\section{Introduction}
Recent findings on the superconductivity in LaFeAsO$_{1-x}$F$_x$\cite{kamihara} and related materials have triggered a great deal of interest in iron compounds because of the possible connection between
the superconductivity and magnetism,\cite{mazin} which undergoes a phase transition from antiferromagnetic to superconducting ground states (and vice versa) tuned by external pressure\cite{takahashi} or chemical doping.\cite{kamihara}
In particular, PbO-type FeSe, which is one of the iron-based superconductors discovered a long time ago,\cite{hsu} has attracted attention as a key material for elucidating the superconducting mechanism,
because of its extremely simple structure (composed only of the superconducting FeSe$_4$ layer) and its excellent response to external pressure.\cite{mizuguchi}
Among all similar materials, FeSe shows the greatest enhancement of its $T\rm_c$ at high pressure:\cite{margadonna}
$T\rm_c$ varies from 9 K (at ambient pressure) to 37 K (at 6 GPa), indicating a growth rate as high as 4.5 K/GPa; as a result, using FeSe, it is possible to demonstrate the strong correlation between the structural parameter and $T\rm_c$.

The maximum $T\rm_c$ value of iron pnictides is apparently attained when the Fe$X_4$ ($X$: anion) tetrahedron assumes a regular shape;\cite{lee} however, this rule is not applicable to FeSe,
because the FeSe$_4$ tetrahedron is distorted from the regular shape,\cite{margadonna} while $T\rm_c$ increases significantly with application of pressure.
Although several studies have investigated FeSe subjected to high pressure,\cite{medvedev,garbarino} the pressure dependence of $T\rm_c$, particularly above 6 GPa, is controversial because of difficulties in measurements under high pressure conditions.
For example, in one of these studies, superconductivity appeared to exist above 20 GPa, even though the phase transition from tetragonal to hexagonal (non-superconductive) was completed at 12.4 GPa.\cite{braithwaite}
This discrepancy is attributable mainly to the following two reasons: the ambiguous definition of $T\rm_c$ and the large anisotropic compressibility of the layered structure.\cite{margadonna}
FeSe does not show Meissner diamagnetism at $T\rm_c^{onset}$, which denotes the beginning of the resistivity drop; therefore, there is no guarantee that a kink in the resistivity immediately represents to a signature of superconductivity.
Therefore, $T\rm_c$ should be decided by zero resistance temperatures. Moreover, the hydrostaticity of pressure is essential to obtain the precise pressure dependence of $T\rm_c$,
because FeSe has an inhomogeneous compressibility,\cite{margadonna} which stems from the layered structure stacked loosely by a van der Waals interaction (see upper inset of Fig. 1).
To overcome all these problems, we used a cubic-anvil-type high-pressure apparatus\cite{mori} that ensures quasihydrostaticity by the isotropic movement of anvil tops,
even after the liquid pressure-transmitting medium solidifies at low temperature and high pressure; using this apparatus, we reconfirmed the $T\rm_c$-$P$ (pressure) phase diagram of FeSe.

With this background, in this study, we measured the electrical resistivity of a high-quality FeSe polycrystal at pressures ranging from 0 GPa to 16 GPa and evaluated the variation in $T\rm_c$ and the electronic state, both of which are closely related to the anion position.
A precise evaluation of zero-resistivity temperature shows that the pressure dependence of $T\rm_c$ has a slightly distorted dome-shaped curve with the maximum $T\rm_c$ (30 K) in the range of 0 $<$ P $<$ 11.5 (GPa)
and that the temperature dependence of resistivities above $T\rm_c$ changes dramatically at around 2 GPa, suggesting the existence of different types of superconductivities at high pressure.
We found a striking correlation between $T\rm_c$ and anion (selenium) height, which was obtained by a direct comparison with previous report,\cite{margadonna} $T\rm_c$ varies with the anion height.
Moreover, this relation is broadly applicable to other ferropnictides, indicating that the high-temperature superconductivity in these materials only appears around the optimum anion height ($\sim$1.38$\rm\AA$).
We suggest that the anion height should be considered as a key determining factor of $T\rm_c$ of iron-based superconductors containing various anions.
\section{Method}
FeSe has a simple tetragonal structure that is composed only of edge-shared FeSe$_4$ tetrahedral layers; however, it is difficult to fabricate a good-quality superconducting FeSe sample,
because a large amount of excess iron is absolutely imperative for the occurrence of superconductivity\cite{mcqueen} and extreme caution is required to prevent the formation of magnetic impurities from easily oxidizable iron.
Polycrystalline samples of FeSe used in this study were prepared by a solid-state reaction using Fe (99.9$\%$, Kojundo-Kagaku) and Se (99.999$\%$, Kojundo-Kagaku) powders.
The powders were mixed in a molar ratio of 100:99 (nominal composition of FeSe$_{0.99}$) in an argon-filled glove box and sealed in an evacuated quartz tube.
Then, the powders were sintered at 1343 K for 3 days, annealed at 693 K for 2 days, and finally quenched in liquid nitrogen. Further details of sample preparation are described in Ref. 12.
The quality of the obtained sample was verified by powder X-ray diffraction using an X-ray diffractometer with a graphite monochromator (MultiFlex, Rigaku);
the results confirmed that the sample quality was similar to that of previously reported high-quality samples.\cite{mcqueen}
The electrical resistivity and magnetic susceptibility of the sample were measured using a physical property measurement system (PPMS, Quantum Design) magnetic property measurement system
(MPMS, Quantum Design), respectively.

As shown in Fig. 1, in our sample, zero resistivity and Meissner effect were observed simultaneously at 7 K at ambient pressure.
In order to evaluate the precise pressure dependence of $T\rm_c$, we defined both $T\rm_c^{offset}$ (determined from the zero-resistivity temperature) and $T\rm_c^{onset}$
(determined from the cross point of two extrapolated lines drawn for the resistivity data around $T\rm_c$).
Electrical resistivity measurements were performed in the cubic-anvil-type apparatus\cite{mori} with Daphne 7474 oil\cite{murata} as the liquid pressure-transmitting medium,
which ensured precise measurements up to 16 GPa under nearly hydrostatic conditions in this study.
Pressure was calibrated using a calibration curve that was previously obtained by observations of several fixed-pressure points (Bi, Te, Sn, ZnS) at room temperature.
The resistivity measurements were performed by a conventional dc four-probe method, as shown in the lower inset of Fig. 1, with an excitation current of 1 mA.
The samples used in these experiments had dimensions of 1.0$\times$0.4$\times$0.2 mm$^3$.
\begin{figure}
\includegraphics[width=1.0\linewidth,keepaspectratio]{fig1s.eps}
\caption{\label{fig1}(Color online) Temperature dependence of magnetic susceptibility  (top main panel) and electrical resistivity (bottom main panel) of polycrystalline FeSe at ambient pressure.
The top and bottom insets show the crystal structure of FeSe and the setting of the sample in the high-pressure apparatus (see text for details), respectively.}
\end{figure}
\section{RESULTS AND DISCUSSION}
The left panel of Fig. 2 shows the temperature dependence of electrical resistivities under application of external pressures ranging from 0 GPa (ambient pressure) to 16 GPa.
With the application of pressure (ambient pressure to 16 GPa), the room-temperature resistivity decreases by a factor of more than 3; it reaches a minimum at 10 GPa and subsequently increases between 10 GPa and 16 GPa.
In the pressure range from 0 GPa to 6 GPa, $T\rm_c$ (both $T\rm_c^{onset}$ and $T\rm_c^{offset}$) increases rapidly but not monotonically; further, the resistivity curves gradually change shape from the one at ambient pressure (see top right panel of Fig. 2).
Meanwhile, as shown in the bottom right panel of Fig. 2, $T\rm_c^{offset}$ suddenly vanishes at 11.5 GPa; this disappearance is attributed to a rapid enhancement of resistivities between 11 GPa and 11.5 GPa.
Although $T\rm_c^{onset}$ still remains above 11.5 GPa, it disappears completely at 16 GPa. Figure 3 shows the pressure dependence of $T\rm_c^{onset}$, $T\rm_c^{offset}$, and the width of the superconducting transition,
$\Delta T\rm_c$ (= $T\rm_c^{onset}$ - $T\rm_c^{offset}$). Beautiful but slightly distorted dome-shaped curves are observed as cuprates\cite{palee} and heavy fermions.\cite{sidorov1}
However, the pressure dependence of  $\Delta T\rm_c$ shows a complicated trend. At low pressures up to 2 GPa,
$\Delta T\rm_c$ increases exponentially, indicating a salient broadening of the transition width, whereas $T\rm_c^{offset}$ increases gradually.
Thereafter,  $\Delta T\rm_c$ decreases moderately but increases again above 9 GPa, resulting in a dome-shaped $T\rm_c$ curve. In the following paragraph, we shed light on the details of the abovementioned behaviors,
in comparison with those reported in previous studies, to elucidate the nature of iron-based superconductors.
\begin{figure}
\includegraphics[width=1.0\linewidth]{fig2s.eps}
\caption{\label{fig2}(Color online) Temperature dependence of resistivity   at ambient and several other pressures (left panel: 0$\sim$16 GPa, top right panel: 0$\sim$8 GPa,
and bottom right panel: 9$\sim$16 GPa) for polycrystalline sample of FeSe.}
\end{figure}

The most striking feature in the low-pressure region ($<$ 2 GPa) is that $T\rm_c^{offset}$ has a relatively flat plateau; that is, an increase in $T\rm_c$ almost levels off between 1 and 1.5 GPa.
A similar behavior was also observed during the measurements of DC magnetization\cite{miyoshi} and electrical resistivity\cite{masaki} of FeSe by using high-pressure piston-cylinder units; therefore, it is probably an important characteristic of FeSe.
A previous $^{77}$Se-NMR measurement\cite{imai} showed that an antiferromagnetic spin fluctuation is significantly enhanced in the plateau region and that there exists a possibility of a magnetic phase transition or spin freezing.
The superconductivity in iron-based compounds is thought to be closely related to a neighboring antiferromagnetic ordered phase; however, tetragonal FeSe exhibits superconductivity without any elemental substitution;
therefore, its antiferromagnetic ground state remains undiscovered. Probably, in the case of FeSe, the nesting of a Fermi surface would improve temporarily up to $\sim$2 GPa by application of external pressure,
resulting in the enhancement of antiferromagnetic instabilities such as would provide constraints on the enhancement of $T\rm_c$.
After that, the nesting condition would worsen with the application of further pressure, and "high-temperature" superconductivity would appear.
The appearance of pressure-induced superconductivity adjacent to a magnetic-ordered phase is a characteristic feature of exotic superconductors such as CeRh$_2$Si$_2$,\cite{movshovich} CeNi$_2$Ge$_2$,\cite{steglich} and
CeIn$_3$,\cite{mathur} with superconductivity appearing around a quantum critical point.
\begin{figure}
\includegraphics[width=1.0\linewidth]{fig3s.eps}
\caption{\label{fig3}(Color online) Pressure dependence of $T\rm_c^{onset}$ (open circle), $T\rm_c^{offset}$ (closed circle, top main panel) and width of superconducting transition
$\Delta T\rm_c$ (= $T\rm_c^{onset}$ - $T\rm_c^{offset}$) (bottom main panel). The solid lines are obtained by connecting the data points.}
\end{figure}

Figure 4 shows an enlarged view of the resistivities around $T\rm_c$ between 1 and 6 GPa.
We can distinguish the gradual change in the shape of resistivity curves: with increasing pressure, the temperature dependence curve of resistivity changes from nearly quadratic to linear.
In particular, the change between 2 GPa and 3 GPa is drastic, implying a phase transition between different superconducting states.
The possibility of two kinds of superconducting phases has also been reported by Sidorov $et$ $al.$,\cite{sidorov2} indicated by the jump in $d\rho/dT$.
A linear dependence of electrical resistivities on temperature is commonly observed in cuprate superconductors\cite{fiory,nakamura,mackenzie} and is considered to be one of the primary indicators of non-Fermi-liquid behavior;
an incoherent scattering of fermion quasiparticles via magnetic interactions leads to resistivity of the form  $\rho (T) =  \rho_0 + AT^\alpha $   where $\rho_0$, $A$ and $\alpha$ are arbitrary constants, however, no linear term is expected according to conventional Fermi-liquid theory.
It should be noted that in our study, the non-Fermi-liquid behavior was observed even in the plateau region (because  = 1.6$\sim$1.2 between 1 and 2 GPa), indicating the development of a spin fluctuation.
The temperature dependence of resistivity in the high-temperature superconducting phase ($>$3 GPa) of FeSe is highly reminiscent of the linear temperature dependence observed in high-$T\rm_c$ cuprates,
interpreted as a "strange metal" phase,\cite{anderson} where it is ascribed to antiferromagnetic spin fluctuations.
Similar behaviors have also been reported for other ferropnictides, ex. Ba(Fe,Co)$_2$As$_2$,\cite{ahilan} implying that antiferromagnetic spin fluctuations
and superconductivity are closely related to each other in iron-based compounds, as discussed in the context of heavy fermion and cuprate superconductors.
\begin{figure}
\includegraphics[width=1.0\linewidth]{fig4s.eps}
\caption{\label{fig4}(Color online) Enlarged view of resistivity  around $T\rm_c$ between 1 and 6 GPa. The dotted lines are guides to the eye, showing the dependence of $T$ and $T^2$. For simplicity, we have not shown the data at 1.5 GPa.}
\end{figure}

For applied pressures greater than 3 GPa, $T\rm_c$ shows the dome-shaped curve, with maximum $T\rm_c$ = 30.02 K at 6 GPa, whereas between 3 GPa and 9 GPa,  $\Delta T\rm_c$ continues to decline steadily.
As has been noted previously, the shape of the Fe$X_4$ tetrahedron is closely related to the value of $T\rm_c$; in the case of iron pnictides, $T\rm_c$ appears to attain maximum values
when the As-Fe-As bond angles come close to 109.47$\rm^ o$,\cite{lee} which corresponds to a regular tetrahedron. However, this rule is not applicable to FeSe.\cite{margadonna}
Therefore, we focus on the relationship of $T\rm_c$ with "Se height," which is the distance of the anion from the nearest iron layer.
Figure 5 shows the pressure dependence of $T\rm_c^{offset}$ and Se height (inversely scaled), obtained from Ref. 6.
Astonishingly, $T\rm_c^{offset}$ varies linearly with the Se height, even in the plateau in the low-pressure region.
Although there is a subtle shift in the pressure dependence, which may be due to the difference in ways of applying pressures (cubic or diamond anvil), there is a clear correlation between both parameters.
Furthermore, $T\rm_c^{offset}$ is inversely proportional to the magnitude of the Se height, as can be observed from the inset of Fig. 4, indicating that the smaller the Se height, the more enhanced is $T\rm_c$.
However, this seems to be contradictory to the behavior observed in other pnictides: in other pnictides, it is observed that $T\rm_c$ is higher when the pnictogen is located at greater heights in the crystal structures;\cite{lee}
this behavior is also supported by the theoretical aspect.\cite{kuroki}
In any case, FeSe is a suitable material for demonstrating the importance of anion position as discussed below, which is inherently linked to the mechanism of superconductivity in iron-based compounds.
\begin{figure}
\includegraphics[width=1.0\linewidth]{fig5s.eps}
\caption{\label{fig5}(Color online) Pressure dependence of $T\rm_c^{offset}$ and Se height $h\rm_{Se}$ (inversely scaled), as obtained from Ref. 6. The inset shows $T\rm_c^{offset}$ as a function of the Se height. The dotted line is a guide to the eye.}
\end{figure}

On application of further pressure up to 9 GPa, $T\rm_c^{offset}$ reduces monotonically to lower temperatures and disappears completely above 11.5 GPa; then, the superconducting transition becomes less sharp,
as is indicated by the broadening of the transition width  $\Delta T\rm_c$. After the disappearance of $T\rm_c^{offset}$, the resistivity over the entire temperature range would improve greatly with increasing pressure,
indicating the occurrence of the metal-semiconductor transition. At $\sim$9 GPa, tetragonal FeSe starts being transformed from a tetragonal structure to a hexagonal (NiAs-type) structure,
and this structure undergoes a transition from a metallic superconducting state to the semiconducting state.\cite{margadonna}
A recent synchrotron X-ray study on FeSe at various pressures\cite{braithwaite} has revealed that the structural transition to the hexagonal phase is completed at around 12.4 GPa,
which is consistent with the fact that all traces of superconductivity (see bottom right panel of Fig. 2) completely vanish by 16 GPa, without any trace of an anomalous decrease in resistivity.
Therefore, the remarkable increase in transition width  $\Delta T\rm_c$ above 9 GPa corresponds to the transition to the hexagonal phase, and this corresponds to the closure of the superconducting dome.
The observed onset of $T\rm_c$ above 11.5 GPa indicates a subtle fraction of the superconducting phase, which may no longer manifest Meissner diamagnetism.
\begin{figure}
\includegraphics[width=1.0\linewidth]{fig6s.eps}
\caption{\label{fig6}(Color online) $T\rm_c$ as a function of anion height ($h\rm_{anion}$) for various iron (and nickel)-based superconductors, as obtained from Ref. 29 (triangle: FeSe, circle: other pnictides).
Lanthanides ($Ln$) indicate $Ln$FeAsO (1111 system). 111, 122, and 42226 represent LiFeAs, Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$, and Sr$_4$Sc$_2$Fe$_2$P$_2$O$_6$,\cite{ogino} respectively.
The yellow line shows the fitting result by the Lorenz function.}
\end{figure}

We now turn to consider, in a more universal sense, the nature of the iron-based superconductivity in FeSe with respect to pressure tuning of $T\rm_c$, which is the focus area in this study.
Figure 6 shows the maximum $T\rm_c$ as a function of anion height ($h\rm_{anion}$) for various iron-based superconductors.\cite{horikin,ogino}
In this study, we successfully derived the $T\rm_c$-$h\rm_{anion}$ diagram of iron (partially nickel)-based superconductors.
The clear correlation between $T\rm_c$ and $h\rm_{anion}$ is a certain indicator of the importance of anion positions in these iron-based superconductors.
As shown in Fig. 6, the anion height dependence of $T\rm_c$ is well described by a Lorenz curve (yellow line).
As the value of anion height increases, $T\rm_c$ of the iron-based superconductors starts to increase dramatically up to $\sim$55 K at a height of 1.38$\rm\AA$, which corresponds to the optimum value of a 1111 system.
However, above the optimum anion height (1.38$\rm\AA$), $T\rm_c$ decreases rapidly with increasing $h\rm_{anion}$, going through our measured FeSe region (1.42$\sim$1.45$\rm\AA$);
finally, the value of $h\rm_{anion}$ becomes equal to that of non-superconducting FeTe (1.77$\rm\AA$).\cite{bao}
It should be noted that superconductors with direct substitution in the Fe$X_4$ tetrahedral layer or a large deviation from a divalent state (Fe$^{2+}$), e.g., an alkali metal element or Co doping samples of a 122 system
or chalcogen-substituted 11-system, are not particularly suitable for this trend.
This is probably due to (1) the considerable disorder in the Fe layers; (2) a large gap among anion heights of different anions, for example, in FeSe$_{1-x}$Te$_x$,
$T\rm_c$ appears to be dominated only by the Fe-Se distance ($T\rm_c$ $\sim$ 14 K at $h\rm_{anion}$ = 1.478$\rm\AA$, which is consistent with the Lorenz curve)\cite{lehman,tegel}; or
(3) coexistence of strong magnetic fluctuation and superconductivity.\cite{lumsden1,lumsden2,parker}
We thus conclude that the appearance of "high-temperature" superconductivity in iron compounds is confined to a specific area that is around the optimum anion height (1.38$\rm\AA$), which corresponds to the radius of arsenic at ambient pressure.
It has been proposed,\cite{kuroki} on the basis of solutions of Eliashberg equations, that the critical temperature of iron pnictides is inherently linked to their structural parameters, particularly pnictogen heights and the $a$-axis lattice parameter.
The result obtained in this study that the pressure evolution of $T\rm_c$ varies with the anion height, as shown in Fig. 5, is in good agreement with the theoretical prediction;
however, the length of the $a$-axis of FeSe monotonically decreases with increasing pressure,\cite{margadonna} which suppresses the enhancement of $T\rm_c$.
An interesting aspect of FeSe, as observed from Fig. 6, is that $T\rm_c$ does not exhibit this trend above 1.43$\rm\AA$ (corresponding to the pressure range of 0$\sim$2 GPa),
which clearly indicates that the system attains a different electronic state below the characteristic pressure ($\sim$2 GPa).
This could occur concomitantly with the reconstruction of a Fermi surface; the shapes of the resistivity curves above $T\rm_c$ change clearly at around 2 GPa, as pointed out above (see Fig. 4),
which implies a significant transformation to the non-Fermi-liquid state. It has been previously suggested that there is a difference in the superconducting gap symmetries of arsenic and phosphide:\cite{fletcher}
a full-gap strong coupling $s$-wave for high-$T\rm_c$ arsenide compounds and nodal low-$T\rm_c$ for phosphide compounds, which is widely perceived in many studies.
A theoretical approach\cite{kuroki} has suggested that the pairing symmetry of iron pnictides is determined by the pnictogen heights between a high-$T\rm_c$ nodeless gap for high $h\rm_{anion}$ or a low-$T\rm_c$ nodal gap for low $h\rm_{anion}$, corresponding to the left-hand side of the Lorenz curve shown in Fig. 6.
Although FeSe is located on the right-hand side, i.e., in a region of extremely high $h\rm_{anion}$, it is highly probable that FeSe shows two or more different types of superconductivities under application of external pressures,
as is the case with pnictides. The extremely soft crystal structure of FeSe enables the control of $h\rm_{anion}$ in a wide range and the superconducting mechanism can be switched by the application of modest pressure.
It may be interesting to explore the gap symmetry of FeSe at high pressure ($\sim$6 GPa) by NMR or muon spin rotation and whether there is any difference between the gap symmetry of FeSe and those of other iron-based superconductors.
\section{SUMMARY}
In this study, the precise pressure dependence of the electric resistivity of FeSe was measured in the pressure range of 0-16.0 GPa at temperatures of 4-300 K by using a cubic-anvil-type high-pressure apparatus.
$T\rm_c$ estimated from zero-resistivity temperature shows a slightly distorted dome-shaped curve, with the maximum $T\rm_c$ = 30 K in the range of 0 $< P <$ 11.5 (GPa), which is lower than those in previous studies.
The temperature dependence of resistivity above $T\rm_c$ changes dramatically at around 2$\sim$3 GPa; the shapes of the resistivity curves change to linear shapes, which is one of the primary indicators of non-Fermi-liquid behavior;
this behavior strongly suggests a phase transition between different superconducting states. A striking correlation is found between $T\rm_c$ and anion (selenium) height: the lower the Se height, the more improved is $T\rm_c$.
Moreover, this relation is broadly applicable to other iron pnictides, indicating that the high-temperature superconductivity in these materials appears only around the optimum anion height ($\sim$1.38$\rm\AA$).
On the basis of these results, we suggest that anion height should be considered as a key determining factor of $T\rm_c$ in iron-based superconductors containing various anions.

\vspace{5mm}
We would like to acknowledge Dr. K. Kuroki (The Univ. of Electro-Communications) for helpful discussions. This work was supported by "High-Tech Research Center"
Project for Private University from the Ministry of Education, Culture, Sports Science and Technology (MEXT),
Grant-in-Aid for Yong Scientists (B) (No. 20740202) and a Grant-in-Aid for Specially Promoted Research from MEXT, and the Asahi Glass Foundation.


\section{Introduction}

Semileptonic tau decays represent a clean benchmark to study the hadronization properties of QCD due to the fact that half of the process is purely electroweak and, therefore,
free of uncertainties at the required precision \cite{Braaten:1991qm, Braaten:1988hc, Braaten:1988ea, Braaten:1990ef, Narison:1988ni, Pich:1989pq, Davier:2005xq, Pich:2013??}.
At the (semi-)inclusive level this allows to extract fundamental parameters of the Standard Model, most importantly the strong coupling $\alpha_S$ \cite{Davier:2008sk,
Beneke:2008ad, Pich:2011bb, Boito:2012cr}. Tau decays containing Kaons have been split into the Cabibbo-allowed and suppressed decays \cite{Barate:1999hj, Abbiendi:2004xa}
rendering possible determinations of the quark-mixing matrix element $|V_{us}|$ \cite{Maltman:2008ib, Antonelli:2013usa} and the mass of the strange quark \cite{Chetyrkin:1998ej, Pich:1999hc,
Korner:2000wd, Kambor:2000dj, Chen:2001qf, Gamiz:2002nu, Gamiz:2004ar, Baikov:2004tk, Gamiz:2007qs} at high precision.\\

At the exclusive level, the largest contribution to the strange spectral function is given by the $\tau^-\to (K\pi)^-\nu_\tau$ decays ($\sim 42\%$). The corresponding differential
decay width was measured by the ALEPH \cite{Barate:1999hj} and OPAL \cite{Abbiendi:2004xa} collaborations, and recently the B-factories BaBar \cite{Aubert:2007jh} and Belle
\cite{Epifanov:2007rf} have published increased accuracy measurements. These high-quality data have motivated several refined studies of the related observables \cite{Jamin:2006tk,
Moussallam:2007qc, Jamin:2008qg, Boito:2008fq, Boito:2010me} allowing for precise determinations of the $K^\star(892)$ pole parameters because this resonance gives the most
of the contribution to the dominating vector form factor. These were also determined for the $K^\star(1410)$ resonance and the relative interference of both states was
characterized, although with much less precision than in the case of the $K^\star(892)$ mass and width.

In order to increase the knowledge of the strange spectral function, the $\tau^-\to \left(K\pi\pi(\pi)\right)^-\nu_\tau$ decays have to be better understood (they add up to
one third of the strange decay width), the $\tau^-\to K^- \eta \nu_\tau$ and $\tau^-\to (K\pi)^- \eta \nu_\tau$ decays being also important for that purpose. The $K^-\eta$
mode is also very sensitive to the $K^\star(1410)$ resonance contribution and may be competitive with the $\tau^-\to (K\pi)^-\nu_\tau$ decays in the extraction of its
parameters. This is one of the motivations for our study of the $\tau^-\to K^- \eta^{(\prime)} \nu_\tau$ decays in this article. We will tackle the analysis of the
$\tau^-\to \left(K\pi\right)^-\pi/\eta\,\nu_\tau$ decays along the lines employed in other three-meson \cite{GomezDumm:2003ku, Dumm:2009kj, Dumm:2009va, Dumm:2012vb} and
one-meson radiative tau decays \cite{Guo:2010dv, Roig:2013??} elsewhere.

The $\tau^-\to K^- \eta \nu_\tau$ decays were first measured by CLEO \cite{Bartelt:1996iv} and ALEPH \cite{Buskulic:1996qs} in the '90s. Only very recently Belle \cite{Inami:2008ar}
and BaBar \cite{delAmoSanchez:2010pc} managed to improve these measurements reducing the branching fraction to essentially half of the CLEO and ALEPH results and achieving a
decrease of the error at the level of one order of magnitude. Belle \cite{Inami:2008ar} measured a branching ratio of $(1.58\pm0.05\pm0.09)\cdot10^{-4}$ and BaBar
\cite{delAmoSanchez:2010pc} $(1.42\pm0.11\pm0.07)\cdot10^{-4}$, which combined to give the PDG average $(1.52\pm0.08)\cdot10^{-4}$ \cite{Beringer:1900zz}. The related decay
$\tau^-\to K^- \eta^\prime \nu_\tau$ has not been detected yet, although an upper limit at the $90\%$ confidence level was placed by BaBar \cite{Lees:2012ks
.

Belle's paper \cite{Inami:2008ar} cites the few existing calculations of the $\tau^-\to K^- \eta \nu_\tau$ decays based on Chiral Lagrangians \cite{Pich:1987qq, Braaten:1989zn,
Li:1996md, Aubrecht:1981cr} and concludes that `further detailed studies of the physical dynamics in $\tau$ decays with $\eta$ mesons are required' (see also, e.g.
Ref.~\cite{Actis:2010gg})\footnote{Very recently, the $\tau^-\to K \pi/\eta \nu_\tau$ decays have been studied \cite{Kimura:2012}. However, no satisfactory description of the
data can be achieved in both decay channels simultaneously.}.
Our aim is to provide a
more elaborated analysis which takes into account the advances in this field since the publication of the quoted references more than fifteen years ago. The considered
$\tau^-\to K^- \eta^{(\prime)} \nu_\tau$ decays are currently modeled in TAUOLA \cite{Jadach:1990mz, Jadach:1993hs}, the standard Monte Carlo generator for tau lepton decays,
relying on phase space. We would like to provide the library with Resonance Chiral Lagrangian-based currents \cite{Shekhovtsova:2012ra, Nugent:2013hxa} that can describe well
these decays for their analyses and for the characterization of the backgrounds they constitute to searches of rarer tau decays and new physics processes.

Our paper is organized as follows: the hadronic matrix element and the participating vector and scalar form factors are defined in section \ref{M.e. and decay width}, where
the differential decay distribution in terms of the latter is also given. These form factors are derived within Chiral Perturbation Theory ($\chi PT$) \cite{Weinberg:1978kz,
Gasser:1983yg, Gasser:1984gg} including resonances ($R\chi T$) \cite{Ecker:1988te, Ecker:1989yg} in section \ref{FFs}. Three different options according to treatment of
final-state interactions in these form factors are discussed in section \ref{FSI} and will be used in the remainder of the paper. In section \ref{Pred Keta}, the
$\tau^-\to K^-\eta\nu_\tau$ decay observables are predicted based on the knowledge of the $\tau^-\to (K\pi)^-\nu_\tau$ decays. These results are then improved in section
\ref{Fit Keta} by fitting the BaBar and Belle $\tau^-\to K^-\eta\nu_\tau$ data. We provide our predictions on the $\tau^-\to K^-\eta^\prime\nu_\tau$ decays in section
\ref{Pred Ketap} and present our conclusions in section \ref{Concl}.

\section{Matrix elements and decay width}\label{M.e. and decay width}
We fix our conventions from the general parametrization of the scalar and vector $K^+\eta^{(\prime)}$ matrix elements \cite{Gasser:1984ux}:
\begin{equation}\label{general definition}
 \left\langle\eta^{(\prime)} \Big|\bar{s}\gamma^\mu u\Big| K^+\right\rangle=c^V_{K\eta^{(\prime)}}\left[\left(p_{\eta^{(\prime)}}+p_K\right)^\mu f_+^{K^+ \eta^{(\prime)}}(t)
+(p_K-p_{\eta^{(\prime)}})^\mu f_-^{K ^+\eta^{(\prime)}}(t)\right]\,,
\end{equation}
where $t=(p_K-p_{\eta^{(\prime)}})^2$. From eq. (\ref{general definition}) one has
\begin{equation}\label{m.e. fmas fmenos}
 \left\langle K^-\eta^{(\prime)} \Big|\bar{s}\gamma^\mu u\Big| 0\right\rangle=c^V_{K\eta^{(\prime)}}\left[\left(p_{\eta^{(\prime)}}-p_K\right)^\mu f_+^{K^- \eta^{(\prime)}}(s)
-q^\mu f_-^{K ^-\eta^{(\prime)}}(s)\right]\,,
\end{equation}
with $q^\mu=\left(p_{\eta^{(\prime)}}+p_K\right)^\mu$, $s=q^2$ and $c^V_{K\eta^{(\prime)}}=-\sqrt{\frac{3}{2}}$. Instead of $f_-^{K^- \eta^{(\prime)}}(s)$ one can use
$f_0^{K^- \eta^{(\prime)}}(s)$ defined through
\begin{equation}\label{definition f0}
 \left\langle 0 \Big|\partial_\mu(\bar{s}\gamma^\mu u)\Big| K^-\eta^{(\prime)}\right\rangle=i(m_s-m_u)\left\langle 0 \Big|\bar{s}u\Big| K^-\eta^{(\prime)}\right\rangle\equiv i\Delta_{K\pi}c^S_{K^-\eta^{(\prime)}}f_0^{K ^-\eta^{(\prime)}}(s)\,,
\end{equation}
with
\begin{equation}
c^S_{K^-\eta}= -\frac{1}{\sqrt{6}}\,,\quad c^S_{K^-\eta^\prime}= \frac{2}{\sqrt{3}}\,,\quad \Delta_{PQ}=m_P^2-m_Q^2\,.
\end{equation}
The mass renormalization $m_s-\bar{m}$ in $\chi PT$ (or $R\chi T$)
needs to be taken into account to define $f_0^{K ^-\eta^{(\prime)}}(s)$ and $\bar{m}=(m_d+m_u)/2$ has been introduced.
We will take $\Delta_{K\pi}\Big|^{QCD}=\Delta_{K\pi}$, which is an excellent approximation. From eqs.~(\ref{m.e. fmas fmenos}) and (\ref{definition f0}) one gets
\begin{equation}\label{Had m.e.}
 \left\langle K^-\eta^{(\prime)} \Big|\bar{s}\gamma^\mu u\Big| 0\right\rangle=\left[\left(p_{\eta^{(\prime)}}-p_K\right)^\mu +\frac{\Delta_{K \eta^{(\prime)}}}{s}q^\mu\right]c^V_{K^-\eta^{(\prime)}}f_+^{K^-\eta^{(\prime)}}(s)+
\frac{\Delta_{K \pi}}{s}q^\mu c^S_{K^-\eta^{(\prime)}} f_0^{K^-\eta^{(\prime)}}(s)\,,
\end{equation}
and the normalization condition
\begin{equation}\label{condition origin}
 f_+^{K^-\eta^{(\prime)}}(0)=-\frac{c^S_{K^-\eta^{(\prime)}}}{c^V_{K^-\eta^{(\prime)}}}\frac{\Delta_{K\pi}}{\Delta_{K\eta^{(\prime)}}}f_0^{K^-\eta^{(\prime)}}(0)\,,
\end{equation}
which is obtained from
\begin{equation}
 f_-^{K ^-\eta^{(\prime)}}(s)=-\frac{\Delta_{K\eta^{(\prime)}}}{s}\left[\frac{c^S_{K^-\eta^{(\prime)}}}{c^V_{K^-\eta^{(\prime)}}}\frac{\Delta_{K\pi}}{\Delta_{K\eta^{(\prime)}}}f_0^{K^-\eta^{(\prime)}}(s)+f_+^{K^-\eta^{(\prime)}}(s)\right]\,.
\end{equation}

In terms of these form factors, the differential decay width reads
\begin{eqnarray} \label{spectral function}
& & \frac{d\Gamma\left(\tau^-\to K^-\eta^{(\prime)}\nu_\tau\right)}{d\sqrt{s}} = \frac{G_F^2M_\tau^3}{32\pi^3s}S_{EW}\Big|V_{us}f_+^{K^-\eta^{(\prime)}}(0)\Big|^2
\left(1-\frac{s}{M_\tau^2}\right)^2\\
& & \left\lbrace\left(1+\frac{2s}{M_\tau^2}\right)q_{K\eta^{(\prime)}}^3(s)\Big|\widetilde{f}_+^{K^-\eta^{(\prime)}}(s)\Big|^2+\frac{3\Delta_{K\eta^{(\prime)}}^2}{4s}q_{K\eta^{(\prime)}}(s)\Big|\widetilde{f}_0^{K^-\eta^{(\prime)}}(s)\Big|^2\right\rbrace\,,\nonumber
\end{eqnarray}
where
\begin{eqnarray}\label{definitions}
& &  q_{PQ}(s)=\frac{\sqrt{s^2-2s\Sigma_{PQ}+\Delta_{PQ}^2}}{2\sqrt{s}}\,,\quad \sigma_{PQ}(s)=\frac{2q_{PQ}(s)}{\sqrt{s}}\theta\left(s-(m_P+m_Q)^2\right)\,,\nonumber\\
& & \Sigma_{PQ}=m_P^2+m_Q^2\
,\quad \widetilde{f}_{+,0}^{K^-\eta^{(\prime)}}(s)=\frac{f_{+,0}^{K^-\eta^{(\prime)}}(s)}{f_{+,0}^{K^-\eta^{(\prime)}}(0)}\,,
\end{eqnarray}
and $S_{EW} = 1.0201$ \cite{Erler:2002mv} represents an electro-weak correction factor.

We have considered the $\eta-\eta^\prime$ mixing up to next-to-leading order in the combined expansion in $p^2$, $m_q$ and $1/N_C$ \cite{Kaiser:1998ds, Kaiser:2000gs}
(see the next section for the introduction of the large-$N_C$ limit of QCD \cite{'tHooft:1973jz, 'tHooft:1974hx, Witten:1979kh} applied to the light-flavoured mesons).
In this way it is found that $\Big|V_{us}f_+^{K^-\eta}(0)\Big|=\Big|V_{us}f_+^{K^-\pi^0}(0)\mathrm{cos}\theta_P\Big|$, $\Big|V_{us}f_+^{K^-\eta^\prime}(0)\Big|=\Big|V_{us}f_+^{K^-\pi^0}(0)\mathrm{sin}\theta_P\Big|$,
where
$\theta_P=(-13.3\pm1.0)^\circ$ \cite{Ambrosino:2006gk}.

The best access to $\Big|V_{us}f_+^{K^-\pi^0}(0)\Big|$ is through semi-leptonic Kaon decay data. We will use the value $0.21664\pm0.00048$ \cite{Beringer:1900zz, Antonelli:2010yf}.
Eq.~(\ref{spectral function}) makes manifest that the unknown strong-interaction dynamics is encoded in the tilded form factors, $\widetilde{f}_{+,0}^{K^-\eta^{(\prime)}}(s)$
which will be subject of our analysis in the following section. We will see in particular that the use of $\widetilde{f}_{+,0}^{K^-\eta^{(\prime)}}(s)$ instead of the
untilded form factors yields more compact expressions that are symmetric under the exchange $\eta\leftrightarrow\eta^\prime$, see eqs.(\ref{RChT VFFs}) and (\ref{RChT SFFs def}).

\section{Scalar and vector form factors in $\boldsymbol{\chi PT}$ with resonances}\label{FFs}
Although there is no analytic method to derive the $\widetilde{f}_{+,0}^{K^-\eta^{(\prime)}}(s)$ form factors directly from the QCD Lagrangian, its symmetries are nevertheless
useful to reduce the model dependence to a minimum and keep as many properties of the fundamental theory as possible.

$\chi PT$ \cite{Weinberg:1978kz, Gasser:1983yg, Gasser:1984gg}, the effective field theory of QCD at low energies, is built as an expansion in even powers of the ratio between
the momenta or masses of the lightest pseudoscalar mesons over the chiral symmetry breaking scale, which is of the order of one GeV. As one approaches the energy region where
new degrees of freedom -the lightest meson resonances- become active, $\chi PT$ ceases to provide a good description of the Physics (even including higher-order corrections
\cite{Bijnens:1999sh, Bijnens:1999hw, Bijnens:2001bb}) and these resonances must be incorporated to the action of the theory. This is done without any ad-hoc dynamical
assumption by $R\chi T$ in the convenient antisymmetric tensor formalism that avoids the introduction of local $\chi PT$ terms at next-to-leading order in the chiral expansion
since their contribution is recovered upon integrating the resonances out \cite{Ecker:1988te, Ecker:1989yg}. The building of the Resonance Chiral Lagrangians is driven by the
spontaneous symmetry breakdown of QCD realized in the meson sector, the discrete symmetries of the strong interaction and unitary symmetry for the resonance multiplets. The
expansion parameter of the theory is the inverse of the number of colours of the gauge group, $1/N_C$. Despite $N_C$ not being small in the real world, the fact that phenomenology
supports this approach to QCD \cite{Manohar:1998xv, Pich:2002xy} hints that the associated coefficients of the expansion are small enough to warrant a meaningful perturbative
approach based on it. At leading order in this expansion there is an infinite number of radial excitations for each resonance with otherwise the same quantum numbers that are
strictly stable and interact through local effective vertices only at tree level.

The relevant effective Lagrangian for the lightest resonance nonets reads~\footnote{We comment on its extension to the infinite spectrum predicted in the $N_C\to\infty$ limit
in the paragraph below eq.~(\ref{JOP FFs}).}:
\begin{eqnarray}
\label{eq:ret} {\cal L}_{\rm R\chi T}   & \doteq   & {\cal L}_{\rm kin}^{\rm V,S}\, + \, \frac{F^2}{4}\langle u_{\mu} u^{\mu} + \chi _+\rangle \, + \,
\frac{F_V}{2\sqrt{2}} \langle V_{\mu\nu} f_+^{\mu\nu}\rangle\,+\,i \,\frac{G_V}{\sqrt{2}} \langle V_{\mu\nu} u^\mu u^\nu\rangle \,+\, c_d \langle S u_{\mu} u^{\mu}\rangle
\,+\, c_m \langle S \chi_ +\rangle\,,\nonumber\\
\label{lagrangian}
\end{eqnarray}
where all coupling constants are real, $F$ is the pion decay constant and we follow the conventions of Ref.~\cite{Ecker:1988te}. Accordingly, $\langle \rangle$ stands for
trace in flavour space, and $u^\mu$, $\chi_+$ and $f_+^{\mu\nu}$ are defined by
\begin{eqnarray}
u^\mu & = & i\,u^\dagger\, D^\mu U\, u^\dagger \,,\nonumber \\
\chi_\pm & = & u^\dagger\, \chi \, u^\dagger\, \pm u\,\chi^\dagger\, u\,, \nonumber \\
f_\pm^{\mu\nu} & = & u^\dagger\, F_L^{\mu\nu}\, u^\dagger\, \pm
u\,F_R^{\mu\nu}\, u \ ,
\end{eqnarray}
where $u$ ($U=u^2$), $\chi$ and $F_{L,R}^{\mu\nu}$ are $3\times 3$ matrices that contain light pseudoscalar fields, current quark masses and external left and right currents,
respectively. The matrix $V^{\mu\nu}$ ($S$) includes the lightest vector (scalar) meson multiplet \footnote{In the $N_C\to\infty$ limit of QCD the lightest scalar meson
multiplet does not correspond to the one including the $f_0(600)$ (or $\sigma$ meson) \cite{Cirigliano:2003yq}, but rather to the one including the $f_0(1370)$ resonance.}, and ${\cal L}_{\rm kin}^{\rm V,S}$ stands for these resonances kinetic term.
We note that resonances with other quantum numbers do not contribute to the considered processes (like the axial-vector and pseudoscalar resonances, which have the wrong parity).

The computation of the vector form factors yields
\begin{equation} \label{RChT VFFs}
 \tilde{f}_+^{K^-\eta}(s)=\frac{f_+^{K^-\eta}(s)}{f_+^{K^-\eta}(0)}=1+\frac{F_V G_V}{F^2}\frac{s}{M_{K^\star}^2-s}\,=\frac{f_+^{K^-\eta^\prime}(s)}{f_+^{K^-\eta^\prime}(0)}=\, \tilde{f}_+^{K^-\eta^\prime}(s)\,,
\end{equation}
because $f_+^{K^-\eta}(0)=\cos\theta_{P}$ and $f_+^{K^-\eta^\prime}(0)=\sin\theta_{P}$. We recall that the normalization of the $K\pi$ vector form factor, $f_+^{K^-\pi}(0)$,
was pre-factored in eq.~(\ref{spectral function}) together with $|V_{us}|$.

The strangeness changing scalar form factors and associated S-wave scattering within $R\chi T$ have been investigated in a series of papers by Jamin, Oller and Pich
\cite{Jamin:2000wn, Jamin:2001zq, Jamin:2006tk,Jamin:2006tj} (see also Ref.~\cite{Bernard:1990kw}). The computation of the scalar form factors gives:
\begin{eqnarray}\label{RChT SFFs}
&& \tilde{f}_0^{K^-\eta}(s)=\frac{f_0^{K^-\eta}(s)}{f_0^{K^-\eta}(0)}=\frac{1}{f_0^{K^-\eta}(0)}\left[\cos\theta_{P}f_0^{K^-\eta_8}(s)\Big|_{\eta_8\to\eta}+2\sqrt{2}\mathrm{sin}\theta_Pf_0^{K^-\eta_1}(s)\Big|_{\eta_1\to\eta}\right]\,,\;\;\;\;\;\;\;\;\\
&& \tilde{f}_0^{K^-\eta^\prime}(s)=\frac{f_0^{K^-\eta^\prime}(s)}{f_0^{K^-\eta^\prime}(0)}=\frac{1}{f_0^{K^-\eta^\prime}(0)}\left[\mathrm{cos}\theta_Pf_0^{K^-\eta_1}(s)\Big|_{\eta_1\to\eta^\prime}-\frac{1}{2\sqrt{2}}\mathrm{sin}\theta_Pf_0^{K^-\eta_8}(s)\Big|_{\eta_8\to\eta^\prime}\right]\,,\nonumber
\end{eqnarray}
and can be written in terms of the $f_0^{K^-\eta_8}(s)$, $f_0^{K^-\eta_1}(s)$ form factors computed in Ref.\cite{Jamin:2001zq}:
\begin{eqnarray}\label{JOP FFs}
 f_0^{K^-\eta_8}(s) & = & 1+\frac{4 c_m}{F^2(M_S^2-s)}\left[c_d(s-m_K^2-p_{\eta_8}^2)+c_m(5m_K^2-3m_\pi^2)\right]+\frac{4c_m(c_m-c_d)}{F^2M_S^2}(3m_K^2-5m_\pi^2)\,,\nonumber\\
 f_0^{K^-\eta_1}(s) & = & 1+\frac{4c_m}{F^2(M_S^2-s)}\left[c_d(s-m_K^2-p_{\eta_1}^2)+c_m2m_K^2\right]-\frac{4c_m(c_m-c_d)}{F^2 M_S^2}2m_\pi^2\,,
\end{eqnarray}
where, for the considered flavour indices, $S$ should correspond to the $K^\star_0(1430)$ resonance. Besides $f_0^{K^-\pi}(0)=f_+^{K^-\pi}(0)$ (see the comment below
equation~(\ref{RChT VFFs})) it has also been used that
\begin{eqnarray}
 f_0^{K^-\eta}(0) & = & \cos\theta_{P}\left(1+\frac{\Delta_{K\eta}+3\Delta_{K\pi}}{M_S^2}\right)+2\sqrt{2}\sin\theta_P\left(1+\frac{\Delta_{K\eta}}{M_S^2}\right)\,,\nonumber\\
 f_0^{K^-\eta^\prime}(0) & = & \cos\theta_{P}\left(1+\frac{\Delta_{K\eta}}{M_S^2}\right)+\sin\theta_P\left(1+\frac{\Delta_{K\eta}+3\Delta_{K\pi}}{M_S^2}\right)\,.
\end{eqnarray}
Indeed, using our conventions, the tilded scalar form factors become simply
\begin{equation}\label{RChT SFFs def}
  \tilde{f}_0^{K^-\eta}(s)=\frac{f_0^{K^-\eta}(s)}{f_0^{K^-\eta}(0)}=1+\frac{c_d c_m}{4 F^2}\frac{s}{M_S^2-s}\,=\frac{f_+^{K^-\eta^\prime}(s)}{f_0^{K^-\eta^\prime}(0)}=\, \tilde{f}_0^{K^-\eta^\prime}(s)\,,
\end{equation}
that is more compact than eqs.~(\ref{RChT SFFs}), (\ref{JOP FFs}) and displays the same symmetry $\eta\leftrightarrow\eta^\prime$ than the vector form factors in
eq.~(\ref{RChT VFFs}).

The computation of the leading order amplitudes in the large-$N_C$ limit within $R\chi T$ demands, however, the inclusion of an infinite tower of resonances per set of
quantum numbers \footnote{We point out that there is no limitation in the $R\chi T$ Lagrangians in this respect. In particular, a second multiplet of resonances has been
introduced in the literature \cite{SanzCillero:2002bs, Mateu:2007tr} and bi- and tri-linear operators in resonance fields have been used \cite{GomezDumm:2003ku,
RuizFemenia:2003hm, Cirigliano:2004ue, Cirigliano:2005xn, Cirigliano:2006hb, Kampf:2011ty}.}. Although the masses of the large-$N_C$ states depart slightly from the
actually measured particles \cite{Masjuan:2007ay} only the second vector state, i.e. the $K^\star(1410)$ resonance, will have some impact on the considered decays. Accordingly,
we will replace the vector form factor in eq.~(\ref{RChT VFFs}) by
\begin{equation} \label{RChT VFFs2Res}
 \tilde{f}_+^{K^-\eta^{(\prime)}}(s)=1+\frac{F_V G_V}{F^2}\frac{s}{M_{K^\star}^2-s}+\frac{F_V^\prime G_V^\prime}{F^2}\frac{s}{M_{K^\star\prime}^2-s}\,,
\end{equation}
where the operators with couplings $F_V^\prime$ and $G_V^\prime$ are defined in analogy with the corresponding unprimed couplings in eq.~(\ref{lagrangian}).

If we require that the $f_+^{K^-\eta^{(\prime)}}(s)$ and $f_0^{K^-\eta^{(\prime)}}(s)$ form factors vanish for $s\to\infty$ at least as $1/s$ \cite{Lepage:1979zb,
Lepage:1980fj}, we obtain the short-distance constraints
\begin{equation}\label{shortdistance}
 F_V G_V + F_V^\prime G_V^\prime= F^2\,,\quad 4 c_d c_m = F^2\,,\quad c_d-c_m=0\,,
\end{equation}
which yield the form factors
\begin{eqnarray} \label{FFs with short distance constraints}
& & \tilde{f}_+^{K^-\eta}(s)=\frac{M_{K^\star}^2+\gamma s}{M_{K^\star}^2-s}-\frac{\gamma s}{M_{K^\star\prime}^2-s}= \tilde{f}_+^{K^-\eta^\prime}(s)\,,\\
& & \tilde{f}_0^{K^-\eta}(s) = \frac{M_S^2}{M_S^2-s}\,=\,\tilde{f}_0^{K^-\eta^\prime}(s)\,,\nonumber
\end{eqnarray}
where $\gamma=-\frac{F_V^\prime G_V^\prime}{F^2}=\frac{F_VG_V}{F^2}-1$ \cite{Jamin:2006tk, Jamin:2008qg, Boito:2008fq, Boito:2010me}. We note that we are disregarding the
modifications introduced by the heavier resonance states to the relation (\ref{shortdistance}) and to the definition of $\gamma$.
\section{Different form factors according to treatment of final-state interactions}\label{FSI}
The form factors in eqs.(\ref{FFs with short distance constraints}) diverge when the exchanged resonance is on-mass shell and, consequently, cannot represent the underlying
dynamics that may peak in the resonance region but does not certainly show a singular behaviour. This is solved by considering a next-to-leading order effect in the large-$N_C$
counting, as it is a non-vanishing resonance width \footnote{Other corrections at this order are neglected. Phenomenology seems to support that this is the predominant
contribution.}. Moreover, since the participating resonances are not narrow, an energy-dependent width needs to be considered. A precise formalism-independent definition of
the off-shell vector resonance width within $R\chi T$ has been given in Ref.~\cite{GomezDumm:2000fz} and employed successfully in a variety of phenomenological studies. Its
application to the $K^*(892)$ resonance gives
\begin{eqnarray}\label{K^* width predicted}
 \Gamma_{K^*}(s) & = & \frac{G_V^2 M_{K^*} s}{64 \pi F^4}
\bigg[\sigma_{K\pi}^3(s)+ \mathrm{cos}^2\theta_P \sigma_{K\eta}^3(s) + \mathrm{sin}^2\theta_P\sigma_{K\eta^\prime}^3(s)\bigg] \,,
\end{eqnarray}
where $\sigma_{PQ}(s)$ was defined in eq.~(\ref{definitions}). Several analyses of the $\pi\pi$ \cite{SanzCillero:2002bs, Pich:2001pj, Dumm:2013zh} and $K\pi$ \cite{Jamin:2008qg,
Boito:2008fq, Boito:2010me} form factors where the $\rho(770)$ and $K^\star(892)$ prevail respectively, have probed the energy-dependent width of these resonances with precision.
 Although the predicted width \cite{Guerrero:1997ku} turns to be quite accurate, it is not optimal to achieve a very precise description of the data and, instead, it is
better to allow (as we will do in the remainder of the paper) the on-shell width to be a free parameter and write
\begin{eqnarray}\label{K^* width}
 \Gamma_{K^*}(s) & = & \Gamma_{K^*}\frac{s}{M_{K^*}^2}\frac{\sigma_{K\pi}^3(s)+\mathrm{cos}^2\theta_P \sigma_{K\eta}^3(s) + \mathrm{sin}^2\theta_P\sigma_{K\eta^\prime}^3(s)}{\sigma_{K\pi}^3(M_{K^*}^2)}\,,
\end{eqnarray}
where it has been taken into account that at the $M_{K^*}$-scale the only absorptive cut is given by the elastic contribution.

In the case of the $K^\star(1410)$ resonance there is no warranty that the $KP$ ($P=\pi$, $\eta$, $\eta^\prime$) cuts contribute in the proportion given in eqs.(\ref{K^* width predicted})
and (\ref{K^* width}). We will assume that the lightest $K\pi$ cut dominates and use throughout that
\begin{equation}\label{Kstarprimewidth}
 \Gamma_{K^{\star\prime}}(s)\,=\,\Gamma_{K^{\star\prime}}\frac{s}{M_{K^{\star\prime}}^2}\frac{\sigma_{K\pi}^3(s)}{\sigma_{K\pi}^3(M_{K^{\star\prime}}^2)}\,.
\end{equation}
The scalar resonance width can also be computed in $R\chi T$ similarly \cite{Ecker:1988te, GomezDumm:2000fz}. In the case of the $K^\star_0(1430)$ it reads
\begin{equation}\label{Gamma S computed}
 \Gamma_{S}(s)\,=\,\Gamma_{S_0}\left(M_S^2\right)\left(\frac{s}{M_S^2}\right)^{3/2}\frac{g(s)}{g\left(M_S^2\right)}\,,
\end{equation}
with
\begin{eqnarray}\label{g(s)}
 g(s) & = & \frac{3}{2}\sigma_{K\pi}(s)+\frac{1}{6}\sigma_{K\eta}(s)\left[\mathrm{cos}\theta_P\left(1+\frac{3\Delta_{K\pi}+\Delta_{K\eta}}{s}\right)+2\sqrt{2}\mathrm{sin}\theta_P\left(1+\frac{\Delta_{K\eta}}{s}\right)\right]^2\nonumber\\
& & +\frac{4}{3}\sigma_{K\eta^\prime}(s)\left[\mathrm{cos}\theta_P\left(1+\frac{\Delta_{K\eta^\prime}}{s}\right)-\frac{\mathrm{sin}\theta_P}{2\sqrt{2}}\left(1+\frac{3\Delta_{K\pi}+\Delta_{K\eta^\prime}}{s}\right)\right]^2\,.
\end{eqnarray}

At this point, different options for the inclusion of the resonances width arise. The most simple prescription is to replace $M_R^2-s$ by $M_R^2-s-iM_R\Gamma_R(s)$ in
eqs.~(\ref{FFs with short distance constraints}). We shall call this option `dipole model', or simply `Breit-Wigner (BW) model'. One should pay attention to the fact that
analyticity of a quantum field theory imposes certain relations between the real and imaginary parts of the amplitudes. In particular, there is one between the real and
imaginary part of the relevant two-point function. At the one-loop level its imaginary part is proportional to the meson width but the real part (which is neglected in this
model) is non-vanishing. As a result, the Breit-Wigner treatment breaks analyticity at the leading non-trivial order.

Instead, one can try to devise a mechanism that keeps the complete complex two-point function. Ref.~\cite{Guerrero:1997ku} used an Omn\`es resummation of final-state
interactions in the vector form factor that was consistent with analyticity at next-to-leading order. The associated violations were small and consequently neglected in their
study of the $\pi\pi$ observables. This strategy was also exported to the $K\pi$ decays of the $\tau$ in Refs.~\cite{Jamin:2006tk, Jamin:2008qg} where it yielded
remarkable agreement with the data. We will call this approach to the vector form factor `the exponential parametrization' (since it exponentiates the real part of the
relevant loop function) and refer to it by the initials of the authors who studied the $K\pi$ system along these lines, `JPP'.

A decade after, a construction that ensures analyticity of the vector form factor exactly was put forward in Ref.~\cite{Boito:2008fq} and applied successfully to the study of
the $K\pi$ tau decays. It is a dispersive representation of the form factor where the input phaseshift, which resums the whole loop function in the denominator of
eq.~(\ref{FFs with short distance constraints}), is proportional to the ratio of the imaginary and real parts of this form factor. This method also succeeded in its
application to the di-pion system \cite{Dumm:2013zh}, where it was rephrased in a way which makes chiral symmetry manifest at next-to-leading order. We will name this method
`dispersive representation' or `BEJ', by the authors who pioneered it in the $K\pi$ system.

We would like to stress that the Breit-Wigner model is consistent with $\chi PT$ only at leading order, while the exponential parametrization (JPP) and the dispersive
representation (BEJ) reproduce the chiral limit results up to next-to-leading order and including the dominant contributions at the next order \cite{Guerrero:1998hd}.

In the dispersive approach to the study of the di-pion and Kaon-pion systems it was possible to achieve a unitary description in the elastic region that could be extended up
to $s_{inel}=4m_K^2$ (the $4\pi$ cut, which is phase-space and large-$N_C$ suppressed is safely neglected) and $s_{inel}=(m_K+m_\eta)^2$, respectively. Most devoted studies
of these form factors neglect -in one way or another- inelasticities and coupled-channel effects beyond $s_{inel}$ in them \footnote{See, however, Ref.~\cite{Moussallam:2007qc},
which includes coupled channels for the $K\pi$ vector form factor.}, an approximation that seems to be supported by the impressive agreement with the data sought. However,
this overlook of the problem seems to be questionable in the case of the $\tau^-\to K^-\eta^{(\prime)}\nu_\tau$ decays where we are concerned with the first (second)
inelastic cuts.

An advisable solution may come from the technology developed for the scalar form factors that were analyzed in a coupled channel approach in Refs.~\cite{Jamin:2001zq, Jamin:2001zr,
Jamin:2006tj} (for the strangeness-changing form factors) \footnote{We will use these unitarized scalar form factors instead of the one in
eq.~(\ref{FFs with short distance constraints}) in the JPP and BEJ treatments (see above).} and \cite{Guo:2012ym, Guo:2012yt} (for the strangeness-conserving ones) unitarizing
$SU(3)$ and $U(3)$ (respectively) $\chi PT$ with explicit exchange of resonances \cite{Guo:2011pa}. However, given the large errors of the $\tau^-\to K^-\eta\nu_\tau$ decay
spectra measured by the BaBar \cite{delAmoSanchez:2010pc} and Belle \cite{Inami:2008ar} Collaborations and the absence of data on the $K^-\eta^{\prime}$ channel we consider
that it is not timely to perform such a cumbersome numerical analysis in the absence of enough experimental guidance \footnote{One could complement this poorly known sector
with the information from meson-meson scattering on the relevant channels \cite{GomezNicola:2001as}. Our research at next-to-leading order in the $1/N_C$ expansion treating
consistently the $\eta-\eta^\prime$ mixing \cite{Kaiser:1998ds, Kaiser:2000gs, Escribano:2010wt} is in progress.}. For this reason we have attempted
to obviate the inherent inelasticity of the $K\eta^{(\prime)}$ channels and tried an elastic description, where the form factor that defines the input phaseshift is given by
eq.~(\ref{FFs with short distance constraints}) with $\Gamma_{K^\star}(s)$ defined analogously to $\Gamma_{K^{\star\prime}}(s)$, i.e., neglecting the inelastic cuts. We
anticipate that the accord with data supports this procedure until more precise measurements demand a better approximation.

Let us recapitulate the different alternatives for the treatment of final-state interactions that will be employed in sections \ref{Pred Keta}-\ref{Pred Ketap} to study the
$\tau^-\to K^-\eta^{(\prime)}\nu_\tau$ decays. The relevant form factors will be obtained from eqs.(\ref{FFs with short distance constraints}) in each case by:
\begin{itemize}
 \item Dipole model (Breit-Wigner): $M_R^2-s$ will be replaced by $M_R^2-s-iM_R\Gamma_R(s)$ with $\Gamma_{K^\star}(s)$ and $\Gamma_S(s)$ given by eqs. (\ref{K^* width}) and (\ref{Gamma S computed}).
 \item Exponential parametrization (JPP): The Breit-Wigner vector form factor described above is multiplied by the exponential of the real part of the loop function. The
unitarized scalar form factor \cite{Jamin:2001zq} will be employed. The relevant formulae can be found in appendix \ref{app}.
 \item Dispersive representation (BEJ): A three-times subtracted dispersion relation will be used for the vector form factor. The input phaseshift will be defined using the
vector form factor in eq.~(\ref{FFs with short distance constraints}) with $\Gamma_{K^\star}(s)$ including only the $K\pi$ cut and resumming also the real part of the loop
function in the denominator. The unitarized scalar form factor will be used \cite{Jamin:2001zq}. More details can be found in appendix \ref{app}.
\end{itemize}

\section{Predictions for the $\boldsymbol{\tau^-\to K^-\eta\nu_\tau}$ decays}\label{Pred Keta}
We note that eqs.(\ref{FFs with short distance constraints}) also hold for the $\tilde{f}_{+,0}^{K^-\pi}(s)$ form factors (see eq.~(\ref{spectral function}) and comments below,
as well). Therefore, in principle the knowledge of these form factors in the $K\pi$ system can be transferred to the $K\eta^{(\prime)}$ systems immediately, taking thus
advantage of the larger statistics accumulated in the former and their sensitivity to the $K^\star(892)$ properties. This is certainly true in the case of the vector form
factor in its assorted versions and in the scalar Breit-Wigner form factor. However, in the BEJ and JPP scalar form factor one has to bear in mind (see appendix \ref{app:both})
that the $KP$ ($P=\pi^0,\,\eta,\,\eta^\prime$) scalar form factors are obtained solving the coupled channel problem which breaks the universality of the $\tilde{f}_0^{K^-P}(s)$
form factors as a result of the unitarization procedure. As a consequence, our application of the $\tilde{f}_0^{K^-\eta^{(\prime)}}(s)$ form factors to the
$\tau^-\to K^-\eta^{(\prime)}\nu_\tau$ decays will provide a test of the unitarized results. Taking into account the explanations in Ref.~\cite{Jamin:2001zq} about the
difficult convergence of the three-channel problem (mainly because of the smallness of the $K\eta$ contribution and its correlation with the $K\eta^\prime$ channel) this
verification is by no means trivial, specially regarding the $K\eta^\prime$ channel, where the scalar contribution is expected to dominate the decay width.

In this way, we have predicted the $\tau^-\to K^-\eta\nu_\tau$ branching ratio and differential decay width using the knowledge acquired in the $\tau^-\to (K\pi)^-\nu_\tau$
decays. Explicitly:
\begin{itemize}
 \item In the dipole model, we have taken the $K^\star(892)$, $K^\star(1410)$ and $K_0^\star(1430)$ mass and width from the PDG \cite{Beringer:1900zz} -since this compilation
employs Breit-Wigner parametrizations to determine these parameters- and estimated the relative weight of them using $\gamma=\frac{F_VG_V}{F^2}-1$ (see discussion at the end
of section \ref{FFs}) \cite{Ecker:1988te}. In this way, we have found $\gamma=-0.021\pm0.031$.
 \item In the JPP parametrization, we have used the best fit results of Ref.~\cite{Jamin:2008qg} for the vector form factor. The scalar form factor has been obtained from the
solutions (6.10) and (6.11) of Ref.~\cite{Jamin:2001zq} \footnote{The relevant $f_0^{K^-\eta^{(\prime)}}(s)$ unitarized scalar form factors have been coded using tables kindly
provided by Matthias Jamin.}. The scalar form factors have also been treated alike in the BEJ approach.
 \item In the BEJ representation, one would use the best fit results of Ref.~\cite{Boito:2010me} to obtain our vector form factor. However, we have noticed the strong dependence
on the actual particle masses of the slope form factor parameters, $\lambda_+^\prime$ and $\lambda_+^{\prime\prime}$. Ref.~\cite{Boito:2010me} used the physical masses in their
study of $\tau^-\to K_S\pi^-\nu_\tau$ data. On the other hand we focus on the $\tau^-\to K^-P\nu_\tau$ decays. Consequently, the masses should correspond now to $K^-\pi^0$
instead of to $K_S\pi^-$. Noteworthy, both the $K^-$ and $\pi^0$ are lighter than the $K_S$ and $\pi^-$ and the corresponding small mass differences, given by isospin breaking,
are big enough to demand for a corresponding change in the $\lambda_+^{\prime(\prime)}$ parameters. Accepting this, the ideal way to proceed would be to fit the BaBar data on
$\tau^-\to K^-\pi^0\nu_\tau$ decays \cite{Aubert:2007jh}. Unfortunately, these data are not publicly available yet. For this reason, we have decided to fit Belle data on the
$\tau^-\to K_S\pi^-\nu_\tau$ decay using the $K^-$ and $\pi^0$ masses throughout. The results can be found in table \ref{Tab:Fake fit}, where they are confronted to the best
fit results of Ref.~\cite{Boito:2008fq}~\footnote{We display the results of this reference instead of those in Ref.~\cite{Boito:2010me} because we are not using information
from $K_{\ell3}$ decays in this exercise. Differences are, nonetheless, tiny.}, both of them yield $\chi^2/dof=1.0$ and are given for $s_{cut}=4$ GeV$^2$, although the
systematic error due to the choice of this energy scale is included in the error estimation. We will use the results in the central column of table \ref{Tab:Fake fit} to give
our predictions of the $\tau^-\to K^-\eta\nu_\tau$ decays based on the $K\pi$ results.
\begin{table*}[h!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Parameter& Best fit with fake masses & Best fit \cite{Boito:2008fq}\\
\hline
$\lambda_+^\prime\times 10^{3}$&$22.2\pm0.9$& $24.7\pm0.8$\\
$\lambda_+^{\prime\prime}\times 10^{4}$&$10.3\pm0.2$& $12.0\pm0.2$\\
$M_{K^\star}$ (MeV)&$892.1\pm0.6$& $892.0\pm0.9$\\
$\Gamma_{K^\star}$ (MeV)&$46.2\pm0.5$& $46.2\pm0.4$\\
$M_{K^{\star\prime}}$ (GeV)&$1.28\pm0.07$& $1.28\pm0.07$\\
$\Gamma_{K^{\star\prime}}$ (GeV)&$0.16^{+0.10}_{-0.07}$& $0.20^{+0.06}_{-0.09}$\\
$\gamma$&$-0.03\pm0.02$& $-0.04\pm0.02$\\
\hline
\end{tabular}
\caption{\small{Results for the fit to Belle $\tau^-\to K_S\pi^-\nu_\tau$ data \cite{Epifanov:2007rf} with a three-times subtracted dispersion relation including two vector
resonances in $\widetilde{f}_+^{K\pi}(s)$, according to eq.~(\ref{FFs with short distance constraints}) and resumming the loop function in the denominator (see
appendix \ref{app:BEJ}), as well as the scalar form factor \cite{Jamin:2001zq}. The middle column is obtained using the masses of the $K^-$ and $\pi^0$} mesons and the last
column using the $K_S$ and $\pi^-$ masses actually corresponding to the data.}\label{Tab:Fake fit}
\end{center}
\end{table*}
\end{itemize}

Proceeding this way we find the differential decay distributions for the three different approaches considered using eq.~(\ref{spectral function}).
This one is, in turn, related to the experimental data by using
\begin{equation}\label{theory_to_experiment}
 \frac{dN_{events}}{dE}\,=\,\frac{d\Gamma}{dE}\frac{N_{events}}{\Gamma_\tau BR(\tau^-\to K^-\eta\nu_\tau)}\Delta E_{bin}\,.
\end{equation}
We thank the Belle Collaboration for providing us with their data \cite{Inami:2008ar}. This was not possible in the case of the BaBar Collaboration \cite{delAmoSanchez:2010pc}
because the person in charge of the analysis left the field and the data file was lost. We have, however, read the data points from the paper's figures and included this
effect in the errors. The number of events after background subtraction in each data set are $611$ (BaBar) and $1365$ (Belle) and the corresponding bin widths are $80$ and $25$
MeV, respectively. In Fig.\ref{fig:Pred_Keta} we show our predictions based on the $K\pi$ system according to BW, JPP and BEJ. In this figure we have normalized the BaBar data
to Belle's using eq.~(\ref{theory_to_experiment}). A look at the data shows some tension between both measurements and we notice a couple of strong oscillations of isolated
Belle data points which do not seem to correspond to any dynamics but rather to an experimental issue or to underestimation of the systematic errors \footnote{We have also
realized that the first two Belle data points, with non-vanishing entries, are below threshold, a fact which may indicate some problem in the calibration of the hadronic
system energy or point to underestimation of the background.}. In this plot there are also shown the corresponding one-sigma bands obtained neglecting correlations between
the resonance parameters and also with respect to other sources of uncertainty, namely $|V_{us}f_+^{K^-\pi^0}(0)|$ and $\theta_P$, whose errors are also accounted for. The
corresponding branching ratios are displayed in table \ref{Tab:Pred_Keta}, where the $\chi^2/dof$ is also shown. We note that the error correlations corresponding to the fit
results shown in table \ref{Tab:Fake fit} have been taken into account in BEJ's branching ratio of table \ref{Tab:Pred_Keta}.

It can be seen that the BW model gives a too low decay width and that the function shape is not followed by this prediction, as indicated by the high value of the $\chi^2/dof$
that is obtained. On the contrary, the JPP and BEJ predictions yield curves that compare quite well with the data already. Moreover, the corresponding branching fractions are
in accord with the PDG value within errors. Altogether, this explains the goodness of the $\chi^2/dof$, which is $1.5\leftrightarrow 1.9$. Besides, we notice that the error bands
are wider in the dispersive representation than in the exponential parametrization, which may be explained by the larger number of parameters entering the former and the more
complicated correlations between them that were neglected in obtaining Fig.~\ref{fig:Pred_Keta} and the JPP  result in table \ref{Tab:Pred_Keta}.

From these results we conclude that quite likely the BW model is a too rough approach to the problem unless our reference values for $\gamma$ and the $K^\star(1410)$
resonance parameters were a bad approximation. We will check this in the next section. On the contrary, the predictions discussed above hint that JPP and BEJ are appropriate
for the analysis of $\tau^-\to K^-\eta\nu_\tau$ data that we will pursue next.
\begin{figure}[h!]
\begin{center}
\vspace*{1.25cm}
\includegraphics[scale=0.75]{Bands.pdf}
\caption{\label{fig:Pred_Keta} \small{BaBar (blue) \cite{delAmoSanchez:2010pc} and Belle (red) \cite{Inami:2008ar} data for the $\tau^-\to K^-\eta\nu_\tau$ decays are
confronted to the predictions obtained in the BW (dotted), JPP (solid) and BEJ (dashed) approaches (see the main text for details) which are shown together with the
corresponding one-sigma error bands in yellow, light blue and light green, respectively.}}
\end{center}
\end{figure}

\begin{table*}[h!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Source& Branching ratio & $\chi^2/dof$\\
\hline
Dipole Model (BW)&$\left(0.78^{+0.17}_{-0.10}\right)\cdot 10^{-4}$& $8.3$\\
JPP&$\left(1.47^{+0.14}_{-0.08}\right)\cdot 10^{-4}$& $1.9$\\
BEJ&$\left(1.49\pm0.05\right)\cdot 10^{-4}$& $1.5$\\
Experimental value&$\left(1.52\pm0.08\right)\cdot 10^{-4}$& -\\
\hline
\end{tabular}
\caption{\small{Predicted branching ratio of the $\tau^-\to K^-\eta\nu_\tau$ decays according to the different approaches used (see the items above eq.~(\ref{theory_to_experiment})
for details). The corresponding $\chi^2/dof$ values are also given and the PDG branching fraction is given for reference.}}\label{Tab:Pred_Keta}
\end{center}
\end{table*}

\section{Fits to the $\boldsymbol{\tau^-\to K^-\eta\nu_\tau}$ BaBar and Belle data}\label{Fit Keta}
We have considered different fits to the $\tau^-\to K^-\eta\nu_\tau$ data. In full generality we have assessed that the data is not sensitive either to the low-energy region
or to the $K^\star(892)$ peak region. This is not surprising, since the threshold for $K^-\eta$ production opens around $1041$ MeV which is some $100$ MeV larger than
$M_{K^\star}+\Gamma_{K^\star}$, a characteristic energy scale for the $K^\star(892)$ region of dominance. This implies first that the fits are unstable under floating $M_{K^\star}$
and $\Gamma_{K^\star}$ (which affects all three approaches) and second that the slopes of the vector form factor, which encode the physics immediately above threshold, can not
be fitted with $\tau^-\to K^-\eta\nu_\tau$ data (this only concerns BEJ). We have considered consequently fits varying only the $K^\star(1410)$ mass and width and $\gamma$ and
sticking to the reference values discussed in the previous section for the remaining parameters in every approach.

Our best fit results for the branching ratios are written in table \ref{Tab:Fit_Keta}, where the corresponding $\chi^2/dof$ can also be read. These are obtained with the best fit
parameter values shown in table \ref{Tab:Fit_results}, which can be compared to the reference values, which were used to obtain the predictions in the previous section, that
are recalled in table \ref{Tab:Fit_Reference}. The corresponding decay distributions with one-sigma error bands attached are plotted in Fig.~\ref{fig:Fit_Keta}.

These results show that the BW model does not really provide a good approximation to the underlying physics for any value of its parameters and should be discarded. Oppositely,
JPP and BEJ are able to yield quite good fits to the data with values of the $\chi^2/dof$ around one. This suggests that the simplified treatment of final state interactions
in BW, which misses the real part of the two-meson rescatterings and violates analyticity by construction, is responsible for the failure.

A closer look to the fit results using JPP and BEJ in tables \ref{Tab:Fit_Keta} and \ref{Tab:Fit_results} shows that:
\begin{itemize}
 \item Fitting $\gamma$ alone is able to improve the quality of both approaches by $15\leftrightarrow 20\%$. The fitted values are consistent with the reference ones (see table
\ref{Tab:Fit_Reference}): in the case of BEJ at one sigma, being the differences in JPP slightly larger than that only. This is satisfactory because both the
$\tau^-\to (K\pi)^-\nu_\tau$ and the $\tau^-\to K^-\eta\nu_\tau$ decays are sensitive to the interplay between the first two vector resonances and contradictory results would
have casted some doubts on autoconsistency.
 \item When the $K^\star(1410)$ parameters are also fitted the results improve by $\sim13\%$ in JPP and by $\sim33\%$ in BEJ. This represents a reduction of the $\chi^2/dof$
by $\sim26\%$ in JPP and by $\sim50\%$ in BEJ. It should be noted that the three-parameter fits do not yield to physical results in BW. Specifically, $K^\star(1410)$ mass and
width tend to the $K^\star(892)$ values and $|\gamma|$ happens to be one order of magnitude larger than the determinations in the literature. Therefore we discard this
result. We also notice that although the branching ratios of both JPP and BEJ (which have been obtained taking into account the parameter fit correlations) are in agreement
with the PDG value, the JPP branching ratios tend to be closer to its lower limit while BEJ is nearer to the upper one. It can be observed that the deviations of the
three-parameter best fit values with respect to the default ones lie within errors in BEJ, as it so happens with $\Gamma_{K^{\star\prime}}$ in JPP. However, there are small
tensions between the reference and best fit values of $M_{K^{\star\prime}}$ and $\gamma$ in JPP.
\end{itemize}

These results are plotted in Fig.~\ref{fig:Fit_Keta}. Although the BW curve has improved with respect to Fig.~\ref{fig:Pred_Keta} and seems to agree well with the data in the
higher-energy half of the spectrum, it fails completely at lower energies. On the contrary, JPP and BEJ provide good quality fits to data which are satisfactory along the
whole phase space. We note that JPP goes slightly below BEJ and its error band is again narrower possibly due to having less parameters. BEJ errors include the systematics
associated to changes in $s_{cut}$ which is slightly enhanced with respect to the $K\pi$ case.

Despite the vector form factor giving the dominant contribution to the decay width, the scalar form factor is not negligible and gives $\sim(3\leftrightarrow4)\%$
of the branching fraction in the JPP and BEJ cases. In the BW model this contribution is $\sim7\%$.

\begin{table*}[h!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Source & Branching ratio & $\chi^2/dof$\\
\hline
Dipole Model (BW) (Fit $\gamma$)&$\left(0.96^{+0.21}_{-0.15}\right)\cdot10^{-4}$& $5.0$\\
Dipole Model (BW) (Fit $\gamma$, $M_{K^{\star\prime}}$, $\Gamma_{K^{\star\prime}})$ &Unphysical result& -\\
JPP (Fit $\gamma$)&$\left(1.50^{+0.19}_{-0.11}\right)\cdot 10^{-4}$& $1.6$\\
JPP (Fit $\gamma$, $M_{K^{\star\prime}}$, $\Gamma_{K^{\star\prime}})$&$\left(1.42\pm0.04\right)\cdot 10^{-4}$& $1.4$\\
BEJ (Fit $\gamma$)&$\left(1.59^{+0.22}_{-0.16}\right)\cdot 10^{-4}$& $1.2$\\
BEJ (Fit $\gamma$, $M_{K^{\star\prime}}$, $\Gamma_{K^{\star\prime}})$ &$\left(1.55\pm0.08\right)\cdot 10^{-4}$& $0.8$\\
Experimental value&$\left(1.52\pm0.08\right)\cdot 10^{-4}$& -\\
\hline
\end{tabular}
\caption{\label{Tab:Fit_Keta} \small{The branching ratios and $\chi^2/dof$ obtained in BW, JPP and BEJ fitting $\gamma$ only and also the $K^\star(1410)$ parameters are
displayed. Other parameters were fixed to the reference values used in section \ref{Pred Keta}. The PDG branching fraction is also given for reference.}}
\end{center}
\end{table*}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c}
\hline
\backslashbox{Fitted value}{Approach}&Dipole Model (BW)&JPP&BEJ\cr
\hline
$\gamma$&$-0.174\pm0.007$&$-0.063\pm0.007$&$-0.041\pm0.021$\cr
\hline
$\gamma$&Unphysical&$-0.078^{+0.012}_{-0.014}$&$-0.051^{+0.012}_{-0.036}$\cr
$M_{K^{\star'}}$ (MeV) &best fit&$1356\pm11$&$1327^{+30}_{-38}$\cr
$\Gamma_{K^{\star'}}$ (MeV) &parameters&$232^{+30}_{-28}$&$213^{+72}_{-118}$\cr
\hline
\end{tabular}
\caption{\label{Tab:Fit_results} \small{The best fit parameter values corresponding to the different alternatives considered in table \ref{Tab:Fit_Keta} are given. These can
be compared to the reference values, which are given in table \ref{Tab:Fit_Reference}. BEJ results for the mass and width of the $K^{\star}(1410)$ correspond to pole values,
while JPP figures are given for the model parameter as in the original literature.}}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c}
\hline
\backslashbox{Reference value}{Approach}&Dipole Model (BW)&JPP&BEJ\cr
\hline
$\gamma$&$-0.021\pm0.031$&$-0.043\pm0.010$&$-0.029\pm0.017$\cr
$M_{K^{\star'}}$ (MeV) & $1414\pm15$ &$1307\pm17$&$1283\pm65$\cr
$\Gamma_{K^{\star'}}$ (MeV) & $232\pm21$ &$206\pm49$&$163\pm68$\cr
\hline
\end{tabular}
\caption{\label{Tab:Fit_Reference} \small{Reference values (used in section \ref{Pred Keta}) corresponding to the best fit parameters appearing in table \ref{Tab:Fit_results}.
Again BEJ results are pole values and JPP ones are model parameters. The latter are converted to resonance pole values in section \ref{Concl}, where the determination of the
$K^{\star}(1410)$ pole parameters is given.}}
\end{center}
\end{table}

The JPP model values appearing in tables \ref{Tab:Fit_results} and \ref{Tab:Fit_Reference} can be translated to pole values along the lines discussed in
Ref.~\cite{Escribano:2002iv}. This yields $M_{K^{\star\prime}}=1332^{+16}_{-18}\,,\,\Gamma_{K^{\star\prime}}=220^{+26}_{-24}$ for the best fit values and
$M_{K^{\star\prime}}=1286^{+26}_{-28}\,,\,\Gamma_{K^{\star\prime}}=197^{+41}_{-45}$ for the reference values, where all quantities are given in MeV. Remarkable
agreement is found between our best fit values in the JPP and BEJ cases, since the latter yields $M_{K^{\star\prime}}=1327^{+30}_{-38}\,,
\,\Gamma_{K^{\star\prime}}=213^{+72}_{-118}$. From the detailed study of the $\pi\pi$, $K\pi$ (in the quoted literature) and $K\eta$ systems (in this paper) within JPP and BEJ,
one can conclude generally that the dispersive form factors allow a better description of the data while the exponential parametrizations lead to the determination of the
resonance pole values with smaller errors. Both things seem to be due to the inclusion of the subtraction constants as extra parameters in the fits within the dispersive
representations.

\begin{figure}[h!]
\begin{center}
\vspace*{1.25cm}
\includegraphics[scale=0.75]{BandsFit.pdf}
\caption{\label{fig:Fit_Keta} \small{BaBar (blue) \cite{delAmoSanchez:2010pc} and Belle (red) \cite{Inami:2008ar} data for the $\tau^-\to K^-\eta\nu_\tau$ decays are
confronted to the best fit results obtained in the BW (dotted), JPP (solid) and BEJ (dashed) approaches (see the main text for details) which are shown together with the
corresponding one-sigma error bands in light green, pink and orange, respectively. The BW curve corresponds to the one-parameter fit while the JPP and BEJ ones correspond
to three-parameter fits.}}
\end{center}
\end{figure}

\section{Predictions for the $\boldsymbol{\tau^-\to K^-\eta^\prime\nu_\tau}$ decays}\label{Pred Ketap}
We can finally profit from our satisfactory description of the $\tau^-\to K^-\eta\nu_\tau$ decays and predict the $\tau^-\to K^-\eta^\prime\nu_\tau$ decay observables, where
there is only the upper limit fixed at ninety percent confidence level by the BaBar Collaboration \cite{Lees:2012ks}, $BR<4.2\cdot10^{-6}$. We have done this for our best fit
results in the BW (one-parameter fit) JPP and BEJ (three-parameter fits) cases. The corresponding results are plotted in Fig.~\ref{fig:Ketap} and the branching ratios can be
read from table \ref{Tab:Pred_Ketap}. In the figure we can see that the decay width is indeed dominated by the scalar contribution \footnote{In principle, both the scalar and
vector $K\eta^\prime$ form factors are suppressed since they are proportional to $\sin\theta_P$. However, the unitarization procedure of the scalar form factor enhances it
sizeably \cite{Jamin:2001zq} due to the effect of the coupled inelastic channels.} \footnote{The suppression of the vector contribution makes that the predicted values using
information from the $K\pi$ system and the one-parameter fits with JPP and BEJ are very similar to the results in table \ref{Tab:Pred_Ketap}. For this reason we do not show them.}.
In fact, the vector form factor contributes in the range $(9\leftrightarrow15)\%$ to the corresponding branching ratio. Although we keep the BW prediction for reference, we do
not draw the associated (large) error band for the sake of clarity in the figure taking into account its wrong description of the $K\eta$ system shown in the previous
section. As the scalar form factor dominates the decay width and we are using the same one in JPP and BEJ, the differences between them are tiny (and the errors, of order one
third, are the same in table \ref{Tab:Pred_Ketap}). As expected from the results in the $\tau^-\to K^-\eta\nu_\tau$ decays, BEJ gives the upper part of the error band while
JPP provides the lower one. We are looking forward to the discovery of this decay mode to verify our predictions. A priori one may forecast some departure from it because of
the effect of the poorly known elastic and $K\eta$ channels in meson-meson scattering, which affects the solution of the coupled system of integral equations and specially
the value of the $K^-\eta^\prime$ scalar form factor, that is anyway suppressed to some extent.
\begin{figure}[h!]
\begin{center}
\vspace*{1.25cm}
\includegraphics[scale=0.75]{SpectrumKetap.pdf}
\caption[]{\label{fig:Ketap} \small{The predicted $\tau^-\to K^-\eta^\prime\nu_\tau$ decay width according to BW (green, its big uncertainty is not shown for clarity of the
figure), JPP (blue with lower band in red) and BEJ (blue with upper part in pink) is shown. In these last two the scalar form factor corresponds to Ref.~\cite{Jamin:2006tj},
which is represented by the author's initials, JOP, in the figure's legend. The corresponding vector form factor contributions, which are subleading are plotted
in orange (solid), blue (dashed) and purple (dotted).}}
\end{center}
\end{figure}

\begin{table*}[h!]
\begin{center}
\begin{tabular}{|c|c|}
\hline
Source & Branching ratio\\
\hline
Dipole Model (BW) (Fit)&$(1.45^{+3.80}_{-0.87})\cdot10^{-6}$\\
JPP (Fit)&$(1.00^{+0.37}_{-0.29})\cdot10^{-6}$\\
BEJ (Fit)&$(1.03^{+0.37}_{-0.29})\cdot10^{-6}$\\
Experimental bound&<$4.2\cdot10^{-6}$ at $90\%$ C.L.\\
\hline
\end{tabular}
\caption{\label{Tab:Pred_Ketap} \small{Predicted branching ratios for the $\tau^-\to K^-\eta^\prime\nu_\tau$ decays. The BaBar upper limit is also shown \cite{Lees:2012ks}.}}
\end{center}
\end{table*}

In Fig.~\ref{fig:Correlation} we also plot the correlation between the $\tau^-\to K^-\eta\nu_\tau$ and $\tau^-\to K^-\eta^\prime\nu_\tau$ branching ratios according to the best fit JPP
result at one sigma. The correlations between the parameters are neglected. Since the vector (scalar) form factor dominates the former (latter) decays and their parameters are
independent the plot does not show any sizeable correlation between both measurements, as expected. As a result, if new data on the $\tau^-\to K^-\eta^\prime\nu_\tau$ decays
demand a more careful determination of the $f_0^{K^-\eta^\prime}(s)$ unitarized form factor this will leave almost unaffected the results obtained for the $\tau^-\to K^-\eta\nu_\tau$
channel.

\begin{figure}[!h]
\begin{center}
\vspace*{1.25cm}
\includegraphics[scale=1.25]{Correlation.pdf}
\caption[]{\label{fig:Correlation} \small{The correlation between the $\tau^-\to K^-\eta\nu_\tau$ and $\tau^-\to K^-\eta^\prime\nu_\tau$ branching ratios is plotted according
to the best fit JPP result at one sigma. Correlations between the parameters are neglected. According to expectations, no sizable correlation between both decay modes is
observed.}}
\end{center}
\end{figure}
\section{Conclusions}\label{Concl}
Hadronic tau decays are an ideal scenario to learn about the non-perturbative character of the strong interactions in rather clean conditions. In this work, we have studied
the $\tau^-\to K^-\eta^{(\prime)}\nu_\tau$ decays motivated by the recent measurements performed by the BaBar \cite{delAmoSanchez:2010pc, Lees:2012ks} and Belle Collaborations
\cite{Inami:2008ar}. These decays allow the application of the knowledge acquired in the study of $\tau^-\to (K\pi)^-\nu_\tau$ decays. In particular, the $K\eta$ decay is
sensitive to the parameters of the $K^\star(1410)$ resonance and to its interplay with the $K^\star(892)$ meson, while the $K\eta^\prime$ decay is an appropriate place to
test the unitarization of the strangeness-changing scalar form factors in three coupled-channel case.

We have defined with detail the (tilded) scalar and vector form factors and we have gone through the steps of their calculation within Chiral Perturbation Theory including
the lightest resonances as explicit degrees of freedom and showed that the results are written in a more compact way using the tilded form factors. Then we have discussed
different options according to the treatment of final-state interactions. Specifically, there is the dipole Breit-Wigner (BW) model, which neglects the real part of the two-meson
loop function violating analyticity at next-to-leading order; there is the exponential parametrization (JPP) where this real part of the loop is resummed through an Omn\`es
exponentiation, which violates analyticity at the next order; and there is the dispersive representation (BEJ), which resums the whole loop function in the denominators,
where analyticity holds exactly.

In our case, an additional difficulty is that the elastic approach is not valid in any region of the phasespace, since the $K\pi$ channel is
open well below the $K\eta^{(\prime)}$ channels. In JPP this is not an issue, since one simply adds the corresponding contribution of these channels to the width and real part
of the loop function. However, in BEJ it prevents an approach which does not include inelasticities and the effect of coupled channels. Being conscious of this, we have
nevertheless attempted a dispersive representation of the $K\eta^{(\prime)}$ vector form factors were the input phaseshift is obtained using the elastic approximation and, to
our surprise, it has done an excellent job in its confrontation to the $K\eta$ data. In the light of more accurate measurements it may become necessary to improve this
treatment in the future. Very good agreement has also been found using JPP but BW has failed in this comparison. In the JPP and BEJ fits to the $K\eta$ channel the scalar
form factor was obtained solving dispersion relations for the three-body problem.

We have checked that the $K\eta^{(\prime)}$ channels are not sensitive either to the $K^\star(892)$ parameters or to the slopes of the form factor, $\lambda_+^{\prime(\prime)}$
(BEJ). We have borrowed this information from the $K\pi$ system. This task was straightforward in BW and JPP although in BEJ we noticed that the $\lambda_+^{\prime(\prime)}$
parameters were sensitive to isospin breaking effects that we had to account for. Once this was done we could fit the $K^\star(1410)$ resonance pole parameters and its relative
weight with respect to the $K^\star(892)$ meson, $\gamma$. Our results for these, with masses and widths in MeV, are
\begin{equation}
 M_{K^{\star\prime}}\,=\,1327^{+30}_{-38},\quad\Gamma_{K^{\star\prime}}\,=\,213^{+72}_{-118},\quad \gamma\,=\,-0.051^{+0.012}_{-0.036}\,,
\end{equation}
in the dispersive representation (BEJ) and
\begin{equation}
 M_{K^{\star\prime}}\,=\,1332^{+16}_{-18},\quad\Gamma_{K^{\star\prime}}\,=\,220^{+26}_{-24},\quad \gamma\,=\,-0.078^{+0.012}_{-0.014}\,,
\end{equation}
for the exponential parametrization (JPP). Our determination of these parameters has shown to be competitive with its extraction from the $\tau^-\to (K\pi)^-\nu_\tau$ decays.
To illustrate this point, we average the JPP and BEJ determinations from the $K\pi$ \cite{Jamin:2008qg, Boito:2010me} and $K\eta$ systems, respectively, to find
\begin{equation}
 M_{K^{\star\prime}}\,=\,1277^{+35}_{-41},\quad\Gamma_{K^{\star\prime}}\,=\,218^{+95}_{-66},\quad \gamma\,=\,-0.049^{+0.019}_{-0.016}\,,
\end{equation}
from $K\pi$
and
\begin{equation}
 M_{K^{\star\prime}}\,=\,1330^{+27}_{-41},\quad\Gamma_{K^{\star\prime}}\,=\,217^{+68}_{-122},\quad \gamma\,=\,-0.065^{+0.025}_{-0.050}\,,
\end{equation}
from $K\eta$. We have thus opened an alternative way of determining these parameters. New, more precise data on the $\tau^-\to (K\pi)^-\nu_\tau$ and
$\tau^-\to K^-\eta\nu_\tau$ decays will make possible a more accurate determination of these parameters.

Finally we have benefited from this study of the $\tau^-\to K^-\eta\nu_\tau$ decays and applied it to the $\tau^-\to K^-\eta^{\prime}\nu_\tau$ decays, were our predictions
respect the upper limit found by BaBar and hint to the possible discovery of this decay mode in the near future.

In this way we consider that we are in position of providing TAUOLA with theory-based currents that can describe well the $\tau^-\to K^-\eta^{(\prime)}\nu_\tau$ decays,
based on the exponential parametrization developed by JPP and the dispersive representation constructed by BEJ.

To conclude, differential distributions of hadronic tau decays provide important information for testing diverse form factors and extracting the corresponding parameters
increasing our knowledge of hadronization in the low-energy non-perturbative regime of QCD. It will be interesting to see if our predictions for the $\tau^-\to K^-\eta^{\prime}\nu_\tau$
decays are corroborated and if more precise data on the $\tau^-\to K^-\eta\nu_\tau$ decays demand a more refined treatment. Finally, we emphasize the need of giving pole
resonance parameters irrespective of the approach employed, either in a theorists' article or in a publication by an experimental collaboration.