diff --git a/draft.tex b/draft.tex
index 7bc39ea..58eb033 100644
--- a/draft.tex
+++ b/draft.tex
@@ -433,21 +433,23 @@
 Moreover, the differential decay width can receive additional complex amplitudes from signal-like backgrounds,  
 \textit{e.g.} $K\pi$ S-wave from a non-resonant decay and/or a scalar resonance (see detailed discussion in Ref.~\cite{Hurth:2017hxg}). 
 These contributions are in general expected to be small~\cite{Aaij:2015oid,Aaij:2016flj}, 
-and in the proposed formalism these are introduced in an identical manner for muons and electrons. 
-Therefore, in this constrained framework these effects are even further suppressed and can then be neglected.
+and in the proposed formalism they benefit from the same description between the muon and electron mode.
+%these are introduced in an identical manner for muons and electrons. 
+Therefore, in this constrained framework these effects are even further suppressed and can then be neglected
+for the scope of this work.
 
 Another important test to probe the stability of the model consists in changing the 
 description of the non-local hadronic effects in the generation of the pseudo-experiments.
 In this way we analyse potential issues that can rise if the truncation 
 $\mathcal{H}_\lambda[z^n]$ is not a good description of nature.
-We proceed as follows: we generate toys with non-zero coefficients for 
+We proceed as follows: we generate ensembles with non-zero coefficients for 
 $\mathcal{H}_\lambda[z^3]$ and $\mathcal{H}_\lambda[z^4]$, and we perform the fit 
 with $\mathcal{H}_\lambda[z^2]$.
 We vary the choice of the $\mathcal{H}_\lambda[z^{(3,4)}]$ generated parameters, 
 including a \textit{provocative} set of values that minimises the tension with the $P_5'$ 
 anomaly~\cite{Aaij:2015oid}, while keeping $\WC_9^{(\mu)}$ and
 $\WC_{10}^{(\mu)}$ at their SM values.
-Despite the mis-modelling of the non-local hadronic effects in the fit results, we observe 
+Despite the mis-modelling of the non-local hadronic effects in the fit, we observe 
 that the determination of $\Delta\WC_9$ and $\Delta\WC_{10}$ is always unbiased, 
 thanks to the relative cancellation of all the shared parameters between the two channels.
 It is worth emphasizing that a hypothetical determination of the individual  
@@ -460,13 +462,13 @@
 all the available information from $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays for a precise 
 determination of LFU-breaking differences of WCs, \textit{i.e.} $\Delta\WC_9$ 
 and $\Delta\WC_{10}$.
-This relies on a shared parametrisation of the local (form-factors) and non-local ($\mathcal{H}_\lambda[q^2]$)
+This relies on a shared parametrisation of the local (form-factors) and non-local ($\mathcal{H}_\lambda(q^2)$)
 hadronic matrix elements between the muonic and electronic channels, 
 that in turn enables the determination of the observables of interest free from any theoretical uncertainty.  
 Figure~\ref{fig:allComponents} illustrates the usefulness of the newly-proposed observables by combining 
-the different information from angular analysis to branching ratio measurements. 
+the different information from angular analysis and branching ratio measurements. 
 Due to the inclusiveness of the approach, the expected sensitivity surpasses any 
-of projection for the current anomalous measurements alone given the benchmark points.
+of projection for the current $R_{K^{*}}$ and $P^{\prime}_{5}$ anomalies alone, given the benchmark points.
 Therefore, this novel formalism can be the \textit{holy grail} to observe 
 NP in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays in near future.
 
@@ -511,20 +513,20 @@
 
 An extension of the physics case of the proposed method
 is to investigate the sensitivity to the chirality-flipped counterparts of the usual Wilson coefficients, 
-\textit{i.e.} $\WC^{\prime (\mu)}_9$ and $\WC^{\prime(\mu)}_{10}$.
+\textit{i.e.} $\WC^{\prime \,(\mu)}_9$ and $\WC^{\prime \,(\mu)}_{10}$.
 Following the formalism discussed in this letter, the primed WCs are examined by considering in addition to the 
-\texttt{BMP}$_{\WC_{9,10}}$ three different modified NP scenarios for the muon only:
-$\WC_{9,10}^{\prime(\mu)} = \WC^{\prime \rm{SM}}_{9,10} = 0$;
-$\WC^{\prime (\mu)}_9 = \WC^{\prime (\mu)}_{10} = 0.3$; 
-and $\WC^{\prime (\mu)}_9 = - \WC^{\prime (\mu)}_{10} =  0.3$.  
-Notice that for the electron mode the $\WC_{9,10}^{\prime(e)}$ is set and fixed to the SM value $\WC^{\prime \rm{SM}}_{9,10} = 0$.
+\texttt{BMP}$_{\WC_{9,10}}$ three different modified NP scenarios:
+$\WC_{9,10}^{\prime \,(\mu)} = \WC^{\prime \,\rm{SM}}_{9,10} = 0$;
+$\WC^{\prime \,(\mu)}_9 = \WC^{\prime \,(\mu)}_{10} = 0.3$; 
+and $\WC^{\prime \,(\mu)}_9 = - \WC^{\prime \,(\mu)}_{10} =  0.3$.  
+Notice that for the electron mode the $\WC_{9,10}^{\prime \,(e)}$ is set and fixed to the SM value $\WC^{\prime \,\rm{SM}}_{9,10} = 0$.
 
 \begin{figure}[b]
 \includegraphics[width=.4\textwidth]{plots/ellipses_CpMu_Hz.pdf} 
 \caption{%
-    Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and 
-	$\WC'^{(\mu)}_{10}$ for different non-local hadronic parametrisation models for a NP scenario 
-	with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$.
+    Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{\,(\mu)}_9$ and 
+	$\WC'^{\,(\mu)}_{10}$ for different non-local hadronic parametrisation models for a NP scenario 
+	with $\WC'^{\,(\mu)}_9 = \WC'^{\,(\mu)}_{10} = 0.3$.
     The contours correspond to $3\,\sigma$ statistical-only uncertainty bands evaluated with
 	the expected statistics after \lhcb Run II.
     \label{fig:Cp_Hz}
@@ -532,24 +534,25 @@
 \end{figure}
 
 Figure~\ref{fig:Cp_Hz} shows the fit results for different order of the analytic expansion for 
-the non-local hadronic contribution for a NP scenario with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$, 
-and yields corresponding to the \lhcb Run II expected statistics. 
-The dependency on the determination of $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ on the 
+the non-local hadronic contribution for the NP scenario with $\WC'^{\,(\mu)}_9 = \WC'^{\,(\mu)}_{10} = 0.3$, 
+and yields corresponding to the expected \lhcb Run II statistics. 
+The dependency on the determination of $\WC'^{\,(\mu)}_9$ and $\WC'^{\,(\mu)}_{10}$ on the 
 order of the expansion clearly saturates after $\mathcal{H}_\lambda[z^3]$ and allows a measurement
 of the primed Wilson coefficients for the muon decay channel $B^{0} \to K^{*0} \mumu$ independent 
 on the theoretical hadronic uncertainty.
 %
-Figure~\ref{fig:Cp} shows the prospects for the sensitivity to the $\WC'^{(\mu)}_9$ and 
-$\WC'^{(\mu)}_{10}$ Wilson coefficients corresponding to the expected statistics at 
+Figure~\ref{fig:Cp} shows the prospects for the sensitivity to the $\WC'^{\,(\mu)}_9$ and 
+$\WC'^{\,(\mu)}_{10}$ Wilson coefficients corresponding to the expected statistics at \lhcb
+upgrades $50\,$fb$^{-1}$ and $\,300\,$fb$^{-1}$.
 Note that only with the full capability of the LHCb experiment it is possible 
 to start disentangling the different NP hypotheses. 
 %
 \begin{figure}[t]
 \includegraphics[width=.4\textwidth]{plots/ellipses_CpMu.pdf} 
 \caption{%
-    Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$
-    for three NP scenarios: (blue) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0$, (orange) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$ 
-    and (magenta) $\WC'^{(\mu)}_9 = - \WC'^{(\mu)}_{10} = 0.3$.
+    Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{\,(\mu)}_9$ and $\WC'^{\,(\mu)}_{10}$
+    for three NP scenarios: (blue) $\WC'^{\,(\mu)}_9 = \WC'^{\,(\mu)}_{10} = 0$, (orange) $\WC'^{\,(\mu)}_9 = \WC'^{\,(\mu)}_{10} = 0.3$ 
+    and (magenta) $\WC'^{\,(\mu)}_9 = - \WC'^{\,(\mu)}_{10} = 0.3$.
     The contours correspond to $3\,\sigma$ statistical-only uncertainty bands expected for the LHCb Upgrade (dotted)  
     $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics.
     \label{fig:Cp}