Topological first-order solitons in a gauged $CP(2)$ model with the Maxwell-Chern-Simons action
R. Casana N. H. Gonzalez-Gutierrez E. da Hora




rodolfo.casana@gmail.com 

neyver.hgg@gmail.com 

edahora.ufma@gmail.com
^* †Departamento de Física, Universidade Federal
do Maranhão, 65080-805, São Luís, Maranhão, Brazil,
^Coordenadoria Interdisciplinar de Ciência e
Tecnologia, Universidade Federal do Maranhão, 65080-805, São Lu
ís, Maranhão, Brazil.


We verify the existence of radially symmetric first-order solitons in a
gauged CP(2) scenario in which the dynamics of the Abelian gauge field is
controlled by the Maxwell-Chern-Simons action. We implement the standard
Bogomol'nyi-Prasad-Sommerfield (BPS) formalism, from which we obtain a
well-defined lower bound for the corresponding energy (i.e. the Bogomol'nyi
bound) and the first-order equations saturating it. We solve these
first-order equations numerically by means of the finite-difference scheme,
therefore obtaining regular solutions of the effective model, their energy
being quantized according the winding number rotulating the final
configurations, as expected. We depict the numerical solutions, whilst
commenting on the main properties they engender.


11.10.Kk, 11.10.Lm, 11.27.+d
Topological first-order solitons in a gauged CP(2) model with the
Maxwell-Chern-Simons action
    E. da Hora
    December 30, 2023
=============================================================================================





§ INTRODUCTION


Time-independent solutions of highly nonlinear equations are of great
importance and interest in many areas of physics and mathematics <cit.>. 
In field theory such a highly nonlinear equations arise naturally, i.e, the
second-order Euler-Lagrange equations, which can be quite hard to solve.
However, under very special circumstances, genuine field solutions can be
obtained via a particular set of two coupled first-order differential
equations, namely the Bogomol'nyi-Prasad-Sommerfield (BPS) ones <cit.>,
the resulting configurations minimizing the energy of the overall system.

In the context of gauged (2+1)-dimensional models, these first-order
solutions are called vortices. In particular, magnetic vortices were
verified to occur in the Maxwell-Higgs electrodynamics <cit.>. Also, it
was demonstrated that electrically charged vortices emerge from both the
Chern-Simons-Higgs <cit.> and the Maxwell-Chern-Simons-Higgs scenarios 
<cit.>.

Moreover, other important examples of first-order solutions include the ones
arising from nonlinear sigma models (NLσM) <cit.> in the
presence of a gauge field, which have been widely applied in the study of
different aspects of field theory and condensed matter physics <cit.>.

In this sense, the existence of topological solitons in a O(3) nonlinear
sigma model endowed by the Maxwell action was demonstrated in <cit.>. Moreover, in the Refs. <cit.>, the
authors studied the O(3) nonlinear sigma model gauged by the Chern-Simons
term, establishing the existence of both topological and nontopological
configurations. The gauged O(3) sigma model with the gauge field dynamics
ruled by both the Maxwell and the Chern-Simons terms was also studied <cit.>.

Topological solitons also appear in the CP(N-1) models whose importance is
due to the fact that they present some fundamental properties (such as
asymptotic freedom, confinement, nontrivial vacuum structure giving rise to
stable instantons, etc.) typically inherent to the Yang-Mills theories <cit.>.

In a recent work, the existence of first-order vortices in a gauged CP(2)
model whose gauge field is ruled by the Maxwell action was proposed <cit.>. This hypothesis was confirmed in the Ref. <cit.>, where
the first-order formalism was implemented in a clear way, giving rise to a
well-defined lower-bound for the total energy (i.e.the Bogomol'nyi bound)
and to the corresponding first-order differential equations.

In the sequel, some of us have considered a gauged CP(2) scenario endowed
by the Maxwell term multiplied by a nontrivial dielectric function, the
resulting noncanonical model supporting nontopological first-order vortices
with no quantized magnetic flux <cit.>. Furthermore, some of us have
also calculated the topological first-order vortices inherent to a gauged CP(2) model in the presence of the Chern-Simons term (instead of the
Maxwell one), see the Ref. <cit.>.

In this context, the aim of the present manuscript is to investigate the
existence of first-order solitons arising from a gauged CP(2) model in
which the dynamics of the gauge sector is ruled simultaneously by both the
Maxwell and the Chern-Simons terms. Therefore, in order to introduce our
results, this work is organized as follows: in the next Sec. II, we define
the gauged Maxwell-Chern-Simons CP(N-1) model, focusing our attention on
those time-independent solitons possessing radial symmetry. We then
particularize our investigation to the case N=3, from which we develop the
corresponding first-order framework via the usual prescription (i.e.
requiring the minimization the total energy), this way finding the resulting
lower-bound for the energy itself and the first-order equations saturating
it. In the Sec. III, we solve the first-order equations numerically by means
of the finite-difference algorithm, from which we depict the numerical
solutions, whilst commenting on the main properties they engender. We end
our work in the Section IV, in which we point out our final observations and
perspectives regarding future contributions.



§ THE OVERALL MODEL


We begin our manuscript by considering the gauged CP(N-1) model <cit.> in the presence of the usual Chern-Simons term, the corresponding
Lagrange density standing for 

    ℒ_0=ℒ_MCS+( P_abD_μϕ _b)
    ^∗P_acD^μϕ _c-U( |ϕ|) 
    ,

where

    ℒ_MCS=ℒ_M+ℒ_CS,

with

    ℒ_M=-1/4F_μνF^μν and  
    ℒ_CS=-κ/4ϵ ^ρμνA_ρF_μν
    .

Here, F_μν=∂ _μA_ν-∂ _νA_μ is the
standard electromagnetic field strength tensor and D_μϕ_a=
∂_μϕ _a-igA_μQ_abϕ _b stands for the
corresponding covariant derivative. Also, P_ab=δ _ab-h^-1ϕ
_aϕ _b^∗ is a projection operator introduced for the sake of
convenience, whilst Q_ab represents a real and diagonal charge matrix.
In the present case, the CP(N-1) field is assumed to satisfy ϕ_a^∗ϕ _a=h.

The corresponding Euler-Lagrange equations for the gauge and scalar sectors
are, respectively,

    ∂ _μF^μρ-κ/2ϵ ^ρμνF_μν=J^ρ

and

    2P_adD_μ( P_dcD^μϕ _c) -P_adD_μD^μϕ _d=-P_ad∂ U/∂ϕ _d^∗,

where 

    J^μ=ig[ ( P_abD^μϕ _b) ^∗(
    P_acQ_cdϕ _d) -h.c.]

stands for the current 4-vector (conserved). Here, h.c. means Hermitian
conjugate.

We look for the time-independent solutions arising from (<ref>). In this
sense, it is instructive to write down the Gauss law for static
configurations, i.e., (here, B=F_12 is the magnetic field) 

    ∂ _j∂ _jA_0+κ B=2g^2A^0( P_acQ_cdϕ
    _d) ^∗P_abQ_bmϕ _m,

the corresponding solutions possessing both magnetic and electric fields.

In addition, the time-independent Ampère's law can be written as

    ∂ _kB+κ∂ _kA_0=-ϵ _kjJ_j,

with

    J_k=ig[ ( P_abD_kϕ _b) ^∗(
    P_acQ_cdϕ _d) -h.c.] ,

the equation of motion for the static scalar sector reading 

    2P_adD_k( P_dcD_kϕ _c) -P_adD_kD_kϕ _d
          -P_ad∂ U/∂ϕ _d^∗+2g^2(
    A_0) ^2P_adQ_dbP_bcQ_ceϕ _e
          -g^2( A_0) ^2P_adQ_dbQ_beϕ _e=0.


It can be shown that the Lagrange density (<ref>) does not support
solitonic configurations satisfying a well-defined first-order framework.
However, the existence of planar first-order solitons carrying both magnetic
and electric fields becomes possible via the introduction of a neutral
scalar field Ψ into the Lagrange density (<ref>), the resulting
model being described by 

    ℒ   =   ℒ_MCS+( P_abD_μϕ _b)
    ^∗P_acD^μϕ _c+1/2∂ _μΨ∂
    ^μΨ
          -g^2Ψ ^2( P_abQ_bdϕ _d) ^∗P_acQ_ceϕ _e-U( |ϕ |,Ψ) ,

U( |ϕ |,Ψ) standing for the potential describing the
scalar-matter self-interaction (to be determined later).

Here, it is worthwhile to point out that the introduction of such a neutral
field is a well-established prescription supporting the consistent study of
first-order configurations in the presence of the Chern-Simons term. This
prescription was firstly used in the context of the
Maxwell-Chern-Simons-Higgs theories <cit.>, the main motivation
coming from supersymmetric arguments. It was also implemented successfully
in connection to nonlinear sigma models <cit.>, being
also used to describe charged solitons arising from Lorentz-violating
scenarios <cit.>.

Along of the remain of this manuscript, we consider the N=3 case, this way
reducing our study to the CP(2) scenario. We then focus our attention on
those time-independent solutions presenting radial symmetry which are
implemented via the standard Ansatz

    A_0=A_0( r)  and  A_i=-1/gr^2ϵ
    _ijx_jA( r) ,


    ( 
    [ ϕ _1; ϕ _2; ϕ _3 ]
    ) =h^1/2( 
    [ e^in_1θsinα( r) cosβ( r); e^in_2θsinα( r) sinβ( r);          e^in_3θcosα( r) ]
    ) ,

where ϵ _ij stands for the bidimensional Levi-Civita tensor (with 
ϵ _12=+1), x_i=r(cosθ ,sinθ ) is the position
vector (written in polar coordinates) and the integers n_1, n_2 and n_3 represent the winding numbers (vorticities) of the corresponding
scalar solutions.



Now, it is important to highlight that, concerning the combination between
the charge matrix Q_ab and the winding numbers n_1, n_2 and n_3, there are two possible choices supporting the existence of
topological solitons: (i) Q=λ _3/2 and n_1=-n_2=n, and (ii) Q=λ _8/2 and n_1=n_2=n (both ones with n_3=0, λ
_3 and λ _8 being the diagonal Gell-Mann matrices, i.e., λ _3=(1,-1,0) and √(3)λ _8=
(1,1,-2)). Nevertheless, in <cit.>, the author have demonstrated
that these two combinations simply mimic each other, this way existing only
one effective scenario. Therefore, in this manuscript, we investigate only
the case defined by n_1=-n_2=n, n_3=0 and 

    Q_ab=1/2λ _3=1/2diag( 1,-1,0) 
    .


Furthermore, given the Ansatz (<ref>) and (<ref>) and the
conventions stated above, one gets that nonsingular solutions possessing
finite-energy are attained by those profile functions α (r), A(r)
and A_0(r) obeying the boundary conditions

    α (r=0)=0,  A(r=0)=0,  A_0^'(r=0)=0,

and

    α (r→∞ )→π/2,  
    A(r→∞ )→ 2n,  A_0(r→∞
    )→ 0,

where prime denotes the derivative with respect to the radial coordinate r
. Here, it is important to point out that the boundary conditions for α (r) and A(r) are well-established in the literature <cit.>
. In addition, we verify the conditions for the electric potential A_0(r)
in the Sec. III below.



It is also instructive to write down the equation of motion for the
additional profile function β( r), i.e. (here, we have
defined H_±(r)=( A_0) ^2±Ψ ^2, for the sake of
simplicity)

    d^2β/dr^2+( 1/r+2αdα/
    dr) dβ/dr+g^2/4H_-(r)sin ^4αsin( 4β)  
          -sin ^2αsin( 4β) /r^2( n-A
    /2) ^2=0,

whose two possible solutions are

    β _1=π/4+π/2k or  β _2=π/2k,

with k∈ℤ. These solutions define a priori two different
scenarios. However, the expressions arising from β (r)=β _2 can
be obtained directly from those inherent to β (r)=β _1 via the
transformations α→α /2 and h→ 4h. Hence,
we conclude that these two a priori different solutions are
phenomenologically equivalent, this way existing only one effective
scenario. So, in what follows, we only consider the case β (r)=β
_1.



We now look for first-order differential equations supporting radially
symmetric solutions of the model (<ref>). As it is usual, we proceed
the minimization of the corresponding energy, the starting-point being the
energy-momentum tensor inherent to (<ref>), i.e., 

    𝒯_ρσ   =   -F_ρμF_σ^μ+∂
    _ρΨ∂ _σΨ +( P_abD_ρϕ _b)
    ^∗P_acD_σϕ _c
          +( P_abD_σϕ _b) ^∗P_acD_ρϕ
    _c-g_ρσℒ_N,

where 

    ℒ_N   =   ℒ_M+( P_abD_μϕ _b)
    ^∗P_acD^μϕ _c+1/2∂ _μΨ∂
    ^μΨ
          -g^2Ψ ^2( P_abQ_bdϕ _d) ^∗P_acQ_ceϕ _e-U

stands for the nontopological part of the Lagrange density (<ref>).
Thus, one gets the radially symmetric energy density to be 

    ε   =   1/2B^2+U+1/2( dA_0/dr)
    ^2+1/2( dΨ/dr) ^2
          +h[ ( dα/dr) ^2+W/4r^2(
    2n-A) ^2sin ^2α]  
          +h[ ( dβ/dr) ^2+g^2W/4H_+(r)
    ] sin ^2α,

where we have introduced the auxiliary function 

    W( α ,β) =1-sin ^2αcos ^2( 2β) ,

the magnetic field B(r) being given by

    B(r)=1/grdA/dr,

according the Ansatz (<ref>) and (<ref>). Here, U=U(α
,Ψ ) represents self-interacting potential.



We now choose the particular solution 

    β( r) =β _1=π/4+π/2k,

from which we get W(α ,β =β _1)=1, the energy density (<ref>)being reduced to 

    ε   =   1/2B^2+U+h( dα/dr) ^2+
    h/4r^2( 2n-A) ^2sin ^2α
          +1/2( dA_0/dr) ^2+1/2( 
    dΨ/dr) ^2+g^2h/4H_+sin ^2α,

which can be rewritten as 

    ε   =   1/2( B∓√(2U)) ^2+h[ 
    dα/dr∓( 2n-A) /2rsinα] ^2
          +1/2( dΨ/dr±dA_0/dr) ^2+
    g^2h/4( Ψ± A_0) ^2sin ^2α
          ± B√(2U)± h( 2n-A) /rdα/drsinα
          ∓dΨ/drdA_0/dr∓g^2h/2Ψ A_0sin
    ^2α.




Here, it is useful to note that, in view of (<ref>) and (<ref>), the
time-independent Gauss law (<ref>) can be written in the form (note
that such equation remains the same after the introduction of the neutral
scalar field Ψ), 

    d^2A_0/dr^2+1/rdA_0/dr+κ B=g^2h
    /2A_0sin ^2α.

It allows the last term in (<ref>) can be rewritten as 

    g^2h/2A_0Ψsin ^2α =Ψd^2A_0/dr^2+
    Ψ/rdA_0/dr+κΨ B,

consequently, the energy density (<ref>) being reduced to be 

    ε   =   1/2( B∓√(2U)) ^2+h[ 
    dα/dr∓( 2n-A) /2rsinα] ^2
          +1/2( dΨ/dr±dA_0/dr) ^2+
    1/4hg^2( Ψ± A_0) ^2sin ^2α
    
    [0.15cm]
          ± B( √(2U)-hgcosα -κΨ)  
          ∓h/rd/dr[ ( 2n-A) cosα]
    ∓1/rd/dr( rΨdA_0/dr) .

Due to the boundary conditions, the last term in (<ref>) (i.e. the total
derivative r^-1( rΨ A_0^') ^') gives
null contribution to the total energy, so we neglect it. However, the
penultimate total derivative provides a nonvanishing contribution to the
overall energy.

The potential U(|ϕ| ,Ψ) is determined by choosing 

    U( α ,Ψ) =1/2( ghcosα +κΨ) ^2,

so, the resulting total energy being given by 

    E    =   ± 4π hn+2π∫_0^∞r{1/2[ 
    /B∓( ghcosα +κΨ) ]
    ^2.  
                +1/2( dΨ/dr±dA_0/dr
    ) ^2+g^2h/4( Ψ± A_0) ^2sin ^2α
                +. h[ dα/dr∓(
    2n-A) /2rsinα] ^2} dr.




The expression in (<ref>) reveals that the energy of the effective model
supports a well-defined lower-bound given by 

    E_bps=± 4π hn=∓ 2π∫_0^∞dr rh/rd/dr
    [ ( 2n-A) cosα] ,

which is attained by those fields satisfying the set of differential
equations 

    dα/dr=±( 2n-A) /2rsinα,


    B=±( hgcosα +κΨ) ,


    dΨ/dr=∓dA_0/dr and  Ψ =∓ A_0,

which therefore stand for the first-order equations inherent to the
effective radially symmetric model.

Here, it is interesting to highlight that the last two equations are
automatically satisfied by Ψ =∓ A_0, the resulting configurations
being then obtained via the two remaining first-order equations, i.e. 

    dα/dr=±( 2n-A) /2rsinα,


    B=1/grdA/dr=± hgcosα -κ A_0,

which must be solved together with the Gauss law (<ref>),

    d^2A_0/dr^2+1/rdA_0/dr+κ B=g^2h
    /2A_0sin ^2α.


In addition, the reader can verify that the solutions for n<0
(anti-vortices) can be obtained directly from the ones for n>0 (vortices)
via the transformations n→ -n, α→α, A→ -A and A_0→ -A_0.



Moreover, another quantity commonly referred when studying vortices is the
magnetic flux Φ _B they give rise. In the present case, the resulting
flux reads 

    Φ _B=2π∫_0^∞rB( r) dr=4π/gn
    .

In this case, whether we consider 

    ρ (r)=g^2h/2A_0sin ^2α

as the electric charge density, the magnetic flux Φ _B (<ref>)
can be verified to be proportional to the total electric charge 𝒬
, i.e. 

    𝒬=2π∫_0^∞rρ (r)dr=κΦ _B,

where we have integrated the Gauss law (<ref>).



§ FIRST-ORDER SOLUTIONS


We now proceed the analysis of the differential equations (<ref>), (
<ref>), (<ref>) themselves in view of the boundary conditions (<ref>) and (<ref>). It is useful to check a priori whether the
solutions attain the boundary values (<ref>) and (<ref>)
correctly. In order to verify such a convergence, we proceed to solve the
differential equations (<ref>), (<ref>) and (<ref>) around
those values. By considering n>0, we calculate the approximate solutions
near the origin to be 

    α (r)    ≈   C_nr^n,
    
    A(r)    ≈   g/2( gh-κ A_0(0)) r^2,


    A_0(r)≈ A_0(0)-κ/4( gh-κ A_0(0))
    r^2,

where C_n and A_0(0)≡ A_0(r=0) are positive constants to be
determined numerically for each value of the vorticity n, the last
solution verifying the boundary value for A_0(r=0) appearing in (<ref>).

A similar procedure reveals that the asymptotic profiles are (here, the
positive constant C_∞ also depends on n ) 

    α (r)    ≈   π/2-C_∞e^-M_αr, 
    
    A(r)    ≈   2n-2MC_∞re^-M_Ar,


    A_0(r)≈2M/gC_∞e^-Mr,

where the parameters 

    M_α=M_A=M=1/2( √(2g^2h+κ ^2)-|κ|)

represent the masses of the corresponding bosons (the relation M_α/M_A=1 defining the BPS point). In this case, it is easy to verify that,
in the limit g≫κ, the masses behave like those arising from a
gauged CP(2) theory endowed by the Maxwell term only <cit.>, 

    M≈ g√(h/2).

On the other hand, when g≪κ, the masses mimic the value obtained
from a gauged CP(2) model in the presence of the Chern-Simons action alone 
<cit.>:

    M≈g^2h/2κ.


In what follows, we proceed the numerical integration of the equations (<ref>), (<ref>), (<ref>) according the conditions (<ref>)
and (<ref>). We solve the corresponding system by means of
finite-difference algorithm, from which we depict the resulting profiles in
the figures 1, 2, 3, 4, 5 and 6. Here, we choose h=κ =1 and g=2,
plotting the solutions for n=1 (solid black line), n=2 (dashed blue
line) and n=3 (dotted red line).



The figures 1 and 2 show the numerical solutions to the profile functions α (r) and A(r),respectively, from which we see that these two
solutions reach the boundary values in a monotonic way, as expected. In
particular, the gauge function obeys A(r→∞ )→ 2n.

In the Figure 3, we depict the numerical profile of the scalar potential A_0(r), the corresponding structures being lumps centered at r=0. It is
interesting to note how A_0(0)≡ A_0(r=0) depends on n, its
value increasing as the vorticity itself increases until reaching the
maximum value

    A_0^max(r=0)=gh/κ,

for sufficiently large values of n (see the Figure 4).

The Figure 5 presents the solutions for the magnetic field B(r), including
the ones obtained for n=7 (long-dashed orange line) and n=12 (dotted
green line). In this case, for n=1 , the profile is a well-defined lump
centered at the origin. On the other hand, for n>1, the solutions stand
for rings also centered at r=0, their amplitudes (radii) decreasing
(increasing) as the vorticity increases.



It is particularly interesting to consider the solution forB(r) coming
from (<ref>). It reads 

    B(r)≈ b_0+1/4( κ ^2b_0-2( C_1)
    ^2gh) r^2,

for n=1, and 

    B(r)≈ b_0+κ ^2b_0/4r^2,

for n≥ 2, where we have defined 

    b_0=gh-κ A_0(0).


In this sense, we get that value of B(r=0), i.e.

    B(r=0)=b_0=gh-κ A_0(0),

which explains the behavior of the magnetic field at the origin: as nincreases, the value of A_0(r=0) increases too (see the Figure 4),
whilst the difference b_0=gh-κ A_0(0) decreases, see the Figure
6. In particular, for sufficiently large values of the vorticity, one gets
that A_0(r=0)→ gh/κ and B(r=0)=b_0→ 0.

The solutions to the electric field E(r) appear in the Figure 7, from
which we see that the resulting configurations are rings centered at r=0,
their amplitude and radii increasing as the vorticity increases, with both E(r=0) and E(r→∞ ) vanishing.

We depict the numerical profiles to the energy distribution ε
_bps(r) in the Figure 8, from which see that the resulting structures
are similar to the magnetic ones, i.e. for n=1 (n>1) the solution is a
lump (ring) centered at r=0. Thus, it is instructive to study the behavior
of ε _bps(r) near the origin which results to be

    ε _bps(r)≈ 2( C_1) ^2h+( b_0)
    ^2+ε _2r^2,

from n=1, and

    ε _bps(r)≈( b_0) ^2+3/4(
    b_0κ) ^2r^2,

for n≥ 2. Here, we have defined

    ε _2=3/4( b_0κ) ^2-2/3
    ( C_1) ^4h+gh/2( C_1) ^2[ (
    A_0(0)) ^2g-4b_0] ,

for the sake of convenience.

We therefore note that, as n increases, ε _bps(r=0)vanishes. In particular, for n=1, one gets that ε
_bps(r=0)=2( C_1) ^2h+( b_0) ^2, see the
Figure 9.



§ FINAL COMMENTS AND PERSPECTIVES


We have shown the existence of radially symmetric first-order solitons in a
gauged CP(2) model endowed by the Maxwell and the Chern-Simons terms
simultaneously.

The point to be raised here is that, in order to implement the Bogomol'nyi
prescription correctly, we have introduced a neutral scalar field into the
original Maxwell-Chern-Simons model (<ref>), such a modification being a
well-known procedure usually performed when the dynamics of the gauge sector
is controlled by the composite Maxwell-Chern-Simons action. Such a procedure
also works in the Lorentz-violating extensions of the simplest
Maxwell-Chern-Simons-Higgs theory and of the O(3)-sigma model with the
Maxwell-Chern-Simons term <cit.>.

We also have calculated the self-interacting potential engendering
self-duality, from which we also have determined the lower-bound for the
overall energy (i.e. the Bogomol'nyi bound) and the corresponding
first-order differential equations supporting topological solitons.
Moreover, we have verified that the resulting Bogomol'nyi bound is
proportional to the topological charge of the model, both ones being
quantized according the vorticity n, as expected. In addition, we have
rewritten the total electric charge 𝒬 in terms of the magnetic
flux Φ _B, which is also quantized, see the equations (<ref>) and
(<ref>).

We are now studying the existence of nontopological solitons in a gauged CP(2) scenario in the presence of the Chern-Simons and of the
Maxwell-Chern-Simons terms, from which we hope interesting results to be
reported in a future contribution.


This work was supported by the Brazilian Government via the Conselho
Nacional de Desenvolvimento Científico e Tecnológico (CNPq), the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
(CAPES), and the Fundação de Amparo à Pesquisa e ao Desenvolvimento
Científico e Tecnológico do Maranhão (FAPEMA). In particular, RC
thanks the support from the grants CNPq/306385/2015-5,
FAPEMA/Universal-00782/15 and FAPEMA/Universal-01131/17, NHGG acknowledges
the full support from CAPES (postgraduate scholarship), and EH thanks the
support from the grants CNPq/307545/2016-4 and CNPq/449855/2014-7.


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