xpression as that valid for the standard asymptotic behavior. The Virasoro charges are then easily integrated to yield $$\label{once}Q_{\pm}[T^{\pm}]=\frac{2}{l}\left(  1\pm\frac{1}{\mu l}\right)
\int T^{\pm}f_{\pm\pm}d\phi\$$ (up to additive constants). The details will be given in [@HMTfuture]. What happens is that the diverging pieces associated with the slower fall-off $h_{--}$ or $h_{++}$ disappear in $\delta Q_{\pm}[T^{\pm}]$ so that $Q_{\pm}$ is given by (\[once\]), and hence the charges acquire no correction involving the terms associated with the relaxed behavior. One can then view $h_{--}$ (or $h_{++}$), which cannot be gauged away, as defining a kind of hair.“ This situation is analogous to the one found for a scalar field with mass $m$ in the range $m_{ \rm BF}^{2}<m^{2}<m_{\rm BF}^{2}+1/l^{2}$ (where $m_{\rm BF}$ is the Breitenlohner-Freedman bound [@BF]). There are then two possible admissible behaviors (two branches”) for the scalar field, and the analysis proceeds as here when only the branch with slower behavior is switched on [@HMTZ3][^1].

Under an asymptotic conformal transformation (\[Asympt KV\]), $f_{++}$ and $f_{--}$ are straightforwardly found to transform as $$\begin{aligned}
\delta_{\eta}f_{++}  &  =2f_{++}\partial_{+}T^{+}+T^{-}\partial_{-}%
f_{++}+T^{+}\partial_{+}f_{++}
  -l^{2}\left(  \partial_{+}T^{+}+\partial_{+}^{3}T^{+}\right)
\!/2\ ,\label{deltaf++}\\
\delta_{\eta}f_{--}  &  =2f_{--}\partial_{-}T^{-}+T^{-}\partial_{-}%
f_{--}+T^{+}\partial_{+}f_{--}
  -l^{2}\left(  \partial_{-}T^{-}+\partial_{-}^{3}T^{-}\right)  \!/2\ .
\label{deltaf--}%\end{aligned}$$ On shell, one verifies that $$\partial_{+}f_{--}=0=\partial_{-}f_{++}%$$ and so (\[deltaf++\]) and (\[deltaf–\]) reduce to $$\begin{aligned}
\delta_{\eta}f_{++}  &  =2f_{++}\partial_{+}T^{+}+T^{+}\partial_{+}%
f_{++}-\frac{l^{2}}{2}\left(  \partial_{+}T^{+}+\partial_{+}^{3}T^{+}\right)
,\label{delta2h++}\\
\delta_{\eta}f_{--}  &  =2f_{--}\partial_{-}T^{-}+T^{-}\partial_{-}%
f_{--}-\frac{l^{2}}{2}\left(  \partial_{-}T^{-}+\p