text:
\section{Introduction}
The $V-A$ structure of the  charged currents is considered a
firmly established fact in weak interaction physics.
The evidence for $V-A$ is very strong for the leptonic couplings,
e.g. by the measurement of the Michel parameters of the muon decay.
However, this is not as clear for the hadronic currents due to our inability to
perform a precise calculation of the hadronic matrix elements.
Consequently, it is hard to exclude admixtures of different helicities in hadronic charged currents.

Over the last ten years, heavy-quark symmetries became a very useful tool in the calculation
of hadronic matrix elements involving heavy quarks.
In particular, they may help to perform the analogue of a Michel parameter analysis for the hadronic charged currents.
Making use of the detailed data from the flavor factories the semileptonic heavy quark decays
may serve as a sensitive test for  possible ``wrong-helicity'' contributions.

In a recent analysis we extracted limits on a right handed admixture from the wealth of data
on inclusive semileptonic decays \cite{Feger:2010qc}.
Despite the large amount of data for the inclusive semileptonic decays and the precise theoretical tools,
it turns out that the exclusive  decays $B \to D^{(\ast)}\ell \bar\nu$
can be much more sensitive to wrong helicity admixtures than the inclusive decays.

In this paper we expand on this idea,
including also scalar and tensor components for the hadronic current.
The method we propose can be most easily explained in the Isgur-Wise limit for the decays $B \to D^{(\ast)}\ell \bar\nu$,
where only a single form factor appears,
of which  the normalization is known.
Starting from this limit, the corrections may be considered
and the method can be systematically improved.

Beyond the Isgur-Wise limit a large number of form factors appears,
most of which are not well studied.
However, based on the detailed analysis of the vector and axial-vector form factors through lattice and
QCD sum rule calculations one may perform a stringent test at least for a possible right-handed admixture.

The paper is organized as follows.
In the next section we study a possible new physics (NP) contribution
in the Isgur-Wise limit to demonstrate how the proposed method works.
After that we will calculate the corresponding radiative corrections,
followed by a section dealing with the calculation of bounds on right-handed
admixtures regarding the contemporary experimental results as well as lattice and non-lattice calculations
to be able to provide a comparison to the standard model (SM) results.
Finally we discuss our results and give a prospect into possible additional measurements.

\section{New Physics contributions in the Isgur-Wise limit}\label{New Physics contributions in the IW limit}
As has been pointed out above,
the $V-A$ structure of the leptonic current is well established and
hence we do not modify this current.
However, the hadronic current may contain a contribution from ``new physics''
and hence the effective Hamiltonian to be considered is
\begin{equation}
H_{\text{eff}}= \frac{4 G_F V_{cb}}{\sqrt{2}} \ J_{h,\mu}  (\bar e \gamma^\mu P_L \nu_e) ,
\end{equation}
where $J_{h,\mu}$ is the generalized hadronic current
and $P_{L,R}= (\openone  \mp \gamma_5)/2$ is the projector of negative/positive chirality.
The modifications in the hadronic current can be considered on the basis of an effective field theory approach
and one obtains (see eg. \cite{Dassinger:2008as})
\begin{eqnarray}
 J_{h,\mu} &= & c_L \bar c  \gamma_\mu P_L b + c_R \bar c  \gamma_\mu P_R b \nonumber \\
  &+&  g_L \bar c \left( i \tensor D_\mu \right) P_L b
 + g_R \bar c \left( i \tensor D_\mu \right) P_R b \nonumber \\
 &+& d_L  i \partial^\nu (\bar c\, i \sigma_{\mu\nu} P_L b)
 + d_R\, i \partial^\nu (\bar c\, i \sigma_{\mu\nu} P_R b) ,\label{hadcur}
\end{eqnarray}
where $D_\mu$ is the QCD covariant derivative and
$ f \tensor D_\mu g = f(D_\mu g) - (D_\mu f)g$
represents the left and right derivatives.
Note that the first line in (\ref{hadcur})
corresponds to dimension-3 operators with dimensionless couplings $c_L$ and $c_R$,
while the second and third lines are dimension-4 operators with dimensionful couplings $g_{L/R}$ and
$d_{L/R}$. There
are two other dimension-4 operator of the form $(m_b + m_c) \bar c \gamma_\mu b$ and
$(m_b - m_c) \bar c \gamma_\mu \gamma_5 b$, which are related by the Gordon identities
\begin{eqnarray}
 (m_b + m_c) \bar c \,\gamma_\mu b
 &=& \bar c \, i \tensor D_\mu b \nonumber\\
  &&+ i \partial^\nu \left( \bar c (- i \sigma_{\mu \nu}) b \right)  , \label{GIV}\\
 - (m_b - m_c) \bar c \,\gamma_\mu\gamma^5 b
 &=& \bar c \, i \tensor D_\mu \gamma^5b \nonumber \\
  &&+ i \partial^\nu \left( \bar c (- i \sigma_{\mu \nu}) \gamma^5 b
 \right)  . \label{GIPV}
\end{eqnarray}
Hence these operators are redundant
and do not need to be considered seperately as it has been done in \cite{Dassinger:2008as}.
Likewise, the pseudotensor is not independent of the tensor due to the relation
\begin{equation}
 \bar c \sigma_{\mu \nu} \gamma_5 b = -\frac i2 \epsilon_{\mu \nu \alpha \beta}  \bar c \sigma^{\alpha \beta} b ,
\end{equation}
but it is convenient to keep this operator explicitely.

From the effective field theory analysis performed in \cite{Dassinger:2008as}
all these operators originate from dimension-6 operators parametrizing physics beyond the standard model.
From this one obtains
\begin{equation}
 \begin{split}
c_L \sim 1_{\text{SM}} &+  {\mathcal O} \left(\frac{v^2}{\Lambda^2} \right), \
c_R \sim   {\mathcal O} \left(\frac{v^2}{\Lambda^2} \right) , \\
&g_{L/R},\, d_{L/R} \sim  \frac{1}{v}  {\mathcal O} \left(\frac{v^2}{\Lambda^2} \right) ,
\end{split}
\end{equation}
where $\Lambda$ is the scale of ``new physics''
and $v$ is the electroweak vacuum expectation value.

We shall use this hadronic current to evaluate the semileptonic widths for $B \to D^{(*)} \ell \bar{\nu}$.
We first study the Isgur-Wise limit where the relevant kinematic quantity is the product of the four
velocities of the initial and final state hadrons
\begin{equation}\label{wdef}
 w = v \cdot v' = \frac{m_B^2 + m_{D^{(\ast)}}^2-q^2}{2 m_B m_{D^{(\ast)}}} \ ,
\end{equation}
where $q^2 \equiv (p - p')^2$,
and $p$ and $p'$ refer to the four-momenta
of the $B$ and $D^{(\ast)}$ mesons, respectively.
The differential exclusive decay rates for the $D$ and $D^\ast$ mesons
can as usual be expressed in terms of the hadronic form factors $\mathcal G (w)$
and $\mathcal F(w)$, respectively.
Thus the differential decay rates read
\begin{eqnarray}
    \frac{d\Gamma}{dw} &=& G_0 (w) |V_{cb}|^2\,\frac{w-1}{w+1}(1+r)^2|\mathcal G (w)|^2,\label{diffdecrateI}\\
    \frac{d\Gamma^{*}}{dw} &=& G_0^{*} (w) |V_{cb}|^2\,|\mathcal F (w)|^2,\label{diffdecrateII}
\end{eqnarray}
where we defined the factors
\begin{equation}
G_0^{(\ast)} (w) =  \frac{G_F^2m_B^5 }{48\pi^3}  r_{(\ast )}^3 \sqrt{w^2 -1} (w+1)^2 \ ,
\end{equation}
containing the kinematical and normalization coefficients to streamline the notation and
$r_{(\ast)}= m_{D^{(\ast)}}/m_B$.
The form factors $\mathcal F (w)$ and $\mathcal G (w)$
are related to matrix elements of the hadronic current,
which in our case also contains the new physics effects represented
by the couplings $c_R$, $d_{L/R}$ and $g_{L/R}$.

The Isgur-Wise limit is taken by letting $m_b,m_c  \to \infty $
with $m_c / m_b \sim {\mathcal O}(1)$.
In this case both the charm and the bottom quark in the hadronic current
have to be replaced by static quarks $h_{v',c}$ and $h_{v,b}$;
to leading order,
the hadronic current matches onto
\begin{eqnarray}
 J_{h,\mu} &=& c_L \bar{h}_{v',c} \gamma_\mu P_L h_{v,b} + c_R \bar{h}_{v',c} \gamma_\mu P_R h_{v,b}\nonumber\\
  &&+ g_L (m_b v_\mu + m_c v'_\mu) \bar{h}_{v',c} P_L h_{v,b}\nonumber\\
  &&+ g_R (m_b v_\mu + m_c v'_\mu) \bar{h}_{v',c} P_R h_{v,b} \nonumber\\
  &&+ d_L  (m_b v^\nu - m_c v^{\prime \nu}) (\bar{h}_{v',c}i \sigma_{\mu\nu} P_L h_{v,b})\nonumber\\
  &&+ d_R  (m_b v^\nu - m_c v^{\prime \nu}) (\bar{h}_{v',c} i \sigma_{\mu\nu} P_R h_{v,b})  ,\label{hadcurII}
\end{eqnarray}
and all the relevant matrix element
can be expressed in terms of the Isgur-Wise function $\xi (w)$
\cite{Isgur:1989vq,Isgur:1989ed,Shifman:1987rj},
which is normalized to $\xi(1) \equiv 1$ at zero recoil
\begin{eqnarray}
\frac{\langle D(v')| \bar{h}_{v',c} h_{v,b} | B(v)\rangle}{\sqrt{m_Bm_D}} &=& (1 + w) \xi(w)\label{BtoD} , \\
\frac{\langle D(v')| \bar{h}_{v',c} \gamma^\mu h_{v,b} | B(v)\rangle}{\sqrt{m_Bm_D}} &=& (v+v')^\mu \xi (w)  ,  \\
\frac{\langle D(v')| \bar{h}_{v',c} \sigma^{\mu \nu}  h_{v,b} | B(v)\rangle}{\sqrt{m_B m_D}}&&\nonumber\\
      =  i (v_\mu'v_\nu  &-& v_\nu' v_\mu) \xi(w) ,
\end{eqnarray}
for the $B \to D\,\ell\,\bar\nu$ decay and
\begin{widetext}
\begin{eqnarray}
\frac{\langle D^\ast(v',\epsilon)|\bar{h}_{v',c} \gamma_5 h_{v,b}| B(v)\rangle}{\sqrt{m_B m_{D^\ast}}}
&=&  (\epsilon^\ast \cdot v) \xi (w) \ ,  \\
\frac{\langle D^\ast (v',\epsilon)|\bar{h}_{v',c} \gamma^\mu h_{v,b}|  B(v)\rangle}{\sqrt{m_Bm_{D^\ast}}}
&=& i   \varepsilon^{\mu\nu\alpha\beta}\epsilon^\ast_\nu v_\alpha'v_\beta \xi(w)\ , \\
\frac{\langle D^\ast (v',\epsilon)|\bar{h}_{v',c}  \gamma^\mu \gamma_5 h_{v,b}| B(v)\rangle}{\sqrt{m_Bm_{D^\ast}}}
&=&  \left[ (1+w) \epsilon^{\ast\mu} + v^\mu (v \cdot \epsilon^\ast) \right] \xi(w)\ ,  \\
\frac{\langle D^\ast(v',\epsilon)|\bar{h}_{v',c}  \sigma^{\mu\nu} h_{v,b}|B(v)\rangle}{\sqrt{m_B m_{D^\ast}}}
&=&\ \varepsilon^{\mu\nu\kappa\tau}  \epsilon^\ast_\kappa (v'+v)_\tau \xi (w) \ .  \label{BtoDst}
\end{eqnarray}
\end{widetext}
for the $\bar B \to D^\ast\ell\bar\nu$ decay.
The decay rate for the semileptonic $\bar B\to D^{(\ast)}\ell \bar\nu$ decays
in the Isgur-Wise limit can thus be expressed solely
in terms of the Isgur Wise function, even in the presence of ``new physics''.
The form factors become
\begin{eqnarray}
    |\mathcal G(w)|^2 &=& \frac{w+1}{w-1}(1+r)^{-2}\, \mathcal A(w) |\xi(w)|^2\ , \\
    |\mathcal F(w)|^2 &=& \left(\mathcal B^T(w) + \mathcal B^L(w)\right) |\xi(w)|^2,
\end{eqnarray}
where we have separated the rate for $B \to D^\ast$
into the contributions for longitudinally and transversely polarized $D^\ast$ mesons.
Hence we end up with
\begin{eqnarray}
\frac{d\Gamma^{B \to D \ell \bar{\nu}}}{dw}  &=& G_0 (w)
 |V_{cb}|^2  \mathcal A (w)  | \xi (w) |^2\ , \\
\frac{d\Gamma^{B \to D^\ast_T \ell \bar{\nu}}}{dw}   &=&G_0^\ast (w)
 |V_{cb}|^2 \mathcal B^T (w)  | \xi (w) |^2\ , \\
\frac{d\Gamma^{B \to D^\ast_L \ell \bar{\nu}}}{dw}   &=&G_0^\ast (w)
 |V_{cb}|^2 \mathcal B^L (w)  | \xi (w) |^2 \ ,\\
 \frac{d\Gamma^{B \to D^\ast  \ell \bar{\nu}}}{dw}   &=& G_0^\ast (w)
 |V_{cb}|^2 \nonumber\\
  && \times\left[ \mathcal B^T (w) + \mathcal B^L (w) \right] | \xi (w) |^2 \ ,
\end{eqnarray}
where the coefficient functions $\mathcal A (w)$, $\mathcal B^T$
and $\mathcal B^L$ contain the dependence on the new physics couplings.
\begin{eqnarray}
\mathcal A (w) &=&\frac{w-1}{w+1}\bigl\lbrack c_+ (1+r) -m_B  d_+ (r^2-2rw+1) \nonumber\\
  &&+2 m_B r g_+  (w+1)\bigr\rbrack^2\ , \label{NP-BtoDpar} \\
\mathcal B^T (w) &=& 2\lbrack1- 2 r_\ast w + (r_\ast)^2 \rbrack \Bigl
\lbrace \lbrack c_- +d_- m_B (r_\ast-1)\rbrack^2 \nonumber\\
&&+\frac{w-1}{w+1}\bigl\lbrackc_+ + d_+ m_B (r_\ast+1)\bigr\rbrack^2 \Bigr
\rbrace\ , \label{NP-BtoDsT} \\
\mathcal B^L (w) &=& \Bigl\lbrace c_- (r_\ast-1)+2 g_- m_B r_\ast (w-1) \nonumber\\
&&+d_- m_B  \lbrack(r_\ast)^2 -2  r_\ast  w +1\rbrack\Bigr\rbrace^2\ , \label{NP-BtoDsL}
\end{eqnarray}
where we define the combinations of coupling constants
\begin{equation}
\begin{split}
c_\pm \equiv (c_L &\pm c_R), \ d_\pm \equiv (d_L\pm d_R) , \\
&  g_\pm \equiv (g_L \pm g_R).
\end{split}
\end{equation}
The expressions of the standard model are retrieved by setting $c_\pm = 1$
and all other couplings zero,
and we recover the standard model case as
\begin{eqnarray}
\mathcal A_{\text{SM}}   (w) &=&\frac{w-1}{w+1} (1+r)^2\ , \\
\mathcal B^T_{\text{SM}}  (w) &=&   \frac{4w}{w+1} \lbrack1- 2 r_\ast w + r_\ast^2 \rbrack\ ,  \\
\mathcal B^L_{\text{SM}} (w) &=&   (r_\ast-1)^2 \ .
\end{eqnarray}
In the context of the extraction of $|V_{cb}|$ from exclusive decays
the measured $w$ spectrum is extrapolated to  $w =1$
and hence one studies the observables
\begin{eqnarray} \label{Ms}
M (w) &=&  \frac{d\Gamma^{B \to D \ell \bar{\nu}}}{dw}  \, \frac{1}{G_0}
\, \frac{1}{\mathcal A_{\text{SM}} (w)}\ , \\
M^\ast_T(w)  &=&  \frac{d\Gamma^{B \to D^\ast_T \ell \bar{\nu}}}{dw}  \, \frac{1}{G_0^\ast}
\frac{1}{\mathcal B^T_{\text{SM}} (w)}\ , \\
M^\ast_L (w) &=&  \frac{d\Gamma^{B \to D^\ast_L \ell \bar{\nu}}}{dw}  \, \frac{1}{G_0^\ast}
\frac{1}{ \mathcal B^L_{\text{SM}} (w)}\ ,  \\
M^\ast (w) &=&  \frac{d\Gamma^{B \to D^\ast  \ell \bar{\nu}}}{dw}  \, \frac{1}{G_0^\ast}
\frac{1}{\mathcal B^L_{\text{SM}}  (w) + \mathcal B^T_{\text{SM}} (w)}  .
 \label{Mss}
\end{eqnarray}
These observables become in the standard model
in all three cases just $| V_{cb}| ^2 | \xi (w) |^2$.
Extrapolating $M(w)$ and/or  $M^\ast(w)$ to the zero-recoil point $w=1$
and making use of the normalization of the Isgur Wise function allows us to extract $|V_{cb}|$ model independently.
Expanding around $w=1$ and using
\begin{equation}\label{ExpIWF}
\xi (w) =  \xi (1) \left[ 1- \rho_{\text{\text{IW}}}^2 (w-1) + \dots \right] \ ,
\end{equation}
we may also obtain information on the slope of the Isgur-Wise function by performing the
corresponding expansion of the expressions \eqref{Ms}--\eqref{Mss}.
In the standard model we obtain:
\begin{eqnarray}
 \rho_{\text{\text{IW}}}^2 &=& - \frac{1}{2 M(1)}\left.  \frac{\partial M(w)}{\partial w} \right|_{w=1} \nonumber\\
    &=& - \frac{1}{2M_T^\ast(1)} \left.  \frac{\partial M_T^\ast(w)}{\partial w} \right|_{w=1} \nonumber  \\
    &=& -   \frac{1}{ 2M_L^\ast(1)} \left.  \frac{\partial M_L^\ast(w)}{\partial w} \right|_{w=1}\nonumber\\
    &=& - \frac{1}{2  M^\ast(1)} \left.  \frac{\partial M^\ast(w)}{\partial w} \right|_{w=1}\ .
\end{eqnarray}
However, the presence of the new physics contributions will change these relation
and allows us to re-interpret the measured observables in terms of a possible ``new physics'' contribution.
In fact,
inserting the general expressions into \eqref{Ms}--\eqref{Mss},
we get
\begin{eqnarray} \label{MsNP}
M (w) &=&  \frac{ \mathcal A (w) }{ \mathcal A_{\text{SM}} (w)} | V_{cb}| ^2 | \xi (w) |^2\ , \\
M^\ast_T(w)  &=&  \frac{ \mathcal B_T (w)}{\mathcal B^T_{\text{SM}} (w)} | V_{cb}| ^2 | \xi (w) |^2 \ ,  \\
M^\ast_L (w) &=&   \frac{ \mathcal B_L (w)}{ \mathcal B^L_{\text{SM}} (w)} | V_{cb}| ^2 | \xi (w) |^2 \ , \\
M^\ast (w) &=&   \frac{ \mathcal B_T (w)+ \mathcal B_L (w) }{\mathcal B^T_{\text{SM}} (w) + \mathcal B^L_{\text{SM}} (w)} | V_{cb}| ^2 | \xi (w) |^2 \ .
 \label{MssNP}
\end{eqnarray}
At the zero-recoil point $w=1$ this becomes
\begin{eqnarray}
 M (1) &=&  | V_{cb}| ^2 | \xi (1) |^2 \biggr[c_+ - m_B d_+ \frac{(r-1)^2}{r+1} \nonumber \\
  && + 4 m_B g_+ \frac{r}{r+1} \biggr]^2 \ , \\
 M_T^\ast (1) &=& | V_{cb}| ^2 | \xi (1) |^2 \left[c_- + (r^\ast-1) m_B d_- \right]^2\nonumber\\
  &=& M_L^\ast (1) = M^\ast (1)\ ,
\end{eqnarray}
leading to
\begin{eqnarray}
&-& \frac{1}{2 M(1)}\left.  \frac{\partial M(w)}{\partial w} \right|_{w=1} =  \rho_{\text{IW}}^2  \nonumber\\
&&  + \frac{2 r m_B (d_+ + g_+)}{m_B d_+ (r-1)^2 - (r+1) c_+ - 4 r m_B g_+}  ,  \\
&-& \frac{1}{2 M_T^\ast(1)}\left.  \frac{\partial M_T^\ast(w)}{\partial w} \right|_{w=1} \nonumber\\
&&=  \rho_{\text{IW}}^2 + \frac{1}{4} \left[1 -
 \left(\frac{c_+ +(r_\ast + 1) m_B d_+}{c_- + (r_\ast - 1)m_B d_-}\right)^2 \right]  , \\
&-& \frac{1}{2 M_L^\ast(1)}\left.  \frac{\partial M_L^\ast(w)}{\partial w} \right|_{w=1} \nonumber\\
&&= \rho_{\text{IW}}^2 - \frac{2 r_\ast m_B (d_- - d_-)}{(r_\ast -1)(c_- + (r_\ast-1)d_-)}  , \\
&-& \frac{1}{2 M^\ast(1)}\left.  \frac{\partial M^\ast(w)}{\partial w} \right|_{w=1} = \rho_{\text{IW}}^2 \nonumber\\
&&+ \frac{1}{6}\biggl\lbrace 1- \left(\frac{c_++\dPm_B(r_\ast+1)}{c_-+\dMm_B (r_\ast-1)}\right)^2  \nonumber\\
&& + \frac{4m_Br_\ast (d_- - g_-)}{(r_\ast-1)\lbrackc_-+d_- m_B (r_\ast-1)\rbrack}\biggr\rbrace\ ,
\end{eqnarray}
when we calculate the slopes.
Note that the current analyses are performed for the total
$B \to D^\ast\,\ell\,\bar\nu$ rate
without the decomposition into longitudinal and transversal polarizations.

\section{Radiative Corrections}
Up to this point we have been relying on the Isgur-Wise limit.
However, from lattice as well as from QCD sum rule studies we know
that the relations obtained in the infinite mass limit may have corrections of
the order of ten percent,
making a sensitive test on the basis of the formulae in the last section impossible.

The first class of corrections are the perturbative QCD corrections
which break the Isgur Wise symmetry.
The dimension-3 operators, i.e.
the vector $\bar{c} \gamma_\mu b$  and the axial vector
$\bar{c} \gamma_\mu \gamma_5 b$
are both conserved in the massless lepton limit
and hence do not have an anomalous dimension.
The dimension-4 operators
\begin{align}
  \mathcal{O}^{\text{S}} &= \bar c i \tensor D_\mu b\ , \\
  \mathcal{O}^{\text{PS}} &= \bar c i \tensor D_\mu \gamma^5 b\ , \\
  \mathcal{O}^{\text{T}}&= i \partial^\nu \bar c (- i \sigma_{\mu \nu}) b\ , \\
  \mathcal{O}^{\text{PT}}&= i \partial^\nu \bar c (- i \sigma_{\mu \nu}) \gamma^5 b \ ,
\end{align}
mix under renomalization and also have anomalous dimensions \cite{Dassinger:2008as}.
Note that $\mathcal{O}^{\text{T}}$ and $\mathcal{O}^{\text{PT}}$ are not independent due to the relation
$\sigma_{\mu \nu} \gamma_5 = \frac{i}{2} \epsilon_{\mu \nu \alpha \beta} \sigma^{\alpha \beta}$. However,
for our purposes it is useful to keep both operators.

Gathering the four operators and the coupling constants in columns
\begin{equation}
\begin{split}
&\vec{{\mathcal O}} (\mu) =  \begin{pmatrix}
    \mathcal{O}^{\text{S}} (\mu) \\
    \mathcal{O}^{\text{PS}} (\mu) \\
    \mathcal{O}^{\text{T}} (\mu) \\
    \mathcal{O}^{\text{PT}} (\mu)
 \end{pmatrix} \ , \\
&\vec{G} (\mu) =  \begin{pmatrix}
    g_{\text{S}} (\mu) \\
    g_{\text{PS}} (\mu) \\
    g_{\text{T}} (\mu) \\
    g_{\text{PT}} (\mu)
 \end{pmatrix}  =
 \begin{pmatrix}
    g_L(\mu) + g_R (\mu)  \\
    g_R(\mu) - g_L (\mu)  \\
    d_L(\mu) + d_R (\mu)  \\
    d_R(\mu) - d_L (\mu)
 \end{pmatrix}\ ,
\end{split}
\end{equation}
we can write the hadronic current \eqref{hadcur} as
\begin{equation}
J_{h,\mu} = c_L \, \bar c  \gamma_\mu P_L b + c_R \, \bar c  \gamma_\mu P_R b + \vec{G} (\mu) \cdot \vec{{\cal O}}(\mu) \ .
\end{equation}

The one-loop anomalous dimension matrix can be calculated and becomes
\begin{equation}
  \gamma = \frac{\alpha_s}{4\pi} C_F
\begin{pmatrix}
 6 & 0 & 4 & 0 \\
 0 & 6 & 0 & 4 \\
 0 & 0 & 2 & 0 \\
 0 & 0 & 0 & 2
\end{pmatrix}
\end{equation}
and the solution of the renormalization group equation can be written as
\begin{widetext}
\begin{equation}\label{wilsons_full}
    \begin{pmatrix}
	g_{\text{S}} (\mu) \\
	g_{\text{PS}} (\mu) \\
	g_{\text{T}} (\mu) \\
	g_{\text{PT}} (\mu)
    \end{pmatrix}
    =
\begin{pmatrix}
 g_{\text{S}} (\Lambda ) \left(\frac{\alpha _s(\Lambda )}{\alpha _s(\mu )}\right){}^{\frac{3 C_F}{\beta _0}} \\
 g_{\text{PS}} (\Lambda ) \left(\frac{\alpha _s(\Lambda )}{\alpha _s(\mu )}\right){}^{\frac{3 C_F}{\beta _0}} \\
 \left(g_{\text{T}} (\Lambda )+ g_{\text{S}} (\Lambda )
  \left(\left(\frac{\alpha _s(\Lambda )}{\alpha _s(\mu )}\right){}^{\frac{2 C_F}{\beta _0}}-1\right)\right) \left(\frac{\alpha _s(\Lambda )}{\alpha _s(\mu )}\right){}^{\frac{C_F}{\beta _0}}
   \\
 \left( g_{\text{PT}} (\Lambda )+ g_{\text{S}} (\Lambda ) \left(\left(\frac{\alpha _s(\Lambda )}{\alpha _s(\mu )}\right){}^{\frac{2 C_F}{\beta _0}}-1\right)\right) \left(\frac{\alpha _s(\Lambda )}{\alpha _s(\mu
   )}\right){}^{\frac{C_F}{\beta _0}}
\end{pmatrix}\ .
\end{equation}
\end{widetext}
Note that his result is compatible with the Gordon Identities \eqref{GIV} and \eqref{GIPV}.
The left hand side of these identities only have the anomalous dimension of the masses
\begin{equation}
\mu \frac{\partial}{\partial \mu} m (\mu) = m (\mu)  \gamma_m (\alpha_s (\mu))
\end{equation}
with $\gamma_m = 6 \alpha_s C_F/(4 \pi)$
and hence we may check
\begin{eqnarray}
\gamma_m   (m_b &+& m_c) \bar c \,\gamma_\mu b = \gamma \cdot \begin{pmatrix}
    \mathcal{O}^{\text{S}} (\mu) \\
    0 \\
    \mathcal{O}^{\text{T}}(\mu) \\
    0
 \end{pmatrix} \nonumber\\
&=&  \frac{6 \alpha_s}{4\pi} C_F \left[ \bar c \, i \tensor D_\mu b + i \partial^\nu \left( \bar c (- i \sigma_{\mu \nu}) b \right) \right]
\end{eqnarray}
and likewise for the axial vector.

The $\mu$ dependence of the operators
has to be cancelled by the corresponding dependence of the matrix elements of the operators.
Here we focus on the matrix elements at the specific kinematic point $v = v'$ or $w =1$.
At this point,
all possible Dirac structures can be expressed in terms of the four matrices \cite{Mannel:1994kv}
\begin{equation}
\openone = \frac{1}{2} (1 + \@ifnextchar[{\fmsl@sh}{\fmsl@sh[0mu]}{v}) ,  \ s_\mu = \frac{1}{4} (1 + \@ifnextchar[{\fmsl@sh}{\fmsl@sh[0mu]}{v}) \gamma_\mu \gamma_5 (1 + \@ifnextchar[{\fmsl@sh}{\fmsl@sh[0mu]}{v})
\end{equation}
with $v\cdot s = 0$.
To this end, we can write
\begin{eqnarray}
\bigl\langle c(p_c &=& m_c v) \bigl| \bar c  \gamma_\mu b \bigr| b (p_b = m_b v) \bigr\rangle \nonumber\\
&&  = \eta_{\text{V}} \, v_\mu  \bar{u}_c (v)  u_b (v)\ ,  \\
\bigl\langle c(p_c &=& m_c v) \bigl| \bar c  \gamma_\mu \gamma_5 b \bigr| b (p_b = m_b v) \bigr\rangle\nonumber\\
 &&= \eta_{\text{A}}  \, \bar{u}_c (v) s_\mu u_b (v)  \ ,
\end{eqnarray}
while out of the dimension-4 operators we have a priori four additional matrix elements.
The matrix element of $\mathcal{O}^{\text{T}}$ vanishes at non-recoil,
while the one of $\mathcal{O}^{\text{PT}}$ does not.
We choose to use $\mathcal{O}^{\text{S}}$ and $\mathcal{O}^{\text{PS}}$,
which have nonvanishing matrix elements at $w = 1$,
\begin{eqnarray}
&&\left\langle c(p_c = m_c v) \left| \bar c \, i \tensor D_\mu b \right| b (p_b = m_b v) \right\rangle \biggr|_\mu   \nonumber\\
&=& v_\mu  \bar{u}_c (v)  u_b (v) \, (m_b + m_c) \eta_\text{S} (\mu)  \ ,  \\
&&\left\langle c(p_c = m_c v) \left| \bar c \, i \tensor D_\mu \gamma^5 b \right| b(p_b = m_b v) \right\rangle\biggr|_\mu \nonumber\\
&=& \bar{u}_c (v) s_\mu u_b(v) \, (m_b - m_c) \eta_\text{PS}(\mu) \, .
\end{eqnarray}
The matrix element of the nonvanishing pseudotensor operator $\mathcal{O}^{\text{PT}}$
can according to the Gordon identity \eqref{GIPV} be expressed through
the axial vector times $(m_b-m_c)$ masses in the corresponding $\overline{\text{MS}}$ scheme.
Thus it can be expressed by
\begin{eqnarray}
  &&\left\langle c(p_c = m_c v) \left| i \partial^\nu \left( \bar c (- i \sigma_{\mu \nu}) \gamma^5 b \right)\right| b (p_b = m_b v) \right\rangle \biggr|_\mu\nonumber\\
  &=& \bar{u}_c (v) s_\mu u_b(v) \, (m_b - m_c) \eta_\text{PT}(\mu) \\
  &=&  - (\overline{m}_b-\overline{m}_c) (\eta_{\text{A}} +\eta_\text{PS}) \, \bar{u}_c (v) s_\mu u_b (v) \,.\label{ptme}
\end{eqnarray}
\begin{widetext}
The matrix elements of the vector and axial-vector currents
are known at two-loops in the full phase space,
while the matrix elements of the dimension-4 currents
are calculated here only at the non-recoil point
$w = 1$.  The result is
\begin{eqnarray}
\eta_{\text{A}} &=& 1+ \frac{\alpha_s}{4\pi}C_F \left[-8+3\frac{m_b+m_c}{m_b-m_c}\log\frac{m_b}{m_c} \right]\ ,  \\
\eta_{\text{V}} &=&   1+\frac{\alpha_s}{4\pi} C_F \left[-6+3\frac{m_b+m_c}{m_b-m_c}\log\frac{m_b}{m_c} \right]\ , \\
\eta_{\text{S}} (\mu)  &=& 1 +  \frac{\alpha_s }{4\pi}C_F
\Bigg[ - 3 \log\left(\frac{\mu^2}{m_b m_c} \right)
 + 6 \frac{m_b^2+m_c^2 }{m_b^2-m_c^2}\log \frac{m_b}{m_c}-10 \Bigg] \ , \\
\eta_{\text{PS}}(\mu) &=& \frac{\alpha_s}{4\pi}C_F \left[ -2 \log\left(\frac{\mu^2}{m_b m_c}\right) + 2 \frac{m_b + m_c}{m_b - m_c} \log\frac{m_b}{m_c} -4\right]\ , \\
 \eta_{\text{PT}}(\mu)&=& 1+ \frac{\alpha_s}{4\pi} C_F \left[ - \log\left(\frac{\mu^2}{m_b m_c}\right) + 4 \frac{m_b+m_c}{m_b-m_c}\log\frac{m_b}{m_c}-8 \right]\ .
\end{eqnarray}
\end{widetext}
Note that the difference in $|\eta_{\text{PS}}+\eta_{\text{A}}| \neq |\eta_{\text{PT}}|$
in \eqref{ptme} is due to the $\alpha_s$ pieces of the $\overline{\text{MS}}$ masses.
It is easy to check that the $\mu$ dependence cancels between the matrix elements
and the Wilson coefficients in the order $\alpha_s$.
However, the renormalization group flow resumms potentially large logarithms of the form
$(\alpha_s / \pi)^n  \ln^n (\Lambda^2 / \mu^2)$
and hence there will be a residual $\mu$ dependence.
Looking at the structure of the matrix element coefficients $\eta_X(\mu)$,
$X\in\{\text{A, V, S, PS, PT}\}$, of the dimension four operators,
a natural choice is $\mu_0^2 = m_b m_c$ and hence we insert this scale for our numerical study.
This includes that the couplings generated at some high scale $\Lambda$
have to be evolved down to this small scale.

Numerically we obtain using the values $m_b = 4.2$~GeV,
$m_c = 1.3$~ GeV (and thus $\mu_0 \approx 2.34$~GeV), and $C_F = 4/3$ as well as $\alpha_s(\mu_0) \approx 0.281 $ \cite{Bethke:2009jm} for $N_f = 3$
\begin{subequations}
\begin{alignat}{2}
    \eta_{\text{V}}  &\approx 1+0.0713\, \alpha_s &\, &\approx 1.02 \ , \label{etaV}\\
    \eta_{\text{A}} &\approx 1-0.1409\, \alpha_s &\, &\approx 0.96 \ , \label{etaA}\\
    \eta_\text{S} (\mu_0) &\approx 1 - 0.1562\, \alpha_s &\, &\approx 0.96 \ ,\label{etaS}\\
    \eta_\text{PS} (\mu_0) &\approx 0+0.0476\, \alpha_s &\, &\approx 0.01 \ , \label{etaPS}\\
    \eta_\text{PT} (\mu_0) &\approx 1 + 0.0951\, \alpha_s &\, &\approx 1.03 \ . \label{etaPT}
\end{alignat}
\end{subequations}
Comparing the results for the vector and axial vector coefficients
Eqs. \eqref{etaV} and \eqref{etaA}, respectively,
with the  form factors  $\mathcal G(1)$ and $\mathcal F(1)$
from lattice or non-lattice calculations,
as discussed in more detail in the next section,
the results from our calculation can be
assumed as a first approximation for these form factors.
Following the same line the results
Eqs. \eqref{etaS}--\eqref{etaPT} can
be considered as first approximations for
scalar, pseudoscalar and pseudotensor form factors values at the non-recoil point.

\section{Constraints on right-handed admixtures}
In this section we shall discuss the bounds on possible admixtures to the standard model current.
In contrast to section \ref{New Physics contributions in the IW limit}
we will perform the analysis not in the Isgur-Wise Limit, and hence we have to deal with the
form factor values at zero recoil. From lattice simulations as well as from QCD sum rules
we know the normalizations for the vector- and the axial-vector form factors, and hence we can
- off the Isgur Wise limit - only
study a possible right-handed admixture to the weak hadronic currents,
which shows up to be the best candidate for sizable contributions \cite{Dassinger:2007pj}.
To be able to extract the strength of the right-handed admixture in the weak currents of exclusive decays,
we start from the  exclusive differential decay rates \eqref{diffdecrateI} and \eqref{diffdecrateII}
of the $D$ and $D^\ast$ mesons, respectively.
All information about the right-handed admixture is contained in the form factors $\mathcal F(w)$ and $\mathcal G(w)$.
Like for the Isgur-Wise function $\xi(w)$ we may extrapolate the form factors to the point $w=1$
and perform an expansion around this point to express the value for any other $w$
by a small correction of order $\Lambda_{\text{QCD}}/m_Q$,
where we use $m_Q$ generically for $m_b$ or $m_c$ respectively.
The form factors $\mathcal F(w)$ and $\mathcal G(w)$ can be expanded as
\begin{eqnarray}
    \mathcal F(w) &=& \mathcal F(1) \nonumber \\
    &\times& \left[ 1 - \rho_\ast^2 (w-1) + c_\ast(w-1)^2 + \dots \right] \label{Fexpansion} ,\\
    \mathcal G(w) &=& \mathcal G(1) \nonumber \\
  &\times& \left[ 1 - \rho^2(w-1) + c(w-1)^2 + \dots\right]  ,  \label{Gexpansion}
\end{eqnarray}
where the slopes
\begin{eqnarray}
    \rho_\ast^2 &=&  -\frac{1}{\mathcal F(1)} \frac{\partial\mathcal F(w)}{\partial w} \bigg|_{w=1} \ , \\
    \rho^2 &=&  -\frac{1}{\mathcal G(1)} \frac{\partial\mathcal G(w)}{\partial w} \bigg|_{w=1} ,
\end{eqnarray}
describing the linear corrections as well as higher order corrections $c$ and $c_\ast$
introducing a possible correction induced by a non-zero curvature.
Note that the slopes differ from $\rho_{\text{\text{IW}}}$ in the Isgur-Wise limit
introduced in section \ref{New Physics contributions in the IW limit},
since they include the contributions from the coefficient functions $\mathcal A(w)$
for the $B \to D^\ast$ decay and $\mathcal B^T(w)$ and $\mathcal B^L(w)$
for the $B \to D$ decay respectively.
Thus, if we evaluate the expansions \eqref{Fexpansion} and \eqref{Gexpansion} up to the first order of magnitude,
the whole information on the right-handed admixtures is contained in the slopes.
Additionally we find the $\rho$ and $\rho_\ast$ to differ from each other,
such that we have the opportunity to calculate a constraint on the right-handed admixture
by comparing the slopes of the two decay modes.
For the $\bar B \to D^\ast \ell \bar \nu$ decay this implies
\begin{equation}
    \rho_\ast^2 = \rho_{\text{SM}}^2 + \frac{R_1^2(1)}{6} \left[1-\left(\frac{c_+}{c_-}\right)^2\right]\ ,
\end{equation}
where $\rho_{\text{SM}}$ denotes the terms known from the standard model.
In contrast to that the value for $\rho$ concerning $B \to D\,\ell\,\bar\nu$ is left untouched,
since the axial vector component vanishes by parity reasons,
as implied by \eqref{BtoD}--\eqref{BtoDst}.
Therefore we may set $\rho \equiv \rho_{\text{SM}}$ and obtain
\begin{equation}
    \left(\frac{c_+}{c_-}\right)^2 = 1 - 6\, \frac{\rho_\ast^2-\rho^2}{R_1^2(1)}
\end{equation}
as a measure for the strength of the right-handed admixture.
Using the averaged results
\begin{eqnarray}
        R_1 = 1.41\pm 0.049 \ &,& \ R_2 = 0.844\pm0.027 \ ,\\
       \rho^2 = 1.18\pm 0.06 \ &,& \ \rho_\ast^2 = 1.24\pm0.04 \ ,
\end{eqnarray}
from the Heavy Flavor Averaging Group \cite{TheHeavyFlavorAveragingGroup:2010qj}
we obtain
\begin{equation}
    \frac{c_+}{c_-} = 0.90\pm 0.09\ ,
\end{equation}
as an estimate for the strength of the right-handed admixture.
Note that we have not included any possible correlations between the errors,
but rather made a naive estimation of the error bars.

Another constraint is given by the fact,
that for the non recoil point $w=1$ the $B \to D\ell\bar\nu$
transition is completely dominated by the vector current,
while in contrast to that the $B\to D^\ast\ell\bar\nu$
transition is proportional to the axial vector current.
Thus, if we include a right-handed admixture,
it is contained in the current experimental results \cite{TheHeavyFlavorAveragingGroup:2010qj},
such that we obtain
\begin{align}
   c_+ |V_{cb}|\, \mathcal G(1) &= (42.3\pm1.5)\times 10^{-3}\ ,\\
   c_- |V_{cb}|\, \mathcal F(1) &= (36.04\pm0.52)\times 10^{-3}\ .
\end{align}
Using lattice data \cite{Bernard:2008dn,Aubin:2004qz,Mackenzie},
\begin{equation}
    \mathcal G(1) = 1.074 \pm 0.024 , \ \mathcal F(1) = 0.908 \pm 0.017 ,
\end{equation}
we find
\begin{equation}
    \frac{c_+}{c_-} = 0.99 \pm 0.05\ ,
\end{equation}
while a calculation using the non-lattice values \cite{Uraltsev:2003ye,Gambino:2010bp}
\begin{equation}
    \mathcal G(1) = 1.02 \pm 0.04 , \ \mathcal F(1) = 0.86 \pm 0.02 ,
\end{equation}
gives us
\begin{equation}
    \frac{c_+}{c_-} = 0.99 \pm 0.06 ,
\end{equation}
which is in both cases compatible with the standard model value $c_+/c_- \equiv 1$.
Again we have calculated the errors using the assumption,
that no sizable correlations between the experimental measurements and the theoretical values occur.

\section{Summary and Conclusions}
While the left-handedness of the weak interaction is in good agreement
with the data taken from purely left-handed leptonic processes \cite{Michel:1949qe,Bouchiat:1957zz}
the situation for the hadronic interactions is still unclear.
On general grounds one would not expect new physics to show up in a charged current interaction,
but this may as well be a false prejudice.
In this paper we have computed the effect of non-standard couplings for the exclusive semileptonic $\bar B\to D^{(\ast)}\ell \bar\nu$ transition,
which have been introduced in the same way we used for the inclusive semileptonic
$\bar B \to X_c \ell\bar\nu$ decays in \cite{Dassinger:2008as}
using operators of higher dimensions using the most general possible parametrization.
The corresponding dimension six operators then allow new physics effects in charged currents in the hadronic current,
while the leptonic current is left untouched.

Applying the extended hadronic current including the standard left-handed coupling
as well as the additional right-handed coupling and right- and left-handed vector and scalar couplings
we have calculated the differential decay rates $d\Gamma/dw$ in the Isgur-Wise limit.
Therefore we have introduced new hadronic form factors corresponding to the Dirac-structure of the currents.
The calculation of the differential decay rate also provides us with information about the slopes $\rho$ and $\rho^\ast$
describing the deviation of the differential rate from the zero recoil point at $w = 1$.

The main corrections are the perturbative QCD effects, which are sizable and have to be taken into account.
Due to the vanishing anomalous dimension of the left and the right handed currents the QCD effects are finite for these currents;
however, additional work is required to compute the virtual corrections to the scalar and tensor currents,
which renormalize under QCD.
Within this paper we have computed the vertex corrections for each occurring current
up to one-loop order including new quark-quark-gluon-boson vertices stemming from the (pseudo)scalar components. Yet the unknown form factors normalizations are still missing and have to be obtained by other methods in order to include these additional structurces, which however are believed to be surpressed.

Comparing the calculated slopes using experimental as well as lattice and non-lattice data
we have been able to calculate bounds on right handed admixtures.
The comparison of the slopes for $B \to D$ and $B \to D^\ast$ decays
gives us the result $c_+/c_- = 0.90 \pm 0.09$ using only experimental data.
Furthermore we have used the opportunity to calculate by comparing the experimental results with
lattice and non-lattice data.
Here we obtain $c_+/c_- = 0.99 \pm 0.05$ for lattice
and $c_+/c_- = 0.99 \pm 0.06$ for non-lattice data.
Thus all results are in good agreement with each other and with the purely left-handed standard model current, where $c_+ = c_- = 1$.

\begin{acknowledgments}
We are grateful to Benjamin Dassinger for useful comments. S.T. acknowlegdes helpful discussions with Robert Feger.
This work is supported by the German research foundation DFG under contract MA118/10-1
and by the German Ministry of Research (BMBF),
contracts 05H09PSF.
\end{acknowledgments}


text:
\section{Introduction}
Let $\mathcal{H}$ denote the class of analytic functions in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$, and  $\mathcal{LU}$ denote the subclass of $\mathcal{H}$ consisting of all locally univalent functions, namely $\mathcal{LU}=\{f\in\mathcal{H}: f'(z)\ne 0, z\in\mathbb{D}\}$. The Schwarzian derivative for a locally univalent function $f\in\mathcal{LU}$ is defined by
$$S_f(z)=\left[\frac{f''(z)}{f'(z)}\right]^{'}-\frac{1}{2}\left[\frac{f''(z)}{f'(z)}\right]^2$$
and the Schwarzian norm (the hyperbolic sup-norm) of $f\in\mathcal{LU}$ is defined by
$$||S_f||=\sup\limits_{z\in\mathbb{D}}(1-|z|^2)^2|S_f(z)|.$$
For a locally univalent function $f\in\mathcal{LU}$, Nehari \cite{Nehari-1949} proved that if $||S_f||\leq 2$ then the function $f$ is univalent in $\mathbb{D}$. On the other hand, for a univalent function $f$, $||S_f||\leq 6$ (see \cite{Nehari-1949}). Both of the constants $2$ and $6$ are best possible.\\

The Schwarzian norm has a significant meaning in the theory of quasiconformal mappings and Teichm\"{u}ller space (see \cite{Lehto-1987}). A mapping $f:\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}$ of the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ is said to be $k$-quasiconformal ($0\le k<1$) mapping if $f$ is a sense preserving homeomorphism of $\widehat{\mathbb{C}}$ and has locally integrable partial derivatives on $\mathbb{C}\setminus\{f^{-1}(\infty)\}$ with $|f_{\bar{z}}|\le k|f_z|$ a.e.. On the other hand, the theory of Teichm\"{u}ler space $\mathcal{T}$ can be identified with the set of Schwarzian derivatives of analytic and univalent functions on $\mathbb{D}$ with quasiconformal extensions to $\widehat{\mathbb{C}}$. It is known that $\mathcal{T}$ is a bounded domain in the Banach space of analytic functions in $\mathbb{D}$ with finite hyperbolic sup-norm (see \cite{Lehto-1987}). Key results connecting the Schwarzian derivative and quasiconformal mappings are given in the following theorem.

\begin{customthm}{A}\label{Thm-q-0001}\cite{Ahlfors-Weill-1962, Kuhnau-1971}
If $f$ extends to a $k$-quasiconformal ($0\le k<1$) mapping of the Riemann sphere $\widehat{\mathbb{C}}$ then $||S_f||\le 6k$. Conversely, if $||S_f||\le 2k$ then $f$ extends to a $k$-quasiconformal mapping of the Riemann sphere $\widehat{\mathbb{C}}$.
\end{customthm}

Let $\mathcal{A}$ denote the class of functions $f$ in $\mathcal{H}$ normalized by $f(0)=0$, $f'(0)=1$. Therefore, every function $f$ in $\mathcal{A}$ has the Taylor series expansion of the form
\begin{equation}\label{q-00001}
f(z)=z+\sum\limits_{n=2}^{\infty}a_nz^n.
\end{equation}
Let $\mathcal{S}$ be the set of all functions $f$ in $\mathcal{A}$ that are univalent in $\mathbb{D}$. A function $f\in\mathcal{A}$ is called starlike (respectively, convex) if the image $f(\mathbb{D})$ is a starlike domain with respect to the origin (respectively, convex). The classes of all starlike and convex functions that are univalent are denoted by $\mathcal{S^*}$ and $\mathcal{C}$, respectively. It is well known that a function $f$ in $\mathcal{A}$ is starlike (respectively, convex) if and only if ${\rm Re\,} [zf'(z)/f(z)] > 0$ (respectively, ${\rm Re\,} [1 + zf''(z)/f'(z)] > 0$) for $z\in\mathbb{D}$.
A function $f$ in $\mathcal{A}$ is said to be starlike of order $\alpha$, $0\le \alpha< 1$ if ${\rm Re\,} [zf'(z)/f(z)] > \alpha$ for $z\in\mathbb{D}$ and is said to be convex of order $\alpha$, $0\le \alpha< 1$ if ${\rm Re\,} [1 + zf''(z)/f'(z)] > \alpha$ for $z\in\mathbb{D}$. The class of all starlike and convex functions of order $\alpha$ is denoted by $\mathcal{S}^*(\alpha)$ and $\mathcal{C}(\alpha)$, respectively. Clearly, a function $f$ in $\mathcal{A}$ belongs to $\mathcal{C}(\alpha)$ if and only if $zf'\in\mathcal{S}^*(\alpha)$. For further information on these classes, we refer to \cite{Duren-1983,Goodman-book-1983}.\\

In this article, we are concerned with two different classes of functions $\mathcal{G}(\beta)$, $\beta>0$ and $\mathcal{F}(\alpha)$, $-1/2\le \alpha\le 0$ defined by
$$\mathcal{G}(\beta)=\Bigg\{f\in \mathcal{A}:{\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)<1+\frac{\beta}{2}\quad\text{for}~z\in\mathbb{D}\Bigg\}$$
and
$$\mathcal{F}(\alpha)=\Bigg\{f\in \mathcal{A}:{\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)>\alpha\quad\text{for}~z\in\mathbb{D}\Bigg\}.$$
In 1941, Ozaki \cite{Ozaki-1941} introduced the class $\mathcal{G}:=\mathcal{G}(1)$ and proved that functions in $\mathcal{G}$ are univalent in $\mathbb{D}$. Later on, Umezawa \cite{Umezawa-1952} studied the class $\mathcal{G}$ and showed that functions in $\mathcal{G}$ are convex in one direction, i.e. every $f\in\mathcal{G}$ maps $|z|=\rho<r$ for every $\rho$ near $r$ into a contour which may be cut by every straight line parallel to this direction in not more than two points. Moreover, functions in $\mathcal{G}$ are starlike in $\mathbb{D}$ (see \cite{Jovanovic-Obradovic-1995}, \cite{Ponnusamy-Rajasekaran-1995}).
Thus, the class $\mathcal{G}(\beta)$ is included in $\mathcal{S}^*$ whenever $\beta\in (0,1]$. One can easily show that functions in $\mathcal{G}(\beta)$ are not univalent in $\mathbb{D}$ for $\beta>1$. For $0<\beta \le 2/3$, the class $\mathcal{G}(\beta)$ was studied by Uralegaddi {\it et al.} \cite{Uralegaddi-Ganigi-Sarangi-1994}. Later, the full class was studied in \cite{Ali-Vasudevarao-2016, Owa-Nishiwaki-2002, Owa-Srivastava-2002, Ponnusamy-Vasudevarao-2007}. On the other hand, for $\alpha=0$, $\mathcal{F}(0)=:\mathcal{C}$ is the class of convex functions. For $\alpha=-1/2$, functions in the class $\mathcal{F}(-1/2)$ are not necessarily starlike but are convex in some direction and so are close-to-convex. Here, we recall that a function $f\in\mathcal{A}$ is called close-to-convex if $f(\mathbb{D})$ is close-to-convex domain, i.e. the complement of $f(\mathbb{D})$ in $\mathbb{C}$ is the union of closed half lines with pairwise disjoint interiors. Pfaltzgraff et al. \cite{Pfaltzgraff-Reade-Umezawa-1976} proved that $\mathcal{F}(\alpha)$ contains non-starlike functions for all $-1/2 \le \alpha < 0$.\\

Although the fundamental work on Schwarzian  derivative in connection with geometric functions theory have been done in \cite{Ahlfors-Weill-1962, Kuhnau-1971, Nehari-1949}); not much research has been done on Schwarzian  derivative for various subclasses of univalent functions. In connection with Teichm\"{u}ller spaces, it is an interesting problem to estimate the norm of the Schwarzian derivatives for typical subclasses of univalent functions. For the class of convex functions $\mathcal{C}$, the Schwarzian norm satisfies $||S_f||\le 2$ and the estimate is sharp. This result was proved repeatedly by many researchers (see \cite{Lehto-1977, Nehari-1976, Robertson-1969}). In 1996, Suita \cite{Suita-1996} studied the class $\mathcal{C}(\alpha)$, $0\le \alpha\le 1$ and using the integral representation of functions in $\mathcal{C}(\alpha)$ proved that the Schwarzian norm satisfies the following sharp inequality
$$||S_f||\le
\begin{cases}
2 & \text{ if }~ 0\le \alpha\le 1/2,\\
8\alpha(1-\alpha) & \text{ if }~ 1/2< \alpha\le 1.
\end{cases}
$$

A function $f$ in $\mathcal{A}$ is said to be strongly starlike (respectively, strongly convex) of order $\alpha$, $0<\alpha\le 1$ if $|\arg\{zf'(z)/f(z)\}|<\pi\alpha/2$ (respectively, $|\arg\{1+zf''(z)/f'(z)\}|<\pi\alpha/2$) for $z\in\mathbb{D}$. The class of all strongly starlike and strongly convex functions of order $\alpha$ is denoted by $\mathcal{S}^*_{\alpha}$ and $\mathcal{K}_{\alpha}$, respectively. In 1989, Mocanu \cite{Mocanu-1989} proved that $\mathcal{K}_{\gamma(\beta)}\subset \mathcal{S}^*_{\beta}$ for $0<\beta<1$ where
$$
\gamma(\beta):=\frac{2}{\pi} \arctan\left[\tan \frac{\pi\beta}{2} + \frac{\beta}{(1+\beta)^{(1+\beta)/2} (1-\beta)^{(1-\beta)/2}\cos (\pi\beta/2)} \right].
$$
In other words, $\mathcal{K}_{\alpha}\subset \mathcal{S}^*_{\gamma^{-1}(\alpha)}$ for $0<\alpha< 1$, where $\gamma^{-1}$ denotes the
inverse function of $\gamma:[0, 1]\to [0, 1]$. Note that $\gamma(\beta)$ increases from $0$ to $1$ when $\beta$ varies from $0$ to $1$.
For $0<\alpha<1$, Fait {\it et al.} \cite{Fait-Krzyz-Zygmunt-1976} proved that every function $f$ in $\mathcal{S}^*_{\alpha}$ can be extended to a $\sin(\pi\alpha/2)$-quasiconformal mapping of $\widehat{\mathbb{C}}$.
Therefore, by Theorem \ref{Thm-q-0001}, it easily follows that $||S_f||\le 6\sin(\pi\alpha/2)$ which was pointed out by Chiang \cite{Chiang-1991}. By Theorem \ref{Thm-q-0001}, this also implies that a function $f\in\mathcal{K}_{\alpha}$ extends to a $\sin(\pi\gamma^{-1}(\alpha)/2)$-quasiconformal mapping of $\widehat{\mathbb{C}}$ and satisfies $||S_f||\le 6\sin(\pi\gamma^{-1}(\alpha)/2)$. Later, Kanas and Sugawa \cite{Kanas-Sugawa-2011} studied the Schwarzian norm for the class $\mathcal{K}_{\alpha}$ of strongly convex functions by means a different method and proved the sharp inequality $||S_f||\leq 2\alpha$ for $f\in\mathcal{K}_{\alpha}$ which by Theorem \ref{Thm-q-0001}, also shows that every functions in $\mathcal{K}_{\alpha}$ can be extended to an $\alpha$-quasiconformal mapping of $\widehat{\mathbb{C}}$. Here we note that $\sin(\pi\gamma^{-1}(\alpha)/2)<\alpha$ when
$0 <\alpha< 0.3354$ and so Kanas and Sugawa \cite{Kanas-Sugawa-2011} obtained a better bound than in \cite{Fait-Krzyz-Zygmunt-1976} when $\alpha> 0.3355$.\\

A function $f\in \mathcal{A}$ is said to be uniformly convex function if every circular arc (positively oriented) of the form $\{z\in\mathbb{D}: |z -\xi|=r\}$, $\xi\in\mathbb{D}$, $0<r<|\xi|+1$ is mapped by $f$ univalently onto a convex arc. The class of all uniformly convex functions is denoted by $\mathcal{UCV}$. In particular, $\mathcal{UCV}\subset\mathcal{K}$.
It is well known that (see \cite{Goodman-book-1983}) a function $f\in \mathcal{A}$ is uniformly convex if and only if
$${\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)> \left|\frac{zf''(z)}{f'(z)}\right|~~\text{for}~ z\in\mathbb{D}.$$
Kanas and Sugawa \cite{Kanas-Sugawa-2011} proved that the Schwarzian norm satisfies $||S_f||\le 8/\pi^2$ for all $f\in\mathcal{UCV}$ and the estimate is sharp. In 2012, Bhowmik and Wirths \cite{Bhowmik-Wirths-2012} studied the class of concave functions $\mathcal{C}o(\alpha)$ for $1\le \alpha\le 2$ and obtained the sharp estimate $||S_f||\le 2(\alpha^2-1)$ for $f\in\mathcal{C}o(\alpha)$. They also proved that $f$ extends to an $(\alpha^2-1)$-quasiconformal mapping of $\widehat{\mathbb{C}}$ when $1\le \alpha<\sqrt{2}$.\\

In this article, our main aim is to find estimates for the modulus of the Schwarzian derivative for functions in $\mathcal{G}(\beta)$, $\beta>0$ and $\mathcal{F}(\alpha)$, $-1/2\le \alpha\le 0$. These results will yield a sharp estimate of the Schwarzian norm for functions in these classes, which will help us to comment on quasiconformal extension of these functions. This also lead us to find a pair of two-point
distortion conditions of such mappings.

\section{Main Results}

Let $\mathcal{B}$ be the class of analytic functions $\omega:\mathbb{D}\rightarrow\mathbb{D}$ and $\mathcal{B}_0$ be the class of Schwarz functions $\omega\in\mathcal{B}$ with $\omega(0)=0$. The Schwarz's lemma states that a function $\omega\in\mathcal{B}_0$ satisfies $|\omega(z)|\le |z|$ and $|\omega'(0)|\le 1$. The equality occurs in any one of the inequalities if and only if $\omega(z)=e^{i\alpha}z$, $\alpha\in\mathbb{R}$. A natural extension of Schwarz lemma, known as Schwarz-Pick lemma, gives the estimate $|\omega'(z)|\le (1-|\omega(z)|^2)/(1-|z|^2)$, $z\in\mathbb{D}$ when $\omega\in\mathcal{B}$. In 1931, Dieudonn\'{e} \cite{Dieudonne-1931} first obtained the exact region of variability of $\omega'(z_0)$ for a fixed $z_0\in\mathbb{D}$ over the class $\mathcal{B}_0$.

\begin{lem}[Dieudonn\'{e}'s lemma]\cite{Dieudonne-1931, Duren-1983}
Let $\omega\in\mathcal{B}_0$ and $z_0\ne 0$ be a fixed point in $\mathbb{D}$. The region of variability of $\omega'(z_0)$ is given by
\begin{equation}\label{q-00005}
\left|\omega'(z_0)-\frac{\omega(z_0)}{z_0}\right|\le \frac{|z_0|^2-|\omega(z_0)|^2}{|z_0|(1-|z_0|^2)}.
\end{equation}
Moreover, the equality occurs in \eqref{q-00005} if and only if $\omega$ is a Blaschke product of degree $2$ fixing $0$.
\end{lem}

The Dieudonn\'{e}'s lemma is an improvement of the Schwarz's lemma as well as Schwarz-Pick lemma. Here, we remark that a Blaschke product of degree $n\in\mathbb{N}$ is of the form
$$B(z)=e^{i\theta}\prod_{j=1}^{n}\frac{z-z_j}{1-\bar{z_j}z},\quad z,z_j\in\mathbb{D}, ~\theta\in\mathbb{R}.$$
The Dieudonn\'{e}'s lemma will play a crucial role to prove our main results.\\

Before we state our main result, let us recall another important and useful tool known as the differential subordination technique. Many problems in geometric function theory can be solved in a simple and sharp manner with the help of differential subordination.
A function $f\in\mathcal{H}$ is said to be subordinate to another function $g\in\mathcal{H}$ if there exists an analytic function $\omega\in\mathcal{B}_0$ such that $f(z)=g(\omega(z))$ and it is denoted by $f\prec g$. Moreover, when $g$ is univalent, $f\prec g$ if and only if $f(0)=g(0)$ and $f(\mathbb{D})\subset g(\mathbb{D})$. In terms of subordination, the classes $\mathcal{G}(\beta)$ and $\mathcal{F}(\alpha)$ can be defined as:
\begin{equation}\label{q-00010}
f\in\mathcal{G}(\beta)\iff 1+\frac{zf''(z)}{f'(z)}\prec \frac{1-(1+\beta)z}{1-z}
\end{equation}
and
\begin{equation}\label{q-00015}
f\in\mathcal{F}(\alpha)\iff 1+\frac{zf''(z)}{f'(z)}\prec \frac{1+(1-2\alpha)z}{1-z}.
\end{equation}

\begin{thm}\label{Thm-q-0005}
For $\beta>0$, let $f\in\mathcal{G}(\beta)$ be of the form \eqref{q-00001}. Then the Schwarzian derivative and Schwarzian norm satisfy the inequalities
$$|S_f(z)|\le  \frac{\beta(2+\beta)}{2(1-|z|)^2} \quad\text{and}\quad ||S_f||\le 2\beta(2+\beta).$$
Moreover, the equality occurs in both the inequalities for the function $f_0(z)$ defined by
 $$f_0(z)=\frac{1-(1-z)^{1+\beta}}{1+\beta}.$$
\end{thm}

\begin{proof}
For $\beta>0$, let $f\in\mathcal{G}(\beta)$ be of the form \eqref{q-00001}. Then from \eqref{q-00010}, we have
$$1+\frac{zf''(z)}{f'(z)}\prec \frac{1-(1+\beta)z}{1-z}$$
and so there exists an analytic function $\omega:\mathbb{D}\rightarrow\mathbb{D}$ with $\omega(0)=0$ such that
$$1+\frac{zf''(z)}{f'(z)}= \frac{1-(1+\beta)\omega(z)}{1-\omega(z)}.$$
A simple computation gives
$$\frac{f''(z)}{f'(z)}=-\frac{\beta\omega(z)}{z(1-\omega(z))}$$
and therefore,
\begin{align}\label{q-000001}
S_f(z)&=\left[\frac{f''(z)}{f'(z)}\right]^{'}-\frac{1}{2}\left[\frac{f''(z)}{f'(z)}\right]^2\\
&=-\beta\left[\frac{z\omega'(z)-\omega(z)}{z^2(1-\omega(z))^2}+\frac{(2+\beta)\omega^2(z)}{2z^2(1-\omega(z))^2}\right].\nonumber
\end{align}
Let us consider the transformation $\displaystyle\zeta(z)=\omega'(z)-\frac{\omega(z)}{z}$. By \eqref{q-00005}, the function $\zeta$ varies over the closed disk
$$|\zeta|\le \frac{|z|^2-|\omega|^2}{|z|(1-|z|^2)}$$
for fixed $|z|<1$. Using the transformation of $\zeta(z)$ in \eqref{q-000001}, we obtain
$$S_f(z)=-\beta\left[\frac{\zeta(z)}{z(1-\omega(z))^2}+\frac{(2+\beta)\omega^2(z)}{2z^2(1-\omega(z))^2}\right]$$
and hence
\begin{align*}
|S_f(z)|&\le \beta\left[\frac{|\zeta(z)|}{|z||1-\omega(z)|^2}+\frac{(2+\beta)|\omega^2(z)|}{2|z|^2|1-\omega(z)|^2}\right]\\
&\le\beta\left[\frac{|z|^2-|\omega(z)|^2}{|z|^2(1-|z|^2)|1-\omega(z)|^2}+\frac{(2+\beta)|\omega(z)|^2}{2|z|^2|1-\omega(z)|^2}\right].
\end{align*}
For $0<s:=|\omega(z)|\le |z|<1$, we have
\begin{align}\label{q-000005}
|S_f(z)|&\le \beta\left[\frac{|z|^2-s^2}{|z|^2(1-|z|^2)(1-s)^2}+\frac{(2+\beta)s^2}{2|z|^2(1-s)^2}\right]\\
&=\beta~\frac{2|z|^2+s^2(\beta-\beta |z|^2-2|z|^2)}{2|z|^2(1-|z|^2)(1-s)^2}=\beta g(s),\nonumber
\end{align}

where $$g(s)=\frac{2|z|^2+s^2(\beta-\beta |z|^2-2|z|^2)}{2|z|^2(1-|z|^2)(1-s)^2}, \quad 0< s\le |z|.$$
Therefore,
\begin{equation*}
g'(s)=\frac{2|z|^2+s(\beta-\beta |z|^2-2|z|^2)}{|z|^2(1-|z|^2)(1-s)^3}.
\end{equation*}
We claim that $g'(s)>0$ for $0<s\le |z|$. If $\beta-\beta |z|^2-2|z|^2\ge 0$, then clearly $g'(s)>0$. If $\beta-\beta |z|^2-2|z|^2<0$, then using $0< s\le |z|$, we have
\begin{align*}
2|z|^2+s(\beta-\beta |z|^2-2|z|^2)&\ge 2|z|^2+|z|(\beta-\beta |z|^2-2|z|^2)\\
&= |z|(1-|z|)(\beta+2|z|+\beta|z|)>0.
\end{align*}
This lead us to conclude that $g'(s)>0$ for all $0<s\le |z|$. Thus, $g(s)$ attain its maximum at the point $s=|z|$ and so from \eqref{q-000005}, we have
$$|S_f(z)|\le \beta g(|z|)= \frac{\beta(2+\beta)}{2(1-|z|)^2}.$$
Therefore,
\begin{align*}
||S_f||&=\sup\limits_{z\in\mathbb{D}}(1-|z|^2)^2|S_f(z)|\le \frac{\beta(2+\beta)}{2}\sup\limits_{z\in\mathbb{D}}(1+|z|)^2\\
&=2\beta(2+\beta).
\end{align*}
To show that the estimates are sharp, let us consider the function $f_0(z)$ defined by
$$f_0(z)=\frac{1-(1-z)^{1+\beta}}{1+\beta}.$$
A simple computation gives
$$ S_{f_0}(z)=-\frac{\beta (2+\beta)}{2(1-z)^2}$$
and
$$
||S_{f_0}||=\sup\limits_{z\in\mathbb{D}}(1-|z|^2)^2|S_{f_0}(z)|=\beta (2+\beta)\sup\limits_{z\in\mathbb{D}}\frac{(1-|z|^2)^2}{2|1-z|^2}.
$$
On the positive real axis, we note that
$$ \beta (2+\beta)\sup\limits_{0<r<1}\frac{(1-r^2)^2}{2(1-r)^2}=2\beta (2+\beta).$$
Thus, $$||S_{f_0}||=2\beta (2+\beta).$$
\end{proof}

\begin{cor}
If $f\in\mathcal{G}:=\mathcal{G}(1)$ be of the form \eqref{q-00001}, then the Schwarzian norm satisfies the sharp inequality $$||S_f||\le 6.$$
\end{cor}

Theorem \ref{Thm-q-0005} and Theorem \ref{Thm-q-0001} immediately gives the following result for functions in $\mathcal{G}(\beta)$.
\begin{cor}
Let $0<\beta<\sqrt{2}-1$, and $f\in\mathcal{G}(\beta)$ be of the form \eqref{q-00001}. Then $f$ extends to an $\beta(2+\beta)$-quasiconformal mapping.
\end{cor}

\begin{thm}\label{Thm-q-0010}
For $-1/2\le \alpha\le 0$, let $f\in\mathcal{F}(\alpha)$ be of the form \eqref{q-00001}. Then the Schwarzian derivative $S_f(z)$ satisfies the following sharp inequality
$$|S_f(z)|\le \frac{2(1-\alpha)}{(1+\alpha)}\frac{(1+\alpha-\alpha|z|^2)}{(1-|z|^2)^2}\quad\text{for}~z\in\mathbb{D}.$$
Moreover, the equality occurs for some fixed $z_0\in \mathbb{D}$ with $-1<z_0<1$ for the function $f_{z_0}$ defined by
$$1+\frac{zf_{z_0}''(z)}{f_{z_0}'(z)}=\frac{1+(1-2\alpha)\omega_{z_0}(z)}{1-\omega_{z_0}(z)},$$
where
$$\omega_{z_0}(z)=-\frac{z(z-b)}{1-bz}\quad \text{with}\quad b=\frac{z_0(2+\alpha-\alpha z_0^2)}{1+\alpha+z_0^2-\alpha z_0^2}.$$
\end{thm}
\begin{proof}
For $-1/2\le \alpha\le 0$, let $f\in\mathcal{F}(\alpha)$.  Then from \eqref{q-00015}, we have
$$1+\frac{zf''(z)}{f'(z)}\prec \frac{1+(1-2\alpha)z}{1-z}.$$
Thus, there exists an analytic function $\omega:\mathbb{D}\rightarrow\mathbb{D}$ with $\omega(0)=0$ such that
$$1+\frac{zf''(z)}{f'(z)}= \frac{1+(1-2\alpha)\omega(z)}{1-\omega(z)}.$$
A simple computation gives
$$\frac{f''(z)}{f'(z)}=\frac{2(1-\alpha)\omega(z)}{z(1-\omega(z))}$$
and hence,
\begin{align}\label{q-000010}
S_f(z)&=\left[\frac{f''(z)}{f'(z)}\right]^{'}-\frac{1}{2}\left[\frac{f''(z)}{f'(z)}\right]^2\\
&=2(1-\alpha)\frac{\omega'(z)}{z(1-\omega(z))^2}-2(1-\alpha)\frac{\omega(z)-\alpha\omega^2(z)}{z^2(1-\omega(z))^2}.\nonumber
\end{align}
Let us consider the transformation $\displaystyle\zeta(z)=\omega'(z)-\frac{\omega(z)}{z}$. By \eqref{q-00005}, the function $\zeta$ varies over the closed disk
$$|\zeta|\le \frac{|z|^2-|\omega|^2}{|z|(1-|z|^2)}$$
for fixed $|z|<1$. Using the transformation $\zeta(z)$ in  \eqref{q-000010}, we obtain
$$S_f(z)=2(1-\alpha)\frac{\alpha\omega^2(z)}{z^2(1-\omega(z))^2}+2(1-\alpha)\frac{\zeta(z)}{z(1-\omega(z))^2}$$
and consequently,
\begin{align}\label{q-000015}
|S_f(z)|&\le 2(1-\alpha)\frac{-\alpha|\omega(z)|^2}{|z|^2|1-\omega(z)|^2}+2(1-\alpha)\frac{|\zeta(z)|}{|z||1-\omega(z)|^2}\\
&\le 2(1-\alpha)\frac{-\alpha|\omega(z)|^2}{|z|^2|1-\omega(z)|^2}+2(1-\alpha)\frac{|z|^2-|\omega^2(z)|}{|z|^2(1-|z|^2)|1-\omega(z)|^2}.\nonumber
\end{align}
For $0<s:=|\omega(z)|\le |z|<1$, we have
\begin{align}\label{q-000020}
|S_f(z)|&\le 2(1-\alpha)\frac{-\alpha s^2}{|z|^2(1-s)^2}+2(1-\alpha)\frac{|z|^2-s^2}{|z|^2(1-|z|^2)(1-s)^2}\\
&=2(1-\alpha)\frac{|z|^2-s^2(1+\alpha-\alpha |z|^2)}{|z|^2(1-|z|^2)(1-s)^2}=h(s),\nonumber
\end{align}
where $$h(s)=2(1-\alpha)\frac{|z|^2-s^2(1+\alpha-\alpha |z|^2)}{|z|^2(1-|z|^2)(1-s)^2},\quad 0<s\le |z|.$$
Therefore,
\begin{equation*}
h'(s)=\frac{4(1-\alpha)}{|z|^2(1-|z|^2)}\frac{|z|^2-s(1+\alpha-\alpha |z|^2)}{(1-s)^3}.
\end{equation*}
Thus, $h'(s)=0$ implies that $s=|z|^2/(1+\alpha-\alpha |z|^2)$, which lies in $(0,|z|)$ as
$$\frac{|z|^2}{1+\alpha-\alpha |z|^2}<|z| \iff (1-|z|)(1+\alpha(1+|z|))>0.$$
A simple calculation shows that $h'(0)>0$ and $h'(|z|)<0$. This lead us to conclude that the function $h$ attain its maximum at $|z|^2/(1+\alpha-\alpha |z|^2)$. Consequently, from \eqref{q-000020}, we have
$$|S_f(z)|\le h\left(\frac{|z|^2}{1+\alpha-\alpha |z|^2}\right)= \frac{2(1-\alpha)}{(1+\alpha)}\frac{(1+\alpha-\alpha|z|^2)}{(1-|z|^2)^2}.$$

To show that the estimate \eqref{q-000015} is sharp, let $z_0\in\mathbb{D}$ with $-1<z_0<1$ be fixed and consider the function $f_{z_0}$ defined by
\begin{equation}\label{q-000025}
1+\frac{zf''_{z_0}(z)}{f'_{z_0}(z)}=\frac{1+(1-2\alpha)\omega_{z_0}(z)}{1-\omega_{z_0}(z)},
\end{equation}
where
$$\omega_{z_0}(z)=-\frac{z(z-b)}{1-bz}\quad \text{with}\quad b=\frac{z_0(2+\alpha-\alpha z_0^2)}{1+\alpha+z_0^2-\alpha z_0^2}.$$
Considering $b$ as a function of $z_0$ in $(-1,1)$, we note that for each $-1/2\le \alpha\le 0$,
$$b'(z_0)=\frac{(1-|z_0|^2)\{\alpha^2(1-|z_0|^2)+(2\alpha+1)+(1+\alpha+\alpha|z_0|^2)\}}{(1+\alpha+z_0^2-\alpha z_0^2)^2}>0,~~ z_0\in (-1,1).$$
Also, $b(-1)=-1$ and $b(1)=1$. This lead us to conclude that $|b|<1$ and so $\omega_{z_0}$ is a Blaschke product of degree $2$ fixing $0$. This also shows that $f_{z_0}\in\mathcal{F}(\alpha)$ for every fixed $z_0\in (-1,1)$.\\

For such an $f_{z_0}$, we compute $S_{f_{z_0}}(z_0)$ using  \eqref{q-000010} as follows
\begin{align*}
S_{f_{z_0}}(z_0)
&=\left[\frac{f''_{z_0}(z)}{f'_{z_0}(z)}\right]^{'}-\frac{1}{2}\left[\frac{f''_{z_0}(z)}{f'_{z_0}(z)}\right]^2~\Bigg\vert_{z=z_0}\\
&=2(1-\alpha)\left[\frac{\omega'_{z_0}(z)}{z(1-\omega_{z_0}(z))^2} -\frac{\omega_{z_0}(z) -\alpha\omega_{z_0}^2(z)}{z^2(1-\omega_{z_0}(z))^2}\right] ~\Bigg\vert_{z=z_0}\\
&=-\frac{2(1-\alpha)}{(1+\alpha)}\frac{(1+\alpha-\alpha z_0^2)}{(1-z_0^2)^2}.
\end{align*}
Therefore,
\begin{equation}\label{q-000030}
|S_{f_{z_0}}(z_0)|=\frac{2(1-\alpha)}{(1+\alpha)}\frac{(1+\alpha-\alpha z_0^2)}{(1-z_0^2)^2}.
\end{equation}
This completes the proof.
\end{proof}

\begin{thm}\label{Thm-q-0015}
For $-1/2\le \alpha\le 0$, let $f\in\mathcal{F}(\alpha)$ be of the form \eqref{q-00001}. Then the Schwarzian norm $||S_f(z)||$ satisfies
$$||S_f||\le \frac{2(1-\alpha)}{1+\alpha}$$
and the estimate is best possible.
\end{thm}
\begin{proof}
For $f\in\mathcal{F}(\alpha)$, from Theorem \ref{Thm-q-0010}, we have
$$|S_f(z)|\le \frac{2(1-\alpha)}{(1+\alpha)}\frac{(1+\alpha-\alpha|z|^2)}{(1-|z|^2)^2}.$$
Therefore,
\begin{align*}
||S_f||&=\sup\limits_{z\in\mathbb{D}}(1-|z|^2)^2|S_f(z)|\\
&\le\frac{2(1-\alpha)}{(1+\alpha)} \sup\limits_{z\in\mathbb{D}}(1+\alpha-\alpha|z|^2)=\frac{2(1-\alpha)}{1+\alpha}.
\end{align*}
To show that the estimate is best possible, we consider the function $f_{z_0}(z)$, $-1<z_0<1$ defined by \eqref{q-000025}. Then from \eqref{q-000030}, we have
$$(1-|z_0|^2)^2|S_{f_{z_0}}(z_0)|=\frac{2(1-\alpha)(1+\alpha-\alpha z_0^2)}{(1+\alpha)}\rightarrow \frac{2(1-\alpha)}{1+\alpha}\quad\text{as}~z_0\rightarrow 1^-.$$
This shows that the estimate is best possible.\\[2mm]
\end{proof}

If we choose $\alpha=0$ in Theorem \ref{Thm-q-0015}, we get the Schwarzian norm  estimate for the class of convex functions, which was first proved by Nehari \cite{Nehari-1976}.
\begin{cor}
If $f\in\mathcal{F}(0)=:\mathcal{C}$ be of the form \eqref{q-00001}, then the Schwarzian norm satisfies the sharp inequality $$||S_f||\le 2.$$
\end{cor}

For $z_1,z_2\in\mathbb{D}$, let the hyperbolic metric $\lambda(z_1,z_2)$ be defined by
$$\lambda(z_1,z_2)=\frac{1}{2}\log\frac{1+\rho(z_1,z_2)}{1-\rho(z_1,z_2)},\quad\text{where}~ \rho(z_1,z_2)=\left|\frac{z_1-z_2}{1-\bar{z_1}z_2}\right|.$$
We also define the following quantity for an analytic and locally univalent function $f$ in $\mathbb{D}$:
$$\Delta_f(z_1,z_2):=\frac{|f(z_1)-f(z_2)|}{(1-|z_1|^2)(1-|z_2|^2)\sqrt{|f'(z_1)||f'(z_2)|}},\quad z_1,z_2\in \mathbb{D}.$$

In order to prove the two point distortion theorem for the classes $\mathcal{G}(\beta)$ and $\mathcal{F}(\alpha)$, we need to state the most classical result about two point distortion, which was proved by Chuaqui {\it et al.} \cite{Chuaqui-Duren-Ma-Mejia-Minda-Osgood-2011}.

\begin{customthm}{B}\label{Thm-q-0020}\cite[Theorem 1.]{Chuaqui-Duren-Ma-Mejia-Minda-Osgood-2011}
Let $f$ be analytic and locally univalent in $\mathbb{D}$ and suppose that the bound $||S_f||\le 2(1+\delta^2)$ holds for some $\delta>0$. Then
\begin{equation}\label{q-000035}
\Delta_f(z_1,z_2)\ge \frac{1}{\delta}\sin\left(\delta\lambda(z_1,z_2)\right)
\end{equation}
for all $z_1,z_2\in\mathbb{D}$ with $\lambda(z_1,z_2)\le \pi/\delta$, and
\begin{equation}\label{q-000040}
\Delta_f(z_1,z_2)\le \frac{1}{\sqrt{2+\delta^2}}\sinh\left(\sqrt{2+\delta^2}~\lambda(z_1,z_2)\right)
\end{equation}
for all $z_1,z_2\in\mathbb{D}$. Each of the inequalities \eqref{q-000035} and \eqref{q-000040} is sharp; for each pair of points $z_1$ and $z_2$ in the specified range, equality occurs for some function $f$ with $||S_f||\le 2(1+\delta^2)$. Equality holds in \eqref{q-000035} precisely for $f=T\circ F\circ \sigma$ and in \eqref{q-000040} for $f=T\circ G\circ \sigma$, where $F$ and $G$ are defined by
\begin{equation}\label{q-000045}
F(z)=\left(\frac{1+z}{1-z}\right)^{i\delta}\quad\text{and}\quad G(z)=\left(\frac{1+z}{1-z}\right)^{\sqrt{2+\delta^2}},
\end{equation}
$\sigma$ is the M\"{o}bius automorphism of $\mathbb{D}$ with $\sigma(z_1)=0$ and $\sigma(z_2)>0$, and $T$ is an arbitrary M\"{o}bius transformation. For each such function $f$, equality holds along the entire (admissible portion of the) hyperbolic geodesic through $z_1$ and $z_2$. Conversely, if either inequality holds for all points $z_1$ and $z_2$ in the specified range, then $||S_f||\le 2(1+\delta^2)$.
\end{customthm}

Now, in view of Theorem \ref{Thm-q-0005}, Theorem \ref{Thm-q-0015} and Theorem \ref{Thm-q-0020}, we obtain a pair of two-point distortion theorems for functions in $\mathcal{G}(\beta)$ and $\mathcal{F}(\alpha)$ for certain range of $\beta$ and $\alpha$:
\begin{cor}
Let $\beta>\sqrt{2}-1$ and $f\in\mathcal{G}(\beta)$ be of the form \eqref{q-00001}. Then
$$\Delta_f(z_1,z_2)\ge \frac{1}{\sqrt{\beta^2+2\beta-1}}\sin\left(\sqrt{\beta^2+2\beta-1}~\lambda(z_1,z_2)\right)$$
for all $z_1,z_2\in\mathbb{D}$ with $\lambda(z_1,z_2)\le \pi/\sqrt{\beta^2+2\beta-1}$ and
$$\Delta_f(z_1,z_2)\le \frac{1}{1+\beta}\sinh\left((1+\beta)\lambda(z_1,z_2)\right)$$
for all $z_1,z_2\in\mathbb{D}$. Both inequalities are sharp.
\end{cor}
\begin{cor}
Let $-1/2\le \alpha<0$ and $f\in\mathcal{F}(\alpha)$ be of the form \eqref{q-00001}. Then
$$\Delta_f(z_1,z_2)\ge \sqrt{\frac{1+\alpha}{-2\alpha}}\sin\left(\sqrt{\frac{-2\alpha}{1+\alpha}}~\lambda(z_1,z_2)\right)$$
for all $z_1,z_2\in\mathbb{D}$ with $\lambda(z_1,z_2)\le \pi\sqrt{(1+\alpha)/(-2\alpha)}$ and
$$\Delta_f(z_1,z_2)\le \sqrt{\frac{1+\alpha}{2}}\sinh\left(\sqrt{\frac{2}{1+\alpha}}~\lambda(z_1,z_2)\right)$$
for all $z_1,z_2\in\mathbb{D}$. Both inequalities are sharp.
\end{cor}

\vspace{4mm}
\textbf{Data availability:} Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.\\

\textbf{Authors contributions:}  All authors contributed equally to the investigation of the problem and the order of the authors is given alphabetically according to the surname. All authors read and approved the final manuscript.\\

\textbf{Acknowledgement:} The second named author thanks the University Grants Commission for the financial support through UGC Fellowship (Grant No. MAY2018-429303).

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