11institutetext:  Aix Marseille Univ, CNRS, CNES, Laboratoire d’Astrophysique
de Marseille, Marseille, France
∗11email<EMAIL_ADDRESS>22institutetext: Department of Earth,
Atmospheric and Planetary Sciences, MIT, 77 Massachusetts Avenue, Cambridge,
MA 02139, USA 33institutetext: Astrophysics Research Centre, Queen’s
University Belfast, BT7 1NN, UK 44institutetext: ONERA, The French Aerospace
Lab BP72, 29 avenue de la Division Leclerc, 92322 Chatillon Cedex, France
55institutetext: Université Côte d’Azur, Observatoire de la Cote d’Azur, CNRS,
Laboratoire Lagrange, France 66institutetext: SETI Institute, Carl Sagan
Center, 189 Bernado Avenue, Mountain View CA 94043, USA 77institutetext:
Institute of Astronomy, Charles University, Prague, V Holešovičkách 2,
CZ-18000, Prague 8, Czech Republic 88institutetext: Department of Mathematics,
Tampere University of Technology, PO Box 553, 33101, Tampere, Finland
99institutetext: IMCCE, Observatoire de Paris, 77 avenue Denfert-Rochereau,
F-75014 Paris Cedex, France 1010institutetext: Astronomical Observatory
Institute, Faculty of Physics, Adam Mickiewicz University, Słoneczna 36,
60-286 Poznań, Poland 1111institutetext: Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
1212institutetext: European Space Agency, ESTEC - Scientific Support Office,
Keplerlaan 1, Noordwijk 2200 AG, The Netherlands 1313institutetext: TMT
Observatory, 100 W. Walnut Street, Suite 300, Pasadena, CA 91124, USA
1414institutetext: Space sciences, Technologies and Astrophysics Research
Institute, Université de Liège, Allée du 6 Août 17, 4000 Liège, Belgium
1515institutetext: Sección Física, Departamento de Ciencias, Pontificia
Universidad Católica del Perú, Apartado, Lima 1761, Perú 1616institutetext:
Institute of Physics, University of Szczecin, Wielkopolska 15, 70-453
Szczecin, Poland 1717institutetext: European Southern Observatory (ESO),
Alonso de Cordova 3107, 1900 Casilla Vitacura, Santiago, Chile

# Closing the gap between Earth-based and interplanetary mission observations:
Vesta seen by VLT/SPHERE ††thanks: Based on observations made with ESO
Telescopes at the Paranal Observatory under programme ID 199.C-0074 (PI: P.
Vernazza),††thanks: Reduced images of Table 3 are only available in electronic
form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/

R. JL. Fétick 11 L. Jorda 11 P. Vernazza 11 M. Marsset 2233 A. Drouard 11 T.
Fusco 1144 B. Carry 55 F. Marchis 66 J. Hanuš 77 M. Viikinkoski 88 M. Birlan
99 P. Bartczak 1010 J. Berthier 99 J. Castillo-Rogez 1111 F. Cipriani 1212 F.
Colas 99 G. Dudziński 1010 C. Dumas 1313 M. Ferrais 1414 E. Jehin 1414 M.
Kaasalainen 88 A. Kryszczynska 1010 P. Lamy 11 H. Le Coroller 11 A. Marciniak
1010 T. Michalowski 1010 P. Michel 55 L.M. Mugnier 44 B. Neichel 11 M. Pajuelo
99 15 15 E. Podlewska-Gaca 1010 16 16 T. Santana-Ros 1010 P. Tanga 55 F.
Vachier 99 A. Vigan 11 O. Witasse 1212 B. Yang 1717

(July 2018)

###### Abstract

Context.  Over the past decades, several interplanetary missions have studied
small bodies in situ, leading to major advances in our understanding of their
geological and geophysical properties. These missions, however, have had a
limited number of targets. Among them, the NASA Dawn mission has characterised
in detail the topography and albedo variegation across the surface of asteroid
(4) Vesta down to a spatial resolution of $\sim$20 m/pixel scale.

Aims.  Here our aim was to determine how much topographic and albedo
information can be retrieved from the ground with VLT/SPHERE in the case of
Vesta, having a former space mission (Dawn) providing us with the ground truth
that can be used as a benchmark.

Methods.  We observed Vesta with VLT/SPHERE/ZIMPOL as part of our ESO large
programme (ID 199.C-0074) at six different epochs, and deconvolved the
collected images with a parametric point spread function (PSF). We then
compared our images with synthetic views of Vesta generated from the 3D shape
model of the Dawn mission, on which we projected Vesta’s albedo information.

Results.  We show that the deconvolution of the VLT/SPHERE images with a
parametric PSF allows the retrieval of the main topographic and albedo
features present across the surface of Vesta down to a spatial resolution of
$\sim$20–30 km. Contour extraction shows an accuracy of $\sim$1 pixel (3.6
mas). The present study provides the very first quantitative estimate of the
accuracy of ground-based adaptive-optics imaging observations of asteroid
surfaces.

Conclusions.  In the case of Vesta, the upcoming generation of 30-40 m
telescopes (ELT, TMT, GMT) should in principle be able to resolve all of the
main features present across its surface, including the troughs and the
north–south crater dichotomy, provided that they operate at the diffraction
limit.

###### Key Words.:

Techniques: high angular resolution – Techniques: image processing – Methods:
observational – Minor planets, asteroids: individual: (4) Vesta

##  1 Introduction

The surface topography of Vesta, the second largest main belt asteroid after
the dwarf planet Ceres, has been characterised in detail by the framing camera
(FC) (Sierks et al., 2011) on board the NASA Dawn mission. The Dawn FC mapped
$\sim$80% of Vesta’s surface with image scales of $\sim$20 m/pixel. These
images revealed the complex topography (Russell et al., 2012; Jaumann et al.,
2012; Marchi et al., 2012; Schenk et al., 2012) summarised below:

  * •

The south polar region consists of two overlapping impact basins (Rheasilvia &
Veneneia) and a central mound whose height rivals that of Olympic Mons on
Mars. Part of the outer perimeter of the Rheasilvia basin is delimited by a
steep scarp. We note that both the impact basin and the central peak were
first detected by Thomas et al. (1997) using the Hubble Space Telescope.

  * •

The surface is characterised by regions with elevated topography ($\sim$20
km), for example the Vestalia Terra whose southernmost part merges into the
rim area of the Rheasilvia basin (Jaumann et al., 2012).

  * •

Numerous troughs are present across Vesta’s surface, especially in the
equatorial and northern regions. Equatorial troughs have lengths that vary
from 19 to 380 km and that can be as wide as 15 km, whereas the most prominent
northern trough is 390 km long and 38 km wide (Jaumann et al., 2012).

  * •

Vesta’s cratering record shows a strong north–south dichotomy with Vesta’s
northern terrains being significantly more cratered than the southern ones
(Marchi et al., 2012).

The images of the Dawn mission also allowed the production of a high-
resolution map of the albedo across Vesta’s surface, revealing the greatest
variation of normal albedo of any asteroid yet observed; (the normal albedo
varies between $\sim$0.15 and $\sim$0.6; Reddy et al. 2012; Schröder et al.
2014).

Here we present a new set of ground-based images of Vesta acquired with
VLT/SPHERE/ZIMPOL as part of our ESO large programme (ID 199.C-0074; Vernazza
et al. 2018). These observations, through direct comparison with the Dawn in
situ measurements, were performed with the aim of testing the ultimate
resolution achieved for images acquired with this new-generation adaptive-
optics (AO) system (Beuzit et al., 2008; Fusco et al., 2006, 2016) and the
robustness of our deconvolution algorithm, which uses a synthetic point spread
function (PSF) as input. These observations were also used to determine which
of the geologic features discovered by Dawn (see above) can already be
identified from the ground, and to place a size limit above which these
features can be retrieved. Finally, these observations were used to produce an
albedo map that could be directly compared with that based on the images of
the Dawn FC (Schröder et al., 2014). In summary, Vesta was used as the
benchmark target for our large programme, allowing us to test and ultimately
validate our different techniques of image analysis.

Figure 1: Four of the 25 Moffat PSFs (left) used for deconvolution, and their
associated OTF (right). Colours indicate different values of $\alpha$, and
line styles show different values of $\beta$. Grey curves in the left and
right panels are respectively the PSF and OTF representing the diffraction
limit of the SPHERE 8m pupil at $\lambda=646$ nm.

Figure 2: Top: VLT/SPHERE/ZIMPOL observation on 2018 July 10${}^{\text{th}}$
deconvolved using 25 different Moffat parametric PSFs. Each image of Vesta is
cropped to $200\times 200$ pixels for best enlargement, and scaled by its
maximum intensity for optimum image dynamics. Bottom: Corresponding FWHM and
Strehl ratio for the 25 PSFs. Values indicate the FWHM and Strehl ratio
respectively, while the colour map (dark for low values, white for high
values) helps to visualise the diagonal trend discussed in the text.

##  2 Observations and data reduction

We observed (4) Vesta with the VLT/SPHERE/ZIMPOL instrument (Beuzit et al.,
2008; Thalmann et al., 2008; Schmid et al., 2012) at six different epochs
between May 20, 2018, and July 10, 2018 (see Table 3 for a complete list of
the observations). The data reduction protocol is the same for all the targets
in our large programme. We refer the reader to Vernazza et al. (2018) for a
description of the procedure. A subset of the Vesta images after pipeline
reduction and before deconvolution are shown in Fig. 7.

##  3 Deconvolution method

###  3.1 Recurrent deconvolution artefacts with the observed PSF

Our large programme was initially designed as follows. Each asteroid
observation was followed by the acquisition of a stellar PSF, using a star of
same apparent magnitude as the asteroid. The asteroid image (cropped to
$500\times 500$ pixels) and its associated PSF (same pixel size as the image)
were then fed to the MISTRAL algorithm (Mugnier et al., 2004) for the
deconvolution process. It appeared that for nearly $50\%$ of the observations,
the deconvolution resulted in strong artefacts at the asteroid edges,
highlighted by the presence of a bright corona (see upper left panel of Fig. 2
for an example). We note that edge issues resulting from the deconvolution
process have already been reported in the literature (Marchis et al., 2006).

The problem was slightly improved using myopic deconvolution (Conan et al.,
1998) with the MISTRAL algorithm. In this mode, the gradient-descent algorithm
estimates simultaneously the couple {object+PSF}. Specifically, the PSF
observed during the night was used to compute an average optical transfer
function (OTF), i.e. the Fourier transform of the PSF, and all the PSFs
observed during the large programme were used to compute a variance of the OTF
around the average. Using this approach, the artefact intensity was reduced
but still visible. This seemed to indicate that the deconvolution issues came
from the PSF itself. We further inferred that the deconvolution artefacts may
arise from a mismatch between the observed stellar PSF and the true PSF during
the asteroid observation. If the shape of the observed PSF is too far from
that of the true PSF, the myopic deconvolution cannot completely correct this
issue. Moreover, myopic (or blind) deconvolution estimates simultaneously the
asteroid image and the PSF, leading to an optimisation process that depends on
a large number of parameters. Consequently, the mathematical problem of myopic
(or blind) deconvolution might be degenerated (Blanco & Mugnier, 2011) and the
minimisation algorithm might converge towards the incorrect solution. Reducing
the number of parameters for the PSF is thus critical to avoid degeneracy and
deconvolution issues.

Figure 3: Comparison of the VLT/SPHERE deconvolved images of Vesta (left) with
synthetic projections of the Dawn model produced with OASIS and with albedo
information from Schröder et al. (2014) (middle), and the same projection
without albedo information but with a latitude/longitude coordinate grid for
reference (right). All coordinates are given in the ‘Claudia’ system (Russell
et al., 2012). No albedo data is available from Dawn for latitudes above
$30^{\circ}N$ (orange line). Finally, some of the main structures that can be
identified in both the VLT/SPHERE images and the synthetic ones are
highlighted: craters are embedded in squares and albedo features in circles.
The square of the intensity is shown for the left and middle columns to
highlight the surface features. Fig. 3 _(continued)_ Deconvolved images for
the two last epochs (phase 0.43 and 0.60) show a clear-dark-clear border on
the left. This deconvolution artefact seems to be the outset of the corona
artefact discussed in the text.

###  3.2 Use of a parametric PSF for the deconvolution

To overcome the deconvolution issues we faced, we started using parametric
PSFs instead of the stellar PSFs. The advantage of a parametric PSF lies in
the flexibility of its parameters, which can be adjusted to better estimate
the true PSF during our asteroid observation. We chose to model these
synthetic PSFs using a Moffat profile (Moffat, 1969) as such functions are
widely used in astronomy to reproduce the sharp coherent peak and wide wings
of PSFs in AO observations (Andersen et al., 2006; Sánchez et al., 2006).
Additionally, the Moffat function has the advantage that it depends on only
two parameters. It is defined as

$M(r)=\frac{\beta-1}{\pi\alpha^{2}}\frac{1}{(1+r^{2}/\alpha^{2})^{\,\beta}},$
(1)

where $r$ is the radius to the PSF centre, $\alpha$ is a scale factor, and
$\beta$ is the Moffat power law index. Setting $\beta>1$ is necessary to have
a finite energy. Under this condition, the multiplicative factor
$(\beta-1)/\pi\alpha^{2}$ in the expression ensures that the total PSF energy
is equal to unity.

As a next step we allowed the two Moffat parameters $\alpha$ and $\beta$ to
vary within their realistic range of values:
$(\alpha,\beta)\in\leavevmode\nobreak\ ]0,+\infty[\leavevmode\nobreak\
\times\leavevmode\nobreak\ ]1,+\infty[$. We then determined a reasonable range
of values from a visual inspection of the images:
$(\alpha,\beta)\in[2.5,4.5]\times[1.3,1.7]$, where $\alpha$ is given in
pixels. Figure 1 shows 4 of the 25 Moffat PSFs tested for the deconvolution of
our Vesta images (for respectively the minimum and the maximum values of
$\alpha$ and $\beta$), and highlights what we consider to be a wide range of
PSF shapes for the deconvolution process.

After scanning the full range of $\alpha$ and $\beta$, two systematic trends
were identified among the deconvolved Vesta images: a large $\alpha$ value in
tandem with a small $\beta$ value leads to the corona artefact that regularly
occurred using observed PSFs (see 3.1), and a small $\alpha$ value in tandem
with a large $\beta$ value leads to under-deconvolution, meaning that the
image blurring is only partially attenuated and the deconvolution is not
complete (see Fig. 2).

Theses trends can be physically well understood when considering the fact that
the full width at half maximum (FWHM) and Strehl ratio of the Moffat PSF are
functions of $\alpha$ and $\beta$, where the FWHM can be written as

$\text{FWHM}=2\alpha\sqrt{2^{1/\beta}-1}$ (2)

and the Strehl ratio as

$\text{Strehl
ratio}=\frac{M(0)}{A(0)}=\frac{\beta-1}{\pi\alpha^{2}}\frac{1}{A(0)},$ (3)

with $A(0)$ being the diffraction pattern induced by the pupil (the Airy
pattern for a circular non-obstructed aperture) at the centre of the PSF
$r=0$.

The diagonal behaviour observed for the deconvolution with respect to the
$\alpha$ and $\beta$ parameters (Fig. 2) makes sense when looking at the FWHM
of the corresponding PSF and Strehl ratios in Fig. 2 (bottom figure). For
large FWHMs (upper left corner)– or equivalently low Strehl ratios–
deconvolution artefacts appear (corona artefact). This implies that the corona
artefacts we encountered during the deconvolution process with observed PSFs
most likely occurred either because the observed PSFs were over-estimating the
FWHM of our asteroid observations or under-estimating their Strehl ratio.
Inversely, for small FWHMs–or large Strehl ratios– the Vesta images appear
under-deconvolved. The diagonal from bottom left to upper right shows rather
stable values for both the FWHM and the Strehl ratio, in agreement with the
deconvolution visual quality highlighted in Fig. 2. Given the deconvolution
trend as a function of the Strehl ratio (Fig. 2), we always choose $\alpha$
and $\beta$ parameters such that the object’s edges were as sharp as possible
without reaching the point where the bright corona effect appears (Table 1,
Fig. 2). This method has already been applied to the images acquired for (16)
Psyche (Viikinkoski et al., 2018) and is now systematically applied to all
targets within our large programme. In the present case, a quantitative
justification (i.e. the accuracy of our deconvolved Vesta images with respect
to those of the Dawn mission) for the choice of the parameters
$(\alpha,\beta)$ is given in Appendix C.

Table 1: Parameters $\alpha$ and $\beta$ chosen for each epoch Epoch |  $\alpha$ (pixel) | $\beta$
---|---|---
2018-05-20 06:41:53 | 4.0 | 1.5
2018-06-08 05:27:05 | 4.0 | 1.5
2018-07-10 01:59:52 | 4.0 | 1.5
2018-07-10 02:16:38 | 4.0 | 1.4
2018-06-08 07:14:40 | 3.0 | 1.5
2018-06-08 07:58:51 | 3.0 | 1.5

In summary, the deconvolution with a Moffat PSF converges in practice toward
satisfactory results. Within our large programme, we therefore started
systematically using a parametric PSF with a Moffat shape to deconvolve our
images. The deconvolved images of Vesta are shown in Fig. 3.

Figure 4: Vesta contours computed for the synthetic images produced with OASIS (light blue line) and VLT/SPHERE after deconvolution (orange line). Contours are plotted over the VLT/SPHERE images for comparison. Epoch (UTC) |  $\chi$ [pix] |  $\chi_{\mathrm{Limb}}$ [pix] |  $\chi_{\mathrm{Term}}$ [pix] |  $\mathcal{A}_{O}$ [pix2] |  $\mathcal{A}_{D}$ [pix2]
---|---|---|---|---|---
2018-05-20 06:41:53 | 0.83 | 0.41 | 0.99 | 19272 | 19414 (+0.73%)
2018-06-08 05:27:05 | 1.50 | 0.65 | 1.27 | 22673 | 23382 (+3.12%)
2018-06-08 07:14:40 | 0.55 | 0.40 | 0.54 | 22762 | 22920 (+0.70%)
2018-06-08 07:58:51 | 0.60 | 0.42 | 0.71 | 22362 | 22379 (+0.52%)
2018-07-10 01:59:52 | 1.37 | 0.39 | 1.73 | 21017 | 21224 (+0.98%)
2018-07-10 02:16:38 | 0.72 | 0.34 | 0.87 | 21223 | 21402 (+0.84%)
Average ${}^{\pm\text{std}}$ | $0.93^{\pm 0.37}$ | $0.44^{\pm 0.10}$ | $1.02^{\pm 0.39}$ |  | ($+1.15^{\pm 0.89}\%$)
Table 2: Results on the contour extraction for OASIS and deconvolved images.
Deconvolved images considered here are the ones from Fig. 3 obtained for the
$(\alpha,\beta)$ parameters in Table 1. The $\chi$, $\chi_{\mathrm{Limb}}$ and
$\chi_{\mathrm{Term}}$ errors are explained in the text. Columns
$\mathcal{A}_{O}$ and $\mathcal{A}_{D}$ are respectively the area encircled by
OASIS and the deconvolution contours. Percentages inside parentheses show the
relative error on the estimated deconvolved area with respect to the OASIS
area.

##  4 VLT/SPHERE compared to Dawn

In this section we determine how much topographic and albedo information can
be retrieved from the ground with VLT/SPHERE in the case of Vesta, having a
former space mission (Dawn) providing us with the ground truth that we can use
as benchmark.

###  4.1 OASIS synthetic images

The NASA/Dawn in situ mission provided high-resolution images of Vesta used to
generate a global shape model (Preusker et al., 2016). The images also
provided the rotational parameters of the objects with a high accuracy
(Konopliv et al., 2014). Knowing the ephemeris of Vesta and the position of
the Earth in the J2000 Equatorial frame, it is thus possible to generate
unconvolved synthetic images of Vesta at the time of the SPHERE observations.
We used the OASIS tool (Jorda et al., 2010) developed and used in the frame of
the Rosetta mission to create these images. The tool takes into account the
pixel scale of the instrument, the viewing and illumination conditions of the
object, its global shape, its rotational parameters, and the Hapke parameters
describing its bi-directional reflectance properties. It rigorously accounts
for cast shadows and considers the geometric intersection between the
triangular facets of the shape model and the pixels to calculate reflectance
values for each pixel. We used the Hapke (1986) function built with a one-
parameter Henyey-Greenstein phase function and the parameters of Li et al.
(2013) to describe the reflectance of the surface. These synthetic images
represent the resolution limit that could have been achieved with a turbulent-
less atmosphere and perfect optics. Finally, OASIS images are multiplied by
the albedo map from Schröder et al. (2014) derived from the Dawn mission and
covering the southern part of Vesta up to $30^{\circ}$N (see Fig. 3, middle
column).

Figure 5: Albedo map of Vesta constructed from VLT/SPHERE images (bottom)
compared to the Dawn Framing Camera 2 (FC2) map derived in situ (top; Schröder
et al. 2014). The two maps are in equidistant cylindrical projection. The Dawn
map only shows the region of Vesta covered by SPHERE to facilitate the
comparison. The north region of the Dawn map, with the latitude $\phi$ in the
$[0^{\circ},+30^{\circ}]$ range, contains several pixels with no available
information. Those pixels are left black.  Figure 6: Histogram of pixel values
for the SPHERE (continuous black) and Dawn (dotted grey) albedo maps. The
SPHERE map exhibits a slightly narrower range of albedo values with respect to
Dawn, owing to its lower spatial resolution and residual blurring of the
images due to imperfect deconvolution.

###  4.2 Contour extraction and comparison

In order to perform a quantitative comparison between observed and synthetic
OASIS images, we performed a comparison between contour plots. Contours are
important for 3D shape reconstruction algorithms. The extraction of these
contour plots was performed in several steps, described below. For the
observed images, a low threshold $T_{1}$ was estimated by fitting the
histogram of the pixels in a box of $21\times 21$ pixels around the minimum
pixel of the image. For the synthetic images, the low threshold was set to
zero. A high threshold $T_{2}$ was then calculated as the maximum pixel value
after removing the 10 % highest pixel values. The contour level to be used was
defined as $T=T_{1}+\zeta\ (T_{2}-T_{1})$, where $\zeta$ is a parameter in the
interval $[0,1]$ allowing the value of the contour level to be set with
respect to the high and low thresholds. Here $\zeta=0.3$ was chosen. The image
was then converted into a triangular mesh. For this, the Cartesian coordinates
of the vertices were defined by the coordinates $(X=i,Y=j)$ of the pixel
$(i,j)$, complemented by its value $Z=P_{ij}$. Each block of $2\times 2$
pixels allowed us to define two triangles. The resulting set of triangles
represents the image as a triangular mesh. Finally, the intersection of the
triangular mesh with the plane $Z=T$ defined the contour plot, represented
here as a set of connected 2D points.

Let us call $\\{D_{i}\\}_{i\in[1,I]}$ and $\\{M_{j}\\}_{j\in[1,J]}$
respectively the set of points defining the deconvolved contour and the OASIS
model contour. We defined the root mean square metric as

$\chi=\sqrt{\frac{\sum_{i}\lambda_{i}\cdot
d(D_{i},\\{M_{j}\\}_{j})^{2}}{\sum_{i}\lambda_{i}}},$ (4)

where the sum index $i$ runs over all the contour points of the deconvolved
image, $d(D_{i},\\{M_{j}\\}_{j})$ is the Euclidean distance between the point
number $i$ and its orthogonal projection onto the OASIS contour, $\lambda_{i}$
is the pixel weighting factor. We chose $\lambda_{i}$ to be the average
distance between $D_{i}$ and its neighbours $D_{i-1}$ and $D_{i+1}$ as

$\lambda_{i}=\frac{1}{2}\left[d(D_{i},D_{i-1})+d(D_{i},D_{i+1})\right]$ (5)

such that an eventual cluster of close points in the contour set does not
induce highly localised weighting in the $\chi$ norm. Since the points are
nearly homogeneously separated, the correction has little impact on the $\chi$
norm with respect to a simple weighting $\lambda_{i}=1$. Finally, forcing
$\lambda_{i}=0$ for some points allows us to select a specific area such as
the limb or terminator for contour comparison.

Extraction of contours from the synthetic and deconvolved images shows good
agreement (see Fig. 4 for contour visualisation). The match between the
contours is better at the limb ($\chi_{\text{Limb}}=0.44$ pixels) than at the
terminator ($\chi_{\text{Term}}=1.02$ pixels). Indeed the OASIS facets
illumination is much more sensitive to angular errors in the orientation of
the facets for nearly tangent solar rays than for normal rays. Moreover,
intensity errors in the synthetic images at the terminator translate into
larger errors in the contour position because the intensity gradient is
smaller there compared to the large jump occurring at the limb.

The average contour error, when taking into account all contour points, is
$\chi=0.93$ pixels, showing that sub-pixel contour resolution is achievable on
asteroid deconvolved images.

The area encircled by the deconvolved contour is larger than the OASIS contour
with a relative area difference of $+1.15\%$. This error might be due to
residual blurring that was not fully removed by the deconvolution, and to
difficulties in extracting the OASIS contour near the terminator. The
resulting error on the volume would be $\sim+1.73\%$. All results concerning
the contours are summarised in Table 2.

###  4.3 Reconnaissance of Vesta’s main topographic features

It appears that most of the main topographic features present across Vesta’s
surface can already be recognised from the ground (Fig. 3). This includes the
south pole impact basin and its prominent central mound, several D$\geq$25 km
sized craters and Matronalia Rupes including its steep scarp and its small and
big arcs (Krohn et al., 2014). From these observations, we can determine a
size limit of $\sim$30 km for the features that can be resolved with
VLT/SPHERE (i.e. features that are 8-10 pixels wide). This detection limit
should be in principle sufficient to recognise the north–south crater
dichotomy detected by Dawn (Marchi et al., 2012; Vincent et al., 2014),
according to the fact that the northern hemisphere hosts 70% of the craters
with D$\geq$30 km (Liu et al., 2018). There are several plausible explanations
as to why we were not able to resolve this dichotomy. First, we mostly imaged
the southern hemisphere, due to Vesta’s spin axis orientation. Second, the
atmospheric blurring (not entirely removed by the AO) is another limiting
factor. Third, the albedo variation across the surface is large, leading to a
confusion between the shade of the craters and local albedo variation. This is
clearly highlighted in the synthetic images generated by OASIS with and
without albedo information (see middle and right column in Fig. 3). Fourth, we
do not observe the equatorial troughs of Vesta, which are 15 km wide and thus
below the detection limit of our images. It follows that future generation
telescopes with mirror sizes in the 30-40 m range should in principle be able
to resolve the main features present across Vesta’s surface, provided that
they operate at the diffraction limit.

Finally, we attempt to constrain the detection rate for the largest craters
with diameter greater than 40 km. Except the large Rheasilvia basin, Vesta
hosts 21 such craters (Liu et al., 2018) and 9 of them were covered by our
observations. Of these nine craters, we could identify seven of them (see Fig.
3) in our images, implying a detection rate of $\sim$80 %. The two missing
craters are Marcia and Calpurnia, which are part of the snowman crater chain
that is located at $\sim 13^{\circ}$ north latitude and $\sim 195^{\circ}$
eastern longitude. The non-detection of the snowman crater chain is not
surprising when looking at the two synthetic OASIS images (with and without
albedo information) as the crater chain can barely be recognised in the
synthetic images with albedo information (see Fig. 5).

###  4.4 Reconnaissance of Vesta’s main albedo features

To further test the reliability of the observation and deconvolution
procedure, we built an albedo map of Vesta from the deconvolved SPHERE images,
and compared this map to that of Schröder et al. (2014) from in situ Dawn
measurements. We used a subsample of 19 high-quality images from the six
epochs of VLT/SPHERE observations.

Owing to the limited number of geometrical views probed by SPHERE, it is not
always possible to differentiate shadows from true albedo variations without
prior knowledge about the local surface topography. This information is very
well constrained in the case of Vesta, and can be retrieved from the OASIS
model. However, this is not true for most targets in our observing programme,
and for almost all asteroids in general. We therefore purposely did not use
any prior information on Vesta’s topography when building the map in order to
evaluate our ability to retrieve albedo information from the SPHERE
observations alone.

First, we corrected the illumination gradient present in the SPHERE images,
which depends on the local incidence, reflection, and phase angle. This was
performed by fitting a second-order polynomial surface to the disc (image
intensity) of Vesta. This method provides satisfactory results for Earth-based
observations of asteroids (Carry et al., 2008, 2010). Across our observing
programme, we also found that it provides better results than using a
scattering law (Lommel-Seeliger, Hapke; Li et al. 2015) when little is known
about the local topography of the object. Using one of the above-mentioned
scattering laws with a low-resolution shape model indeed results in
artificially ‘paved’ images owing to the resolution, i.e. the size of the
facets, in the model. The use of a polynomial fit, which qualitatively mimics
the Lommel-Seeliger laws, to the disc of the object results in smoother
photometric correction without artificial discontinuities. However, it cannot
properly fit the illumination of local terrain features, such as craters and
mountains.

For each SPHERE image, we then defined a region of interest (ROI) containing
the set of pixels to be projected on the map. As discussed in Section 3, an
over-deconvolution usually enhances the brightness of the image regions with
strong luminosity gradient, such as the asteroid border. To avoid including
pixels affected by this effect, we only considered pixels contained in the
central region of the asteroid, i.e. 20 pixels away from the asteroid contour
(limb and terminator).

Next, the longitude and latitude of each pixel contained in the ROI was
measured using a projection of the OASIS model (Section 4.1). For other
targets in our programme, this would be performed using a lower resolution
model derived with the ADAM software (Viikinkoski et al., 2015).

Each pixel value was then projected on the map, using an equidistant
cylindrical projection. The individual maps from the different epochs of
observation were then combined using the overlapping regions to balance their
brightness level. Specifically, the combined map was calculated as a weighted
average of the individual maps, where each pixel was attributed a Gaussian
weight inversely proportional to the projected surface area it covers, and
inversely proportional to its projected distance from the sub-Earth point.
Finally, we normalised the combined map with the global average albedo of
Vesta in the Johnson V band (centred on 540 nm) in order to allow a direct
comparison with the albedo map of Schröder et al. (2014).

The resulting map is shown and compared to that of Schröder et al. (2014) in
Fig. 5. The Dawn map displays only the region of Vesta seen by SPHERE in order
to facilitate the comparison. Other pixels were set to zero.

The SPHERE map exhibits a wide range of albedo values (typically
AV,N=0.34–0.45), close to that of Dawn (mostly AV,N=0.30–0.46), but slightly
narrower owing to its lower spatial resolution and residual blurring due to
imperfect deconvolution (Fig. 6). The peak of the two distributions is
slightly offset, due to the different shape of the distributions: The SPHERE
distribution appears even on both sides of the peak, whereas the low-end tail
of the Dawn distribution is wider than the high-end one, possibly because
multiple species with different albedos are spatially resolved. Most of the
main albedo features present in the Dawn map can also be identified in the
SPHERE map. We find that surface features down to 20 km in size can be
identified from the SPHERE map.

Only a few inconsistencies between the two maps are found. The most obvious
example is the presence of a dark region located near $\lambda$=80∘,
${\phi}=-50^{\circ}$ on the SPHERE map, whereas Dawn finds no such albedo
variation at this location. This feature most likely results from the presence
of shadows in the SPHERE images caused by irregular terrains rather than true
albedo variations. The brightness of the southernmost region of Vesta on the
map for $\lambda\in[180^{\circ},270^{\circ}]$ and ${\phi}<-60^{\circ}$ is
enhanced with SPHERE compared to Dawn. This is due to the illumination of the
Rheasilvia central peak, which could not be accurately corrected. The Dawn map
contains very localised regions with very low albedo values (A${}_{\rm
V,N}<$0.25). While these regions are also seen with SPHERE, they display
higher albedos (A${}_{\rm V,N}>$0.30) due to the lower spatial resolution of
the map that attenuates the high-frequency albedo variations. Additional
differences between the two maps may arise from the different filters used
during the observations: ZIMPOL N_R ($\lambda$=589.2–702.6 nm) versus DAWN FC2
clear filter ($\lambda$=438–965 nm).

##  5 Conclusion

In this article, we evaluated how much topographic and albedo information can
be retrieved from the ground with VLT/SPHERE in the case of the asteroid (4)
Vesta. This object can be used as a benchmark for ground-based observations
since the Dawn space mission provided us with ground truth information. We
observed (4) Vesta with VLT/SPHERE/ZIMPOL as part of our ESO large programme
at six different epochs, and deconvolved the collected images with a
parametric PSF. We then compared our VLT/SPHERE images with synthetic images
of Vesta produced with the OASIS software that uses the 3D shape model of the
Dawn mission as input and on which we re-projected the albedo map of the Dawn
mission (Schröder et al., 2014). We further produced our own albedo map of
Vesta from the SPHERE images alone, and without using any prior information
about the surface topography of Vesta. We compared this map to that of
Schröder et al. (2014) to evaluate our ability to differentiate true albedo
variegation from shadows and regions with enhanced illumination due to
imperfect photometric correction of the SPHERE images.

We show that the deconvolution of the VLT/SPHERE images with a parametric PSF
allows the retrieval of the main topographic and albedo features present
across Vesta’s surface down to a spatial resolution of $\sim$20–30 km. Contour
extraction shows a precision of $\sim$1 pixel, with a strong difference
between the estimation of the limb (precision of $\sim$0.5 pixel) and the
terminator (precision of $\sim$1 pixel). The consequent relative error on the
estimated area is $\leq 2\%$.

The present study demonstrates for the very first time the accuracy of ground-
based AO imaging observations of asteroids with respect to in situ
observations. Future generation telescopes (ELT, TMT, GMT) could use Vesta as
a benchmark to catch all of the main features present across its surface
(including the troughs and the north–south crater dichotomy), provided that
these telescopes operate at the diffraction limit.

Data retrieval: As soon as papers for our large programme are accepted for
publication, we will make the corresponding reduced and deconvolved AO images
and 3D shape models publicly available at http://observations.lam.fr/astero/.

###### Acknowledgements.

This work was supported by the French Direction Générale de l’Armement (DGA)
and Aix-Marseille Université (AMU). P. Vernazza, A. Drouard, and B. Carry were
supported by CNRS/INSU/PNP. J. Hanuš was supported by the grant 18-09470S of
the Czech Science Foundation and by the Charles University Research Programme
No. UNCE/SCI/023. This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under grant agreement No.
730890. This material reflects only the authors’ views and the Commission is
not liable for any use that may be made of the information contained herein.
The authors thank S. Schröder for providing his reconstructed albedo map based
on the Dawn images and for his very fruitful comments as a referee.

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##  Appendix A Observation table

Table 3: List of Vesta images. For each observation, the table gives the epoch, the airmass, the distance to the Earth $\Delta$ and the Sun $r$, the phase angle $\alpha$, and the angular diameter $D_{\mathrm{a}}$. All observations were performed on the VLT/SPHERE instrument, with N_R filter, and 80 seconds of exposure time. Date | UT | Airmass | $\Delta$ | $r$ | $\alpha$ | $D_{\mathrm{a}}$
---|---|---|---|---|---|---
|  |  | (AU) | (AU) | (°) | (″)
2018-05-20 | 6:41:53 | 1.01 | 1.25 | 2.15 | 15.8 | 0.579
2018-05-20 | 6:43:24 | 1.01 | 1.25 | 2.15 | 15.8 | 0.579
2018-05-20 | 6:44:56 | 1.01 | 1.25 | 2.15 | 15.8 | 0.579
2018-05-20 | 6:46:25 | 1.01 | 1.25 | 2.15 | 15.8 | 0.579
2018-05-20 | 6:47:56 | 1.01 | 1.25 | 2.15 | 15.8 | 0.579
2018-06-08 | 5:27:05 | 1.01 | 1.16 | 2.15 | 6.8 | 0.624
2018-06-08 | 5:28:36 | 1.01 | 1.16 | 2.15 | 6.8 | 0.624
2018-06-08 | 5:30:08 | 1.01 | 1.16 | 2.15 | 6.8 | 0.624
2018-06-08 | 5:31:38 | 1.00 | 1.16 | 2.15 | 6.8 | 0.624
2018-06-08 | 5:33:07 | 1.00 | 1.16 | 2.15 | 6.8 | 0.624
2018-06-08 | 7:14:40 | 1.08 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 7:16:10 | 1.09 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 7:17:42 | 1.09 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 7:19:12 | 1.09 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 7:20:41 | 1.10 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 7:58:51 | 1.19 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 8:00:22 | 1.19 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 8:01:51 | 1.20 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 8:03:21 | 1.20 | 1.16 | 2.15 | 6.7 | 0.624
2018-06-08 | 8:04:52 | 1.21 | 1.16 | 2.15 | 6.7 | 0.624
2018-07-10 | 1:59:52 | 1.04 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:01:24 | 1.03 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:02:56 | 1.03 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:04:25 | 1.03 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:05:56 | 1.03 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:16:38 | 1.02 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:18:09 | 1.02 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:19:39 | 1.02 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:21:09 | 1.02 | 1.19 | 2.16 | 10.9 | 0.608
2018-07-10 | 2:22:39 | 1.02 | 1.19 | 2.16 | 10.9 | 0.608

##  Appendix B Additional figure

Figure 7: Top: Reduced Vesta images after pipeline. Bottom: Same images after
deconvolution.

##  Appendix C Optimisation of the PSF parameters

Using OASIS synthetic images with albedo, we were also able to determine the
optimal values of $\alpha$ and $\beta$ for the deconvolution by calculating
the root square error between the deconvolved SPHERE/ZIMPOL images and the
synthetic images (OASIS+albedo model). The error writes as

$\epsilon(\alpha,\beta)=\sqrt{\sum_{p}(\text{DEC}_{p}(\alpha,\beta)-a_{p}\cdot\text{OAS}_{p})^{2}},$
(6)

where the sum runs over the pixels, OAS is the OASIS model multiplied by the
albedo $a_{p}$ corresponding to each pixel, and DEC is our deconvolved image.
When writing $\text{DEC}_{p}(\alpha,\beta)$ we recall that the deconvolution
depends on the PSF parameters $(\alpha,\beta)$. The minimisation of the error
$\epsilon(\alpha,\beta)$ identifies the optimal values for the $\alpha$ and
$\beta$ parameters and thus the most accurate deconvolution in the sense of
the norm given by Eq.6. Figure 9 shows the evolution of $\epsilon$ with
respect to the PSF parameters $\alpha$ and $\beta$ for the six epochs. The
minimum is reached in the narrow range $\alpha\simeq 3.5-4$ and $\beta\simeq
1.5-1.6$. Figure 9 also shows that the error increases dramatically when the
corona artefact appears, and that the minimum error is reached for sharp
deconvolutions just before the corona effect.

Thus, the $(\alpha,\beta)$ parameters empirically estimated in Sect. 3.2 are
close to the actual optimal values that minimise the $\epsilon$ criteria. Near
such optimal values, only the corona effect may corrupt the deconvolution
result at high $\alpha$ and low $\beta$. Otherwise the deconvolution zone is
stable since the deconvolution quality evolves continuously with the
parameters (see Figs. 2 and 9).

Importantly, we believe that it is possible to further improve the
deconvolution procedure using even more accurate PSF models, with a shape
closer to that of the AO corrected PSF (paper in preparation on the AO PSF
modelling). Finally, PSF parameters (whatever PSF model is used) may be
estimated automatically by a myopic deconvolution algorithm, such as Mugnier
et al. (2004) and Blanco & Mugnier (2011).

Figure 8: Correlation between OASIS (including albedo) and deconvolved image
pixel intensities. Each intensity map has been divided by its average value
for normalisation. Each of the six graphs corresponds to an epoch of
observation. The linear fit of each cloud of points is also shown (orange
line). The correlation percentage between OASIS and deconvolution intensities
is written at the top of each graph. Figure 9: Root square error between
deconvolved images and OASIS+albedo model. Each of the six subfigures
corresponds to one of the observing epoch. Evolution of the error with respect
to the $\alpha$ and $\beta$ parameters is then visible. For each epoch,
optimal values of $\alpha$ and $\beta$ are the ones that minimise the error.
Invalid pixels (missing albedo) were removed for error computation. However
all pixels (inside and outside the asteroid) are considered to take into
account the residual blurring after eventual incomplete deconvolution. Errors
have also been divided by the flux of each Vesta observation so they reach
comparable values between the different epochs.


# A model-independent description of $B\rightarrow D\pi\ell\nu$ decays

Erik J. Gustafson  Fermi National Accelerator Laboratory, Batavia, Illinois,
60510, USA  Florian Herren  Fermi National Accelerator Laboratory, Batavia,
Illinois, 60510, USA  Physics Institute, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Ruth S. Van de Water
Fermi National Accelerator Laboratory, Batavia, Illinois, 60510, USA  Raynette
van Tonder  Department of Physics, McGill University, 3600 rue University,
Montréal, Québec, H3A 2T8, Canada  Michael L. Wagman  Fermi National
Accelerator Laboratory, Batavia, Illinois, 60510, USA

###### Abstract

We introduce a new parameterization of $B\rightarrow D\pi\ell\nu$ form factors
using a partial-wave expansion and derive bounds on the series coefficients
using analyticity and unitarity. This is the first generalization of the
model-independent formalism developed by Boyd, Grinstein, and Lebed for $B\to
D\ell\nu$ to semileptonic decays with multi-hadron final states, and enables
data-driven form-factor determinations with robust, systematically-improvable
uncertainties. Using this formalism, we extract the form-factor parameters for
$B\to D_{2}^{\ast}(\to D\pi)\ell\nu$ decays in a model-independent way from
fits of data from the Belle Experiment, and, for the first time, study the
two-pole structure in the $D\pi$ S-wave in semileptonic decays employing
lineshapes from unitarized chiral perturbation theory.

††preprint: FERMILAB-PUB-23-662-T ZU-TH 67/23

Motivation —  Experimental measurements of tree-level semileptonic $B$-meson
decays enable theoretically clean determinations of the Cabibbo-Kobayashi-
Maskawa (CKM) matrix elements $|V_{ub}|$ and $|V_{cb}|$, allowing for
sensitive tests of the Standard Model by overconstraining the CKM unitarity
triangle [1, 2, 3]. Further, $|V_{ub}|$ and $|V_{cb}|$ are parametric inputs
to predictions for loop-level flavor-changing processes that are sensitive to
new high-scale physics beyond the reach directly detectable by the LHC [4, 5].

A major challenge for both inclusive and exclusive determinations of
$|V_{ub}|$ is suppressing the CKM-favored $B\rightarrow X_{c}\ell\nu$
background, which exhibits a similar experimental signature and is
$\mathcal{O}(100)$ times more abundant than $B\rightarrow X_{u}\ell\nu$
decays. The background subtraction process is further complicated by the
orbitally excited states, collectively referred to as $D^{**}$, whose
kinematic distributions remain poorly understood and branching fractions
exhibit uncertainties of approximately 20% [6]. In measurements performed by
the Belle and Belle II Collaborations, the remaining “gap” between the sum of
all considered exclusive modes and the inclusive $B\rightarrow X\ell\nu$
branching fraction, comprising unmeasured non-resonant $B\rightarrow
X_{c}\ell\nu$ decays, is generally treated in simulation by assuming a
composition of equal parts of $B\rightarrow D^{(*)}\eta\ell\nu$ decays, as
prescribed in Ref. [7]. Because neither experimental evidence nor theoretical
predictions exist for $B\rightarrow D^{(*)}\eta\ell\nu$ decays, a 100%
uncertainty is assumed for the corresponding branching fractions. For these
reasons, the $X_{c}\ell\nu$ modeling uncertainty is hard to quantify and
becomes dominant for studies of inclusive $B\rightarrow X_{c/u}\ell\nu$ decays
[7, 8, 9, 10, 11].

Exclusive measurements relying on tagged methods, in which machine learning
algorithms are employed to fully reconstruct the companion $B$ meson through
exclusive decay modes [12, 13], do not rely as directly on $X_{c}\ell\nu$
modeling as inclusive analyses. However, significant differences in these
reconstruction algorithms’ performances between data and simulation is
accounted for by performing a calibration using a decay with a well known
branching fraction: inclusive $B\rightarrow X\ell\nu$ [14]. This calibration,
in turn, becomes a leading source of systematic error for tagged analyses [15,
16, 17]. In addition, the limited knowledge of $B\rightarrow
D^{\ast\ast}\ell/\tau\nu$ branching fractions and form factors are large
systematic uncertainties in studies of rare processes such as $B\rightarrow
K\nu\nu$ at Belle II or $R(D^{\ast})$ at the LHCb experiment [18, 19].

The most commonly used description of $B\rightarrow D^{\ast\ast}\ell\nu$
decays is the Leibovich-Ligeti-Stewart-Wise (LLSW) parameterization [20, 21]
with central values from the fit given in Refs. [22, 23]. This
parameterization includes a single $D_{0}^{\ast}$ resonance. Studies in the
context of unitarized chiral perturbation theory (UChPT), however, have shown
that the scalar member of the $D^{\ast\ast}$ family, the $D_{0}^{\ast}(2300)$,
is an overlap of two states with poles near $(2.1-i0.1)$ and $(2.45-i0.13)$
GeV [24, 25, 26]. Consequently, the S-wave lineshape is not described by a
simple Breit-Wigner distribution, but has a more complex structure. This
conclusion is supported by lattice quantum chromodynamics (LQCD) results from
a coupled-channel analysis performed by the Hadron Spectrum Collaboration [27]
and a reinterpretation [28] of the partial-wave analysis of $B^{+}\rightarrow
D^{-}\pi^{+}\pi^{+}$ decays by LHCb [29]. Further, in Ref. [30], Le Yaouanc,
Leroy, and Roudeau point out that in the fits to the LLSW parametrization tail
effects from the $D^{\ast}$ resonance are omitted and consequently
overestimates the $D_{0}^{\ast}$ contribution.

To address these and other limitations of existing parameterizations, in this
letter we develop the first model-independent description of resonant and non-
resonant $B\rightarrow D\pi\ell\nu$ decays based on analyticity and unitarity.
We then apply our formalism, which accommodates arbitrary lineshapes, to fit
experimental spectrum measurements and draw conclusions about the pole
structure of the S-wave channel.

Form-factor parameterization —  Semileptonic $B\rightarrow D\pi\ell\nu$ decays
are characterized by five kinematic variables: the momentum transfer square
$q^{2}$,111It is sometimes useful to instead consider the dependence on the
recoil parameter $w=(M_{B}^{2}+M_{D\pi}^{2}-q^{2})/(2M_{B}M_{D\pi})$. the
helicity angle of the charged lepton $\cos\theta_{l}$, the helicity angle of
the $D$ meson $\cos\theta$, the azimuthal angle between the $\ell\nu$ and
$D\pi$ planes $\chi$, and the invariant mass of the hadronic system
$M_{D\pi}$.

Form-factor decompositions for charged-current semileptonic decays involving
two final state hadrons have been performed for $B\rightarrow\pi\pi\ell\nu$
decays [31] and involve a partial-wave decomposition in $\cos\theta$ to
disentangle contributions from different hadronic resonances. Following a
similar strategy, we express the $B\rightarrow D\pi\ell\nu$ hadronic matrix
elements as

$\begin{split}\left\langle
D(p_{D})\pi(p_{\pi})|V_{\mu}|B(p_{B})\right\rangle&=\frac{2i}{M_{B}+M_{D\pi}}\epsilon_{\mu\nu\rho\sigma}p_{D\pi}^{\rho}p_{B}^{\sigma}\sum_{l>0}L^{(l),\nu}V^{(l)}(q^{2},M_{D\pi}^{2})~{},\\\
\left\langle
D(p_{D})\pi(p_{\pi})|A_{\mu}|B(p_{B})\right\rangle&=M_{D\pi}\frac{q^{\mu}}{q^{2}}\sum_{l\geq
0}L^{(l),\nu}q_{\nu}A_{0}^{(l)}(q^{2},M_{D\pi}^{2})+\sum_{l>0}\left(L^{(l),\mu}-\frac{L^{(l),\nu}q_{\nu}}{q^{2}}q^{\mu}\right)A_{1}^{(l)}(q^{2},M_{D\pi}^{2})\\\
&\hskip
10.0pt+\left[(p_{B}+p_{D\pi})^{\mu}-\frac{M^{2}_{B}-M^{2}_{D\pi}}{q^{2}}q^{\mu}\right]\sum_{l\geq
0}L^{(l),\nu}q_{\nu}A_{2}^{(l)}(q^{2},M_{D\pi}^{2})~{}.\end{split}$ (1)

The vector $L^{(l)}$ is related to the angular momentum of the final-state
hadron system in the $B$-meson rest frame and fulfills

$L_{\mu}^{(l)}p^{\mu}_{D\pi}=0,\;\;L_{\mu}^{(0)}q^{\mu}=1,\;\;L_{\mu}^{(l)}q^{\mu}=M_{B}W^{l}P_{l}(\cos\theta),\\!\\!$
(2)

where $W=|\vec{q}||\vec{p_{D}}|/(M_{B}M_{D\pi})$ and $P_{l}$ are the Legendre
polynomials.

For $l=0$ only two form factors — $f_{0}$ and $f_{+}$ — contribute ($V^{(l)}$
and $A_{1}^{(l)}$ do not contribute to S-wave decays since they directly
couple the weak current to the angular momentum of the hadronic system), and
we define them as Boyd, Grinstein, and Lebed (BGL) do for $B\rightarrow
D\ell\nu$ decays [32, 33, 34]:

$A_{0}^{(0)}=\frac{M_{B}^{2}-M^{2}_{D\pi}}{M_{D\pi}}f_{0}~{},\quad
A_{2}^{(0)}=f_{+}~{}.$ (3)

For $l>0$, we re-write the form factors in Eq. (1) analogously:

$\begin{split}V^{(l)}&=\frac{M_{B}+M_{D\pi}}{2}g_{l}\,,\quad
A_{0}^{(l)}=\mathcal{F}_{2,l}\,,\quad A_{1}^{(l)}=\frac{1}{2}f_{l}\,,\\\
A_{2}^{(l)}&=\frac{1}{\lambda_{B}}\left[M_{D\pi}(M_{B}^{2}-M_{D\pi}^{2})\mathcal{F}_{1,l}-(p_{D\pi}\cdot
q)f_{l}\right]~{},\end{split}$ (4)

where the threshold factor
$\lambda_{B}=M_{B}^{4}+M_{D\pi}^{4}+q^{4}-2(M_{B}^{2}M_{D\pi}^{2}+M_{D\pi}^{2}q^{2}+q^{2}M_{B}^{2})$.
With this choice (and a suitably defined polarization vector $L_{\mu}^{(l)}$),
the $l=1$ and $l=2$ terms match the standard expressions for $B\rightarrow
D^{\ast}\ell\nu$ and $B\rightarrow D_{2}^{\ast}\ell\nu$ decays, respectively
[35, 36].

Using Eqs. (1), (3), and (4), it is straightforward to derive the
$B\rightarrow D\pi\ell\nu$ differential decay rate. After integrating over all
angles, the double differential decay rate for massless leptons can be written
on a single line:

$\frac{\mathrm{d}^{2}\Gamma}{\mathrm{d}M^{2}_{D\pi}\mathrm{d}q^{2}}=\frac{G_{F}^{2}|V_{cb}|^{2}}{(4\pi)^{5}}M_{B}\left(W\frac{\lambda_{B}}{M_{B}^{2}}\frac{4|f_{+}|^{2}}{3}+M_{D\pi}^{2}\sum_{l>0}W^{2l+1}\left[\frac{4(M_{B}^{2}-M_{D\pi}^{2})^{2}}{3(2l+1)}\frac{|\mathcal{F}_{1,l}|^{2}}{\lambda_{B}}+\frac{l(l+1)}{(2l+1)}q^{2}\left(|g_{l}|^{2}+\frac{|f_{l}|^{2}}{\lambda_{B}}\right)\right]\right)~{}.$
(5)

The fully general five-fold differential decay rate allowing for interference
effects between different partial waves is provided in the supplemental
material [37].

Unitarity bounds —  Model-independent constraints on the $B\rightarrow
D\pi\ell\nu$ form factors arise from analyticity and unitarity. We begin with
the two-point functions

$\Pi_{(J)}^{L/T}(q)\equiv i\int d^{4}x\ e^{iq\cdot
x}\,\left<0\right|J^{L/T}(x)\,J^{L/T}(0)\left|0\right>,$ (6)

where $J^{\mu}$ denotes a $b\rightarrow c$ flavor-changing vector or axial-
vector current and $L/T$ denotes the component longitudinal or transverse to
$q^{\mu}$. Susceptibilities $\chi_{(J)}^{L/T}$ are defined from derivatives of
$\Pi_{(J)}^{L/T}(q)$ as

$\begin{split}&\chi^{L}_{(J)}(Q^{2})\equiv\frac{\partial\Pi_{(J)}^{L}}{\partial
q^{2}}\Big{|}_{q^{2}=Q^{2}}=\frac{1}{\pi}\int_{0}^{\infty}\mathrm{d}q^{2}\frac{\operatorname{Im}\Pi_{(J)}^{L}(q^{2})}{(q^{2}-Q^{2})^{2}}~{},\\\
&\chi^{T}_{(J)}(Q^{2})\equiv\frac{1}{2}\frac{\partial^{2}\Pi_{(J)}^{T}}{\partial(q^{2})^{2}}\Big{|}_{q^{2}=Q^{2}}=\frac{1}{\pi}\int_{0}^{\infty}\mathrm{d}q^{2}\frac{\operatorname{Im}\Pi_{(J)}^{T}(q^{2})}{(q^{2}-Q^{2})^{3}}~{},\end{split}$
(7)

and are related to integrals over the imaginary part of $\Pi_{(J)}^{L/T}(q)$
via dispersion relations. The susceptibilities $\chi^{L/T}_{(V/A)}$ are
perturbatively calculable for spacelike $q^{2}$ and have been computed to
${\mathcal{O}}(\alpha_{S}^{2})$ at $Q^{2}=0$ in Ref. [38].

Separately, the optical theorem relates
$\operatorname{Im}\Pi_{(J)}^{T}(q^{2})$ to a sum of squared amplitudes for all
intermediate states that can appear between the currents in Eq. (6). This sum
includes terms with the matrix element
$\left<\overline{B}D\pi|J^{L/T}|0\right>$, which is related to $|\left\langle
D\pi|J^{L/T}|B\right\rangle|^{2}$ by crossing symmetry and can therefore be
parameterized by the form-factor decomposition in Eq. (1). The contribution of
the $B\rightarrow D\pi\ell\nu$ channel to the dispersion relations in Eq. (7)
is then given by evaluating the phase space integrals arising in the sum over
states,

$\begin{split}\operatorname{Im}\Pi_{A}^{L}\Big{|}_{D\pi}&=\frac{1}{32\pi^{3}}\frac{M_{B}^{4}}{q^{4}}\int_{(M_{D}+m_{\pi})^{2}}^{(\sqrt{q^{2}}-M_{B})^{2}}\mathrm{d}M_{D\pi}^{2}\,\left(M_{D\pi}^{2}\sum_{l>0}\frac{W^{2l+1}}{2l+1}|\mathcal{F}_{2}^{(l)}|^{2}+\frac{(M_{B}^{2}-M^{2}_{D\pi})^{2}}{M_{B}^{2}}|f_{0}|^{2}\right)~{},\\\
\operatorname{Im}\Pi_{A}^{T}\Big{|}_{D\pi}&=\frac{1}{96\pi^{3}}\frac{M_{B}^{4}}{q^{2}\lambda_{B}}\int_{(M_{D}+m_{\pi})^{2}}^{(\sqrt{q^{2}}-M_{B})^{2}}\mathrm{d}M_{D\pi}^{2}\,\left(M_{D\pi}^{2}\sum_{l>0}\frac{W^{2l+1}}{2l+1}\left(\frac{|\mathcal{F}_{1}^{(l)}|^{2}}{q^{2}}+\frac{l+1}{l}|f^{(l)}|^{2}\right)+\frac{|f_{+}|^{2}}{q^{2}M_{B}^{2}}\right)~{},\\\
\operatorname{Im}\Pi_{V}^{T}\Big{|}_{D\pi}&=\frac{1}{96\pi^{3}}\frac{M_{B}^{4}}{q^{2}}\int_{(M_{D}+m_{\pi})^{2}}^{(\sqrt{q^{2}}-M_{B})^{2}}\mathrm{d}M_{D\pi}^{2}\,M_{D\pi}^{2}\sum_{l>0}W^{2l+1}\frac{l+1}{l(2l+1)}|g^{(l)}|^{2}~{}.\end{split}$
(8)

Positivity of the squared amplitudes appearing in the sum over states implies
$\operatorname{Im}\Pi_{J}^{L/T}\Big{|}_{D\pi}\leq\operatorname{Im}\Pi_{J}^{L/T}$.
Inequalities for the $B\rightarrow D\pi\ell\nu$ form factors can then be
derived from this inequality by inserting Eq. (8) and the perturbative
expression for $\operatorname{Im}\Pi_{J}^{L/T}$. These so-called unitarity
bounds provide $q^{2}$-dependent constraints that should be incorporated in
determinations of the $B\rightarrow D\pi\ell\nu$ form factors. Conveniently,
the unitarity bounds apply only to either the single form
factor$\mathcal{F}_{2}^{(l)}$, the pair of form factors
$\mathcal{F}_{1}^{(l)}/f^{(l)}$, or the single form factor $g^{(l)}$ rather
than more general linear combinations. A parameterization of the
$q^{2}$-dependence of the form factors is required to concretely specify how
the bounds are imposed; we turn to this next.

Coupled-channel S-wave and $z$-expansion —  The model-independent
parameterization and bounds presented in previous sections make no assumptions
about the number, energies, or lineshapes of possible resonances. To render
fitting the measured $B\rightarrow D\pi\ell\nu$ decay spectra to our
parameterization more tractable, it is helpful to include additional
theoretical information and make some plausible assumptions.

The semileptonic $B$-decay form factors can be factorized into a part
describing the short-distance weak decay and a part encoding the long-ranged
final-state interactions between the hadrons [39, 40]:

$f^{(l)}(q^{2},M_{D\pi}^{2})=\hat{f}^{(l)}(q^{2},M_{D\pi}^{2})g^{(l)}(M_{D\pi}^{2})~{}.$
(9)

The weak-interaction contribution to the form factors of QCD resonances is
approximately independent of $M_{D\pi}$ [41]:

$\hat{f}^{(l)}(q^{2},M_{D\pi}^{2})\approx\tilde{f}^{(l)}(q^{2})+\mathcal{O}((M_{R}^{2}-M_{D\pi}^{2})/M_{B}^{2})~{},$
(10)

Indeed, studies of $B\rightarrow\pi\pi(K)$ in the context of light cone sum
rules (LCSR) [42] and recent LQCD studies of the $B\rightarrow\rho$ form
factors [43] point towards the smallness of the neglected contributions.

The interaction potentials relevant for coupled-channel S-wave $D\pi$, $D\eta$
and $D_{s}K$ scattering have been determined to next-to-leading order in UChPT
[44] and the $S$-matrix has been obtained from those results in Ref. [24]. To
connect the semileptonic-decay form factors to the known $S$-matrix, we make
use of the dispersion relation [45]

$\begin{split}&\operatorname{Im}{\vec{f}(q^{2},M_{D\pi}^{2}+i\epsilon)}\\\
&=T^{\ast}(M_{D\pi}^{2}+i\epsilon)\Sigma(M_{D\pi}^{2})\vec{f}(q^{2},M_{D\pi}^{2}+i\epsilon)~{},\end{split}$
(11)

where the form-factor $f$ is a vector in channel space and stands for either
$f_{+}$ or $f_{0}$. Here, $T$ is the $T$-matrix, and $\Sigma$ contains the
relevant phase-space factors and is defined in Ref. [45]. The solution of Eq.
(11) is well known and given by the Muskhelishvili-Omnès (MO) matrix $\Omega$
[45, 46]

$\begin{split}&\vec{f}(q^{2},M_{D\pi}^{2})=\Omega(M_{D\pi}^{2})\vec{P}(q^{2},M_{D\pi}^{2})~{},\\\
&\operatorname{Im}{\Omega(s+i\epsilon)}=\frac{1}{\pi}\int_{s_{\mathrm{thr}}}^{\infty}\frac{T^{\ast}(s^{\prime})\Sigma(s^{\prime})\Omega(s^{\prime})}{s^{\prime}-s-i\epsilon}\mathrm{d}s^{\prime}~{},\end{split}$
(12)

where, $\vec{P}$ are boundary functions and we neglect higher order terms in
$M_{D\pi}$ following the same arguments as for Eq. (10). A numerical algorithm
to solve the above integral equation is outlined in Ref. [47].

The $q^{2}$-dependent remainder of a given form factor $f$ defined in Eq.
(10), as well as the boundary functions $\vec{P}$ in Eq. (12) can be expanded
as [32, 33, 34]

$\tilde{f}_{l}(q^{2})=\frac{1}{\phi^{(f)}_{l}(q^{2})B_{f}(q^{2})}\sum_{i=0}^{\infty}a^{(f)}_{li}z^{i}$
(13)

where $B_{f}$ is a Blaschke product taking into account the poles of all
subthreshold $B_{c}$ resonances for a given channel and $\phi^{(f)}_{l}$ is
the relevant outer function. Additional details on the derivation and
numerical calculation of the outer functions can be found in the supplemental
material [37] and Ref. [48]. The variable $z$ is given by

$z(q^{2},q^{2}_{0})=\frac{q^{2}_{0}-q^{2}}{(\sqrt{q^{2}_{+}-q^{2}}-\sqrt{q^{2}_{+}-q_{0}^{2}})^{2}}$
(14)

where $q^{2}_{+}=(M_{B}+M_{D}+m_{\pi})$ and ranges from $0$ to $0.06$ for
$q^{2}_{0}=0$ GeV2 . In terms of the coefficients $a^{(f)}_{li}$ the unitarity
bounds in Eq. (8) take the form

$\sum_{i,l}|a^{(f)}_{li}|^{2}<1~{},$ (15)

allowing for an easy integration of the form factors in a fit, including
priors on the coefficients. Recently discussed problems associated to lower-
lying branch cut at $q^{2}=(M_{B}+M_{D})^{2}$ can be incorporated as outlined
in Refs. [49, 50, 51, 52].

Experimental fits —  To test our new $B\rightarrow D\pi\ell\nu$ form-factor
description and extract the coefficients of the $z$ expansion from data, we
proceed in two steps. First, we fit the measured $w$\- and
$\cos\theta$-dependence of the $B\rightarrow D_{2}^{\ast}\ell\nu$ differential
decay width [53] and $B^{0}\rightarrow D_{2}^{\ast-}\pi^{+}$ branching
fraction [6] constraining the $a_{li}^{(f)}$ with Gaussian priors centered at
zero with unit width. We employ the least-squares fitting package `lsqfit` and
use the augmented $\chi_{\rm aug}^{2}$ defined in Refs. [54, 55] to assess the
goodness of fit. Additional numerical inputs are taken from Refs. [56, 57, 58,
59] as discussed in the supplemental material [37]. The loose constraints help
the fit converge more quickly but have little impact on the final results
since the magnitudes of the resulting $z$-coefficients are all of order a
tenth or smaller.

As shown in Fig. 1, we find a harder, i.e., enhanced for high values of
$q^{2}$, $D_{2}^{\ast}$ $w$-spectrum than Refs. [22, 23] and also better
describe the data.

Figure 1: Normalized $B\rightarrow D_{2}^{\ast}\ell\nu$ $w$-spectrum. The
black data points are from Ref. [53]. The blue solid curve with error band is
our fit result, while the orange dashed curve and band are from Refs. [22,
23]. The $\chi_{\rm aug}^{2}/\text{dof}=6.4/12$ and $Q=0.9$.

The most likely reason for this is the greater flexibility in the model
independent approach employed here in comparison to the heavy-quark effective
theory (HQET) based approach of these works.

Next, we fit the $B\rightarrow D\pi\ell\nu$ $M_{D\pi}$-spectrum measured
recently by Belle [60] using the $z$-expansion coefficients from the first fit
as priors to constrain the shape the $D_{2}^{\ast}$ form factors. Following
Refs. [61, 30], we parameterize the $D^{\ast}$ and $D^{\ast}_{2}$ lineshapes
by a Breit-Wigner distribution and with Blatt-Weisskopf damping factors [62,
63]. In contrast to Ref. [30] we allow the Blatt-Weisskopf radius to be
determined in the fit. As shown in Fig. 2, our form-factor parameterization
provides a good description of the data over the entire invariant-mass range.

Figure 2: Fit of the measured $M_{D\pi}$-spectrum [60] using the $z$-expansion
to parameterize $D_{2}^{\ast}$ and S-wave form factors. The $\chi^{2}_{\rm
aug}/\text{dof}=124.4/133$ and $Q=0.69$. Only data for the more precise
$B^{+}$ mode is shown.

Our fit to the $B\rightarrow D\pi\ell\nu$ invariant-mass spectrum can be used
to make predictions for related quantities. Figure 3 shows the predicted
partial-wave contributions to the $q^{2}$-spectrum. After integrating over the
momentum transfer, we obtain for the D-wave channel $\mathrm{Br}(B\rightarrow
D_{2}^{\ast}(\rightarrow D\pi^{\pm})\ell\nu)=(1.90\pm 0.11)\times 10^{-3}$,
which is larger than Belle’s determination in Ref. [60]. This is because the
smooth falling function employed by Belle to describe the seemingly
nonresonant contributions overlaps with the $D_{2}^{\ast}$ resonance, whereas
in our description, the S-wave and $D^{\ast}$ components are negligible near
the resonance. For the S-wave contribution, we obtain
$\mathrm{Br}(B\rightarrow(D\pi)_{S}\ell\nu)=(1.03\pm 0.27)\times 10^{-3}$,
which agrees with the arguments made in Ref. [30] but is smaller than the
branching fraction usually assigned in experimental analyses.

Figure 3: Predicted partial-wave decomposition of the $B\rightarrow
D\pi\ell\nu$ $q^{2}$-spectrum (dashed and dotted curves with error bands) and
their total (solid curve with error band) from the fit in Fig. 2.

Implications and outlook —  We present the first model-independent description
of $B\rightarrow D\pi\ell\nu$ decays based on unitarity and analyticity of the
relevant form factors and the factorization of final-state interactions. This
constitutes the first generalization of the BGL parameterization to multi-
hadron final states and provides the first step towards a model-independent
study of semileptonic $B$-meson decays into higher resonances and non-resonant
final states. Our framework does not include any assumptions about lineshapes
of resonances and is extendable to other decay processes with charmed mesons
in the final state such as $B\rightarrow D^{\ast}\pi\ell\nu$ or
$B_{s}\rightarrow DK\ell\nu$. Further, it is also valid for final states with
more than two hadrons, and can be combined with other known $b\rightarrow c$
form factors in a global fit to obtain constraints on less well-known form
factors (see e.g Ref. [64]). By replacing the $D$ meson by a pion, the
unitarity bounds can be applied to $B\rightarrow\pi\pi\ell\nu$ decays,
including the phenomenologically interesting $B\rightarrow\rho\ell\nu$
channel, which is the target of first LQCD calculations beyond the narrow-
width limit [43], as well as non-resonant backgrounds, which constitute the
dominant systematic uncertainty [65].

Taking into account recent theoretical considerations and measurements of
$B\rightarrow D\pi\ell\nu$ decays by Belle, we provide precise predictions for
semileptonic decays into the broad two-pole structure in the S-wave and
determine the form-factor parameters for $B\rightarrow D_{2}^{\ast}\ell\nu$
decays from data. This marks the first time in which a three-component
hypothesis consisting of S-wave contributions, $D^{\ast}$ virtual
contributions and $D_{2}^{\ast}$ contributions, is compared to the measured
$M_{D\pi}$-spectrum. Previous works either do not include all three components
simultaneously [53, 66, 22] or do not compare to the measured
$M_{D\pi}$-spectra [30]. We demonstrate, in contrast to existing literature,
that our treatment of the S-wave is compatible with the $M_{D\pi}$-spectrum
measured by Belle and thus is favored over previous models that assume a
single, broad S-wave state, the $D_{0}^{\ast}(2300)$. While more careful
studies need to be conducted, the change in the shape for $B\rightarrow
D_{2}^{\ast}\ell\nu$ decays, as well as the inclusion of the virtual
$D^{\ast}$ contribution lead to an overall harder $q^{2}$-spectrum,
potentially resolving some of the discrepancies seen in inclusive analyses at
high $q^{2}$ [10, 7].

The coupled channel nature of the S-wave contribution enables us to obtain
predictions for $B\rightarrow D\eta\ell\nu$ and $B\rightarrow D_{s}K\ell\nu$
decays purely based on measurements of $B\rightarrow D\pi\ell\nu$ decays. For
the $D\eta$ S-wave contribution we obtain
$\mathrm{Br}(B\rightarrow(D\eta)_{S}\ell\nu)=(1.9\pm 1.7)\times 10^{-5}$, two
orders of magnitude too small to constitute a sizeable portion of the
semileptonic gap. Consequently, both approaches utilized by the Belle and
Belle II Collaborations in recent measurements to fill the semileptonic gap in
terms of $B\rightarrow D^{(\ast)}\eta\ell\nu$ decays, either via a broad
S-wave resonance or equidistributed in phase space, are ruled out.

Additional theoretical work, such as LCSR computations of the S-wave form
factors [67], would greatly improve the results presented in this letter.

Future experimental measurements of the $q^{2}$\- and $\cos\theta_{l}$-spectra
of $B\rightarrow D_{2}^{\ast}\ell\nu$ decays by Belle II with the already
available data set, as well as updated angular analyses of $B^{0}\rightarrow
D^{0}\pi^{-}\pi^{+}$ and $B^{0}\rightarrow D^{0}\pi^{-}K^{+}$ decays by LHCb,
would improve the precision of the form factors presented in this letter. In
the long term, a full partial-wave analysis of $B\rightarrow D\pi\ell\nu$
decays is required to ultimately determine the exact composition of the $D\pi$
spectrum in semileptonic decays. Additionally, the final state interactions
between $D$ mesons and light hadrons can be tested by measuring femtoscopic
correlation functions at the ALICE experiment [68]. This result could provide
a direct, orthogonal test of the S-wave two-pole structure in heavy ion
collisions [69].

##  I Acknowledgments

We thank Dmitri Liventsev and Frank Meier for discussions and providing us
with the data from Refs. [53] and [60], respectively, and Miguel Albaladejo
for providing us with code to reproduce the results of Ref. [24]. We also
thank Hank Lamm, Andreas Kronfeld and Luka Leskovec for helpful discussions on
the subject matter and Bastian Kubis for helpful feedback on the draft. RvT
thanks Fermilab and the Karlsruhe Institute of Technology for hospitality
during the completion of this work. Fermilab is operated by Fermi Research
Alliance, LLC under contract number DE-AC02-07CH11359 with the United States
Department of Energy. FH acknowledges support by the Alexander von Humboldt
foundation. RvT acknowledges support by the Natural Sciences and Engineering
Research Council of Canada.

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## Supplemental material

###  .1 Fivefold differential decay rate

The fivefold differential decay rate for $B\rightarrow D\pi\ell\nu$ decays is
given by

$\frac{\mathrm{d}\Gamma_{B\rightarrow
D\pi\ell\nu}}{\mathrm{d}M_{D\pi}^{2}\mathrm{d}q^{2}\mathrm{d}\cos\theta\mathrm{d}\cos\theta_{l}\mathrm{d}\chi}=\frac{G_{F}^{2}|V_{cb}|^{2}}{M_{B}}\frac{1}{(4\pi)^{6}}\left(1-\frac{m_{l}^{2}}{q^{2}}\right)W\sum_{a,b}\left|\mathcal{M}_{ab}\right|^{2}~{},$
(16)

where $a,b$ label the partial waves and $G_{F}$ is Fermi’s constant and the
matrix.

The matrix element squared is given by

$\left|\mathcal{M}_{ab}\right|^{2}=\left\langle
D(p_{D})\pi(p_{\pi})|V_{\mu}-A_{\mu}|B(p_{B})\right\rangle\left\langle
D(p_{D})\pi(p_{\pi})|V_{\nu}-A_{\nu}|B(p_{B})\right\rangle
L^{\mu\nu}(p_{l},p_{\nu})$ (17)

where

$L^{\mu\nu}(p_{l},p_{\nu})=\mathrm{Tr}\left(\gamma^{\mu}P_{L}(\not{p}_{l}-m_{l})\gamma^{\nu}P_{L}\not{p}_{\nu}\right)$
(18)

is the leptonic tensor and $P_{L}=(\mathbb{1}-\gamma_{5})/2$.

Inserting Eq. (1), we obtain

$\begin{split}\left|\mathcal{M}_{ab}\right|^{2}&=M_{B}^{2}M_{D\pi}^{2}(q^{2}-m_{l}^{2})W^{a+b}\Bigg{\\{}\left[\frac{\tilde{\mathcal{F}}_{1,a}\mathcal{F}^{\ast}_{1,b}}{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})q^{2}}+\frac{m_{l}^{2}}{q^{4}}\mathcal{F}_{2,a}\mathcal{F}^{\ast}_{2,b}-\frac{f_{a}f^{\ast}_{b}}{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}-g_{a}g^{\ast}_{b}\right]P_{a}^{0}P_{b}^{0}\\\
&-\left(1-\frac{m_{l}^{2}}{q^{2}}\right)\frac{\tilde{\mathcal{F}}_{1,a}\mathcal{F}^{\ast}_{1,b}}{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})q^{2}}\cos^{2}\theta_{l}P_{a}^{0}P_{b}^{0}\\\
&+\left(\frac{f_{a}f^{\ast}_{b}}{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}+g_{a}g^{\ast}_{b}\right)\left[P_{a-1}^{0}P_{b-1}^{0}+\frac{P_{a-1}^{1}P_{b-1}^{1}}{ab}\right]-\left(1-\frac{m_{l}^{2}}{q^{2}}\right)g_{a}g^{\ast}_{b}\frac{P_{a}^{1}P_{b}^{1}}{ab}(1-\cos^{2}\theta_{l})\\\
&-\left(1-\frac{m_{l}^{2}}{q^{2}}\right)\left(\frac{f_{a}f^{\ast}_{b}}{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}-g_{a}g^{\ast}_{b}\right)\frac{P_{a}^{1}P_{b}^{1}}{ab}\cos^{2}\chi(1-\cos^{2}\theta_{l})\\\
&-\frac{f_{a}g^{\ast}_{b}+g_{a}f^{\ast}_{b}}{\sqrt{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}}\left[P_{a-1}^{0}P_{b-1}^{0}+\frac{P_{a-1}^{1}P_{b-1}^{1}}{ab}\right]\cos\theta_{l}+\left(\frac{f_{a}g^{\ast}_{b}+g_{a}f^{\ast}_{b}}{\sqrt{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}}+\frac{m_{l}^{2}}{q^{4}}\frac{\tilde{\mathcal{F}}_{1,a}\mathcal{F}^{\ast}_{2,b}+\mathcal{F}_{2,a}\mathcal{F}^{\ast}_{1,b}}{\sqrt{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}}\right)\cos\theta_{l}P_{a}^{0}P_{b}^{0}\\\
&+i\left(1-\frac{m_{l}^{2}}{q^{2}}\right)\frac{g_{a}f_{b}^{\ast}-f_{a}g_{b}^{\ast}}{\sqrt{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}}\frac{P_{a}^{1}P_{b}^{1}}{ab}(1-\cos^{2}\theta_{l})\cos\chi\sin\chi\\\
&+\left(1-\frac{m_{l}^{2}}{q^{2}}\right)\left[\frac{\tilde{\mathcal{F}}_{1,a}f_{b}^{\ast}}{\sqrt{q^{2}}\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}\frac{P_{a}^{0}P_{b}^{1}}{b}+\frac{f_{a}\mathcal{F}^{\ast}_{1,b}}{\sqrt{q^{2}}\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}\frac{P_{a}^{1}P_{b}^{0}}{a}\right]\cos\theta_{l}\sin\theta_{l}\cos\chi\\\
&+i\left(1-\frac{m_{l}^{2}}{q^{2}}\right)\left[\frac{\tilde{\mathcal{F}}_{1,a}g_{b}^{\ast}}{\sqrt{q^{2}}\sqrt{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}}\frac{P_{a}^{0}P_{b}^{1}}{b}-\frac{g_{a}\mathcal{F}^{\ast}_{1,b}}{\sqrt{q^{2}}\sqrt{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}}\frac{P_{a}^{1}P_{b}^{0}}{a}\right]\cos\theta_{l}\sin\theta_{l}\sin\chi\\\
&-\left[\left(\tilde{\mathcal{F}}_{1,a}g_{b}^{\ast}+\frac{m_{l}^{2}}{q^{2}}\mathcal{F}_{2,a}f_{b}^{\ast}\right)\frac{P_{a}^{0}P_{b}^{1}}{b}+\left(g_{a}\mathcal{F}^{\ast}_{1,b}+\frac{m_{l}^{2}}{q^{2}}f_{a}\mathcal{F}^{\ast}_{2,b}\right)\frac{P_{a}^{1}P_{b}^{0}}{a}\right]\frac{\cos\chi\sin\theta_{l}}{\sqrt{q^{2}}\sqrt{\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}}\\\
&-i\left[\left(\tilde{\mathcal{F}}_{1,a}f_{b}^{\ast}+\frac{m_{l}^{2}}{2q^{2}}\mathcal{F}_{2,a}g_{b}^{\ast}\right)\frac{P_{a}^{0}P_{b}^{1}}{b}-\left(f_{a}\mathcal{F}^{\ast}_{1,b}+\frac{m_{l}^{2}}{2q^{2}}g_{a}\mathcal{F}^{\ast}_{2,b}\right)\frac{P_{a}^{1}P_{b}^{0}}{a}\right]\frac{\sin\chi\sin\theta_{l}}{\sqrt{q^{2}}\lambda(M_{B}^{2},M_{D\pi}^{2},q^{2})}\Bigg{\\}}~{},\end{split}$
(19)

where we have introduced
$\tilde{\mathcal{F}}_{1,i}=(M_{B}^{2}-M_{D\pi}^{2})\mathcal{F}_{1,i}$ for
brevity. The $P_{a}^{i}$ are the associated Legendre polynomials with
arguments of $\cos\theta$. Note, that this expression is also valid for
massive leptons and incorporates all interference terms between different
partial waves.

###  .2 Unitarity bounds and outer functions

The derivation of outer functions from the contributions to the unitarity
bounds in Eq.(8) is complicated by the appearance of integrals of the form

$\mathcal{I}^{(l,i)}(q^{2})=\int_{(M_{D}+m_{\pi})^{2}}^{(\sqrt{q^{2}}-M_{B})^{2}}\mathrm{d}M_{D\pi}^{2}\,M_{D\pi}^{2}W^{2l+1}\lambda^{i}(M_{B}^{2},M_{D\pi}^{2},q^{2})g^{(l)}(M_{D\pi}^{2})~{}.$
(20)

In the narrow-width limit these integrals can be evaluated analytically and
for $l=1$ lead to the standard expressions of Ref. [34].

For more complicated $g^{(l)}$ no simple analytic solution exists.
Consequently the contribution to the outer functions from the integrals needs
to be computed numerically through the equation [48]

$\tilde{\phi}^{(f)}_{l}(q^{2})=e^{i\varphi}\exp\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}t\frac{e^{it}+z}{e^{it}-z}\log|\mathcal{I}^{(l,i)}(e^{it})|\right)~{}.$
(21)

Here $\varphi$ is an arbitrary phase and the full outer function
$\phi^{(f)}_{l}$ is the product of the appropriate
$\tilde{\phi}^{(f)}_{l}(q^{2})$, as well as constants and simple functions of
$z$ that can be derived following Ref. [34].

###  .3 Fit of experimental data

To fit to the available data from the Belle experiment [53, 60] we need to
parametrize the lineshapes of the $D^{\ast}$ and $D_{2}^{\ast}$. To this end,
we write the $M_{D\pi}$-dependent part of form factors such as $g^{(l)}$ in
the decomposition of Eq. (9) as

$g^{(l)}(M_{D\pi})=\frac{g_{l}}{(M_{D\pi}^{2}-M_{R,l}^{2})+iM_{R,l}\Gamma_{R}(M_{D\pi}^{2})}X^{(l)}(|\vec{p}_{D}|r_{\mathrm{BW}},|\vec{p}_{D,0}|r_{\mathrm{BW}})~{}.$
(22)

Here, $M_{R,l}$ are the respective nominal resonance masses, $g_{l}$ is the
coupling of the resonances to a $D$-meson and a pion, $\Gamma_{R}$ the energy-
dependent width and the $X^{(l)}$ are Blatt-Weisskopf damping factors. We take
the masses, widths and branching ratios for the $D^{\ast}$ and $D_{2}^{\ast}$
from Ref. [6] and determine $g_{l}$ from them. The damping factors are given
by

$X^{(1)}(z,z_{0})=\sqrt{\frac{1+z_{0}^{2}}{1+z^{2}}}~{},\qquad
X^{(2)}(z,z_{0})=\sqrt{\frac{9+3z_{0}^{2}+z_{0}^{4}}{9+3z^{2}+z^{4}}}~{}.$
(23)

The $D$-meson three-momentum is evaluated in the resonance center-of-mass
frame, $|\vec{p}_{D,0}|$ denotes the $D$-meson three-momentum for
$M_{D\pi}=M_{R}$. The Blatt-Weisskopf radius $r_{\mathrm{BW}}$ is treated as a
parameter in the fit with a prior of $4\pm 1$ GeV-1, where the central value
corresponds to the value chosen by the LHCb experiment for their amplitude
analysis of $B^{-}\rightarrow D^{+}\pi^{-}\pi^{-}$ decays [29].

Uncertainties in the scattering phase shifts from Ref. [24] are obtained using
resampling techniques. In particular, we compute our results using an ensemble
of 150 samples provided by the authors of Ref. [24] that describe the combined
$1\sigma$ confidence interval of the parameters fitted in that work [56],
compute statistical uncertainties from the variation of our results across
these resampled ensembles, and combine these uncertainties in quadrature with
the other uncertainties appearing in our analysis.

To obtain the P-Wave form factors, we integrate the five-fold differential
decay rate over $M_{D\pi}$ and match the four-fold differential decay rate to
the one obtained from the LQCD determination of the $B\rightarrow
D^{\ast}\ell\nu$ form factors by the Fermilab/MILC collaboration [57]:

$\int\mathrm{d}M_{D\pi}^{2}\frac{\mathrm{d}^{5}\Gamma}{\mathrm{d}M_{D\pi}^{2}\mathrm{d}q^{2}\mathrm{d}\cos\theta\mathrm{d}\cos\theta_{l}\mathrm{d}\chi}=\mathrm{Br}(D^{\ast}\rightarrow
D\pi)\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}q^{2}\mathrm{d}\cos\theta\mathrm{d}\cos\theta_{l}\mathrm{d}\chi}\Bigg{|}_{\text{FNAL/MILC}}~{}.$
(24)

In a first step, we obtain the D-Wave form factor parameters by performing a
simultaneous fit to the $w$\- and $|\cos\theta|$\- spectra from Ref. [53], as
well as the world averages for the nonleptonic rates for $B^{0}\rightarrow
D_{2}^{\ast-}\pi^{+}$ and $B^{0}\rightarrow D_{2}^{\ast-}K^{+}$ [6]. We
require regularity at $q^{2}=0$

$\begin{split}A_{1}^{(l)}(0,M_{D\pi}^{2})&=M_{D\pi}A_{0}^{(l)}(0,M_{D\pi}^{2})\\\
&-(M^{2}_{B}-M^{2}_{D\pi})A_{2}^{(l)}(0,M_{D\pi}^{2})~{},\end{split}$ (25)

connecting the longitudinal form factor $A_{2}$, which appears in the
nonleptonic decays under consideration, to the other three form factors. We
subtract the contribution of $B_{c}$ meson states for which the leptonic decay
constants are known from the susceptibilities to strengthen the unitarity
bounds following Refs. [58, 59] and truncate the $z$-expansion of the form
factors at linear order in $z$.

We then perform a fit to the $M_{D\pi}$-spectra of Ref. [60] with the S-, P-
and D-Wave form factor parameters entering the fit with priors set to the
previously extracted values and covariances, masses and widths constrained by
their world averages and $r_{\mathrm{BW}}$ treated as described above. Both
charge modes are fitted simultaneously, unlike in experimental fits to the
mass spectrum, with the sole difference being the exact values of the involved
masses. Since $D^{\ast 0}\rightarrow D^{+}\pi^{-}$ decays are not allowed on-
shell, this leads to slightly different shapes in the $M_{D\pi}$ spectrum at
low masses. As a cross-check, we also considered the $M_{D\pi}$-spectra of
Ref. [53], finding good agreement and compatible results with the main fit.