11institutetext: Théodore Cherrière 22institutetext: Johann Radon Institute
for Computational and Applied Mathematics (RICAM), Austrian Academy of
Science, A-4040 Linz, Altenbergerstraße 69, 22email:
<EMAIL_ADDRESS>
This work is associated with FWF-funded CRC F90 (CREATOR – Computational
Electric Machine Laboratory), project 08.

# Towards topology optimization of a hybrid - excited machine using recursive
material interpolation

Théodore Cherrière

###### Abstract

Hybrid-excited electrical machines aim to combine the advantages of permanent
magnet machines (high efficiency and torque density) with those of separately
excited machines (ease of flux-weakening at high speed). These machines are of
interest to electric vehicles, and only parametric approaches are available in
the literature for their optimization. This work proposes a more general
topology optimization methodology by extending the formalism of density
methods. The difficulty lies in integrating the numerous natures of materials
(conductors, permanent magnets, ferromagnetic material, air…) without strongly
deconvexifying the optimization problem, which leads to non-physical results
with unsatisfactory performance. To address this issue, a recursive material
interpolation is introduced. The hybrid-excited rotors optimized by this
approach are compared with those of existing techniques, demonstrating a clear
superiority of the recursive interpolation.

##  1 Introduction

Topology optimization is a promising tool for designing innovative
electromagnetic actuators, overcoming state-of-the-art structures. Unlike
conventional optimization methodologies relying on a geometric parametrization
of an initial structure arbitrarily chosen by the designer from experience
Cherriere:Cisse2021 , topology optimization is nonparametric and thus aims to
extend the space of possible results. Initially developed in mechanical
engineering Cherriere:Bendsoe1989 , topology optimization was progressively
applied to electromagnetic actuators from the 1990s Cherriere:Dyck1996 .
Topology optimization is now mature for single-material problems and is
implemented in many commercial software.

Among the various topology optimization techniques (cf Cherriere:Lucchini2022
for a detailed state of the art), density methods are the most widely used.
They rely on a so-called ”density” field $\rho:\Omega\rightarrow\mathcal{D}$
over the design region $x\in\Omega$, from which the physical properties at
each point $x\in\Omega$ are defined by interpolating the candidate materials’
properties. Each vertex of the interpolation domain $\mathcal{D}$ is
associated with a candidate material, so to consider more than two materials,
the dimension of $\mathcal{D}$ must be increased. The literature generally
considers hypercubic domains Cherriere:Bruyneel2011  so that
$\mathcal{D}=[0,1]^{n}$, with $n\in\mathbb{N}^{*}$, as illustrated in Figure 1
for $n=1$ and $2$. Note that despite its name, there is generally no physical
interpretation of intermediate values of $\rho$.

Figure 1: Interpolation of single/bi-material (left) and multi-material
(right) with density methods.

We consider an objective function $f$ to be maximized, which generally depends
on a physical state $a\in H^{1}_{0}(\Omega)$ that implicitly depends on the
design field $\rho$ by solving the physical equations. The optimization
problem is then written as

$\text{find}\quad\quad\rho_{\text{opt}}=\arg\displaystyle\min_{\rho:\Omega\rightarrow\mathcal{D}}f(a(\rho)).$
(1)

Since the sensitivity $\mathrm{d}_{\rho}f$ can be computed efficiently by the
adjoint method Cherriere:Cea1986 , we can solve (1) by a projected gradient
descent Cherriere:Calamai1987 . Unfortunately, hypercubic domains are not
suitable for all optimization problems since the resulting interpolation of
physical properties is often non-monotonic, as shown in Figure 1, which
strongly dexonvexifies the problem and leads to uninteresting local optima.
One solution is to extend $\mathcal{D}$ to the set of convex polytopes
Cherriere:Cherriere2022 . These polytopes allow materials of the same nature
(typically, conductors or magnets with different phases or orientations) to
lie on the same plane and the other materials to be placed on an orthogonal
axis. However, this solution is limited in practice to low dimensions
($\dim\mathcal{D}\leq 3$) because of the complexity of the projection, shape
function computation, and the difficulty of visualization to select
appropriate higher-dimensional domains. This dimension limitation prevents
more materials from being considered, as in the case of hybrid-excited
electric machines that aim to combine the flexibility of wound-field
excitation with the efficiency of permanent magnets (PM) for traction
applications Cherriere:Zhu2019 .

In this context, this work presents a generalization of the interpolation
framework required for multi-material density-based topology optimization. It
relies on recursive interpolations supported on subdomains up to three
dimensions. The implementation is freely available Cherriere:Cherriere2024Code
and is briefly described in Section 2. This formalism is then applied to the
topology optimization of a hybrid-excited machine rotor and compared with
existing techniques in the literature in Section 3.

##  2 Interpolation framework

Let us first consider $n_{m}$ candidate materials distributed over a single
polytope $\mathcal{D}$. Each material $i$ is associated with the $i$-vertex of
$\mathcal{D}$. The interpolation of a material property $\kappa$, possibly
dependent on the physical state $a$ in the case of non-linear materials, is
denoted $\tilde{\kappa}(a,\rho)$. This interpolation is constructed using
generalized barycentric coordinates of $\mathcal{D}$ Cherriere:Cherriere2022 .
We denote $\omega_{i}(\rho)$ the barycentric coordinate associated with vertex
$i$, which is 1 when $\rho$ corresponds to the coordinates of vertex $i$, and
0 when $\rho$ corresponds to the coordinates of the other vertices. These
barycentric coordinates partition the unit, and their computation is automatic
Cherriere:Floater2014 . Therefore, a generic interpolation is defined by:

$\tilde{\kappa}(\rho,a)=\sum_{i=1}^{n_{m}}\omega_{i}(\rho)\kappa_{i}(a).$ (2)

In (2), another interpolation $\tilde{\kappa}_{i}$ on a new sub-domain can
replace any material property $\kappa_{i}$. Figure 2 (left) gives a convenient
representation of a possible hierarchical interpolation domain. Projection
onto such a structure is simply reduced to projection onto each low-
dimensional sub-domain, which is easy to handle. The optimization variables
are then dissociated and defined on each sub-domain.

From a computational point of view, this structure is a rooted tree denoted as
$\mathcal{T}$, the branches being products by shape functions, the nodes being
sums, and the leaves being material properties. Using Neveu’s notation
Cherriere:Neveu1986 , the set of children associated to the node $l$ is
$\mathcal{C}(l)=\\{[l,n]~{}|~{}n\in\mathbb{N}^{*},[l,n]\in\mathcal{T}\\}$,
with $[\alpha,\beta]$ the concatenation of the lists of indices $\alpha$ and
$\beta$. An example is shown in Figure 2 (right).

Figure 2: Hierarchical $\mathcal{D}$: 3D visualization (left) and associated
rooted tree with Neveu’s labels (right)

Each subdomain $\mathcal{D}_{l}$ is associated with a specific optimization
variable $\rho_{l}$. The evaluation of the interpolated physical property
$\tilde{\kappa}_{\emptyset}$ can be performed recursively by applying the
chain rule with $l=\emptyset$:

$\tilde{\kappa}_{l}(a,\rho)=\left\\{\begin{array}[]{ll}\kappa_{l}(a)&\quad\text{if~{}}\mathcal{C}(l)=\emptyset\\\
\sum_{i\in\mathcal{C}(l)}\omega_{i}(\rho_{l})\tilde{\kappa}_{i}(a,\rho)&\quad\text{else.}\end{array}\right.$
(3)

It is necessary to compute the derivatives with respect to $a$ to solve the
nonlinear physical problem using Newton’s method and the derivative with
respect to $\rho$ to compute the sensitivities of $f$ within the gradient
descent framework. The evaluation of $\partial_{a}\tilde{\kappa}$ is
straightforward by replacing $\tilde{\kappa}$ and $\kappa$ by
$\partial_{a}\tilde{\kappa}$ and $\partial_{a}\kappa$ in (3), respectively.
The computation of $\partial_{\rho}\tilde{\kappa}$ is a little bit more
involved since the optimization variables are dissociated in each subdomain
$\mathcal{D}_{l}$. Each partial derivative $\partial_{\rho_{l}}\tilde{\kappa}$
can therefore be evaluated recursively for all $l\in\mathcal{T}$ by calling
drho_k$(\emptyset,1)$ defined in Algorithm 1.

Algorithm 1 drho_k($l,k$) : Evaluate $\partial_{\rho}\tilde{\kappa}$

Compute $\mathcal{C}(l)$

$\displaystyle\partial_{\rho_{l}}\tilde{\kappa}\leftarrow
k\sum_{i\in\mathcal{C}(l)}\tilde{\kappa}_{l}(a,\rho)\frac{\mathrm{d}\omega_{i}}{\mathrm{d}\rho_{l}}$
{$\tilde{\kappa}_{l}(a,\rho)$ is computed with (3)}

for $i\in\mathcal{C}(l)$ do

drho_k($i,k\omega_{i}(\rho_{l})$)

end for

A general open-source Matlab${}^{\text{\textregistered}}$ implementation is
provided in Cherriere:Cherriere2024Code , including elements for defining
interpolation domains and functions to be interpolated, as well as evaluation,
derivation, and projection routines. An example of application to electrical
engineering is given in the next section. This implementation also allows for
the consideration of penalty functions, common in topology optimization to
eliminate intermediate materials Cherriere:Bendsoe1999 , but not used in this
paper.

##  3 Topology optimization of a hybrid-excited rotor

###  3.1 Problem definition

The topology optimization of rotors for hybrid-excited machines
Cherriere:Zhu2019  is a typical problem where several materials’ natures
coexist. These machines combine ferromagnetic material, air, PMs, and
adjustable direct-current (DC) electrical conductors. The roles of the DC
supply are (i) to reinforce the flux created by the magnets at low speeds to
generate greater torque and (ii) to weaken the magnets’ flux at high speeds to
reduce the back-electromotive force.

We consider a rotor pole with 16 different candidate ”materials”: one ideal
magnet with a remanent field of $1\text{\,}\mathrm{T}$ and 12 discrete
orientations, two electrical conductors with opposite current density equal to
$\pm$10\text{\,}\mathrm{A}\mathrm{/}\mathrm{m}\mathrm{m}^{2}$$, iron-silicon
steel with standard anhysteretic nonlinear behavior Cherriere:Cherriere2023_M
and air. Several interpolation domains $\mathcal{D}$ are compared and plotted
in Figure 3, one being defined recursively with the formalism described in
Section 2. The optimization problem reads:

$\text{find}\quad\displaystyle\rho^{*}=\min_{\rho\in\mathcal{D}}-\left(\frac{\gamma+1}{2}\phi^{+}(a(\rho))-\frac{\gamma-1}{2}\phi^{-}(a(\rho))\right),$
(4)

with $\gamma\in[-1,1]$ an arbitrary coefficient, $\phi^{+}$ and $\phi^{-}$ the
outward magnetic fluxes produced by the rotor pole when the supply current is
positive and negative, respectively. The physical state $a$ is here the
component along $z$ of the magnetic vector potential, which verifies the
classical equations of 2D magnetostatics Cherriere:Cherriere2023_M . The
physical quantities interpolated by (2) are magnetic polarization and current
density. The optimization region is a rotor pole of the same size as that of
the BMWi3 motor, discretized on a mesh of 9663 elements. The stator influence
is neglected and is replaced by air. The optimization algorithm is a
regularized projected gradient descent with density filtering that is detailed
in Cherriere:Cherriere2024Compumag . It stops after 500 iterations or if the
design variables stagnate when the norm of the relative difference between two
consecutive iterations is less than $10^{-4}$.

Hexadecagon (2D) | Diamond (3D) | Recursive domain (6D)
---|---|---
|  |
Figure 3: 3D representations of the interpolation domains with the materials’
color scale.

###  3.2 Optimization results

Optimizations are performed for a regular distribution of $\gamma\in[-1,1]$
and homogeneous initializations in the different interpolation domains shown
in Figure 3.

According to (4), the outwards flux should be maximized when $\gamma=1$, and
the inwards flux should be maximized when $\gamma=-1$. Then, the associated
optimized rotors contain only PMs arranged in a well-known Halbach structure
Cherriere:Zhu2001 , because the magnets generate higher flux than electrical
conductors with reasonable current.

When $\gamma=0$, the flux must be maximized outwards with a positive current
feeding and inwards with a negative current feeding. Then, the optimized
result contains no magnets (that cannot reverse the direction of their flux)
but rather conductors and steel to carry the magnetic flux. These single-
excited optimized rotors obtained with integer $\gamma$ are shown in Figure 4,
and the color scale to interpret the materials is given in the previous Figure
3.

$\gamma=-1$ | $\gamma=0$ | $\gamma=1$
---|---|---
|  |
$\phi^{+}=-$23.1\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$ | $\phi^{+}=$7.3\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$ | $\phi^{+}=$23.1\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$
$\phi^{-}=-$23.1\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$ | $\phi^{-}=-$7.3\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$ | $\phi^{-}=$23.1\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$
Figure 4: Non-hybridized optimized designs obtained with integers $\gamma$ and
the recursive interpolation domain (results obtained with the other domains
are similar). $\phi$ is normalized on $1\text{\,}\mathrm{m}$ axial length.

Hybrid excited rotors are obtained with intermediate (non-integer) values of
$\gamma$. In particular, a ”hybridization indicator”, denoted $sd_{0}$, can be
defined by the normalized signed distance to points where
$|\phi^{+}|=|\phi^{-}|$. It is maximum when $|\phi^{+}|$ is maximum and
$|\phi^{-}|=0$, or vice versa, corresponding to an ideal hybrid-excited rotor.
This indicator is plotted in the $(\phi^{+},\phi^{-})$ plane in Figure 5,
where the Pareto front based on recursive interpolation reaches higher values
than the others, and stays between the physical lower and upper bounds of the
flux.

Figure 5: Pareto front obtained for $\gamma\in[-1,1]$. The step size between
$\gamma$ values is $10^{-3}$.

The best structures according to $sd_{0}$ indicator are plotted in Figure 6.
We note that the recursive interpolation domain leads to a more performing
hybrid-excited rotor. Moreover, this design is symmetric and contains fewer
”gray” intermediate materials, highlighting the non-convergence in the
optimizations relying on the other domains. The structure obtained by
recursive interpolation is close to a machine with parallel hybrid excitation
Cherriere:Hwang2018 , in which the flux created by the windings is in parallel
with the flux created by the magnets. This structure is more controllable than
serial double-excitation, in which the flux generated by the windings passes
through the magnet, whose permeability is very low. Once again, the remaining
magnets are arranged in a Halbach array, which maximizes their flux.

Hexadecagon | Diamond | Recursive domain
---|---|---
$\gamma=-0.032$ | $\gamma=-0.068$ | $\gamma=0.174$
|  |
$\phi^{+}=-$0.3\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$ | $\phi^{+}=-$0.6\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$ | $\phi^{+}=$12.1\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$
$\phi^{-}=$-6.5\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$ | $\phi^{-}=-$8.1\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$ | $\phi^{-}=$0.4\text{\,}\mathrm{m}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$$
$sd_{0}=$26\text{\,}\%$$ | $sd_{0}=$32\text{\,}\%$$ | $sd_{0}=$50\text{\,}\%$$
Figure 6: Optimized designs that maximize the hybridization indicator $sd_{0}$
for the three different interpolation domains.

##  4 Conclusion

The recursive interpolation proposed in this article offers more flexibility
than the interpolation domains currently available in the literature and can
be adapted to a wide variety of applications. In particular, the formalism is
suitable for the topology optimization of double-excitation machines, which is
original.

Further work will focus on the rotor’s mechanical integrity and include
additional alternating current conductors for applications such as stators of
hybrid-excited flux-switching machines Cherriere:Gaussens2014 . The long-term
objective is to build a comprehensive toolbox for optimizing electromagnetic
actuators without initial information on the geometry and requiring as few
tuning parameters as possible.

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# Towards establishing lepton flavour universality violation in
$\bar{B}\to\bar{K}^{*}\ell^{+}\ell^{-}$ decays

Andrea Mauri<EMAIL_ADDRESS>Nicola Serra<EMAIL_ADDRESS>Rafael Silva
Coutinho<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
Winterthurer Strasse 190, 8057 Zürich, Switzerland

###### Abstract

Rare semileptonic $b\to s\ell^{+}\ell^{-}$ transitions provide some of the
most promising frameworks to search for new physics effects. Recent analyses
of these decays have indicated an anomalous behaviour in measurements of
angular distributions of the decay $B^{0}\to K^{*}\mu^{+}\mu^{-}$ and lepton-
flavour-universality observables. Unambiguously establishing if these
deviations have a common nature is of paramount importance in order to
understand the observed pattern. We propose a novel approach to independently
and complementary probe this hypothesis by performing a simultaneous amplitude
analysis of $\bar{B}^{0}\to\bar{K}^{*0}\mu^{+}\mu^{-}$ and
$\bar{B}^{0}\to\bar{K}^{*0}e^{+}e^{-}$ decays. This method enables the direct
determination of observables that encode potential non-equal couplings of
muons and electrons, and are found to be insensitive to nonperturbative QCD
effects. If current hints of new physics are confirmed, our approach could
allow an early discovery of physics beyond the standard model with LHCb run II
data sets.

††preprint: ZU-TH-15/18

Flavour changing neutral current processes of B meson decays are crucial
probes for the standard model (SM), since as yet undiscovered particles may
contribute to these transitions and cause observables to deviate from their SM
predictions Grossman and Worah (1997); Fleischer (1997); London and Soni
(1997); Ciuchini  _et al._ (1997). The decay mode
$\bar{B}\to\bar{K}^{*}\ell^{+}\ell^{-}$ is a prime example (i.e.,
$\ell=\mu,e$), which offers a rich framework to study from differential decay
widths to angular observables. An anomalous behaviour in angular and branching
fraction analyses of the decay channel
$\bar{B}^{0}\to\bar{K}^{*0}\mu^{+}\mu^{-}$ has been recently reported Aaij
_et al._ (2016a); Wehle  _et al._ (2016); Aaij  _et al._ (2013a, 2014a),
notably in one of the observables with reduced theoretical uncertainties,
$P^{\prime}_{5}$ Aaij  _et al._ (2013b); Descotes-Genon  _et al._ (2016).
Several models have been suggested in order to interpret these results as new
physics (NP) signatures Gauld  _et al._ (2014); Buras and Girrbach (2013);
Altmannshofer and Straub (2013); Crivellin  _et al._ (2015); Hiller and
Schmaltz (2014); Biswas  _et al._ (2015); Gripaios  _et al._ (2015).
Nonetheless, the vectorlike nature of this pattern could be also explained by
non-perturbative QCD contributions from $b\to sc{\bar{c}}$ operators (i.e.,
charm loops) that are able to either mimic or camouflage NP effects Jäger and
Martin Camalich (2013, 2016); Ciuchini  _et al._ (2016). Nonstandard
measurement in ratios of $b\to s\ell^{+}\ell^{-}$ processes \- such as of
$R_{K}$ Aaij  _et al._ (2014b) and $R_{K^{*}}$ Aaij  _et al._ (2017a) \-
indicate a suppression of the muon channel which is also compatible with the
$P^{\prime}_{5}$ anomaly. In this case an immediate interpretation of lepton
flavour universality (LFU) breaking is suggested due to the small theoretical
uncertainties in their predictions Hiller and Kruger (2004); Bordone  _et al._
(2016). Whilst the individual level of significance of the present anomalies
is still inconclusive, there is an appealing nontrivial consistent pattern
shown in global analysis fits Capdevila  _et al._ (2018); Altmannshofer  _et
al._ (2017); Hurth  _et al._ (2017a); Ciuchini  _et al._ (2017); Alok  _et
al._ (2017).

The formalism of b decays is commonly described within an effective field
theory Altmannshofer  _et al._ (2009), which probes distinct energy scales;
with regimes classified into short-distance (high energies) perturbative and
noncalculable long-distance effects. These can be parametrised in the weak
Lagrangian in terms of effective operators with different Lorentz structures,
$\mathcal{O}_{i}$, with corresponding couplings $\mathcal{C}_{i}$ \- referred
to as Wilson coefficients (WC). Only a subset of the operators that are most
sensitive to NP is examined in this work Bobeth  _et al._ (2017), i.e.
$\mathcal{O}_{7}$ (virtual photon exchanges), $\mathcal{O}_{9,10}$ (vector and
axial currents) and corresponding right-handed couplings with flipped
helicities. In this framework NP effects are incorporated by introducing
deviations in the WCs Ali  _et al._ (1995) from their SM predictions, i.e.,
$\mathcal{C}_{i}=\mathcal{C}^{\mathrm{SM}}_{i}+\mathcal{C}^{\mathrm{NP}}_{i}$.
For instance, the anomalous pattern seen in semileptonic decays can be
explained by a shift in the coefficient $\mathcal{C}_{9}$ only, or
$\mathcal{C}_{9}$ and $\mathcal{C}_{10}$ simultaneously Capdevila  _et al._
(2018); Altmannshofer  _et al._ (2017); Hurth  _et al._ (2017a). A direct
experimental determination of the WCs is currently bounded by sizeable
uncertainties that arise from nonfactorisable hadronic matrix elements that
are difficult to assess reliably from first principles. Some promising
approaches suggest to extract this contribution from data-driven analyses
Blake  _et al._ (2017); Hurth  _et al._ (2017b) and by exploiting analytical
properties of its structure Bobeth  _et al._ (2017). However, these models
still have intrinsic limitations, in particular in the assumptions that enter
in parametrisation of the dilepton invariant mass distribution.

In this article, we propose a new model-independent approach that from a
simultaneous unbinned amplitude analysis of both
$\bar{B}^{0}\to\bar{K}^{*0}\mu^{+}\mu^{-}$ and
$\bar{B}^{0}\to\bar{K}^{*0}e^{+}e^{-}$ decays can, for the first time,
unambiguously determine LFU-breaking from direct measurements of WCs. This
work builds on the generalisation of Ref. Bobeth  _et al._ (2017), but it is
insensitive to the model assumptions of the parametrisation. This effect
relies on the strong correlation between the muon and electron modes imposed
by the lepton-flavour universality of the hadronic matrix elements.
Furthermore, in this method the full set of observables (e.g $R_{K^{*}}$,
$P^{\prime}_{5}$ and branching fraction measurements) available in
$\bar{B}\to\bar{K}^{*}\ell^{+}\ell^{-}$ decays is exploited, providing
unprecedented precision on LFU in a single analysis.

Consider the differential decay rate for
$\bar{B}\to\bar{K}^{*}\ell^{+}\ell^{-}$ decays (dominated by the on-shell
$\bar{K}^{*0}$ contribution) fully described by four kinematic variables; the
di-lepton squared invariant mass, $q^{2}$, and the three angles
$\vec{\Omega}=(\cos\theta_{\ell},\cos\theta_{K},\phi)$ Altmannshofer  _et al._
(2009). The probability density function ($p.d.f.$) for this decay can be
written as

$p.d.f.=\frac{1}{\Gamma}\frac{\textrm{d}^{4}\Gamma}{\textrm{d}q^{2}\textrm{d}^{3}\Omega},\
\quad\text{with}\quad\Gamma=\int_{q^{2}}\textrm{d}q^{2}\frac{\textrm{d}\Gamma}{\textrm{d}q^{2}}\,,$
(1)

with different $q^{2}$intervals depending on the lepton flavour under study.
For a complete definition of
$\textrm{d}^{4}\Gamma/(\textrm{d}q^{2}\textrm{d}^{3}\Omega)$ we refer to
Bobeth  _et al._ (2008); Altmannshofer  _et al._ (2009) and references
therein. It is convenient to explicitly write the WC dependence on the decay
width by the transversity amplitudes ($\lambda=\perp,\parallel,0$) as Bobeth
_et al._ (2017)

$\displaystyle{\cal{A}}_{\lambda}^{(\ell)\,L,R}$ $\displaystyle=$
$\displaystyle{\cal{N}}_{\lambda}^{(\ell)}\ \bigg{\\{}(C^{(\ell)}_{9}\mp
C^{(\ell)}_{10}){\cal{F}}_{\lambda}(q^{2})$
$\displaystyle+\frac{2m_{b}M_{B}}{q^{2}}\bigg{[}C^{(\ell)}_{7}{\cal{F}}_{\lambda}^{T}(q^{2})-16\pi^{2}\frac{M_{B}}{m_{b}}{\cal{H}}_{\lambda}(q^{2})\bigg{]}\bigg{\\}}\,,$

where ${\cal{N}}_{\lambda}^{(\ell)}$ is a normalisation factor, and
${\cal{F}}^{(T)}_{\lambda}(q^{2})$ and $\mathcal{H}_{\lambda}(q^{2})$ are
referred to “local” and “nonlocal” hadronic matrix elements, respectively. The
${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factors, while
$\mathcal{H}_{\lambda}(q^{2})$ encode the aforementioned nonfactorisable
hadronic contributions and are described using two complementary
parametrisations Bobeth  _et al._ (2017); Hurth  _et al._ (2017b) \- for
brevity only a subset of results is shown for the latter approach. In the
following this function is expressed in terms of a “conformal” variable
$z(q^{2})$ Bobeth  _et al._ (2017); Boyd  _et al._ (1995); Bourrely  _et al._
(2009), with an analytical expansion truncated at a given order $z^{n}$
(herein referred to as $\mathcal{H}_{\lambda}[z^{n}]$), after removing
singularities related to the $J/\psi(1S)$ and $\psi(2S)$. Further information
about the formalism is given in appendix A. One of the drawbacks of this
expansion is that a priori there is no physics argument to justify the order
of the polynomial to be curtailed at - which in turn currently limits any
claim on NP sensitivity.

In order to overcome these points, we investigate the LFU-breaking hypothesis
using direct determinations of the difference of Wilson coefficients between
muons and electrons, i.e.,

$\Delta\mathcal{C}\xspace_{i}=\widetilde{\mathcal{C}}_{i}^{(\mu)}-\widetilde{\mathcal{C}}_{i}^{(e)}\,,$
(3)

where the usual WCs $\mathcal{C}_{i}^{(\mu,e)}$ are renamed as
$\widetilde{\mathcal{C}}_{i}^{(\mu,e)}$, since an accurate disentanglement
between the physical meaning of $\mathcal{C}\xspace_{i}^{(\mu,e)}$ and the
above-mentioned hadronic pollution cannot be achieved at the current stage of
the theory Chrzaszcz  _et al._ (2018). The key feature of this strategy is to
realise that all hadronic matrix elements are known to be lepton-flavour
universal, and thus are shared among both semileptonic decays. This benefits
from the large statistics available for
$\bar{B}^{0}\to\bar{K}^{*0}\mu^{+}\mu^{-}$ decays that is sufficient to enable
the determination of these multispace parameters. Note that an amplitude
analysis of the electron mode only has been previously disregarded, given the
limited data set in either LHCb or Belle experiments. In a common framework
the hadronic contributions are treated as nuisance parameters, while only the
Wilson coefficients $\widetilde{\mathcal{C}\xspace}_{9}^{(\mu,e)}$ and
$\widetilde{\mathcal{C}\xspace}_{10}^{(\mu,e)}$ are kept separately for the
two channels. For consistency the WC $\widetilde{C}_{7}$ is also shared in the
fit and fixed to its SM value, given its universal coupling to photons and the
strong constraint from radiative $B$ decays Paul and Straub (2017). In the
following, all the right-handed WCs are fixed to their SM values, i.e.,
$\mathcal{C}\xspace_{i}^{\prime\,(\mu,e)}=0$, while sensitivity studies on the
determination of the WCs $\mathcal{C}\xspace_{9}^{\prime\,(\mu)}$ and
$\mathcal{C}\xspace_{10}^{\prime\,(\mu)}$ are detailed in Appendix B.

Signal-only ensembles of pseudoexperiments are generated with sample size
corresponding roughly to the yields foreseen in LHCb run-II [$8\,$fb-1] and
future upgrades [$50\,$-$\,300\,$fb-1] Aaij  _et al._ (2017b), and Belle II
[$50\,$ab-1]. These are extrapolated from Refs. Aaij  _et al._ (2016a, 2017a);
Wehle  _et al._ (2016) by scaling, respectively, with luminosity and
$\sigma_{b\bar{b}}\propto\sqrt{s}$ for LHCb, where $s$ denotes the designed
centre-of-mass energy of the $b$-quark pair, and exclusively with luminosity
for Belle II. Note that for brevity most of the results are shown for the
representative scenario of LHCb run II. The studied $q^{2}$range corresponds
to $1.1\,^{2}\leq q^{2}\leq 8.0\,^{2}$ and $11.0\,^{2}\leq q^{2}\leq
12.5\,^{2}$ for the muon mode and $1.1\,^{2}\leq q^{2}\leq 7.0\,^{2}$ for the
electron mode in LHCb; while in Belle II the same kinematic regions are
considered for both semileptonic channels, namely $1.1\,^{2}\leq q^{2}\leq
8.0\,^{2}$ and $10.0\,^{2}\leq q^{2}\leq 13.0\,^{2}$. This definition of
$q^{2}$ranges are broadly consistent with published results, and assumes
improvements in the electron mode resolution for LHCb Lionetto  _et al._
(2018).

Within the SM setup the Wilson coefficients are set to
$\mathcal{C}^{\rm{SM}}_{9}=4.27$, $\mathcal{C}^{\rm{SM}}_{10}=-4.17$ and
$\mathcal{C}^{\rm{SM}}_{7}=-0.34$ (see Bobeth  _et al._ (2017) and references
therein), corresponding to a fixed renormalisation scale of $\mu=4.2\,$GeV.
This baseline model is modified for two NP benchmark points (BMP),
$\Delta\mathcal{C}\xspace_{9}=-1$ and
$\Delta\mathcal{C}\xspace_{9}=-\Delta\mathcal{C}\xspace_{10}=-0.7$, referred
to, respectively, as BMP${}_{\mathcal{C}\xspace_{9}}$ and
BMP${}_{\mathcal{C}\xspace_{9,10}}$, where NP is inserted only in the case of
muons, i.e., $\mathcal{C}\xspace_{i}^{(e)}=\mathcal{C}\xspace_{i}^{\rm{SM}}$.
These points are favoured by several global fit analyses with similar
significance Capdevila  _et al._ (2018); Altmannshofer  _et al._ (2017); Hurth
_et al._ (2017a).

An extended unbinned maximum likelihood fit is performed to these simulated
samples, in which multivariate Gaussian terms are added to the log-likelihood
to incorporate prior knowledge on the nuisance parameters. In order to probe
the model-independence of the framework, the nonlocal hadronic parametrisation
is modified in several ways (see Appendix A for a detailed discussion), i.e.,

  1. i.

baseline $\mathcal{H}_{\lambda}[z^{2}]$ SM prediction Bobeth  _et al._ (2017)
included as a multivariate Gaussian constraint;

  2. ii.

no theoretical assumption on $\mathcal{H}_{\lambda}[z^{2}]$ and with free-
floating parameters;

  3. iii.

higher orders of the analytical expansion of $\mathcal{H}_{\lambda}[z^{n}]$ up
to $z^{3}$ and $z^{4}$ \- free floating;

  4. iv.

and re-parametrisation of the nonlocal hadronic matrix elements as proposed in
Ref. Hurth  _et al._ (2017b), i.e., including them as multiplicative factors
to the corresponding leading hadronic terms.

On the other hand, form factors parameters are taken from Bharucha  _et al._
(2016) and, in order to guarantee a good agreement between Light-Cone Sum
Rules Ball and Braun (1998); Khodjamirian  _et al._ (2007) and Lattice results
Becirevic  _et al._ (2007); Horgan  _et al._ (2014), their uncertainties are
doubled with respect to Ref. Bharucha  _et al._ (2016).

Figure 1 shows the fit results for several alternative parametrisations of the
nonlocal hadronic contribution for the BMP${}_{\mathcal{C}\xspace_{9}}$
hypothesis, with yields corresponding to LHCb run II scenario. We observe that
the sensitivity to $\widetilde{\mathcal{C}\xspace}_{9}^{(\mu,e)}$ is strongly
dependent on the model assumption used for the nonlocal matrix elements.
Nonetheless, it is noticeable that the high correlation of the
$\widetilde{\mathcal{C}}_{9}^{(\mu)}$ and $\widetilde{\mathcal{C}}_{9}^{(e)}$
coefficients is sufficient to preserve the true underlying physics at any
order of the series expansion $\mathcal{H}_{\lambda}[z^{n}]$ and without any
parametric theoretical input, i.e., the two-dimensional pull estimator with
respect to the LFU hypothesis is unbiased.

Figure 1: Two-dimensional sensitivity scans for the pair of Wilson
coefficients $\widetilde{\mathcal{C}}_{9}^{(\mu)}$ and
$\widetilde{\mathcal{C}}_{9}^{(e)}$ for different nonlocal hadronic
parametrisation models evaluated at BMP${}_{\mathcal{C}\xspace_{9}}$, and with
the expected statistics after LHCbrun II. The contours correspond to 99%
confidence level statistical-only uncertainty bands and the dotted black line
indicates the LFU hypothesis.

We note that, as commonly stated in the literature (see e.g., recent review in
Ref. Capdevila  _et al._ (2017)), the determination of
$\mathcal{C}\xspace_{10}^{(\mu,e)}$ is insensitive to the lack of knowledge on
the nonlocal hadronic effects. Nevertheless, its precision is still bounded to
the uncertainties on the form factors, that are found to be the limiting
factor by the end of run II.

Figure 2: Two-dimensional sensitivity scans for the proposed observables
$\Delta\mathcal{C}\xspace_{9}$ and $\Delta\mathcal{C}\xspace_{10}$ for
different nonlocal hadronic parametrisation models evaluated at (top)
BMP${}_{\mathcal{C}\xspace_{9}}$ and (bottom)
BMP${}_{\mathcal{C}\xspace_{9,10}}$, and with the expected statistics after
LHCbrun II. The contours correspond to 99% confidence level statistical-only
uncertainty bands.

The sensitivity to the two benchmarklike NP scenarios using the proposed
pseudo-observables $\Delta\mathcal{C}\xspace_{i}$ is shown in Fig. 2. We
quantify the maximal expected significance with respect to the SM to be $4.6$
and $5.3\,\sigma$ for BMP${}_{\mathcal{C}\xspace_{9}}$ and
BMP${}_{\mathcal{C}\xspace_{9,10}}$, respectively. Realistic experimental
effects are necessary to determine the exact sensitivity achievable.
Nevertheless, these results suggest that a first observation (with a single
measurement) of LFU breaking appears to be feasible with the expected recorded
statistics by the end of LHCb run II. Furthermore, it is interesting to
examine the prospects for confirming this evidence in the upcoming LHCb/Belle
upgrades Albrecht  _et al._ (2017). Figure 3 summarises the two-dimensional
statistical-only significances for the designed luminosities. Both LHCb
Upgrade and Belle II experiments have comparable sensitivities (within
$8.0-10\,\sigma$), while LHCb High-Lumi has an overwhelming significance.
These unprecedented data sets will not only yield insights on this phenomena
but also enable a deeper understanding of the nature of NP - insensitive to
both local and nonlocal hadronic uncertainties.

Figure 3: Two-dimensional sensitivity scans for the proposed observables
$\Delta\mathcal{C}\xspace_{9}$ and $\Delta\mathcal{C}\xspace_{10}$ for the two
considered NP scenarios: (green) BMP${}_{\mathcal{C}\xspace_{9}}$ and (red)
BMP${}_{\mathcal{C}\xspace_{9,10}}$. The contours correspond to 99% confidence
level statistical-only uncertainty bands expected for the (dashed) BelleII 50
ab-1 and LHCbUpgrade (dotted) $50\,$fb-1 and (solid) $\,300\,$fb-1 statistics.

Experimental resolution and detector acceptance/efficiency effects are not
considered in this work, as these would require further information from
current (nonpublic) or planned B-physics experiments. Nevertheless, the
precision on this measurement can remain unbiased either by parametrising
these effects in the amplitude model and/or even recomputing the angles or the
$q^{2}$ variables constraining the $B$ invariant mass Lionetto  _et al._
(2018). Moreover, the differential decay width can receive additional complex
amplitudes from signal-like backgrounds, e.g., $K\pi$ S-wave from a
nonresonant decay and/or a scalar resonance Becirevic and Tayduganov (2013).
These contributions are determined to be small Aaij  _et al._ (2016a, b), and
in the proposed formalism they benefit from the same description between the
muon and electron mode (see detailed discussion in Ref. Chrzaszcz  _et al._
(2018)). Therefore, such contribution does not dilute the expected sensitivity
of the measurement.

Another important test to probe the stability of the model consists in
analysing potential issues that can rise if the truncation
$\mathcal{H}_{\lambda}[z^{n}]$ is not a good description of nature. We proceed
as follows: we generate ensembles with nonzero coefficients for
$\mathcal{H}_{\lambda}[z^{3}]$ and $\mathcal{H}_{\lambda}[z^{4}]$, and we
perform the fit with $\mathcal{H}_{\lambda}[z^{2}]$. Despite the mismodelling
of the nonlocal hadronic effects in the fit, we observe that the determination
of $\Delta\mathcal{C}\xspace_{9}$ and $\Delta\mathcal{C}\xspace_{10}$ is
always unbiased, thanks to the relative cancellation of all the shared
parameters between the two channels. It is worth mentioning that a
hypothetical determination of the individual
$\widetilde{\mathcal{C}\xspace}_{9}^{(\mu,e)}$ and
$\widetilde{\mathcal{C}\xspace}_{10}^{(\mu,e)}$ WCs can also produce a shift
in their central values that mimics the behaviour of NP Chrzaszcz  _et al._
(2018).

In conclusion, we propose a clean and model-independent method to combine all
the available information from $\bar{B}\to\bar{K}^{*}\ell^{+}\ell^{-}$ decays
for a precise determination of LFU-breaking differences of WCs, i.e.,
$\Delta\mathcal{C}\xspace_{9}$ and $\Delta\mathcal{C}\xspace_{10}$. This
relies on a shared parametrisation of the local (form factors) and nonlocal
($\mathcal{H}_{\lambda}[z^{n}]$) hadronic matrix elements between the muonic
and electronic channels, that in turn enables the determination of the
observables of interest free from any theoretical uncertainty. In addition,
this simultaneous analysis is more robust against experimental effects such as
mismodeling of the detector resolution, since most parameters are effectively
determined from the muon mode. This would be an important benefit for LHCb
where the electron resolution is significantly worse than that of muons.
Figure 4 illustrates the usefulness of the newly-proposed observables by
combining the different information from angular analysis to branching ratio
measurements. Due to the inclusiveness of the approach, the expected
sensitivity surpasses any of the projections for the foreseen measurements of,
e.g., $R_{K^{*}}$ or $P^{\prime}_{5}$ alone - given the benchmark points.
Therefore, this novel formalism can be the most immediate method to observe
unambiguously NP in $\bar{B}\to\bar{K}^{*}\ell^{+}\ell^{-}$ decays.

A promising feature of this framework is the possibility to extend the
analysis to include other decay channels involving flavour changing neutral
currents. For instance, the charged decay
$\bar{B}^{+}\to\bar{K}^{*+}\ell^{+}\ell^{-}\xspace$ undergoes the same physics
and is easily accessible at the $B$-factories, while other rare semi-leptonic
decays such as $B^{+}\to K^{+}\ell^{+}\ell^{-}\xspace$ and
$\Lambda_{b}\to\Lambda^{(*)}\ell^{+}\ell^{-}$ have a different phenomenology
but access the same NP information in terms of WC description. Thus, an
unbinned global simultaneous fit to all data involving $b\to
s\ell^{+}\ell^{-}$ transitions is a natural and appealing extension of this
work. Moreover, the parameter space of the investigated WCs can also be
broadened to incorporate direct measurement of the right-handed
$\mathcal{C}\xspace_{i}^{\prime}$ \- currently weakly constrained by global
fits Capdevila  _et al._ (2018); Altmannshofer  _et al._ (2017); Hurth  _et
al._ (2017a).

Figure 4: Sensitivity to BMP${}_{\mathcal{C}\xspace_{9,10}}$ scenario for the
expected statistics after the LHCbrun II. The relative contribution (68, 95,
99% confidence level contours) of each step of the analysis is shown in
different colours, together with the result of full amplitude method proposed
in this article.

## Acknowledgments

We acknowledge useful contributions from Michele Atzeni, Gino Isidori, Danny
van Dyk and Patrick Owen. This work is supported by the Swiss National Science
Foundation (SNF) under Contracts No. 173104 and No. 174182.

##  Appendix A Formalism

The nonlocal hadronic matrix elements $\mathcal{H}_{\lambda}(q^{2})$ are
investigated using two complementary parametrisations Bobeth  _et al._ (2017);
Hurth  _et al._ (2017b).

The nominal parametrisation Bobeth  _et al._ (2017) is obtained through the
mapping

$q^{2}\mapsto
z(q^{2})\equiv\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{0}}}\,,$
(4)

where $t_{+}=4M_{D}^{2}$ and
$t_{0}=t_{+}-\sqrt{t_{+}(t_{+}-M^{2}_{\psi(2S)})}$, which leads to the
functions $\mathcal{H}_{\lambda}(z)$ that are characterised by two
singularities at $z_{J/\psi}$ and $z_{\psi(2S)}$. These can be expressed as

$\mathcal{H}_{\lambda}(z)=\frac{1-zz^{*}_{J/\psi}}{z-z_{J/\psi}}\frac{1-zz^{*}_{\psi(2S)}}{z-z_{\psi(2S)}}\hat{\mathcal{H}}_{\lambda}(z)\,,$
(5)

where the functions $\hat{\mathcal{H}}_{\lambda}(z)$ are analytical and can be
Taylor-expanded around $z=0$ as

$\hat{\mathcal{H}}_{\lambda}(z)=\Big{[}\sum^{n}_{k=0}\alpha_{k}^{(\lambda)}z^{k}\Big{]}\mathcal{F}_{\lambda}(z)\,.$
(6)

Several orders of the polynomials are studied in the text. Note that any
additional order $k$ introduces a complex parameter, $\alpha_{k}^{(\lambda)}$,
for each of the polarisations $\lambda=\perp,\parallel,0$. These nuisance
parameters can be either free floated in the fit (nominal configuration
labelled as $\mathcal{H}_{\lambda}[z^{2},...,z^{4}]$) or Gaussian constrained
to their SM prediction (labelled as $\mathcal{H}_{\lambda}[z^{2}]$ with theo.
priors in the plots).

Finally, the nonlocal hadronic matrix elements are reparametrised following
Ref. Hurth  _et al._ (2017b), in which these nonlocal hadronic contributions
are included as multiplicative factors, leading to a reformulation of the
amplitudes of Eq. Towards establishing lepton flavour universality violation
in $\bar{B}\to\bar{K}^{*}\ell^{+}\ell^{-}$ decays as

$\displaystyle{\cal{A}}_{\lambda}^{(\ell)\,L,R}={\cal{N}}_{\lambda}^{(\ell)}\
\bigg{\\{}$ $\displaystyle(C^{(\ell)}_{9}\mp
C^{(\ell)}_{10}){\cal{F}}_{\lambda}(q^{2})\big{[}1+a_{\lambda}+b_{\lambda}\frac{q^{2}}{\mathrm{\scalebox{0.8}{6
GeV${}^{2}$}}}\big{]}$ (7)
$\displaystyle+\frac{2m_{b}M_{B}}{q^{2}}C^{(\ell)}_{7}{\cal{F}}_{\lambda}^{T}(q^{2})\bigg{\\}}\,,$

where $a_{\lambda}$ and $b_{\lambda}$ are complex coefficients Gaussian
constrained around zero.

##  Appendix B Right-handed Wilson coefficients

An extension of the physics case of the proposed method is to investigate the
sensitivity to the chirality-flipped counterparts of the usual Wilson
coefficients, i.e., $\mathcal{C}\xspace^{\prime(\mu)}_{9}$ and
$\mathcal{C}\xspace^{\prime(\mu)}_{10}$. Following the formalism discussed in
this article, the primed WCs are examined by considering in addition to the
BMP${}_{\mathcal{C}\xspace_{9,10}}$ three different modified NP scenarios for
the muon only:
$\mathcal{C}\xspace_{9,10}^{\prime(\mu)}=\mathcal{C}\xspace^{\prime\rm{SM}}_{9,10}=0$;
$\mathcal{C}\xspace^{\prime(\mu)}_{9}=\mathcal{C}\xspace^{\prime(\mu)}_{10}=0.3$;
and
$\mathcal{C}\xspace^{\prime(\mu)}_{9}=-\mathcal{C}\xspace^{\prime(\mu)}_{10}=0.3$.
Notice that for the electron mode the $\mathcal{C}\xspace_{9,10}^{\prime(e)}$
is set and fixed to the SM value
$\mathcal{C}\xspace^{\prime\rm{SM}}_{9,10}=0$.

Figure 5: Two-dimensional sensitivity scans for the pair of Wilson
coefficients $\mathcal{C}\xspace^{\prime(\mu)}_{9}$ and
$\mathcal{C}\xspace^{\prime(\mu)}_{10}$ for different nonlocal hadronic
parametrisation models for a NP scenario with
$\mathcal{C}\xspace^{\prime(\mu)}_{9}=\mathcal{C}\xspace^{\prime(\mu)}_{10}=0.3$.
The contours correspond to 99% confidence level statistical-only uncertainty
bands evaluated with the expected statistics after LHCbrun II.  Figure 6: Two-
dimensional sensitivity scans for the pair of Wilson coefficients
$\mathcal{C}\xspace^{\prime(\mu)}_{9}$ and
$\mathcal{C}\xspace^{\prime(\mu)}_{10}$ for three NP scenarios: (blue)
$\mathcal{C}\xspace^{\prime(\mu)}_{9}=\mathcal{C}\xspace^{\prime(\mu)}_{10}=0$,
(orange)
$\mathcal{C}\xspace^{\prime(\mu)}_{9}=\mathcal{C}\xspace^{\prime(\mu)}_{10}=0.3$
and (magenta)
$\mathcal{C}\xspace^{\prime(\mu)}_{9}=-\mathcal{C}\xspace^{\prime(\mu)}_{10}=0.3$.
The contours correspond to 99% confidence level statistical-only uncertainty
bands expected for the LHCb Upgrade (dotted) $50\,$fb-1 and (solid)
$\,300\,$fb-1 statistics.

Figure 5 shows the fit results for different order of the analytic expansion
for the nonlocal hadronic contribution for a NP scenario with
$\mathcal{C}\xspace^{\prime(\mu)}_{9}=\mathcal{C}\xspace^{\prime(\mu)}_{10}=0.3$,
and yields corresponding to the LHCbrun II expected statistics. The dependency
on the determination of $\mathcal{C}\xspace^{\prime(\mu)}_{9}$ and
$\mathcal{C}\xspace^{\prime(\mu)}_{10}$ on the order of the expansion clearly
saturates after $\mathcal{H}_{\lambda}[z^{3}]$ and allows a measurement of the
primed Wilson coefficients for the muon decay channel $B^{0}\to
K^{*0}{\Pmu^{+}\Pmu^{-}}\xspace$ independent on the theoretical hadronic
uncertainty. Figure 6 shows the prospects for the sensitivity to the
$\mathcal{C}\xspace^{\prime(\mu)}_{9}$ and
$\mathcal{C}\xspace^{\prime(\mu)}_{10}$ Wilson coefficients corresponding to
the expected statistics at the LHCb upgrade with $50$ and $300\,$fb-1. Note
that only with the full capability of the LHCb experiment it is possible to
start disentangling the different NP hypotheses.

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