within the LHCb acceptance and pass the $\tau$ flight requirement. On
their own, the reconstructed decay angles do not provide enough signal and
background separation to reliably measure the twelve parameters that describe
the $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ signal. As such, a multivariate
classifier is included as a fourth dimension to provide sufficient separation.

Figure 6: Reconstructed angular distributions of the signal and several
categories of background. All distributions are normalised to have equal
integral.

To minimise dependence on any underlying model, the classifier is designed to
avoid any input variables that relate directly to the $B$ decay kinematics;
variables describing the $3\pi$ system are thus preferred. The most
appropriate variables include the reconstructed proper lifetime and the
invariant mass of the $3\pi$ system, and the invariant mass of the
$\pi^{+}\pi^{+}$ and $\pi^{+}\pi^{-}$ combinations. These variables provide
discrimination between $\tau$ candidates and charm mesons, due to their
different lifetimes and decay properties.

Figure 7: Input variables used in the multivariate classifier.

A gradient boosted decision tree (BDT) classifier is trained using the scikit-
learn package scikit-learn . The total $B\to D^{*-}D_{s}^{+}(X)$ background
sample described above is used to train the BDT to favour $B^{0}\to
D^{*-}\tau^{+}\nu_{\tau}$ decays. The $D_{s}^{+}$ sample is used as it is the
largest category of background remaining after the $\tau$ flight requirement.
The distributions for each input variable in signal and background are shown
in Fig. 7. The area under the classifier Receiver Operating Characteristic
(ROC) curve is 0.84, and the performance is illustrated in Fig. 8 (a) where
the BDT distributions are shown for both in signal and background. The
classifier is applied to all generated signal and background samples, and
those events with classifier decision values above zero are retained for use
in the fit and shown in Fig. 8. This selection requirement is 80% efficient on
signal while rejecting 70% of background. The feed-down and signal BDT
distributions are confirmed to be almost identical.

(a)(b)

Figure 8: (a) BDT distributions for $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$ signal
(S) and $B\to D^{*-}D_{s}^{+}(X)$ background (B) in the test and training
samples. (b) BDT distributions for signal and all categories of background for
BDT$>0$.

###  4.2 $\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}\bar{\nu}_{\tau}$ component
of the signal

The $B^{0}\to
D^{*-}(\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}\bar{\nu}_{\tau})\nu_{\tau}$
decay, where the neutral pion is not reconstructed, is not a background but
rather contributes to the total signal. This decay mode differs from the
three-prong signal only in the $\tau$ decay and thus has the same $I_{X}$
coefficients. To benefit from the presence of this mode in the signal, a
dedicated sample is generated and processed in a manner identical to the
principal $\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu}_{\tau}$ signal. Small
differences in the reconstructed decay angles are observed, due to the
additional missing momentum in the
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}\bar{\nu}_{\tau}$ decay. The BDT
distributions are also not equivalent for the two cases, since the invariant
mass variables used in the BDT differ. Following Ref. PhysRevD.97.072013 , a
total signal sample is created from both the
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu}_{\tau}$ and
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}\bar{\nu}_{\tau}$ samples, using
$f_{3\pi}=78\%$ as a relative fraction.

###  4.3 Final signal plus background sample

To create a realistic total dataset that includes both signal and background
events, the generated signal and background samples are summed using the
relative fractions listed in Tab. 4; the values used are based upon those
measured in LHCb data PhysRevD.97.072013 . The signal comprises
$f_{\text{sig}}=11.8\%$ of the total sample, while the total feed-down
contribution is 11% of the signal fraction (1.3% in total). The majority of
the sample is composed of $B\to D^{*-}D_{s}^{+}(X)$ background decays (62.1%)
with smaller fractions assigned to the $B\to D^{*-}D^{+}(X)$ and $B\to
D^{*-}D^{0}(X)$ backgrounds.

Mode | Fraction | Value
---|---|---
$B^{0}\to D^{*-}(\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu}_{\tau})\nu_{\tau}$ | $f_{3\pi}\times f_{\text{sig}}$ |  0.78 (fixed) $\times$ 0.118
$B^{0}\to D^{*-}(\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}\bar{\nu}_{\tau})\nu_{\tau}$ | $(1-f_{3\pi})\times f_{\text{sig}}$ |  0.22 (fixed) $\times$ 0.118
$B\to D^{*-}D_{s}^{+}(X)$ | $f_{D_{s}^{+}}$ | 0.621
$B\to D^{*-}D^{+}(X)$ | $f_{D^{+}}$ | 0.152
$B\to D^{*-}D^{0}(X)$ | $f_{D^{0}}$ | 0.057 (fixed)
$B^{+}\to D^{**}(\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu}_{\tau})\nu_{\tau}$ | $f_{D^{**}}\times f_{3\pi}\times f_{\text{sig}}$ |  0.11 (fixed) $\times$ 0.78 (fixed) $\times$ 0.118
$B^{+}\to D^{**}(\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}\bar{\nu}_{\tau})\nu_{\tau}$ | $f_{D^{**}}\times(1-f_{3\pi})\times f_{\text{sig}}$ |  0.11 (fixed) $\times$ 0.22 (fixed) $\times$ 0.118
$B\to D^{*-}\pi^{+}\pi^{+}\pi^{-}(X)$ | $(1-f_{\text{sig}}-f_{D_{s}^{+}}-f_{D^{+}}-f_{D^{0}}-f_{D^{**}})$ |  0.039 (constrained by $\sum f=1$)
Table 4: Fractions used to construct the total data sample. The fractions that
are fixed in the fit are labelled so.

###  4.4 Four-dimensional fit

To reliably measure the signal fraction $f_{\text{sig}}$ and $I_{X}$
coefficients, a four-dimensional binned maximum-likelihood fit to the total
dataset is performed using the PDF

$\begin{split}P((\cos\theta_{D},\cos\theta_{L},\chi)_{\text{Reco}},\text{BDT})&=f_{\text{sig}}h_{\text{sig}}\\\
&+f_{D_{s}^{+}}h_{D_{s}^{+}}+f_{D^{+}}h_{D^{+}}+f_{D^{0}}h_{D^{0}}+f_{D^{**}}h_{D^{**}}\\\
&+(1-f_{\text{sig}}-f_{D_{s}^{+}}-f_{D^{+}}-f_{D^{0}}-f_{D^{**}})h_{\text{prompt}}\,,\end{split}$
(10)

where $h_{\text{sig}}$ represents the signal PDF, defined in (6). In the fit,
$f_{\text{sig}}$, $f_{D_{s}^{+}}$, and $f_{D^{+}}$ freely vary, as do the
eleven $I_{X}$ coefficients within $h_{\text{sig}}$. The fractions $f_{D^{0}}$
and $f_{D^{**}}$ are fixed, matching the procedure in Ref. PhysRevD.97.072013
. The prompt fraction is constrained such that the fractions sum to unity.

The twelve signal template histograms $h_{I_{X}}$ are created following a
procedure that is essentially identical to that described in Sec. 3, but where
the additional BDT dimension is included and a signal sample containing both
the $\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\bar{\nu}_{\tau}$ and
$\tau^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}\bar{\nu}_{\tau}$ modes is used. For
each background mode $\beta$, a template $h_{\beta}$ is created by filling a
four-dimensional histogram with the
($(\cos\theta_{D},\cos\theta_{L},\chi)_{\text{Reco}}$,BDT) distribution of the
total $\beta$ sample. Large generated samples exceeding one million events are
used to create all of the templates, in order to avoid statistical
uncertainties on the template bin contents. The number of bins
$n_{\text{bins}}$ in each fit dimension is chosen to be equal and is
determined using
$n_{\text{bins}}=\text{Int}\left[\left(N_{\text{sig}}\Big{/}{25}\right)^{\frac{1}{4}}+\frac{1}{2}\right]$,
where $N_{\text{sig}}$ is the anticipated number of signal events in the
sample. The bin boundaries are not uniform, but are chosen such that each bin
is populated with approximately 25 signal events. An alternative binning based
on 50 signal events per bin gives consistent results.

##  5 Expected precision

###  5.1 Hadron collider scenario

The four-dimensional fit is applied to datasets corresponding to three
scenarios: $N_{\text{sig}}=8000$, $N_{\text{sig}}=40,000$ and
$N_{\text{sig}}=100,000$. These are calculated by extrapolating the yield
measured in Ref. PhysRevD.97.072013  to 9 fb-1 (Runs $1+2$), 23 fb-1 (Runs
1–3), and 50 fb-1 (Runs 1–4), as anticipated by LHCb
TheLHCbCollaboration:2320509 . The 23 and 50 fb-1 scenario yield expectations
account for additional improvements in the upgrade LHCb detector performance
relative to LHCb in Run 1 and 2. The signal and background fractions listed in
Tab. 4 are used to create the data sample for each of these cases, maintaining
the same signal purity throughout.

#### 9 fb$\mathbf{{}^{-1}\ (N_{\text{sig}}=8000)}$:

The 4D template fit functions stably in this lowest-statistics case, and the
resulting fit projections are shown in Fig. 9. The signal fraction is measured
to be $f_{\text{sig}}=0.116\pm 0.010$ (8.6% relative uncertainty), which
agrees with the input value $f_{\text{sig}}=0.118$. The $I_{X}$ values are
measured with large uncertainties but remain compatible with the true values,
as shown in Fig. 10. Using the $I_{X}$ results for this sample, the derived
value of $F_{L}(D^{*})$ is $0.368\pm 0.047$, where the uncertainty quoted is
statistical only. It will thus be possible to make a competitive measurement
of $F_{L}(D^{*})$ using the 9 fb-1 data.

Figure 9: One-dimensional projections of the $N_{\text{sig}}=8,000$ binned
fit, where the solid points represent the data and the filled histograms
represent each fit component. The total $B^{0}\to D^{*-}\tau^{+}\nu_{\tau}$
signal, given by the sum of all twelve angular terms, is shown in red.

#### 23 fb$\mathbf{{}^{-1}\ (N_{\text{sig}}=40,000)}$:

Increasing the signal yield by a factor five gives a strong improvement in the
$I_{X}$ measurements, as shown in Fig. 10; the derived value of $F_{L}(D^{*})$
is $0.489\pm 0.018$. The signal fraction is $f_{\text{sig}}=0.117\pm 0.003$
(2.6% relative uncertainty), which has reduced more than a naive
$\sqrt{N_{\text{sig}}}$ scaling due to the larger number of bins in the 23
fb-1 fit relative to the 9 fb-1 fit. Both fits operate with binning schemes
that require approximately 25 signal events per bin, and the increased number
of bins in the 23 fb-1 fit provides greater differentiation between the
$I_{X}$ signal components and the backgrounds.

#### 50 fb$\mathbf{{}^{-1}\ (N_{\text{sig}}=100,000)}$:

With the largest dataset, the $I_{X}$ values are measured with absolute
statistical uncertainties in the range 0.01–0.06, as shown in Fig. 10. The
statistical correlation matrix for the fit is provided in App. C; the other
fit scenarios show a consistent pattern of parameter correlations. The signal
fraction is measured to be $f_{\text{sig}}=0.116\pm 0.002$ (1.5% relative
uncertainty). Given the considerable reduction in $I_{X}$ uncertainty between
the 23 and 50 fb-1 scenarios, it is well motivated to continue performing
measurements of this type during Run 4 of the LHC. This is highlighted by the
derived value of $F_{L}(D^{*})$ value, which is found to be $0.446\pm 0.010$.

###  5.2 Fit stability validation

To demonstrate the stability and accuracy of the three fit scenarios, many
pseudo-experiments (“toys”) based on the fits are run. Using the template PDFs
and the yields from the 9, 23, and 50 fb-1 fits, toy datasets are generated
where the number of events is independently determined in each bin according
to Poisson variations of the bin content. The template fit is applied to each
toy dataset, and pull distributions are created for all freely varying fit
parameters. All pull distributions have mean values close to zero and widths
close to unity, as expected for an unbiased fit returning the appropriate
uncertainties.

Figure 10: Comparison of the signal-only unbinned parametric fit to the truth-
level angles (orange) and the LHCb scenario template fits. The fit
uncertainties dominate the results, but systematic uncertainties are included
in the error bars.

###  5.3 $B$-factory scenario

Complementary to LHCb, the Belle II experiment SHWARTZ2015233 ; Kou:2018nap
can use an anticipated 50 ab-1 dataset to measure the angular coefficients in
$B\to D^{*}\tau\nu_{\tau}$ decays. The $b\bar{b}$ production cross section is
much lower in $e^{+}e^{-}$ collisions compared to $pp$, but the well-defined
initial state and the absence of other tracks in $B{\kern
1.99997pt\overline{\kern-1.99997ptB}{}}$ events give considerable advantages
when reconstructing final states with missing particles. By reconstructing the
second $B$ meson, the momentum of a signal $B^{0}$ meson can be well defined
in spite of the missing neutrino.

To estimate the performance of the template fit at an $e^{+}e^{-}$ experiment,
the signal and background samples generated to emulate the LHCb resolution and
acceptance are reprocessed using the true $B$ meson four-vector in the decay
angle calculations rather than relying on the estimation in Eq. (5). Vertex
and track momentum resolutions are assumed to be similar, so an overall 10-20%
advantage in angular resolution is determined.

The three-prong $\tau$ decay mode has not yet been used in a $B$-factory
semitauonic analysis, so the signal yields at Belle II must be approximated.
Belle have used the $\tau^{+}\to\pi^{+}\nu_{\tau}$ and
$\tau^{+}\to\rho^{+}\nu_{\tau}$ modes with hadronic tagging to measure the
$\tau$ polarisation Hirose:2016wfn . Signal yields of $N(B^{0}\to
D^{*-}\tau^{+}\nu_{\tau})=88\pm 11$ and $N(B^{+}\to
D^{*0}\tau^{+}\nu_{\tau})=210\pm 27$ are reported in 0.77 ab-1 of data.
Assuming near-perfect track finding efficiency at Belle II, such that the
three-prong modes are reconstructed with a similar efficiency as the one-
prong, a total tagged sample of $N_{\text{sig}}(B^{0}+B^{+})\approx 7000$
three-prong events is estimated in 50 ab-1 of Belle II data.

In Ref. Hirose:2016wfn , the Belle Collaboration reported a signal purity of
18.6%. Although the combinatorial background at Belle II and LHCb differ, the
$B$ backgrounds generated in Sec. 4 are still the most important. Thus, a data
sample is created containing 7000 signal events with 18.6% purity, where the
relative background fractions remain the same as those used in Sec. 4.

#### Results for 50 ab-1 of $\mathbf{e^{+}e^{-}}$ data
$\mathbf{(N_{\text{sig}}=7000)}$:

The four-dimensional template fit to the $B$-factory sample is performed in
$((\cos\theta_{D},\cos\theta_{L},\chi)_{\text{Reco}},\text{BDT})$ variable
space, where the decay angles are calculated using the true $B$ meson four-
vector to mimic the benefit of the hadronic tagging. The number of bins in
each dimension is chosen in the same manner as the LHCb scenario fits. The
signal fraction is measured to be $f_{\text{sig}}=0.195\pm 0.014$ (7.0%
relative uncertainty) and is consistent with the input value. The
uncertainties on the $I_{X}$ measurements are compared to the 23 fb-1 LHCb
scenario in Fig. 11. Even though the $B$-factory signal yield is lower, the
overall $I_{X}$ precision is competitive due to the higher purity and
constraint on the initial state from the tagging of the other $B$ decay.

Figure 11: Comparison of absolute $I_{X}$ coefficient statistical
uncertainties in the $N_{\text{sig}}=40,000$ hadron collider template fit
(navy) and the $N_{\text{sig}}=7,000$ $B$-factory fit (green). The average
uncertainties over all $I_{X}$ coefficients are indicated by the dotted lines.

###  5.4 Systematic uncertainties

The dominant systematic uncertainty comes from the assumed accuracy of the
templates used to model the background. Measured branching fractions are used
to define the contribution from each background decay, so these are varied
within their uncertainties to determine the appropriate uncertainty.
Similarly, fixed fractions are used to define the feed-down contribution,
which has not yet been confirmed experimentally and thus a 40% variation
around $f_{D^{**}}=0.11$ is used. Smaller variations in the angular
coefficient measurements are seen when the number of bins in the weighting
procedure is varied from the default $30^{3}$ binning. The total systematic
uncertainties are found to be small relative to the statistical uncertainties,
even in the highest yield case. The systematic uncertainties are shown to
modestly increase the error bars in Fig. 10.

Further systematic effects not quantified here include those due to limited MC
statistics and imperfect simulation of experimental data. The Beeston-Barlow
Barlow:1993dm  method can account for the first effect, while the second must
be addressed through the development of control channels such as $B^{0}\to
D^{*-}D_{s}^{+}$ decays at LHCb or an inclusive study of $D_{(s)}\to 3\pi X$
decays at BESIII.

###  5.5 Determination of $R(D^{*})$

It is possible to convert the measured signal fraction $f_{\text{sig}}$ to a
value of $\mathcal{B}(B^{0}\to D^{*-}\tau^{+}\nu_{\tau})$ and thus $R(D^{*})$
if $\epsilon_{\text{sig}}$, the total signal efficiency, is known. Typically,
$\epsilon_{\text{sig}}$ is calculated from simulated samples that accurately
model detector and selection inefficiencies. However, the simulated sample
used to estimate $\epsilon_{\text{sig}}$ must be generated using a specific
model, and so $\epsilon_{\text{sig}}$ is unavoidably model-dependent. This
leads to an additional systematic uncertainty that must be considered in the
extrapolation to $R(D^{*})$, particularly if measurements of the $I_{X}$
coefficients diverge from their SM expectations.

It should be noted that the model-independent strategy cannot be competitive
with the accuracy with which model-dependent fits can measure $R(D^{*})$. For
example, the statistical uncertainty on $f_{\text{sig}}$ in the 9 fb-1 LHCb
scenario is 8.6%. The statistical uncertainty of a model-dependent fit to the
same dataset may be under 3%, based on an extrapolation of the statistical
uncertainty in Ref. PhysRevD.97.072013 . The inferior precision is due to the
fact that (1) the model-independent fit has twelve parameters to describe the
signal rather than one overall yield, and (2) the angular variables are less
discriminating between signal and background than the model-dependent
variables currently used such as $q^{2}$. To confirm these assertions, a test
is performed using the generated 9 fb-1 LHCb dataset, where all of the $I_{X}$
coefficients are fixed and the signal-background separation is artificially
improved by 20%. In this test, the fit uncertainty on $f_{\text{sig}}$ indeed
reduces from 8.6% to 3.3%.

##  6 Conclusion

A model-independent method for measuring the angular coefficients of $B^{0}\to
D^{*-}\tau^{+}\nu_{\tau}$ decays is demonstrated. Reconstruction bias and
resolution effects caused by the missing neutrino are handled using a binned
template fit to the decay angles, with a BDT classifier included to improve
signal-background separation. A realistic background mixture is introduced,
and the template fit is found to be statistically unbiased and model-
independent even with the current LHCb statistics (Run 1 + 2). Due to the
cleaner $e^{+}e^{-}$ environment and a better-constrained $B$ meson
reconstruction, Belle II should perform competitively despite lower signal
yields.

The template fit is directly applicable to the isospin partner decay
$B^{+}\to{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}^{*0}\tau^{+}\nu_{\tau}$, if the
neutral vector meson can be efficiently reconstructed. By extension, template
angular analysis of the pseudoscalar semitauonic decays $B^{+}\to{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\tau^{+}\nu_{\tau}$ and
$B^{0}\to D^{-}\tau^{+}\nu_{\tau}$ is motivated for experimental reasons.
Here, the decay rate depends only on $\cos\theta_{L}$,

$\frac{d^{2}\Gamma}{dq^{2}d(\cos\theta_{L})}=a+b\cos\theta_{L}+c\cos^{2}\theta_{L}\,,$
(11)

where $a$, $b$, and $c$ are the $q^{2}$-dependent angular coefficients
Becirevic:2019tpx . However, a full angular analysis would enable the
measurement of $I_{X}$ coefficients in the $D^{*-}$ feed-down in addition to
the $(a,b,c)$ coefficients of the pseudoscalar modes.

The template procedure should be applied as a null test to $B\to D^{(*)}l\nu$
($l\in\\{e,\mu\\}$) decays, since they are also governed by Eq. (1). And in
all cases, ${C\\!P}$ conservation, which could be violated if additional NP
processes interfere with the single SM amplitude London:2019wpw , can be
verified by splitting according to the $\tau$ lepton charge. Finally, the
template method is ideal for angular analysis that searches for right-handed
currents in $B_{s}^{0}\to K^{*-}\mu^{+}\nu_{\mu}$ and
$B^{+}\to\rho^{0}\mu^{+}\nu_{\mu}$ decays, which suffer similar complications
from neutrinos in the final state.

##  Appendix A Decay angle definitions

In this work, $\theta_{D}$ is defined as the angle between the direction of
the ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ meson and the
direction opposite that of the $B^{0}$ meson in in the $D^{*-}$ meson rest
frame. The angle $\theta_{L}$ is defined as the angle between the direction of
the $\tau^{+}$ lepton and the direction opposite that of the $B^{0}$ meson in
in the mediator ($W^{+}$) rest frame. The angle $\chi$ is the angle between
the plane containing the $\tau^{+}$ and $\nu_{\tau}$ and the plane containing
the ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ and pion from the
$D^{*-}$ in the $B^{0}$ rest frame. The three decay angles are displayed
graphically in Fig. 12. Explicitly, the decay angles are defined following the
definitions in Ref. Aaij2013

$\displaystyle\cos\theta_{D}$
$\displaystyle=\Big{(}\hat{p}_{D^{0}}^{(D^{*-})}\Big{)}\cdot\Big{(}\hat{p}_{D^{*-}}^{(B^{0})}\Big{)}=\Big{(}\hat{p}_{D^{0}}^{(D^{*-})}\Big{)}\cdot\Big{(}-\hat{p}_{B^{0}}^{(D^{*-})}\Big{)}\,,$
(12) $\displaystyle\cos\theta_{L}$
$\displaystyle=\Big{(}\hat{p}_{\tau^{+}}^{(W^{+})}\Big{)}\cdot\Big{(}\hat{p}_{W^{+}}^{(B^{0})}\Big{)}=\Big{(}\hat{p}_{\tau^{+}}^{(W^{+})}\Big{)}\cdot\Big{(}-\hat{p}_{B^{0}}^{(W^{+})}\Big{)}\,,$
(13) $\displaystyle\cos\chi$
$\displaystyle=\Big{(}\hat{p}_{\tau^{+}}^{(B^{0})}\times\hat{p}_{\nu_{\tau}}^{(B^{0})}\Big{)}\cdot\Big{(}\hat{p}_{D^{0}}^{(B^{0})}\times\hat{p}_{\pi^{-}}^{(B^{0})}\Big{)}$
(14)

where the $\hat{p}_{X}^{(Y)}$ are unit vectors describing the direction of a
particle $X$ in the rest frame of the system $Y$. In every case the particle
momenta are first boosted to the $B^{0}$ rest frame. In this basis, the
angular definition for the $\bar{B}^{0}$ decay is a $CP$ transformation of
that for the $B^{0}$ decay.

Figure 12: Diagram of the three $B^{0}\to D^{*-}\tau^{+}\nu$ decay angles.

##  Appendix B Background models

Mode | EvtGen models
---|---
$B^{0}\to D^{*-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$ |  $B^{0}$ (PHSP)
$B^{0}\to D^{*-}(\omega\to\pi^{+}\pi^{-}\pi^{0})\pi^{+}$ |  $B^{0}$ (PHSP),
|  $\omega$ (OMEGA_DALITZ)
$B^{0}\to D^{*-}\pi^{+}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ |  $B^{0}$ (PHSP)
$B^{+}\to D^{*-}\pi^{+}\pi^{+}\pi^{-}\pi^{+}$ |  $B^{+}$ (PHSP)
Table 5: $B^{+}\to D^{*-}\pi^{+}\pi^{+}\pi^{-}(X)$ modes generated in order to construct the prompt background sample. Mode | EvtGen models
---|---
$B^{0}\to D^{*-}D_{s}^{+}$ |  $B^{0}$ (SVS)
$B^{0}\to D^{*-}(D_{s}^{*+}\to D_{s}^{+}\gamma)$ |  $B^{0}$ (SVV_HELAMP 0.4904 0.0 0.7204 0.0 0.4904 0.0),
|  $D_{s}^{*+}$ (VSP_PWAVE)
$B^{0}\to D^{*-}(D_{s}^{*+}\to D_{s}^{+}\pi^{0})$ |  $B^{0}$ (""),
|  $D_{s}^{*+}$ (VSS)
$B^{0}\to D^{*-}(D_{s1}(2460)^{+}\to D_{s}^{+}\gamma)$ |  $B^{0}$ (SVV_HELAMP 1 0 1 0 1 0),
|  $D_{s1}(2460)^{+}$ (VSP_PWAVE)
$B^{0}\to D^{*-}(D_{s1}(2460)^{+}\to(D_{s}^{*+}\to D_{s}^{+}\gamma)\pi^{0})$ |  $B^{0}$ (""),
|  $D_{s1}(2460)^{+}$ (PHSP),
|  $D_{s}^{*+}$ (VSP_PWAVE)
$B^{+}\to(D_{1}(2420)^{0}\to D^{*-}\pi^{+})D_{s}^{+}$ |  $B^{+}$ (SVS),
|  $D_{1}(2420)^{0}$ (VVS_PWAVE 0 0 0 0 1 0)
$B^{+}\to(D_{1}(2420)^{0}\to D^{*-}\pi^{+})(D_{s}^{*+}\to D_{s}^{+}\gamma)$ |  $B^{+}$ (SVV_HELAMP 0.48 0.0 0.734 0.0 0.48 0.0),
|  $D_{1}(2420)^{0}$ (""),
|  $D_{s}^{*+}$ (VSP_PWAVE)
$B^{+}\to(D_{1}(2420)^{0}\to D^{*-}\pi^{+})(D_{s}^{*+}\to D_{s}^{+}\pi^{0})$ |  $B^{+}$ (""),
|  $D_{1}(2420)^{0}$ (""),
|  $D_{s}^{*+}$ (VSS)
$B^{+}\to(D_{1}(2420)^{0}\to D^{*-}\pi^{+})(D_{s1}(2460)^{+}\to D_{s}^{+}\gamma)$ |  $B^{+}$ (SVV_HELAMP 1.0 0.0 1.0 0.0 1.0 0.0),
|  $D_{1}(2420)^{0}$ (VVS_PWAVE 0 0 0 0 1 0),
|  $D_{s1}(2460)^{+}$ (VSP_PWAVE)
$B^{+}\to(D_{1}(2420)^{0}\to D^{*-}\pi^{+})(D_{s1}(2460)^{+}\to(D_{s}^{*+}\to D_{s}^{+}\gamma)\pi^{0})$ |  $B^{+}$ (""),
|  $D_{1}(2420)^{0}$ (""),
|  $D_{s1}(2460)^{+}$ (PHSP),
|  $D_{s}^{*+}$ (VSP_PWAVE)
Table 6: Background modes generated to construct the $B\to D^{*-}D_{s}^{+}(X)$ sample. In all cases, $D^{*-}\to{{\kern 1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\pi^{-}$ is generated according to the VSS model, and ${{\kern 1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\to K^{+}\pi$ is generated according to phase space. Where indicated, "" means that the same model was used as in the previous row. Mode | EvtGen models
---|---
$D_{s}^{+}\to(\eta\to\pi^{+}\pi^{-}\pi^{0})\pi^{+}$ |  $D_{s}^{+}$ (PHSP),
|  $\eta$ (ETA_DALITZ)
$D_{s}^{+}\to(\omega\to\pi^{+}\pi^{-}\pi^{0})\pi^{+}$ |  $D_{s}^{+}$ (SVS),
|  $\omega$ (OMEGA_DALITZ)
$D_{s}^{+}\to(\eta\to\pi^{+}\pi^{-}\pi^{0})(\rho^{+}\pi^{+}\pi^{0})$ |  $D_{s}^{+}$ (SVS),
|  $\eta$ (ETA_DALITZ),
|  $\rho^{+}$ (VSS)
$D_{s}^{+}\to(\omega\to\pi^{+}\pi^{-}\pi^{0})(\rho^{+}\pi^{+}\pi^{0})$ |  $D_{s}^{+}$ (PHSP),
|  $\omega$ (OMEGA_DALITZ),
|  $\rho^{+}$ (VSS)
$D_{s}^{+}\to(\rho^{0}\to\pi^{+}\pi-)(\rho^{0}\to\pi^{+}\pi-)\pi^{+}$ |  $D_{s}^{+}$ (PHSP),
|  $\rho^{0}$ (VSS)
$D_{s}^{+}\to(\omega\to\pi^{+}\pi^{-}\pi^{0})\pi^{+}\pi^{+}\pi^{-}$ |  $D_{s}^{+}$ (PHSP),
|  $\omega$ (OMEGA_DALITZ)
$D_{s}^{+}\to(\eta^{\prime}\to(\eta\to\pi^{+}\pi^{-}\pi^{0})\pi^{+}\pi^{-})\pi^{+}$ |  $D_{s}^{+}$ (PHSP),
|  $\eta^{\prime}$ (PHSP),
|  $\eta$ (ETA_DALITZ)
$D_{s}^{+}\to(\eta^{\prime}\to(\rho^{0}\to\pi^{+}\pi^{-})\gamma)\pi^{+}$ |  $D_{s}^{+}$ (""),
|  $\eta^{\prime}$ (SVP_HELAMP 1.0 0.0 1.0 0.0),
|  $\rho^{0}$ (VSS)
$D_{s}^{+}\to(\eta^{\prime}\to(\eta\to\pi^{+}\pi^{-}\pi^{0})\pi^{+}\pi^{-})(\rho^{+}\to\pi^{+}\pi^{0})$ |  $D_{s}^{+}$ (SVS),
|  $\eta^{\prime}$ (PHSP),
|  $\eta$ (ETA_DALITZ),
|  $\rho^{+}$ (VSS)
$D_{s}^{+}\to(\eta^{\prime}\to(\rho^{0}\to\pi^{+}\pi^{-})\gamma)(\rho^{+}\to\pi^{+}\pi^{0})$ |  $D_{s}^{+}$ (""),
|  $\eta^{\prime}$ (SVP_HELAMP 1.0 0.0 1.0 0.0),
|  $\rho^{0,+}$ (VSS)
Table 7: $D_{s}^{+}$ decays generated for constructing the $B\to D^{*-}D_{s}^{+}(X)$ sample. Where indicated, "" means that the same model was used as in the previous row. Mode | EvtGen models
---|---
$B^{-}\to D^{*-}D^{0}\bar{K}^{0}$ |  $B^{-}$ (PHSP)
$B^{-}\to D^{*-}(D^{*0}\to D^{0}\pi^{0})\bar{K}^{0}$ |  $B^{-}$ (PHSP),
|  $D^{*0}$ (VSS)
$B^{-}\to D^{*-}(D^{*0}\to D^{0}\gamma)\bar{K}^{0}$ |  $B^{-}$ (PHSP),
|  $D^{*0}$ (VSP_PWAVE)
$B^{0}\to D^{*-}D^{0}K^{+}$ |  $B^{0}$ (PHSP)
$B^{0}\to D^{*-}(D^{*0}\to D^{0}\pi^{0})K^{+}$ |  $B^{0}$ (PHSP),
|  $D^{*0}$ (VSS)
$B^{0}\to D^{*-}(D^{*0}\to D^{0}\gamma)K^{+}$ |  $B^{0}$ (PHSP),
|  $D^{*0}$ (VSP_PWAVE)
$B^{0}\to D^{*-}(D^{*+}\to D^{0}\pi^{+})\bar{K}^{0}$ |  $B^{0}$ (PHSP),
|  $D^{*+}$ (VSS)
$B^{0}\to D^{*-}D^{+}\bar{K}^{0}$ |  $B^{0}$ (PHSP)
$B^{-}\to D^{*-}D^{+}K^{-}$ |  $B^{0}$ (PHSP)
Table 8: Background modes generated to construct the $B\to D^{*-}D^{0}(X)$ and $B\to D^{*-}D^{+}(X)$ samples. In all cases, $D^{*-}\to{{\kern 1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\pi^{-}$ is generated according to the VSS model, and ${{\kern 1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\to K^{+}\pi$ is generated according to phase space. Mode | EvtGen models
---|---
$D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}$ |  $D^{0}$ (PHSP)
$D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}\pi^{0}$ |  $D^{0}$ (PHSP)
$D^{+}\to K_{s}^{0}\pi^{+}\pi^{+}\pi^{-}$ |  $D^{+}$ (PHSP)
$D^{+}\to\pi^{+}\pi^{+}\pi^{-}\pi^{0}$ |  $D^{+}$ (PHSP)
Table 9: $D^{0,+}$ decays generated for constructing the $B\to D^{*-}D^{0,+}(X)$ sample. Mode | EvtGen models
---|---
$B^{+}\to(D_{1}(2420)^{0}\to D^{*-}\pi^{+})\tau^{+}\nu_{\tau}$ |  $B^{+}$ (ISGW2),
|  $D_{1}(2420)^{0}$ (VVS_PWAVE 0.0 0.0 0.0 0.0 1.0 0.0)
$B^{+}\to(D_{2}^{*}(2460)^{0}\to D^{*-}\pi^{+})\tau^{+}\nu_{\tau}$ |  $B^{+}$ (ISGW2),
|  $D_{2}^{*}(2460)^{0}$ (TVS_PWAVE 0.0 0.0 1.0 0.0 0.0 0.0)
Table 10: $B^{+}\to D^{**0}\tau^{+}\nu_{\tau}$ modes generated for
constructing the feed down background sample. As these modes have not been
observed, they are added together in equal proportion to create the total
sample.

##  Appendix C Correlation matrix

| $I_{1s}$ | $I_{2c}$ | $I_{2s}$ | $I_{3}$ | $I_{4}$ | $I_{5}$ | $I_{6c}$ | $I_{6s}$ | $I_{7}$ | $I_{8}$ | $I_{9}$ | $f_{\text{sig}}$ | $f_{D_{s}^{+}}$ | $f_{D^{+}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$I_{1s}$ |  -1.00 | -0.26 |  -0.86 |  -0.21 | -0.17 |  -0.00 |  -0.03 |  -0.23 |  -0.00 | -0.01 | -0.00 |  -0.16 |  -0.45 | -0.26
$I_{2c}$ | -0.26 |  -1.00 | -0.35 | -0.26 | -0.07 | -0.07 | -0.04 |  -0.01 |  -0.00 |  -0.01 |  -0.02 |  -0.13 | -0.08 |  -0.14
$I_{2s}$ |  -0.86 | -0.35 |  -1.00 |  -0.02 | -0.05 |  -0.11 |  -0.06 |  -0.10 |  -0.00 | -0.01 | -0.02 |  -0.08 |  -0.20 | -0.04
$I_{3}$ |  -0.21 | -0.26 |  -0.02 |  -1.00 | -0.19 | -0.03 | -0.21 | -0.22 | -0.01 |  -0.03 | -0.00 | -0.14 |  -0.39 | -0.44
$I_{4}$ | -0.17 | -0.07 | -0.05 | -0.19 |  -1.00 |  -0.05 | -0.01 |  -0.04 | -0.00 |  -0.01 |  -0.01 | -0.45 | -0.01 | -0.17
$I_{5}$ |  -0.00 | -0.07 |  -0.11 | -0.03 |  -0.05 |  -1.00 | -0.09 | -0.36 | -0.01 | -0.00 |  -0.01 | -0.08 |  -0.01 |  -0.04
$I_{6c}$ |  -0.03 | -0.04 |  -0.06 | -0.21 | -0.01 | -0.09 |  -1.00 | -0.13 |  -0.01 | -0.03 | -0.00 |  -0.08 |  -0.03 |  -0.03
$I_{6s}$ |  -0.23 |  -0.01 |  -0.10 | -0.22 |  -0.04 | -0.36 | -0.13 |  -1.00 | -0.01 | -0.00 |  -0.00 |  -0.20 | -0.05 | -0.00
$I_{7}$ |  -0.00 |  -0.00 |  -0.00 | -0.01 | -0.00 | -0.01 |  -0.01 | -0.01 |  -1.00 | -0.07 | -0.35 | -0.00 |  -0.01 | -0.00
$I_{8}$ | -0.01 |  -0.01 | -0.01 |  -0.03 |  -0.01 | -0.00 | -0.03 | -0.00 | -0.07 |  -1.00 | -0.05 | -0.01 | -0.01 |  -0.00
$I_{9}$ | -0.00 |  -0.02 | -0.02 | -0.00 |  -0.01 |  -0.01 | -0.00 |  -0.00 | -0.35 | -0.05 |  -1.00 |  -0.01 |  -0.00 | -0.01
$f_{\text{sig}}$ |  -0.16 |  -0.13 |  -0.08 | -0.14 | -0.45 | -0.08 |  -0.08 |  -0.20 | -0.00 | -0.01 |  -0.01 |  -1.00 |  -0.08 | -0.24
$f_{D_{s}^{+}}$ |  -0.45 | -0.08 |  -0.20 |  -0.39 | -0.01 |  -0.01 |  -0.03 | -0.05 |  -0.01 | -0.01 |  -0.00 |  -0.08 |  -1.00 | -0.79
$f_{D^{+}}$ | -0.26 |  -0.14 | -0.04 | -0.44 | -0.17 |  -0.04 |  -0.03 | -0.00 | -0.00 |  -0.00 | -0.01 | -0.24 | -0.79 |  -1.00
Table 11: Statistical correlation matrix for the $N=100,000$ hadron collider
binned fit. All other fits show a consistent pattern of correlations.

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## Acknowledgements

We wish to thank Daniel Craik, Dean Robinson, Michele Papucci, and Tom Blake
for their advice and assistance. We are particularly indebted to Marco Fedele
for several fruitful exchanges. The work is supported by the Science and
Technology Facilities Council (STFC, United Kingdom) and the Centre National
de la Recherche Scientifique (CNRS, France).


# Enhanced strange metallicity from Coulomb repulsion

Andrew Hardy  Department of Physics, University of Toronto, 60 St. George
Street, Toronto, ON, M5S 1A7 Canada  Center for Computational Quantum Physics,
Flatiron Institute, New York, New York 10010, USA  Olivier Parcollet  Center
for Computational Quantum Physics, Flatiron Institute, New York, New York
10010, USA  Université Paris-Saclay, CNRS, CEA, Institut de Physique
Théorique, 91191, Gif-sur-Yvette, France  Antoine Georges  Collège de France,
11 place Marcelin Berthelot, 75005 Paris, France  Center for Computational
Quantum Physics, Flatiron Institute, New York, New York 10010, USA  CPHT,
CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay,
91128 Palaiseau, France  DQMP, University of Geneva, 24 quai Ernest-Ansermet,
1211 Geneva, Switzerland  Aavishkar A. Patel  Center for Computational Quantum
Physics, Flatiron Institute, New York, New York 10010, USA

###### Abstract

We solve a model of electrons with Hubbard-$U$ Coulomb repulsion and a random
Yukawa coupling to a two-dimensional bosonic bath, using an extended dynamical
mean field theory scheme. Our model exhibits a quantum critical point, at
which the repulsive component of the electron interactions strongly enhances
the effects of the quantum critical bosonic fluctuations on the electrons,
leading to a breakdown of Fermi liquid physics and the formation of a strange
metal with ‘Planckian’ ($\mathcal{O}(k_{B}T/\hbar)$) quasiparticle decay rates
at low temperatures $T\rightarrow 0$. Furthermore, the eventual Mott
transition that occurs as the repulsion is increased seemingly bounds the
maximum decay rate in the strange metal. Our results provide insight into low-
temperature strange metallicity observed in proximity to a Mott transition, as
is observed, for instance, in recent experiments on certain moiré materials.

††preprint: APS/123-QED

Introduction: Understanding the transport properties of itinerant electron
systems proximate to a quantum critical point (QCP) remains a major open
problem in condensed matter physics. These systems could possibly describe the
enigmatic strange metals, which have linear-in-temperature resistivity at low
temperatures $T\rightarrow 0$ [1]. Strange metal behavior has been observed in
a wide variety of strongly correlated systems [2, 3, 4, 5, 6, 7, 8], the most
iconic of these being proximate to a putative two-dimensional (2D)
antiferromagnetic critical point, as seen in Fe- and also possibly CuO- based
superconductors [2, 3, 4, 9, 10, 11]. Since strange metals are often the
parent states out of which unconventional pairing emerges, understanding them
is a prerequisite for a comprehensive theory of correlated-electron
superconductivity.

The properties of low-temperature strange metals are well described by the
“marginal Fermi liquid” (MFL) phenomenology [12, 13, 14], in which a
particular scaling form of the electron self energy gives rise to the observed
linear-in-energy quasiparticle decay and transport scattering rates. Recently,
simple models of two-dimensional metallic QCPs [15, 16] with spatially random
fermion-boson Yukawa interactions [17, 18], were proposed as a generic
mechanism for the MFL phenomenology of strange metals, within a perturbative
Eliashberg approach.

The Eliashberg approach utilized in Refs. [17, 18] is justifiable only in the
limit of a large number of fermion flavors [19], which eliminates important
physical features of realistic SU(2) electrons, such as their Hubbard-$U$
Coulomb repulsion. Recent observations of quantum criticality and strange
metals [8, 20] proximate to Mott transitions in 2D transition metal
dichalcogenide moiré bilayers [21, 22, 23] have underscored the need to
incorporate the effects of such repulsion into models of strange metal QCPs.
In this Letter, we address this issue using a model of SU(2) electrons with
both Hubbard-$U$ Coulomb repulsion and spatially random Yukawa interactions
mediated by 2D bosons, that we solve using an extended dynamical mean field
theory (EDMFT) method [24, 25, 26].

Using this model, we find that not only do the MFL and strange metal behaviors
associated with the QCP survive into non-perturbative regimes, but that they
are strongly enhanced by the repulsive interactions, leading to ‘Planckian’
quasiparticle decay rates $\sim\alpha_{0}k_{B}T/\hbar$ [27], where
$\alpha_{0}\sim\mathcal{O}(1)$ and reaches its largest value just before the
repulsion-driven Mott transition.

Model: We consider the following Hamiltonian $H=H_{c}+H_{Y}$ that we will use
to set up a single-site EDMFT impurity problem:

$\begin{gathered}H_{c}=-\sum_{ij,\sigma}\left(t_{ij}+\mu\delta_{ij}\right)c_{i\sigma}^{\dagger}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow},\\\
H_{Y}=\sum_{ir}\frac{{g^{\prime}}_{ir}}{\sqrt{V}}(-1)^{i}\vec{S}_{i}\cdot\vec{\phi}_{r},~{}~{}\ll{g^{\prime}}_{ir}g^{\prime}_{jr^{\prime}}\gg={g^{\prime}}^{2}\delta_{ij}\delta_{rr^{\prime}}.\end{gathered}$
(1)

Here, $i,j,r$ label the sites of a 2D lattice with volume $V$ occupied by the
SU(2) electrons and the $3$-component bosonic field $\vec{\phi}$ that
represents antiferromagnetic (Néel) order. $H_{c}$ describes the conventional
Hubbard model, which is augmented by the Yukawa term $H_{Y}$ that couples the
electron spin $\vec{S}_{i}$ at each site $i$ to all sites $r$ of the 2D
bosonic bath via spatially-random interactions $g^{\prime}_{ir}$, which are
independent random Gaussian variables with zero mean and variance
${g^{\prime}}^{2}$. The 2D bosonic bath has a conventional imaginary-time
($\tau$) Lagrangian

$\begin{gathered}\mathcal{L}_{\vec{\phi}}=\sum_{r}\frac{1}{2}\left[|\partial_{\tau}\vec{\phi}_{r\tau}|^{2}+|\nabla\vec{\phi}_{r\tau}|^{2}+m_{b}^{2}|\vec{\phi}_{r\tau}|^{2}\right]\\\
-i\sum_{r}\lambda_{r\tau}\left[|\vec{\phi}_{r}(\tau)|^{2}-3\kappa\right],\end{gathered}$
(2)

which contains the standard kinetic and mass terms, as well as a boson self-
interaction that is expressed in the non-linear sigma model fashion using the
Lagrange multiplier field $\lambda$ [28].

In the thermodynamic limit of large volume $V$, we obtain a saddle point by
averaging over the random coupling $g^{\prime}_{ir}$ using replicas
(Supplementary Information), as in the Sachdev-Ye-Kitaev models [29]. This
introduces the bilocal fields
$D(\tau-\tau^{\prime})=(1/(3V))\sum_{r}\langle\vec{\phi}_{r\tau}\cdot\vec{\phi}_{r\tau^{\prime}}\rangle$
and
$\Pi(\tau-\tau^{\prime})=({g^{\prime}}^{2}/(3V))\sum_{i}\langle\vec{S}_{i\tau}\cdot\vec{S}_{i\tau^{\prime}}\rangle$.
Here $D(\tau-\tau^{\prime})$ is the local (i.e. spatially averaged) propagator
of the bosonic bath, and $\Pi(\tau-\tau^{\prime})$ is the Lagrange multiplier
field that enforces its definition and functions as the spatially-uniform
self-energy of the bosons $\vec{\phi}$. The bilocal field
$D(\tau-\tau^{\prime})$ mediates a dynamical local spin-spin interaction for
the electrons, which enters into the effective local impurity action
$S_{\mathrm{imp}}$ of the EDMFT scheme [24, 25, 26]:

$\displaystyle S_{\mathrm{imp}}=\int
d\tau\left[\sum_{\sigma}c_{\sigma\tau}^{\dagger}\left(\partial_{\tau}-\mu\right)c_{\sigma\tau}+Un_{\uparrow\tau}n_{\downarrow\tau}\right]+$
(3) $\displaystyle\int d\tau
d\tau^{\prime}\left[\sum_{\sigma}\Delta\left(\tau-\tau^{\prime}\right)c_{\sigma\tau}^{\dagger}c_{\sigma\tau^{\prime}}-\frac{{g^{\prime}}^{2}D\left(\tau-\tau^{\prime}\right)}{2}\vec{S}_{\tau}\cdot\vec{S}_{\tau^{\prime}}\right].$

The hybridization $\Delta(\tau-\tau^{\prime})$ maps the local problem to the
full dispersing lattice [30]; we use the hybridization function
$\Delta(\tau-\tau^{\prime})=t^{2}G(\tau-\tau^{\prime})$, which is a generic
choice for metallic systems with local propagator $G$ 111This hybridization
function is formally exact only for a Bethe lattice, or for all-to-all and
random $t_{ij}$ [30]. However, since we are considering metallic systems with
a finite and non-singular density of states, it should continue to produce the
correct qualitative behavior even for a 2D system., and is computed self-
consistently [32]. The dynamical interaction $D(\tau-\tau^{\prime})$ is also
determined self-consistently by solving the bosonic problem with effective
action $S_{\mathrm{bath}}=\int d\tau\mathcal{L}_{\vec{\phi}}-(1/2)\int d\tau
d\tau^{\prime}\Pi(\tau-\tau^{\prime})\sum_{r}\vec{\phi}_{r\tau}\cdot\vec{\phi}_{r\tau^{\prime}}$,
with
$\Pi(\tau-\tau^{\prime})=({g^{\prime}}^{2}/3)\langle\vec{S}_{\tau}\cdot\vec{S}_{\tau^{\prime}}\rangle$
determined from the impurity problem. Finally, the electron self energy
$\Sigma$ is given by
$\Sigma(i\omega_{n})=i\omega_{n}+\mu-\Delta(i\omega_{n})-G^{-1}(i\omega_{n})$,
where $\omega_{n}\equiv(2n+1)\pi T$ are fermionic Matsubara frequencies.
Unless otherwise specified, we will work with $t=1$, and non-perturbative
interaction strengths $U=2t$, ${g^{\prime}}^{2}=2t$. The impurity problem is
solved using a continuous time Quantum Monte Carlo based on a double expansion
in hybridization and spin-spin interaction (CT-SEG) [33, 34, 35].

We solve the bosonic problem by taking the saddle point of $S_{\mathrm{bath}}$
with respect to $\lambda$ 222This approximation is valid for 3-component
bosons in 2D at or above their upper critical dimension, requiring a dynamical
critical exponent $z\geq 2$ [28], which we shall subsequently show is the case
here. We also approximate the lattice boson dispersion using a quadratic
dispersion and a cutoff $\Lambda_{b}$, which does not affect the low-energy
properties that we will be interested in. This yields the convenient analytic
expression

$D(i\Omega_{n})=\frac{1}{4\pi}\ln(\frac{\tilde{m}^{2}_{b}+\Omega_{n}^{2}-\Pi(i\Omega_{n})+\Lambda_{b}^{2}}{\tilde{m}^{2}_{b}+\Omega_{n}^{2}-\Pi(i\Omega_{n})}),$
(4)

with bosonic Matsubara frequencies $\Omega_{n}\equiv 2n\pi T$, and the
constraint $D(\tau=0)=\kappa$ that is satisfied by adjusting
$\tilde{m}_{b}^{2}\equiv m_{b}^{2}-i\lambda$.

In order to probe strange metal physics, we need to tune the system to a QCP
at which the bosons are gapless and coupling to them can destroy the electron
quasiparticles [37]. The value of $\kappa$ controls the value of
$\tilde{m}_{b}^{2}$, and therefore the value of the boson gap
$m^{2}=\tilde{m}_{b}^{2}-\Pi(0)$. The QCP can then be identified as a critical
$\kappa=\kappa_{c}$, for which $m^{2}(T\rightarrow 0,\kappa\leq\kappa_{c})=0$
but $m^{2}(T\rightarrow 0,\kappa>\kappa_{c})\neq 0$. Fig. 1 shows our
numerical result for $m^{2}$ as a function of $\kappa$ and $T$, which
establishes the existence of such a QCP in the bosonic sector. The frequency
dependent part of $\Pi(i\Omega_{n})$ is $\sim|\Omega_{n}|^{y}$ with $y<1$ at
low frequencies (Supplementary Information), which implies a dynamical
critical exponent of $z>2$ for the quadratically dispersing 2D bosons.

Figure 1: Plot of the effective boson mass $m^{2}=m^{2}_{b}-\Pi(0)$ as a
function of the tuning parameter $\kappa$. The dashed line indicates a “phase
boundary” beyond which the bosons develop quasi-long range order and $m^{2}$
becomes small. The phase boundary extrapolates to a QCP ($\kappa=\kappa_{c}$)
as $T\rightarrow 0$.

Marginal Fermi liquid: We determine the fate the electron quasiparticles upon
approach to such a QCP by analyzing the electron self-energy
$\Sigma(i\omega_{n})$. In a Fermi liquid (FL) with well-defined quasiparticles
with quasiparticle residue $Z$, $-\Sigma(i\omega_{n})\approx
i\omega_{n}(Z^{-1}-1)+iA(\pi^{2}T^{2}-\omega_{n}^{2})$, leading to a
quasiparticle decay rate that scales quadratically in $\omega,~{}T$ upon
analytic continuation to real frequencies $\omega$. At the first Matsubara
frequency $\omega_{1}=\pi T$, $-\mathrm{Im}[\Sigma(i\omega_{1})]\approx
T(Z^{-1}-1)$ [38]. In contrast, the breakdown of FL physics and the
destruction of quasiparticles at low energies is characterized by a non-Fermi
liquid behavior of $-\mathrm{Im}[\Sigma(i\omega_{n}\gg T)]\sim\omega_{n}^{x}$,
with $x<1$. Additionally, at the first Matsubara frequency $\omega_{1}=\pi T$,
$-\mathrm{Im}[\Sigma(i\omega_{1})]\sim T^{x}$ [37]. The special case of
$x\rightarrow 1^{-}$ is termed a MFL [12], where
$-\mathrm{Im}[\Sigma(i\omega_{n}\gg
T)]\sim\omega_{n}\ln(\Lambda_{f}/\omega_{n})$ and
$-\mathrm{Im}[\Sigma(i\omega_{1})]\sim T\ln(\Lambda_{f}/T)$. For real
frequencies, MFL behavior leads to the linear-in-$\omega$ and linear-in-$T$
quasiparticle decay rates observed experimentally in strange metals [14].

Figure 2: The self-energy -$\operatorname{Im}[\Sigma(i\omega_{n})]$ upon
approaching criticality. Lighter colors indicate increasing $\kappa$ and
correspondingly, a decreasing $m^{2}$. The inset shows the fit to Eq. 5. The
points lie on the $y=x$ line, indicating an excellent fit.

We expect that $\Sigma(i\omega_{n})$ will cross over from FL to MFL behavior
as the QCP is approached by tuning $\kappa$. We therefore consider a
‘universal’ scaling ansatz

$-\operatorname{Im}[\Sigma(i\omega_{n})]=\gamma\omega_{n}\ln(\frac{\Lambda_{f}}{M+|\omega_{n}|})+\tau^{-1};$
(5)

here the MFL scaling is modified by an infrared cutoff $M$, which accounts for
the presence of a finite boson gap and which should vanish as $T\rightarrow 0$
at the QCP. Away from the QCP, $M>0$, and $-\mathrm{Im}[\Sigma(i\omega_{n})]$
is an analytic function of $\omega_{n}$, as in a FL. The quasiparticle decay
rate arising from thermal fluctuations, which should vanish at $T=0$, is given
by $-\text{Im}[\Sigma(0)]=\tau^{-1}$. The ultraviolet cutoff $\Lambda_{f}$ and
the effective coupling constant $\gamma$ should be independent of $m^{2}$.
This scaling form is motivated by the perturbative calculation of
$\Sigma(i\omega_{n})$ (Supplementary Information). As shown in Fig. 2, we
obtain an excellent fit to Eq. 5 for sizeable $g^{\prime},~{}U$, thereby
establishing MFL behavior in a non-perturbative regime.

Figure 3: a) Plot of the fermion quasiparticle residue $Z$ as a function of
the effective boson mass $m^{2}=\tilde{m}^{2}_{b}-\Pi(0)$. The fit lines are
of the form $Z=(1+a_{1}\log(a_{2}/m^{2}))^{-1}$ b) The IR cut-off $M$ in Eq.5,
which is linearly proportional to $m^{2}$. The fit lines are linear fits. c)
$-\imaginary[\Sigma(\omega_{1})]$ as a function of $T$, showing a deviation
from the expected linear FL behavior upon approaching the QCP (reducing
$m^{2}$), with the temperature scale below which
$-\mathrm{Im}[\Sigma(i\omega_{1})]\propto T$ reducing as the QCP is
approached. The fit lines are linear fits. d) The quasiparticle decay rate
$\tau^{-1}(T)$, which transitions from a $T^{2}$ to a linear-in-$T$ dependency
upon approaching the QCP. The fit lines are of the form
$\tau^{-1}=\alpha_{0}T+\alpha_{1}T^{2}$. The values of $m^{2}$ shown in panels
c) and d) correspond to the lowest temperatures.

The behavior of various parameters associated with Eq. 5 is shown in Fig. 3.
Fig. 3a) shows the quasiparticle residue $Z$, computed through the extracted
parameters as
$Z=(1-\partial_{\omega_{n}}\imaginary(\Sigma(i\omega_{n}))^{-1}|_{\omega_{n}\rightarrow
0}=(1+\gamma\log(\lambda_{f}/M))^{-1}$. The decrease of $Z$ as $m^{2}$ is
reduced indicates the destruction of quasiparticles as the QCP is approached.
Fig. 3b) shows that the IR cutoff $M$ decreases linearly with $m^{2}$,
extrapolating to $M\rightarrow 0$ as $m^{2}\rightarrow 0$, thereby
establishing quantum critical behavior in the electron sector as well. An
independent check that does not involve Eq. 5 is shown in Fig. 3c), where we
observe a clear deviation from the FL scaling of
$-\imaginary[\Sigma(i\omega_{1},T)]\propto T$ as $m^{2}$ is reduced and the
QCP is approached. The behavior of $\gamma$ and $\Lambda_{f}$ is provided in
the Supplementary Information, where we find that they are rather insensitive
to $m^{2}$.

The quasiparticle decay rate $\tau^{-1}$ is shown in Fig. 3d). The
$T$-dependence of $\tau^{-1}$ is fit well by
$\tau^{-1}(T)=\alpha_{0}T+\alpha_{1}T^{2}$. As $m^{2}$ is decreased and the
QCP is approached, The coefficient of the $T$-linear term $\alpha_{0}$ grows
by orders of magnitude, and $\tau^{-1}(T)$ crosses over to a linear scaling
with $T$ with $\alpha_{0}\sim\mathcal{O}(1)$, which is reminiscent of the
‘Planckian’ decay rates observed in experiments on strange metals [27]. Since
the $T$-linear quasiparticle decay rate at the QCP arises out of spatially
random interactions in Eq. 1 that do not conserve momentum, we expect it to
translate into a $T$-linear transport scattering rate and lead to strange
metallic $T$-linear resistivity at the QCP, following the reasoning of Ref.
[17].

We finally turn to the main result of this Letter. We consider
$\Sigma(i\omega_{n})$ for systems tuned to criticality with increasing values
of $U$. Fig. 4 a) shows $-\mathrm{Im}[\Sigma(i\omega_{n})]$ with its value at
$g^{\prime}=0$ subtracted off, thereby isolating the contribution of the
critical bosonic fluctuations to the self energy. This contribution increases
significantly with increasing $U$, leading to a strong enhancement of strange
metal and MFL behavior as shown in Fig. 4b, c. The slope $\alpha_{0}$ of the
$T$-linear quasiparticle decay rate $\tau^{-1}(T)\approx\alpha_{0}T$ increases
rapidly as $U$ is increased, as does the coefficient $\gamma$ of the MFL
frequency dependence in Eq. 5.

Increasing $U$ beyond $U\approx 2.5t$ (at ${g^{\prime}}^{2}=2t$) leads to a
Mott transition before criticality can be reached at low $T$ (Supplementary
Information), which seemingly establishes a bound on the maximum value of
$\alpha_{0}$, that is somewhat larger than $1$. This is reminiscent of the
“stability bounds” on $T$-linear decay rates noted empirically in models of
systems with well-defined quasiparticles and attractive electron-phonon
interactions [39], but arises here in a different context of quantum critical
systems with poorly defined quasiparticles and with a combination of
attractive (Yukawa) and repulsive (Hubbard) interactions.

Figure 4: a) The contribution of the critical bosonic fluctuations to
$-\mathrm{Im}[\Sigma(i\omega_{n})]$ for different values of $U$ and fixed
$g^{\prime}$. b) The slope $\alpha_{0}$ of the $T$-linear quasiparticle decay
rate as a function of $U$. c) The coefficient $\gamma$ of the fit to Eq. 5 as
a function of $U$, which determines the strength of the MFL.

Outlook: We have solved a model of electrons with a spatially random Yukawa
coupling to a 2D bosonic bath. We find a quantum critical point that gives
rise to marginal Fermi liquid and strange metal behaviors, but in a non-
perturbative regime unlike in previous works that utilized Eliashberg theory.
Our most striking result is that these behaviors are strongly enhanced by the
Hubbard-$U$ repulsive interactions between the electrons. This Coulomb
repulsion allows for large $T$-linear quasiparticle decay rates in the strange
metal, that are around the ‘Planckian’ value of $k_{B}T/\hbar$, with the
largest values occuring just before the Mott transition. The amplification by
Coulomb repulsion of the quantum critical bosonic fluctuations that lead to
strange metal behavior might have implications for strange metals recently
observed near Mott transitions in moiré materials [8].

Several interesting directions exist for future studies: recent work [40] has
studied superconductivity arising from random Yukawa interactions in an
Eliashberg approach without repulsion. It would be interesting to examine if
the repulsion-enhanced quantum critical bosonic fluctuations described in this
Letter can lead to an amplification of superconductivity that exceeds any
suppression by the bare repulsion itself. It would also be interesting to
investigate if these models can produce crossovers to ‘bad metals’ with
$\mathcal{O}(\hbar/e^{2})$ resistivities at high temperatures, and to examine
the transport properties of such regimes.

Random Yukawa models have also been studied recently without assumptions of
disorder self-averaging [41, 42], in which it was found that the quantum
critical point is replaced by a gapless boson phase that also gives rise to
strange metal behavior. It would be interesting to explore the effects of the
Hubbard-$U$ repulsion in such a scenario using multi-site DMFT methods.
Finally, the dynamical cluster approximation [43], which restores momentum-
space structure and thereby a finite average antiferromagnetic coupling in the
effective impurity problem, can be applied to our problem to explore the
interplay between pseudogap and strange metal phases.

###### Acknowledgements.

Acknowledgments A. H. acknowledges support from a NSERC Graduate Fellowship
(PGS-D) and a pre-doctoral fellowship at the Flatiron Institute while this
work was being completed. The algorithms used in this study were implemented
using the TRIQS library [44, 45, 46]. The Flatiron Institute is a division of
the Simons Foundation.

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ξin 1,2,3,4 See pages ξ of Supp.pdf