[a,c]André H. Hoang

# Inclusive $\tau$ Hadronic Decay Rate in a Renormalon-Free Gluon Condensate
Scheme

Miguel A. Benitez-Rathgeb  Diogo Boito  Matthias Jamin  Christoph Regner

###### Abstract

In a recent work by some of us it was shown that the long-standing discrepancy
between the QCD perturbation series for the inclusive hadronic tau decay rate
computed in the CIPT and FOPT expansion approaches can be understood from the
fact that CIPT has an infrared (IR) sensitivity that it not compatible with
the standard form of the operator production expansion (OPE). For concrete IR
renormalon models for the QCD Adler function the resulting CIPT-FOPT
discrepancy, the asymptotic separation, can be calculated analytically from
the Borel representation of the CIPT series expansion. If the known
perturbative corrections for the QCD Adler function at the 5-loop level
already have a sizeable contribution from the asymptotic behavior related to
the gluon condensate (GC) renormalon, the asymptotic separation is dominated
by that renormalon. This implies that the CIPT expansion can be reconciled
with FOPT, when a renormalon-free scheme for the GC is adopted. In this talk
we discuss such a renormalon-free scheme for the GC, which involves
perturbative subtractions in analogy to using short-distance quark mass
schemes instead of the pole mass. Using a concrete realistic high-order Borel
model for the Adler function consistent with the known corrections up to 5
loops and containing a sizeable GC renormalon contribution, we show that the
CIPT-FOPT discrepancy can be avoided when switching to the renormalon-free GC
scheme. At the same time, the perturbative convergence of $\tau$ hadronic
spectral funtion moments strongly sensitive to the GC OPE corrections is
considerably improved. We show that these improvements may lead to higher
precision for strong coupling determinations.

##  1 Introduction

Extractions of the strong coupling $\alpha_{s}$ from inclusive hadronic $\tau$
decay spectral function moments represents one of the most precise methods to
obtain this fundamental parameter of QCD from experimental data. The
determinations are based on weighted finite-energy sum rule integrals
(spectral function moments) over the experimental inclusive $\tau$ hadronic
spectral functions compared to the corresponding theory predictions. The
inclusive $\tau$ hadronic decay width
$R_{\tau}\equiv\frac{\Gamma[\tau^{-}\to{\rm
hadrons}~{}\nu_{\tau}(\gamma)]}{\Gamma[\tau^{-}\to
e^{-}\overline{\nu}_{e}\nu_{\tau}(\gamma)]}$ is a particular case of these
spectral function moments. In the chiral limit the theoretical spectral
function moments can be dissected into the following individual contributions,

$\displaystyle R^{(W)}(s_{0})\,=\,\frac{3}{2}\,S_{\rm
ew}\,|V_{ud}|^{2}\Big{[}\,\delta^{\rm
tree}_{W}+\delta^{(0)}_{W}(s_{0})+\sum_{d\geq
4}\delta^{(d)}_{W}(s_{0})+\delta_{W}^{(\rm DV)}(s_{0})\Big{]}\,,$ (1)

where $S_{\rm ew}$ denotes electroweak corrections and $s_{0}$ is the upper
bound of the weighted spectral function integrations. $\delta^{\rm tree}_{W}$
corresponds to the tree level contribution while $\delta^{(0)}_{W}(s_{0})$
represents the perturbative QCD corrections. The index $W$ refers to the
weight function $W(x)$ which satisfies $W(1)=0$ and typically is a polynomial
in $x$. For the inclusive decay rate, which is called the kinematic moment,
this polynomial is $W_{\rm kin}=(1-x)^{3}(1+x)=1-2x+2x^{3}-x^{4}$ and we have
$s_{0}=m_{\tau}^{2}$. The term $\delta^{(d)}_{W}(s_{0})$ denotes non-
perturbative higher dimensional corrections in the operator product expansion
(OPE). The last term in Eq. (1), $\delta_{W}^{(\rm DV)}(s_{0})$, stands for
the duality violation (DV) contributions, which are suppressed for weight
functions with $\frac{d}{dx}W(1)=0$ and not further considered throughout this
talk.

The QCD corrections are given by contour integrals of the form
($x\equiv\frac{s}{s_{0}}$)

$\delta^{(0)}_{W}(s_{0})\,=\,\frac{1}{2\pi
i}\,\,\ointctrclockwise_{|s|=s_{0}}\\!\\!\frac{{\rm
d}s}{s}\,W({\textstyle\frac{s}{s_{0}}})\,\hat{D}(s),$ (2)

involving the (reduced) partonic Adler function $\hat{D}(s)$ for invariant
mass squared $s$,

$\displaystyle\hat{D}(s)$ $\displaystyle\,=\,$
$\displaystyle\,\sum\limits_{n=1}^{\infty}c_{n,1}\,\big{(}{\textstyle\frac{\alpha_{s}(-s)}{\pi}}\big{)}^{n}\,,$
(3) $\displaystyle\,=\,$
$\displaystyle\,\sum\limits_{n=1}^{\infty}\,\big{(}{\textstyle\frac{\alpha_{s}(s_{0})}{\pi}}\big{)}^{n}\,\sum\limits_{k=1}^{n}k\,c_{n,k}\,\ln^{k-1}({\textstyle\frac{-s}{s_{0}}})\,.$
(4)

The contour integral can be analytically related to an integration along the
real positive $s$-axis with $s_{0}$ being the upper bound. The conventional
integration path along the circle with $|s|=s_{0}$ may be deformed as long as
it remains in the perturbative region. The nonperturbative terms
$\delta^{(d)}_{W}(s_{0})$ can be obtained from an analogous integral over the
Adler function’s OPE corrections
($a(\mu^{2})=\alpha_{s}(\mu^{2})\beta_{0}/4\pi$)

$\displaystyle D^{\rm
OPE}(s)\,=\,\frac{2\pi^{2}}{3}\,\frac{1-\frac{22}{81}\,a(-s)+\ldots}{s^{2}}\,\langle\bar{G}^{2}\rangle+\sum_{d=6}^{\infty}\frac{1}{(-s)^{d/2}}\sum_{i}C_{d,i}(\alpha_{s}(-s))\langle\bar{\cal
O}_{d,\gamma_{i}}\rangle\,,$ (5)

where the term that is leading in the power expansion in $\Lambda_{\rm
QCD}^{2}/s$ involves the renormalization scheme-invariant GC matrix element
$\langle\bar{G}^{2}\rangle=\langle\Omega|(\frac{\alpha_{s}}{\pi}+\ldots)G^{\mu\nu}G_{\mu\nu}|\Omega\rangle$.
For spectral function moments with a weight function without a quadratic
$x^{2}$ term, such as $W_{\rm kin}$, the GC correction is very strongly
suppressed (and even vanishing when the higher order corrections in its Wilson
coefficient are neglected).

In the theory integrals a prescription for setting the renormalization scale
$\mu$ has to be adopted and the two most commonly considered prescriptions are
known as fixed-order (FOPT), where $\delta^{(0)}_{W}(s_{0})$ is an expansion
in powers of $\alpha_{s}(s_{0})$ (see Eq. (4)), and contour-improved
perturbation theory (CIPT) [1], where the expansion is in powers of
$\alpha_{s}(-s)$ before doing the contour integration (see Eq. (3)). When
determining the strong coupling from $\tau$ hadronic spectral function
moments, the discrepancy of the results obtained from using either FOPT or
CIPT has been discussed controversially [2, 3] and up to now constituted one
of the dominant uncertainties in the determination of $\alpha_{s}$ from
hadronic $\tau$ decays. In general, CIPT leads to smaller values for
$\delta^{(0)}_{W}(s_{0})$ than FOPT, which typically results in larger
extracted values for $\alpha_{s}(m_{\tau}^{2})$ when the CIPT expansion is
employed.

##  2 Asymptotic Separation

In a recent work two of us have provided a conceptual as well as quantitative
analysis of the observed CIPT-FOPT discrepancy. In contrast to previous
studies [6, 2, 3], where it was assumed that the Borel representations for the
FOPT and CIPT expansion series are identical, it was shown in Ref. [4, 5] that
their Borel representations are in fact intrinsically different in the
presence of IR renormalons. In particular, the expressions for the two non-
equivalent representations are given by

$\displaystyle\delta_{W,{\rm Borel}}^{(0),{\rm FOPT}}(s_{0})$
$\displaystyle={\rm PV}\int_{0}^{\infty}\\!\\!{\rm d}u\,\,\frac{1}{2\pi
i}\,\ointctrclockwise_{|x|=1}\frac{{\rm
d}x}{x}\,W(x)\,B[\hat{D}](u)\,e^{-\frac{u}{a(-xs_{0})}}\,,$ (6)
$\displaystyle\delta_{W,{\rm Borel}}^{(0),{\rm CIPT}}(s_{0})$
$\displaystyle=\int_{0}^{\infty}\\!\\!{\rm d}\bar{u}\,\,\frac{1}{2\pi
i}\,\ointctrclockwise_{{\cal C}_{x}}\frac{{\rm
d}x}{x}\,W(x)\,\big{(}{\textstyle\frac{\alpha_{s}(-xs_{0})}{\alpha_{s}(s_{0})}}\big{)}\,B[\hat{D}]\Big{(}{\textstyle\frac{\alpha_{s}(-xs_{0})}{\alpha_{s}(s_{0})}}\bar{u}\Big{)}\,e^{-\frac{\bar{u}}{a(s_{0})}}\,,$
(7)

where $B[\hat{D}](u)$ is the Borel function of the real-valued Euclidean Adler
function $\hat{D}(-s_{0})$ with respect to an expansion in
$\alpha_{s}(s_{0})$, in the sense that the $u$-Taylor series of
$B[\hat{D}](u)$ gives the series in powers of $\alpha_{s}(s_{0})$ through the
Borel integral

$\displaystyle\hat{D}(-s_{0})=\int^{\infty}_{0}\mathrm{d}u\,B[\hat{D}](u)\,\mathrm{e}^{-\frac{u}{a(s_{0})}}\,.$
(8)

The previously known FOPT Borel representation in Eq. (6) includes the
commonly employed principal value (PV) prescription to obtain a well-defined
result for the value of the Borel representation (briefly called the Borel
sum). This is due to the IR renormalon cuts (or poles) contained in
$B[\hat{D}](u)$ along the positive real $u$-axis. The novel Borel
representation for the CIPT series in Eq. (7) implies a different
regularization of the non-analytic IR renormalon cuts because
$\frac{\alpha_{s}(-xs_{0})}{\alpha_{s}(s_{0})}\bar{u}$ is complex-valued. Due
to the fact that the complex-valued coupling $\alpha_{s}(-xs_{0})$ is now part
of the argument of the Borel function $B[\hat{D}]$, the path ${\cal{C}}_{x}$
needs to be deformed away from the circle $|x|=1$ to account for the modified
singularity structure when the CIPT Borel sum is evaluated.

Figure 1:  Graphical representation of the integration paths in the complex
Borel plane involved for the FOPT and CIPT approach for the cases of a UV
renormalon ($p<0$, left panel) and an IR renormalon ($p>0$, right panel). The
red zig-zag lines denote renormalon cuts starting at $u=p$. Figures taken from
Ref. [4].

The Borel representations are formally related through the complex-valued
change of variables $u=\bar{u}\,\alpha_{s}(-xs_{0})/\alpha_{s}(s_{0})$, which
(as illustrated in Fig. 1 for ${\rm Im}[\alpha_{s}(-xs_{0})]>0$) leads to a
difference between $\delta_{W,{\rm Borel}}^{(0),{\rm FOPT}}(s_{0})$ and
$\delta_{W,{\rm Borel}}^{(0),{\rm CIPT}}(s_{0})$ in the presence of IR
renormalon cuts. This difference can be computed analytically for any concrete
expression of the Borel function $B(u)$ and is dubbed the asymptotic
separation (AS). The asymptotic separation describes the observed discrepancy
between the CIPT and FOPT expansions for spectral function moments very well,
as is shown for example for the kinematic moment in the left panel in Fig. 2
as a function the order. Note that the results shown in Fig. 2 up to order 5
(i.e. 6-loops for the Adler function) correspond to the QCD corrections used
in state-of-the-art phenomenological analyses,111This includes the concretely
known 5-loop corrections up to $c_{4,1}$ [12] and an estimate for the 6-loop
coefficient $c_{5,1}$. while the orders beyond are based on a Borel model for
$B[\hat{D}](u)$ following Ref. [6]. The FOPT Borel sum (red horizontal line)
and the CIPT Borel sum (blue horizontal line), which is the sum of the FOPT
Borel sum and the asymptotic separation, are computed from the Borel model as
well. The Borel model provides a good approximation for the true Borel
function [3] and relies on the proposition that the asymptotic behavior of the
gluon condensate (GC) renormalon has a sizeable contribution to the known QCD
corrections up to 5-loops and therefore contains a GC contribution with a
sizeable norm $N_{g}$. The known QCD corrections to the Adler function are
fully consistent with this proposition. In this context the asymptotic
separation is numerically by far dominated by the contribution from the GC
renormalon.

One can also construct conceptual toy Borel models for $B[\hat{D}](u)$
containing only the GC renormalon, which in the context of the standard form
of the OPE and the canonical renormalon calculus formalism must lead to
convergent spectral function moments for weight functions such as $W_{\rm
kin}$ (because for these spectral function moments all OPE corrections vanish
identically). It was shown in Ref. [4, 5] that for these moments
$\delta^{(0)}_{W}(s_{0})$ is indeed convergent in the FOPT expansion, while
the CIPT expansion series is divergent. This implies that the OPE corrections
that need to be added to $\delta^{(0)}_{W}$ in the CIPT approach differ from
those in the FOPT approach and, more importantly, do not have the standard
form assumed in the literature. In other words, the CIPT expansion and the
CIPT Borel representations given in Eq. (7) are not consistent with the
standard form of the OPE as given in Eq. (5).

From these results one can conclude that the consistency of the CIPT expansion
can be reconciled (at least for the most part) if the GC renormalon is removed
from the Adler function. This can be achieved through a change of scheme for
the GC matrix element, which is in close analogy to the well-known use of
short-distance mass renormalization schemes in favor of the pole mass for
quark mass-sensitive observables [7, 8].

Figure 2:  Perturbative series for the kinematic moment for
$s_{0}=m_{\tau}^{2}$ in the FOPT and CIPT expansions as a function of order
$n$ in the $\overline{\rm MS}$ GC scheme (left panel) and the RF GC scheme for
$R=0.8\,m_{\tau}$ and $N_{g}=0.64$ (right panel) using the Borel model in
section 6 of Ref. [6] for $n>5$. The red and blue horizontal lines represents
the Borel sums (BSs) of the FOPT and CIPT series, respectively.

##  3 Renormalon-Free Gluon Condensate Scheme

In Ref. [9] a suitable renormalon-free (RF) GC matrix element scheme was
constructed in two steps. Calling the original GC matrix element
$\langle\bar{G}^{2}\rangle$ (which still contains the ${\cal O}(\Lambda_{\rm
QCD}^{4})$ renormalon) ${\overline{\rm MS}}$ GC scheme, one can first define
an $R$-scale dependent renormalon-free GC matrix element $\langle
G^{2}\rangle(R^{2})$ by the relation

$\displaystyle\langle\bar{G}^{2}\rangle\equiv\langle
G^{2}\rangle(R^{2})-R^{4}\sum_{\ell=1}N_{g}~{}r_{\ell}^{(4,0)}\bar{a}^{\ell}(R^{2})\,.$
(9)

The term $N_{g}$ refers to the universal norm of the gluon condensate and $R$
takes the role of an IR factorization scale. The coefficients
$r_{\ell}^{(4,0)}$ contain the divergent asymptotic series related to a pure
${\cal O}(\Lambda_{\rm QCD}^{4})$ renormalon and, using the $C$-scheme for the
strong coupling [10] (indicated by the bar), can be given in closed form,
$r_{\ell}^{(4,0)}=(\frac{1}{2})^{\ell+4\hat{b}_{1}}\,\frac{\Gamma(\ell+4\hat{b}_{1})}{\Gamma(1+4\hat{b}_{1})}$.
Numerically the $C$-scheme for the strong coupling is very close to the
$\overline{\rm MS}$ scheme, see the appendix of Ref. [9] for details. However,
the definition of Eq. (9) is somewhat inconvenient for actual phenomenological
applications, since the dependence on $R$, which cancels between the
subtraction series and $\langle G^{2}\rangle(R^{2})$, would generate a
sizeable intermediate $R$-dependence. It is therefore suitable to define, in a
second step a scale-invariant GC matrix element $\langle G^{2}\rangle^{\rm
RF}$, called the RF GC scheme, by the relation

$\displaystyle\langle G^{2}\rangle(R^{2})$ $\displaystyle\equiv$
$\displaystyle\langle G^{2}\rangle^{\rm RF}+N_{g}\,\bar{c}_{0}(R^{2})\,.$ (10)

The function $\bar{c}_{0}(R^{2})$ satisfies the same $R$-evolution equation as
the subtraction series on the RHS in Eq. (9) and thus of $\langle
G^{2}\rangle(R^{2})$. Because the subtraction series contains a pure ${\cal
O}(\Lambda_{\rm QCD}^{4})$ renormalon this $R$-evolution equation is a
convergent series [11] and can even be given in closed form [9]. A suitable
choice is ($\hat{b}_{1}=\beta_{1}/2\beta_{0}^{2}$)

$\bar{c}_{0}(R^{2})\equiv-\frac{R^{4}\,e^{-\frac{2}{\bar{a}(R^{2})}}}{(\bar{a}(R^{2}))^{4\hat{b}_{1}}}\,{\rm
Re}\left[\,e^{4\pi\hat{b}_{1}i}\,\Gamma\Big{(}\\!-4\hat{b}_{1},-\frac{2}{\bar{a}(R^{2})}\,\Big{)}\,\right]\,,$
(11)

which is the FOPT Borel sum of the subtraction series itself using the common
PV prescription.

The perturbative Adler function in the RF GC scheme then has the form

$\displaystyle\hat{D}^{\rm RF}(s,R^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{s^{2}}\,\Big{[}1+\bar{c}_{4,0}^{(1)}\bar{a}(-s)\Big{]}\,\frac{2\pi^{2}}{3}\,N_{g}\,\bar{c}_{0}(R^{2})\,+\,\sum_{\ell=1}^{\infty}\,\bar{c}_{\ell}\,\bar{a}^{\ell}(-s)\,\qquad$
(12)
$\displaystyle\,-\,\Big{[}1+\bar{c}_{4,0}^{(1)}\,\bar{a}(-s)\Big{]}\,\frac{2\pi^{2}}{3}\,N_{g}\,\frac{R^{4}}{s^{2}}\sum\limits_{\ell=1}^{\infty}\Big{(}\frac{1}{2}\Big{)}^{\ell+4\hat{b}_{1}}\,\frac{\Gamma(\ell+4\hat{b}_{1})}{\Gamma(1+4\hat{b}_{1})}\,\bar{a}^{\ell}(R^{2})\,,$

where the term involving $\bar{c}_{0}(R^{2})$ must be treated strictly as a
tree-level term (i.e. not being reexpanded again and numerically evaluated in
the $C$-scheme for the strong coupling). For the perturbation series, either
using the CIPT or the FOPT renormalization scale setting prescription, it is
mandatory to consistently expand and truncate the other terms in
$a^{\ell}(-s)$ or $a^{\ell}(s_{0})$, i.e. using the strong coupling at a
common renormalization scale.222This expansion may be carried out using the
usual $\overline{\rm MS}$ strong coupling scheme. We have done so in our
numerical analyses below. This ensures the systematic removal of the GC
renormalon from the Adler function. The GC OPE correction in the RF GC scheme
adopts the standard form

$\displaystyle\delta D^{\rm OPE,RF}_{4,0}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{s^{2}}\frac{2\pi^{2}}{3}\,\Big{[}1-\frac{22}{81}\,\bar{a}(-s)\Big{]}\,\langle
G^{2}\rangle^{\rm RF}\,.$ (13)

A graphical illustration of the impact of the subtraction of the gluon
condensate renormalon on the perturbative series in the RF GC scheme is given
by the right panel in Fig. 2 for the case of the kinematic moment, for
$R=0.8\,m_{\tau}$ and considering again the Borel model for the orders beyond
the 6-loop level. The Borel model provides the concrete value $N_{g}=0.64$ for
the GC renormalon norm. In the RF GC scheme the CIPT and FOPT series are
perfectly compatible and both approach the FOPT Borel sum given by the FOPT
Borel representation in Eq. (6).

Figure 3:  FOPT and CIPT expansion series in the $\overline{\rm MS}$ GC scheme
(left panel) as well as in the RF GC scheme for $R=0.8\,m_{\tau}$ and
$N_{g}=0.64$ (right panel) for the case of the kinematic moment with
$\alpha_{s}(m_{\tau}^{2})=0.315$ and including renormalization scale
variations. The vertical line indicates the model dependence starting at order
six. The space between the dashed lines corresponds to the ambiguity of the
FOPT Borel sum. Figures taken from Ref. [9].

In Fig. 3 the results of Fig. 2 are shown once more, but this time we include
the error bands arising from renormalization scale variations. The latter are
determined by expanding in powers of $\bar{a}^{\ell}(-\xi s)$ (for CIPT) and
$\bar{a}^{\ell}(\xi s_{0})$ (for FOPT) and using variations in the interval
$1/2\leq\xi\leq 2$. While the scale variation bands do not overlap at the six-
loop level in the $\overline{\rm MS}$ GC scheme, they do so in the RF GC
scheme.

##  4 Strong coupling predictions in the RF GC scheme

The CIPT-FOPT discrepancy has been one of the dominant sources of uncertainty
in $\alpha_{s}$ determinations from experimental data on the inclusive
hadronic $\tau$ decay spectra. We can study the impact the RF GC can have on
strong coupling determinations from data through the following simplistic toy
analysis: We take the spectral function moment FOPT Borel sum of Eq. (6) based
on the Borel model mentioned above for the input value
$\alpha_{s}(m_{\tau}^{2})=0.315$ as a proxy for the perturbative part of the
“experimental" moment values and determine the strong coupling from the 6-loop
series (as it is common in current state-of-the-art strong coupling
determinations). To estimate the theoretical uncertainty we account for
renormalization scale variations (using the prescription of Fig. 3), an
uncertainty in the estimate of the 6-loop coefficient $c_{5,1}$ and variations
for $R$ in the range $0.7m_{\tau}\leq R\leq 0.9m_{\tau}$ (see Ref. [9] for
details). We consider the spectral function moments for the weight functions
$W_{\rm kin}(x)$, $W_{2}(x)=(1-x)^{3}=1-3x+3x^{2}-x^{3}$ and
$W_{3}(x)=2(1-x)$. For the spectral function moments associated to the weight
functions $W_{\rm kin}(x)$ and $W_{3}(x)$ the GC OPE correction is highly
suppressed (because the weight functions do not contain a quadratic term
$x^{2}$). Spectral function moments of this kind have been the basis of all
recent high precision determinations of $\alpha_{s}$ from hadronic $\tau$
decay data. On the other hand, for the spectral function moment associated to
the weight function $W_{2}(x)$ the GC OPE correction unsuppressed (because
$W_{2}$ contains a quadratic term $x^{2}$).

The results are shown in Fig. 4. The left panel shows to the resulting
extracted values for $\alpha_{s}(m_{\tau}^{2})$ using perturbation theory in
the unsubtracted $\overline{\rm MS}$ GC scheme. For the kinematic moment and
the one associated to the weight function $W_{3}$ we see the well-known CIPT-
FOPT discrepancy, where the values for $\alpha_{s}(m_{\tau}^{2})$ obtained
from CIPT and FOPT are not compatible and where the result for
$\alpha_{s}(m_{\tau}^{2})$ from the CIPT expansion comes out way too large.
For the moment associated to the weight function $W_{2}$ the results are
compatible, but the errors are very large as well, reflecting the large
perturbative uncertainties for moments where the GC OPE corrections is
unsuppressed. The right panel shows the results obtained using the RF GC
scheme. The CIPT and FOPT results for $\alpha_{s}(m_{\tau}^{2})$ obtained from
the moments for $W_{\rm kin}$ and $W_{3}$ are now fully compatible and
consistent with the input value $\alpha_{s}(m_{\tau}^{2})=0.315$. This has
been achieved due to the significant modification of the CIPT expansion when
the RF GC scheme is employed. Furthermore, the large uncertainties for
$\alpha_{s}(m_{\tau}^{2})$ obtained from the moment associated to the weight
function $W_{2}$, which were large for CIPT as well as FOPT in the
$\overline{\rm MS}$ GC scheme, have decreased significantly. The results show
that in the RF GC scheme we can achieve two improvements. First, the CIPT-FOPT
discrepancy in strong coupling determinations can be avoided and, second,
high-precision analyses can be carried out with spectral function moments
where the GC OPE corrections is unsuppressed.

Figure 4:  Strong coupling results determined from the toy analysis described
in Sec. 4. The input value for the ‘experimental’ moments is
$\alpha_{s}(m_{\tau}^{2})=0.315$. Figures taken from Ref. [9].

##  5 Conclusions

In the recent work of Refs. [4, 5] it was shown that the Borel representations
for the FOPT and CIPT hadronic $\tau$ spectral function moments are
intrinsically different. In the presence of IR renormalons the different
analytic properties inherent to the FOPT and CIPT expansions results in a
systematic and computable disparity in their respective Borel sums, which we
called the asymptotic separation. From dedicated studies of toy Borel models
for the Adler function it was shown that the CIPT expansion is not compatible
with the standard form of the operator product expansion. Given that the major
contribution to the CIPT-FOPT discrepancy is related to the leading gluon
condensate renormalon, we have shown in Ref. [9] that the discrepancy between
the two approaches can be reconciled in an IR subtracted perturbation theory.
In particular, we defined a new renormalon-free gluon condensate scheme [9]
and observed that in this scheme the perturbative series indeed converge to
the same Borel sum given by the FOPT Borel representation. By considering a
toy model extraction of the strong coupling, we showed that the RF GC scheme
will lead to strong coupling determinations from hadronic $\tau$ decays where
the CIPT-FOPT discrepancy does not arise any more. The results shown in this
talk were based on the concrete value $N_{g}=0.64$ for gluon condensate
renormalon norm which was obtained from a concrete realistic Borel model for
the Adler function. In practice $N_{g}$ can only be determined approximately.
In Ref. [13] we have carried out complete phenomenological strong coupling
determinations in the RF GC scheme, accounting also for the uncertainty in the
GC renormalon norm $N_{g}$, which turn out to give a relatively small
contribution to the final error in $\alpha_{s}(m_{\tau}^{2})$.

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A new class of holographic dark energy models in LRS Bianchi Type-I
Anirudh Pradhan1, Vinod Kumar Bhardwaj2, Archana Dixit3, Syamala Krishnannair4

1,2,3Department of Mathematics, Institute of Applied Sciences and Humanities,
GLA University, Mathura-281 406, Uttar Pradesh, India

4Department of Mathematical Sciences, Faculty of Science, Agriculture and
Engineering,University of Zululand, Kwadlangezwa 3886, South Africa

1E-mail<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

In this paper, we examine the (LRS) Bianchi-type-I cosmological model with
holographic dark energy. The exact solutions to the corresponding field
equations are obtained by using generalized hybrid expansion law (HEL). The
EoS parameter $\omega$ for DE is found to be time dependent and redshift
dependent and its exiting range for derived model is agreeing well with the
current observations. Here we likewise apply two mathematical diagnostics, the
statefinder (r, s) and $\omega_{d}-\omega^{{}^{\prime}}_{d}$ plane to
segregate HDE model from the $\Lambda CDM$ model. Here the
$\omega_{d}-\omega^{{}^{\prime}}_{d}$ diagnostic trajectories is the good tool
to classifying the dynamical DE model. We found that our model lies in both
thawing region and freezing region. We have also construct the potential as
well as dynamics of the quintessence and tachyon scalar field. Some physical
and geometric properties of this model along with the physical acceptability
of cosmological solution have been discussed in detail.

Keywords : LRS Bianchi-I HDE models; Statefinders;
$\omega_{d}-\omega_{d}^{{}^{\prime}}$ plane; Scalar field DE models
PACS: 98.80.-k

##  1 Introduction

Motivated by the discovery of “black hole thermodynamics [1, 2] Hooft proposed
the well known holographic principle (HP)” [3]. As a advanced form of “Plato’s
cave ”, the HP describes that all of the info confined in a bulk of space may
be visualized as a hologram, which correlates to a concept positioning on the
frontier of that space. Afterward, Susskind [4] suggest the exact string
hypothesis thought of the rule. Besides, Maldacena [5] recommended the notable
“ AdS/CFT correspondence” in 1997, which is the best acknowledgment of
Holographic Principle. Presently it notable that Holographic Principle is
normally used to be an essential idea of quantum gravity.

According to the basic concepts of quantum gravitational, we can also
investigate the nature of DE, by HDE principle. As per this rule, the ‘degrees
of freedom’ in a constrained structure must be finite and not measure with the
volume of the system [6]. As we mention above that HP is the essential part of
“quantum gravity” and has significant potential to solve several long-lasting
problems in different fields. The HP can also be used to solve the DE problem.
The HP reveal us that all the physical measures of the universe, like energy
density of DE ($\rho_{d}$), can be depicted by certain amounts on the limit of
the universe. This indicates that only two physical measures “the diminished
Planck mass $M_{p}$ and the cosmological length scale $L$”, can be utilized to
develop the expression of $\rho_{d}$. In view of the dimensional analysis, we
have

$\rho_{d}=c_{1}M_{p}^{4}+c_{2}M_{p}^{4}L^{-2}+c_{3}L^{4-}+........$ (1)

here $c_{1},c_{2},c_{3}$ are constants. Author [7] suggest that the $c_{1}$
expression is not viable with Holographic principle. According to the HP, the
local quantum field (LQF) theory should not be a decent characterization for a
black hole. Particularly, the traditional estimate $\rho_{d}\approx
c_{1}M^{4}_{P}$ the (LQF) this explanation should not include any theory. The
LQF theory attains a non-trivial ‘UV cutoff’ $\Lambda$. As a consequence of
the QF theory, the vacuum fluctuation calculated is $\rho_{d}\approx
c_{1}M_{4}^{p}L^{-2}$. The suggestion of HDE was depend on the concept of (QF)
theory, that a short distance cut-off is associated to a long-distance cut-off
due to the limit set by the establishment of a black hole [8]-[14].

We compared only second term and neglecting other terms then we get HDE as

$\rho_{d}=3C^{2}M^{2}_{p}L^{-2}$ (2)

where $C$ is other constant. In Eq.(1) $c_{1}$ term is not present in this
expression. It is important to note that this expression of $\rho_{d}$ is
obtained by dimensional analysis and applying HP, rather than adding a DE term
to the Lagrangian. Because of this remarkable feature HDE theory distinguished
from other DE theories. In the cosmological context Fischler and Susskind [15]
suggested a new version of the holographic concept in the cosmological
evolution. In the framework of the DE problem, HP provides a relation
$\rho=H^{2}$. Granda and Olivers suggested a holographic density of the type
$\rho\approx\alpha H^{2}+\beta\dot{H}$ in [16], where $\alpha$ and $\beta$ are
constants. They contend that their new DE model describes the universe’s
accelerated expansion and it is consistent with current observational
findings. Bianchi-type models are the most basic and accurate anisotropic
models, completely describing anisotropic effects. Chimento et al. [17] looked
at a Universe with an interacting dark sector and a decoupling component that
may be used to simulate a radiation term. [18] discusses a model of
interacting dark matter and dark energy based on a modified holographic Ricci
cutoff in a spatially flat FRW spacetime. In [19], a universe with interacting
dark matter, holographic dark energy, and dark radiation is examined for FRW
spacetime. FRW universe with interacting dark matter, modified holographic
Ricci dark energy, and a decoupled baryonic component was studied by Chimento
et al. [20]. Chimento et al. [21] also introduced scalar field models through
internal space structure generalization of the quintom cosmological scenario.

In this manner numerous cosmologist have proposed the various aspects of these
spatially homogeneous and anisotropic Bianchi-type models in various modified
theories [22]-[29]. Zadeh [30] examined the astronomical advancement of THDE
in Bianchi type-I model with various IR cut-offs. Author [31, 32, 33] dealt
with different anisotropic DE cosmological models . Authors [34, 35] worked on
the general class of anisotropic cosmological models in particular theories.

In the literature, anisotropic Bianchi type-I, III, and V DE models with the
standard perfect fluid have been also widely researched [36]-[42]. Sarkar and
Mahanta [43] developed a HDE cosmological model in Bianchi type-I space time
with quintessence. Adhav et al. [44] investigated an homogeneous and
anisotropic Bianchi type-I universe field with interacting dark matter and
HDE. A similar way Pradhan and Amirhashchi [45] constructed a accelerating DE
model and investigated some new precise solutions of Einstein’s field
conditions in a spatially and anisotropic Bianchi type-V space-time.

The Bianchi type-I string cosmological model with bulk viscosity was examined
by Tiwari and Sonia [46] . Many cosmologists [47, 48, 49] have explored the
Bianchi type-I models from a different perspective. Now we can say that
Spatially homogeneous and anisotropic cosmological models play an important
role in the description of large-scale phenomena in the cosmos. Among the
numerous dynamical DE models, the cosmological constant is the most important
component but it suffers from ‘cosmic coincidence and fine tuning’ problems.
As a result of this explanation, various dynamical DE models have been
recommended and they were given as family of ‘scalar field models (
quintessence, k-essence, phantom, ghost, etc.), Chaplygin gas as well as
holographic DE (HDE)’ models. The authors [50] concluded that the modified
structure scalars play an important role in the dynamics of compact object.
The authors [51] investigated the oscillatory behaviour of the anisotropy
Bianchi -I spacetime. and emphasized the presence of a residue in the energy
density in a late time isotropic universe coming from its past anisotropic
behaviour. Renyi HDE correspondence is discussed in higher dimensional Kaluza-
Klein cosmology [52].

In this manuscript we have chosen LRS Bianchi types-I (LRSBT-I) homogeneous
and isotropic model. This model is very important to study space-times where
anisotropy occurs at early stage and isotropy at later stage of the evolution
of the universe. We have developed various cosmological parameters to describe
the dynamics of universe. These parameters are evaluated in terms of $t$ and
redshift $z$. Our work is outlined in the following manner: In Sec. $2$, we
formulate the basic field equations for LRSBT-I metric. The solutions and the
physical behaviour of the cosmological parameters are discussed in Sec. $3$.
In Sect. $4$, we have discussed the state finder diagnosis. In Sec. $5$, we
have studied the distinct regions in $\omega_{d}-\omega_{d}^{{}^{\prime}}$
plane. In Sec. $6$, we have constructed the DE scalar field models for tachyon
and quintessence. The nature of the deceleration parameter ($q$) is discussed
in Sec. $7$. We have concluded the results in the last Sec. $8$.

##  2  LRS Bianchi type-I universe and Basic Field Equations

In this cosmological model we consider the LRS Bianchi type I metric of the
form

$ds^{2}=-dt^{2}+A(t)^{2}dx^{2}+B(t)^{2}\left(dy^{2}+dz^{2}\right),$ (3)

where the metric coefficients “$A$ and $B$” are function of $t$ only.

In general relativity, Einstein’s field equation with the cosmological
constant:

$R_{ij}-\frac{1}{2}g_{ij}R+\Lambda g_{ij}=-\left(T_{ij}+\bar{T}_{ij}\right)$
(4)

The notations have their typical significance. The new HDE for actual
interpretation and the energy momentum tensor for matter can be composed as :
$T_{ij}=\rho_{m}u_{i}u_{j}$,
$\bar{T}_{ij}=(\rho_{d}+p_{d})u_{i}u_{j}+g_{ij}p_{d}$, where $\rho_{m}$ and
$\rho_{d}$ address the matter energy density and new HDE density, and $p_{d}$
stands for the HDE pressure. The field equations given by Eq. (4) along with
the energy momentum tensors defined above for the LRSBT-I metric of three
independent equations:

$2\frac{\ddot{B}}{B}+\frac{\dot{B}^{2}}{B^{2}}=-p_{d}+\Lambda,$ (5)
$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}=-p_{d}+\Lambda,$
(6)
$2\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}^{2}}{B^{2}}=\rho_{m}+\rho_{d}+\Lambda.$
(7)

Now, as indicated in Refs. [53, 54], the HDE density with IR cut-off is
defined as:

$\rho_{d}=3M^{2}_{p}(kH^{2}+m\dot{H}),$ (8)

where $k$ and $m$ are the dimensionless parameters and must satisfy the
restrictions imposed by the current observational framework with
$M^{2}_{p}=8\pi G=1$. The energy conservation law is determined as
$T^{ij}_{;j}=0$, gives
$\rho_{m}+\dot{\rho_{d}}+3H(\rho_{m}+\rho_{d}+p_{d})=0$. The energy
conservation law for matter and dark energy are, separately expressed as
$\dot{\rho_{m}}+3H\rho_{m}=0$ and $\dot{\rho_{d}}+3H(\rho_{d}+p_{d})=0$
respectively. Here, $H$ is the Hubble parameter defined by
$H=\frac{\dot{a(t)}}{a(t)}$, where the dot represents a time derivative. The
Hubble parameter varies with time, not with space, being the Hubble constant
$H_{0}$ is the current value. The relative expansion of the universe is
parametrized by a dimensionless scale factor $a$. The Hubble constant
($H_{0}$) is important as it is needed to estimate the size and the age of the
Universe. $H$ indicates the rate at which the universe is expanding, from the
pre mordial “Big Bang”.

##  3 Solutions and Physical Behaviour of the Model

Here, we have three equations Eq. (5)- Eq. (7) with five unknown parameters
namely A, B, $\rho_{d}$, $\rho_{m}$ and $\Lambda$. For solving these
equations, we need two more physical assumption. Our first assumption has to
taken in Eq. (8) and the second assumption is the scale factor $a(t)$ which is
defined as [55, 56] :

$a=t^{\alpha}e^{\beta t}$ (9)

here $\alpha$ and $\beta$ are constant. The scale factor is the fundamental
function that characterises the global features of our Universe. For the
analysis of various problems, an exact equation for this scale factor as a
function of physical and conformal time is desirable [57]. In cosmology, the
scale factor is a parameter that represents how the universe’s size is
changing in relation to its current size.

This results have a linear relationship between the directional Hubble
parameters as $H_{1}=nH_{2}$ where $n$ is positive constant. Here
$A=\left(t^{\alpha}e^{\beta t}\right)^{\frac{3n}{(n+2)}}$,
$B=\left(t^{\alpha}e^{\beta t}\right)^{\frac{3}{(n+2)}}$, then we obtained
$H_{1}=\frac{\dot{A}}{A}=\frac{3n}{(n+2)}\left(\frac{\alpha}{t}+\beta\right)$,
$H_{2}=\frac{\dot{B}}{B}=\frac{3}{(n+2)}\left(\frac{\alpha}{t}+\beta\right)$
are the directional Hubble Parameters. The Hubble Parameter is derived as:

$H=\left(\frac{\alpha}{t}+\beta\right)$ (10)

By using Eq. (9) we obtained deceleration parameter
$q=\frac{\alpha}{(\alpha+\beta t)^{2}}-1$. This defines the relation
$\beta=\dfrac{1}{13.8}[\left(\frac{\alpha}{0.27}\right)^{1/2}-\alpha]$,
between $\alpha$ and $\beta$, by using $t_{0}=13.8Gry$ $q_{0}=-0.73$, we get
the corresponding value of $\beta$ [58, 59] [See Table - 1]. From this table,
we observe that $\alpha$ increases with the increase of $\beta$. For
$0<\alpha\leq 1$, $\beta$ increases but for $\alpha>1$, $\beta$ starts
decreasing. In all figures we have considered the three sets of ($\alpha$,
$\beta$) as ($0.25$, $0.0516$), ($0.50$, $0.0624$) and ($0.75$, $0.0665$)
respectively. It is worth mentioned that for $0<\alpha<1$, our models show the
transitioning scenario whereas for $\alpha\geq 1$, the models are in
accelerating phase only.

Using Eq. (9) in (8), we obtain the HDE of the model as:

$\rho_{d}=\frac{3\left(k(\alpha+\beta t)^{2}-\alpha m\right)}{t^{2}}.$ (11)

By using Eqs. (10)-(11) in energy conservation law
$\dot{\rho_{d}}+3H(1+\omega_{d})\rho_{d}=0$, we get the EoS parameter for the
model as:

$\omega_{d}=-\frac{2\alpha(m-k(\alpha+\beta t))}{3(\alpha+\beta
t)\left(k(\alpha+\beta t)^{2}-\alpha m\right)}-1.$ (12)

Solving Eqs. (5)-(7) and using Eqs. (9)-(11), the expressions for different
cosmological parameters, $p_{d}$, $\rho_{m}$ and $\Lambda$ are obtained as

$p_{d}=\frac{3\left(k(\alpha+\beta t)^{2}-\alpha
m\right)}{t^{2}}\left(-\frac{2\alpha(m-k(\alpha+\beta t))}{3(\alpha+\beta
t)\left(k(\alpha+\beta t)^{2}-\alpha m\right)}-1\right),$ (13)
$\rho_{m}=\frac{2}{t^{2}}\left(\frac{\alpha(-k)(n+2)^{2}+9\alpha^{2}(n-1)+9\beta^{2}(n-1)t^{2}+3\alpha(6\beta
nt+n-6\beta t+2)}{(n+2)^{2}}+\frac{\alpha m}{\alpha+\beta t}\right),$ (14)

$\Lambda=\frac{1}{t^{2}}\left(2\alpha k+3\left(\alpha m-k(\alpha+\beta
t)^{2}\right)-\frac{2\alpha m}{\alpha+\beta
t}-\frac{6\alpha}{n+2}+\frac{27(\alpha+\beta t)^{2}}{(n+2)^{2}}\right).$ (15)

The matter density parameter $\Omega_{m}$ and HDE density parameter
$\Omega_{d}$ are given by $\Omega_{m}=\frac{\rho_{m}}{3H^{2}}$ and
$\Omega_{d}=\frac{\rho_{d}}{3H^{2}}$.

$\displaystyle\Omega_{m}=\frac{\alpha(-k)(n+2)^{2}+18\alpha^{2}(n-1)+18\beta^{2}(n-1)t^{2}+6\alpha(6\beta
nt+n-6\beta t+2)}{3(2\alpha+\beta t)^{2}(n+2)^{2}}+\frac{2\alpha
m}{3(\alpha+\beta t)^{3}}$ (16) $\Omega_{d}=k-\frac{\alpha m}{(\alpha+\beta
t)^{2}}$ (17)

(a) (b) .

Figure 1: Variation of EoS parameter $\omega_{d}$ versus $t$ and $z$

The redshift of a comoving cosmological source will change with time along the
expansion of the universe. The monitoring of the redshift was first proposed
by Sandage in 1962 [60], as a useful tool for testing of the cosmology. The
expansion of universe can be directly measured by the redshift drift.
Therefore, redshift is projected as a valuable addition to other DE probes
[61, 62, 63]. Now in our derived model, we use Hubble horizon as a IR cut-off,
where $L=H^{-1}$. In general, the transformation between scale factor and
redshift is given by $a=\frac{a_{0}}{(1+z)}$, where $a_{0}$ is the present
value of $a$. The redshift transformations are useful in validation of models
with observational data. Here we obtained cosmological parameters in terms of
$t$ as well as redshift in this paper.

The evolution of EoS parameter $\omega_{d}$ versus $t$ and $z$ for different
values of $\alpha$ and $\beta$ are shown in fig.1(a)$\&$ 1(b). The EoS
parameter exists in both quintessence and phantom regions and also crosses the
phantom divided line (PDL) ($\omega_{d}=-1$). Also, the latest cosmological
data from SNIa (Supernova Legacy Survey, Gold Sample of Hubble Space
Telescope) [64, 65], CMB (WMAP, BOOMERANG) [66] and large scale structure
(SDSS) [67] rule out that $\omega_{d}=-1$, they moderately support dynamically
developing DE crossing the PDL (see [68, 69, 70]). In cosmology, a redshift is
an increase in the wavelength of electromagnetic radiation, with a
corresponding fall in frequency and photon energy (such as light). Negative
redshift, or blueshift, is the total opposite of positive redshift, with a
reduction in wavelength and concomitant rise in frequency and energy. In our
LRS Bianchi type-I model, $\omega_{d}=-1$, in late time as $z\to-1$ which
suggests that matter in the Universe behaves like a perfect fluid initially.
Later on the model is similar to a dark energy model and behaves like a
quintessence model and finally approaches -1 then entering the phantom region.

As a consequence, our DE model agrees well with both well-established
simulated predictions and current findings.The dynamical features of the model
with HEL is most general and interesting feature as obtained from the power
law or the exponential expansion. Fig.1(b), shows that the EoS parameter
approaches towards $\Lambda$CDM ($\omega_{d}=-1$) in late time.

(a) (b) ‘

Figure 2: Variation of energy density parameters.

Figure 2 describe the behaviour of energy density parameters with respect to
time and redshift $z$ respectively. In the late time the total energy density
$\Omega\to 1$ of the universe approaches to unity at low redshift region as
clearly shown in Figure 2(a) $\&$ 2(b).

(a) (b)

Figure 3: Variation of cosmological constant $(\Lambda)$.

Figures $3a$ $\&$ $3b$ depict the behavior of the ‘cosmological constant
$\Lambda$’ versus $t$ and redshift $z$ respectively and show the universe’s
expanding mode. We observed that the cosmological parameter is the positive
decreasing function of time and redshift $(z\to-1$) and approaches towards
zero in late time. Which is a good agreement with recent SNe Ia observation
[71, 72]. Authors [73, 74, 75] propose the existence of a positive
cosmological constant with a magnitude of $\Lambda(Gh/c3)\approx 10^{-123}$.
The most impressive clarification for these info is that the dark energy
compares to a positive cosmological constant. These findings on the amplitude
and redshift of type Ia supernovas show that our universe is accelerating,
with cosmological density produced by the cosmological expansion. The outcome
suggests a very minute positive value having magnitude $10^{-123}$.

##  4 Statefinder

The Statefinder is a geometrical diagnostic and allows us to characterize the
properties of dark energy in a model independent manner. The Statefinder is
dimensionless and is constructed from the scale factor of the Universe and its
time derivatives only. In our model we demonstrate that the Statefinder
diagnostic can effectively differentiate between different forms of dark
energy.

The stability of DE models can also be examined through the statefinder
diagnostic pair $(r,s)$, which gives us an idea related to the dynamical
nature of the model. The statefinder parameters $(r,s)$ [76, 77, 78] are a
pair of parameters that explore the dynamics of the universe’s expansion by
using the higher derivatives of the scale factor. In fact, the trajectories in
the $(r,s)$ plane corresponding to different cosmological models show
qualitative behaviour.

$r=\frac{\dddot{a}}{aH^{3}}=\frac{\alpha(-3\alpha-3\beta t+2)}{(\alpha+\beta
t)^{3}}+1$ (18) $s=\frac{r-1}{3(-\frac{1}{2}+q)}=\frac{\alpha(-3\alpha-3\beta
t+2)}{3(\alpha+\beta t)^{3}\left(\frac{1}{2}-\frac{\alpha}{(\alpha+\beta
t)^{2}}\right)}$ (19)

In the derived model, the ($r,s$) trajectories divide the region into two
parts for the various value of the $\alpha$ and $\beta$ as shown in figure 4a.
The region $r<1$ ,$s>0$ shows the quintessence era and $r>1$ ,$s<0$ shows the
Chaplygin Gas (CG)region. The region $r=1$ ,$s=2/3$ describing the HDE and
region $r=1$ ,$s=0$ approaches to $\Lambda$CDM model. In figure 4b, the
trajectories $(r,q)$ show the evolution of the universe. Our model starts with
SCDM ($(r,q)=(1,0.5)$), $\Lambda$CDM ($(r,q)=(1,0)$) and tends to SS
($(r,q)=(1,-1)$) model as shown in figure by the dots.

(a) (b)

Figure 4: Plots of $r$ versus $s$ and $r$ versus $q$ .

##  5 $\omega_{d}-\omega_{d}^{{}^{\prime}}$ plane

The EoS parameter of dark energy $\omega_{d}=p_{d}/\rho_{d}$, plays a key role
in observational cosmology. According to the distinct region in the
$\omega_{d}-\omega^{\prime}_{d}$ plane, Caldwell and Linder [79, 80]
categorized the quintessence models defined by the quantities $\omega_{d}$ and
$\omega^{\prime}_{d}$ and divide into two parts ”thawing” and ”freezing,”
region. Some researchers have recently examined the evolution of quintessence
DE models on the $\omega_{d}-\omega^{\prime}_{d}$ plane, where
$\omega^{\prime}_{d}$ is the time derivative of $\omega_{d}$ with respect to
$\log a$.

$\omega^{\prime}_{d}=\frac{d\omega_{d}}{d\log a}=\frac{\alpha
t\left(3\alpha^{2}k^{2}-3km(\alpha-\beta t)(\alpha+\beta t)-2\beta
m^{2}(\alpha+\beta t)^{2}\right)}{3(\alpha+\beta t)^{3}(\alpha
k-m(\alpha+\beta t))^{2}}$ (20)

In our derived model the quintessence region asymptotically approaches to
zero. Later this analysis was extended by many researchers to other dynamic DE
models. Clearly, a fixed point at (-1,0) is the $\Lambda$ CDM in the
$\omega_{d}-\omega^{\prime}_{d}$ plane.

Finally we analyzed the evolutionary behavior of the EoS parameter of the HDE
in figure $5$. The trajectories lies in both freezing and thawing regions for
three values of parameters $\alpha$ and $\beta$ as clearly shown in figure.

Figure 5: Plot of $\omega_{d}-\omega^{\prime}_{d}$.

##  6 Scalar Field DE Models

###  6.1 Tachyon scalar field model of HDE

To depicts the accelerated expansion and evaluation of universe, the tachyon
field (TF) is one of the DE candidates [81, 82]. Some tachyon condensate by an
powerful Lagrangian density [83, 84] recommends a tachyon field is given by

$L=-V(\phi)\sqrt{1+\partial_{i}\phi\ \partial^{i}\phi}$ (21)

Tachyon is observed the most important DE constituent which explain the
accelerated expansion of the universe. The EoS parameter of the tachyon DE
matter distribution lies between $-1$ and $0$ [85]. The ‘energy density
$\rho_{d}$’ and ‘pressure $p_{d}$’ associated with Scalar field ($\phi$) and
‘scalar potential $V(\phi)$’ for tachyon matter distribution are read as:

$p_{d}=\left(1-\dot{\phi}^{2}\right)^{1/2}V(\phi)$ (22)
$\rho_{d}=\frac{V(\phi)}{\sqrt{1-\dot{\phi}^{2}}}$ (23)

Here, $\dot{\phi}^{2}$ is the scalar field and $V(\phi)$ is the potential
respectively.
From Eqs. (22) - (23) and using Eqs. (11) and (13), the scalar $\phi$ and
potential $V(\phi)$ are determined as:

$\phi=\int\sqrt{\frac{2}{3}}\sqrt{-\frac{\alpha(m-k(\alpha+\beta
t))}{(\alpha+\beta t)\left(k(\alpha+\beta t)^{2}-\alpha m\right)}}\,dt+c_{1}$
(24) $V(\phi)=\sqrt{3}\sqrt{\frac{\left(k(\alpha+\beta t)^{2}-\alpha
m\right)\left(-2\alpha k+3\left(k(\alpha+\beta t)^{2}-\alpha
m\right)+\frac{2\alpha m}{\alpha+\beta t}\right)}{t^{4}}}$ (25)

(a) (b)

Figure 6: Plots of $\dot{\phi}$ and $V(\phi)$ for tachyon field .

We likewise present an extensive examination of the cosmological results of
homogeneous tachyon matter coinciding with non-relativistic matter and
radiation, with an emphasis on the converse square and exponential potentials
for the tachyonic scalar field. Figures $6a$ and $6b$ show that the scalar
field $\phi$ and potential $V(\phi)$ both remains positive as time advances in
our determined model. The dynamical behaviour of the tachyon field associated
with an exponential potential is presented, and we demonstrate that the pre-
inflation dynamics in modified loop cosmology and standard loop quantum
cosmology are remarkably similar.

###  6.2 Quintessence scalar field model of HDE

Quintessence is a conceptual formulation of DE characterized by homogeneous
variable ‘Scalar field and the potential’, which are essential for the
expansion of the universe [86]. For the quintessence scalar $\phi$, the
general formulation of action is as follows:

$S=\int{d^{4}x\sqrt{-g}\left(-V(\phi)-\frac{1}{2}g^{ij}\partial_{i}\phi\partial_{j}\phi\right)}$
(26)

The EoS parameter in the quintessence model indicates that the accelerated
expansion of the universe exists for $-1\leq\omega\leq-\frac{1}{3}$ [83]. For
the quintessence model, the relations of $\rho_{d}$ and $p_{d}$ in the form of
‘$\phi$’ and ‘$V(\phi)$’ are established as [69]:

$p_{d}=\frac{1}{2}\left(\dot{\phi}^{2}-2V(\phi)\right)$ (27)
$\rho_{d}=\frac{1}{2}\left(\dot{\phi}^{2}+2V(\phi)\right)$ (28)

From Eqs. (11), (13), (27) and (28), the scalar $\phi$ and potential $V(\phi)$
for the ‘quintessence model’ are obtained as

$\phi=\int\sqrt{\frac{3\left(-\frac{2\alpha(m-k(\alpha+\beta
t))}{3(\alpha+\beta t)\left(k(\alpha+\beta t)^{2}-\alpha
m\right)}-1\right)\left(k(\alpha+\beta t)^{2}-\alpha
m\right)}{t^{2}}+\frac{3\left(k(\alpha+\beta t)^{2}-\alpha
m\right)}{t^{2}}}\,dt+c_{2}$ (29)
$V(\phi)=\frac{k\left(3\alpha^{3}+3\beta^{3}t^{3}+\alpha^{2}(9\beta
t-1)+\alpha\beta t(9\beta t-1)\right)+\alpha m(-3\alpha-3\beta
t+1)}{t^{2}(\alpha+\beta t)}$ (30)

(a) (b)

Figure 7: Plots of $\dot{\phi}$ and $V(\phi)$ for quintessence field .

Figures $7a$ and $7b$ depict the growth of the ‘scalar field $\phi$’ and ‘
potential $V(\phi)$’ with time $t$ for all three values of $\alpha$ and
$\beta$ in our derived model. The Scale field and potential diminish with time
and tends to be low positive at the present epoch.

##  7 Deceleration parameter for the Model

From the Table-1, we observe that $\alpha$ increases with the increase of
$\beta$. For $0<\alpha\leq 1$, $\beta$ increases but for $\alpha>1$, $\beta$
starts decreasing. It is worth mentioned that for $0<\alpha<1$, our models
show the transitioning scenario whereas for $\alpha\geq 1$, the models are in
accelerating phase only.

Figure $8(a)$ depicts the variation of DP ($q$) with time $t$. Because DP
($q$) lies between the range $-1<q<0$, SNe Ia data reveals the expansion of
the present universe in accelerating mode. As a result, the value of the
decelerating parameter matches recent data.

We have demonstrated in Fig. $8(b)$ that expansion of $q(z)$ takes place with
the change of signature flipping from a decelerating to an accelerating phase.
We also observed that $q$ tends to $-1$ as $z$ approaches to $-1$. SNe Ia data
shows the progression from previous deceleration to current acceleration. The
transition point $z_{t}=0.46\pm 0.13$ at $(1\;\sigma)$ c.l. was investigated
in 2004 by the HZSNS group [74]. In 2007, this value of $z_{t}$ was improved
by $z_{t}=0.43\pm 0.07$ at $(1\;\sigma)$ c.l. [87]. SNLS first year data set
[88] combined with ESSENCE supernova data [89], provide a progress redshift
$z_{t}\sim 0.6(1\;\sigma)$ in improved agreement with the flat $\Lambda$CDM
model ($z_{t}=(2\Omega_{d}/\Omega_{m})^{\frac{1}{3}}-1\sim 0.66$).

Again $q(z)$ is reconstructed by the joined $(SNIa+CC+H_{0})$, and obtained
the transition redshift $z_{t}={0.69}^{+0.09}_{-0.06},{0.65}^{+0.10}_{-0.07}$
and ${0.61}^{+0.12}_{-0.08}$ within $(1\sigma)$ [90] and found consistent with
past outcomes [91, 92, 93, 94, 95] including the $\Lambda CDM$ expectation
$z_{t}\approx 0.7$. $0.6\leq z_{t}\leq 1.18$ ($2\sigma$, joint examination )
[96] is the other limit of transition redshift.

From the figure $8(b)$, the transitioning redshifts occur at $z_{t}\cong
0.6921$ (for $\alpha=0.25$ & $\beta=0.9516$); $z_{t}\cong 0.4443$ (for
$\alpha=0.50$ & $\beta=0.0624$), and $z_{t}\cong 0.995$ (for $\alpha=0.75$ &
$\beta=0.0665$) for above three cases which are found to be well consistent
with the recent 36 observational Hubble data (OHD) provides the redshift range
$0.07\leq z\leq 2.36$ [97]. Comprising 740 SNIa with JLA indicates the
redshift range ‘$0.01\leq z\leq 1.30$’. Obviously, our results are in good
accordance with those reported in Refs. [74, 89, 96, 97].

(a) (b)

Figure 8: Plots of Deceleration parameter $q$ versus $t$ and redshift $z$.  Table 1: The behaviour of physical parameter $\alpha$ | $\beta$ | $q$ | $\omega$ | $\Omega_{d}$ | $\Omega_{m}$ | $\Omega$ | $r$ | $s$ | $\omega_{d}^{{}^{\prime}}$
---|---|---|---|---|---|---|---|---|---
0.25 | 0.0516124 | 1.00372 | 4.93611 | -0.115795 | 2.06244 | 1.94665 | 6.33415 | 3.52981 | 59.0776
0.5 | 0.0623788 | 0.280993 | -0.711735 | 0.232561 | 0.863276 | 1.09584 | 1.25779 | -0.392361 | 0.232003
0.75 | 0.0664251 | -0.0377516 | -0.494967 | 0.386196 | 0.539123 | 0.925319 | 0.293123 | 0.438168 | -0.224348
1. | 0.0669928 | -0.222349 | -0.53636 | 0.475172 | 0.400258 | 0.87543 | 0.0385825 | 0.443653 | -0.124511
1.25 | 0.0653375 | -0.344267 | -0.589897 | 0.533937 | 0.326571 | 0.860507 | -0.0173261 | 0.401661 | -0.0699145
1.5 | 0.0621031 | -0.431397 | -0.635868 | 0.575933 | 0.282205 | 0.858139 | -0.00564742 | 0.359906 | -0.041409
1.75 | 0.0576721 | -0.497056 | -0.673521 | 0.607581 | 0.253154 | 0.860735 | 0.0304176 | 0.324149 | -0.0255545
2. | 0.0522939 | -0.54846 | -0.704391 | 0.632358 | 0.232952 | 0.86531 | 0.0744813 | 0.294247 | -0.0161907
2.25 | 0.0461414 | -0.589887 | -0.729989 | 0.652325 | 0.218254 | 0.87058 | 0.119843 | 0.269189 | -0.0103853
2.5 | 0.0393408 | -0.624037 | -0.751496 | 0.668786 | 0.207177 | 0.875963 | 0.163705 | 0.248003 | -0.00664547
2.75 | 0.0319872 | -0.65271 | -0.769794 | 0.682606 | 0.198589 | 0.881195 | 0.204962 | 0.229904 | -0.00416254
3. | 0.0241546 | -0.677148 | -0.785542 | 0.694385 | 0.191773 | 0.886159 | 0.243268 | 0.214284 | -0.00247481

##  8 Conclusion

In this paper we have developed an anisotropic dark energy cosmological model
in the framework of the LRS Bianchi type-I metric. The anisotropic behaviour
of the model is simulated by taking HSF scale factor. In this model we are
able to get the exact solutions to the field equations by using the hybrid
scale factor. This manuscript depends on the HP and dimensional analysis and
explored various parameters in the context of HDE, like EoS parameter, density
parameter, and cosmological constant. In addition, the state finder
trajectories and $\omega_{d}-\omega_{d}{{}^{\prime}}$ plane are also
diagnosed. We also analyzed the Hubble parameter for HDE, for both,
theoretical and observational aspects.

The main findings of our model are as follows.

  * •

The EoS parameter, which is a function of time and redshift, has been
displayed in Figs. $1(a)$ and $1(b)$. Now in our derived model, we have
obtained a quintessence and phantom region and crosses the Phantom divided
line for three values of ($\alpha$, $\beta$) are shown in Figs. $1(a)$ and
$1(b)$. From the graph it is clear that the universe is under the influence of
dark energy as the equation of state predicted accelerated expansion phase

  * •

Figs. $2a$ $\&$ $2b$ depict the overall density parameter, which has been
matched with $\Lambda CDM$ and approaches to Unity i.e. $\Omega\equiv 1$ for
the different values of ($\alpha$, $\beta$).

  * •

Figures $3a$ $\&$ $3b$ depict the evolution of the cosmological constant
versus $t$ and $z$ respectively. Here $\Lambda$ is a positive decreasing
function and approaches towards zero.

  * •

As illustrated in Figs. $4a$ and $4b$, the trajectories of ‘ $(r,s)$ and
$(r,q)$’ for various values of $\alpha$ and $\beta$ have been displayed.
During the early time, the statefinder trajectory $(r,s)$ plane is separated
into two regions: “ $r<1,s>0$ quintessence region and the $r>1,s<0$ Chaplygin
Gas”, and its approaches to $\Lambda$CDM $(r,s)=(1,0)$ in the late time. The
trajectory $(r,q)$ evolves from the region of (SCDM) in the $(r-q)$ plane and
approaches the SS model in late time.

  * •

In Fig. $5$ we have analyzed the evolutionary behaviour of the EoS parameter
of the HDE models by constructing the $\omega_{d}-\omega_{d}^{{}^{\prime}}$
plane and found that $\omega_{d}^{{}^{\prime}}<0$ for $\omega_{d}<0$ for the
HDE model implying the plane to lie in the freezing region as well as
$\omega_{d}^{{}^{\prime}}>0$ for $\omega_{d}<0$ thawing region.

  * •

We have also constructed the potential as well as dynamics of the quintessence
and tachyon scalar field see Figs $6$ & $7$.

  * •

Figure $8(b)$ shows the transition redshift from decelerated to accelerated
expansion occurs at $z_{t}\cong 0.6921$ (for $\alpha=0.25$ & $\beta=0.9516$);
$z_{t}\cong 0.4443$ (for $\alpha=0.50$ & $\beta=0.0624$), and $z_{t}\cong
0.995$ (for $\alpha=0.75$ & $\beta=0.0665$) for above three cases which are in
good agreement with those reported in Refs. [74, 89, 96, 97].

As a result, our findings support recent observational data that revealed the
current isotropic state. This demonstrates that our model closely matches
current observations.

## Credit authorship contribution statement

Anirudh Pradhan: Ideas, Formulation and writing- original draft preparation.
Vinod Kumar Bhardwaj: Application of mathematical formulation, computation &
drawing figures. Archana Dixit: Writing- reviewing, validation & editing.
Symala Krishannair: Final draft preparation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

The author (AP) thanks the IUCAA, Pune, India for providing the facility under
visiting associateship. We are so grateful to the reviewers for their many
valuable suggestions and comments that significantly improved the paper.

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