Time-dependent diffusive interactions of dark matter and dark energy in the
context of $k-$essence cosmology

Abhijit Bandyopadhyay111Email<EMAIL_ADDRESS>and Anirban
Chatterjee222Email<EMAIL_ADDRESS>
Department of Physics

Ramakrishna Mission Vivekananda Educational and Research Institute

Belur Math, Howrah 711202, India

###### Abstract

We investigated the scenario of time-dependent diffusive interaction between
dark matter and dark energy and showed that such a model can be accommodated
within the observations of luminosity distance - redshift data in Supernova Ia
(SNe Ia) observations. We obtain constrains on different relevant parameters
of this model from the observational data. We consider a homogeneous scalar
field $\phi(t)$ driven by a $k-$essence Lagrangian of the from $L=V(\phi)F(X)$
with constant potential $V(\phi)=V$, to describe the dynamics of dark energy
in this model. Using the temporal behaviour of the FRW scale factor, the
equation of state and total energy density of the dark fluid, extracted from
the analysis of SNe Ia (JLA) data, we have obtained the time-dependence of the
$k-$essence scalar field and also reconstructed form of the function $F(X)$ in
the $k-$essence Lagrangian.

##  1 Introduction

In this work, we explore a scenario of interacting dark matter (DM) and dark
energy (DE), where the DE is represented by a homogeneous scalar field $\phi$
with its dynamics driven by a $k$-essence Lagrangian with constant potential
and DM is assumed to be a perfect fluid. In the late time phase of cosmic
evolution, we can neglect the effect of radiation and baryonic matter to the
total energy density of the universe. Thus for the late time phase, dark
energy and dark matter are the dominant sectors to study the new consequences.
Motivations behind consideration of such classes of a unified model of DM and
DE is to address the issue of coincidence of observed present-day dark energy
and dark matter densities using a dynamical relation between dark matter and
dark energy [1, 2]. The interaction between DM and DE is assumed to happen
through velocity diffusion of particles of dark matter fluid in the background
of field $\phi$. In an earlier work [3], we have studied implications of such
a diffusive interaction with constant (time independent) diffusion
coefficient. In this work we revisit interacting model of DE and DM with a
time-dependent diffusion coefficient. We also explore the implications of
measurements of luminosity distances and redshifts in Supernova Ia (SNe Ia)
observations [4, 5, 6, 7] in the constraining different parameters for this
model.

The basic framework of the model of DM-DE interactions has been discussed in
detail in [3, 8]. We, first, indulge in a brief recollection of the model with
an emphasis on the consideration of the time-dependence of the diffusion
coefficient. We assume dark matter and dark energy to be perfect fluids and
neglect contribution of radiation and baryonic matter during the late time
phase of cosmic evolution (probed in SNe Ia observations). In a flat FRW
spacetime background, conservation of total energy momentum tensor for the
dark fluid (DE + DM) takes the form

$\displaystyle\Big{[}\dot{\rho}_{\rm dm}+3H\dot{\rho}_{\rm
dm}\Big{]}=-\Big{[}\dot{\rho}_{\rm de}+3H(\rho_{\rm de}+p_{\rm
de})\Big{]}=Q(t)$ (1)

where $\rho_{\rm dm}$ and $\rho_{\rm de}$ are the energy densities of dark
matter and dark energy fluid respectively. $p_{\rm de}$ is the pressure of the
dark energy fluid and we take dark matter as pressureless dust.
$H\equiv\dot{a}/a$ is the Hubble parameter, where $a(t)$ is the scale factor
corresponding to Friedman-Robertson-Walker (FRW) spacetime background. Here
$t$ is regarded as a dimensionless time parameter with $t=1$ corresponding to
present epoch. The quantity $Q(t)$ in Eq. (1), is a measure of rate of energy
transfer between the fluid dark matter and dark energy caused by a diffusion
at the instant of time $t$. We parametrize time dependence of the quantity
$Q(t)$ in terms of a parameter $k$

$\displaystyle Q(t)=Q_{0}\big{[}a(t)\big{]}^{k}$ (2)

where scale factor $a(t)$ has been taken to be normalised to unity at present
epoch, $a(t=1)=1$, and $Q_{0}$ is the value of $Q(t)$ at present epoch. Using
the results of a comprehensive analysis of recently released “Joint Light-
curve Analysis” (JLA) data [5] consisting of 740 SNe Ia events as performed in
[6], the observed temporal behaviour of the quantities: the scale factor
$a(t)$, the total energy density $(\rho_{\rm dm}+\rho_{\rm de})(t)$ and the
equation of state $\omega(t)$ have been extracted over a time domain
$0.44<t<1$ accessible in SNe Ia observations. Using these observed time
evolutions and the chosen form Eq. (2) of parametrisation of $Q(t)$ in Eq. (1)
we obtained the range of values of $k$ for which the interacting DE-DM model
with such a time-dependent diffusion term $Q(t)$, may be accommodated within
the scheme of luminosity distance - redshift measurements of SNe Ia
observation. The methodology of obtaining such constraints has been discussed
comprehensively in Sec. 2.

We have assumed that, the diffusive interaction, considered here, occurs in
the background of $k-$essence scalar field $\phi$, whose dynamics is driven by
a non-canonical Lagrangian $L=V(\phi)F(X)$, where
$X=(1/2)g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$. We take the potential
$V(\phi)$ to be constant, which ensures existence of a scaling relation of the
form $X(dF/dX)^{2}=Ca^{-6}$, where $C$ is a constant [9, 10]. We then identify
the stress energy tensor corresponding to the non-canonical Lagrangian with
that of the dark energy fluid and consider the $k-$essence scalar field
$\phi=\phi(t)$ to be homogeneous. Use of the scaling relation, then enables us
to establish relations between time derivative of scalar field and energy
density and pressure of dark energy fluid. Using the above relations along
with the observed temporal behaviour of the quantities like $a(t)$,
$\omega(t)$ and $\rho_{\rm dm}+\rho_{\rm de}$ we reconstructed the temporal
behaviour of the $k-$essence scalar field $\phi$ as well as the form of the
function $F(X)$. The results are shown for different benchmark values of the
parameter $k$ with its allowed domain as obtained earlier. The quantity $Q(t)$
plays an important role to study the interaction between DM and DE sectors. We
have also shown the energy transferred in these two sectors for some chosen
values of the diffusion parameters. This has been described in detail in Sec.
3.

##  2 Observational constraints on time-dependent diffusive DE-DM interacting
model

We use the measurements of luminosity distance and redshift in SNe Ia
observations for redshift values up to $z\approx 1$ to extract features of
late time phase of cosmic evolution. These data are instrumental in obtaining
the temporal behaviour of FRW scale factor $a(t)$. There exist different
compilations of SNe Ia data from different surveys. These include high
redshift ($z\sim 1$) projects viz. Supernova Legacy Survey (SNLS) ([11],[12]),
the ESSENCE project [13], the Pan-STARRS survey ([14],[15], [16]);
intermediate redshift ($0.05<z<0.4$) projects viz SDSS-II supernova surveys
([17],[18],[19], [20],[21]) and small redshift programmes such as surveys like
Harvard-Smithsonian Center for Astrophysics survey [22], the Carnegie
Supernova Project ([23],[24],[25]), the Lick Observatory Supernova Search [26]
and the Nearby Supernova Factory [27]. Various other compilations of SNe Ia
data may be found in [28]. All such surveys include nearly one thousand SNe Ia
events. In this work we used the results from analysis of “Joint Light-curve
Analysis” (JLA) data ([4],[5],[6],[7]) consisting of 740 data points from 3
years survey of SDSS, five-year SNLS survey and 14 very high redshift
$0.7<z<1.4$ SNe Ia from space-based observations with the HST[29]. The
methodology of analysis of JLA data has been described in detail in [6, 30]
where a $\chi^{2}$ function for the JLA data is defined as

$\displaystyle\chi^{2}_{\rm SN}(\alpha,\beta)$ $\displaystyle=$
$\displaystyle\sum_{i,j}\left(\mu_{{\rm obs}}^{(i)}-\mu_{\rm
th}^{(i)}\right)\left(\Sigma^{-1}\right)_{ij}\left(\mu_{\rm
obs}^{(j)}-\mu_{\rm th}^{(j)}\right)$ (3)

where $i$ and $j$ run from 1 to 740 corresponding to 740 SNe Ia events
contained in the JLA data set [5]. $\mu_{{\rm obs}}^{(i)}$ is the observed
value of distance modulus at a redshift $z_{i}$ corresponding to $i^{\rm th}$
entry of the JLA data set and $\mu_{{\rm th}}^{(i)}$ is the corresponding
theoretical estimate expressed through an empirical relation expressed in
terms of various parameters. $\Sigma$ is the total covariant matrix given in
terms of statistical and systematic uncertainties (see [6] for details). In
[6], Wang et. al. have performed the marginalisation of the $\chi^{2}$
function of Eq. (3), over the parameters of the theory to obtain best-fit
values of the parameters. The estimated value of the quantity
$E(z)=H(z)/H_{0}$ at the best-fit has been shown in left panel of Fig. 5 of
[6], where $H=\dot{a}/a$ is the Hubble parameter at redshift $z$ and $H_{0}$
is its value at the present epoch ($z=0$). We have used the results of
analysis obtained by Wang et. al. in [6] along with the relations
$H=\dot{a}/a$ and $1/a=1+z$ (value of the FRW scale factor at present epoch is
normalised to unity) and write $dt=-\frac{dz}{(1+z)H_{0}E(z)}$ which on
integration gives

$\displaystyle\frac{t(z)}{t_{0}}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{H_{0}t_{0}}\int_{z}^{0}\frac{dz^{\prime}}{(1+z^{\prime})E(z^{\prime})}$
(4)

where $t_{0}$ is the time denoting the present epoch. We use the function
$E(z)$ as obtained at best-fit from analysis of JLA data in [6] in Eq. (4),
and obtain $t$ as a function of $z$ by performing the integration numerically.
We then eliminate $z$ from the obtained $z$ \- $t(z)$ dependence and the the
equation $1/a=1+z$ to obtain scale factor $a(t)$ as a function of $t$. The
obtained temporal profile may be used to find the time-dependence of the
Hubble parameter $H\equiv\dot{a}/a$ which directly govern the cosmological
dynamics through Friedmann equations. In extreme right panel of Fig. 1, we
have shown the observed temporal behaviour of $H$ in terms of a newly
introduced time parameter $\tau$, defined as

$\displaystyle\tau=\ln a$ (5)

Note that present epoch corresponds to $\tau=0$ as the scale factor at present
epoch is normalised to unity.

The two independent Friedmann equations governing dynamics of late time cosmic
evolution in FRW spacetime background are

$\displaystyle H^{2}=\frac{8\pi G}{3}(\rho_{\rm
de}+\rho_{dm})\,\quad;\quad\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\left[(\rho_{\rm dm}+\rho_{\rm de})+3p_{\rm de}\right]\,.$ (6)

Here we considered a flat spacetime background (zero curvature constant) and
neglect contributions from radiation and baryonic matter during late time
phase of cosmic evolution. Using Eq. (6), the equation of state of total dark
fluid may be expressed as

$\displaystyle\omega\equiv\frac{p_{\rm de}}{\rho_{\rm de}+\rho_{\rm dm}}$
$\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{\ddot{a}a}{\dot{a}^{2}}-\frac{1}{3}$ (7)
$\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{a(a^{\prime}HH^{\prime}+H^{2}a^{\prime\prime})}{(a^{\prime}H)^{2}}-\frac{1}{3}$

where in the last expression, time variable has been changed from $t$ to
$\tau$ and ′ corresponds to derivatives with respect to $\tau$. We use the
time dependence of the scale factor $a(\tau)$ extracted from the analysis of
JLA data to obtain the temporal behaviour of the equation of state
$\omega(\tau)$ of the dark fluid. The obtained time-dependence has been shown
in the extreme left panel of Fig. (1).

Transforming the time variable from $t$ to $\tau$ and using
$\omega=\frac{p_{\rm de}}{\rho_{\rm de}+\rho_{\rm dm}}$ in the total
continuity equation (Eq. (1)) of the dark fluid we obtain

$\displaystyle\frac{d}{d\tau}\ln(\rho_{\rm de}+\rho_{\rm dm})$
$\displaystyle=$ $\displaystyle-3(1+\omega(\tau))\,,$ (8)

and performing integration over $\tau$ we obtain

$\displaystyle\rho(\tau)$ $\displaystyle\equiv$ $\displaystyle\Big{[}\rho_{\rm
de}+\rho_{\rm dm}\Big{]}_{\tau}={\Big{[}\rho_{\rm de}+\rho_{\rm
dm}\Big{]}_{0}}\exp\left[-3\int^{\tau}_{\tau^{\prime}=0}(1+\omega(\tau^{\prime}))d\tau^{\prime}\right]$
(9)

We use the obtained $\tau$-dependence of the function $\omega(\tau)$ as
extracted from SNe Ia data, in Eq. (9) to obtain temporal behaviour of the
total density ($\rho_{\rm de}+\rho_{\rm dm}$) of the dark fluid. We have shown
this dependence in the middle panel of Fig. (1).

Figure 1:  Plot of $\omega$, $\rho/(\rho^{0}_{\rm de}+\rho^{0}_{\rm dm})$ and
$H$ as a function of time parameter $\tau$ as extracted from JLA data.

As already mentioned in Sec. 1, we assume the quantity $Q(t)$ in Eq. (1),
which is a measure of rate of energy transfer between the fluid dark matter
and dark energy, to be time-dependent and parametrise the time dependence as

$\displaystyle Q(t)$ $\displaystyle=$ $\displaystyle
Q_{0}\Big{[}a(t)\Big{]}^{k}$ (10)

where $k$ is a constant. In terms of the time parameter $\tau$, the above
equation reads $Q(\tau)=Q_{0}e^{k\tau}$ and Eq. (1) takes the following form,

$\displaystyle\frac{d{\rho}_{\rm dm}}{d\tau}+3\rho_{\rm
dm}=Q_{0}\frac{e^{k\tau}}{H(\tau)}$ (11)

We assume a trial solution of Eq. (11) for $\rho_{\rm dm}(\tau)$ as

$\displaystyle\rho_{\rm dm}(\tau)$ $\displaystyle=$
$\displaystyle\Big{[}\rho_{\rm de}+\rho_{\rm
dm}\Big{]}_{0}\sum_{i=0}^{\infty}\alpha_{i}\tau^{i}$ (12)

and express the temporal behaviour of $H(\tau)$ as extracted from JLA data in
the form of polynomial of the form

$\displaystyle\frac{1}{H(\tau)}$ $\displaystyle=$
$\displaystyle\sum_{i}\gamma_{i}\tau^{i}$ (13)

where, the coefficients $\gamma_{i}$’s are obtained by fitting the above
polynomial with the temporal behaviour of reciprocal of the function $H(\tau)$
shown in the right panel of Fig. (1). The obtained best-fit values of the
parameters $\gamma_{i}$’s are given in Tab. 1.

$\gamma_{0}=$ | 1.00118 | $\gamma_{2}=$ | -0.321574 | $\gamma_{i}=0$
---|---|---|---|---
$\gamma_{1}=$ | 0.449751 | $\gamma_{3}=$ | -0.149526 | for $i>3$
Table 1:  Values of $\gamma_{i}$’s in Eq. (13) providing best fit to the
values of $\frac{1}{H(\tau)}$ extracted from analysis of JLA data

We use the Eqs. (12) and (13) in Eq. (11) to obtain

$\displaystyle\sum_{i=0}^{\infty}i\alpha_{i}\tau^{i-1}+3\sum_{i=0}^{\infty}\alpha_{i}\tau^{i}=\beta_{0}\sum_{i=0}^{3}\gamma_{i}\tau^{i}\sum_{j=0}^{\infty}\frac{k^{j}\tau^{j}}{j!}\,.$
(14)

where,

$\displaystyle\beta_{0}\equiv Q_{0}/[\rho_{\rm de}+\rho_{\rm dm}]_{0}$ (15)

is the ratio of value of $Q$ to that of total dark fluid energy density at
present epoch. Equating the coefficients of $\tau^{i}$ from both sides of Eq.
(14) we obtain

$\displaystyle\alpha_{i+1}$ $\displaystyle=$
$\displaystyle\beta_{0}\sum_{n=0}^{i}\frac{\gamma_{n}k^{i-n}}{(i-n)!(i+1)}-\frac{3\alpha_{i}}{i+1}$
(16)

We note from Eq. (12) that

$\displaystyle\alpha_{0}$ $\displaystyle=$ $\displaystyle\frac{\rho^{0}_{\rm
dm}}{[\rho_{\rm de}+\rho_{\rm dm}]_{0}}$ (17)

is the fraction of dark matter energy density contribution to the total dark
fluid density at present epoch which has the mathematically allowed domain as
$0\leqslant\alpha_{0}\leqslant 1$. However, WMAP [31] and Planck [32]
measurements established that, the observed value of the fraction $\alpha_{0}$
is $\sim 0.27$. For given values of $\alpha_{0}$, $k$, $\beta_{0}$ one may
find $\alpha_{i}$’s ($i>0$) using Eq. (16). The series {$\alpha_{i}$} will
always converge as $\gamma_{i}$’s are zero for $i>3$ (see Tab. 1) and $(i+1)$
appears in the denominator of the recursion relation (Eq. (16)). Using the
evaluated series {$\alpha_{i}$}, we use Eq. (12) to obtain the values of
$\rho_{\rm dm}(\tau)$ at any any $\tau$, corresponding to any given set of
values of ($\alpha_{0}$, $k$, $\beta_{0}$). Since, $|\tau|<1$ and the series
{$\alpha_{i}$} converges, the numerical value of $\rho_{\rm dm}(\tau)$ will
have negligible contribution from the terms above certain order in the
summation series of Eq. (12).

The temporal behaviour of energy density $\Big{[}\rho_{\rm dm}+\rho_{\rm
de}\Big{]}_{\tau}$ of the total dark fluid has already been obtained directly
from the analysis of JLA data (middle panel of Fig. 1). The estimated value of
the dark matter density $\rho_{\rm dm}(\tau;\alpha_{0},k,\beta_{0})$ computed
from Eq. (12) for given sets of values of ($\alpha_{0}$, $k$, $\beta_{0}$) is
subject to the constraint

$\displaystyle 0<\rho_{\rm dm}(\tau;\ \alpha_{0},k,\beta_{0})<\Big{[}\rho_{\rm
dm}+\rho_{\rm de}\Big{]}_{\tau}$ (18)

The above constraints put limits on the range of allowed values of
$\alpha_{0}$, $k$, $\beta_{0}$. The range of values of the parameters
($\alpha_{0}$, $k$, $\beta_{0}$) for which the condition in Eq. (18) is
satisfied, corresponds to the values of the parameters for which the scenario
of interacting DE-DM with time-dependent diffusion coefficient may by
accommodated with the luminosity distance - redshift measurements of the SNe
Ia events in the JLA data. The constraints on parameters ($\alpha_{0}$, $k$,
$\beta_{0}$), thus obtained, are presented in Figs. 2 and 3. In Fig. 2 we have
shown the allowed domain of $\alpha_{0}-k$ parameter space for 4 different
benchmark values of the parameter $\beta_{0}$ (viz. 0.1, 1.0, 5.0, 10.0). In
Fig. 3, the allowed domain in $\beta_{0}-k$ parameter space is shown for
$\alpha_{0}=0.27$.

Figure 2:  Allowed regions in $\alpha_{0}-k$ parameter space for different
benchmark values of $\beta_{0}$.  Figure 3:  Allowed region in $\beta_{0}-k$
parameter space for $\alpha_{0}=0.27$.

##  3 Realisation in terms of $k$-essence cosmology

We also try to realise the diffusive interaction between DE and DM with time-
dependent diffusion coefficient, in terms of a $k-$essence scalar field $\phi$
representing the dynamics of dark energy. The scalar field $\phi$ plays the
role of background medium in which diffusion takes place.

###  3.1 Theoretical framework of $k$-essence model

The minimally coupled(with gravitational field $g_{\mu\nu}$) action for a
$k-$essence scalar field $\phi$ is written as,

$\displaystyle S_{k}$ $\displaystyle=$ $\displaystyle\int d^{4}x\sqrt{-g}{\cal
L}(X,\phi)$ (19)

where kinetic term
$X\equiv\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$, $g$ is
determinant of the metric $g_{\mu\nu}$ and $\nabla_{\mu}$ represents covariant
derivative associated with metric $g_{\mu\nu}$. The total action for
gravitational field $g_{\mu\nu}$ and $k-$essence scalar field is express as,

$\displaystyle S_{k}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[-\frac{R}{16\pi G_{N}}+{\cal L}(X,\phi)\right]$ (20)

where $G_{N}$ is Newton’s gravitational constant. We obtain the expression for
energy momentum tensor for the $k-$essence field by varying the action with
respect to the field $g_{\mu\nu}$:

$\displaystyle T_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$ (21)
$\displaystyle=$ $\displaystyle\frac{\partial{\cal L}}{\partial
X}\nabla_{\mu}\phi\nabla_{\nu}\phi-g_{\mu\nu}{\cal L}$

the energy-momentum tensor Eq. (21) may be written in the form of that of a
perfect fluid as

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}$ (22)

where $u_{\mu}$ is the effective four-velocity given by

$\displaystyle u_{\mu}$ $\displaystyle=$ $\displaystyle{\rm
sgn}(\partial_{0}\phi)\frac{\nabla_{\mu}\phi}{\sqrt{2X}}\,,$ (23)

the pressure $p$ of the fluid is the Lagrangian density,

$\displaystyle p$ $\displaystyle=$ $\displaystyle{\cal L}(\phi,X)$ (24)

and the energy density is

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle 2X\frac{\partial
p}{\partial X}-p$ (25)

We use the form of $k$-essence model Lagrangian as

$\displaystyle{\cal L}(\phi,X)=V(\phi)F(X)$ (26)

For the $k$-essence Lagrangian of the form ${\cal L}(\phi,X)=V(\phi)F(X)$, the
pressure and energy density of the fluid (whose energy momentum is equivalent
to that of the $k$-essence field) may be written from Eqs. (24) and (25) as

$\displaystyle p$ $\displaystyle=$ $\displaystyle V(\phi)F(X)$ (27)
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle V(\phi)(2XF_{X}-F)$ (28)

where $F_{X}=dF/dX$. In this way we can relate the dark energy fluid with
$k$-essence scalar field $\phi$.

###  3.2 Diffusion in the background of $k$-essence scalar field

We assume the field $\phi$ to be spatially homogeneous ($\phi=\phi(t)$) and
the potential $V(\phi)$ to be constant ($V$). These respectively imply

$\displaystyle X=(1/2)\dot{\phi}^{2}$ (29)

and existence of a scaling relation of the form

$\displaystyle XF_{X}^{2}=Ca^{-6}\,,\quad\mbox{where $C$ is a constant}$ (30)

Using the time-dependences of various cosmological parameters extracted from
SNe Ia data as described in Sec. 2 we obtained the constraints on the temporal
behaviour of the field $\phi$ and the form of the function $F(X)$ for
different modes of time dependences of the diffusion coefficient,
characterised in terms of the parameter $k$. The methodology of obtaining such
results are described below.

Figure 4:  Variation of the scalar field $\phi$ with time parameter $\tau$ and
$t/t_{0}$.

Using Eqs. (27) , (28) and (30) we can write,

$\displaystyle X$ $\displaystyle=$ $\displaystyle\frac{a^{6}(\rho_{\rm
de}+p_{\rm de})^{2}}{4CV^{2}}$ (31)

Using Eq. (29) in Eq. (31) and changing the time parameter from $t$ to $\tau$
(as given in Eq. (5)) we obtain

$\displaystyle\left[\frac{\sqrt{2C}V}{(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0})}\right]\left(\frac{d\phi}{d\tau}\right)$ $\displaystyle=$
$\displaystyle\frac{a^{3}}{H}\left[\frac{\rho_{\rm de}}{(\rho_{\rm
dm}^{0}+\rho_{\rm de}^{0})}+\frac{p_{\rm de}}{(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0})}\right]$ (32)

which on Integration gives

$\displaystyle\left[\frac{\sqrt{2C}V}{(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0})}\right](\phi(\tau)-\phi_{0})$ $\displaystyle=$
$\displaystyle\int_{\tau^{\prime}=0}^{\tau}d\tau^{\prime}\left[\frac{a^{3}(\tau^{\prime})}{H(\tau^{\prime})}\left(\frac{\rho_{\rm
de}(\tau^{\prime})}{\rho_{\rm dm}^{0}+\rho_{\rm de}^{0}}+\frac{p_{\rm
de}(\tau^{\prime})}{\rho_{\rm dm}^{0}+\rho_{\rm de}^{0}}\right)\right]$ (33)

The right hand side of Eq. (33) contains the quantities $a$ and $H$ whose
observed $\tau-$dependences has already been obtained from the analysis of JLA
data as discussed in Sec. 2. Since, $\omega=p_{\rm de}\Big{/}(\rho_{\rm
dm}+\rho_{\rm de})$, the last term in the right hand side of the above
equation, $p_{\rm de}(\tau)\Big{/}(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})$ may
be expressed as $\omega(\tau)\rho(\tau)\Big{/}(\rho_{\rm dm}^{0}+\rho_{\rm
de}^{0})$. This $\tau-$dependence is known from observation as the dependences
$\omega(\tau)$ and $\rho(\tau)\Big{/}(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})$
are separately known from observation as shown earlier (middle panel of Fig.
1). To evaluate the other remaining term of Eq. (33), $\rho_{\rm
de}(\tau)\Big{/}(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})$, we may compute dark
matter density $\rho_{\rm dm}(\tau;\alpha_{0},k,\beta_{0})\Big{/}(\rho_{\rm
dm}^{0}+\rho_{\rm de}^{0})$ from Eq. (12), corresponding to a set of values of
parameters ($\alpha_{0},k,\beta_{0}$) within their corresponding domains
allowed from JLA data as depicted in Fig. 2. The dark energy density
$\rho_{\rm de}$ may also be evaluated at a given ($\alpha_{0},k,\beta_{0}$)
value as

$\displaystyle\frac{\rho_{\rm de}(\tau;\alpha_{0},k,\beta_{0})}{(\rho_{\rm
dm}^{0}+\rho_{\rm de}^{0})}$ $\displaystyle=$
$\displaystyle\frac{\Big{[}\rho_{\rm dm}+\rho_{\rm
de}\Big{]}_{\tau}}{(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})}-\frac{\rho_{\rm
dm}(\tau;\alpha_{0},k,\beta_{0})}{(\rho_{\rm dm}^{0}+\rho_{\rm de}^{0})}$ (34)

The $\tau$-dependence of the $k-$essence field, thus computed performing the
integration in Eq. (33), would be dependent on the parameter values
$\alpha_{0}$, $k$ and $\beta_{0}$. To depict the temporal behaviour of the
scalar field $\phi$ we have set $\beta_{0}=1$ and fix $\alpha_{0}$ at its
experimental observed value at 0.27 (see discussion after Eq. (17)). The
obtained $\tau$ dependence of the scalar field $\phi$ for different benchmark
values of the parameter $k$ are shown in Fig. 4. We have shown the time
dependence in terms of both the time parameters $\tau=\ln a(t)$ and $t$. We
find that, for any value of the diffusion parameter $k$, the time dependence
of the $k-$essence scalar field $\phi$ may be fitted in terms of polynomial of
degree 2 as

$\displaystyle\phi(t/t_{0},k)-\phi_{0}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{1}(k)\,(t/t_{0}-1)+\varepsilon_{2}(k)\,(t/t_{0}-1)^{2}$
(35)

where, $\phi_{0}(k)$ is the value of the field at present epoch ($t=t_{0}$),
and the coefficients $\varepsilon_{1}(k)$ and $\varepsilon_{2}(k)$ are
functions of $k$. We have also computed the values of these coefficients at
different values of the parameter $k$ and $k-$dependences of the coefficients
$\varepsilon_{1,2}$ are illustrated in Fig. 5.

Figure 5:  Variation of the coefficients $\varepsilon_{1},\varepsilon_{2}$ (in
arbitrary units) with $k$.

Finally, in the context of realisation of time dependent diffusive DE-DM
interactions in terms of dynamics of a $k-$essence field, we have also find
the form of the function $F(X)$ appearing in the $k-$essence Lagrangian. We
mentioned below how this is done using the inputs from the observational data.
Using Eq. (27) , (28) and the equation $\omega(\tau)=p_{\rm
de}(\tau)/\rho(\tau)$ we may write

$\displaystyle F(X)V$ $\displaystyle=$
$\displaystyle\omega(\tau)\frac{\rho(\tau)}{(\rho_{\rm de}^{0}+\rho_{\rm
dm}^{0})}$ (36)

and Eq. (31) may be rewritten in the form

$\displaystyle XV_{1}$ $\displaystyle=$ $\displaystyle
a^{6}\left[\frac{\rho_{\rm de}(\tau;\alpha_{0},\beta_{0},k)}{\rho_{\rm
de}^{0}+\rho_{\rm dm}^{0}}+\frac{\omega(\tau)\rho(\tau)}{\rho_{\rm
de}^{0}+\rho_{\rm dm}^{0}}\right]^{2}$ (37)

where $V_{1}=\frac{4CV^{2}}{(\rho_{\rm de}^{0}+\rho_{\rm dm}^{0})^{2}}$ is a
constant. Using Eq. (34) and observed $\tau$ dependences of $a(\tau)$,
$\omega(\tau)$, $\rho(\tau)$ as extracted from JLA data in Eqs. (36) and (37,)
we compute values of both the quantities $F(X)V$ and $XV_{1}$ at different
values of $\tau$ corresponding to any given set of values of
$(\alpha_{0},\beta_{0},k)$. We then eliminate $\tau$ from the Eqs. (36) and
(37) to obtain dependence of $F(X)V$ as a function of $XV_{1}$. Thus form of
the function $F(X)$ has been extracted from observational data upto the
undetermined constants $V$ and $V_{1}$. We have shown the variation of $F(X)$
with $X$ in fig. 6 for different values of $k$ evaluated at $\beta_{0}=1$ and
$\alpha_{0}=0.27$.

Figure 6:  Variation of reconstructed $F(X)V$ with $XV_{1}$ for different
values of $k$. $V$ and $V_{1}$ are constants (See text for details).

Figure. 7 depicts the diffusive activity between the dark sectors. The energy
has been transferred from dark matter to dark energy for the chosen benchmark
values of the diffusion parameter $\beta_{0}$ (for fixed $k$ and
$\alpha_{0}$). For $\beta_{0}=0$, the continuity equation of dark matter gives
$\rho_{\rm dm}\propto\frac{1}{a^{3}}$ which corresponds to the non-interacting
scenario between these two sectors. We also choose two non-zero (positive)
values of $\beta_{0}$ to realise the interacting nature of this model. The
figure shows for the larger values of $\beta_{0}$ corresponds to the higher
amount of transferred energy from dark matter to dark energy. The figure also
represents dark matter density is always lower than dark energy density for
the higher values of $\beta_{0}$ at any epoch.

Figure 7:  Temporal behaviour of the energy density of dark matter and dark
energy for the benchmark values of diffusion parameters
($k,\alpha_{0},\beta_{0}$).

##  4 Conclusion

In this work, we study the scenario of interacting DE and DM, with a time-
dependent diffusive interaction between them. We have shown that such a model
can be accommodated within the observations of luminosity distance - redshift
data in SNe Ia events. We obtain the constrains on different relevant
parameters of this model from the observational data. The two parameters of
the model which are relevant in the context of this work are $Q_{0}$ and $k$.
They parametrize the rate of exchange of energy due to diffusion from dark
matter to dark energy in a form $Q_{0}[a(t)]^{k}$ appearing in the non-
conservation equation (1). For convenience, in stead of working with the
parameter $Q_{0}$, we have chosen $Q_{0}/(\rho_{\rm de}^{0}+\rho_{\rm
dm}^{0})\equiv\beta_{0}$ as the parameter. Besides $\beta_{0}$ and $k$, the
parameter $\alpha_{0}\equiv\rho_{\rm dm}^{0}/(\rho_{\rm de}^{0}+\rho_{\rm
dm}^{0})$, which is approximately the fractional of contribution of dark
matter dark density to the total density at present epoch also appears in the
framework of our analysis.

We have shown the temporal behaviour of the Hubble parameter, $H$, total
equation of state ($\omega$) and total energy density $(\rho)$ of the dark
fluid as extracted from JLA data in Fig. 1. We have exploited these
dependences to obtain the constraints on the above mentioned parameters
$\beta_{0},k,\alpha_{0}$. The obtained constraints are depicted in Figs. 2 and
3. The value of the parameter $\alpha_{0}$ is however, independently
determined from WMAP and PLANCK experiments as $\alpha_{0}\approx 0.27$. The
results show that, if we choose this value of $\alpha_{0}(\approx 0.27)$, the
allowed values of the parameter $k$ has a lower limit. For example, for
different values of $\beta_{0}$, the obtained values of the lower limit of $k$
are given in Tab. 2.

values of $\beta_{0}$ | 0.1 | 1.0 | 5.0 | 10.0
---|---|---|---|---
lower limit of $k$ | -6.13 | -1.11 | 4.76 | 9.89
Table 2:  Lower limit of allowed values of parameter $k$ for different values
of $\beta_{0}$ with $\alpha_{0}=0.27$.

In addition to this, in the context of the interacting DE-DM model considered
here along with the constraints on relevant parameters of the model from
observational data, we also address certain issues related to $k$-essence
scalar field model of dark energy. We consider a homogeneous scalar field
$\phi$ driven by a $k-$essence Lagrangian (with constant potential) to
represent dynamics of dark energy. We assume, in DE-DM interacting scenario
considered here, the diffusion from dark matter takes place in the background
medium of the scalar field. Using the observational features of the
cosmological parameters as extracted from the JLA data as inputs, we find
constraints on the time-dependence of the field $\phi$. The existence of
scaling relation (Eq. 30) owing to constancy of the potential $V$ in the
$k-$essence Lagrangian, also enables us to obtain the form of the function
$F(X)$ appearing in the Lagrangian. We found that the temporal behaviour of
the scalar field, in this context, may be very convincingly accommodated
within a profile
$\phi(t/t_{0},k)-\phi_{0}(k)=\varepsilon_{1}(k)(t/t_{0}-1)+\varepsilon_{2}(k)(t/t_{0}-1)^{2}$,
where coefficients $\varepsilon_{1,2}$ are dependent on the (diffusion)
parameter $k$ and these dependences are also obtained using inputs from
observation (See Fig. 5). The obtain form of the function $F(X)$ for different
values of the parameter $k$ are depicted in Fig. 6. Diffusive nature of DM and
DE components have been achieved through the parametrize form of $Q(t)$, fig.
7 represents the validity of our model in both non-interacting and interacting
scenarios.

##  5 Acknowledgements

The authors are grateful to the referee for very helpful suggestions. A.C.
would like to thank University Grants Commission (UGC), India for supporting
this work by means of NET Fellowship (Ref.No.22/06/2014(i)EU-V and
Sr.No.2061451168).

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# Gauge Parameter Independence of the Higgs Pole Mass in the Multi-Higgs model

Chungku Kim<EMAIL_ADDRESS>Department of Physics, Keimyung University, Daegu
704-701, Korea

###### Abstract

We have investigated the gauge parameter dependence of the Higgs pole masses
in the multi-Higgs model in two different types of the gauge fixing
conditions. It turns out that the Higgs pole masses of the multi-Higgs model
when expressed in terms of the Lagrangian parameters, are independent of the
gauge parameter for both cases.

###### pacs:

11.15.Bt, 12.38.Bx

##  I
INTRODUCTION

Recently, gauge field theory which contains several Higgs field became of
interest to solve the questions of the neutrino mass, the matter-antimatter in
the universe and the nature of the dark matterMulti . In obtaining the
physical results from these models, the pole mass plays an important role in
the process where the characteristic scale is close to the mass shellPole . It
was shown that the pole mass is infrared finiteInfrared  and invariant under
the renormalization groupRG . In case of the quantized gauge field theory, we
need to add a gauge-fixing termsGauge  which contain a gauge parameters and
the physical quantities such as the pole mass should be independant of these
gauge parameters. In case of the standard model which have single Higgs boson,
the gauge independence of the Higgs pole mass for the Higgs boson was shown in
Ref.parameter . In this paper, we will investigate the gauge parameter
dependence of the Higgs pole masses in multi-Higgs models in two different
types of the gauge fixing conditions.

##  II GAUGE DEPENDENCE OF THE POLE MASS IN MULTI-HIGGS MODEL

In the broken symmetry phase of the multi-Higgs model, the Higgs pole masses
$M_{i}^{2}$ are determined from the conditionQuiros

$\left[\det\Pi_{pq}(p^{2})\right]_{p^{2}=-M_{i}^{2}}=0,$ (1)

where the renormalized inverse Higgs propagator $\Pi_{pq}(p^{2})$ is obtained
from the renormalized effective action in the broken symmetry phase $\Gamma$
as

$\Pi_{pq}(p^{2})=\left[\frac{\delta^{2}\Gamma}{\delta h_{p}\delta
h_{q}}\right]_{h=0}.$ (2)

where $h$ is the classical field corresponding to the Higgs field $H$ of the
Lagrangian density. Then the gauge parameter $\xi^{\alpha}$, corresponding to
$\alpha-th$ gauge boson, dependence of the Higgs pole mass can be obtained by
taking $\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}$ to Eq.(1) to obtain
as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\left[\det\Pi(p^{2})\right]_{p^{2}=-M_{i}^{2}}=-(\xi^{\alpha}\frac{\partial
M_{i}^{2}}{\partial\xi^{\alpha}})\left[\frac{\delta}{\delta
p^{2}}\det\Pi(p^{2})\right]_{p^{2}=-M_{i}^{2}}+\left[\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\det\Pi(p^{2})\right]_{p^{2}=-M_{i}^{2}}$
(3) $\displaystyle=$ $\displaystyle-(\xi^{\alpha}\frac{\partial
M_{i}^{2}}{\partial\xi^{\alpha}})\left[\frac{\delta}{\delta
p^{2}}\det\Pi(p^{2})\right]_{p^{2}=-M_{i}^{2}}+\left[Tr\\{\Gamma^{-1}(p^{2})\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\Pi(p^{2})\\}\det\Pi(p^{2})\right]_{p^{2}=-M_{i}^{2}}$
$\displaystyle=$ $\displaystyle-(\xi^{\alpha}\frac{\partial
M_{i}^{2}}{\partial\xi^{\alpha}})\left[\frac{\delta}{\delta
p^{2}}\det\Pi(p^{2})\right]_{p^{2}=-M_{i}^{2}}+\left[Tr\\{adj(\Pi(p^{2})\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\Pi(p^{2})\\}\right]_{p^{2}=-M_{i}^{2}},$

where $adj(\Pi(p^{2}))$ is the adjoint of $\Pi(p^{2})$ given as

$adj\\{\Pi(p^{2})\\}=\Pi^{-1}(p^{2})\det\Pi(p^{2}).$ (4)

Note that although $\left[\det\Pi_{pq}(p^{2})\right]_{p^{2}=-M_{i}^{2}}$
vanishes, $adj\\{\Pi(p^{2})\\}$ is nonzero when $p^{2}=-M_{i}^{2}$. It follows
that if $\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\Pi(p^{2})$ have a
form

$\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\Pi(p^{2})=A_{1}\Pi+\Pi
A_{2},$ (5)

for some matrices $A_{1}$and $A_{2}$, then
$\left[Tr\\{adj(\Pi(p^{2})\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\Pi(p^{2})\\}\right]_{p^{2}=-M_{i}^{2}}$
term of the last line of Eq.(3) is proportional to
$\left[Tr\\{\det\Pi(p^{2})\\}\right]_{p^{2}=-M_{i}^{2}}$ and vanishes due to
Eq.(1). As a result, we obtain

$\xi^{\alpha}\frac{\partial M_{i}^{2}}{\partial\xi^{\alpha}}=0,$ (6)

if Eq.(5) is satisfied.

In order to test whether Eq.(5) is satisfied or not in given gauge fixing
term, let us start from the Nielsen identityNielsen  for the gauge parameter
$\xi^{\alpha}$ in the symmetric phase of the multi-Higgs model given asDelCima

$\xi^{\alpha}\frac{\partial\Gamma^{SYM}(\phi)}{\partial\xi^{\alpha}}+C_{i}^{\alpha}(\phi)\frac{\delta\Gamma^{SYM}}{\delta\phi_{i}}=0\text{
(no sum on }\alpha\text{),}$ (7)

where $\phi_{i}$ is the classical field corresponding to the scalar field
$\Phi_{i}$ of the Lagrangian density and $\Gamma^{SYM}(\phi)$ is the effective
action in the symmetric phase and the index $i$ runs both the internal and the
space-time variables. In the broken symmetry phase, the gauge fixing can be
divided into two classes which we call class I and class II type respectively.
In case of the class I type of gauge fixing, the gauge fixing term in the
broken symmetry phase $L_{g.f.}$ is obtained from that of the symmetric phase
$L_{g.f.}^{SYM}$ as

$L_{g.f.}(H_{i},v_{i})=\left[L_{g.f.}^{SYM}(\Phi_{i})\right]_{\Phi_{i}=H_{i}+v_{i}},$
(8)

where $v_{i}$ is the vacuum expectation value(VEV) for the scalar field
$\Phi_{i}$ and $H_{i}$ is the corresponding Higgs field. The typical case is
the non-linear $\overline{R_{\xi}}$ gaugeBar  and the Feynman gauge also
belongs to this class. In case of the class II type of the gauge fixing, the
gauge fixing term in the broken symmetry phase do not have a corresponding
term in the symmetric phase as in Eq.(8) and is obtained by identifying the
gauge parameter $u_{i}$ of the symmetric phase with the VEV in the broken
symmetric phase as

$L_{g.f.}(H_{i},v_{i})=\left[L_{g.f.}^{SYM}(\Phi_{i},u_{i})\right]_{u_{i}=v_{i},\Phi_{i}=H_{i}+v_{i}}.$
(9)

The typical case is the $R_{\xi}$ gauge fixing scheme of the form

$L_{g.f.}=\frac{1}{2\xi^{\alpha}}(\partial_{\mu}G_{\mu}^{\alpha}-gv_{i}T_{ij}^{\alpha}\Phi_{j})^{2}.$
(10)

in the broken symmetry phase which do not have a corresponding term in the
symmetric phase satisfying Eq.(8). In order to obtain this gauge fixing term
in the broken symmetry phase, we choose a gauge fixing term in the symmetric
phase of the form

$L_{g.f.}^{SYM}(\Phi_{i},u_{i})=\frac{1}{2\xi^{\alpha}}(\partial_{\mu}G_{\mu}^{\alpha}-gu_{i}T_{ij}^{\alpha}\Phi_{j})^{2},$
(11)

where $u_{i}$ plays a role of the gauge parameter in the symmetric phaseu .
Then, in the broken symmetry phase, we should substitute $v$ for$\ u$ to
obtain Eq.(9). Since $u_{i}$ is a gauge parameter in the symmetric phase, it
also satisfies a Nielsen identity of the formu

$u_{i}\frac{\partial\Gamma^{SYM}(\phi)}{\partial
u_{i}}+C_{ij}(\phi)\frac{\delta\Gamma^{SYM}}{\delta\phi_{j}}=0\text{ (no sum
on }\alpha\text{).}$ (12)

Now, let us consider the gauge dependence of the VEVs $v_{p}$ corresponding to
p-th Higgs boson which satisfy

$\left[\frac{\delta\Gamma^{SYM}}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p}}=0.$
(13)

In case of the class I type of gauge fixing, by taking
$\xi^{\alpha}\frac{\delta}{\delta\xi^{\alpha}}$ to Eq.(13) and by using
Eq.(7), we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\left[\frac{\delta\Gamma^{SYM}}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p}}=\xi^{\alpha}\frac{\partial
v_{j}}{\partial\xi^{\alpha}}\left[\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{p}\delta\phi_{j}}\right]_{\phi_{p}=v_{p}}+\left[\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p}}$
(14) $\displaystyle=$ $\displaystyle\left[\xi^{\alpha}\frac{\partial
v_{j}}{\partial\xi^{\alpha}}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{p}\delta\phi_{j}}-\frac{\delta
C_{j}^{\alpha}}{\delta\phi_{p}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{j}}-C_{j}^{\alpha}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{p}\delta\phi_{j}}\right]_{\phi_{p}=v_{p}}.$

The second term of the last line of this equation vanishes due to Eq.(13) and
we obtain

$\xi^{\alpha}\frac{\partial v_{j}}{\partial\xi^{\alpha}}=C_{j}^{\alpha}(v).$
(15)

In case of the class II type of gauge fixing, since $u_{i}$ should be
identified as a VEV $v_{i}$ in the broken symmetry phase, Eq.(8) should be
modified as

$\left[\frac{\delta\Gamma^{SYM}}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}=0.$
(16)

This modification of the VEV causes the anomalous behavior of the gamma
function of the VEVgamma1 gamma2 . By taking
$\xi^{\alpha}\frac{\delta}{\delta\xi^{\alpha}}$ to the Eq.(16) and by using
Eqs.(7) and (12), we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\left[\frac{\delta\Gamma^{SYM}}{\delta\phi_{p}}\right]_{\phi_{p}=u_{p}=v_{p}}$
(17) $\displaystyle=$ $\displaystyle\xi^{\alpha}\frac{\partial
v_{i}}{\partial\xi^{\alpha}}\left[\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{p}\delta\phi_{i}}+\frac{\partial}{\partial
u_{i}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}+\left[\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}$
$\displaystyle=$ $\displaystyle\xi^{\alpha}\frac{\partial
v_{i}}{\partial\xi^{\alpha}}\left[\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{p}\delta\phi_{i}}-\frac{1}{u_{i}}\frac{\delta
C_{ik}}{\delta\phi_{p}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{k}}-\frac{1}{u_{i}}C_{ik}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{p}\delta\phi_{k}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}$
$\displaystyle-\left[\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{p}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{i}}+C_{i}^{\alpha}(\phi)\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{p}\delta\phi_{i}}+\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}.$

Then, applying Eq.(16) to last line of above equation, we obtain

$\xi^{\alpha}\frac{\partial
v_{i}}{\partial\xi^{\alpha}}-\xi^{\alpha}\frac{\partial
v_{k}}{\partial\xi^{\alpha}}\left[\frac{1}{u_{k}}C_{ki}(\phi)\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}=\left[C_{i}^{\alpha}(\phi)\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}.$
(18)

Having obtained the gauge dependence of the VEVs in case of the class I and II
type of gauge fixing given in Eqs.(15) and (18), let us consider the gauge
dependence of the inverse Higgs propagator in the broken symmetry phase. In
case of the class I type of gauge fixing, the inverse Higgs propagator in the
broken symmetry phase is gven by

$\Pi_{pq}(p^{2})=\left[\frac{\delta^{2}\Gamma}{\delta h_{p}\delta
h_{q}}\right]_{h_{i}=0}=\left[\frac{\delta^{2}\Gamma^{SYM}(\phi)}{\delta\phi_{p}\delta\phi_{q}}\right]_{\phi_{i}=v_{i}},$
(19)

from which we obtain

$\displaystyle\xi^{\alpha}\frac{\delta}{\delta\xi^{\alpha}}\Pi_{pq}(p^{2})$
(20) $\displaystyle=$ $\displaystyle\xi^{\alpha}\frac{\partial
v_{i}}{\partial\xi^{\alpha}}\left[\frac{\delta^{3}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}\delta\phi_{q}}\right]_{\phi_{p}=v_{p}}-\left[\frac{\delta^{2}}{\delta\phi_{p}\delta\phi_{q}}\\{C_{i}^{\alpha}(\phi)\frac{\delta\Gamma^{SYM}}{\delta\phi_{i}}\\}\right]_{\phi_{p}=v_{p}}=\xi^{\alpha}\frac{\partial
v_{i}}{\partial\xi^{\alpha}}\left[\frac{\delta^{3}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}\delta\phi_{q}}\right]_{\phi_{p}=v_{p}}$
$\displaystyle-\left[\frac{\delta^{2}C_{i}^{\alpha}(\phi)}{\delta\phi_{p}\delta
h_{q}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{k}}+\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{p}}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{q}}+\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{q}}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}}+C_{i}^{\alpha}(\phi)\frac{\delta^{3}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}\delta\phi_{q}}\right]_{\phi_{p}=v_{p}}$
$\displaystyle=$ $\displaystyle-\left[\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p}}\Pi_{qi}(p^{2})-\left[\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{q}}\right]_{\phi_{p}=v_{p}}\Pi_{pi}(p^{2}),$

where we have used Eqs.(7) and (15). Now, consider the case of the class II
type of gauge fixing where the inverse Higgs propagator in the broken symmetry
phase is gven by

$\Pi_{pq}(p^{2})=\left[\frac{\delta^{2}\Gamma}{\delta h_{p}\delta
h_{q}}\right]_{h_{i}=0}=\left[\frac{\delta^{2}\Gamma^{SYM}(\phi)}{\delta\phi_{p}\delta\phi_{q}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}.$
(21)

Then we obtain

$\displaystyle\xi^{\alpha}\frac{\partial}{\partial\xi^{\alpha}}\Pi_{pq}(p^{2})$
$\displaystyle=$ $\displaystyle\xi^{\alpha}\frac{\partial
v_{i}}{\partial\xi^{\alpha}}\left[\frac{\delta^{3}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}\delta\phi_{q}}+\frac{\partial}{\partial
u_{i}}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{p}\delta\phi_{q}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}-\left[\frac{\delta^{2}}{\delta\phi_{p}\delta\phi_{q}}\\{C_{i}^{\alpha}(\phi)\frac{\delta\Gamma^{SYM}}{\delta\phi_{i}}\\}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}$
$\displaystyle=$ $\displaystyle\xi^{\alpha}\frac{\partial
v_{i}}{\partial\xi^{\alpha}}\left[\frac{\delta^{3}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}\delta\phi_{q}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}$
$\displaystyle-\xi^{\alpha}\frac{\partial
v_{k}}{\partial\xi^{\alpha}}\left[\frac{1}{u_{k}}\\{\frac{\delta^{2}C_{ki}}{\delta\phi_{p}\delta\phi_{q}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{i}}+\frac{\delta
C_{ki}}{\delta\phi_{p}}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{q}\delta\phi_{i}}+\frac{\delta
C_{ki}}{\delta\phi_{q}}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}}+C_{ki}\frac{\delta^{3}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}\delta\phi_{q}}\\}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}$
$\displaystyle-\left[\frac{\delta^{2}C_{i}^{\alpha}(\phi)}{\delta\phi_{p}\delta\phi_{q}}\frac{\delta\Gamma^{SYM}}{\delta\phi_{i}}+\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{p}}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}}+\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{q}}\frac{\delta^{2}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}}+C_{i}^{\alpha}(\phi)\frac{\delta^{3}\Gamma^{SYM}}{\delta\phi_{i}\delta\phi_{p}\delta\phi_{q}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}$
$\displaystyle=$ $\displaystyle-\left[\frac{\xi^{\alpha}}{v_{k}}\frac{\partial
v_{k}}{\partial\xi^{\alpha}}\frac{\delta C_{ki}}{\delta\phi_{p}}+\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}\Pi_{qi}(p^{2})-\left[\frac{\xi^{\alpha}}{v_{k}}\frac{\partial
v_{k}}{\partial\xi^{\alpha}}\frac{\delta C_{ki}}{\delta\phi_{q}}+\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{p}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}\Pi_{pi}(p^{2}),$

where we have used Eqs.(7),(12) and (18). From Eqs.(20) and (22), we can see
that both the class I and II gauge fixing satisfy Eq.(5) with

$A_{1ij}^{T}=A_{2ij}=-\left[\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{j}}\right]_{\phi_{p}=v_{p}},$ (23)

in case of class I type of gauge fixing and

$A_{1ij}^{T}=A_{2ij}=-\left[\frac{\xi^{\alpha}}{v_{k}}\frac{\partial
v_{k}}{\partial\xi^{\alpha}}\frac{\delta C_{ki}}{\delta\phi_{j}}+\frac{\delta
C_{i}^{\alpha}(\phi)}{\delta\phi_{j}}\right]_{\phi_{p}=v_{p},u_{i}=v_{i}}$
(24)

in case of class II type of gauge fixing. As a result, the Higgs pole masses
of the multi-Higgs model are gauge independent in both the class I and II type
of the gauge fixing scheme.

##  III
DISCUSSIONS

In this paper, we have investigated the gauge parameter dependence of the
Higgs pole masses in the multi-Higgs model in two different types of the gauge
fixing conditions. It turns out that the Higgs pole masses in case of the
multi-Higgs model when expressed in terms of the Lagrangian parameters, are
independent of the gauge parameter. Extension of this method in case of the
pole mass for the gauge bosons in the multi-Higgs model seems necessarily and
this work is in progress.

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