# Pion mass dependence in $D\pi$ scattering and the $D_{0}^{*}(2300)$
resonance from lattice QCD

Haobo Yan<EMAIL_ADDRESS>School of Physics, Peking University, Beijing
100871, China  Chuan Liu<EMAIL_ADDRESS>School of Physics, Peking
University, Beijing 100871, China  Center for High Energy Physics, Peking
University, Beijing 100871, China  Collaborative Innovation Center of Quantum
Matter, Beijing 100871, China  Liuming Liu<EMAIL_ADDRESS>Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  University
of Chinese Academy of Sciences, Beijing 100049, China  Yu Meng
<EMAIL_ADDRESS>School of Physics, Zhengzhou University, Zhengzhou 450001,
China  Hanyang Xing  Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  University of Chinese Academy of Sciences, Beijing
100049, China

###### Abstract

Lattice QCD results for isospin $I=\frac{1}{2}$ $D\pi$ scattering are
presented. Utilizing a series of $N_{\text{f}}=2+1$ Wilson-Clover ensembles
with pion masses of $m_{\pi}\approx 132,208,305$ and $317$ MeV, various two-
particle operators are constructed and the corresponding finite-volume spectra
are determined. The $S$ and $P$-wave scattering phase shifts are then
extracted using the Lüscher approach. A clear trend for the motion of the
$D_{0}^{*}(2300)$ pole is identified. With the physical pion mass
configurations also included, this calculation constitutes the first lattice
calculation in which the pion mass dependence of the $D_{0}^{*}(2300)$ pole is
investigated and the scattering lengths are extrapolated/interpolated to the
physical pion mass in $D\pi$ scattering.

_Introduction._ — Over two decades, both experimental and theoretical efforts
have been devoted to the study of scalar resonance $D_{0}^{*}(2300)$, which
has been discovered and verified by many experiments [1, 2, 3, 4, 5]. Still,
various puzzles remain. The scalar resonance $D_{0}^{*}(2300)$ lies so close
to its strange partner $D_{s0}^{*}(2317)$, which is attributed within the
quark model to the strong coupling to the threshold [6]. There are also
suggestions of a two-pole structure in this particular channel from unitarized
chiral perturbation theory [7]. However, this assertion needs to be tested at
different quark masses which is impossible in real-world experiments. It is
therefore crucial to study $D\pi$ scattering systematically within the non-
perturbative framework such as lattice QCD in which pion mass dependence could
be analyzed. This is what we would like to address in this letter.

Furthermore, the study of the $D\pi$ two-meson system also provides a pivotal
input to the more complicated three-body $DD\pi$ system. An example that has
drawn much attention recently is the doubly-charmed tetraquark candidate,
$T_{cc}(3875)\to DD^{*}\to DD\pi$ discovered by the LHCb collaboration [8, 9].
There have been several lattice calculations on $DD^{*}$ scattering recently,
claiming the state that is related to $T_{cc}$ could be identified [10, 11,
12, 13]. However, the actual lowest $T_{cc}$ decay channel is a three-body
channel and the large pion mass used in these lattice studies might obscure
such complexity by lifting the $DD\pi$ channel beyond the $DD^{*}$ threshold.
The more complete approach thus involves the three-body scattering problem on
the lattice and relies on the $D\pi$ two-body scattering amplitudes as an
input, see recent development on the three-body quantization conditions [14].

To search for hadronic resonances on the lattice, the state-of-the-art method
is to study the scattering amplitudes of the decay products by applying the
Lüscher’s formulation to the finite-volume energy levels. The scattering
amplitudes are then analytically continued to the complex energy plane. The
poles found in these scattering amplitudes would correspond to various hadrons
found in the experiments. There are, however, indirect extraction of the
$I=\frac{1}{2}$ $D\pi$ scattering parameters. In Ref. [15], the scattering
length is extracted from the lattice-determined form factor by a combined fit
of the scattering length and the Omnès subtraction parameters. In Ref. [16]
the scattering length of $I=\frac{1}{2}$ $D\pi$ at physical $m_{\pi}$ was
obtained within chiral perturbation theory with low-energy constants
determined from other channels. There are also calculations of the scattering
length from phenomenological models, see review [17] and also [18, 19, 20,
21].

Direct lattice QCD calculations on $I=\frac{1}{2}$ $D\pi$ scattering were
conducted on an $N_{\text{f}}=2$, $m_{\pi}\approx 266$ MeV lattice [22] and a
resonance pole was found. Afterward, the Hadron Spectrum Collaboration
performed a $D\pi,D\eta,D_{s}\bar{K}$ coupled-channel analysis on an
$N_{\text{f}}=2+1$ anisotropic lattice ensemble at $m_{\pi}\approx 391$ MeV
and found $D_{0}^{*}$ to be a bound state [23]. They also performed a single-
channel analysis at $m_{\pi}\approx 239$ MeV and found a resonance [24]. A
calculation at the $SU(3)$ point is also performed [25]. However, focusing on
the physical strange sector, the pole positions determined by the lattice
studies are all lower than the PDG average value [26]. The pole position from
these lattice results also moves in the opposite direction, i.e. away from the
experimental value as $m_{\pi}$ decreases. Therefore, it is crucial to examine
the movement of the $D_{0}^{*}$ pole at different $m_{\pi}$ values,
particularly its physical value. Additionally, the experimental measurements
are presented as the bare parameters of the relativistic Breit-Wigner(RBW),
which could be largely different from the pole positions for broad resonances
such as $D_{0}^{*}(2300)$. A more consistent practice is to translate not only
the lattice results but also the parameters from the experiments to the pole
positions.

In this letter, we report lattice results on $D\pi$ scattering using a set of
symmetric $N_{\text{f}}=2+1$ Wilson-clover lattice ensembles at
$m_{\pi}\approx 132,208,305$ and $317$ MeV to further examine the movement of
the $D_{0}^{*}$ pole in the complex plane. The pole positions are then
compared to the experimental measurements. The scattering lengths are
interpolated/extrapolated towards the physical pion mass point, where a
comparison between lattice and experiment is made.

_Finite volume spectra._ — Lattice QCD calculations are performed using Monte
Carlo sampling of the QCD path integral. The gauge field configurations in
this work are generated with $N_{\text{f}}=2+1$ flavors of quarks using the
tadpole-improved tree-level Symanzik gauge action and the tadpole-improved
tree-level Clover fermions [27]. The gauge fields are periodic in all
directions and the fermions are periodic in space and anti-periodic in time.
There are plenty of studies based on these ensembles, see, for e.g., [28, 29,
30, 31, 32, 33, 34, 35, 36]. The simulation is performed on $m_{\pi}\approx
132,208,305$ and $317$ MeV with the details listed in Tab. 1. Among these
configurations, F32P21, F48P21, and F32P30, F48P30 are two couples that share
the same pion mass and lattice spacing but different volumes in order to
obtain more kinematic points in the finite-volume spectra, rendering a more
stable and precise determination of the scattering parameters. A finer lattice
H48P32 is used to estimate the lattice discretization effects.

Table 1: Gauge configurations used in this work. The errors are purely statistical. configuration | volume |  $a$/fm |  $m_{\pi}$/MeV | $N_{\text{confs}}$
---|---|---|---|---
C48P14 | $48^{3}\times 96$ | $0.10530(18)$ | $131.5(1.8)$ | $132$
F32P21 | $32^{3}\times 64$ | $0.07746(18)$ | $206.8(2.1)$ | $459$
F48P21 | $48^{3}\times 96$ | $0.07746(18)$ | $208.12(70)$ | $222$
F32P30 | $32^{3}\times 96$ | $0.07746(18)$ | $305.81(71)$ | $567$
F48P30 | $48^{3}\times 96$ | $0.07746(18)$ | $304.98(50)$ | $201$
H48P32 | $48^{3}\times 144$ | $0.05187(26)$ | $317.00(68)$ | $274$

To constrain the phase shifts more effectively near the threshold, many
operators need to be constructed to interpolate the full spectra in a
particular channel. The operators will interpolate the vacuum to the $D\pi$
system with desired quantum numbers, and so the possible resonances.

Since the physical process happens on a cubic spacetime, the rotational
symmetries are broken into the cubic group, and its subgroups are further
reduced by the Lorentz boost. The finite-volume spectra are also subduced into
irreducible representations (irreps) of the little groups. We consider
reference frames with a center of mass (CM) three-momenta $\vec{P}$= [0,0,0],
[0,0,1], [0,1,1], [1,1,1], [0,0,2] in the unit of $\frac{2\pi}{L}$, which
corresponds to the cubic group $O_{h}$ and the little groups
$C_{4v},C_{2v},C_{3v},C_{4v}$, respectively. We consider all irreps with
leading partial waves up to $P$-wave for $D\pi$ scattering. Note that since
$m_{D}\neq m_{\pi}$, $S$\- and $P$-wave can mix in moving frames where parity
is no longer a good quantum number. For the $D\pi$ system, up to $P$-wave, the
irreps $A_{1}$ of $C_{4v},C_{2v},C_{3v}$ contain both $S$ and $P$-wave; irreps
$T_{1}^{-}$ of $O_{h}$, $E_{2}$ of $C_{4v},C_{3v}$ and $B_{1},B_{2}$ of
$C_{2v}$ house only $P$-wave; $A_{1}^{+}$ of $O_{h}$ houses only $S$-wave. The
explicit forms of operators used in this work are listed in the Supplemental
Material [37].

We have developed a Mathematica package OpTion (Operator construcTion) that
can construct general $N$-hadron operators. By setting the type of hadrons,
the desired quantum numbers, the number of derivatives, and the maximum
relative momentum, the operator set can be obtained by one line of
straightforward commands. The formalism of the operator construction will be
described in detail by an independent publication [38].

The single meson mass and the finite-volume spectra can be obtained from the
spectral decomposition from the two-point correlation matrices using the
interpolating operators $O_{i}$,

$C_{ij}(t)=\sum_{t^{\prime}}\langle
O_{i}(t+t^{\prime})O_{j}^{\dagger}(t^{\prime})\rangle_{T}$ (1)

where $t^{\prime}$ loops over all time slices to enhance the signal. Since the
single mesons $D$ and $\pi$ are stable and are the lowest states of their
quantum number, a single quark-bilinear operator suffices to extract the
energy.

The distillation method [39] is used to reduce the cost of generating
propagators, which are calculated only in the subspace spanned by the $N_{v}$
lowest eigenvectors of the covariant Laplacian operator. The quark fields are
smeared by the distillation operator such that the high-lying modes are
suppressed. And $C_{ij}(t)$’s are then estimated using relevant perambulators.
The $N_{v}$ applied in this work is $100,200$ for ensembles with volume
$32^{3}$ and $48^{3}$, respectively. It has been checked that the spectra are
insensitive to $N_{v}$ and the error would slightly increase with smaller
values of $N_{v}$.

The finite-volume energy spectra in each irrep can be obtained by solving a
generalized eigenvalue problem (GEVP) [40] of the correlation matrices,

$C(t)v_{n}(t,t_{0})=\lambda_{n}(t,t_{0})C(t_{0})v_{n}(t,t_{0}),$ (2)

where the reference time $t_{0}=20$ for H48P32 and $t_{0}=10$ for other
configurations. Changing $t_{0}$ hardly affects the resulting spectra. The
energy levels of the $n^{\text{th}}$ excited states are extracted by fitting
the eigenvalues $\lambda_{n}(t,t_{0})$

$\lambda_{n}(t,t_{0})=(1-A_{n})\operatorname{e}^{-E_{n}(t-t_{0})}+A_{n}\operatorname{e}^{-E_{n}^{\prime}(t-t_{0})}.$
(3)

Non-negligible thermal pollution is found in the correlation functions. And
the effect is addressed by generalizing the ”weighting and shifting” method in
Ref. [41]. We refer to the detailed derivation and the solution in the
Supplemental Material [37].

The finite-volume spectra for at $m_{\pi}\approx 305$ MeV are shown in Fig. 1.
The spectra at $132,208$, and $317$ MeV can be found in the Supplemental
Material [37]. There are two volumes at this pion mass. The red, purple, and
green curves represent the $D\pi$, $D^{*}\pi$, and the $D\pi\pi$ non-
interacting levels. The transparent lines indicate the thresholds. The black
points are the energy levels obtained by fitting the eigenvalues of the
correlation matrices. The orange bands are predictions from Lüscher’s equation
which will be explained in the next section.

Figure 1: $I=\frac{1}{2}$ $D\pi$ finite-volume spectra at $m_{\pi}\approx 305$
MeV for irreps with leading partial waves being $S$\- or $P$-wave. The black
and gray points are the energy levels. The red, purple, and green curves
indicate the non-interacting levels of the $D\pi$, $D^{*}\pi$, and the
$D\pi\pi$ thresholds, respectively. Only black levels are used in the
scattering analysis. The orange bands are the solutions of the Lüscher’s
equations.

For two-body scattering near the threshold, partial waves with lower $l$
dominate. For the irreps that contain $P$-wave, there is a level much lower
than the $D\pi$ threshold and is consistent among different irreps,
corresponding to a vector $D^{*}$ bound state. For $[000]T_{1}^{-}$, the
lowest $D\pi$ partial wave is the $P$-wave. The first excited states are not
far from the non-interacting levels, indicating a small interaction in the
$P$-wave. For irreps having $S$-wave as the leading partial wave, we find the
$D\pi$ scattering energy levels are substantially lower than their
corresponding non-interacting ones, indicating a strong attractive
interaction.

High-lying levels beyond the lowest non-interacting $D\pi\pi$ in the rest
frame, or the lowest non-interacting $D^{*}\pi$ level in moving frames are
discarded, and only the black points are used in the following scattering
analysis.

_Scattering analysis._ — The finite volume spectra can be translated into the
infinite-volume scattering phase shifts via Lüscher’s quantization condition,
which can be written in the general form as [42],

$\operatorname{det}[M(E,\vec{P};L)-\cot\delta(E)]=0,$ (4)

where $M(E,\vec{P};L)$ is the appropriate linear combination of Lüscher’s zeta
functions. The form of the equation varies in different moving frames and
irreps. The scattering $t$-matrix, which is related to the scattering phase
shift $\delta$ by

$t_{l}=\frac{\sqrt{s}}{2}\frac{1}{k\cot\delta_{l}-ik},$ (5)

can then be deduced from the lattice.

The determinant in Eq. 4 contains an infinite number of partial waves. Since
we mainly focus on the energy region near the threshold where the $l$-wave is
suppressed by $k^{2l}$, the high partial waves can be safely discarded and the
equation then becomes finite-dimensional. The partial wave cutoff is taken to
be $P$-wave in this work.

The cutoff Lüscher’s equation is still an under-constrained problem, one way
to proceed is to parameterize the scattering phase shifts in terms of models
and extract the model parameters from the lattice spectra. To extract both the
$S$\- and $P$-wave phase shifts, we do a correlated global fit for energy
levels below the lowest inelastic thresholds of the irreps with leading
partial waves up to $P$-wave, i.e. the irreps shown in Fig. 1. The number of
energy levels is $17\times 2=34$ for $m_{\pi}\approx 305$ and $208$ MeV, but
is $17$ for $132$ and $317$ MeV, since there are two volumes for
$m_{\pi}\approx 305$ and $208$ MeV.

To estimate the model dependencies, we apply many different parametrizations.
The main results are reported by using the effective range (ERE) expansion,

$k^{2l+1}\cot\delta_{l}=\frac{1}{a_{l}}+\frac{1}{2}r_{l}k^{2}+P_{2}k^{4}+\mathcal{O}(k^{6}),$
(6)

where for $l=0$, $a_{0}$ is the scattering length and $r_{0}$ is the effective
range. The scattering momentum $k$ is related to the scattering energy by

$E(\vec{k})=\sqrt{m_{D}^{2}+Z_{D}\vec{k}^{2}}+\sqrt{m_{\pi}^{2}+Z_{\pi}\vec{k}^{2}}.$
(7)

where $Z_{X}$ is calculated from the single-particle energy levels. The
difference of $Z_{X}$ from $1$ is a measure of the discretization effect, and
this information is used to estimate the systematic error, see the
Supplemental Material [37] for details. The fit results are tabulated in Tab.
2. Other parametrizations to describe the lattice data and the details about
the fitting technique can also be found in the Supplemental Material [37]. In
general, little model dependency is found.

Table 2: The fitting result for the $S$\- and $P$-wave of $I=\frac{1}{2}$ $D\pi$ scattering using the ERE expansion parametrization. $m_{\pi}/\text{MeV}$ | $a_{0}/\text{fm}$ | $r_{0}/\text{fm}$ | $a_{1}/\text{fm}^{3}$ | $r_{1}/\text{fm}^{-1}$
---|---|---|---|---
$317$ | $3.9(1.2)$ | $-0.488(80)$ | $0.0325(85)$ | $28(10)$
$305$ | $3.36(82)$ | $-0.734(59)$ | $0.0288(70)$ | $32(11)$
$208$ | $0.663(56)$ | $-1.15(14)$ | $0.020(21)$ | $127(237)$
$132$ | $0.44(17)$ | $-1.5(1.9)$ | $0.011(36)$ | $702(591)$

The energy spectra for continuous volume can then be calculated from the
parameterized Lüscher’s formula, which is also plotted as the orange bands in
Fig. 1. The solution from the Lüscher’s formula describes the data well.

The scattering amplitudes for the configurations used in this work are plotted
in Fig. 2. There are $4$ bands for $m_{\pi}\approx 305,317$ MeV and $2$ for
$m_{\pi}\approx 208,132$ MeV. The bands with the same color represent the
results from various parametrizations at each pion mass. The width of each
band corresponds to the statistical errors. The difference between the bands
is estimated as the systematical error due to different parameterizations. We
observe little parametrization dependence by the plots. The energy levels used
to constrain the amplitudes are shown in the panel below. The gray curves are
reconstructed from the PDG-averaged RBW parameters. In the left panel, the
$S$-wave phase shift $\delta_{0}$ is shown. The phase shift of all $m_{\pi}$’s
rises from $0$ to $180$ degree, indicating a nontrivial structure. In the
middle panel, the normalized $S$-wave cross section $\rho^{2}|t_{0}|^{2}$ is
plotted, and there is indeed an enhancement above the $D\pi$ threshold. In the
right panel, the $P$-wave phase shift $\delta_{1}$ is plotted, which is small
compared to the $S$-wave in the elastic zone, as expected.

Figure 2: The $S$-wave scattering phase shift $\delta_{0}$, the normalized
cross section $\rho^{2}|t_{0}|^{2}$, and $P$-wave scattering phase shift
$\delta_{1}$ of our calculations and the reconstruction of the experimental
measurement. The bands with the same color are the results of different
parametrizations at each pion mass. The width of each band corresponds to the
statistical errors. The lower panel shows the energy levels used to constrain
the amplitude. The red, yellow, cyan, and purple bands are results from
$m_{\pi}\approx 132,208,305$ and $317$ MeV. The gray bands are reconstructed
from the PDG value.

We have also attempted to fit the spectra separately for $S$\- and $P$-wave.
We use $[000]A_{1}^{+}$ to fit the $S$-wave and $[000]T_{1}^{-}$ to fit the
$P$-wave. The results are consistent with that of the combined fit described
above.

_Pole positions._ — To determine if the attractive force in a given channel is
strong enough to form nontrivial structures in a way that minimizes the model
dependence, we analytically continue the scattering amplitude by pushing the
energy to the complex plane, $s=E_{\text{CM}}^{2}$. There is a branch cut
separating the amplitudes into two Riemann sheets starting from the elastic
scattering threshold to the positive infinity. The experiment and the lattice
calculation can detect only the real axis of the complex plane. If there is an
isolated pole near the real axis, the scattering phase shifts, and the cross
section would signal a sharp change close to the pole position. The purpose of
this section is to find the complex poles of the amplitudes and determine
their properties.

The two Riemann sheets are distinguished by the sign of the imaginary part of
the scattering momentum $\operatorname{Im}k$. The sheet with
$\operatorname{Im}k>0$ is the physical sheet while the other is the un-
physical sheet. The $t$-matrix near a pole behaves as
$t\sim\frac{c^{2}}{s-s_{0}}$ where $s_{0}$ is the pole position. The mass $m$
and the width $\Gamma$ of the resonance can be read from the pole position by
$\sqrt{s_{0}}=m-\frac{i}{2}\Gamma$.

To compare the results from different lattice groups, we focus on the mass of
the structure subtracted by spin-averaged mass of $D$ and $D^{*}$, as was done
in Ref. [22],

$\sqrt{s_{0}^{\prime}}=\sqrt{s_{0}}-\frac{1}{4}(m_{D}+3m_{D^{*}}),$ (8)

which we refer to as the spin-average-subtracted pole position.

For the $S$-wave of the $D\pi$ scattering, we found a virtual state for
$m_{\pi}\approx 305$ and $317$ MeV, and a resonance pole for $m_{\pi}\approx
132$ and $208$ MeV, from all parametrizations. These poles are identified to
the scalar $D_{0}^{*}$ found in the experiment. The spin-average-subtracted
pole positions on the Riemann sheet are shown as the covariance ellipses and
the red error bars in Fig. 3. The poles from different parametrizations at the
same $m_{\pi}$ are represented by the overlapping ellipses. The small
deviations among different ellipses indicate that the parameterization
dependence of the pole positions is small compared to the statistical error.
The red error bars are determined from all possible pole positions from
different parametrizations and add the discretization error in quadrature. The
results at $m_{\pi}\approx 317$ MeV and $m_{\pi}\approx 305$ MeV are close to
each other. Because the two results share similar pion mass but rather
different lattice spacings, the lattice discretization effect is therefore
small for lattice spacing $a\lesssim 0.07746$ fm. In the figure, we have also
presented the spin-average-subtracted pole positions from other lattice
calculations which were done at $m_{\pi}\approx 266,239$ and $391$ MeV. [22,
23, 24]. The plot also indicates results from the experimental side that
describe $D_{0}^{*}$ by RBW. Given that the physical $D_{0}^{*}$ resonance is
broad, a simple RBW description differs significantly from the pole positions
of the analytically continued amplitudes. Therefore, we continue the RBW
lineshape to the complex plane, find the pole position, and subtract the
physical spin average. These results are the blue points. The solid blue data
are from the PDG average value [26] and the transparent blue data are the
experimental measurements [1, 3, 4, 5] it uses.

Figure 3: The poles on the Riemann sheet for $S$-wave of $I=\frac{1}{2}$
$D\pi$ scattering. The poles from this work are the covariance ellipses and
the red points that wrap over them. The blue points are the experimental
measurements [1, 3, 4, 5]. The solid blue point is the PDG average [26]. The
orange points are the results from other lattice calculations[22, 23, 24].

Combining the results from the previous lattice calculations [22, 23, 24], an
interesting running behavior of the $D_{0}^{*}$ pole as a function of
$m_{\pi}$ is emerging: The pole starts at $m_{\pi}\approx 391$ MeV on the real
axis and becomes lighter as $m_{\pi}$ decreases. At some point in $317$ MeV
$\lesssim m_{\pi}\lesssim 391$ MeV, the pole penetrates the Riemann sheet and
passes to the un-physical sheet where it becomes a virtual state. As $m_{\pi}$
further decreases, the pole evolves to the complex plane at $305$ MeV
$\lesssim m_{\pi}\lesssim 266$ MeV and becomes a resonance, and the pole stops
moving leftwards but turns rightwards, with mass increasing instead of
decreasing. Finally, the pole runs towards the position of the experimental
measurements. The pole at the physical pion mass is consistent with the
experimental measurements albeit with a large uncertainty.

For the $P$-wave, a bound state is found and corresponds to the vector
$D^{*}$. The mass of the vector meson is consistent with what is obtained from
a single-meson operator. The results are $f$ for $m_{\pi}\approx 132,208,305$
and $317$ MeV, respectively. No simple movement of the pole position under
$m_{\pi}$ is observed. Note that although $D^{*}$ should be a resonance in
nature. In our physical configurations, the hyperfine splitting
$m_{D^{*}}-m_{D}$ becomes smaller than $m_{\pi}$ due to the lattice
discretization effect, making $D^{*}$ still a bound state.

_Chiral interpolations of $\frac{1}{a_{0}}$._— In this section, we interpolate
the $S$-wave scattering length to the physical pion mass
$m_{\pi}^{\text{phys}}=135$ MeV. The systematic error of the scattering length
from the model dependence is included by wrapping the spread of results of
$\lim_{k\to 0}k\cot\delta_{0}$. The discretization error is also included. We
use a polynomial form to describe the chiral behavior,

$\frac{1}{a_{0}(m_{\pi})}=c_{0}+c_{1}m_{\pi}^{2}+c_{2}m_{\pi}^{4}.$ (9)

Similar extrapolations were done in Ref. [12]. The fit is shown in Fig. 4,
where the results from other lattice results Ref. [22, 23, 24] are also shown.
The fitting results are $c_{0}=4.2(1.1)\ \text{fm}^{-1}$, $c_{1}=-80(35)\
\text{MeV}^{-2}\cdot\text{fm}^{-1}$, $c_{2}=407(242)\
\text{MeV}^{-4}\cdot\text{fm}^{-1}$. The blue band is the reconstructed curve
using the fitted parameters. The prediction at physical pion mass is the blue
point on the axis. Adding to Eq. 9 an $a^{2}$-term gives a measure of the
discretization error. The error is added in quadrature and the final result is
$\frac{1}{a_{0}(m_{\pi}^{\text{phys}})}=2.89(58)\ \text{fm}^{-1}$.

Figure 4: The chiral interpolation of the inverse of the $I=\frac{1}{2}$
$D\pi$ scattering length. The fit takes data from this work, shown as the red
data. The orange points are from previous lattice works [22, 23, 24]. The blue
band is the reconstructed fitting model. The blue error bar is the prediction
at physical pion mass.

Since there is no direct experimental report of the scattering length, one can
deduce it from analytically continuing the RBW amplitudes and taking the limit
of energy to the threshold. The result from [22] is also presented in the same
way. The scattering lengths in this work together with previous lattice
results [22, 23, 24], the deduced PDG average [26], and the experimental
measurements [1, 3, 4, 5] are shown in Fig. 5. Limited by only one or two
values in previous lattice calculations, the dependence of the pion mass is
hardly recognized, which leads to poor knowledge of the discrepancy with
experiments. Given the red points in this work, a clearly visible $m_{\pi}$
dependency is identified and the discrepancy with the PDG value is also
greatly alleviated. Our result is consistent with the BaBar and Bell
experiments within the range of error.

Figure 5: The comparison of $\frac{1}{a_{0}}$ from this work (red), the
previous lattice results [22, 23, 24] (orange), and the experiments [26, 1, 3,
4, 5] (blue). The experimental values are obtained by analytically continuing
the RBW amplitudes from the measured parameters.

_Conclusions._ — Using a series of $N_{\text{f}}=2+1$ Wilson-Clover
configurations, the $S$\- and $P$-wave $I=\frac{1}{2}$ $D\pi$ scattering
phases are studied at $m_{\pi}\approx 132,208,305$ and $317$ MeV within
Lüscher’s formalism. Various single-meson and two-meson interpolating
operators are constructed, which leads to stable excited states energy spectra
from variational analysis of the correlation matrices.

A strong attractive interaction in irreps that contain $S$-wave is observed.
The $P$-wave is small in the threshold region. The scattering phase shifts are
extracted by solving the Lüscher’s equation. Several parameterizations are
utilized to analyze the model dependency. A broad rising curve is observed for
all pion masses. The $D_{0}^{*}$ found on our configurations are virtual
states at $m_{\pi}\approx 305,317$ MeV and are resonance states at
$m_{\pi}\approx 132,208$ MeV. Together with the results from previous works,
we arrive at the following picture for the movement of the $D_{0}^{*}$ pole
when $m_{\pi}$ varies: $D_{0}^{*}$ pole is a bound state at $m_{\pi}\gtrsim
391$ MeV, it then evolves into a virtual state at somewhere in $317$ MeV
$\lesssim m_{\pi}\lesssim 391$ MeV, finally becomes a resonance at $305$ MeV
$\lesssim m_{\pi}\lesssim 266$ MeV. Surprisingly, the pole moves to the left
at first, but then turns around and moves to the right towards the
experimental measurements. The trajectory of the $D_{0}^{*}$ pole as a
function of pion mass also provides an essential input for the
phenomenological models.

The scattering lengths are interpolated to the physical pion mass, which is
then compared to the deduced experimental measurements. There is a discrepancy
between the result and the PDG-averaged result. Future calculations performed
at finer lattice spacings and more pion masses would be valuable to further
clarify these issues. In any case, the results presented in this letter on the
$D\pi$ scattering amplitude near the threshold could be utilized for future
$DD\pi$ three-body studies.

We found only one pole near the threshold from the parameterizations we
investigated, the missing of the two-pole structure is probably due to the
lack of consideration of the coupled-channel effects. To solve the puzzle of
the mass ordering between $D_{0}^{*}$ and $D_{s0}^{*}$, we are currently
performing a similar investigation in $DK$ scattering.

H. Yan is grateful to L. An, F. Guo, B. Hörz, S. Prelovšek, and J. Wu for
valuable discussions, and extends appreciation to members of CLQCD, including
Y. Chen, X. Feng, P. Sun, for collaborations and comments. Software `Chroma`
[43] and `QUDA` [44, 45, 46] are used to generate the configurations and solve
the perambulators. H. Yan, C. Liu, and Y. Meng acknowledge support from NSFC
under Grant No. 12293060, 12293061, 12293063, 12305094, 11935017. Part of the
simulations were performed on the High-performance Computing Platform of
Peking University, the Southern Nuclear Science Computing Center (SNSC), and
the SongShan supercomputer at Zhengzhou National Supercomputing Center.

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## Supplemental Material

###  .1 Table of operators

For the construction of the single-hadron operators, the helicity strategy
proposed by the Hadron Spectrum Collaboration [47, 48] is applied. For the
construction of the two-hadron operators, we propose a generalized version of
the projection formula put forward by Prelovsek et. al [49],

$O_{\Gamma,\mu}^{[|k|]}(\vec{P})=\sum_{R\in
G}T_{\mu,\mu}^{\Gamma}(R)RO_{1}(\vec{k})O_{2}(\vec{P}-\vec{k})R^{\dagger},$
(10)

where $\vec{k}$ designates the relative momentum between the two single
hadrons. $T_{\mu,\mu}^{\Gamma}$ is the diagonal matrix elements $\mu,\mu$ of
the irrep $\Gamma$. $O_{1,2}$ are single-hadron operators of the form
$\bar{\psi}\Gamma\psi$ with $\bar{\psi}$ and $\psi$ being light or charm
quarks and the gamma matrices transform as scalar and vector.

It is useful to do a linear transformation of the operator set if possible. We
perform the transformation in the irreps in the CM frame to let the operators
transform in a certain partial wave, as constructed by the partial-wave method
in Ref. [49]. In this way, the operators mainly couple to the low partial
waves we are concerned.

The operators need to be further projected to isospin $I=\frac{1}{2}$,

$O_{D^{(*)}\pi}^{I=\frac{1}{2}}=2D^{0}\pi^{+}-\sqrt{2}D^{(*)+}\pi^{0},$ (11)

The single-hadron operators appeared with different electric charges had the
same flavor structure and differed only in isospin.

Each operator is then a linear combination of quark bilinear or the product of
two bilinears. The operators are represented as

$\begin{cases}O_{\text{one}}(\vec{P})=\sum_{i}\eta_{i}D_{\mu_{i}}(\vec{P}),\\\
O_{\text{two}}(\vec{P})=\sum_{i}\eta_{i}D_{\mu_{i}}(\vec{p}_{i})\pi(\vec{P}-\vec{p}_{i}),\end{cases}$
(12)

and are uniquely identified by the parameters $\eta_{i}$, $\mu_{i}$ and
$\vec{p}_{i}$ for one- and two-meson operators, respectively.
$\mu_{i}\in\\{0,5,x,y,z,5x,5y,5z\\}$ is the Cartesian gamma matrices and
corresponds to
$\gamma_{0},\gamma_{5},\gamma_{x},\gamma_{y},\gamma_{z},\gamma_{5}\gamma_{x},\gamma_{5}\gamma_{y},\gamma_{5}\gamma_{z}$.
$\vec{p}_{i}$ is the relative momenta. For convenience, the parameters are
denoted by

$\eta_{\mu_{i}}^{\alpha_{i}},$ (13)

where $\alpha_{i}$ one-to-one corresponds to the momentum vector. One
direction notation in $\alpha_{i}$ means one unit of momentum in that
direction. For example, $\alpha=yz\to\vec{p}=[011]$,
$\alpha=-2x\to\vec{p}=[-200]$, etc. Operators for isospin $I=\frac{1}{2}$ for
$\vec{d}=[000]$ is shown in Tab. 3; $\vec{d}=[001]$ is shown in Tab. 4;
$\vec{d}=[011]$ is shown in Tab. 5; $\vec{d}=[111]$ is shown in Tab. 6;
$\vec{d}=[002]$ is shown in Tab. 7.

We have also tested that adding to the operator set the covariant derivatives
makes the resulting energy levels plateau a little earlier, but the signals
become significantly worse. As a result, the operators with covariant
derivatives are not applied in this work.

Table 3: Operators for $\vec{d}=[000]$ used to interpolate the $D\pi$ system. Each irrep contains both one-meson type and two-meson type operators. The symbols $\eta_{\mu_{i}}^{\alpha_{i}}$ are defined in the context. Each operator consists of a linear combination of one- or two-hadron quark bilinears. $\eta$ denotes the coefficients of certain operators. $\mu$ indicates the Cartesian gamma matrices. $\alpha$ denotes the momentum vector of the single hadron or the relative momentum vector of the two hadrons. irrep | type | operator
---|---|---
$A_{1}^{+}$ | one | $(+1)^{0}_{0}$
two | $(+1)^{0}_{5}$
$(+1)^{z}_{5},(+1)^{y}_{5},(+1)^{x}_{5},(+1)^{-z}_{5},(+1)^{-y}_{5},(+1)^{-x}_{5}$
$(+1)^{yz}_{5},(+1)^{xz}_{5},(+1)^{xy}_{5},(+1)^{-y,z}_{5},(+1)^{-x,z}_{5},(+1)^{-x,y}_{5},(+1)^{y,-z}_{5},(+1)^{x,-z}_{5},(+1)^{x,-y}_{5},(+1)^{-y,-z}_{5},(+1)^{-x,-z}_{5},(+1)^{-x,-y}_{5}$
$T_{1}^{-}$ | one | $(+1)^{0}_{z}$
two | $(+1)^{z}_{5},(-1)^{-z}_{5}$
$(+1)^{xz}_{5},(+1)^{-x,z}_{5},(+1)^{yz}_{5},(+1)^{-y,z}_{5},(-1)^{x,-z}_{5},(-1)^{-x,-z}_{5},(-1)^{y,-z}_{5},(-1)^{-y,-z}_{5}$
$(+1)^{y}_{x},(-1)^{-y}_{x},(+1)^{-x}_{y},(-1)^{x}_{y}$
$T_{1}^{+}$ | one | $(+1)^{0}_{5z}$
two | $(+1)^{0}_{z}$
$(+1)^{z}_{z},(+1)^{y}_{z},(+1)^{x}_{z},(+1)^{-z}_{z},(+1)^{-y}_{z},(+1)^{-x}_{z}$
$(-2)^{z}_{z},(+1)^{y}_{z},(+1)^{x}_{z},(-2)^{-z}_{z},(+1)^{-y}_{z},(+1)^{-x}_{z}$
Table 4: Operators for $\vec{d}=[001]$ used to interpolate the $D\pi$ system. Notations as in Tab. 3. irrep | type | operator
---|---|---
$A_{1}$ | one | $(+1)^{z}_{0}$
$(+1)^{z}_{z}$
two | $(+1)^{0}_{5}$
$(+1)^{z}_{5}$
$(+1)^{-x}_{5},(+1)^{x}_{5},(+1)^{-y}_{5},(+1)^{y}_{5}$
$(+1)^{xz}_{5},(+1)^{-x,z}_{5},(+1)^{yz}_{5},(+1)^{-y,z}_{5}$
$(+1)^{-z}_{5}$
$(+1)^{2z}_{5}$
$E_{2}$ | one | $(+1)^{z}_{y}$
$(+1)^{z}_{5x}$
two | $(+1)^{y}_{5},(-1)^{-y}_{5}$
$(+1)^{yz}_{5},(-1)^{-y,z}_{5}$
$(+1)^{0}_{x}$
$(+1)^{z}_{x}$
Table 5: Operators for $\vec{d}=[011]$ used to interpolate the $D\pi$ system. Notations as in Tab. 3. irrep | type | operator
---|---|---
$A_{1}$ | one | $(+1)^{yz}_{0}$
$(+1)^{yz}_{y},(+1)^{yz}_{z}$
two | $(+1)^{0}_{5}$
$(+1)^{yz}_{5}$
$(+1)^{y}_{5},(+1)^{z}_{5}$
$(+1)^{xy}_{5},(+1)^{xz}_{5},(+1)^{-x,y}_{5},(+1)^{-x,z}_{5}$
$(+1)^{xyz}_{5},(+1)^{-x,yz}_{5}$
$(+1)^{y}_{x},(-1)^{z}_{x}$
$B_{1}$ | one | $(+1)^{yz}_{y},(-1)^{yz}_{z}$
$(+1)^{yz}_{5x}$
two | $(+1)^{y}_{5},(-1)^{z}_{5}$
$(+1)^{xy}_{5},(-1)^{xz}_{5},(+1)^{-x,y}_{5},(-1)^{-x,z}_{5}$
$(+1)^{y}_{x},(+1)^{z}_{x}$
$B_{2}$ | one | $(+1)^{yz}_{x}$
$(+1)^{yz}_{5y},(-1)^{yz}_{5z}$
two | $(+1)^{xy}_{5},(+1)^{xz}_{5},(-1)^{-x,y}_{5},(-1)^{-x,z}_{5}$
$(+1)^{x}_{5},(-1)^{-x}_{5}$
$(+1)^{xyz}_{5},(-1)^{-x,yz}_{5}$
$(+1)^{y}_{y},(-1)^{z}_{z}$
$(+1)^{y}_{z},(-1)^{z}_{y}$
$(+1)^{0}_{y},(-1)^{0}_{z}$
$(+1)^{yz}_{y},(-1)^{yz}_{z}$
Table 6: Operators for $\vec{d}=[111]$ used to interpolate the $D\pi$ system. Notations as in Tab. 3. irrep | type | operator
---|---|---
$A_{1}$ | one | $(+1)^{xyz}_{0}$
$(+1)^{xyz}_{x},(+1)^{xyz}_{y},(+1)^{xyz}_{z}$
two | $(+1)^{0}_{5}$
$(+1)^{xyz}_{5}$
$(+1)^{x}_{5},(+1)^{y}_{5},(+1)^{z}_{5}$
$(+1)^{yz}_{5},(+1)^{xz}_{5},(+1)^{xy}_{5}$
$(+1)^{x}_{y},(-1)^{x}_{z},(+1)^{y}_{z},(-1)^{y}_{x},(+1)^{z}_{x},(-1)^{z}_{y}$
$(+1)^{xy}_{x},(-1)^{xy}_{y},(+1)^{yz}_{y},(-1)^{yz}_{z},(+1)^{xz}_{z},(-1)^{xz}_{x}$
$E_{2}$ | one | $(+1)^{xyz}_{x},(-1)^{xyz}_{y}$
$(+1)^{xyz}_{5x},(+1)^{xyz}_{5y},(-2)^{xyz}_{5z}$
two | $(+1)^{x}_{5},(-1)^{y}_{5}$
$(+1)^{yz}_{5},(-1)^{xz}_{5}$
$(+1)^{0}_{x},(+1)^{0}_{y},(-2)^{0}_{z}$
$(+1)^{xyz}_{x},(+1)^{xyz}_{y},(-2)^{xyz}_{z}$
$(+1)^{x}_{x},(+1)^{y}_{y},(-2)^{z}_{z}$
$(+1)^{yz}_{x},(+1)^{xz}_{y},(-2)^{xy}_{z}$
Table 7: Operators for $\vec{d}=[002]$ used to interpolate the $D\pi$ system. Notations as in Tab. 3. irrep | type | operator
---|---|---
$A_{1}$ | one | $(+1)^{2z}_{0}$
$(+1)^{2z}_{z}$
two | $(+1)^{0}_{5}$
$(+1)^{2z}_{5}$
$(+1)^{z}_{5}$
$(+1)^{xz}_{5},(+1)^{-x,z}_{5},(+1)^{yz}_{5},(+1)^{-y,z}_{5}$
$E_{2}$ | one | $(+1)^{2z}_{y}$
$(+1)^{2z}_{5x}$
two | $(+1)^{yz}_{5},(-1)^{-y,z}_{5}$
$(+1)^{0}_{x}$
$(+1)^{z}_{x}$
$(+1)^{yz}_{x},(+1)^{-y,z}_{x}$

###  .2 Thermal pollutions

It is found that the effective mass of some eigenvalues does not plateau at
large time, but keeps rising or dropping. We found such behavior is due to the
non-negligible thermal pollution which can be explained by considering the
decomposition of the correlation matrix. We attribute the effect to the
elements whose source and sink are both $D\pi$-like two-body operators with
the same momentum structure:

$\displaystyle C_{[D\pi]_{i},[D\pi]_{j}}(t)=\langle
D_{\vec{k}}^{-}(t)\pi_{\vec{P}-\vec{k}}^{-}(t)D_{\vec{k}}^{+}(0)\pi_{\vec{P}-\vec{k}}^{+}(0)\rangle_{T}$
(14)
$\displaystyle\supset\frac{1}{Z_{T}}\left[e^{-E_{\pi}(\vec{P}-\vec{k})T}\langle\pi_{\vec{P}-\vec{k}}^{-}|D_{\vec{k}}^{-}(t)\pi_{\vec{P}-\vec{k}}^{-}(t)|D_{\vec{k}}^{+}\rangle\langle
D_{\vec{k}}^{+}|D_{\vec{k}}^{+}(0)\pi_{\vec{P}-\vec{k}}^{+}(0)|\pi_{\vec{P}-\vec{k}}^{-}\rangle\right.$
$\displaystyle+\left.e^{-E_{D}(\vec{k})T}\langle
D_{\vec{k}}^{-}|D_{\vec{k}}^{-}(t)\pi_{\vec{P}-\vec{k}}^{-}(t)|\pi_{\vec{P}-\vec{k}}^{+}\rangle\langle\pi_{\vec{P}-\vec{k}}^{+}|D_{\vec{k}}^{+}(0)\pi_{\vec{P}-\vec{k}}^{+}(0)|D_{\vec{k}}^{-}\rangle\right]$
$\displaystyle=\frac{1}{Z_{T}}|z_{\vec{k}}^{D}|^{2}|z_{\vec{P}-\vec{k}}^{\pi}|^{2}\left[e^{-E_{\pi}(\vec{P}-\vec{k})T-(E_{D}(\vec{k})-E_{\pi}(\vec{P}-\vec{k}))t}+e^{-E_{D}(\vec{k})T-(E_{\pi}(\vec{P}-\vec{k})-E_{D}(\vec{k}))t}\right]$
$\displaystyle\approx\frac{1}{Z_{T}}|z_{\vec{k}}^{D}|^{2}|z_{\vec{P}-\vec{k}}^{\pi}|^{2}e^{-E_{\pi}(\vec{P}-\vec{k})T-(E_{D}(\vec{k})-E_{\pi}(\vec{P}-\vec{k}))t},$

where $z_{\vec{k}}^{X}\equiv\langle
X_{\vec{k}}^{+}|X_{\vec{k}}^{+}|\Omega\rangle$. In the last step, the second
thermal state is discarded since it is sub-leading, compared to the first
term.

We generalize the ”weighting and shifting” method as described in Ref. [41] to
the $D\pi$ system to eliminate the leading thermal state pollution. The
leading thermal pollution can be removed by constructing the weighted-shifted
correlation matrices,

$\displaystyle\tilde{C}_{ij}(t)$
$\displaystyle=e^{(E_{D}(\vec{k})-E_{\pi}(\vec{P}-\vec{k}))t}C_{ij}(t)-e^{(E_{D}(\vec{k})-E_{\pi}(\vec{P}-\vec{k}))(t+1)}C_{ij}(t+1).$
(15)

Most elements have no such thermal state but the procedure does not affect
their behavior under $t$. The spectra should be shifted up by
$(E_{D}(\vec{k})-E_{\pi}(\vec{P}-\vec{k}))$ to compensate for the weighting
procedure. The weighted-shifted correlation matrices $\tilde{C}_{ij}(t)$ can
then be sent through the usual GEVP process, yielding the right energy levels.
By doing correlated fits of these eigenvalues, towers of energy spectra in
different irreps can be obtained.

###  .3 Finite-volume spectra at other pion masses

The lattice spectra and the predictions from the Lüshcer’s analysis for
$I=\frac{1}{2}$ $D\pi$ scattering at $m_{\pi}=317,208$, and $132$ MeV are
shown in Fig. 6, 7, 8, respectively. The fact that the first few non-
interacting levels of $m_{\pi}=132$ MeV increase as $L$ increases is a
reflection of the discretization artifact since the configuration is coarser
than the others.

Figure 6: $I=\frac{1}{2}$ $D\pi$ finite-volume spectra at $m_{\pi}\approx 317$
MeV. Descriptions as in Fig. 1. Figure 7: $I=\frac{1}{2}$ $D\pi$ finite-volume
spectra at $m_{\pi}\approx 208$ MeV. Descriptions as in Fig. 1. Figure 8:
$I=\frac{1}{2}$ $D\pi$ finite-volume spectra at $m_{\pi}\approx 132$ MeV.
Descriptions as in Fig. 1.

###  .4 Discretization errors

The dispersion relations are examined and well-described via
$E_{X}=\sqrt{m_{X}^{2}+Z_{X}\vec{k}^{2}}$, where $\vec{k}$ is the scattering
momentum. We find $Z_{\pi}$ of all configurations is close to $1$. However,
$Z_{D}$ and $Z_{D^{*}}$ is statistically smaller than $1$, indicating a non-
negligible discretization effect. The extracted value of the meson masses,
$Z_{D}$ and $Z_{\pi}$ are tabulated in Tab. 8

Table 8: The meson mass and coefficients in the dispersion relation of the configurations. configurations |  $m_{\pi}$/MeV |  $m_{D}$/MeV |  $m_{D^{*}}$/MeV | $Z_{\pi}$ | $Z_{D}$ | $Z_{D^{*}}$
---|---|---|---|---|---|---
C48P14 | $131.5(1.8)$ | $1903.30(62)$ | $1987.85(84)$ | $1.0090(55)$ | $0.8643(88)$ | $0.870(14)$
F32P21 | $206.8(2.1)$ | $1901.3(1.1)$ | $2007.2(2.2)$ | $1.0409(84)$ | $0.9283(48)$ | $0.905(16)$
F48P21 | $208.12(70)$ | $1900.71(48)$ | $2006.3(1.2)$ | $1.0157(49)$ | $0.9302(31)$ | $0.9360(45)$
F32P30 | $305.81(71)$ | $1965.7(1.0)$ | $2076.6(1.5)$ | $0.9977(56)$ | $0.9083(74)$ | $0.9027(93)$
F48P30 | $304.98(50)$ | $1966.20(57)$ | $2074.1(1.1)$ | $1.0054(24)$ | $0.9262(41)$ | $0.9243(70)$
H48P32 | $317.00(68)$ | $1979.41(85)$ | $2108.9(2.0)$ | $1.0115(41)$ | $0.9551(66)$ | $0.935(15)$

The coefficients $Z_{X}$ show more deviation from $1$ at coarser
configurations and less at finer configurations, which reflects the degree of
the discretization artifact. The discretization error is addressed by
comparing the difference between the corrected dispersion relation and the
continuum dispersion relation. In other words, the difference of the
observables by the $Z_{X}$ measured on the lattice and $1$ in Eq. 7. The
errors are found to be small and are added in quadrature with the
parametrization errors.

###  .5 Parametrization variations and the fitting details

Besides the ERE used in the main context, we also used the K-matrix
parametrization, which provides many flexible forms for each partial wave. For
elastic scattering, the $t$-matrix is related to the $K$-matrix by

$t_{l}^{-1}=\frac{1}{(2k)^{2l}}K_{l}^{-1}+I_{l},$ (16)

where the Chew-Mandelstam phase space $I_{l}$ is taken to be $-i\rho$. The
$K$-matrix can be parametrized to be

$K_{l}(s)=\frac{g_{l}^{2}}{m_{l}^{2}-s}+\sum_{n}\gamma_{l}^{(n)}s^{n}$ (17)

or

$K_{l}(s)=\frac{1}{\sum_{n=0}c_{l}^{(n)}s^{n}}.$ (18)

The number of parameters can be varied to give many different
parametrizations. The $K$-matrix with pole parametrizations is used to fit the
attractive $I=\frac{1}{2}$ channel. We do not apply the RBW parameterization
here since the RBW applies only to resonances that are narrow and there are no
relevant thresholds or other nearby resonances. However, the resonance poles
$D_{0}^{*}$ and $D_{1}$ are either broad or close to the threshold with the
unphysical pion mass. The parametrizations that yield pole positions too deep
that touch the left-hand cut are eliminated from this work.

With appropriate parametrization of the scattering phase shifts, the lattice
spectra below the inelastic threshold in any irrep can be solved from
Lüscher’s formula can be solved given the specific parameters. What we need to
solve is the opposite: given the lattice spectra in many irreps, the
parameters are determined by minimizing the $\chi^{2}$, as in Ref. [41],

$\chi^{2}(\\{\vec{a}\\})=\sum_{L}\sum_{\begin{subarray}{c}\vec{P}\Gamma n\\\
\vec{P}^{\prime}\Gamma^{\prime}n^{\prime}\end{subarray}}[E_{\text{CM}}(L;\vec{P}\Gamma
n)-E_{\text{CM}}^{\text{det}}(L;\vec{P}\Gamma
n;\\{\vec{a}\\})]\operatorname{Cov}^{-1}(L;\vec{P}\Gamma
n;\vec{P}^{\prime}\Gamma^{\prime}n^{\prime})[E_{\text{CM}}(L;\vec{P}^{\prime}\Gamma^{\prime}n^{\prime})-E_{\text{CM}}^{\text{det}}(L;\vec{P}^{\prime}\Gamma
n^{\prime};\\{\vec{a}\\})],$ (19)

where $\vec{a}$ represents the list of the parameters that need to be
determined. $\operatorname{Cov}(L;\vec{P}\Gamma
n;\vec{P}^{\prime}\Gamma^{\prime}n^{\prime})$ is the covariance matrix between
the energy levels. Note that the covariance between different irrep of the
same configuration is not zero, but the covariance between different
configurations is zero. $E_{\text{CM}}=\sqrt{E^{2}-|\vec{P}|^{2}}$ is the CM
energy fitted from the GEVP eigenvalues, while $E_{\text{CM}}^{\text{det}}$ is
the solution of the Lüscher’s equation given the parameters.

When accounting for the discretization errors, Eq. 7 is used. This corrected
dispersion relation moves the non-interacting levels from the singularities of
the conventional zeta function. In order not to miss any energy levels, the
energy range is divided into small ranges with a step of $0.0005$ to search
for the solutions of the Lüscher’s equation.

Various types of parametrizing the phase shifts are tested to measure the
model dependence. As seen from Fig. 2, there is little statistical error from
the parametrization. The parametrizations we have used in this work for
$I=\frac{1}{2}$ $D\pi$ are tabulated in Tab. 9. The parametrizations that
generate poles in the left-hand region or have large unreasonable
$\chi^{2}/\text{d.o.f}$ are not presented and are used.

Table 9: Parametrizations used for $I=\frac{1}{2}$ $D\pi$ scattering analysis. configurations |  $S$-wave parametrizations |  $P$-wave parametrizations | $\chi^{2}/\text{d.o.f}$
---|---|---|---
F32P30 / F48P30 | $k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.10$
$k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}+P_{2,0}k^{4}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.14$
$k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}+P_{2,0}k^{4}$ | $K_{1}=\frac{g_{1}^{2}}{m_{1}^{2}-s}$ | $1.55$
$K_{0}=\frac{g_{0}^{2}}{m_{0}^{2}-s}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.10$
F32P21 / F48P21 | $k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.94$
$k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}+P_{2,0}k^{4}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.76$
$k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}$ | $K_{1}=\frac{g_{1}^{2}}{m_{1}^{2}-s}$ | $2.01$
$k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}+P_{2,0}k^{4}$ | $K_{1}=\frac{g_{1}^{2}}{m_{1}^{2}-s}$ | $1.82$
H48P32 | $k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.17$
$k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}+P_{2,0}k^{4}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.27$
C48P14 | $k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.46$
$k\cot\delta_{0}=\frac{1}{a_{0}}+\frac{1}{2}r_{0}k^{2}+P_{2,0}k^{4}$ | $k^{3}\cot\delta_{1}=\frac{1}{a_{1}}+\frac{1}{2}r_{1}k^{2}$ | $1.63$


# Generalized D-Forms Have No Spurious Creases

Erik D. Demaine111Partially supported by NSF CAREER award CCF-0347776, DOE
grant DE-FG02-04ER25647, and AFOSR grant FA9550-07-1-0538. and Gregory N.
Price222Partially supported by an NSF Graduate Fellowship.
MIT Computer Science and Artificial Intelligence Laboratory
32 Vassar St., Cambridge, MA 02139, USA
<EMAIL_ADDRESS>
###### Abstract

A convex surface that is flat everywhere but on finitely many smooth curves
(or _seams_) and points is a _seam form_. We show that the only creases
through the flat components of a seam form are either between vertices or
tangent to the seams. As corollaries we resolve open problems about certain
special seam forms: the flat components of a D-form have no creases at all,
and the flat component of a pita-form has at most one crease, between the
seam’s endpoints.

##  1 Introduction

Given any metric space $S$ with the topology and local geometry required of
the surface of a convex three-dimensional body, there is exactly one convex
body up to isometry whose surface has the intrinsic geometry of $S$. This is
the endpoint of a line of research pursued in the middle of the last century
by Alexandrov and Pogorelov [Ale50], and it implies a strong correspondence
between the geometry of a convex body and the intrinsic geometry of its
surface. On the other hand, the exact nature of this correspondence is not yet
well understood—which properties in the surface geometry imply what properties
in the body geometry, and vice versa.

For example, if $S$ is a _D-form_ , obtained by sewing together two smooth
convex shapes of the same perimeter, then even for this special case the most
basic questions are open. These forms were invented by an artist [Wil] and
introduced into the literature in [PW01]. The latter study poses three
problems: (1) when is the D-form the convex hull of a space curve, (2) when
are the two pieces free of creases, and (3) how can one compute the D-form
numerically from the two shapes. A later treatment [DO07] suggests an informal
argument for Problem 1 (arguing that the D-form is always the convex hull of
its seam) and leaves Problems 2 and 3 open. The same book [DO07] introduces
also a related special case where $S$ is obtained by sewing up a single smooth
convex shape along its boundary in one seam, calling these _pita-forms_ and
suggesting, based on paper experiments, that pita-forms might never have
creases.

We resolve Problems 1 and 2: both D-forms and pita-forms are always the convex
hull of their seams, and (excluding the seam) D-forms are always free of
creases but a pita-form may have one crease. Our results apply to a natural
generalization of both D-forms and pita-forms, the _seam form_ , which roughly
consists of intrinsically flat pieces joined along finitely many seams.
Because the original sources of the problem are stated informally, we first
introduce precise definitions that we believe capture the intuitive picture.
Then we show the following theorems:

######  Theorem 1.

Every three-dimensional convex body is the convex hull of the nonflat points
on its surface.

######  Corollary 2.

Every seam form is the convex hull of its seams and vertices.

######  Theorem 3.

In a flat component of a seam form, every crease lies on a line segment
composed of creases, and each endpoint of such a segment is either a strict
vertex or a point of tangency to a seam.

######  Corollary 4.

The flat components of a D-form are without creases; in the flat component of
a pita-form, the only crease(s) make up the line segment between the endpoints
of the seam.

Intuitively the line segment between a pita-form’s endpoints in Corollary 4
should be thought of as one “crease”; it is a consequence of our definitions,
below, that this segment may be arbitrarily subdivided into several segments
we call creases.

Problem 3, to efficiently compute the three-dimensional shape of a D-form or
seam form from its two-dimensional intrinsic geometry, has now been largely
resolved. To make this problem well posed, one needs a finite representation
of the input geometry, which is most naturally done by a piecewise-linear or
polyhedral approximation. With considerable effort, the problem of
reconstructing a three-dimensional convex polyhedron from its intrinsic
geometry can be reduced to the solution of a high-dimensional ordinary
differential equation [BI06]. The numerical solution of this equation appears
to be achievable efficiently in practice, and is provably achievable within
pseudopolynomial time [KPD09].

We introduce terminology in Section 2, prove Theorem 1 and its Corollary 2 in
Section 3, and prove Theorem 3 and its Corollary 4 in Section 4. In Section 5
we describe counterexamples that show the necessity of some of the hypotheses
in our results.

##  2 Background and Notation

For us a surface is a metric 2-manifold embedded in $\mathbb{R}^{3}$. The
surface is $C^{k}$ if the manifold and its embedding are $C^{k}$. The surface
is piecewise-$C^{k}$ if it can be decomposed as a complex of vertices, $C^{k}$
open edges, and $C^{k}$ open regions.

A good surface is a piecewise-$C^{2}$ surface. A good surface $S$ therefore
decomposes into a union of $C^{2}$ surfaces $S_{i}$, called pieces, $C^{2}$
edges $\gamma_{j}$, which we call semicreases, and vertices. If $S$ is itself
$C^{1}$ everywhere on a semicrease, we call it a proper semicrease; otherwise
it is a crease. (This conservative definition of crease, where some parts may
be $C^{1}$ but not $C^{2}$, only broadens our characterization of creases in
seam forms.)

A point on a surface is flat if it has a neighborhood isometric to a region in
the plane. A surface or part of a surface is called flat if all of its points
are flat.

A surface $S$ is convex if $S\subseteq\partial X$ for some bounded convex body
$X$ in $\mathbb{R}^{3}$. A normal to a convex body $X$ at a point $x$ is a
unit vector $n$ with $n\cdot x=\sup_{x^{\prime}\in X}n\cdot x^{\prime}$. The
relation between points on the boundary of $X$ and their normals is
traditionally called the Gauss map, though it need not be a map—one point may
have many normals. We write $G(x)$ for the normals at $x$, and $G(U)$ for all
the normals to any point in $U\subset X$. Observe that $G(x)$ is always a
convex subset of the sphere.

A consequence of Gauss’ celebrated Theorema Egregium [Gau02] is that a convex
surface $U$ is flat just if $G(U)$ has zero area. If $G(x)$ has positive
spherical area, then we call $x$ a strict vertex. The $C^{2}$ condition
prevents a strict vertex $x$ from being on a semicrease or a piece, so for
good surfaces, strict vertices are indeed vertices.

Figure 1: A D-form, constructed by sewing together two ellipses (at left). The
solid line is the seam, and the dotted lines are the false edges from
projection into the page.

A seam form $S$ is a good convex surface in which each piece $S_{i}$ is flat.
It is simple to verify that such a surface decomposes uniquely into maximal
connected flat open subcomplexes, which we call flat components, and some
leftover semicreases and vertices, the connected components of which we call
seams.

A simple seam form is a seam form in which each flat component is isometric to
a convex plane region, and a simple smooth seam form is a simple seam form in
which these regions have smooth ($C^{\infty}$) boundary. A simple smooth seam
form with one flat component is called a pita form, and with two flat
components is called a D-form. See Figures 1 and 3.

Given a convex body $X$ and a point $x\in X$, we say that a line $\ell$
traverses $x$ if $x\in\ell$ and some open neighborhood of $x$ on $\ell$ is
contained in $X$. An extreme point of $X$ is a point $x\in X$ not traversed by
any line.

##  3 Convex Hull

In this section we prove Theorem 1, that every three-dimensional convex body
is the convex hull of the nonflat points on its surface.

From convex geometry we have the following characterization of the minimal set
from which a convex body can be recovered as the convex hull: we need only the
extreme points.

######  Theorem 5 (Minkowski’s Theorem).

Every convex body in $\mathbb{R}^{n}$ is the convex hull of its extreme
points.

The proof is a straightforward induction on the dimension of the body and can
be found as Corollary 1.4.5 in Schneider’s textbook [Sch93].

It remains to describe the extreme points of a seam form. To do so we begin
with the following proposition:

######  Proposition 6.

If $p$ is an extreme point of the convex body $X$, then for every open
neighborhood $U$ of $p$ in $X$, some hyperplane has $p$ strictly on one side
and all of $X\setminus U$ strictly on the other.

###### Proof.

This is Lemma 1.4.6 in [Sch93]. For completeness we give the proof.

Because $p$ is an extreme point, it cannot by definition be the convex
combination of any two other points in $X$. Therefore it is not the convex
combination $a_{1}x_{1}+\dotsb+a_{k}x_{k}$, with all $a_{i}>0$, of any $k$
other points in $X$, because otherwise we would have
$p=(1-a_{k})(\frac{a_{1}}{1-a_{k}}x_{1}+\dotsb+\frac{a_{k-1}}{1-a_{k}}x_{k-1})+a_{k}x_{k}$,
and certainly the convex combination
$\frac{a_{1}}{1-a_{k}}x_{1}+\dotsb+\frac{a_{k-1}}{1-a_{k}}x_{k-1}$ is a point
in $X$. In other words, $p$ lies outside the convex hull of
$X\setminus\\{p\\}$, and consequently outside the convex hull $Y$ of
$X\setminus U$.

Now $Y$ is itself a convex body, and $p$ a point outside it. By the Separating
Hyperplane Theorem, some hyperplane strictly separates them, and because
$X\setminus U\subset Y$, it strictly separates $p$ and $X\setminus U$ as
required. ∎

######  Proposition 7.

On the surface of a convex body, there are no flat extreme points.

###### Proof.

Suppose some extreme point $p$ of a convex body $X$ was flat, with a
neighborhood $S\subset\partial X$ isometric to a plane region. Let $U$ be an
open neighborhood of $p$ in $X$ with $U\cap\partial X\subset S$. Let the
hyperplane $H$ guaranteed by Proposition 6 separate $X$ into convex bodies $C$
and $Y$ with $p\in C$, and let $D=C\cap\partial X$. Because $C\subset U$, we
have $D\subset S$ so that $D$ is flat.

Now consider the normals to $X$ along the portion $D$ of its surface. Let $d$
be the distance from $p$ to $H$, and let $r$ be the maximum distance from the
projection of $p$ onto $H$ to any point in $H\cap X$. Then any plane through
$p$ and making an angle at most $\theta=\tan^{-1}(d/r)$ to $H$ fails to
intersect $H\cap X$ and therefore fails to intersect $Y$. Therefore the
normals to these planes, covering a spherical area of $2\pi(1-\cos\theta)>0$,
all are normals to $X$ somewhere on $D$. This gives $G(D)$ a positive area,
contradicting that $D\subset S$ is flat. ∎

Theorem 1 is now immediate from Theorem 5 and Proposition 7, and Corollary 2
follows.

##  4 Creases

In this section we prove Theorem 3, characterizing the possible creases of a
seam form.

######  Proposition 8.

Let $\gamma$ be a crease in a flat component of a seam form $S$. Then $\gamma$
lies on a line segment $[p,q]$ between endpoints $p$ and $q$ that lie on
seams, and the whole segment is composed of creases and vertices.

###### Proof.

Let $S_{1}$ and $S_{2}$ be the open pieces bordered by $\gamma$ in the
decomposition of the good surface $S$, and let $x\in\gamma$ be a point at
which $S$ is not $C^{1}$. Then $S_{1}$ and $S_{2}$ are $C^{2}$ surfaces, so
they have normals $n_{1}$ and $n_{2}$ at $x$, and because $S$ is not $C^{1}$
at $x$, these normals are distinct. Therefore $G(x)$ contains at least two
distinct vectors.

By Proposition 7, $x$ must be traversed by some line $\ell$, so that $\ell\cap
S=[p,q]$ for some $p$ and $q$. Necessarily $\ell$ is perpendicular to all of
$G(x)$, so for each $y\in[p,q]$ and each normal $n\in G(x)$, $n\cdot y=n\cdot
x=\sup_{x^{\prime}\in X}n\cdot x^{\prime}$ and $n$ is a normal of $y$.
Therefore each $G(y)$ contains $G(x)$, and so like $G(x)$ has at least two
distinct vectors.

The multiple normals in $G(x)$ and hence in each $G(y)$ determine a unique
perpendicular line, so that no other line may traverse any point of $[p,q]$.
In particular no line traverses $p$ or $q$, so by Proposition 7, these points
are not flat and must lie on seams or vertices.

At the same time, because a $C^{1}$ surface has only one normal at each point,
no point of $[p,q]$ can be on a $C^{2}$ piece or a semicrease. The whole
segment is therefore made up of creases and (nonstrict) vertices. Because a
crease is defined from a cell-complex decomposition, only one crease runs
through a given point, so because $\gamma$ runs through $x$ it must be one of
the creases making up $[p,q]$. ∎

In order to analyze the Gauss map at seam and vertex points, we introduce some
additional notation. Let $x\in S$ be incident to a 1- or 2-cell $C$, a
(semi)crease or piece. Then we define

$G_{C}(x)=\varliminf_{U\ni x}\overline{G(C\cap U)}$

as the Gauss map at $x$ on $C$. For comparison, observe that
$G(x)=\varliminf_{U\ni x}\overline{G(U)}$ because the relation $G$ is closed,
and in particular $G_{C}(x)\subseteq G(x)$.

###### Proof of Theorem 3.

Let $\gamma$ be a crease in a flat component of a seam form $S$. By
Proposition 8, $\gamma$ lies on a segment $[p,q]$ composed of creases and
vertices and whose endpoints lie on seams. It remains to prove that if an
endpoint, say $p$, lies on a seam and is not a strict vertex, then the seam is
tangent to $[p,q]$.

Let $G_{\gamma}(p)$ be the great circular arc $mn$, and let the pieces of $S$
bordering $\gamma$ be $S_{1}$ and $S_{2}$. By continuity, $G_{S_{1}}(p)\ni m$
and $G_{S_{2}}(p)\ni n$ (possibly after exchanging the names $m,n$), and
because $m\neq n$ the Gauss map at $p$ on at least one of the cells $C$
surrounding $p$ from $S_{1}$ to $S_{2}$ apart from $\gamma$ must be a
positive-length spherical curve in order to complete the path from $m$ to $n$.
If $p$ is not a strict vertex, then $G(p)$ is a convex spherical shape of zero
area, so it is a great circular arc, and $G_{C}(p)\subseteq G(p)$ is also a
great circular arc. If $C$ is a piece, then $G_{C}(p)$ is either a singleton
or a curve not lying on a great circle, because a great-circle Gauss map makes
parallel rule lines that cannot converge at $p$. Therefore $C$ is a
semicrease. Because $G(C)$ must be more than a single point, $C$ is a crease,
and to make the Gauss map lie within the arc $G(p)$, $C$ must be tangent to
$[p,q]$ as required. Finally, because $p$ is the endpoint of the intersection
of the line $pq$ with $S$, the crease $C$ must not be a line segment, so by
Proposition 8, it is actually part of the seam and the proof is complete. ∎

Of course, in a convex plane region, no line segment in the interior is
tangent to the boundary, from which follows a corollary about simple seam
forms.

######  Corollary 9.

In a simple seam form, every crease in a flat component is on a line segment
between two strict vertices.

Finally, in a simple smooth seam form such as a pita-form or a D-form, the
requirement of smoothness sharply limits the possible configurations. By
(local) convexity, no vertex can be incident to three or more semicreases as
part of its seam, and a vertex through which a seam passes cannot be a strict
vertex. Consequently a pita-form must have a single path for its seam and just
two strict vertices located at the seam’s endpoints, and a D-form must have a
single cycle for its seam and no strict vertices. Corollary 4 follows.

##  5 Counterexamples

We have required the flat components of a D-form to be convex. We could relax
this requirement, requiring instead only that the metric space resulting from
joining the two components be locally convex, and the Alexandrov-Pogorelov
theorem would still guarantee a unique convex embedding in three-dimensional
space. Of course Corollary 2 would still guarantee that the resulting body
would be the convex hull of its seam, but it turns out that Corollary 4, whose
conditions would no longer be satisfied, really would fail in its conclusion:
one can construct a “D-form” under this relaxed definition which contains
creases in its flat components. Indeed it is not hard to construct such an
example, if one keeps in mind Theorem 3 that the offending crease must be
tangent to a seam; see Figure 2.

Figure 2: A “D-form” with a relaxed convexity condition. The solid dark line
is the seam, the dotted lines are false edges from projection onto the page,
and the solid light lines are the creases through a flat component. Actual
D-forms have no such creases.

For pita-forms, we have concluded in Corollary 4 that a pita-form may have at
most one crease. Indeed this is tight, and it is easy to construct an example
pita-form with a crease; see Figure 3. This possibility of creases therefore
represents a real difference from D-forms. It represents also a contrast from
the appearance of the natural paper experiments, which led the authors first
introducting pita-forms ([DO07]) to suggest that pita-forms might never have
creases; in fact, once one is familiar with Corollary 2, it is clear that the
same experiments really would have to be creased if only the paper were
behaving ideally.

Figure 3: A typical pita-form. A crease runs between the endpoints of the
seam.

For some theorems of the same flavor as our results here, one might hope to
obtain proofs by showing that the desired properties hold of convex polyhedra,
which are relatively concrete and amenable to reasoning, and then that they
carry over to general convex bodies as limits of polyhedra. Indeed, this is
the approach taken in [DO07] to argue for Corollary 2 for the case of D-forms.
Unfortunately this approach does not hold as widely as one might like. In
particular, Corollary 9 cannot be proven by a limiting argument of the obvious
form, even when restricted to D-forms. When each flat component of the D-form
is approximated by a sequence of polygons, it is possible for the dihedral
angles inside the components to approach positive limits, even when the angles
of the polygonal approximations are required to converge to zero. For example,
in the D-form obtained from two circular disks (which is just the double cover
of a disk), the components may be approximated by regular $n$-gons for
increasing $n$, and the resulting approximations to the D-form may be
antiprisms of two smaller $n$-gons and $2n$ triangles. In this approximation
sequence, the dihedral angles between each $n$-gon and its $n$ neighboring
triangles approach $\pi/4$, not zero, even though they lie inside the flat
components. For this and other reasons we have chosen direct proofs that
attack the general case of convex bodies.

##  6 Acknowledgments

We thank Jonathan Kelner, Joseph O’Rourke, and Johannes Wallner for helpful
discussions, and the anonymous referees for helpful comments.

## References

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  * [BI06] Alexander I. Bobenko and Ivan Izmestiev, Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes, Annales de l’Institut Fourier, in press. arXiv:math.DG/0609447.
  * [DO07] Erik D. Demaine and Joseph O’Rourke, Geometric Folding Algorithms, Cambridge University Press, Cambridge, 2007. Pages 352–354.
  * [Gau02] Gauss, Carl Friedrich, General Investigations of Curved Surfaces, 1827, Morehead and Hiltebeitel, tr., Princeton, 1902.
  * [KPD09] Daniel Kane, Gregory N. Price, and Erik D. Demaine, A pseudopolynomial algorithm for Alexandrov’s theorem, Algorithms and Data Structures Symposium (WADS) 2009. arXiv:0812.5030.
  * [PW01] Helmut Pottmann and Johannes Wallner, Computational Line Geometry, Springer-Verlag, Berlin, 2001. Page 418\.
  * [Sch93] Rolf Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993\.
  * [Wil] Tony Wills, DForms: 3D forms from two 2D sheets, Bridges: Mathematical Connections in Art, Music, and Science, Reza Sarhangi and John Sharp, eds, London, pp. 503–510.