# AFLOW for alloys

Cormac Toher<EMAIL_ADDRESS>Department of Materials Science and
Engineering, The University of Texas at Dallas, Richardson, Texas 75080, USA
Department of Chemistry and Biochemistry, The University of Texas at Dallas,
Richardson, Texas 75080, USA  Stefano Curtarolo<EMAIL_ADDRESS>Materials
Science, Electrical Engineering, and Physics, Duke University, Durham NC,
27708

###### Abstract

Many different types of phases can form within alloys, from highly-ordered
intermetallic compounds, to structurally-ordered but chemically-disordered
solid solutions, and structurally-disordered (i.e. amorphous) metallic
glasses. The different types of phases display very different properties, so
predicting phase formation is important for understanding how materials will
behave. Here, we review how first-principles data from the AFLOW repository
and the aflow++ software can be used to predict phase formation in alloys, and
describe some general trends that can be deduced from the data, particularly
with respect to the importance of disorder and entropy in multicomponent
systems.

amorphous, alloy, disorder, entropy, phase

## Introduction

A wide variety of phases can form within alloys [1], from highly-ordered
intermetallic compounds [2], to structurally-ordered and chemically-disordered
solid solutions [3, 4, 5, 6, 7], and structurally-disordered (i.e. amorphous)
metallic glasses [8, 9, 10, 11, 12] (Figure 1). Different phase-types lead to
very different properties: intermetallics are hard but brittle; solid
solutions display solid-solution strengthening and can even overcome the
strength-ductility trade-off [13, 14]; while metallic glasses can be processed
using thermoplastic blow-molding [15]. Metallic engineering materials
typically contain multiple phases forming a carefully-engineered
microstructure: small plates or needles of harder intermetallic or ceramic
compounds (e.g. cementite (Fe3C) in steels, or the gamma phase in Ni-
superalloys) embedded in a softer, more ductile matrix. The dispersal of the
intermetallic phases in the matrix is controlled by factors such as the
annealing time and quench rate, which limits the extent of diffusion, and can
lead to materials with good combinations of strength and toughness. Other
microstructures can be formed by rapid unmixing processes such as spinodal
decomposition [16, 17], where the internal stresses at the interface between
phases impede the motion of dislocations, increasing hardness [18].

Figure 1: Order and disorder in alloy space. Phases formed in alloys range
from structurally disordered metallic glasses (formed when components have a
large range of ionic radii — descriptor $\delta$), to ordered intermetallic
compounds, and solid solutions (formed when components have a large ratio of
mixing entropy $\Delta S_{\mathrm{mix}}$ to mixing enthalpy $\Delta
H_{\mathrm{mix}}$). Figure is inspired by Figure 1 in Ref. 19.

Due to the large effect of phase composition on alloy properties, predicting
which phases are most likely to form under specific conditions, and which
processing pathways can give rise to desired combinations of phases, is an
important challenge. Experimental investigation of phase formation can be
expensive and slow, due to the time needed for atoms to diffuse into the
equilibrium microstructure. Therefore, computational methods have been
developed to predict phase formation, including CALPHAD (calculation of phase
diagrams) [20, 21, 22, 7] that interpolates/extrapolates from existing
experimental data [23, 24], molecular dynamics simulations using either
empirical force-fields or ab initio calculations [25], and first-principles
calculations of formation enthalpies that can be combined with cluster
expansion [26, 27, 28] or machine-learning [29] to generate large structure-
energy ensembles for use in Monte Carlo simulations [30, 31, 32] or
statistical thermodynamics models [33].

Figure 2: Enthalpy and entropy gains. (a) Expectation value of enthalpy gain
$\left<{\Delta
H\left[N|\left\\{1,\cdots\\!,N\\!\\!-\\!\\!1\right\\}\right]}\right>$ and
ideal entropy contributions as functions of the number of elements, $N$. Even
at room temperature, entropy eventually overwhelms enthalpy in determining the
phase stability of multi-component systems. (b) Enthalpy gain distribution for
$N=2,3,4$; gains decrease sharply with increasing number of elements. Figure
is adapted from Figure 1 in Ref. 34.

The large data sets required to train cluster expansion or machine-learning
models require automated management of first-principles calculations, such as
that performed by AFLOW [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47].
AFLOW generates input files for density functional theory (DFT) calculations
[48, 49, 50] by decorating structural prototypes from the in-built Library of
Crystallographic Prototypes [51, 52, 53], monitors the calculations and
automatically detects and corrects any errors, and finally extracts relevant
information from the output files. Finally, the extracted data is formatted
and added to the AFLOW database [54, 55, 56, 57], which is accessible using
the AFLUX search-API and AFLOW REST-API [56, 58]. The AFLOW database currently
contains over 3.5 million entries, covering binary (1738), ternary (30,289)
and quaternary (150,659) systems. Here, we review how AFLOW data can be used
to predict the formation of disordered materials, and to make general
statements about the types of phases that will form in multi-component
materials.

## The inevitability of disorder

The central role of entropy in the formation of multi-component systems can be
demonstrated using large computational data sets. Analysis [34] of formation
enthalpies for 202,261 binary, 974,808 ternary and 432,840 quaternary entries
from the AFLOW.org ab-initio materials repository with the associated tools
[58] shows that the gain in formation enthalpy from adding additional elements
decreases rapidly with increasing number of elements, while the
configurational entropy continues to grow monotonically.

The formation enthalpy “gain” [34] ${\Delta
H\left[N|\left\\{1,\cdots\\!,N\\!\\!-\\!\\!1\right\\}\right]}$ of an
$N$-element ordered compound with respect to combinations of
$\\{1,\cdots\\!,N\\!-\\!1\\}$-element ordered sub-components can be defined as
the energetic distance of the enthalpy of the $N$-element compound,
$H\left[N\right]$, below the $\\{1,\cdots\\!,N\\!-\\!1\\}$ convex-hull hyper-
surface $H_{\rm hull}\left[\\{1,\cdots\\!,N\\!\\!-\\!\\!1\\}\right]$ generated
from its $\\{1,\cdots\\!,N\\!\\!-\\!\\!1\\}$-element components:

${\Delta
H\left[N|\left\\{1,\cdots\\!,N\\!\\!-\\!\\!1\right\\}\right]}\equiv\left\\{\begin{array}[]{ll}H_{\rm
hull}\left[\\{1,\cdots\\!,N\\!\\!-\\!\\!1\\}\right]-H\left[N\right]&\mbox{if
$H\left[N\right]<H_{\rm
hull}\left[\\{1,\cdots\\!,N\\!\\!-\\!\\!1\\}\right]$};\\\
0&\mbox{otherwise}.\end{array}\right.$

For binary compounds, $\Delta H\left[2|1\right]$ is equivalent to the usual
formation enthalpy; for $N>2$, the formation enthalpies can be written as sums
of the recursive gains. Physically, ${\Delta
H\left[N|\left\\{1,\cdots\\!,N\\!\\!-\\!\\!1\right\\}\right]}$ represents the
enthalpy gained by creating an ordered $N$-species compound out of all the
combinations of ordered $\left\\{1-,\cdots\\!,N\\!-\\!1\right\\}$-ones. The
expected enthalpy gains can be extracted from the metal alloy phase diagrams
generated with AFLOW.org data.

The expectation value of the enthalpy gain $\left<\Delta H\right>$ is shown in
Figure 2(a) for $N=2,3,4$, with the cohesive energy of the elemental reference
phase shown for $N=1$. $\left<\Delta H\right>$ is $\sim$181 meV/atom for
binaries, reducing to $\sim$7 meV/atom for ternaries, and effectively
disappearing ($<0.1$ meV/atom) for quaternaries. The frequency distributions
of enthalpy gains for all systems are displayed in Figure 2(b). 588 (59%) of
binary systems show enthalpy gain, forming a total of 1995 binary compounds;
2237 (16%) of ternaries form a total of 3040 ternary compounds, and just 426
(0.3%) of quaternaries form quaternary compounds. 434 binary, 192 ternary but
only 10 quaternary systems have an enthalpy gain exceeding 100 meV/atom.
Therefore, as the number of elements in an alloy increases, the likelihood of
finding new materials consisting of single-phase intermetallic compounds drops
rapidly. Instead, multi-component materials are dominated by disorder,
consisting of solid-solutions, metallic glasses and multiple separate phases
[59].

Figure 3: Descriptor for glass forming ability. The glass forming ability of
metal alloy systems can be predicted from the spread of formation energies of
relevant ordered structures. (a) Different structural phases with similar
energies compete against each other during solidification, frustrating
crystallization and promoting glass formation. (b) A broad distribution of
energies implies a low glass forming ability, while a narrow distribution
indicates a high glass forming ability. (c) The GFA of intermediate
stoichiometries can be estimated by generating pseudostructures $\phi$ that
are linear combinations of the real structures $\psi$ — particularly useful
for ternary alloys. Panels (a) and (b) adapted from Figure 1 of Ref. 47, and
panel (c) adapted from Figure 1 of Ref. 60.

While this analysis was performed for alloys composed of 45 different metallic
elements, this finding is expected to generalize to other systems such as
high-entropy ceramics [61, 62, 5], although formation enthalpy corrections
[63, 64] to account for DFT inaccuracies will be required. Mixing different
types of bonding can generate additional enthalpic stabilization, increasing
the threshold number of elements at which entropy dominates. Meanwhile,
reciprocal systems might be able to reduce the threshold, due to the
additional entropic stabilization of every equivalent sublattice.

## Glass formation

Metallic glasses are amorphous materials, lacking an ordered crystal lattice
and its associated defects, providing them with a combination of excellent
mechanical properties [65] and plastic-like processability [66, 15, 67],
rendering them of great interest for a range of commercial and industrial
applications [68, 69, 70].

Figure 4: Outline of the method to predict order-disorder transitions. (a)
Statistical ensemble is constructed: atomic configurations are generated and
their multiplicity is calculated. Their energies are then estimated using
cluster expansion with AFLOW energies serving as the training set. (b) The
GQCA model is solved for the population vector ${\bf P}({\bf X},T)$. (c)
$T_{\mathrm{c}}$ at equi-composition is evaluated as the maximum temperature
derivative point of the order parameter $\alpha({\bf X}^{\mathrm{(ec)}},T)$.
(d) Boundary lines are found assuming $D_{\mathrm{KL}}({\bf
X},T_{\mathrm{c}})\approx D_{\mathrm{KL}}({\bf
X}^{\mathrm{(ec)}},T^{\mathrm{(ec)}}_{\mathrm{c}})$. Figure is adapted from
Figures 1 and 2 in Ref. 33.

A model based on the competition between different ordered structures has been
successfully used to predict the glass forming ability (GFA) of metallic
glasses [71, 60]. The model is based on the ansatz that, at a particular
composition, if an alloy system has a large number of structures that are
similar to each other in energy, but different from each other structurally,
then competition between the ordered phases during solidification will
frustrate crystallization and promote glass formation (Figure 3(a, b)). Atomic
environment [72, 73] comparisons determine the similarity of ordered
crystalline phases, enabling the formulation of a quantitative descriptor that
can be applied to the entire AFLOW.org database. The different structures are
weighted according to a Boltzmann distribution to create the GFA descriptor.
The model has been applied to AFLOW data for binary and ternary alloy systems,
and successfully predicts 73% of the glass forming compositions for a set of
16 experimentally characterized binary alloy systems. Notably, the model also
predicts that 17% of binary alloy systems should have a GFA greater than CuZr,
a well-known glass-former. Therefore, there are likely to be many more glass-
forming alloy compositions discovered in the future.

A limitation of this model is the granularity of available calculated
formation enthalpy data: calculations are performed for specific compositions
that have structures with reasonably-sized primitive cells. For compositions
with 3 or more elements, this becomes a major issue, which can be addressed by
also considering competition between structures with different
stoichiometries. Compositions with stoichiometries in-between those of common
ordered structures can be considered as consisting of combinations of local
structures with different compositions [60], representing local stoichiometric
fluctuations within the sample. Pseudo-structures can then be constructed as
linear combinations of the formation enthalpies and atomic environments of the
real structures (Figure 3(c)). The GFA descriptor can then be calculated using
these pseudo-structures for intermediate stoichiometries, allowing prediction
of glass formation across the entire composition space [60].

## Solid solutions and high-entropy alloys

In solid solutions, different elements occupy the same crystallographic site
on a lattice. High-entropy alloys are solid solutions with multiple (usually
at least 4 or 5) components forming a single phase [3, 4, 74, 75, 76, 13],
typically on a simple fcc or bcc lattice. Similar to the method described
above for predicting glass formation, materials that form high-entropy single-
phases can be rapidly identified using a descriptor (EFA, entropy-forming-
ability) based on the distribution of energies for a set of ordered
configurations based on a particular lattice [62]. A narrow distribution of
energies indicates miscibility leading to the formation of a high-entropy
single phase, while a broader distribution corresponds to preferred ordered
configurations leading to phase separation.

The EFA descriptor is qualitative, only indicating which compositions are most
likely to form solid solutions. Directly predicting the order-disorder
transition temperatures from first-principles for alloy systems requires Monte
Carlo simulations combined with ab initio calculations [30, 31, 32], which can
be computationally expensive. To accelerate the prediction of phase diagrams,
AFLOW data was used to formulate a descriptor for the order-disorder
transition temperature [33], incorporating formation enthalpy calculations
into a mean field statistical mechanics model, and making use of order
parameters for predicting the transition temperature of a multi-component
system into a solid solution phase.

The main steps for method to predict the transition temperature are [33]:

i. AFLOW data for binary alloy systems are leveraged to train cluster
expansion (CE) [26, 77] models (Figure 4(a)), within the Alloy Theoretic
Automated Toolkit (AT-AT) [28] implementation. The zero temperature energies
$\varepsilon_{j}$ of atomic configurations based on derivative structures of
fcc and bcc lattices are then estimated using the CE models.

ii. The atomic configurations are then incorporated into the generalized
quasi-chemical approximation (GQCA) mean field statistical mechanical model
[78, 79]. GQCA is particularly suitable for solid solutions, where long-range
order is not expected to be important and the material is spatially
homogeneous.

The thermodynamic potential $\Phi$ within the GQCA model is given by

$\Phi=N\cdot\left(\sum_{j}\varepsilon_{j}P_{j}-Ts-\sum_{k}\mu_{k}X_{k}\right),$
(1)

where $X_{k}$ is the global atomic fraction of element $k$, $\mu_{k}$ is the
chemical potential of $k$, and $s$ is entropy, which in the large-system limit
becomes

$s=k_{\mathrm{B}}\left(-\sum_{k}X_{k}\log_{e}X_{k}-\frac{1}{n}\sum_{j}P_{j}\log_{e}(P_{j}/\tilde{P}_{j})\right)$
(2)

Minimizing the thermodynamic potential subject to the constraints
$\sum_{j}P_{j}=1$ and $\sum_{j}P_{j}x_{kj}=X_{k}$, where $x_{kj}$ is the
atomic fraction of element $k$ in configuration $j$, gives the temperature-
dependent population vector (see Figure 4(b))

$P_{j}=\frac{\tilde{P}_{j}\,e^{-n\beta\left(\epsilon_{j}-\sum_{k}\mu_{k}x_{kj}\right)}}{{\sum_{j}\tilde{P}_{j}\,e^{-n\beta\left(\epsilon_{j}-\sum_{k}\mu_{k}x_{kj}\right)}}},$
(3)

where $\beta=1/k_{\mathrm{B}}T$.

In the high-temperature limit, this becomes the temperature-independent
population vector

$\tilde{P_{j}}=\frac{g_{j}\,\prod_{k}X_{k}^{\,\,n\cdot\,x_{kj}}}{{\sum\limits_{j^{\prime}}g_{j^{\prime}}\,\prod\limits_{k}X_{k}^{\,\,n\cdot\,x_{kj^{\prime}}}}},$
(4)

iii. Comparison of the behavior of the thermodynamic potentials and population
vectors to experimental data leads to the introduction of two order parameters
to estimate the transition temperature and miscibility gap. The first order
parameter is based on the angle between the population vector as a function of
temperature and the population vector for the fully disordered high
temperature limit

$\alpha\left({\bf X},T\right)\equiv{\bf P}\cdot{\bf\tilde{P}}/|{\bf
P}||{\bf\tilde{P}}|.$ (5)

The maximum of the rate of change of this angle as a function of temperature,
max($\partial\alpha/\partial T$), is found to correspond to the order-disorder
transition temperature at equi-atomic concentration (Figure 4(c)). The second
order parameter is the relative entropy (_Kullback-Leibler_ divergence [80])

$D_{\mathrm{KL}}\equiv\frac{1}{n}\sum_{j}P_{j}\log_{e}(P_{j}/\tilde{P}_{j})$
(6)

which is the entropy-loss due to ordering. The surface of constant relative
entropy is used to extrapolate the transition temperature at equi-atomic
concentration to other compositions, and thus generate the order-disorder
transition boundary (Figure 4(d)) to delineate the miscibility gap in the
phase diagram.

Defining a solid-solution-forming alloy as a composition where the order-
disorder transition temperature is lower than the melting temperature (as
obtained from experimental data or CALPHAD predictions), 58 out of 117 ($\sim
49.6\%$) investigated binary alloy systems are expected to form solid
solutions — 56 of these were corroborated by experimental reports. 148 out of
441 investigated ternary systems were predicted to form solutions ($\sim
33.6\%$), while 46% of the 1100 investigated quaternaries and 130 quinaries
are expected to form high-entropy alloys.

## Vibrational Contributions

The works reviewed above focus on the role of configurational entropy in phase
formation. To understand the role of vibrations in phase formation in multi-
component systems, the AFLOW-POCC (Partial Occupation) [39] and AFLOW-APL
(Automatic Phonon Library) [81] modules were combined to calculate the
contribution of vibrations to the Gibbs free energy of formation for 3
recently synthesized [62, 82, 83, 84, 85] high-entropy carbides: (HfNbTaTiV)C,
(HfNbTaTiW)C, and (HfNbTaTiZr)C. The results [81] showed that for (HfNbTaTiV)C
and (HfNbTaTiZr)C, where all of the binary precursors were in the same
rocksalt structure as the multicomponent phase, the net contribution of the
vibrational energy to mixing was negligible. However for (HfNbTaTiW)C, the WC
precursor is not rocksalt, and the vibrational contribution was significant.
Therefore, vibrational contributions are likely to be most significant when
precursors or decomposition products have different structures than the high-
entropy phase.

## Conclusion

Thermodynamic analyses of the AFLOW database leads to the following general
conclusions and predictions about the distribution of different types of
phases in alloy systems: (i) disorder is inevitable for multi-component
systems [34]; (ii) competition between ordered structures plays an important
role in glass formation [71, 60]; (iii) spectral descriptors based on the
energy distribution of ordered structures can be used to predict the
synthesizability of high-entropy phases [62]; (iv) high-entropy alloy
transition temperatures can be predicted from the rate of change of the
population vector as a function of temperature [33]; (v) vibrational
contributions are important when the components have different structures than
the high-entropy phase [81].

## Acknowledgments

We thank Drs. S. Divilov, H. Eckert, C. Oses, D. Hicks, M. Esters, M. Mehl, S.
Griesemer, R. Friedrich and X. Campilongo for insightful discussions. CT
acknowledges support from National Science Foundation (DMR-2219788). SC
acknowledges support from ONR (N000142112515). This work was supported by
high-performance computer time and resources from the DoD High Performance
Computing Modernization Program (Frontier).

## References

  * [1] T. B. Massalski, H. Okamoto, P. R. Subramanian, and L. Kacprzak, eds., _Binary Alloy Phase Diagrams_ (ASM International, Materials Park, Ohio, USA, 1990).
  * [2] S. Curtarolo, D. Morgan, and G. Ceder, _Accuracy of ab initio methods in predicting the crystal structures of metals: A review of 80 binary alloys_ , Calphad 29, 163–211 (2005), doi:10.1016/j.calphad.2005.01.002.
  * [3] M. C. Gao, _Design of High-Entropy Alloys_ , in _High-Entropy Alloys: Fundamentals and Applications_ , edited by M. C. Gao, J.-W. Yeh, P. K. Liaw, and Y. Zhang (Springer, Cham, Switzerland, 2016), chap. 11, pp. 369–398, doi:10.1007/978-3-319-27013-5.
  * [4] E. P. George, D. Raabe, and R. O. Ritchie, _High-entropy alloys_ , Nat. Rev. Mater. 4, 515–534 (2019), doi:10.1038/s41578-019-0121-4.
  * [5] C. Toher, C. Oses, M. Esters, D. Hicks, G. Kotsonis, C. M. Rost, D. W. Brenner, J.-P. Maria, and S. Curtarolo, _High-entropy ceramics: propelling applications through disorder_ , MRS Bull. 47, 194–202, February (2022), doi:10.1557/s43577-022-00281-x.
  * [6] D. B. Miracle, _High entropy alloys as a bold step forward in alloy development_ , Nat. Commun. 10, 1805 (2019), doi:10.1038/s41467-019-09700-1.
  * [7] O. N. Senkov, J. D. Miller, D. B. Miracle, and C. Woodward, _Accelerated exploration of multi-principal element alloys with solid solution phases_ , Nat. Commun. 6, 6529 (2015), doi:10.1038/ncomms7529.
  * [8] A. L. Greer, _Confusion by design_ , Nature 366, 303–304 (1993), doi:10.1038/366303a0.
  * [9] A. L. Greer, _Metallic glasses…on the threshold_ , Materials Today 12, 14–22 (2009), doi:10.1016/S1369-7021(09)70037-9.
  * [10] W. L. Johnson, _Bulk Glass-Forming Metallic Alloys: Science and Technology_ , MRS Bull. 24, 42–56 (1999), doi:10.1557/S0883769400053252.
  * [11] A. L. Greer, _New horizons for glass formation and stability_ , Nat. Mater. 14, 542–546 (2015), doi:10.1038/nmat4292.
  * [12] D. B. Miracle, _A structural model for metallic glasses_ , Nat. Mater. 3, 697–702 (2004), doi:10.1038/nmat1219.
  * [13] Z. Li, K. G. Pradeep, Y. Deng, D. Raabe, and C. C. Tasan, _Metastable high-entropy dual-phase alloys overcome the strength-ductility trade-off_ , Nature 534, 227–230 (2016), doi:10.1038/nature17981.
  * [14] Z. Li, C. C. Tasan, H. Springer, B. Gault, and D. Raabe, _Interstitial atoms enable joint twinning and transformation induced plasticity in strong and ductile high-entropy alloys_ , Sci. Rep. 7, 40704 (2017), doi:10.1038/srep40704.
  * [15] J. Schroers, T. M. Hodges, G. Kumar, H. Raman, A. J. Barnes, Q. Pham, and T. A. Waniuk, _Thermoplastic blow molding of metals_ , Materials Today 14, 14–19 (2011), doi:10.1016/S1369-7021(11)70018-9.
  * [16] J. W. Cahn, _On spinodal decomposition_ , Acta Metall. 9, 795–801 (1961), doi:10.1016/0001-6160(61)90182-1.
  * [17] J. W. Cahn, _On spinodal decomposition in cubic crystals_ , Acta Metall. 10, 179–183 (1962), doi:10.1016/0001-6160(62)90114-1.
  * [18] J. W. Cahn, _Hardening by spinodal decomposition_ , Acta Metall. 11, 1275–1282 (1963), doi:10.1016/0001-6160(63)90022-1.
  * [19] X. Yang and Y. Zhang, _Prediction of high-entropy stabilized solid-solution in multi-component alloys_ , Mater. Chem. Phys. 132, 233–238 (2012), doi:10.1016/j.matchemphys.2011.11.021.
  * [20] L. Kaufman and H. Bernstein, _Computer Calculation of Phase Diagrams with Special Reference to Refractory Metals_ , Refractory Materials (Academic Press, New York, 1970).
  * [21] M. C. Gao and D. E. Alman, _Searching for Next Single-Phase High-Entropy Alloy Compositions_ , Entropy 15, 4504–4519 (2013), doi:10.3390/e15104504.
  * [22] F. Zhang, C. Zhang, S. L. Chen, J. Zhu, W. S. Cao, and U. R. Kattner, _An understanding of high entropy alloys from phase diagram calculations_ , Calphad 45, 1–10 (2014), doi:10.1016/j.calphad.2013.10.006.
  * [23] Thermo-Calc Software, Thermo-Calc Software Database SSOL version 5, http://www.thermocalc.com/products-services/software/thermo-calc/ (2015).
  * [24] European Cooperation in Science and Technology (COST), _507 - Thermochemical database for light metal alloys_ , http://www.cost.eu/COST_Actions/mpns/507 (2011).
  * [25] G. Kresse and J. Hafner, _Ab initio molecular dynamics for liquid metals_ , Phys. Rev. B 47, 558–561 (1993), doi:10.1103/PhysRevB.47.558.
  * [26] D. de Fontaine, _Cluster Approach to Order-Disorder Transformations in Alloys_ , in _Solid State Physics_ , edited by H. Ehrenreich and D. Turnbull (Academic Press, New York, 1994), vol. 47, pp. 33–176, doi:10.1016/S0081-1947(08)60639-6.
  * [27] A. van de Walle and M. D. Asta, _Self-driven lattice-model Monte Carlo simulations of alloy thermodynamic properties and phase diagrams_ , Model. Simul. Mater. Sci. Eng. 10, 521 (2002), doi:10.1088/0965-0393/10/5/304.
  * [28] A. van de Walle, M. Asta, and G. Ceder, _The alloy theoretic automated toolkit: A user guide_ , Calphad 26, 539–553 (2002), doi:10.1016/S0364-5916(02)80006-2.
  * [29] G. L. W. Hart, T. Mueller, C. Toher, and S. Curtarolo, _Machine Learning and Alloys_ , Nat. Rev. Mater. 6, 730–755 (2021), doi:10.1038/s41578-021-00340-w.
  * [30] M. Widom, _Prediction of Structure and Phase Transformations_ , in _High-Entropy Alloys: Fundamentals and Applications_ , edited by M. C. Gao, J.-W. Yeh, P. K. Liaw, and Y. Zhang (Springer, Cham, Switzerland, 2016), chap. 8.
  * [31] W. P. Huhn and M. Widom, _Prediction of A2 to B2 Phase Transition in the High Entropy Alloy Mo-Nb-Ta-W_ , JOM 65, 1772–1779 (2013), doi:10.1007/s11837-013-0772-3.
  * [32] M. Widom, _Entropy and diffuse scattering: comparison of NbTiVZr and CrMoNbV_ , Metall. Mater. Trans. A 47, 3306–3311 (2016), doi:10.1007/s11661-015-3095-x.
  * [33] Y. Lederer, C. Toher, K. S. Vecchio, and S. Curtarolo, _The search for high entropy alloys: A high-throughput ab-initio approach_ , Acta Mater. 159, 364–383 (2018), doi:10.1016/j.actamat.2018.07.042.
  * [34] C. Toher, C. Oses, D. Hicks, and S. Curtarolo, _Unavoidable disorder and entropy in multi-component systems_ , npj Comput. Mater. 5, 69 (2019), doi:10.1038/s41524-019-0206-z.
  * [35] C. Oses, M. Esters, D. Hicks, S. Divilov, H. Eckert, R. Friedrich, M. J. Mehl, A. Smolyanyuk, X. Campilongo, A. van de Walle, J. Schroers, A. G. Kusne, I. Takeuchi, E. Zurek, M. Buongiorno Nardelli, M. Fornari, Y. Lederer, O. Levy, C. Toher, and S. Curtarolo, _aflow++: A C++ framework for autonomous materials design_ , Comput. Mater. Sci. 217, 111889 (2023), doi:10.1016/j.commatsci.2022.111889.
  * [36] S. Curtarolo, W. Setyawan, G. L. W. Hart, M. Jahnátek, R. V. Chepulskii, R. H. Taylor, S. Wang, J. Xue, K. Yang, O. Levy, M. J. Mehl, H. T. Stokes, D. O. Demchenko, and D. Morgan, _AFLOW: An automatic framework for high-throughput materials discovery_ , Comput. Mater. Sci. 58, 218–226 (2012), doi:10.1016/j.commatsci.2012.02.005.
  * [37] C. Toher, C. Oses, J. J. Plata, D. Hicks, F. Rose, O. Levy, M. de Jong, M. Asta, M. Fornari, M. Buongiorno Nardelli, and S. Curtarolo, _Combining the AFLOW GIBBS and elastic libraries to efficiently and robustly screen thermomechanical properties of solids_ , Phys. Rev. Materials 1, 015401 (2017), doi:10.1103/PhysRevMaterials.1.015401.
  * [38] J. J. Plata, P. Nath, D. Usanmaz, J. Carrete, C. Toher, M. de Jong, M. D. Asta, M. Fornari, M. Buongiorno Nardelli, and S. Curtarolo, _An efficient and accurate framework for calculating lattice thermal conductivity of solids: AFLOW-AAPL Automatic Anharmonic Phonon Library_ , npj Comput. Mater. 3, 45 (2017), doi:10.1038/s41524-017-0046-7.
  * [39] K. Yang, C. Oses, and S. Curtarolo, _Modeling Off-Stoichiometry Materials with a High-Throughput Ab-Initio Approach_ , Chem. Mater. 28, 6484–6492 (2016), doi:10.1021/acs.chemmater.6b01449.
  * [40] G. L. W. Hart, S. Curtarolo, T. B. Massalski, and O. Levy, _Comprehensive Search for New Phases and Compounds in Binary Alloy Systems Based on Platinum-Group Metals, Using a Computational First-Principles Approach_ , Phys. Rev. X 3, 041035 (2013), doi:10.1103/PhysRevX.3.041035.
  * [41] J. Carrete, N. Mingo, S. Wang, and S. Curtarolo, _Nanograined Half-Heusler Semiconductors as Advanced Thermoelectrics: An Ab Initio High-Throughput Statistical Study_ , Adv. Func. Mater. 24, 7427–7432 (2014), doi:10.1002/adfm.201401201.
  * [42] O. Levy, M. Jahnátek, R. V. Chepulskii, G. L. W. Hart, and S. Curtarolo, _Ordered Structures in Rhenium Binary Alloys from First-Principles Calculations_ , J. Am. Chem. Soc. 133, 158–163 (2011), doi:10.1021/ja1091672.
  * [43] W. Setyawan and S. Curtarolo, _High-throughput electronic band structure calculations: Challenges and tools_ , Comput. Mater. Sci. 49, 299–312 (2010), doi:10.1016/j.commatsci.2010.05.010.
  * [44] O. Levy, G. L. W. Hart, and S. Curtarolo, _Structure maps for hcp metals from first-principles calculations_ , Phys. Rev. B 81, 174106 (2010), doi:10.1103/PhysRevB.81.174106.
  * [45] O. Levy, R. V. Chepulskii, G. L. W. Hart, and S. Curtarolo, _The New Face of Rhodium Alloys: Revealing Ordered Structures from First Principles_ , J. Am. Chem. Soc. 132, 833–837 (2010), doi:10.1021/ja908879y.
  * [46] O. Levy, G. L. W. Hart, and S. Curtarolo, _Uncovering Compounds by Synergy of Cluster Expansion and High-Throughput Methods_ , J. Am. Chem. Soc. 132, 4830–4833 (2010), doi:10.1021/ja9105623.
  * [47] C. Oses, C. Toher, and S. Curtarolo, _Data-driven design of inorganic materials with the Automatic Flow Framework for Materials Discovery_ , MRS Bull. 43, 670–675 (2018), doi:10.1557/mrs.2018.207.
  * [48] G. Kresse and J. Furthmüller, _Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set_ , Phys. Rev. B 54, 11169–11186 (1996), doi:10.1103/PhysRevB.54.11169.
  * [49] G. Kresse and D. Joubert, _From ultrasoft pseudopotentials to the projector augmented-wave method_ , Phys. Rev. B 59, 1758–1775 (1999), doi:10.1103/PhysRevB.59.1758.
  * [50] R. M. Martin, _Electronic Structure: Basic Theory and Practical Methods_ (Cambridge University Press, U.K., 2004).
  * [51] M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, _The AFLOW Library of Crystallographic Prototypes: Part 1_ , Comput. Mater. Sci. 136, S1–S828 (2017), doi:10.1016/j.commatsci.2017.01.017.
  * [52] D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, _The AFLOW Library of Crystallographic Prototypes: Part 2_ , Comput. Mater. Sci. 161, S1–S1011 (2019), doi:10.1016/j.commatsci.2018.10.043.
  * [53] D. Hicks, M. J. Mehl, M. Esters, C. Oses, O. Levy, G. L. W. Hart, C. Toher, and S. Curtarolo, _The AFLOW Library of Crystallographic Prototypes: Part 3_ , Comput. Mater. Sci. 199, 110450 (2021), doi:10.1016/j.commatsci.2021.110450.
  * [54] M. Esters, C. Oses, S. Divilov, H. Eckert, R. Friedrich, D. Hicks, M. J. Mehl, F. Rose, A. Smolyanyuk, A. Calzolari, X. Campilongo, C. Toher, and S. Curtarolo, _aflow.org: A web ecosystem of databases, software and tools_ , Comput. Mater. Sci. 216, 111808 (2023), doi:10.1016/j.commatsci.2022.111808.
  * [55] S. Curtarolo, W. Setyawan, S. Wang, J. Xue, K. Yang, R. H. Taylor, L. J. Nelson, G. L. W. Hart, S. Sanvito, M. Buongiorno Nardelli, N. Mingo, and O. Levy, _AFLOWLIB.ORG: A distributed materials properties repository from high-throughput ab initio calculations_ , Comput. Mater. Sci. 58, 227–235 (2012), doi:10.1016/j.commatsci.2012.02.002.
  * [56] R. H. Taylor, F. Rose, C. Toher, O. Levy, K. Yang, M. Buongiorno Nardelli, and S. Curtarolo, _A RESTful API for exchanging materials data in the AFLOWLIB.org consortium_ , Comput. Mater. Sci. 93, 178–192 (2014), doi:10.1016/j.commatsci.2014.05.014.
  * [57] C. E. Calderon, J. J. Plata, C. Toher, C. Oses, O. Levy, M. Fornari, A. Natan, M. J. Mehl, G. L. W. Hart, M. Buongiorno Nardelli, and S. Curtarolo, _The AFLOW standard for high-throughput materials science calculations_ , Comput. Mater. Sci. 108, 233–238 (2015), doi:10.1016/j.commatsci.2015.07.019.
  * [58] F. Rose, C. Toher, E. Gossett, C. Oses, M. Buongiorno Nardelli, M. Fornari, and S. Curtarolo, _AFLUX: The LUX materials search API for the AFLOW data repositories_ , Comput. Mater. Sci. 137, 362–370 (2017), doi:10.1016/j.commatsci.2017.04.036.
  * [59] D. B. Miracle and O. N. Senkov, _A critical review of high entropy alloys and related concepts_ , Acta Mater. 122, 448–511 (2017), doi:10.1016/j.actamat.2016.08.081.
  * [60] D. C. Ford, D. Hicks, C. Oses, C. Toher, and S. Curtarolo, _Metallic glasses for biodegradable implants_ , Acta Mater. 176, 297–305 (2019), doi:10.1016/j.actamat.2019.07.008.
  * [61] C. M. Rost, E. Sachet, T. Borman, A. Moballegh, E. C. Dickey, D. Hou, J. L. Jones, S. Curtarolo, and J.-P. Maria, _Entropy-stabilized oxides_ , Nat. Commun. 6, 8485 (2015), doi:10.1038/ncomms9485.
  * [62] P. Sarker, T. Harrington, C. Toher, C. Oses, M. Samiee, J.-P. Maria, D. W. Brenner, K. S. Vecchio, and S. Curtarolo, _High-entropy high-hardness metal carbides discovered by entropy descriptors_ , Nat. Commun. 9, 4980 (2018), doi:10.1038/s41467-018-07160-7.
  * [63] R. Friedrich, D. Usanmaz, C. Oses, A. Supka, M. Fornari, M. Buongiorno Nardelli, C. Toher, and S. Curtarolo, _Coordination corrected ab initio formation enthalpies_ , npj Comput. Mater. 5, 59 (2019), doi:10.1038/s41524-019-0192-1.
  * [64] R. Friedrich, M. Esters, C. Oses, S. Ki, M. J. Brenner, D. Hicks, M. J. Mehl, C. Toher, and S. Curtarolo, _Automated coordination corrected enthalpies with AFLOW-CCE_ , Phys. Rev. Materials 5, 043803 (2021), doi:10.1103/PhysRevMaterials.5.043803.
  * [65] W. Chen, J. Ketkaew, Z. Liu, R. M. Ojeda Mota, K. O’Brien, C. S. da Silva, and J. Schroers, _Does the fracture toughness of bulk metallic glasses scatter?_ , Scr. Mater. 107, 1–4 (2015), doi:10.1016/j.scriptamat.2015.05.003.
  * [66] J. Schroers and N. Paton, _Amorphous metal alloys form like plastics_ , Adv. Mater. Processes 164, 61 (2006).
  * [67] G. Kaltenboeck, M. D. Demetriou, S. Roberts, and W. L. Johnson, _Shaping metallic glasses by electromagnetic pulsing_ , Nat. Commun. 7, 10576 (2016), doi:10.1038/ncomms10576.
  * [68] J. Schroers, _Processing of Bulk Metallic Glass_ , Adv. Mater. 22, 1566–1597 (2010), doi:10.1002/adma.200902776.
  * [69] W. L. Johnson, J. H. Na, and M. D. Demetriou, _Quantifying the origin of metallic glass formation_ , Nat. Commun. 7, 10313 (2016), doi:10.1038/ncomms10313.
  * [70] M. F. Ashby and A. L. Greer, _Metallic glasses as structural materials_ , Scr. Mater. 54, 321–326 (2006), doi:10.1016/j.scriptamat.2005.09.051.
  * [71] E. Perim, D. Lee, Y. Liu, C. Toher, P. Gong, Y. Li, W. N. Simmons, O. Levy, J. J. Vlassak, J. Schroers, and S. Curtarolo, _Spectral descriptors for bulk metallic glasses based on the thermodynamics of competing crystalline phases_ , Nat. Commun. 7, 12315 (2016), doi:10.1038/ncomms12315.
  * [72] J. L. C. Daams and P. Villars, _Atomic environments in relation to compound prediction_ , Eng. Appl. Artif. Intel. 13, 507 (2000), doi:10.1016/S0952-1976(00)00029-4.
  * [73] J. L. C. Daams, J. H. N. van Vucht, and P. Villars, _Atomic-environment classification of the cubic “intermetallic” structure types_ , J. Alloys Compd. 182, 1–33 (1992), doi:10.1016/0925-8388(92)90570-Y.
  * [74] X. Lim, _Mixed-up metals make for stronger, tougher, stretchier alloys_ , Nature 533, 306–307 (2016), doi:10.1038/533306a.
  * [75] B. Gludovatz, A. Hohenwarter, D. Catoor, E. H. Chang, E. P. George, and R. O. Ritchie, _A fracture-resistant high-entropy alloy for cryogenic applications_ , Science 345, 1153–1158 (2014), doi:10.1126/science.1254581.
  * [76] F. von Rohr, M. J. Winiarski, J. Tao, T. Klimczuk, and R. J. Cava, _Effect of electron count and chemical complexity in the Ta-Nb-Hf-Zr-Ti high-entropy alloy superconductor_ , Proc. Natl. Acad. Sci. 113, E7144–E7150 (2016), doi:10.1073/pnas.1615926113.
  * [77] J. M. Sanchez, F. Ducastelle, and D. Gratias, _Generalized cluster description of multicomponent systems_ , Physica A 128, 334–350 (1984).
  * [78] A. Sher, M. van Schilfgaarde, A.-B. Chen, and W. Chen, _Quasichemical approximation in binary alloys_ , Phys. Rev. B 36, 4279 (1987), doi:10.1103/PhysRevB.36.4279.
  * [79] M. A. Berding and A. Sher, _Electronic quasichemical formalism: Application to arsenic deactivation in silicon_ , Phys. Rev. B 58, 3853 (1998), doi:10.1103/PhysRevB.58.3853.
  * [80] S. Kullback and R. A. Leibler, _On Information and Sufficiency_ , Ann. Math. Stat. 22, 79–86 (1951), doi:10.1214/aoms/1177729694.
  * [81] M. Esters, C. Oses, D. Hicks, M. J. Mehl, M. Jahnátek, M. D. Hossain, J.-P. Maria, D. W. Brenner, C. Toher, and S. Curtarolo, _Settling the matter of the role of vibrations in the stability of high-entropy carbides_ , Nat. Commun. 12, 5747 (2021), doi:10.1038/s41467-021-25979-5.
  * [82] T. J. Harrington, J. Gild, P. Sarker, C. Toher, C. M. Rost, O. F. Dippo, C. McElfresh, K. Kaufmann, E. Marin, L. Borowski, P. E. Hopkins, J. Luo, S. Curtarolo, D. W. Brenner, and K. S. Vecchio, _Phase stability and mechanical properties of novel high entropy transition metal carbides_ , Acta Mater. 166, 271–280 (2019), doi:10.1016/j.actamat.2018.12.054.
  * [83] M. D. Hossain, T. Borman, A. Kumar, X. Chen, A. Khosravani, S. R. Kalidindi, E. A. Paisley, M. Esters, C. Oses, C. Toher, S. Curtarolo, J. M. LeBeau, D. Brenner, and J.-P. Maria, _Carbon stoichiometry and mechanical properties of high entropy carbides_ , Acta Mater. 215, 117051 (2021), doi:10.1016/j.actamat.2021.117051.
  * [84] M. D. Hossain, T. Borman, C. Oses, M. Esters, C. Toher, L. Feng, A. Kumar, W. G. Fahrenholtz, S. Curtarolo, D. Brenner, J. M. LeBeau, and J.-P. Maria, _Entropy Landscaping of High-Entropy Carbides_ , Adv. Mater. 33, 2102904 (2021), doi:10.1002/adma.202102904.
  * [85] A. Calzolari, C. Oses, C. Toher, M. Esters, X. Campilongo, S. P. Stepanoff, D. E. Wolfe, and S. Curtarolo, _Plasmonic high-entropy carbides_ , Nat. Commun. 13, 5993 (2022), doi:10.1038/s41467-022-33497-1.


Measurement of the presence of $\mathrm{a}_{1}(1420)$ and
$\mathup{{{\omega}}}(782)$ in
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
at Belle

A. Rabusov1$\star$, D. Greenwald1, and S. Paul1

1 Technical University of Munich, Munich, Germany

*<EMAIL_ADDRESS>
The 17th International Workshop on Tau Lepton Physics
Louisville, USA, 4-8 December 2023 10.21468/SciPostPhysProc.?

## Abstract

We present preliminary results of a partial-wave analysis of
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
using data from the Belle experiment at the KEKB
$\mathup{{{e}}^{\scriptstyle{+}}}\mathup{{{e}}^{\scriptstyle{-}}}$ collider.
We validate our analysis with a model-independent one. We see an
$\mathrm{a}_{1}(1420)$ resonance and a G-parity violating
$1^{-}\,[\mathup{{{\omega}}}(782)\,\mathup{{{\pi}}}]_{\mathrm{P}}$ wave in
tauon decays. Our results will improve models used in simulation studies
necessary for measuring the tauon electric and magnetic dipole moments and
Michel parameters.

##  1 Introduction

Many studies of spin correlation in
$\mathup{{{e}}^{\scriptstyle{+}}}\mathup{{{e}}^{\scriptstyle{-}}}\to\mathup{{{\tau}}^{\scriptstyle{+}}}\mathup{{{\tau}}^{\scriptstyle{-}}}$,
such as measurement of the tauon electric and magnetic dipole moments, analyze
tauon decay to
$\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
[1].111Inclusion of charge-conjugated decays is assumed throughout. However,
lack of knowledge about
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$,
which has never been analyzed for intermediate resonances, limits the
precision of such measurements.

This decay proceeds mostly through the $\mathrm{a}_{1}(1260)$, a broad
unflavored ground-state axial-vector meson [2], whose resonance shape is
poorly known [3, 4, 5]. What other resonances are present and in which amounts
is also poorly known. The COMPASS and VES experiments observed the
$\mathrm{a}_{1}(1420)$, a narrow unflavoured axial-vector meson in pion-proton
scattering [6, 7]. Seeing it in
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
can clarify whether it is a particle or an artifact of
$\mathup{{{K}}^{\scriptstyle{\ast}}}\mathup{{{K}}}$ scattering [8].

To study such matters, we perform a partial-wave analysis (PWA) of
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
using $980\text{\,}\mathrm{f}\mathrm{b}^{-1}$ of data collected by the Belle
experiment [9] at the asymmetric
$\mathup{{{e}}^{\scriptstyle{+}}}\mathup{{{e}}^{\scriptstyle{-}}}$ collider
KEKB [10]. Consult [9] for details of the Belle detector.

##  2 Event selection

Using simulated data, we optimized our event selection to maximize efficiency
and purity without introducing significant uneven detection effieciency across
the decay’s phase space. In each event, we define two hemispheres defined by
the plane perpendicular to the thrust axis of all charged particles and
photons. We require three charged particles be present in one hemisphere, the
signal hemisphere, and one be present in the other, the tag hemisphere.

We train a boosted decision tree (BDT) on simulated data, using ROOT’s TMVA
software [11], to remove events from processes other than
$\mathup{{{e}}^{\scriptstyle{+}}}\mathup{{{e}}^{\scriptstyle{-}}}\to\mathup{{{\tau}}^{\scriptstyle{+}}}\mathup{{{\tau}}^{\scriptstyle{-}}}$.
It looks at the thrust, the sum of charged-particle and photon momenta in the
center-of-momentum (CM) frame, the cosine of the polar angle of the missing
momentum in the CM frame, the missing mass, the energy measured by Belle’s
electromagnetic calorimeter, and the sum of the energies of charged particles
in the CM frame. The thrust is the most discriminating of the variables.

We veto the presence of charged kaons in the signal hemisphere by requiring
the two particles with like charges be consistent with being pions. We veto
the presence of neutral kaons by requiring the mass of each opposite-charge
pion pair in the signal hemisphere be more than
$12\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ from the known
$\mathup{{{K}}^{\scriptstyle{0}}}$ mass [5]. And we reduce the presence of
neutral pions by requiring the sum of photon energies in the signal hemisphere
be below $480\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$.

We find $55\text{\times}{10}^{6}$ events, with a 32% efficiency to find signal
and an 82% purity. This is the largest sample of
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
analyzed to date. It is contaminated with $10\text{\times}{10}^{6}$ background
events, mostly originating from
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\pi}}^{\scriptstyle{0}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
and
$\mathup{{{e}}^{\scriptstyle{+}}}\mathup{{{e}}^{\scriptstyle{-}}}\to\mathit{{q}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{\mkern
4.0mu\overline{\mkern-4.0mu{q}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$,
where
$\mathit{{q}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
can be a $\mathup{{{u}}}$, $\mathup{{{d}}}$, $\mathup{{{s}}}$, or
$\mathup{{{c}}}$ quark. To account for it, we train a neural network on
simulated data to describe the multidimensional distribution of the
background; for more details consult [12, 13]. We use the neural network’s
prediction of the background intensity in our partial-wave analysis.

##  3 Partial-wave analysis

$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
has a seven-dimensional phase-space. We model it in the mass of the three
pions, $m_{3\mathup{{{\pi}}}}$; squared masses of the two
$\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\pi}}^{\scriptstyle{-}}}$ pairs,
$s_{1}$ and $s_{2}$; three Euler angles of the decay plane; and the helicity
angle of the neutrino in the tauon rest frame [14]. One of the Euler angles is
immeasurable because we cannot detect the
$\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$. We average the decay rate over
this angle. We studied the effect of the averaging procedure and found it
introduces no biases [13].

We split the data into bins of $m_{3\mathup{{{\pi}}}}$ and perform a partial-
wave decomposition independently in each bin using the isobar model and tensor
formalism described in [15]. It assumes that the decay proceeds through a
resonance $\mathrm{X}^{-}$ that decays to three charged pions via a sequence
of two-body decays,
$\mathrm{X}^{-}\to\xi^{0}\mathup{{{\pi}}^{\scriptstyle{-}}}$ and
$\xi^{0}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}$,
where $\xi^{0}$ is an isobar. The only requirement on $\mathrm{X}^{-}$ in the
partial-wave decomposition is that its spin anad parity, $J^{P}$, be $0^{-}$,
$1^{+}$, or $1^{-}$; the precense of the last would violate $G$ parity.

We allow $\xi^{0}$ to be $\mathup{{{\rho}}}(770)$, $\mathup{{{\rho}}}(1450)$,
$\mathrm{f}_{0}(500)$, $\mathrm{f}_{0}(980)$, $\mathrm{f}_{0}(1500)$,
$\mathrm{f}_{2}(1270)$, or $\mathup{{{\omega}}}(782)$. We model all but
$\mathrm{f}_{0}(500)$ with the relativistic Breit-Wigner function with masses
and widths the same as in the COMPASS PWA [3] and $\mathrm{f}_{0}(500)$ with
the broad $(\mathup{{{\pi}}}\mathup{{{\pi}}})_{\mathrm{S}}$ component
described in [16]. We allow angular momenta between $\xi^{0}$ and the
remaining pion in $[0,3]$. We denote a partial wave by
$J^{P}[\xi^{0}\mathup{{{\pi}}}]_{L}$ for specific isobar resonances $\xi^{0}$
and $J^{P}[(\mathup{{{\pi}}}\mathup{{{\pi}}})_{j}\mathup{{{\pi}}}]_{L}$ for
generic isobars with spin $j$; $L$ is the total angular momentum of the three
pions.

The preliminary results of the PWA were presented in [17]. Here we present an
update that includes most of the systematic uncertainties. We observe that the
most intense partial wave is the
$1^{+}\,[\mathup{{{\rho}}}(770)\,\mathup{{{\pi}}}]_{\mathrm{S}}$ wave, with a
fit fraction of $(76.42\pm 0.05\pm 3.29)\%$, where the first uncertainty is
statistical, and the second uncertainty is systematic. The next most intense
is the $1^{+}\,[\sigma\,\mathup{{{\pi}}}]_{\mathrm{P}}$ wave with a fit
fraction of $(8.40\pm 0.02\pm 1.16)\%$. The fit fraction of a partial wave is
the integral over $m_{3\mathup{{{\pi}}}}$ of the intensity of that wave alone
divided by the same integral of the intensity of the full PWA model. Fit
fractions need not sum to unity due to interferences between partial waves.
Our results agree with those of CLEO II for
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{0}}}\mathup{{{\pi}}^{\scriptstyle{0}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
[4].

We verify the presence of $\mathrm{a}_{1}(1420)$ using quasi-model-independent
PWA (QMIPWA) [18]. For partial-wave decomposition, we parameterize the
$(\mathup{{{\pi}}}\mathup{{{\pi}}})_{\mathrm{S}}$ wave with a complex-valued
step function and let the fit find the
$(\mathup{{{\pi}}}\mathup{{{\pi}}})_{\mathrm{S}}$ wave amplitude as a function
of $m_{2\mathup{{{\pi}}}}$. To reduce bias caused by the
$1^{+}\,[\mathup{{{\rho}}}(770)\,\mathup{{{\pi}}}]_{\mathrm{S}}$ model, we
similarly parameterize the
$1^{+}\,[\mathup{{{\rho}}}(770)\,\mathup{{{\pi}}}]_{\mathrm{S}}$ wave. We
observe the narrow peak of the $\mathrm{f}_{0}(980)$ in the
$(\mathup{{{\pi}}}\mathup{{{\pi}}})_{\mathrm{S}}$ wave, as shown in Fig. 1(a).

In [19], Mirkes and Urech stated that $1^{-}$ intensity in
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
comes from the $G$-violating decay of
$\mathup{{{\omega}}}(782)\to\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\pi}}^{\scriptstyle{-}}}$,
where $\mathup{{{\omega}}}(782)$ is produced by decay of a
$\mathup{{{\rho}}}(770)$, $\mathup{{{\rho}}}(1450)$, or
$\mathup{{{\rho}}}(1700)$. We free the
$1^{-}\,[(\mathup{{{\pi}}}\mathup{{{\pi}}})_{\mathrm{P}}\,\mathup{{{\pi}}}]_{\mathrm{P}}$
wave in our QMIPWA and observe a narrow peak at
$782\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$, as shown in Fig. 1(b). We
include the $1^{-}\,[\mathup{{{\omega}}}(782)\,\mathup{{{\pi}}}]_{\mathrm{P}}$
wave in the conventional PWA and measure a fit fraction of $[2.95\pm
0.04]\text{\times}{10}^{-03}$, consistent with the prediction of
$4\text{\times}{10}^{-3}$ in [19].

(a)
$1^{+}\,[(\mathup{{{\pi}}}\mathup{{{\pi}}})_{\mathrm{S}}\,\mathup{{{\pi}}}]_{\mathrm{P}}$
wave.

(b)
$1^{-}\,[(\mathup{{{\pi}}}\mathup{{{\pi}}})_{\mathrm{P}}\,\mathup{{{\pi}}}]_{\mathrm{P}}$
wave.

Figure 1: QMIPWA intensities as functions of $m_{2\pi}$ with statistical
uncertainties.

##  4 Conclusion

We see the $\mathrm{a}_{1}(1420)$ and
$1^{-}\,[\mathup{{{\omega}}}(782)\,\mathup{{{\pi}}}]_{\mathrm{P}}$ wave in
tauon decays in both conventional PWA and QMIPWA; this is their first sighting
in tauon decay. We will soon provide an updated model for
$\mathup{{{\tau}}^{\scriptstyle{-}}}\to\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{-}}}\mathup{{{\pi}}^{\scriptstyle{+}}}\mathup{{{\nu}}{}_{\scriptstyle{\\!\tau}}}$
with about 15 partial waves and statistical and systematic uncertainties,
which can be used with the TAUOLA software library for simulating tauon decay
and in future measurements of the tau electric and magnetic dipole moments at
Belle II.

## Acknowledgments

We acknowledge Florian Kaspar for providing his code to train a neural network
on simulated background data, Dmitrii Ryabchikov for providing his code to
resolve ambiguities in QMIPWA, and Stefan Wallner for cross checking our
acceptance correction scheme.

## References

  * [1] E. Kou _et al._ ,  _The Belle II Physics Book_ ,  Progress of Theoretical and Experimental Physics 2019(12) (2019),  10.1093/ptep/ptz106,  123C01.
  * [2] G. Alexander _et al._ ,  _Evidence for the $\tau\to\nu\rho\pi$ decay mode_,  Physics Letters B 73(1), 99 (1978),  https://doi.org/10.1016/0370-2693(78)90181-8.
  * [3] M. Aghasyan _et al._ ,  _Light isovector resonances in ${\pi}^{-}p\rightarrow{\pi}^{-}{\pi}^{-}{\pi}^{+}p$ at $190\text{ }\text{ }\mathrm{GeV}/c$_,  Phys. Rev. D 98, 092003 (2018),  10.1103/PhysRevD.98.092003.
  * [4] D. M. e. Asner,  _Hadronic structure in the decay ${\tau}^{-}\rightarrow{\nu}_{\tau}{\pi}^{-}{\pi}^{0}{\pi}^{0}$ and the sign of the tau neutrino helicity_,  Phys. Rev. D 61, 012002 (1999),  10.1103/PhysRevD.61.012002.
  * [5] P. A. Zyla _et al._ ,  _Rev. of Particle Physics_ ,  PTEP 2020(8), 083C01 (2020),  10.1093/ptep/ptaa104.
  * [6] C. e. Adolph,  _Observation of a new narrow axial-vector meson ${a}_{1}(1420)$_,  Phys. Rev. Lett. 115, 082001 (2015),  10.1103/PhysRevLett.115.082001.
  * [7] D. Ryabchikov,  _Meson spectroscopy at VES and COMPASS_ ,  EPJ Web Conf. 212, 03010 (2019),  10.1051/epjconf/201921203010.
  * [8] M. Mikhasenko, B. Ketzer and A. Sarantsev,  _Nature of the ${a}_{1}(1420)$_,  Phys. Rev. D 91, 094015 (2015),  10.1103/PhysRevD.91.094015.
  * [9] A. Abashian _et al._ ,  _The Belle Detector_ ,  Nucl. Instrum. Meth. A 479, 117 (2002),  10.1016/S0168-9002(01)02013-7.
  * [10] S. Kurokawa and E. Kikutani,  _Overview of the KEKB accelerators_ ,  Nucl. Instrum. Meth. A 499, 1 (2003),  10.1016/S0168-9002(02)01771-0.
  * [11] A. Hocker _et al._ ,  _TMVA - Toolkit for Multivariate Data Analysis_ ,  Tech. rep. (2007), physics/0703039.
  * [12] A. Mathad, D. O’Hanlon, A. Poluektov and R. Rabadan,  _Efficient description of experimental effects in amplitude analyses_ ,  JINST 16(06), P06016 (2021),  10.1088/1748-0221/16/06/P06016,  1902.01452.
  * [13] A. Rabusov, D. Greenwald and S. Paul,  _Partial wave analysis of $\tau^{-}\to\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ at Belle_,  In _41st International Conference on High Energy Physics_ (2022), 2211.11696.
  * [14] J. H. Kühn and E. Mirkes,  _Structure functions in tau decays_ ,  Zeitschrift für Physik C Particles and Fields 67(2), 364 (1995),  10.1007/BF01571299.
  * [15] F. Krinner and S. Paul,  _Hadronic Currents and Form Factors in three-body Semileptonic $\tau$ Decays_ (2021),  2107.04295.
  * [16] I. Kachaev and Ihep,  _Modified S-wave $\pi\pi$ scattering amplitude for multiparticle PWA_ (2023),  2305.11711.
  * [17] A. Rabusov, D. Greenwald and S. Paul,  _Partial-wave analysis of $\tau^{-}\to\pi^{-}\pi^{-}\pi^{+}\nu_{\tau}$ at Belle_,  In _20th International Conference on Hadron Spectroscopy and Structure_ (2023), 2310.09155.
  * [18] F. Krinner, D. Greenwald, D. Ryabchikov, B. Grube and S. Paul,  _Ambiguities in model-independent partial-wave analysis_ ,  Phys. Rev. D 97, 114008 (2018),  10.1103/PhysRevD.97.114008.
  * [19] E. Mirkes and R. Urech,  _Isospin violation in tau — $>$ 3 pi tau-neutrino_,  Eur. Phys. J. C 1, 201 (1998),  10.1007/BF01245809,  hep-ph/9702382.