text:
\section{Introduction}
Optical surveys sampling the sky over time scales of a few days significantly advanced our knowledge of astronomical transients of different origins,
including super-luminous supernovae (SLSNe; \citealt{Quimby11, Chomiuk11, Gal-Yam12}), very fast-rising stellar explosions (e.g. \citealt{Drout14}) and stellar tidal disruptions by super-massive black holes (TDEs; \citealt{Rees88,Komossa15}). Occasionally, a transient is found with properties that seem to defy all previous classification schemes. The event ASASSN-15lh belongs to this category.

ASASSN-15lh  (\citealt{Dong16}) was discovered by the All-Sky Automated Survey for Supernovae (ASAS-SN\footnote{http://www.astronomy.ohio-state.edu/~assassin/index.shtml}) on 14 June 2015 at z=0.2326  ($d=1171$ Mpc for standard \emph{Plank} cosmology). Its extremely large peak luminosity $L_{pk}\sim2\times10^{45}\,\rm{erg\,s^{-1}}$  and the blue, almost featureless spectrum with no apparent sign of H or He (and some spectroscopic resemblance to the SLSN 2010gx) led \cite{Dong16} and \cite{Godoy-Rivera16} to suggest that ASASSN-15lh is the most luminous SLSN ever detected. The very large energy radiated by ASSASN-15lh ($E_{rad}\sim (1.5-2)\times10^{52}\,\rm{erg}$, \citealt{Godoy-Rivera16}), requires extreme properties of the progenitor star  and sources of energy that are different from the standard radioactive decay of $^{56}$Ni that powers normal H-stripped SNe in the local Universe (\citealt{Dong16, Chatzopoulos16,Kozyreva16}). In this context, the double-humped light-curve of ASASSN-15lh has been interpreted by \cite{Chatzopoulos16} as a signature of the interaction of massive SN ejecta $M_{ej}\sim36\,\rm{M_{\sun}}$ with an H-poor circumstellar shell of $M_{CSM}\sim20\,\rm{M_{\sun}}$, possibly supplemented by radiation from a newly-born rapidly-rotating magnetar \citep{Metzger15,Sukhbold16,Bersten16,Dai+16}.

The old, massive $M_{\ast}\sim 2\times 10^{11}\,\rm{M_{\sun}}$ host galaxy of ASASSN-15lh, with limited star formation ($SFR<0.3\,\rm{M_{\sun}yr^{-1}}$; \citealt{Dong16}), is however markedly different from those of core-collapse SNe (e.g. \citealt{Leaman11}) as well as of envelope-stripped SLSNe, which tend to be younger star forming systems with significantly lower stellar mass (\citealt{Lunnan14,Lunnan15,Perley16}). This observation, together with the location of the transient, astrometrically consistent with the host galaxy nucleus, inspired a connection between ASASSN-15lh and the tidal disruption of a star by the host-galaxy super-massive black hole (SMBH, \citealt{Godoy-Rivera16,Brown16,leloudas16,Perley16}). In this context ASASSN-15lh would be the most luminous TDE ever observed, associated to a SMBH with mass $M_{\bullet}\sim10^{8.6}\,\rm{M_{\sun}}$ \citep{Dong16, Godoy-Rivera16} significantly larger than any SMBH presently associated to a TDE (e.g. \citealt{Komossa15}).

It is clear that the luminosity, the spectral properties and the double-humped (i.e., ``Bactrian'') light-curve of ASASSN-15lh, as well as its host galaxy, are unprecedented both in the context of SLSNe and in the context of TDEs.

In this paper we present and discuss the following observational facts: [i] Uncovering of persistent, soft X-ray emission at the location of ASASSN-15lh (Sec. \ref{SubSec:CXO} and Sec. \ref{SubSec:XRT}); [ii] Detection of significant temporal variability at UV wavelengths during the re-brightening phase (Sec. \ref{SubSec:UV}); [iii] Detection of narrow nebular spectral features connected to the host-galaxy nucleus (Sec. \ref{SubSec:Spec}). We propose a scenario where a single physical mechanism can naturally explain the double-humped light-curve of ASASSN-15lh and suggest that its location, very close or coincident with the nucleus of a galaxy that harbors a massive SMBH, is likely the key to unlocking the mysterious nature of the transient (Sec. \ref{Sec:Int}).
Conclusions are drawn in Sec. \ref{Sec:conclusions}.

In our analysis we assume the object's time of first light to be  April 29, 2015, corresponding to a 30-day (rest-frame) rise-time to maximum V-band luminosity \citep{Dong16}. Our main conclusions do not depend on this assumption.
\section{Data Analysis and Results}

\subsection{X-ray analysis- CXO}
\label{SubSec:CXO}

\begin{figure}
\vskip -0.0 true cm
\centering
\includegraphics[scale=0.42]{Spectral_par_BB_17880_joint.eps}
\includegraphics[scale=0.42]{Spectral_par_PL_17880_joint.eps}
\caption{Constraints on the spectral model parameters of the X-ray source detected at the location of ASASSN-15lh, as derived from our second epoch (shaded region) and a joint fit of the third and fourth epoch (contours) of CXO observation. \emph{Upper panel}: absorbed blackbody spectrum. \emph{Lower panel}: absorbed power-law spectrum. In both panels the cross symbol identifies the best fitting model parameters. These observations favor a soft X-ray spectrum with limited intrinsic absorption. }
\label{Fig:Chandra}
\end{figure}

We obtained four epochs of deep X-ray observations of ASASSN-15lh with the Chandra X-ray Observatory (CXO) on November 12, 2015 (exposure of 10 ks), December 13, 2015 (10 ks), February 20, 2016 (40 ks) and August 19, 2016 (30 ks, PI Margutti), corresponding to $\delta t$$=$$129.4$ days, $\delta t$$=$$154.6$ days, $\delta t$$=$$210.5$ and $\delta t$$=$$357.8$ days rest frame since optical maximum light, which occurred on June 5, 2015 \citep{Dong16}. CXO data have been reduced with the CIAO software package (version 4.8) and corresponding calibration files. Standard ACIS data filtering has been applied.

ASASSN-15lh is not detected in our first epoch of observations (ID 17879),  with a 3$\sigma$ count-rate upper limit of $9.98\times 10^{-5}\,\rm{c\,s^{-1}}$ (0.5-8 keV).  The Galactic column density in the direction of the transient is $3.07\times 10^{20}\,\rm{cm^{-2}}$ \citep{Kalberla05}. For an assumed power-law spectrum with  photon index $\Gamma=2$ and Galactic absorption, the unabsorbed 0.3-10 keV flux limit is $F_x<1.1\times 10^{-15}\rm{erg\,s^{-1}cm^{-2}}$ ($L_x<1.8\times 10^{41}\,\rm{erg\,s^{-1}}$).
Our analysis below favors a soft X-ray spectrum with negligible absorption and $\Gamma\sim 3$ or a thermal spectrum with $T\sim0.17\,\rm{keV}$. For these parameters, the unabsorbed 0.3-10 keV flux limit is $F_x<2.0\times 10^{-15}\rm{erg\,s^{-1}cm^{-2}}$ (non-thermal spectrum) and $F_x<8.0\times 10^{-16}\rm{erg\,s^{-1}cm^{-2}}$ (thermal spectrum).

In our second epoch of observation (ID 17880) we find evidence for weak, soft X-ray emission at the location of ASASSN-15lh. We detect two photons with energy $<$1 keV in a $1\arcsec$ region around the transient, corresponding to a 4.3$\,\sigma$ c.l.  detection in the 0.5-1 keV energy range, and to a 3.3$\,\sigma$ c.l. detection in the 0.5-8 keV energy range. We constrain the spectral parameters by using the observed background and the actual instrumental response to simulate the expected emission from a grid of thermal and non-thermal spectral models with a wide range of intrinsic absorption  $NH_{int}=(0-4)\times 10^{22}\,\rm{cm^{-2}}$. The regions excluded at 3$\sigma$ confidence are shaded in Fig. \ref{Fig:Chandra}.

An X-ray source is clearly detected at the location of ASASSN-15lh at the time of our third CXO observation (ID 17881), with count-rate $1.5\times 10^{-4}\,\rm{c\,s^{-1}}$ and significance of 5.2$\sigma$ in the 0.5-8 keV range (4.7$\sigma$ in the 0.5-1 keV energy range). In our fourth epoch (ID 17882) the source is still detected with count-rate $1.7\times 10^{-4}\,\rm{c\,s^{-1}}$ and significance of 4.9$\sigma$ in the 0.5-8 keV range (3.6$\sigma$ in the 0.5-1 keV energy range).

We employ the Cash statistics to fit the spectra (we have a total of six and five photons in a 1$\arcsec$ region around the transient in the third and fourth epoch, respectively), and perform a series of MCMC simulations to constrain the spectral parameters. The analysis of each of the two epochs taken separately points to a soft X-ray spectrum, with limited absorption and no evidence for statistically significant evolution between the two epochs. We thus constrain the X-ray source spectral parameters with a joint spectral fit of the two CXO epochs of observation, where the spectral normalization is allowed to vary from one epoch to the other.

The results are displayed in Fig. \ref{Fig:Chandra}. For an absorbed,  non-thermal power-law spectrum, the best-fitting parameters are $\Gamma=3.0$ and $NH_{int}\sim 10^{19}\,\rm{cm^{-2}}$. The inferred (0.3-10 keV) unabsorbed flux for this model is $F_x=3.6\times 10^{-15}\rm{erg\,s^{-1}cm^{-2}}$ corresponding to $L_x=5.9\times 10^{41}\,\rm{erg\,s^{-1}}$ (third epoch) and $F_x=4.9\times 10^{-15}\rm{erg\,s^{-1}cm^{-2}}$ ($L_x=8.1\times 10^{41}\,\rm{erg\,s^{-1}}$, fourth epoch).
The best-fitting parameters for an absorbed blackbody spectrum are: $T=0.17$ keV, $NH_{int}\sim 10^{18}\,\rm{cm^{-2}}$. The inferred (0.3-10 keV) unabsorbed flux for this model is $F_x=1.2\times 10^{-15}\rm{erg\,s^{-1}cm^{-2}}$, corresponding to $L_x=2.0\times 10^{41}\,\rm{erg\,s^{-1}}$ (third epoch) and $F_x=1.4 \times 10^{-15}\rm{erg\,s^{-1}cm^{-2}}$ ($L_x=2.3\times 10^{41}\,\rm{erg\,s^{-1}}$, fourth epoch). Both spectral models point to a very limited amount of neutral Hydrogen in the host galaxy along our line of sight, consistent with the very low N(HI) inferred by \cite{leloudas16} from Ly-$\rm{\alpha}$ and the very strong high ionization lines (\ion{N}{5} and \ion{O}{6}).

With reference to Fig. \ref{Fig:Chandra} we find that: (i) the X-ray source shows a soft spectrum (most of the allowed parameter space is  at $\Gamma>2$ and $T<1$ keV) with limited intrinsic absorption (of the order of a few $10^{22}\,\rm{cm^{-2}}$ at most).  (ii) There is no evidence for strong temporal and/or spectral variability of the X-ray source.

We first evaluate the possibility that the X-ray emission arises from a population of Low-Mass X-ray Binaries (LMXBs) residing in the early-type host galaxy, using the $L_x-L_{\rm{B}}$ and $L_x-L_{\rm{K}}$ relations by  \cite{Kim04}. For the host-galaxy of ASASSN-15lh \cite{Dong16} measure $M_{\rm{K}}=-25.5$ mag and $M_{\rm{B}}=-19.96$ mag, which imply $L_{x,XRB}=(1-6)\times 10^{40}\,\rm{erg\,s^{-1}}$ (0.3-8 keV). This is a factor $\ge10$ smaller then the measured X-ray emission at the location of ASASSN-15lh (re-calibrated with the same spectral model as \citealt{Kim04} in the 0.3-8 keV band). We conclude that LMXBs are unlikely to be the source of the detected X-rays. We thus envision two possible scenarios: either the X-rays originate from weak AGN activity from the host galaxy nucleus or they are physically connected to the optical/UV transient. In the first case we expect a somewhat stable X-ray emission over the time scale of years, while we anticipate fading if the X-ray emission is directly connected to ASASSN-15lh. Future observations will clarify the origin of the detected high-energy emission. Below we put our results into the context of X-ray emission from known transients (i.e. SNe and TDEs).

The detected emission is softer than the typical X-ray spectrum of SNe associated with Gamma-Ray Bursts ($\Gamma\sim2$, e.g. \citealt{Margutti13}) and normal H-stripped SNe (e.g. \citealt{Chevalier06,Dwarkadas12}), which typically show $\Gamma\sim2$ and a decaying flux with time. A way to sustain luminous X-ray emission over a long time is to invoke the SN shock interaction with a thick medium (see e.g. SN\,2014C, \citealt{Margutti16}). However, the observed X-ray spectrum of H-stripped SNe strongly interacting with the environment is even harder ($T\sim20$ keV), and thus even more different from what we observe at the location of ASASSN-15lh  (e.g. \citealt{Margutti16}). It is thus unlikely that a SN shock interaction with the medium is powering both the X-ray and optical/UV emission from ASASSN-15lh. Finally, compared to the only other X-ray source associated to a SLSN-I so far, the emission at the location of ASASSN-15lh is also  softer and significantly longer lived (Fig. \ref{Fig:XraysLC}): for the SLSN-I SCP06F6, \cite{Levan13} reports $\Gamma\sim2.6 $ (or a thermal spectrum with $T\sim1.6$ keV).

The X-ray properties of ASASSN-15lh are instead more reminiscent of the soft X-ray emission detected in non-jetted TDEs.  Non-jetted TDEs detected with \emph{ROSAT}, \emph{XMM-Newton}, \emph{Chandra} and, more recently, with \emph{Swift} show peak luminosities of $L_x\sim10^{42}-10^{44}\,\rm{erg\,s^{-1}}$ and very soft spectra that later harden with time on a time-scale of years and with initial temperatures $T<0.2$ keV  (e.g. \citealt{Komossa15} for a recent review).

As for the TDEs ASASSN-15oi \citep{Holoien16} and ASASSN-14li \citep{vanVelzen16}, the X-ray emission is more luminous than what expected based on the extrapolation of the optical/UV blackbody model (see Sec. \ref{SubSec:UV}) and a more complex model is needed. In this context ASASSN-15lh would show the most extreme ratio $L_{1,UV}/L_{1,X-rays}\sim 10^5$ (compared to $L_{1,UV}/L_{1,X-rays}\sim 10^{2-3}$ for ASASSN-14li and $L_{1,UV}/L_{1,X-rays}\sim 10^4$ for ASASSN-15oi).

In Fig. \ref{Fig:XraysLC} we put ASASSN-15lh on the X-ray luminosity plane of energetic envelope-stripped core-collapse SNe (i.e. GRB-SNe and SLSNe) and TDEs.  ASASSN-15lh is $\sim1000$ times less luminous than the SLSN-I SCP06F6  and does not experience a similar drop in luminosity. At $\sim$100 days, the X-ray emission at the location of ASASSN-15lh is more luminous than GRB-SNe. However, observations obtained around the same epoch by the ATCA in Fig.  \ref{Fig:Radio} put deep limits to the radio emission from ASASSN-15lh \citep{Kool15,leloudas16}, and rule out the presence of powerful jets seen on axis (most of the parameter space associated with off-axis GRB-like jets is also ruled out).  Also in this case, the luminous and not strongly variable X-ray emission at the location of ASASSN-15lh, which lacks a luminous radio counterpart, seems to be more in line with observations of non-jetted TDEs (recent examples are ASASSN-14li, \citealt{Miller15, Alexander16,Holoien16}; or ASASSN-15oi, \citealt{Holoien16b}).

\subsection{X-ray analysis- Swift/XRT and XMM-Newton}
\label{SubSec:XRT}

\begin{figure*}
\vskip -0.0 true cm
\centering
\includegraphics[scale=0.7]{lcrestGRB_15lh.eps}
\caption{ASASSN-15lh in the X-ray phase space of envelope-stripped SNe and TDEs. For ASASSN-15lh we merged the second and third CXO epoch for the sake of clarity and we show the luminosity of ASASSN-15lh at each epoch for both a thermal (bottom points) and a non-thermal (top points) spectrum. At $\sim 100$ days, ASASSN-15lh is more luminous than any previously detected H-stripped SN with the exception of SCP06F6. Its persistent and soft X-ray emission is more similar to non-relativistic TDEs, like ASASSN-15oi and ASASSN-14li.
The X-ray light-curve of the relativistic TDE SwiftJ2058 overlaps with the TDE SwiftJ1644 \citep{Pasham15} and it is not displayed here for clarity.
References: \cite{Gezari12,Margutti13d,Holoien14,Miller15,Holoien16,Mangano16,Brown16bis}. For SLSNe-I  we updated the sample of \cite{Levan13}. The detailed analysis will appear in Margutti et al., in prep.}
\label{Fig:XraysLC}
\end{figure*}

\begin{figure}
\vskip -0.0 true cm
\centering
\includegraphics[scale=0.6]{RadioLCkcorrSLSNe_15lh.eps}
\caption{ASASSN-15lh (red star) in the context of radio emission from envelope-stripped SNe (grey and black filled circles for GRB-SNe and SLSNe, respectively) and TDEs (blue diamonds), as constrained by observations with ATCA \citep{Kool15}. Radio observations acquired by \citet{leloudas16} three weeks later led to very similar limits and are not displayed here for clarity. The limits on the radio emission from ASASSN-15lh rule out most of the parameter space associated with powerful relativistic jets with kinetic energy $E_k$=$10^{51}\,\rm{erg}$ propagating into a circumburst medium with density  $1\,\rm{cm^{-3}}$ (orange lines, \citealt{vanEerten10}). SwiftJ1644 and SwiftJ2058 are the two relativistic TDEs known to date with radio observations.  References: \cite{Cenko12,Chomiuk12b,Chandra12,vanVelzen13,Margutti13d,Chornock14,Nicholl16,Alexander16,Berger12,Pasham15,Zauderer13}. }
\label{Fig:Radio}
\end{figure}

We re-processed all the X-ray data collected by the \emph{Swift}-XRT \citep{Burrows05} between June 24, 2015 until July 22, 2016 (total exposure time of $\sim$270 ks), following the prescriptions outlined in \cite{Margutti13}. A targeted search for X-ray emission at the location of ASASSN-15lh identifies the presence of a weak X-ray excess with significance of $3\,\sigma$ in the 0.3-5 keV range. The significance is reduced to $2.4\,\sigma$ in the 0.3-10 keV energy range, consistent with the soft X-ray spectrum suggested by the CXO observations. We infer a background subtracted count-rate of $(1.1\pm0.4)\times 10^{-4}\,\rm{c\,s^{-1}}$ (0.3-5 keV), which corresponds to an unabsorbed 0.3-10 keV flux $F_x=(4.1\pm1.5)\times 10^{-15}\,\rm{erg\,s^{-1}cm^{-2}}$ and $F_x=(3.7\pm 1.4)\times 10^{-15}\,\rm{erg\,s^{-1}cm^{-2}}$ for a blackbody and power-law spectral model, respectively, and the best-fitting spectral parameters derived from the CXO data. The average flux inferred from \emph{Swift}-XRT observations is thus consistent with the results from the CXO analysis and suggests that the X-ray source at the location of ASASSN-15lh experienced at most mild temporal variability over the $\sim$1 yr of \emph{Swift} monitoring. We note that flux variations of the order of a factor of a few are consistent with our findings, given the uncertainties affecting both the  \emph{Swift}-XRT and the CXO measurements. A delayed onset of the X-ray emission with respect to the optical emission is also clearly allowed, since \emph{Swift}-XRT data started to be collected after optical maximum light.

Finally, XMM-Newton observed ASASSN-15lh on November 18, 2015 ($\delta t=134.2$ days rest-frame since maximum light), six days after our first CXO epoch, which yielded a non-detection. From the XMM-Newton observations \citet{leloudas16} infer a $95\%$ confidence level flux limit $F_x<2\times 10^{-16}\,\rm{erg\,s^{-1}cm^{-2}}$ (0.3-1 keV). We do not confirm the results from \citet{leloudas16}. Adopting their inferred count-rate limit of 11 source counts in 9 ks of exposure time with EPIC-MOS2, their assumed blackbody spectrum with $T=18$ eV, and following the flux calibration procedure outlined in \citet{leloudas16}, we infer a flux limit which is $\sim100$ times larger.

We re-analyzed the XMM data using standard routines in the Scientific Analysis System (SAS version 15.0.0) and the relative calibration files. We employ a source region of 32$\arcsec$ radius and extract the background from a source-free region on the same chip. No X-ray source is detected at the location of ASASSN-15lh. Our best constraints are derived from observations obtained with EPIC-MOS2, with total exposure time of 9 ks (after removal of time windows contaminated by proton flaring) and a 3$\,\sigma$ count-rate upper limit of $0.002\,\rm{c\,s^{-1}}$ (0.3-10 keV).  For the best fitting spectral models derived from CXO detections, we infer the following unabsorbed 0.3-10 keV flux limits: $F_x<1.4\times 10^{-14}\,\rm{erg\,s^{-1}cm^{-2}}$ and $F_x<1.5\times 10^{-14}\,\rm{erg\,s^{-1}cm^{-2}}$ for the blackbody and the power-law spectrum, respectively. XMM observations do not reach the necessary depth to probe the emission from the X-ray source that we detect with CXO and the stacking of \emph{Swift}-XRT observations. A summary of the results from the X-ray observations of ASASSN-15lh can be found in Table \ref{Tab:Xray}.

\subsection{UV analysis}
\label{SubSec:UV}

\begin{figure}
\vskip -0.0 true cm
\centering
\includegraphics[scale=0.55]{variabilityUV.eps}
\caption{Swift-UVOT observations of ASASSN-15lh reveal the presence of pronounced time variability on short time scales $\Delta t \lesssim 5$ days across the \emph{Swift}-UVOT bands at the time of the re-brightening. Vertical dotted lines mark the times of the CXO observations. }
\label{Fig:UVvar}
\end{figure}

We re-analyzed all the \emph{Swift}-UVOT observations obtained from June 24, 2015 until July 22, 2016 following the prescriptions by \cite{Brown09} and adopting the updated calibration files and revised zero points by \cite{Breeveld11}. Each individual frame has been visually inspected and quality flagged. Observations with insufficient exposure time have been merged  to obtain higher signal-to-noise ratio (S/N) images from which we extracted the final photometry reported in Table \ref{Tab:UVOT}. We corrected for Galactic extinction in the direction of the transient  ($E(B-V)=0.03$ mag, \citealt{Schlafly11}) and subtracted the host galaxy flux component as constrained by \cite{Dong16}. We performed a self-consistent flux calibration, and applied a dynamical count-to flux conversion that accounts for the spectral evolution of ASASSN-15lh, following the procedure outlined in \cite{Margutti14}. Finally, we computed a bolometric light-curve of ASASSN-15lh by integrating the best-fitting blackbody spectra.

A partial collection of the \emph{Swift}-UVOT photometry of ASASSN-15lh has already been presented by \cite{Dong16}, \cite{Godoy-Rivera16}, \cite{Brown16} and \cite{leloudas16}. Here we update the observations and focus on the presence of significant temporal variability that appears at the time of the re-brightening. Figure \ref{Fig:UVvar} shows the presence of pronounced temporal variability across the UVOT bands, and more pronounced at UV wavelengths as first noticed by \cite{Brown16}. The short variability time scale $\Delta t \lesssim 5$ days at $\sim150$ days since first light argues against the interpretation of the SN shock interaction with the surroundings as the main source of energy powering the re-brightening \citep{Chatzopoulos16}. For a typical SN shock velocity $v_{sh}\sim0.2$ c (e.g. \citealt{Margutti14b}, their Fig. 2) we do not expect significant temporal variability on $\Delta t<30$ days at $t\sim150$ days, contrary to what we observe in ASASSN-15lh. This observation motivates us to consider alternative explanations of the UV re-brightening (Sec. \ref{SubSec:Rep}).

\subsection{Late-Time Optical Spectroscopy}
\label{SubSec:Spec}

\begin{figure*}
\vskip -0.0 true cm
\centering
\includegraphics[width=7.2in]{15lh-latespec-v4.eps}
\caption{\emph{Left Panel}: A decomposition of the observed late-time spectrum (black line) obtained on 2016 June 10 ($\delta t$= 301 days rest-frame since maximum light) into a sum (magenta line) of the FAST host-galaxy model that best fits the pre-transient photometry (red line) and a featureless blackbody with temperature inferred from our fits of ASASSN-15lh (blue line), demonstrates that the strongest spectral features can be reasonably attributed to the underlying host galaxy stellar population. The green squares mark the {\it Swift} UVOT photometry after subtraction of the host model and interpolated to the date of observation. The gray circles are the scaled values for the broadband photometry of the host \citep{Dong16}.  We emphasize the lack of unambiguous evidence for broad spectral features associated with the transient. \emph{Inset}: The observed spectrum in the vicinity of the H$\alpha$/[\ion{N}{2}] complex is shown enlarged.  Two clear emission lines are present prior to any correction for the stellar continuum.
\emph{Right Panels}: Zoom-ins to spectral regions of interest in the difference spectrum show the presence of \emph{narrow} emission lines at the expected wavelengths (dotted lines) associated with H$\alpha$, H$\beta$, [\ion{N}{2}] $\lambda\lambda$6548, 6583, and possibly [\ion{S}{2}] and [\ion{O}{3}]~$\lambda$5007 at a common redshift of $z$=0.2320. }
\label{Fig:spec1}
\end{figure*}

We acquired deep multi-epoch optical spectroscopy of ASASSN-15lh, spanning the time range $\delta t=$ 35--350 rest-frame days after maximum light and sampling key points in the late evolution of the transient. A more detailed analysis will be presented in future work (Chornock et al., in prep.). Here we concentrate on an analysis of our highest S/N late-time spectrum, which was acquired well after the second re-brightening and when the underlying emission from the host galaxy stellar population is better revealed.

We observed ASASSN-15lh on 2016 June 10 ($\delta t=301$ days rest-frame since maximum light) using the Low Dispersion Survey Spectrograph (LDSS3C; \citealt{ldssref}) on the 6.5~m Magellan Clay telescope.  We obtained three 1800~s exposures using the VPH-All grism and a 1\arcsec-wide slit near the center of the field of view oriented at a position angle of 128\fdg3, which was close to the parallactic angle  \citep{fil82}. This setup covered the range 3800--10500~\AA\ with a resolution of 8.1~\AA. Standard IRAF\footnote{IRAF is
  distributed by the National Optical Astronomy Observatory,
    which is operated by the Association of Universities for Research
    in Astronomy, Inc., under cooperative agreement with the National
    Science Foundation.} tasks were used to perform two-dimensional image processing.  We used custom IDL scripts to perform flux calibration and correction for telluric absorption using observations of EG131 obtained immediately prior to the object. We took particular care to mitigate the effects of second-order light contamination by combining observations of the standard star taken both with and without an order-blocking filter.  However, small residual contamination at long wavelengths ($\lambda$>8000~\AA) is possible.

The resulting spectrum is shown in black in Figure~\ref{Fig:spec1}.  Numerous stellar absorption features from star light in the host galaxy are visible, as well as two emission peaks near H$\alpha$ (observed wavelengths $\sim$8100~\AA).  Several authors have fit the available pre-outburst host galaxy photometry \citep{melchior15,Dong16,leloudas16} and have found consistent results.  However, the presence of spectral features from the host stellar population has the potential to improve the constraints on the stellar population synthesis, so we used an iterative procedure to incorporate this information while avoiding the flux from the transient.

First, we estimated a best-fit blackbody temperature of $T_{\mathrm{BB}}\approx15,000$~K at the time of observations from the analysis of the UVOT photometry described above.  We then subtracted a scaled blackbody spectrum from the observed spectrum under the constraint that the blackbody-subtracted spectrum had to match the observed colors of the host galaxy to obtain an initial estimate of the host-only spectrum.
We then used the FAST code \citep{kriek09} to fit the host-only spectrum combined with, and normalized by, the broadband $grizyJK$ host photometry \citep{melchior15,Dong16}.  For simplicity, we fixed the metallicity to solar and assumed a \citet{chabrier03} initial mass function and zero internal extinction.  We obtained a satisfactory fit using the \citet{bc03} stellar models and an exponentially-declining star formation law.  The best-fit model has a total stellar mass of 1.2$\times$10$^{11}$~$M_{\sun}$, a current stellar age of 10~Gyr, and an $e$-folding timescale of 2~Gyr, resulting in a current star-formation rate of $\sim$0.8~$M_{\sun}\,\rm{yr^{-1}}$.  These numbers are in broad agreement with those reported previously (e.g, \citealt{Dong16,leloudas16}).
Other choices for the stellar population model produced qualitatively similar results, although usually with smaller current star-formation rates.  Our best fit for the host is plotted in red in Figure~\ref{Fig:spec1}.

We then fitted our observed spectrum as a linear combination of the host galaxy model and a blackbody to find appropriate flux scaling factors. The scaled blackbody is plotted as blue in Figure~\ref{Fig:spec1} and good agreement can be seen with the host-subtracted UVOT $ubv$ photometry (green squares) interpolated to the date of observation. Both the fitted host spectrum and the overplotted host photometry (gray circles) have been scaled by a factor of 0.40 from the values for the whole host, which presumably results from the smaller size of our spectroscopic aperture relative to the host as a whole.  The \citet{bc03} models clearly have narrower features than those visible in our spectrum, so the host template was smoothed with a 10~\AA\ boxcar function to mimic the combined effects of our spectral resolution and the internal velocity dispersion of the host galaxy.  Our results are not very sensitive to the width of this smoothing kernel.  The sum of the scaled blackbody and the smoothed galaxy template is plotted in magenta in Figure~\ref{Fig:spec1} and is a good match to the observed spectrum in black.

\citet{leloudas16} noted two emission peaks near 4000 and 5200~\AA\ in their late-time spectra of ASASSN-15lh.  However,
accurate modeling of the host galaxy stellar component from our late-time spectrum demonstrates that the most prominent broad spectral features detected in the observed (host plus transient) late-time spectra have to be attributed to the underlying continuum from the host galaxy star light (Figure \ref{Fig:spec1}). We do not find unambiguous evidence for broad spectral features associated with the transient at this epoch. Small, broad, low-amplitude discrepancies between the observed spectrum and combined fit (black and magenta lines, respectively) are present, but it is not yet clear if they represent true spectral features of the transient or limitations in the stellar population synthesis modeling.
More observations of the host will be required after the optical transient fades further to more accurately constrain the presence of possible low amplitude broad spectral features in the transient spectrum at late times.

Without any correction for the host galaxy, the spectrum has the two obvious narrow emission features near 8085 and 8111~\AA\ (in air) noted above, which can be clearly associated with H$\alpha$ and [\ion{N}{2}]~$\lambda$6583 at $z$=0.2320 (lower-right panel of Figure~\ref{Fig:spec1})\footnote{This redshift is consistent with that measured from the stellar absorption features.  Note that this value is slightly offset from the redshift $z$=0.2326 measured from narrow UV absorption lines \citep{Dong16,Brown16,leloudas16}. We do not discuss further the implications of this possible velocity offset for the UV absorbers in this work.}.  [\ion{N}{2}]~$\lambda$6548 is blended in the blue wing of H$\alpha$.  H$\beta$ is only visible in emission after subtraction of the host model.
Weaker features also appear to be present in the difference spectrum near the [\ion{S}{2}] doublet and [\ion{O}{3}]~$\lambda$5007. We searched for [\ion{O}{2}] $\lambda$3727 emission and none is visible, but the S/N of the spectrum is not as high at those wavelengths.  Inspection of our spectral sequence reveals that the H$\alpha$/[\ion{N}{2}] lines are present in several of our higher S/N spectra throughout the evolution of the transient, consistent with a constant low-level contribution that is strongly diluted by light from the transient at earlier times.

The peaks of H$\alpha$ and [\ion{N}{2}]~$\lambda$6583 are of comparable height \emph{prior} to subtraction of the host model.  Strong [\ion{N}{2}]/H$\alpha$ is a possible sign of ionization by an AGN-like continuum.  However, after correction for the underlying Balmer absorption in our best-fit host model, the ratio decreases to $\sim$0.5.
This line ratio, combined with weak [\ion{O}{3}]/H$\beta$ and [\ion{S}{2}]/H$\alpha$, is consistent with the nebular emission being powered by star formation instead of AGN activity \citep{kewley06}. We caution that these ratios are sensitive to systematic errors in the modeling of the underlying stellar absorption, and in particular the strength of the stellar Balmer absorption.
If all of the inferred H$\alpha$ emission (flux $\sim$2.7$\times$10$^{-16}$ erg~cm$^{-2}$~s$^{-1}$) is powered by star formation, the inferred rate is $\sim$0.4~$M_{\sun}$~yr$^{-1}$ \citep{Kennicutt98}, in rough agreement with that estimated from the host galaxy stellar population fit.

\citet{leloudas16} reported H$\alpha$ emission from ASASSN-15lh with a full width at half-maximum (FWHM) of 2500~km~s$^{-1}$, but in our data, it is clear that the reported emission feature is just the narrow nebular H$\alpha$ and [\ion{N}{2}] from the host blended together at low S/N or low resolution in their data. Note that in their highest S/N spectra (inset of their Figure 1), the putative H$\alpha$ from the transient is flat-topped or double peaked, consistent with the two strong nebular emission lines of roughly equal height (inset of our Figure~\ref{Fig:spec1}) being blended together.  We also note that \citet{leloudas16} do not attempt to correct for the contribution from the underlying stellar continuum. Therefore, we do not confirm their claim of H$\alpha$ emission from the transient itself and the reported velocity FWHM likely reflects the spacing of the two [\ion{N}{2}] lines, which are each offset by $\sim$1000~km~s$^{-1}$ from the central H$\alpha$ emission.  \citet{Godoy-Rivera16} also report a "bump" near H$\alpha$ at late times, but they do not report a FWHM, so it is not clear if they are are also possibly referring to a noisy detection of the narrow nebular lines.

\subsection{Re-analysis of Early-Time Optical Spectra}
\label{SubSec:Spec2}

In addition, we re-evaluated the early optical spectra of ASASSN-15lh and were unable to confirm the likeness to SLSNe reported by \citet{Dong16}. The  \ion{O}{2} ion, which is commonly observed in SLSNe \citep{Quimby11}, has a number of distinctive absorption features not observed in ASASSN-15lh (Fig. ~\ref{fig:spec-compare}). The strongest two features centered near 4100 and 4400~\AA\ are always observed to be of comparable strength and no reasonable values of temperature or density can change this ratio. ASASSN-15lh only shows the 4100~\AA\ feature (Fig. \ref{fig:spec-compare}, see also \citealt{leloudas16}).  Without the accompanying 4400~\AA\ feature, it is hard to reconcile the proposed association with \ion{O}{2}, and thus the spectroscopic connection to SLSNe is not robust.

The spectral features of ASASSN-15lh trend redward over time toward declining velocities. This is similar to the spectral evolution of supernovae where it is attributed to an expanding and cooling photosphere. However, unlike supernovae, the features of ASASSN-15lh do not show traditional P-Cyg profiles and become increasingly inconspicuous. For example, the +30 day spectrum of SN\,2010gx exhibits pronounced features, whereas the +39 day ASASSN-15lh spectrum is nearly featureless (Fig. \ref{fig:spec-compare}). We explored a variety of  possible ions using the highly parameterized spectrum synthesis tool, \texttt{SYN++} \citep{Thomas11}, to determine whether blending of features could reproduce the spectral features and evolution of ASASSN-15lh, but were unsuccessful.

To our knowledge the only previous examples of spectral features becoming increasingly inconspicuous in the early phases of a supernova involve interaction with dense CSM. SN-CSM interaction can rescale or ``mute'' the line profile relative to the continuum \citep{Branch00}. Most confirmed instances of SN-CSM interaction involve H-rich material that can be readily identified by the presence of H Balmer lines that may be narrow ($<$ 100 km\,s$^{-1}$) to broad ($\sim 10^3$ km\,s$^{-1}$) in width, depending on their origin of formation. The hydrogen-poor SLSN iPTF13ehe exhibited H$\alpha$ Balmer emission with broad and narrow components +251 days post maximum \citep{Yan15}. However, no such lines are observed in ASASN-15lh. Interaction with H-poor CSM ejected by rapidly rotating pulsational pair instability supernovae is possible \citep{Chatzopoulos12,Chatzopoulos16}, but the spectroscopic consequences of such interaction are poorly understood \citep{Chatzopoulos13}, and the timescales of variability observed in the UV strongly favor against this scenario (section \ref{SubSec:UV}).

A luminous central source over-ionizing expanding ejecta is a speculative, though attractive, scenario that may explain the spectroscopic evolution of ASASSN-15lh toward a featureless continuum. As there is no precedent for this scenario, the specific spectral signatures are unclear. Certainly, the ionizing photons must be extremely energetic for no strong optical or UV lines to be observable. An analogous phenomenon may be variable UV absorption commonly seen in Seyfert galaxies \citep{Maran96,Crenshaw00}. In some cases variability in the form of absorption components appearing and disappearing, or decreasing outflow velocities \citep{Gabel03}, can result from changes in the ionizing flux \citep{Kraemer02}. ASASSN-15lh may be an extreme version of these processes.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{15lh-faircompare-v2.pdf}

\caption{Placing early-phase optical spectra of ASASSN-15lh in context with SLSNe. (a) The \ion{O}{2} ion is a signature of SLSNe, and here we show how using the simple assumptions of SYN++ and a photospheric velocity of 19,000 km~s$^{-1}$, the -4 day spectrum of the SLSN 2010gx can be reproduced. (b) By contrast, we cannot reproduce +13 day spectrum of ASASSN-15lh. It clearly misses an accompanying feature around 4400~\AA. (c) Evolution in the spectra are observed. Most conspicuous is the 4100~\AA\ feature that drifts to longer wavelengths. (d) The evolution toward increasingly inconspicuous spectral features is unlike SLSNe that exhibit increasingly stronger spectral features. Here we show the +30 day spectrum of SN\,2010gx, which is unlike the nearly featureless +39 day spectrum of ASASSN-15lh. Data have been retrieved from WISEREP \citep{YG12}, normalized according to the procedure outlined in \citet{Jeffery07} to aid in visual comparison, and were originally published in \citet{Pastorello10} and \citet{Dong16}.}

\label{fig:spec-compare}
\end{figure}

\section{Interpretation}
\label{Sec:Int}

\begin{figure}
\vskip -0.0 true cm
\centering
\includegraphics[width=.5\textwidth]{opacity.pdf}
\caption{Bolometric light curve model (solid line) due to reprocessed luminosity from a central source (dashed line). For this plot we use the spin down power from a magnetar engine as central source as an example. We artificially and abruptly change the ejecta opacity at the time shown by the vertical dashed line to approximate the effect of an ionization break-out.  The object's time of first light is assumed to occur 30 days before the light curve peak.  The bolometric light curve of ASASSN-15lh that we derived in Sec. \ref{SubSec:UV} is shown for comparison with blue triangles. }
\label{Fig:Opacity}
\end{figure}

\subsection{The ``Reprocessing Picture"}
\label{SubSec:Rep}
Although the mechanisms behind SLSNe powered by a stellar-mass compact object$-$such as a magnetar$-$ and the tidal disruption and accretion of a star by a SMBH do differ significantly, the basic physical process driving the light curves of these events may be similar.
A central source of UV/X-ray radiation (an accreting SMBH or the pulsar wind nebula of a rapidly spinning neutron star, NS) is absorbed by a dense column of gas, and downgraded into optical radiation, where the lower opacity allows the radiation to more readily escape.
Such a ``reprocessing" picture has been applied to explain both TDEs (\citealt{Loeb97,Guillochon13,Metzger16}) and SLSNe (\citealt{Kasen10,Woosley10, Metzger14}).

Consider the characteristic timescale of the central engine in a magnetar-powered SLSN and TDE scenarios.  In a magnetar-powered SLSN, the central engine lifetime is the magnetic dipole spin-down timescale of the magnetar:
\begin{equation}
t_{\rm sd} \simeq 17\,{\rm d}(M/1.4M_{\odot})^{3/2}B_{13}^{-2}P_{\rm ms}^{2},
\label{eq:tsd}
\end{equation}
where $M$, $P = P_{\rm ms}$ ms, and $B_{\rm d} = 10^{13}B_{13}\,{\rm G}$ are, respectively, the mass, initial spin period, and dipole surface magnetic field strength of the magnetar (e.g.~\citealt{Spitkovsky06}).  The maximum energy of the engine is limited to the rotational energy of the NS,
\begin{equation}
E_{\rm rot} = I\Omega^{2}/2 \simeq 2.5\times 10^{52}{\rm erg}\,\,(M/1.4M_{\odot})^{3/2}P_{\rm ms}^{-2}\,,
\label{eq:Erot}
\end{equation}
which can vary from $\sim 3\times 10^{52}-10^{53}$ erg for the minimum value of the spin-period set by the mass shedding limit, depending on the mass of the NS (\citealt{Metzger15}).
In order to simultaneously explain the large radiated energy and duration of ASASSN-15lh with a magnetar, we require a maximally spinning neutron star ($P_{\rm ms} \lesssim 1$) and a relatively weak magnetic field $B_{\rm d} \sim 10^{12}-10^{13}$ G (e.g., \citealt{Metzger15}, see also \citealt{Dong16,Chatzopoulos16, Bersten16, Sukhbold16}).

In the TDE scenario, the engine lifetime is uncertain, but is commonly attributed to the fall-back time of the most tightly bound stellar debris following the disruption (e.g., \citealt{Guillochon13,Stone13}),
\begin{equation}
t_{\rm fb} \approx 410\,{\rm d}\left(\frac{M_{\bullet}}{10^{8}M_{\odot}}\right)^{1/2}\left(\frac{m_{\star}}{M_{\odot}}\right)^{1/5},
\label{eq:tfb}
\end{equation}
where $M_{\bullet}$ and $m_{\star}$ are the mass of the SMBH and the star, respectively, and we have assumed a stellar mass-radius relationship $R_{\star} \simeq R_{\odot}(m_{\star}/M_{\odot})^{4/5}$ appropriate to lower main-sequence stars.  The maximum radiated energy is that liberated by the accretion of the half of the stellar mass which remains bound to the SMBH,
\begin{equation}
E_{\rm acc} = \eta (m_{\star}/2)c^{2} \simeq 9\times 10^{52}\,{\rm erg}\left(\frac{\eta}{0.1}\right)\left(\frac{m_{\star}}{M_{\odot}}\right),
\end{equation}
where the radiative efficiency for geometrically thin accretion varies from $\eta \approx 0.04-0.42$, depending on the spin of the SMBH and its orientation relative to the angular momentum of the accreting gas.

In the TDE scenario, the energetics of ASASSN-15lh are reasonably accommodated by the accretion of a solar-mass star.  However, the high mass SMBH $M_{\bullet} \sim 6\times 10^{8}M_{\odot}$ inferred from the host of ASASSN-15lh, would appear to predict a long duration of the transient $t_{\rm fb} \gtrsim$ 2 year, inconsistent with the much shorter observed decay time of the first peak of a few weeks. This inconsistency could be solved by considering that a main-sequence star can only be disrupted by such a massive black hole if the SMBH is \emph{spinning} in a prograde direction with respect to the orbit of the disrupted star (e.g. \citealt{Kesden12}).  Precession of the star during the phase of tidal compression due to the BH spin may substantially enhance the spread in the energy distribution of the stellar debris as compared to the Newtonian case, by partially aligning the direction of the hydrodynamic bounce with the velocity vector of the star (\citealt{Metzger16,leloudas16}).  More tightly bound debris has a shorter orbital period, which could significantly speed-up the flare evolution timescale as compared to the Newtonian gravity estimate in Equation \ref{eq:tfb}.  Though promising, general relativistic numerical simulations are needed to confirm this possibility.

In addition to possibly speeding up the flare evolution, the high BH spin required to explain ASASSN-15lh as a TDE would (i) naturally result in a large value of the accretion efficiency $\eta$, accounting for its high luminosity; (ii) possibly aid in the process of debris circularization by inducing precession of the stellar debris streams (e.g.~\citealt{Dai15}, \citealt{Hayasaki+16}, \citealt{Bonnerot+16}).  Precession of the streams out of the orbital plane due to misaligned BH spin could also help make the geometry of the reprocessing material relatively spherical (e.g., \citealt{Guillochon15}), consistent with the low measured optical polarization of ASASSN-15lh reported by \cite{Brown16}.

\subsection{X-rays Escape and the Ionization Break-Out}
In both the magnetar SLSN and TDE scenarios, UV/X-ray radiation from the central source may ionize its way through the ejecta at late times.  This process can result in the direct escape of UV/soft X-ray radiation while having an indirect influence on the observed optical light curve by changing the ejecta opacity (Sec. $\ref{SubSec:LC}$).

Approximating the ejecta as a homogeneously expanding sphere of mass $M_{\rm ej}$, velocity $v_{\rm ej} = 10^{9}v_{9}$ cm s$^{-1}$, and radius $R_{\rm ej} = v_{\rm ej}t$,  the neutral column density is
\begin{equation}
\Sigma (t)\simeq \frac{M_{\rm ej}}{(4\pi/3)R_{\rm ej}^{2}m_p} \approx 1.7\times 10^{24}\,{\rm cm^{-2}} f_n\left(\frac{M_{\rm ej}}{M_{\odot}}\right)v_9^{-2}\left(\frac{t}{150{\rm d}}\right)^{-2},
\end{equation}
where $f_n$ is the neutral fraction.  This is much higher than the inferred X-ray absorption column of $NH_{int}< 3\times 10^{22}$ cm$^{-2}$ towards ASASSN-15lh, requiring a very low neutral fraction if the X-ray source is related to the optical transient. This is consistent with the very low N(HI) inferred by \cite{leloudas16} from Ly-$\rm{\alpha}$.

The ejecta from TDEs and SLSNe are expected to have markedly different chemical composition. In a TDE the ejecta has nearly solar composition (e.g.~\citealt{Kochanek16a}) and the escape of soft X-rays is inhibited primarily by the bound-free opacity of neutral helium (\citealt{Metzger16,Roth15}).  By contrast, in a H-poor SLSN, X-rays are blocked more severely by neutral oxygen and carbon (\citealt{Metzger14}).

A central engine with an UV/X-ray luminosity $L$ releases an energy $L \times t$ in ionizing radiation on a timescale $t$.  If the ejecta contains a mass fraction $X_Z$ of elements with atomic number $Z = 8Z_8$, the radiation ionizes its way through the ejecta on a timescale
\begin{eqnarray}
t_{\rm ion} \approx  \left\{
\begin{array}{lr}
120\,{\rm d}\,M_3^{3/4} v_9^{-5/4}T_{5}^{-0.2}\left(\frac{X_{A}}{0.1}\right)^{1/4}\left(\frac{L t}{10^{52}{\rm ergs}}\right)^{-1/4}Z_8^{3/4}
,
(\eta_{\rm thr} \ll 1) \\
110\,{\rm d}\,M_3 v_9^{-3/2}T_{5}^{-0.4}\left(\frac{X_{A}}{0.1}\right)^{1/2}\left(\frac{L t}{10^{52}{\rm ergs}}\right)^{-1/2}Z_8^{3/2},
(\eta_{\rm thr} \gg 1), \\
\end{array}
\right.
\label{eq:tbo}
\end{eqnarray}
where $M_3 \equiv M_{\rm ej}/(3M_{\odot})$ and $T_{5} = T/10^{5}$ K is the temperature of electrons in the recombination layer and
\begin{eqnarray}
\eta_{\rm thr} \approx 0.7\left(\frac{L t}{10^{52}{\rm erg}}\right)^{-1}M_{3}v_9^{-1}\left(\frac{X_{A}}{0.1}\right)T_{5}^{-0.8}Z_8^{3}
\label{eq:etaA}
\end{eqnarray}
is the ratio of absorptive and scattering opacity in the ejecta \citep{Metzger14}.

For typical parameters and an engine similar to ASASSN-15lh with $L t \sim 10^{52}$ ergs, we have $t_{\rm ion} \sim $ month in the case of a He-rich composition ($Z = 2$) of a TDE-like scenario.  By contrast, for a CO-rich composition of an exploding massive star ($Z = 8$), we have $t_{\rm ion} \sim $ months, making break-out harder to achieve.  In the latter case, X-ray break-out is even less likely considering that the K-shell valence electrons of oxygen have a binding energy of $\sim 1$ keV, while the {\it measured} keV X-ray luminosity of ASASSN-15lh $\leq 10^{42}$ erg s$^{-1}$ is much less than the optical/UV luminosity (in other words, the true value of $L t$ to use in Equation \ref{eq:tbo} should be much lower than $10^{52}$ erg).

We conclude that an ionization break-out could allow the escape of X-rays in the TDE scenario, but is probably not sufficient to do so in the case of a H-poor supernova given the observed soft X-ray spectrum.

\subsection{The double-humped light-curve of ASASSN-15lh}
\label{SubSec:LC}

The ionization of the ejecta reduces the bound-free opacity, allowing the escape to the observer of UV and X-ray radiation with energies \emph{above} the ionization threshold. This process is unlikely to explain the observed UV re-brightening by itself, as even the highest frequency UV bands of \emph{Swift}-UVOT are below the first ionization energies of the most abundant elements (H, He, C, O). However, an ionization break-out may have an indirect effect on the light-curve via the continuum opacity.

At early times the  ejecta is largely neutral and the opacity at optical frequencies is dominated by electron scattering, while the opacity at UV frequencies is dominated by line transitions of metals. However, once the ejecta becomes ionized by the central engine, the electron scattering opacity will increase, while the UV opacity will decrease as the ionized atoms have fewer bound-bound transitions.  Therefore, following ionization break-out we expect a shift of the peak of the spectral energy distribution from optical to UV frequencies. The appeal of  this model is that a \emph{single} central-engine timescale would naturally reproduce the double-peaked temporal structure of ASASSN-15lh, which has no analogue in previously observed TDE or SLSN light curves. As a comparison, the TDE model invoked by \cite{leloudas16} combines two luminosity mechanisms, which result into two different timescales.

While accurate modeling, beyond the scope of this paper, is necessary to understand if this effect alone can quantitatively explain the observations of ASASSN-15lh, here we consider a toy model to illustrate the basic principles. For illustrative purposes we use the spin down luminosity of a magnetar as the central source of ionizing photons. In particular,
we consider a magnetar light-curve with parameters $P = 1$ ms and $B_{\rm d} = 3\times 10^{12}$ G, similar to that described in \cite{Metzger15}, and a total ejecta mass of $M_{\rm ej} = 10M_{\odot}$.  However, we artificially change the grey opacity from $\kappa_i = 0.02$ cm$^{2}$ g$^{-1}$ to $\kappa_f = 0.2$ cm$^{2}$ g$^{-1}$ at a time corresponding to ionization break-out of about 50 days.  As shown in Fig. \ref{Fig:Opacity}, this produces a minimum/flattening in the bolometric light curve, similar to that observed in ASASSN-15lh.  Although we have applied the model to a magnetar for concreteness (and since the process of debris circularization in TDEs remains uncertain), a similar result applies to the TDE case if the central UV/X-ray accretion power smoothly rises on a timescale of a few weeks and then decays $\propto t^{-5/3}$ at later times.  We also caution that a simple change in the grey opacity is unlikely to accurately predict the effect of wavelength-dependent opacity change created by an ionization break-out.
\section{Summary and Conclusions}
 \label{Sec:conclusions}
We presented evidence for luminous, soft and persistent X-ray emission at the location of ASASSN-15lh, and discussed its origin in the context of multi-wavelength observations of the transient, which include constraints on its radio emission and early and late-time optical spectroscopy. Our re-analysis of early-time spectra does not confirm the robust association of ASASSN-15lh with SLSNe claimed by previous studies, and invites us to be open-minded about the nature of ASASSN-15lh. Late-time spectra reveal the emergence of \emph{narrow} emission features from the host galaxy, while we associate the most prominent  \emph{broad} spectral features to the underlying stellar population. No clear evidence is found for broad spectral features associated with the transient at late times.

We propose a model that explains the double-peaked temporal structure of ASASSN-15lh in the optical/UV band as originating from the temporal evolution of the ejecta opacity, which changes as a result of persistent ionizing flux from a long-lived central source (either a magnetar or an accreting SMBH). We speculate that  the evolution of ASASSN-15lh towards a featureless spectrum also results from the presence of a persistent central source of ionizing photons. The exceptionally long active time-scale and high luminosity of the ionizing central source powering ASASSN-15lh (i.e. months) is most likely the key physical property that distinguishes ASASSN-15lh from all the TDEs and SLSNe discovered so far.

The optical/UV spectral evolution of ASASSN-15lh, its peculiar re-brightening and the presence of soft and persistent X-ray emission are indeed unprecedented among SLSNe and TDEs and suggest two scenarios: (i) either ASASSN-15lh is the first member of a class of stellar explosions with extreme properties that are intrinsically rare or that have been overlooked because of the very close location to the host-galaxy nucleus or, alternatively,  (ii) ASASSN-15lh results from refreshed nuclear activity of the host-galaxy SMBH.

In the first scenario the detected X-ray emission is physically unrelated to the transient and most likely originates from the host galaxy nucleus. We thus expect no fading of the X-ray source over the time scales of years.

Instead, if the X-ray emission is physically associated with the optical/UV transient, then ASASSN-15lh is unlikely to originate from a stellar explosion and an association with the activity of the host nucleus is favored.  In this case, ASASSN-15lh would be a TDE from the most massive spinning SMBH observed to date.  The fast initial decay timescale of the transient is challenging to understand based on the fall-back timescale of the disrupted star in Newtonian gravity, possibly suggesting that BH spin plays a key role in enhancing the energy spread of the disrupted star.  ASASSN-15lh and similar events discovered in the future would then constitute direct probes of matter under strong gravity around
very massive, dormant, spinning SMBH in galaxies. We emphasize that this scenario predicts significant temporal evolution of the X-ray emission over the next few years, as we expect a TDE to have a non-negligible impact on the inner part of the accretion disk even in the case of a pre-existing weak AGN.

Continued deep X-ray monitoring of ASASSN-15lh will constrain the temporal evolution of the X-ray source and its fading, revealing in this way if the X-ray source is indeed physically related to the optical/UV transient.  Future X-ray observations thus hold the keys to unveil the true nature of ASASSN-15lh.

\acknowledgments
R. M. acknowledges partial support from the James Arthur Fellowship at NYU during the completion of this project and the Research Corporation for Science Advancement. BDM gratefully acknowledges support from the NSF (AST-1410950, AST-1615084), NASA Astrophysics Theory Program (NNX16AB30G), the Alfred P. Sloan Foundation, and the Research Corporation for Science Advancement.   G.M. acknowledges the financial support from the UnivEarthS Labex programof Sorbonne Paris Cite (ANR10LABX0023 and ANR11IDEX000502).
The scientific results reported in this article are based on observations made by the Chandra X-ray Observatory under program GO 17500103, PI Margutti, observations IDs 17879, 17880, 17881, 17882.
This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile.

\textit{Facilities:} \facility{Magellan:Clay}

\bibliographystyle{apj}


text:
\section{introduction}

A physical system sometimes shows a peculiar fractal structure
of the spectrum if the system has two scales of periods.
In the nearest-neighbor (NN) tight-binding model with a magnetic
flux, Hofstadter obtained one-particle spectra of butterfly-shape
and discovered a multi-fractal structure.\cite{a}
Experimentally, it may be a challenging theme to observe the
spectrum of the fractal structure.
Actually, interesting phenomena have been observed in experiments of
lateral super lattices.\cite{b,c,ca}
The system with the cosine potential has been studied well
theoretically.\cite{d}
In the case of the antidot array, however, the potential term
includes a large number of the cosine functions with different wave
lengths.\cite{fgk}
The steeper the potential of the antidot is, the harder to solve the
eigenvalue problem is.
Silberbauer\cite{silb} and K\"uhn {\it et al}.\cite{kuhn}
studied a steep antidots potential problem.
Huang {\it et al}. \cite{hgm} solved the periodic short-range potential
problem in a finite system numerically. Analytic properties of the problem
are unknown.

In the present paper, we study eigenvalue problems of the two-dimensional
electron systems  defined on an infinite plane
with the periodic short-range potentials in the
lowest Landau level (LLL) analytically and numerically.
We find Hofstadter-type bands and flat bands.
The former bands satisfy a duality relation.
Namely, the spectrum at one value of $t$ is connected with that of $1/t$,
where $t$ is the flux $\Phi$ penetrating the unit cell of the periodic
potential normalized by the unit flux $\Phi_0$, {\it i.e.} $t=\Phi/\Phi_0$.
Our duality is concerned with flux and is different from the Aubry and
Andre's duality,\cite{ahk} which is concerned with  hopping strengths
in the NN tight-binding model.
We also study the connection between a periodic short-range potential
problem and a tight-binding model.

The magnetic von Neumann lattice\cite{e,ha} is a representation of the
quantum system in a uniform magnetic field and has  quite useful
properties in studying the system with  periodic potentials.
In particular, its lattice structure varies and
we can select the suitable one in accordance with
the potential lattice.
Using a momentum representation of the von Neumann lattice,\cite{i}
we give a proof of the duality and show
energy spectra in the periodic potential problems.

In a two-dimensional system under a uniform magnetic field $B$,
 the von Neumann lattice basis
 is formed by the  direct product of harmonic
oscillator eigenfunctions
$\vert f_l\rangle$ of relative coordinates ($\xi,\eta$) and
coherent states $\vert \alpha_{mn} \rangle$ of guiding center coordinates
($X,Y$).
The coherent states are defined by
\begin{equation}
(X+iY)\vert\alpha_{mn}\rangle=az_{mn}\vert\alpha_{mn}\rangle,
\end{equation}
where $a=\sqrt{2\pi\hbar/eB}$, $z_{mn}=m\omega_x+n\omega_y$
 ($m,n \in {\bf Z}$) and
$\omega_x ,\omega_y$ are complex numbers that satisfy
${\rm Im}[\omega_x^*\omega_y]=1$.\cite{pb}
For our purpose, it is convenient to use the Fourier transformed basis
as
\begin{equation}
\vert l,{\bf p}\rangle=\frac{1}{\beta({\bf p})}\sum_{m,n}e^{ip_xm+ip_yn}
\vert f_l \otimes \alpha_{mn}\rangle,
\end{equation}
where $\beta({\bf p})$ is a normalization factor defined by
\begin{equation}
\beta({\bf p})=\left(2{\rm Im}\tau\right)^{1\over4}
e^{i{\tau\over4\pi}p_y^2}
\vartheta_1({p_x+\tau p_y\over2\pi}\vert\tau),
\end{equation}
and $\tau=-\omega_x / \omega_y$.
For $\tau=i$, the von Neumann lattice becomes a square lattice.
For $\tau=e^{i2\pi /3}$, it becomes a triangular lattice.
The Fourier transformed basis is an orthonormal set of extended states
and obeys a nontrivial boundary condition
\begin{equation}
\vert l,{\bf p}+2\pi {\bf N}\rangle=e^{-i\phi (p,N)}
\vert l,{\bf p}\rangle ,
\end{equation}
where $\phi (p,N)=\pi (N_x+N_y)-N_y p_x$.

If the magnetic von Neumann lattice has a periodicity
commensurate with the periodicity of the external potential $V({\bf x})$,
the one-body potential problem becomes easy to treat.
This happens when $t$ is equal to $q/p$
with relatively prime integers  $p,q$.
Let us consider an arbitrary regular lattice of short-range
potentials of $t=q/p$:
\begin{equation}
V({\bf x})=a^2V_0\sum_N\delta^{(2)}(z+aN_x q \omega_x+a {N_y\over p} \omega_y).
\label{peri}
\end{equation}
The matrix element of the potential in the LLL is given by
\begin{eqnarray}
& &\langle 0,{\bf p}\vert V({\bf x})\vert 0,{\bf p'}\rangle  \\
&=&\frac{V_0}{q} \sum_{r_p,r_q,{\bf N}} \beta (p_x-2\pi{r_p\over p},p_y)
\beta^* (p_x-2\pi({r_p\over p}+{r_q\over q}),p_y)  \nonumber \\
&\times &(2\pi)^2
\delta (p_x' -p_x+2\pi({r_q\over q}+ N_x))
\delta (p_y' -p_y +2\pi N_y) e^{i \phi(p',N)}, \nonumber
\end{eqnarray}
where $r_p=0,1,...,p-1$ and  $r_q=0,1,...,q-1$.
Here, we study the eigenvalue equations in the LLL.
The eigenvalue equation becomes
\begin{equation}
D^\dagger D\psi=\epsilon \psi,
\label{eig1}
\end{equation}
where a $p\times q$ matrix $D$ is defined by
\begin{equation}
(D({\bf p}))_{r_p r_q}=\beta^*(p_x-2\pi({r_p\over p}+{r_q\over q}),p_y),
\label{dd}
\end{equation}
and  $D^\dagger D$ is a $q\times q$ Hermitian matrix.
The magnetic Brillouin zone (MBZ)
is the region where $\vert p_x\vert<\pi/q$ and $\vert p_y\vert<\pi$.
Each band has a $p$-fold degeneracy.
Consequently, the fundamental region of the MBZ is the region where
$\vert p_x\vert<\pi/pq$ and $\vert p_y\vert<\pi$.

We should note that the rank of $D^\dagger D$ is min($p,q$) generally.
If $p<q$,  the band splits into $p$ subbands and
one flat band which  corresponds to the  zero modes of $D$.
The number of the zero modes is $q-p$.
By a linear transformation, $\psi'=D\psi$,
the eigenvalue equation (\ref{eig1}) becomes
\begin{equation}
DD^\dagger\psi'=\epsilon\psi'.
\label{eig2}
\end{equation}
$DD^\dagger$ is a $p\times p$ Hermitian matrix.
Equation (\ref{eig1}) is equivalent to Eq.~(\ref{eig2}) except for
the zero modes.
Apparently, Eq.~(\ref{eig2}) is obtained by
taking the complex conjugate and
interchanging $p$ and $q$ of Eq.~(\ref{eig1}).
Therefore, there exists a duality relation between two problems of
$t=q/p>1$ and $t=p/q<1$. Actually, the energy spectra $E$ obey the
relation
\begin{equation}
E(t)={1\over t}E({1\over t}),
\label{dual}
\end{equation}
except for the flat bands.
If $p>q$, Eq.~(\ref{dual}) is also obtained.
Thus, the duality relation is proved in an arbitrary regular lattice of
short-range potentials.
The self-dual point is $t=1$ and the physical quantity has
critical behavior near the point.

Before solving  the eigenvalue problems numerically,
we clarify the connection between a periodic
short-range potential problem and a tight-binding model in the LLL.
Let us suppose a square lattice potential
$V({\bf x})=V_0 {\rm exp}(2\pi i(mx+ny)/b)$, where
$t=(b/a)^2=q/p$.
The potential energy term in the second quantized form  becomes
\begin{eqnarray}
&V_0&\int_{\rm MBZ}{d^2p\over(2\pi)^2}
\sum_{r_q}c^\dagger_{r_q}({\bf p})c_{r_q+m}({\bf p})
\nonumber\\
&\times&e^{-{\pi\over2t}(m^2+n^2)+in(p_x-{2\pi\over t}
r_q)-i{m\over t}(p_y+\pi n)}
\label{potpot}
\end{eqnarray}
where
\begin{equation}
c_{r_q}({\bf p})=b_0(p_x+\pi
-2\pi{p\over q}r_q,p_y)e^{-i\pi{p\over q}r_q}
\end{equation}
and $b_l({\bf p})$ is the annihilation operator of the state
$\vert l,{\bf p}\rangle$.
Equation (\ref{potpot}) is equivalent to
the tight-binding Hamiltonian with the following hopping term:
\begin{equation}
V_{\rm hop}=
V_0e^{-{\pi\over2t}((m_2-m_1)^2+(
n_2-n_1)^2)+i{\pi\over t}(m_2+m_1)(n_2-n_1)},
\end{equation}
where $m_2-m_1=m$ and $n_2-n_1=n$.
This hopping term has a flux  per  unit cell
of a square lattice $2\pi/t$ and the hopping strength varies with $t$.
If  $(m,n)=(\pm1,0)$, $(0,\pm1)$, Eq.~(\ref{potpot})  becomes
the NN tight-binding Hamiltonian.
In a square lattice of short-range potentials,
the summation is taken over all $(m,n)$ with an equal weight.
Therefore,
the result becomes a tight-binding Hamiltonian with finite-range
hopping terms
\begin{equation}
{1\over t}\sum_{m,n}c^\dagger(m_2,n_2)V_{\rm hop}(m_2,n_2;m_1,n_1)
c(m_1,n_1).
\end{equation}
The hopping range is about $\sqrt{t}$ and the number of relevant terms
increases linearly with  $t$.

In a regular lattice,
the one-particle spectrum is given by the
solution of Eq.~(\ref{eig1}).
Since the matrix element of  $D^\dagger D$ is given in the analytic form,
it is easy to solve numerically.
Figure 1 shows the spectra for the square and triangular lattices.
The points at $E=0$ correspond to the original LLL.
There are two marked structures in these figures :
Hofstadter-type bands and a large gap above the flat bands in
$t>1$.
The origin of the large gap is easily understood
in a dilute potential limit  $t \rightarrow \infty$.
In this limit, the potential approaches  one short-range
potential. Its spectrum
consists of a bound state trapped at the potential and a flat band.
The bound state has the energy $E=V_0$ and the flat band has the energy $E=0$.
In a dense potential limit $t \rightarrow 0$, the potential
approaches  constant $V_0 /t$.
Therefore, the asymptotic form of the spectrum in
$t \rightarrow 0$ and $t \rightarrow \infty$
except for the zero modes
is given by
\begin{equation}
E(t)\sim V_0 (1+\frac{1}{t}),
\label{asym}
\end{equation}
which satisfies the duality relation (\ref{dual}).

To check the duality, we
replace $E$ with $Et$ for $t<1$ and replace $t$ with
$2-1/t$ otherwise in Fig.~1.
Then the figures become  symmetric with respect to
$t=1$ owing to the duality.
The results for the square lattice  are shown in Fig.~2.
The duality is clear in this figure.
The patterns in Figs.~1 and 2 are
very similar to that of the NN and next-nearest-neighbor (NNN)
tight-binding model on the square and the triangular lattice.
However, in contrast with the NN or NNN tight-binding model, a periodicity
with respect to the flux does not exist in our model. Instead,
the duality between $t$ and $1/t$ does exist.

As an example of an irregular lattice, we  study a honeycomb lattice
potential numerically.
A honeycomb lattice  can be regarded as a sum of
two triangular lattices.
This property causes the following eigenvalue equation at
$2t={\rm (flux\; per\; hexagon)}/\Phi_0=q/p$:
\begin{equation}
(D_1^\dagger D_1 +D_2^\dagger D_2)\psi=\epsilon \psi,
\end{equation}
where $D_1$ and $D_2$ are defined by
\begin{equation}
D_1=D(p_x,p_y),\quad
D_2=D(p_x-{2\pi\over3p},p_y+{2\pi q\over3}).
\end{equation}
Since both $D_1^\dagger D_1$ and $D_2^\dagger D_2$ are positive definite,
the zero modes must satisfy both $D_1\psi=0$ and $D_2\psi=0$.
If $q-2p>0$, the zero modes exist and the LLL splits into $2p$ subbands
and one flat band.
Otherwise, the LLL splits into $q$ subbands.
This system does not satisfy the duality relation (\ref{dual}).
However, the asymptotic behavior of the one-particle energy spectrum
is also given  by Eq.~(\ref{asym})
from arguments similar to that of  the regular lattice.
The numerical result of the spectrum is shown in Fig.~3.
The spectrum also has a Hofstadter-type structure and a large gap
above the flat bands.

Next we study the possibility of observing
the band structure obtained above.
The gap above the flat bands may be observed in the
magnetoresistance experiment.\cite{ca}
We assume that the antidot potential has a height
$\tilde V_0$ and an area of the base $r_0^2$.
$V_0$ in Eq.~(\ref{peri}) is related to $\tilde V_0$ and $r_0$ as
$a^2V_0=r_0^2\tilde V_0$.
The magnitude of the large gap above the flat bands is of the order
of $V_0$ and the correction of the finite size effect of the antidot to
the energy is estimated as $V_0{\rm O}(r_0/b)$ where $b$ is a lattice
constant of the antidot array.
Therefore the gap above the flat bands should be observed in
$r_0\ll b$.
In the current experiments,\cite{c}
$r_0/b\approx 0.25$, which is enough to observe the gap.
Taking the Landau level mixing into consideration,
the eigenvalue equation for the potential of Eq.~(\ref{peri})
has the form
\begin{equation}
\sum_{l'}[E_l\delta_{ll'}+{V_0\over q}D^{(l)\dagger}D^{(l')}]
\psi^{(l')}=\epsilon\psi^{(l)}.
\end{equation}
Because of the $(q-p)$-zero modes of $D^{(l)}$ for $p<q$,
the $l$-th Landau band has a flat band with the energy $E_l$.
Therefore, the flat bands remain flat in the presence
of the Landau level mixing effect in the short-range periodic
potential problem.
The correction of the Landau level mixing to the energy above the
large gap is estimated by the second-order perturbative calculation as
$E_{\rm LL}\approx (V_0)^2/\hbar\omega_c$. From $V_0\gg E_{\rm LL}$,
we obtain the condition $\hbar\omega_c\gg\tilde V_0(r_0/a)^2$ for
the gap to survive.
Using the realistic values\cite{b,ca}
$m=0.07m_e$ and $\tilde V_0=0.3$[meV],
this condition becomes $2\pi\hbar^2/m r_0^2\gg\tilde V_0$.
This is satisfied in the current experiments.
So far, conditions for observation are satisfied.
However, the parameter $t$ in current experiments, {\it e.g.}
$B=5$[T], $b=200$[nm] and $t\approx50$,
is too large to observe the large gap, the Hofstadter-type bands, and
the critical behavior near $t=1$.
Thus we hope that the experiment will be made in a finer lattice
of the antidot array.

In summary,
we solved various periodic potential
problems in the LLL using a momentum representation of
the von Neumann lattice.
In a periodic array of short-range potentials, the energy spectrum
has three remarkable structures.
(i) The spectrum has a Hofstadter-type structure, which is commonly
seen in the periodic potential problems in a magnetic field.
(ii) The duality relation exists for a regular lattice
potential of Eq.~(\ref{peri}).
A honeycomb lattice is an irregular lattice and obeys a duality
relation asymptotically.
(iii) There is a large gap above flat bands\cite{aoki} that comes from
zero modes of the matrix $D$ of Eq.~(\ref{dd}).
These structures are universal in the periodic short-range potential
problem in the LLL.
The conditions for the observation of these structures were obtained.
In addition, the equivalence between two problems of a square lattice of
short-range potentials and a tight-binding model with a inverse flux
was shown.

This work was partially supported by the special Grant-in-Aid
for Promotion of Education and Science in Hokkaido University
provided by the Ministry of Education, Science, Sports and Culture,
the Grant-in-Aid for Scientific Research (07640522), and the
Grant-in-Aid for International Scientific Research (Joint Research
07044048) from
the Ministry of Education, Science, Sports and Culture, Japan.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Generalized distribution amplitudes (GDAs) \cite{Diehl98,muller94}
are important nonperturbative objects for understanding how quarks
and gluons form hadrons in hard exclusive processes. They describe
the soft transition $q \bar{q} \rightarrow h \bar{h}$ or $g g
\rightarrow h \bar{h}$, in the kinematical regime where the
invariant mass of the hadron pair $h\bar{h}$ is small compared to
the hard scale of the process. The typical processes to which the
GDAs formalism can be applied are the two photon process
$\gamma^*\gamma \rightarrow h \bar{h}$~\cite{Diehl00,dkv03,apt04}
and hadron pair eletroproduction $\gamma^* p\rightarrow p^\prime h
\bar{h}~$\cite{Polyakov99,lppsg}, for which factorization theorems
have been proved~\cite{freund00}. In Ref.~\cite{ls05} we gave an
analysis of the process $e^+e^-\rightarrow Z \rightarrow \pi \pi
\gamma$ at the $Z$-pole energy, with small pion pair invariant mass.
The subprocess $Z \rightarrow \pi \pi \gamma$ can be factorized into
the convolution of a hard coefficient and the generalized
distribution amplitude of two pions (2-pion GDA). In the case of
charged pion pair production, the process is sensitive to the
$C$-odd parts of 2-pion GDAs, and provide an opportunity to access
them in $e^+e^-$ annihilation. In this work we extend the analysis
in Ref.~\cite{ls05} to a generalized process $e^+e^-\rightarrow
\gamma^*\, \textrm{or} \, Z^*\rightarrow\pi\pi\gamma$, in the
kinematical region where $\sqrt{s}$ is large but much below the
$Z$-pole. Like the case of the production through $Z$ boson analyzed
in Ref.~\cite{ls05}, a similar factorization holds for the case of
the production through a virtual photon, which gives the dominant
contribution for neutral pion pair production in the kinematical
region we discuss here. Therefore, the investigation of the process
$e^+e^-\rightarrow \gamma^*\rightarrow\pi^0\pi^0\gamma$ can provide
detailed information about 2-pion GDAs, especially their $C$-even
part. We also discuss the $\gamma Z$ interference term, which
contains the weak mixing angle $\sin \theta_W$ and can be used as an
alternative approach to extract its value. The Bremsstrahlung
process $e^+e^-\rightarrow\gamma^*\gamma\rightarrow\pi^+\pi^-\gamma$
dominates in the case of charged pion production, and its
interference with $e^+e^-\rightarrow \gamma^ * \rightarrow
\pi^+\pi^-\gamma$ is a valuable tool for studying the process
$\gamma^ * \rightarrow \pi\pi\gamma$ at the amplitude level.

\section{Analysis of the process
$\gamma^*\rightarrow\pi\pi\gamma$}\label{gammagamma}

We first study the subprocess of pion pair production
\begin{equation}
\gamma^*(q)\rightarrow \pi(p_1)+\pi(p_2)+
\gamma(q^\prime)\label{gamma}
\end{equation}
in the framework of GDAs. It is convenient to investigate this
subprocess in the pion pair center of mass frame (shown in
Fig.~\ref{kine}a), where the kinematics is expressed as (choosing
the $z$ axis along the momentum of the real photon)
\begin{eqnarray}
q&=&\frac{Q}{\sqrt{2}}v+\frac{Q}{\sqrt{2}}v^\prime,~~~
q^\prime=\frac{Q^2-W^2}{\sqrt{2}Q}v^\prime,\nonumber \\
P&=&\frac{Q}{\sqrt{2}}v+\frac{W^2}{\sqrt{2}Q}v^\prime,
\nonumber \\
p_1&=&\frac{\zeta Q}{\sqrt{2}}v+\frac{(1-\zeta)W^2}
{\sqrt{2}Q}v^{\prime}+\frac{\Delta_T}{2},\nonumber\\
p_2&=&\frac{(1-\zeta)Q}{\sqrt{2}}v+\frac{\zeta
W^2}{\sqrt{2}Q}v^{\prime}-\frac{\Delta_T}{2}
\end{eqnarray}
respectively. Here $v$ and $v^\prime$ are two lightlike vectors
which satisfy: $v^2=v^{\prime 2}=0$, $v\cdot v^\prime=1$, $W$ is the
invariant mass of the pion pair and $q^2=Q^2=s$, $P$ is the total
momentum of the pion pair. The skewness parameter $\zeta$ is the
momentum fraction of plus momentum carried by $\pi(p_1)$ with
respect to the pion pair:
\begin{equation}
\zeta=\frac{p_1^+}{P^+}=\frac{1+\beta \cos
\theta}{2},~~~~\beta=\sqrt{1-\frac{4m_\pi^2}{W^2}}.
\end{equation}

\begin{figure}

\begin{center}
\scalebox{1.05}{\includegraphics{kine.eps}} \caption{\small The
kinematics of pion pair production. (a) the c.m. frame of the pion
pair, (b) the c.m. frame of $e^+e^-$. } \label{kine}
\end{center}

\end{figure}

\begin{figure*}

\begin{center}
\scalebox{0.88}{\hspace{-0.5cm}\includegraphics{pionpair.eps}}
\caption{\small (a) Diagram of the process $e^+e^-\rightarrow V^*
\rightarrow \pi \pi \gamma$, where $V$ can be a photon or $Z$ boson.
(b) Diagram of the DVCS process $e\pi\rightarrow  e \pi \gamma$, the
crossing related channel of the process depicted in (a).}
\label{pionpair}
\end{center}

\end{figure*}

In the kinematical region $W^2 \ll s$, the process given in
Eq.~(\ref{gamma}) can be analyzed in the framework of GDAs, which
has been applied in Ref.~\cite{ls05} to the case where the virtual
photon is replaced by the $Z$ boson at $\sqrt{s}=M_Z$. From
Fig.~\ref{pionpair}a and the one with the two photon vertices
interchanged, the hadronic tensor of the process can be calculated
as~\cite{Diehl98,ls05}
\begin{eqnarray}
T^{\mu\nu}_{\gamma^*\rightarrow \pi\pi\gamma}&=&i\int
d^4xe^{-iq\cdot x}\langle
\pi(p_1)\pi(p_2)|TJ_{em}^\mu(x)J_{em}^\nu(0)|0\rangle
\nonumber\\
&=&-g^{\mu\nu}_\perp\sum_q \frac{e^2 e^2_q}{2} \int_0^1 dz
\frac{2z-1}{z(1-z)}\Phi_q^{\pi\pi}(z,\zeta,W^2),\nonumber\\\label{gammatensor}
\end{eqnarray}
which has the same form as that of the two photon
process~\cite{Diehl98,Diehl00}, and $\Phi_q(z,\zeta,W^2)$ is the GDA
defined as~\cite{Diehl98}:
\begin{eqnarray}
\Phi_q(z,\zeta,W^2)&=&\int \frac{dx^-}{2\pi} e^{iz(P^+
x^-)}\nonumber\\
&&\times\langle
\pi(p_1)\pi(p_2)|\bar{\psi}(x^-)\gamma^+\psi(0)|0\rangle,
\end{eqnarray}
where $z=k^+/P^+$ is the momentum fraction of the quark with
respect to the hadronic system.

As discussed in Refs.~\cite{Diehl98,Polyakov99}, a useful constraint
on the GDAs comes from charge conjugation invariance of the strong
interactions:
\begin{equation}
\Phi_q^{\pi\pi}(z,\zeta,W^2)=-\Phi_q^{\pi\pi}(1-z,1-\zeta,W^2).
\end{equation}
Following this constraint one can define the $C$-even and $C$-odd
parts (in \cite{Polyakov99} they are also called isoscalar and
isovector parts respectively) of 2-pion GDA as~\cite{Diehl00}
\begin{equation}
\Phi_q^{\pm}(z,\zeta,W^2)=\frac{1}{2}[\Phi_q^{\pi\pi}(z,\zeta,W^2)
\pm \Phi_q^{\pi\pi}(z,1-\zeta,W^2)],\label{cpm}
\end{equation}
where the superscript $\pi \pi$ represents $\pi^0 \pi^0$ or $\pi^+
\pi^-$. Then
\begin{equation}
\Phi_q^{\pi\pi}(z,\zeta,W^2)=\Phi_q^{+}(z,\zeta,W^2)+\Phi_q^{-}(z,\zeta,W^2).\label{cdecomp}
\end{equation}
The properties of $\Phi_q^{\pm}(z,\zeta,W^2)$ under the interchanges
$z \rightarrow 1-z$ and $\zeta \rightarrow 1-\zeta$ can be easily
derived from (\ref{cdecomp})~\cite{Polyakov99}:
\begin{eqnarray}
\Phi_q^{+}(z,\zeta,W^2)&=&-\Phi_q^{+}(1-z,\zeta,W^2)\nonumber\\
&=&\Phi_q^{+}(z,1-\zeta,W^2),\label{ceven}\\
\Phi_q^{-}(z,\zeta,W^2)&=&\Phi_q^{-}(1-z,\zeta,W^2)\nonumber\\
&=&-\Phi_q^{-}(z,1-\zeta,W^2).\label{codd}
\end{eqnarray}
The $\zeta \rightarrow 1-\zeta$ exchange corresponds to the
interchange of the two pions, thus in the case in which the final
two pions are $\pi^0 \pi^0$ we get
\begin{equation}
\Phi_q^{\pi^0 \pi^0}(z,\zeta,W^2)=\Phi_q^{\pi^0
\pi^0}(z,1-\zeta,W^2).
\end{equation}
Therefore there is only a $C$-even part for $\Phi_q^{\pi^0
\pi^0}(z,\zeta,W^2)$. In the following we use
$\Phi_q^{\pm}(z,\zeta,W^2)$ to represent the $C$-even/odd part of
$\Phi_q^{\pi^+\pi^-}(z,\zeta,W^2)$ respectively. Isospin invariance
implies~\cite{Diehl00}
\begin{eqnarray}
\Phi_q^{\pi^0\pi^0}(z,\zeta,W^2)=\Phi_q^{+}(z,\zeta,W^2).
\end{eqnarray}
Thus according to (\ref{gammatensor}), only
$\Phi_q^{+}(z,\zeta,W^2)$ contributes to the process
$\gamma^*\rightarrow \pi\pi\gamma$. Here we only consider the
contributions coming from the $u$ and $d$ quark. Another consequence
of isospin invariance is
\begin{eqnarray}
\Phi^{+}_u=\Phi^{+}_d,~~~~\Phi^{-}_u=-\Phi^{-}_d.
\end{eqnarray}

Eq.~(\ref{gammatensor}) shows that the process $e^+e^- \rightarrow
\gamma^* \rightarrow \pi\pi\gamma$ can be also applied to probe
$\Phi^{+}$, just like the two photon process $\gamma^* \gamma
\rightarrow \pi \pi$, which has been proposed~\cite{Diehl00} to
access $\Phi^{+}$. Interest on the process $e^+e^- \rightarrow
\gamma^* \rightarrow \pi\pi\gamma$ relies also on the fact that this
process is related to its crossed channel, the deeply virtual
Compton scattering process (DVCS) $e \pi \rightarrow e\pi\gamma$
(shown in Fig.~\ref{pionpair}b), and is referred to as timelike
DVCS. In the factorization of the DVCS process, the nonperturbative
objects are the generalized parton distributions (GPDs)
\cite{dm,dvcs}, which are important for understanding the parton
angular momenta inside the nucleon~\cite{ji97}. The simplest DVCS
process is $e \pi \rightarrow e\pi\gamma$. However it is difficult
to use the pion as the target, which makes the pion GPDs difficult
to be experimentally accessed. It has been shown~\cite{teryaev} that
there is a crossing relation that connects GPDs and GDAs. Therefore
to measure the process $e^+e^- \rightarrow \gamma^* \rightarrow
\pi\pi\gamma$, the crossed channel of the DVCS process, can also
give experimental indication about the GPDs of the pion. In the
kinematical region $W^2\ll s \ll M_Z^2$, the suitable facilities to
investigate this process are $B$-factories. Finally, we remark that
there is a suggestion~\cite{ps05b} of using the transition
distribution amplitude (TDA)~\cite{ps05} formalism to describe the
process $e^+e^-\rightarrow\pi\pi\gamma$.

\section{Phenomenology about the $e^+e^-\rightarrow \pi\pi\gamma$ process}

\subsection{Calculation of the cross-section}
 In this section we study the phenomenology associated
with the process \begin{equation}
 e^+(l)+e^-(l^\prime) \rightarrow V^*(q) \rightarrow
\pi(p_1)+\pi(p_2)+\gamma(q^\prime),
\end{equation}
where the intermediate vector boson $V$ can be a photon or a $Z$
boson, based on the analysis given in Section \ref{gammagamma} and
in Ref.~\cite{ls05}. We will consider the helicity of the electron
in the analysis. It is useful to define two Lorentz invariant
variables:
\begin{equation}
x=\frac{2q\cdot q^\prime}{q^2}, \;\;\;\; y=\frac{l\cdot
q^\prime}{q\cdot q^\prime}.
\end{equation}
 The
differential cross-section for the process $e^+e^-\rightarrow
V^*\rightarrow\gamma\pi\pi$ is expressed as
\begin{eqnarray}
d\sigma_{e^+e^-\rightarrow V^* \rightarrow\gamma\pi\pi}
&=&\frac{\beta
W(s-W^2)}{64(2\pi)^5s^2}|\mathcal{A}_{e^+e^-\rightarrow
V^* \rightarrow \pi\pi\gamma}|^2\nonumber\\
&&\times dWd\Omega^* d\Omega^\prime,\label{csgandz}
\end{eqnarray}
in which ($|\Vec p^*|,\Omega^*$) is the momentum of pion 1 in the
c.m. frame of the pion pair, and $\Omega^\prime$ is the angle of the
photon in the rest frame of the $Z$ boson (that is, the c.m. frame
of $e^+e^-$).

The amplitude of the process can be calculated from
\begin{eqnarray}
&&\mathcal{A}_{e^+e^-\rightarrow V^* \rightarrow
\gamma\pi\pi}=\sum_{i,j}\bar{v}(l^\prime)V^\mu
u(l) \epsilon^{(i)*}_\mu\nonumber\\
&&\hspace{1cm}\times\frac{1}{s-M_V^2}\mathcal{A}_{i,j}^{V^*\rightarrow
\gamma\pi\pi},
\end{eqnarray}
where $V^\mu=e\gamma^\mu(1+\lambda_e\gamma_5)$ for lepton-$\gamma$
vertex, and
$V^\mu=e\gamma^\mu(c^l_V-\gamma_5c^l_A)(1+\lambda_e\gamma_5)/\sin
2\theta_W$ for lepton-$Z$ boson vertex, with $\lambda_e=\pm 1$ the
helicity of the electron. The amplitude of $V^*\rightarrow
\pi\pi\gamma$ can be calculated from
\begin{equation}
\mathcal{A}_{i,j}^{V^*\rightarrow
\gamma\pi\pi}=\epsilon^{(i)}_\alpha\epsilon^{\prime (j)*}_\beta
T^{\alpha\beta}_V(z,\zeta,W^2),
\end{equation}
where $\epsilon$ and $\epsilon^\prime$ are the polarization vectors
of the intermediate virtual boson (photon or $Z$) and the real
photon, respectively. In our reference frame these vectors have the
form:
\begin{eqnarray}
\epsilon^{(\pm)}_\mu&=&\left(0,\frac{\mp
1}{\sqrt{2}},\frac{-i}{\sqrt{2}},0\right
),\epsilon_\mu^0=\left(\frac{|\Vec
q|}{\sqrt{s}},0,0,\frac{q_0}{\sqrt{s}}\right ),\nonumber\\
\epsilon^{\prime(\pm)}_\mu&=&\left(0,\frac{\mp
1}{\sqrt{2}},\frac{-i}{\sqrt{2}},0\right ).
\end{eqnarray}

Using the notation of Ref.~\cite{bjm99} the cross-section of the
process can be written as:
\begin{eqnarray}
\frac{d\sigma}{dW^2 d\cos \theta d\phi dy}=\frac{\alpha^3 \beta
(s-W^2)}{32\pi
s^4}\sum_{ab}L_{\mu\nu}^{ab}\mathcal{W}_{ab}^{\mu\nu}\chi_{ab}.\label{csgz}
\end{eqnarray}
The indices $a$, $b$ can be $\gamma$ for the photon or $Z$ for the
$Z$ boson. The relative propagator factors $\chi_{ab}$ are given
by
\begin{eqnarray}
\chi_{\gamma \gamma}&=&1,\label{gg}\\
\chi_{\gamma
Z}&=&\chi_{Z\gamma}=\frac{1}{\sin^2(2 \theta_W)}\frac{s}{s-M_Z^2},\label{gz}\\
\chi_{ZZ}&=&(\chi_{\gamma Z})^2.\label{zz}
\end{eqnarray}
which correspond to the contributions from $\gamma$, $\gamma Z$
interference and $Z$ boson. The lepton tensors are:
\begin{eqnarray}
L_{ab}^{\mu\nu}=-2sC_{ab}I_1(y)g_\perp^{\mu\nu}-isD_{ab}I_2(y)\epsilon_\perp^{\mu\nu}.
\label{ltensor}
\end{eqnarray}

\begin{figure}

\begin{center}
\scalebox{0.9}{\includegraphics{csection.eps}} \caption{\small The
$\gamma$ (solid line), $\gamma Z$ interference term (dashed line)
and $Z$ (dotted line) contributions to the helicity-independent
differential cross-section of the process $e^+e^-\rightarrow V^*
\rightarrow \pi^0\pi^0\gamma$ at $W=0.7\,\textrm{GeV}$,
$\theta=10^\circ$.} \label{csection}
\end{center}

\end{figure}

The notation used in the above equation is
\begin{eqnarray}
C_{\gamma\gamma}&=&1,\\
 C_{\gamma_Z}&=&C_{Z\gamma}=(c^l_V-c^l_A\lambda_e),\\
 C_{ZZ}&=&(c^{l2}_V+c^{l2}_A)-2c^l_Vc^l_A\lambda_e,\\
 D_{\gamma\gamma}&=&\lambda_e,\\
 D_{\gamma_Z}&=&D_{Z\gamma}=(c^l_V\lambda_e-c^l_A),\\
 D_{ZZ}&=&(c^{l2}_V+c^{l2}_A)\lambda_e-2c^l_Vc^l_A,\\
I_1(y)&=&(\frac{1}{2}-y+y^2),\\
I_2(y)&=&(1-2y).
\end{eqnarray}

 The hadron tensors appearing in Eq.~(\ref{csgz}) are:
\begin{eqnarray}
W^{\mu\nu}_{\gamma\gamma}&=& -g_\perp^{\mu
\nu}\left | \sum_qe_q^2V_q(\cos\theta,W^2)\right |^2, \label{wgg} \\
 W^{\mu\nu}_{\gamma Z}&=& -g_\perp^{\mu
\nu}\textrm{Re}\left \{\left [ \sum_q e_q c_V^q V_q^*(\cos\theta,W^2)\right ]\right.\nonumber\\
&&\left. \times \left [ \sum_q e_q^2 V_q(\cos\theta,W^2)\right ] \right\}\nonumber\\
&&+i\epsilon_\perp^{\mu
\nu}\textrm{Im}\left \{\left [ \sum_q e_q a_V^q V_q^*(\cos\theta,W^2)\right ]\right.\nonumber\\
&&\left. \times \left [ \sum_q e_q^2 A_q(\cos\theta,W^2)\right ] \right\}, \label{wgz} \\
W^{\mu\nu}_{Z Z}&=& -g_\perp^{\mu \nu} \left \{ \left | \sum_q e_q
c_V^q
V_q(\cos\theta,W^2)\right |^2 \right. \nonumber\\
&&+\left .\left | \sum_q e_q c_A^q A_q(\cos\theta,W^2)\right |^2 \right \} \nonumber\\
&&-2i\epsilon_\perp^{\mu\nu}\textrm{Im}\left \{\left [ \sum_q e_q
c_V^q V_q^*(\cos\theta,W^2)\right ]\right.\nonumber
\\
&& \times \left .\left [ \sum_q e_q c_A^q
A_q(\cos\theta,W^2)\right ]\right \},
\end{eqnarray}
where
\begin{eqnarray}
\hspace{-0.5cm}V_q(\cos\theta,W^2)&=& \int_{0}^{1}
dz\frac{2z-1}{z(1-z)}\Phi_q^+(z,\zeta,W^2),\\
\hspace{-0.5cm}A_q(\cos\theta,W^2)&=& \int_{0}^{1}
dz\frac{1}{z(1-z)}\Phi_q^-(z,\zeta,W^2).
\end{eqnarray}
The vector and axial-vector couplings to the $Z$ boson appearing in
the above equations are given by:
\begin{eqnarray}
c^i_V &=& T_3^i-2Q^i\sin^2\theta_W, \label{cqv}\\
c^i_A &=& T_3^i,\label{cqa}
\end{eqnarray}
where $Q^i$ denotes the charge and $T_3^i$ is the weak isospin of
particle $i$ (i.e., $T_3^i=+1/2$ for $i=u$, and $-1/2$ for
$i=e^-$, $d$, $s$).

\subsection{Results for the neutral case}

In the previous subsection we have given the expression for the
cross-section of the general process $e^+e^-\rightarrow
\gamma^*\,\textrm{or} \,Z^* \rightarrow \pi \pi \gamma$, in terms of
2-pion GDAs. There are interesting phenomenological implications
coming from Eq.~(\ref{csgandz}). As demonstrated in
Ref.~\cite{ls05}, measuring the charged pion pair production in the
kinematical regime $\sqrt{s}=M_Z$, where the main contribution comes
from $Z$ boson, can provide an opportunity to access the $C$-odd
part of 2-pion GDAs $\Phi^-_q$. In this work, however, we consider a
different kinematical region $W^2 \ll s\ll M_Z^2$. Consequently a
different phenomenology emerges: the production through $\gamma^*$
is dominant over the $\gamma Z$ interference and $Z$ boson terms,
which is a result coming from the factors given in Eqs.~(\ref{gg}),
(\ref{gz}) and (\ref{zz}).

To avoid the competing contribution from the Bremsstrahlung process,
in this subsection we limit ourselves to the case of neutral pion
pair production. Therefore the terms containing $A_q$ in the
hadronic tensor (\ref{ltensor}) do not contribute to the
cross-section, and the calculation is simplified. Based on
(\ref{csgandz}), we calculate the $\gamma$, $\gamma Z$ interference
and $Z$ contributions to the unpolarized differential cross-section
of the process $e^+e^-\rightarrow \gamma^* \,\textrm{or}\,Z^*
\rightarrow \pi^0\pi^0\gamma$ as functions of $\sqrt{s}$. The
results are shown in Fig.~\ref{csection} by a solid line, dashed
line and dotted line, respectively. For $\Phi_q^+$, which is needed
in the calculation, we adopt the simple model given in
\cite{Diehl00}. The curves show that the contribution from the
production through $\gamma^*$ is about several orders of magnitude
larger than those through $\gamma Z$ interference and $Z$ boson, in
the regime $W^2 \ll s \ll M_Z^2$. Therefore it would be suitable to
investigate $\Phi^+_q$ through the process $e^+e^- \rightarrow
\gamma^* \rightarrow \pi^0\pi^0 \gamma$, in the regime where
$\sqrt{s}$ is several GeV, where the cross-section is relatively
large.

\begin{figure}

\begin{center}
\scalebox{0.9}{\includegraphics{neutralw.eps}} \caption{\small The
$W$-dependent differential cross-section of the process
$e^+e^-\rightarrow \gamma^* \rightarrow \pi^0\pi^0\gamma$ at
$\sqrt{s}=10$ GeV and $\theta=20^\circ$.} \label{neutralw}
\end{center}

\end{figure}

\begin{figure}

\begin{center}
\scalebox{0.9}{\includegraphics{rs.eps}} \caption{\small The
$\sqrt{s}$ dependent ratio of the helicity dependent and
helicity-independent cross-section of the process $e^+e^-\rightarrow
V^* \rightarrow \pi^0\pi^0\gamma$.} \label{rs}
\end{center}

\end{figure}

\begin{figure*}

\begin{center}
\scalebox{1.3}{\includegraphics{bremint.eps}} \caption{\small Upper
panel: the differential cross-sections of the Bremsstrahlung
process, at  $\sqrt{s}=5$ (solid line) and 10 GeV (dashed line).
Lower panel: contributions to the differential cross-sections coming
from the unpolarized interference term between the Bremsstrahlung
process and the process $e^+e^-\rightarrow \gamma^* \rightarrow
\pi^+\pi^-\gamma$ at $\sqrt{s}=5$ (solid line), 10 GeV (dashed line)
and helicity dependent interference term (dotted line) at
$\sqrt{s}=5$ GeV. The first column: the $W$ dependent differential
cross-sections at $\theta=20^\circ$, $\phi=180^\circ$, $y=0.3$; the
second column: the $\theta$ dependent differential cross-sections at
$W=0.7 \,\textrm{GeV}$, $\phi=180^\circ$, $y=0.3$; the third column:
the $\phi$ dependent differential cross-sections at $W=0.7
\,\textrm{GeV}$, $\theta=20^\circ$, $y=0.3$; The fourth column: the
$y$ dependent differential cross-sections at $W=0.7 \,\textrm{GeV}$,
$\theta=20^\circ$, $\phi=180^\circ$.} \label{bremint}
\end{center}

\end{figure*}

The $\gamma Z$ interference term is also interesting because it
contains the weak mixing angle $\sin\theta_W$. To isolate this term
one needs to go beyond the helicity-independent cross-section, that
is, to consider the helicity dependent cross-section of the process.
The helicity dependent differential cross-section is dominated by
the $\gamma Z$ interference term:
\begin{eqnarray}
&&\frac{d\sigma(+\lambda_e)-d\sigma(-\lambda_e)}{2dW^2 d\cos
\theta}\nonumber\\
&=&-\frac{\alpha^3 \beta (s-W^2)}{6 s^3}\chi_{\gamma
Z}c_A^l\lambda_e\nonumber\\
&&\times(e_uc_V^u+e_dc_V^d)(e_u^2+e_d^2)|V_u(\cos\theta,W^2)|^2,
\end{eqnarray}
since the  $Z$ boson term can be ignored compared to that of the
$\gamma Z$ interference term.

 As discussed previously, the unpolarized differential
cross-section is dominated by the virtual photon production, which
reads:
\begin{eqnarray}
&&\frac{d\sigma(+\lambda_e)+d\sigma(-\lambda_e)}{2dW^2 d(\cos
\theta)}\nonumber\\
&=&\frac{\alpha^3 \beta (s-W^2)}{12
s^3}(e_u^2+e_d^2)^2|V_u(\cos\theta,W^2)|^2.
\end{eqnarray}
The $W$-dependence of this cross-ection is shown in
Fig.~\ref{neutralw}, for $\sqrt{s}=10$ GeV and $\theta=20^\circ$.

The ratio of the helicity dependent and helicity-independent
cross-section, that is, the helicity asymmetry of the process is
($\lambda_e=+1$)
\begin{eqnarray}
R&=&\frac{\sigma(+\lambda_e)-\sigma(-\lambda_e)}{\sigma(+\lambda_e)+\sigma(-\lambda_e)}\nonumber\\
&=&-2c_A^l\chi_{\gamma
Z}\frac{e_uc_V^u+e_dc_V^d}{e_u^2+e_d^2}\nonumber\\
&=&\frac{\frac{9}{10}-2\sin^2
\theta_W}{\sin^2(2\theta_W)}\frac{s}{s-M_Z^2},\label{ratio}
\end{eqnarray}
which depends on $s$ and $\sin\theta_W$. Therefore the measurement
on this helicity asymmetry provides an alternative approach to
extract the value of weak mixing angle $\sin\theta_W$. In
Fig.~\ref{rs} we show the $\sqrt{s}$ dependent ratio $R$ which is
given in (\ref{ratio}). For $\sqrt{s}=10\,\textrm{GeV}$, which is
the typical c.m. energy of CLEO or $B$-factories, such as BaBar and
Belle, this ratio is about 0.8\%, which is a sizeable value.

\subsection{Results for the charged case}

In the case of charged pion pair production, besides the process
$e^+e^- \rightarrow \gamma^* \rightarrow \pi^+ \pi^- \gamma$, the
Bremsstrahlung process $e^+e^- \rightarrow \gamma^* \gamma
\rightarrow \pi^+ \pi^- \gamma$ also needs to be considered. The
cross-section of the former process is the same as that of neutral
pion production, since only $\Phi_q^+$ contributes. The differential
cross-section of the Bremsstrahlung process has the form
\begin{eqnarray}
\frac{d \sigma^B }{ d W^2 d ( \cos \theta) d\phi dy}
        &=& \frac{ \alpha^3 \beta^3 (s - W^2) }{ 16 \pi s^2 W^2}
        |F_\pi(W^2)|^2 \nonumber\\
        &&\hspace{-2cm}\times( (1-2x(1-x)) \frac{I_1 (y)}{y(1-y)}\sin^2 \theta\nonumber\\
        &&\hspace{-2cm}+4x(1-x)\cos^2
        \theta\nonumber\\
        &&\hspace{-2cm}+ \sqrt{ x ( 1 - x ) }
        ( 1 - 2 x ) \frac{ I_2 (y) }{ \sqrt{ y ( 1 - y ) } } \sin 2
        \theta \cos \phi \nonumber\\
        &&\hspace{-2 cm}- x (1-x) 2  \sin^2 \theta \cos 2 \phi)
        ,\label{bremcs}
\end{eqnarray}
where $F_\pi(W^2)$ is the timelike pion form factor.

It is interesting to investigate the interference
\cite{brodskylecture,ls05} between $e^+e^- \rightarrow \gamma^*
\rightarrow \pi^+ \pi^- \gamma$ and the Bremsstrahlung process,
because it can be applied to study the subprocess $\gamma^*
\rightarrow \pi^+ \pi^- \gamma$ at the amplitude level, similar to
the case $e\gamma\rightarrow e \pi \pi$ that has been discussed in
Refs.~\cite{Diehl00,lps06}. Knowledge of the process $\gamma^*
\rightarrow \pi^+ \pi^- \gamma$ at the amplitude level can help to
determine not only the magnitude of $\Phi^+_q$, but also its phase.
The form of the interference term reads
\begin{eqnarray}
\frac{d\sigma^I}{dW^2 d(\cos \theta) d\phi dy}&=&
            \frac{\alpha^3\beta^2(s - W^2)}{16\pi s^2\sqrt{s}W}\nonumber\\
            &&\hspace{-2.5cm}\times(\sqrt{x(1-x)}
            \textrm{Re}\{F_\pi^* V\}\cos \theta \nonumber\\
            &&\hspace{-2.5cm}+[\frac{xI_1 (y)}{\sqrt{y(1-y)}}-(1-x)\sqrt{y(1-y)}]\nonumber\\
            &&\hspace{-2.5cm}\times\textrm{Re}\{F_\pi^* V\}\sin\theta \cos \phi \nonumber\\
            &&\hspace{-2.5cm}+\lambda_e
            x\frac{I_2 (y)}{2\sqrt{y(1-y)}}\textrm{Im}\{F_\pi^* V\} \sin \theta \sin
            \phi),\label{intf}
\end{eqnarray}
where $V=\sum_{q} V_q$ for $q=u,d$. This interference term can be
obtained from taking the difference of the cross-section in the
exchange $(\theta, \phi)\rightarrow (\pi-\theta, \pi+\phi)$. The
last term in Eq.~(\ref{intf}) is the helicity dependent term, which
can probe $\textrm{Im}\{F^* V \}$, and which together with the
helicity-independent terms can be used to reconstruct the full
complex amplitude of $\gamma^*\rightarrow \pi^+\pi^-\gamma$, in the
region where $F_\pi(W^2)$ is known. Conversely, these measurements
could also be used in order to obtain information about the pion
timelike form factor $F_\pi(W^2)$.

In Fig.~\ref{bremint} we give the curves of the differential
cross-sections, coming from the Bremsstrahlung process and its
interference with the process $e^+e^- \rightarrow \gamma^*
\rightarrow \pi^+ \pi^- \gamma$, respectively. For $F_\pi(W^2)$ we
use the parametrization $N=1$ given in Ref.~\cite{ks90}. We show the
cross-sections for $\sqrt{s}= 5$ (solid line) and 10 GeV (dashed
line) in Fig.~\ref{bremint}. In the later case the terms containing
$1-x$ have sizeable contributions. In the lower panel of
Fig.~\ref{bremint} we also show the contribution from the helicity
dependent interference term (dotted line) for $\sqrt{s}=5$ GeV.

Fig.~\ref{bremint} shows that the contribution coming from the
Bremsstrahlung process is $1 \sim 2$ orders of magnitude larger than
that from the interference term. According to the shape of $F_\pi$,
the $W$-dependent curves of both the Bremsstrahlung and interference
contributions have peaks in the region $W\sim m_\rho$. In the case
of $\theta$-dependence the curves of Bremsstrahlung and
(unpolarized) interference contributions increase with $\theta$ from
$0^\circ$ to $90^\circ$, and then decrease. The $\phi$-dependent
curve of the Bremsstrahlung contribution has a large value at
$\phi=180^\circ$ and a small value at $\phi=0^\circ$, while that of
the (unpolarized) interference contribution has a large (absolute)
value at $\phi=0^\circ$ and a small (absolute) value at
$\phi=180^\circ$. The $y$-dependence of the interference
contribution shows that the curves have maximum magnitude in the
region $y\rightarrow 0$ or 1 (due to the factor $\sqrt{y(1-y)}$ in
the denominators of some terms in (\ref{intf})), which is therefore
the suitable region in order to study the interference term. A
similar situation occurs in the case of the Bremsstrahlung process.

\section{Summary}

We have given a detailed analysis of the process $e^+e^-\rightarrow
\gamma^*\,\textrm{or} \, Z^*\rightarrow\pi\pi\gamma$, in the
kinematical region where $\sqrt{s}$ is large compared to the
invariant mass of the pion pair but much below the $Z$-pole. A
factorization similar to which has been shown in Ref.~\cite{ls05}
holds for the $\gamma$ case: the amplitude of the process can be
expressed as the convolution between the hard scattering $\gamma^*
\rightarrow q \bar{q} \gamma$ and 2-pion GDAs which describe the
soft transition $q \bar{q} \rightarrow \pi \pi$. Thus the
investigation on the process $e^+e^-\rightarrow
\gamma^*\rightarrow\pi\pi\gamma$ can provide detailed information
about 2-pion GDAs, especially their $C$-even part. The experimental
measurement of this process can also provide experimental indication
of pion GPDs, by virtue of the fact that this process is the crossed
channel of the DVCS process $e\pi \rightarrow e \pi \gamma$, and
there is a crossing relation between GPD and GDA. We studied the
production of both neutral and charged pion. In the former case the
$\gamma Z$ interference term can be applied to extract the value of
the weak mixing angle $\sin \theta_W$ through the measurement of the
helicity asymmetry in the process $e^+e^-\rightarrow
\pi^0\pi^0\gamma$. In the last case the interference between the
Bremsstrahlung process and the process $e^+e^-\rightarrow
\gamma^*\rightarrow \pi^+\pi^-\gamma$ is useful in order to study
the subprocess $\gamma^*\rightarrow \pi^+\pi^-\gamma$ at the
amplitude level.

\begin{acknowledgments}
This work is supported by Fondecyt (Chile) under Projects
No.~3050047 and No.~1030355.
\end{acknowledgments}