11institutetext: Departamento de Física, CCEN, Universidade Federal da
Paraíba, Caixa Postal 5008, 58051-900, João Pessoa, PB, Brazil
22institutetext: Departamento de Física, Universidade Federal Rural de
Pernambuco, 52171–900, Recife, PE, Brazil

# Electronic properties of single and double napped carbon nanocones

F.A. Gomes 11 V.B. Bezerra 11 J.R.F Lima 22 F. Moraes 1122

(Received: date / Revised version: date)

###### Abstract

In this paper we study the electronic properties of carbon nanocones with one
and two nappes, with pentagonal and heptagonal defects in their lattices. We
use the continuum model, which is based on a Dirac-like Hamiltonian with the
topological defects described by localized non-Abelian gauge field fluxes. We
develop a geometrical approach that can describe the two nappes of the double
cone surface simultaneously, by extending the radial coordinate to the
complete set of real numbers. We show that, for some combinations of different
nanocones, forming the double conical surface, the local density of states
near the apex of the cone does not vanish at the Fermi energy and presents a
strong dependence on the angular momentum. We also obtain the energy spectrum
for finite-sized nanocones and verify that it depends on the choice of
topological defect on the surface, which suggests that a double nanocone can
be used to control the electronic transport in carbon-based electronic
devices.

##  1 Introduction

Carbon nanomaterials have attracted a great deal of attention in the last
years due to their unusual physical properties and the wide range of potential
applications Tagmatarchis . The boom in the study of these materials occurred
after the discovery of fullerenes in 1985 KROTO  and carbon nanotubes in 1991
Iijima . Since then, various carbon nanomaterials have been obtained, such as
nanocones GE , graphene Novoselov , nanoscrolls doi:10.1021/nl900677y , onions
Ugarte  and nanotori Liu . In particular, carbon nanocones were first observed
experimentally in 1992 as endcaps of carbon nanotubes Iijima2 ; Iijima3 , and
in 1994 as free-standing structures Maohui ; GE . Carbon nanocones with cone
angles of 19∘, 39∘, 60∘, 85∘ and 113∘ have already been observed Krishnan .
The carbon nanocone is a result of the introduction of a topological defect
called disclination in a planar graphene sheet, resulting from the
substitution of a hexagon by either a pentagon or a heptagon in the graphene
lattice.

The study of the electronic properties of carbon nanocones revealed an
enhancement in the local density of electronic states (LDoS) in the vicinity
of the cone apex, which was obtained by first principles PhysRevLett.78.2811 ;
PhysRevB.61.8496 ; PhysRevLett.86.5970 , tight-binding PhysRevB.49.7697 ;
PhysRevB.52.6015 ; PhysRevB.64.195419  and continuum model
lammert2000topological ; lammert2004graphene  calculations. This enhancement
can be used in applications of carbon nanocones in field emission 0957  and
scanning probes PhysRevLett.86.5970 , for instance. A carbon nanocone-based
electronic rectification device was also proposed doi:10.1063/1.3684276 , and
it was demonstrated that carbon nanocones are excellent thermal rectifiers
doi:10.1063/1.3049603 , which makes them a promising practical phononic
device. Carbon nanocones have also been suggested as cheaper and more easily
produced alternative to carbon nanotubes for applications as gas storage
devices doi:10.1021/jp2069094  and as capsules for drug delivery
doi:10.1021/nn800395t .

Another kind of conical structure that has been attracting attention in the
last few years is the double cone, where two cones are connected by their
apex. Theoretical studies indicate interesting properties like a high
dependence of the electronic states on the angular momentum of the electrons,
both for the classical and for non-relativistic quantum dynamics problem
kowalski2013dynamics . In Ref. 1751-8121-50-6-065302 , it is presented the
relativistic quantum problem of a charged particle restricted to a double cone
with and without a magnetic field. Even though these studies have shown
interesting properties for the double cone structure, the literature for
carbon double nanocones is still very scarce. In a computational simulation of
carbon double nanocones it was indicated that its experimental realization is
possible, since its formation energy is slightly lower than the energy of a
single cone lopes2015theoretical . These results motivated us to explore other
possible conical defects in carbon structures and investigate their
correspondent electronic properties.

In this paper, we study the electronic properties of carbon nanocones for
single and double conical surfaces. We develop and use a continuum model to
describe the double conical surfaces, which is an extension of the continuum
model developed in Ref. lammert2004graphene  for nanocones with only one
nappe. Even though this model is limited to low-energy electronic states near
the Fermi energy, it is very useful in long-distance physics, in which
situation it is difficult to deal with first-principles methods due to the
great number of atoms. The approach used here allows us to explore many
possible combinations of defects in the nanocone with one or two nappes. We
focus our attention in pentagonal and heptagonal defects and verified that the
connection of two carbon nanocones by their vertexes, creating a double
conical surface, brings up new electronic properties that could be used in
future applications.

The paper is organized as follows. In Section 2 we build the continuum model
for a double conical surface, based on the effective Dirac equation for a
graphene sheet, with localized gauge fluxes which describe the defects needed
to create the nanocone. We solve the effective Dirac equation in Section 3,
and investigate the LDoS and the energy spectrum in Section 4, showing that it
has a direct dependence on the angular momentum and the kind of defects that
are present on the surface. The paper is summarized and concluded in Section
5.

##  2 The continuum model for a double carbon nanocone

In this section we will develop a continuum model to describe a double carbon
nanocone. The starting point is the effective Dirac equation for low-energy
states in graphene. In this model, the lattice of the structure disappears and
the pentagons and heptagons in the structure are included in the Dirac
equation as localized fictitious gauge fluxes. This model was first proposed
in Ref. PhysRevLett.69.172  to investigate the electronic structure of
fullerenes, and has been widely used to describe carbon-based nanostructures
lammert2000topological ; lima ; jonasjap ; Cunha2015 .

The low-energy electrons in graphene are modeled as massless Dirac fermions
obeying the effective equation

$-i\hbar v_{f}\sigma^{\mu}\partial_{\mu}\Psi=E\Psi,$ (1)

where $v_{f}$ is the Fermi velocity and the pseudospin operator $\sigma^{\mu}$
are the usual Pauli matrices, which act on the spinor
$\Psi=(\Psi_{A},\Psi_{B})^{T}$. These components are labeled with respect to
the sub-lattices, named $A$ and $B$ shown in figure 1. Each component $A$ and
$B$ has two subcomponents related to two independent points, $K$ and
$K^{\prime}$, in the first Brillouin zone (FBZ), in reciprocal space. These
are the points where the band crosses the Fermi level. In other words, the
low-energy excitations are centered around these two points. Corner points of
the FBZ at 120$\degree$ from them are equivalent. We notice an important
covariance when the coordinate frame ($\hat{e}_{\mu}$, defined for $K$) is
rotated counterclockwise by $\theta$. Under this rotation, the wavefunction is
acted upon by $exp(\frac{i\theta\sigma^{3}}{2})$, which preserves the Dirac
Hamiltonian. We observe the same behavior for $K^{{}^{\prime}}$, but the
matching frame is rotated by 180$\degree$ from the corresponding one at K. In
order to avoid this inconvenience, the $K^{{}^{\prime}}$ frame is rotated in
order to coincide with the $K$ frame. The dispersion relation is the same in
these two points, and this degree of freedom is called K-spin
neto2009electronic . The Pauli matrices acting on the pseudospin and on the
K-spin are labeled by $\sigma^{\mu}$ and $\tau^{\mu}$, respectively.

Figure 1: (Color online) In (a) we see the hexagonal structure of graphene
built by two triangular sublattices $A$ and $B$. The lattice vectors are
$\vec{a_{1}}$ and $\vec{a_{2}}$. (b) is the correspondent Brillouin zone with
reciprocal-lattice vectors $\vec{b_{1}}$ and $\vec{b_{2}}$ in reciprocal
space. The Dirac points are localized at the points $K$ and $K^{{}^{\prime}}$.

In this paper, we focus our attention on graphene conical surfaces with one
and two nappes. The dynamics of quantum particles on this kind of surface has
been studied in recent years due to its high dependence on the angular
momentum kowalski2013dynamics  and also to the possibility of experimental
realization lopes2015theoretical .

While a single cone can be simply described in spherical coordinates
$(r,\theta,\phi)$ by $\theta=const.$, the double cone requires two values for
$\theta$. In order to work with this surface, we modify the spherical
coordinate system to keep a constant $\theta$ as the double cone equation.
1751-8121-50-6-065302 . We do this by extending the radial coordinate to the
whole set of the real numbers. The differential distance vector in this case
is given by

$\displaystyle d\vec{r}=dl\ \hat{e}_{l}+\left|l\right|\sin\theta d\phi\
\hat{e}_{\phi},$ (2)

where $l\in\mathbb{R}$ is the new radial coordinate. The square of the line
element is then

$ds^{2}=d\vec{r}.d\vec{r}=dl^{2}+\left|l\right|^{2}\alpha^{2}d\phi^{2},$ (3)

where $\alpha=\sin\theta$.

Figure 2: Coordinate system for a double cone surface.

The tetrad formalism is specially useful birrell1984quantum ;
misner1973gravitation  to deal with the relation (3) and the Dirac equation
(1). Using the tetrad formalism, we introduce at each point a set of locally
inertial coordinates and a set of orthonormal vectors, $e_{\mu}^{a}$, fixing
the transformation between the local (Latin indexes) and the general (Greek
indexes) coordinates. These vectors are found by the relation
$g_{\mu\nu}(x)=e^{a}_{\mu}(x)e^{b}_{\nu}(x)\eta_{ab}$, where
$g_{\mu\nu}=diag(1,\left|l\right|^{2}\alpha^{2})$ and $\eta_{ab}=diag(1,1)$.
In our problem they are given by

$e^{\mu}_{a}=\left(\begin{array}[]{cc}1&0\\\
0&\frac{1}{\alpha\left|l\right|}\end{array}\right).$ (4)

The $\sigma^{\mu}$ matrices in (1) are related with the flat space matrices
$\sigma^{a}$ by $\sigma^{\mu}=e^{\mu}_{a}\sigma^{a}$. Therefore

$\displaystyle\sigma^{l}$ $\displaystyle=$ $\displaystyle\sigma^{1},$ (5)
$\displaystyle\sigma^{\phi}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{\alpha\left|l\right|}.$

In order to give the shape of a cone to the flat hexagon-tiled surface of
graphene one can substitute one of its hexagons for a pentagon. This leads to
the appearance of a Aharonov-Bohm-like phase in the wavefunction of quantum
particles moving around it furtado2008geometric . This effect can be
incorporated into the Hamiltonian of the continuum model by the addition of
two fictitious non-Abelian gauge fields piercing the surface at the location
of the pentagon. One of them promotes the frame rotation covariance and the
other one, the $K$ spin rotation invariance to local symmetries. The procedure
is similar to the electromagnetic gauge potential transformation used to
compensate a change of wave function phase in the Aharonov-Bohm effect
aharonov1961further . In the case described in this article a spinorial
connection is added to the covariant derivative in order to accomplish the
frame rotation covariance 1751-8121-50-6-065302 . Thus, we introduce the spin
connection, $\Omega_{\mu}=-\frac{1}{8}w_{\mu,a,b}[\sigma^{a},\sigma^{b}]$, in
the covariant derivative $\nabla_{\mu}=\partial_{\mu}+\Omega_{\mu}$, where
$w_{\mu,a,b}$ is the spinorial connection. This gauge field can be explicitly
expressed as

$\Omega_{\phi}=-\frac{i}{2}\alpha\sigma^{3}.$ (6)

where $\sigma^{3}$ is the third Pauli matrix acting on the components of the
spinor related to the sublattices A and B. To promote $K$ spin rotation
invariance, we introduce the also non-Abelian gauge field, for one and two
defects, given by lammert2004graphene

$a_{\phi}=-\frac{3}{2}\frac{\left(\alpha-1\right)}{\alpha\left|l\right|}\tau^{2},\quad\text{for
one defect,}$ (7)

and

$\displaystyle
a_{\phi}=\frac{-3\left(\alpha-1\right)}{2\alpha\left|l\right|}\left[1-\frac{2}{3}\left(n-m\right)\right]\tau^{3},$
(8) for two defects.

We see in the relation (8), for two defects, a dependence with $n$ and $m$,
which are the parametric coordinates (n,m) of the defects’ centers in the
graphene lattice, as defined in lammert2004graphene .

The continuum model developed in this section allows us to investigate the
electronic properties of a variety of graphene conical surfaces. In the next
section, we use this approach to solve the free particle problem in this
geometric approach.

##  3 Free particle problem

In this section, we solve the effective Dirac equation developed in the last
section for the case of a free particle constrained to a carbon conical
surface. With this solution, we will be able to evaluate the influence of the
surface’s geometry on its electronic properties. We start by rewriting the
effective Dirac equation (1) including now the non-Abelian gauge fields,

$\displaystyle\left[\sigma^{i}p_{i}-i\hbar\sigma^{i}\Omega_{i}-\hbar\sigma^{i}a_{i}-\frac{E}{v_{f}}\right]\psi=0.$
(9)

Choosing to work with the quadratic form of this equation, we apply on it the
operator

$\displaystyle\left[\sigma^{j}p_{j}-i\hbar\sigma^{j}\Omega_{j}-\hbar\sigma^{j}a_{j}+\frac{E}{v_{f}}\right]$
(10)

and after some calculations, we get

$\displaystyle p_{i}$ $\displaystyle p_{i}$
$\displaystyle\psi-2i\hbar\Omega_{i}p_{i}\psi-2\hbar
a_{i}p_{i}\psi-\hbar^{2}\Omega_{i}\Omega_{i}\psi$ $\displaystyle+$
$\displaystyle 2$ $\displaystyle
i\hbar^{2}\Omega_{i}a_{i}\psi+\hbar^{2}a_{i}a_{i}\psi-\frac{E^{2}}{v_{f}^{2}}=0.$
(11)

Using the $\sigma^{\mu}$ matrices (5), the spinorial connection (6) and
considering that $a_{i}$ only has component in $\phi$, we get

$\displaystyle-$ $\displaystyle\frac{\partial^{2}}{\partial
l^{2}}\psi-\frac{1}{l}\frac{\partial}{\partial
l}\psi-\frac{1}{l^{2}\alpha^{2}}\frac{\partial^{2}}{\partial\phi^{2}}\psi+i\frac{\sigma^{3}}{\alpha
l^{2}}\frac{\partial}{\partial\phi}\psi+\frac{2ia_{\phi}}{\alpha\left|l\right|}\frac{\partial}{\partial\phi}\psi$
(12) $\displaystyle-$
$\displaystyle\frac{1}{4l^{2}}\psi+\frac{a_{\phi}\sigma^{3}}{\left|l\right|}\psi+{a_{\phi}}^{2}\psi-k^{2}\psi=0,$

where we defined

$k^{2}=\left(\frac{E}{\hbar v_{f}}\right)^{2}.$ (13)

The solutions of the above equation have the form

$\psi(l,\phi)=e^{ij\phi}\psi_{j}(l),$ (14)

where $\psi_{j}(l)$ is a function of the coordinate $l$ only and the angular
momentum quantum number $j$ is an integer plus one-half.

We diagonalize Eq. (12) taking into account that the only operators acting on
the $K$ spin are $\tau^{2}$ and $\tau^{3}$ from the gauge fields (7) and (8),
respectively. Similarly, the matrix $\sigma^{3}$ that acts on the pseudospin
is replaced by its eigenvalue $\sigma=\pm 1$, where the plus (minus) sign is
related to the sub-lattice A (B). Equation (12) then becomes

$\displaystyle-$ $\displaystyle\frac{\partial^{2}}{\partial
l^{2}}\psi_{j}-\frac{1}{l}\frac{\partial}{\partial
l}\psi_{j}+\frac{j^{2}}{l^{2}\alpha^{2}}\psi_{j}-\frac{j\sigma}{\alpha
l^{2}}\psi_{j}-\frac{2j\tau a^{{}^{\prime}}_{\phi}}{\alpha l^{2}}\psi_{j}$
(15) $\displaystyle+$
$\displaystyle\frac{1}{4l^{2}}\psi_{j}+\frac{a^{{}^{\prime}}_{\phi}\sigma\tau}{l^{2}}\psi_{j}+\frac{{a^{{}^{\prime}}_{\phi}}^{2}}{l^{2}}\psi_{j}-k^{2}\psi_{j}=0,$

where $\tau=\pm 1$,

$\displaystyle
a^{{}^{\prime}}_{\phi}=-\frac{3}{2}\frac{\left(\alpha-1\right)}{\alpha},\quad\text{for
one defect,}$ (16)

and

$\displaystyle
a^{{}^{\prime}}_{\phi}=-\frac{3}{2}\frac{\left(\alpha-1\right)}{\alpha}\left[1-\frac{2}{3}\left(n-m\right)\right],$
(17) for two defect,

in order to make explicit the radial coordinate ($l$) dependence due to the
gauge fields. We reorganize the terms in Eq. (15) and rewrite it in the form

$\displaystyle\frac{\partial^{2}}{\partial
l^{2}}\psi_{j}+\frac{1}{l}\frac{\partial}{\partial
l}\psi_{j}-\frac{\nu^{2}}{l^{2}}\psi_{j}+k^{2}\psi_{j}=0,$ (18)

where

$\nu^{2}=\left(\frac{j}{\alpha}-\tau
a^{{}^{\prime}}_{\phi}-\frac{\sigma}{2}\right)^{2}.$ (19)

Equation (18) is clearly a Bessel equation, whose solution is a combination of
Bessel functions of first and second kinds, respectively. Since we are
including the origin ($l=0$), we do not consider the Bessel function of second
kind because it diverges there. Thus the solution is written as

$\psi_{j}(l)=CJ_{\nu}(kl),$ (20)

where $C$ is a normalization constant and $J_{\nu}(kl)$ is the Bessel function
of first kind. It is important to notice that the results obtained here
reproduce the known results for flat graphene and for the singly napped carbon
nanocone lammert2004graphene ; lammert2000topological ; bueno2012landau . If
we consider $\alpha=1$, the gauge fields (7) and (8) vanish giving rise to a
Bessel function of index $j-\frac{1}{2}$, as expected for the flat case.

At this point, we notice that the system of coordinates used here has the
advantage of allowing us to deal with two single cones, joined by their
apexes, in a unified form. Since we have two single nanocones in the same
structure, we expect to reproduce the known results for a single carbon
nanocone, but in a domain twice as large. This is exactly what we obtain in
this section. The solution is in agreement with the known results for a single
cone, but now it is dependent on the extended radius $l$.

This unified system of coordinates brings us the possibility of putting
together different conical structures and, in this way, combine their
properties. In the next section we will explore the electronic properties of
some of these combinations.

##  4 Electronic Properties

Figure 3: Conical defects on a graphene surface. (a), (b) and (c) are
pentagonal defects, wherein (b) has parametric coordinate (1,1) and (c) (2,0).
Structures (d), (e) and (f) have heptagonal defects, wherein (e) has
parametric coordinate (1,1) and (f), (2,0).

The model developed in this article allows us to study a variety of conical
surfaces. The first parameter that gives us freedom to model different
deformed surfaces is the parameter $\alpha$ which appears in the metric (3)
and is given by

$\displaystyle\alpha=1+\frac{\lambda}{2\pi},$ (21)

where $\lambda$ is the angle of the inserted ($\lambda>0$) or removed
($\lambda<0$) slice from the flat graphene lattice in order to make the cone,
in the procedure known as the Volterra process puntigam1997volterra .

The hexagonal symmetry of the graphene sheet requires that $\lambda$ be a
multiple of $\frac{\pi}{3}$. Moreover, square or octagonal defects are rare to
occur due to the high deformation they promote and consequently due to their
high formation energy lammert2004graphene . Therefore, we consider here only
pentagonal ($\lambda=-\frac{\pi}{3}$) and heptagonal ($\lambda=\frac{\pi}{3}$)
defects.

Type of defect | $\alpha$ | $a^{{}^{\prime}}_{\phi}$ | $\nu_{0}$
---|---|---|---
1 pentagon | $5/6$ | $3/10$ | $-1/5$
2 pentagons (1,1) ($n\not\equiv m(mod3)$) | $2/3$ | $1/4$ | $0$
2 pentagons (2,0) ($n\equiv m(mod3)$) | $2/3$ | $3/4$ | $-1/2$
1 heptagon | $7/6$ | $-3/4$ | $-2/7$
2 heptagons (1,1) ($n\not\equiv m(mod3)$) | $4/3$ | $-1/8$ | $-1/4$
2 heptagons (2,0) ($n\equiv m(mod3)$) | $4/3$ | $-3/8$ | $-1/2$
Table 1: Values of $\alpha$ and $a^{{}^{\prime}}_{\phi,i}$ for the different
structures with pentagonal and heptagonal defects, as illustrated in Fig. 3.
Also the values of $\nu_{0}$, which are the values of $\nu$ that minimize
$2\nu+1$ on the given class of the cone.

Another possibility, is to consider two defects in the same structure. The
gauge field (8) gives the influence of the defects in terms of their
parametric coordinates $(n,m)$. It is known that the effects of topological
defects in a graphene surface are dependent on the relative position of the
defects. This sensitivity can be classified by a $n-m$ combination rule
lammert2004graphene . Two classes of defects are sorted depending whether
$n-m$ is a multiple of three or not. We choose one example of each class. For
the class $n\not\equiv m(mod3)$, we choose the structure (1,1). For the class
$n\equiv m(mod3)$, we choose the structure (2,0). So, we consider here six
different cones, three with pentagonal and three with heptagonal defects,
which are shown in Fig. 3. The values of the parameters $\alpha$ and
$a^{{}^{\prime}}_{\phi}$ for each cone considered here are given in Table 1.
The double conical surfaces that we consider are combinations of the cones
shown in Fig. 3. The combinations with the same cone in each nappe is shown in
Fig. 4.

Figure 4: Double conical surfaces built by juxtaposition of the surfaces shown
in figure 3.

###  4.1 Local density of states

One important quantity to characterize the electronic and transport properties
of a solid is the density of states (DoS). It gives us information about the
number of states available to be occupied per interval of energy at each
energy level. Thus, it shows us the conducting characteristics of the
material, if it is a metal, insulator or semiconductor.

Since we are focusing our attention in surfaces with conical defects, which,
in the continuum limit, present singular curvature, we will look more
carefully to the local density of states (LDoS) near the apex of the surface.
As pointed out in Ref. lammert2004graphene , the LDoS in the continuum theory
can be written, by dimensional analysis, as

$\displaystyle\left.\frac{dn}{dE}\right|_{l}=\frac{f_{2}(\left|k\right|l)}{\pi\hbar
v_{f}l},$ (22)

where $f_{2}(\left|k\right|l)$ is a scale function determined from the
solution of the effective Dirac equation obtained in the previous section.

Figure 5: Schematic diagram for the density of states near the apex of the
cone, according to equations (22) and (24). We plot the LDoS for cones with
pentagonal defects in (a), (c) and (e), where (a) is for single nappe
structures, (c) for two identical nappes and (e) for the combination of
different nappes with the structure (2,0). We follow the same sequence for
heptagonal structures in (b), (d) and (f).

In order to obtain an expression for $f_{2}(\left|k\right|l)$, we initially
consider a large conical surface with boundary at $\ell=L$ and take
$L\rightarrow\infty$. Taking this condition into account, we impose a boundary
condition that ensures that the radial component of the current density
vanishes at the boundary, i.e., $\psi_{A}(L)\psi_{B}(L)=0$. This condition
leads us to the scale function

$\displaystyle
f_{2}(\left|k\right|l)=\frac{kl}{2\alpha}\sum_{\nu}\left|J_{\nu}(kl)\right|^{2}.$
(23)

Since we are interested in the LDoS near the apex of the cone, we consider the
asymptotic form of the Bessel function as $l\rightarrow 0$ in Eq. (23). Then,
we obtain

$\displaystyle
f_{2}(\left|k\right|l)=\frac{kl}{2\alpha}\frac{\left|kl\right|^{2\nu_{0}}}{2^{2\nu_{0}}\Gamma(\nu_{0}+1)^{2}}.$
(24)

As k is proportional to the energy E (see Eq. 13), the LDoS will be
proportional to $\left|E\right|^{2\nu_{0}+1}$, where $\nu_{0}$ is the value of
$\nu$ that minimizes $2\nu+1$ on the given cone class. For a given class, the
allowed values of $\nu$ (19) will depend on the parameters $j$, $\tau$ and
$\sigma$. The combination of these parameters that minimizes $\nu$, defines
$\nu_{0}$. The values of $\nu_{0}$ for all cones considered here are in Table
1.

The LDoS in terms of the energy is plotted in Fig. 5. We consider structures
with pentagonal and heptagonal defects, in single and double carbon
nanoconical surfaces, combining different defected structures. It is important
to mention that, for all the cones considered here, the values of the angular
momentum quantum number $j$ that minimizes $2\nu_{0}+1$ is always $\pm 1/2$,
which are the states with zero orbital angular momentum. This is due to the
fact that we are looking at the LDoS near the cone’s apex, and these states
are more concentrated in this region.

The LDoS for cones of a single nappe with pentagonal and heptagonal
topological defects are plotted in Fig. 5 $(a)$ and $(b)$, respectively. As
can be seen, the LDoS shows a direct dependence with the type of defect on the
surface. In Fig. 5 $(a)$ we reproduce the results obtained in Ref.
lammert2004graphene , which shows that the cone with (1,1) pentagonal defect
has the same linear LDoS near the Fermi energy as planar graphene, while in
the structure (2,0) the LDoS is nearly flat and nonzero at the Fermi energy.
As can be seen in Fig. 5 $(b)$, for the case of heptagonal topological
defects, the flattening of the LDoS near the Fermi energy for the structure
(2,0) also occurs, but the linear LDoS of planar graphene is not reproduced by
any of the cones. The enhancement of the LDoS observed in the structure (2,0)
occurs because the wave function becomes more strongly concentrated at the
apex as $2\nu+1$ decreases. The class that presents the minimum of $2\nu+1$,
when we compare $\nu_{0}$ for all classes, is $n\equiv m(mod3)$. We are
studying as example of this class, the structure (2,0). In this structure
$\alpha=2/3$ and $a^{{}^{\prime}}_{\phi}=3/4$, this leads to $\nu_{0}=-1/2$,
and as consequence $2\nu_{0}+1=0$. The same happens for the structure (2,0)
with two heptagonal defects, where $\alpha=4/3$ and
$a^{{}^{\prime}}_{\phi}=-3/8$. In these cases, the topological defects
maintain the conducting characteristic (i.e. semimetal) of the structure.

The LDoS of the cones with two nappes are plotted in Figs. 5 $(c)$ \- $(f)$.
In Figs. 5 $(c)$ and $(d)$ we consider the cones with the same topological
defect in each nappe. The LDoS in this case does not change its behavior, and
the result is just the double of the single nappe structure, as expected. A
more interesting result comes when the double cone is the combination of two
nappes with different defects in the structure, as illustrated in Figs. 5
$(e)$ and $(f)$. In this case, the LDoS is the combination of the behavior of
the two different nappes. We choose as example, combinations where at least
one nappe is a (2,0) structure. As a result, the (2,0) structure leads to a
nonzero LDoS at the Fermi energy, and therefore metallic behavior, in all
cases.

###  4.2 Energy spectrum: finite nanocones

If we consider finite cones with size $l=L$, the wave function for $l\geq L$
is equal to zero. So, the boundary condition at the edge of the cone is given
by

$\psi_{j}(L)=CJ_{\nu}(kL)=0.$ (25)

Then, the argument of the above Bessel function, $kL$, has to be a zero of
this function. This gives rise to a discrete energy spectrum given by

$E_{n,j}=j_{\nu,n}\frac{\hbar v_{F}}{L}$ (26)

where $j_{\nu,n}$ is the $n$’th zero of the Bessel function of order $\nu$.

Figure 6: The energy spectrum for the pentagonal and heptagonal finite conical
surfaces considered here. For all the cases, we consider $L=50$ nm and
$\nu=\nu_{0}$.

In Fig. 6 we plot the energy spectra for the finite case of the conical
surfaces studied in the previous section. We consider spectra corresponding to
the lowest value of $j$, which gives a wave function that is more concentrated
in the cone apex. For this case, $\nu=\nu_{0}$. It is possible to see that the
energy spectrum of the cone depends on the topological defects present on the
surface, even though different cones may have the same energy spectrum, as for
the case of two pentagons and two heptagons $n=m(mod3)$. We remark that the
size of the cone, $L$, can be used to control the interval between subsequent
allowed values of energy.

For the case of a double conical surface we have two values for $L$, one for
each cone. Since the energy spectrum depends on $L$ and on the topological
defect in the cone, there are many possible combinations of cones with the
most varied energy spectra. Thus, one can choose two specific cones in a
double conical surface in order to control the electronic transport from one
to the other. From the classical point of view, the two cones are connected
only by the states with zero angular momentum. In the quantum regime, the
lowest angular momentum states are more concentrated in the cones’
junction/apex, so they will have a higher probability to move between the
cones. In the continuum limit approach that we are using, the apexes of the
cones are only a point and we do not have the effects of the connection
between the cones. So, further analysis with other approaches, such as tight
binding and first principle calculations, are necessary in order to understand
the transport properties in the double conical surface.

##  5 Concluding remarks

The study of double carbon nanocone’s electronic properties is the purpose of
this work. In the continuum model, the nanocones electronic properties are
modeled by an effective Dirac equation with the topological defects in the
lattice represented by localized gauge fluxes. Each kind of defect is
represented by a specific gauge field and multiple defects can be combined in
a different net flux. This freedom allows us to explore a variety of defected
surfaces, namely, nanocones with one and two pentagons or heptagons in the
structure. We use a geometrical approach that makes possible to describe the
two cones simultaneously. This approach extends the radial coordinate to the
whole set of real numbers.

The LDoS for different combinations of nanocones in the double conical
structure is investigated. It is shown that the class of nanocones with two
defects and parametric coordinates $n\equiv m(mod3)$ leads to an enhancement
of the apical wave function concentration, resulting in a non-vanishing LDoS
at the Fermi energy. We also obtain the energy spectra for finite nanocones,
which depends both on their size and on the topological defects on the cone.
So, the combination of two nanocones in a double conical surface with suitable
topological defects can be used, for instance, to control the electronic
transport from one cone to the other.

The approach developed in this work can be used to explore the electronic
properties of other Dirac materials with topological defects. It would be
interesting, for future applications of carbon-based devices, to investigate
how the electric and magnetic fields affect the electronic properties of
double conical surfaces. We expect to explore this problem in forthcoming
publications.

Acknowledgements: This work was partially supported by Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), CAPES and FACEPE.

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# Spaces where all bijections are morphisms

Lucas H. R. de Souza

###### Abstract

Here we classify all topological spaces where all bijections to itself are
homeomorphisms. As a consequence, we also classify all topological spaces
where all maps to itself are continuous.

Analogously, we classify all measurable spaces where all bijections to itself
are measurable with measurable inverse. As a consequence, we also classify all
measurable spaces where all maps to itself are measurable.

††Mathematics Subject Classification (2010). Primary: 54F65, 28A05; Secondary:
54A10, 54A25, 54C05.††Keywords: Cofinite topology, cocountable topology,
cardinal, countable-cocountable $\sigma$-algebra, bijections.

##  1 Introduction

We have trivially that topological spaces with the trivial or the discrete
topology have the property that all bijections to itself are homeomorphisms.
Consider the following family of topological spaces:

######  Definition 1.

Let $\kappa$ be a cardinal number and $X$ a topological space. The co-$\kappa$
topology on $X$ is the topology where the open sets are the empty set and the
subsets of $X$ whose complement has cardinality less than $\kappa$. We say
that a topological space is cocardinal if it has the co-$\kappa$ topology for
some cardinal $\kappa$.

This topology is not defined for $1<\kappa<\aleph_{0}$. Note that the
co-$\aleph_{0}$ topology is the cofinite topology (see p. 49 of [3] for more
information about these topological spaces) and the co-$\aleph_{0}^{+}$
topology is the cocountable topology (if $\alpha$ is a cardinal, then we
denote by $\alpha^{+}$ its successor cardinal, see Chapters I.10 to I.13 of
[1] for cardinal numbers). Note also that the co-$1$ topology is the trivial
topology and the co-$\kappa$ topology is the discrete topology if $\kappa$ is
bigger than the cardinality of $X$.

It is easy to see that, for cocardinal topologies, all bijective maps are
homeomorphisms. Our objective is to show that this property actually
characterizes the cocardinal topologies:

######  Theorem 1.

Let $X$ be a topological space. Then $Homeo(X)=bij(X)$ if and only if $X$ is
cocardinal.

Here $Homeo(X)$ is the set of all homeomorphisms from $X$ to $X$ and $bij(X)$
is the set of all bijective maps from $X$ to $X$.

As a consequence, we have the following:

######  Corollary 1.

Let $X$ be a topological space. Then $\mathcal{C}(X,X)=Map(X,X)$ if and only
if $X$ has the trivial topology or the discrete topology.

Here $\mathcal{C}(X,X)$ is the set of continuous maps from $X$ to $X$ and
$Map(X,X)$ is the set of all maps from $X$ to $X$.

We have also an analogous version of this theorem for measurable spaces:

######  Definition 2.

Let $\kappa$ be a cardinal number and $X$ a measurable space. The
$\kappa$-co-$\kappa$ algebra on $X$ is the $\sigma$-algebra where the
measurable sets are the subsets of $X$ with cardinality less than $\kappa$ or
its complement has cardinality less than $\kappa$. We say that a measurable
space is cocardinal if it has the $\kappa$-co-$\kappa$ algebra for some
cardinal $\kappa$.

This $\sigma$-algebra is not defined for $1<\kappa\leqslant\aleph_{0}$. Note
that the $\aleph_{0}^{+}$-co-$\aleph_{0}^{+}$ algebra is the countable-
cocountable algebra (see Examples 3.3 of [2] for the proof that this is
actually a $\sigma$-algebra). Note also that the $1$-co-$1$ algebra is the
trivial $\sigma$-algebra and the $\kappa$-co-$\kappa$ algebra is the discrete
$\sigma$-algebra if $\kappa$ is bigger than the cardinality of $X$.

If $X$ is a measurable space, let $Iso(X)$ be the set of all measurable maps
from $X$ to $X$ with measurable inverse.

######  Theorem 2.

Let $X$ be a measurable space. Then $Iso(X)=bij(X)$ if and only if $X$ is
cocardinal.

As a consequence, we have the following:

######  Corollary 2.

Let $X$ be a measurable space. Then $Mor(X,X)=Map(X,X)$ if and only if $X$ has
the trivial $\sigma$-algebra or the discrete $\sigma$-algebra.

Here $Mor(X,X)$ is the set of measurable maps from $X$ to $X$.

######  Acknowledgements.

The author was partially supported by the joint FAPEMIG/CNPq program ”Programa
de Apoio à Fixação de Jovens Doutores no Brasil”, Grant Number BPD-00721-22
(FAPEMIG), 150784/2023-6 (CNPq).

##  2 Proof of Theorem 1

For a topological space $X$, consider $Closed(X)$ as the set of closed sets of
$X$. For a set $X$, we denote the cardinality of $X$ by $\\#X$.

######  Lemma 1.

Let $X$ be a topological space such that $Homeo(X)=bij(X)$. Let $F\in
Closed(X)$ such that $\\#F<\\#X$. If $F^{\prime}\subseteq X$ such that
$\\#F=\\#F^{\prime}$, then $F^{\prime}\in Closed(X)$.

###### Proof.

If $X$ is infinite and $\\#F<\\#X$, then $\\#(X-F)=\\#X$. Analogously,
$\\#(X-F^{\prime})=\\#X$. So $\\#(X-F)=\\#(X-F^{\prime})$ (this occurs
trivially for a finite set $X$). So there exists a bijection $f:X\rightarrow
X$ such that $f(F)=F^{\prime}$ and $f(X-F)=X-F^{\prime}$. Since
$Homeo(X)=bij(X)$, then $f$ is a homeomorphism, which implies that
$F^{\prime}$ is closed. ∎

######  Lemma 2.

Let $X$ be a topological space such that $Homeo(X)=bij(X)$. Let $F\in
Closed(X)$ such that $\\#F<\\#X$. If $F^{\prime}\subseteq X$ such that
$\\#F>\\#F^{\prime}$, then $F^{\prime}\in Closed(X)$.

###### Proof.

Suppose that $F$ is finite.

Let $x\in F$. Then $\\{x\\}=\bigcap\\{G\subseteq X:x\in G,\\#G=\\#F\\}$. By
the lemma above, all subsets $G$ are closed, which implies that $\\{x\\}$ is
closed and, since $F$ is finite, every subset of $F$ is also closed.

Suppose that $F$ is infinite.

Let $G\subseteq F$ such that $\\#G=\\#F^{\prime}$. Let $x\in F-G$. We have
that $\\#(F-\\{x\\})=\\#F$. By the lemma above, $F-\\{x\\}$ is closed. But
$G=\bigcap_{x\in F-G}F-\\{x\\}$, which implies that $G$ is closed. Since
$\\#G=\\#F^{\prime}$, then, by the lemma above, $F^{\prime}$ is closed. ∎

######  Lemma 3.

Let $X$ be an infinite set, $x\in X$ and $F\subset X$ such that $\\#F=\\#X$.
Then there exists $U,V\subseteq X$ such that $U\cap V=\\{x\\}$,
$\\#U=\\#V=\\#(X-F)$ and $\\#(X-U)=\\#(X-V)=\\#F$.

###### Proof.

Suppose that $\\#(X-F)<\aleph_{0}$.

Since $X$ is infinite, there exists $U^{\prime}$ and $V^{\prime}$ disjoint
subsets of $X$ such that $x\notin U^{\prime}\cup V^{\prime}$ and
$\\#U^{\prime}=\\#V^{\prime}=\\#(X-F)-1$.

Suppose that $\aleph_{0}\leqslant\\#(X-F)<\\#X$.

Since $X$ is infinite, there exists $U^{\prime}$ and $V^{\prime}$ disjoint
subsets of $X$ such that $\\#U^{\prime}=\\#V^{\prime}=\\#(X-F)$. We have that
$\\#(X-U^{\prime})=\\#X=\\#F$ (since $\\#(X-F)<\\#X$) and, analogously,
$\\#(X-V^{\prime})=\\#X=\\#F$.

Suppose that $\\#(X-F)=\\#X$.

Take $U^{\prime}$ and $V^{\prime}$ disjoint subsets of $X$ such that
$U^{\prime}\cup V^{\prime}=X-(F\cup\\{x\\})$ and
$\\#U^{\prime}=\\#V^{\prime}=\\#(X-F)$. We have that $\\#(X-U^{\prime})=\\#F$
(since $F\subset X-U^{\prime}$ and $\\#F=\\#X$) and, analogously,
$\\#(X-V^{\prime})=\\#F$.

In all cases, take $U=U^{\prime}\cup\\{x\\}$ and $V=V^{\prime}\cup\\{x\\}$. We
have that $\\#U=\\#V=\\#(X-F)$, $U\cap V=\\{x\\}$ and
$\\#(X-U)=\\#(X-V)=\\#X=\\#F$. ∎

######  Theorem 1.

Let $X$ be a topological space. Then $Homeo(X)=bij(X)$ if and only if $X$ is
cocardinal.

###### Proof.

We already know that, for cocardinal spaces, $Homeo(X)=bij(X)$ holds.

Let $\kappa=\sup\\{\kappa^{\prime}:\exists F\in Closed(X):F\neq
X,\\#F=\kappa^{\prime}\\}$.

Suppose that $\kappa=0$.

If $F$ is closed in $X$ with $F\neq X$, then $\\#F=0$. Then $X$ has the
trivial topology.

Suppose that $\kappa>0$.

Suppose that $X$ is finite.

There exists $F\in Closed(X)$ such that $F\neq X$ and $\\#F=\kappa$. By Lemma
2, every point is a closed set. Since $X$ is finite, it follows that $X$ has
the discrete topology.

Suppose that $X$ is infinite.

Suppose that $\kappa\neq\\#X$ or $\max\\{\kappa^{\prime}\\!:\\!\exists
F\\!\in\\!Closed(X):F\neq X,\\#F=\kappa^{\prime}\\}\neq\\#X$.

Let $\alpha<\kappa$. There exists cardinal $\kappa^{\prime}$ such that
$\alpha\leqslant\kappa^{\prime}\leqslant\kappa$ and there exists $F\in
Closed(X)$ such that $F\neq X$ and $\\#F=\kappa^{\prime}$. By Lemma 2, every
subset of $X$ with cardinality $\alpha$ is a closed set in $X$. If there
exists a closed set $G$ with cardinality $\kappa$, then $\kappa\neq\\#X$, by
the assumptions on $\kappa$. Then every subset of $X$ with cardinality
$\kappa$ is closed (by Lemma 1). Since there is no closed set $G\neq X$ with
cardinality bigger than $\kappa$, it follows that $X$ has the co-$\kappa^{+}$
topology. If there is no closed set different from $X$ with cardinality
$\kappa$, it follows that $X$ has the co-$\kappa$ topology.

Suppose that $\kappa=\max\\{\kappa^{\prime}:\exists F\in Closed(X):F\neq
X,\\#F=\kappa^{\prime}\\}=\\#X$.

There exists $F\in Closed(X)$ such that $F\neq X$ and $\\#F=\\#X$. Let $x\in
X$. By Lemma 3, there exists $U_{x},V_{x}\subseteq X$ such that $U_{x}\cap
V_{x}=\\{x\\}$, $\\#U_{x}=\\#V_{x}=\\#(X-F)$ and
$\\#(X-U_{x})=\\#(X-V_{x})=\\#F$. Then there exists $f,g\in bij(X)$ such that
$f(U_{x})=X-F$ and $g(V_{x})=X-F$. Since $bij(X)=Homeo(X)$, $U_{x}$ and
$V_{x}$ are open sets, which implies that $\\{x\\}$ is an open set as well.
Then $X$ has the discrete topology.

In all cases we get that $X$ is cocardinal. ∎

######  Corollary 1.

Let $X$ be a topological space. Then $\mathcal{C}(X,X)=Map(X,X)$ if and only
if $X$ has the trivial topology or the discrete topology.

###### Proof.

It is easy to see that if $X$ has the trivial topology or the discrete
topology, then $\mathcal{C}(X,X)=Map(X,X)$.

Let $X$ with the co-$\kappa$ topology, for $\kappa$ a cardinal with
$\aleph_{0}\leqslant\kappa<\\#X^{+}$. Then there exists $F\subset X$ that is
not closed. Let $x,y\in X$, with $x\neq y$. In this case, take a map
$f:X\rightarrow X$ defined as $f(F)=x$, $f(X-F)=y$. Since $X$ has the
co-$\kappa$ topology, with $\kappa>1$, then $\\{x\\}\in Closed(X)$. But
$F=f^{-1}(x)$, which implies that $f$ is not continuous. Thus
$\mathcal{C}(X,X)\neq Map(X,X)$.

Since $\mathcal{C}(X,X)=Map(X,X)$ implies $Homeo(X)=bij(X)$, then this
corollary follows from the theorem above. ∎

##  3 Proof of Theorem 2

######  Lemma 4.

Let $(X,\Sigma)$ be a measurable space such that $Iso(X)=bij(X)$. Let
$F\in\Sigma$ such that $\\#F<\\#X$. If $F^{\prime}\subseteq X$ such that
$\\#F=\\#F^{\prime}$, then $F^{\prime}\in\Sigma$.

###### Proof.

Analogous to Lemma 1. ∎

######  Lemma 5.

Let $(X,\Sigma)$ be a measurable space such that $Iso(X)=bij(X)$. Let
$F\in\Sigma$ such that $\\#F<\\#X$. If $F^{\prime}\subseteq X$ such that
$\\#F>\\#F^{\prime}$, then $F^{\prime}\in\Sigma$.

###### Proof.

Suppose that $F$ is finite.

Let $x\in F$ and $y\in X-F$. Then $\\{x\\}=\bigcap_{z\in
F-\\{x\\}}(F-\\{z\\})\cup\\{y\\}$. By the lemma above, for every $z\in
F-\\{x\\}$, the set $(F-\\{z\\})\cup\\{y\\}\in\Sigma$ , which implies that
$\\{x\\}\in\Sigma$ and, since $F$ is finite, every subset of $F$ is also in
$\Sigma$. By the lemma above, we get that $F^{\prime}\in\Sigma$.

Suppose that $F$ is infinite.

Let $G\subset F$ such that $\\#G=\\#F^{\prime}$. We have that #(F - G) = #F.
Take $U^{\prime}$ and $V^{\prime}$ disjoint subsets of $F$ such that
$\\#U^{\prime}=\\#V^{\prime}=\\#F$ and $U^{\prime}\cup V^{\prime}=F-G$. Take
$U=U^{\prime}\cup G$ and $V=V^{\prime}\cup G$. We have that $\\#U=\\#V=\\#F$.
So, by the lemma above, $U,V\in\Sigma$. But $U\cap V=G$, which implies that
$G\in\Sigma$. Since $\\#G=\\#F^{\prime}$, then, by the lemma above,
$F^{\prime}\in\Sigma$. ∎

######  Lemma 6.

Let $(X,\Sigma)$ be a measurable space such that $Iso(X)=bij(X)$. Let
$F\in\Sigma$ such that $\\#F=\\#X$ and $\\#(X-F)=\\#X$. If
$F^{\prime}\subseteq X$ such that $\\#F>\\#F^{\prime}$, then
$F^{\prime}\in\Sigma$.

###### Proof.

Take $G\subseteq F$ such that $\\#G=\\#F^{\prime}$. Take $G^{\prime}=(X-F)\cup
G$. We have that $\\#G^{\prime}=\\#(X-G^{\prime})=\\#X$. Take a bijection
$f:X\rightarrow X$ such that $f(F)=G^{\prime}$ and $f(X-F)=X-G^{\prime}$.
Since $Iso(X)=bij(X)$, then $G^{\prime}\in\Sigma$, which implies that $G=F\cap
G^{\prime}\in\Sigma$. ∎

######  Theorem 2.

Let $(X,\Sigma)$ be a measurable space. Then $Iso(X)=bij(X)$ if and only if
$X$ is cocardinal.

###### Proof.

It is easy to see that, for cocardinal spaces, $Iso(X)=bij(X)$ holds.

Let $\kappa=\sup\\{\kappa^{\prime}\neq\\#X:\exists
F\in\Sigma:\\#F=\kappa^{\prime}\\}$.

Suppose that $\kappa=0$.

Let $F\in\Sigma$ with $F\neq X$. If $\\#F>0$, then $\\#F=\\#X$. Analogously,
$\\#(X-F)=\\#X$. By the previous lemma, if $x\in X$, then $\\{x\\}\in\Sigma$,
contradicting the fact that $\kappa=0$. So $\\#F=0$. Then $X$ has the trivial
$\sigma$-algebra.

Suppose that $\kappa>0$.

Suppose that $X$ is finite.

There exists $F\in\Sigma$ such that $F\neq X$ and $\\#F=\kappa$. By Lemma 5,
every point is a measurable set. Since $X$ is finite, it follows that $X$ has
the discrete $\sigma$-algebra.

Suppose that $X$ is infinite.

Let $\alpha<\kappa$. There exists a cardinal $\kappa^{\prime}$ such that
$\alpha\leqslant\kappa^{\prime}\leqslant\kappa$ (or
$\alpha\leqslant\kappa^{\prime}<\kappa$, if $\kappa=\\#X$) and there exists
$F\in\Sigma$ such that $\\#F=\kappa^{\prime}$. By Lemma 5, every subset of $X$
with cardinality $\alpha$ is a measurable set in $X$.

Suppose that $\kappa\neq\\#X$.

If there exists a measurable set $G$ with cardinality $\kappa$, then every
subset of $X$ with cardinality $\kappa$ is measurable (by the fact that
$\kappa\neq\\#X$ and Lemma 4). Let $G$ be a measurable set with cardinality
bigger than $\kappa$. By the definition of $\kappa$, $\\#G=\\#X$.

Suppose that there is no measurable set $G$ with $\\#G=\\#X$ and
$\\#(X-G)=\\#X$. If there is no measurable set with cardinality $\kappa$, then
$X$ has the $\kappa$-co-$\kappa$ algebra. If there is a measurable set with
cardinality $\kappa$, then $X$ has the $\kappa^{+}$-co-$\kappa^{+}$ algebra.

Suppose that there is a measurable set $G$ with $\\#G=\\#X$ and
$\\#(X-G)=\\#X$. Let $\beta$ be a cardinal such that
$\kappa\leqslant\beta<\\#X$. Take $G^{\prime}\subseteq G$ such that
$\\#G^{\prime}=\beta$. By Lemma 6, $G^{\prime}\in\Sigma$. By the definition of
$\kappa$, we get that $\beta=\kappa$ and then $\\#X=\kappa^{+}$. We also get
that all subsets of $X$ with cardinality $\kappa$ are measurable. Let
$H\subset X$ with $\\#H>\kappa$. Then $\\#H=\\#X$. If
$\\#(X-H)\leqslant\kappa$, then $X-H\in\Sigma$, which implies that
$H\in\Sigma$. If $\\#(X-H)=\\#X$, then take a bijection $g:X\rightarrow X$
such that $g(G)=H$ and $g(X-G)=X-H$. Since $Iso(X)=bij(X)$, then $H\in\Sigma$.
So we get that $X$ has the discrete $\sigma$-algebra.

Suppose that $\kappa=\\#X$.

Suppose that there is no measurable set $G$ with $\\#G=\\#X$ and
$\\#(X-G)=\\#X$. Since for every $\alpha<\kappa$ and every $H\subset X$ with
$\\#H=\alpha$, $H\in\Sigma$, then $X$ has the $\kappa$-co-$\kappa$ algebra.

Suppose that there is a measurable set $G$ with $\\#G=\\#X$ and
$\\#(X-G)=\\#X$. If $H$ is another subset of $X$ with $\\#H=\\#X$ and
$\\#(X-H)=\\#X$, then $H\in\Sigma$, by the same argument as in the case
$\kappa\neq\\#X$. So every subset of $X$ is in $\Sigma$, i. e., $X$ has the
discrete $\sigma$-algebra.

In all cases we get that $X$ is cocardinal. ∎

######  Corollary 2.

Let $X$ be a measurable space. Then $Mor(X,X)=Map(X,X)$ if and only if $X$ has
the trivial $\sigma$-algebra or the discrete $\sigma$-algebra.

###### Proof.

Analogous to Corollary 1. ∎

## References

  * [1] K. Kunen, Set Theory. Studies in Logic. College Publications, 2011
  * [2] R. L. Schilling, Measures, Integrals and Martingales. Cambridge University Press, 2005.
  * [3] L. A. Steen and J. A. Seebach Jr., Counterexamples in Topology. Holt, Rinehart and Winston Inc., 1970.


# Measuring the Breaking of Lepton Flavour Universality in $B\to
K^{*}\ell^{+}\ell^{-}$

Nicola Serra<EMAIL_ADDRESS>Rafael Silva Coutinho
<EMAIL_ADDRESS>Danny van Dyk<EMAIL_ADDRESS>Physik-
Institut, Universität Zürich, Winterthurer Strasse 190, 8057 Zürich,
Switzerland

###### Abstract

We propose measurements of weighted differences of the angular observables in
the rare decays $B\to K^{*}\ell^{+}\ell^{-}$. The proposed observables are
very sensitive to the difference between the Wilson coefficients
$\mathcal{C}_{9}^{(e)}$ and $\mathcal{C}_{9}^{(\mu)}$ for decays into
electrons and muons, respectively. At the same time, the charm-induced
hadronic contributions are kinematically suppressed to $\lesssim 7\%(4\%)$ in
the region $1\,^{2}\leq q^{2}\leq 6\,^{2}$, as long as LFU breaking occurs
only in $\mathcal{C}^{\text{$(\ell)$}}_{9}$. This suppression becomes stronger
for the region of low hadronic recoil, $q^{2}\geq 15\,^{2}$.

††preprint: EOS-2016-03, ZU-TH-12/16

##  I Introduction

In this letter, we investigate the suitability of new observables to measure
the breaking of Lepton-Flavour Universality (LFU) in rare $b\to s\ell\ell$
transitions. While measurements Aaij  _et al._ (2014); Wilson  of the ratio
$R_{K}$,

$\displaystyle R_{K}\equiv\frac{\mathcal{B}(B\to
K\mu^{+}\mu^{-})}{\mathcal{B}(B\to Ke^{+}e^{-})}\,,$ (1)

for dilepton masses $1\,^{2}\leq q^{2}\leq 6\,^{2}$, hint toward LFU breaking,
there has been no unambiguous discovery of such effects. It has been proposed
Hiller and Schmaltz (2015) to expand such measurements to the decays $B\to
X_{s}\ell^{+}\ell^{-}$, $B\to K^{*}\ell^{+}\ell^{-}$ and
$B_{s}\to\phi\ell^{+}\ell^{-}$, introducing similar ratios $R_{X_{s}}$,
$R_{K^{*}}$ and $R_{\phi}$, respectively. Analysing LFU breaking in angular
observables of the decay $B\to K^{*}\ell^{+}\ell^{-}$ has been proposed in
Altmannshofer and Straub (2015a), and more recently studied in Capdevila  _et
al._ (2016). Within this letter we propose observables that can be used to
accurately measure the size of this breaking, specifically in the decays $B\to
K^{*}\ell^{+}\ell^{-}$. Our study focuses on observables in which charm-
induced long-distance contributions can be kinematically suppressed.

The exclusive decays $B\to K^{*}\ell^{+}\ell^{-}$ for $\ell=e,\mu$ are
governed by the effective field theory for flavour-changing neutral $b\to
s\\{\gamma,\ell^{+}\ell^{-}\\}$ transitions; see e.g. Bobeth  _et al._ (2013).
The theory’s Hamiltonian density to leading power in $G_{F}$ is

$\displaystyle{\cal{H}}^{(\ell)}_{\rm eff}$
$\displaystyle=-\frac{4\,G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\,\Bigg{\\{}\,\left[\frac{\alpha_{e}}{4\pi}\,\sum_{i}\mathcal{C}^{\text{$(\ell)$}}_{i}(\mu)\mathcal{O}_{i}^{(\ell)}\right]$
(2)
$\displaystyle+\left[\frac{e}{(4\pi)^{2}}\sum_{j}\mathcal{C}^{\text{}}_{j}(\mu)\mathcal{O}_{j}\right]+\left[\sum_{k}\mathcal{C}^{\text{}}_{k}(\mu)\mathcal{O}_{k}\right]$
$\displaystyle+\mathcal{O}\left({(V_{ub}V_{us}^{*})/(V_{tb}V_{ts}^{*})}\right)+\text{h.c.}\Bigg{\\}}\,,$

where $\mathcal{C}^{\text{}}_{i}(\mu)$ denotes the Wilson coefficients at the
renormalisation scale $\mu$, and $\mathcal{O}_{\\{i,j,k\\}}$ denotes a basis
of dimension-6 field operators. The index $i$ iterates over all semileptonic
operators, $i=9,9^{\prime},10,10^{\prime},S,S^{\prime},P,P^{\prime},T,T5$,
which are dependent on the final state lepton flavour $\ell=e,\mu$. The
indices $j$ and $k$ iterate over the radiative ($j=7,7^{\prime}$), and the
four-quark and chromomagnetic operators ($k=1,\dots,6,8$), respectively. The
most relevant operators read

$\displaystyle\mathcal{O}_{7(7^{\prime})}$
$\displaystyle=\frac{m_{b}}{e}\\!\left[\bar{s}\sigma^{\mu\nu}P_{R(L)}b\right]F_{\mu\nu}\,,$
(3) $\displaystyle\mathcal{O}_{9(9^{\prime})}$
$\displaystyle=\left[\bar{s}\gamma_{\mu}P_{L(R)}b\right]\\!\left[\bar{\ell}\gamma^{\mu}\ell\right]\,,$
$\displaystyle\mathcal{O}_{10(10^{\prime})}$
$\displaystyle=\left[\bar{s}\gamma_{\mu}P_{L(R)}b\right]\\!\left[\bar{\ell}\gamma^{\mu}\gamma_{5}\ell\right]\,,$

where a primed index indicates a flip of the quarks’ chiralities with respect
to the unprimed, SM-like operator.

Hadronic matrix elements of the semileptonic operators are parametrized in
terms of form factors, which can be determined using non-perturbative methods
such as lattice QCD (see e.g. Horgan  _et al._ (2014)) and QCD sum rules (see
e.g. Bharucha  _et al._ (2015)). However, hadronic matrix elements of the
correlator between four-quark operators
$\mathcal{O}_{i}\sim\left[\bar{s}\Gamma_{i}b\right]\,\left[\bar{q}\Gamma_{i}^{\prime}q\right]$,
$i=1,\dots,6$ as well as the chromomagnetic operator $\mathcal{O}_{8}$ on the
one hand, and the electromagnetic current on the other, are more complicated
to estimate. These non-local matrix elements contribute to the transition
amplitudes $A_{\lambda}$, with $\lambda=0,\perp,\parallel$, through shifts
$\mathcal{C}^{\text{$(\ell)$}}_{9}\mapsto\mathcal{C}^{\text{$(\ell)$}}_{9}+h_{9,\lambda}(q^{2})$
and $\mathcal{C}^{\text{}}_{7}\mapsto\mathcal{C}^{\text{}}_{7}+h_{7,\lambda}$.
Note that the shifts to $\mathcal{C}^{\text{}}_{9}$ are explicitly dependent
on $q^{2}$, the momentum transfer to the lepton pair.

Within ratios of observables for either $\ell=\mu$ or $\ell=e$ final states,
the non-local contributions $h_{9,\lambda}(q^{2})$ do not cancel completely.
However, within differences of angular observables they can be kinematically
suppressed.111In reference Capdevila  _et al._ (2016) the authors propose
observables $Q_{i}$, $T_{i}$ and $B_{i}$, which are constructed from
differences of the principal angular observables $J_{i}$ in $\mu$ and $e$
final states. Of the proposed observables, the
$B_{i}\equiv(J_{i}^{(\mu)}-J_{i}^{(e)})/J_{i}^{(e)}$ observables are closest
to what we propose here. However, their normalisation to the electron-mode
angular observable lifts the kinematic suppression of the charm-induced long-
distance contributions that we aim for.

The remainder of this letter is structured as follows: We propose the new
observables in section II. Their numerical evaluations and theoretical
uncertainties are discussed in section III. Thereafter, we study the
experimental feasibility of their measurements for both future Belle-II and
LHCb data sets in section IV, before we conclude in section V.

##  II Measures of LFU Breaking

The angular PDF for $B\to K^{*}(\to K\pi)\ell^{+}\ell^{-}$ decays is well
known in the literature, and we use the conventions specified in Altmannshofer
_et al._ (2009). There, the CP-averaged and normalized angular observables are

$S_{i}(q^{2})\equiv\frac{4}{3}\,\frac{J_{i}(q^{2})+\overline{J}_{i}(q^{2})}{{\mathrm{d}}\Gamma\,/{\mathrm{d}}q\,^{2}+{\mathrm{d}}\overline{\Gamma}\,/{\mathrm{d}}q\,^{2}}\,,$
(4)

where unbarred quantities stem from the decay
$\bar{B}\to\bar{K}^{*}\ell^{+}\ell^{-}$, and the bar indicates CP conjugation.
For the definitions of the $J_{i}$, see Kruger  _et al._ (2000); Kruger and
Matias (2005); Altmannshofer  _et al._ (2009); Bobeth  _et al._ (2013).
222Note that the definition of the angular observables does not account for
purely QED-induced modifications to the overall angular distribution; see
Huber  _et al._ (2015); Gratrex  _et al._ (2016) for recent discussions.  Here
and throughout the rest of this letter, we will refer to $S_{i}^{(\ell)}$ and
$\Gamma^{(\ell)}$ as one of the angular observables or the decay width for the
$\ell$ final state, respectively.

All spin-averaged observables can be expressed in terms of sesquilinear
combinations of up to 14 transversity amplitudes when working in the full
basis of dimension-six semileptonic operators Bobeth  _et al._ (2013). For the
discussions at hand, however, we restrict our study to the operators
$\mathcal{O}_{9,10}$. In this case, all observables can be expressed in terms
of only $7$ transversity amplitudes,

$\displaystyle\frac{A_{0}^{L(R)}}{N\sqrt{\beta_{\ell}}}$
$\displaystyle=-\left[\left(\mathcal{C}^{\text{eff,$(\ell)$}}_{9,0}\mp\mathcal{C}^{\text{}}_{10}\right)F_{V,0}\\!+\\!\frac{2m_{b}}{M_{B}}\mathcal{C}^{\text{eff}}_{7,0}F_{T,0}\\!\right]$
(5) $\displaystyle\frac{A_{\parallel}^{L(R)}}{N\sqrt{\beta_{\ell}}}$
$\displaystyle=-\left[\left(\mathcal{C}^{\text{eff,$(\ell)$}}_{9,\parallel}\mp\mathcal{C}^{\text{}}_{10}\right)F_{V,\parallel}\\!+\\!\frac{2m_{b}M_{B}}{q^{2}}\mathcal{C}^{\text{eff}}_{7,\parallel}F_{T,\parallel}\\!\right]$
$\displaystyle\frac{A_{\perp}^{L(R)}}{N\sqrt{\beta_{\ell}}}$
$\displaystyle=\left[\left(\mathcal{C}^{\text{eff,$(\ell)$}}_{9,\perp}\mp\mathcal{C}^{\text{}}_{10}\right)F_{V,\perp}\\!+\\!\frac{2m_{b}M_{B}}{q^{2}}\mathcal{C}^{\text{eff}}_{7,\perp}F_{T,\perp}\\!\right]$

as well as $A_{t}$. The latter is not relevant to the discussions at hand.
Note that our convention for the normalization constant $N$

$N\equiv
G_{F}\alpha_{e}V_{tb}V_{ts}^{*}\sqrt{\frac{q^{2}\sqrt{\lambda}}{3\cdot
2^{10}\pi^{5}M_{B}^{3}}}\,,$ (6)

differs from, e.g., the normalization $N^{\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bobeth:2012vn}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ as
used in reference Bobeth  _et al._ (2013):
$N^{\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bobeth:2012vn}{\@@citephrase{(}}{\@@citephrase{)}}}}}=\sqrt{\beta_{\ell}}N$.
Our choice ensures that the normalization is universal for all lepton
flavours.

We propose to measure weighted differences of angular observables,

$D_{i}(q^{2})\equiv\frac{{\mathrm{d}}\mathcal{B}^{(e)}\,}{{\mathrm{d}}q\,^{2}}S^{(e)}_{i}(q^{2})-\frac{{\mathrm{d}}\mathcal{B}^{(\mu)}\,}{{\mathrm{d}}q\,^{2}}S^{(\mu)}_{i}(q^{2})\,.$
(7)

Assuming LFU breaking only333Note that lepton-universal NP effects are not
precluded here.  in the Wilson coefficient $\mathcal{C}^{\text{}}_{9}$, we
obtain for the indices $i=4,5,6s$ the expressions

$\displaystyle\frac{-D_{4}(q^{2})}{\sqrt{2}\,N^{2}\,\tau_{B}}$
$\displaystyle=\operatorname{Re}\left[\Delta_{9^{2}}^{(3)}\right]F_{V,\parallel}F_{V,0}$
(8)
$\displaystyle-\operatorname{Re}\left[\Delta_{9}^{(3)}\dots\right]+\mathcal{O}\left({\Delta_{\beta}^{(3)}}\right)\,,$
$\displaystyle\frac{D_{5}(q^{2})}{4\,\sqrt{2}\,N^{2}\,\tau_{B}}$
$\displaystyle=\operatorname{Re}\left\\{\Delta_{9}^{(2)}\mathcal{C}^{\text{}}_{10}\right\\}F_{V,\perp}F_{V,0}$
$\displaystyle+\mathcal{O}\left({\Delta_{\beta}^{(2)}}\right)$
$\displaystyle\frac{D_{6s}(q^{2})}{8\,N^{2}\,\tau_{B}}$
$\displaystyle=\operatorname{Re}\left\\{\Delta_{9}^{(2)}\mathcal{C}^{\text{}}_{10}\right\\}F_{V,\parallel}F_{V,\perp}$
$\displaystyle+\mathcal{O}\left({\Delta_{\beta}^{(2)}}\right)$

where $\tau_{B}$ is the lifetime of the $B$ mesons, and the dots indicate an
unsuppressed expression linear in the non-local contributions $h_{9,0}(q^{2})$
and $h_{9,\parallel}(q^{2})$. Moreover, we introduce

$\displaystyle\Delta_{9^{l}}^{(k)}$
$\displaystyle\equiv\beta_{e}^{k}\left[\mathcal{C}^{\text{$(e)$}}_{9}\right]^{l}-\beta_{\mu}^{k}\left[\mathcal{C}^{\text{$(\mu)$}}_{9}\right]^{l}\,,$
(9) $\displaystyle\Delta_{\beta}^{(k)}$
$\displaystyle\equiv\beta_{e}^{k}-\beta_{\mu}^{k}\,.$

We find that eq. (8) holds up to corrections of order
$\beta_{e}^{3}-\beta_{\mu}^{3}$ (for $D_{4}$) and
$\beta_{e}^{2}-\beta_{\mu}^{2}$ (for $D_{5,6s}$). Note that $D_{4}$ is free of
hadronic contributions in the term $\propto\Delta_{9^{2}}^{(3)}$, but not free
of them in the linear term $\Delta_{9}^{(3)}$. For the full results, see eqs.
(17)–(19). The expressions eq. (8) hold _in the entire $q^{2}$ spectrum_,
since no explicit expression for the hadronic two-point correlation functions,
$h_{9,\lambda}(q^{2})\equiv\mathcal{C}^{\text{eff,($\ell$)}}_{9,\lambda}(q^{2})-\mathcal{C}^{\text{$(\ell)$}}_{9}$,
have been used. We emphasize that this also holds in between the two vetoes
for the $J/\psi$ and $\psi(2S)$ charmonia.

Figure 1: The $q^{2}$ dependence of the suppression terms
$\Delta_{\beta}^{(k)}$ for $k=2$ (lower blue curve) and $k=3$ (upper orange
curve), respectively, on a log scale. The windows in which either QCD-improved
factorization or the operator product expansion at low recoil can be applied
are indicated by the shaded area on the left and right, respectively.

For $q^{2}\geq 1\,^{2}$, the suppressed terms in eq. (8) scale with
$\Delta_{\beta}^{(3)}<6.6\%$ and $\Delta_{\beta}^{(2)}<4.5\%$, respectively.
For the low recoil region, these terms further shrink down to $<0.5\%$ and
$<0.3\%$, respectively. Therefore, from a theoretical point of view, the low
recoil region would be ideal for our proposed analysis. However, at LHCb the
experimental analysis of the $e^{+}e^{-}$ final state is more challenging for
large $q^{2}$.

While our approach is – in principal – also applicable to the angular
observables $S_{i}$ with $i=7,8,9$, we remind that any observation of non-
vanishing values for these observables already constitutes a sign of NP.

Figure 2: The $q^{2}$ behaviour of $R_{K^{*}}$ for our benchmark point
$\mathcal{C}^{\text{$(e)$}}_{9}=\mathcal{C}^{\text{SM}}_{9}=\mathcal{C}^{\text{$(\mu)$}}_{9}+1$
(blue shaded area) and the SM (orange line). The illustrated uncertainty is
due to our incomplete knowledge of the CKM matrix elements, the form factors,
and the charm-induced long-distance contributions. The SM uncertainty too
small to be shown.

Note that the proposed observables are not independent of $R_{K^{*}}$, the
ratio of the decay rates into $\mu$ versus $e$, since:

$\displaystyle D_{i}(q^{2})$
$\displaystyle=\left[\frac{{\mathrm{d}}\mathcal{B}^{(e)}\,}{{\mathrm{d}}q\,^{2}}+\frac{{\mathrm{d}}\mathcal{B}^{(e)}\,}{{\mathrm{d}}q\,^{2}}\right]$
(10)
$\displaystyle\times\left[\omega^{(e)}(q^{2})S^{(e)}_{i}(q^{2})-\omega^{(\mu)}(q^{2})S^{(\mu)}_{i}(q^{2})\right]\,,$

with $\mathcal{B}^{(\ell)}$ the branching ratio of $B\to
K^{*}\ell^{+}\ell^{-}$, and weights

$\displaystyle\omega^{(e)}(q^{2})$
$\displaystyle\equiv\frac{1}{1+R_{K^{*}}(q^{2})}\,,$ (11)
$\displaystyle\omega^{(\mu)}(q^{2})$
$\displaystyle\equiv\frac{R_{K^{*}}(q^{2})}{1+R_{K^{*}}(q^{2})}\,.$

Integration over $q^{2}$ from $a$ to $b$ then yields

$\int_{a}^{b}{\mathrm{d}}q\,^{2}\,D_{i}(q^{2})\equiv\langle
D_{i}\rangle_{a,b}\\\ =\langle\mathcal{B}^{(e)}\rangle_{a,b}\,\langle
S^{(e)}_{i}\rangle_{a,b}-\langle\mathcal{B}^{(\mu)}\rangle_{a,b}\,\langle
S^{(\mu)}_{i}\rangle_{a,b}\,.$ (12)

We also wish to comment on opportunities for decays other than $B\to K^{*}(\to
K\pi)\ell^{+}\ell^{-}$:

  1. 1.

The decays $B_{s}\to\phi(\to K^{+}K^{-})\ell^{+}\ell^{-}$, with $\ell=e,\mu$
can be described by the same angular PDF as $B\to K^{*}\ell^{+}\ell^{-}$
decays. Thus a generalization of the $D_{i}$ observables to the $B_{s}$ decay
is obvious. However, measurements of the theoretically most interesting
observables $S_{5,6s}$ will require flavour tagging. Notice that the
feasibility for this measurement in LHCb is both limited by the observed yield
in Run-I and the low tagging power capability. Moreover, a flavour-tagged
analysis at Belle II is very difficult, since the production of $B_{s}$ pairs
at the $\Upsilon(5S)$ resonance does not occur through eigenstates of the
charge-conjugation operators (unlike production of $B_{d}$ pairs at the
$\Upsilon(4S)$).

  2. 2.

The observables $D_{i}$ can be generalized to the entire phase space of the
$K\pi$ final state, i.e., to $K\pi$ masses outside the window that usually is
associated with an on-shell $K^{*}(892)$. As for the $K^{*}(892)$, any
significant deviation from zero, relative to the branching ratio, is a
definite sign of LFU breaking, and thus a signal of NP. However, at the
present time, the theoretical understanding of hadronic effects in $B\to
K\pi\ell^{+}\ell^{-}$ is not well-enough developed for us to produce numerical
estimates for small values of $q^{2}$.

  3. 3.

The decays $\Lambda_{b}\to\Lambda(\to N\pi)\ell^{+}\ell^{-}$ give rise to 10
angular observables Böer  _et al._ (2015). Amongst these observables, $K_{1c}$
and $K_{4s}$ permit a suppression of the charm-induced non-local matrix
elements in the same fashion as shown in eq. (8). However, at the present
time, measurements of the muon final state are affected by large statistical
uncertainties, and no measurements for the electron final state are available.

##  III Numerical Results

In order to show that the observables $D_{4,5,6s}$ are indeed sensitive to LFU
breaking, we evaluate them at large hadronic recoil in one bin $1\,^{2}\leq
q^{2}\leq 6\,^{2}$, which we denoted as $\langle\cdot\rangle_{1,6}$. Our
numerical calculations are carried out using the EOS software van Dyk  _et
al._ (2016a), which has been modified for this purpose van Dyk  _et al._
(2016b). The evaluation of $B\to K^{*}\ell^{+}\ell^{-}$ observables in the
large recoil region implements the results of the framework of QCD-improved
factorization Beneke  _et al._ (2001, 2005). The uncertainties on the $D_{i}$
arise dominantly from uncertainties of the CKM Wolfenstein parameters and our
incomplete knowledge of the $B\to K^{*}$ form factors. The numerical input
values, their sources and their prior PDFs are listed in table 1. In the SM,
i.e., for
$\mathcal{C}^{\text{$(e)$}}_{9}=\mathcal{C}^{\text{$(\mu)$}}_{9}=\mathcal{C}^{\text{SM}}_{9}$,
we obtain

$\displaystyle\langle R^{\text{SM}}_{K^{*}}\rangle_{1,6}$
$\displaystyle=0.997^{+0.0005}_{-0.0004}\,,$ (13) $\displaystyle\langle
D^{\text{SM}}_{4}\rangle_{1,6}$ $\displaystyle=(+2.9^{+1.1}_{-1.7})\cdot
10^{-10}$ $\displaystyle(^{+39}_{-60}\%)\,,$ $\displaystyle\langle
D^{\text{SM}}_{5}\rangle_{1,6}$ $\displaystyle=(-1.1^{+2.0}_{-2.0})\cdot
10^{-10}$ $\displaystyle(\pm 176\%)\,,$ $\displaystyle\langle
D^{\text{SM}}_{6s}\rangle_{1,6}$ $\displaystyle=(+4.4^{+1.7}_{-1.5})\cdot
10^{-10}$ $\displaystyle(^{+40}_{-33}\%)\,.$

The large relative uncertainties in the SM are to be expected, since for
lepton-universal models the short-distance contributions on the right-hand
side of eq. (8) are small compared to the correction that involve the hadronic
contributions $h_{9,\lambda}$.

Figure 3: Comparison of the $q^{2}$-integrated observables $D_{i}$ in the SM
(blue circles) with their values in the benchmark point (red squares) for the
bin $1\,^{2}\leq q^{2}\leq 6\,^{2}$. The smaller (red) error marks correspond
to the pure form factor uncertainties, while the larger (red) error marks
correspond to the uncertainties as obtained for the data driven scenario; see
the text for more information. The experimental measurement of branching ratio
$\mathcal{B}(B\to K^{*}\mu^{+}\mu^{-})$, including the experimental
uncertainty, is shown for comparison as the grey band.

However, in the case of LFU breaking, a sizeable leading short-distance term
can reduce the relative size of he non-local hadronic uncertainties. For
comparison, we define a benchmark point
$\mathcal{C}^{\text{$(e)$}}_{9}=\mathcal{C}^{\text{SM}}_{9}=\mathcal{C}^{\text{$(\mu)$}}_{9}+1$.
This point is favoured by several global analyses of data on $b\to
s\ell^{+}\ell^{-}$ processes; see e.g. Beaujean  _et al._ (2014);
Altmannshofer and Straub (2015b); Descotes-Genon  _et al._ (2016); Hurth  _et
al._ (2016). For our benchmark point (BMP) we obtain

$\displaystyle\langle R^{\text{BMP}}_{K^{*}}\rangle_{1,6}$
$\displaystyle=0.86^{+0.02}_{-0.01}\,,$ (14) $\displaystyle\langle
D^{\text{BMP}}_{4}\rangle_{1,6}$ $\displaystyle=(+2.4^{+2.00}_{-0.95})\cdot
10^{-9}$ $\displaystyle(^{+84}_{-40}\%)\,,$ $\displaystyle\langle
D^{\text{BMP}}_{5}\rangle_{1,6}$ $\displaystyle=(-2.7^{+0.31}_{-0.28})\cdot
10^{-8}$ $\displaystyle(^{+12}_{-11}\%)\,,$ $\displaystyle\langle
D^{\text{BMP}}_{6s}\rangle_{1,6}$ $\displaystyle=(-1.7^{+0.21}_{-0.26})\cdot
10^{-8}$ $\displaystyle(^{+12}_{-15}\%)\,,$

where the uncertainties of $D_{5,6s}$ are now dominated by the parametric CKM
and form factor uncertainties, while $D_{4}$ still shows large charm-induced
uncertainties. A comparison between eq. (13) and eq. (14) clearly shows that
the observables $D_{4,5,6s}$ are very sensitive to LFU-breaking NP effects,
with relative enhancements (for the central values only) of

$\displaystyle\frac{\langle D^{\text{BMP}}_{4}\rangle_{1,6}}{\langle
D^{\text{SM}}_{4}\rangle_{1,6}}$ $\displaystyle\simeq 8\,,$ (15)
$\displaystyle\frac{\langle D^{\text{BMP}}_{5}\rangle_{1,6}}{\langle
D^{\text{SM}}_{5}\rangle_{1,6}}$ $\displaystyle\simeq 200\,,$
$\displaystyle\frac{\langle D^{\text{BMP}}_{6s}\rangle_{1,6}}{\langle
D^{\text{SM}}_{6s}\rangle_{1,6}}$ $\displaystyle\simeq 40\,.$

At the same time, it shows that the relative uncertainty is reduced slightly
for $D_{4}$, and strongly for $D_{5,6s}$. This decrease in (relative)
uncertainty emerges, since the impact of the non-local hadronic matrix
elements is reduced compared to the now leading contributions from form
factors and $\mathcal{C}^{\text{$(e)$}}_{9}-\mathcal{C}^{\text{$(\mu)$}}_{9}$.

Parameter | prior | unit | source
---|---|---|---
CKM Wolfenstein parameters
$\lambda$ | $0.2253\pm 0.0006$ | — | Bona  _et al._ (2006)
$A$ | $0.806\pm 0.020$ | — | Bona  _et al._ (2006)
$\bar{\rho}$ | $0.132\pm 0.049$ | — | Bona  _et al._ (2006)
$\bar{\eta}$ | $0.369\pm 0.050$ | — | Bona  _et al._ (2006)
Quark masses
$\overline{m}_{c}(\overline{m}_{c})$ | $1.275\pm 0.025$ |  | Olive  _et al._ (2014)
$\overline{m}_{b}(\overline{m}_{b})$ | $4.18\pm 0.03$ |  | Olive  _et al._ (2014)
$B\to K^{*}$ power correction parameters
$r_{0,\perp,\parallel}$ | $1.00\pm 0.45$ | — | this work
Table 1:  Numerical inputs for the calculations of the electron and muons
components of the observables $D_{i}$. The power corrections $r_{\chi}$,
$\chi=0,\perp,\parallel$ are scaling factors to the dipole form factors
$\cal{T}$ as introduced in Beneke  _et al._ (2005). The prior distributions
for all listed parameters are Gaussian, and the given intervals correspond to
their central $68\%$ probability intervals. We do not list the $B\to K^{*}$
form factor parameters here, which are taken from a simultaneous fit to Light-
Cone Sum Rule and lattice QCD results Bharucha  _et al._ (2015), including
their correlation matrix. For the data-driven scenario, we also use $B\to K$
form factor parameters as obtained in Khodjamirian  _et al._ (2010).

We wish to further illustrate the usefulness of the newly-proposed observables
by studying a data-driven scenario (DDS). For this, we carry out a Bayesian
fit involving a free-floating $1.5\leq\mathcal{C}^{\text{$(\mu)$}}_{9}\leq
5.5$, while we fix all other Wilson coefficients to their SM values. The
likelihood is comprised from the LHCb measurement of $R_{K}$ Aaij  _et al._
(2014), a recent preliminary result for $R_{K}$ by the BaBar collaboration
Wilson , as well as the LHCb results for $P^{\prime}_{5}$ Aaij  _et al._
(2016), an angular observables in the decay $B\to K^{*}\mu^{+}\mu^{-}$ that
exhibits reduced sensitivity to hadronic form factors. We then proceed to
produce posterior-predictive distributions for $R_{K^{*}}$ and $D_{4,5,6s}$,
which can be summarized as

$\displaystyle\langle R^{\text{DDS}}_{K^{*}}\rangle_{1,6}$
$\displaystyle=0.85\pm 0.04\,,$ (16) $\displaystyle\langle
D^{\text{DDS}}_{4}\rangle_{1,6}$ $\displaystyle=(2.4^{+2.1}_{-1.5})\cdot
10^{-9}\,,$ $\displaystyle\langle D^{\text{DDS}}_{5}\rangle_{1,6}$
$\displaystyle=(-3.1^{+0.9}_{-1.2})\cdot 10^{-8}\,,$ $\displaystyle\langle
D^{\text{DDS}}_{6s}\rangle_{1,6}$ $\displaystyle=(-2.1^{+0.8}_{-0.7})\cdot
10^{-8}\,,$

which corresponds to qualitatively the same type of enhancements as in eq.
(15). A comparison of all our numerical results for the $D_{4,5,6s}$ is
depicted in figure 3.

##  IV Experimental Feasibility

Figure 4: Projected statistical-only significance as a function of the
extrapolated yields in LHCb for the (blue) $\langle D_{5}\rangle_{1,6}$ and
(orange) $\langle D_{6s}\rangle_{1,6}$ observables, obtained from the (solid
line) method-of-moments and (dash-dotted line) likelihood fit. The red lines
indicate the combined significance of $\langle D_{5}\rangle_{1,6}$ and
$\langle D_{6s}\rangle_{1,6}$.  Figure 5: Projected statistical-only
significance as a function of the extrapolated yields in Belle-II for the
(blue) $\langle D_{5}\rangle_{1,6}$ and (orange) $\langle D_{6s}\rangle_{1,6}$
observables, obtained from the (solid line) method-of-moments and (dash-dotted
line) likelihood fit. The red lines indicate the combined significance of
$\langle D_{5}\rangle_{1,6}$ and $\langle D_{6s}\rangle_{1,6}$.

A series of signal-only ensembles of pseudoexperiments is generated to
investigate the minimum amount of data required to claim an observation of NP
in these observables. The simulation is performed without considering any
experimental effects, i.e., background contributions, acceptance, resolution
or bin migration. Similarly to the numerical calculations, the toy model is
implemented using the EOS framework independently for each final state flavour
at $1\,^{2}\leq q^{2}\leq 6\,^{2}$. Pseudoexperiments are generated with
sample sizes corresponding roughly to the yields for the current and
forthcoming data taking periods available at LHCb and Belle II. 444Notice that
effects of potential improvements (e.g. in the electron detection efficiency
or reconstruction of $K^{*0}$ through $K_{S}\pi^{0}$), and of analysing
additional signal channels (e.g. $B^{+}\to K^{*+}\ell^{+}\ell^{-}$) are not
investigated here.  These yields are extrapolated (rounded to the nearest
50/500) from the values reported in Aaij  _et al._ (2014, 2016) and Abdesselam
_et al._ (2016) by scaling the luminosities and the $b\bar{b}$ cross section
$\sigma_{b\bar{b}}$. For LHCb, we scale $\sigma_{b\bar{b}}\propto\sqrt{s}$,
while for Belle $\sigma_{b\bar{b}}\propto s$, where $s$ denotes the designed
centre-of-mass energy of the $b$-quark pair. In particular, the significance
for the range of $[3\,$-$\,50]\,$ and $[1\,$-$\,50]\,{\rm{ab}}^{-1}$ are
examined for LHCb and Belle II, respectively. Note that the relative yields
between electrons and muons are fixed in the pseudoexperiments. Ensembles with
other sample sizes are also generated to test the scaling of the
uncertainties, though only a representative subset of the results obtained are
shown here.

A convenient strategy to determine the $S^{({\ell})}_{i}(q^{2})$, $\ell=e,\mu$
observables is to utilise the principal angular moments Beaujean  _et al._
(2015). Although this approach provides an approximately $15\%$ worse
precision on the measurement compared to the likelihood fit Aaij  _et al._
(2016), as a proof-of-principle for the novel observables this is more robust
(e.g. against mismodelling of the PDF) and insensitive to the choice of the
estimated signal yield. Nevertheless, for completeness the results from an
unbinned maximum likelihood fit are also reported. Note that the observables
in both approaches correspond to the average $\langle
S^{({\ell})}_{i}\rangle_{1,6}$, obtained by summing over each toy candidate
for a given experiment.

Since the signal yield projections for the decay $B\to K^{*0}e^{+}e^{-}$ in
both experiments indicate limited datasets, the stability of the likelihood
fit is enforced by simplifying the differential decay rate. This is achieved
by applying folding techniques to specific regions of the three-dimensional
angular space, as detailed in Ref. Aaij  _et al._ (2013). Notice that the
angular analysis is performed separately for each lepton flavour. It is worth
emphasizing that, despite potential benefits on the experimental side, to
examine both final states simultaneously the choice to share/constrain angular
observables across different flavours should be in general avoided – unless
otherwise strongly motivated. For instance, assuming
$\mathcal{C}^{\text{$(e)$}}_{9}=\mathcal{C}^{\text{SM}}_{9}=\mathcal{C}^{\text{$(\mu)$}}_{9}+1$,
$F_{L}^{(\mu)}$ is reduced by $10\%$ compared to $F_{L}^{(e)}$.

Furthermore, it has been shown that an alternative approach to weighting each
event by the inverse of its efficiency is given by the calculation of the
unfolding matrix using the method of moments Beaujean  _et al._ (2015). This
is of particular interest in a simultaneous determination of the expressions
$\omega^{(\ell)}(q^{2})$ (see eq. (11)) and $S^{(\ell)}_{i}(q^{2})$, in which
a shared unfolded parametrisation can be used. Further advantages regarding
the impact of common systematic effects are experiment dependent, and
therefore, not discussed in this note.

In order to obtain the profile of the statistical-only significance for the
extrapolated signal yields, both SM and NP simulated ensembles are examined.
The resulting $\langle D_{i}\rangle_{1,6}$ observables are fitted, and the
distance between the SM prediction and the fit results is calculated in units
of the toy measurements’ standard deviations. Additional constraints on the
likelihood fit were necessary in order to ensure the stability of the $\langle
D_{6s}\rangle_{1,6}$ determination. In particular, the values of the
longitudinal polarisation of the $K^{*0}$ meson and the transverse
polarisation asymmetry are constrained within uncertainties to the theory
predictions.

Figures 4 and 5 summarise the expected sensitivity to benchmark-like NP
effects for the proposed observables $\langle D_{5}\rangle_{1,6}$ and $\langle
D_{6s}\rangle_{1,6}$. Projections for $\langle D_{4}\rangle_{1,6}$ are not
shown here since these are of limited usefulness. Further studies with
realistic experimental effects are necessary to determine the exact
sensitivities achievable. An extrapolation of the precision estimated here
suggests that such measurements appear to be feasible, albeit that the
confirmation of benchmark-like NP effects independently in each observable is
only possible with the full capability of the experiments. Furthermore, the
correlation between these observables can be estimated and used in a combined
significance. Based on our extrapolations, a first evidence of NP in the LHCb
measurement and considering only this novel approach can be achieved with
$1300$ $B\to K^{*}e^{+}e^{-}$ signal events, which can currently not be
expected to be recorded before the end of LHCb Run-II. Similar sensitivity in
the Belle-II measurement can be achieved with $1500$ signal events, which
corresponds roughly to an integrated luminosity of $50\,$ab-1. Note that the
possibility of the proposed approach to go beyond the usual theory upper bound
of $q^{2}<6.0\,$GeV raises interesting prospects for Belle-II: first, to
increase their sensitivity due to stronger suppression of the charm-induced
contributions; and second, to record more events than currently considered.

##  V Conclusion

Recent measurements of $b\to s\ell^{+}\ell^{-}$ transitions show an
interesting pattern of deviations with respect to SM predictions. In
particular, the anomalous LHCb and Belle measurements of the observable
$P_{5}^{\prime}$, and the LHCb measurement of the LFU-probing ratio $R_{K}$
can be simultaneously explained with NP contributions to the Wilson
coefficients $\mathcal{C}^{\text{$(\mu)$}}_{9}$ and/or
$\mathcal{C}^{\text{$(\mu)$}}_{10}$. This generated large attention in the
flavour physics community, in particular concerning long-distance charm-
induced effects, which might be able to explain the deviation in
$P_{5}^{\prime}$.

Here, we proposed a new set of observables $D_{i}$ ($i=4,5,6s$), sensitive to
LFU-breaking effects in the decays $B\to K^{*}\ell^{+}\ell^{-}$. These
observables are branching-ratio-weighted averages of differences (with respect
to the final-state lepton flavour) of the angular observables $S_{4,5,6s}$. In
the presence of the LFU-breaking NP effects in
$\mathcal{C}^{\text{$(\ell)$}}_{9}$, their theoretical uncertainties are
dominated by $B\to K^{*}$ form factor and CKM parameter uncertainties, while
non-local hadronic contributions are kinematically suppressed. This allows
predictions in the NP scenarios that can be systematically improved as our
knowledge of the $B\to K^{*}$ form factors and CKM Wolfenstein parameters
improves. As examples we discussed here one benchmark point, as well as a
data-driven scenario based on a fit of the observable $R_{K}$ and
$P^{\prime}_{5}$. All these scenarios have peculiar patterns of deviations of
the observables $D_{i}$ with respect to SM predictions (with reduced
theoretical uncertainties). Therefore these new observables, in addition to
providing sensitivity to discover NP with LFU-breaking effects, are useful to
disentangle the different scenarios and are crucial to test the mutual
consistency across different measurements. It is important to highlight that
these observables are also independent of other LFU-breaking measurements,
e.g. $R_{K}$ or $R_{\phi}$. Hence, these can be included in global fits, which
improves the potential sensitivity to LFU-breaking NP effects.

The $D_{i}$ observables can be measured at the LHCb (and its upgrade) or at
the Belle II experiments, either by performing likelihood fits of the angular
distribution of the decays $B\to K^{*}\ell^{+}\ell^{-}$ or by using the method
of moments. We found that in order to obtain $3\sigma$ evidence for NP in only
these observables and using the method of moments, roughly 1500 $B\to
K^{*}e^{+}e^{-}$ signal events are necessary in either experiment.

Our approach can be generalized for other decays to $K\pi\ell^{+}\ell^{-}$
final states. Here as well, a significant deviations from zero of the $D_{i}$
observables, relative in size to the branching ratio, would be a clear sign of
NP. However the theoretical and experimental knowledge of the $K\pi$ invariant
mass region outside the $K^{*0}(892)$ is not yet sufficient to provide solid
numerical predictions.

###### Acknowledgements.

This work is supported by the Swiss National Science Foundation under grant
PP00P2-144674. We thank Marcin Chrzaszcz, Akimasa Ishikawa, Patrick Owen,
Vincenzo Vagnoni, and Simon Wehle for careful reading of the manuscript and
valuable comments. D.v.D. is grateful to the Mainz Institute for Theoretical
Physics (MITP) for its hospitality and its partial support during the
completion of this work.

##  Appendix A Additional Formulae

The full expressions for the observables $D_{4}$ through $D_{6s}$ in the basis
of SM-like operators, assuming real valued Wilson coefficients and LFU-
breaking only in the coefficient $\mathcal{C}^{\text{}}_{9}$, read:

$\displaystyle\frac{-D_{4}(q^{2})}{\sqrt{2}\,N^{2}\,\tau_{B}}$
$\displaystyle=\left[\Delta_{9^{2}}^{(3)}\right]F_{V,\parallel}F_{V,0}-\operatorname{Re}\left[\Delta_{9}^{(3)}(h_{9,\parallel}+h_{9,0})\right]F_{V,\parallel}F_{V,0}$
(17)
$\displaystyle-\frac{2m_{b}}{M_{B}}\operatorname{Re}\left[\Delta_{9}^{(3)}(\mathcal{C}^{\text{}}_{7}+h_{7,0})^{*}\right]F_{T,0}F_{V,\parallel}-\frac{2m_{b}M_{B}}{q^{2}}\operatorname{Re}\left[\Delta_{9}^{(3)}(\mathcal{C}^{\text{}}_{7}+h_{7,\parallel})^{*}\right]F_{T,\parallel}F_{V,0}$
$\displaystyle+\Delta_{\beta}^{(3)}\frac{2m_{b}^{2}}{s}\operatorname{Re}\left[(\mathcal{C}^{\text{}}_{7}+h_{7,0})(\mathcal{C}^{\text{}}_{7}+h_{7,\perp})^{*}\right]F_{T,0}F_{T,\parallel}$
$\displaystyle+\Delta_{\beta}^{(3)}\left\\{\left|\mathcal{C}^{\text{}}_{10}\right|^{2}+\operatorname{Re}\left[h_{9,0}h^{*}_{9,\parallel}\right]\right\\}F_{V,0}F_{V,\parallel}$
$\displaystyle+\Delta_{\beta}^{(3)}\frac{2m_{b}M_{B}}{q^{2}}\operatorname{Re}\left[(\mathcal{C}^{\text{}}_{7}+h_{7,\parallel})h^{*}_{9,0}\right]F_{T,\parallel}F_{V,0}+3\Delta_{\beta}^{(3)}\frac{2m_{b}}{M_{B}}\operatorname{Re}\left[(\mathcal{C}^{\text{}}_{7}+h_{7,0})h^{*}_{9,\parallel}\right]F_{T,0}F_{V,\parallel}\,,$

and

$\displaystyle\frac{D_{5}(q^{2})}{2\,\sqrt{2}\,N^{2}\,\tau_{B}}$
$\displaystyle=2\operatorname{Re}\left[\mathcal{C}^{\text{}}_{10}(\Delta_{9}^{(2)})^{*}\right]F_{V,\perp}F_{V,0}-\Delta_{\beta}^{(2)}\operatorname{Re}\left[\mathcal{C}^{\text{}}_{10}(h_{9,0}+h_{9,\perp})^{*}\right]F_{V,\perp}F_{V,0}$
(18)
$\displaystyle-\Delta_{\beta}^{(2)}\frac{2m_{b}}{M_{B}}\operatorname{Re}\left[\mathcal{C}^{\text{}}_{10}(\mathcal{C}^{\text{}}_{7}+h_{7,0})^{*}\right]F_{T,0}F_{V,\perp}-\Delta_{\beta}^{(2)}\frac{2m_{b}M_{B}}{q^{2}}\operatorname{Re}\left[\mathcal{C}^{\text{}}_{10}(\mathcal{C}^{\text{}}_{7}+h_{7,\perp})^{*}\right]F_{T,\perp}F_{V,0}\,,$

and

$\displaystyle\frac{D_{6s}(q^{2})}{4\,N^{2}\,\tau_{B}}$
$\displaystyle=2\operatorname{Re}\left[\mathcal{C}^{\text{}}_{10}(\Delta_{9}^{(2)})^{*}\right]F_{V,\parallel}F_{V,\perp}-\Delta_{\beta}^{(2)}\operatorname{Re}\left[\mathcal{C}^{\text{}}_{10}(\Delta_{9,\parallel}+\Delta_{9,\perp})^{*}\right]F_{V,\parallel}F_{V,\perp}$
(19)
$\displaystyle-\Delta_{\beta}^{(2)}\frac{2m_{b}M_{B}}{q^{2}}\operatorname{Re}\left[\mathcal{C}^{\text{}}_{10}(\mathcal{C}^{\text{}}_{7}+\Delta_{7,\perp})^{*}\right]F_{T,\perp}F_{V,\parallel}-\Delta_{\beta}^{(2)}\frac{2m_{b}M_{B}}{q^{2}}\operatorname{Re}\left[\mathcal{C}^{\text{}}_{10}(\mathcal{C}^{\text{}}_{7}+\Delta_{7,\parallel})^{*}\right]F_{T,\parallel}F_{V,\perp}\,.$

The form factors $F_{V,\lambda}$ and $F_{T,\lambda}$ for polarizations
$\lambda=0,\perp,\parallel$ are introduced _ad hoc_ in eq. (5). The form
factors for the vector and axialvector currents are expressed as

$\displaystyle F_{V,\perp}$
$\displaystyle=\sqrt{2}\frac{\sqrt{\lambda}}{M_{B}+M_{K^{*}}}V\,,$ (20)
$\displaystyle F_{V,\parallel}$
$\displaystyle=\sqrt{2}(M_{B}+M_{K^{*}})A_{1}\,,$ (21) $\displaystyle F_{V,0}$
$\displaystyle=\frac{(M_{B}^{2}-M_{K^{*}}^{2}-q^{2})(M_{B}+M_{K^{*}})A_{1}}{2M_{K^{*}}\sqrt{q^{2}}}$
(22) $\displaystyle\quad-\frac{\lambda
A_{2}}{2M_{K^{*}}(M_{B}+M_{K^{*}})\sqrt{q^{2}}}\,.$

The form factors for the tensor current are expressed as

$\displaystyle F_{T,\perp}$
$\displaystyle=\sqrt{2}\frac{\sqrt{\lambda}}{M_{B}}T_{1}\,,$ (23)
$\displaystyle F_{T,\parallel}$
$\displaystyle=\sqrt{2}\frac{M_{B}^{2}-M_{K^{*}}^{2}}{M_{B}}A_{1}\,,$ (24)
$\displaystyle F_{T,0}$
$\displaystyle=\frac{M_{B}(M_{B}^{2}+3M_{K^{*}}^{2}-q^{2})T_{1}}{2M_{K^{*}}\sqrt{q^{2}}}$
(25) $\displaystyle\quad-\frac{\lambda
T_{1}}{2M_{K^{*}}(M_{B}^{2}-M_{K^{*}}^{2})\sqrt{q^{2}}}\,.$

Here $V$, $A_{1,2}$ and $T_{1,2,3}$ are the form factors in the common
parametrization (see e.g. Bobeth  _et al._ (2013) for their definitions).

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