diff --git a/draft.tex b/draft.tex
index 9f99547..b9e8333 100644
--- a/draft.tex
+++ b/draft.tex
@@ -199,7 +199,7 @@
 %allows a pertubative calculation~\cite{PhysRevD.8.3633,Politzer:1974fr,Gross:1998jx}.}
 and non-calculable long-distance effects. 
 These can be parametrised in the weak Lagrangian in terms of 
-effective vertex operators with different Lorentz structures, $\mathcal{O}_i$, 
+effective operators with different Lorentz structures, $\mathcal{O}_i$, 
 with corresponding couplings $\mathcal{C}_i$ - referred to as Wilson coefficients (WC). 
 %induced couplings $\mathcal{C}_i$ - referred to as Wilson coefficients (WC) - 
 %and effective vertex operators with different Lorentz structure, $\mathcal{O}_i$.
@@ -210,7 +210,7 @@
 %hereafter only a subset of the Wilson coefficients (WC) $C_i$ for the basis of
 %dimension-six field operators $O_i$ is used for the weak Lagrangian~\cite{Bobeth:2017vxj}.  
 In this framework NP effects are incorporated 
-by introducing deviations in the WCs~\cite{Ali:1994bf},  
+by introducing deviations in the WCs~\cite{Ali:1994bf} from their SM predictions,  
 {\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$. 
 %For instance, whilst the individual level of significance of the present anomalies is still inconclusive, 
 %there is an appealing non-trivial consistency shown in 
@@ -222,11 +222,12 @@
 or $\mathcal{C}_9$ and $\mathcal{C}_{10}$ simultaneously~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. 
 A direct experimental determination of the WCs is currently 
 bounded by sizeable uncertainties that arise from 
-non-factorisable hadronic contributions that are difficult to assess reliably from first principles. 
-Some promising approaches suggest to extract these elements 
-either from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} 
+non-factorisable hadronic matrix elements that are difficult to assess reliably from first principles. 
+Some promising approaches suggest to extract this contribution 
+from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw}  
 %or by using the analytical and dispersive properties of these correlators~\cite{Bobeth:2017vxj}. 
-or by exploring properties (\textit{e.g.} analyticity) of the structure of these functions~\cite{Bobeth:2017vxj}. 
+%and by exploring properties (\textit{e.g.} analyticity) of the structure of these functions~\cite{Bobeth:2017vxj}. 
+and by exploiting analytical properties of its structure~\cite{Bobeth:2017vxj}. 
 However, these models still have intrinsic limitations, in particular 
 in the assumptions that enter in parametrisation of the di-lepton invariant mass distribution. 
 
@@ -242,7 +243,7 @@
 %when examining directly the difference in Wilson coefficients.
 Furthermore, in this method the full set of observables  
 (\textit{e.g} $R_{K^{*}}$, $P^{\prime}_{5}$ and branching fraction measurements) available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$  
-decays is exploited, providing unprecedented precision on LFU in a single measurement.
+decays is exploited, providing unprecedented precision on LFU in a single analysis.
 %and therefore, most stringent constraints on LFU for a single measurement can be expected.  
 
 %Let us consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$ 
@@ -282,22 +283,25 @@
 where ${\cal{N}}_{\lambda}^{(\ell)}$ is a normalisation factor, and 
 ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ and $\mathcal{H}_\lambda(q^{2})$ 
 are referred to ``local'' and ``non-local'' hadronic matrix elements, respectively.  
-While the ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factor parameters set from~\cite{Straub:2015ica}\footnote{In order 
+While the form factors ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ can be taken from~\cite{Straub:2015ica}\footnote{In order 
 to guarantee a good agreement between Light-Cone Sum Rules~\cite{Ball:1998kk,Khodjamirian:2006st} 
 and Lattice results~\cite{Becirevic:2006nm,Horgan:2013hoa}, 
 %sure full agreement among the LCSRs and Lattice results.
 uncertainties on the form factors parameters are doubled with respect to Ref.~\cite{Straub:2015ica}},
 the $\mathcal{H}_\lambda(q^{2})$ encode the aforementioned non-factorisable
-hadronic contribution and
+hadronic contributions and
 are described using two complementary parametrisations~\cite{Bobeth:2017vxj,Hurth:2017sqw} - 
 for brevity only a subset of results is shown for the latter approach.
-In the following this function is expressed in terms of a ``conformal'' 
+In the following
+%~\footnote{After removing below-threshold poles, \textit{i.e.} $J/\psi(1S)$ and $\psi(2S)$, }
+this function is expressed in terms of a ``conformal'' 
 variable $z(q^{2})$~\cite{Bobeth:2017vxj,Boyd:1995cf,Bourrely:2008za}, 
 with an analytical expansion truncated at a given order  
-$z^n$ (herein referred to as $\mathcal{H}_\lambda[z^n]$). 
+$z^n$ (herein referred to as $\mathcal{H}_\lambda[z^n]$), 
+after removing singularities related to the $J/\psi(1S)$ and $\psi(2S)$. 
 Some of the drawbacks of this expansion is that \textit{a-priori} there is 
-no physics argument to justify the order of the polynomial to be curtailed at 
-or even if this series will ever converge - 
+no physics argument to justify the order of the polynomial to be curtailed at - 
+%or even if this series will ever converge - 
 which in turn currently limits any claim on NP sensitivity. 
 
 In order to overcome these points, we investigate the LFU-breaking
@@ -307,9 +311,9 @@
 \Delta \WC_i = \widetilde{\mathcal{C}}_i^{(\mu)} - \widetilde{\mathcal{C}}_i^{(e)}\,,
 \end{equation}
 where the usual WCs $\mathcal{C}_i^{(\mu,e)}$ are renamed as $\widetilde{\mathcal{C}}_i^{(\mu,e)}$, 
-since a precise disentanglement between the physical 
-meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution is
-impossible at the current stage of the theoretical knowledge~\cite{Mauri:2018}.
+since an accurate disentanglement between the physical 
+meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution 
+cannot be achieved at the current stage of the theory~\cite{Mauri:2018}.
 The key feature of this strategy is to realise that all hadronic matrix elements 
 are known to be lepton-flavour universal, and thus are shared among both semileptonic decays.
 This benefits from the large statistics available for $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ decays 
@@ -343,7 +347,8 @@
 and assumes improvements in the electron mode resolution for LHCb~\cite{Lionetto:XX}.  
 
 Within the SM setup the Wilson coefficients are set to  
-$\mathcal{C}^{\rm{SM}}_9 = 4.27$, $\mathcal{C}^{\rm{SM}}_{10} = - 4.17$ and $\mathcal{C}^{\rm{SM}}_7 = -0.34$.
+$\mathcal{C}^{\rm{SM}}_9 = 4.27$, $\mathcal{C}^{\rm{SM}}_{10} = - 4.17$ and $\mathcal{C}^{\rm{SM}}_7 = -0.34$ (see~\cite{Bobeth:2017vxj} and references therein), 
+corresponding to a fixed renormalisation scale of $\mu = 4.2\,$GeV.
 This baseline model is modified in the case of muons for two NP benchmark points (BMP), 
 \textit{i.e.} $\WC_9^{(e)} = \WC^{\rm{SM}}_9 = \WC^{(\mu)}_9 + 1$ and 
 %{\color{red} $\WC^{\rm{NP}(\mu)}_9 = - 1$ }
@@ -391,7 +396,8 @@
 the model assumption used for the non-local matrix elements. 
 Nonetheless, it is noticeable that the high correlation of the  
 $\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ coefficients 
-is sufficient to preserve the true underlying physics at any order of the series expansion $\mathcal{H}_\lambda[z^n]$,  
+is sufficient to preserve the true underlying physics at any order of the series expansion $\mathcal{H}_\lambda[z^n]$
+and without any parametric theoretical input,  
 \textit{i.e.} the two-dimensional pull estimator with respect to the LFU hypothesis is unbiased. 
 %
 \begin{figure}[t]
@@ -426,7 +432,7 @@
 %\end{center}
 \end{figure}
 
-The sensitivity to the two benchmark-like NP scenarios using the proposed observables $\Delta \WC_i$
+The sensitivity to the two benchmark-like NP scenarios using the proposed pseudo observables $\Delta \WC_i$
 is shown in Fig.~\ref{fig:DeltaC9C10}. 
 %Fig.~\ref{fig:DeltaC9C10} shows the sensitivity to the two NP scenarios, NP$_{\WC_9}$ 
 %and NP$_{\WC_9-\WC_{10}}$ in terms of the two model-independent LFU-breaking 
@@ -478,7 +484,7 @@
 applied. 
 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}, 
+These contributions are determined to be small~\cite{Aaij:2015oid,Aaij:2016flj}, 
 and in the proposed formalism they benefit from the same description between the muon and electron mode.
 Therefore, in this constrained framework these effects are even further suppressed and can then be neglected
 for the scope of this work.
@@ -511,7 +517,7 @@
 This relies on a shared parametrisation of the local (form-factors) and non-local ($\mathcal{H}_\lambda[z^n]$)
 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.  
+free from any parametric theoretical uncertainty.  
 In addition, this simultaneous analysis is more robust against 
 experimental effects such
 as mismodeling of the detector resolution, since most parameters are
@@ -551,8 +557,8 @@
 \end{figure}
 
 
-%We acknowledge useful contributions from Gino Isidori, Danny van Dyk and Patrick Owen.
-%This work is supported by the Swiss National Science Foundation (SNF) under contracts 173104 and 174182.   
+We acknowledge useful contributions from Gino Isidori, Danny van Dyk and Patrick Owen.
+This work is supported by the Swiss National Science Foundation (SNF) under contracts 173104 and 174182.   
 
 
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