# Analyzing leptonic decays of heavy quarkonia in covariant confined quark
model

Stanislav Dubnička1,
Anna Zuzana Dubničková2,
Mikhail A. Ivanov3,
Andrej Liptaj1‡
Akmaral Tyulemissova3,4
Zhomart Tyulemissov3,4,5

###### Abstract

The leptonic decay widths of heavy quarkonia are complementary observables to
the semileptonic decays of heavy mesons and provide additional insight into
the possible lepton flavor universality breaking. We analyze the former in the
framework of the covariant confined quark model, introducing a novel approach
for the description of the radially excited quarkonia states, which is based
on constituent quark mass running and the orthogonality of the states. We are
able to describe the experimental observations by our approach within the
Standard model, which does not support new physics explanations.

1 Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia.

2 Faculty of Mathematics, Physics and Informatics,

Comenius University, Bratislava, Slovakia.

3 Bogoliubov Laboratory of Theoretical Physics,

Joint Institute for Nuclear Research, Dubna, Russia.

4Institute of Nuclear Physics, Ministry of Energy

of the Republic of Kazakhstan, Almaty, Kazakhstan.

5Al-Farabi Kazakh National University

Almaty, Kazakhstan

<EMAIL_ADDRESS>
PACS: 13.20.Gd, 12.39.Ki

Keywords: heavy quarkonia, leptonic decays, covariant confined quark model

##  1 Introduction

The heavy quarkonia $J/\psi,\Psi(2s),\Upsilon(\text{n}s)_{\text{n}=1,2,3}$ are
among the most simple realizations of quantum chromodynamics (QCD) and
naturally play a role in various physics areas (quark masses, determination of
$\alpha_{s}$, confinement, quark-gluon plasma and QCD in general [1, 2, 3, 4,
5, 6, 7]). Here we study their leptonic decays which have the advantage of
rather clean experimental signatures and can be compared to theoretical
predictions at the level which has a discriminating power. This becomes useful
when the lepton flavor universality question is addressed, where interesting
developments are ongoing. Indeed, several experiments have measured these

$R_{D^{(*)}}=\frac{\Gamma(B\to D^{(*)}\tau\nu_{\tau})}{\Gamma(B\to
D^{(*)}\ell\nu_{\ell})},\;R_{K^{(*)}}=\frac{\Gamma(B\to
K^{(*)}\tau^{+}\tau^{-})}{\Gamma(B\to K^{(*)}\ell^{+}\ell^{-})},\;\ell=e,\mu,$

and other semileptonic ratios and have seen deviations from the Standard Model
(SM), e.g. [8, 9]. Nevertheless, with time passing, new measurement arrived
where these deviations seem to progressively shrink [10, 11, 12, 13, 14] (for
[14] see addendum). To shed an additional light on this topic the leptonic
decays of heavy quarkonia come handy. As it was under quite general
assumptions analyzed in [15], a new physics (NP) contribution to the $b\to
c\tau\nu$ transition ought to influence also the $c\bar{c}\to\tau^{+}\tau^{-}$
and $b\bar{b}\to\tau^{+}\tau^{-}$ processes from where the motivation to study
the latter.

The analysis of the leptonic decays of the $J/\psi$, $\Psi(2s)$ and
$Y\left(\text{n}s\right)$ states has been addressed in numerous works. Already
decades ago, the quark model description of various hadronic decays was given
in [16], where the often cited Van Royen-Weisskopf formula was presented: it
predicts the electromagnetic decay width
$\Gamma_{\Upsilon\to\ell\ell}^{\text{EM}}$ be proportional to
$\left(1+2x_{\ell}^{2}\right)\left(1+4x_{\ell}^{2}\right)^{1/2}$,
$x_{\ell}=m_{\ell}/m_{\Upsilon}$. Significantly later, a highly cited paper
[17] investigates quarkonium Schrödinger wave function at origin for several
common potentials, which provides essential inputs for evaluating various
quarkonia observables. The reference [18] already deals with the leptonic
universality breaking and proposes a light Higghs-like CP-odd particle $A^{0}$
with a mass similar to $\Upsilon$ for the breaking to happen. Coming back to
potential models, one can mention [19], which uses a semirelativistic
Hamiltonian, linear confinement and pQCD one-loop short distant potential to
arrive to leptonic decay widths which are, for the most part, in a good
agreement with data. Similarly, phenomenological potentials are studied in the
non-relativistic QCD approach to describe heavy quarkonia in [20], and again,
leptonic decay widths are well described. Higher excited heavy quarkonia
states are, in an analogical way, analyzed in [21]. A nice review of
charmonium states mentioning several theoretical approaches is given in [22].
The author analyses various properties of these states from the perspective of
the underlying theory of QCD, the first QCD correction to leptonic decay of
the charmonium in a non-relativistic picture is given in the Eq. (23) of the
text. An excellent investigation of possible NP contributions to $R_{D^{(*)}}$
using leptonic decays of quarkonia is, within the effective field theory (EFT)
approach, presented in [23]. Here the authors assume that the NP enters
through gauge-invariant effective operators of dimension six and that only the
$\tau$ sector is affected. They present a list of all relevant four-fermion
operators, analyze the decays (semileptonic and leptonic) in various NP
scenarios and show which of them are favored and which are not. A more recent
publication [24] is based on the Bethe-Salpeter method. The authors use a
relativistic wave function combined with a simple kernel to asses relativistic
effects in the heavy quarkonia decays, which are, as they claim, large. Many
authors address short-distant corrections to the leptonic decay widths of
quarkonia which can be computed at various precision order. This topic is
covered in works [25, 26, 27] and references therein. An important branch of
the theoretical investigation is represented by the QCD on the lattice. The
work [28] uses non-relativistic QCD at NLO to describe $\Upsilon$ spectrum,
the electronic decay widths are presented too. A relativistic action is used
in [29], where $J/\Psi$ is investigated and its various parameters, including
the width to two electrons, are calculated, and also in [30], where the
authors address the ground state $\Upsilon(1s)$ and compute also
$\Gamma_{\Upsilon\to e^{+}e^{-}}$.

In this text we are interested in the leptonic decay widths of heavy quarkonia
in the framework of the covariant confined quark model (CCQM), which is an EFT
approach based on quark-hadron interaction Lagrangian. The ground states
$J/\Psi$ and $\Upsilon(1s)$ were already addressed in [31]. Here, however, our
research is extended in two ways: we include into the consideration also
higher excited states and we apply a novel approach to describe these states.
Also, the here presented analysis is based on updated values of the model
parameters and thus numbers for ground states are modified too.

The text is organized as follows: we start by presenting very briefly the
CCQM, including references to our previous texts. Next we address the
transition amplitudes $V\to\gamma$, their expression within the CCQM and their
gauge invariance. By comparing the Lorentz structures we identify and compute
the appropriate form factors and predict decay widths of ground states. Then
we present the novel approach to describe radial excitations: we implement
quark mass running and introduce the orthogonality requirement for the excited
states. With this in hand, we compute leptonic decay widths for higher vector
states.

##  2 CCQM and $V\to\gamma\to\ell\ell$ amplitude

###  2.1 Model

The CCQM has been presented in great details in several works, see e.g. [31,
32]. To keep the text short and focus on the novel ideas, we give only a brief
characteristic of the model and provide the reader with references.

  * •

The CCQM is based on an invariant quark-hadron non-local interaction
Lagrangian written in the case of mesons as

$\displaystyle\mathcal{L}_{\text{int}}$
$\displaystyle=g_{M}M(x)J_{M}(x)+\text{H.c.},$ $\displaystyle J_{M}(x)$
$\displaystyle=\int dx_{1}\int
dx_{2}\>F_{M}(x;x_{1},x_{2})\bar{q}_{2}(x_{2})\Gamma_{M}q_{1}(x_{1}),$

where $g_{M}$ is the coupling between the mesonic field $M$ and the quark
current $J_{M}$, $F_{M}$ is a vertex function (see hereunder) and $\Gamma_{M}$
the appropriate string of Dirac matrices which corresponds to the spin of the
meson. The vertex function has two components: a delta function which matches
the barycenter of quarks to the meson position and the remaining part, which
is chosen to have an exponential form in the momentum space mainly for
computational reasons

$\displaystyle F_{M}(x;x_{1},x_{2})$
$\displaystyle=\delta(x-\omega_{1}x_{1}-\omega_{2}x_{2})\Phi_{M}\left[\left(x_{1}-x_{2}\right)^{2}\right],$
$\displaystyle\omega_{1,2}$ $\displaystyle=m_{1,2}/\left(m_{1}+m_{2}\right),$
$\displaystyle\Phi_{M}\left[\left(x_{1}-x_{2}\right)^{2}\right]$
$\displaystyle=\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik(x_{1}-x_{2})}\tilde{\Phi}(-k^{2}),$
$\displaystyle\tilde{\Phi}(-k^{2})$ $\displaystyle=e^{k^{2}/\Lambda_{M}^{2}}.$

Here $\Lambda_{M}$ is a free parameter of the model, $m_{i}$ are quark masses
and the minus sign in the argument of $\tilde{\Phi}$ indicates that the
exponent becomes negative in the Euclidean region so that the expression has
an appropriate fall-off behavior without ultraviolet divergences. Besides
hadron-related parameters $\Lambda_{M}$ the model comprises as parameters
constituent quark masses. In situations where the decaying particle is very
heavy we implement an infrared confinement ([31, 33]) to avoid decays to
quarks, it depends on an additional parameter $\lambda_{\text{cutoff}}$. We do
not use this mechanism in the present work.

  * •

The CCQM includes both, hadronic and mesonic fields as elementary, one
therefore needs to address the question of the double counting. To avoid the
latter we use the so-called compositeness condition, which is expressed in
terms of the derivative of the meson mass operator $\Pi_{M}^{{}^{\prime}}$

$Z_{M}=1-g_{M}^{2}\Pi_{M}^{{}^{\prime}}(m_{M}^{2})=0,$ (1)

where $Z_{M}$ is the renormalization constant. The meaning is that the overlap
between the physical state and the corresponding bare state represented by
$Z_{M}$ is set to zero, i. e. the physical state does not contain the bare
state and can be therefore interpreted as bound, for details see [31, 34]. As
a result quarks exist only as virtual particles which are responsible for the
interaction, i.e. an initial state meson fluctuates into a quark pair prior to
interaction and is possibly re-created from quarks in the final state. The
equality in (1) is reached by tuning the value of $g_{M}$, which is in this
way determined. The model does not contain gluons, their action is effectively
taken into the account by the parameter-dependent vertex function.

  * •

The inclusion of the electromagnetic (EM) interaction into a theory with a
non-local Lagrangian is not trivial. For the free parts of the Lagrangian the
usual minimal substitution is applied, it defines
$\mathcal{L}_{\text{int}}^{\text{EM}_{1}}$. For the interaction part an
approach inspired by [35] is used, i.e. the quark fields are multiplied by a
gauge field exponential

$q_{i}(x)\to e^{-ie_{q_{i}}I(x_{i},x,P)}q_{i}(x),\quad
I(x_{i},x,P)=\int_{x}^{x_{i}}dz_{\mu}A^{\mu}(z),$ (2)

where $P$ is a path between $x_{i}$ and $x$. Since only derivatives of $I$
appear in the perturbative expansion, the latter is path-independent

$\frac{\partial}{\partial x^{\mu}}I(x,y,P)=A_{\mu}(x).$

The corresponding Lagrangian terms $\mathcal{L}_{\text{int}}^{\text{EM}_{2}}$
are then generated by the expansion of the gauge exponential (2) in orders of
$A^{\mu}$, see [31, 36]. In a diagrammatic representation the term
$\mathcal{L}_{\text{int}}^{\text{EM}_{1}}$ generates photon line attached to
the quark line, the $\mathcal{L}_{\text{int}}^{\text{EM}_{2}}$ part
corresponds to a photon attached to the non-local vertex.

With this defined, we evaluate the appropriate Feynman diagrams and get
observable predictions. We use the Schwinger representation of the quark
propagators and perform computations analytically with the exception of the
last step, which is the integration over the space of Schwinger parameters,
which we do numerically.

###  2.2 Amplitude

Figure 1: Diagrams describing the $V\to\gamma$ transition.

The relevant quark model diagrams of the $V\to\gamma$ transition are shown in
Fig. 1. The corresponding Feynman one-loop integrals read

$\displaystyle M^{\mu\nu}(p)$
$\displaystyle=N_{c}g_{V}\\!\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\Big{\\{}\Phi_{V}(-k^{2})\,\text{tr}\Big{[}\gamma^{\mu}S(k+\tfrac{1}{2}\,p)\gamma^{\nu}S(k-\tfrac{1}{2}\,p)\Big{]}$
$\displaystyle\quad-\,\int\limits_{0}^{1}d\tau\Phi^{\prime}_{V}(-z_{\tau})(2\,k+\tfrac{1}{2}\,p)^{\mu}\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}\Big{\\}},$
(3) $\displaystyle z_{\tau}$
$\displaystyle=\tau(k+\tfrac{1}{2}\,p)^{2}+(1-\tau)k^{2}\,.$

Here $N_{c}=3$ is the number of colors, $g_{V}$ is the coupling of the vector
meson with its constituent quarks, $S$ denotes the quark propagator and
$\Phi^{\prime}_{V}$ is the derivative of the vertex function with respect to
its argument. In this paper we consider the charmonium ($c\bar{c}$-state) and
bottomonium ($b\bar{b}$-state). The constituent quark propagator is written as

$S(k)=\frac{1}{m_{q}-\not\\!k},$

where $m_{q}$ is the constituent quark mass. We use the Lorentz index $\mu$ in
relation to the photon field whereas $\nu$ corresponds to the vector meson
field. With the simple Gaussian form for the vertex function one has

$\Phi_{V}(-k^{2})=\exp(s\,k^{2}),\quad\Phi^{\prime}_{V}(-k^{2})=-s\exp(s\,k^{2}),\quad\text{where}\quad
s=1/\Lambda_{V}^{2}\,.$ (4)

Here the parameter $\Lambda_{V}$ characterizes the size of the vector meson.
One is allowed to use different choices for $\Phi_{V}$ as long as it falls off
sufficiently fast in the ultraviolet region of Euclidean space to render the
Feynman diagrams ultraviolet finite [37].

###  2.3 Gauge invariance

As the first step in our study, let us prove that the amplitude $M^{\mu\nu}$
given by Eq. (3) is gauge invariant, i.e. $M^{\mu\nu}p_{\mu}=0$. The loop (or
bubble) integral contracted with $p_{\mu}$ can be written as

$\displaystyle\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\Phi_{V}\left(-k^{2}\right)\,\text{tr}\Big{[}\\!\not\\!p\,S(k+\tfrac{1}{2}\,p)\gamma^{\nu}S(k-\tfrac{1}{2}\,p)\Big{]}$
$\displaystyle\quad=\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\Phi_{V}\left(-k^{2}\right)\,\Big{\\{}\text{tr}\Big{[}S(k+\tfrac{1}{2}\,p)\gamma^{\nu}\Big{]}-\text{tr}\Big{[}S(k-\tfrac{1}{2}\,p)\gamma^{\nu}\Big{]}\Big{\\}}\,.$
(5)

The integrand of the tadpole integral contracted with $p_{\mu}$ may be
transformed in the following way

$\displaystyle-\int\limits_{0}^{1}d\tau\Phi^{\prime}_{V}\Big{(}-\tau(k+\tfrac{1}{2}\,p)^{2}-(1-\tau)k^{2}\Big{)}\left(2\,kp+\tfrac{1}{2}\,p^{2}\right)\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}\Big{\\}}$
$\displaystyle\quad=-\frac{\Phi_{V}\Big{(}-(k+\tfrac{1}{2}\,p)^{2}\Big{)}-\Phi_{V}\Big{(}-k^{2}\Big{)}}{k^{2}-(k+\tfrac{1}{2}p)^{2}}\,2\,\left(kp+\tfrac{1}{4}\,p^{2}\right)\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}$
$\displaystyle\quad=2\,\Big{[}\Phi_{V}\Big{(}-(k+\tfrac{1}{2}\,p)^{2}\Big{)}-\Phi_{V}\Big{(}-k^{2}\Big{)}\Big{]}\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}\,.$

Finally, we use the identity

$\int d^{4}kf(k)\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}=\frac{1}{2}\int
d^{4}k\left(f(k)-f(-k)\right)\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]},$

which is valid for any well behaved function $f(k)$. We come to the final
expression for the tadpole integral contracted with the external momentum $p$.
One has

$\displaystyle-\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\int\limits_{0}^{1}d\tau\Phi^{\prime}_{V}\Big{(}-\tau(k+\tfrac{1}{2}\,p)^{2}-(1-\tau)k^{2}\Big{)}\left(2\,kp+\tfrac{1}{2}\,p^{2}\right)\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}$
$\displaystyle\quad=\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\Big{[}\Phi_{V}\Big{(}-(k+\tfrac{1}{2}\,p)^{2}\Big{)}-\Phi_{V}\Big{(}-(k-\tfrac{1}{2}\,p)^{2}\Big{)}\Big{]}\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}$
$\displaystyle\quad=-\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\Phi_{V}\left(-k^{2}\right)\,\Big{\\{}\text{tr}\Big{[}S(k+\tfrac{1}{2}\,p)\gamma^{\nu}\Big{]}-\text{tr}\Big{[}S(k-\tfrac{1}{2}\,p)\gamma^{\nu}\Big{]}\Big{\\}}\,.$
(6)

It is readily seen that the expression given by Eq. (5) is exactly canceled
out by the one given in Eq. (6). Thus, we have $M^{\mu\nu}p_{\mu}=0$. It is
interesting to see what is going on in the case of a null momentum $p=0$. If
$p=0$ then the integral corresponding to the tadpole diagram may be
transferred to the one corresponding to the loop diagram by using the
integration by parts. One has

$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}i}\,\Big{\\{}\Phi_{V}\left(-k^{2}\right)\text{tr}\Big{[}\gamma^{\mu}S(k)\gamma^{\nu}S(k)\Big{]}-2k^{\mu}\,\Phi^{\prime}_{V}\left(-k^{2}\right)\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}\Big{\\}}$
$\displaystyle\quad=\int\frac{d^{4}k}{(2\pi)^{4}i}\,\frac{\partial}{\partial
k^{\mu}}\Big{\\{}\Phi_{V}\left(-k^{2}\right)\text{tr}\Big{[}\gamma^{\nu}S(k)\Big{]}\Big{\\}}=0.$
(7)

In the general case the invariant amplitude of the $V\to\gamma$ transition is
written as

$M^{\mu\nu}(p)=G_{V}(p^{2})\,g^{\mu\nu}+F_{V}(p^{2})\,p^{\mu}p^{\nu},$

where the form factor $G_{V}$ has a dimension of mass squared whereas the form
factor $F_{V}$ is dimensionless. The gauge invariance gives the relation
between the two form factors

$G_{V}(p^{2})=-p^{2}F_{V}(p^{2})$ (8)

valid for any $p^{2}$ value. From Eq. (7) follows that both form factors do
not have singularity at $p^{2}=0$. In the case of decay
$V\to\ell^{+}\ell^{-}$, only the first form factor $G_{V}$ gives a
contribution to the decay amplitude due to the conservation of the vector
lepton current.

###  2.4 Results

By using the standard technique of integration over the loop momentum _k_ one
can finalize the two-fold integrals in the expressions for $V\to\gamma$ form
factors. One can obtain the following expressions for the integrals from the
loop diagram

$\displaystyle G_{V}^{{\rm\,loop}}(p^{2})=$
$\displaystyle\qquad\frac{3g_{V}}{(4\pi)^{2}}\\!\int\limits_{0}^{\infty}\\!\frac{dt\,t}{(s+t)^{2}}\int\limits_{0}^{1}\\!d\alpha\,e^{z_{\alpha}}\Big{\\{}4\,m_{q}^{2}+\frac{4}{s+t}+\Big{[}1-\frac{4\,t^{2}}{(s+t)^{2}}\left(\tfrac{1}{2}-\alpha\right)^{2}\Big{]}p^{2}\Big{\\}},$
$\displaystyle F_{V}^{{\rm\,loop}}(p^{2})=$
$\displaystyle\qquad\frac{3g_{V}}{(4\pi)^{2}}\\!\int\limits_{0}^{\infty}\\!\frac{dt\,t}{(s+t)^{2}}\int\limits_{0}^{1}\\!d\alpha\,e^{z_{\alpha}}\Big{\\{}-2+\frac{8\,t^{2}}{(s+t)^{2}}\left(\tfrac{1}{2}-\alpha\right)^{2}\Big{\\}},$
$\displaystyle
z_{\alpha}=-t\left[m_{q}^{2}-\alpha(1-\alpha)\,p^{2}\right]+\frac{st}{s+t}\left(\tfrac{1}{2}-\alpha\right)^{2}p^{2}.$

In the case of the tadpole diagram, one has

$\displaystyle G_{V}^{{\rm\,tad}}(p^{2})$
$\displaystyle=\frac{3g_{V}}{(4\pi)^{2}}\\!\int\limits_{0}^{\infty}\\!\frac{dt\,s}{(s+t)^{2}}\int\limits_{0}^{1}\\!d\tau\,e^{z_{\tau}}\Big{[}-\frac{4}{s+t}\Big{]},$
$\displaystyle F_{V}^{{\rm\,tad}}(p^{2})$
$\displaystyle=\frac{3g_{V}}{(4\pi)^{2}}\\!\int\limits_{0}^{\infty}\\!\frac{dt\,s}{(s+t)^{2}}\int\limits_{0}^{1}\\!d\tau\,e^{z_{\tau}}\Big{[}-\Big{(}1-\frac{2s\tau}{s+t}\Big{)}\frac{s\tau}{s+t}\Big{]},$
$\displaystyle z_{\tau}$
$\displaystyle=-tm_{q}^{2}+s\tau\left(1-\frac{s\tau}{s+t}\right)\frac{p^{2}}{4}.$

Then we define the full $V\to\gamma$ form factors as

$\displaystyle G_{V}(p^{2})$
$\displaystyle=G_{V}^{{\rm\,loop}}(p^{2})+G_{V}^{{\rm\,tad}}(p^{2})\,,$
$\displaystyle F_{V}(p^{2})$
$\displaystyle=F_{V}^{{\rm\,loop}}(p^{2})+F_{V}^{{\rm\,tad}}(p^{2})\,.$

Their numerical values are calculated by using the FORTRAN codes which require
the NAG library.

The model parameters were determined by fitting calculated quantities of basic
processes to available experimental data or lattice simulations (for details,
see Ref. [31]). Here, we take the values of the constituent quark masses
$m_{q}$ from an updated fit made in [38]. This fit improves the description of
new data on the exclusive $B$-meson and heavy baryon decays. Note that in the
fit, the infrared cutoff parameter $\lambda$ has been kept fixed. The updated
numerical values of the constituent quark masses and the cutoff parameter
$\lambda$ are given in Table 1.

$m_{u}$ | $m_{s}$ | $m_{c}$ | $m_{b}$ | $\lambda$ |
---|---|---|---|---|---
$0.241$ | $0.428$ | $1.67$ | $5.04$ | $0.181$ | GeV
Table 1: Model parameters: quark masses and cutoff parameter $\lambda$ (all in
GeV).

Size parameters of charmonium and bottomonium are

$\Lambda_{J/\psi}=2.795(1)~{}{\rm GeV},\qquad\Lambda_{\Upsilon}=4.03(4)~{}{\rm
GeV}.$

The masses of ground and radially excited states of charmonium and bottomonium
are given in Table 2.

$J/\Psi$ | $\Psi(2s)$ |  | $\Upsilon(1s)$ | $\Upsilon(2s)$ | $\Upsilon(3s)$
---|---|---|---|---|---
$3096.900(6)$ | $3686.10(6)$ |  | $9460.30(26)$ | $10023.26(31)$ | $10355.2(5)$
Table 2: Masses of charmonia and bottomonia from PDG [39] (in MeV).

First, we have checked numerically that the equation (8) is satisfied with
high accuracy (6 digits calculated with relative error $10^{-5}$). The term
proportional to $p^{\mu}$ does not contribute to the amplitude of the
$V\to\gamma\to\ell^{-}\ell^{+}$ decay due to the conservation of the lepton
current. Therefore we will use the form factor
$G_{V}(p^{2})=-p^{2}F_{V}(p^{2})$ on the mass shell of the vector meson
$p^{2}=m_{V}^{2}$ for the calculation of the decay width. One has

$\Gamma(V\to\ell^{+}\ell^{-})=\frac{4\pi\alpha^{2}}{3}\,m_{V}\Big{(}\frac{Q_{V}f_{V}}{m_{V}}\Big{)}^{2}\sqrt{1-4x_{\ell}^{2}}\,(1+2x_{\ell}^{2}),$

where $f_{V}\equiv G_{V}(m_{V}^{2})/m_{V}$, $x_{\ell}=m_{\ell}/m_{V}$ and the
quark charge factor $Q_{V}=(e_{u}-e_{d})/\sqrt{2}=1/\sqrt{2}$ for the
$\rho^{0}$ meson, $Q_{V}=e_{c}=2/3$ for $\psi$ mesons, and $Q_{V}=e_{b}=-1/3$
for $\Upsilon$ mesons. Note that in our notation the leptonic constant $f_{V}$
coincides with the weak leptonic constant and has the dimension of mass. The
results as predicted by the CCQM are summarized in Table 3. The theoretical
errors correspond to the propagated errors of hadronic parameters
$\Lambda_{V}$, which, on their turn, get the error from the fit of the CCQM to
decay width experimental data.

Quantity | $J/\psi$ | $\Upsilon(1s)$
---|---|---
$f_{V}$ (MeV) | 415.4 | 715
| CCQM | Expt. | CCQM | Expt.
$B(V\to\tau^{+}\tau^{-})$ (%) |  |  | $2.46(7)$ | $2.60(10)$
$B(V\to\mu^{+}\mu^{-})$ (%) | $5.964(40)$ | $5.961(33)$ | $2.48(7)$ | $2.48(5)$
$B(V\to e^{+}e^{-})$ (%) | $5.964(40)$ | $5.971(32)$ | $2.48(7)$ | $2.38(11)$
Table 3: Leptonic branching fractions of $J/\psi$ and $\Upsilon(1s)$ compared
to experimental numbers.

##  3 New approach to radial excitations

###  3.1 First step: running quark mass in a loop

We assume that the binding energy of the radial excitations of a quarkonium
$q\bar{q}$ is equal to the binding energy $E$ of the ground state $V_{0}$:
$E=m_{V_{0}}-2m_{q_{0}}$. This means that the value of the constituent quark
mass is for an excited state $V^{\prime}$ equal to

$m_{q^{\prime}}=(m_{V^{\prime}}-E)/2.$ (9)

The ground state quark masses are taken from Table 1. We find the values of
the size parameters $\Lambda_{V}$ by fitting the model to the experimental
data on the leptonic decays $V\to\ell^{+}\ell^{-}$. The size parameters and
branching fractions predicted under this assumption are shown in Tabs. 4 and
5.

| $J/\psi$ | $\Psi(2s)$
---|---|---
$\Lambda_{V}$(GeV) | 2.795 | 1.68
| CCQM | Expt. | CCQM | Expt.
$\mathcal{B}(V\to\tau^{+}\tau^{-})$(%) | - |  | 0.31(1) | 0.31(4)
$\mathcal{B}(V\to\mu^{+}\mu^{-})$(%) | 5.964(40) | 5.961(33) | 0.80(2) | 0.80(6)
$\mathcal{B}(V\to e^{+}e^{-})$(%) | 5.964(40) | 5.971(32) | 0.80(2) | 0.79(2)
Table 4: Charmonia: size parameters and leptonic branching fractions with running constituent quark masses. | $\Upsilon(1s)$ | $\Upsilon(2s)$ | $\Upsilon(3s)$
---|---|---|---
$\Lambda_{V}$(GeV) | 4.03 | 2.80 | 2.485
| CCQM | Expt. | CCQM | Expt. | CCQM | Expt.
$\mathcal{B}(V\to\tau^{+}\tau^{-})$(%) | 2.46(7) | 2.60(10) | 1.90(0.5) | 2.00(21) | 2.17(3) | 2.29(30)
$\mathcal{B}(V\to\mu^{+}\mu^{-})$(%) | 2.48(7) | 2.48(5) | 1.92(0.5) | 1.93(17) | 2.18(3) | 2.18(21)
$\mathcal{B}(V\to e^{+}e^{-})$(%) | 2.48(7) | 2.38(11) | 1.92(0.5) | 1.91(16) | 2.18(3) | 2.18(20)
Table 5: Bottomonia: size parameters and leptonic branching fractions with
running constituent quark masses.

###  3.2 Second step: orthogonality of the radial excitations

We assume that different radial excitations cannot pass into each other
through a quark loop. We call this assumption as the orthogonality condition.
To realize this condition we choose the vertex functions in the following form

$\Phi_{n}(-k^{2})=\Big{(}1+\sum\limits_{m=1}^{n}c_{m+n-1}s_{n}^{m}k^{2m}\Big{)}\Phi_{V}(-k^{2}),$
(10)

where the function $\Phi_{V}(-k^{2})$ is given by Eq. (4). Then the
orthogonality condition implies that

$\left(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\\!\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\Phi_{n}(-k^{2})\,\Phi_{m}(-k^{2})\text{tr}\Big{[}\gamma^{\mu}S(k+\tfrac{1}{2}\,p)\gamma^{\nu}S(k-\tfrac{1}{2}\,p)\Big{]}=0.$
(11)

This condition will allow us to determine the numerical values of the
coefficients $c_{m+n-1}$ in the allowed region of $p^{2}$ for certain
quarkonium spectrum. Here, we consider two radial excitations of the
charmonium, $J/\psi$ and $\Psi(2S)$, and three for the bottomonium:
$\Upsilon(1S)$, $\Upsilon(2S)$ and $\Upsilon(3S)$. We find the values of
coefficients $c_{i}$ which solve (11) for charmonia and bottomonia separately,
the solution depends on the quarkonium mass. This dependence is shown in Fig.
2.

Figure 2: The dependence of coefficients of the orthogonal vertex functions on
the quarkonium mass for charmonia and bottomonia.

###  3.3 Results and discussion

| $J/\psi$ | $\Psi(2s)$
---|---|---
$\Lambda_{V}$(GeV) | 2.795 | 0.463
$c_{i}$ |  |  | $c_{1}=0.802$
| CCQM | Expt. | CCQM | Expt.
$\mathcal{B}(V\to\tau^{+}\tau^{-})$(%) | - |  | 0.31(1) | 0.31(4)
$\mathcal{B}(V\to\mu^{+}\mu^{-})$(%) | 5.964(40) | 5.961(33) | 0.81(2) | 0.80(6)
$\mathcal{B}(V\to e^{+}e^{-})$(%) | 5.964(40) | 5.971(32) | 0.81(2) | 0.79(2)
Table 6: Charmonia: size parameters and leptonic branching fractions in the new approach with running constituent quark masses and orthogonal vertex functions. | $\Upsilon(1s)$ | $\Upsilon(2s)$ | $\Upsilon(3s)$
---|---|---|---
$\Lambda_{V}$(GeV) | 4.03 | 3.77 | 3.01
$c_{i}$ |  | $c_{1}=1.979,\;c_{2}=4.083,\;c_{3}=2.235$
| CCQM | Expt. | CCQM | Expt. | CCQM | Expt.
$\mathcal{B}(V\to\tau^{+}\tau^{-})$(%) | 2.46(7) | 2.60(10) | 1.92(0.5) | 2.00(21) | 2.17(3) | 2.29(30)
$\mathcal{B}(V\to\mu^{+}\mu^{-})$(%) | 2.48(7) | 2.48(5) | 1.93(0.5) | 1.93(17) | 2.18(3) | 2.18(21)
$\mathcal{B}(V\to e^{+}e^{-})$(%) | 2.48(7) | 2.38(11) | 1.93(0.5) | 1.91(16) | 2.18(3) | 2.18(20)
Table 7: Bottomonia: size parameters and leptonic branching fractions in the
new approach with running constituent quark masses and orthogonal vertex
functions.

With quark masses defined by (9) and modified vertex functions (10), where the
coefficients are determined by (11), we settle, in an optimization procedure,
the value of size parameters $\Lambda_{V}$ so as to get the best description
of leptonic decay widths. We then get our results summarized in Tabs. 6 and 7.
The difference between the theory and the experimental values is, in terms of
standard deviations, shown in Table 8.

|
$\left|\mathcal{B}_{\text{expt}}-\mathcal{B}_{\text{CCQM}}\right|/\sigma\left(\mathcal{B}_{\text{expt}}-\mathcal{B}_{\text{CCQM}}\right)$
---|---
| $J/\psi$ | $\Psi(2s)$ |  | $\Upsilon(1s)$ | $\Upsilon(2s)$ | $\Upsilon(3s)$
$\tau^{+}\tau^{-}$ |  |  $\ll$1 |  | 1.15 | 0.38 | 0.40
$\mu^{+}\mu^{-}$ | 0.06 | 0.16 |  |  $\ll$1 |  $\ll$1 |  $\ll$1
$e^{+}e^{-}$ | 0.12 | 0.71 |  | 0.77 | 0.12 |  $\ll$1
Table 8: The difference between CCQM values and the experiment expressed in
terms of standard deviations.

As readily seen, only one difference slightly exceeds $1\sigma$ deviation,
others are within the $1\sigma$ interval. In this sense we are able to
describe the experimental measurements within our CCQM SM approach. Taking
into the account the relation with the semileptonic decays mentioned in the
Introduction, our results do not give support for new physics phenomena being
involved in the latter.

On the other hand, the results presented in Tabs. 6 and 7 were reached as fits
where parameters were tuned and not as predictions. This is natural, since we
are presenting here a novel approach where the orthogonal vertex functions
were presented for the first time with their parameters not settled until now.
It is certainly desirable to analyze other processes in future, where heavy
quarkonia or possible other particles with their excited states play a role.
This will ensure that the number of parameters is significantly smaller than
the number of observables, i.e. the model will become more constrained with
more predictive power. Additionally, as part of our outlook, the CCQM can be
extended to include new physics EFT operators, similarly to what has been
presented in [23]. Then the corresponding Wilson coefficients can be analyzed
for semileptonic and leptonic decays and conclusions about their significance
drawn from the comparison to data.

## Acknowledgment

The research has been funded by the Science Committee of the Ministry of
Science and Higher Education of the Republic of Kazakhstan (Grant No.
AP19678771). Zh.T.’s work is supported by the JINR grant of young scientists
and specialists No. 24-301-06. The authors S.D., A.Z.D and A.L. acknowledge
the support of VEGA grant No. 2/0105/21.

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# Surjectivity of linear operators and semialgebraic global diffeomorphisms

Francisco Braun∗, Luis Renato Gonçalves Dias† and Jean Venato-Santos‡ ∗
Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São
Carlos, São Paulo, Brazil<EMAIL_ADDRESS>†,‡ Faculdade de
Matemática, Universidade Federal de Uberlândia, 38408-100 Uberlândia, Minas
Gerais, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove that a $C^{\infty}$ semialgebraic local diffeomorphism of
$\mathbb{R}^{n}$ with non-properness set having codimension greater than or
equal to $2$ is a global diffeomorphism if $n-1$ suitable linear partial
differential operators are surjective. Then we state a new analytic conjecture
for a polynomial local diffeomorphism of $\mathbb{R}^{n}$. Our conjecture
implies a very known conjecture of Z. Jelonek. We further relate the
surjectivity of these operators with the fibration concept and state a general
global injectivity theorem for semialgebraic mappings which turns out to unify
and generalize previous results of the literature.

###### Key words and phrases:

Global injectivity, Linear operators, Semialgebraic mappings, Local fibrations

###### 2010 Mathematics Subject Classification:

Primary: 14R15; Secondary: 47A99, 14P10, 14D06

∗ Partial support provided by the grants 2019/07316-0 and 2020/14498-4 of the
São Paulo Research Foundation (FAPESP)

† Partial support provided by the Grant 304163/2017-1 of the National Council
for Scientific and Technological Development – CNPq

‡ Support provided by the Grant APQ-02056-21 of Fapemig-Brazil

##  1\. Introduction

Let $F=(f_{1},\ldots,f_{n}):\mathbb{K}^{n}\to\mathbb{K}^{n}$,
$\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, be a differentiable mapping. We
denote by $J(F)$ the Jacobian determinant of $F$. We say that _$F$ is proper
at $y\in\mathbb{K}^{n}$_ if there exists a neighborhood $V$ of $y$ such that
$F^{-1}(\overline{V})$ is compact. We denote by $S_{F}$ the set of points at
which $F$ is not proper. This is the _non-properness set_ of $F$, also known
as _Jelonek’s set_ of $F$.

It is known that $S_{F}$ is a semialgebraic set when $F$ is a semialgebraic
mapping, see for instance [16, Theorem 6.4]. Thus, in this case, it makes
sense to consider the dimension and the codimension of $S_{F}$.

The following conjecture was raised by Jelonek in [16]:

######  Conjecture 1.1 (Jelonek’s conjecture).

Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a polynomial mapping with nowhere
vanishing $J(F)$ and such that $\operatorname{{codim}}S_{F}\geq 2$. Then $F$
is a bijective mapping.

In his paper, Jelonek proved that Conjecture 1.1 is true under the additional
assumption $\operatorname{{codim}}(S_{F})\geq 3$, the proof working for the
semialgebraic case as well (although the conclusion is just the injectivity of
$F$), and also that it is true for $n=2$, see [3, Proposition 2.13] for a
semialgebraic version in $n=2$. Much more appealing is the fact, proved in the
same paper, that Conjecture 1.1 implies the famous Jacobian conjecture in
$\mathbb{C}^{n}$: “A polynomial mapping $F:\mathbb{C}^{n}\to\mathbb{C}^{n}$
with $J(F)=1$ is an automorphism.”

We remark that the Pinchuk [18] mappings $P:\mathbb{R}^{2}\to\mathbb{R}^{2}$
constructed in order to disprove the _real Jacobian conjecture_ , that a
polynomial mapping $\mathbb{R}^{n}\to\mathbb{R}^{n}$ with nowhere zero
Jacobian determinant is injective, satisfy
$\operatorname{{codim}}\left(S_{P}\right)=1$ (see yet a rational
parametrization of $S_{P}$ for a specific Pinchuk mapping in [8]), and so the
assumption on $\operatorname{{codim}}\left(S_{F}\right)\geq 2$ is necessary in
Conjecture 1.1, and it remains to investigate it in dimension $n\geq 3$.

On the other hand, for a fixed $C^{\infty}$ mapping
$F=(f_{1},\ldots,f_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$ (resp. an entire
analytic mapping $F=(f_{1},\ldots,f_{n}):\mathbb{C}^{n}\to\mathbb{C}^{n}$) and
for an $i\in\\{1,2,\ldots,n\\}$, let
$\Delta_{i}^{F}:C^{\infty}(\mathbb{R}^{n})\to C^{\infty}(\mathbb{R}^{n})$
(resp. $\Delta_{i}^{F}:E_{n}\to E_{n}$) be the operator defined by

(1) $\Delta_{i}^{F}(g)=J(f_{1},\ldots,f_{i-1},g,f_{i+1},\ldots,f_{n}),$

where $C^{k}(M)$ denotes the space of $C^{k}$ functions defined on a given
$C^{k}$ manifold $M$, and $E_{n}$ denotes the space of entire analytic
functions on $\mathbb{C}^{n}$ with the topology of uniform convergence on
compact subsets.

Stein [19, 20, 21], for $n=2$, and Krasiński and Spodzieja [17], for $n\geq
2$, proved that for any polynomial mapping $F:\mathbb{C}^{n}\to\mathbb{C}^{n}$
with $J(F)=1$, the image $\Delta_{i}^{F}(E_{n})$ is dense in $E_{n}$ for $n-1$
indices $i\in\\{1,\ldots,n\\}$ if, and only if, $F$ is a polynomial
automorphism. Actually, for $n=2$, it was proved in [21] that
$\Delta_{i}^{F}(E_{2})$ is closed, hence $F$ is a polynomial automorphism if
and only if $\Delta_{i}^{F}$ is surjective for $i=1$ or $2$ in the
bidimensional case. So the following conjecture, stated for $n=2$ in [19],
implies the Jacobian conjecture:

######  Conjecture 1.2.

Let $F:\mathbb{C}^{n}\to\mathbb{C}^{n}$ be an entire analytic mapping such
that $J(F)=1$. Then the image $\Delta_{i}^{F}(E_{n})$ is dense in $E_{n}$ for
$n-1$ different indices $i\in\\{1,\ldots,n\\}$.

The following analogous statement in the real $C^{\infty}$ case was recently
proved false in [4] for $n\geq 3$ (in $\mathbb{R}^{2}$ it is true: see for
instance [5]):

> Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a $C^{\infty}$ mapping with
> nowhere vanishing $J(F)$. If
> $\Delta_{i}^{F}\left(C^{\infty}(\mathbb{R}^{n})\right)=C^{\infty}(\mathbb{R}^{n})$
> for $n-1$ indices $i\in\\{1,\ldots,n\\}$, then $F$ is injective.

The first main result of this paper is to consider the real $C^{\infty}$ case
patching together assumptions on the operators $\Delta_{i}^{F}$ and on the set
$S_{F}$ in the following theorem. In order to consider $S_{F}$ we assume $F$
to be semialgebraic.

######  Theorem 1.3.

Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a $C^{\infty}$ semialgebraic
mapping with nowhere vanishing $J(F)$ and such that
$\operatorname{{codim}}S_{F}\geq 2$. Then $F$ is a bijective mapping if and
only if
$\Delta_{i}^{F}\left(C^{\infty}(\mathbb{R}^{n})\right)=C^{\infty}(\mathbb{R}^{n})$
for $n-1$ indices $i\in\\{1,\ldots,n\\}$.

We remark that in the preceding analogous polynomial case in $\mathbb{C}^{n}$,
the set $S_{F}$ is empty or a hypersurface, according to [15, Theorem 3.8],
hence its _real_ codimension is greater than or equal to $2$. Thus our
assumption in Theorem 1.3 is the natural real semialgebraic counterpart of the
preceding complex results.

We also observe that the counterexample constructed in [4] is not
semialgebraic, but we can (and will) construct a semialgebraic one, see
Example 3.1. This supports the assumption on $S_{F}$ in Theorem 1.3.

A similar result than the above ones in the polynomial case is Theorem 4.1 of
[17] that for any polynomial mapping $F:\mathbb{K}^{n}\to\mathbb{K}^{n}$,
$\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, such that $J(F)=1$, then $F$ is a
polynomial automorphism if, and only if,
$\Delta_{i}^{F}(\mathbb{K}[x_{1},\ldots,x_{n}])=\mathbb{K}[x_{1},\ldots,x_{n}]$
for $n-1$ different indices $i\in\\{1,\ldots,n\\}$. The bijective mapping of
$\mathbb{R}^{2}$ given by $F(x,y)=\big{(}x(x^{2}+1),y(y^{2}+1)\big{)}$ does
not satisfy the hypothesis of this result but does satisfy the ones of Theorem
1.3.

Related to the above conjectures, we state the following

######  Conjecture 1.4.

Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a polynomial mapping with nowhere
vanishing $J(F)$ and such that $\operatorname{{codim}}S_{F}\geq 2$. Then
$\Delta_{i}^{F}(C^{\infty}(\mathbb{R}^{n}))=C^{\infty}(\mathbb{R}^{n})$ for
$n-1$ indices $i\in\\{1,\ldots,n\\}$.

By Theorem 1.3, this conjecture implies the Jelonek’s one 1.1, and so the
Jacobian conjecture as well.

Our second main result is Theorem 1.5 below. It provides topological
properties that certain leaves of foliations related to $F$ must satisfy when
$\Delta_{i}^{F}$ is surjective. This theorem is also the main tool for the
proof of Theorem 1.3.

######  Theorem 1.5.

Let $F=(f_{1},\ldots,f_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a $C^{\infty}$
mapping with nowhere vanishing $J(F)$ and such that there exists
$i\in\\{1,\ldots,n\\}$ with
$\Delta_{i}^{F}(C^{\infty}(\mathbb{R}^{n}))=C^{\infty}(\mathbb{R}^{n})$. Let
$j\in\\{1,\ldots,n\\}\setminus\\{i\\}$ fixed. Then for any nonempty connected
component $L$ of
$\bigcap_{l\in\\{1,\ldots,n\\}\setminus\\{i,j\\}}f^{-1}_{l}(c_{l})$ it holds:

  1. (i)

The non-empty fibers of the restriction $f_{j}|_{L}:L\to\mathbb{R}$ are
connected.

  2. (ii)

$L$ is diffeomorphic to $\mathbb{R}^{2}$.

######  Remark 1.6.

It is clear that the operator $\Delta_{i}^{F}$ as defined in (1) does not
depend on the component $f_{i}$ of the mapping $F=(f_{1},\ldots,f_{n})$.
Actually, by considering $F_{i}=(f_{1},\ldots,f_{i-1},f_{i+1},\ldots,f_{n})$,
we can denote by $\Delta_{i}^{F_{i}}$ the same operator defined in (1).
Further, we do not need to assume that the _submersion_
$F_{i}\colon\mathbb{R}^{n}\to\mathbb{R}^{n-1}$ is part of a local
diffeomorphism $F$ in order to state Theorem 1.5. But the assumption on the
surjectivity of $\Delta_{i}^{F_{i}}$ immediately guarantees the existence of a
function $f_{i}\in C^{\infty}_{n}$ such that
$J(f_{1},\ldots,f_{i-1},f_{i},f_{i+1},\ldots,f_{n})\equiv 1$, for instance. So
we do not miss generality in the statement of Theorem 1.5.

Actually, for $n=3$, we can characterize the surjectivity of $\Delta_{i}^{F}$
as

######  Corollary 1.7.

Let $F=(f_{1},f_{2},f_{3}):\mathbb{R}^{3}\to\mathbb{R}^{3}$ be a $C^{\infty}$
mapping with nowhere vanishing $J(F)$ and let $i\in\\{1,2,3\\}$. Then
$\Delta_{i}^{F}(C^{\infty}(\mathbb{R}^{3}))=C^{\infty}(\mathbb{R}^{3})$ if and
only if for $j,k\in\\{1,2,3\\}\setminus\\{i\\}$, $j\neq k$, and for any
connected component $L_{j}$ and $L_{k}$ of nonempty fibers $f_{j}$ and $f_{k}$
respectively, the sets $f_{k}^{-1}(c)\cap L_{j}$ and $f^{-1}_{j}(c)\cap L_{k}$
are connected for any $c\in\mathbb{R}$.

The sufficient condition above is Theorem 2 of [4]. That this condition is
necessary follows directly from (i) of Theorem 1.5. See also an
$n$-dimensional version of this in [5].

The proofs of theorems 1.5 and 1.3 are presented in Section 2.

We end the paper in Section 4 bringing the concept of local fibration to the
subject of the paper. In Section 4, we discuss how this topological concept is
related to the surjectivity of the operators defined in (1). Also, motivated
by the works of Byrnes and Lindquist [6] and Campbell [9], we show that if
$F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a semialgebraic local homeomorphism
such that either $F$ or $F_{i}$ (as above defined) is a local fibration on its
range, then $F$ is injective, see Theorem 4.2.

##  2\. Proof of theorems 1.3 and 1.5

For a given mapping $F=(f_{1},\ldots,f_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$
and for any positive integer $i\leq n$, we will denote by
$F_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n-1}$ the mapping with coordinate
functions $(f_{1},\ldots,f_{i-1},f_{i+1},\ldots,f_{n})$.

We begin gathering the following straightforward properties:

######  Lemma 2.1.

Let $F=(f_{1},\ldots,f_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a local
homeomorphism. Let $i\in\\{1,\ldots,n\\}$ fixed and $R$ be a connected
component of a nonempty fiber of $F_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n-1}$.
Then

  1. (a)

$R$ is homeomorphic (diffeomorphic if $F$ is $C^{1}$) to $\mathbb{R}$ and its
both ends are unbounded.

  2. (b)

The function $f_{i}$ is strictly monotone along $R$.

  3. (c)

If $F$ is $C^{\infty}$, the integral curves of $\Delta^{F}_{i}$ are the
connected components of the nonempty fibers of $F_{i}$.

In particular, for any positive integer $k\leq n$, the non-empty connected
components of $\cap_{j=1}^{k}f_{\sigma(j)}^{-1}(c_{j})$ are unbounded for any
injective mapping $\sigma:\\{1,\ldots,k\\}\to\\{1,\ldots,n\\}$. We remark that
this could not be the case if
$(f_{\sigma(1)},\ldots,f_{\sigma(k)}):\mathbb{R}^{n}\to\mathbb{R}^{k}$ is just
a _submersion_ (not coming from coordinate functions of a local homeomorphism)
with $n\geq 3$, see for instance [10, Section II]: there exists a submersion
$g\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$ with a fiber having one connected
component diffeomorphic to $S^{1}$.

A key point in our paper is the following characterization of surjectivity of
vector fields given in [12].

######  Lemma 2.2 (Part of Theorem 6.4.2 of [12]).

Let $M$ be a $C^{\infty}$ manifold and $X:C^{\infty}(M)\to C^{\infty}(M)$ be a
vector field. Then $X(C^{\infty}(M))=C^{\infty}(M)$, if and only if

  1. (i)

No integral curve of $X$ is contained in a compact subset of $M$.

  2. (ii)

For all compact $K\subset M$, there exists a compact $K^{\prime}\subset M$
such that every compact interval on an integral curve of $X$ with endpoints in
$K$ is contained in $K^{\prime}$.

We first provide the

###### Proof of Statement (i) of Theorem 1.5.

The proof presented here is an adaptation to a bidimensional submanifold ($L$)
of the proof given in [13, Proposition 1.4] in the plane. Lemma 2.1 will be
used throughout the proof. Observe that $L$ is invariant by $\Delta_{i}^{F}$:
the integral curves of $\Delta_{i}^{F}$ in $L$ are given by the connected
components of the fibers of $f_{j}|_{L}$. This is a regular foliation of $L$.
Below, when referring to the flow of $\Delta_{i}^{F}$ we mean this foliation,
i.e. the flow of $\Delta_{i}^{F}$ restricted to $L$.

In the proof we will denote by $\gamma_{p}=\gamma_{p}(t)$ the integral curve
of $\Delta_{i}^{F}$ passing through $p\in L$, i.e., such that
$\gamma_{p}(0)=p$. Further, we will say that a continuous path $\sigma$ in $L$
is _tangent_ to a given integral curve $\gamma$ at a point $q$ when $\sigma$
is not tranversal to $\gamma$ at $q$. Finally, we will say that a continuous
injective path $\sigma$ in $L$ is _ideal_ if it has at most finitely many
points of tangency with the flow of $\Delta_{i}^{F}$.

We begin proving that _any two points of $L$ can be joined by an ideal path_.
Indeed, for a given $p\in L$, we define $Z(p)=\\{q\in L\ |\ \textnormal{ there
exists an ideal path from }p\textnormal{ to }q\\}$. It is enough to prove that
$Z(p)=L$.

By the flow box theorem, [7, Lemma 3, pag 69], it follows that $Z(p)$ is non-
empty. _$Z(p)$ is open:_ Indeed, let $x\in Z(p)$ and denote by $\sigma_{1}$ an
ideal path joining $p$ to $x$. Applying the flow box theorem to the point $x$,
we get a neighborhood of $x$ such that any point $y$ of it can be joined to
$x$ by an ideal path $\sigma_{2}$. Clearly the concatenation of $\sigma_{1}$
and $\sigma_{2}$ is an ideal path connecting $p$ to $y$. _$Z(p)$ is closed:_
Indeed, let $\\{x_{n}\\}$ be a sequence in $Z(p)$ converging to $x$. Similarly
as above, for an $n$ big enough, we manage to construct an ideal path from $p$
to $x$, passing by $x_{n}$, and so $x\in Z(p)$. So $Z(p)=L$ as we wanted.

Now we pass to the proof of the result. Acting by contradiction, we suppose
that for some $c\in\mathbb{R}$ the set ${f_{j}|_{L}}^{-1}(c)=L\cap
f_{j}^{-1}(c)$ is not connected. We let $R_{1}$ and $R_{2}$ be two distinct
connected components of ${f_{j}|_{L}}^{-1}(c)$ and we consider $p_{1}\in
R_{1}$ and $p_{2}\in R_{2}$. So $R_{k}=\gamma_{p_{k}}$, $k=1,2$. We denote by
$\Omega(p_{1},p_{2})$ the (non-empty) set of ideal paths $\sigma:[0,1]\to L$
such that $\sigma(0)=p_{1}$ and $\sigma(1)=p_{2}$. We then fix a path
$\sigma\in\Omega(p_{1},p_{2})$ that _minimizes the number of tangencies with
the flow of $\Delta_{i}^{F}$_.

_We claim that _$R_{1}\cap\sigma([0,1])=\\{p_{1}\\}$ and
$R_{2}\cap\sigma([0,1])=\\{p_{2}\\}$__. If, for instance,
$R_{1}\cap\sigma([0,1])\neq\\{p_{1}\\}$, let $t_{1}\in(0,1]$ be the greatest
time $t$ such that $\sigma(t)\in R_{1}$. We consider $\lambda$ to be the new
path formed by the concatenation of $R_{1}$ from $p_{1}$ to $\sigma(t_{1})$
and by $\sigma([t_{1},1])$. Then by applying the flow box theorem in this
interval of $R_{1}$, we can modify $\lambda$ to a new path
$\overline{\sigma}\in\Omega(p_{1},p_{2})$ without tangency points from $0$ to
$\sigma(t)$, for a $t>t_{1}$ close enough of $t_{1}$. Since $\sigma$ must have
a tangency point in $(0,t_{1})$, it follows that $\overline{\sigma}$ has less
tangencies than $\sigma$, a contradiction with the choice of $\sigma$. The
proof for $R_{2}$ is analogous. The claim is proved. So, in the reasoning
below, we can fix our attention to the open interval $(0,1)$.

Since $f_{j}\circ\sigma$ has a maximum or a minimum point in $(0,1)$, it
follows that the finite set

$\\{s\in(0,1)\ |\ \sigma\textnormal{ is tangent to
}\gamma_{\sigma(s)}\textnormal{ at some point of }\sigma([0,1])\\}$

is not empty and we can let $s_{0}>0$ be its minimum. We have two
possibilities: (i) $\sigma$ is tangent to $\gamma_{\sigma(s_{0})}$ at
$\sigma(s_{0})$ or (ii) $\sigma$ is transversal to $\gamma_{\sigma(s_{0})}$ at
$\sigma(s_{0})$.

Below we will show that assuming either (i) or (ii) leads to a contradiction.
This will finish the proof.

First we assume (i) and let
$T:=\\{s\in(0,s_{0})\mid\\#(\gamma_{\sigma(s)}\cap\sigma([0,1]))\geq 2\\}$,
where as usual $\\#Z$ denotes the cardinality of a set $Z$. By continuous
dependence, it follows that $s\in T$ for all $s<s_{0}$ close enough to
$s_{0}$. Thus $T\neq\emptyset$ and we may consider $\widetilde{s}=\inf T$.

If $\widetilde{s}=0$, then $\widetilde{s}\notin T$. If $\widetilde{s}>0$, we
have that $\gamma_{\sigma(\widetilde{s})}$ is transversal to $\sigma([0,1])$
by the choice of $s_{0}$. Then, it follows by the flow box theorem that
$\widetilde{s}\notin T$. In any case, so, we have that $\tilde{s}\notin T$.

We are going to apply Lemma 2.2 to show that $\Delta_{i}^{F}$ is not
surjective, the aimed contradiction. Let the compact set $K:=\sigma([0,1])$
and let $K^{\prime}$ be any given compact set. The integral curve
$\gamma_{\sigma(\widetilde{s})}$ is not bounded in any direction by Lemma 2.1.
So we consider a tubular neighborhood $N$ of it escaping from $K^{\prime}$ in
both directions. Then for any $t$ such that $\sigma(t)\in N$ it follows that
the integral curve $\gamma_{\sigma(t)}$ stays in $N$ before it can eventually
return to $K$. In particular, by the definition of $\widetilde{s}$, there does
exist $s\in T$ close enough to $\widetilde{s}$ such that the orbit
$\gamma_{\sigma(s)}$ cuts at least twice the compact $K$ but in between it
escapes $K^{\prime}$. By Lemma 2.2 it follows that $\Delta_{i}^{F}$ is not
surjective. So (i) cannot be true.

On the other hand, if (ii) is in force, let $s_{1}>s_{0}$ be such that
$\sigma(s_{1})\in\gamma_{\sigma(s_{0})}$ and $\sigma$ is tangent to
$\gamma_{\sigma(s_{0})}$ at $\sigma(s_{1})$. We assume that $s_{1}$ is the
smallest $s\in(s_{0},1)$ with this property.

We define now the continuous path $\widetilde{\sigma}$ being the concatenation
of the the following three paths: the path $\sigma([0,s_{0}])$, the path
$\gamma_{\sigma(s_{0})}$ from $\sigma(s_{0})$ to $\sigma(s_{1})$, and the path
$\sigma([s_{1},1])$. By using once more the flow box theorem in a tube around
this interval of $\gamma_{\sigma(s_{0})}$, we can approximate the path
$\widetilde{\sigma}$ to a path $\overline{\sigma}\in\Omega(p_{1},p_{2})$ with
less tangencies with the flow of $\Delta_{i}^{F}$ than the path $\sigma$, a
contradiction with the choice of $\sigma$. ∎

For our next proof we recall the following consequence of [22, Proposition
2.7]:

######  Proposition 2.3.

Let $M\subset\mathbb{R}^{n}$ be a $C^{\infty}$ submanifold of dimension $m+1$
and $g\colon M\to\mathbb{R}^{m}$ be a $C^{\infty}$ submersion. If the fibers
of $g$ are diffeomorphic to $\mathbb{R}$ and closed in $\mathbb{R}^{n}$, then
$M$ is diffeomorphic to $\mathbb{R}^{m+1}$.

###### Proof of Statement (ii) of Theorem 1.5.

By Statement (i) of the theorem, together with (a) of Lemma 2.1, it follows
that the non-empty fibers of the restriction $f_{j}|_{L}:L\to\mathbb{R}$ are
diffeomorphic to $\mathbb{R}$. Also, they are clearly closed in
$\mathbb{R}^{n}$. Since $f_{j}|_{L}(L)$ is an interval (and so diffeomorphic
to $\mathbb{R}$), it follows from Proposition 2.3 that $L$ is diffeomorphic to
$\mathbb{R}^{2}$. ∎

The following result of [14] will be used in the proof of Theorem 1.3.

######  Theorem 2.4 (Theorem 1.1 of [14]).

Let $F=(f_{1},\ldots,f_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a $C^{2}$
semialgebraic mapping with nowhere vanishing $J(F)$ and such that
$\operatorname{{codim}}S_{F}\geq 2$. If for all $i,j\in\\{1,\ldots,n\\}$ with
$i\neq j$, any nonempty connected component of
$\bigcap_{l\in\\{1,\ldots,n\\}\setminus\\{i,j\\}}f^{-1}_{l}(c_{l})$ is
diffeomorphic to $\mathbb{R}^{2}$, then $F$ is bijective.

###### Proof of Theorem 1.3.

Assume that $F$ is a global diffeomorphism and let any $i\in\\{1,\ldots,n\\}$.
For a given $\psi\in C^{\infty}(\mathbb{R}^{n})$, let
$\ell(x)=\int_{0}^{x_{i}}\psi\circ F^{-1}(z)J(F^{-1})(z)ds$, with
$z=(x_{1},\ldots,x_{i-1},s,x_{i+1},\ldots,x_{n})$. Since for any $g\in
C^{\infty}(\mathbb{R}^{n})$ it holds

$\displaystyle\partial_{i}(g\circ F^{-1})(x)$
$\displaystyle=J\left((f_{1},\ldots,f_{i-1},g,f_{i+1},\ldots,f_{n})\circ
F^{-1}\right)(x)$ $\displaystyle=\Delta_{i}^{F}(g)(F^{-1}(x))J(F^{-1})(x),$

it follows that defining $g=\ell\circ F\in C^{\infty}(\mathbb{R}^{n})$, we
have $\Delta^{F}_{i}(g)=\psi$. This proves the surjectivity of
$\Delta_{i}^{F}$.

On the other hand, we now assume that
$\Delta_{i}^{F}(C^{\infty}(\mathbb{R}^{n}))=C^{\infty}(\mathbb{R}^{n})$ for
$n-1$ indices $i\in\\{1,\ldots,n\\}$ and let distinct
$j,k\in\\{1,\ldots,n\\}$. It follows by (ii) of Theorem 1.5 that any nonempty
connected component of
$\bigcap_{l\in\\{1,\ldots,n\\}\setminus\\{i,j\\}}f^{-1}_{l}(c_{l})$ is
diffeomorphic to $\mathbb{R}^{2}$. Now the result follows by Theorem 2.4. ∎

##  3\. A semialgebraic counterexample

######  Example 3.1.

The example we are going to construct here is strongly based on the
construction presented in [4]. The idea is to consider $C^{\infty}$
semialgebraic functions in place of the $C^{\infty}$ ones appearing in that
paper. The most difficult part is that we have to modify also the functions
$g_{h}$ of [4]: this is a family of $C^{\infty}$ functions satisfying a list
of properties. What happens is that properties (c) and (d) in this list
implies that the function $g_{h}^{\prime\prime\prime}$ is flat at the point
$1$ without being identically zero. This is impossible if $g_{h}$ is
$C^{\infty}$ and semialgebraic, see for instance [2, Proposition 2.9.5]. Since
the proofs in [4] use the specific $g_{h}$, when we modify them, we have to
provide new proofs.

Let $L>0$ and fix $A:\mathbb{R}\to(-L,L)$ and $E:\mathbb{R}\to(0,\infty)$ two
$C^{\infty}$ semialgebraic diffeomorphisms. For instance, we can take
$A(x)=Lx/\sqrt{1+x^{2}}$ and $E(x)=x+\sqrt{1+x^{2}}$. Further, for any $h>L$,
we define the polynomial

$g_{h}(z)=-\frac{h}{50}z(2z+3)(2z-3)(3z^{2}+7).$

Then for $h_{2}>h_{1}>L$, we define the $C^{\infty}$ semialgebraic mapping
$F:\mathbb{R}^{3}\to\mathbb{R}^{3}$ by

$F(x,y,z)=\left((A(x)+g_{h_{1}}(z))E(y),(A(x)+g_{h_{2}}(z))E(y),(1-z^{2})E(y)\right).$

We claim that (i) $J(F)$ is nowhere zero, (ii) $F$ is not injective and (iii)
$\Delta_{1}^{F}$ and $\Delta_{2}^{F}$ are surjective. Indeed, straightforward
calculations give

$J(F)(x,y,z)=\frac{h_{1}-h_{2}}{50}E(y)^{2}E^{\prime}(y)A^{\prime}(x)\left(36z^{6}-59z^{4}+60z^{2}+63\right),$

which is nowhere zero as the polynomial $m(z)=36z^{6}-59z^{4}+60z^{2}+63$ is
positive (to see this, take $z^{2}=t$ and observe that the resulting degree
$3$ polynomial has derivative strictly positive, and so it has only one real
zero, that must be negative as the leading and the constant coefficients of
$m(z)$ are positive). Moreover, $F(x,y,3/2)=F(x,y,-3/2)$ for any
$x,y\in\mathbb{R}$. So it remains to prove (iii). From Corollary 1.7, by
defining

$f_{h}(x,y,z)=(A(x)+g_{h}(z))E(y),$

with $h>L$, it is enough to prove that for any connected component $L_{i}$ of
$f_{i}^{-1}(c_{i})$, $i=h,3$, it holds that $L_{h}\cap f_{3}^{-1}(c_{3})$ and
$L_{3}\cap f_{h}^{-1}(c_{h})$ are connected sets.

This is the most involving part. We begin by calculating the connected
components of $f_{h}^{-1}(c)$ and of $f_{3}^{-1}(c)$ for any $c\in\mathbb{R}$:
see the complete frame in Table 1, where, similarly as in [4], when we write,
for instance, $A(x)=-g_{h}(z)$, $z<-1$, we mean the largest interval contained
in $z<-1$ where this expression makes sense.

| $f_{h}^{-1}(c)$ | $f_{3}^{-1}(c)$
---|---|---
$c<0$ |  $2$ components |  $2$ components:
$A(x)=cE^{-1}(y)-g_{h}(z)$, $z<1$ |  $E^{-1}(y)=(1-z^{2})/c$, $z<-1$
$A(x)=cE^{-1}(y)-g_{h}(z)$, $z>1$ |  $E^{-1}(y)=(1-z^{2})/c$, $z>1$
$c=0$ |  $3$ components: |  $2$ components:
$A(x)=-g_{h}(z)$, $z<-1$ |
| $z=-1$
$A(x)=-g_{h}(z)$, $-1<z<1$ |
| $z=1$
$A(x)=-g_{h}(z)$, $z>1$ |
$c>0$ |  $2$ components: |  $1$ component:
$A(x)=cE^{-1}(y)-g_{h}(z)$, $z<-1$ |
|  $E^{-1}(y)=(1-z^{2})/c$, $-1<z<1$
$A(x)=cE^{-1}(y)-g_{h}(z)$, $z>-1$ |
Table 1. Connected components of fibers

Clearly the intersection of each of the two connected components of
$f_{3}^{-1}(0)$ with the fibers of $f_{h}$ are connected or empty (use the
fact that the function $y\to cE^{-1}(y)+cte$ is injective). Analogously, since
$z=-1$ and $z=1$ do not occur both in the same connected component of
$f_{h}^{-1}(c)$, it follows that $f_{3}^{-1}(0)$ intersected with any of them
is also connected.

Each one of the other required intersections is a solution of the system of
equations

$E^{-1}(y)=(1-z^{2})/c_{3},\ \ \ \ A(x)=c_{h}E^{-1}(y)-g_{h}(z),$

for $z$ in _only one_ of the intervals

$I_{1}=(-\infty,-1),\ \ \ \ I_{2}=(-1,1),\ \ \ \ I_{3}=(1,\infty).$

They are all connected by

######  Lemma 3.2.

Let $\alpha\in\mathbb{R}$ and let $k_{\alpha}:\mathbb{R}\to\mathbb{R}$ be
defined by $k_{\alpha}(z)=\alpha(1-z^{2})-g_{h}(z)$. Then the sets
$k_{\alpha}^{-1}(-L,L)\cap I_{i}$, $i=1,2,3$ are connected. In particular, for
each $c_{3}\neq 0$, the solution of the system $E^{-1}(y)=(1-z^{2})/c_{3}$,
$A(x)=k_{\alpha}(z)$, $z\in I_{i}$, form a connected set of $\mathbb{R}^{3}$
for $i=1,2,3$.

###### Proof.

The polynomial function

$k_{\alpha}(z)=\alpha(1-z^{2})-g_{h}(z)=\frac{6h}{25}z^{5}+\frac{h}{50}z^{3}-\frac{63h}{50}z+\alpha(1-z^{2})$

is such that
$k_{\alpha}^{\prime\prime\prime}(z)=\frac{72h}{5}z^{2}+\frac{3h}{25}>0$,
$\forall z$, implying that $k_{\alpha}^{\prime\prime}(z)$ is strictly
increasing. Since $k_{\alpha}^{\prime\prime}$ is a polynomial of degree $3$,
it has exactly one real root, say $q_{0}$. Therefore,
$k_{\alpha}^{\prime}(z)=\frac{6h}{5}z^{4}+\frac{3h}{50}z^{2}-2\alpha
z-\frac{63h}{50}$ is strictly decreasing in $(-\infty,q_{0})$ and strictly
increasing in $(q_{0},\infty)$. So, since this is a polynomial of degree $4$
with positive leader coefficient and
$k_{\alpha}^{\prime}(0)=-\frac{63h}{50}<0$, it follows that
$k_{\alpha}^{\prime}(z)$ has exactly two simple zeros, being one negative, say
$p^{-}$, and the other one positive, say $p^{+}$. As a consequence,
$k_{\alpha}(z)$ has a local maximum at $p^{-}$ and a local minimum at $p^{+}$,
being strictly increasing in $(-\infty,p^{-})\cup(p^{+},\infty)$ and strictly
decreasing in $(p^{-},p^{+})$. Therefore, since $h>L$ and $k_{\alpha}(\pm
1)=\mp h$, it follows that $k_{\alpha}^{-1}(-L,L)\cap I_{i}$, $i=1,2,3$, are
connected sets. ∎

##  4\. Fibrations and geometric properties of the operators $\Delta_{i}^{F}$

Let $M$ and $N$ be topological spaces and $f:M\to N$ be a continuous mapping.
We recall that $f$ is said to be a _local fibration at_ $t_{0}\in N$ if there
exist a neighborhood $V$ of $t_{0}$ and a homeomorphism $h:f^{-1}(t_{0})\times
V\to f^{-1}\left(V\right)$ such that $f\circ h=\pi_{2}$, where
$\pi_{2}:f^{-1}(t_{0})\times V\to V$ is the canonical projection. The set of
points of $N$ where $f$ is not a local fibration is called the _bifurcation
set of $f$_ and it is denoted by $B(f)$. We say that $f$ is a _local
fibration_ if $B(f)=\emptyset$. We further say that $f$ is a _local fibration
on its range_ if $B(f)\cap f(M)=\emptyset$. Finally, when it is possible to
take $V=N$ for some $t_{0}\in N$ above, we say that $f$ is a _trivial
fibration_.

Next result provides a relation between the local fibration concept and the
surjectivity of the operators $\Delta^{F}_{i}$ associated to a given
$C^{\infty}$ local diffeomorphism $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$:

######  Proposition 4.1.

Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a $C^{\infty}$ mapping with nowhere
vanishing $J(F)$ and $i\in\\{1,\ldots,n\\}$. If $F_{i}$ is a local fibration
on its range then
$\Delta_{i}^{F}\left(C^{\infty}(\mathbb{R}^{n})\right)=C^{\infty}(\mathbb{R}^{n})$.

###### Proof.

We denote by $\gamma_{x}$ the integral curve of $\Delta_{i}^{F}$ passing
through $x$. We suppose on the contrary that
$\Delta_{i}^{F}\left(C^{\infty}(\mathbb{R}^{n})\right)\neq
C^{\infty}(\mathbb{R}^{n})$. Since from (a) and (c) of Lemma 2.1 the integral
curves of $\Delta_{i}^{F}$ are unbounded, it follows from Lemma 2.2 that there
exist $p_{1},p_{2}\in\mathbb{R}^{n}$ and sequences
$\\{x_{k}\\}\subset\mathbb{R}^{n}$ and
$\\{s_{k}\\},\\{t_{k}\\}\subset\mathbb{R}$, with $0<s_{k}<t_{k}$, such that

(2) $x_{k}\to p_{1},\ \gamma_{x_{k}}(t_{k})\to p_{2}\textnormal{ and
}|\gamma_{x_{k}}(s_{k})|\to\infty.$

By using the flow box theorem together with (2) (consider also (b) of Lemma
2.1) it follows that $p_{1}$ and $p_{2}$ are in distinct integral curves of
$\Delta_{i}^{F}$, say $\gamma_{p_{1}}$ and $\gamma_{p_{2}}$, respectively.
Setting $c=F_{i}(p_{1})$ it follows by continuity and by Lemma 2.1 that
$\gamma_{p_{1}}$ and $\gamma_{p_{2}}$ are distinct connected components of
$F_{i}^{-1}(c)$.

Since $F_{i}$ is a local fibration at $c$, there exist an open neighborhood
$V$ of $c$ and a homeomorphism $h:F_{i}^{-1}(c)\times V\to F_{i}^{-1}(V)$ such
that $F_{i}\circ h=\pi_{2}$. The open sets $h\left(\gamma_{p_{1}}\times
V\right)$ and $h\left(\gamma_{p_{2}}\times V\right)$ do not intersect each
other. This is a contradiction with (2), because for $k$ big enough the curve
$\gamma_{x_{k}}$ connects these two sets but will also be contained in one of
them. ∎

The converse of Proposition 4.1 is not true. Let, for instance, the polynomial
mapping $F(x,y,z)=\left(P(x,y),z\right)$, where
$P:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is the specific above-mentioned Pinchuk
mapping considered in [8]. The operator $\Delta_{3}^{F}=J(P)\partial_{3}$ is
clearly surjective, but $F_{3}$ can not be a fibration at its range because
there are points in $P(\mathbb{R}^{2})$ with different number of pre-images,
according to [8].

Now let $F:\mathbb{K}^{n}\to\mathbb{K}^{n}$ be a polynomial mapping for
$\mathbb{K}=\mathbb{C}$ (resp. a rational mapping for $\mathbb{K}=\mathbb{R}$)
with nowhere vanishing $J(F)$. It is proved in [6] for $\mathbb{K}=\mathbb{C}$
(resp. in [9] for $\mathbb{K}=\mathbb{R}$) that if $F$ is proper when
restricted to its image, then $F$ is an invertible mapping. I.e., if
$S_{F}\cap F(\mathbb{K}^{n})=\emptyset$ then $F$ is invertible. Bearing in
mind that in this case $B(F)=S_{F}$ and motivated by these results and by the
subject of our paper, we state the following:

######  Theorem 4.2.

Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a semialgebraic local
homeomorphism. If either $B(F)\cap F(\mathbb{R}^{n})=\emptyset$ or
$B(F_{i})\cap F_{i}(\mathbb{R}^{n})=\emptyset$ for some $i\in\\{1,\ldots
n\\}$, then $F$ is an injective mapping.

Before proving it, we recall a classical equation involving Euler
characteristics, see for example [11, Corollary 2.5.5]:

######  Lemma 4.3.

Let $M$ and $N$ be topological spaces. Let $f\colon M\to N$ be a local
fibration such that $N$ and $R=f^{-1}(t)$, for some $t\in N$, are homotopy
equivalent to finite CW-complexes. Then the three Euler characteristics
$\chi(M)$, $\chi(N)$ and $\chi(R)$ are defined and related as follows

$\chi(M)=\chi(N)\chi(R).$

A semialgebraic set is homeomorphic to a simplicial complex (which is a CW-
complex), see for instance [2]. So, since the image of a semialgebraic set by
a semialgebraic mapping is again a semialgebraic set, we can apply the above
formula and deliver the

###### Proof of Theorem 4.2.

For any $y\in F_{i}(\mathbb{R}^{n})$ (resp. $y\in F(\mathbb{R}^{n})$), the $d$
connected components of $F^{-1}_{i}(y)$ (resp. $F^{-1}(y)$) are homeomorphic
to $\mathbb{R}$ by (a) of Lemma 2.1 (resp. are points). By Proposition 4.3,

$1=\chi(F_{i}(\mathbb{R}^{n}))d\ \ \ \ (\textnormal{resp.
}1=\chi(F(\mathbb{R}^{n}))d).$

Since $\chi(F_{i}(\mathbb{R}^{n}))\in\mathbb{Z}$ (resp.
$\chi(F(\mathbb{R}^{n}))\in\mathbb{Z}$), it follows that $d=1$. This provides
the injectivity of $F$ since the $i$-th component of $F$ is monotone along
each nonempty fiber of $F_{i}$ by (b) of Lemma 2.1 (resp. trivially). ∎

The next result follows direct by Proposition 4.2 since an injective
polynomial $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is bijective, see [1].

######  Corollary 4.4.

Let $F=(f_{1},\ldots,f_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a polynomial
mapping with nowhere vanishing $J(F)$. If either $B(F)\cap
F(\mathbb{R}^{n})=\emptyset$ or $B(F_{i})\cap F_{i}(\mathbb{R}^{n})=\emptyset$
for some $i\in\\{1,\ldots n\\}$, then $F$ is invertible.

In the light of the preceding two results, we finish the paper with an open
problem. If we do not assume that $F$ is semialgebraic, then Proposition 4.2
may be not true, as the non-injective local diffeormorphism
$F(x,y,z)=\left(e^{x}\cos y,e^{x}\sin y,z\right)$ illustrates: here $F_{3}$ is
a local fibration on its range. It is then natural to ask:

> Is it true that a given $C^{k}$ (resp. $C^{0}$) mapping
> $F=(f_{1},\ldots,f_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$, with nowhere zero
> $J(F)$ (resp. locally homeomorphism), such that $F_{i}$ is a local fibration
> on its range for $n-1$ indices $i\in\\{1,\ldots,n\\}$ is necessarily
> injective?

Since $F_{i}$ being a locally fibration on its range implies the surjectivity
of $\Delta_{i}^{F}$ (in the $C^{\infty}$ case), this would also provide an
alternative to the false claim recalled in the introduction section “If
$\Delta_{i}^{F}(C_{n}^{\infty})=C_{n}^{\infty}$ for $n-1$ indices
$i\in\\{1,\ldots,n\\}$, then $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$, a
$C^{\infty}$ mapping with nowhere vanishing $J(F)$, is injective.”

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